488 TRANSACTIONS OF SECTION A. 



Multiplying these by P and by Q respectively, adding, and noting that the 

 equation P/i + Qy =;R is satisfied by the integral in question, we have 



R^* + ^^-(p^ + y|-^R|'V?>'^fp|-'+Q?^" + i^?")=o. 



oz ou\ ox oy ozJ dv\ ox oy oz' 



The coefficients of ^? and J^ vanish identicallv ; and thus wo have 



GU 00 "^ 



OS 



as a relation which must be satisfied in order that the given partial equation may 

 have an integral given by 



There are three distinct ways in which the relation can be satisfied. 

 Firstly, the quantity „^ may vanish identically. We then know that 

 <f) (z, ti, V) does not contain z explicitly. Thus 



^ (.r, y, s) = a function of u and « only. 



The proposition, that the form of/ in the general integral f{u, v) = can be 

 chosen so as to give -^ =/(«, v), is true in this case ; and a method of determining 

 the form of / is indicated by the course of the analysis. This is the customarily 

 accepted result. Examples are so frequent that none need be adduced here. 



Socondly, the quantity ^^ may vanish, not identically, but only concur- 

 rently with ^=0. In that case <f> (z, u, v) does contain z explicitly; and the 

 proposition, that the form of/ in f(ie, v)-0 can be chosen so as to make 

 ^ = .f{'<) ^)> 19 not valid, As an example, given by Chrystal, consider the 

 equation 



[l + {z-x-y)i]p + q^.2, 

 obviously satisfied by 



we can take 



yl/ = z~x-y-zO; 



u = 2y-z, v-y + 2{z-x-y)^, 



80 



The quantity .Jr vanishes only concurrently with 0=0; the general 



integral cannot be specialised so that/ (?«, v) = \|r. In this example, one hypothesis 

 made in the course of the general proof is not justified. All initial values that 

 satisfy >//■ = constitute a branch-place for i', so that v is not a regular function in 

 the vicinity of those values. 



Thirdly, the quantity .^? may not vanish at all ; in that case the relation 



can only be satisfied if R = 0. Again, the general integral cannot be specialised 

 so that/(i<, v) = \|/-. As an example, consider the equation 



xp-\-2yq = 2(z - -) , 



^ y 



obviously satisfied by 



f = = -'^%0; 



y 



