Chapter IV: Part 4

(f) _Malignant Disease of the Uterus._--The varieties of malignant
disease met with in the uterus are sarcoma, carcinoma and
chorion-epithelioma malignum. Sarcomata may occur in the body and in
the neck. They occur at an earlier age than carcinomata. Marked
enlargement and haemorrhage are the symptoms. The differential
diagnosis is microscopic. Extirpation of the uterus is the only chance
of prolonging life. The age at which women are most subject to
carcinoma (cancer) of the uterus is towards the decline of sexual
life. Of 3385 collected cases of cancer of the uterus 1169 occurred
between 40 and 50, and 856 between 50 and 60. In contradistinction to
fibroid tumours it frequently arises after the menopause. It may be
divided into cancer of the body and cancer of the neck (cervix).
Cancer of the neck of the uterus is almost exclusively confined to
women who have been pregnant (Bland-Sutton). Predisposing causes may
be injuries during delivery. The symptoms which induce women to seek
medical aid are haemorrhage, foetid discharge, and later pain and
cachexia. An unfortunate belief amongst the public that the menopause
is associated with irregular bleeding and offensive discharges has
prevented many women from seeking medical advice until too late. It
cannot be too widely understood that cancer of the cervix is in its
early stages a purely local disease, and if removed in this stage
usually results in cure. So important is the recognition of this fact
in the saving of human life that at the meeting of the British Medical
Association in April 1909 the council issued for publication a special
appeal to medical practitioners, midwives and nurses, and directed it
to be published in British and colonial medical and nursing journals.
It will be useful to quote here a part of the appeal directed to
midwives and nurses: "Cancer may occur at any age and in a woman who
looks quite well, and who may have no pain, no wasting, no foul
discharge and no profuse bleeding. To wait for pain, wasting, foul
discharge or profuse bleeding is to throw away the chance of
successful treatment. The early symptoms of cancer of the womb
are:--(1) bleeding which occurs after the change of life, (2) bleeding
after sexual intercourse or after a vaginal douche, (3) bleeding,
slight or abundant, even in young women, if occurring between the
usual monthly periods, and especially when accompanied by a
bad-smelling or watery blood-tinged discharge, (4) thin watery
discharge occurring at any age." On examination the cervix presents
certain characteristic signs, though these may be modified according
to the variety of cancer present. Hard nodules or definite loss of
substance, extreme friability and bleeding after slight manipulation,
are suspicious. Epithelial cancer of the cervix may assume a
proliferating ulcerative type, forming the well-known "cauliflower"
excrescence. The treatment of cancer of the cervix is free removal at
the earliest possible moment. Cancer of the body of the uterus is rare
before the 45th year. It is most frequent at or subsequent to the
menopause. The majority of the patients are nulliparae (Bland-Sutton).
The signs are fitful haemorrhages after the menopause, followed by
profuse and offensive discharges. The uterus on examination often
feels enlarged. The diagnosis being made, hysterectomy (removal of the
uterus) is the only treatment. Cancer of the body of the uterus may
complicate fibroids. Chorion-epithelioma malignum (deciduoma) was
first described in 1889 by Sanger and Pfeiffer. It is a malignant
disease presenting microscopic characters resembling decidual tissue.
It occurs in connexion with recent pregnancy, and particularly with
the variety of abortion termed hydatid mole. In many cases it destroys
life with a rapidity unequalled by any other kind of growth. It
quickly ulcerates and infiltrates the uterine tissues, forming
metastatic growths in the lung and vagina. Clinically it is recognized
by the occurrence after pregnancy of violent haemorrhages, progressive
cachexia and fever with rigors. Recent suggestions have been made as
to chorion-epithelioma being the result of pathological changes in the
lutein tissue of the ovary. The growth is usually primary in the
uterus, but may be so in the Fallopian tubes and in the vagina. A few
cases have been recorded unconnected with pregnancy. The virulence of
chorion-epithelioma varies, but in the present state of our knowledge
immediate removal of the primary growth along with the affected organ
is the only treatment.

_Diseases of the Fallopian Tubes._--The Fallopian tubes or oviducts
are liable to inflammatory affections, tuberculosis, sarcomata,
cancer, chorion-epithelioma and tubal pregnancy. Salpingitis
(inflammation of the oviducts) is nearly always secondary to septic
infection of the genital tract. The chief causes are septic
endometritis following labour or abortion, gangrene of a myoma,
gonorrhoea, tuberculosis and cancer of the uterus; it sometimes
follows the specific fevers. When the pus escapes from the tubes into
the coelom it sets up pelvic peritonitis. When the inflammation is
adjacent to the ostium it leads to the matting together of the tubal
fimbriae and glues them to an adjacent organ. This seals the ostium.
The occluded tube may now have an accumulation of pus in it
(pyosalpinx). When in consequence of the sealing of the ostium the
tube becomes distended with serous fluid it is termed hydrosalpinx.
Haematosalpinx is a term applied to the non-gravid tube distended with
blood; later the tubes may become sclerosed. Acute septic salpingitis
is ushered in by a rigor, the temperature rising to 103 deg., 104 deg.
F., with severe pain and constitutional disturbance. The symptoms may
become merged in those of general peritonitis. In chronic disease
there is a history of puerperal trouble followed by sterility, with
excessive and painful menstruation. Acute salpingitis requires
absolute rest, opium suppositories and hot fomentations. With urgent
symptoms removal of the inflamed adnexa must be resorted to. Chronic
salpingitis often renders a woman an invalid. Permanent relief can
only be afforded by surgical intervention. Tuberculous salpingitis is
usually secondary to other tuberculous infections. The Fallopian tubes
may be the seat of malignant disease. This is rarely primary. By far
the most important of the conditions of the Fallopian tubes is tubal
pregnancy (or ectopic gestation). It is now known that fertilization
of the human ovum by the spermatozoon may take place even when the
ovum is in its follicle in the ovary, for oosperms have been found in
the ovary and Fallopian tubes as well as in the uterus. Belief in
ovarian pregnancy is of old standing, and had been regarded as
possible but unproved, no case of an early embryo in its membranes in
the sac of an ovary being forthcoming, until the remarkable case
published by Dr Catherine van Tussenboek of Amsterdam in 1899
(Bland-Sutton). Tubal pregnancy is most frequent in the left tube; it
sometimes complicates uterine pregnancy; rarely both tubes are
pregnant. When the oosperm lodges in the ampulla or isthmus it is
called tubal gestation; when it is retained in the portion traversing
the uterine wall it is called tubo-uterine gestation. Wherever the
fertilized ovum remains and implants its villi the tube becomes turgid
and swollen, and the abdominal ostium gradually closes. The ovum in
this situation is liable to apoplexy, forming tubal mole. When the
abdominal ostium remains pervious the ovum may escape into the
coelomic cavity (tubal abortion); death from shock and haemorrhage
into the abdominal cavity may result. When neither of these
occurrences has taken place the ovum continues to grow inside the
tube, the rupture of the distended tube usually taking place between
the sixth and the tenth week. The rupture of the tube may be
intraperitoneal or extraperitoneal. The danger is death from
haemorrhage occurring during the rupture, or adhesions may form, the
retained blood forming a haematocele. The ovum may be destroyed or may
continue to develop. In rare cases rupture may not occur, the tube
bulging into the peritoneal cavity; and the foetus may break through
the membranes and lie free among the intestines, where it may die,
becoming encysted or calcified. The tubal placenta possesses foetal
structures, the true decidua forming in the uterus. The signs
suggestive of tubal pregnancy before rupture are missed periods,
pelvic pains and the presence of an enlarged tube. When rupture takes
place it is attended in both varieties with sudden and severe pain and
more or less marked collapse, and a tumour may or may not be felt
according to the situation of the rupture. There is a general "feeling
of something having given way." If diagnosed before rupture, the sac
must be removed by abdominal section. In intraperitoneal rupture
immediate operation affords the only chance of saving life. In
extraperitoneal rupture the foetus may occasionally remain alive until
full term and be rescued by abdominal section, if the condition is
recognized, or a false labour may take place, accompanied by death of
the foetus.

