432 MR. 0. HEAVISIDE ON THE FORCES, STRESSES, AND 
and an example of (13) is 
VVVVA = V.VA - V 2 A, 
or 
curPA = V cliv A - V 3 A,. 
which is an important formula. 
Other important formulae are the next three. 
cliv PA = P cliv A + AV.P,. 
(23) 
(24) 
P being scalar. Here note that AV.P and AVP (the latter being the scalar product 
of A and VP) are identical. This is not true when for P we substitute a vector. 
Also 
cliv YAB = B curl A — A curl B;.(_25) 
which is an example of (12), noting that both A and B have to be differentiated. 
And 
curl VAB = BV.A + A div B - AV.B - B cliv A. 
(26) 
This is an example of (13). 
§ 9. When one vector D is a linear function of another vector E, that is, con¬ 
nected by equations of the form 
Di = c 11 E 1 + c 12 E 2 + c 13 E 3 , I 
D 2 - ~t CooBo + c 23 E 3 , >,.(27) 
H.3 = C 31 E 1 + C 32 E 2 + c 33 E 3'J 
in terms of the rectangular components, we denote this simply by 
I) = cE, 
(28) 
where c is the linear operator. The conjugate function is given by 
D' = c'E,.(29) 
where D' is got from D by exchanging c 13 and c 21 , &c. Should the nine coefficients 
reduce to six by c 12 = c 21 , &c., I) and D are identical, or D is a self-conjugate or sym¬ 
metrical linear function of E. 
But, in general, it is the sum of D and D' which is a symmetrical function of E, 
and the difference is a simple vector-product. Thus 
X) = c 0 E + V€E,1 
D' - c 0 E - V€E, J 
(30) 
where c 0 is a self-conjugate operator, and € is the vector given by 
