or THE MOTIONS OF A SOLID BODY. 2^7 



7^;^: = ^' ■ (f + i) ■■ but since ,'Hr'-*^-/«. 

 and Bq'^ + .^r'« = ~^, we have d ^ = 



If we substitute in this equation the value of dt already 

 found, we shall be able to find 4' in terms ofp' : and we shall 

 thus obtain the three angles 9, <p, and ^ in terms of p', q\ 

 and /, which will also be derived from the time t. Hav- 

 ing therefore computed in this manner the values of these 

 angles, with regard to the plane of x' and y' which has 

 been considered, it will be easy to deduce from them, by 

 spherical trigonometry, the similar quantities which belong 

 to any other plane, and of which the determination will 

 introduce two new independent quantities, which, with 

 the three already mentioned, and that which belongs to 

 the fluemt of x^, will constitute the six independent quan- 

 tities required in the complete solution of the problem: 

 but the investigation is obviously simplified by referring 

 it to the fixed plane of greatest rotatory power. 



Scholium. The position of the three principal axes 

 with respect to the body being supposed to be known, if 

 we are acquainted with that of the momentary axis of ro- 

 tation for any instant, and with the angular velocity of ro- 

 tation, we shall have the values of p, q, and r, for the 

 given time, since their values are equal to the products of 

 the angular velocity into the cosines of the angles formed 

 by the momentary axis with the principal axis : hence we 

 shall obtain the values of/?', q\ and r, which are propor- 

 tional to the sines of the angles formed by the principal 

 axes with the plane of greatest rotatory power, which is 



