Chapter XII: Part 12

Experiment also, however, gives a result which, at least at first view, might not have been anticipated from the mode of illustration we have adopted. It is found that the resistance arising from friction does not at all depend on the magnitude of the surface of contact; but provided the nature of the surfaces and the amount of pressure remain the same, this resistance will be equal, whether the surfaces which move one upon the other be great or small. Thus, if the moving body be a flat block of wood, the face of which is equal to a square foot in magnitude, and the edge of which does not exceed a square inch, it will be subject to the same amount of friction, whether it move upon its broad face or upon its narrow edge. If we consider the effect of the pressure in each case, we shall be able to perceive why this must be the case. Let us suppose the weight of the block to be 144 ounces. When it rests upon its face, a pressure to this amount acts upon a surface of 144 square inches, so that a pressure of one ounce acts upon each square inch. The total resistance arising from friction will, therefore, be 144 times that resistance which would be produced by a surface of one square inch under a pressure of one ounce. Now, suppose the block placed upon its edge, there is then a pressure of 144 ounces upon a surface equal to one square inch. But it has been already shown, that when the surface is the same, the friction must increase in proportion to the pressure. Hence we infer that the friction produced in the present case is 144 times the friction which would be produced by a pressure of one ounce acting on one square inch of surface, which is the same resistance as that which the body was proved to be subject to when resting on its face.

These two laws, that friction is independent of the magnitude of the surface, and is proportional to the pressure when the quality of the surfaces is the same, are useful in practice, and _generally_ true. In very extreme cases they are, however, in error. When the pressure is very intense, in proportion to the surface, the friction is somewhat _less_ than it would be by these laws; and when it is very small in proportion to the surface, it is somewhat _greater_.

(323.) There are two methods of establishing by experiment the laws of friction, which have been just explained.

First. The surfaces between which the friction is to be determined being rendered perfectly flat, let one be fixed in the horizontal position on a table T T′, _fig. 176._; and let the other be attached to the bottom of a box B C, adapted to receive weights, so as to vary the pressure. Let a silken cord S P, attached to the box, be carried parallel to the table over a wheel at P, and let a dish D be suspended from it. If no friction existed between the surfaces, the smallest weight appended to the cord would draw the box towards P with a continually increasing speed. But the friction which always exists interrupts this effect, and a small weight may act upon the string without moving the box at all. Let weights be put in the dish D, until a sufficient force is obtained to overcome the friction without giving the box an accelerated motion. Such a weight is equivalent to the amount of the friction.

The amount of the weight of the box being previously ascertained, let this weight be now doubled by placing additional weights in the box. The pressure will thus be doubled, and it will be found that the weight of the dish D and its load, which before was able to overcome the friction, is now altogether inadequate to it. Let additional weights be placed in the dish until the friction be counteracted as before, and it will be observed, that the whole weight necessary to produce this effect is exactly twice the weight which produced it in the former case. Thus it appears that a double amount of pressure produces a double amount of friction; and in a similar way it may be proved, that any proposed increase or decrease of the pressure will be attended with a proportionate variation in the amount of the friction.

Second. Let one of the surfaces be attached to a flat plane A B, _fig. 177._, which can be placed at any inclination with an horizontal plane B C, the other surface being, as before, attached to the box adapted to receive weights. The box being placed upon the plane, let the latter be slightly elevated. The tendency of the box to descend upon A B, will bear the same proportion to its entire weight as the perpendicular A E bears to the length of the plane A B (286.). Thus if the length A B be 36 inches, and the height A E be three inches, that is a twelfth part of the length, then the tendency of the weight to move down the plane is equal to a twelfth part of its whole amount. If the weight were twelve ounces, and the surfaces perfectly smooth, a force of one ounce acting up the plane would be necessary to prevent the descent of the weight.

In this case also the pressure on the plane will be represented by the length of the base B E (286.), that is, it will bear the same proportion to the whole weight as B E bears to B A. The relative amounts of the weight, the tendency to descend, and the pressure, will always be exhibited by the relative lengths of A B, A E, and B E.

This being premised, let the elevation of the plane A B be gradually increased until the tendency of the weight to descend just overcomes the friction, but not so much as to allow the box to descend with accelerated speed. The proportion of the whole weight, which then acts down the plane, will be found by measuring the height A E, and the pressure will be determined by measuring the base B E. Now let the weight in the box be increased, and it will be found that the same elevation is necessary to overcome the friction; nor will this elevation suffer any change, however the pressure or the magnitude of the surfaces which move in contact may be varied.

Since, therefore, in all these cases, the height A E and the base B E remain the same, it follows that the proportion between the friction and pressure is undisturbed.

