178 THE ROYAL SOCIETY OF CANADA 



If we eliminate 62 between (22) and (23), there results the equation 



dp\\/ dpdq\\/ 



VX/ dq\'K/ 



which, on simplification, reduces to 

 (24) dnog\ ^_Q 



dpdq 

 This equation however expresses the fact that one application of 

 the Laplace a — transformation to (A) leads to an equation with 

 vanishing invariant. (Forsyth, Theory of Differential Equations, 

 Part IV, Vol. VI, p. 133). The integration of (A) can therefore be 

 effected by the method of Laplace; and thence the equations sought 

 can be found from (6). 



Equation (24) is known as Liouville's equation, and its general 

 integral is (Forsyth, Differential Equations, Vol. VI, p. 143). 



^_g a'{p)^'{q) 

 \a(p) + 0(q)Y 

 Equation (A) now becomes 



(^') ^'^ -2 "''^' e 



dpdq ia+0y 

 If we effect the transformation 



~p = a{p), g = i9(5) 

 on this equation and drop the bars in the resulting equation, we obtain 

 (B) d-'d _ 2 ^ 



dpdq (p+qY ' 

 To (B) apply the Laplace cr — transformation 



- dd 



d= — ; 



dp 



dp p+q 



