Attractions of Spheroids of every Description. 
constant angles 9 and «ar, which determine the position of the 
attracted point in the surface of the sphere, that d is of the 
variable angles 9' and s' ; in other words, v is what d becomes 
at the attracted point. Perspicuity requires that we distinguish 
two cases : the first is, when d is a rational and integral func- 
tion of cos. 9', sin. 9' cos. s', sin. 9' sin. s' ; or, p/, s/i — f 2 . 
COS. s', v/r — p/ a . sin. Txd ; the second is, when d is any other 
function of the sines and cosines of the angles 9" and s'. 
In the first case, Lagrange transforms d into a function of 
7, f . cos. <p, \S l — 7* . sin. <pf which transformation 
he shews to be always possible ; and having substituted dy . 
d<p for df . dis', he integrates the formula i J * - ‘ as he 
had done in No. 7 for the case when d is a constant quantity, 
by a method entirely similar to that employed in No. 3 of my 
paper : and hence he proves the truth of the equation 
when r = a. Thus then Laplace’s theorem is rigorously 
proved for an extensive class of spheroids ; and in this point 
also the investigations of Lagrange coincide with the conclu- 
sions obtained in my paper. 
With regard to the general case, when u' is any function of 
the sines and cosines of the angles 9 ' and s', it is not easy to 
discover what are the precise sentiments of Lagrange. From 
his saying that the formula J * v ' d J j JSt j s always integrable 
when d is a rational and integral function of p/, V 1 — p/ a . 
cos. tj', V 1 — p/ a . sin. s', are we to understand that the me- 
thod, which follows in No. 9, is to be confined exclusively to 
* See No. 3 of the preceding paper. 
MDCCCXII. G 
