THEORY OF ELECTROMAGNETISM. 777 
In place of 57 (2) 
[y\Jv -f- dV^]a + u — 0 . (27). " 
112. It is well to point out what the exact physical significance of is. It 
implies the existence of a membranous stress, i.e., a stress such as a perfectly flexible 
membrane could exhibit. 
To investigate the properties of such a stress in a membrane coincident with the 
actual surface, let % , j', Jc be three mutually perpendicular unit vectors, so that 
[equation (l), § 107] the two latter are parallel to the surface. Consider an elemen¬ 
tary triangle in the surface at the point under consideration, whose vector edges, taken 
in the positive direction round the triangle, are yj ’, — yj + zk' = clp and — zk'. Let 
yFy, F, and zF~ be the forces exerted by the rest of the membrane on the triangle 
across these three faces respectively. Since all other forces on the element are of a 
higher order of smallness than these three, we have as the equation of motion 
F = — yF tJ — zF z 
= - F ;/ Sy dp + F,S/d dp 
= - F,S k' (i dp') - F..S \) (i dp) 
= - y dp), 
where <E>/ is a linear vector function of a vector. This equation does not completely 
determine <t>/ since i'dp is not perfectly arbitrary, but confined to a plane. The 
arbitrary part of <3?/ having no physical bearing on the problem in hand may be chosen 
at will. For present purposes it is convenient to define <E>/ completely by the 
equation 
q>/fc) = — FyS/fw — F.S/w, 
where <y is a perfectly arbitrary vector. This gives 
cE>,V =0.. (28). 
[This is not always the most convenient way of choosing the arbitrary part of <E>/ as, 
for instance, in the study of surface tension.] 
Since the membrane is perfectly flexible F (/ and F, are parallel to the tangent plane 
and, therefore, <E>/ only operates on vectors in the plane, and strains them in the 
plane. Thus <E>/ has four disposable scalars. 
Calling the side of dp towards which i’dp points the negative side, the above 
amounts to saying that the force exerted across the element dp by the part of the 
membrane on the positive side on the part on the negative side is — <£>/ (■ idp'). [The 
direction round any closed curve on the membrane, which is that of positive rotation 
* As with, equation (6), § 100, this may be put in the form v a ~b = [UrdV4]«_6 
MLCCCCXIl.—A. 5 G 
