4-6 Mr. J. Cockle on a new Imaginary in Algebra. 



its use would not be confined to the discussion of the functions 

 which I have proposed to call tessarines. It would, as it seems 

 to me, be capable of other applications, and would tend to 



generalize all processes to which the imaginary symbol \^ — \ 

 has been yet applied. Thus, by way of illustration, consider 

 the expression 



^w ±ix ±jy ikg 



If we vary, in every possible way, the order of the signs in 

 the above; add the different values, so obtained, together; and 

 expand the sum ; it will be seen that the result is free from 

 imaginaries. Hence, that sum may, in all cases, be used instead 

 of the resulting series. The finite expression for the series, so 

 obtained, would in some instances be found useful. 



It now only remains to make one remark respecting the 



notation which I have adopted. In taking i to represent V' — 1, 

 I think that I acted under an impression that it had been so 

 used anterior to the quaternion theory*. The use of^" and k 

 followed that of /, and seemed to offer an easier and better 

 mode of con^paring results with that theory than I should 

 otherwise have had. In order, however, in future to avoid 

 confusion, and the misapprehensions which may arise from 

 employing like symbols for unlike purposes, I shall use a, /3, 

 and y in place of f, j and k respectively. Under this notation, 

 a tessarine {t) will be written 



w + ax + ^y + yz, 

 where 



a2=-l, «^ = y, -^^ = 72, ya=-/3; 



and also, if the view taken in this paper be correct, 



(3^=1, and /3y = «. 



In my endeavours to bring space of three dimensions under 

 the dominion of a new species of ordinary algebra, I may per- 

 haps be permitted to disclaim anything like dogmatism. I 

 should wish all my views respecting the new symbol to be re- 

 garded in the light of suggestions. And if, regarded as sug- 

 gestions, they should have the effect of directing attention to 

 the theory of congeneric surd equations, they will not be with- 

 out their utility. 



2 Church-Yard Court, Temple, 

 November 23, 1848. 



Postscript, 7th December 1848. Perhaps I shall be per- 

 mitted to add the following few lines on the subject : — 



* Sir W. R. Hamilton has noticed this in the Phil. Mag. S. 3, vol. xxv. 



