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Uskal; a few smaller streams, such as the Masmár, the 1. Spring Balances.—The general principle of this class Mahkárá, &c.; and the Banjár, Hálon, and Jamunia, trib- of balances is that when an elastic body is acted upon by a utaries of the Narbadá, which drain a portion of the upper weight suspended from it, it undergoes a change of form, plateau. Bálághát contains very extensive forests, but which, cæteris paribus, is the greater the greater the weight. they do not produce timber of any great value. They teem The simplest form of the spring balance is a straight spiral with wild animals, from the great bison to the fox; 470 of hard steel (or other kind of elastic) wire, suspended by beasts and venomous snakes were killed in 1867–68, a total its upper end from a fixed point, and having its lower end reward of £156 being paid under this head. The district bent into a hook, from which, by means of another hook contained in 1868 an assessed area of 1462:08 square miles crossing the first, the body to be weighed is suspended, or 935,731 acres, of which 214,587 acres were under culti- matters being arranged so that even in the empty instruvation; 488,510 grazing lands; 116,938 culturable, but not ment the axis of the spiral is a plumb-line. Supposing a actually under cultivation ; 115,696 unculturable waste. body to be suspended at the lower hook, it is clear that the The census report of 1872 returned the area at 2608 square point where the hooks intersect each other will descend miles. The census of 1866 showed a population of 170,964. from the level it originally occupied, and that it must fall This had in 1872 increased to 195,008, residing in 37,192 through a certain height h before it can, by itself, remain houses and 781 villages; average number of persons per at rest. This height, provided the spiral was not strained square mile, 74:77; per village, 249-69; per house, 5.24. beyond its limit of elasticity (i.e., into a permanent change Of the total population, 131,176 or 67.27 per cent. were of form), is proportional to the weight P of the body, and Hindus; 2934 or 1:50 per cent. Mahometans ; 39 Buddh- consequently has to the mass M the relation h=cgM, ists; 11 Christians; 60,848 or 31.20 per cent. of unspeci- where c is a constant and g the acceleration of gravity fied religions of aboriginal or imperfectly Hinduized types. Hence, supposing in a first case, h and M to have been i
Since 1867 considerable encouragement has been given and M', and in a second case, WI and M"', we have h':h!!:: to the cultivating tribes of Ponwárs, Kunbís, Marárs, &c., GM':g''M''; and it is only as long as g is the same that of the low country to immigrate, and take up lands in the we can say "':':: M': M". Spring balances are very upland tracts. By this means a large quantity of jungle C
D lands has lately come under cultivation. The acreage under the principal crops grown in the district is returned as follows:-rice, 188,312 acres; wheat, 585; other food grains, 8770; oil-seeds, 3436; sugar, 505; fibres, 100; tobacco, 638; total, 202,346 acres.
Iron smelted by the Gonds; gold exists in the beds of some of the rivers, but not in sufficient quantities to repay the labor of washing. There are no regularly made roads in the district. Five passes lead from the low country to the highlands, viz., the Bánpur Ghát, the Warai Ghát, the Pancherá Ghát, the Bhondwá Ghát, and the Ah
В. nadpur Ghát. For revenue purposes the district is divided into two subdivisions, the Búrhá Tahsil, and the Paraswara Tahsíl. In 1868-69 the total revenue of the Bálághát district amounted to £11,746, of which £6754, or 57 per cent., was from land. For the protection of person and property, Government maintained, in 1868, 115 policemen, at a total cost of £1156, 168. In 1868 only two towns in the district had upwards of 2000 inhabitants, viz., Hattá, population, 2608, and Lanjí, population, 2116. About 60 years ago the upper part of the district was an impenetrable waste. About that time one Lachhman Náik established the first villages on the
P Paraswárá plateau, on which there are about 30 flour. ishing settlements. But a handsome Buddhist temple of cut stone, belonging to some remote period, is suggest
Fig. 1.--Diagram illustrating Chain Balance. ive of a civilization which had disappeared before historic times.
