118 
2. The notation is a doubly dual one, i. e., from a given theorem and proof 
a dual theorem and proof may be derived by correspondence, and each of these 
has its dual by another correspondence, so that every theorem is of four-fold in- 
terpretation. ; 
3. As illustrative applications we have taken the extensions, to n-fold 
algebra, of Green’s theorem connecting integration through aspaee with integration 
over the boundary of that space (the laws of multiplication undetermined) ; the 
theory of determinants in any algebra; quaternions, and four-fold space. 
4, The alternate processes lead in quaternions to formulas that are almost 
> and the two notations are 
identical with those of Prof. Shaw’s ‘‘ A Processes,’ 
readily convertible. The advantages of our notation are that it pertains to a 
general theory and that its developments are easy and natural rather than arbi- 
trary and labored. 
Il. DEFINITIONS. 
5. We consider a function, ¢(p,, po, .. -. pn), of n variables, and substitu- 
tions, s, s,’, efc., that permute these variables among themselves. 
6. We let (s) stand for the assemblage (s,, s,,... 8m), (s’), stand for (s,’, 
8’, . - 8m’), and (t)=(s) (s’), stand for (t,, 2, . . . tmm’), where ty s =r &/’, 7 — 
Dies «My 1D, . on’, and um (r—1) 4247, Hay. 
7. We further denote, by +s, the substitution s, with the factor 1 or —1, 
according as s involves an even or an odd number of transpositions, and by e(s), 
the fraction which is the ratio of the excess of the number of positive over the 
number of negative substitutions in (s) to the whole number of substitutions in (s). 
When (s) forms a ‘‘group” we have e(s) = 1, — 1, or 0, the latter value in all 
cases where the group contains both positive and negative substitutions. 
] r=m 
8. We denote by A(s), the alternate process, [~ frihs sy, This process per- 
formed on any operand 6 before which it is placed, gives as a result a sum of 
terms, = + 0s,, divided by the number of terms in the sum, where gs, is the 
function @ with its variables rearranged by the substitution s,, 
9. When (s) includes all substitutions of the n variables, so that m= | n, the 
corresponding process is denoted by A. When a process pertains to the group 
of |m substitution of m given variables (m not— n), it is denoted by A with the 
affected variables correspondingly marked. 
10. A function @ is alternate as to (s) when +8s,* 90, r—1, 2,...m™. 
11. A function © is alternate as to (s) for the arrangements (s’) when every 9g’,’ is 
alternate as to (s). 
OE. 
