128 
[22 
22. 
ON ALGEBRAICAL COUPLES. 
[From the Philosophical Magazine, voi. xxvil. (1845), pp. 38—40.] 
It is worth while, in connection with the theory of quaternions and the researches 
of Mr Graves {Phil. Mag. [Yol. xxvi. (1845), pp. 315—320]), to investigate the pro 
perties of a couple ix+jy in which i, j are symbols such that 
i 2 = ai + Gj, 
ij = a'i + G'j, 
ji = 7 i+ Bj, 
f = ry 'i+B'j. 
ix +jy ix x +jy x = iX +jY, 
X = a xx x + a!xy x + 7 x x y y'yy x , 
Y = G'xx x + G'xy x + B'x x y + B'yy x . 
Imagine the constants a, G... so determined that ix +jy may have a modulus of 
the form K (x + \y) (x + py) ; there results one of the four following essentially inde 
pendent systems 
A. # = ^ ( 8 V + 7 A + ft) i ~ ~j, 
V = P + Bj, 
j 2 = — \pBi + (7 + A + pB)j, 
IX + \Y = ^ (7 -1- AS) (x + Ay) (x x + Ay x ), 
X + pY = - (7 + pB) (x + py) (x + py x ). 
\ p 
If- 
then