
ELEMENTARY PRINCIPLES OF QUATERNIONS. 189 
That rule may be presented under a form in which it applies also to the 
vector V(pza) of the product of two vectors p, w, generally not perpendicular 
to each other. 
The vector of the product is C | Vip a) 
V(pa) = (Tp To sin oa) gU-. 
Through a common origin, O, we draw p and 
w in their directions OA, OB. At O weerect a 
perpendicular C’OC to the plane AOB. 
Then we take for the positive part of the 
direction V(pa) the direction OC, so situated 
that if an observer, standing in O, puts his eye 
into C, and directs it towards the opening of 
the angle AOB (not exceeding 180°, and not 
negative), he will see the multiplier p towards his /eft, and the multiplicand a 
towards his right. 
This verified, we will have to take 

eo On 
according to the coinciding of V(pa) with either + Uz, or (— Uz); namely, 
Ur has been determined in such a way that its expression in function of “a, 6, c, 
x,y, 2, makes it to coincide with the positive half of the axis, as for example, 
2, when Up coincides with the axis +2, and Uo with the axis +y; the angle 
between w and Uc being not greater than a right angle. 
Tf, therefore, we originally did choose the positive halves of the axis in such 
a way that the direction of axis +z is taken im respect with the axis + a, + y, 
by the same rule as that by which V’pa) is determined in its positive part in 
respect with Up and Uo, then we have Ur coinciding with V(pa), and vice 
versd, and 
. g=+1. 
We may, without loss of generality, admit this sole value, provided the 
axes are determined in their respective arrangements according to the above 
tule. The positive half of one of the axes, when they are considered in the 
cyclical order, 2, y, z, 2, y, &c., in respect with its two preceding ones, will 
be in the same relation for any set of three consecutive ones. 
Determination of the Value of J. 
We have to make a choice between the values + 1 and —1. As long as 
not more than two vector factors are concerned in a product, the decision 
between + 1 and — 1 might be dispensed with ; but the indetermination, when 
