of certain differential Expressions , &c. 
Let « = 3, then, 
73. 
n =4, E 
&c. 
t-v _____ 8C , (zw-i) B 
(2 m + 7)c ' 2m -+ 7 1 
I2D . (2JM — 3) C 
(2m + 9) c 
2m + 9 
Since, by the preceding forms, the coefficients A, B, can always 
be expressed in finite algebraic terms, and in terms involving 
/R d^J'-^-y the problem, that of expanding (1 -\-e'* — 2<?'.cos. 
is resolved in its most extensive sense. A and B, however, can be 
determined most easily, in certain values of the index and 
mathematicians have therefore given methods for deriving A', B'> 
(index 2 - w * 1 =±= 1) from A, B, (index j . A method as eligible 
as any, depends on a problem similar to the preceding ; thus, we 
may determine A', B', from A, B, by deducing the integrals of 
V2m-i dQ, . CO s. 0 . dQ, from those of 
Wzm+i m dQ } yzm+i _ C os. e . db ; or, since 
Att = a zm + l . (l-ftf') 2 ” 1 * 1 . R zm+i'dQ, 
— = a 2m+l . (i+e'Y m +‘ ,R zm + l .cos.O .dl, 
2 
AV 
B'- 
a Zm + l ,-(i + g') 2m+1 
a ’ 1 • C 1 -\- e ') z 
.R zm - 1 do, 
and, since cos. 8 =2x * — 1 (x = cos.y)- 
By substituting, in form (g ), page 263, for 2 n, 1, we have 
fx* R 2m +L 
(2m + 2) e z —(2m-\-i) 
jRzm-i dQ (X=l) 
(2m-f 3) e z 
.yR 2m + i j&-j- 
(zm-j-i) (i—e z ) 
(2m + 3) e* 
__ ( 2 m+i)^- 2 ( 2 m+i) rpzm-t-i i 
“ (2m + 3 )e* J ^ 9_+ ' 
./(ar- 1) R 2m+I db 
2 . 2m + 1 I— 
2m -f 3 
^0. 
Nn 
MDCCCIV. 
