FLUXES OF ENERGY IN THE ELECTROMAGNETIC FIELD. 
429 
§ 7 . Quantities being divided into scalars and vectors, I denote the scalars, as 
usual, by ordinary letters, and put the vectors in the plain black type, known, I 
believe, as Clarendon type, rejecting Maxwell’s German letters on account of their 
being hard to read. A special type is certainly not essential, but it facilitates the 
reading of printed complex vector investigations to be able to see at a glance which 
quantities are scalars and which are vectors, and eases the strain on the memory. 
But in MS. work there is no occasion for specially formed letters. 
Thus A stands for a vector. The tensor of a vector may be denoted by the same 
letter plain; thus A is the tensor of A. (In MS. the tensor is A 0 .) Its rectangular 
scalar components are A 1? A 3 , A 3 . A unit vector parallel to A may be denoted by 
Aj, so that A = AA X . But little things of this sort are very much matters of taste. 
What is important is to avoid as far as possible the use of letter prefixes, which, 
when they come two (or even three) together, as in Quaternions, are very confusing. 
The scalar product of a pair of vectors A and B is denoted by AB, and is defined 
to be 
AB = A l B l + AML + A 3 B 3 = AB cos AB = BA .(6) 
The addition of vectors being as in the polygon of displacements, or velocities, or 
forces ; he., such that the vector length of any closed circuit is zero; either of the 
vectors A and B may be split into the sum of any number of others, and the multi¬ 
plication of the two sums to form AB is done as in common algebra; thus 
(a t b) (c + d) = ac + ad + be + t>d. 
— C& “I - d-f- cb “l - db. 
If N be a unit vector, NN or N 3 = I ; similarly A 3 = A 3 for any vector. 
The reciprocal of a vector A has the same direction ; its tensor is the reciprocal of 
the tensor of A. Thus 
AA-' = ^=1; 
and 
AB- 1 = B- 1 A = 4 ~ cos AB . (g) 
B r> 
The vector product of a pair of vectors is denoted by YAB, and is defined to be 
the vector whose tensor is AB sin AB, and whose direction is perpendicular to 
the plane of A and B. Or 
YAB - i (A 3 B 3 - A 3 B 3 ) + j (A 3 B 3 - A 3 B 3 ) + k (A 1 B 3 - A^BQ - - VBA, ... (9) 
where i, j, k, are any three mutually rectangular unit vectors. The tensor of YAB is 
Y 0 AB; or 
V 0 AB = AB sin AB. 
(10) 
