120 
MAJOR P. A. ]\IACMABON OX A CERTAIX CLASS OF 
and the condensed form, thence derivable, which has the form 
1 
1 — + I' {Jc — 1) — A: (Z; — ])” + . . . + { —h {k — !)”■ ^ .. . se,, 
The latter generating' function occurs in the Theory of the Composition of 
Numbers. The corresponding redundant form is not unique (this will appear in the 
sequel, l)ut that given above is one of the most useftd. 
Art. 14 . The second one was founded on the relation 
(X„X„ X3...X„) = (1, X.1, Xg,, 
C fj Xgj, 
1, 1, c 
I, 1, 1, 1 
• X,1) (.Tj, .To, .Tg . . .'a:„) 
• X„2 j 
. x.„ 
'•/iS 
leading to the condensed form 
1 — 2.ri - — 1) — - O^Pa. — 1) “ 1) 
• ■ . t) (Ns (^43 1 ) . , . (A„, n — \ ■ 1 ) 
* ^7} — \ '^’k_ 
wherein the numbers a, . are in ascending order of magnitude. 
These particular cases gave rise to dual interpretations in arithmetic. 
Art. 15 . The general theorem, as so far developed, apparently only admits of a 
single interpretation. 
Regarding the product 
(«i^i -h + • ■ • + (&!«] + 63.T0 + .. . + h„x„y-^ .. . (ngTi +%T,j+ .. . +?^„.T„)^^ 
the coefficient of 
. . . . T,/" 
may be interpreted in the theory of permutations. 
Considering the permutations of the quantities which form the product 
.Tp 5 .r./^ . . . xj’', 
the coefficient indicates the number of permutations which possess the property that 
