TRANSACTIONS OP THE SECTIONS. 5 



whose general tlieory has not, so far as I know, been given. In fact the auxiliary 

 groups on which they are constructed are of a kind entirely new, of which- a brief 

 account may be seen at the end of my memoir " On the Theory of Groups and 

 many- valued Functions," in the forthcoming volume of the 'Memoirs of the 

 Literary and Philosophical Society of Manchester,' 1861. 

 These auxiliaries are, 



1234 1234 



.,. 2-1432 2^14^3 .,,s 



W 32423^2 3^12^ ^ ■' 



43^21% 4=3^21. 



The peculiarity of these groups (h, h') is that the four elements therein are 

 affected with different exponents, which are essential to the groups, and cannot dis- 

 appear from them or from their eqviivalents. 



Thus (h) and (h') give the two groups 



12345678 12345678 



21436587 21436587 



43127865 43128756 



34218756 34217865 



65871234 65781243 



56782143 56872134 



7865.3421 87653412 



87564312 78564321 



which are equivalent to the two gi'oups at page 5 above quoted. 



Thm toe have, by a direct tactical ^jrocess, found the 12 regidar square roots of the 

 substitution 21436587. 



What has just been done is a case of a more general theorem. 



If we define that 



l2«-i = l, 22"-' =2, 32«-i=3, 42<'-i=4, 



and that the addition of the same nimiber to all the exponents of the elements of a 

 substitution makes no change in the substitution, we find that the two following 

 are true gi-oups, by the usual test, that the product of any two substitutions of the 

 group is a substitution of the gi'oup 



H. W. 



1234 1234 



• 2''143„ 2«14''3 



384012 3"412" 



43a2i« 4''3"21 



c. g. 3" 4"12 . 2°143"=42°-> 3" 21"=43" 21", 



3°4«12.43"21"=21"4"3^«-'=21''4"3=2"143". 



We may substitute in either H or H' for each one v of the four elements a group 

 W of 2a— 2 powers made with 2a— 2 elements, and for «« the residt of a— 1 cyclical 

 pennutations of the vertical rows of u. The constracted is always a grouped group, 

 which is no group of powers, nor equivalent to that built on the auxiliarj' with 

 a= 1, if we take a such that it shall not be prime to 2a — 2, that is, if we take for 

 a any even value. 



When a is odd, H and H' are still groups, and proper auxiliaries ; but I believe 

 that the gTouped group consti'ucted will always be equivalent to the one formed by 

 taking a=l, that is, the two gToups will differ neither in the number nor in the 

 orders of the circidar factors of their substitutions. 



There is but one square root of imity in the gTOup given by H when a =4, of 

 which the grmip contains twelve 6th roots of the 12th order, with all their j)otvers. 



It is perfectly easy to write out by a direct tactical method the eighteen cube 

 roots, and the eighteen 6th roots of the substitution (^ = 231564897, all of the 9th 

 order, considered in page 6 of last year's Report. 



