552 PROFESSOR CHRYSTAL ON LAGRANGE'S RULE FOR THE 



identity in x, y, z; but this is not in general true, as may be shown by taking a case at 

 random. 



The first example of Lagrange's Rule, given in Forsyth (§ 189), is the equation 



xp+yq = z (B). 



Now, 



yz-x i = (A) 



is a particular solution, as may be readily verified. Here we may take u=x\z, v=y\z. 

 Since ^=yz — x 2 , we find 



8(\p-,u,v) yz — x 2 



S(x,y^) yz 3 



which does not vanish identically, but only as a consequence of (A). 



The fallacy thus disclosed consists in a confusion between a conditional equation and 

 an identity — an error that has occurred very often in the history of mathematics. The 

 cause of the defects in the various demonstrations lies somewhat deeper, as we shall point 

 out by and by. 



The following attempt at a rigorous demonstration was given more than a year ago, 

 in a correspondence with Mr Forsyth on the subject. Owing to pressure of other work, 

 I have not found time to publish it until now. Meanwhile, M. Goursat's Legons sur 

 V Integration des Equations aux Derivees Partielles du Premier Ordre have been pub- 

 lished, and there I find the first rigorous demonstration of Lagrange's Eule with which 

 I am acquainted. His demonstration depends on the thorough discussion of the passage 

 from the ordinary linear equation to Jacobi's homogeneous linear equation ; so that it is 

 totally different from mine. The form in which the latter now appears owes something 

 to the private criticisms of Mr Forsyth, and something to my perusal of M. Goursat's 

 able and interesting Legons. 



I give at the end of this paper a short historical account of the earlier demonstrations 

 of Lagrange's Rule, all of which are more or less defective. 



Proof of Lagrange's Rule for the Solution of a Linear Partial Differential Equation. 



Let us take, for simplicity, the case of two independent variables ; and let the 



equation be 



Pp + Qq = R (1), 



where P, Q, R are single valued functions of x, y, z, and p = Bz/Bx, q = Bz/By, as usual. 

 By a solution of (l) we understand an equation of the form 



#W) = ( 2 X 



which leads to one or more determinations of z as a finite, continuous, single-valued 



