with a Determination of the Limit beyond which it fails. 331 
SCHOLIA. 
1. Throughout the general reasonings in the foregoing discussion, we have 
considered the squares which enter one of the proposed factors to be different in 
value from those which enter the other. When, however, this is not the case, 
and the two factors are identical, it may be proper here to observe that, in using 
any of the preceding formule for the product, one, at least, of the root-quantities 
furnished by the multiplier, or one of those furnished by the multiplicand, must 
be taken with the minus sign; otherwise, in the product, only the first row of 
the results will be significant ; and thus no decomposition of the proposed square 
will be obtained. But there is no need for any model formula in this case; for 
it is pretty obvious that the square of a polynomial, formed by the sum of any 
number of squares, may be expressed by a polynomial of the same number of 
squares, without any limitation : thus, 
(3 aE a ER VS 
(@ji— 23 +434+454+ .... +0,)?+(22,0,) + (22,0,)? + (24,0,) + .... +(20,0,)?. 
The same thing evidently holds when the factors, instead of being identical, are 
such, that the several squares in the one have the same common ratio to those 
in the other. And formule might be determined, of like generality with this, 
which would exhibit the product when only a partial number of these ratios are 
equal. 
For instance, the product of sixteen squares by sixteen, will be expressed by 
the sum of the squares of the sixteen rows of combinations which follow ;_ pro- 
vided there exist these eight equal ratios, viz. : 
OLDE NTE en: 
Sw Cig 2s cai Up OE a (1), 
among the proposed quantities, 
(874¢? 40? eye 8 Ee ey 4 2)”) \ (2) 
(8° + 400? +0? +0? +2? +y" +2° +87 +0 40 +0? 40? +07 ty) +27) fo 
