-.30_
= H(P2" =~ 202001 = (£:° 4 202" = 483780 = 484 + 4185)p0s
= 2(p1® = 2¢5) (#1701 = Pabos + Pa1 = #1P11) + 2(£1 For = $3%1a
= PaPas 4 by + 1°Puy — 322 Petor + £2P02 — FaPsa = Pates +
2titetor + 2tatetas
= 24to1 + abas = tobss ~ febas + Pus o
) In case the letters of only sequence appear in the sum,
the P's have only one index number between the commas in the
parentheses. For example, Zagga;® = P,,(1,2) = S,S0; = i1 .
Here S, = :Zay, 'S,s =.Zay®, Sy = Tas®
Therefore Sa4a;% = s38, = S5 = £3P 3ps »
Similarly Zagaye,® = Fg,(1,2) = 3[S,%5,, = S8, = 5,8, + 3,,]
= 3s,%sp = 5, - 25,54 + 25,
= tape — 404 «
Sums of this type are special cases of the most general form

given above, *

The method used in this paper'has an advantage over the
step by step processes (see Cullis, and Junker) because a;y 3
tunction way be expressed in terms of s's without knowledge of
expressions of lower orders, The expression for F in terms of the
S's can be written down directly from formula (13), and the S's
can te transformed immediately into s's in accordance with their
definitions, which can then be expressed in terms of #'s by use
of formula (18). A formula for P in terws of the'p's could be
derived, but it is complicated and not of special value. The
easier way is to substitute for each s in terms of p's after P
has been expressed in terms of s's,

As an example of the use of the preceding theorems, we
shall express in terms of s's the set of functions called the A's

A Dresden solves this special problem by another method, See Annals

of Mathematios, series 2, Vol, 24, P. 227, and Vol, 25, p. 71,