i6 Mr. Ivory on the Method of computing 
ff 
\ r—p ) 
!— I 
p* . y ' . d^tl . dm 
f 
~ff 
r— p]* 1 . p* . y . dfx .' . dm' 
f 
* + I 
and because y' is a function of the variable angles Q r and -sr', or 
of p/ and -a/ ; and y is a constant quantity ; therefore, if u' be 
put to denote a function of the angles Q' and both the in- 
tegrals in the general term will be obtained by investigating 
the integral 
f t P ( r — 1 . p a • v 1 . df . dm 
j r 1 — 2fp . 7 + p 2 ji±i 
for the whole surface of the sphere, and in the particular cir- 
cumstance of r = p, or r — p — o. 
3. The formula which is now to be considered cannot be 
integrated without limiting the symbol u to denote a particular 
function, or class of functions. But Laplace's demonstration 
will be completely overturned, if it shall be shown that, in 
any hypothesis for v , the formula in question has a finite value 
when r — p — o: for then the only reason which he can be 
supposed to assign for rejecting such terms in the value of 
; namely, that they contain a vanishing factor, must be 
allowed to be inconclusive. We shall henceforth suppose that 
v f denotes a rational and integral function of ^ , s / 1 — ^' 2 . 
cos. -et', v/ 1 — • sin. st', which are three rectangular co-or- 
dinates of a point in the surface of a sphere ; a supposition 
which in effect embraces the whole extent of Laplace's 
method. 
The demonstration which Laplace has given of his funda- 
mental theorem is independent on the function y, being drawn 
entirely from the nature of the algebraic expression of the 
distance between the attracted point and a molecule of the 
