120 
Note.—If (s) bea group this condition means practically for all the arrange- 
ments of (f). ; 
Cor. 2. A(t) * 6 A(s) > (or e(s’) * ©) when o is alternate (or symmetric) as to (s’). 
Theor. 3. If (s) be a group, then’ A(s) @ is an alternate function of the group (s) 
for all arrangements of the variables. 
Note.—A(s) ‘@ is an alternate function of the group (s) only for those ar- 
rangements s,.... that satisfy s (s) —(s)s. These include the group (s). 
Theor. 4. If (s) be a group, and (s’) be any assemblage contained in (s), then, 
A(s) 9 = A(s) * A(e’) 9 = Ale) Ale’) 9 « 
15. These are the principal theorems of the subject. We note some im- 
portant special cases where the processes are those that pertain to all the substitu- 
tions of given numbers (variables or places). 
16. Let A’ affect m’ given numbers, let A” affect m’’ other given numbers, 
and so on. Then A’ A”... is a process whose factors are commutative and 
whose substitutions form a group (s), consisting of substitutions that permute 
each set of variables (or the variables in each set of places) among themselves. 
One complementary assemblage (s’), such that (s) (s’) forms the complete group 
of |” substitutions then consists of the substitutions that leave each set of variables 
(or the variables in each set of places) in their original order among themselves. 
Any element s’ ,’ of this assemblage may be replaced by any product s; 8’ ,’ with- 
out charging the assemblage as’a complement of (s). 
We then have from th. 2. 
Theor. 2’. Ly ae 
Ag=Aw) * A’ AY... Oa ty A AY... 
poe 
In this expansion of A © in terms of minor A’s, all terms may be made posi- 
tive by replacing every negative s’ ,’ by its product by a transposition of (s). 
(a). Ao = A(y > (or O) when o is alternate (or symmetric) as to (s) for all ar- 
rangements of the variables. 
Note.—In particular, if 0 be symmetric as to certain variables (or places) for all 
arrangements of the variables, then A o = 0. 
(b). Ago = A’ A”... 9 (ore) * A’ A”... 4%) when ¢ is alternate (or sym- 
metric) as to (s’). 
IV. LINEAR ALTERNATES. 
17. A function ¢p is said to be linear when ¢ (x p+ yq)—zop+yoq, 
where », y are ordinary numbers (scalars). 
18. A function  (p,, po, - - + Pm) is a linear alternate of m‘ order, when it is 
linear as to each of its variables. and the interchange of any two variables changes 
its sign. 
