- 
Sir Witt1am Rowan Hamirton’s Researches respecting Quaternions. 275 
Its imponential, by (402), will then take the form 
Pp '¢=logp+i,(0 +2nz), (408) 
n denoting here any positive or negative whole number, or zero; and log 
denoting the (real and) natural or Napierian logarithm of the positive (or abso- 
lute) number «; or in other words, that determined (real) number, whether 
positive or negative or null, which satisfies the equation 
= P(log,). (409) 
42. Substituting this expression (408) for the imponential of a quaternion in 
the general expression (392) fora power of a set, we find, for a power of a quater- 
nion q, with another quaternion q’ as the exponent of that power, the expression, 
gy =Pri{7 lgu+7i,(6 + 2nz)t; (410) 
which, however, it is not generally allowed to resolve into the two factors, 
p(q’ log w) and p{q’t,(@ + 2n7)}, because q’ and q’7, are not, in general, condi- 
rectional quaternions ; if this latter name be given to quaternions which have 
vector units equal or opposite, so that in each case they are commutative with 
each other, as factors in multiplication. But if we change the exponent q’, in 
t 
(410), to any numerical fraction, ‘, where s and ¢ denote whole numbers, then 
this resolution into factors is allowed, and the formula becomes 
t 
: t t. 
= P} log + i,(0 + 2nn)f 
t (10 | 2ina 
= P(-l - P5Z2%( — + — 
(<log #4) iG + —\ 
; .. \ (40 , ina 
= “(cos + 2, sin) & + a) 2 (411) 
and thus it will be found that the chief results of the thirty-sixth and thirty- 
seventh articles, respecting certain powers and roots of a quaternion, are repro- 
duced under a simpler and more general aspect ; for instance, the square root of 
a quaternion is now given under the form 
q' = pw (cos +7, sin) G+ nn) SSGe in (cos 5 + 2, sin a) (412) 
But in the particular case where the original quaternion, q, reduces itself to a 
negative number, ¢ = w= — », so that its amplitude, @, is some odd multiple 