_Diseases of the Ovaries and Parovarium._--The ovaries undergo
striking changes at puberty, and again at the menopause, after which
there is a gradual shrinkage. One or both may be absent or malformed,
or they are subject to displacements, being either undescended,
contained in a hernia or prolapsed. Either of these conditions, if a
source of pain, may necessitate their removal. The ovary is also
subject to haemorrhage or apoplexy. Acute inflammations (oophorites)
are constantly associated with salpingitis or other septic conditions
of the genital tract or with an attack of mumps. The relation of
oophoritis to mumps is at present unknown. Acute oophoritis may
culminate in abscess but more usually adhesions are formed. The
surgical treatment is that of pyosalpinx. Chronic inflammation may
follow acute or be consequent on pelvic cellulitis. Its constant
features are more or less pain followed by sterility. The ovary may be
the seat of tuberculosis, which is generally secondary to other
lesions. Suppuration and abscess of the ovary also occur.
Perioophoritis, or chronic inflammation in the neighbourhood, may also
involve the gland. The cause of cirrhosis of the ovaries is unknown,
though it may be associated with cirrhotic liver. The change is met
with in women between 20 and 40 years of age, the ovaries being in a
shrunken, hard, wrinkled condition. Under ovarian neuralgia are
grouped indefinite painful symptoms occurring frequently in neurotic
and alcoholic subjects, and often worse during menstruation. The
treatment, whether local or operative, is usually unsatisfactory. The
ovary is frequently the seat of tumours, dermoids and cysts. Cysts may
be simple, unilocular or multilocular, and may attain an enormous
size. The largest on record was removed by Dr Elizabeth Reifsnyder of
Shanghai, and contained 100 litres of fluid, and the patient
recovered. The operation is termed ovariotomy. Dermoid cysts
containing skin, bones, teeth and hair, are of frequent growth in the
ovary, and have attained the weight of from 20 to 40 kilogrammes. In
one case a girl weighed 27 kilogrammes and her tumour 44 kilogrammes
(Keen). Papillomatous cysts also occur in the ovary. Parovarian and
Gartnerian cysts are found, and adenomata form 20% of all ovarian
cysts. Occasionally the tunic of peritoneum surrounding the ovary
becomes distended with serous fluid. This is termed ovarian hydrocele.
Ovarian fibroids occur, and malignant disease (sarcoma and carcinoma)
is fairly frequent, sarcoma being the most usual ovarian tumour
occurring before puberty. Carcinoma of the ovary is rarely primary,
but it is a common situation for secondary cancer to that of the
breast, gall-bladder or gastro-intestinal tract. The treatment of all
rapidly-growing tumours of the ovary is removal.

_Diseases of the Pelvic Peritoneum and Connective Tissue._--Women are
excessively liable to peritoneal infections. (1) Septic infection
often follows acute salpingitis and may give rise to pelvic
peritonitis (perimetritis), which may be adhesive, serous or purulent.
It may follow the rupture of ovarian or dermoid cysts, rupture of the
uterus, extra uterine pregnancy or extension from pyosalpinx. The
symptoms are severe pain, fever, 103 deg. F. and higher, marked
constitutional disturbances, vomiting, restlessness, even delirium.
The abdomen is fixed and tympanitic. Its results are the formation of
adhesions causing abnormal positions of the organs, or chronic
peritonitis may follow. The treatment is rest in bed, opium, hot
stupes to the abdomen and quinine. (2) Epithelial infections take
place in the peritoneum in connexion with other malignant growths. (3)
Hydroperitoneum, a collection of free fluid in the abdominal cavity,
may be due to tumours of the abdominal viscera or to tuberculosis of
the peritoneum. (4) Pelvic cellulitis (parametritis) signifies the
inflammation of the connective tissue between the folds of the broad
ligament (mesometrium). The general causes are septic changes
following abortion, delivery at term (especially instrumental
delivery), following operations on the uterus or salpingitis. The
symptoms are chill followed by severe intrapelvic pain and tension,
fever 100 deg. to 102 deg. F. There may be nausea and vomiting,
diarrhoea, rectal tenseness and dysuria. If consequent on parturition
the lochia cease or become offensive. On examination there is
tenderness and swelling in one flank and the uterus becomes fixed and
immovable in the exudate as if embedded in plaster of Paris. The
illness may go to resolution if treated by rest, opium, hot stupes or
icebags and glycerine tampons, or may go on to suppuration forming
pelvic abscess, which signifies a collection of pus between the layers
of the broad ligament. The pus in a pelvic abscess may point and
escape through the walls of the vagina, rectum or bladder. It
occasionally points in the groin. If the pus can be localized an
incision should be made and the abscess drained. The tumours which
arise in the broad ligament are haematocele, solid tumours (as
myomata, lipomata and sarcomata), and echinnococcus colonies
(hydatids).

