THEORY OF ELECTROMAGNETISM. 
691 
and that B, C are fluxes, and H, A intensities, it will follow that 
4ttC = VVH, B' = VV'A'. 
Prop. IV. If t be a flux SVrc/s = SWcA'.— Proved by applying eq. (4), as we 
applied eq. (3) to prove Prop. III. As particular cases notice that 
SVDcfe = SV'DW, SVCtA = SV'CW.(8). 
9. Intimately connected with these two classes of vectors are two classes of linear 
vector funct ions of a vector. 
In the following statements, as indeed throughout the paper, a will denote an 
intensity, and r a flux. 
Class I. of Linear Vector Functions of a Vector. 
(Examples :—The reciprocal of any function of Class II. ; ordinary stress, cf>, d>; 
heat and electric conductivity, y, It -1 ; specific inductive capacity, K ; magnetic 
permeability, f). 
42 being of this class, the three allied symbols, 42, 42', 42", are connected by the 
equations 
S<r a Cla/jds = Sc rfD.'aiids' — Scr„' Q 'crCds ~) 
42' = m~ l y42y', H' = J 
. (9) 
a a and ai being any two intensities. 
Class II. of Linear Vector Functions of a Vector. 
(Example.—The reciprocal of any function of Class I., e.g., electric resistance, R). 
T being of this class, the three allied symbols T, T', T" are connected by the 
equations 
SrAAcA = St^'T Ciidf = Sr/'T'V/'cfe' 
T' = mx~ l Ty -1 , T" = 
r a and r b being any two fluxes. 
Of course, it is understood that 42' and T' are not, as usual, the conjugates of 
42 and T. Note, that if 42 or T is self-conjugate, then 42''and 42" or T' and T" are 
also self-conjugate. The first and second of each of the sets of equations (9) and (10) 
may be taken as the definitions of 42', 42", T', T". The third and fourth equations of 
each set can easily be proved by equations (5) and (6) to follow. 
4 t 2 
