210 



CHAPTER VIII. 



To prove that this substitution satisfies relation 209) for m = 3, 

 consider it to be expressed in the notation used for the general 

 substitution [] 2 of 206. The condition 209) built for the sub- 

 stitution 211) therefore becomes 



14 

 14 



23 

 23 



14 

 13 



23 



24 



1 3 

 14 



24 



23 



13 

 13 



24 

 24 



12 



12 



- 3 (mod 2). 



The left member may be written (mod 2): 



n 







32 







W 42 ^43 



24 



13 14 



# 2 9 ^23 ^"24 



Upon expanding according to the elements of the first column the 

 determinant on the left of the following identity 



C*-, 



"11 



"21 "22 



"23 



a 1 .: 



Dfq 



U "42 "43 "44 



we obtain the first three terms in the above expression together with 



14 



42 



12 



It remains to show that the sum of these terms together with 

 zero - Upon applying the Abelian relations (mod 2), 



aa 



i2i 



34 



"43 <*44 



24 



"43 "44 



"11 "12 



"21 "22 



"21 "22 



"41 "42 



"32 "84 



"42 "44 



"23 "24 



"33 "34 



12 14 



22 24 



21 22 



"31 "32 



the sum is seen to be congruent to zero (mod 2). The substitutions 

 211) therefore belong to J?. Their number equals the order 



of the quaternary Abelian group $^.(4, 2") ( 115), which was shown 

 above to be holoedrically isomorphic with the group of the sub- 

 stitutions 211) leaving ^ 3 fixed. We prove in the next section that 

 this number equals the total number of substitutions belonging to 

 Jr (m = 3) and leaving i/ 3 fixed. It follows that the substitutions 

 211) include the following substitutions of Jj' not altering q m : 



