234 Sir Witt1am Rowan Hamitton’s Researches respecting Quaternions. 
1] 
as in the general form (155); and may regard the one mwmeral couple (m;',m;') as 
the symbolical product of the other two. If we simplify the formule by assuming 
the five constant coefficients of derivation which still remain disposable, namely, 
a, ¢, c’, d, d’, as follows: 
SFG, 6S 15 d= Gd =o (179) 
we shall then have 
Xo (ae a,) = (a, a,) ) Xi (a a,) = (—a, a, > (180) 
or more concisely, 
DG Ss NSS SSR A (181) 
this last symbol being here the same characteristic of derivation of an ordinal 
couple which was considered in former articles of this paper. And the equation 
for the multiplication of two numeral couples will then reduce itself to the follow- 
ing form : 
(mM M;) (MM m,) = (m5 mM, — MM, MM, + mM), m,) (182) 
which agrees with that assigned in the earlier Essay. (See Vol. X VII. page 403.) 
With the same values of the coefficients of derivation, and consequently with the 
same values of the coefficients of multiplication likewise, we may write also, as 
in that Essay (compare the page just cited), a formula for the division of one 
numeral couple by another, namely : 
(anos m2 wl = Ce yS ae mi +m, mi mi’ Mo my erat \ (183) 
(Mm; m,) m+m mi, + my 
It is not necessary, and it would detain us too long from the main subject of 
this memoir, to consider here any other and less simple formule of the same sort, 
which may be obtained for the same case of couples, by any other systems of co- 
efficients of derivation and multiplication, which satisfy the same conditions of 
detachment, assigned in the present article. 
20. It may be instructive, however, to consider here the same case of couples, 
as an exemplification of some other general formule which have been already 
given in this Essay. Writing, for abridgment, 
api Os a, a as ae ae (184) 
and in like manner, 
a, a= GQ, ,5 ane a=—a bee (185) 
