408 Mr. Cockle on the Symbols of Algebra^ 



algebraic in its origin, nor is it to be considered as other than 

 algebraic in its laws of combination with other symbols ; and 

 if in first approaching the subject of impossible quantity some 

 mysteries present themselves and some difficulties arise, it 

 may be worth considering whether our estimate of the range 

 of ordinary algebra has not been too limited, and whether its 

 own inherent powers, when sufficiently developed, may not 

 explain the mystery and clear away the difficulty. With these 

 views it will not excite surprise if I state here, that I regard 

 paragraph 8 of the Rev. Charles Graves's paper on Triple 

 Algebra {supra, pp. 123, 124) as a valuable contribution to 

 ordinary algebra. 



I take the opportunity of adding one or two remarks on 

 tessarines, premising that I shall use i' f and ^ to denote the 

 imaginaries which enter into those expressions. 



1. The product of two tessarines of the form 



Yi'x -\-j'y + k'z 



is of the same form, and the moduli and amplitudes of the fac- 

 tors and product are related in the same manner, and the latter 

 may be constructed as readily, as if the factors and the product 

 were quaternions. 



2. The product of the two tessarines 



2'^i +/j/i + A'zi, and i'x^-^j'yc^+Wzcif 

 will be of the form 



provided that 



But, if the two systems of values ^-j, yj, ^j, and oe^^y^, z^, re- 

 spectively satisfy the condition 



f-a:^-z^=0, (w.) 



and if, moreover, 



■^1 "^a 



then (w.) may be satisfied*. But (w.) represents a right- 



• If we multiply the two equations that result from substituting the two 

 systems of values of .r i/ and z respectively, we shall have after reduction, &c. 



which suggests matter for future observation. With reference to the sub- 

 ject of Imaginary Geometry {supj-a, p. 132-135), and indeed of analytical 

 geometry in general, I may remark that I propose to call the real primary 

 axis Cthat of jr) the axe, the unreal secondary axis (that of 5^) the^erp^, and 

 the impossible tertiary axis (that of 2) the norme. Thus, in the equation 



