294 Sir Witt1Am Rowan Hamirton’s Researches respecting Quaternions. 
** Division [, Submultiplication|.— By (55), 
$ncos 4+ sin 0(¢ cos p+) sin ¢ cosy +h sin # sin ~)} x 
$c cos 9—p- sin 0 (i cos 6 +/ sin ¢ cos~+h sin psiny)}=1. (57) 
«The reciprocal of a quaternion may be found by changing the modulus to its reci- 
procal, and then either changing the amplitude to its negative, or else the direction to its 
opposite; this latter change (of direction, rather than of amplitude,) agreeing better than 
the former with the construction in fig. 1. Accordingly, in that figure or in this, in 
which r represents the direction of multiplier, and may be called the multiplier-point, r’ 
multiplicand point, and r” product point, if we prolong re’ and rr” till they meet in rR’, the 
point diametrically opposite tor; then, in the triangle rR‘ Rr” nr’, the point rR’, with ampli- 
tude 9’, will be equal to the product of x‘ as multiplier, with amplitude 0, and r” as multi- 
plicand, with amplitude 6”, by the theorems already established. We may, therefore, 
return from product to multiplicand, by multiplying by reciprocal of multiplier. But it is 
natural to call this return division [submultiplication]. To divide [or rather to sub- 
multiply] is, therefore, to multiply by the reciprocal of the proposed divisor, if this 
reciprocal be determined by the rule assigned above. These definitions and theorems 
respecting division of quaternions lead us to put the equation (4) under the form 
w —in—jJy—khz 
1 +ia!+7y' +h2=... = — 
uu! + ta! + 7y! + wee Tye 
(wl + ta" + jy" + hz”); (58) 
and so conduct us not only to the relation ew’ =(w?+a*>+y*?+2°)"(ww! + vx" + yy” +22”), 
which we had already, but also to these others, which can likewise be deduced easily from 
the equations of multiplication, (5) and (6), 
Va(wt+et+y+2) (wa! —aw" + zy” —y2") | 
y= (we +a? +y? +27)! (wy! — yw"! + a2" —z2"); 
(59) 
a= (we tart yet 2?) (we — zw" + ya" —a2y"). | 
The modulus of the quotient is the quotient of the moduli. 
ju’ COS 0” tau sin 0” (i cos p+) sin cos p +h sin gsin p) j 
nw cosO + sin# (écos@+/sin p cos~+hsin ¢ sin p) (60) 
= - cos (6"—0) +4 sin(6”— 6) (icosg+jsin ¢ cos P+ sin @ sin yp). | 
Ml lu 
*¢ Codirectional quaternions may be divided by each other, by division of moduli and 
subtraction of amplitudes; and diametrically opposite quaternions may be treated as codi- 
rectional, by changing an amplitude to its negative. A quaternion divided by itself gives 
unity, under the form 1 +720 +70 +20. 
‘© Raising to any real Power.—The transformation (54) of the v* power of a quater- 
nion is now seen to hold good, if the exponent vy be any real quantity. 
