OF FLUIDS ON THE MOTION OF PENDULUMS. [37] 



The general equations (75), (76), as well as the equations of condition (79), may be 

 satisfied by taking 



X^e'*'"'' sin OF^r), x * = e^'sinfl F % (r) (80) 



Substituting in (75), (76), and (79), we get 



F^ + jrJi'W-^ito-O, (81) 



F t "(r) + J Fi{r) -^F s (r) - m 2 F 2 (r) = 0, . . . . (82) 



F 1 (a) + F a (a) = ac, F{(fi) + F 2 \a) = c, .... (83) 

 besides which we have the condition that the velocity shall vanish at an infinite distance. 



28. The integral of (81) is 



F 1 (r) = - + Br (84) 



r 



The integral of (82) cannot be obtained in finite terms. 



To simplify the latter equation, assume F 2 (r) = ^3 (r). Substituting in (82), and inte- 

 grating once, we get 



F 3 "(r) + - F 3 '(r) - m*F 3 (r) = (85) 



r 



It is unnecessary to add an arbitrary constant, because such a constant, if introduced, might 

 be got rid of by writing F 3 (r) + C for F 3 (r). 



To integrate (85) by series according to ascending powers of r, let us first, instead of (85), 

 take the equation formed from it by multiplying the second term by 1 - 8. Assuming in this 

 new equation F a (r) = Ax* + B t aP + ..., and determining the arbitrary indices a, /3...and the 

 arbitrary constants A t , B t ...so as to satisfy the equation, we get 



_ s v . . m 2 r 2 mV . 



^ (r) ^^ 1+ 2l2^)- f 2, 4 ( 2 -^)(4-^) + -^ 



+ A " T Jl + 2(2~7I) + 2.4 (2 +3) (4 + S) + "'» 

 = (A ) + A /l + Ajlogr)\l + — +— 3 + ...] 



+ terms involving J* S 3 ... 

 In this expression 





'a + rs — 71 — tz, *i + 



£, =1-' + 2" l + 3- 1 ... +1" 1 (86) 



