Ix 



tiative, may be obtained by agreeing to denote by the geome- 

 trical symbol ba the vector j3 — a, which is the difference of 

 two other vectors a and ]3 drawn to the two points a and B, 

 tVom any common origin ; so that ba is the vector to a from a. 

 Denoting also by the symbol cba the quaternion cb X ba, 

 which is the product of the two vectors cb and ba ; by dcba 

 the continued product DC 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 



■ , (31), and , (32) 



AC ^ V.ABC ^ 



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 d on the base abc of a tetrahe- 

 dron : the sextuple area of this tetrahedron abcd being 

 expressed in the same notation by the symbol s . abcd. 



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 : 



Aa + da) = {l+j+~ + .. )fa ; (33) 



the only new circumstance being, that in interpreting or 

 transforming the separate terms, for example, the term ^d^, 

 of the resulting development of the function f{a -f- da), if a 

 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, S.a^ = 2aSa, we have been obliged to 

 write 

 d.a^ = a.da-f da.a; (."4) g.a' = a.ca ~|- Sa.a. (35). 