extensively used for the weighing of the cheaper articles BALANCE. For the measurement of the “mass” of of commerce and other purposes, where a high degree of (i.e., of the quantity of matter contained in) a given body precision is not required. In this class of instruments
, to we possess only one method, which, being independent of combine compactness with relatively considerable range, any supposition regarding the nature of the matter to be the spring is generally made rather strong; and sometimes measured, is of perfectly, general applicability. The the exactitude of the reading is increased by inserting, bemethod—to give it at once in its customary form-consists tween the index and that point the displacement of which in this, that after having fixed upon a unit mass, and serves to measure the weight, a system of levers or toothed procured a sufficiently complete set of bodies representing wheels, constructed so as to' magnify into convenient visieach a known number of mass-units (a “set of weights”), bility the displacement corresponding to the least difference we determine the ratio of the weight of the body under of weight to be determined. Attempts to convert the examination to the weight of the unit piece of the set, and spring balance into a precision instrument have scarcely identify this ratio with the ratio of the masses. Machines ever been made; the only case in point known to the writer constructed for this particular modus of weighing are called is that of an elegant little instrument constructed by balances. Evidently the weight of a body as determined Professor Jolly, of Munich, for the determination of the by means of a balance-and it is in this sense that the specific gravity of solids by immersion, which consists of a term is always used in everyday life, and also in certain long steel-wire spiral, suspended in front of a vertical sciences, as, for instance, in chemistry—is independent of strip of silvered glass bearing a millimetre scale. To read the magnitude of the force of gravity; what the merchant off the position of equilibrium of the index on the scale, or chemist) calls, say, a “pound” of gold is the same at the observing eye is placed in such a position that the eye, the bottom as it is at the top of Mont Blanc, although its its image in the glass, and the index are in a line, and the real weight, i.e., the force with which it tends to fall, is point on the scale noted down with which the index appargreater in the former than it is in the latter case.
ently coincides. To any person acquainted with the elements of mechan- 2. Chain Balances. This invention of Wilhelm Weber's ics, numerous ideal contrivances for ascertaining which of having never, so far as we know, found its way into actual two bodies is the heavier, and for even determining the practice, we confine ourselves to an illustration of its prinratio of their weights, will readily suggest themselves; but ciple. Imagine a flexible string to have its two ends atthere would be no use in our noticing any of these many tached to the two fixed points C and D (fig. 1), forming the conceivable balances, except those which have been actually terminal points of a horizontal line CD shorter than the realized and successfully employed. These may be con- string. Suppose two weights to be suspended, the one at a veniently arranged under six heads.
point A, the other at a point B of the string; the form of VOL. III.-111
the polygon CDBA will depend, cæteris paribus, on the remains at rest in its normal position, and, if brought ont ratio of the two weights. Assuming, for simplicity's sake, of it, will return to it, being in stable equiüibrium. This at CA to be equal to DB, then, if the weights are equal, say, once suggests two modes of constructing the instrument each-P units, the line AB will be horizontal. But if and two corresponding methods of weighing. now, say, the weight at B be replaced by a heavier weight First Method.-We so construct our instrument that Q, the point A will ascend through a height h, the point while l' is constant, " can be made to vary and its ratio to B will descend through a lesser height W in accordance l' be measured. In order then to determine an unknown with equation Ph=QK', and the angle between what is now weight P', we suspend it at the point pivot A; we then the position of rest of the base line A'B', and the original take a standard weight P' and, by shifting it forwards and line AB will depend on the ratio of P: Q. The exact backwards on AB, find that particular position of the point measurement of this angle would be difficult
, but it would of suspension B, at which pri exactly counterpoises P'. We be easy to devise very exact means for ascertaining whether
," or not it was horizontal, and, if not, whether it slanted
then read off
and have P' =P in But, practically, the down the one way or the other; and thus the instrument body to be weighed cannot be directly suspended from A, but might serve to determine whether P was equal to, or greater must be placed in a pan suspended from A, and consequently or less than, Q; and this obviously is all that is required the weight Po of the pan and its appurtenances would to convert the contrivance into an exact balance.
always have to be deducted from the total weight P', as 3. Lever Balances. This class of balances, being more found by the experiment, to arrive at the weight of the extensively used than any other, forms the most important object p=P-Po. Hence, what is actually done in pracdivision of our subject. There is a great variety of lever tice is so to shape the right arm that its back coincides balances; but they are all founded upon the same princi- with the line AB, and to lay down on it a scale, the deples, and it is consequently expedient to begin by summing grees of which are equal to one another, and to l' (or some up these into one general theory.
