34 



The Rev. Thos. P. Kirkman 



which contains 60 values, because (£<; is not symmetrical in 

 456 nor in any pair of them (art. 2). & d is one of our 

 sought functions. 



We next take any seventh unmarked equivalent, 241563, 

 for G., d , and by the same five 0's we obtain five groups more, 

 G 2 e G 2e i- • • Go £5 , to be marked out as all giving the function 

 and values ($? d ". ($ 2d 1S another of our sought functions. 



Thus we go on till we have ten standards selected from 

 groups unmarked, each heading, as follows, a line of five 

 groups G ie marked out by it. 



2 = a/3yC()($. 



G f! =(234561): 235164; 236514; 234615 

 G M = (241563): 251364; 241653; 261345 

 G 3d =(2463i5): 254631; 265143; 245361 

 £*<* = (245613): 256341; 264135; 246531 

 G 5d = (34256i): 352164; 362514; 342615 

 G 6d =(345 l62 ) : 35 62I 4; 3 6 45 21 ; 346125 

 G 7d = (345 62 !): 356142; 346512; 365124 

 G 8 d=(435 2 6i): 536124; 634512; 436215 

 G 9d = (435612): 536241; 634125; 436521 

 G 1 o ( i = (45 6 3 I 2): 564231; 645123; 465321 



Of these standards the last only is so symmetrical under 2 

 that a substitution C of I m gives 



C6i<w = <Bio* (art. 3). 



And every substitution of I <+1 is such a C. 



This G^J =(456312) gives under 2 the function, 8>o, 



a/3ycco aftycoo 



6104=123456 + 456123 



+ 231456 + 564123 



+ 312456 + 645123 



+ 132456 + 465123 



+ 321456 + 654123 



+ 213456 + 546123 



which is six times repeated in Otoa' ' ar) d has therefore twelve 



terms and ten values. There are but ten ways of making 



236145; 2 35 6 4i- 



2 6i534; 251643. 



2645^; 256134. 



265314; 245163. 



362145; 352641. 



365241; 3546i2. 



354261; 364215- 



635^2; 534621. 



635214; 534i62. 



654213; 546i3 2 - 



