THEORY OF ELECTROMAGNETISM. 
775 
ds on the left being taken, as usual, twice, but on the right only once. That we may 
substitute S Spifc (i'WV/), or SS p{ [{^/}] a + j (t'Wt'V/) for [SSp/ {<£/} (fYf / V 1 , )] a + b is 
obvious, from the fact that the space derivatives of Bp' involved are only the tangential 
ones, which are the same for both regions bounded, because there is no slipping. 
The boundary of the surface in question, like the boundary of any volume, must be 
supposed to involve not only the geometrical boundary, but also any lines of dis¬ 
continuity on it. With this meaning for the boundary, we have, by equation (3), § 5, 
jf s d P ; {</>/} v,' ds = - j ss p’t; (Up) + jjss^,,' (h'wv,') ds. . (9). 
The geometrical meaning of s/Vt'V/ should be noticed. By equation (3), § 5, we 
have 
=\i’dp .(10), 
from which, by limiting the portion of surface considered to an element bounded by 
lines of curvature, it can easily be deduced that 
'i/WV/ = i' ( r~ l + r'~ l ) .(11), 
where r, r are the principal radii of curvature, reckoned positive or negative, 
according as the centre of the corresponding curvature is in the region a or b 
\i' — U vd,~\. Thus — ^‘ i / V^ v V 1 / , or Yi'V'.i' may be called the vector curvature of the 
surface at the point.* 
Since = 0, we see by equation (11) that equation (9) may be written 
ffsSp,'{<(>/} V,'efe' = + . (12), 
ough this last simplification is not needed for our purposes. 
111. We are now in a position to see what alterations l s occasions in the various 
equations given above. This may be done in the following semi-tabular form.t 
Add to right of 45 (9) 
— j" [ SSd [ d VZJ„ + 6 ds + IjsSp'^/ (i'Yi'Vi) ^ s ' ~ [ SS p'<f> s ' ( i'dp ) . . (13). 
* Using, for the moment, the notation of Tait’s ‘Quaternions,’ 3rd edition, §§ 296, et seq., for p and 
dashes, it seems to me that lucidity would be gained by calling p" the vector curvature of the curve at 
the point considered. Thus the vector curvature of a curve is a vector whose tensor is the ordinary 
scalar curvature, and which is drawn from the point on the curve in question towards the centre of 
curvature. By analogy, I would call the vector, drawn from a point on a surface towards the concave 
side, and equal in magnitude to the sum of the principal curvatures, the vector curvature of the surface 
at the point. 
f In what follows “ 45 (9) ” stands for “ § 45, equation (9).” 
