extensive Class of Spheroids. 51 
u = 1: for the fluent in question is equal to — - * 
1 A x i — tt . z + «+ 1 
(1 -(!,•)*+ 1 
d «+I C (») 
j a quantity which is evanescent at both 
the limits. 
If we consider — 1_ as a symbolical representation of O', 
d^o 
the equation (1) will be included in the equation ( 2) ; whence 
it is easy to infer that whatever is proved of by the help 
of the equation (2) may be transferred to C ^ by putting 
n — o; a remark that will enable us to consult brevity, and 
of which we shall freely avail ourselves. 
3. It is now proposed to find the value of in a series 
of the powers of p,.* The equation (2), by expanding its last 
term, will become 
• / • « \ Ml) dC® » / , \ d(i C (f ) 
2 (Z + 1 ) C — 2p,.^— -f (l — p,).— — 0 : 
let the series 
A w H-*' + A (,) |/- 2 + A' 2 V'-+.... + A< s) . i/- 22 ... + &c. 
be assumed as equivalent to C 1 ' 1 ; then by substituting and 
equating the coefficient of p l ~ 25 to 0, we shall get 
A ( s ) (i)—2s+2 ( l—ZS+l ) (s_l) . 
2S (zi — 2S -f l) 
and, by putting s = 1,5 = 2, &c. successively, we shall hence 
be able to determine the proportions of all the coefficients to 
the first one A^, which must be investigated from other 
considerations. Now is the coefficient of “7 in the ex- 
r l "r 1 
* Mec. Celeste, Liv. 3e, No. 15. 
H 2 
