4<2 Mr. J. Cockle on a new Imaginary in Algebra. 



tity. That equation is to be considered, rather as an evidence 

 of the nature of the quantity which I am discussing, than as a 

 guide (or impediment) to us in its symbolic application ; and 

 although it merits further consideration, yet I do not feel 

 called upon to bestow that consideration here*, inasmuch as 

 in the theory of tessarines J is not affected with a radical sign, 

 and it consequently becomes unnecessary, for the purpose which 

 I have in view, to enter upon the subject of equations expressed 

 by means of radicals. But I am about to point out another 

 anomaly, the reverse of that which occurs in (3.) : it is that 

 on the supposition that J^ is equal to unity, 

 (H-i)(l-i) = 1-1=0; 

 that is to say, the (unaccented) zero may be considered as the 

 product of two impossible factors, neitherf of which vanishes. 

 It can, however, be at once shown that this anomaly cannot 

 lead us into error; for assuming the equation 



^ a^-b^=[a^jh){a-jb), .... (4.) 

 and bearing in mind that a tessarine cannot vanish unless all 

 its constituents are zero, we see that neither a+jb nor a—jb 

 can vanish, unless a~0 and b = 0. Suppose that a=0 = 6, 

 then the equation (4.) becomes an identical one, and no error 

 is introduced. On the other hand, imagine that a^— 6^ should 

 vanish from a becoming equal to b (both a and b being differ- 

 ent from zero), then the right-hand side of (4.) would become 



(a +»(«-»; 

 but, bearing in mind the fundamental property of tessarines, 

 we should be in no danger of inferring that one of these fac- 

 tors must be zero, and consequently we should introduce no 

 error into our investigations. It may be said, Is zero, then, 

 decomposable into non-vanishing factors? Impossible. I reply, 

 true, the factors are impossible : they are so by their origin 

 and nature. 



4. Of the Interpretation of' the Symbol in Geometry. 

 In this field, I am about to indicate what (I hope) will be my 



* If for no other purpose, the accent on the zero is useful for the pur- 

 pose of denoting an impossible equation. If the accented zero is different 

 from the arithmetical zero (and there are indications of a difference), what 

 is the square (0'^) ? a negation of a negation? I think that we must regard 

 0'" as identical with 0', when n is positive, and, consequently, 0'~" as iden- 

 tical with 0'~\ and then n is negative. Zero is not supposed to be in- 

 cluded in these values of n. 



t If one such factor be zero, the other must be infinite. But this is in- 

 consistent with the forms of the factors. It is on a consideration of this 

 kind that my theory of congeneric surd equations is based. (See Mecha- 

 nics' Magazine, vols, xlvii., xlviii. and xlix.) 