(324.) The law that friction is proportional to the pressure, has been questioned by the late professor Vince of Cambridge, who deduced from a series of experiments, that although the friction increases with the pressure, yet that it increases in a somewhat less ratio; and from this it would follow, that the variation of the surface of contact must produce some effect upon the amount of friction. The law, as we have explained it, however, is sufficiently near the truth for most practical purposes.

(325.) There are several circumstances regarding the quality of the surfaces which produce important effects on the quantity of friction, and which ought to be noticed here.

This resistance is different in the surfaces of different substances. When the surfaces are those of wood newly planed, it amounts to about half the pressure, but is different in different kinds of wood. The friction of metallic surfaces is about one fourth of the pressure.

In general the friction between the surfaces of bodies of different kinds is less than between those of the same kind. Thus, between wood and metal the friction is about one fifth of the pressure.

It is evident that the smoother the surfaces are the less will be the friction. On this account, the friction of surfaces, when first brought into contact, is often greater than after their attrition has been continued for a certain time, because that process has a tendency to remove and rub off those minute asperities and projections on which the friction depends. But this has a limit, and after a certain quantity of attrition the friction ceases to decrease. Newly planed surfaces of wood have at first a degree of friction which is equal to half the entire pressure, but after they are worn by attrition it is reduced to a third.

If the surfaces in contact be placed with their grains in the same direction, the friction will be greater than if the grains cross each other.

Smearing the surfaces with unctuous matter diminishes the friction, probably by filling the cavities between the minute projections which produce the friction.

When the surfaces are first placed in contact, the friction is less than when they are suffered to rest so for some time; this is proved by observing the force which in each case is necessary to move the one upon the other, that force being less if applied at the first moment of contact than when the contact has continued. This, however, has a limit. There is a certain time, different in different substances, within which this resistance attains its greatest amount. In surfaces of wood this takes place in about two minutes; in metals the time is imperceptibly short; and when a surface of wood is placed upon a surface of metal, it continues to increase for several days. The limit is larger when the surfaces are great, and belong to substances of different kinds.

The velocity with which the surfaces move upon one another produces but little effect upon the friction.

(326.) There are several ways in which bodies may move one upon the other, in which friction will produce different effects. The principal of these are, first, the case where one body _slides_ over another; the second, where a body having a round form _rolls_ upon another; and, _thirdly_, where an axis revolves within a hollow cylinder, or the hollow cylinder revolves upon the axis.

With the same amount of pressure and a like quality of surface, the quantity of friction is greatest in the first case and least in the second. The friction in the second case also depends on the diameter of the body which rolls, and is small in proportion as that diameter is great. Thus a carriage with large wheels is less impeded by the friction of the road than one with small wheels.

In the third case, the leverage of the wheel aids the power in overcoming the friction. Let _fig. 178._ represent a section of the wheel and axle; let C be the centre of the axle, and let B E be the hollow cylinder in the nave of the wheel in which the axle is inserted. If B be the part on which the axle presses, and the wheel turn in the direction N D M, the friction will act at B in the direction B F, and with the leverage B C. The power acts against this at D in the direction D A, and with the leverage D C. It is therefore evident, that as D C is greater than B C, in the same proportion does the power act with mechanical advantage on the friction.

(327.) Contrivances for diminishing the effects of friction depend on the properties just explained, the motion of rolling being as much as possible substituted for that of sliding; and where the motion of rolling cannot be applied, that of a wheel upon its axle is used. In some cases both these motions are combined.

If a heavy load be drawn upon a plane in the manner of a sledge, the motion will be that of sliding, the species which is attended with the greatest quantity of friction; but if the load be placed upon cylindrical rollers, the nature of the motion is changed, and becomes that in which there is the least quantity of friction. Thus large blocks of stone, or heavy beams of timber, which would require an enormous power to move them on a level road, are easily advanced when rollers are put under them.

When very heavy weights are to be moved through small spaces, this method is used with advantage; but when loads ore to be transported to considerable distances, the process is inconvenient and slow, owing to the necessity of continually replacing the rollers in front of the load as they are left behind by its progressive advancement.

The wheels of carriages may be regarded as rollers which are continually carried forward with the load. In addition to the friction of the rolling motion on the road, they have, it is true, the friction of the axle in the nave; but, on the other hand, they are free from the friction of the rollers with the under surface of the load, or the carriage in which the load is transported. The advantages of wheel carriages in diminishing the effects of friction is sometimes attributed to the slowness with which that axle moves within the box, compared with the rate at which the wheel moves over the road; but this is erroneous. The quantity of friction does not in any case vary considerably with the velocity of the motion, but least of all does it in that particular kind of motion here considered.

