SOLUTION OF A LINEAR PARTIAL DIFFERENTIAL EQUATION. 557 



It will be seen that the solution z — x—y is given in the Lagrangian form, viz., v = 0, if we consider 

 Xi(x,u,v) = or x 2 (y,u,v) = 0, but in the form z — u = 0, if we consider x£z,u,v) = 0. 



Ex. 2. Consider 



z — 2x — 2/ = 0, 



which is a solution of the same differential equation as before. Here, in order to avoid the infinite 

 value of v, we choose 



u=x + y , v=(z-2x-y)/x(z — x—y). 

 We then find 



Xi(x,w,v)=x 2 vj(l~vx) ; 

 X2 (y,u,v)=(u - y) 2 v/(l -vu + vy) ; 

 X s (z,u,v)=(z— u) 2 v/(l — vu+vz) . 



In this instance the solution in question is given in the Lagrangian form in all three cases, viz., 

 v = 0. 



Ex. 3. \},(x,y,z)=zy—x 2 = ; 



xp + yq = z. 



u=xfz , v=yjz . 



Xi(%,w,v)=x(v — u 2 )uv ; 



X 2 {y,u,,v)=y(v -u 2 )/v 2 ; 



Xa&upy^zfy — u^/v. 



In this case also the solution is given in Lagrange's form, v — u 2 — 0, in all three cases. 

 Ex. 4. 



\Js(x,y,z)=z-x-y: 



{1+ J(z-x-y)}p + q = 2. 



u==2y — z, v=y + 2 J(z— x— y) . 



Xi(x,u,v)=2 + v -u — x — 2 J(l+v — u—x) ; 



X 2 (y,u,v)^i(v-y) 2 ; 



X 3 (z,u,v)=i(2v -u-zf. 



Here the solution z — x — y = cannot be put into the Lagrangian form at all. This is in agreement 

 with our demonstration; for the relation z — x — y = gives a locus of branch-points of the coefficient 

 of p. The solution in question is therefore a singular solution of the linear equation. It sometimes 

 happens that the solutions classified as singular for the purposes of the above demonstration are 

 particular or limiting cases of a Lagrangian solution ; but this is not the case in the example just 

 given. 



Historical Remarks on the Previous Demonstrations of Lagrange's Rule. 



The first germ of the Eule itself was given by Lagrange in § 52 of his memoir " Sur 

 les Integrates Particulieres des Equations Differentielles " (Nouv. Mem. d. I Ac. d. Berl., 

 1774).* It is given in a complete and general form in Art. V. of his memoir "Sur dif- 

 ferentes Questions d' Analyse," &c. {ibid., 1779) ;t and is there accompanied by a verifi- 

 cation that the Lagrangian integral satisfies the differential equation. He returns to the 

 subject {ibid., 1785) \ in a memoir entitled " Methode Generale pour integrer les Equations 

 aux Differences Partielles." In so doing, he says : — " Mais, comme cette methode n'y 



* (Euvres, t. iv. p. 82. t (Humes, t. iv. p. 624. \ CEuvres, t. v. p. 543. 



