T-ROFESSOr. (E IT. DARWTX OX EIJRPSOTDAL IIAR.MOXTC AXALVSTS. 
521 
2 
2? +”i 
i + s 1 
i — -s: 
n 
P(2-1) + -3L^'[32 
+ y8--- 
~ - 6S + 0 - .r {t~ + 2S - I) + 2T] ] 
H/3 (S + 1) - cos 2<^} . (69). 
We know that is derivaiile from by mnltijrlication l)y 1/C/, and we have 
found in (33), § 9, 
V = ] -f /3(S + 1) + + 8S + 5 + - T]. 
(C/) 
hfence multiplying (69) liy —— we liave 
\ C ; } 
CE'’ = ^rVl rW;' ’ + I'®*- + + */3-[- S' + 26S + 4-2 
-f .^=(SS=-2S+ 1)-2T| ;+/3A.'y/: n^(S + 1)-C0S'2.^> . (ro) 
s 
I have also obtained this result by dii’eet development. It may he thought 
surprising that the last term is now tlie same in l)oth formulm, notwithstanding the 
difference in the earlier stages, but if the reader will go through the analysis he will 
see how this has been brought about. The forimdre (69) and (70) also hold true when 
.s = 3 (as I have verified), notwltlistandlng tiie foct that P" is not to ])e derived from 
^3^ by a factor. 
The next stej) is tlie integration with resj^ect to cj). 
liave 
CO.S 
sin 
V (s 
oo.s 
sin 
I 
Therefoi'e 
~ 4 i 4’cos 2 .n'(^ -|- /3[(/n_2 + /b+ 2 ) cos 2(f) dr cos 2(s — !)</> 
i /E + 2 Cos2(.v + !)</>] + /3'[-g(^A_2)^ + irips + ^y + (jA-4 + 26.1 -1 
d" 2b-2ib + 2) cos A(f) rb it,-2^6+ 2 cos 2.y(/) rk {p,-i + i(/6_o)/) cos 2(.s‘ — 2 )<f) 
± (p., + 4 4- i ( p- + 2 )") cos 2 (.s- 4- 2) (i] . 
Also 
(C/)- or (S/b 
1 — P cos 2(f) 
liave the same forms with accented y/s. 
Accordingly, Avitli unaccented y/s, we have to multiply this expression bv 
^'OL, CXOVTT.—A. 3 X 
