THEORY OF ELECTROMAGNETISM. 
773 
make this supposition. l s will be assumed a function, then, of d ( „ d^, 0, TV, TV As 
throughout the rest of this paper, we assume that there is no slipping at the surface. 
This leads to a relation between TV and TV Let i,j, k as usual stand for a set of 
mutually perpendicular unit vectors, which are, however, functions of the position of a 
point. Let 
i — U v a , i = TJvJ .(1); 
j and k are thus parallel to the surface. 
We have 
T'oj = tiSi'YiSio) + S (j$ka> + Jc&j(o) Sj x Yk 
= tiSico (yt ) 2 + S (jSko) + kSj(o) S xjx^> 
where the summation sign implies that i, j, k are to be changed cyclically. Since 
there is no slipping the strains in the surface of each region bounded are the same or 
XJ = xJ> Xoh = Xb Jc 
Hence, putting 
2 V = „ 2r = = [¥WU : .(2), 
it at once follows that 
Tv_ 5 &> = — 2rSA) + 2 i {SjrSjoj + SkTSkco) .(3). 
Thus TV and TV can be expressed in terms of the independent variables T' and T. 
That these last are independent is seen thus. The deformation in the neighbourhood 
of a point on the bounding surface requires for complete specification a knowledge of 
the following three things : (l) the pure strain of an element of the surface, (2) the 
displacement of a point in the region a near to the element of surface relative to the 
latter when purely strained, (3) a similar displacement in the region b. These three 
are independent, and each requires three scalars to specify it—nine in all, the same 
number as is required to specify T r and T. Thus l s is a function of the variables 
d n , d 6 , T r and T. 
103. The part of Sl s depending on STV and STq is — SST^GVC —_SSr r V//“ where 
G stands for *G. Put now 
* We may dispense with T altogether thus. Put ^ + Vr( ) = n. Then with the meaning 
of n G explained on page 103 of former paper, 
r v = Y^nGr, a = n a - V r V ( )/2, 
i.e., G is the pure part, and rV/2 the rotation-vector of nd. 
