202 
Mr. Babbage on the analogy between the 
fo—fx 
•tycix =fx whose solution is fa = ‘ .... y ~ ^ if fx .fax — 1 it 
apparently becomes infinite, but by subtracting the equation 
fx . ypocx -\~fx . fax . tyocx == fx .fax (which is deduced from 
the former by putting ocx for x and multiplying by fx) we 
have since cdx = x and also/x .fax= 1, o=fx—fx .fax, and 
s i 
the solution becomes a vanishing fraction whose value is 
<pX—fx . tpaX—faX . <px 
= ‘ /* . fa/ far let ca = - X and fx=fx = t then the 
general solution of the equation ^ + 'I' ( — x) =s 1 
9(— + 
is +*= L ?F3+?r • 
In a similar manner the solutions of the equations 
^ x — \}/ (— x) — x, and fa —fx . fax \{fx .focx^=. i when 
ocx = x are 
x<px — <px + <p( — x) <px -f -jx . (pxx 
“ <px + <K—x) and ^ X —fx.<p*x+f*J7fx 
Let fa + fax=fx if uxzxzx this is on ly possible whenfx=f (x, ux) 
* i a 
then ?»■*• -fix , «) 
fa = — 
f>X + f)xX 
Let fa — ^ocxx=fx and xx—x this is only possible when fx= 
{x-—ux) xf(x, ocx ), then 
[x— ax) f(x, ax) . (px+tpx + Qux 
fa = I - L- 
<px + tyax 
The above are sufficient as examples, but the same reason- 
ing has led me to the following curious proposition. When- 
ever the method of elimination apparently fails, the real value of 
the vanishing fraction will give the general solution of the equa- 
tion. This principle puts us in possession of the general 
solutions of several classes of equations, for besides the cases 
