430 ME. O. HEAVISIDE OH THE FORCES, STRESSES, AND 
Its components are iVAB, jVAB, kVAB. 
In accordance with the definitions of the scalar and vector products, we have 
i 2 = I. J 2 = h k 2 =l;] 
ij = 0, jk = 0, ki = 0; } .(11) 
Vij = k, Vjk = i, Vki = j ; J 
and from these we prove at once that 
V (a + b) (c + d) = Vac + Vad + Vbc + Vbd, 
and so on, for any number of component vectors. The order of the letters in each 
product has to be preserved, since Yab = — Yba. 
Two very useful formulae of transformation are 
and 
AVBC = SVGA ----- CVAB 
A x (B 2 C 3 - B 3 Co) + Ao (BgCj - B 1 Cg) + A 3 (B L C 3 - B^) ; 
VAVEC = B.CA - C.AB, ] 
= B(CA) -C(AB) 
• ( 12 ) 
• (13) 
Here the dots, or the brackets in the alternative notation, merely act as separators, 
separating the scalar products CA and AB from the vectors they multiply. A space 
would be equivalent, but would be obviously unpractical. 
As ~ is a scalar product, so in harmony therewith, there is the vector product Y -. 
JD 
Since YAB = — VBA, it is now necessary to make a convention as to whether the 
denominator comes first or last in Y-. Say therefore, VAB -1 . Its tensor is 
B 
TT A _ A . A 
’ o b B sm 
(14) 
§ 8. Differentiation of vectors, and of scalar and vector functions of vectors with 
respect to scalar variables is done as usual. Thus, 
A = iAj + jA, + kAg. 
d . 
^ AB — AB -f- BA. 
d 
-77 AVBC - AVBC + AVBC + AVBC. 
(15) 
The same applies with complex scalar differentiators, e.</., with the differentiator 
0 d 
dt = dt + qV ’ 
