Ix 
trative, may be obtained by agreeing to denote by the geome- 
trical symbol Ba the vector (3 — a, which is the difference of 
two other vectors a and 3 drawn to the two points a and B, 
from any common origin ; so that Ba is the vector to B froma. 
Denoting also by the symbol cpa the quaternion cB X Ba, 
which is the product of the two vectors cB and Ba; by pcBa 
the continued product pc X cB X Ba, and so on: the fore- 
going equations of central surfaces may be transformed, and a 
great number of geometrical processes and results expressed 
under concise and not inelegant forms. For example, the 
symbols 
V.ABC S. ABCD 
——, (3), and ee? (32) 
will denote, in length and in direction, the perpendiculars let 
fall, respectively, from the summit B on the base ac of a tri- 
angle, and from the summit p on the base asc of a tetrahe- 
dron: the sextuple area of this tetrahedron ascp being 
expressed in the same notation by the symbol s. agcp. 
The developments (15) and (16), with a great number of 
others, may be included in a formula which corresponds to 
Taylor’s theorem, namely, the following : 
di. 
flat da) = (14545 4+---)fas (33) 
the only new circumstance being, that in interpreting or 
transforming the separate terms, for example, the term }d?fa, 
of the resulting development of the function f(a + da), ifa 
and its differential da denote vectors, we must in general em- 
ploy new rules of differentiation, having indeed a very close 
affinity to the known rules, but modified by the non-commu- 
tative character of the operation of multiplication in this cal- 
culus of vectors or of quaternions. It is thus that, instead of 
writing d.a? = 2ada, d.a” = 2ada, we have been obliged to 
write 
da? =a.da+ da.a; (54) 8.a? = a.da+ da.a. (35). 
