the Attractions of Spheroids of every Description. 25 
spheroid above the sphere ; the value of the same function, 
relatively to that excess will be =f— 
therefore, 
r 
dm 
■ zra .y+ 
dm 
, *•> 
/ • 
v=ztz.£ + f%. 
3 r ' J / 
Let p # = a . ( 1 -f « .y) be the radius of the spheroid drawn 
to the molecule dm ; then the thickness of the molecule will 
be = a . a .y, and dm — a. . a 3 .y' . df . ^ : again, if we ex- 
pand — into a series of terms containing the descending powers 
of r, as Laplace has done,* we shall have 
7 = T • + £ Q (l) + 1 • Q (2) + &C. 
denoting generally such a function of p, and -us as satisfies 
his equation in partial fluxions : and if we farther put J* . 
dm — a .a 3 . y' . d\d . dzs' = a . a* . U^, we shall get 
V= 4 f 4 + - • 7 * {U C0) + t • U(0 + 7 • U ( 2 ) J r &c.} : 
and will satisfy the same equation in partial fluxions that 
does. 
Moreover suppose r to vary and equate the fluxions of ~ 
and of the series equal to it ; and after having multiplied by 
r, the result will be as follows : 
f 3 
2 = -L . 0 (o) 4 
a 
F 
, 0 ) 
,(*) 
4* &c. 
but — ra.«/~\ ./ 3 — ~ f— ~ a" ; therefore, by substitution 
we shall readily get 
2 /z 
-i-7 + T.Q <0, + i. 2 Q(' ) + ^. 3 Q^ + &c. 
MDCCCXII, 
* Liv. $e, No. 9. 
E 
