202 REPORT—1885. 
cosines of 0,0, measured from O,, then the values of £, n, depending on the 
initial velocity are— 
Inds 
| ee Z 7 ame 
i Aer? (v ’) 
f= — EE 7 (tr) 0S ee 
fee (2t-r) 
while the values depending on the initial displacement are— 
Ps ef (st-r) 
wr 
nv — _ lnnds », dye 
nf! = — OS ft (bt r) Jot vig, agen 
un — Ul—n?)ds of4,_ 
(er) 
From this it follows that the vibration at O, arising from that at O,, 
lies in the plane through O,O and the axis of z, and is perpendicular to 
the radius 0,0; and if ¢ be the angle between the axis of z and the line 
0,0, 6 that between 0,0 and the wave normal, the value of this dis- 
placement is — 
ry a gre Bie 
7 € + cos 0) sin of ( r) ; or GOL) 
Hence if (bt) = sin as bt, 
cdS . Qa 
=> b} 6 ———s — . . 
a @ + cos ) sin @ cos i C r) (82) 
and the total effect at O will be found by integrating this over the whole 
wave front. 
We have thus found the complete expression for the law of: disturb- 
ance in the secondary wave, and can see in what way it involves @ and ¢, 
and how its phase is related to that of the disturbance over the primary 
wave. 
The theory of diffraction given by Fresnel, and applied by him to 
points in the neighbourhood of the principal wave normal, is thus fully 
justified, since for such points 9 is small, and cos@ therefore approxi- 
mately unity, while ¢ is nearly constant. The expression shows that an 
addition of a quarter period must be made to the phase; but this will not 
affect the form of the diffraction pattern obtained. 
But the results of the investigation are of even more importance in 
their bearing on the relation between the position of the plane of polari- 
sation and the direction of vibration of plane polarised light. For con- 
sider a ray diffracted in a direction making an angle 0 with the incident 
wave normal, and let the plane containing the incident and diffracted 
ray be called the plane of diffraction, and let the directions of vibration 
