[171 
62 ON LAPLACES METHOD FOR THE ATTRACTION OF ELLIPSOIDS, 
or, reducing, 
(i + 1) (2i + 5) (a 2 + b 2 + c 2 ) U i+1 
o\f dUi i r.dUi dUA 
/n‘ , -i \ (dU t dU{ d U{\ „ 
- (2l + 1) l d^ + d& + dy) = 0 - 
Now Ui is a function of a + 6, /3 + 6, y + 0; if, therefore, for any particular value 
of i, Ui is independent of 6, it is clear that Ui must be a function of the differences 
dU■ dU■ dU- 
of these quantities, and we shall have - = 0; and this being so, the 
clCL dp d f y 
equation of differences, and consequently U{ +1 , will be independent of 6. But U 0 is 
independent of 6, hence U lt U. 2) &c. are all of them independent of 0, or v is inde 
pendent of 6; i.e. for ellipsoids having the same foci for their principal sections, and 
acting on the same external point, the potentials, and therefore the attractions, are 
proportional to the masses, which is Maclaurin’s theorem for the attractions of ellipsoids 
upon an external point. 
The foregoing equation, omitting the term which vanishes, gives 
dUi _dUi dU■> 
/n . „ ( aUi j „dUi dUi\ 
-<* + *>(“-ST+Wd5-* + <»-3c) 
0W.= 
(2i: + 1) ( aS 1ST + P %' + ^ + H g + / 3 + , r) p » 
(i + 1) (2 i + 5) (a 2 + 6 2 + c 2 ) 
It may be remarked that this equation, with the assistance of the equation 
= 0, 
dUj dUi dUj 
da d/3 dy 
gives 
(i + 1)(2i + 6 )(^ + ^ + 
dUu 
— — (2 i + f) 
dUi^,dUi^ dUi' 
CL —j h 0 —jt—|- C —j— 
da db dc / 
- (2i +1) (2 (a ^ + /3 + 7 + fV*) = ((2t + f) (2i + 1) - (2i +1) (2i + ©) ¡7,- = 0, 
which is as it should be. 
Write 
Ui = 
Qi 
(a 2 + b 2 + c 2 Y + i ’