the calculus of functions. 201 
\ 
may obtain another value of p which will also satisfy the given 
equation ; it will be 
% S °> f 
This on trial will be found to agree with the condition, and 
%, <p and (p are arbitrary functions 
1 
The equation we are considering will also be satisfied by 
making $ (x,y) = a ~~ or more generally by the constant 
quantity a multiplied by any fraction whose numerator and 
denominator become equal when x is put lory : such are the 
following. 
+ * y + y* „ * (y* + * z ) V- 2 + — 5y 3 
a ■ r,“T‘T^ > a - rrr— 5 a ~ j 7 > cxu 
•Problem XI. 
Given the equation 
ax + by 
Assume (x, y ) = px + qy then we have 
4 **(iv,y) —p {px + qy)+q (px -f qy) = (/> + ?) (px +qy) 
and + 3,3 (>, y ) = ( p + qf (px + qy ) 
and generally 
"fay) = (p + qT~\l> ® + qy) 
hence p. (p + qf~ l ~ a and q (p -f q) n ~~ l = b, which gives for 
the values of p and q 
a 
“n— 1 
Inl- 
and q = — L_ 
(„ + *)— 
This is a very limited solution not containing even an arbi- 
trary constant, it might easily be rendered more general, but 
