ELEMENTARY PRINCIPLES OF QUATERNIONS. 195 
§ 5. Generalisation of the Rule of Multiplication according to the Distributive Law, applied to 
Vectors and to Quaternions. 
We have to show that any expressions for p and a, formed by vector-addition, 
may be treated by the distributive law, when multiplied one by the other. 
We consider the result above 
po = — (av + by + cz) 
+ i(bz — cy) + j(ca — az) + k(ay — bz). 
We replace a, b,c, 2, y, z respectively by a’+ a”, 0° +6", (+ 0", a+ x", 
y+y’, = +2’ in the second member, and we effect the multiplication of 
(a + a”) (a + v0”), &e., &e., 
by the distributive law applied to scalar multiplication. 
If we designate by 
Pe. 5 
the vectors 
ia’ + 9b! + ke’, ia’ + jy’ + ke’, 
and two other similar expressions with two dashes (”), then the result of the 
substitution of a + a”, &c., in the expression of pa, will be composed of the 
aggregate of terms which represent respectively the products 
PG, pm, p @, Pe 
But in reality they are not in that order; the vectors, as for example, are in 
three groups or components parallel respectively to 7, 7, & But neither 
addition, nor even vector addition, is changed in its result if we change the 
relative order of the terms to be added. We therefore add together, first the 
scalars belonging to pw, and then the vectors belonging to it; then we group 
the terms belonging to p’w”, and so on. 
In the first member we have evidently also, for similar reasons, 
p=i(a +a) +5 + 0’) + he + 0% 
= (ia + 9b + ke’) + (th" + 7b" + ke") 
TO Oe ke 
and equally 
o-ort+a. 
Therefore we arrive at the result 
(+p )lat+toa)=patpu +p'at+po, 
VOL. XXVII. PART I. 3 E 
