of certain differential Expressions , &c. 
267 
if x be cosine of cos. nQ == 
(2x 4 — l) K — n . (2X * — 1 )*“* + ~~ 3 - • l)* _4 -r-&C.} 
Example. Suppose zm -j- 1 = — 3, 
• A — 1 
‘ it a 3 . (1 -f e') 3 
integral of dx J ). (*=i) 
S' dd 
l-^T- Now, by form ( e ), page 262, / being 
consequently, A 
/(■ 
2 /(!) 
(a 3 ( i + e') 3 ) ’ i—e z a 3 . ( i + e') (i — e’) 2 
_ 2 /(0 
(«+ 6 ) (*_&) a . w ’ 
and, to compute this quantity, the series (6), page 244, or the 
series (9), page 249, may be used ; that is, if be a y/ it is 
most commodious to employ series (6), if a s/ it is most com- 
modious to employ series (9). 
To determine B, 
Bw I rcos.e . <?9 , cos. 0 . dQ __ {zx 1 — i) zdx 
2 a 3 . (1 + e ') 3 1 J R 3 ’ Ut RJ * V(i— x*) . R 3 ’ 
Q 
x being the cosine of — . 
Now, by form (£■), page 263, 
f* dx 1 — f dx 
u /v'(i-* 1 ) .W — e\[i-e z yj V J (i-e*x z y 
and r rf * ___ — /(») . 
hence, 
rcos. 8 ri6 2 |z4 -<?*) r f \ _£ r 
,7 R 3 <?*. J_e a J V 1 / e z j v '(i—x 1 )(i—e' 2 -jc z )‘ 
(1 +O(i+o » 1 \ (1+0* r dx _ 
e'. (x— <?T 
Hence, calling (from <r=o to *=1 ) F( 1 ), 
**+&* 2 /( 1 ) 1 
we have B 
ab . («— ,b) z (« + &) * it 
ab . (« + 6) 