convenient submultiple or multiple of l') in length, and so Theory of the Lever Balance (fig. 2).— In developing to adjust P, and number the scale, that when the sliding the “theory" of a machine, the first step, always is and weight p? is suspended at the zero-point, it just countermust be that we substitute for the machine as it is a poises the pan ; so that when now it is shifted successively fictitious machine, which, while it closely corresponds in its to the points 1, 2, 3 . p, it balances exactly 1, 2, 3 ... working to the actual thing, is free from its defects. In p units of weight placed in the pan. This is the principle this sense what now follows has to be understood. Imagine 1 of the common steel-yard, which, on account of the rapidity
of its working, and as it requires only one standard weight,
B is very much used in practice for rough weighings, but Ae
which, when carefully constructed and adjusted, is susceptible of a very considerable degree of precision. In the case of a precision steel-yard, it is best so to distribute the mass of the beam that the right arm balances the left one +the pan, to divide that arm very exactly into, say, only
10 equal parts, and instead of one sliding weight of pii 0
B units to use a set of standards weighing P", * P", do
P", robo P!!!, &c. . The great difficulty is to ensure to the heavier sliding weights a sufficiently constant position on the beam. To show the extent to which this difficulty can
be overcome it may be stated that in an elegant little steelpi
yard, constructed by Mr. Westphal of Celle (for the deterLP 'S
mination of specific gravities), which we had lately occaw
sion to examine, even the largest rider, which weighs about 10 grammes, was so constant in its indications that, when suspended in any notch, it always produced the same effect to within less than goboth of its value.
Second Method.- We so construct our instrument that
both l' and I have constant values, and are nearly or ex109 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
actly equal to each other, and provide it with pans, whose Fig. 2.—Diagram illustrating the theory of the Lover Balance.
weights po and per' are so adjusted against each other that p' l=po!!", and, consequently, the empty instrument is
at rest in its normal position. We next procure a suffian inflexible beam suspended from a stand in such a manner ciently complete set of weights, i.e., a set which, by properly that, while it can rotate freely about a certain horizontal combining the several pieces with one another, enables us axis fixed in its position with respect to both the stand and to build up any integral multiple of the smallest difference the beam, and passing through the latter somewhere above of weight o we care to determine, a set, for instance, which its centre of gravity, it cannot perform any other motion. virtually contains any term of the series 0·001, 0·002, Imagine the beam at each end to be provided with a 0.003 .
.. 100.000 grammes.
In order now to vertical slit, and each slit to be traversed by a rigid line determine an unknown weight p', we place it, say, in the fixed in the beam in such a situation that both lines are left pan, and then, by a series of trials, find that combinaparallel to, and in one and the same plane with, the axis tion of standards p, which, when placed in the right pan, of rotation; and suppose the mass of the beam to be so establishes equilibrium to within £d. Evidentlydistributed that the line connecting the centre of gravity 8 with its projection O on the axis of rotation stands per
1 pendicular on that plane. Suppose now two weights, P' and p!', to be suspended by means of absolutely flexible strings, In the case of purely relative weighings, there is nothing to the former from a point A on the rigid line in the left, the other from a point B on the rigid line in the right slit, and hinder us from adopting i units (e.g.
, grammes) as our clearly, whatever may be the effect, it will not depend on unit of mass, and simply to identify the relative value of p the length of the strings
. Hence we may replace the two with the number p'. But even if we want to know the weights by two material points situated in A and B, and absolute value of p in true grammes, we need not know weighing P' and P' respectively. But two such points are equivalent, statically, to one point (weighing P + P') the numerical value of All we have to do is, after situated somewhere in D within the right line connecting A with B. Suppose the beam to be arrested in its having determined the value of p' in terms of to reverse "normal position" (by, which we mean that position in which AB stands horizontal and the line SO is a plumb- the positions of object and standards, and, in a similar line), and then to be released, the statical effect will manner, to ascertain the value pı" which now counterpoises depend on the situation of the point D, and this situation, the unknown weight p' lying in the right pan. Obviously supposing the ratio l':1 to be given, on the ratio P': P'.
in p'=puta whence(p?)? -p''p,'', and p' =V p'R".