In certain cases, where it is of great importance to remove the effects of friction, a contrivance called _friction-wheels_, or friction-rollers, is used. The axle of a friction-wheel, instead of revolving within a hollow cylinder, which is fixed, rests upon the edges of wheels which revolve with it; the species of motion thus becomes that in which the friction is of least amount.

Let A B and D C, _fig. 179._, be two wheels revolving on pivots P Q with as little friction as possible, and so placed that the axle O of a third wheel E F may rest between their edges. As the wheel E F revolves, the axle O, instead of grinding its surface on the surface on which it presses, carries that surface with it, causing the wheels A B, C D, to revolve.

In wheel carriages, the roughness of the road is more easily overcome by large wheels than by small ones. The cause of this arises partly from the large wheels not being so liable to sink into holes as small ones, but more because, in surmounting obstacles, the load is elevated less abruptly. This will be easily understood by observing the curves in _fig. 180._, which represent the elevation of the axle in each case.

(328.) If a carriage were capable of moving on a road without friction, the most advantageous direction in which a force could be applied to draw it would be parallel to the road. When the motion is impeded by friction, it is better, however, that the line of draught should be inclined to the road, so that the drawing force may be expended partly in lessening the pressure on the road, and partly in advancing the load.

Let W, _fig. 181._, be a load which is to be moved upon the plane surface A B. If the drawing force be applied in the direction C D, parallel to the plane A B, it will have to overcome the friction produced by the pressure of the whole weight of the load upon the plane; but if it be inclined upwards in the direction C E, it will be equivalent to two forces expressed (74.) by C G and C F. The part C G has the effect of lightening the pressure of the carriage upon the road, and therefore of diminishing the friction in the same proportion. The part C F draws the load along the plane. Since C F is less than C E or C D the whole moving force, it is evident that a part of the force of draught is lost by this obliquity; but, on the other hand, a part of the opposing resistance is also removed. If the latter exceed the former, an advantage will be gained by the obliquity; but if the former exceed the latter, force will be lost.

By mathematical reasoning, founded on these considerations, it is proved that the best angle of draught is exactly that obliquity which should be given to the road in order to enable the carriage to move of itself. This obliquity is sometimes called the _angle of repose_, and is that angle which determines the proportion of the friction to the pressure in the second method, explained in (323.). The more rough the road is, the greater will this angle be; and therefore it follows, that on bad roads the obliquity of the traces to the road should be greater than on good ones. On a smooth Macadamised way, a very slight declivity would cause a carriage to roll by its own weight: hence, in this case, the traces should be nearly parallel to the road.

In rail roads, for like reasons, the line of draught should be parallel to the road, or nearly so.

(329.) When ropes or cords form a part of machinery, the effects of their imperfect flexibility are in a certain degree counteracted by bending them over the grooves of wheels. But although this so far diminishes these effects as to render ropes practically useful, yet still, in calculating the powers of machinery, it is necessary to take into account some consequences of the rigidity of cordage which even by these means are not removed.

To explain the way in which the stiffness of a rope modifies the operation of a machine, we shall suppose it bent over a wheel and stretched by weights A B, _fig. 182._, at its extremities. The weights A and B being equal, and acting at C and D in opposite ways, balance the wheel. If the weight A receive an addition, it will overcome the resistance of B, and turn the wheel in the direction D E C. Now, for the present, let us suppose that the rope is perfectly inflexible; the wheel and weights will be turned into the position represented in _fig. 183._ The leverage by which A acts will be diminished, and will become O F, having been before O C; and the leverage by which B acts will be increased to O G, having been before O D.

But the rope not being inflexible will yield partially to the effects of the weights A and B, and the parts A C and B D will be bent into the forms represented in _fig. 184._ The form of the curvature which the rope on each side of the wheel receives is still such that the descending weight A works with a diminished leverage F O, while the ascending weight resists it with an increased leverage G O. Thus so much of the moving power is lost, by the stiffness of the rope, as is necessary to compensate this disadvantageous change in the power of the machine.

CHAP. XX.

ON THE STRENGTH OF MATERIALS.

(330.) Experimental enquiries into the laws which regulate the strength of solid bodies, or their power to resist forces variously applied to tear or break them, are obstructed by practical difficulties, the nature and extent of which are so discouraging that few have ventured to encounter them at all, and still fewer have had the steadiness to persevere until any result showing a general law has been obtained. These difficulties arise, partly from the great forces which must be applied, but more from the peculiar nature of the objects of those experiments. The end to which such an enquiry must be directed is the development of a _general law_; that is, such a rule as would be rigidly observed if the materials, the strength of which is the object of enquiry, were perfectly uniform in their texture, and subject to no casual inequalities. In proportion as these inequalities are frequent, experiments must be multiplied, that a long average may embrace cases varying in both extremes, so as to eliminate each other’s effects in the final result.

