40 E. B. Wilson, 
In case there is only one root +1, it may be assumed that the 
dyadic takes the form 
(79) alot ale'+ BR +B trl tylO+ oll +0le+ele+..., 
in which all the shearing terms occur: for if any of them were 
absent, a reduction to two spaces of lower dimensions could be 
effected as in the case of two pairs of roots. This may be factored. 
In case 2 = 2, 
(80) ala'+ea\e’+ lf =(ale’ — B\B’) (ala’'+ a|s'— BIB) 
where each of the factors is a square root of J. In case m= 8, 
(80’ ale’ + alp’+818'+ Bly +rl7 = 
(a\a’—B|8'+ Bly’ +77) (ale’+ ale’ — Bla +717) 
with similar remarks. Again in case m = 4, the factors are 
(80") ale’ +elP'+B\F+6ly+y7ly+y7|0+6|0 =(ele’— Ble + 
By -+y|7'—d|0 = 2y|0'—B]0) (cle al8 — Ble +77 —7 le ole 
If the root were —1, the factors would be respectively 
—ala'+ a|8'— B|p’=(ale’— B|8')(— al\a’+ a |P'+ BB), 
(80”) ala’ talp'— pie +Bly—rly’ 
=(al\e’— B|B’— Bly’ + yl7’)(—eala’+ al 6+ 6|8’— |r’), 
—alae+alp’— BIP+ Bly —yl7r+7|0— 6|0 = (ale’— B|B'— Bly 
+y|y'—-6|d'+ 2y|0’— Bd’) (—a| a’ +a|p'+8| 8’ —y|y’— y|0+- 4| 0). 
Although this method of factoring could be carried on to higher 
dimensions, it is better to proceed in another way, which at the 
same time will indicate how the factors may be obtained if they 
are not evident. Consider, for example, the case of seven dimen- 

sions, where 
(79) ale tala +ale+elytyrly + rio +66 + dle + ele 
rel Solo + oly + aly’ 
and note that the two expressions 
(81) ale’ +alp’—BIB t+yrly trl —dé|0 +elf +e|o 
— O19 Flay’ 
ale’ — BIB + Bly +yiy—d\|o+dle tele —C|6 +o|n' +y\7 
are obviously square roots of I and that their product has the form 
(79°): ale +a) 8 = BIA Bly aa yly' Pyle 4-010 10 eae 
+ elt) + 6g + Sin + nin’ + alo + 01g. 
This fails to be identical with (79') owing to the extra terms ~|0’ 
+6|¢’. Nevertheless it belongs to the same type of dyadic as that. 
In fact it is true that when a dyadic has been reduced to the form 
(64), which in matricular expression means that the only terms oc- 
curring are those of the main diagonal and some (the shearing 
terms) along the diagonal next above it, then the addition of any 
terms in the half-square whose diagonal is constructed of the # terms 
alert | Bitwulyst..., 1S =, or of the: p-tvterme 

