22 E. B. Wilson, 
#,—1 consists of a single term containing all the antecedents. The 
product #,@,_; is therefore null. In this case @, is evidently also 
null, and equation (28) must therefore hold for all cases. 
The demonstration just given for the relation between # and 
@,—, would evidently apply with only insignificant alterations to 
establishing a similar relation between Py and Pp—2, Pze and Py—s, 
and so on. Hence the general formula 
(29) (Pie Pn—-k = Pn [n—% 
On taking the conjugate of each side, the formula 
(29°) P;; (Pn—k)e = Dy, Ly 
is found. In case ® has no degree of nullity, these equations may 
neysolved. / Hence 
(30) Pi, = Dy, (Po Je or DP, = (Pn—k)e | Bn. 
The formulas represented in (29’) look much like the successive 
double powers of the formula for k=1, which is 
(28') P (Pn—1)e= Py I. : 
If the ordinary rules of forming successive double powers be applied 
formally, the result is 
(PD Pn—1, eo a P, Ls, (PD Pn—1, c)3 a P,, I;, ses 
Ors Py (Pn—1)o, — p As. P:, (Pr—1)3, Chama p hee 56 
A comparison with (29‘) would apparently yield the result 
(31) (Pn—1)o = Py Py_-2, (Pn—1)s == Pp, Drs, ae 
The justification for such procedure, however, would involve the 
discussion of double powers of order greater than x. 
Invariant Properties of Dyadics. 
10. The scalar invariants.—Let a dyadic # be written as the 
sum of any number of dyads: 
=alat+PBlAt+yly+.... 
Suppose that the vertical bar which serves to keep the elements 
of the dyads apart, be removed and the sign of the combinatorial 
product be inserted in its place (again the value of having a sign 
such as < for the combinatorial product is brought out) so that 
the antecedent and consequent of the dyad coalesce into a scalar. 
The sum of these scalars, taken with the propery sign, obtained from 
each dyad will be called the scalar of the dyadic and written? 
(32) (1-1¢=aetBetyyt.... 

1 The sign is negative when the negative sign is called for in equa- 
yin 
tion (9 
