S i <p i = 



-A/ 



Sj<Pj = 



-B, 



Skcpk — 



-a. 



RIGID BODY ABOUT A FIXED POINT. 279 



square roots of the moments of inertia about them ; * so that, if i, j, k be taken 

 as unit vectors in the directions of its axes respectively, we have 



(22), 



A, B, C, being the principal moments of inertia. Consequently 



<p s =-{AiSi s + B/S/g + Ck§k g } .... (23). 

 Thus the equation (21) for n breaks up, if we put 



>j = %w x + ju. 2 + ku 3 



into the three following scalar equations 



A« x + (C - B)« a « 3 = 0, 

 B<5 2 +(A-C) W3Wl = 0, ■ 

 Cw 3 +(B-A) Wl « 2 = 0,. 



which are the same as those of Euler. Only, it is to be understood that the 

 equations just written are not primarily to be considered as equations of rotation. 

 They rather express, with reference to fixed axes in the initial position of the body, 

 the motion of the extremity, <o x , w 2 , » 3 , of the vector corresponding to the instan- 

 taneous axis in the moving body. If, however, we consider w v « 2 , w 3 as standing 

 for their values in terms of w, x, y, z (§ 27 below), or any other coordinates 

 employed to refer the body to fixed axes, they are the equations of motion. 

 Similar remarks apply to the equation which determines £, for if we put 



£ = %zr x + j*r 2 + kzr 3 , 



(20) may be reduced to three scalar equations of the form 



ri + ft~i) 



ar 2^ r 3 ~ 



24. Euler's equations in their usual form are easily deduced from what pre- 

 cedes. For, let 



whatever be f ; that is, let <p represent with reference to the moving principal 

 axes what <p represents with reference to the principal axes in the initial position 

 of the body, and we have 



= qtq- 1 = q^i^-'Oq- 1 



* For further information about this equation, see Hamilton, Proc. B. I. A. 1847, and Elements 

 of Quaternions, p. 755. Also Tait, Quaternions, § 367. 



