ELEMENTARY PRINCIPLES OF QUATERNIONS. 179 
as a new vector, having its origin in common with that of a, and its extremity in 
common with that of G, in its position constructed by addition, namely in B,’. 
Let p be that vector ; we have thus 
p=at B 
in the full meaning of a vector equation, namely 
Tp =T (a + 8) = AB,’ in length, 
Up= U (a + B) in direction, 

A 
parallel to AB,’, and in the same direction. 
In like manner we might construct 6 + a, and then it may be seen easily 
that 8 + ais equal to a + 8, namely, equal in length, and of the same direction ; 
but what is different is the origin of 8 + a, which is that of 6 arbitrarily given, 
whereas the origin of a + @ is that of a; but position does not enter into the 
expression of a vector. 
Subtraction may be defined in like manner, or may be reduced to addition, 
by the remark that, as for example, 
a— fp’ =a + TB U(—8). 
We may generalise these constructions for three or more vectors; and form 
any polynomial vector - sum equal to a vector p. 
In these definitions of vector - addition and subtraction the signs + and — 
have received an extended meaning, which returns to its original algebraic 
meaning when the vectors a and 8 are parallel. 
The addition and subtraction of quaternions consist in adding scalar to 
scalar, vector to vector, each according to its appropriate rule. 
Conversely, a given vector p may always be decomposed into the vector-sum 
of three components, parallel to three given directions. 
Let Ua, UB, Uy, be the versors of the given directions. Then if a, b, c, 
designate the Cartesian co-ordinates of the extremity of vector p, in respect to 
its own origin, and parallel to axes of direction given by Ua, UB, Uy, we have 
p=aUa+bUB+cUy. 
In the case of the axes 2, y, z, a vector p may be compounded of terms 
parallel to them, so that we have— 
p=ta+ jb + ke. 
The tensor of p is evidently the length of the diagonal of the rectangular 
parallelepiped construed with za, jb, kc. Therefore by definition (10), Tp or 
Tia +76 + ke) = J/a’+ +e’. 
VOL. XXVII. PART II. 3A 
