Art. 6. — R. Kaibara 



2. Let «, h, c (a^-b^c) be the semi-axes of the ellipsoid, <t, 

 its density, and w, the angular velocity (about the axis c); then, 

 if we write 



/•" ds 



P = abc —. ., , N ,-' 



where 



(«- + , 

 /•° ds 



p ds 



R = abc ^T", ^^, ' 



Û ^ 



J = V'(«"'' + s) {^' + s)(c~ + s), 



the condition for the equilibrium of the ellipsoid is expressed by 

 the well-known equations: 



where 



d\P-iè) = b'-(Q-O) = c^B, 

 'A-a 



(1) 



From these equations, Ave get 



(2) 



this represents the relation to be fulfilled by the axes a, h, c. 



For the sake of brevity, we shall denote the ratios of axes and 

 eccentricities of the principal sections thus: — 



_ c _^ cr—c' 



^ '^ ~ a "^ ~~ 





a 



'■^= b— 



V ä- — b'- 



(3) 



' " a " a 



It can be easily proved that P, Q, R satisfy the identities 



P+Q + i? = 2, 1 



a-P-\-J)'Q-\-rR = S,j 

 ds 



(4) 



where S = abc J -^. 



' 



By combining (-1) with (1), various forms of expression can be 



