Mr. J. Cockle on a new Imaginary in Algebra. 39 



ordinary (unaccented) zero which represents a certain state of 

 quantity*. 

 Let 



0'=l + <;-; (1.) 



multiply both sides of this equation by 1 — V7, and we have 



0'x(l->/;')=(l + y>-)(l-v'i); . . . (2.) 

 but 



l--/;'=2-(l + Vi) = 2-0'; 

 hence, substituting on the left-hand side of (2.), and multiply- 

 ing together the factors which compose its right, we obtain 



0'x(2-0') = l-J. (3.)t 



In (1.) substitute unity for J; the result is 

 0'=14-1; 

 hence J is not equal to unity, and consequently 1 —j is not 

 equal to zero. Acting upon this, let us proceed to obtain, 

 from the symbols of ordinary algebra, the most general ex- 

 pression for a quantity different from zero. We may attain 

 our end as follows. Let a, w, and oe be any real quantities 

 whatever, positive, negative or zero, and let /= ^ — 1 ; also 

 suppose that 



W = MJ-a+( )( ), 



yw — a/ \x—a} 



and 



V /a—a\/a—a\ 



X = ^-a+(, )( I; 



Xw—a/ \x—a' 



then 



W + /X 

 is the most general expression of ordinary algebra for a quan- 

 tity different from zero, and may be made to take any given 

 value whatever, excepting zero. Now, as we have seen, 



1 —j= some quantity other than zero ,- 



hence, those who would maintain the adequacy of ihe ordinary 

 notation to express any relation whatever, possible or impos- 

 sible, must sustain the equation 



l-j=W + iX; 



* Peacock (Third Report of British Association), pp. 232,233, &c., and 

 also p. 268. The accented zero is discontinuous; and a remark of Pro- 

 fessor J. R. Young, on impossible equations, in the Mechanics' Magazine 

 (vol. xlix. p. 463) suggests the characteristic that, to 0', the right-hand 

 side of (1.) can make no approximation in terms of the ordinary symbols of 

 algebra. 



t The product \/J x Vj has the double form J and v^'^ ; I here take 



the former value, on the principle that ^—1 x v^— I is equal to —1. 



