GROUPS AND MANY-VALUED FUNCTIONS. 397 



I hope soon to have the honour of presenting to the 

 Literary and Philosophical Society of Manchester 

 a second Memoir^ iu which this Theory of Grouped 

 Groups will be discussed in detail. 



Here I shall merely reraarkj that it will often be found 

 necessary, in handling the theorem H, (37), to include in 

 the number tv of equivalent auxiliary groups certain of 

 those above described, which are not identical, at least as 

 operators, with groups of the ordinary form, in which the 

 same element can, by modification of vertical rows, be 

 made to shew the same exponent, (39), whether unity or 

 some other, wherever it appears. 



Note on Art. 39, page 321, o/" the Memoir on the 

 Theory of Groups. 



I have neglected to observe, that if the group {g') of page 

 319 be transformed, with the understanding that the square 

 groups of the fourth order, written at page 321, are to be 

 substituted for the elements, the transformed group {g") will 

 be 123, 23"i'*, 3^12*, 13^2^, 3^21^, 213, which differs from {g') 

 by the addition of 3 to the last exponent of each triplet, the 

 definitions being of course in this case i^^i, 2^ = 2, 3^ = 3. 

 The two last written triplets of page 321 should be 3^2 1^ and 

 aSjSg^ The reduction of exponents named in the fourth line of 

 page 322 proceeds by the addition of the same number to every 

 exponent in a vertical row of the auxiliary, the defijiitions being 

 i*+i= I, 2*+^= 2, &c., if ^ be the number of vertical rows in 

 the square groups to be substituted. 



T. P. K. 



