of certain differential Expressions, &c. 2 65 
and, consequently, J x ■ finally depends on the integrals of 
Rrffl, and of -f-. 
The expressions hitherto given, are analytical. By the intro- 
duction of geometrical language, there arise forms such as 
d&y/ ( i—e\ sin. 2 9),d9</( 1 — e 2 cos. * 0 ), d9<f 1 1+ (cos. 0 )* }> 
d9y/ (1 -\-m . cos. 9 ), d9y/ [m. cos. 9-\-i), 
(cos. 9) n . 1 -\-m cos. 9 } ' — r— ; the integration of which depends 
on that of dx J ( , and of 7 7) » as mi g ht easil y 
be shewn. I shall, however, omit the proof, and only observe, that 
this variety of expression, by rendering obscure, or remote, the 
origin of differential expressions, is rather an inconvenience than a 
benefit to science. 
Before I quit this subject, I wish to shew how, from the pre- 
ceding integrals and methods, the coefficients in the series A-f B. 
cos. 0-fC.cos. 2 0-f &c. the expansion of (a 2 -\-b 2 — 2^6 .cos. 
may be determined and computed. 
{ <,+*>-**. cos. « X + £ - S cos. 
=(if^j- =e') a 2m+I | i+«'* — ie\ cos. 6 1 * =A-|-B . cos. 6 + C. 
cos. consequently, a? m+l .f{ i-\-e ,a . — 2<?'.c os^y^ir d9 = 
A 9 -f B sin. 
C . sin. 
-f- &c. Let 
A7t — 2 ^« C0B * ^)~ T ~ d 9 t (when 9 is put = %'), 
Now, i-\-e' 2 -*- 2^. cos. 9 — i-\-e ' 2 — 2<?'| e (cos. -i) — i | 
= ( 1+C ').{ 1 _ ^(cos.l)-}, let = cos. 4 = * 
l -f e' 2 — 2 e'. cos. 0 = ( l +e ') 9 { 1 — <? 2 x 2 J, 
M m a 
! 
