Chapter II: Part 2

The nature of the triple point depends upon the relation of the line at infinity to the cubic before inversion. Thus the line at infinity may cut the cubic in three distinct points all real, or one real and two imaginary, in one real and two coincident points (an ordinary tangent), or in three coincident points (an inflectional tangent). Hence the quartic may have at the triple point three distinct tangents all real, or one real and two imaginary, one real and two coincident, or all coincident.

This quartic may be generated in a manner similar to that used for the curves already discussed. We showed in the section on nodal cubics that a system of conics through A, B, C, D, and a projective pencil of rays with its vertex at A generate by the intersection of corresponding elements a cubic with a node at A. Invert the whole figure from A and then project:—the pencil of rays remains a pencil; the system of conics becomes a system of unicursal cubics having a common node at A and passing through five other common points; the cubic inverts and projects into a quartic with a triple point at A, passing through the five other common points of the system of cubics.

The three points of inflection of a nodal cubic lie on a right line. Inverting:—there are three points on a circular quartic with a triple point whose osculating circles pass through the triple point, and these three points lie on a circle through the triple point. Let these three points be designated by A, B, and C. The lines from the triple point O to the points A, B, C, and the common chord of the osculating circles at two of them form a harmonic pencil. Through one of these points, A, and the triple point draw a circle touching the quartic; the point of contact is on the common chord of the osculating circles at B and C.

From theorems which we have already proved for a system of cubics having a common node and passing through five others fixed points, we can infer other theorems for a system of quartics having a common triple point and passing through seven other fixed points. For example, any conic through the common double point and two of the fixed points is cut by the cubics in pairs of points which determine at the node a pencil in involution. Hence any cubic having its node at the common triple point and passing through any four of the fixed points is cut by the quartics in pairs of points which determine at the common triple point a pencil in involution. Again, the pairs of tangents to the cubics at the common double point form a pencil in involution, the two cuspidal tangents being the foci of the pencil. Inverting:—the line at infinity (which passes through two of the fixed points, i. e. the circular points) cuts the system of circular quartics in pairs of points in involution. Projecting:—a line through any two of the seven fixed points cuts the system of quartics in pairs of points in involution. Since the line at infinity touches the inverse of a cuspidal cubic, it follows that any line through two of the fixed points will touch two of the quartics of the system; these points of contact are therefore the foci of the involution.

Other theorems on such a system of quartics will be given in the next section.

SYSTEMS OF QUARTICS THROUGH SIXTEEN POINTS.

Let U and V represent a system of quartics through sixteen points. Since the discriminant of quartic is of the 27th degree in the coefficients it follows that there are 27 values of k for which the discriminant vanishes, and hence 27 quartics of the system which have double points. As in case of cubics these 27 points are called the critic centres of the system. Let the equation of the system of quartics be written

u₄ + u₃ + u₂ + u₁ + u₀ = 0.

In a manner similar to that employed for cubics, we find the equation of the polar cubics of the origin with respect to the system to be

u₃ + 2u₂ + 3u₁ + 4u₀ = 0.

The polar conics of the origin are given by

u₂ + 3u₁ + 6u₀ = 0;

and the polar lines of the origin, by u₁ + 4u₀ = 0.

The origin may be any point in the plane and hence we conclude that only one quartic of the system passes through a given point and that the polar cubics of any point form a system through nine points. The polar conics of any point form a system through four points and the polar lines meet in a point.

If one of the critic centres be taken for origin, we can readily see that such a point is also a critic centre on each of its systems of polar curves. It is thus at a vertex of the self-polar triangle of its system of polar conics and the opposite side of the triangle is the common polar line of the critic centre with respect to each of the systems of curves. The tangents at the node of the nodal quartic coincide with those of its polar cubic and these we know coincide with the lines which constitute its polar conic.

If two of the sixteen basal points coincide, such a point is a critic centre. The argument is the same as for a system of cubics. We can also see that two of the basal points of each of its systems of polar curves coincide at the critic centre. The sixteen basal points of the system of quartics may unite two and two so that it is possible to draw a system of quartics touching eight given lines each at a fixed point.

If three of the basal points of our system of quartics coincide, all the quartics have at such a point a common point of inflection and a common inflectional tangent. The demonstration is the same as that already given for cubics. The system of polar cubics of such a point also have this point for a common point if inflection and the same tangent for a common inflectional tangent. I prefer to show this analytically for the sake of the method. The equation of the system of quartics having the origin for a common point of inflection and the axis of y for a common inflectional tangent may be written

u₄ + u₃ + {(B + kB₁)xy + (C + kC₁)y²} + (A + kA₁)y = 0.

The equation of the polar cubics of the origin is therefore,

u₃ + 2{(B + kB₁)xy + (C + kC₁)y²} + 3(A + kA₁)y = 0,

which proves the proposition. The properties of the system of polar conics of such a point are therefore the same as those already proved for cubics. One quartic of the system has a double point at the common point of inflection of the others.

When four basal points coincide they give rise either to a common point of undulation or a common double point on all the quartics of the system. The equation of the system having a common point of undulation may be written

u₄ + (A + kA₁)x²y + (B + kB₁)xy² + (C + kC₁)y³

+ (D + kD₁)xy + (E + kE₁)y² + (F + kF₁)y = 0.

There is one value of k for which the last term vanishes, and hence the origin is a critic centre. The polar cubics of the point of undulation break up into a system of conics through four points and the common tangent at the common point of undulation. For the equation of the polar cubics is

y{(A + kA₁)x² + (B + kK₁)xy + (C + kC₁)y²

+ 2(D + kD₁)x + 2(E + kE₁)y + (F + kF₁)} = 0.

The system of polar conics of the origin consequently breaks up into the line y = 0 and a pencil meeting in a point. The common tangent at the common point of undulation is also the common polar line of the point of undulation.

