678 
MR. G. F. C. SEATTLE ON PROBLEMS IN' ELECTRIC CONVECTION. 
experienced by a unit of positive electricity in motion at the same speed as the rest of 
the system. This mechanical force experienced by the unit charge consists not only 
of the part due to the existence of electric force in the field, but also of a part due to 
the fact that the moving unit charge is acted on like a current element by the magnetic 
induction. 
Mathematical A bbreviations. 
2. Certain mathematical forms occur so frequently in the theory of electro¬ 
magnetism that it is convenient to have some compact method of indicating them. 
The following are the abbreviations which will be employed in this essay. 
(1.) The vector quantity whose components are A x , A 2 , A 3 , will be written A in 
clarendon type, while its magnitude without regard to direction will be denoted by A. 
(2.) The scalar quantity AB cos 6 = A 1 B i + A 3 B 3 -j- A 3 B 3 , where 6 is the’angle 
between A and B, is called the Scalar Product of A and B, and is denoted by SAB. 
Of course SAB = SBA. If A and B are parallel and in the same sense, SAB = AB 
simply. If they are perpendicular to each other, SAB = 0. 
(3.) The vector C, whose components are 
Ci = A 3 B 3 — A 3 B 3 C 3 = A 3 Bj — AjBg C 3 = A x B 3 — A 3 B ls 
is called the Vector Product of A and B, and is denoted by C = A AB. If 6 be the 
angle between A and B, then C = AB sin 6 . Moreover, C is perpendicular to both 
A and B, and its positive direction is such that right-handed rotation about C carries 
A to B. It is plain that VAB = — VBA, and that if A and B are parallel, then 
VAB = 0. 
(4.) If D = VAB, then VC VAB stands for VCD. By working out the three 
components of VCD, it is easily found that 
VCVAB = ASBC - BSCA. 
(5.) If D = VAB, then SCVAB stands for SCD. 
(6.) The vector A, whose components are 
rf'P d >P dV 
where ''P is any scalar 
dx’ dy 5 dz’ 
quantity, is called the Slope of 'P and is denoted by A = V v P. The vector A points 
in the direction in which 'P increases most rapidly, and is normal to the surface 
= constant. 
(7.) The value of the surface integral of the normal component (reckoned outwards) 
of a vector A, when applied to any infinitesimal closed surface, is 
d A 1 d A 3 dA 3 
—I— —I—- 
dx dy dz 
per unit volume of the enclosed space. This is called the Divergence of A, and will 
be denoted by div A. It is a scalar quantity. 
