4 $ 
Attractions of Spheroids of every Description. 
same as that in No. 9 of Lagrange's memoir, becomes un- 
satisfactory and undeserving the name of proof, except when 
all the functions r (o \ &c. are finite quantities at 
both the limits, and likewise for every intermediate value of 7; 
which will not be the case unless v be a rational and integral 
function of |j/, s/ 1 — jj/* . cos. ■&', v/r — (x /l . sin. ts . Luckily, 
however, the author's own formulas suggest a clear and satis- 
factory way of determining this point without any transforma- 
tion or the help of difficult integrations. 
Lagrange has proved in the most incontestible manner, 
that the theorem of Laplace cannot be true unless the fol- 
lowing equation likewise take place, viz. 
i s + a (§ ) = — snra'.v. 
and hence, it is plain, we shall be able to discover what func- 
tion u is of the sines and cosines of the angles 9 and cr, by 
considering in what manner these quantities enter into the 
equivalent expression on the left-hand side. Let x == cos. 9 
— V 1 — [x 1 . cos. sr, z = V 1 — . sin. si ; x'= cos. 9 f 
= p/, y — V 1 — f 2 . cos. w', z' = Vi — . sin. V ; then * 
y = + Vi — f . V 1 — [x /a . cos. [V — vr) = xx'-\-yy'-{-zz' ; 
and, by substitution, we shall get 
1 1 
J V r 3 -— zra . (xx'-\- j/+ a z ’ 
therefore is a function of x, y, z ; and ■§■ . ~ -f a 
will likewise be a function of the same quantities : but 
i • * + a (£) —fj‘ { i ■ 7 + a (~3r)} • • 
d\)J . dus ' ; 
* See No. 3 of my paper. 
G 2 
