Sir Wizi1am Rowan Hamiuton’s Researches respecting Quaternions. 293 
“Tf @=7, so that we have to extract the n" root of the quaternion (48), when w= —n, 
we have still ¢ and yf left undetermined, but the formula is now 
(2 He LD ih a )r 
(—1+10 +j0+ hO)> = = "cos (tcos@ +) sing cosp +A sing siny). (50) 
For example, the square root of —1 may have any arbitrary direction, provided that it is 
a pure imaginary with modulus =1 ; 
(—1+i0+j0+20)' =i cosg tj sin g cosy +h sing sin y. (51) 
‘“* Exponent any positive quantity. The power is 
yrcos( “8 - 2pr +usin( 9 + pr ) (i cos @+/ sin ¢ cosp+h sin @ sin ), (52) 
pm Be P ae aie t 
it— be any positive fraction ; and it is natural to define that the power with incommen- 
surable exponent 
$cos 0+ sin 0 (é cos ¢+/sin p cos +h sin ¢ sin W) }” (53) 
is the limit of the power with exponent =, if v be limit of *; hence, generally, the power 
(53) is 
p’ cos (v8+ 2vpr) +p" sin (v8+2vpr) (tcos@+jsingcos~+hsin@psiny); (54) 
at least, if »y be positive. The reason for this last restriction is, that we have not yet con- 
sidered division, at least in the present letter, which I am aiming to make complete in 
itself, so far as it goes. 
“ Multiplication of codirectional Quaternions.— If, in fig. 1, we conceive R’ to approach 
to R, then, in general, r” will approach either to R or to the point diametrically opposite: 
and, in the first case, 0” will tend to become the sum of § and 6’; but, in the second case, 
the sum of theirsupplements. In each case we may treat 6” as = 0+ 6’, if we treat R” as 
coinciding with r, or ¢” and y as equal to @ and. Thus, generally, 
$u cos 6 +p sin ™ (i cosp¢+/sin ¢ cos+hksin o sin) } 
x $y’ cos 6+ p/sin 8 (i cos p+) sin ¢ cos P+hAsin @ sin L) } 
= pp! cos (0+ 6’) + py’ sin (0+ 6’) (¢ cos +7 sin @ cos W+hsin psin JL) ; (55) 
which accordingly agrees with the equations of multiplication (5) and (6), whatever p, py’, 
0, 8', p, and may be. (Indeed, if 6’+6=7, the position of r” is undetermined ; but this 
is indifferent, because its amplitude is now =7, and the product is a pure real negative.) 
For example, by making ¢=0, we fall back on the old and well-known theorem of ordi- 
nary imaginaries, that 
(4 cos 0+iu sin 8) (u/ cos 6 + iu’ sin 6’) = up! cos (8+ 6’) + tu’ sin (0+0). (56) 
