50 



cone (45), and was studied by both Poinsot and Mac CuUagh. 

 But it would far exceed the limits of the present communica- 

 tion, if the author were to attempt here to call into review the 

 labours of all the eminent men who, since the time of Euler, 

 have treated, in their several ways, of the rotation of a solid 

 body. He desires, however, before he concludes tliis sketch, 

 to show how his own methods may be employed to assign the 

 values of the three principal moments, and the positions of 

 the three principal axes of inertia ; although it has not been 

 necessary for him, so far, on the plan which he has pursued, 

 to make any use of those axes. 



VI. Let us, then, inquire under what conditions the body 

 can continue to revolve, with a constant velocity, round a per- 

 manent axis of rotation. The condition of such a double per- 

 manence, of both the direction and the velocity of rotation, is 

 completely expressed, on the present plan, by the one diffe- 

 rential equation, 



1 = 0; (47) 



that is, in virtue of the formula (23), by 



V.iy = Q', (48) 



or, on account of (28) and (36), by this other equation, 



(<T+«)t = 0, (49) 



where a is the characteristic of operation lately employed, 

 and 5 is a scalar coefficient, which must, if possible, be so de- 

 termined as to allow the following symbolic expression for 

 the sought permanent axis of rotation, namely, 



t = (<T+5)-'0, (50) 



to give a value different from zero, or to represent an actual 

 vector t, and not a null one. Now if we assumed any actual 

 vector K, such that 



(<7 + 5)<=K, (51) 



