778 
MR. A. McAULAY ON THE MATHEMATICAL 
round i is, of course, considered positive. Thus, for such a. closed curve, i dp points 
inwards. This is the reason for taking the positive and negative sides as just defined. 
It also accounts for the sign given to <£/, since the latter is thus brought into harmony 
with the sign of the linear vector function which represents an ordinary stress.] This 
stress will be called the stress <E>/. 
113. We now seek the force and couple per unit surface due to the stress <£>/. For 
this purpose, first take a finite portion of the surface. The force exerted by the 
stress on any portion of the surface is 
—[<*>/ (Up) = - fjV(iji'v;) ds' 
by equation (3), § 5. Hence the force per unit surface due to the stress is 
- &a (W). 
Again, the couple for a finite portion of the surface round any arbitrary origin is 
- fv*/ (i -d P ') = - 
Hence (by the force per unit surface just obtained) the couple per unit surface 
= - (ivi't) = y - v^v = v&x 
by equation (28). Assuming, which will be the case if there be no other couple per unit 
surface, as is certainly true in our case, that there is no such stress couple per unit 
surface, w r e see that V£<!>/£ = 0 or <E>/ is self-conjugate. Thus <j>/ is of exactly the 
same type as </>/ and has three disposable coordinates only. [It is not necessary to 
assume this couple zero since the problem may be treated in an exactly similar 
manner to that of general stress (former paper, p. 106, et seq.)d\ 
114. Now suppose <E>/ is an “external’’ stress in the actual surface under con¬ 
sideration. The part of SQS q due to it will be JJ’SSp'cff/ ds' — JSSp'dff ( i'dp ), 
so that the only way in which the expression (13) is affected by these new terms is 
that </>/ must be changed into <£/ + <£>/• Similarly for all the subsequent expressions 
in which (/>/ occurs. This shows that <f>/ is a stress of the kind contemplated. 
The bearing of this on capillary phenomena will not be discussed here, because this 
is foreign to the objects of the present paper. It was necessary in this paper to show 
the general results flowing from the existence of l s . 
It should, however, be remarked that this stress, though not affecting the 
mechanical action of the field on a body as a whole, would affect the strains of a 
body, and probably be sometimes comparable in this effect with the similar effects 
resulting from the dependence of l on strain. 