BIBLIOGRAPHY.--Albutt, Playfair and Eden, _System of Gynaecology_
(1906); McNaughton Jones, _Manual of Diseases of Women_ (1904);
Bland-Sutton and Giles, _Diseases of Women_ (1906); C. Lockyer,
"Lutein Cysts in association with Chorio-Epithelioma," _Journal of
Obstetrics and Gynaecology_ (January, 1905); W. Stewart McKay,
_History of Ancient Gynaecology_; Hart and Barbour, _Diseases of
Women_; Howard Kelly, _Operative Gynaecology_. (H. L. H.)

GYONGYOSI, ISTVAN [STEPHEN] (1620-1704), Hungarian poet, was born of poor but noble parents in 1620. His abilities early attracted the notice of Count Ferencz Wesselenyi, who in 1640 appointed him to a post of confidence in Fulek castle. Here he remained till 1653, when he married and became an assessor of the judicial board. In 1681 he was elected as a representative of his county at the diet held at Soprony (Oedenburg). From 1686 to 1693, and again from 1700 to his death in 1704, he was deputy lord-lieutenant of the county of Gomor. Of his literary works the most famous is the epic poem _Muranyi Venus_ (Caschau, 1664), in honour of his benefactor's wife Maria Szecsi, the heroine of Murany. Among his later productions the best known are _Rozsa-Koszoru_, or Rose-Wreath (1690), _Kemeny-Janos_ (1693), _Cupido_ (1695), _Palinodia_ (1695) and _Chariklia_ (1700).

The earliest edition of his collected poetical works is by Dugonics
(Pressburg and Pest, 1796); the best modern selection is that of
Toldy, entitled _Gyongyosi Istvan valogatott poetai munkai_ (Select
poetical works of Stephen Gyongyosi, 2 vols., 1864-1865).

GYOR (Ger. _Raab_), a town of Hungary, capital of a county of the same name, 88 m. W. of Budapest by rail. Pop. (1900) 27,758. It is situated at the confluence of the Raab with the Danube, and is composed of the inner town and three suburbs. Gyor is a well-built town, and is the seat of a Roman Catholic bishop. Amongst its principal buildings are the cathedral, dating from the 12th century, and rebuilt in 1639-1654; the bishop's palace; the town hall; the Roman Catholic seminary for priests and several churches. There are manufactures of cloth, machinery and tobacco, and an active trade in grain and horses. Twenty miles by rail W. S. W. of the town is situated Csorna, a village with a Premonstratensian abbey, whose archives contain numerous valuable historical documents.

Gyor is one of the oldest towns in Hungary and occupies the site of the Roman _Arabona_. It was already a place of some importance in the 10th century, and its bishopric was created in the 11th century. It was a strongly fortified town which resisted successfully the attacks of the Turks, into whose hands it fell by treachery in 1594, but they retained possession of it only for four years. Montecucculi made Gyor a first-class fortress, and it remained so until 1783, when it was abandoned. At the beginning of the 19th century, the fortifications were re-erected, but were easily taken by the French in 1809, and were again stormed by the Austrians on the 28th of June 1849.

About 11 m. S.E. of Gyor on a spur of the Bakony Forest lies the famous Benedictine abbey of Pannonhalma (Ger. _St Martinsberg_; Lat. _Mons Sancti Martini_), one of the oldest and wealthiest abbeys of Hungary. It was founded by King St Stephen, and the original deed from 1001 is preserved in the archives of the abbey. The present building is a block of palaces, containing a beautiful church, some of its parts dating from the 12th century, and lies on a hill 1200 ft. high. The church has a tower 130 ft. high. In the convent there are a seminary for priests, a normal school, a gymnasium and a library of 120,000 vols. The chief abbot has the rank of a bishop, and is a member of the Upper House of the Hungarian parliament, while in spiritual matters he is subordinate immediately to the Roman curia.

GYP, the pen name of SIBYLLE GABRIELLE MARIE ANTOINETTE RIQUETI DE MIRABEAU, Comtesse de Martel de Janville (1850- ) French writer, who was born at the chateau of Koetsal in the Morbihan. Her father, who was the grandson of the vicomte de Mirabeau and great-nephew of the orator, served in the Papal Zouaves, and died during the campaign of 1860. Her mother, the comtesse de Mirabeau, in addition to some graver compositions, contributed to the _Figaro_ and the _Vie parisienne_, under various pseudonyms, papers in the manner successfully developed by her daughter. Under the pseudonym of "Gyp" Madame de Martel, who was married in 1869, sent to the _Vie parisienne_, and later to the _Revue des deux mondes_, a large number of social sketches and dialogues, afterwards reprinted in volumes. Her later work includes stories of a more formal sort, essentially differing but little from the shorter studies. The following list includes some of the best known of Madame de Martel's publications, nearly seventy in number: _Petit Bob_ (1882); _Autour du mariage_ (1883); _Ce que femme veut_ (1883); _Le Monde a cote_ (1884), _Sans voiles_ (1885); _Autour du divorce_ (1886); _Dans le train_ (1886); _Mademoiselle Loulou_ (1888); _Bob au salon_ (1888-1889); _L'Education d'un prince_ (1890); _Passionette_ (1891); _Ohe! la grande vie_ (1891); _Une Election a Tigre-sur-mer_ (1890), an account of "Gyp's" experiences in support of a Boulangist candidate; _Mariage civil_ (1892); _Ces bons docteurs_ (1892); _Du haut en bas_ (1893); _Mariage de chiffon_ (1894); _Leurs ames_ (1895); _Le Coeur d'Ariane_ (1895); _Le Bonheur de Ginette_ (1896); _Totote_ (1897); _Lune de miel_ (1898); _Israel_ (1898); _L'Entrevue_ (1899); _Le Pays des champs_ (1900); _Trop de chic_ (1900); _Le Friquet_ (1901); _La Fee_ (1902); _Un Mariage chic_ (1903); _Un Menage dernier cri_ (1903); _Maman_ (1904); _Le Coeur de Pierrette_ (1905). From the first "Gyp," writing of a society to which she belonged, displayed all the qualities which have given her a distinct, if not pre-eminent, position among writers of her class. Those qualities included an intense faculty of observation, much skill in innuendo, a mordant wit combined with some breadth of humour, and a singular power of animating ordinary dialogues without destroying the appearance of reality. Her Parisian types of the spoiled child, of the precocious schoolgirl, of the young bride, and of various masculine figures in the gay world, have become almost classical, and may probably survive as faithful pictures of luxurious manners in the 19th century. Some later productions, inspired by a violent anti-Semitic and Nationalist bias, deserve little consideration. An earlier attempt to dramatize _Autour du mariage_ was a failure, not owing to the audacities which it shares with most of its author's works, but from lack of cohesion and incident. More successful was _Mademoiselle Eve_ (1895), but indeed "Gyp's" successes are all achieved without a trace of dramatic faculty. In 1901 Madame de Martel furnished a sensational incident in the Nationalist campaign during the municipal elections in Paris. She was said to have been the victim of a kidnapping outrage or piece of horseplay provoked by her political attitude, but though a most circumstantial account of the outrages committed on her and of her adventurous escape was published, the affair was never clearly explained or verified.

