200 
Mr. Babbage on the analogy between the 
then fax will b e fax + vepax 
and fax will be fax -j- vcpax 
and these being substituted we have 
(I — fx . fax ) — (fx . QaX+fax . <px)v — (px . (pax . V 2 
now in consequence of the two equations given above, the first 
term in both numerator and denominator vanishes, and divid- 
ing the remainder by v and then making v — o, we have for 
the value of the expression when fx — fx .fa.x = o and also 
b s 
fx .fax — 1 
fx—fx .fax <px—fx . faX—faX . <px 
i 1 * * s 
I —fx .fax ’ fx . (pax -j- fax . <px 
where <p and q> are arbitrary. 
If we take the expression 
fx—fx .fax+fa .fax . fa 1 X 
8 * X 
1 +fa fa x •/*** 
and if fx and fx are so assumed that the numerator and deno- 
minator both vanish, by a similar mode of treatment to that 
already pointed out, we shall find its real value to be 
(px—fx . (pax— fax . <bX — fx .fax . <pax — fax .fax . <px—fc .fax . (poPx 
i i i i i * 
fax .fa z x . I px + fx . fa 2 X . pax -f- fx .fax .fa 2 x 
And similarly if we supposed both numerator and denomi- 
nator of the fraction 
fa— fa -fa x + &c - ± fa • fax . . . fa n ~~ z x .fa n 1 x 
IX X 
i dzfa . fax . . .fa n l x 
to vanish, it would yet retain a value contairiing arbitrary 
functions. 
It. will be needless to multiply examples, as the mode of 