The materials of which structures and works of art are composed are liable to so many and so considerable inequalities of texture, that any rule which can be deduced, even by the most extensive series of experiments, must be regarded as a mean result, from which individual examples will be found to vary in so great a degree, that more than usual caution must be observed in its practical application. The details of this subject belong to engineering, more properly than to the elements of mechanics. Nevertheless, a general view of the most important principles which have been established respecting the strength of materials will not be misplaced in this treatise.

A piece of solid matter may be submitted to the action of a force tending to separate its parts in several ways; the principal of which are,--

1. To a _direct pull_,--as when a rope or wire is stretched by a weight. When a tie-beam resists the separation of the sides of a structure, &c.

2. To a direct pressure or thrust,--as when a weight rests upon a pillar.

3. To a transverse strain,--as when weights on the ends of a lever press it on the fulcrum.

(331.) If a solid be submitted to a force which draws it in the direction of its length, having a tendency to pull its ends in opposite directions, its strength or power to resist such a force is proportional to the magnitude of its transverse section. Thus, suppose a square rod of metal A B, _fig. 185._, of the breadth and thickness of one inch, be pulled by a force in the direction A B, and that a certain force is found sufficient to tear it; a rod of the same metal of twice the breadth and the same thickness will require double the force to break it; one of treble the breadth and the same thickness will require treble the force to break it, and so on.

The reason of this is evident. A rod of double or treble the thickness, in this case, is equivalent to two or three equal and similar rods which equally and separately resist the drawing force, and therefore possess a degree of strength proportionate to their number.

It will easily be perceived, that whatever be the section, the same reasoning will be applicable, and the power of resistance will, in general, be proportional to its magnitude or area.

If the material were perfectly uniform throughout its dimensions, the resistance to a direct pull would not be affected by the length of the rod. In practice, however, the increase of length is found to lessen the strength. This is to be attributed to the increased chance of inequality.

(332.) No satisfactory results have been obtained either by theory or experiment respecting the laws by which solids resist compression. The power of a perpendicular pillar to support a weight placed upon it evidently depends on its thickness, or the magnitude of its base, and on its height. It is certain that when the height is the same, the strength increases with every increase of the base, but it seems doubtful whether the strength be exactly proportional to the base. That is, if two columns of the same material have equal heights, and the base of one be double the base of the other, the strength of one will be greater, but it is not certain whether it will exactly double that of the other. According to the theory of Euler, which is in a certain degree verified by the experiments of Musschenbrock, the strength will be increased in a greater proportion than the base, so that, if the base be doubled, the strength will be more than doubled.

When the base is the same, the strength is diminished by increasing the height, and this decrease of strength is proportionally greater than the increase of height. According to Euler’s theory, the decrease of strength is proportional to the square of the height; that is, when the height is increased in a two-fold proportion, the strength is diminished in a four-fold proportion.

(333.) The strain to which solids forming the parts of structures of every kind are most commonly exposed is the lateral or transverse strain, or that which acts at right angles to their lengths. If any strain act obliquely to the direction of their length it may be resolved into two forces (76.), one in the direction of the length, and the other at right angles to the length. That part which acts in the direction of the length will produce either compression or a direct pull, and its effect must be investigated accordingly.

Although the results of theory, as well as those of experimental investigations, present great discordances respecting the transverse strength of solids, yet there are some particulars, in which they, for the most part, agree; to this it is our object here to confine our observations, declining all details relating to disputed points.

Let A B C D, _fig. 186._, be a beam, supported at its ends A and B. Its strength to support a weight at E pressing downwards at right angles to its length is evidently proportional to its breadth, the other things being the same. For a beam of double or treble breadth, and of the same thickness, is equivalent to two or three equal and similar beams placed side by side. Since each of these would possess the same strength, the whole taken together would possess double or treble the strength of any one of them.

When the breadth and length are the same the strength obviously increases with the depth, but not in the same proportion. The increase of strength is found to be much greater in proportion than the increase of depth. By the theory of Galileo, a double or treble thickness ought to increase the strength in a four-fold or nine-fold proportion, and experiments in most cases do not materially vary from this rule.

If while the breadth and depth remain the same, the length of the beam, or rather, the distance between the points of support, vary, the strength will vary accordingly, decreasing in the same proportion as the length increases.

From these observations it appears, that the transverse strength of a beam depends more on its thickness than its breadth. Hence we find that a broad thin board is much stronger when its edge is presented upwards. On this principle the joists or rafters of floors and roofs are constructed.

If two beams be in all respects similar, their strengths will be in the proportion of the squares of their lengths. Let the length, breadth, and depth of the one be respectively double the length, breadth, and depth of the other. By the double breadth the beam doubles its strength, but by doubling the length half this strength is lost. Thus the increase of length and breadth counteract each other’s effects, and as far as they are concerned the strength of the beam is not changed. But by doubling the thickness the strength is increased in a four-fold proportion, that is, as the square of the length. In the same manner it may be shown, that when all the dimensions are trebled, the strength is increased in a nine-fold proportion, and so on.

