48 E. By Wilson, 
Analogously to (91) the formula for Ee) ) is 
(92) ZQ—S[(Py By) aelat(ayeay) Bs B+ (a 3 aB)yly 
3 (i vey) al\B+(eBBy)aly—(B va v) Ble 
— (abu Bly +7 af) y\a-— (a7 eB) 7B} 
And 
Oh = a5 Ai 7 Aim Ay q Cy Cl Wp| Cj. O'm a 2 Dp all ae 
6 BR EE EAE rae et fap ve ruta 
After making the necessary substitutions and reductions, it ap- 
pears that 
(92°) “| Yj Ul Aim 
=e) — aos Lin (Gg ULAim | el @; 
| Am 7 Int An m | 
where the double accent on ¥ means that / and m cannot be equal 
among themselves nor equal to 7 or 7, Moreover the equal results 
obtained from /=a, m—6 and from /=6, m=a have been ac- 
counted for. The formula for Z( is the obvious generalisation of 
the results for 3!) and Z@). The result tor = may be stated in 
words: To find the coefficients of «@;| @; in =, form a determinant 
of the #th order from the matrix of 2 by taking as the main 
diagonal aj; and any combination (not permutation) of k—1 of the 
elements in the main diagonal of 2 excluding a,j, aj; and add the 
determinants of all possible combinations. 
With the aid of these dyadics it is possible to express the in- 
variants of the product of a dyadic and /—26\6. Let 
(85) X= 2 1—2 o|0) 
Then 
(93). Xs = 2s — 2626, Xrs—Ly5—= 2 6 5 6, k= 2,3). ee 
Xn = — Qn 
A similar result could be obtained for the product 
(94) XO Oreo = 2 |) 
Here however, the expansion by the binomial theorem is 
(2 o|6 == 2. 7la)y, (6) ottlr+4 is * Grou 
and hence ‘ 
(94°) Xks=2rs— 205) 6-27 AMT +4AD Ese (Le—2 x 6t| 67) 
The term Qe & (J:-2% 6t| 67) could be treated as Q0% (Ih_-1% 6|T) 
was treated; and the invariant dyadic which resulted would be of 
the second type, that is, the antecedents would be of the form e« ~ 
and the consequents of the form «. In like manner for products 
with more complex square roots of 7, would yield invariant dyadics 
of higher types. The study of these dyadics will not be taken up 
at this time. 

