the calculus of functions. 247 
From the first of these we may eliminate d ~j~- by means of 
the differential of the second, and from the result 
may be eliminated by means of the differential of the third. 
And by continuing this process, observing that $ (<2?, a p y) = 
^ ( x »y) we shall find 
'+(*.*) = 
this partial differential equation must be solved, and the arbi- 
trary functions which enter into its integral, must be made to 
satisfy the conditions of the Problem. 
Ex. Let p=4}3 then {x,y) = and the solution of 
the resulting partial differential equation will be 
^ (x, y) = f py — 7 <py + sin x. py + cos x. py 
II IS S3 I* 
hence 
d^{s, uy) _____ ^ fyay -f cos x . pay — sin x . (pay 
dx 11 is 11 14 
the first condition to be satisfied is 
(py = pay 
II SI 
which is readily fulfilled by making py = (p {y, ay , ay, a 6 y), 
zs z 
the next condition is 
py = » — pay 
12, 12, 
This must be solved by Prob. VIII. Part I., and we shall have 
py = {—y + «y — »'y + °?y) <p (7> «y> “50 
12 a 
the third and fourth conditions are 
py — — pay and py = pay 
S3 14 14 13 
In the second of these put ay fory, and it becomes pay=.pa 2 y, 
14 13 
