124] 



MOTION OF A SOLID OF REVOLUTION. 



187 



in the usual notation of elliptic integrals. If we eliminate t 

 between (5) and (9), and then integrate with respect to 0, 

 we find 



-^F*ft 



(10), 



the origin of x being taken to correspond to the position = 0. 

 The path can then be traced, in any particular case, by means of 

 Legendre s Tables. See the curve marked I in the figure. 



If, on the other hand, the solid does not make a complete 

 revolution, but oscillates through an angle a on each side of the 

 position 6 = 0, the proper form of the first integral of (3) is 



frs.rffl-E^ ; (11), 



\ sin 2 aj 



where 

 If we put 

 this gives 



whence 



sur a = 



ABQ ft) 2 



A-B 

 sin 6 sin a sin - 



(12). 



sin 2 a 



siu a 



(13). 



Transforming to ^r as independent variable, in (5), and integrating, 

 we find 



x = -g- sin a . .F(sin a, ^r) - -y^cosec a . #(sin a, yfr 



Qco 

 y = j- cos &amp;gt;|r 



The path of the point is here a sinuous curve crossing the line 

 of the impulse at intervals of time equal to a half-period of the 

 angular motion. This is illustrated by the curves III and IV of the 

 figure. 



There remains a critical case between the two preceding, where 

 the solid just makes a half-revolution, 6 having as asymptotic 



