320 Mr. Younc on an Extension of a Theorem of Euler, 
vered; and, by having thus marked each pair with the same number, the means 
of readily revising the comparisons will be secured. The result will be, that all 
the terms, except the vertical rows of squares, will disappear from the develop- 
ments ; and it is obvious that these squares, sixty-four in number, constitute the 
product of those originally proposed : therefore, 
(s $e? fut fot fw? $a? $y? +2) x 
(2? +@ 4+vU 4ev4+uv4+r7 47 47)= 
(Se + !” + ul” + vy”? + mw! + gz!” +y'? + ze) 
or, as the theorem may be more concisely expressed, 
3,(0) X %( 0’ = 2,( 0”). 
Although the complete verification of this theorem has been actually exhi- 
bited as above, yet, as before briefly noticed, such verification was not absolutely 
necessary in order to produce confidence in the truth of the proposition. We 
have only to contemplate the internal constitution of the eight rows in the eight- 
square form here presented, in order to perceive that the conclusion just arrived 
at is a necessary consequence of that constitution. For, upon examining that 
form more minutely, we find that not only do the consecutive pairs of binomials 
in every row conform to the four-square model,—a degree of conformity which, 
as before observed, it is essential to secure,—but further, that in each row every 
pair of binomials, whether consecutive or not, involves two sets of four letters, 
such that if all the other letters in the entire form be suppressed, or be replaced 
by so many zeros, the model for four squares will be complied with by the 
remaining expressions. Now, as in every set of such remaining expressions the 
double products, furnished by development as above, would, as we know from 
the property of the four-square formula, cancel one another, it necessarily follows 
that the double products supplied by each of the eight rows of the preceding 
form must be cancelled by like double products arismg from the other rows. It 
is solely because of this uniform agreement with the four-square model, when the 
quantities which compose a pair of binomials in any row are throughout supposed 
to be zero, that the eight-square form is admissible ; for it is obvious that, if by 
means of these zeros any reduction of the advanced to the inferior form presented 
results not in conformity with the model for that inferior form, the double pro- 
ducts furnished by such results could not cancel. 
