SOLUTION OF A LINEAR PARTIAL DIFFERENTIAL EQUATION. 561 



so that ^ does not vanish as a mere consequence of (E). Also, if we replace x and y in 

 >//■ by their values in terms of z, we get 



from which z does not disappear of itself. 



There is in this demonstration the same fallacy of confusion between identical and 

 conditional equality that was pointed out in the former one. It is certainly very strange 

 that this proof, occurring in a book so widely read as the Calcul des Fonctions, should 

 not have been questioned before. 



Jacobi, in his " Dilucidationes de iEquationum Differentialium Vulgarum Systematis," 

 &c. (Crelles Jour., 1842, Werke, Bd. iv. p. 150), refers to Lagrange's work as follows : — 

 "111. Lagrange (Acad. Ber. a., 1779, pp. 152-160) sequationum differentialium primi 

 ordinis linearium solutionem, hoc est reductionem ad sequationes vulgares, primum obiter 

 et adumbrata tantum demonstratione dedit. De ilia demonstratione pretiosa alio loco 

 mini agendum erit.* Aliam postea dedit demonstrationem in commentatione" (" Methode 

 Generale," &c, Acad. Berl. a., 1785, pp. 174-190). 



It does not appear from this that Jacobi had read Lagrange very closely. The 

 " Demonstratio Adumbrata" is a perfectly clear and good proof so far as it goes, while the 

 " Alia Demonstratio" is fallacious, as we have seen. 



Jacobi's own method is very beautiful. He first considers the " Homogeneous" 

 equation, 



Z I +X .|+- •••+ x '' S f > =° < A >> 



where Z, X 1; . . . . , X n are functions of z, x x , . . . . , x n ; and shows that every integral 



of (A) is of the form 



H(/ 1 ,/ 2 ,....,/„) = (B), 



where 



f\ = a vfl~ a ^ ■ ■ ■ ■ >fn~ a n (C) 



is the integral system of 



dx 1 /X 1 = dx 2 /X 2 — .... = dx n /X n = dz/Z .... (D). 



He then shows that the equation 



x >!+ x ^+----+ x ..lr z (E) 



can be transformed into (A) by supposing 



f(z,x u , X n ) = a (F), 



where a is an arbitrary constant, to be a solution of (E), and replacing Bz/Bx^ .... by 

 their values - (&f 'Sa; 1 )-^(^/*/ Sz), .... 



So far as proving that every possible solution of (E) is contained in (B) is concerned, 



* I have not been able to find tins " alium locum." 



