282 
structed with three conterminous edges, each equal to the 
unit of length, and touching at the vertex a the three circles 
ABC, ACD, ADE, which have respectively for chords the three 
remote sides of the pentagon, and are not now homospheric 
circles. And because, in general, in this notation, the equa- 
tion 
S.ABCDEA = 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=se+va=Te X va. (20) 
The factor Tq is always a positive, or rather an absolute 
(or signless) 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 vq 
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 formule : 
sti as; VE= =v; (21) 
rons pop 0 Oi i Ofe (22) 
ruil 
