Sir Witt1am Rowan Hamirton’s Researches respecting Quaternions. 277 
ture already established in this paper, and may say that, in the general expression 
ga=wt+iatyy + kz, the three coefficients, x, y, z, which multiply respectively 
the three coordinate characteristics, 7, j, k, are the three coorDINATES of the qua- 
ternion, and that the square root 7 of the sum of their squares is the Rapius 
of the same quaternion. We shall also say that the point r, on the surface of 
the unit sphere, which constructs or represents the direction of the vector unit in 
its expression, is at once the REPRESENTATIVE POINT of that vector unit, 7, and 
also (in a similar sense) the representative point of the quaternion q itself. 
On the general Logarithms of a Set, and especially on those of a Quaternion. 
43. Though we cannot enter here at any length into the theory of dogarithms 
of sets, yet it is obvious that if we make 
y=) (418) 
the general expression (392) for a power of a set gives this inverse expression for 
the exponent q’ : 
Sof =F = —-3 (419) 
in which expression, however, for a logarithm ofa set, under the form of a frac- 
tion, the numerator and the denominator are to be regarded as separately subject 
to that indeterminateness, whatever it may be, which arises in the return from 
the exponential of a set to the set itself, or in the passage from a set g to its impo- 
nential p~'g. ‘Thus in the case of quaternions, the general logarithm of the 
quaternion q’’, to the base q, may, by (419) and (408), be written thus : 
pene ae + te (e” a aa) (4 
og uw +2, (0 + 2n7) 
It involves, therefore, ¢vo arbitrary and independent whole numbers, n' and n, 
in its expression, as happens in the theories of John T. Graves, Esq., Professor 
Ohm, and others, respecting the general logarithms of ordinary imaginary quantities 
20) 
to ordinary imaginary bases; and also in that theory of the general logarithms of 
numeral couples, with other numeral couples for their bases, which was published 
by the present author (as part of the Essay already several times cited, on Conju- 
gate Functions and Algebraic Couples, and on Algebra as the Science of Pure 
Time), in the seventeenth volume of the Transactions of this Academy. 
