Page:Encyclopædia Britannica, Ninth Edition, v. 3.djvu/281

left, aud F iu a certain fixed point B on the right edge, and, consequently, the statical condition of the balance is 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 C will depend only on P and P", and supposing P to remain constant it will change its position whenever P" changes its value.

Fio. 7. Diagram illustrating theory of Precision Balance.

The point C will in general lie outside of the axis of rotation, and conse quently there will in general be only two positions of the beam in which it can remain at rest, namely, first, that posi tion in which C 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 angle of free play which the beam has at its disposal. The second of the two positions, if it is within this angle, can easily be found experimentally, because it is the position of stable equilibrium, which the beam, when left to itself in any but the first position, will always by itself tend to assume. The first position, viz., that of unstable equilibrium, is practically beyond the reach of experimental determina tion. Hence the points A, B, and S must be situated so that, at least whenever P7 = FT exactly or very nearly, the beam has a definite position of stable equilibrium, and that this position is within the angle of free play. To formulate these conditions mathematically, assume a system of rectangular co-ordinates, X, Y, Z, to be connected with the beam, so that the axis of the Z coincides with the central edge and the origin with the projection O of the centre of gravity on that edge, while the Y-axis passes through the centre of gravity. Let the values of the co-ordinates of the points A, B, S, C (imagined to be situated as indicated by the figure) be as follows:—

Point A B S C x = - 1 +1" x, y= h h" s y (The z s are evidently of no practical consequence.) To find X Q and y we need only again apply the reasoning which helped us in the case of the similar problem regarding the ideal instrument. Assuming, then, first, gravity, to act parallel to Y, we have (F + P" + W) x 9 - FT - P7. Assuming, secondly, gravity to act parallel to X, we have (F + p" + W)y = P A + P"A" + Ws, .: for the distance of the common centre of gravity C of the system from the axis of rotation, r= Jx* + yf, and for the angle a through which the needle, supposing it to start from the zero-point, must turn to reach its position of stable equilibrium—

p /;// p/// x r t r t, , , If, in particular cases, one or more of the points A, B, S should lie above the X-axis, we need only consider the respective ordinates as being in themselves negative, and the equations (as can easily be shown) remain in force. Taking equation 3, together with what was said before, we at once see that if a balance is to be at all available for what it has been made for, and supposing two of the co-ordinates h , h" to have been chosen at random, the third must be chosen so that, at least whenever F ex actly or nearly counterpoises P", Ws + P A + F A" > 0. For if it were = 0, then, in case of P7 = P 7," the balance would have no definite position of equilibrium, and if it were negative, y would be negative, and the position of stable equilibrium would lie outside the angle of free play. Obviously, the best thing the maker can do is so to adjust the balance that h = h" = and I = I", because then the customary method of weighing (see above) assumes its greatest simplicity, and, especially, the factor with which the deviation of the needle has to be multiplied to convert it into the corresponding excess of weight present on the respective pan assumes its highest degree of relative con stancy. We speak of a degree of constancy because this factor can never be absolutely constant, for the simple reason that no beam is absolutely inflexible, and consequently h as well as h" is a function of P , and P" of the form h = A + yP, where y has a very obvious meaning. What is actually done in the adjusting of the best instruments is so to place the terminal edges that, for a certain medium value of P + P", h + h" = 0, so that the sensibility of the balance is about the same when the pans are empty as when they are charged with the largest weights they are intended to carry. The condition I = I" also cannot be fulfilled absolutely in practice, but mechanicians now- a-days have no difficulty in reducing the difference y - 1 to less than 0000 and even a greater value would create no serious inconvenience. We shall therefore now assume our balance to be exactly equal-armed ; and, substituting for h + h" the symbol 2h, and under standing it to be that (small) value which corresponds to the charge, substitute for equation 3 the simpler expression {{center|{{missing table]]}} which, on the understanding that P" = P + A, and that A is a very small weight, gives the tangent-value corre sponding to P and A. Sometimes it is convenient to look upon the pans (weighing p each) as forming part and parcel of the beam ; the equation then assumes the form—

tana = wvT^ (5) where p = P p u .

In a precision balance the sensibility, i.e., the tangent- value of the deviation produced by A = 1, which is

_ , A WY + 2ph must have a pretty considerable value, and at the same time ought to be as nearly as possible independent of the charge. Hence what the equation (4) indicates with refer ence to a balance to be constructed is, that, so far as these two qualities are concerned, 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