Mr. James Cockle on a new Imaginary in Algebra. 437 



pp. 364—367 of the current (49th) volume of the Mechanics' 

 Magazine ; I shall rather go on with the development of the 

 properties of expressions which involve it. 



For convenience, I shall denote the product of i and j by k; 

 I shall assume that,/ 2 is equal to unity. I say assume because, 

 although I have reasons (not, however, free from objection,) 

 for making such a supposition, yet I wish to reserve to myself 

 full liberty to modify the assumption, and to discuss ab initio 

 the symbol/ 4 and its symbolic relation toj. Then the follow- 

 ing system of relations will, together with the second equation 

 given in this letter, furnish us with all the conditions requi- 

 site for the formation of a theory of these three imaginaries. 

 That system is — 



{/=£> j^-h ld——j. 



Almost the first thing that strikes us, in examining the ex- 

 pression for a tessarine, is, that, under the above system, the 

 product of two or more tessarines is a tessarine. Let us 

 confine our attention to the case in which two only are mul- 

 tiplied together, and let the factors be t and t\ where 



t="w + ix+jy + kz t 



and 



t' = «/ + *V +jy' + Icz 1 . 

 iuvh no i 

 By actual multiplication, we obtain 



tt' = ww' -f « S W +J 2 j/y 4- k*zz' "J 

 + ifpud + nix/) +jk(yz l + zy' ) 

 +i(wy +^tt/) + ik{xz' + zaf) 

 + k(wz' -f zw') + y(xy' +yx') . 

 Suppose that 



ww' — xx 1 +yy* — zz f = «/',"*] 



wx' + x'w'+yz' + zy'=x", I 



ivy' -i-y > w'—xz , —zx , =y", j 



loz' + zw 1 + xy' +yx' == z", J 

 and also that 



id" + ix" +jy" + kz" = t" ; 



then we have, under the foregoing system of relations, 



tt' = t"; 



and, if we had three or more such factors as t, the result would 

 still be a tessarine. 



The next point that arises is, — what is the tessarine law of 

 moduli ? or, more generally, what modular or quasi-modular 