When the four coincident basal points form a common double point on the quartic, it is not difficult to show that two of the quartics are cuspidal at this point. The polar cubics of the common double point form a system having the same point for common double point. The tangents to the quartics at the common node constitute the system of polar conics and form a pencil in involution. Twelve of the sixteen basal points may unite in three groups of four each and the system of quartics is then trinodal and passes through four other fixed points. This is the system obtained by inverting a system of conics through four points and then projecting.

A few special cases should be noticed here. If the four fixed points and two of the nodes lie on a conic, this conic together with the two lines from the third node to the first two constitute a quartic of the system. If the four fixed points lie on a line, the quartic then consists of this line and the sides of the triangle formed by the nodes. If the three nodes and three of the fixed points lie on a conic, the system of quartics then consists of this conic and a system of conics through the three nodes and the fourth fixed point. A special case of a system of quartics with three nodes is a system of cubics having a common node and passing through five other fixed points together with a line through two of them.

If a fifth basal point be moved up to join the four at the common node, the quartics have one tangent at the common node common to all. If six basal points coincide they have both tangents at the node common to all. In this case one of the quartics has a triple point at the common node of the others. If seven basal points coincide, one of these tangents is an inflectional tangent as well. If eight points coincide, both are inflectional tangents.

When nine of the basal points of a system of quartics coincide, the quartics have a common triple point. This is nicely shown by inverting a system of nodal cubics from the common node. The inverse curves form a system of quartics having a triple point and passing through seven other fixed points. The common triple point on two quartics counts for nine points of intersection and the seven others make the requisite sixteen. From our knowledge of a system of cubics having a common node it is readily inferred that three of the quartics must each break up into a nodal cubic and a right line through the node. If the seven fixed points of the system of quartics lie on a cubic having a node at the common triple point, the system of quartics then consists of this cubic and a pencil of lines through the node. If two of the seven fixed points lie on a line through the common triple point, the system of quartics then consists of this right line and a system of cubics through the other five points and having a common node at the common triple point.

The system of cubics having a common node may have one, two, or three of the other basal points at infinity; and these may be all distinct or two or three of them coincident. Whence we infer that if the system of quartics have ten coincident basal points, one of the tangents at the triple point is common to all the quartics of the system. If eleven basal points coincide, two of the triple-point tangents are common to all the quartics. If twelve coincide, all three triple-point tangents are common. These triple-point tangents may be all distinct, two coincident, or all three coincident.

If thirteen basal points coincide, the system of quartics then consists of the three fixed lines joining the multiple point to the other three, together with a pencil of lines through the multiple point. If fourteen points coincide, two lines are fixed and these with any two lines of the pencil form a quartic of the system. If fifteen points coincide, only one line is fixed and each quartic consists of this line and any other three of the pencil. When all sixteen points coincide, any four lines through it form a quartic of the system.

In this paper cubic and quartic curves only are considered. I expect in a future paper to extend the methods herein developed to curves of still higher degrees. Many of the present results can be generalized and stated for a unicursal curve of the nth degree. I have purposely omitted all consideration of focal properties of these curves. There are also many special forms of interest which do not properly belong to a general treatment of the subject.

NOTE A.

The theorem concerning the three points on a conic A, B, and C, whose osculating circles pass through a fourth point O on the conic, is due to Steiner. From the properties of the harmonic polars of the points of inflection on a nodal cubic we may infer many other theorems concerning the points A, B, and C on a conic. Let the cubic be projected into a circular cubic and then inverted from the node. Its points of inflection A₁, B₁, C₁ invert into the points A, B, and C. The harmonic polar of A₁ inverts into the common chord O P of the circles osculating the conic at B and C; and similarly for the other harmonic polars.

The pencil O {A B P C} is harmonic. Any circle through A and O meets the conic in S and T so that the pencil O {A S P T} is harmonic. The two circles through O and tangent to the conic at S and T intersect on O P. If two circles be drawn through O and A intersecting the conic one in S and T and the other in U and V, the circles O S U and O T V intersect on O P; so also the circles O S V and O T U. But one circle can be drawn through O and A and tangent to the conic; its point of contact is on O P. Let l, m, and n be three points on the conic on a circle through O. Draw the circles O A l, O A m, and O A n intersecting the conic again in l₁, m₁, n₁; l₁, m₁, n₁, are also on a circle through O, and the circles through l, m, n and l₁, m₁, n₁ intersect on O P.

NOTE B.

From the fundamental property of the Cissoid of Diocles we can obtain by inversion an interesting theorem concerning the parabola. In the figure of the Cissoid given in Salmon’s H. P. C. Art. 214, A M₁ = M R, whence A M₁ = A R - A M; or A R = A M + A M₁. Inverting from the cusp and representing the inverse points by the same letters, we have for the parabola

1 1 1
——— = ——— + ————— .
A R A M A M₁

This result is interpreted as follows:—draw the circle of curvature at the vertex of a parabola; this circle is tangent to the ordinate B D which is equal to the abscissa A D; draw a line through A cutting the circle in R, the ordinate B D in M, and the parabola in M₁; then

1 1 1
——— = ——— + ————— .
A R A M A M₁

Draw the circle with centre at D and radius A D; any chord of the parabola through the vertex is cut harmonically by the parabola, the circle, and the double ordinate through D.

Footnotes:

[1] See note A.

[2] A few of the results of this section are due to the late Mr. H. B. Hall.

[3] See note B.

Foreign Settlements in Kansas.

A CONTRIBUTION TO DIALECT STUDY IN THE STATE.

Explanatory.—Some years ago when the subject of dialect study in Kansas, or rather of Kansas dialect, was mentioned, Mr. Noble Prentis, a gentleman who is warranted in speaking with authority on Kansas, was inclined to think that he settled the question in short order by declaring that there is no Kansas dialect. Probably the majority of intelligent citizens of the state would turn off the subject with the same reply. In the sense of a mode of speech common to the inhabitants of Kansas and peculiar to them, Mr. Prentis was indeed right. There is no vocabulary, at least no extensive vocabulary, by which the native of Kansas may be recognized in the American Babel. We have no distinctive pronunciation by which we may be known from the inhabitants of Nebraska or set apart from the citizens of Missouri. The verb fails to agree with its subject and the participle is deprived of its final ‘g’ with about equal frequency in Western Kansas and Eastern Colorado.

