of homogeneous "Ellipsoids. 
361 
D 
By the theory of equations RR' = — — : and, by substitu- 
tion, the last expression of will become 
=//{ 
dM 
dp 
co s.p sin. -p -f 2 cos. *p sin. p^dp . dq . 
It is remarkable, that the last expression of A (l) does not 
contain either of the quantities b or c ; for these do not enter 
into the function M : and hence we are to conclude that the 
value of is independent on these co-ordinates, and is the 
same for all points situated within the same principal section 
of the ellipsoid. Another inference is, that all the other co- 
efficients A^, A^, &c. of the expansion of the force A are 
severally equal to 0, as is plain from the law which connects 
those quantities with one another, and with At 1 ': on this ac- 
count the expansion alluded to will be reduced to its first term, 
and we shall have, simply, 
A = A( l) xa. 
The same considerations likewise suggest a new analytical 
expression of Al l} ; which, on account of its simplicity, and its 
immediate dependence on the figure and equation of the solid, 
seems to deserve the preference to every other : for, since it 
has been shewn that the value of A^ is independent on the 
co-ordinates b and c, we may exterminate these quantities from 
the formula (4) ; and thus 
^(i) rr 2 x .dy. dx _ 
JJ | x* + y*+ Z* 
the fluent to be extended to the whole of the surface of the 
principal section made by the plane of y and 2. 
The same reasoning that has been applied to the determi- 