( 4 ا) - p
for which expression, if the two arms are very nearly of secondly, gravity to act parallel to the line OB, we have equal length, we may safely substitute p'-1 (p!'+''). Or, instead of at once finding the counterpoise for p in stand
(W+A). 00-W. OS; ards, we may first counterpoise it by means of shot or other
AL" material placed in the opposite pan, and then find out the
W number of grammes p'' which has to be substituted for pl to again establish absolute equilibrium. Evidently p'ap!!! Obviously, the right way of graduating the limb is to This (in reference to the ideal machine meant to be realized) place the marks so that their radial projections on the tanis the theory of the common balance as we see it working in gent to the circle at the zero-point divide that line into every grocer's shop, and also that of the modern precision parts of equal length. In the ordinary balance where !" balance, which, in fact, is nothing but an equal-armed beam and scales refinedly constructed. In the case of the
is a constant, the factor
has a constant value, whicb latter class of balances the inconvenience involved in the can be determined by one experiment with a known 4use of very small weights may be avoided (and is generally always supposing that in the instrument used the requireavoided) by dividing the right arm of the beam, or rather ments of our theory were exactly fulfilled. In good prethe line AB, into lo equal parts, and determining differ- cision-balances they are fulfilled, to such an extent at least, ences of less than, say, 0.01 gramme by means of a sliding that although the factor named is not absolutely constant, weight possessing that value. But evidently, instead of but a function of P, it can be looked upon as a relative condividing the whole length of the right arm, it is better to stant, so that by determining the deviations produced by a divide some portion of it which is so situated that the rider given 4, say 4-1 milligramme, for a series of charges can be shifted from the very zero to the “10," and so to (i.e., values of P''), one is enabled to readily convert de adjust the rider, that when it is shifted successively from viations of the needle, as read off on the scale, into differ0 to 1, 2, 3 ...n it is the same as if 1, 2, 3 n tenths ences of weight. This method is very generally followed of its weight were placed in the right pan. The rider in in the exact determinations of weights as required in chemthis case must, of course, form part and parcel of the beam. ical assaying, in the adjusting of sets of weights, &c. Only, It is singular that none of our precision-balance makers instead of letting the needle come to rest and then reading have ever thought of this very obvious improvement on the off its position, what is done is to note down 2, 3, 4 ...n customary system. In the very excellent instrument made consecutive excursions of the needle, and from the readings by Messrs. Becker and Company of New York, this, it is (a, ag, ag, ay ..an) to calculate the position on where the true, is realized partially in a rider weighing 12 milli- needle would come to rest if it were allowed to do so. It grammes and a beam divided into 12 equal parts (instead being understood that the readings must be taken as posiof 10 and 10 respectively); but this does not enable one to tive or negative quantities according as they lie to the
left shift the rider to where it would indicate from 0 to say to or to the right of the zero-point, as might be identified with or of a milligramme. Whichever of these modes of any of the sums— weighing we may adopt, we must have an arrangement to see whether the balance is in its normal position, and it is
} (a +az), } (Ag +as), } (am-stan), desirable also to have some means to enable us, in the but clearly it is much better to calculate as by taking the course of our trials, to form at least an idea as to the addi- mean of these quantities, thustional weight which would have to be added to the stand
q+am+2 (a+ag .. +On-1), ards on the pan (or to be taken away) in order to establish
2 (n-1) equilibrium. To define the normal position, all that is re- and it is also easily seen that to eliminate as much as quired is to provide the beam with a sufficiently long needle,” the axis of which is parallel to the line os, and possible the influences of the resistance of the air and which plays against a circular limb fixed to the stand and let us at once add by anticipation of what ought to constructed so that the upper edge of the limb coincides be reserved for a subsequent paragraph) of the friction very nearly with the path of the point of the vibrating in the pivots of the balance, it is expedient to let n be an needle, and to graduate the limb so that, as fig: 2 shows, confined to small A's, and it is easy to conceive a balance
odd number. Theoretically this, method is, of course, not the zero point indicates the normal position of the beam. in which the limb is so graduated that it gives directly the In order to see how the graduation must be made to be as convenient as possible a means for translating deviations weight of an object placed in the right pan; this is the of the needle into differences of weight, let us assume the principle of the Tangent Balance, a class of instruments balance to be charged with Pgrammes from A and with which used to be very generally employed for the weighP” +4 grammes from B, and P and p' to satisfy the ing of letters, parcels, &c., but is now almost entirely superequation P' V=P' W. The two weights P' and p'
