108 Proceedings of Royal Society of Edinburgh. [sess. 
the variation of the pot. en., does not enter in all its generality. 
A linear vector function, such as <£, can only be split up in one way 
into what may be called a pure part, <jf>, and a rotational part, Ye ( ). 
For the first part 6, and for the last 3, independent scalars must be 
assigned, and it is only the first 6 that have anything to do with 
the pot. en. Expressed in physical language : — A stress can only 
be split up in one way , into two stresses, of which the first is an 
ordinary stress, producing no couple per unit volume, and the second 
is a couple-stress. The latter is quite independent of the pot. en. 
Here, by a “ couple-stress,” is meant a stress which produces on any 
interface, whatever be the direction of its normal, a tangential 
force. Shortly stated, the above may be put : — The part of the 
stress to which the couple per unit of volume is due, is independent 
of the pot. en. This is strictly analogous to the well-known 
corresponding strain theorem that the rotation of an element 
is independent of the pot. en.* 
We may anticipate here by saying that this part of the stress is 
therefore also independent of the strain, though of course other 
means enable us to determine it. In fact, in order that an element 
should not have infinite angular acceleration, it is easy to see that on 
the whole it must be subject to zero couple per unit volume. In 
other words, if be the given external couple per unit volume 
of the unstrained body 
m + 2me = 0 (17) 
is the equation which gives the part of <£ we have called e. Here 
m stands for d<s'/d<s, so that by equations (9) and (7) above 
6m = S£ 1 £ 2 4 s x£ix£ 2X£3 = S ^i^3 S X^iX^ 2 X & ] 
= S^ 3 S^^3 \ • • (18). 
= SV 1 V 2 V 3 S pip 2 p 3 J 
If § be the given external force per unit volume of the unstrained 
body, we must have for equilibrium 
g + = 0 (19), 
or 
g + mfoV i-W'c) = 0 (20). 
* That there is at least very grave doubt whether any such thing as a 
molecular couple, and therefore that a stress-couple exists (even in the case of 
magnetism), I hope to show on some future occasion. It is well to learn, how- 
ever, the nature of the phenomenon, if it possibly exist. 