GYPSUM, a common mineral consisting of hydrous calcium sulphate, named from the Gr. [Greek: gypsos], a word used by Theophrastus to denote not only the raw mineral but also the product of its calcination, which was employed in ancient times, as it still is, as a plaster. When crystallized, gypsum is often called selenite, the [Greek: selenites] of Dioscorides, so named from [Greek: selene], "the moon," probably in allusion to the soft moon-like reflection of light from some of its faces, or, according to a legend, because it is found at night when the moon is on the increase. The granular, marble-like gypsum is termed alabaster (q.v.).

Gypsum crystallizes in the monoclinic system, the habit of the crystals being usually either prismatic or tabular; in the latter case the broad planes are parallel to the faces of the clinopinacoid. The crystals may become lenticular by curvature of certain faces. In the characteristic type represented in fig. 1, f represents the prism, l the hemi-pyramid and P the clinopinacoid. Twins are common, as in fig. 2, forming in some cases arrow-headed and swallow-tailed crystals. Cleavage is perfect parallel to the clinopinacoid, yielding thin plates, often diamond-shaped, with pearly lustre; these flakes are usually flexible, but may be brittle, as in the gypsum of Montmartre. Two other cleavages are recognized, but they are imperfect. Crystals of gypsum, when occurring in clay, may enclose much muddy matter; in other cases a large proportion of sand may be mechanically entangled in the crystals without serious disturbance of form; whilst certain crystals occasionally enclose cavities with liquid and an air-bubble. Gypsum not infrequently becomes fibrous. This variety occurs in veins, often running through gypseous marls, with the fibres disposed at right angles to the direction of the vein. Such gypsum when cut and polished has a pearly opalescence, or satiny sheen, whence it is called satin-spar (q.v.).

Gypsum is so soft as to be scratched even by the finger-nail (H = 1.5 to 2). Its specific gravity is about 2.3. The mineral is slightly soluble in water, one part of gypsum being soluble, according to G. K. Cameron, in 372 parts of pure water at 26 deg. C. Waters percolating through gypseous strata, like the Keuper marls, dissolve the calcium sulphate and thus become permanently hard or "selenitic." Such water has special value for brewing pale ale, and the water used by the Burton breweries is of this character; hence the artificial dissolving of gypsum in water for brewing purposes is known as "burtonization." Deposits of gypsum are formed in boilers using selenitic water.

Pure gypsum is colourless or white, but it is often tinted, especially in the alabaster variety, grey, yellow or pink. Gypsum crystallizes with two molecules of water, equal to about 21% by weight, and consequently has the formula CaSO4.2H2O. By exposure to strong heat all the water may be expelled, and the substance then has the composition of anhydrite (q.v.). When the calcination, however, is conducted at such a temperature that only about 75% of the water is lost, it yields a white pulverulent substance, known as "plaster of Paris," which may readily be caused to recombine with water, forming a hard cement. The gypsum quarries of Montmartre, in the north of Paris, were worked in Tertiary strata, rich in fossils. Gypsum is largely quarried in England for conversion into plaster of Paris, whence it is sometimes known as "plaster stone," and since much is sent to the Staffordshire potteries for making moulds it is also termed "potter's stone." The chief workings are in the Keuper marls near Newark in Nottinghamshire, Fauld in Staffordshire and Chellaston in Derbyshire. It is also worked in Permian beds in Cumberland and Westmorland, and in Purbeck strata near Battle in Sussex.

Gypsum frequently occurs in association with rock-salt, having been deposited in shallow basins of salt water. Much of the calcium in sea-water exists as sulphate; and on evaporation of a drop of sea-water under the microscope this sulphate is deposited as acicular crystals of gypsum. In salt-lagoons the deposition of the gypsum is probably effected in most cases by means of micro-organisms. Waters containing sulphuretted hydrogen, on exposure to the air in the presence of limestone, may yield gypsum by the formation of sulphuric acid and its interaction with the calcium carbonate. In volcanic districts gypsum is produced by the action of sulphuric acid, resulting from the oxidation of sulphurous vapours, on lime-bearing minerals, like labradorite and augite, in the volcanic rocks: hence gypsum is common around solfataras. Again, by the oxidation of iron-pyrites and the action of the resulting sulphuric acid on limestone or on shells, gypsum may be formed; whence its origin in most clays. Gypsum is also formed in some cases by the hydration of anhydrite, the change being accompanied by an increase of volume to the extent of about 60%. Conversely gypsum may, under certain conditions, be dehydrated or reduced to anhydrite.

Some of the largest known crystals of selenite have been found in southern Utah, where they occur in huge geodes, or crystal-lined cavities, in deposits from the old salt-lakes. Fine crystals, sometimes curiously bent, occur in the Permian rocks of Friedrichroda, near Gotha, where there is a grotto called the Marienglashohle, close to Rheinhardsbrunn. Many of the best localities for selenite are in the New Red Sandstone formation (Trias and Permian), notably the salt-mines of Hall and Hallein, near Salzburg, and of Bex in Switzerland. Excellent crystals, usually of a brownish colour arranged in groups, are often found in the brine-chambers and the launders used in salt-works. Selenite also occurs in fine crystals in the sulphur-bearing marls of Girgenti and other Sicilian localities; whilst in Britain very bold crystals are yielded by the Kimeridge clay of Shotover Hill near Oxford. Twisted crystals and rosettes of gypsum found in the Mammoth Cave, Kentucky, have been called "oulopholites" ([Greek: oulos], "woolly"; [Greek: pholeos], "cave").

