326 Mr. Youne on an Extension of a Theorem of Euler, 
ment is introduced in another place, for the semi-row commencing with ss’ 
would then be 
ss) — ss + tf’ — tt + uw, — uw + vv’, — 07, 
so that the product of the combinations ¢¢’, w,w’, would no longer cancel the pro- 
duct of tw’, wt, furnished by the fourth row; and thus, as before, the removal 
of one discrepancy necessitates the introduction of another. 
It is this discrepancy, as respects the last four signs in the third row above, 
that causes the signs in the semi-row mentioned at page 324, viz. : 
sw, — ws + af — ta + yw — uy +z — v2, 
and which were in part deduced from them, to involve error; an error which, as 
we have seen, cannot possibly be removed without introducing a like error else- 
where. 
Hence, the sixteen-square formula is, in general, impossible ; and from this 
it follows that the thirty-two square form is also impossible, and so on. For if 
the form for thirty-two had place, then, by reducing these thirty-two to sixteen, 
by employing zeros instead of the remaining quantities, we should be conducted 
to a correct sixteen-square formula, which has been shown, however, to have no 
existence. 
If it be imagined for a moment that an advanced form might have place, and 
yet the next inferior form not be necessarily furnished by it when the requisite 
squares are assumed to be zero, the impression will be removed by observing 
that when, in the form for four, the two dashed and the two undashed quan- 
tities entering into any of the binomials are throughout made zero, two entire 
rows of that form disappear ; that when, in the form for eight, the four dashed 
and the four undashed quantities entering into a pair of binomials in any row, 
are throughout made zero, four entire rows vanish; and, likewise, in the failing 
form for sixteen, and generally, the necessary constitution of the rows renders 
these evanescences unavoidable. 
It is proper that we make the preceding stipulation as to the zero-quantities 
forming bimomials in the same row; for if they be chosen at random, the above 
conclusion will not necessarily follow. In the eight-square formula, for in- 
stance, if our zero-quantities are not selected in reference to this condition, 
only one row will disappear: thus, if the quantities made zero be s, t, 2, 9, 
