Sir Witt1AM Rowan Hamirton’s Researches respecting Quaternions. 231 
of separation R,, we must endeavour to satisfy all the numerical conditions which 
are included in the form, 
OSS ot ot Cs et Gr terra t) = (160) 
The number of these conditions of detachment (160) is n‘, because each of the 
four indices, 7, 7’, s’, t, may receive any one of the 7 values 0, 1,...2—1; and 
they involve only 2° numerical coefficients, or rather their ratios, which are 
fewer by one, to be determined; from which it may at first sight seem to be 
impossible to satisfy all these conditions of detachment, except by making all the 
coefficients of derivation vanish. Yet we shall see that when m = 2, namely, for 
the case of nwmeral couples, the conditions admit of an indeterminate form of 
solution: and for the case m = 4, it will be shown that they can also be satisfied 
by that system of coefficients on which is founded our theory of nwmeral quater- 
nions, and even by a system of coefficients somewhat more general. A more 
complete discussion of the important formula (160) will not be needed for the 
purposes of the present Essay. 
Case of Couples. 
19. If we suppose 2 = 2, then the index s, with respect to which the summa- 
tion is to be performed, can be only 0 or 1; the formula (160) becomes, there- 
fore, in this case, 
nN, +1, 0 Co, at + nN, rl Ci, «, t = C,,, ,0 C,, 0, t + C+, Bek Cc, 1,¢* (161) 
If we suppose also that the two simple or elementary derivations of one ordinal 
couple from another are denoted thus: 
Xo (A> a.) = (gos Ao,,) = (aa, a’a,, ba, + 5a); } 
Xa hosee = Ch Oy) (ca, + ¢’a,, da, + d’a,) 5 
we shall have, by (135), for the 2*= 8 coefficients of derivation of the form ¢ 
the abridged symbols : 
(162) 
7,8,0 
C = 7 ne SHO aS eS i's 
0, 0,0 0, 0,1 p 0, 1,0 0,1,1 f } (163) 
Cicoses, Crt s eyo es Cane: 
And if we employ in like manner these other temporary abridgments, for the 
eight coefficients of multiplication of one numeral couple by another, 
VOL. XXI. 21 
