26 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF 



and so for the others. Now the integral equation is, by hypothesis, equivalent to a 

 partial differential equation of the second order, say to 



F (a, 1), c, f, g, h, I, m, n, v, x, y, z) = ; 



hence when these values are substituted the equation is to be satisfied, and accord- 

 ingly the terms involving the various combinations of the arbitrary functions must 

 disappear. Thus the highest power of </> n or what in effect is the same thing, the 



highest power of < u must disappear of itself, and therefore 

 8 3F 3F , , 8 3F 3F , 3F 3F 



or, with the former notation, 



A. 3 + H&& 4- 



38 

 But the term involving the highest power of < ]2 >.- , which will be of the same 



3 



degree as the highest power of < u , , must also disappear of itself; and this gives 



rise to the equation 



2Af^. + H (i, % + ^.,.) + G (^ + &,,) + 2B^ y + F (^.. -f ^,) + 2C^ S - ; 



and likewise tlie term involving tho highest power of <f>.,a ^ must disappear, 

 leading to the equation 



AT,. 5 + HT,^ + G^T,, + Br,/ + F^, + Cr,* = 0. 



From these we have 



, + 1 Hr, y + J Gij.-) 2 = (i H 2 - AB) T,/ + 2 (i GH - AF) 77^ -f ;(i G 3 - AC) ^ 



-f 1 H^ + 1 Gf r ) (AT;, + 1 Ht;, + 1 Gr?,) 



= (i H - AB) ^ + (i GH - i AF) (^ + f ,) + (i G 2 - AC) fa s ; 



so that, squaring the last, subtracting the product of the first two, reducing, and 

 removing the factor A, we have 



(Discrt. of A) (#.. - ^'f = 0. 

 Similarly, by taking modifications of the first equation in the form 



or in the umn 