But in the same sense it is true that there is no Kansas flora, no Kansas fauna; that is, there is no plant and there is no animal found quite generally in Kansas and found nowhere outside of Kansas. The remark that there is no such thing as a Kansas dialect rests upon a misapprehension of what is meant by the term. In just the same way that we speak of the flora and the fauna of Kansas we may speak of the dialect of Kansas. Yet to avoid popular misapprehension it may be better to speak of dialect in Kansas, rather than of Kansas dialect.

Dialect study involves the observation and description of all facts concerning the natural living speech of men, and especially those points in which the speech of individuals or groups differs from that of the standard literary language as represented in classic writers and classic speakers. Standard literary English is always a little behind the times. It is the stuffed and mounted specimen in the museum. Dialect is the live animal on its native heath. Most people, indeed, will think that their speech does not differ materially from standard English. They say, “We speak near enough alike ‘for practical purposes’. But a thousand years hence the pronunciation of our country may have changed so much that it will seem like another language, and our descendants will write learned theses to prove that we pronounced ‘cough’ like cow or like cuff. A new language will have grown out of an old one, but no one know how it came about. Careful dialect study will help explain it.”

Kansas is a peculiarly favorable field for dialect study. We have here side by side representatives from nearly every state in the Union, and from a dozen foreign countries. The observer has here what elsewhere he must travel over half the world to find. In a district where the people are all natives, the speech is so nearly homogeneous that it is difficult to find any one who recognizes the peculiarities of his own language, but here the contrast of strange tongues strikes us immediately and we become conscious early of the fact that all men do not speak alike.

Study of dialect may be classified under the heads of pronunciation, grammar and vocabulary. Of these the last two are the easiest, and may be carried on by almost any one with pleasure and valuable results. Pronunciation is the most difficult of these matters to study, as competent observation and reports can be made only by one who has made a thorough study of Phonetics. To those who might wish to take up the study of this branch of the subject, Sweet’s Primer of Phonetics, and Grandgent’s “Vowel Measurements” and “German and English Sounds” are recommended.

In the study of dialect vocabularies it may become of the greatest importance to establish the exact locality of a word and the origin of the persons by whom it is used. For instance, in a family of my acquaintance the word ‘slandering’ = sauntering was familiar. It was a great puzzle to me until I learned that some of the children had been in the care of a German maid. The German word ‘schlendern’ suggested the unquestionable source of the peculiar word. As a source of information regarding the origin of the foreign elements of our population when their native speech shall have been forgotten, but when the influence of it will be left in vocabulary and pronunciation I have thought that a map of the state with the location of all the foreign settlements of even quite small size would be of interest and in time of great value. In the following pages I transmit the results of my inquiries so far as received. It is my intention to make the report complete and to publish the map, when as complete as it can be made, in colors. Unexpected difficulties have delayed the work and prevented its being complete. I depended for my information upon the County Superintendents of the State, a class of unusually intelligent and well-informed men and women. But in not a few cases there seems to have been a suspicion in the mind of my correspondent that I might be a special officer of the state trying to locate violations of the law requiring district schools to be conducted in English, and hence information regarding schools in foreign tongue was withheld or given but partially. And in some cases my informants were not well posted. A superintendent by the name of Schauermann in a county containing a town called Suabia, tells me that there are no foreigners in his county. In such cases time must be taken to secure a correct result.

The questions asked were: Locate, and give origin, date and approximate numbers of any settlements—six or more families—of foreigners in your county. Do they still use their language to any extent? Do they have church service and schools conducted in their native tongue? In many replies one or more of these points was neglected so that the information is not yet by any means what I desire to make it. However, for the purpose of dialect study approximate correctness in location is of chief importance, and accuracy as to numbers quite secondary.

Through the aid of ministers and others to whom I have been referred by the superintendents I hope to make this report complete in the following respects: The more exact limits of the settlement; the numbers of those foreign-born; the province as well as land from which they came; the number of churches; the number of schools and the length of time the same are conducted. I solicit the co-operation of everyone interested in this work, and also in the whole subject of dialect study. As intimated above, interested observers can without especial training do a service to science and at the same time find a fascinating pastime for themselves by making collections of words and constructions which they believe to be unusual or new. If any such are sent to the writer they will be duly acknowledged. They should in every case be accompanied by a statement of the age, condition and birth-place of the person using them.

I wish here to call attention to the work of the American Dialect Society which exists to promote this study. It desires as wide a membership as possible, and membership is open to all interested in the subject. The publication of the Society, Dialect Notes, contains reports of word-lists and other studies, and will be an aid to any who wish to undertake similar work. Subscriptions and membership fees should be sent to Mr. C. H. Grandgent, Treas., Cambridge, Mass.

REPORTS BY COUNTIES.

ATCHISON.—Reports no foreigners, by John
Klopfenstein, Supt.

ALLEN.—Swedes and Danes, from 600 to 700, settled
from 1873 to 1880. Have church service, and four to five
months school in Swedish. Grove and Elsmore townships.
Germans in and around Humboldt.

ANDERSON.—Irish in Reeder township, 1860 and 1874.
Germans, 1860 in Putman township, 1880 in Westphalia
township. Have both church and schools in German.

BARBER.—Reports “no foreigners worth making
account of”, by J. O. Hahn, Sup’t.

BARTON.—No report.

BOURBON.—Reports no foreigners.

BROWN.—No report.

BUTLER.—Germans (Prussians), speaking Low German,
in Fairmount and Milton townships. Hold church services
but no schools in German.

