81 
Such a sy8tem exists. For inspection shows the elementary symmetric functions 
of Zi® (i=1--4) are unchanged by the group, and hence Z consists of an infinite 
number of forms. The sequel will show, however, that these are not all the in- 
variant forms of the group and hence we ask for an expression of the entire 
system. 
1. To determine analytic expressions for the system Z. 
Definitions—An 7-lettered form is one whose terms each contain 7 letters. By 
an invariant we shall mean a form invariant under the group. 
An invariant may be expressed as a sum of 
one-lettered forms, 
two-lettered forms, 
three-lettered forms, and 
four-lettered forms. 
Moreover, since the substitutions are linear, the sum of all i-lettered terms of an 
invariant is itself an invariant where 7 is any one of 1, 2, 3, or 4. 
a. Four-lettered forms. 
We shall consider first four-lettered invariants. 
Let F be a four-lettered invariant. Where 
B’ oA Oo” 
F= Via Ae Zi 70 - rhe Z! Ze ‘ ra + a Re Re i Wi Fhe ase sat ine caa 
a+Bty+tdé=a’+ B’+y’+07=............ 
and where any exponent is a positive integer. 
Suppose a is the least exponent, then 
(4 Aaa Li,) “ [a possible four-lettered form ++ a sum of 7-lettered forms. | 
Where i—1--~-3, 
or F==(Z, Z, Zs Zs)“ [1 + 92] (say-) 
Since (Z, Z, Z; Z,) “ is an invariant, so also is 9, + @,, and therefore, by 
remark above, so also is #,. Hence ¢, is susceptible of treatment similar to that 
applied to F, until finally we should have any four-lettered invariant expressed 
as the sum of invariant forms, of which a type is 
(Z, Z, Z, Zi)" [sum of i-lettered forms] i—1--3. 
Hence we need seek no expression for an i-lettered form where i > 3. 
b. Three-lettered invariant forms. 
Let a term of this invariant be 
BAVA 
6—SCIENCE. 
