the calculus of Junctions. 
2 09 
with which it coincides when « (1, 1) = /3 (1, 1) and 
n = m + 1 by their assistance we may solve a variety of pro- 
blems relating to simultaneous functions. 
From ( b ) we have 
''*(£■ A+,' 
.(3 (*»,}') 
- n 
putting in this for <p 
(3 
its value ^ (a?, y ) we have 
from this we may deduce the solution of the following Pro- 
blem. 
Problem XIII. 
Given the equation 
■fy 2 ’ 2 O,j0 = Fx^(x,y) 
make <p = F and take a (a?,y) B+I any homogeneous function 
of the n ith degree, and /3 (a?, y) n a similar function of the 
n th , also let a (1, 1) == /3 (1, 1) then the equation is satisfied by 
making 
Ex. Let 4 / 2 ’ 2 (a?, y) = 
Suppose a (a?, y) H+ ( = af-f- / and /3 (a?, y) n = 2a?, then one solu- 
tion is 4 / (a?, y) = L 
or let « (f, A+ , = (* s + /)?>( 1 ) and Z 3 (£» £>» = 3 * ?{f ) 
2^9 
[HO 