In addition to the use of gypsum in cement-making, the mineral finds application as an agricultural agent in dressing land, and it has also been used in the manufacture of porcelain and glass. Formerly it was employed, in the form of thin cleavage-plates, for glazing windows, and seems to have been, with mica, called _lapis specularis_. It is still known in Germany as _Marienglas_ and _Fraueneis_. Delicate cleavage-plates of gypsum are used in microscopic petrography for the determination of certain optical constants in the rock-forming minerals. (F. W. R.*)

GYROSCOPE AND GYROSTAT. These are scientific models or instruments designed to illustrate experimentally the dynamics of a rotating body such as the spinning-top, hoop and bicycle, and also the precession of the equinox and the rotation of the earth.

The gyroscope (Gr. [Greek: gyros], ring, [Greek: skopein], to see) may be distinguished from the gyrostat ([Greek: gyros], and [Greek: statikos], stationary) as an instrument in which the rotating wheel or disk is mounted in gimbals so that the principal axis of rotation always passes through a fixed point (fig. 1). It can be made to imitate the motion of a spinning-top of which the point is placed in a smooth agate cup as in Maxwell's dynamical top (figs. 2, 3). (_Collected Works_, i. 248.) A bicycle wheel, with a prolongation of the axle placed in a cup, can also be made to serve (fig. 4).

The gyrostat is an instrument designed by Lord Kelvin (_Natural Philosophy_, S 345) to illustrate the more complicated state of motion of a spinning body when free to wander about on a horizontal plane, like a top spun on the pavement, or a hoop or bicycle on the road. It consists essentially of a massive fly-wheel concealed in a metal casing, and its behaviour on a table, or with various modes of suspension or support, described in Thomson and Tait, _Natural Philosophy_, serves to illustrate the curious reversal of the ordinary laws of statical equilibrium due to the _gyrostatic domination_ of the interior invisible fly-wheel, when rotated rapidly (fig. 5).

The toy shown in figs. 6 and 7, which can be bought for a shilling, is acting as a gyroscope in fig. 6 and a gyrostat in fig. 7.

The gyroscope, as represented in figs. 2 and 3 by Maxwell's dynamical top, is provided with screws by which the centre of gravity can be brought into coincidence with the point of support. It can then be used to illustrate Poinsot's theory of the motion of a body under no force, the gyroscope being made kinetically unsymmetrical by a setting of the screws. The discussion of this movement is required for Jacobi's theorems on the allied motion of a top and of a body under no force (Poinsot, _Theorie nouvelle de la rotation des corps_, Paris, 1857; Jacobi, _Werke_, ii. Note B, p. 476).

To imitate the movement of the top the centre of gravity is displaced from the point of support so as to give a preponderance. When the motion takes place in the neighbourhood of the downward vertical, the bicycle wheel can be made to serve again mounted as in fig. 8 by a stalk in the prolongation of the axle, suspended from a universal joint at O; it can then be spun by hand and projected in any manner.

The first practical application of the gyroscopic principle was invented and carried out (1744) by Serson, with a spinning top with a polished upper plane surface for giving an artificial horizon at sea, undisturbed by the motion of the ship, when the real horizon was obscured. The instrument has been perfected by Admiral Georges Ernest Fleuriais (fig. 9), and is interesting theoretically as showing the correction required practically for the rotation of the earth. Gilbert's barogyroscope is devised for the same purpose of showing the earth's rotation; a description of it, and of the latest form employed by Foppl, is given in the _Ency. d. math. Wiss._, 1904, with bibliographical references in the article "Mechanics of Physical Apparatus." The rotation of the fly-wheel is maintained here by an electric motor, as devised by G. M. Hopkins, and described in the _Scientific American_, 1878. To demonstrate the rotation of the earth by the constancy in direction of the axis of a gyroscope is a suggestion that has often been made; by E. Sang in 1836, and others. The experiment was first carried out with success by Foucault in 1851, by a simple pendulum swung in the dome of the Pantheon, Paris, and it has been repeated frequently (_Memoires sur le pendule_, 1889).

A gyroscopic fly-wheel will preserve its original direction in space only when left absolutely free in all directions, as required in the experiments above. If employed in steering, as of a torpedo, the gyroscope must act through the intermediary of a light relay; but if direct-acting, the reaction will cause precession of the axis, and the original direction is lost.

The gyrostatic principle, in which one degree of freedom is suppressed in the axis, is useful for imparting steadiness and stability in a moving body; it is employed by Schlick to mitigate the rolling of a ship and to maintain the upright position of Brennan's monorail car.

Lastly, as an application of gyroscopic theory, a stretched chain of fly-wheels in rotation was employed by Kelvin as a mechanical model of the rotary polarization of light in an electromagnetic field; the apparatus may be constructed of bicycle wheels connected by short links, and suspended vertically.

_Theory of the Symmetrical Top._

1. The physical constants of a given symmetrical top, expressed in
C.G.S. units, which are employed in the subsequent formulae, are
denoted by M, h, C and A. M is the weight in grammes (g) as given by
the number of gramme weights which equilibrate the top when weighed in
a balance; h is the distance OG in centimetres (cm.) between G the
centre of gravity and O the point of support, and Mh may be called the
preponderance in g.-cm.; Mh and M can be measured by a spring balance
holding up in a horizontal position the axis OC in fig. 8 suspended at
O. Then gMh (dyne-cm. or ergs) is the moment of gravity about O when
the axis OG is horizontal, gMh sin [theta] being the moment when the
axis OG makes an angle [theta] with the vertical, and g = 981
(cm./s^2) on the average; C is the moment of inertia of the top about
OG, and A about any axis through O at right angles to OG, both
measured in g-cm.^2.

To measure A experimentally, swing the top freely about O in small
plane oscillation, and determine the length, l cm., of the equivalent
simple pendulum; then

(1) l = A/Mh, A = Mhl.

Next make the top, or this simple pendulum, perform small conical
revolutions, nearly coincident with the downward vertical position of
equilibrium, and measure n, the mean angular velocity of the conical
pendulum in radians / second; and T its period in seconds; then

(2) 4[pi]^2/T^2 = n^2 = g/l = gMh/A;

and f = n/2[pi] is the number of revolutions per second, called the
_frequency_, T = 2[pi]/n is the period of a revolution, in seconds.

Steady motion of the top.

