119 
12. We distinguish between s; * ¢— 0s, and‘ s; 4; viz., the latter function 
involves the symbol s; which is a function of the variables so that a substitution 
on the variables of * s,9, which have the same order as in @, is not equal to the 
same substitution on ¢,,. In facts*s,¢—s, ¢; —=s8;5s°o. 
13. We have also symmetric processes, C(s), symmetric functions as to (s), ete., 
whose definitions are obtained by replacing + s; by s; in the above definitions. 
“ce 
There is a dualty between ‘‘alternate” and ‘‘symmetric” which consists in the 
interchange of corresponding terms. The fraction e(s) is in general its own dual. 
14. There is also a dual interpretation of the substitutions, viz., write for the 
moment ¢ (p,, Po, °° pn) —? : a dy" as where we have the number of a “ varia- 
“cc ” 
ble,” and beneath it, the number of its ‘‘place” in ¢. Ordinary substitutions 
affect the upper line of numbers only, 7. e., the ‘‘ variables.” The same substitu- 
tions on the lower line of numbers only are “‘ 
place” substitutions. The substitu- 
tion s that affects the given number 1, 2, ... mn may be marked 8 or s according as 
it affects variable or place numbers. The dualty arises from these two interpre- 
tations of the substitutions of any process. When the variables of the operand 
that are affected by s occupy the places corresponding to their numbers, we 
have s=3-', and the processes AG. A(s) give the same result provided (s) is a 
substitution group. If, however, the above arrangement of the variables be 
affected by a substitution s’, and the result taken as operand, we have s — ¥ 3-1 9/1, 
so that the two processes A, A to the same group (s) are in eeneeaih different, the 
latter being equivalent to the former to a group that is similar to (s) only. 
lI. THEOREMS. 
[The proofs are too elementary to need insertion.]| 
Theor. 1. If ¢ be alternate (or symmetric) as to (s), then is A(e) 6 = (or e(s) 9). 
Theor. 2. If (t) = (s) (s’), then is A(t) ¢ = A(e) Ae’) * ¢ = Ale’) * Ale) 9. 
Note.—This result shows that the product of two alternate processes is an 
alternate process, and that a process (Ai?) may be expended in terms of a given 
minor process ( A(s)_). 
E.g., A*pqr=} (pAqr—qApr+rApq), A'pqrs=3(Apq Ars+ 
Ars'Apq—Apr‘Aqs—Aqs'Apr+Aps'Aqr+Aqr'Aps), ete. 
These are place expansions. Variable expansions give different results, 
e*g.,A*pqgr=tA (pq r—qp r+ rp) 
=+ (pAqr—Aqpr+Agqr‘p). See art. 16. 
Cor. 1. A(t) * = A(e’) o (Or e(s) * A(s’) 9, when @ is alternate (or symmetric) as 
to (s) for the arrangemints (s’). 
