350 Mr. J. Prescott on the 



OE by c, the component R sin ft has a moment R sin ft . c cos y 

 about OY in the direction in which y increases ; the com- 

 ponent RcosyScos^ has a moment R cos ft cos x . c sin y in 

 the same direction about OY ; and, lastly, the component 

 R cos ft sin x has a moment R cos ft sin x . c cos y about ON 

 in the direction in which x increases, regarding Og as fixed. 

 Thus the couples increasing x and y are respectively 



Re cos ft cos y sin x 

 and Rc(cos/3sin ycos tf + sin ft cosy). 



36. Now let us assume that all the angles involved in the 

 kinetic energy and the couples, except a. and v, are very 

 small. Then the couples become approximately Rc# and 

 ~Rc(y + ft), while the kinetic energy becomes 



Hence 



v=i\{f+(i-xy}+iB{i- v (d-x)y. 



3sc 



. =— A(«-<8)+By{ii-y(«-«)}. 



= 0, 



BU , 



== «B(c-^){z;— y{o-x)}. 



Using x, v, and y as generalized co-ordinates, Lagrange's 

 equations of motion give 



that is, 



d-dv _BU_ 



dt ~dv " U ' 



whence 



B{z)— y{oL— %)}= a constant = Bo) say. 



