324 Mr. Youne on an Extension of a Theorem of Euler, 
the object in view is, to a certain extent, accomplished, for all the like products 
cancel one another. If to this we could add that, as far as they go, these same 
expressions agree with the eight-square model, then we might at once conclude 
the existence of the sixteen-square formula, and might actually complete it by 
deducing the wanting rows from those above, without disturbing any of the signs 
of these latter, as already explained at page 321. 
But the semi-row next in order to the last of the above group, as deduced in 
this way from the preceding expressions, is 
su — ws +ut — te + yu — uy, +2 — v2’, 
which implies a discrepancy ; for the product of the combinations tz’, zv/, does 
not cancel the like product furnished by ¢v’, z,2’, in the third row, the signs of 
both products beg minus. We are forced to conclude, therefore, that, not- 
withstanding the cancelling of all the like products in the partial construction 
above, there is, at least, one set of eight which, in that construction, is out of 
keeping with the model ; and it now remains for us to discover this set, and to 
inquire whether it can possibly be brought into conformity with the model, with- 
out such an interference with the existing signs as would cause products which 
already vanish to re-appear. 
If we refer to our eight-square model, and conceive the semi-rows, 
wl +a + y! +2! 
w+rty +2, 
there employed, to be changed into 
wr, ty, + 2, 
wW,+2,+Y, + Zp 
the modified formula will then agree,—as far as the first four rows, which are all 
that are here exhibited,—with the expressions which, in the above scheme, make 
up the commencing and terminating portions of the first four rows, with this im- 
portant difference, namely, that in the terminating portion of the third row the 
signs are the opposites of those required by the model. This, as we shall pre- 
sently see, is the discrepancy already indicated ; and we proceed to show that its 
character is such that it cannot possibly be removed without the mtroduction of 
a similar discrepancy elsewhere ; that is to say, the refractory signs here alluded 
to cannot be brought into conformity with the law of the model,—as they must 