CHASE.—Russian Mennonites, speaking both Russian
and German, in Diamond Creek township, no church, but a
portion of schooling in German. Germans at Strong City,
with both church and schools in their native tongue.

CHATAUQUA.—Some Norwegians and Swedes, 1870, no location
given. Neither schools nor churches in native tongue.
One colony of ‘Russians’ (Mennonites?), who have also
given up their language.

CHEROKEE.—Weir City, French and Italians, number
considerable. Scammon, Scotch, also in large numbers.
The French and Italians have neither schools nor church
in the native tongue. Germans in Ross, twenty families;
with church originally Lutheran, now Mennonite; school
irregularly during past ten years. Swedes, a few
families in Cherokee township, have entirely given up
Swedish language. The Scotch, French and Italians in
mines or mining industries.

CHEYENNE.—Germans settled in 1885-86 on Hackberry
Creek, 160 persons; in the northeast corner of the
county, 100; on west border of county, north of
Republican river, 120; all with churches and the last
two with occasional schools. Swedes are across the
Republican adjoining last named German settlement,
120, entered 1886, having neither church nor school in
Swedish.

CLARKE.—Reports no foreigners.

CLAY.—No report.

CLOUD.—Canadian French are scattered over much of
the county, with considerable settlements in and around
the towns of Concordia, Clyde, St. Joseph and Aurora. In
all there are churches, in the first three schools also
conducted in French. Norwegians occupy portions of Sibley
and Lincoln townships with two churches in their own
tongue. They number about three hundred. Irish occupy
portions of Solomon and Lyon, the south part of Meredith
and the southeast corner of Grant townships.

COFFEY.—Germans in Liberty and on border of Leroy
and Avon township. Have church service in German.

COMANCHE.—Germans. A few scattered families.

COWLEY.—A few Swedes and Germans, widely scattered.

CRAWFORD.—Irish and French in Grant township;
Swedes in west part of Sherman township, have all
given up their language. Italians, Austrians and other
nationalities in south part of Washington, southeast
part of Sheridan and all over Baker township, especially
in Pittsburg, employed in mining and smelting.

DECATUR.—Swedes in Oberlin township; Mennonites
in Prairie Dog township; Germans in and around Dresden,
with Catholic church; Bohemians in Jennings and Garfield
townships.

DICKINSON.—Germans, 500 in number settled in
1860 in Liberty, Union and Lyon townships. Have three
churches and two schools in German. Also in Wheatland,
Jefferson, Bonner and Ridge townships, one church and a
school. Swedes, 100 settled in 1860 in Center and Hayes
township, with two churches and one school in Swedish.
Irish, several hundred in south part of Banner township.

DONIPHAN.—Germans in Wayne, Marion and southern
part of Center, Burr Oak and Washington townships, with
church service in native tongue. Norwegians in eastern
part of Wolf River township.

DOUGLAS.—There are German settlements in Eudora
township (300), Marion township (600), and Big Springs
township (100), with churches in all and school in
the first. There are about five hundred Germans in
Lawrence, with three German churches. There are smaller
settlements of Germans and Scandinavians at several
points in the county.

EDWARDS.—Germans and Swedes in Kinsley, Jackson
and Trenton townships, have church service in their
mother tongue.

ELK.—Swedes in Painter and Hood townships; Irish
in Falls township; Germans on the border of Elk and Wild
Cat townships. None of these have church or school in
the native tongue, but all use it at home.

ELLIS.—Germans from Russia, settled about 1876
in Catherine, Hartsook, Lookout, Wheatland and Freedom
townships, about 3000 in number—one third of the
population of the county in 1891. They are Catholics,
and have both churches and parochial schools conducted
in German. They are large wheat-growers.

ELLSWORTH.—Germans, (Methodists) in south part of
Valley township; Germans (Lutherans) in north part of
Columbia and Ellsworth townships; Germans (Baptists)
from Prussia, in Green Garden and south west corner of
Empire townships. These all have church service, and
the Lutherans schools in their own tongue. Bohemians in
Valley and Noble townships.

FINNEY.—Reports no foreigners.

FORD.—Germans in Wheatland and Speareville
townships. Have church, and one school conducted in
German.

FRANKLIN.—No report.

GARFIELD.—A few scattered families of Germans.

GEARY.—Irish (Connaught) came into Jackson,
Jefferson and Liberty townships 1855, about 1500 in
number. Germans (Anhalt) about 1500 came into Jefferson,
Milford and Lyon townships in 1862. About 300 English
from Sussex settled in Lyon township in 1870. The
Germans maintain both churches and schools in German.

GOVE.—Swedes in Lewis and south part of Grinnell
and south west corner of Gove townships.

GRAHAM.—A settlement of Canadian French (600)
was made in adjacent parts of Wild Horse and Morelan
townships about 1880. They conduct church service but no
schools in French.

GRANT.—Reports no foreigners.

GRAY.—Reports no foreigners.

GREELEY.—Swedes in the north west part of the
county, have church service and summer school in Swedish.

GREENWOOD.—Norwegians, about 200, in south part of
Salem township, have church in their own tongue. Germans
in Shell Rock township, about 300, also have church in
their own language.

HAMILTON.—Reports no foreigners.

HARPER.—Germans about the town of Harper.
Hungarians south of Bluff City. Both have church service
in German. About 100 French in Odell and Stohrville
townships.

HARVEY.—Germans (Russian Mennonites) from Odessa,
a few from Prussia, the latter speaking Low German. They
settled from 1874 to 1876 in Alta and Garden townships,
in Pleasant and the eastern part of Newton townships, and
about Halstead. They have church and school in German,
but speak Russian also. French in north part of Emma
township, engaged in raising silk worms.

HASKELL.—Reports no foreigners.

HODGMAN.—Germans, about 30 families, settled about
1884 in south east corner of Sterling township; have
preaching in German. Swedes in north west corner of
Marena township.

JACKSON.—Danes in Netawaca and Whiting townships;
Irish in Washington township; neither continue to use
their native tongue.

