the calculus of functions. 203 
This elegant property of homogeneous functions will assist 
us in solving a variety of equations. 
Problem XII. 
Given the equation 
^ n ’ n (x,y) = a{^ (a?,y )} 6 
Determine n from the equation b=.n 1 and also determine 
k — 1 
1 — n 
4 ( 1, 1) from the equation j 4 ( 1 > 1 ) } = a 
Or the given equation will be satisfied by any homogeneous 
1 
function of the degree indicated by b provided the sum of 
all its coefficients is equal to the quantity a. l ~ b 
Ex. Let ^ 3,3 (a?,y) = 8 (4 (a?,y)) 4 
here b = 4, k = 3, therefore n k ~~ 1 = n* = b = 4 and n = + 2 
I 
also a=^-& and ^ ( i, 1 ) = 8 3 = 2 
therefore any of the following quantities will satisfy the equa- 
tion 2 %y, of + y\ wy + y 4 < 27 * — ocy + 
The properties of homogeneous functions are so nearly 
connected with the solution of equations containing simulta- 
neous functions, that it will be convenient to examine into 
them a little farther, and to adopt some means of denoting 
them with brevity. In order to signify that a function of 
several variables has in each of its terms the sum of the 
indices of any two of them always the same, I shall make 
use of a line placed beneath those two variables : thus 
*f/(a?,y) signifies an homogeneous function of x and y ; and 
mdcccxvi. E e 
