OF FLUIDS ON THE MOTION OF PENDULUMS. 



[25] 



Since the operations represented by the two expressions within parentheses are evidently 

 convertible, the integral of this equation is 



f = ^ + f»* (21) 



where \j/ u \^ 2 are the integrals of the equations 



/ d? d 2 1 _d_\ 



/ <P cP Id 1 d\ 

 \d,v 2 dm 3 "nr dw /x dt) * 2 



(22) 

 (23) 



f , / d* d* 1 d\ d\ J_ dyj, 



~y{dl? + dlr* + wd^)~dl) : &d ; & : ' ' * - (24) 



11. By means of the last three equations, the expression for dp obtained from (l6) and 

 (17) is greatly simplified. We get, in the first place, 



1 dp 

 p doe 



but by adding together equations (22) and (23), and taking account of (21), we get 



d 2 ^ _ d?\f, \_d>\f 1 df 2 

 da; 2 d-ar 2 nr dw ft! dt 



On substituting in (24), it will be found that all the terms in the right-hand member of 



the equation destroy one another, except those which contain — — and — ^— , and the equation 



dt dt 



is reduced to 



dp 

 dm 



urdtdur 



The equation (17) may be reduced in a similar manner, and we get finally 



■sr \dtdx dtdur I 



which is an exact differential by virtue of (22). 



(25) 



* If we denote for shortness the operation 



<P <P 1 d 

 di- d a * a d a 

 by D, our equation becomes 



which gives by the separation of symbols 



■bur &-}&•-»*.-• » 



d'l' . 

 so that —j- is composed of two parts, which are separately the 



integrals of (22), (23). Hence we have for the integral of ('20') 

 W m ♦» + +1+ *j * being a function of x and a without t 

 which satisfies the equation D'V = 0. For the equations (22), 



Vol. IX. Paet II. 



(23) will not be altered if we put/\^,d<, fty^dl for \fr lt <^ s , 

 the arbitrary functions which would arise from the integration 

 with respect to t being supposed to be included in ¥. The 

 function ¥, which taken by itself can only correspond to steady 

 motion, is excluded from the problem under consideration by 

 the condition of periodicity. But we may even, independently 

 of this condition, regard (21 ) as the complete integral of (20'), 

 provided we suppose included in (21) terms which would be 

 obtained by supposing \\r at first to vary slowly with the 

 time, employing the integrals of (22) and (23), and then 

 making the rate of variation diminish indefinitely. By treat- 

 ing the symbolical expression in the right-hand member of 



d 

 equation (a) as a vanishing fraction, — being supposed to 



vanish, we obtain in fact D~* ; so that under the convention 

 just mentioned the function ¥ may be supposed to be in- 

 cluded in \]i i + i// 2 . The same remarks will apply to the 

 equation in Section III. which answers to (20'). 



28 