JEFFERSON.—Germans (Swiss) in Delaware, Jefferson
and Kentucky townships, maintaining church but no
schools in German.

JEWELL.—Swedes, widely scattered in Sinclair,
Allen, Ewing and Ezbon townships.

JOHNSON.—No report.

KEARNEY.—No report.

KINGMAN.—A small settlement of Germans in Peters
township, not using German to any extent. A few Irish in
Union township.

KIOWA.—No report.

LABETTE.—Swedes and Norwegians settled in Valley
and Canada townships about 1869. Still speak their
language, but have neither church nor school in it.

LANE.—Reports no foreigners.

LEAVENWORTH.—German, in 1873 in Easton township;
in Fair township in 1876; about 600 in each place. They
have church service and schools in German.

LINCOLN.—Danes settled in Grant township in 1869
and since, 400 in number. Germans settled in Pleasant
township in 1872, with 300, and in Indiana township in
1869 and later with about 375. Danes and Germans have
good schools and churches in native tongue. Bohemians
in Highland township in 1878 with thirty families.
They speak their native tongue, but have no schools or
churches.

LINN.—Reports no foreigners.

LOGAN.—Swedes, about 200, about Page City, in
north part of county. Have church and school both in
Swedish.

LYON.—Welsh, between 1000 and 1500 are located
in and about Emporia, with three churches conducted in
Welsh. There is a settlement of Scandinavians near Olpe
in Centre township.

MARION.—Germans (Russian Mennonites), settled in
Logan, Durham, Lehigh, Risley, Menno, West Branch and
Liberty townships, from 1870 to 1875, some 5000 in number.
They speak both Russian and German, and have church
service and schools in the latter tongue. Bohemians,
about 500 in number are settled in Clark township. They
speak Czech and have church service in that language.
French to the number of 200 settled soon after 1870 on
the border of Grant and Doyle townships. They speak
French still, but have no schools or church service in
the language.

MARSHALL.—Germans (Pommeranians, Hanoverians,
Frisians) to the number of 2000, settled in the west
part of Marysville township from before 1860 to 1870.
They have both church and school in their own tongue.
In the north part of Murray and the south half of
Vermillion townships are 1200 Irish, who use only
English in church and school. They came before 1870.
Bohemians in small numbers occupy the north part of
Guittard, the north west corner of Waterville and the
south part of Blue Rapids townships; Swedes a portion of
the south part of Waterville township. No report as to
their language.

MEADE.—No report.

MIAMI.—Germans occupy the north part of Wea and
the west part of Valley townships, about 200 in each
settlement; the first has a Catholic, the second a
Lutheran church. Irish occupy the north part of Osage
township, also about 200 in number.

MITCHELL.—Germans to the number of 1200 occupy
Pittsburg, Blue Hill, and Carr Creek townships. In the
first there is a church, and a well-attended school
(Catholic) at Tipton.

MONTGOMERY.—Germans to the number of 100 are
settled in and about Independence. They have church
service in German (Lutheran).

MORRIS.—Swedes occupy Diamond Valley, the west
part of Creek, and the north part of Parker townships.
They have several churches and occasionally a school
conducted in Swedish.

MORTON.—Reports no foreigners.

MCPHERSON.—Swedes settled, about 1870, in Union,
Smoky Hill, Harper, New Gottland, Delmore, and portions
of other townships, in large numbers, several thousand.
They have several churches and excellent schools
conducted in Swedish. Germans (Russian Mennonites)
occupy Superior, Turkey Creek, Mound, Lone Tree, King
City, and portions of McPherson and other southern
townships, with several churches and schools. The
Mennonites number about 5000 and settled after 1876.

NEMAHA.—Germans (Swiss) occupy Nemaha and
Washington, and a portion of Richmond townships, with
German churches and schools. Irish are in Clear Creek
and north east corner of Neuchatel townships. Most of
Neuchatel township is occupied by French (Swiss).

NEOSHO.—Germans have a considerable settlement in
the south east corner of Tioga township, with church
service (Lutheran) in German; another in the east part
of Lincoln township, where the language is spoken, but
without church or school. Swedes have settlements in the
north west part of Tioga and the east part of Big Creek
townships; church in the first only, though in both
Swedish is spoken almost exclusively.

NESS.—No report.

NORTON.—Germans to the number of 100 settled about
1880 in Grant township. They have church service in
German.

OSAGE.—Swedes, (700 in number,) settled in
Grant township in 1871, where they have four churches
conducted in Swedish. Welsh settled in 1869 in Arvonia
township, and others in the north part of Superior
township, 700 in number. They have six churches with
services in Welsh. Germans are in the north part of
Scranton and Ridgway townships, 200 in number; French
in the central part of Superior township, 200 strong;
Danes, 200, in north part of Melvern and Olivet
townships; a considerable number of Scotch and Irish in
and near Scranton. Most of these latter are engaged in
coal mining. None of the foreigners have schools—except
Sunday schools—in their native tongue.

OSBORNE.—Germans settled in Bloom township, where
they have both church and school in their mother tongue.

OTTAWA.—Bohemians are located about the border
of Sheridan and Fountain townships; Danes in the south
part of Buckeye township; Irish, arrived about 1885, in
the south part of Chapman township. None of these have
church or school in a foreign tongue.

PAWNEE.—Swedes settled about 1877 in the west part
of Garfield and the north part of Walnut townships, about
500 in all. They speak their native language at home
almost exclusively, and have preaching in it.

PHILLIPS.—Germans occupy Mound and south part of
Dayton townships, with preaching and parochial school
in German. Dutch occupy east part of Prairie View with
adjacent portions of Long Island, Dayton, and Beaver
townships, with preaching in Dutch. Some Danes and Swedes
in Crystal township, and some scattered Poles.

