with a Determination of the Limit beyond which it fails. 321 
When once we are satisfied as to the existence of the eight-square form, we 
need not keep the above considerations constantly before us in the actual con- 
struction of it. After the first four rows are determined, the four that are to 
follow may be readily derived from them, by aid of the property that the like 
products are to cancel. ‘Thus: having resolved to keep the leading signs plus, 
we commence the four rows, to be deduced from those above, with sw’, sx’, sy’, 
sz’, respectively; and, guided by the before-mentioned property, we annex to 
these, with the minus sign, the combinations ws’, xs’, ys’, zs’. To determine 
the sign of xt’, we refer to the combinations s¢’, zw’, in the second row; and as 
these give a minus product, we write down .t’ plus ; so that the product of sw’, 
at’, in the row now in course of formation, may cancel the like product above : 
and by proceeding in this manner, all the wanting rows may be easily completed, 
each term that we insert suggesting the adjacent term. We thus see, when any 
formula of this kind actually exists, how the lower half of it may be deduced from 
the upper, without any further recurrence to a pre-established model. This is a 
circumstance worthy of note, as well as the following particulars respecting the 
several rows : 
1. The same letter is never repeated in the same row. 
2. The same combination of two letters is never repeated throughout the 
group of rows; and, consequently, the same product of two combinations can 
occur only twice. 
3. In any row, the signs of any consecutive combinations, separated by the 
sign minus, may be interchanged without causing any departure from the pre- 
ceding model : for such interchange of signs is merely equivalent to an inter- 
change of places between the two dashed quantities entering the pair, and 
between the two undashed quantities; or it may be regarded as arising from 
making a letter in each of the two combinations negative, as we are, of course, 
at liberty to do, since the signs of the roots of the proposed squares,—and it is 
with these roots we are dealing,—are arbitrary. It must be observed, however, 
that these changes or mterchanges, when made at all, must be made in every 
place where the letters concerned occur, otherwise conformity to the model will 
be destroyed. 
4. It is thus obvious that the four-square and eight-square models admit of a 
great variety of apparently different forms. In each case, however, it is plain 
2u2 
