of homogeneous Ellipsoids. 357 
to the case when the attracted point is without the surface of 
the ellipsoid. 
Let R 1 = x 1 -j- (6 — y) % -{- (c — %)\ then 
A = j R a -[- a ( £ — ^ x )\-k 
A'= | R 2 *4” a “h 2J: ) j 1 : 
and, if the function -L — be expanded into a series, the ter- 
minus generalis of that series will be 
; 1. 3.5.7 .... 2M — x a" (a -f- 2xY— a n (a — 2x) n m 
2. 4.6.8 .... 2 n ‘ R*" + ‘ 
and, hence it is plain, that all the even powers of a will disap- 
appear, and only the odd powers will remain. Now, the ex- 
pansion of the force A cannot contain any of the powers of a , 
excepting those which enter into the series for -L *— f : there- 
fore, supposing the expansion of A to be arranged according 
to the powers of a, it will necessarily be of this form, viz. 
A = At 1 * a -f At 3 ’ a 3 + A* 5 * a 5 +A (7) a 7 -f, &c.: 
where At 1 *, At 3 *, At 5 *, &c. are functions independent of a. The 
first of these coefficients, it is easy to prove, will be determined 
by this formula, 
2 x . dy . dz 
(4) 
j + [b— ;y ) 2 + (c - *) 2 j f 
and, with regard to the rest, they may be all shewn to depend 
on At 1 *, in consequence of an equation in partial fluxions, first 
noticed by La Place, and derived from the nature of the 
functions under consideration. In effect, the truth of the fol- 
lowing formulas will be established by merely performing the 
operations indicated, viz. 
l±±) . ( d ±i) + { d ±i)- 
^ da* ' T" ' db* ' + \ dc % ‘ 
3 A 
MDCCCIX. 
