Page:Encyclopædia Britannica, Ninth Edition, v. 3.djvu/279

material placed in the opposite pan, and tlien find out the number of grammes p" which has to be substituted for p to again establish absolute equilibrium. Evidently p = p". This (in reference to the ideal machine meant to be realised) is the theory of the common balance as we see it working in every grocer s shop, and also that of the modern precision balance, which, in fact, is nothing but an equal-armed beam and scales refinedly constructed. In the case of the latter class of balances the inconvenience involved in the use of very small weights may be avoided (and is generally avoided) by dividing the right arm of the beam, or rather the line AB, into 10 equal parts, and determining differ ences of less than, say, O Ol gramme by means of a sliding weight possessing that value. But evidently, instead of dividing the whole length of the right, arm, it is better to divide some portion of it which is so situated that the rider can be shifted from the very zero to the " 10," and so to adjust the rider, that when it is shifted successively from to 1, 2, 3 ... n it is the same as if 1, 2, 3 . . . n tenths of its weight were placed in the right pan. The rider in this case must, of course, form part and parcel of the beam. It is singular that none of our precision-balance makers have ever thought of this very obvious improvement on the customary system. In the very excellent instrument made by Messrs Becker and Company of New York, this, it is true, is realised partially in a rider weighing 12 milligrammes and a beam divided into 12 equal parts (instead of 10 and 10 respectively) ; but this does not enable one to shift the rider to where it would indicate from to say ^ or ^ of a milligramme. Whichever of these modes of weighing we may adopt, we must have an arrangement to see whether the balance is in its normal position, and it is desirable also to have some means to enable us, in the course of our trials, to form at least an idea as to the additional weight which would have to be added to the standards on the pan (or to be taken away) in order to establish equilibrium. To define the normal position, all that is required is to provide the beam with a sufficiently long " needle," the axis of which is parallel to the line OS, and which plays against a circular limb fixed to the stand and constructed so that the upper edge of the limb coincides very nearly with the path of the point of the vibrating needle, and to graduate the limb so that, as fig. 2 shows, the zero point indicates the normal position of the beam. In order to see how the graduation must be made to be as convenient as possible a means for translating deviations of the needle into differences of weight, let us assume the balance to be charged with P grammes from A and with P" + A grammes from B, and P and P" to satisfy the equation P I = P 7". The two weights F and F being equivalent to one point F + P" in the axis of rotation, the effect is the same as if these two weights did not exist and the beam was only under the influence of two weights, viz., the weight W of the beam acting in S and the weight A acting in B. But this comes to the same as if both W and A were replaced by one point weighing W + A, and situated somewhere at C between, and on a line with, B and S. Hence, suppos ing the beam to be first arrested in its normal position and then to be left to itself, the right arm will go down and not be able by itself to remain at rest before it has reached that position in which C lies vertically below the axis of rotation. Caiteris paribus C will be the nearer to B, and consequently the angle a, through which the beam (and with it the needle) has to turn to assume what now is its position of stable equilibrium, will be the greater the greater A is, and for the same A and W the angle of devia tion will be the greater the less the distance s of the centre of gravity of the beam S is from the axis of rotation. The former proposition enables one in a given case to form an idea of the amount A which has to be taken away from the right pan to establish equilibrium. To find the exact mathematical relation between A and the corresponding angle a, let us remember that the position of C is the same whatever may be the direction of gravity with regard to the beam. Assuming gravity to act parallel to OS, we have (W + A) CC = AZ", where C stands for the pro jection of C on OS. Assuming, secondly, gravity to act parallel to the line OB, we have (W + A). (JO = W. OS;

CC A/" ,.-^=tana = s . . . . (2).

Obviously, the right way of graduating the limb is to place the marks so that their radial projections on the tangent to the circle at the zero-point divide that line into parts of equal length. In the ordinary balance where I" is a If constant, the factor has a constant value, which can be Ws determined by one experiment with a known A always supposing that in the instrument used the requirements of our theory were exactly fulfilled. In good precision balances they are fulfilled, to such an extent at least, that although the factor named is not absolutely constant, but a function of P, it can be looked upon as a relative constant, so that by determining the deviations produced by a given A, say A = 1 milligramme, for a series of charges (i.e., values of P"), one is enabled to readily convert deviations of the needle, as read oft on the scale, into differences of weight. This method is very generally followed in the exact determina tions of weights as required in chemical assaying, in the adjusting of sets of weights, &c. Only, instead of letting the needle come to rest and then reading off its position, what is done is to note down 2, 3, 4 . . . n consecutive excursions of the needle, and from the readings (a v a. 2 , a 3 , a 4 . . . ) to calculate the position a where the needle u ould come to rest if it were allowed to do so. It being understood that the readings must be taken as positive or negative quantities according as they lie to the left or to the right of the zero-point, a might be identified with any of the sums—

but clearly it is much better to calculate by taking the mean of these quantities, thus—

and it is also easily seen that to eliminate as much as possible the influences of the resistance of the air and (let us at once add by anticipation of what ought to be reserved for a subsequent paragraph) of the friction in the pivots of the balance, it is expedient to let n be an odd number. Theoretically this method is. of course, not confined to small A s, and it is easy to conceive a balance in which the limb is so graduated that it gives directly the weight of an object placed in the right pan ; this is the principle of the Tangent Balance, a class of instruments which used to be very generally employed for the weighing of letters, parcels, &c., but is now almost entirely superseded by the spring balance.

After having thus given a general theory of the ideal, let us now pass to the actual instrument. But in doing so we must confine ourselves mainly to the consideration of that particular class of instruments called precision balances, which are used in chemical assaying, for the ad justment of standard weights, and for other exact gravi metric work.

The Precision Balance being, as already said, quite

identical in principle with the ordinary " pair of scales," there is no sharp Hue of demarcation between it and what

is usually called "a common balance," and it is equally