162 REPORT—1885. _ 
du , dv , dw dv dw 
Abd ib pble easel ag 99) Ba ee gaa 
A(T iF dy ci dz ii 7) 
clu dv dw,’ dv dw 
P(t eo 201. <2) 
Nee 55 dy Tee ) le Bie | 
B(= +5) = B, (3 +7) 
dy daz dy da 
du , dw du dw 
prc ae a) I 3 peel ot donk 
a(S 35 7.) : ( dz # 7) 
when «=0. 
The problem now resolves itself into two cases. Let us take the plane 
of incidence as the plane zy, and suppose that the vibrations in the 
incident wave are perpendicular to this, then— 
Casz I.—Light polarised in the plane of incidence, 
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=, =v=v,—0, 
and the conditions are 
w= } 
dw dw ‘ ; : a GLO) 
Ba = By, —_ 
dx Py da 
Now, we have seen that Fresnel originally assumed that the rigidity of 
the ether is the same in all media, and the density different. Green, 
adopting this view, puts B=B,, A=A,,* and the above formule lead him 
to results agreeing with those given by Fresnel’s simple theory for this 
case, while, by making the angle of refraction imaginary, it is shown that 
the wave, when totally reflected, undergoes just the change of phase given 
by Fresnel. 
Case II.—Light polarised at right angles to the plane of incidence, the 
vibrations being therefore in that plane. 
Then w= w,=0, and the surface conditions are 
US Uj, v=), 
du, dv dv du, , dv,\ dv 
At eat — ) 2B = Ab if Se pak | 
ates dy : ( daz i dy Fj am} dy 
du , dv\ _ du, , dv, 
SE aes * OF pe 
We have here four equations to. determine two unknowns, viz. the inten- 
sities of the reflected and refracted rays, and it is clear, therefore, that 
two more quantities must come under consideration. 
Now, in the general case it follows from the equations of motion given 
above that two waves can traverse the medium. In the one of these the vibra- 
tions are transverse, and travel with the velocity. / B/p. This constitutes the 
light-wave. In the other the vibrations are longitudinal, and travel with the 
velocity,/ A/p. In the case before us, then, reflexion gives rise to both 
these, and we have two reflected and two refracted waves. But experi- 
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* The physical meaning of these constants and the relations implied by these 
sonditions will be considered later, see p. 167. 
