the Attractions of Spheroids of every Description. 15 
found, we first substitute a . (1 -J- a ,'y) for p, retaining only 
the quantities of the first order with regard to a ; and then 
combine the two expressions so as to exterminate 1 we 
shall get the following equation instead of that of Laplace, 
viz. 
*v+«(C’ 
• &c.(C). 
2 . In order to find the integrals in the equation (C), we 
must begin with seeking an analytical expression for the value 
of dm, which may be conceived to be a prism standing on art 
indefinitely small portion of the spherical surface, and limited 
in its height by the surface of the -spheroid. Let p denote the 
radius of the spheroid drawn to the molecule dm, and 9’ and 
vr the angles which determine the position of p 7 in like manner 
as 9 and determine the position of p : and, if y’ be put for 
the same function of 9’ and -ad that y is of 0 and -sr, then p'= a . 
(1 + « ./). Suppose 6' and vr', the arcs which determine the 
position of p', to vary ; and the correspondent .fluxion of the 
spherical surface whose radius is p, will be = p s . sin. 9’ . dQ' . 
dvr' = (/ being put for cos. 9') p* . df . dvr ' ; this is the base of 
the prism equal to dm : the height of the prism is plainly = 
p' — p = a . a . (jy # — y ) : therefore dm = a . a . p*. (/ — y ) . 
dpi . dvr ' : and, by substitution, the equation (C) will become 
•§ V ~f» a 
3 
J X Jj ^ r ' • dd • dd 
hi • d • (d — y) • • f?cr . 
&c. (D). 
Since r, the distance of the attracted point from the centre, 
is = p -j- £r, and / 7 = jV — srp . 7 -{- p a ; therefore the ge- 
neral term of the series in the last equation will be 
