332 Mr. Youne on an Extension of a Theorem of Euler, 
Expressions, of which the Sum of the Squares is equal to the Product of the 
Factors (2), when the Conditions (1) have place. 
/ 
ss’ tt! +uu'+ov! +w' an’ +yy' +22’ +8,8) 40 tui po vi ww! +a a) 4 y+2 2, 
1 / 7. 1 tf ‘ 
st! —ts’ +-uv'—vu'+we'—aw'+y7 —zy' +s t, —ts) +u,v) —v u/s a) —a wi sy 2), —2Y, 
tl = af; ‘ / at ane) eo 
su’ —us' +ot! —te’ +yw! —wy'+02' —2a' +s) —u,s' 40 ft, tv) +0 y) yw Ac 0), —2 27, 
/ 
su’ —vs' +tu! —ut! 4w2' —2zu’+a2y' —ya! 40,8) —s,v) +t) tu! wz’, —2wi +o y'-y 2", 
/ 7 2 ate Uy / / , vA banal 
sw!—ws'+at’ —ta’ +uy’ —yu! +20! v7 +8 ,wi—w,s'+a t) ta) ty ui, uy) +0 2) —2v, 
/ : U / / pal, / 
sa! —as! +t! —wt! +yv' —vy! +2u! —uz! +8 2'—2 8/4 wi —w tiv y’, —yv, 44,2, —zu 
sy —ys! +t! te! +02’ —xv! +wu'—uw'+ys\—s y’ +42) —2 ti 40,0) —a vl + uw 
. 1 de / / U ! a7, ie ee hte pia ag ate Qo ’ 
sz’ —2s' +ty! —yt! +0! —wo' ua’ —xu' +82) —2,8) +t), yt wv), 0 we 2 
ss —ss'4tt! tt, +uu!—uu' +v0'—0 v'4+u'w wu +a 2’ —an'+yy, yy! +22 —22’, 
wv 1 / / ’ / t 
st, ts’ +ts’ —s t! ou! —wv'--uv')—0 +a —2 ww 2! —aw' sy 2! —2y’, +2y' -yZ 
5 ‘ 
su’ —u,s'+0 t/t! +t! —vt!, +us’) —su'+-wy'—y o'+2 0 a7’, +20", —a 2! +yw' wy 
1 / J / ‘ (ae Oe a 7 ‘ 
sv’ —v stu’ —w t'+8 0! —vs), 44! —ut! +2 y'—ya', +2 0! -w2' +y 0! —ay! +2! —wz 
Ji ‘ 
sw’ —w,s' +a tte’ +y ju! —uy', +2 02" +s! —sw'+at, ta" +uy! —yu’, +0,2' —20v' 
sar —as'4tw!—w t'+y vo! vy’, uz’ —2 +t owt’, ars), —s x 4u,2' —cu’, +yv' —v,y' 
/ / / / / / / 
sy, —y8' +2,t! tz’, +0a'—a2 '+uw'—wy' +200", —v a'4+u'—wu'+et) —t2' +8,y' -ys' 
se’ —z,s' +ty’ -y t'+0u' 0 v'+a ul —ua' wr’ —0 w'+au! —ua' +28, —s2' +yt) —ty 
Since, in virtue of the conditions (1), the semi-row commencing with s/f’, 
and the six semi-rows which follow it, all vanish, they should be expunged from 
the above form, after they have subserved the purpose of facilitating the con- 
struction of the lower rows, agreeably to the directions at page 321. 
Dismissing, then, these zero-binomials as superfluous, we shall find all the 
double products, arising from the development of the squares of the sixteen rows 
of combinations, to disappear, like as in the forms before established. Those 
products whose like occur in the group here supposed to be expunged, will be 
cancelled by equal products in the group retained: these latter products, though 
not like, being equivalent in virtue of the assumed conditions (1) above. And 
it may be further observed that, after the suppression here recommended, the 
present form may be employed, instead of the more abridged one at page 323, 
to prove the general impossibility of the sixteen-square theorem ; for it may 
be readily shown, in a manner analogous to that there adopted, that the chasm 
