OF FLUIDS ON THE MOTION OF PENDULUMS. [103] 



Now treating A t as a slowly varying parameter, we get from a formula given by Mr Airy, and 

 obtained by the method of the variation of parameters, 



dA t G . , . 



__= _sin«#, (191) 



where / denotes the moment of inertia. In the expression for G we may replace A t _ tj under 

 the integral sign by A t outside it, because A t is supposed to vary so slowly that A t ~t does not 

 much differ from A t while t y is small enough to render the integral of importance. Making 

 this simplification and substituting in (191) we get 



dA t , . r* . , . dt, 



— — -= - c slant / cosn(t - tj — j- , . . . . (192) 



A t at J Q \/*i 



where c = \/(irn') . po*»/" 1 . If then A^ be the initial and A the final value of J„ we get 



from (192) 



A r*[ . r* , . dtx \ , 



log — = c / I sinn t cos n(t - ti) ——\dt (193) 



A J [ J -vhi 



Let now A^ + AA be what A a would become if, while the final arc A and the whole time 

 t remained the same, the motion had been going on for an indefinite time before the epoch 

 from which t is measured, in which case the superior limit in the integral involved in the 

 expression for G would have been oo in place of t. Then 



!°g Jo+ ^ A ° - c /"'(sin nt /"* 'cos n (t - *,) -A) dt, . . (194) 

 A J a [ J V # ij 



whence by subtracting, member from member, equation (193) from equation (194), we get 



A +AA rtj . /•» d *i\j, 



log = c / < sin n t I cos n{t — tj — j- >dt, 



A J I J t V *J 



which becomes after integration by parts 



A + AA c f / 7r , /•» dt 



log ; — — = — \\/ 2*/* . cos nt - cos Znt / cosnt — - 



s A 4>n \ v 2« v J, y/t 



. /•" . d# 1 



+ (2«£ - sin2n<) / sin «£ — — V. . (195) 

 J t y/t) 



Now < is supposed to be very large : in Coulomb's experiments in fact 10 oscillations were 

 observed, so that nt = IO7J-. But when t is at all large the two integrals 



dt r 00 . dt 



cos n tf — j- , I want — 7- 



can be expressed under the forms 



- P sin nt + Q cos nt, P cos ntf + Q sin n#, 



where 



P=n- 1 t~i -1 .3.2- 2 M- 3 rS+ ..., Q-l.fl- 1 *-'*-**- l.S.J 2- s w" 4 ri +..., 



