ON THE MOTION OF PENDULUMS. 41
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 — S. Assuming in this new equation Fz(r) = Alx0-JrBix^-i- ..., and determining the arbitrary indices a, /3... and the arbitrary constants A/} Bf... so as to satisfy the equation, we get
.1. mV -i-+ +
--272^8) 2. 4(2-8) (4-8)
mV2
1 '"}
2 (2 + S) 2 . 4 (2 + S) (4 + S)
s+— s+ mV s .
2 °i^22 4a 2 22 42 62 8 + terms involving S2, 3s — In this expression
8i = rl + z* + a-l...+f* .................. (86).
Putting now
substituting in the above equation, and then making S vanish, we get
n 0 ^ a
~ ~ +22 2 + 2>22 3+" ...... ^ ''
The series in this equation are evidently convergent for all values of r, however great ; but, nevertheless, they give us no information as to what becomes of jP8 (r) when r becomes infinite, and yet one relation between C and D has to be determined by the condition that FQ (r) shall not become infinite with r.
The equation (85) may be integrated by means of descending series combined with exponentials, by assuming
I have already given the integral in this form in a paper, On the