ON OPTICAL THEORIES. 231 
d?u 
getas —9 x A 
m IP + + ae 
PU _ Ha + =H + A 
Pe 
X/ and &’ are each put equal to zero, and the condition A +A=0 is 
assumed ; that is, it is supposed, as we have stated before, that the sphere 
of action of each ether particle on the matter is small compared with the 
dimensions of the element of volume considered. 
An expression is then found for the rate at which work is being done 
on the compound medium, and the condition formed that this expression 
should be a function of the time only. 
So far as the terms depending on the mutual reactions are concerned, 
the rate of increase of the energy is given by 
i= {e (vol.),(A Hw 4B 0 +0 tee) 
(52) 
+> fe (surface) yj. S,, = J (vol.) + J (surface) 
where the = implies that more than one medium may come into con- 
sideration, and the integrals are to extend over the whole volume of each 
separate medium and all the interfaces between the media, these being 
indicated by J (vol.) and J (surf.) respectively. 
Forms are then found for A, B, C which make J (vol.) a complete 
differential coefficient with respect to the time, and at the same time lead 
to linear equations of motion which admit of solution in the form 
—° my+nz+et) Hour possible forms are found, which are given 
elow. 
QQ) —A,=n,(u—- U)+ 03(v — V+) + 02(w — W) } 
=p, 4e-V) _ », do —W) 
Ua a | 
CRN eh a i Aa (53) 
eee prrria <i eg MERIT aye | 
ae ay — V d3(w—W) 
2 ae ) a 
It will be noticed that (1) gives us Helmholtz’s theory ; (3) gives 
us Ketteler’s in the modified form I have suggested; for an isotropic 
medium it is shown that the coefficients o and s vanish. Lommel’s form is 
not included in the above; it is therefore, we see, inconsistent with the 
conservation of energy in the medium. 
But there are other terms in the volume integral J (vol.) which will, 
when combined with suitable terms in the surface integral J (surf.), make 
the whole up to a differential coefficient of the time. 
These terms are given by 
ee day, dA, aAg 
sate Keg ha ayia Wy 
