the calculus of functions. 205 
♦ ifcr’} « 
let oc, (1, 1 ) = /3 ( 1, 1 ) and also let m = n s then this equation 
becomes -^ Z} 2 (oV,y) = <p (1 ) 
this affords another and a more direct solution of Prob. 10. 
for if \J/ 2 ’ 2 (x>y) = a } a general solution is 
<2(1,1) being equal to / 3 (i, 1), this latter condition, however, 
is not absolutely necessary, and if we wish to avoid it, the 
a (x,y) 
- — n 
13 {x ,y) n 
general solution will be 
a 
<P 
» ( 1 , 1 ) 
13 (I, I )/ 
■<P 
i * *WF n 
the following is another solution of the same question 
Assume 
a + bx + cx % + &c. 
by + cxy 
^ fay) = <P< - 
cy 
2, 
a -| -'bx +*^ 21 + &c. 
*by -f* *cxy 
> = K 
'ey* 
2. 
taking the second function on both sides we have 
C 1 
a + b I 
cl 
C 1 
b >K + 1 >K a + &c. 
s i c\ 
J Z J 
n. 
51 
■K + 1 >K* + &c. 
S I 
J 2 J J 
let a = 'a b + b = l b + c + r + r == V + V + &c. 
E e 2 
