PARTIAL DIFFERENTIAL EQUATIONS. 611 



But I am not able to find that it has ever been applied to any equation of the first order. 

 Lacrolv (Vol. il. p. 558) gives something as near to the whole method as can well be imagined. 

 He sees everything except the completely interchangeable character of « and pj; + qy — z ; that 

 he did not see this last may be suspected from his making the restriction that z must only enter 

 in px + qy — z. 



It is to be noted that, so far as equations of t\\Q first order are concerned, the solution takes 

 exactly the same form, even though we can only integrate the transformed equation by reducing 

 it to 



s], {X, Y, Z,J)=0, ^ = 0, 



for the forms of — and are unaltered. There is now one equation more, — ^ = 0, and 



dX dY dA 



one more quantity, A, to eliminate. 



Let the first instance be 



Ax + By + Cx + D=0, 



where each of the four, A, &c. is any function whatever of p, q, and px + qy - z. The transformed 

 equation is obviously of the form Pp + Qq + R = 0, where P, Q, B are all functions of x,y, z. 

 Lagrange has given a laborious method for the integration of z = pq, and Lacroix (Vol. ii. 

 p. 565) does not refer to p. 558, I suppose for the reason just given. The transformed equation is 

 px+ qy - z = xy, of which the integral is 



- xy 



'/(!) 



We may therefore find the general solution oi z - pq from 



Generally, however, the most convenient method is to select an appropriate primary solution, and 

 then to use Lagrange's process. This may be done, if we please, from the common ditf'erentia! 

 equations which integrate the transformed partial ones. These are, in the present case, 



z = xy + hx, y = ax. 



The retransformcd equations are 



px + qy - z = pq + bp, q = up. 



With these, and z = pq, eliminate p and q, which gives 



(ic + ay — by 



z = , so that we have the general solution by eliminating a from 



4 a 



{x + ay + (baf , dz 



z = ^^ ? — ^—^ , and — = 0. 



ia da 



Hut wc may often, most often I think, (irocure the primary Milutioti in ,iii e.isiir manner froni the 

 result of the complete method. Let fz = iiz + b, and we then have 



X = }' + b, // = A' + a, z= XY, 



