i go Mr. Babbage’s essay towards 
and we should find a precisely similar equation for determin- 
ing If we are acquainted with two particular solutions of 
this equation, we may from them derive the general solution 
of the given equation. If, however, the functions a and (3 are 
of such a nature that the two following equations are fulfilled 
eq. (i) becomes identical without assigning any particular 
value to/ or f. (The two conditions are a (x (x, y), fi (x,/)) = x 
i 
and &(*(x,y),G(x,y))=ty). 
It may be curious to enquire whether we can discover any 
forms which will satisfy these equations, for this purpose let us 
assume x (x, y)~a-\-bx~\- cy, and also (3 (x, y ) = a -j- bx -{- cy, 
x x 
this will only lead us to a particular solution, but I shall pre- 
sently show that it may be rendered general. If the two 
conditions already specified are fulfilled, the arbitrary con- 
stants will be determined, and we shall have the following 
equations 
« {x,y) = a-J r bxJ r b ~y 
^{ x >y)=Tri~ bx ~ h y 
which may be thus generalised. Let <p be any function, and 
let <p be the inverse of that function, so that (p<p‘x=x then the 
conditions will be fulfilled, if 
* (*>y) = ( p 1 {a-\-b(px-\--p pyj 
and /3 ( x >y) = ^ — b<px — bpy^ 
Some remarks, however, are necessary on the inverse func- 
tion c p\ If we combine x and constant quantities by any of 
the direct operations, addition, multiplication, elevation of 
powers, &c. the result which is called a function of x admits 
