314 Mr. Youne on an Extension of a Theorem of Euler, 
In instituting an inquiry into the practicability of generalizing the above- 
mentioned theorem of Euler, the first thing that presents itself to the mind 1s, that, 
if the supposed generalization of the theorem existed, it is reasonable to expect 
that something like a uniform law would be observed to prevail among the com- 
ponent terms of the products in the known particular cases; which law might 
serve to suggest the anticipated extension. Let us, then, examine into the con- 
stitution of these products, in the established forms for two and for four squares. 
These products are respectively 
FEMS +A) = (Wl $2 (ye! — ay) 
and 
(wet a? ty?P+2°)(w+ePr+y4+2) = (ww + 00 + yy +22/P + 
(wa! — xw' + yz’ —2y') + 
(wy — yw! + ev’ — x2’) + 
(we! — zw’ + ay’ — yz’). 
In the first of these cases we may observe, that the roots of the resulting 
squares arise from multiplying the roots of the original squares, as follows : 
Yaak 
y te 
yy! ae 
yz — zy or zy’ — yz’, 
the signs being made alternate in the second row. In like manner, as respects 
the resulting squares in the other case, the roots of these also are constructed 
from those of the original squares, in a way so analogous as to furnish indications 
of a general principle ; thus, 
wai ty +2 
w+u+y +2 
wu + va! + yy! +22! 
wa’ — cw! + yz’ — zy’ 
wy’ — yw! + zu! — x2! 
we’ — zw’ + ary’ — yx’, 
in which construction it will be perceived that, guided by the operation in the 
first case, we commence with the symmetrically situated pairs, 