seded by the spring balance. being equivalent to one point P' +P" in the axis of rota
After having thus given a general theory of the ideal, tion, the effect is the same as if these two weights did not let us now pass to the actual instrument. But in doing exist and the beam was only under the influence of two
80 we must confine ourselves mainly to the consideration weights, viz., the weight W of the beam acting in S and of that particular class of instruments called precision the weight á acting in B. But this comes to the same
balances, which are used in chemical assaying, for the adas if both W and A were replaced by one point weighing justment of standard weights, and for other exact graviW+A, and situated somewhere at C, between, and on a
metric work. line with, B and S. Hence, supposing the beam to be first identical in principle with the ordinary.“ pair of scales,"
The Precision Balance being, as already said, quite arrested in its normal position and then to be left to itself
, there is no sharp
line of demarcation between it and what main at rest before it has reached that position in which is usually called “a common balance,” and it is equally Co lies vertically below the axis of rotation. Coteris paribus impossible to name the inventor of the more perfect form Co will be the nearer to B, and consequently the angle a, what is now considered its most perfected form, we may
of the instrument. But taking the precision balance in through which the beam (and with it the needle) has to turn to assume what now is its position of stable equi- safely say that all which distinguishes it from the comlibrium, will be the greater the greater A is
, and for the mon balance proper is, in the main, the invention of the same A and W the angle of deviation will be the greater late Mr. Robinson of London. In Robinson's
, as in most the less the distance 8 of the centre of gravity of the beam modern precision balances, the beam consists of a perforated S is from the axis of rotation. The former proposition en
flat rhombus or isosceles triangle, made in one piece out of ables one in a given case to form an idea of the amount A gun-metal or hard-hammered brass. The substitution for which has to be taken
away from the right pan to establish either of those materials of hard steel would greatly increase equilibrium. To find the exact mathematical relation be the relative inflexibility of the beam, but, unfortunately, tween A and the corresponding angle a, let us remember steel is given to rusting, and, besides, is apt to become that the position of Co is the same whatever may be the di- magnetic, and has therefore been almost entirely abandoned. rection of gravity with regard to the beam. Assuming they considerably diminish its weight (as compared with
The perforations in the beam are an important feature, as gravity to act parallel to OS, we have (W+A) CC. - A1", what that would be if the perforations were filled up) without where C stands for the projection of Co on OS. Assuming, I to any great extent reducing its relative solidity. In fact,
the loss of carrying power which a solid rhombus suffers in its highest place, the central edge is just barely lifted from consequence of the middle portions being cut out, is so its bearing, and the terminal bearings are similarly lifted slight that a very insignificant increase in the size of the from their respective knife-edges, so that the beam is now minor diagonal is sufficient to compensate for it. Why a at rest in its normal position. In other balances, as, for balance beam should be made as light as possible is easily | instance, in the justly celebrated instruments of Mr. Stau. seen; the object (and it is as well here to say at once, the only object) is to diminish the influence of the unavoidable imperfections of the central pivot. To reduce these imperfections to a minimum, the beam in all modern balances is supported on a polished horizontal plane of agate or hard steel fixed to the stand, by means of a perfectly straight "knife-edge,” ground to a prism, of hard steel or agate, which is firmly connected with the beam, 60 that the edge coincides with the intended axis of rotation. In the best instruments the bearing plane is continuous, and the edge rests on it along its entire length; in less expensive instruments the bearing consists of two separate parts, of which the one supports the front end, the other the hind end of the edge. Every complete balance is provided with an “arrestment,” one of the objects of which is, as the name indicates, to enable one to arrest the beam, and, if desired, to bring it back to its normal position; but the most important function of it is to secure to every point of the central edge a perfectly fixed position on its bearing. So far all modern
recision balances agree; but the way in which the pointpivols A and B of our fictitious machine are sought to
Fig. 5.—Becker's Balance. be realized varies very much in different instruments. In Robinson's, and in the best modern balances, the beam dinger of Giessen, Robinson's plane terminal bearings are is provided at its two extremities with two knife-edges sim- replaced by roof-shaped ones (fig. 6), so that their form ilar to the central one (except that they are turned upwards),
alone suffices to secure to them a fixed position which, in intention at least, are parallel to, and in the same
on their knife-edges. Another construction plane as, the central edge; on each knife-edge rests a plane
(which offers the great advantage of being easy
of execution and facilitating the adjustment of Fig. 6.
