PEOFESSOE SYLVESTER ON THE MOTION OF A RIGID BODY 

 C -^— (A— B)u,yj= J cos I* ; 



cosXu,+ cos/a<yj+ cosj'f3=0. 

 Av^dvi -\-Bv2dv-{-Cv3(lv3=0, 



762 



also 



Hence 



and 



Avl+Bvl+Cvl^K, 



a constant, as was to be proved. 



In the case actually under consideration, if ly,, u^, u^ are the angular velocities of the 



associated free body, and r the time corresponding to t, so that dt, dr are the intervals 



of time of the rolling and the free body undergoing the same infinitesimal angular 



displacement of position, we have 



and 



8r 



df= 



Hence 



K 



so that using the notation in ordinary use for the motion of a free body. 



and thus the time t of the rolling ellipsoid is known as an elliptic function in terms of s*^. 



Fui-thermore, by the well-known equations of vis viva and conservation of areas 



applied to the free body whose kinematical exponent is the ellipsoid with semiaxes 



■a, b, c, i. e. whose moments of inertia may be denoted by -^» j5» ^' we have 



m^ aj^ 00^ 



-^ + 7l + -|=M, 

 a-' b^ c^ 



CU^ Ui^ Oi^ — 



I _L 2 I_ 3 T 2 



or r C* 



Consequently if A, B, C are respectively representable by 



h.\t hilt ^1^ 



the multiplicator of ado) is the numerator of the expression above given for dt, becomes 

 a constant, viz. XL^^+jiiM. But this is the case when the density of the eUipsoid is uni- 

 form ; for then . 



A:B:C 



and the determinant 



1_ 

 b* 



J'+C; c'+a'; a'+b' 



le.^a(a-'-P){a^+l')=0. 



