396 Mr. Babbage’s essay towards 
As I shall have frequent occasion to make use of symme- 
trical functions of two or more quantities, I shall for the 
sake of brevity denote this by putting a line over the func- 
tional characteristic; thus q> (xy) represents a symmetrical 
function of x and jy, which it is well known possesses the fol- 
lowing property, 
q [ x >y] = { y> * } 
As we are only considering functional equations of one vari- 
able, this will be sufficient for the present purpose ; it might 
perhaps otherwise be more advisable to put the line over the 
quantities relative to which the function is symmetrical ; thus 
<p | x,y, z,vj is symmetrical relative to z and v, but it is not 
so in respect to the other variables. This would possess 
the advantage of readily designating a function symmetrical 
relative to two quantities in one way, and likewise symmetrical 
with respect to two others, but in a different manner,* thus 
1 1 
<p y, v, zj 
a particular case of this is 
v -f g 4- a x y 
v 3 z 3 — ax z y z 
which is symmetrical in one sense relative to x andy, and in 
a different sense with respect to v and z ; but these belong to 
other enquiries. 
Problem II. 
Required a general solution of the equation i] j x =|«i, 
having given a particular solution of f x =fct x 
• This is not a mere imaginary refinement; I have constantly had occasion to 
make use of fuoctions of many variables which were symmetrical by pairs, when in- 
vestigating the nature of functional equations of more than one variable. 