2. In the popular explanation of the steady movement of the top at a
constant inclination to the vertical, depending on the composition of
angular velocity, such as given in Perry's _Spinning Tops_, or
Worthington's _Dynamics of Rotation_, it is asserted that the moment
of gravity is always generating an angular velocity about an axis OB
perpendicular to the vertical plane COC' through the axis of the top
OC'; and this angular velocity, compounded with the resultant angular
velocity about an axis OI, nearly coincident with OC', causes the axes
OI and OC' to keep taking up a new position by moving at right angles
to the plane COC', at a constant precessional angular velocity, say
[mu] rad./sec., round the vertical OC (fig. 4).

If, however, the axis OC' is prevented from taking up this
precessional velocity, the top at once falls down; thence all the
ingenious attempts--for instance, in the swinging cabin of the
Bessemer ship--to utilise the gyroscope as a mechanical directive
agency have always resulted in failure (_Engineer_, October 1874),
unless restricted to actuate a light relay, which guides the
mechanism, as in steering a torpedo.

An experimental verification can be carried out with the gyroscope in
fig. 1; so long as the vertical spindle is free to rotate in its
socket, the rapidly rotating wheel will resist the impulse of tapping
on the gimbal by moving to one side; but when the pinch screw prevents
the rotation of the vertical spindle in the massive pedestal, this
resistance to the tapping at once disappears, provided the friction of
the table prevents the movement of the pedestal; and if the wheel has
any preponderance, it falls down.

Familiar instances of the same principles are observable in the
movement of a hoop, or in the steering of a bicycle; it is essential
that the handle of the bicycle should be free to rotate to secure the
stability of the movement.

The bicycle wheel, employed as a spinning top, in fig. 4, can also be
held by the stalk, and will thus, when rotated rapidly, convey a
distinct muscular impression of resistance to change of direction, if
brandished.

Elementary demonstration of the condition of steady motion.

3. A demonstration, depending on the elementary principles of
dynamics, of the exact conditions required for the axis OC' of a
spinning top to spin steadily at a constant inclination [theta] to the
vertical OC, is given here before proceeding to the more complicated
question of the general motion, when [theta], the inclination of the
axis, is varying by nutation.

It is a fundamental principle in dynamics that if OH is a vector
representing to scale the angular momentum of a system, and if Oh is
the vector representing the axis of the impressed couple or torque,
then OH will vary so that the velocity of H is represented to scale by
the impressed couple Oh, and if the top is moving freely about O, Oh
is at right angles to the vertical plane COC', and

(1) Oh = gMh sin [theta].

In the case of the steady motion of the top, the vector OH lies in the
vertical plane COC', in OK suppose (fig. 4), and has a component OC =
G about the vertical and a component OC' = G', suppose, about the axis
OC; and G' = CR, if R denotes the angular velocity of the top with
which it is spun about OC'.

If [mu] denotes the constant precessional angular velocity of the
vertical plane COC' the components of angular velocity and momentum
about OA are [mu] sin [theta] and A[mu] sin [theta], OA being
perpendicular to OC' in the plane COC'; so that the vector OK has the
components

(2) OC' = G', and C'K = A[mu] sin[theta],

and the horizontal component

(3) CK = OC' sin[theta] - C'K cos[theta]
= G' sin[theta] - A[mu] sin[theta] cos[theta].

The velocity of K being equal to the impressed couple Oh,

(4) gMh sin[theta] = [mu].CK = sin [theta] (G'[mu] - A[mu]^2 cos [theta]),

and dropping the factor sin[theta],

(5) A[mu]^2 cos[theta] - G'[mu] + gMh = 0, or A[mu]^2 cos[theta] - CR[mu] + An^2 = 0,

the condition for steady motion.

Solving this as a quadratic in [mu], the roots [mu]1, [mu]2 are given by
_ _
G' | 4A^2n^2 |
(6) [mu]1, [mu]2 = -- sec [theta] |1 [+-] [root] (1 - ------- cos [theta]) |;
2A |_ G'^2 _|

and the minimum value of G' = CR for real values of [mu] is given by

G'^2 CR
(7) ------- = cos [theta], -- = 2[root](cos [theta]);
4A^2n^2 An

for a smaller value of R the top cannot spin steadily at the
inclination [theta] to the upward vertical.

Interpreted geometrically in fig. 4

(8) [mu] = gMh sin [theta]/CK = An^2/KN, and [mu] = C'K/A sin [theta] = KM/A,

(9) KM.KN = A^2n^2,

so that K lies on a hyperbola with OC, OC' as asymptotes.

Constrained motion of the gyroscope.

4. Suppose the top or gyroscope, instead of moving freely about the
point O, is held in a ring or frame which is compelled to rotate about
the vertical axis OC with constant angular velocity [mu]; then if N
denotes the couple of reaction of the frame keeping the top from
falling, acting in the plane COC', equation (4) S 3 becomes modified
into

(1) gMh sin [theta] - N = [mu].CK = sin [theta] G'[mu] - A[mu]^2 cos [theta],

(2) N = sin [theta] (A[mu]^2 cos [theta] - G'[mu] + gMh)
= A sin[theta] cos [theta] ([mu] - [mu]1)([mu] - [mu]2);

and hence, as [mu] increases through [mu]2 and [mu]1, the sign of N
can be determined, positive or negative, according as the tendency of
the axis is to fall or rise.

When G' = CR is large, [mu]2 is large, and

(3) [mu]1 [~=] gMh/G' = An^2/CR,

the same for all inclinations, and this is the precession observed in
the spinning top and centrifugal machine of fig. 10 This is true
accurately when the axis OC' is horizontal, and then it agrees with
the result of the popular explanation of S 2.

If the axis of the top OC' is pointing upward, the precession is in
the same direction as the rotation, and an increase of [mu] from [mu]1
makes N negative, and the top rises; conversely a decrease of the
procession [mu] causes the axis to fall (Perry, _Spinning Tops_, p.
48).

If the axis points downward, as in the centrifugal machine with upper
support, the precession is in the opposite direction to the rotation,
and to make the axis approach the vertical position the precession
must be reduced.

Centrifugal machine.

This is effected automatically in the Weston centrifugal machine (fig.
10) used for the separation of water and molasses, by the friction of
the indiarubber cushions above the support; or else the spindle is
produced downwards below the drum a short distance, and turns in a
hole in a weight resting on the bottom of the case, which weight is
dragged round until the spindle is upright; this second arrangement is
more effective when a liquid is treated in the drum, and wave action
is set up (_The Centrifugal Machine_, C. A. Matthey).

Similar considerations apply to the stability of the whirling bowl in
a cream-separating machine.

