22 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



arbitrary function, which would arise through the integration, is definite : we have, 



in fact, 



21 - m-n 

 a -h-r/ + f + - - =0. 



Using this integral, the second equation becomes (on the elimination of 21 m n) 



d .. , . a + //& a 

 - (h 0)4-- ~ = 0. 



dy v y + z 



Combining this with the first equation, we have 



d . , a 111 + I 



(a - 2/i + 1) + - = 0, 



dx v y + z 



so that, as p -f- 1 = 0, we have 



dx \ y + .-: 



Since p -{-1 = implies that z -\- x is SL function of y, we infer that 



n 2h + 1) , 



- = arb. in. ol 11, 



y + - 



when 



-/ -{- x = arb. fn. of y ; 



and consequently an integral that can be associated with the original equation is 

 given by 



a - Ik + 1, 



y + " 



, y), 



where 6 is an arbitrary function. 



Taking next the case p = q, the alternative that arises out of the characteristic 

 equation, we have the three other equations the same as before. It is now necessary 

 to take account of the three equations of identity, which, when the relation p = q is 

 used, are of the form 



dh da /dg dy \ 



dx dy ' ~ P \dy dx J ' 



dx dy ~ " \dy dx I ' 



df da Idc dc 



- = n I 



da; dy f \dy dx 



so that, eliminating the terms in p from the three equations, we have 



