249 



begun above with one other only system of seven cyclical 

 triads, exhausting the duads in seven. Hence r=2, and 

 there are thirty equivalent groups of 7'6*4. And as 



we see by the corollary to my theorem A (art. 9^ '^ Theory of 

 Groups/' Sec), that the group has no derived derangement, 

 i. e., it is maximum. 



Take next the 66 triads : -— 



12436 12590 15738 235^7 267i.5 2748o 340,9 39^46 46^27 5702« eOfir^ 



1093„ 17^60 I6O28 20au 249g« 28O35 37826 45769 4806« 590^8 68^13 



13^25 128-« I4O57 29067 2O849 34658 35O16 458i2 469io 59rzi7 780i9 



18956 1564^ 13679 236„„ 27«39 3ao78 3795o 45^30 56870 6798« 78«45 



la489 138^0 17924 239i8 25cr68 347i« 3589« 48937 56903 67O34 89a,2 



which exhaust the duads of eleven elements 12 '"' oa three 

 times, and which^ disregarding the subindices, fulfil the con- 

 dition, that if ahc, abd, abe, be three triads, cde is a fourth. 

 These 55 triads are formed by a simple cyclical kind of pro- 

 cess from the triad 128. The subindex under 124 shows 

 that 361, o62 and o64 are triads of the system. 



Each of the 55 triads determines a sextuplet. We form 



on (347«i), on (68«i3), on (I9O3,), ou (325^7), on (a5268), on (I2590), 



-.vhich are all the sextuplets containing ISao^. The first, 

 third, and fifth of (347)ai are the three containing al ; the 

 second, fourth, and sixth are the three whose subindices are 

 the circle 34, 47, 73 ; and so on of the rest. The remaining 



