322 Mr. Youne on an Extension of a Theorem of Euler, 
that any one of these varieties, by suitable changes, such as those we have now 
mentioned, will supply any other; so that a single variety may be considered as 
virtually involving all the varieties: when the signs of the combinations are once 
so arranged as to cause the double products in the developments above to cancel, 
we may clearly change these signs in any way we please that does not interfere 
with this essential condition. Even after the partial construction of a formula, 
by aid of a previous model, as noticed at page 315, we may obviously intro- 
duce any changes of this kind that we please, provided we take care that 
all the like products, supplied by the partial form, shall still neutralize one 
another, the changes adverted to being, of course, made in obedience to the 
precept (3). We may then proceed to complete the formula in the way already 
explained. 
Keeping these general principles in view, let us now reflect upon the con- 
struction of the sixteen-square formula. If such a construction be possible, we 
shall necessarily arrive at it by operating with sets of eight, under the guidance 
of the eight-square model, in imitation of the proceeding that conducted us to 
this latter formula from that for four. There can be no question that, if the 
supposed form exists, this is the legitimate and only sure path to it; imasmuch as 
we thus provide for the demands of the subordinate form while carrying on the 
construction of that next in advance,—a provision necessary to the correctness 
of the advanced form, since this must coincide with the former when reduced 
to it by the introduction of the proper number of zeros in place of the general 
symbols. 
In this way, then, we shall obtain sixteen rows of products. These products 
will all differ from one another; and, as in the preceding cases, will be those 
which arise from combining all the proposed letters,—a dashed one with an 
undashed one,—in every possible way. And, assuming these combinations to be 
connected together, as in the former cases, by the signs fitted to cause all the 
double products to cancel, we may apply to the completed form, thus imagined, 
the remarks (1), (2), (3), (4), already made with more especial reference to the 
eight-square formula. 
It is in accordance with these general principles and directions, as far, at 
least, as they can be complied with, that the following partial construction has 
been effected ; and which, to the extent to which it has been carried, satisfies the 
