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



relation or relations exist between the constituents of the fac- 

 tors and those of the product? Seeing that the operations of 

 algebra necessarily and inevitably conduct us to tessarine for- 

 mulae, I cannot doubt but that some relations may be found, 

 such as to enable us to extend the boundaries of analysis. 

 I have not, however, time to pursue this inquiry further now 

 — more particularly as I wish to make one or two remarks 

 upon other parts of the subject of this new imaginary j. 



For instance, I would observe that the product of two or 

 more factors of the form 



is of the same form ; thus 



(w +jy)(wl +jj/) = «/' +jy", 

 where 



w" ^-Wurf + yy 1 , 

 and 



i/'=.'wy'-{-yiio f . 



There are cases in which the impossible quantity j altogether 

 disappears from a result. Thus, if e be the base of the Na- 

 pierian logarithms, we have 



and 



hence 



and 



w 2 . fr 3 v 4 



l.( s *_ s -«) = ^ + JC+&C.. 



•)> 



This last is a very simple case ; but the nature of such a 

 letter as this obliges me to have recourse to illustration rather 

 than extensive investigation. 



I need not tell you how to read the second equation which 

 occurs in this communication; still, I cannot help saying — to 

 use words adopted on another occasion, by our friend Professor 

 J. R. Young (Alg. 4th edit. p. 131), — that we must read that 

 equation as meaning that unity together with the plus \_square-~] 

 root of/ is equal to zero, and not that unity together with a 

 square-root ofj is equal to zero. The whole discussion is in- 

 volved in the correct reading of that fundamental equation. 



By way of conclusion, I will add that, if we had three inde- 



