ELEMENTARY PRINCIPLES OF QUATERNIONS. 177 
directions” to such which are parallel, but drawn towards opposed regions of 
space. 
We agree to represent by (— Up) the versor in the opposed direction of 
_ that which (+ Up) designates, or simply which Up designates. 
A vector, considered in itself, may depend on position, but we agree that 
the expression of a vector is to be independent of position. 
As a consequence of this, an expression like 
wuUp, 
where w is an ordinary quantity, will represent a vector parallel to p; and 
when w, reduced to a number, is positive, that vector wUp will be of the same 
direction as p. It will be of opposed direction when w reduces itself to a 
negative number. 
The sign of equality made use of in connection with vectors is to express by 
agreement the equality of the tensors and the sameness of direction, so that 
p'=p 
- expresses—Ist, that Tp’ =Tp; 2d, that Up’ = Up, which equality expresses by 
agreement : sameness of direction of p’ and p. The two equalities into which 
p’ =p decomposes itself are tacitly founded on the necessary hypothesis that 
LE Wo) =1. 
From what has been agreed to above, we have also U(— p) = — Up. 
In opposition to a vector, we call scalar any ordinary quantity, which is not 
affected by a factor of the versor or vector kind. 
We agree by definition that there is heterogeneity between scalars and 
vectors, so that an expression, composed of the aggregate of a scalar and a 
vector, cannot be equal to zero, unless both elements vanish separately. 
Such an aggregate, the connection being established by the signs + or —, 
is called a quaternion. 
Until we have explained the operation which can give rise to the connection 
between scalar and vector, we cannot give an analytical definition of the 
meaning of the signs + or —, applied to the linking together of the two 
elements. So long as this is not done, these signs may be looked upon in this 
instance as signifying “and” or “ accompanied by.” 
When the sign of equality is used for establishing equality between two 
quaternions, its meaning is by definition, (5), that there is equality between the 
scalars separately, and between the vectors separately. 
Designating a quaternion by g, and by Sq and Vq its scalar and vector 
respectively, we have 
q=nq + Vq. 
