Sir Witt1am Rowan Haminton’s Researches respecting Quaternions. 263 
It is evident from what has been said, that the reciprocal of the reciprocal of a 
numeral set is equal to that set etse/f; and that to divide by such a set is to pre- 
multiply by (or to multiply into) its reciprocal ; thus, generally, 
/ 1 / - / « 
LA Re a ear MSH (320) 
The reciprocal of a quaternion is given by the formula, 
(wi +yy +kz)t = (we +y +2)" (w—ir—jy—hkz). (321) 
In general, the reciprocal of the product of any number of sets is equal to the 
product of the reciprocals of those sets, arranged in the contrary order: thus we 
may write, 
(658929190) =o MN Io ess (322) 
On Powers of a Numeral Set, with whole or fractional Exponents; Square and 
Square Root of a Quaternion ; Indeterminate Expressions, by Quaternions, 
for the Square Roots of Negative Numbers. 
36. The symbol q~', for the reciprocal of a numeral set, is only one of a 
system of symbols of the same sort, which may easily be formed by an adaptation 
of received algebraic notation. For with the notions given already, respecting 
multiplication and division of sets, there is no difficulty in interpreting now, in 
an extended sense, adapted to the present theory, the following usual system of 
equations, 
T=1L 9 =H F=IXF, F=ITXF:-. 
Tee eel 1 (323) 
a eS ne 
q q q q LT ie q q 
and then the well-known equation of the exponential law, 
EXO Mig =O (324) 
will hold good, as in ordinary algebra, the exponents r and s being here supposed 
to denote any two positive or negative whole numbers, or zero. 
These two other usual equations, 
qGyY=e, ()=4, (325) 
will then also hold good for numeral sets, at least when r, s, ¢, and ‘, denote 
whole numbers ; and the latter. of these two formule may be employed as a 
definition to interpret the symbol q:, when the exponent is a numerical fraction ; 
VOL. XXI. 2N 
