Sir Wirt1aM Rowan Haminton’s Fesearches respecting Quaternions. 259 
2, j, k, and to the linear independence, already supposed to exist, between the 
four symbols @, j, /, and 1, we find that the one quaternion formula (308) 
resolves itself into the fowr equations, (303) and (304). And either from the 
four equations thus obtained, or by an application of the law of the modulus to 
the quaternion equation (308), the relation (305) may be obtained. It is worth 
while observing that we may also write the quaternion formula, 
ey.) 
(w + ix + jy + he) (wl + ie” + jy" + kz") (w — iv — jy — kz); (309) 
or, more fully, 
(wp+rty+es) (w,—w"’ +ia7,+jy,+kz,) = 
(w? — 2 —y? — 2°) (ia + yy" + kz’) 
+ 2(00" + yy"! + 22") (tx + jy + kz) 
+ 2Qw fi(y2"— zy") + j(2n" — x2") + k(ry"— ya")}; (310) 
by resolving which one formula, the same separate values for w,, .7,,, y,,. Z,, may 
ut 
be obtained, as from the system of the fowr ordinary equations (302), (303). 
On the Operation of pre-multiplying one numeral Set by another, and on frac- 
tional Symbols for Sets. 
34. Since we have seen that we are not at liberty to assume generally, for adi 
numeral sets, that the commutative formula of multiplication holds good, we 
must (in general) distinguish between tivo modes of combination of two such sets 
with each other, as factors, in some such way as the following. We saw reason, 
in the twenty-second article, to regard an ordinal set, q, as having been generated 
by a certain symbolical multiplication, or complex derivation, from a single stan- 
dard: ordinal relation, a, as from an original operand or derivand ; the operator, 
or symbolical multiplier, having been a numeral set, g. If such an ordinal set, 
q, or g Xa, be again operated on by the new numeral set, q’, as by a new sym- 
bolical multiplier, the result will be a new ordinal set, g’ x (gq X a), which, in 
this theory, admits of being denoted also by (4 X 7) X a; and generally, in the 
same theory, the conditions of detachment entitle us to write the formula 
TXx*=F%x)x4, (311) 
whatever operand set (of the same order) may here be denoted by the symbol q’. 
