11 
direction is perpendicular to both of their's, being obtained 
from that of (3., by making it revolve right-handedly through 
a right angle round a as an axis. These definitions, which 
are compatible with the formule (A) (B) (C), and may serve 
to replace them, will be found sufficient to prove generally, 
and perhaps with somewhat greater geometrical clearness than 
those formulz, the distributive and associative properties of 
quaternion multiplication, which have been already stated to 
exist. They give easily the following corollaries, which are 
of very frequent use in this calculus : 
af + Ba = 28, = — 2 AB cos(A, B) ; (a) 
a — Ba = 2aB, = 2yAB sin (A, B); (b) 
A and B denoting here the dengths of the lines a and B, and 
(A, B) the angle between them ; while y is a vector-unit per- 
pendicular to their plane, and such that a right-handed rota- 
tion, equal to the angle (A, B), performed round y, would 
bring the direction of a to coincide with that of B. For 
example, when B = a, then B = A, (A,B) =0, and 
af'= Pa=a = ~ A’, 
so that the length A of any vector a, in this theory, may be 
expressed under the form 
A=V—2a. (c) 
More generally we have the equation 
af — Ba = 0, (d) 
when the lines a and f are coincident or opposite in direction ; 
while, on the contrary, the condition for their being at right 
angles to each other is expressed by the formula 
ap + Ba = 0. (e) 
These simple principles suffice to give, in a new way, 
algebraical solutions of many geometrical problems, of various 
degrees of difficulty and importance. Thus, if it be required, 
as an easy instance, to determine the length of the resultant 
