545 
1907-8.] Algebra after Hamilton, or Multenions, § 7. 
present case the second order multenions oq, co 2 , . . . . are all commutative 
with each other and with scalars ; and since, therefore, among themselves 
and scalars they obey all the laws of ordinary algebra, we have here 
p = or 
C Pp = e“pe-“ \ 
where w = J(#qq~ l •+ 0'qt 3 _1 + . . . . ) J 
( 68 ) 
This, without doubt, is to be regarded as the standard form of the 
fictor-into-fictor rigid replacement. It seems to me a rather remarkable 
generalisation of a well-known quaternion result. 
The following summarise the connections between rotational Unities (<p) 
and skewlinities (y). 
</> = X + \/( 1 +X 2 ) = eW ( w Vs -1 + • • • •) I / 69 ) 
X = i (</> ~ </> _1 ) = Si©( ), o)' = (sin O.L^-p 1 + sin 0\i 4 i 3 -1 +....)! 
(<p 2 - 2<p cos 0+ 2<£ cos 6' + 1) . . . . = 0 . . . (70) 
with an additional factor <£-+-! when n is odd. [Proved by finding 
^<iW[-l]) from (65).] 
(y 2 + sin 2 0)(x 2 + sin 2 6') ....= 0 . . . .(71) 
with an additional factor y when n is odd. 
When we are given <p alone, we may permit ourselves to interchange q 
and q. Thus the range of d need only be from 0 to ir ; that is, the range 
of each angle Jd, Jd', .... of (68) need only be from 0 to \ir. It is best to 
suppose this permissible convention in connection with (68), for thereby cd 
and (p are each uniquely given by the other. [It is best to ignore the 
limiting cases, corresponding to the semi-revolution of a rigid body, which 
may be supposed effected either in the one direction or the opposite.] 
It may be noticed that since for any second order multenion Kw — — co, 
e~ (a = Ke w . More generally, when q is such that K q= —q and p = e q , then 
p~ l — Kp, 
though in general pKp is not a scalar. 
e^ve w can be showrn, directly, to be of the same order as v by putting 
e w = Lt (1 +n~ 1 co) n ; and thus e“( )e~ M may be shown to be commutative 
7l = oo 
with S 0 . 
(68) suggests that any second order multenion may be transformed by 
p( )p~ 1 to the form there given, though, of course, d, d' must now be un- 
restricted. This is the case. [To convert aqq + 6qq + cqq = q _1 if 1 if 1 
(aq + 6q + cq) to qi 2 J(a 2 + b 2 + c 2 ), remember that q _1 q _1 q _1 behaves like a 
scalar with reference to q, q, q, and therefore put 
p~ l = (aq + bt 2 + cq)q _1 4- J(a 2 + h 2 + c 2 ). 
VOL. XXVIII. 
35 
