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 



Au = Icoa0, Bv = -Isin0, r=6. 



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

 direction of the impulse in space ; the third gives 



sin&amp;lt;9cos&amp;lt;9 = .................. (3). 



We may suppose, without loss of generality, that A &amp;gt; 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 has been determined from (3) and 

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



x = u cos v sin = r cos 2 6 + ^ sin 2 0, 



= (:l~]sin0cos0 = -?6 



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 = o&amp;gt; 2 (1 - & 2 sin 2 0) (7), 



A-B I* 

 where fc 2 = -r-D?y 2 W 



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