(334.) In all structures the materials have to support their own weight, and therefore their available strength is to be estimated by the excess of their absolute strength above that degree of strength which is just sufficient to support their own weight. This consideration leads to some conclusions, of which numerous and striking illustrations are presented in the works of nature and art.

We have seen that the absolute strength with which a lateral strain is resisted is in the proportion of the square of the linear dimensions of similar parts of a structure, and therefore the amount of this strength increases rapidly with every increase of the dimensions of a body. But at the same time the weight of the body increases in a still more rapid proportion. Thus, if the several dimensions be doubled, the strength will be increased in a four-fold but the weight in an eight-fold proportion. If the dimensions be trebled, the strength will be multiplied nine times, but the weight twenty-seven times. Again, if the dimensions be multiplied four times, the strength will be multiplied sixteen times, and the weight sixty-four times, and so on.

Hence it is obvious, that although the strength of a body of small dimensions may greatly exceed its weight, and, therefore, it may be able to support a load many times its own weight; yet by a great increase in the dimensions the weight increasing in a much greater degree the available strength may be much diminished, and such a magnitude may be assigned, that the weight of the body must exceed its strength, and it not only would be unable to support any load, but would actually fall to pieces by its own weight.

The strength of a structure of any kind is not, therefore, to be determined by that of its model, which will always be much stronger in proportion to its size. All works natural and artificial have limits of magnitude which, while their materials remain the same, they cannot surpass.

In conformity with what has just been explained, it has been observed, that small animals are stronger in proportion than large ones; that the young plant has more available strength in proportion than the large forest tree; that children are less liable to injury from accident than men, &c. But although to a certain extent these observations are just, yet it ought not to be forgotten, that the mechanical conclusions which they are brought to illustrate are founded on the supposition, that the smaller and greater bodies which are compared are composed of precisely similar materials. This is not the case in any of the examples here adduced.

CHAP. XXI.

ON BALANCES AND PENDULUMS.

(335.) The preceding chapters have been confined almost wholly to the consideration of the laws of mechanics, without entering into a particular description of the machinery and instruments dependant upon those laws. Such descriptions would have interfered too much with the regular progress of the subject, and it therefore appeared preferable to devote a chapter exclusively to this portion of the work.

Perhaps there are no ideas which man receives through the medium of sense which may not be referred ultimately to matter and motion. In proportion, therefore, as he becomes acquainted with the properties of the one and the laws of the other, his knowledge is extended, his comforts are multiplied; he is enabled to bend the powers of nature to his will, and to construct machinery which effects with ease that which the united labour of thousands would in vain be exerted to accomplish.

Of the properties of matter, one of the most important is its weight, and the element which mingles inseparably with the laws of motion is time.

In the present chapter it is our intention to describe such instruments as are usually employed for determining the weight of bodies. To attempt a description of the various machines which are used for the measurement of time, would lead us into too wide a field for the present occasion, and we shall, therefore, confine ourselves to an account of the methods which have been practised to perfect, to perfect that instrument which affords the most correct means of measuring time, the pendulum.

The instrument by which we are enabled to determine, with greater accuracy than by any other means, the relative weight of a body, compared with the weight of another body assumed as a standard, is the balance.

_London, Pubd. by Longman & Co._]

_Of the Balance._

The balance may be described as consisting of an inflexible rod or lever, called the beam, furnished with three axes; one, the fulcrum or centre of motion situated in the middle, upon which the beam turns, and the other two near the extremities, and at equal distances from the middle. These last are called the points of support, and serve to sustain the pans or scales.

The points of support and the fulcrum are in the same right line, and the centre of gravity of the whole should be a little below the fulcrum when the position of the beam is horizontal.

The arms of the lever being equal, it follows that if equal weights be put into the scales no effect will be produced on the position of the balance, and the beam will remain horizontal.

If a small addition be made to the weight in one of the scales, the horizontality of the beam will be disturbed; and after oscillating for some time, it will, on attaining a state of rest, form an angle with the horizon, the extent of which is a measure of the delicacy or sensibility of the balance.

As the sensibility of a balance is of the utmost importance in nice scientific enquiries, we shall enter somewhat at large into a consideration of the circumstances by which this property is influenced.

In _fig. 187._ let A B represent the beam drawn from the horizontal position by a very small weight placed in the scale suspended from the point of support B; then the force tending to draw the beam from the horizontal position may be expressed by P B, multiplied by such very small weight acting upon the point B.

