224 Mr. Babbage’s essay towards 
or putting A x for x 3 and A 'y for y we have 
F{ A x } A*y, A'/ (x,y), AJA »(. v,y), A 'fi> 2 (x,y ), &c. | = o 
an equation of the required form. 
One important use of this transformation is the solution of 
equations of the form 
F [x,y, 4/ (x,y), vj/n 2 (oiX,y) 3 ^ l ’ z &x,y), &c.| = o 
in which the functions are taken only relative toy, and yet x 
is not altogether constant. It may be transformed into 
F [x,y,^ (x,y),^'> 2 (x,y),^>3(x,y), &c.| =o 
which may be treated as an equation of one variable, oc being 
constant. This is the species of equation alluded to at page 
(198) 
After considering the various equations amongst the higher 
orders of functions, another question presents itself, which 
maybe thus stated. What must be the form of a function 
of ( n ) variables, such that taking the functions relative to 
any or to all of them any number of times, and combining 
these quantities in any manner, the result shall (when all 
these variables are made equal to x) be equal to a given func- 
tion of x ? This question might thus be expressed when there 
are only two variables 
^(x,y)A l3Z [#>y)A 2 ’ 1 (x,y)> &c. ] =f(a) [y—x^ 
this condition obviously enlarges the signification of the func- 
tion \f/, and the solutions ought to be more general. We shall 
accordingly find that some equations, of which without this 
condition we cannot find even a particular solution, are capa- 
ble, when it is added, of very extensive ones. When there are 
more than two variables, the condition may be, that making 
