INTO THE FORM OF THE WAVE-SURFACE OF QUARTZ. 
317 
Now we shall find the whole effect of C 3 is so small, not exceeding y^gth P ar f °f 
the chief terms in the final formula, that we may not merely reject its squares but also 
put B=a 3 in the small terms, except, of course, in the factor B— a 2 itself. 
Resuming the rotation used for the smaller rings we have 
and 
s' 2 =a 3 + - 
4tt 2 C 2 
s" 2 =B- 
a~\~(a 2 — B) 
4tt 2 C 3 
BA 2 (B—a 2 ). 
// // 
sin t =s sm t, sin l =s sm l. 
( 8 ) 
By equation (l) the retardation R is 
T sin t (cot i" — cot i) 
Assuming that the normal to the plate coincides with the optic axis we have for the 
ordinary wave 
B=a 2 cos 3 t'd -b 2 sin 3 i 
So a 2 —B=(a 2 —6 3 ) sin 3 i — (a 3 — ?> 3 )a 3 sin 3 i, neglecting C. 
For the extraordinary wave 
B=ct 3 cos 2 l'-\- lr sin 3 i", a 3 —B= (a 3 —6 3 ) sin 3 t"= (a 3 —6 2 )a 3 sin 2 1 , 
with sufficient accuracy. Hence 
s' 3 =a 2 +K -i 
s" 2 =B — K J 
where 
K=——■. 
« 4 A, 2 (a 2 — b' 2 ) sin 2 t 
To obtain the part of R not involving C we have 
sin 3 i"=B sin 3 1 
= (a 2 cos 3 sin 2 t ") sin 2 l ; 
therefore 
cot 
a sm l 
Similarly 
cot l '=‘ A- a '’ A t } ; 
a sm l 
