and therefore, 



100 



W + V = fj. {»x + /Sy + yz) , 



^W , , , 



-t_ = ^.r — |x« (eu; + |8y -f- yz) , 



ooc 



-^ = fx.y- l^^ (»x + I3y + yz) , 

 jp-=:i^z — iJ.y {»x + /iy + yz) . 



This system of equations (C) is one form for the integral of the 

 partial differential equation {A) ; the quantities a, /3, y, being sup- 

 posed to be eliminated) and W being an arbitrary function of these 

 quantities, of the kind already mentioned. 



Transformation and Development of the Integral. 



3. The system of equations (C) may be transformed into the 

 following : 



dU dU jr , rr dU , ^ dU ,„. 



^x=-^, i.y= -^, V+U=.-^+^^; (D) 



in which C7 is a function of the three independent variables a, (2, z, 

 obtained from the function W by putting 



U=W—iuyz, (E) 



and by considering y as a function of a, jS. Let us now proceed to 

 eliminate a, /3, between the three equations (D), by the theorems 

 which Laplace has given in the second Book of the MScanique 

 CSleste, for the development of functions into series. 



This elimination may be simplified by a proper choice of the 

 coordinates. The rays of an ordinary system being perpendicular to 

 to the surfaces which have for equation 



V = const. , 



