202 
Mr. Babbage’s essay towards 
the problem itself would scarcely have been worth noticing 
had it not been for the very curious results to which it led me. 
The relation between ^ (a?, y) = px qy and Y' ,n (#> y ) 
+ c l) n ~ l + qy) is ve ry remarkable, it appears from 
this, that in the present case in going from the n th to the 
n -I- r L simultaneous function, we have only to multiply by the 
sum of the co-efficients of the original function. On enquir- 
ing a little more minutely into the cause of this circumstance, 
it will be found that it depends on the original function con- 
taining x and y of the same dimensions in all its terms, or 
more generally that the expression of ^{x,y) is homogeneous. 
Let us now ? assume some homogeneous function, and examine 
its second and higher simultaneous functions, let 
t]/ (x, y) = ax n + by^ + c y q x n ~~ q -f- &c. 
the second simultaneous function is 
ty 2,2 (x, y) — a { $ {x,y) } w + b | $ (x, y) ] ” + c { y) j M + &c. 
or i \, 2 > 2 (x,y)= [ (x,y) } ” { a + b^-c+^c. } — iK 1 , 1 ) { ${oc,y) } \ a ) 
If we now take the simultaneous third functions we have 
¥'K x >y) = 4'(i» 0 CV’ 2 ( x >y) n — + (l *) DK 1 * 1 ) W (tjO)*] 71 
henca {x,y)= 
Repeating the same operation we should have 
V’ 4 (p,y) = {'Ki, 0} * 
and generally 
k — I i +. it + &c. 
$ k,k {sc. y) = j ^ C^y) } n x { ^ .(* > 1 ) } == 
^ j k—i 
