extensive Class of Spheroids. 
59 
trplied by cos. no? and clear of p and f ; but the like part of 
(«) / . m\* / _ d n C& d n C'0) 
**\(i 7? 
dpi. 
df 
. cos. ;z (p (which is 
the whole expression of the part of multiplied by cos. nap) 
obtained by the help of the formulas in No. 3, is 
( „) M. 3 . 5 ^, 
\2.4.o.... i—n l T 
therefore by equating the equivalent expressions, we get 
= _£ . 
^ 1 — n - fi . i—n- j-2 z-J-k 
When i—n is odd, make/* + q — i—i; p — q = n; s = j-:: 
then we will obtain 
1. 3.5.. ..1+n 1.J.5 ....i—n , 
2 x — I — : . . cos. mp 
2.4.6.. ..z-fw—i 2.4.6....J— «— 1 rr t 
for the part of O l \ or of the coefficient of 4 — , which is mul- 
r ^ p+i 
tiplied by yy/ . cos, ?z<p : but the like part of / 3 ^ . (1 — „ 
„ d n c(‘) d n C'(') 
(1 — f'Y . — • — . cos. «<p, obtained by the formulas 
djA. djj. 
in No. 3, is 
^ ■ ( £'&£!b )"• • cos - ; 
whence we get, in this case also, 
fl("> — i 
' i — 22 4- 1 . t — 22 -J- 2 /- 4 « ’ 
Now if we write 2^ in the place of ( 3 ^ ; that is, if we 
henceforth put (as in No. 4) 
/3 (m) = 
I ZZ-f- 1 . 1 — ZZ-f 2 . Z — 22 -J- 3 2 - 1 - 2 * ’ 
then all the terras of the expansion we are seeking, will be 
found by making n = 1, n — 2, n = 3, &c. successively, and, 
it will be thus expressed, viz. 
I 2 