Let the centre of gravity of the whole be at G; then the force acting against the former will be G P multiplied into the weight of the beam and scales, and when these forces are equal, the beam will rest in an inclined position. Hence we may perceive that as the centre of gravity is nearer to or further from the fulcrum S, (every thing else remaining the same) the sensibility of the balance will be increased or diminished.

For, suppose the centre of gravity were removed to _g_, then to produce an opposing force equal to that acting upon the extremity of the beam, the distance _g p_ from the perpendicular line must be increased until it becomes nearly equal to G P; but for this purpose the end of the beam B must descend, which will increase the angle H S B.

As all weights placed in the scales are referred to the line joining the points of support, and as this line is above the centre of gravity of the beam when not loaded, such weights will raise the centre of gravity; but it will be seen that the sensibility of the balance, as far as it depends upon this cause, will remain unaltered.

For, calling the distance S G unity, the distance of the centre of gravity from the point S (to which the weight which has been added is referred) will be expressed by the reciprocal of the weight of the beam so increased; that is, if the weight of the beam be doubled by weights placed in the scales, S _g_ will be one half of S G; and if the weight of the beam be in like manner trebled, S _g_ will be one third of S G, and so on. And as G P varies as S G, _g p_ will be inversely proportionate to the increased weight of the beam, and consequently, the product obtained by multiplying _g p_ by the weight of the beam and its load will be a constant quantity, and the sensibility of the balance, as before stated, will suffer no alteration.

We will now suppose that the fulcrum S, _fig. 188._, is situated below the line joining the points of support, and that the centre of gravity of the beam when not loaded is at G. Also that when a very small weight is placed in the scale suspended from the point B, the beam is drawn from its horizontal position, the deviation being a measure of the sensibility of the balance. Then, as before stated, G P multiplied by the weight of the beam will be equal to P′ B multiplied by the very small additional weight acting on the point B.

Now if we place equal weights in both scales, such additional weights will be referred to the point W, and the resulting distance of the centre of gravity from the point W, calling W G unity, will be expressed as before by the reciprocal of the increased weight of the loaded beam. But G P will decrease in a greater proportion than W G: thus, supposing the weight of the beam to be doubled, W _g_ would be one half of W G; but _g p_, as will be evident on an inspection of the figure, will be less than half of G P; and the same small weight which was before applied to the point B, if now added, would depress the point B, until the distance _g p_ became such as that, when multiplied by the weight of the whole, the product would be as before equal to P′ B, multiplied by the before mentioned very small added weight. The sensibility of the balance, therefore, in this case would be increased.

If the beam be sufficiently loaded, the centre of gravity will at length be raised to the fulcrum S, and the beam will rest indifferently in any position. If more weight be then added, the centre of gravity will be raised above the fulcrum, and the beam will turn over.

Lastly, if the fulcrum S, _fig. 189._, is above the line joining the two points of support, as any additional weights placed in the scales will be referred to the point W, in the line joining A and B, if the weight of the beam be doubled by such added weights, and the centre of gravity be consequently raised to _g_, W _g_ will become equal to half of W G. But _g p_, being greater than one half of G P, the end of the beam B will rise until _g p_ becomes such as to be equal, when multiplied by the whole increased weight of the beam, to P B, multiplied by the small weight, which we suppose to have been placed as in the preceding examples, in the scale.

From what has been said it will be seen that there are three positions of the fulcrum which influence the sensibility of the balance: first, when the fulcrum and the points of support are in a right line, when the sensibility of the balance will remain the same, though the weight with which the beam is loaded should be varied: secondly, when the fulcrum is below the line joining the two points of support, in which case the sensibility of the balance will be increased by additional weights, until at length the centre of gravity is raised above the fulcrum, when the beam will turn over; and, thirdly, when the fulcrum is above the line joining the two points of support, in which case the sensibility of the balance will be diminished as the weight with which the beam is loaded is increased.

The sensibility of a balance, as here defined, is the angular deviation of the beam occasioned by placing an additional constant small weight in one of the scales; but it is frequently expressed by the proportion which such small additional weight bears to the weight of the beam and its load, and sometimes to the weight the value of which is to be determined.

This proportion, however, will evidently vary with different weights, except in the case where the centre of gravity of the beam is in the line joining the points supporting the scales, the fulcrum being above this line, and it is therefore necessary, in every other case, when speaking of the sensibility of the balance, to designate the weight with which it is loaded: thus, if a balance has a troy pound in each scale, and the horizontality of the beam varies a certain small quantity, just perceptible on the addition of one hundredth of a grain, we say that the balance is sensible to 1/1152000 part of its load with a pound in each scale, or that it will determine the weight of a troy pound within 1/576000 part of the whole.

