283 



When we operate by the characteristics t and u on a 

 straight line, regarded as a vector, we obtain as the tensor of 

 this line a signless /iz<?«6er expressing its length; and, as the 

 versor of the same line, an imaginary unit, determining its 

 direction. When we operate on the product abc z: ab X bc 

 of two successive lines, regarded as a quaternion, we obtain 

 for the tensor, T • abc, the product of the lengths of the two 

 lines, 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 =: cos B + V^— 1 sin b ; (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 abc. 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 v — I which enters 

 into the expression (23) is perpendicular to the plane of the 

 triangle abc. It is the versor of the vector of that quaternion 

 which is denoted by the same symbol abc ; and it may, there- 

 fore, be replaced by the symbol uv . abc, which we may 

 agree to abridge to w. abc, so that we may establish the sym- 

 bolic equation : 



UVQ = WQ, or simply, uv = w ; (24) 



we may also call wq the vector unit of the quaternion q. The 

 expression (23) suggests also the denoting the amplitude of 

 any quaternion by the geometrical mark for an angle, which 

 notation will also agree with the original conception of such 



