Assuming that the arms are equal in length and that equal loads are carried in each pan, or in other words that the beam is in equilibrium, the forces denoted by the dotted lines p and p', acting vertically, are of course equal. If now a small excess of weight be added to the right hand pan, there will be an increased force on the right-hand arm, which may be represented by p", and the beam will be deflected by an amount which for small additions of weight is propor tional to the excess, and is of course indicated by the pointer and scale.
In using a fine balance, precautions must be observed in its cam and manipulation. The bal ance, which is inclosed in a glass case and kept free from dust. is first leveled in order that the supporting pillar should be vertical. If the pointer of the balance is not at the middle point, or zero (many physicists prefer to renumber the scale so that the numbers run consecutively from the left to the right, instead of the usual ar rangement where the divisions are numbered to right and left from the centre-mark), i.e. the zero or point of rest of the balance does not coincide with the zero of the scale, the true zero must be determined by taking the mean of an even number of swings on one side, and an odd number on the other, and then their mean. The body is placed in one of the pans while both pans and beam are supported. In the other are placed the weights. the usual practice being to begin with the larger weights and work toward the smaller. The weights are of brass, platinum, or alu minium. the smaller weights being in the form of pieces of sheet metal and 'riders' of fine wire. The pans and beams should always be supported while the weights are changed. After the ap proximate weight of the body is ascertained. so that the addition of a centigram will cause the pointer to come to rest on the opposite side of the scale, the 'rider' is brought into play and adjusted at such a position that the pointer comes to rest at zero. A more rapid and usual method is that known as 'weighing by swings,' where points of rest when the weights differ by a certain small unit are obtained as described above. The amount to he added to the smaller amount of the weights will he such a fraction of the small units as will be obtained by taking as the denominator the number of scale divisions between the two rest-points, and as the numera tor the distance of the true zero from the first point. It is often convenient with such a bal
ance to have a table of sensitiveness, from which one can tell at a glance the number of milligrams corresponding to a scale division for any given load. The experimenter can also find the ratio of the two arms of the balance and use it as a correction factor, as well as determine the errors of the set of weights in order to refine his meas urement. He should, in addition, take into con sideration the amount of air displaced by weights and the body being weighed, particularly if their densities differ considerably. An analytical or assay balance will have a sensitiveness of 1-50 or even 1-400 of a milligram. and is used in chemi cal analyses for the exact determination of the mass of various substances, though a sensitive ness of one-half milligram suffices for many ordi nary analytical purposes. In the use of the balance there are many methods. complex and exact, which are used by the physicist, and which will be found described in the more advanced treatises on experimental physics.
Probably the most accurate use of the bal ance is at the International Bureau of Weights and Measures at Sevres, near Paris, where the standard kilograms constructed by the bureau are preserved and compared. Here the balances are of great sensitiveness, and are operated by an observer in an adjoining room. who uses a tele scope to observe the dellections. There is auto matic apparatus to change the weights from one pan in another, and the temperature is main tained constant.
Balances are constructed in many forms for various purposes where it is not at all necessary to employ such refinements as have been de scribed. Those of the apothecary or grocer are familiar examples, while the balance for weigh ing bullion Cl nribines large capacity with con siderable sensitiveness. In small balances use is sometimes made of the bent lever, especially in balances for weighing letters, chemicals, and other small objects. Here a constant, weight is placed on the short or bent arm, and the body of unknown mass on the long arm. A pointer on a graduated scale indicates the weight of the body. The graduations are not even, as the angular dis placement from a position of equilibrium does not increase proportionately with the increase of weight.