2()8 CELESTIAL MECHANICS. I. vil. 29- 



supposed in this proposition to be that of x' and y, and 

 with respect to which the sum of the projections of the 

 areas described by the revolving" radii, multiplied by the 

 masses of the respective particles, is a maximum. We 

 may therefore determine at every instant the intersection 

 of the surface of the body with this plane, and may conse- 

 quently find its situation by the actual conditions of the 

 motion of the body. 



{556, Lemma. The square of the radius 



2 

 of gyration of a sphere is -^ of the square of 



the semidiameter. 



The fluxion of the surface of a sphere is as do: — .y = 



rdx, that of a great circle being da; — , where the sine is 



X, and the cosine y: and at last, when x:=zr, the surface 

 of the hemisphere becomes equal to that of the cor- 

 responding semicylinder (183): the fluxion of the rotatory 

 inertia of the surface will be represented by rdx.y^=:{r^ — 

 x^) rdx:=:r^dx — rx^da:, and the fluent byr'ar — ^rx^ or, 

 for the hemisphere, by ^ r^ which, divided by r^, gives the 

 square of the radius of gyration f r% and the rotatory in- 

 ertia I r^M, M being the content or mass of the surface 

 of which the radius is r. 



If the sphere be now supposed to increase by concen- 

 tric surfaces, the fluxion of the mass will be as r^dr.f, if f 

 be the density, and that of the rotatory inertia as f r*dr . f , 



for^dr 

 and the square of the radius of gyration will be f y. -, 



which, when przl, becomes 4 --^^ — = j.rS and the rota- 

 tory inertia of the homogeneous sphere will be f r^m.] 



