10 
group of them may be collected or associated into one partial 
sum. 
Scalars are multiplied, as well as added, by the known 
rules of ordinary algebra, for the multiplication of real num- 
bers, positive or negative; because the positive unity of the 
system has been assumed to be itself a scalar, and not a vector 
unit. 
For the same reason, to multiply any vector by any scalar 
a, isin general to change its length in a known ratio, and to 
preserve or reverse its direction, according as a is > or < 0; 
the product is therefore a new vector, which may be denoted 
by aa. The same new vector is obtained, under the form aa, 
when we multiply the scalar a by the vectora. Ifa+aand 
b+ 3 be two quaternion factors, of which a and 6 are the scalar 
parts, and a, (3 the vectors, then with a view to preserving the 
distributive character of multiplication, it is natural to define 
that the product may be distributed into the four following 
parts : 
(a+a) (6+) = ab+aB+ab+ap. 
And if the multiplicand vector 3 be decomposed into two parts, 
or summands, one = 3, and in the direction of the multiplier 
a, or in a direction exactly opposite thereto, and the other 
= B., and in a direction perpendicular to the former (so that 
3, and 3, are the projections of 3 on a itself, and on the plane 
perpendicular to a), then it may be farther defined that the 
multiplication of any one vector [3 by another vector a may be 
accomplished by the formula 
af = a(B; + Bs) = a, + aP.:; 
in which, by what has been shewn, the partial product a/3, is 
to be considered as equal to a scalar, namely, the product of 
the lengths ofa and ,, taken with the sign — or +, accord- 
ing as the direction of (3, coincides with, or is opposite to that 
of a; while the other partial product a/3. is a vector, of which 
the length is the product of the lengths of a and B., while its 
