272 Sir Wittram Rowan Haminton’s Researches respecting Quaternions. 
pg, may be called by contrast the amponential function, and denoted by the cha- 
racteristic P~'; thus, we shall suppose p~'g to be such that 
Pye (382) 
or that, more fully, 
q=1+rP'q+s(e 9g) +3? 9) + &e. (383) 
Then, because the function P is such that 
py X Pg =P(y +9) fgg = 47 (384) 
and because, by the associative principle of multiplication, any two whole powers 
of the same numeral set, q, are commutative as factors, that is to say, may 
change their places with each other, without altering the value of the product ; 
we shall have, generally, 
Pf'(9) X PF (9) = PPD) +F(9))> (385) 
because we shall have 
SD) XP) =LD XP (4) (386) 
if the symbols f(q) and /’(q) denote here any combinations of whole powers of 
one common numeral set, g, and of any given numerical coefficients. For exam- 
ple, if a denote a number, we shall have 
pa X Pq = P(a+q). (387) 
We may also deduce, from the formula (385), this other important corollary, 
which is general for numeral sets, and in which the symbol p.sq represents the 
same function as p(sq), while s may, at first, be supposed to denote a whole 
number : 
(Pq)* = P(sq) = P.sq- (388) 
We have, therefore, for any two whole numbers, s and t, the relation 
(p.sq)' = (v.tq)’; (889) 
and, therefore, as an equation of which the second member is, at least, one of the 
values of the first, we have 
(P.sq — Pilg. (390) 
We are thus led to write, as an equation of the same sort, giving an expression 
for, at least, one value of any fractional power of a set, whenever the imponen- 
tial of that set can be discovered, 
y= p({ P19). (391) 
