i§4 Mr. Babbage’s essay towards 
I shall give one example which will illustrate these various 
modifications of the original functions, 
3 > 3> 2 > 2 \ 3 > 2 \ 3 y a,X “1 
, 3, i, 2 , 4.5 3, i f y ~ ___ . /T . 
V{x,y,z,v) =/(«•) L» = /3z~ 7 ' J 
This equation contains the analytical enunciation of the follow- 
ing Problem. 
What must be the form of a function of four quantities 
t]/ (x,y,z,v) such that taking the second function relative to 2 , 
the third relative to x, and the second simultaneous one relative 
toy and v: if in the result ccx be put for y and flz for v, and 
the whole be then considered as a function of x and z, and if 
on this hypothesis the third function be taken relative to z, and 
the second relative to x ; and if yx be now put for % and the 
third function of the expression considered merely as a func- 
tion of x be taken, then it is required that the final result 
shall be equal to fx a given function of x ? 
Symmetrical functions I shall denote as in my former Paper, 
by putting a line over the quantities relative to which they 
1 1 
are symmetrical, thus (<z, y , z, v) is symmetrical relative to 
x and y in one sense, and relative to % and v in another. 
Problem I. 
Required the solution of the functional equation 
= H*x,{3y) 
To avoid repetition a, /3, 7 , &c. unless otherwise mentioned, 
always express known functions, and <p, t[/, % are unknown or 
arbitrary ones. 
Put 4/ (x,y,) = <p ( fa,fy ,) 
