283 
When we operate by the characteristics 1 and uv on a 
straight line, regarded as a vector, we obtain as the tensor of 
this line a signless number expressing its length ; and, as the 
versor of the same line, an imaginary unit, determining its 
direction. When we operate on the product anc = aB X BC 
of two successive lines, regarded as a quaternion, we obtain 
for the tensor, r. aBc, the product of the lengths of the two 
- dines, or the area of the rectangle under them; and for the 
versor of the same product of two successive sides of a triangle 
(or polygon), we obtain an expression of the form 
u.aBc = cosB + Y— | sinB; (23) 
the symbol B in the second member denoting the internal 
__ angle of the figure at the point denoted by the same letter, 
which angle is thus the amplitude of the versor, and at the 
same time (in the sense Of the author’s first communication) 
the amplitude of the quaternion itself, which quaternion is 
here denoted by the symbol asc. In this theory (as was 
shown by the author to the Academy in that first communi- 
cation), there are infinitely many different square roots of 
negative unity, constructed by lines equal to each other, and 
- to the unit of length, but distinguishable by their directional 
(or polar) coordinates: the particular /—1 which enters 
_ into the expression (23) is perpendicular to the plane of the 
4 oe abc. It is the versur of the vector of that quaternion . 
Fs hich i is denoted by the same symbol asc ; and it may, there- 
‘be replaced by the symbol uv.asc, which we may 
ree to abridge to w. aBc, so that we may establish the sym- 
equation: 
4 bat: 
Uvg = wa, orsimply, uv = w; (24) 
ptertes also call wa the vector unit of the quaternion a. The 
on 1) suggests also the denoting the amplitude of 
