Mr. A. Cay ley's Note on the Theory of Permutations. 527 



moist summers to account for the production of gigantic glaciers 

 upon land. This last hypothesis is at variance even with the physical 

 phaenomena in and around the Alps, whilst it is in entire antagonism 

 to the much grander and clearly established distribution of erratics 

 of the North during the glacial period. The effect in each case is 

 commensurate with the cause. The Scandinavian chain, from 

 whence the blocks of central Europe radiated, is of many times larger 

 area than the Alps, and hence its blocks have spread over a much 

 greater space. All the chief difficulties of the problem vanish when 

 it is admitted, that enormous changes of the level of the land in 

 relation to the waters have taken place since the distribution of 

 large erratics ; the great northern glacial continent having subsided, 

 and the bottom of the sea further south having been elevated into 

 dry land, whilst the Alps and Jura, formerly at lower levels, have 

 been considerably and irregularly raised. 



LXXVII. Note on the Theory of Pernmtations. 

 By A. Cay LEY*. 



IT seems worth inquiring whether the distinction made use 

 • of in the theory of determinants, of the permutations of a 

 series of things all of them different, into positive and nega- 

 tive permutations, can be made in the case of a series of things 

 not all of them different. The ordinary rule is well known, 

 viz. permutations are considered as positive or negative ac- 

 cording as they are derived from the primitive arrangement 

 by an even or an odd number of inversions (/. e. interchanges 

 of two things) ; and it is obvious that this rule fails when two 

 or more of the series of things become identical, since in this 

 case any given permutation can be derived indifferently by 

 means of an even or an odd number of inversions. To state 

 the rule in a different form, it will be convenient to enter into 

 some preliminary explanations. Consider a series of n things, 

 all of them different, and let abc... be the primitive arrange- 

 ment; imagine a symbol such as {xyz) [u) {vw)... whereas, y, &c. 

 are the entire series of n things, and which symbol is to be 

 considered as furnishing a rule by which a permutation is to 

 be derived from the primitive arrangement abc... as follows, 

 viz. the {a^yz) of the symbol denotes that the letters x^y^z in 

 the primitive arrangement abc... are to be interchanged x intoj/, 

 y into 2;, z into x. The [u) of the symbol denotes that the 

 letter u in the primitive arrangement abc... is to remain unal- 

 tered. The {vw) of the symbol denotes that the letters <r, y in 

 the primitive arrangement are to be interchanged x into ?/ and 

 y into Xf and so on. It is easily seen that any permutation 



* Communicated by the Author. 



