with a Determination of the Limit beyond which it fails. 325 
be before the products furnished by the particular set of eight now considered 
can cancel,—without introducing discord among other sets of eight, at present 
in harmony with that model, and thus causing products to re-appear which now 
cancel. 
In order to this, we may observe that there are only two different ways of 
proceeding, by which the required modification of the signs of 
wy’ — Ye, an 2m, Pa xZ, 
can be brought about ; first, by supposing two pairs of letters in the expression 
to be minus, as w, w/, and x, x’, or z, 2,5 or, instead of w, w’, taking y’, y', 
with either of the latter pairs; and, secondly, by changing all the signs in the 
semi-row, 
su — us, +oft —tr + wy) -— yw +20 — «7'. 
Suppose we attempt to remove the discrepancy in the former of these ways, 
that is, by simply changing the signs of w,, w’, and x, x’, or Z, 2’. 
If we make no further changes, conformity to the model, for the before-men- 
tioned set of eight, will unquestionably be brought about ; but in order to pre- 
serve the conformity which already exists for the set, 
stt pu tutu tae ty +2, 
§ +t,+u,+v,+w,+4,+Y, + Zp, 
these same changes of signs must also be made in the rows which precede and 
follow the third in the above scheme : but these changes in the other rows intro- 
duce the discrepancy into the first-mentioned set, or rather merely transfer it to 
the neighbouring row in that set. Of course, changing the signs of any quanti- 
ties, throughout any of these formule, can never interfere with any conformity 
or want of conformity to a model that might exist before these changes were 
made. Again, if we attempt the correction by changing all the signs in the 
semi-row, 
su, —us, tut —tv + wy, — yw, +24 — £2Z', 
leaving the other signs unaltered, the desired conformity will also in this way be 
brought about ; and we shall, at the same time, escape the dilemma in which the 
former mode of proceeding placed us; for the second-mentioned set of eight will 
still preserve its conformity to the model, inasmuch as the changing of the signs 
of an entire row can never disturb such conformity. But here again disagree- 
