488 
AR = 1, and representing the two angles at A by NW, NR re- 
spectively, then, if p denote the length of the wave-normal, we 
have sn NW=psin NR. ‘Take x the pole of the tangent 
plane of the wave-surface at R’ (or, what is the same thing, the. 
image of the point W), in respect of the sphere radius AR, 
then x will be the pot on the index-surface corresponding 
to the point #’ of the wave-surface; and let 4K be drawn 
through the point A parallel to R’x. ‘Take AT” perpendicu- 
lar to the plane WAR’ (or, what is the same thing, the plane 
KAR’) as the direction of the refracted vibration, the plane 
KAT" will be the polar plane; and by 4°, the theorem of the 
polar plane, the directions of the incident and reflected vi- 
brations are given as the intersections of the polar plane 
with the wave-fronts or planes through 4 normal to the 
directions of the incident and reflected rays respectively ; 
these intersections are represented in the figure by AZ and 
AT". The relative magnitudes of the vibrations are then 
determined by 2°, the principle of equivalent vibrations, viz., 
considering these vibrations as forces acting in the given di- 
rections AT’, AT, AT” respectively, the refracted vibration 
will be the resultant of the incident and reflected vibrations : 
the terminated lines A 7", A 7, AT” in the figure are taken 
to represent in direction and magnitude the vibrations corre- 
sponding to the refracted ray and to the incident and reflected 
rays respectively, and the lines RV’, Rt, R't’ are drawn 
through the extremities 2’, R, R” of the three rays equal and 
parallel to AT’, AT, and AZ” respectively. Let m', m, m" 
denote the masses of ether set in motion by the three rays re- 
spectively, then, according to Mac Cullagh’s hypothesis of 
equal densities, we have 
geehh _AWocos R'N 
m=m iM: Af cos RN: ~~ as 
(where RN, &c., denote the angles RAN, &c.); or writing 
as before, AR = 1, AW =p, where sin NW =p sin RN, we 
have 
