ELEMENTARY PRINCIPLES OF QUATERNIONS. gy 
In agreeing to this value of f, namely, taking 
(Up)? ee = 7? ea af ; 
we gain the practical rule of multiplication of two conjugate quaternions, and, 
as we will show it in its place, of any two quaternions. 
We add the remark, that in virtue of } = — 1, we have for any vector 
ge Nd) ENG) 
so that the tensor of a quaternion will be expressed now according to 
(Ty = (Sq) — (V9)’, 
and as by the distributive rule the second member expresses the product of 
(So Voix (89 4 Va), 
namely, of 
qkq, or indifferently of (Kq) x q , 
we have the results— 
qKq = (Kq) x ¢ = (Tq). 
We may also remark that the formula 
(S¢ + Vq)(S¢ — V9) = (Se? — (Vay 
defines the analytical character of the signs + , and —, used in connecting a 
scalar and a vector into a quaternion. 
The vector of a product has now become 
A eS 
UpUa = — cospa + Ursinpa. 
It presents the anomaly of giving the value — 1 for the product of two unit 
vectors of the same direction, and the value + 1 for the product of two unit 
vectors opposed in direction, like in 
Up U(— p) =1, 
This seems to be in opposition with what takes place for Cartesian co- 
ordinates, where, as for example, 
Dexa = aa 
ax(—#)=—«'; 
_ but the contradiction ceases when we reflect that these co-ordinates are scalars 
and not vectors, or scalars affected by versor factors. 
VOL. XXVII. PART II. 3D 
