ELEMENTARY PRINCIPLES OF QUATERNIONS. 197 
and as the second member is a scalar, we have to verify if 
e— yy? = a?b? — ab? — B’a’ +078". 
Now, we have by squaring— 
ab? + 2abSaB + S’?a8 
y= + ap’ +V'aB 
+ 2[bS.aVaB + aS.BVaB + abSaB]. 
But Va is perpendicular to both a and 8; therefore 
Savas = 0S. 6Vas = 0: 
‘Therefore c? —y’ must be Si to 
a*b? — ab? — a’ B’ + S*aB — V7aB ; 
and as we have found above 
Sia6—V-o6 = 1748 = (We) xis’ = a's’, 
it follows that the expressions of ¢ and y are respectively the scalar and the 
vector of the product of (a + a) (6 + #); and therefore the distributive law 
applied to the multiplication of two quaternions gives us their product. 
The typical form of a quaternion being 
gq = Tq (cos u + UG sin wu), 
where w is supposed to be between 0° and 180°, gives by multiplication, and 
: . : au ite . me 
comparison, for the irreducible, positive fraction — : 
g* = (Tq)* [cos v + UG sin v], 
where 
v =| (w + 360° N)—360°M ], 
N having to receive one of the 7 values 
=n—l1 i — 
x g pe ty oy Pde 

and M taking one value for each value of N, in order to reduce the value of v, 
so as to be comprised between — 180° and + 180°, 
The system of values N = 0, M = 0 in the case of 
oS eals0., 
may be noticed as being applied in spherical trigonometry. 