We can write equation (1)

(4) N = An^2 sin [theta] - [mu].CK = (A^n^2 - KM.KN) sin [theta]/A,

so that N is negative or positive, and the axis tends to rise or fall
according as K moves to the inside or outside of the hyperbola of free
motion. Thus a tap on the axis tending to hurry the precession is
equivalent to an impulse couple giving an increase to C'K, and will
make K move to the interior of the hyperbola and cause the axis to
rise; the steering of a bicycle may be explained in this way; but K1
will move to the exterior of the hyperbola, and so the axis will fall
in this second more violent motion.

Friction on the point of the top may be supposed to act like a tap in
the direction opposite to the precession; and so the axis of a top
spun violently rises at first and up to the vertical position, but
falls away again as the motion dies out. Friction considered as acting
in retarding the rotation may be compared to an impulse couple tending
to reduce OC', and so make K and K1 both move to the exterior of the
hyperbola, and the axis falls in both cases. The axis may rise or fall
according to the direction of the frictional couple, depending on the
shape of the point; an analytical treatment of the varying motion is
very intractable; a memoir by E. G. Gallop may be consulted in the
_Trans. Camb. Phil. Soc._, 1903.

The earth behaves in precession like a large spinning top, of which
the axis describes a circle round the pole of the ecliptic of mean
angular radius [theta], about 23-1/2 deg., in a period of 26,000
years, so that R/[mu] = 26000 X 365; and the mean couple producing
precession is

(5) CR[mu] sin [theta] = CR^2 sin 23-1/2 deg./(26000 X 365),

one 12 millionth part of 1/2CR^2, the rotation energy of the earth.

5. If the preponderance is absent, by making the C.G coincide with O,
and if A[mu] is insensible compared with G',

(1) N = -G'[mu] sin [theta],

the formula which suffices to explain most gyroscopic action.

Gyroscopic action of railway wheels.

Thus a carriage running round a curve experiences, in consequence of
the rotation of the wheels, an increase of pressure Z on the outer
track, and a diminution Z on the inner, giving a couple, if a is the
gauge,

(2) Za = G'[mu],

tending to help the centrifugal force to upset the train; and if c is
the radius of the curve, b of the wheels, C their moment of inertia,
and v the velocity of the train,

(3) [mu] = v/c, G' = Cv/b,

(4) Z = Cv^2/abc (dynes),

so that Z is the fraction C/Mab of the centrifugal force Mv^2/c, or
the fraction C/Mh of its transference of weight, with h the height of
the centre of gravity of the carriage above the road. A Brennan
carriage on a monorail would lean over to the inside of the curve at
an angle [alpha], given by

(6) tan [alpha] = G'[mu]/gMh = G'v/gMhc.

The gyroscopic action of a dynamo, turbine, and other rotating
machinery on a steamer, paddle or screw, due to its rolling and
pitching, can be evaluated in a similar elementary manner
(Worthington, _Dynamics of Rotation_), and Schlick's gyroscopic
apparatus is intended to mitigate the oscillation.

6. If the axis OC in fig. 4 is inclined at an angle [alpha] to the
vertical, the equation (2) S 4 becomes

(1) N = sin [theta] (A[mu]^2 cos [theta] - G'[mu]) + gMh sin ([alpha] -
[theta]).

Suppose, for instance, that OC is parallel to the earth's axis, and
that the frame is fixed in the meridian; then [alpha] is the
co-latitude, and [mu] is the angular velocity of the earth, the square
of which may be neglected; so that, putting N = 0, [alpha] - [theta] =
E,

(2) gMh sin E - G'[mu] sin ([alpha] - E) = 0,

G'[mu] sin [alpha] G'[mu]
(3) tan E = ------------------------ [~=] ------ sin [alpha].
gMh + G'[mu] cos [alpha] gMh

The barogyroscope.

This is the theory of Gilbert's barogyroscope, described in Appell's
_Mecanique rationnelle_, ii. 387: it consists essentially of a rapidly
rotated fly-wheel, mounted on knife-edges by an axis perpendicular to
its axis of rotation and pointing east and west; spun with
considerable angular momentum G', and provided with a slight
preponderance Mh, it should tilt to an angle E with the vertical, and
thus demonstrate experimentally the rotation of the earth.

Foucault's gyroscope.

In Foucault's gyroscope (_Comptes rendus_, 1852; Perry, p. 105) the
preponderance is made zero, and the axis points to the pole, when free
to move in the meridian.

Generally, if constrained to move in any other plane, the axis seeks
the position nearest to the polar axis, like a dipping needle with
respect to the magnetic pole. (_A gyrostatic working model of the
magnetic compass_, by Sir W. Thomson. British Association Report,
Montreal, 1884. A. S. Chessin, St Louis Academy of Science, January
1902.)

Gyroscopic horizon.

A spinning top with a polished upper plane surface will provide an
artificial horizon at sea, when the real horizon is obscured. The
first instrument of this kind was constructed by Serson, and is
described in the _Gentleman's Magazine_, vol. xxiv., 1754; also by
Segner in his _Specimen theoriae turbinum_ (Halae, 1755). The inventor
was sent to sea by the Admiralty to test his instrument, but he was
lost in the wreck of the "Victory," 1744. A copy of the Serson top,
from the royal collection, is now in the Museum of King's College,
London. Troughton's Nautical Top (1819) is intended for the same
purpose.

The instrument is in favour with French navigators, perfected by
Admiral Fleuriais (fig. 9); but it must be noticed that the horizon
given by the top is inclined to the true horizon at the angle E given
by equation (3) above; and if [mu]1 is the precessional angular
velocity as given by (3) S 4, and T = 2[pi]/[mu], its period in
seconds,

[mu] T cos lat T cos lat
(4) tan E = ----- cos lat = ---------, or E = ---------,
[mu]1 86400 8[pi]

if E is expressed in minutes, taking [mu] = 2[pi]/86400; thus making
the true latitude E nautical miles to the south of that given by the
top (_Revue maritime_, 1890; _Comptes rendus_, 1896).

This can be seen by elementary consideration of the theory above, for
the velocity of the vector OC' of the top due to the rotation of the
earth is

(5) [mu].OC' cos lat = gMh sin E = [mu]1.OC' sin E,

[mu] T cos lat
sin E = ----- cos lat, E = ---------,
[mu]1 8[pi]

in which 8[pi] can be replaced by 25, in practice; so that the
Fleuriais gyroscopic horizon is an illustration of the influence of
the rotation of the earth and of the need for its allowance.

Euler's coordinate angles.

