274 Sir Wirr1aAm Rowan Hamitton’s Researches respecting Quaternions. 
provided that we make 
Po = pins 04 F aulay cw Ses (400) 
where 7’ is any whole number, and 7 is, as usual, the least positive root of the 
numerical equation, 
a sing = 0. 
Hence, the sought émponential of the quaternion ¢/ is 
Pld =p tp +0 (0 + 2n'z) ;5x (401) 
and, in like manner, by suppressing the accents, the imponential of q is found 
to be 
epl¢g=Plu+i(0 + 2nz), (402) 
where @ may be said to be the AMPLITUDE, and p is what we have already called the 
MODULUs of q. 
41. We may also say that cis the imaginary unit, or perhaps, more expressively, 
that it is the vecror uNIT, of the same quaternion g. For in the applications of this 
theory to geometrical questions, this imaginary or vector unit « may be regarded 
as having in general a given direction in space when q is a given quaternion ; 
and if we denote its direction cosines by a, B, y, so that 
ve < a 2 2 9 
a=-, Jove Y=> e+pP+ty=l, (403) 
we may write, generally, by (345), 
cai+jpp8+hy, ?=—1. (404) 
This power of representing ani DIRECTION IN TRIDIMENSIONAL SPACE, by one 
of the quaternion forms of /( —1), is one of the chief peculiarities of the pre- 
sent theory; and will be found to be one of the chief causes of its power, when 
employed as an instrument in researches of a geometrical kind. If a, p, y be con- 
ceived to be the three rectangular coordinates of a point R upon a spheric surface, 
with radius unity, described about the origin of coordinates as centre, we may 
also write, more concisely and, at the same time, not less expressively, 
iSa5 ie — (405) 
A numeral quaternion g may therefore, in general, be thus expressed : 
q = “(cos 6+7,sin 0) ; (406) 
where 
paV( Heb y +2) siete hy=Vv(-1). (407) 
