Functions formed on Groups. 25 



of which each marks out a different group, in the order 



*-*<:> *~*2<;j v*8e » • • • ^SOo 



where G i£ = eG id d~\ got by in I, +1 . 



The 30 standards are written in vertical order in sixes 

 thus, which are to be read as (& d} 2dJ (J$ Mj ® u , Q Ml (5 M) 

 &C, all under afiycee. 



<&& ■ • and <3 ( t+,w 

 are two different sets of 60 values, each value of 12 products. 

 And 



are two sets of 60 values, each on the left algebraically- 

 identical with one on the right ; but the values on the left 

 are found in a different order from their equals on the right. 

 The 30 groups G ie are thus written in the order of the 

 standards G id which mark them out : 



But is this all quite proved ? How do we know, first, 

 that no one of the 30 standards under 2 has fewer than 60 

 values ? And, secondly, how do we know that 



are none of them true under this 2 ? 



