178 _ G. PLARR ON THE ESTABLISHMENT OF THE 
We designate by Kg, pronounced conjugate of g, the quaternion 
We designate by Tq the tensor of g. As q cannot be zero unless both of 
its elements Sg and T (Vq) are zero, we are led to agree by definition (6) to 
the formula 

Tg = / (8g)" + (TVq)™, 
where in anticipation we may state that m and 7 will have to be taken equal 
to units as well as /. 
If we generally take for the definition of a versor the quaternion divided 
by its tensor, then we have for the versor of a quaternion 
so that under another form we have 
g = T9-U¢, and TUg) =A; 
(By this a scalar may be said to have a versor, namely + 1 or — 1, according as 
the sign of the scalar is positive or negative in an absolute way. But we will 
never employ the expression “ versor” in respect to a scalar.) | 
We shall refer directions in space to the directions of the axes of a, y, z, in 
a triply-rectangular system of axes, and we designate, according to use, by 
tJ, k, 
the versors in the direction of the positive halves of the axes 2, y, z, respectively. 
§ 2. Addition and Subtraction of Vectors and Quaternions, and Expressions of Vectors 
Sounded thereon. 
Let a and 8 be vectors representing AA’ and BB’ respectively. Each 
considered separately has its position chosen 
at will. . 
In the expression 
a+ 6 
we agree by definition, (7), of addition that 
the sign + has to represent the operation of 
constructing the term @ following the sign, so _ 
as to put its origin into coincidence with the — 
extremity of the term a preceding the sign +, and drawing £ in its own direction. 
We agree, further, to look upon the binomial expression 
a+ 6 


