198 G. PLARR ON THE ESTABLISHMENT OF THE 
§ 6. Products of Three or more Factors. Associative Property in Multiplication. 
By definition, (17), the product of three factors, vectors or quaternions, sup- 
pose p, g, 7, to be quaternions, is to be made so as to multiply the first multi- 
plicand 7, by its next multiplier g, so as to form the product gr; and 
taking this again for a multiplicand, multiply it by py. The product will then 
bes 
p x (gr) = par. 
The question arises, if the product is the same when one multiplies 7 by the 
product pq? or, in formula, is 
pq x 7, the same as p x qr? 
The affirmative to this question constitutes the associative property in multi- 
plication ; namely, let be three quaternions— 
P= O04 6;¢=]60452 7 =] 02 a7 
We get by the rule of multiplication of two quaternions— 
Sqr = be + SBy 
Var = by + Be + VBy. 
Then 
pxq =(a+ta)(Sqr + Var) 
can again be developed by the rule of multiplication of two ordinary quater- 
nions. 
If we develope the product we get ten terms, which, being grouped con- 
veniently, reduce themselves to the eight terms which one would obtain in 
applying the distributive rule to the expression 
(a+a)(b+ B)(c+y), 
and observing the rule of preserving the vector factors a, 8, y, in their order in 
the partial products, namely, a in the place to the left, then 8, and, thirdly, y in 
the place to the right. 
Seven of these terms will contain not more than two vector factors; only one 
contains three, namely, the product 
a x (By). 
If we effect the product 
(Spq + Vpq) (¢ + y), 
we will find again eight terms, seven of which will be at once declared identical — 
with the seven corresponding ones of the product 
(a + a) (Sqr + Var), 

