The centre of gravity of the beam being properly adjusted, and the equality of the two pans being assured, it is evident that the beam will set itself in a horizontal position when the pans are empty. The balance may still be de fective, however, through the arms not being of precisely equal length. The equality of the arms may be tested in the following manner: Let a mass, P, be placed in one of the pans, and suppose that w is the mass that has to be placed in the other pan in order to secure a perfect balance. Let L be the length of the arm from vehich P is suspended, and 1 be the length of the arm from which w is suspended, as indicated in good balance are so nearly equal that the simple aritlunetic mean of W and Tv is a sufficiently close approximation to the geometric mean re quired by theory.
The sensitiveness of a balance depends largely upon the position of the centre of grav ity of the beam relatively to the central knife edge. Thus, if the arms of the balance are precisely equal, and the beam hangs perfectly horizontal with a weight P in each pan, the angle x, through which the beam turns when the weight in the left-hand pan is increased to P p, may be taken as a measure of the sen sitiveness of the balance. Let S be the weight
of the beam itself, and let the centre of gravity of the beam be at a distance, h, below the central knife-edge when the beam is horizontal. Then, if x is the angle that the beam makes with the horizontal when it comes to rest with P p in the left-hand pan and P in the right hand pan, the theory of the lever gives the equa tion (see Fig. 4)— (P ± P).L.cos x= P.L.cos x h.S.sin x, from which we easily obtain — L X p tan h X S Fig. 3. Then, by the principle of the lever, we have— P X L= w X 1.
Next, let P be placed in the other pan, con nected with the arm whose length is 1, and let W be the mass that must be suspended from the arm of length L, in order to secure a per fect balance. We then have the equation —