assigning certain Portions of a Sphere . 
*57 
of the semicircle : hence, for the whole sphere, 8 s — 
= ~ x (zr ) 3 — -J (diameter) 3 , which is the result the theorem 
announces. 
1. The transformation* of z' x y into z' (^) ^ 9, in the 
present case, may be avoided by originally taking the element 
of the solidity differently; thus, if a be the incremental arc de- 
scribed with radius £, the element of the solidity = z' u % •=. 
(since i : 9 : : £ : 1) %' g g 9 = ^ P — f ^ 9 ; whence 
5 = ff^ r* — f 9, as before. 
2. The intersection of the surfaces of the cylinder and sphere 
is a curve of double curvature, of which the two equations are 
I ' l 
y — (rx — Pf and % = (r 4 — rxf ; and hence it appears, that 
the same curve may likewise be formed by the intersection of 
the surfaces of two cylinders, one perpendicular to the plane 
of x and y on a circular base, the other perpendicular to the 
plane of jr and z on a parabolic base. 
3. The area of the curve of double curvature .= f z' (x 1 -]- fY 
—ff 1 — P — y 4 ) 2 (x 4 -f- y'Y = — P x\ when x — r, 
= r 4 ; this area is the surface of half the cylinder included in 
the hemisphere, and therefore the surface of the two cylinders 
included in the sphere = 8 r\ 
* The method of finding triple and double integrals, is not confined to the solution 
of merely geometrical problems like the present. The attraction of an elliptical spheroid 
depends on the formula f/M xy ; and, by integrating it, M. Lagrange (Mem. de 
Berlin,) and M. Legendre (Mem. de l’Acad. 1788), gave the analytical solution 
of the problem of the attraction of a spheroid, which Maclaurin had solved, in his 
Treatise of the Tides, on purely geometrical principles. 
