204 Mr. Babbage on the analogy between the 
or from F | if/a?, -tyx , . . ^x, x, ax, ax , . . a n — 1 x j = o 
If there should exist any doubt respecting the accuracy of 
this reasoning, it may be confirmed by arriving at the same 
conclusion in rather a different manner. If in the given 
equation we substitute successively ax, ax, . . a n ~ l x for x, 
we shall have the following equations. 
F^-tyx, ipux, . . \pa n — l X, x, ax, . . . a n ~ l x j —o 
F{^ax, $a*X, ... -tyx, x,ax, . . a n ~ 'x^>=0 
See. See. 
F{ -<pa n — J X, $X, . . ■tya. n — z X, x, aX, . . a n — l xj ^=0 
To eliminate ^/ax t -tyalx dec, from these n equations would 
in most cases be excessively troublesome. It may however 
be observed, that ->pax occurs in the second exactly in the 
same manner that if/a? occurs in the first; also ■fyofx is con- 
tained in the third in the same manner as ^a? is contained in 
the first, and similarly with the rest. From this it follows, 
that though no one of the equations is symmetrical relative 
to -\x, -tyax . . . %\/a n — 1 x, yet when all the equations are consi- 
dered together, they are symmetrical relative to if/a?, if tax, 
•^a n — l x ; from this it follows that whether we find by elimi- 
nation if/a?, if tax, . . or if ia. n —'x the result will be the same, 
therefore ;J/a? == if/«a? = &c. — -tya n — J x } and we may determine 
4 /x from the equation 
F | -tyx, if/a?, . . . if ix, x, ax, . . . a n — 1 x } = o (<2) 
It does not follow from hence, that this equation contains 
all the values of if/a?: on the contrary, if the elimination be- 
tween the n equations above written were actually performed, 
it would be found that the equation ( a ) would enter into the 
final result as one of its factors. 
