extensive Class of Spheroids. 
75 
term of the series for the attractive force on a point within 
the spheroid (No. 8), that term will become 
i — 2 
.f/rr. dv! . dm 1 . Q . 
In the first place I say that all the terms in which i is odd 
are evanescent. For s = (i — e —J ) . p/ a nf- ~T • ( 1 — V**) 
i— 2 
cos. 2 is'; whence it follows that s 2 may be expanded into a 
series of this form, viz. A ( °^ -J- A^ l) . cos. 2or'~J- . cos. 4,^ 
„ . . A^ . cos. 2 n & f . . &c. ; of which the general term is A^ 
. cos. 2,nm', and if we combine this quantity with the expansion 
of (No. 5), there will result one term, and only one, in- 
dependent of sines and cosines, viz. 
0 (2 «) , d 2n c(') /yv ■"cC 1 ’) .(*) 1,1, 
& •(!— V-h - ~in ’ A 
all the other terms, produced by the multiplication, contain sines 
or cosines of variable angles ; on which account they vanish 
when they are integrated with regard to d-®' between the re- 
quired limits : since s contains no other power of [j/ but f*, it is 
i — 2 
plain that every coefficient of the developement of s 2 , as A^ n \ 
will be an even function of f, or will contain only even powers 
of that quantity: and, because i is odd, therefore C^\ and all 
its fluxions of the even orders, will be odd functions of p/: 
upon the whole then the quantity under the double sign of 
integration will be an odd function of p/; or it will be an assem- 
blage of the odd powers of that quantity : therefore the in- 
tegral, between the limits p/ = 1 and p/ = — - 1 , is equal to 
L 2 