POTTAWATOMIE.—Germans, about 2500, in west half of
Mill Creek and adjacent portions of Sherman and Vienna
townships, also in Pottawatomie and adjacent portions of
Union, Louisville, and St. George townships. There are a
few families in Wamego and St. Mary’s Mission. They have
several schools and churches conducted in German. Swedes
occupy the whole of Blue Valley and the west border of
Greene townships, and have a small settlement in St.
Mary township, numbering in all 1200. They have church
service and a parochial school conducted in Swedish.
Irish, to the number of 2000 occupy Clear Creek, Emmet,
St. Mary and the border of St. Clere townships. French
(Canadian), numbering 200, are found in the north part
of Mill Creek and in Union townships, also a few about
St. Mary’s Mission.

PRATT.—Reports no foreigners.

RAWLINS.—Germans in north east part of county with
church and school in German. Swedes in east part of
county, Bohemians and Hungarians in north and north east
portion.

RENO.—Germans, about 300, came in 1880 to north
east corner of Little River township, and about 200 to
south east corner of Sumner township; also a settlement
in the west part of Hayes township; Dutch, about 350,
came 1878 into Haven township; Russians are settled
in Salt Creek and Medford townships. All have church
service and schools in their native tongue. There are
also a few French and Danes in the county.

REPUBLIC.—No report.

RICE.—There is a considerable settlement of
Germans in Valley township, also Pennsylvania Germans in
the west part of Sterling township, with German churches
in both. There are also some Germans in the town of
Lyons, with a German church.

RILEY.—Swedes, about 2500, occupy Jackson, Swede
Creek and adjacent portions of Mayday, Center, Fancy
Creek and Sherman townships. They have church services
and summer schools in their own tongue. Bohemians and
Germans, about 500 together, occupy the north east part
of Swede Creek township.

ROOKS.—Germans, 10 families, settled 1880 in north
part of Northhampton township. Bohemians, 10 families,
located in north part of Logan township in 1879. French,
about 30 families, south west corner of Logan, and same
number in Twin Mound township, settled in 1878, speak
French and have preaching in that tongue. The Germans
have church service in German.

RUSH.—Germans (Russian Mennonites) are located as
follows: in Big Timber township 75 families, in Illinois
township 25 families, in Pioneer township 50 families,
in Lone Star township 50 families, in Banner township
25 families, in Garfield township 25 families, in Belle
Prairie township 30 families. In each township there is
one church or more, but no German schools (?). Bohemians
are found in Banner and Garfield townships, about 25
families in each.

RUSSELL.—No report.

SALINE.—Germans, (Bavarians and Swabians) about
200, in Gypsum and south part of Ohio townships; Swedes,
3000 to 4000, in Washington, Smolan, Falun, Liberty and
Smoky View, and adjacent parts of Spring Creek, Summit
and Walnut townships, also in Salina. The Swedes came
in 1868. Both Germans and Swedes have preaching and the
latter have schools in their tongue.

SCOTT.—No report.

SEDGWICK.—Germans, 3000 to 4000, settled from
1874-82 in Sherman, Grand River, Garden Plain, Attica
and Union townships. Also about 2000 Germans in the
city of Wichita. In both places schools and churches in
German. Russians, Italians, French and Scandinavians
are represented, a few hundred each, in Wichita. In the
country townships a few Dutch and Swedes.

SEWARD.—Reports no foreigners.

SHAWNEE.—Germans (Moravians) in Rossville
township, speak their native tongue almost exclusively,
but have neither school nor preaching in German.

SHERIDAN.—No reports.

SHERMAN.—Germans, 20 families about the center of
the county. Swedes, 10 families in north east corner
and 25 families in south west corner. Both Germans and
Swedes have schools and preaching in their native tongue.

SMITH.—Germans in west part of Swan and Cedar
townships, and on border of Harvey and Banner townships,
in both churches, and in the first schools, in German.
Dutch, in the south half of Lincoln township, have
church but no schools.

STAFFORD.—Germans in Hayes and Cooper townships,
three hundred in number, with two churches having
service in German.

STANTON.—A few scattered Germans.

STEVENS.—No report.

SUMNER.—No report.

THOMAS.—A few foreigners scattered about the
country; all anglicised.

TREGO.—No report.

WABAUNSEE.—Germans and some Swedes in Kaw,
Newbury, Mill, Farmer, Alma and Washington townships,
with both preaching and schools in the mother tongue.

WALLACE.—Swedes, to the number of 300, have
settled since 1888 in the south west corner of the
county. They have church and schools in Swedish.

WASHINGTON.—Germans in Franklin, Charleston,
Hanover and north part of Sherman townships, have both
church and schools (6) conducted in German. Bohemians
are numerous in Little Blue township; French about
midway in Sherman township; Irish in Barnes, south part
of Sherman and Koloko townships.

WICHITA.—No report.

WILSON.—Swedes have settled since 1870 in Colfax
township. They have preaching but no schools in Swedish.

WOODSON.—No report.

WYANDOTTE.—Germans, 150, in north west corner of
Prairie township; Swedes, 350, in Kansas City, Kas.;
both have church service in the native language. Welsh,
200, in Rosedale, and Irish about midway in Wyandotte
township.

SUMMARIES.

There are German settlements of thirty or more persons
in the following counties: Allen, Anderson, Butler,
Chase, Chautauqua, Cherokee, Cheyenne, Coffey, Comanche,
Cowley, Crawford, Decatur, Dickinson, Doniphan, Douglas,
Edwards, Elk, Ellis, Ellsworth, Ford, Garfield, Geary,
Greenwood, Harper, Harvey, Hodgeman, Jefferson, Kingman,
Leavenworth, Lincoln, Marion, Marshall, Miami, Mitchell,
Montgomery, McPherson, Nemaha, Neosho, Norton, Osage,
Osborne, Phillips, Pottawatomie, Rawlins, Reno, Rice,
Riley, Rooks, Rush, Saline, Sedgwick, Shawnee, Sherman,
Smith, Stafford, Stanton, Thomas, Wabaunsee, Washington,
Wyandotte. Total, 60.

