226 
Mr. Babbage’s essay towards 
It is evident that particular values of <p are 
oc — y -f- z. and z 
many others might be mentioned, but it is desirable to deter- 
mine <p more generally 
Since <p (x,y, z) = 2 £y = xT\ 
it is evident that <p ( x, x, z)=z% 
and since this is independent on any particular value of x we 
have 
<P (y, v, z) = z 
that is to say, that whatever quantity is substituted for x, if the 
same quantity is also substituted for y, the result will be equal 
to z. Now let v = (p (x,y } z), it becomes 
<p { <p (#>y, *0, <p (y> y, z),z] z=z 
but this expression is nothing more than the second simulta- 
neous function relative to x and y, and may be therefore 
more concisely expressed thus 
<p 2 > 2 > I (x,y, z) =z 
in which equation, since it does not vary relative to z, that 
quantity may be considered as a constant ; and the equation 
(p 2 > 2 (x i y) = z = constant 
being solved, we have only to substitute instead of the various 
constant quantities arbitrary functions of z: thus then the solu- 
tion of the equation 
0 v >y > *D = ^ Cy — ^ 
is reduced to that of 
$ 2 > 2 (x,y) = constant 
and we have only to refer to Problem ( 10) for its general 
solution. 
Let us apply this to the solution of the equation of this 
Problem 1 ( x , y) = fn (x, y) Q = x'} 
