In the great majority of analytical determinations, particularly where we are concerned with obtaining percentages, we can ignore the error introduced by the buoyant effect of the air, but where this is not permissible we can correct for it with sufficient accuracy as indicated above in formula (10) and as discussed further in § 74. 60. Standard of Mass.-The fundamental standard of mass which has been adopted by the principal governments of the world is the International Prototype Kilogram, which is a mass of platinum-iridium made in 1887 and deposited in the International Bureau of Weights and Measures near Paris. It represents as exactly as possible the mass of 1000 cubic centimeters of water at 4°C. Two authentic copies of this standard are kept in a vault at the National Bureau of Standards, Washington. They are known as National Prototype Standards of Mass, and are used only when needed to verify the secondary standards of the Bureau. The Unit of Mass that is almost invariably employed in laboratory work, however, is not the kilogram but the gram, which is the one-thousandth part of the kilogram. This unit is consequently equal to the mass of one cubic centimeter of water at 4° C.

It must be mentioned that our standard masses are almost always spoken of as "weights"; also the results of a weighing are always spoken of as if they were "weights," notwithstanding the fact that a weighing performed on an analytical balance is not a determination of weight, but a determination of mass. The usage of the term weight to express mass as well as weight (i.e., the force with which a given mass is attracted by the earth) is not only confusing but misleading. However, the usage seems to be too firmly established to seek to correct it.

61. The Equal-arm Balance. It follows from theoretical considerations similar to those developed in the preceding part of this chapter that the equal-arm balance is the simplest and most effective type of instrument for making precise measurements of mass. In consequence of this theoretical advantage and the further advantage of practical utility, the construction of the equal-arm balance has reached a very high degree of perfection. Essentially an equal-arm balance consists of a metal beam mounted at its center upon a prism knife-edge and carrying at each end a terminal prism-edge. The central prism-edge rests

upon an agate plane (when the balance is in use) and the terminal prism-edges each support a suspension which carries a pan, the one pan serving to carry the object, the other pan serving to carry the reference masses. The beam carries a long pointer which plays over a divided scale when the balance is in operation. In addition to the foregoing parts there is a mechanism for supporting the beam when not in use, also another mechanism for similarly supporting the pans. The details of assembly are shown herewith in Fig. 8.

The column T, which is anchored to the floor F, carries the agate plane A. The beam B is mounted at its center upon the central prism-edge D; at the ends of the beam B are the terminal prism-edges C and C' which carry

the small agate planes P and P'. These latter serve to mount specially constructed stirrups S and S' which carry the balance pans N and N' through the bow strings O and O'. In order to protect the bearing edges of the prisms D, C and C' from wear, balances are provided with two sets of rests known respectively as "beam rests" and "pan rests." The beam rests R and R' are attached to the movable arms E and E' which are hinged to the column T and are operated by means of the milled head H placed at the center of the balance just below the floor F of the balance. When the head H is rotated it turns a cam which raises a rod passing through the column T; this rod in turn raises the arms E and E' until the studs R and R' engage corresponding studs in the beam B and cause enough of a movement to bring the central prism-edge clear of the agate plane A; at the same time the stirrups S and S' are engaged by the ends of the arms E and E' and lifted clear of the terminal prisms C and C'. This whole arrangement is so disposed that each time the beam is raised and lowered the several parts are brought back into exactly the same alignment.

The pan rests L and L' are controlled by means of a small push button K placed to one side of H. When the button is pushed in, the rod to which it is attached moves a lever (not shown in diagram) which causes the arms carrying L and L' to drop so that the pans are free to move up and down. At all times when the balance is not swinging the pan rests should be in their raised position so as to keep the pans from moving about and so protect the terminal prisms C and C' from needless wear.

While we shall give a set of rules in § 68 for the use of the balance, we would drive home the idea at this juncture that in operating a balance it should always be handled so that edges of the prisms are brought down gently and lifted gently from their bearings. If this care is not exercised and the prismedges are brought down roughly it will only take a month or so before the balance will be unfit for use.

Besides the parts already discussed there should be mentioned the pointer W which is attached to the beam and which plays over the graduated scale V. If the scale is not already numbered it should be numbered as illustrated in Fig. 8.

The weight G is a small weight which can be moved up or down on the pointer W when it is desired to increase or decrease the sensibility of the balance. The rider rod I and the rider hook J are used to manipulate the "rider," which is a small piece of platinum wire bent in such form as to hang on the beam of the balance and be easily movable from one position to another on the beam. The rider is used in the final adjustment of weights in a weighing when the necessary adjustment lies within 5 mg., thus obviating the troublesome manipulation of the very small 1 and 2 mg. weights. The levelling screws X and X' are used to level the balance as indicated by the levelling plate U. The adjusting screw Y on the beam is used for the purpose of

5 In accordance with the recommendation of Kohlrausch, Leitfaden der praktischen Physik, 44, Leipzig (1896). It is also an advantage to have a scale with red lines, as pointed out by C. M. Clark, J. A. C. S. 32, 884 (1910).

adjusting the equilibrium position of the balance so that the needle shall rest sensibly at the middle point of the scale V.

62. General Theory of the Balance. Reduced to its elements, an equal-arm balance consists of a rigid beam supported horizontally at its center on a prism-edge and loaded at its ends. The center of gravity of the balance must be below the central prismedge, since if it were above, the beam would be in unstable equilibrium, and if the center of gravity were at this point, the equilibrium would be neutral.

Sensibility. The sensibility of a balance or the angle a through which the balance turns for a given excess of weight, w, in the

one pan or the other depends upon the following factors (ignoring friction): on w, the given excess of weight; on 1, half the length of the beam; on W, the weight of the beam; and on l', the distance of the center of gravity of the balance below the central prismedge, according to the relationship:

Proof. Taking moments about B, assuming both arms equal,

Multiplying both sides of the last equality by (BM) we get

or since for small angles, such as we encounter in the swings of a balance, tan a = a, we have finally

From this equation it follows that:

1. The longer the beam, the greater the sensitiveness 2. The lighter the beam, the greater the sensitiveness

3. The smaller the distance between the point of suspension of the balance and the center of gravity, the greater the sensitiveness.

Unfortunately the preceding conditions for maximum sensitiveness conflict with one another in the mechanical construction of a balance. Thus long arms are incompatible with minimum weight. The length of the arms is also practically limited by another condition, namely, the time of swing of the balance, which must not be excessive. If a balance is regarded as a compound pendulum, it can be shown that its time of vibration is given by the expression

g

=

weight of pan with load,

acceleration due to gravity, and W, 1, and l', have the same significance as before.

From this equation it will be seen that a limit is soon reached beyond which it is undesirable to increase the sensitiveness of a balance by increasing the length of its arms, l, for its time of vibration becomes excessive, and the process of weighing very tedious. A time swing of 12 to 15 seconds is about the maxi