A.—_MATHEMATICAL AND PHYSICAL SCIENCES 21 
method of attack is to a great extent geometrical and independent of any 
hypothesis as to the nature of a medium. 
The developments which had taken place in analytical mathematical 
methods beginning with the work of the Bernoullis on strings which led 
to Fourier’s work and Lagrange’s treatment of dynamical problems made 
it possible to submit the hypothesis that light is due to the vibrations of 
an elastic medium to a more rigorous analysis. ‘The earliest investigation 
of this kind is due to Cauchy. In Cauchy’s treatment the elastic medium 
is supposed to consist of small particles or molecules which act on each 
other, and the further hypothesis is made that the force between any two 
particles is along the line joining the two points which are taken to 
represent the two particles. As the same problem was discussed by 
Green in a more general way in 1837 it is unnecessary to refer to Cauchy’s 
results in detail. 
The hypothesis which Green made with respect to the mutual actions 
of portions of the elastic medium was that they possessed a work function. 
He investigated the form of this function and proved that when the medium 
is isotropic and homogeneous it involves two constants, and that, if trans- 
verse waves are propagated in the medium independently of normal 
waves, the velocity of propagation of normal waves must be either in- 
definitely great or indefinitely small. He further proved that if the 
elastic medium is stable the velocity of propagation of normal waves in it 
must be indefinitely great. 
The difference between two isotropic homogeneous media is assumed 
to be a difference between their densities, and on this assumption the 
relations between the amplitudes of the incident, the transmitted, and the 
reflected waves are obtained when waves are incident on a surface 
separating two such media. For waves polarised in the plane of incidence 
the relations are the same as Fresnel’s, and for waves polarised per- 
pendicularly to the plane of incidence the relations are very approxi- 
mately the same as Fresnel’s except when the index of refraction is great. 
The difference between Cauchy’s hypothesis as to the nature of the 
mutual actions of the medium and Green’s hypothesis has been referred to 
above ; another important difference in their treatments is that Cauchy 
assumes that the direction of the disturbance in the medium is parallel 
to the plane of polarisation, while Green, in accordance with Fresnel’s 
view, assumes that this direction is perpendicular to the plane of 
polarisation. 
Green’s investigation is of special interest, as it is the first where 
Lagrange’s dynamical method is used for the treatment of a physical 
problem, and where the advantages of using a general dynamical principle 
as the basis of the argument rather than hypotheses which involve the 
assumption of particular modes of action are recognised. 
In 1839 Green applied the same method of treatment to the investiga- 
1 The assumption that the difference between two isotropic homogeneous 
media is a difference in the elastic constants leads to results which do not agree 
with the observed facts. 
