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, is in general to change its length in a known ratio, and to 

 preserve or reverse its direction, according as a is > or < ; 

 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 vector a. If a + « and 

 b-\-^he two quaternion factors, of which a and b are the scalar 

 parts, and a, j3 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 + i3) = a6 + aj3-|-a6 + «j3. 



And if the multiplicand vector |3 be decomposed into two parts, 

 or summands, one = /3i and in the direction of the multiplier 

 a, or in a direction exactly opposite thereto, and the other 

 = /So, and in a direction perpendicular to the former (so that 

 j3i and j32 are the projections of j3 on a itself, and on the plane 

 perpendicular to a), then it may be farther defined that the 

 multiplication of any one vector j3 by another vector « may be 

 accomplished by the formula 



a/3 = a(i3i + i32) = aj3i+ai3,; 

 in which, by what has been shewn, the partial product aj3i is 

 to be considered as equal to a scalar, namely, the product of 

 the lengths of a and j3i, taken with the sign — or -f-j accord- 

 ing as the direction of j3i coincides with, or is opposite to that 

 of a ; while the other partial product a^^ is a vector, of which 

 the length is the product of the lengths of a and /3-2, while its 



