GRAVITATIONAL STABILITY OF Till. KAHTII. 



179 



function ol r which is such that i*f H is finite at all points within r = a, including the 

 origin r = 0. We seek next to determine E in the form 



(21) 



where E,(r) is a certain function of r which is such that " is finite at all points 

 within ? = a, including the origin r = 0, and F, is a spherical solid harmonic of 

 degree n. The equations of motion of type (15) then become three equations of the 

 type 



in which we must have 



= 0, 



3it 



. . (22) 

 (23) 



It appears on trial that we can obtain a solution in which 



U = , + Mj + 3, V = f, + t',+ t' 3 , W = 



where u,, r,, w, satisfy the system of equations 



. . (24) 



*. . . . (25) 



also tig, v,, w a satisfy the system of equations 



, , s _ . 



' o ' "5 " 



3x dy 3z 

 and Us, r,, tw 3 are a complementary solution of the system of equations 



3x 3y 3z 



7. The sets of functions u u v lt w l and u 3 , v t , 



2 A 2 



(26) 



. . . . (27) 



can be any particular solutions 