Scandinavians in settlements of thirty or over are
found in: Allen, Chautauqua, Cherokee, Cheyenne, Cloud,
Cowley, Crawford, Decatur, Dickinson, Doniphan, Edwards,
Elk, Gove, Greeley, Greenwood, Hodgeman, Jackson,
Jewell, Labette, Lincoln, Logan, Lyon, Marshall, Morris,
McPherson, Neosho, Osage, Ottawa, Pawnee, Phillips,
Pottawatomie, Rawlins, Riley, Saline, Sedgwick, Sherman,
Wabaunsee, Wallace, Wilson, Wyandotte. Total, 40.

Settlements of Slavonic peoples, Bohemians, Poles,
Russians, or Hungarians, in: Decatur, Ellsworth, Harper,
Lincoln, Marshall, Ottawa, Phillips, Rawlins, Reno,
Riley, Rooks, Rush, Sedgwick, Washington. Total, 14.

Settlements of Irish have been made in: Anderson, Cloud,
Crawford, Dickinson, Doniphan, Elk, Geary, Jackson,
Kingman, Marshall, Miami, Nemaha, Osage, Ottawa,
Pottawatomie, Washington, Wyandotte. Total, 17.

French are found in settlements of thirty or more in:
Cherokee, Cloud, Crawford, Doniphan, Graham, Harper,
Harvey, Nemaha, Osage, Pottawatomie, Rooks, Sedgwick,
Washington. Total, 13.

Italians are in Cherokee, Crawford, Sedgwick. Total, 3.

Welsh in Lyon, Osage and Wyandotte. Total, 3.

Dutch in Phillips, Reno, Sedgwick. Total, 3.

Scotch are reported from Cherokee and Osage. Total, 2.

English in Geary and Doniphan. Total, 2.

The following counties report that there are no
settlements of people of foreign birth within their
borders: Atchison, Barber, Bourbon, Clarke, Finney,
Grant, Gray, Hamilton, Haskell, Lane, Linn, Morton,
Pratt, Seward. Total, 14.

No reports have been secured from the following
counties: Barton, Brown, Clay, Franklin, Johnson,
Kearney, Kiowa, Meade, Ness, Republic, Russell, Scott,
Sheridan, Stevens, Sumner, Trego, Wichita, Woodson.
Total, 18.

Seventy-four of our Kansas counties report settlements of citizens of foreign birth in numbers above 30. In so many cases there is no report or estimate of numbers that it is not worth while to give summaries. Probably there are not actually ten counties that have not such settlements.

Church services in a foreign tongue are held as follows:
Allen S.,[4] Anderson G., Butler G., Chase G., Cheyenne
G., Cherokee G., Cloud F. S., Coffey G., Decatur G.,
Dickinson G. S., Doniphan G., Douglas G., Edwards G. S.,
Ellis G., Ellsworth G., Ford G., Geary G., Graham F.,
Greeley S., Greenwood G. S., Harper G. Hung., Harvey
G., Hodgeman G., Jefferson G., Leavenworth G., Lincoln
G. Du., Logan S., Lyon W. G., Marion G. Boh., Marshall
G., Miami G., Mitchell G., Montgomery G., Morris S.,
McPherson S. G., Nemaha G., Neosho G. S., Norton G.,
Osage S. Welsh, Osborne G., Pawnee S., Phillips G.
Du., Pottawatomie G. S., Rawlins G., Reno G. Du. Rus.,
Rice G., Riley S., Rooks F. G., Rush G., Saline G. S.,
Sedgwick G., Sherman G. S., Smith G. Du., Stafford G.,
Wabaunsee G., Wallace S., Washington G. Wilson S.,
Wyandotte G. S. Total, 58.

This total of fifty-eight counties in which church service is held in a foreign tongue does not at all indicate the number of such churches. In many of the reports received the number is not given, or merely in the plural. These very incomplete reports indicate one hundred thirty-eight such churches; it is safe to say that the number is nearly double this.

More interesting is the number of schools conducted in a
foreign tongue. The counties having them are: Allen S.,
Anderson G., Chase G., Cheyenne G., Cherokee, G., Cloud
F., Dickinson G. S., Douglas G., Ellis G., Ellsworth G.,
Ford G., Geary G., Greeley S., Harvey G., Leavenworth
G., Lincoln G. S., Logan S., Marion G., Marshall G.,
Mitchell G., Morris S., McPherson S. G., Nemaha G.,
Osborne G., Phillips G., Pottawatomie G. S., Rawlins G.,
Reno G. Du. Rus., Riley S., Rush G., Saline S., Sedgwick
G., Sherman G. S., Smith G. Du., Wabaunsee G., Wallace
S., Washington G.
Total, 37.

The number of separate schools in a foreign language so far as reported is seventy-four, and here, too, it is safe to say that the actual number is much larger.

EXPLANATION.

The spaces indicating settlements are in many cases too small to admit a complete description of the inhabitants, and accordingly they have been marked by races rather than by nationalities and tribes. “German” is made to do duty for all inhabitants of Germany whether Low or High, as well as for Austrians, German Swiss, and Russo-German Mennonites. The last are reported simply as Mennonites, but are, I believe, in all cases of German origin. “Scandinavian” is used instead of Swede, Norwegian and Dane, because in some cases the distinction was not made in the reports, and in order to limit the number of colors on the map which is to come. In the case of Scotch I have been unable to secure information whether they are Highlanders or Lowlanders, and in case of Irish, to what extent, if at all, they speak the old Irish language.

W. H. CARRUTH.

B - Bohemians (in a few cases other Slavs)

G - Germans (including Dutch and Russian Mennonites)

S - Scandinavians (Danes, Swedes and Norwegians)

I - Irish W - Welsh It - Italians F - French

H - Hungarians]

Footnote:

[4] G = German, S = Scandinavian, F = French, W = Welsh, Du = Dutch.

