( 551 ) 



XX. — A Demonstration of Lagrange's Rule for the Solution of a Linear Partial 

 Differential Equation, with some Historical Remarks on Defective Demonstrations 

 hitherto Current. By Gr. Chrystal, Professor of Mathematics, University of Edin- 

 burgh. 



(Read June 15, 1891.) 



It seems strange that a principle so fundamental and so widely used as Lagrange's 

 Rule for Solving a Linear Differential Equation should hitherto have been almost 

 invariably provided with an inadequate demonstration. I noticed several years ago that 

 the demonstrations in our current English text-books were apparently insufficient ; but, as 

 the method by which I treated Linear Partial Differential Equations in my lectures did 

 not involve the use of them, it did not occur to me to analyse them closely with a view 

 to discovering in what the exact nature of the defect consisted. The consideration of 

 certain special cases recently led me to examine the matter more closely, and I was 

 greatly surprised to find that most of the general demonstrations given are vitiated by a 

 very obvious fallacy, and in point of fact do not fit the actual facts disclosed by the 

 examination of particular cases at all. 



The point where the demonstrations fail is in showing that every solution of the 

 differential equation Pp + Qq = R must be of the form f(u,v) = ; where u = a, v = h 

 are two independent integrals of the Auxiliary System dx/P = dy/Q = dzfR. 



In Forsyth's Differential Equations (§ 186), for example, a demonstration is given, 

 which is briefly this. Let 



■yjr(x,yt) = .... (A) 



(B); 



be any solution of 









Yp + Qq- 



= R 



then, it can be shown that 









S\fr S\fs 



s^ 





'Ex Sy 



Sz 





Su Su 



Su 





Sx Sy 



Sz 





Sv Sv 



Sv 





Sx Sy 



SV 



= 



(C). 



From (C) it is inferred that there must be a functional relation between -^, u, v, so 

 that ^(x,y,z) = F(u,v). 



Now, this application of Jacobi's well-known theorem is not justified, unless the 

 equation (C) be an identity in x, y, z. The premises, however, requires merely that (C) 

 shall be satisfied in consequence of (A). Not only is it not proved that (C) is an 



VOL. XXXVI. PART II. (NO. 20). 4 



