PKOFESSOE G. II. DARWIN OX ELLIPSOIDAL HAIEMOXIC ANALYSIS. 
497 
We cannot in the })resent case nse E/ as an abridged notation, because it is infinite 
as involving .s'' — I in the denominator. 
It is easy to verify that the other coefficients of and S' are, in fact, reproduced 
in the transformation. 
Lastly when s = 0, we have only cosine functions. As before" 
C = I — cos 2(j) — (1 + cos icf)) -j- /3pb (1 — b/3 cos 2(f)) cos 2rf) 
+ /S'U'^ cos -icf) 
= 1 - iV/3^ - WA + ... 
(T I 
This must be equal to , and therefore — = 1 — 
t i It 
Now 
Hence 
Since in this case E; = 
J2 -2 — i [b ) A i- — ii‘8 (h ^1 [b 1} • 
]). = 1 + + 1 ^) — 3 ] 
t (i +1) 
. . . . (38). 
- 1 
(38). 
Thus the general formula again holds good. 
]t is easy to verify that the other coefficients of (C are in fact reproduced. 
The principal use of the transforming factors, determined in this section, is that it 
will enable us to avoid some tedious analysis hereafter. 
§ 10. The Functions of the Second Kind. 
The second continued fraction of § G terminates because 
c,;,_p2u + 2][q5fi-2n+ I} = 0 
when n = h: ('*'~*'~ 1)5 since one of the two factors then assumes the form 
{%, i + Ij. 
Hence it follows that the equation for determining a is the same as Ijefore; but w’e 
cannot on that account assume that tlie q coefficients vanish when their suffixes are 
greater than i. 
In considering the P-functions it was immaterial whether or not we regarded them 
as vanishing, because vanishes if t is greater tlian i. But tlie Q-functions do not 
vanish in this case, and therefore we must postulate the existence of q’s with suffix 
D-reater than i. 
o 
3 s 
VOL. CXCVII.-A. 
