with a Determination of the Limit beyond which it fails. 327 
and s‘, ¢’, 2’, y’, the row marked (w”) will be the only row that will vanish. 
We may hence notice, in passing, that four squares multiplied by four squares 
may be made to produce seven squares ; but not, in general, either six squares 
or five. 
We shall only further remind the reader that, when any formula of the kind 
here discussed actually exists, and that we have partially constructed it, to an 
extent however limited, in strict conformity to the subordinate model, that con- 
struction must be correct as far as it goes; and must admit of completion with- 
out disturbing the signs in the partial form. Whenever, therefore, such com- 
pletion is shown to be impossible, we may infer that the supposed form has no 
existence ; and from this consideration, by first excluding the four combinations 
here shown to be refractory from the above group, we may arrive, somewhat 
differently, at the conclusion already deduced. We shall again advert to this 
presently. 
Returning now to the four-square and eight-square formule, we may make 
the following inferences, viz. : 
1. Certain coefficients may be introduced in connexion with the original 
squares, which coefficients will reappear in the corresponding squares of the pro- 
duct. This will readily be seen by taking the case of four squares with the 
suitable coefficients, 
w+ Vb. +Ve.y + V (be). 2 
w+vVb.c +Vveo.y + ¥(be).z 
ww’ + bra!’ + cyy’ + bez’ =v" 
Vb(w2!’ — rw’ + czy — cyz’) = Vb.2" 
Ve(wy — yw’ + bza’ — bre’) = Ve.y” 
V (be)(we’ — zw’ + xy’ — yx’) = V (be). 2", 
which shows, as indeed was before proved by Lagrange, that the product of 
(w? + bx? + cy”? + bez?) 
and 
(w? + ba’? + cy* + bez’) 
is of the same form as each of the factors; that is 
(w” + br”? + cy”? + bez?) (w? + ba? + cy? + bez’) =u"? + ba’? + cy!” + be2”. 
VOL. XXI. 2x 
