292 Sir Witrtam Rowan Hammmton’s Researches respecting Quaternions. 
result respecting the product of two pure imaginaries, coincident in direction with each 
other. In general, the square of a quaternion may be obtained by squaring the modulus 
and doubling the amplitude ; that is, the square of 
cos A+ sin (7 cos ¢+/jsin ¢ cosy +h sin ¢ sin W), (43) 
may always be thus expressed : 
pw cos 264 pw? sin 20(7 cos ¢ +7 sin @ cos P+A sin ¢ sin W) ; (44) 
for instance, 
(tcos ¢+/sin ¢ cosf+h sin gsiny)?=—1; (45) 
although, when 0 >5 6 <7, it is supposed, in the construction, that we treat cos 20 as 
= cos (27—26); sin 20 cos ¢ as = sin (27—24) cos (w—¢); sin 20 sin @ cos ¥ as = sin 
(27—29) sin (w— ) cos (r+); and sin 20 sing sin ~ as = sin (27—26) sin (7—@) sin 
(7+); all which is evidently allowed. 
“Cubing a Quaternion.—The cube may always be found by cubing the modulus, and 
tripling the amplitude. 
** Raising to any whole Power.—The n* power of the quaternion (43) is the fol- 
lowing, if x be a positive whole number: 
pu” cos nO +n" sin nO(i cos p+) sin ¢ cosy +h sin ¢ sin w), (46) 
** Extracting a Root.—The n'* root has, in general, m, and only m, values, included 
under the form 
: 2 ee 
preos T°PF 4 sin =e —- 
(tcos P+j sin  cosy+hsin ¢ sind). (47) 
** Roots of Reals.—If 6=0, so that we have to extract the n‘* root of a positive real 
quantity, w, considered as the quaternion 
w +20 +70 +h0 =w, (48) 
» and y remain entirely undetermined, in the formula 
(1 +70 +70 +hO)* =u" cos PF 4. sin = (écosp+jsin pcospy+ksingsiny). (49) 
For example, unity, considered as 14+20+70+40, has not only itself as a cube root, but 
(The 
2 
also every possible quaternion which has its modulus =1, and its amplitude = = 
amplitude = corresponds merely to quaternions with directions opposite to those with the 
amplitude = and direction is here indifferent.) But unity has only two square roots, 
+=14704704+20. 
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