INTO THE FORM OF THE WAVE-SURFACE OF QUARTZ. 
311 
Let 
sin i 
■ s Sill l 
sin i" — 6 ,// sin t 
S =S — or, 
l = i —e. 
( 2 ) 
We shall see that a-/s' is always less than Ys’oot so although R contains <j as 
a factor the squares of c t/s may be neglected, while e/i' is smaller still. 
From (2) we obtain 
e cos L=a sin i. . 
Then 
x // , / 6 e ° 
cot i — cot i = . „ - . = . 0 -, 
sur u s " snr c sm i cos i 
Whence 
Again 
R=T sin i (cot i "— cot l) = 
s ' 2 
To¬ 
cos L 
dx= 
i 
a 
'c'2 
R cos i' 
T 
I have examined the effect of retaining the squares of cr and e in finding this 
relation, and find that the above equation is true within one part in ten thousand. 
For the successive rings we have R=w\, where n is an integer. 
So 
where T is measured in millimetres. 
D: 
n cos l 
(3) 
All the nine theoretical formulae I have examined for the relative retardation near 
the axis in quartz take one or other of the following forms :— 
D 2 =P 1 2 sin 4 <£+D 0 2 ] 
or >.(4) 
D 2 =P 3 2 sin 4 <j?> + D 0 2 cos 4 <j>J 
where P : and P 3 are constants which have certain values assigned to them. 
D 0 is the value of D when <f>= 0, and is known from the rotation. If p be the 
rotation, measured in degrees, of a millimetre plate of quartz p=180D 0 . 
If the plate he cut strictly at right angles to the optic axis <f)=i and the diameter 
of each ring is 2i. Using the equation sin <jf>=a sin t, where l/a is the ordinary index, 
and eliminating D between (3) and (4) we have 
w 2 cos 2 <^)=T 2 (P 1 2 a 4 sin 4 i-f-D 0 2 ) 
n 2 cos 2 </>=T 2 (P 3 2 a 4 sin 4 i-J-D 0 2 cos 4 <f>) 
