122 Dr. Hargreave on Differential 



For the proposed equation is simply 



which is an obvious transformation of 

 (j>(x,y,p) = 0. 



This theorem has long been known and usefully employed 

 with regard to one particular pair of correlated functions, viz. p 

 and px—y, which give w = w; and this instance has the pecu- 

 liarity, that if the substitution be effected twice, we are remitted 

 to the original equation. This will always be the case when u, 

 v y and w, expressed in terms of x, y, and p, are similar in form 

 to the expression of x, y, and p in terms of u } v, and w. 



The theorem may be stated more generally in this form :— If 

 the equation 



4>(u m , v m , w m )=0 

 be integrable for any pair of correlated functions u m and v m , 

 then 



<f>(u n , v n ,w n ) = 



is also integrable for every other known pair of correlated func- 

 tions u n and v n . 



If, therefore, we know a few pairs of correlated functions, 

 there is theoretically no limit to the number of integrable forms 

 which may be deduced from a single integrated equation. The 

 substitution of u, v, and w for x, y, and p respectively may be 

 repeated as often as we please, and the results may be varied 

 indefinitely by crossing them with other sets of correlated 

 functions. 



A very general process of integration for equations of the 

 first order is that by which we are enabled to solve 



(j){x,y,p)=0 

 whenever (f> is linear with regard to x and y, or linear with regard 

 to y and p. The forms 



y + ^f 1 p+f 2 p = 0, 



p + yf l x+f 2 x=0 

 are always soluble. We may now therefore assert that the forms 



v +uf i w+fc i w = 0, 



w + vf Y u +f 2 u = 



are soluble when u and v are correlated functions of x and y } and 

 w is v ! -r-u f . 



It will be apparent from the nature of the process, that it 

 cannot be made very useful as a means of integrating any par- 

 ticular equation proposed for solution, as we have no means of 