the instrument) is to give to the terminal edges
the form of circular rings, the planes of which stand parallel to the central edge, and from which the pans are suspended directly by sharp hooks, so that the points A' and B' coincide with A and B respectively. In either case the terminal bearings are independent of the arrestment, which must consequently be provided with some extra arrangement, by means of which the beam, when the central edge is lifted from its support, is steadied and held fast in its normal position. In second and third class instruments even the central edge is made independent of the
arrestment, by letting it work in a semi-cylindrical or, what Fig. 3.-Oertling's Balance. End of Beam.
is better, a roof-shaped bearing, which, by its form, assigns
to it (in intention at least) a definite position. agate or steel bearing, with which is firmly connected a In order now to develop a complete theory of the precision bent wire or stirrup, provided at its lower end with a cir- balance, let us first imagine an instrument, which, for discular hook, the plane of which stands perpendicular to the tinctness, we will assume to be constructed on Robinson's corresponding knife-edge; -and from this hook the pan is model, the knife-edges and bearings, &c., being exactly and suspended by means of a second hook crossing the first, mat- absolutely what they are meant to be, except that the ters being arranged so that, supposing both end-bearings to terminal edges, while still parallel to the axis of rotation, be in their proper places and to lie horizontally, the work are slightly shifted out of their proper places. Supposing ing points A and B' of the two hook-and-eye arrangements such a balance were charged with P'=p'o+p' from the are vertically below the intended point-pivots A and B left, and P"=p'+p' from the right knife-edge, –and on the edges. In this construction it is an important func- it is clear that in this case also the charges may be assumed tion of the arrestment to assign to each of the two terminal to be concentrated, -p' in a certain fixed point A on the bearings a perfectly constant position on its knife-edge. How left, and p in a certain fixed point B on the right edge, this is done a glance at figs. 3 and 4 (of which the former and, consequently, the statical condition of the balance is is taken from an excellent instrument constructed by L. the same as if the weights W, P',P" were all concentrated
in one fixed point C. (fig. 7), the position of which, in regard to the beam, is independent of the extent to which the latter may have turned, and independent of the direction of gravity. It is also easily seen that in a given beam the position of Co will depend only on P' and P', and supposing P' to remain constant, it will change its position whenever P!! changes its value. The point Č, will in general lie outside of the axis of rotation, and consequently there will in general be only two positions of the beam in which it can remain at rest, namely, first, that position in which Co lies vertically above, and, secondly, that position in which it lies vertically below the axis of rotation.
Only one of these two positions can possibly lie within the Fig. 4.--Becker's Balanco. End of Beam.
angle of free play which the beam has at its disposal. The
second of the two positions, if it is within this angle, can Dertling of London, and the latter from an equally good easily be found experimentally, because it is the position balanca, represented in fig. 5, made by Messrs. Becker & of stable equilibrium, which the beam, when left to itself Co., of New York) shows better than any verbal explana- in any but the first position, will always by itself tend to tion. But what cannot be seen from these sketches is that assume. The first position, viz., that of unstable equilibthe range of the arrestment is regulated, and its catching rium, is practically beyond the reach of experimental de contrivances are placed, so that when the arrestment is at termination. Hence the points A, B, and $ must be situ
sted so that, at least whenever P'l' - Pin exactly or very quently h' as well as "' is a function of P', and P'l of Dearly, the beam has a definite position of stable equilib- the form h= ho+yP, where y has a very obvious meaning. rium, and that this position is within the angle of free What is actually done in the adjusting of the best instruplay. To formulate ihese conditions mathematically, as- ments is so to place the terminal edges that, for a certain sume a system of rectangular co-ordinates, X, Y, Z, to be medium value of P/+P', h' +'' = 0, so that the sensiconnected with the beam, so that the axis of the Z coin bility of the balance is about the same when the pans are cides with the central edge and the origin with the pro- empty as when they are charged with the largest weights jection 0 of the centre of gravity on that edge, while the they are intended to carry. The condition l'=\" also Y-axis passes through the centre of gravity. Let the cannot be fulfilled absolutely in practice, but mechanicians values of the co-ordinates of the points A, B, S, Co now-a-days have no difficulty in reducing the difference (imagined to be situated as indicated by the figure) be as 1"
- 1 to less than £ sotoo, and even a greater value follows: Point 8 Co
would create no serious inconvenience. We shall there I=-1 +1" 0
fore now assume our balance to be exactly equal-armed;
