Sir Wit11Am Rowan Hamuvron’s Researches respecting Quaternions. 273 
The simplicity of this equation may now induce us to extend it, as we propose to 
do, by definition, to the cases where the exponent of the power, instead of being 
a numerical fraction, is an incommensurable number, or even a numeral set. We 
shall, therefore, write generally 
qg” = P(g Pg); (392) 
and thus we shall have a general expression for any power of a numeral set, 
through the help of the characteristics of the exponential and imponential thereof. 
Application to Quaternions ; Amplitude and Vector Unit ; Coordinates, 
Radius, and Representative Point. 
40. On applying these general principles to the case of a quaternion, we have 
first, by (387), 
pg = P(w + ia + jy + kz) = pw.r(ia + jy + kz); (393) 
and then, if we use the notations (345), and attend to the connexion already 
established between quaternions and couples, we find that 
P(ix +yy + kz) = P(r) = cosr +esmr; C= —1; (394) 
where cosr and sin7 denote, as usual, the cosine and sine of 7, so that, in the 
theory of couples, the following equation holds good : 
p(0, r) = (cos7, sinr). (395) 
(Compare the earlier Essay, where the functional sign F was used instead of p). 
Thus the exponential of a quaternion q is expressed generally, with these nota- 
tions, by the formula, 
Pq = Pw.(cosr + csinr). (396) 
Reciprocally the imponential p™ q’, of any other quaternion, gq’, is to be found by 
comparing this formula (396) with the expression of that quaternion g’, when put 
under the form, 
gr=w+lr=p (cof +csin€@), (397) 
where 
w= (w?+r”’), tan = — (398) 
We find, in this manner, that we may suppose 
g=Pg garg, (399) 
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