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[814 
814. 
ON DOUBLE ALGEBRA. 
[From the Proceedings of the London Mathematical Society, vol. xv. (1884), pp. 185—197. 
Read April 3, 1884.] 
1. I consider the Double Algebra formed with the extraordinary symbols, or 
“ extraordinaries ” x, y, which are such that 
x 2 = ax -f hy, 
xy = cx + dy, 
yx = ex+fy, 
y 2 =gx+hy, 
or, as these equations may also be written, 
X y 
(a, b) 
(c, d) 
(e, f) 
(?» h ) 
where a, b, c, d, e, f, g, h are ordinary symbols, or say coefficients; all coefficients being 
commutative and associative inter se and with the extraordinaries x, y. 
The system depends in the first instance on the eight parameters a, b, c, d, e, f, g, h; 
but we may, instead of the extraordinaries x, y, consider the new extraordinaries con 
nected therewith by the linear relations f = ax + /3y, y = yx + by, where the coefficients 
a, /3, y, 8 may be determined so as to establish between the eight parameters any 
four relations at pleasure (or, what is the same thing, a, ¡3, y, S are what I call 
“ apoclastic ” constants): and the number of parameters is thus properly 8—4, =4. 
2. The extraordinaries here considered are not in general associative; differing 
herein from the imaginaries of Peirce’s Memoir, “Linear Associative Algebra” (1870),