The Great Spirit Spring Mound.

BY E. H. S. BAILEY.

The “Waconda” or Great Spirit Spring, which is situated in Mitchell County, Kansas, about two miles east of Cawker City, has been described in detail by G. E. Patrick in vol. vii, p. 22, Transactions of the Kansas Academy of Science. An analysis of the water, and of the rock forming the mound on which the spring is located, is also given.

The spring is upon a conical, limestone mound 42 feet in height, and 150 feet in diameter at the top. The pool itself is a nearly circular lake about 50 feet in diameter, 35 feet deep, and the water rises to within a few inches of the top of the basin. There is a level space on all sides of the spring so wide that a carriage can be readily driven around it.

There is but little indication of organic matter in the water of the large spring, though there is a slimy white deposit adhering to the bottom and sides, but the water is colorless, clear, and transparent. The excess of water, instead of overflowing the bank, escapes by numerous small fissures, from 10 to 20 feet down on the sides, especially on the side away from the bluff. In these lateral springs there is an abundance of green algæ, and a whitish scum, which seems to be detached from the bottom and to float to the surface. This has a slimy, granular feeling suggesting in a very marked manner hydrated silica.

The mound is situated within about 200 feet of a limestone bluff, which rises perhaps 20 feet above the level of the spring. The natural inference would be that the harder material of the mound protected it from the erosion which carried away the rock in the valley of the Solomon on the south, and the rock between the spring and the bluff.

Is it not possible however that the mound has been really made by the successive deposits from the spring? Although the mound is plainly stratified, this need not interfere with the theory, for the water may have been intermittent in its flow. The rock is very porous, and on being ground to a thin section is shown to be concretionary in structure.

An analysis of the water of the spring (loc. cit.) showed that it contained over 1120 grains of mineral matter per gallon, of which 775 grains were sodium chloride and 206 grains sodium sulphate, with 66 grains of magnesium sulphate, 41 grains of magnesium carbonate, and 31 grains of calcium carbonate. An analysis by the author shows that there are 0.874 grains of silica.

Samples of the rock composing the mound, and of the adjoining bluff were secured, and comparative analyses made, with the following results:

GREAT
COUNTRY SPIRIT
ROCK. MOUND.

Silica and insoluble residue 2.14 4.10
Oxides of Iron and Alumina 3.22 [5]2.66
Sulphuric Anhydride .00 0.34
Carbon Dioxide 40.90 39.10
Calcium Oxide 51.90 49.28
Magnesium Oxide 0.63 1.15
Water and organic matter, undetermined 1.21 [6]3.37
—————— ——————
100.00 100.00

Specific gravity 2.52 2.79

The rocks are entirely different in appearance and structure, that of the mound being twice as hard as that of the bluff. The former contains much organic matter as is shown by blackening when it is heated in a tube and giving off the characteristic odor. The iron is practically of the ferrous variety, probably combined with carbonic acid, and the rock contains traces of chlorides. The particular sample taken was at some distance from the spring, and had been thoroughly exposed to the weather.

The rock of the mound is of just such a character as might have been built up by deposition from the water, as it contains the least soluble constituents of the water. The process of solidification would have been assisted by the silica in the water, forming insoluble cementing silicates, as noticed by Prof. Patrick. The analysis given above shows that there is abundant silica in the water for this purpose.

Mention has been made of the organic growth in the adjacent springs. The mixed scum on being heated changes from a dull green to a vivid grass-green, and if ignited it swells up and emits an ill-smelling vapor, which is evidently nitrogenous in its character. A grayish white ash is left, which contains much carbonate of lime. This is evidently freshly deposited, as it is entangled in the algæ in granular lumps.

A specimen of the white scum, noticed above, only slightly mixed with the green algæ, was analyzed. The acid solution of the ash contains 1.26 per cent of soluble silica. This was of course a combined silica, probably calcium silicate, which becomes the cementing material in the rock. In another sample of ash, after removing all the substances soluble in hot water, the residue was found to contain 76.46 per cent of silica.

The siliceous residue from the scum was examined by Dr. S. W. Williston. It consists mostly of diatoms. He recognized

Navicula— 2 species
Nitzschia— 2 species
Asteronella— 1 species.

All three genera are found both in fresh and salt or brackish water.

The green material consists essentially of Oscillaria and Confervæ. If the scum is allowed to stand for a short time a very strong sulphuretted odor is developed, strangely suggestive of salt water marshes or mud flats; and indeed the same odor is noticed in the vicinity of the spring. No characteristic salt water organisms, that should occasion this peculiar odor have, however, yet been observed here. A more extended and special study of the organic life of these interior salt water marshes and springs would be of great interest.

Footnotes:

[5] Mostly FeO, and so calculated.

[6] With alkalies.

On Pascal’s Limaçon and the Cardioid.

BY H. C. RIGGS.

The inverse of a conic with respect to a focus is a curve called Pascal’s Limaçon. From the polar equation of a conic, the focus being the pole, it is evident that the polar equation of the limaçon may be written in the form:

e 1
r = ——— cos_x_ + ——— ;
p p

where e and p are constants, being respectively the eccentricity and semi-latus rectum of the conic.

From the above equation it is readily seen that the curve may be traced by drawing from a fixed point O on a circle any number of chords and laying off a constant length on each of these lines, measured from the circumference of the circle. The point O is the node of the limaçon; and the fixed circle, which I shall call the base circle, is the inverse of the directrix of the conic. This is readily shown as follows:—the polar equation of the directrix is r = p/(e cos_x_). Hence the equation of its inverse is r = (e cos_x_)/p, which is the equation of the base circle of the limaçon.

If the conic which we invert be an ellipse, the point O will be an acnode on the Limaçon; if the conic be a hyperbola, the point O is a crunode. If the conic be a parabola, O is then a cusp and the inverse curve is called the Cardioid.

The limaçon may also be traced as a roulette.