402 Mr. Oliver Heaviside on the 
Vector Product.—We define VAB thus, 
VAB=7(A,B;—A;B,) +j(A3B,— AiB;) + &(ArB,— A,B), (3) 
and call VAB the vector product of A and B. Its magnitude 
is that of A x that of B x the sine of the angle between them. 
Its direction is perpendicular to A and to B with the usual 
conventional relation between positive directions of translation 
and of rotation (the vine system). Thus, Vyw=k; Vjk=1; 
Vki=j. Notice that VAB=—VBA, the direction being 
reversed by reversing the order of the letters; or by ex- 
changing A and B in (3) we negative each term. 
Hlamilton’s VY. The operator 
= da *4 dy +k @ tue 2 fol cok anaes (4) 
may, since the differentiations are scalar, be treated as a vector, 
of course with either a scalar or a vector to follow it. If it 
operate on a scalar P we have the vector 
(OE ete ag: cake 
VP=i 7 TS dy +k ee ent Aare Pad i5 (5) 
whose three components are dP /dzx, &. If it operate on a 
vector A, we have, by (2), the scalar product 
todaAgte OAs 4 dhe 
VAs tm hae 
and, by (8), the vector product 
ise GAs a) (dA, es) dA, dA, | 
vva=i(G! —)+i(G- Ss +h(S2 - a). (7) 
The scalar product VA is the divergence of the vector A, the 
amount leaving the unit volume, if it be a flux. The vector 
product (7) is the curl of A, which will occur below. There 
are three remarkable theorems relating to V, viz. 
P,—P,=\, VPds, | rr 
\Ads=\\ BdS,...iic! ft. beeen 
SV CaS =\\vCde . 2. 
Starting with P, a single-valued scalar function of position, 
the rise in its value from any point to another is expressed in 
(8) as the line-integral, along any line joining the points, of 
V Pds, the scalar product of VP, and ds the vector element 
of the curve. 
Then passing from an unclosed to a closed curve, let A be 
any vector function of position (single-valued of course). Its 
