with a Determination of the Limit beyond which it fails. 317 
have the form for eight squares already developed in full. The following seems 
to be the easiest way of writing out the developments in question : 
(ss’ + dt! + uw’ + ov! + ww’ 4+ re’ + yy +27) = 
(ss’)? + 2ss‘(tt + uu! + vo! + ww! 4 va’ + yy’ + 22’) 
+ (tt)? + 2tt (uu! + vv! + wo! + wa! + yy! + 22’) 
+ (uu')? + 2uu' (vv! + wo! + va’ + yy! + 22’ 
+ (vv')? + 2vv'(wo! + wa’ + yy! + 22’ 
+ (ww') + 2wu! (av! + yy! + 22 
+ (a)? + 2are'(yy! + 22’ 
+ (yy) + 2yy"(22’) 
+ (22’)*. 
(st! — ts’ + wo! — vu’ + wa! — aw! + ye — 2 P= 
( st’)? + 2st’(— ts’ + wv! — vu’ + wa’! — aw! + yx — zy’) 
+ (ts’)? — 2ts'(uv’ — vu! + wa! — vw! + yz’ — zy’) 
+ (uv)? + Quv'(— vu! + we! — rw! + yz — zy’) 
+ (vu’)? — 2vu' (wa! — rw! + yz — zy’) 
+ (we P + Qwe'(— ew’ + yz! — zy’) 
+ (xw')? — 2aw'(y2! — zy’) 
+ (yz)? + 2yz!(— 4) 
+ (zy')’. 
(su’ — us’ + vt! — to’ + yw’ — wy + 22’ — zr’ P= 
(sze’)? 4 Qse!(— us’ + vt’ — to’ + yw! — wy! + az — 22’) 
+ (us')? — 2us'(vt! — to’ + yw! — wy! + az’ — za’) 
+ (vt)? + 2vt!(— te’ + yw! — wy! + xz’ — zz’) 
+ (tv)? — 2te' (yw! — wy’ + x2 — 22’) 
+ (yw')? + 2yw'(— wy! + x2! — 22") 
+ (wy!) — 2uy!(w2! — za’) 
+ (x2')? + 247 (— 22’) 
+ (20) 