The nearer the centre of gravity of a balance is to its fulcrum the slower will be the oscillations of the beam. The number of oscillations, therefore, made by the beam in a given time (a minute for example), affords the most accurate method of judging of the sensibility of the balance, which will be the greater as the oscillations are fewer.

Balances of the most perfect kind, and of such only it is our present object to treat, are usually furnished with adjustments, by means of which the length of the arms, or the distances of the fulcrum from the points of support, may be equalised, and the fulcrum and the two points of support be placed in a right line; but these adjustments, as will hereafter be seen, are not absolutely necessary.

The beam is variously constructed, according to the purposes to which the balance is to be applied. Sometimes it is made of a rod of solid steel; sometimes of two hollow cones joined at their bases; and, in some balances, the beam is a frame in the form of a rhombus: the principal object in all, however, is to combine strength and inflexibility with lightness.

A balance of the best kind, made by Troughton, is so contrived as to be contained, when not in use, in a drawer below the case; and when in use, it is protected from any disturbance from currents of air, by being enclosed in the case above the drawer, the back and front of which are of plate glass. There are doors in the sides, through which the scale-pans are loaded, and there is a door at the top through which the beam may be taken out.

A strong brass pillar, in the centre of the box, supports a square piece, on the front and back of which rise two arches, nearly semicircular, on which are fixed two horizontal planes of agate, intended to support the fulcrum. Within the pillar is a cylindrical tube, which slides up and down by means of a handle on the outside of the case. To the top of this interior tube is fixed an arch, the terminations of which pass beneath and outside of the two arches before described. These terminations are formed into Y _s_, destined to receive the ends of the fulcrum, which are made cylindrical for this purpose, when the interior tube is elevated in order to relieve the axis when the balance is not in use. On depressing the interior tube, the Y _s_ quit the axis, and leave it in its proper position on the agate planes. The beam is about eighteen inches long, and is formed of two hollow cones of brass, joined at their bases. The thickness of the brass does not exceed 0·02 of an inch, but by means of circular rings driven into the cones at intervals they are rendered almost inflexible. Across the middle of the beam passes a cylinder of steel, the lower side of which is formed into an edge, having an angle of about thirty degrees, which, being hardened and well polished, constitutes the fulcrum, and rests upon the agate planes for the length of about 0·05 of an inch.

Each point of suspension is formed of an axis having two sharp concave edges, upon which rest at right angles two other sharp concave edges formed in the spur-shaped piece to which the strings carrying the scale-pan are attached. The two points are adjustable, the one horizontally, for the purpose of equalising the arms of the beam, and the other vertically, for bringing the points of suspension and the fulcrum into a right line.

Such is the form of Troughton’s balance: we shall now give the description of a balance as constructed by Mr. Robinson of Devonshire Street, Portland Place:--

The beam of this balance is only ten inches long. It is a frame of bell-metal in the form of a rhombus. The fulcrum is an equilateral triangular prism of steel one inch in length; but the edge on which the beam vibrates is formed to an angle of 120°, in order to prevent any injury from the weight with which it may be loaded. The chief peculiarity in this balance consists in the knife-edge which forms the fulcrum bearing upon an agate plane throughout its whole length, whereas we have seen in the balance before described that the whole weight is supported by portions only of the knife-edge, amounting together to one tenth of an inch. The supports for the scales are knife-edges each six tenths of an inch long. These are each furnished with two pressing screws, by means of which they may be made parallel to the central knife-edge.

Each end of the beam is sprung obliquely upwards and towards the middle, so as to form a spring through which a pushing screw passes, which serves to vary the distance of the point of support from the fulcrum, and, at the same time, by its oblique action to raise or depress it, so as to furnish a means of bringing the points of support and the fulcrum into a right line.

A piece of wire, four inches long, on which a screw is cut, proceeds from the middle of the beam downwards. This is pointed to serve as an index, and a small brass ball moves on the screw, by changing the situation of which the place of the centre of gravity may be varied at pleasure.

The fulcrum, as before remarked, rests upon an agate plane throughout its whole length, and the scale-pans are attached to planes of agate which rest upon the knife-edges forming the points of support. This method of supporting the scale-pans, we have reason to believe, is due to Mr. Cavendish. Upon the lower half of the pillar to which the agate plane is fixed, a tube slides up and down by means of a lever which passes to the outside of the case. From the top of this tube arms proceed obliquely towards the ends of the balance, serving to support a horizontal piece, carrying at each extremity two sets of Y _s_, one a little above the other. The upper Y _s_ are destined to receive the agate planes to which the scale-pans are attached, and thus to relieve the knife-edges from their pressure; the lower to receive the knife-edges which, form the points of support, consequently these latter Y _s_, when in action, sustain the whole beam.

