ON  THE MOTION OF PENDULUMS.                          95
equation is reduced to Q = Q'. It is now no longer necessary to distinguish between t_2 and £_a', and between t_t and £/, which may be supposed equal. Also m : m^f :: S : $', where S, S' are the specific gravities of the brass and ivory spheres respectively. Substituting in the equation Q = Q'_t and solving with respect to k, we get
This equation contains the algebraical definition of that function k of which the numerical value is determined by combining, in Bessel's manner, the results obtained with the four pendulums. Since the equation is linear so far as regards k, k_l, &c., we may consider separately the different parts of which these quantities are composed, and add the results. For the part which relates to the spheres, regarded as suspended by infinitely fine wires, we have &'_2 = &_2 and k\=k_l) since the radii of the two spheres were equal, or at least so nearly equal that the difference is insensible in the present enquiry. We get then from (159)
/ *1p   _ f 2£
&=*%! ;.* ..................... (160),
,   .   ,         .                                 *»""*!
which gives
Since i_2>^i and k«>k_l} the equations (161) shew that the value of k determined by Bessel's method is greater than the factor which relates to the short pendulum, which was a seconds' pendulum nearly, and even greater than that which relates to the long pendulum, as has been already remarked in Art. 6.
If k_s be the factor relating to either sphere oscillating once in a second, and if the effect of the confinement of the air be neglected, we have from the formula (148)
^-i:*,-*:*.-!::^:^:!,
and in Bessel's experiments ^ = 1*001, £_2 = V721, 2a = 2*143 in English inches. We thus get from either of the equations (160) or (161), on substituting 0-116 for vV, A = 0-786. The value of the factor k_g, which relates to the sphere of the same size, swung as a seconds' pendulum, is only 0*694, and k_l may be regarded as equal to k_g. The formula (148) gives &_2 — 0'755.