554 



PROFESSOR CHRYSTAL ON LAGRANGE'S RULE FOR THE 



We shall now prove the following : — 



Prop. II. If any solution of (I) can he put into the form 



x-g(u,v) = ' (9), 



then the relation between (x,y,z) involved in this solution must make P = ; in like 

 manner, if any solution can be put into the form 



y-h(u,v) = (10), 



it must make Q = ; and, if any solution can be put into the form 



z-k(u,v) = (11), 



it must make R = 0. 



Let B denote differentiation of the functions g, h, k on the supposition that u and v 



are replaced by their values as functions of (x,y,z), say 



u = u(x,y^), v = v(x,y,z) (13), 



(x,y,z) being supposed unrestricted so far as the differentiation is concerned. Then 



to 



f z =guu z +g v v z , 



from which it is evident that Bg/Bz cannot be infinite in consequence of (9) ; for if g u and 

 g v were infinite that would involve a relation of the form f(u,v) = 0, and u z and v z are 

 finite by our hypotheses. It must be observed, however, that Bg/Bz may vanish. 

 From (9) we have 



p Ji-to)/to q =- s -i /to 



p V Sx) I Sz' q Sy 



Hence (l) or (8) is equivalent to 



I -to -to _to 



Sx Sy Sz 



11% Vj h 



u z 



Sy/ Sz 



to 



Sz 



= 



(14). 



Since g is a function of u and v, 



and (14) reduces to 



to 



Sx 



to 



Sy 



to 



Sz 



u x 



Uy 



u z 



v x 



Vy 



v z 



^0; 



Uy U Z 



'*" 



Since BgjBz± go , the last equation requires in any case that 



Hence, by (7), it follows that P = 0. 



Vy V Z 



(15). 



(16). 



