with a Determination of the Limit beyond which it fails. 323 
essential condition which it is our object to impress upon the completed form :— 
all the like products furnished by the combinations here exhibited cancel one 
another. If the anticipated form could be completed, then, as we have thus got 
more than half the requisite rows, such completion might now be effected simply 
by aid of these rows, and without any further reference to a guiding model, as 
already sufficiently explained at page 321. But this we shall find to be impossi- 
ble; for although, as just stated, all the like products supplied by the partial 
construction really cancel one another, yet there is inherent in that construction 
a want of conformity with the eight-square model, as respects one of the sets of 
eight involved, which no modifications of signs can remove ; and which, by over- 
ruling the combinations yet to be formed, precludes the cancelling of the like 
products which remain to be compared. This will be clearly seen presently. 
Partial Construction of the sixteen-square Formula. 
+t ¢utvg¢u te ty tts tt ewtvtuteaty +2Z 
stt+utvtut+ertytets tt tutvt+u,tatyt7 
ss’ Ht! uu'+ovv' +ww' aa’ +yy' +22! +8,5) +60 4aul+0 0! +w we! baa ty y +22, 
st’ ts’ +uv'—vu'+wa'! —aw'+yz' —zy’ +s t) ts) +u 0! —v uw a! —a wy 2) —2,y/', 
sul —us' tut! —to’ t+yw! —wy'+02' —ca' +s 0) —10,8' +0 t tv! +00 ' —y wi +20) az, 
sv’ —vs' +tu! —ut! +w2 —2w'+0y! —ye!' +08) —s 0! +t, tu! 40 ,2/, —2w' +a yy 
sw’—ws'+at! ta’ +a’ —yu! +20! —v7 +8 w'—w sia t! tv) ty, —wy) +0 Zz) —2v' 
sa! —as! +tw' —wt! +yv! —vy! +2u! —uz' +8,x/—2 8/4! —w tiv y', —y 0, +,2) —2u' 
sy —ys! +2t! —tz! 40a —av! 4wu'—uw'+y s/—s y' +t,2), —2,t) 40,0" —2 vo! tw ww 
sz’ —2s' +ty! —yt! +0! —wo! +ur’—ru' +s,z/ —z 8' +t,y/, —y ft +w v) —v wi ta ua’ 
ssi —ss4t,t’ tt, +uu'—uu' +vv' 
st, —t,s’ +s’ —s t'+vu/, —w v'4+uv 
su’ —u s'+v t/t! +t" —vt +us' —s u! 
sv, —v s'+tu’ —u t'+s8,v' —vs' +t,u'—ut’.* 
, 
,— 0 
/ 
/ 
4 
—U/u 
Now, it appears from examining these expressions, that, as before remarked, 
* These rows are exhibited complete in the scholium at the end of this paper, where it is shown 
that, under certain conditions, the sixteen-square formula has place. 
mM 
“~ y 