7. In the ordinary treatment of the general theory of the gyroscope,
the motion is referred to two sets of rectangular axes; the one Ox,
Oy, Oz fixed in space, with Oz vertically upward and the other OX, OY,
OZ fixed in the rotating wheel with OZ in the axis of figure OC.

The relative position of the two sets of axes is given by means of
Euler's unsymmetrical angles [theta], [phi], [psi], such that the
successive turning of the axes Ox, Oy, Oz through the angles (i.)
[psi] about Oz, (ii.) [theta] about OE, (iii.) [phi] about OZ, brings
them into coincidence with OX, OY, OZ, as shown in fig. 11,
representing the _concave_ side of a spherical surface.

The component angular velocities about OD, OE, OZ are

(1) [.[psi]] sin [theta], .[theta], .[phi] + .[psi] cos [theta];

so that, denoting the components about OX, OY, OZ by P, Q, R,

(2) P = .[theta] cos [phi] + .[psi] sin [theta] sin [phi],
Q = -.[theta] sin [phi] + .[psi] sin [theta] cos [phi],
R = .[phi] + .[psi] cos [theta].

Consider, for instance, the motion of a fly-wheel of preponderance Mh,
and equatoreal moment of inertia A, of which the axis OC is held in a
light ring ZCX at a constant angle [gamma] with OZ, while OZ is held
by another ring zZ, which constrains it to move round the vertical Oz
at a constant inclination [theta] with constant angular velocity [mu],
so that

(3) .[theta] = 0, .[psi] = [mu];

(4) P = [mu] sin [theta] sin [phi],
Q = [mu] sin [theta] cos [phi],
R = .[phi] + [mu] cos [theta].

With CXF a quadrant, the components of angular velocity and momentum
about OF, OY, are

(5) P cos [gamma] - R sin [gamma], Q, and A(P cos [gamma]
- R sin [gamma]), AQ,

so that, denoting the components of angular momentum of the fly-wheel
about OC, OX, OY, OZ by K or G', h1, h2, h3,

(6) h1 = A(P cos [gamma] - R sin [gamma]) cos [gamma] + K sin [gamma],

(7) h2 = AQ,

(8) h3 = -A(P cos [gamma] - R sin [gamma]) sin [gamma]
+ K cos [gamma];

and the dynamical equation

dh3
(9) --- - h1Q + h2P = N,
dt

with K constant, and with preponderance downward

(10) N = gMh cos zY sin [gamma]
= gMh sin [gamma] sin [theta] cos [phi],

reduces to

d^2[phi]
(11) A -------- sin [gamma]
dt^2

+ A[mu]^2 sin [gamma] sin^2 [theta] sin [phi] cos [phi]
+ A[mu]^2 cos [gamma] sin [theta] cos [theta] cos [phi]
- (K[mu] + gMh) sin [theta] cos [phi] = 0.

The position of relative equilibrium is given by

(12) cos [phi] = 0, and sin [phi]

K[mu] + gMh - A[mu]^2 cos [gamma] cos [theta]
= ---------------------------------------------.
A[mu]^2 sin [gamma] sin [theta]

For small values of [mu] the equation becomes

d^2[phi]
(13) A -------- sin [gamma] - (K[mu] + gMh) sin [theta] cos [phi] = 0,
dt^2

so that [phi] = 1/2[pi] gives the position of stable equilibrium, and
the period of a small oscillation is 2[pi] [root]{A sin [gamma]/(K[mu] +
gMh) sin [theta]}.

In the general case, denoting the periods of vibration about [phi] =
1/2[pi], -1/2[pi], and the sidelong position of equilibrium by
2[pi]/(n1, n2, or n3), we shall find

sin [theta]
(14) n1^2 = ------------- {gMh + K[mu] - A[mu]^2 cos ([gamma] - [theta])},
A sin [gamma]

sin [theta]
(15) n2^2 = ------------- {-gMh - K[mu] + A[mu]^2 cos ([gamma] + [theta])},
A sin [gamma]

(16) n3 = n1 n2/[mu] sin [theta].

The first integral of (11) gives

/d[phi]\^2
(17) 1/2A ( ------ ) sin [gamma]
\ dt /

+ 1/2A[mu]^2 sin [gamma] sin^2 [theta] sin^2 [phi]
- A[mu]^2 cos [gamma] sin [theta] cos [theta] sin [phi]
+ (K[mu] + gMh) sin [theta] sin [phi] - H = 0,

and putting tan (1/4[pi] + 1/2[phi]) = z, this reduces to

dz
(18) -- = n [root]Z
dt

where Z is a quadratic in z^2, so that z is a Jacobian elliptic
function of t, and we have

(19) tan (1/4[pi] + 1/2[phi]) = C(tn, dn, nc, or cn)nt,

according as the ring ZC performs complete revolutions, or oscillates
about a sidelong position of equilibrium, or oscillates about the
stable position of equilibrium [phi] = [+-]1/2[pi].

Suppose Oz is parallel to the earth's axis, and [mu] is the diurnal
rotation, the square of which may be neglected, then if Gilbert's
barogyroscope of S 6 has the knife-edges turned in azimuth to make an
angle [beta] with E. and W., so that OZ lies in the horizon at an
angle E.[beta].N., we must put [gamma] = 1/2[pi], cos [theta] = sin
[alpha] sin [beta]; and putting [phi] = 1/2[pi] - [delta] + E, where
[delta] denotes the angle between Zz and the vertical plane Z[zeta]
through the zenith [zeta],

(20) sin [theta] cos [delta] = cos [alpha], sin [theta] sin [delta]
= sin [alpha] cos [beta];

so that equations (9) and (10) for relative equilibrium reduce to

(21) gMh sin E = KQ = K[mu] sin [theta] cos [phi]
= K[mu] sin [theta] sin ([delta] - E),

and will change (3) S 6 into

K[mu] sin [alpha] cos [beta]
(22) tan E = ----------------------------,
gMh + K[mu] cos [alpha]

a multiplication of (3) S 6 by cos [beta] (Gilbert, _Comptes rendus_,
1882).

Changing the sign of K or h and E and denoting the revolutions/second
of the gyroscope wheel by F, then in the preceding notation, T
denoting the period of vibration as a simple pendulum,

K[mu] sin [alpha] cos [beta] F sin [alpha] cos [beta]
(23) tan E = ----------------------------- = ----------------------------,
gMh - K[mu] cos [alpha] 86400 A/T^2C - F cos [alpha]