60 



MR. IVORY ON HOMOGENEOUS ELLIPSOIDS 



We thus obtain these two equations, 



B-/ 1 



1 + A' 



2' 



c-/ 



1 + X' 



f2» 



and from these we deduce 



/=B~^ 



+ X' 



1' 



f=C-j 



lK' 



(2.) 



which coincide with the equations of Lagrange. 



It is next requisite to substitute for the symbols A, B, C, what they stand for. The 

 values given in the Mdcanique Cdeste are in a convenient form for this purpose, viz. 



x^ dx 



d¥ = 



V{i +\^x^-),{\ + X'2:r2)' 



. 3M /»^ ,_, ^ SM r' 



dF 



+ A2 ^2' 



_ 3M_ /»! 



~ F Jo i 



In these expressions M is the mass of the ellipsoid ; therefore if we put ^ for the den- 

 sity, we shall have 



^ = l|£.(i+x^)*(i+x'2)*. 

 These several values being substituted in the equations (2.), the result will be 

 9= 4^-' ^ 





(1 +X«^2)4(1 + A'^J^T 

 — -i/L±5 f^ >'^-3x^d x{l -x^) 



(3.) 



+ \>^-Jo (1+^2 ^9)i (1 _^^n _j,«)4 J 



Here 7 stands for the proportion of the intensities of the centrifugal and attractive 

 forces ; it depends only on the kind of matter of which the spheroid is formed, and 

 the velocity of rotation. 



4. The equations (3.) comprehend all ellipsoids that are susceptible of an equili- 

 brium on the supposition of a centrifugal force. To begin with the more simple case 

 of the spheroid of revolution, let X = X' = /; and the two equations will coincide in 

 one, viz. 



9 -Jo ^J\ +l^xy ' ^ ^^ 



which expresses the relation between q and /, in a spheroid of revolution having its 

 semi-axes equal to k and k\/\ + /^. 



From the equation (4.) we learn that q will be known when / is given, or that every 

 spheroid of a determinate form requires an appropriate velocity of rotation. 



