78 ME. C. J. HARGREAVE ON THE CALCULATION OF ATTRACTIONS, 
4. This equation was not integrated ; but by a skilful use of its properties, the 
problem of attractions was greatly simplified by Laplace. He laid down a theorem, 
respecting the surfaces of all spheroids of small deviation, that their radii vectores 
might be developed into series, every term of which would satisfy the above equa- 
tion ; and he also gave a method of expansion. By means of this theorem, the pro- 
blem could be solved for spheroidal bodies which differ but little from spheres ; but 
its generality has been greatly restricted by the researches of subsequent writers*, by 
whom it has been shown that it is true only for bodies whose radii are expressible in 
rational and integral functions of f , a/(1 — f 2 ) cos $>',*/( 1 — f 2 ) sin <£>'. Among these 
are the ellipsoid and elliptical spheroid, and a large class of other spheroids. In these 
papers I have adopted a different proceeding ; I integrate the equation itself generally, 
and determine the arbitrary functions contained in the integrals by the circumstances 
of the problem itself. In consequence of the peculiar form which V n then takes, V 
may be found by effecting the operations indicated, which are only explicit integra- 
tions. 
5. I shall now proceed to integrate this equation. 
Consider p and <p as functions of two new variables X and Y, to be determined 
from the equations, 
dx = ^(d<p + kdp) 
where k and k' are the roots of the equation (1 — f 2 ) k 2 -f 1 — 2 = 0. These roots 
J- — f 4 
are ~ ^ , whence we obtain 
l - f 4 
X = (p + i nA 11 ! log and Y = - \ ^ -1 log . . (4.) 
d 2 K_(dXy^P_ dXdY cPP, (dY\2 ££ * X d P d* Y d P n 
dv? — \dfx) dX 2 "P * dp dp dXd Y “P \dfuj d Y 2 d Y? dX "P d ^ d Y’ 
- _ 1 _ 2 d*V n 1 JP dP\ 
- (I - F 2 ) 2 dX 2 T" (1 - ^ dXd Y (1 - d Y 2 "T" (1 - f* 2 ) 2 \dX ~ rfY/ ; 
dXdY n , dYdV n (dV n d P \ 
— dp dX djtt rf Y ~ 1 - p 2 VrfX dY/ ’ 
rf 2 P„. _ C*X\2d 2 P re dXdY (dYVd 2 P n d 2 XdP re tfY dP„ 
df ~ \d<pj dX 2 * d<? d? dXd Y \d<p/ dY 2 dp 2 dX df dY ’ 
d* P d 2 P d 2 P 
- dX 2 "T - z dXdY "*■ d Y 2 
Substituting these in (3.), we obtain 
4 rf 2 P 
dXdY + w (w + 1) P re = 0. 
* See two articles by Mr. Ivory in the Philosophical Transactions, 1812. 
