278 PROFESSOR TAIT ON THE ROTATION OF A 



<7« = 22 (7), 



7S = it • • ... (17), 



<Pn=Z (18). 



We have only to eliminate £ and >?, and we get 



2? = 2?- 1 (?- x 7!?) • ... (19), 



in which <? is now the only unknown ; 7, if variable, being supposed known in 

 terms of q and t. It is hardly conceivable that any simpler, or more easily inter- 

 pretable, equation for q can be presented until symbols are devised far more com- 

 prehensive in their meaning than any we yet have. 



22. Before entering into considerations as to the integration of this equation, 

 we may investigate some other consequences of the group of equations in § 21. 

 Thus, for instance, differentiating (17), we have 



n + n = il + qi , 



and, eliminating q by means of (7) 



yqri + 2yq = qnl + 2ql 



whence 



t = V£» + q~ l n ; 



which gives, in the case when no forces act, the forms 



e = v^-^ (20), 



and 



(as { = <pn) 



<ph = — V. y\$n ..... (21). 



To each of these the term q~ l yq, or q~^q, must be added on the right, if forces 

 act. 



23. It is now desirable to examine the formation of the function <p. By its 

 definition (1G) we have 



<Pg = 2 • m (aSag — a 2 g) , 



= — 2 . muYug . 

 Hence 



-S# g = 2.™(TVag) 2 , 



so that — Sf <£f is the moment of inertia of the body about the vector f , multiplied 

 by the square of the tensor of f . Thus the equation 



evidently belongs to an ellipsoid, of which the radii- vectores are inversely as the 



