200 G. PLARR ON THE ESTABLISHMENT OF THE 
The first members are the expressions entering into a x By, and the last 
members are those which enter into a8 x y. 
The identity of a x By and a@ x y is therefore established. Therefore also 
the identity of p x gr and of pq x 7. 
Applying the associative property to the product of four vector factors, 
aBy5, we liken a to p, B tog, and yd to 7. Then we have 
(a8) x (yd) = a x [B x (y8)]. 
So that if two quaternions af, yd are the result of the products of two vectors 
each, their product may be formed as if the product of the four vectors had to 
be made according to the general definition of multiplication, namely, we have 
(a8) x (v8) = 4 x [B x (70)] = «Bys. 
General Remark.—The product of any number of vector, or quaternion 
factors, may be indicated irrespectively of the grouping together, or the in- 
dicating of the intermediate products. 
As, for example, the product of n factors a,, a, in «,, may be looked upon 
as formed by two factors 
(aya, ome! Se Gy) x (On 41 En42 Ser Piis a.) ? 
for any value of / comprised in the enumerationh =1, 2,...n—1,n; the 
last value giving to the second factor the value = one. 
We may also remark the following theorem, founded on the associative 
property 
Kq Kp x pq = Kq x (Kp x p)q = (Tq) (Tpy’, 
because Kp x p isascalar = (Tp)’, and K(pq) x (pq) = (Tpq)’ = (Tp)? x (Tq). 
Therefore 
K(pq) = Kq Kp. 
§ 7. Division, and the Definition of a Quaternion by that Operation. 
We dejine, (18), division by likening the dividend A to the product of the 
_ divisor B, as multiplicand, by the quotient C, as multiplier, A, B, C being 
quaternions generally. 
This gives by definition | 
(b= A 
We multiply both members of the identity KB = KB respectively by CB 
and by A. This gives 
(CB) x KB = AKBe 