When the lever is freed from a notch in which it is lodged, a spring is allowed to act upon the tube we have mentioned, and to elevate it. The upper Y _s_ first meet the agate planes carrying the scale-pans and free them from the knife-edges. The lower Y _s_ then come into action and raise the whole beam, elevating the central knife-edge above the agate plane. This is the usual state of the balance when not in use: when it is to be brought into action, the reverse of what we have described takes place. On pressing down the lever, the central knife-edge first meets the agate plane, and afterwards the two agate planes carrying the scale-pans are deposited upon their supporting knife-edges.

A balance of this construction was employed by the writer of this article in adjusting the national standard pound. With a pound troy in each scale, the addition of one hundredth of a grain caused the index to vary one division, equal to one tenth of an inch, and Mr. Robinson adjusts these balances so that with one thousand grains in each scale, the index varies perceptibly on the addition of one thousandth of a grain, or of one-millionth part of the weight to be determined.

It may not be uninteresting to subjoin, from the Philosophical Transactions for 1826, the description of a balance perhaps the most sensible that has yet been made, constructed for verifying the national standard bushel. The author says,--

“The weight of the bushel measure, together with the 80 lbs. of water it should contain, was about 250 lbs.; and as I could find no balance capable of determining so large a weight with sufficient accuracy, I was under the necessity of constructing one for this express purpose.

“I first tried cast iron; but though the beam was made as light as was consistent with the requisite degree of strength, the inertia of such a mass appeared to be so considerable, that much time must have been lost before the balance would have answered to the small differences I wished to ascertain. Lightness was a property essentially necessary, and bulk was very desirable, in order to preclude such errors as might arise from the beam being partially affected by sudden alterations of temperature. I therefore determined to employ wood, a material in which the requisites I sought were combined. The beam was made of a plank of mahogany, about 7O inches long, 22 inches wide, and 2-1/4 thick, tapering from the middle to the extremities. An opening was cut in the centre, and strong blocks screwed to each side of the plank, to form a bearing for the back of a knife-edge which passed through the centre. Blocks were also screwed to each side at the extremities of the beam on which rested the backs of the knife-edges for supporting the pans. The opening in the centre was made sufficiently large to admit the support hereafter to be described, upon which the knife-edge rested.

“In all beams which I have seen, with the exception of those made by Mr. Robinson, the whole weight is sustained by short portions at the extremities of the knife-edge; and the weight being thus thrown upon a few points, the knife-edge becomes more liable to change its figure and to suffer injury.

“To remedy this defect, the central knife-edge of the beam I am describing was made 6 inches, and the two others 5 inches long. They were triangular prisms with equal sides of three fourths of an inch, very carefully finished, and the edges ultimately formed to an angle of 120°.

“Each knife-edge was screwed to a thick plate of brass, the surfaces in contact having been previously ground together; and these plates were screwed to the beam, the knife-edges being placed in the same plane, and as nearly equidistant and parallel to each other as could be done by construction.

“The support upon which the central knife-edge rested throughout its whole length was formed of a plate of polished hard steel, screwed to a block of cast iron. This block was passed through the opening before mentioned in the centre of the beam, and properly attached to a frame of cast iron.

“The stirrups to which the scales were hooked rested upon plates of polished steel to which they were attached, and the under surfaces of which were formed by careful grinding into cylindrical segments. These were in contact with the knife-edges their whole length, and were known to be in their proper position by the correspondence of their extremities with those of the knife-edges. A well imagined contrivance was applied by Mr. Bate for raising the beam when loaded, in order to prevent unnecessary wear of the knife-edge, and for the purpose of adjusting the place of the centre of gravity, when the beam was loaded with the weight required to be determined, a screw carrying a movable ball projected vertically from the middle of die beam.

“The performance of this balance fully equalled my expectations. With two hundred and fifty pounds in each scale, the addition of a single grain occasioned an immediate variation in the index of one twentieth of an inch, the radius being fifty inches.”

From the preceding account it appears that this balance is sensible to 1/1750000 part of the weight which was to be determined.

We shall now describe the method to be pursued in adjusting a balance.

1. To bring the points of suspension and the fulcrum into a right line.

Make the vibrations of the balance very slow by moving the weight which influences the centre of gravity, and bring the beam into a horizontal position, by means of small bits of paper thrown into the scales. Then load the scales with nearly the greatest weight the beam is fitted to carry. If the vibrations are performed in the same time as before, no further adjustment is necessary; but if the beam vibrates quicker, or if it oversets, cause it to vibrate in the same time as at first, by moving the adjusting weight, and note the distance through which the weight has passed. Move the weight then in the contrary direction through double this distance, and then produce the former slow motion by means of the screw acting vertically on the point of support. Repeat this operation until the adjustment is perfect.

2. To make the arms of the beam of an equal length.