195 



so that the coefficient of 



. x x y x z x is J(a 3 + 2a/3 2 - 2a - /3) COS 0 - 0 

 making these substitutions, we have 

 1 15 fi ' 



+ 



_ £L a 3 + 2/3 2 a) aj,% + (S(a 3 + 2/3*a) - 4a - 2/3 sin 2 *) 

 -4 8 r ( 



+ sin 2 * (2a - p) I^j sin (0 - 0) 

 _J.L./L ( a s + 2a/3 2 - 2a - /3 sin 2 *) a?^*, cos 0 - 0 



and the whole coefficient of sin 0 - 0 will be the quantity just found, 

 minus the quantity first found above, that is, it will be 



^ 2 |t 1 (aS + 2/32a) ^ + (3(a3 + Wa) Binh (4a " 2/3)) ^ 



4- sin 2 * (2a - /3) V j - | a r^j 



call this, for shortness, — M y , and the coefficient of cos 0 - 0, just 

 given, - -i- if Then the equation for w l will become 



dt*> x C-B 1 - . 1 , T - — - 



— + — - — w 3 a> 2 = — M y sin 0 - 0 - —N cos 0-0, 

 dt A. -A J± 



and in like manner 



dw A - C 1 • . 1 



— + — — iv 3 u} x = - — Wsm 6 - e + — m x cos 0 - 0 



at Jd Jj H 



where M x is what M y becomes when x x and y x are interchanged. In 

 order that these may be integrated, it will be necessary to express 0 and 

 0 in terms of t ; but before doing so it may be well to make a remark 

 upon the quantities 2(m x x \), &c, which occur continually. We may 

 always choose the axes so as to make 



2 (mx x ) = 0, 2 (mx x y x ) = 0, &c. ; 



but not so as to satisfy any further conditions, such as 



2 (mx x *y x ), 2 (mx l 2 z 1 - 0. 



But there is one case in which a certain class of these quantities will 

 always vanish. Whenever the body is perfectly symmetrical in term 



