RIGID BODY ABOUT A FIXED POINT. 281 



25. However they are obtained, such equations as those of § 23 were shown 

 long ago by Euler to be integrable as follows. 

 Putting 



2fu 1 u. z u z dt = s , 



we have 



A^ 2 = A^ 2 + (B - C)s 



with other two equations of the same form. Hence 



ds 



2dt 



Jfa 



Cr^ n B - C Y /^ C-AV/ n , A - B Y 



(°i + ST') (®« + "~B~V (°» + ~C~ S ) ' 



so that t is known in terms of s by an elliptic integral. Thus, finally, n or £ may 

 be expressed in terms of t ; and in some of the succeeding investigations for q 

 we shall suppose this to have been done. It is with this integration, or an 

 equivalent one, that most writers on the farther development of the subject have 

 commenced their investigations. 



26. By § 16, 7 is evidently the vector moment of momentum of the rigid 

 body ; and the kinetic energy is, as in § 17, 



But 



so that when no forces act 



But, by (17), we have also 



— \ 2 • inir- = — JSs/ . 



Ssy = S . q- l tqq~ x yq = S>j£ , 



S^- J £ = S*pn = - h 2 . 



T£ = Ty , or Tcpv — Ty , 



so that we have, for the equations of the cones described in the initial position of 

 the body by n and £, that is, for the cones described in the moving body by the 

 instantaneous axis and by the perpendicular to the invariable plane, 



wp + ^sw-^o, 



W{<pnf + 7 2 S^ = 0. 



This is on the supposition that 7 and h are constants. If forces act, these 

 quantities are functions of t, and the equations of the cones then described in the 

 body must be found by eliminating t between the respective equations. The 

 final results to which such a process will lead must, of course, depend entirely 

 upon the way in which t is involved in these equations, and therefore no general 

 statement on the subject can be made. 



27. Recurring to our equations for the determination of q, and taking first the 

 case of no forces, we see that, if we assume n to have been found (as in § 25) by 

 means of elliptic integrals, we have to solve the equation 



VOL. XXV. PART II. 4 C 



