44 The Rev. Thos. P. Kirkman on 



Taking now 0.^=251364, above given, we obtain by H fi 

 G 2 rf, G^ . . Go f5 to be marked out by G 2d . These are 261 534, 

 241563, 261345, 251643, 241635. G 2 i has no symmetry 

 under 2 in 23 or in 456, wherefore §., a is a function of 6 

 terms and of 60 values. We retain O d and (3 2rf as won 

 functions. 



There are 60- 12 = 48 groups not marked out. Taking 

 G D = 234561, which has no substitution 6 in common with 

 I 12 , as is clear by a glance at the substitutions beginning in 

 our paradigm with 1 , we name as above the 1 1 0's of I 12 , and 

 obtain 1 1 groups G E1 , G E2 . . G E11 to be marked out by G p . 

 Then taking at random, as we require them, three more 

 standards, G 2D , G 3D , G 4D , we obtain three more elevens, 

 G 2 Ei, & c -j G 3E] , &c, G 4E1 , &c, to be marked out. 



These elevens are all written below, headed by the 

 standards which mark them out. 



The reader, if he likes to construct (art. 1) the last three 

 standards, will satisfy himself that by the absence of sym- 

 metry under, /3/3 and yyy, they are all proved to have 60 

 values. 



Thus we have demonstrated that there are, 



under n/3/3yyy, 



six distinct functions constructible, namely, 



(3d =465213, which has 6 terms and 10 values, 

 (3o d = 251364, which has 6 terms and 60 values, 



