106 Proceedings of Royal Society of Edinburgh. [sess. 
If Q be any function of an independent variable vector cr, we 
already know (Tait’s Quaternions , 3rd ed., § 480) that 
dQ= -SdovV.Q (12). 
A similar theorem in a bolds. If Q be any function of an inde- 
pendent variable linear vector function of a vector <j> (whether 
general or self-conjugate) 
dQ= - Qi^dcfj^a^ (13). 
From this it is not hard to show that a is an invariant. That is, 
it is a symbol, independent in meaning of the three mutually per- 
pendicular unit vectors, i,j , &, used in defining it. 
I now proceed to a few examples of the use, in the subject of 
Elasticity , of the symbols introduced. 
Assuming uniformity of temperature throughout, let us investigate 
the general equations connected with the state of an elastic body. 
The two usual assumptions will not be made (1) that the strain is 
small, and (2) that there is no molecular couple. Let d<s be the 
volume of an element in some standard state — say the initial state. 
We assume that the pot. en. of any finite volume of the body is of the 
form fffwd s, where w depends only on the state as to strain of the 
body at the point. (Notice that it is not here assumed, as I 
believe is invariably done, that w depends only on the coordinates 
of pure strain at the point. It may, e.g ., so far as we know at 
present, involve also the space-derivatives of those coordinates. 
We do, however, assume that it is not a function of the state of the 
body at a point that is at a finite distance from the point under 
consideration. ) 
Our first object will be to get, not the equations of motion, but 
the equations connecting stress and strain. As will be seen, these 
two problems are not necessarily identical. The plan adopted is to 
assume the body to be strained in the most general manner. Then 
impose the most general small additional strain, and use the prin- 
ciple that for any finite portion of the body 
Jff&wds = (work done by stresses on boundary of portion) 
- (work done by stresses throughout volume of portion) (14) 
Let p be the coordinate vector of the element ds in its standard 
