with a Determination of the Limit beyond which it fails. 315 
we +a',w+a,andy’+7,y+2; 
then proceed to the symmetrically placed pairs, 
w+ty,wt+y,anda’+ 7,242; 
and finally to the remaining symmetrical pairs, 
w+2,w+z,and a’ +y,r4+y; 
it being observed, in reference to the last four of these pairs, that the first row 
of results are exhibited in the leading row of the complete form. 
Still submitting to the guidance of the same principle of construction, and 
proceeding to the case of eight squares, the following results present themselves : 
s+ 4+wv+v’4+w4+e4+yt+ 2 
Ste ieee we hl y Elz 
ss) ttt +-uu' + v0! + ww’ + re’ + yy +22 (s’”) 
st — ts’ + uv’ — vu’ + wa’ — rw! + y2' — 2/ (t”) 
su’ —us’ + 0t — te + yw! — wy + 42! — 28 (u’’) 
su’ — vs’ + tu’ — ul’ + we! — zw’ + xy — yr’ (v’) 
sw’ — ws’ + xt — te’ + uy! — yu 42! — v2’ (w’’) 
sa’ — as’ + tw’ — wt! + yr!’ — vy! +2 — uZ (2’) 
sy’ — ys’ + 2t — tz’ + uw — uw!’ +02 — av’ (y’’) 
sz’ — zs’ + ty’ — yt’ + ow’ — we! + usr! — xu’. Ge) 
Now, just as, in the former case, we commenced the rows in order with 
w+2z,w + y, and w + 2, in succession, so here we begin with s +¢+ u + v, 
stitw-+a,ands+t+y-+ =z, in succession. 
When these several sets of four are disposed of, agreeably to the above general 
form for fours, then the set of four which remains, after the imagined suppression 
of these, is at each step to be treated in the same way. And whether we are to 
take the several rows of these second sets of results with the same signs that they 
have in the pre-established form for four, or with all the signs in any one of 
those rows changed,—which is, of course, allowable, without any violation of that 
form,—is to be determined by this circumstance, viz.: we are to take care that 
the two consecutive binomials, or pairs of results, in each row for eight, all con- 
form to the general model for four. Attending to this arrangement, the conse- 
cutive pairs of binomials in any row, as in the fifth, for example, 
