*j 6 Mr. Ivory on the Attractions of an 
nothing (No. 4). Therefore all the terms are evanescent 
when i is odd. 
Again I say that all the terms are evanescent when i is even. 
i—2 
except when it is = 2. For in this case s 2 will be an inte- 
ger power, and it will comain a finite number of terms which 
may be generally represented thus, viz. 
( 1 — [jl'* )‘ 2 . M^. cos. %nm' ; 
M 1 ^ being a rational and integral function of : and this 
quantity when combined with the developement of O^, will 
produce one term, and only one, clear of sines and cosines, 
viz. 
( 1 ■ gf ■//**» • • gf • M ( "> . V. Mi 
now since p/* is the greatest pow r er in 5, the greatest power 
i—2 
in 5 2 will be p /“ 2 ; therefore ( 1—p/ 2 ) 2 " . M (w) . cannot con- 
tain any power of f greater than i — 2, nor M^any greater 
than i— 2« — 2, which number the dimensions of cannot 
pass: but greater than i— 2/z — 2, denotes the dimen- 
sions of 
a^c'O') 
djj! 
2 n 
: therefore, by a property of this sort of inte- 
grals already demonstrated (No. 4), the preceding quantity 
is evanescent. Therefore all those terms of the series are 
evanescent in which i is an even number ; but from this the 
case of /= a, when the term assumes a particular form, must 
be excepted. 
If now we reject all the terms that have been proved to be 
evanescent, we shall have, for a point within or in the surface 
of the ellipsoid, 
