si 6 Mr. Babbage's essay towards 
making = a our equation becomes 
i i 
w— »m w— m 
F | v, ( a<pv), (a <p a v), . . . 
which may be solved by Prob. XIX. Part I. 
Problem XX. 
Given the equation 
$ 2 > i (*,y) ~^ 1 ’ 2 i x >y) 
This equation, containing no simultaneous function, is diffe- 
rent from any we have yet solved, and requires the application 
of a peculiar artifice. 
In my former Paper, in order to reduce the equation -Afx—ux 
to one of the first order, I made use of the substitution q> f q>x 
for \|/.r : an analogous one must be employed on the present 
occasion ; let us suppose 
$(.x,y) = <p'f{<px,<py) 
the effect of this will be very similar to that of the one just 
alluded to, and its great utility will be evident by considering 
its result in the various orders of the same function, thus 
+*’ ‘ ( x >y)=$'f (QQf@ x ,ty),M= < py(f(<P x ,<Py),<Py) 
= f' /*■ 1 (<P x, <py) 
\ x >y)= -<P/(<P*> <p<p'f(<px, <?y))=<p'f(<px,f(<px, <py)) 
= <Py) 
^-*(x,y)=<p'f(ip<pf(<px, %), <P<Pf(<px,<$y))=<pf(J(tpx, <py ), 
/(<&><&))=$ f~* (<p*, <&) 
and continuing the same substitutions we shall find 
p>‘(x,y) = <p'/3> >((px, <py) and ^>3(x,y)=ip’f>, }(<px, <py ) 
