40 Mr. J. Cockle on a 7ie'w Imaginary in Algebra. 



but this equation cannot be sustained; for, if we deduce from 

 it the value of^, and substitute that value in (1.), we arrive at 



0'=l+-v/l-W-zX, 

 in place of which we may write 



0' = l+a+f/3; 

 this last equation gives /3 = 0, whence we infer that X = 0; 

 consequently, a being real, and the positive square root being 

 taken, we arrive at 



0'= 1 +«= unity together with a positive number', 

 which cannot be. Hence j is not of the form W + fX, and 

 yet it is different from zero, as will be seen on substituting 

 zero for J in (1.). In short, J is a i\\XQ.x\\.\\.y sui generis — an 

 impossible quantity — a quantity the very conception of the 

 existence of which involves the equation (3.). And, of this last- 

 mentioned equation, it may be added, that it is to be regarded 

 as one of the principal symbolic decompositions which the 

 theory ofj involves, and is not to be confounded with the ap- 

 parently similar equation 



Ox('^— 0) = W+2X. 



Thus there are impossibilities not capable of being expressed 

 either by zero, or by quantities of the form W -f /X, unlimited 

 as are the values which may be given to W and X. And, 

 although infinity be among such values, it becomes necessary 

 to have recourse to the new symbol j to indicate the impossi- 

 bility implied in (1.). And^" would be useful, were it only to 

 indicate such impossibilities. 



2. Of the Value of its Square. 



We are now arrived at a topic, the discussion of which will, 

 perhaps, assist in showing that the apparent difficulties, con- 

 nected with the theory ofj, are not such as to justify us in re- 

 jecting it as unsuited to the purposes of analysis, or as inca- 

 pable of becoming the symbol of a calculus. In the fact, that 

 j has a real square, we have something like a key to the me- 

 thod of rendering available this anomalous symbol. Starting 

 from our fundamental equation, slightly changed in its form, 

 we have 



+ -/;■= -1; 



on cubing both sides of this equation, we obtain 

 whence* we infer that 



* The present case is distinguishable from that alhuled to in the last 

 note. 



