282 



strucled with three conterminous edges, each equal to the 

 unit of length, and touching at the vertex a the three circles 

 ABC, ACD, ADE, which havc respectively for chords the three 

 remote sides of the pentagon, and are not now homosphaeric 

 circles. And because, in general, in this notation, the equa- 

 tion 



S . ABCDEA =r S . BCDEAB (19) 



holds good, it follows that for any rectilinear pentagon (in 

 space) the five triangular pyramids constructed on the fore- 

 going plan, with the five corners of the pentagon for their 

 respective vertices, have equal volumes. 



Besides the characteristics s and v, which serve to de- 

 compose a quaternion q into two -parts, of distinct and deter- 

 mined kinds, the author frequently finds it to be convenient 

 to use two other characteristics of operation, t and u, which 

 serve to decompose the same quaternion into two factors, of 

 kinds equally distinct and equally determinate; in such a 

 manner that we may write generally, with these character- 

 istics, for any quaternion q, 



Q = SQ 4- VQ = TQ X UQ, (20) 



The factor tq is always a positive, or rather an absolute 

 (or *2>«fcs) number; it is what was called by the author, 

 in his first communication on this subject to the Academy, 

 the modulus, but he has since come to prefer to call it the 

 TENSOR of the quaternion q: and he calls the other factor uq 

 the VERSOR of the same quaternion. As the scalar of a sum 

 is the sum of the scalars, and the vector of a sum is the sum 

 of the vectors, so the tensor of a product is the product of the 

 tensors, and the versor of a product is the product of the ver- 

 sors; relations or properties which may be concisely expressed 

 by the formulae : 



82 = ss ; \'S, — ^v ; (21) 



T n = rrx; u n = n u . (22) 



