204 
Mr . Babbage's essay towards 
as it may be convenient also to denote the sum of the two 
indices, I shall place it underneath on the outside of the 
parenthesis; thus then the expression \p (#,y) denotes any 
q 
homogeneous function of oc and y of the q th degree. A func- 
tion of three variables x, y, and z, may be homogeneous, with 
respect to two of them (a? and y) in one sense, and also rela- 
tive to y and z in another ; but it does not from thence follow 
that it will be homogeneous relative to all three, such a func- 
tion would be denoted thus 
^(x,y,z) 
~ - Jp, q 
a particular case of this expression is x? -f- xyz = (x,y , z) 
_ — 2 , 2 
This notation being premised, we have the following theorems 
relative to homogeneous functions. 
>y)y 0) 
l=y - 1 
= X {+(1,1)' * ( 2 ) 
And generally if we have any homogeneous function of the 
n th degree, and instead of x and y we substitute any other func- 
tion whatever as \p (<r, y), then we shall have the following 
equation 1 
+ (x,y), = i) (s) 
1 i i n 
N f a (£* y) n 'j 
Assume *(x,y) = 9 ) 
call the latter member, for the sake of brevity, K, and take the 
second simultaneous function on both sides ; in this case a(x,y) 
" — 71 
will become a ( l , 1 ) K» by eq. (3), and for the same reason /3 (x,y ) m 
will become 0 ( l, 1 ) K", and consequently we shall have 
