436 Mr. James Cockle on a new Imaginary in Algebra. 



In what follows, I propose to confine the term unreal to the 

 imaginary quantities of ordinary algebra. When the word 

 impossible is used, it must be understood as referring exclu- 

 sively to expressions involving the new symbol. So that 

 imaginary quantities constitute a genus which includes two 

 species, viz. unreal quantities and impossible quantities. 



Let 



l+* 2 =0, 



then / is the simplest representation of unreal quantity : and, if 



i + v7=o, 



then j is the simplest representation of impossible quantity. 

 It is to be borne in mind that, in the latter equation, the ra- 

 dical is to be considered as essentially affected with the sign + . 

 Let iv, x, y, z be any real quantities, positive, negative, or 

 zero j also let 



w + ix+jy + ijz=t, 



then t I call a tessarine, and ?c, .r, y, z its constituents. The 

 latter term I have adopted from the quaternion theory of Sir 

 W. R. Hamilton. 



This being premised, I shall first proceed to prove, that, 

 when a tessarine vanishes, its constituents are simultaneously 

 zero. 



Suppose then that / = 0, and solve this last equation with 

 respect toj; the result is 



j = - (w + ix) +(y + iz) ; 



that is to say, y is capable of being linearly expressed in terms 

 of i and of real quantities, for we may readily reduce the above 

 expression for j to the form 



j=u + ip. 



But, as may be proved from the principles of my theory of 

 congeneric surd equations*, there is no linear relation between 

 i and j. Hence a and /3 must each be of the form 0-^-0 ; and 

 we see, in consequence, that when t = 0, we have, necessarily, 



w>=o, x=0, y=0 f and #=0. 



The symbol 7 is not capable of being identified with any of 

 the imaginaries of the quaternion theory. It is a deduction 

 from the principles of algebra rather than an invention. I shall 

 not repeat my argument in favour of its admission as a new 

 and independent symbol, because I think that I have put the 

 question of the propriety of such admission beyond dispute at 



* Mechanics' Magazine, vol. xlvii. pp. 151,307,331,409; vol. xlviii. 

 p. 181 ; vol. xlix. p. 364. 



