OF FLUIDS ON THE MOTION OF PENDULUMS. [75] 



values of l 2 - l t and // — /,' got from (158) and the similar equation relating to the ivory 

 sphere be substituted. The result is 



&■ - mil I'gjij) ic(i - 5*'^ - '«■ o - * *'.)} 



»» »» to 



mm m J 



This equation is of the form 



i> + Qto, + Rm? = V + Q'to, + .R'to, 2 , 



and P — f, and Rm*, R'rn* may be neglected, so that the equation is reduced to Q = Q'. 

 It is now no longer necessary to distinguish between t 2 and t 2 , and between t x and </, which 

 may be supposed equal. Also m : tri :: S : &, where S, S' are the specific gravities of the 

 brass and ivory spheres respectively. Substituting in the equation Q = Q', and solving with 

 respect to k, we get 



, * * (Sk\ - S'k,) - t* (Sk\ - S"k,) 



"^ w-w-o < 159 > 



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 pen- 

 dulums. Since the equation is linear so far as regards k, k u &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 k' 2 = k 2 

 and k\ = &„ 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) 



■ ^"g*' , 



la — l\ 



which gives 



K -— rtj K — — K 2 K 2 **** ni 



/ 2 = 71 ~ Vz TV \1°U 



If 2 G] t/ 2 — &l 



Since t 2 >t 1 and k 2 >k,, the equations (l6l) 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) 



*!-£ :k,-^:k s -^::t 1 h:t 2 i:l, 



and in Bessel's experiments f, m 1*001, 4 = 1*721, 2 a = 2*143 in English inches. We thus 

 get from either of the equations (160) or (161), on substituting 0*1 16 for <\/n', k = 0*786. 

 The value of the factor k„ which relates to a sphere of the same size, swung as a seconds' pen- 

 dulum, is only 0*694, and k x may be regarded as equal to k,. The formula (148) gives 

 k 2 = 0*755. 



34—2 



