G8 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF 



so that, as 6 is a function of u only, we have 



where A and B are independent of 17, that is, are arbitrary functions of u. We thus 

 have the theorem*- 



Jfp (?{) awJ (j (u) denote any functions of u satisfying the equation 



^ + g a + 1 = 0, 



/tJ //' be defined us a function of x, y, :., ly the equation 



z = u + xj>(u) + yq(u], 

 also if v denote 



/; 6c(V/ arbitrary functions of u, then v satisfies the equation 



V : v + K--V = 0. 



Tlie connection between this solution and the general solution indicated in 39 

 can be established as in the corresponding case of the potential equation ( 34, 36). 



,. . f)-6-' , 3-y 3f 



Application to ^~ \ 4- = u, ^ . 

 * * ox 3 c>f- ot 



-11. The j)recediug examples have led, in each case, to a solution which (though 

 not the most general) was expressible in a finite form. We now take one other 



example, viz., 



y l + I + c = 0, 



which can be regarded as the equation for the variable conduction of heat in two 



* This result can also be expressed (see the paper already quoted, 33, note) in a form symmetrical 

 as regards the three variables, as follows : 



If 2>, q, r, le tliree arbitrary functions of u such that 



and if u la determined as a function of x, y, z, by the equation 



au = xp () + yq () + zr (u), 

 a being any constant, also if v denote 



(ii) e> to' + 111' + s'-') ( i>'- + 'i' 1 + i-'*f "" + \i, ( u ) e ~ to' + 



xp' yq zr 



where and ty denote arbitrary functions of u, then v satisfies the equation 



V 2 t> + A = 0. 



