1890-91.] Mr A. M‘Aulay on Quaternion Differentiation. 119 
According to him, the properties of the medium depend on six 
independent constants for each point, called the coefficients of spec, 
ind. cap. These coefficients will themselves he functions of the 
state of the medium, and therefore in particular of its strain. 
Assume with Maxwell that 
®=-v», = (70), 
where K is not a mere scalar, hut a self-conjugate linear vector 
function of a vector. K is itself a function of the position of a point 
and also of the strain at the point. Here v is for obvious reasons put 
for Maxwell’s V. Assume further, that if W be the pot. en. of the 
field ; D,o- he the volume and surface density of electricity ; and the 
rest of the notation be identical with Maxwell’s, 
= / ■■■■(' >’ 
and * . \ 
W = ffvads + fffvDds + ifffSQ&s . . . (72), 
D=-SV3) o- = [S2)UV] a + [S2)Uv] 6 . . (73), 
the last occurring only at a surface of discontinuity in !£), UV point- 
ing away from the region of the corresponding 2), and the two 
regions hounded by the surface being denoted by the suffixes 
a and b. In future, such expressions as [ ] a + [ ] 6 will, for brevity, 
be written [ ] a+6 All our integrals are supposed to extend 
throughout all space * though, as K = 0 for conductors, these may 
be excluded. The boundaries of space are the surface at infinity 
and all surfaces of discontinuity in 3) or 
To find the mechanical results flowing from the above assumptions, 
let p be the present (whether strained or not) vector coordinate of 
any point, and let the medium be (additionally) strained by a small 
displacement Srj, vanishing at infinity. Let SW be the increment 
in W Then if 8W can be expressed in the form 
8W = -///Shield? ..... (74), 
* In taking this for the form of W, and operating upon it as follows, we are 
following Helmholtz for the particular case when K is a mere scalar. See 
Wiss. Abh., equation (2 d), p. 805. For the various assumptions above see 
Maxwell’s Electricity and Magnetism, part i. 
