with a Determination of the Limit beyond which it fails. 333 
thus left cannot be filled up so as to satisfy the conditions necessary to the exis- 
tence of the theorem in its general form. 
2. If we glance at the developments exhibited at length for the eight-square 
form, or contemplate the leading binomials in the several rows of that above, we 
shall immediately perceive that the squares of these leading binomials furnish all 
the double products which cancel those supplied by the square of the first row of 
combinations ; and that such must always be the case whatever number of squares 
enter the original factors. We infer, therefore, that 
(ait apt+ajt----+75) (YitMetyst----Yn)= 
(2yY tyr oey s+ pspes, +L nYn) (Li Ys— of) H(LYs-Bey, ) + she HLyYn—Tny)) +P» 
where Pp, is some function of the combinations, into which, however, neither «, 
nor ¥, can enter, or, at least, can enter only to be mutually neutralized, since all 
the combinations involving these are evidently implied in the other terms. 
Suppose, now, that 2,=0, and y,=0; then, similarly, 
(a+ajtait....4+0,) (W+ystyit----+y¥,)= 
(LY OY AD Yh oo ALY a) +( LoL Yo) (LY 52 Yo PH «LY nnn) TP + 
But, on the same supposition, the preceding equation gives, for the first member 
of this, the value 
(Yt LYy+ Cyst. 0. -+ Mya) + Pp 
since P, is not affected by the supposition. Hence 
P, = (1243 — LyYo)” + (Loy — LiYo)” ++ «© - + (Len — TnYo)’ + Po 5 
Similarly, 
Py = (LY4— 2yYg) + (C5 — Veg)? He + (Yn — Try)? + Pop 
and so on. And thus, by supposing successively 7,=0, y,=0; 7,=0, y,=0; 
&c., up to 7, =0, y, = 0; we shall arrive at p, =0; so that, returning, by succes- 
sive substitutions, to the original equation, we have finally 
(MAG + Het. ++ Fan) (YF Yet Yat-- + +In)= 
(LY, + Loot LYgt +--+ 4nYn)? + 
(LiY2— 2.) + (MYs— Vsti) +--+ (MYn — Unf) + 
(L2Ys— Uso) + (Leys — LsYo) +--+ + (LoYn — LnYo)? + 
(2 3Ys— Lis)” + (Ys — VY)? ++ s+ (L5Yn— Las) + «+ (LaYn— FnYn)’s 
