Functions given by Groups. 319 



The bars under S show that the terms in It+i are purely- 

 tactical, the elements being merely permutable positions, 

 not of necessity magnitudes or quantities of any kind ; and 

 the terms are not yet algebraic products, such as 



i«2V4>'5y6'y+ i"2'^3^5^6V+ i''2'^3'^6V5''+ i''2^3V6'^5'^ + 

 €tc., products of powers of x\X^x^ . . x^,. 



Such products arise out of the substitutions of a group 

 only when the group is placed, as we say, under S, and 

 thus is made suddenly into a function of algebraic quantities. 



The Index group is always a woven group {M.M. p. 331), 

 in which no element under any repeating cluster ever 

 changes place with one under a different exponent. It is 

 never a grouped group, nor a woven grouped group {M.M. 

 p. 304 — 331). And it is never transitive, i.e.^ no element 

 •can be found in every vertical row. 



7. Every group G^ in (A„) becomes instantly an algebraic 

 function 0„, of Q values, (Ba included, when placed under, 

 or, as we say, crowned with, almost any S. The very few 

 cases in which the number of values is <Q need not here 

 be noticed. 



We need not, in reading a function (Ba, so formed by S 

 <2 times written, repeat the indices in all the L terms of the 

 rectangle ; but we ascribe to every element, from top to 

 bottom of each vertical row, the exponent standing in S over 

 the row, and this in each of the Q rectangles. We can thus 

 dictate the function (B^ and all its Q — i values, which are 

 the Q — I derivates under S of G^. Thus every rectangle 

 of the line G^ under S in (H„), is a sum of L products of 

 powers. We often speak of G^ and its Q - i values as the 

 Q values of the function (B^, or of G^ under 2. 



When S has n different exponents, each of our Q 

 equivalents in (A J under S becomes a O-valued function 

 distinct from the Q - i others, and the Q^ values of these Q 

 functions are all different. Thus functions are easily 



