123-124] MOTION OF A SOLID OF REVOLUTION. 185 



the axis of x coinciding with the line of the resultant impulse 

 (/, say) of the motion ; and let 6 be the angle which the line Ox 

 (fixed in the solid) makes with x. We have then 



The first two of equations (2) merely express the fixity of the 

 direction of the impulse in space ; the third gives 



= .................. (3). 



We may suppose, without loss of generality, that A > B. If 

 we write 20 = S-, (3) becomes 



which is the equation of motion of the common pendulum. Hence 

 the angular motion of the body is that of a 'quadrantal 

 pendulum/ i.e. a body whose motion follows the same law in 

 regard to a quadrant as the ordinary pendulum does in regard to 

 a half-circumference. When 6 has been determined from (3) and 

 the initial conditions, x, y are to be found from the equations 



x = u cos 6 - v sin 6 = - cos 2 6 + - sin 2 0, 



the latter of which gives 



y = -?-0 (6), 



as is otherwise obvious, the additive constant being zero since the 

 axis of x is taken to be coincident with, and not merely parallel 

 to, the line of the impulse /. 



Let us first suppose that the body makes complete revolutions, 

 in which case the first integral of (3) is of the form 



2 = w 2 (1 - k* sin 2 0) (7), 



, 72 A-B I 2 

 Where k -ABQ'^ (8) " 



Hence, reckoning t from the position 6 0, we have 



