VII.] 



INDUCTION. 



145 



Origin and 

 four proximates. 



gBCD AbcD 



I 



ABCd— AlicD— A&CD 



ABtD aBcD 



Six 



mediates. 



abCD 



Obverse and 

 four ultimates. 



AbCd 



Ahcd 



I 

 abcD — ahcd — oBcd 



aBCd 



ahCd. 



ABcd 



It will be seen that the four proximates are respectively 

 obverse to the four ultimates, and that the mediates form 

 three pairs of obverses. Every proximate or ultimate is 

 distant I and 3 respectively from such a pair of mediates. 



Aided by this system of nomenclature Professor Clifford 



proceeds to an exhaustive enumeration of types, in which 



it is impossible to follow him. The results are as follows : — 



I -fold statements i type \ 



2 >; » 



4 „ » 19 » M59 



3 T » 



7 » 



8-fold statements y8 „ 



Now as each seven-fold or less-than-seven-fold statement 



is complementary to a uine-fold or more-than-nine-fold 



statement, it follows that the complete number of types 



will be 159 X 2 + 78 = 396. 



It a})pears then that the types of statement concerning 

 four classes are only about 26 times as numerous as those 

 concerning three classes, fifteen in number, although the 

 number of possible combinations is 256 times as great. 



Professor Clifford informs me that tlie knowledge of tlie 

 I)ossible groupings of subdivisions of classes which he 

 obtained by this inquiry has been of service to him in 

 some applications of hyper-elliptic functions to which he 

 lias subsequently been led. Professor Cayley has since 

 (;x]>ressed his opinion that this line of investigation should 

 1)('. followed out, owing to the bearing of the theory of 

 compound combinations upon the higher geometry.^ It 

 .seems likely that many unexpected points of connection 



' Procecdinfjx of the Manchester Literary and Philosophical Society, 

 6t!i Febnuiry, iti/y, vol. xvi., p. 113. 



