210 
Mr. Babbage on the analogy between the 
The analogy between the various orders of functions which 
contain many variables, and that branch of the integral cal- 
culus which relates to partial differentials , is too apparent to 
require illustration. I shall therefore proceed to show that a 
functional equation admits of three species of solutions, ist. 
The complete solution , this contains as many arbitrary junctions as 
the nature of the given equation will admit. 2d. The particular 
case , this contains all solutions which are less general than the 
complete solution , and which are only particular cases of it. If 
they contain arbitrary functions, I have called them, for the 
sake of convenience, general solutions. 3d. The particular solu- 
tion, this is a solution which satisfies the equation, and may or may 
not contain arbitrary functions ; its peculiar property is, that it is 
found from one part only of the equation and independent of the 
rest; thus if we have the equation 
— y} • F{x,y,4/(x,y), ^ 2 , \x,y) &c.| 
= (0)} . F {x,y,v}/(.z,y),&c.)} 
—1 
\|/(x, y ) = <p [<px — <py) is a particular solution, for it satisfies 
the equation by making % ty 2 ,t (x,y) — y—o and -ty 2 > 2 (x,y) — 
(p\o) = 0, and is totally independent of the rest of the equa- 
tion contained in the given function F and F, provided only 
1 
that it does not make either of them infinite, whether it is 
contained in the complete solution, I have not yet ascertained ; 
but I am rather of opinion, that it will be found not to be in- 
cluded in it. In both parts of my Essay towards the Calcu- 
lus of Functions, I have used the expression, particular solu- 
tion, instead of particular case, this arose from not having 
taken a sufficiently extensive view of the calculus ; it would 