and, substituting for h' +'' the symbol 2h, and under(The e's are evidently of no practical consequence.) To standing it to be that (small
) value which corresponds to find i, and y we need only again apply the reasoning which the charge, substitute for equation 3 the simpler expression helped us in the case of the similar problem regarding the
tan a= ideal instrument. Assuming, then, first, gravity to act
W80 +2Ph parallel to Y, we have (P’+P" #W) 2.-P!"–PV. which, on the understanding that P"-P'+4, and that
A is a very small weight, gives the tangent-value corresponding to P and A. Sometimes it is convenient to look upon the pans (weighing Po each) as forming part and parcel of the beam ; the equation then assumes the form
AL tan a=
W's' +2ph where p=P-. Po
In a precision balance the sensibility, i.e., the tangent-
must have a pretty considerable value, and at the same -B
time ought to be as nearly as possible independent of the
charge. Hence what the equation (4) indicates with referiyo
ence to a balance to be constructed is, that, so far as these PI two qualities are concer
cerned, we may choose the weight of the beam as we like; and in regard to the sensibility which
the instrument is meant to have when charged to a certain Y P+P +W+A
extent, we have even the free choice of the arm-length, because, whatever I or W be, if only the centre of gravity of the empty beam is brought to the proper distance from the central edge, we can give to the sensibility any value we please. What is actually done is so to construct the
beam that its centre of gravity lies decidedly lower than Fig. 7.—Diagram illustrating theory of Precision Balance. one would ever care to have it, and then to connect with the
beam a small movable weight (called the “bob”) in such a Assuming, secondly, gravity to act parallel to X, we have manner that it can be shifted up and down along a wire, (P'+P'it w) y=P'W + Þ"L" + W8, .. for the distance the axis of which coincides with the Y-axis, and thus the of the common centre of gravity Co of the system from value 80 of the distance of the centre of gravity of the beam the axis of rotation, r=V7?+yo?, and for the angle a
from the central edge be caused to assume any value, from a through which the needle, supposing it to start from the certain maximum down to nothing, and even a little beyond zero-point, must turn to reach its position of stable equi- nothing. As to the relative independence of the sensibility librium
of the charge, equation 5 shows that a given balance will Yo PN-P'l'
possess this quality in the higher a degree the less the distan a
tance h of the central edge is from the plane of the two
terminal ones, and, supposing h to be constant (i. e., the If, in particular cases, one or more of the points A, B, S, adjustment to be finished), the less the initial sensibility a, should lie above the X-axis, we need only consider the exhibited by the empty instrument. Passing from one respective ordinates as being in themselves negative, and balance to the other, but supposing h and ag to remain conthe equations (as can easily be shown) remain in force. stant, we readily see that the sensibility is the more nearly Taking equation 3, together with what was said before, we independent of the charge p in the pans, the greater the at once see that if a balance is to be at all available for arm-length l is. From what has been said above, it would what it has been made for, and supposing two of the appear that by means of a balance provided with a gravityco-ordinates h', h' to have been chosen ai random, the bob, we could attain any degree of precision we liked, but third must be chosen so that, at least whenever Pex- evidently this is not possible practically, because in the actly or nearly counterpoises P'!, W8, +P'H' + P'H'>0. actual instrument neither the knife-edges and their bearings For if it were = 0, then, in case of PV =P!!!, the balance nor the arrestment are what we have hitherto supposed Fould have no definite position of equilibrium, and if it them to be; and, consequently, both l' and I'' as well as h, Kere negative, yo would be negative, and the position of instead of being constants, are variable quantities. Obvistable equilibrium would lie outside the angle of free play. ously, the non-constancy of the ratio !!!!" is the most Obriously, the best thing the maker can do is so to adjust important point, and to this point we shall therefore conthe balance that h'=h" = 0 and l' =!!!, because then the fine our attention. Let us imagine that the imaginary customary method of weighing (see above) assumes its balance hitherto considered has been charged equally on greatest simplicity, and, especially, the factor with which both sides (with P-Po+p), so that its normal position is the deviation of the needle has to be multiplied to convert its position of rest, and then assume, first, that the middle it into the corresponding excess of weight present on the edge (which hitherto has been an absolutely rigid line) is respective pan assumes its highest degree of relative con- now à narrow and slightly, but irregularly, curved rough stancy. We speak of a degree of constancy because this surface. The effect will be, that, supposing the balance to factor can never be absolutely constant, for the simple be repeatedly arrested and made to vibrate, the axis of reason that no beam is absolutely inflexible, and conse- rotation, instead of being constant, will shift irregularly