20 E. B. Wilson, 
(24°) (DPQ ...), = Py Fy Qe... . 
As a corollary it is seen that (@”), = (@,)" = @f. The formula given 
in article 5 for the reciprocal of a dyadic may be used to extend 
this result to negative exponents in case # is complete. 
If @ and ¥ are homologous dyadics, the developments of article 6 
show that the expansion of 
(P+ pn = gr a n pnr-1 py ies) gu—2 pe o ... tt gn 
may be carried on by the ordinary binomial theorem. It the dyadics 
are not homologous, this will no longer be true: the second term, 
for instance will consist of 7 terms in which # occurs ~—1 times 
and # once, but the rearrangement which permits of writing 
n @”—1 P will be impossible. There is, however, a binomial theorem 
for the Ath of a sum, namely, 
(25) (P+ Pj = Oy + Gir P+ Gyo Xe +... + H, 
k 
The proof consists in considering the expansion of (+ ®) x. As 
the commutative and associative laws hold for double products, it 
is possible to write — 
k (k—1) rae are 
2! ae eee 

k k kb 1 
p G\x = @x px xP 
(Disnaeey) ee en Cs She 
=k! Oh, +h. (A—-1)! Bei P+ kh (k—-1).(k-2)! Geol W+t.... 
1 
On dividing through by hl the theorem is proved. It will be 
noticed that the usual binomial coefficients are lacking in the bi- 
nomial theorem for double products. 
9. Conjugate dyadics.—The conjugate of a given primary dyadic 
is a dyadic which satisfies the condition 
(26) epfh=Go, & =(—1)"—1(ala+ Blie+...) fS=—alat+Ale+... 
for all values of the quantity g. It is denoted by a subscript c. 
The dyadic %, is not primary, but of the class m—1. The neces- 
sity for the negative sign arises when 7 is even, because then 
ao =—oae. In the definition of conjugates for dyadics of the second 
and higher classes, the factor (—1)"—1 is applied only in the case 
of dyadics of odd class; for it is only in such cases that the re- 
versed of the order of the factors changes the sign. The idem- 
factors: J, 1, J3, ..., Ina; tp =1, which are mentioned imjariclers 
and which are the appropriate idemfactors respectively for elements 
of the first, second, third, ... (w—1)st, and mth classes, as may be 
