1883.] of Dark Rings in Quartz. 59 



assuming cf> is the same for both waves ; and putting A + B = 2d 2 

 in (8). It may be shewn however that no one of these approxima- 

 tions can produce an error of as much as half a minute in the 

 present case. 



If we put <£ = we obtain 



R = 27rG 

 L a\ ' 



and it is not difficult to see that p/180 = R/X where p is the 

 rotation measured in degrees. Hence p/180 = 27rGL/a i X*. 



The above calculation is taken partly from MacCullagh and 

 partly from Yerdet. I have thought it better to give it, as 

 MacCullagh does not give equation (8) and Verdet's result is 

 affected by a slight numerical error. I have used MacCullagh's 

 notation. 



If in the above formula (8) we put R=nX where n is an integer 

 and insert the known values of the constants we get a series of 

 values of <p corresponding to the dark rings. Then the formula 

 sin = a sin <£>' gives the value of <£' which is the angular radius 

 of the ring outside the quartz. The first ring is given by n = 3. 



The thickness T of the piece of quartz I used was by a 

 careful measure found to be 24'00 mm. and L= T sec <p. The 

 value of G is given on MacCullagh's theory by the formula 

 27rC/a 4 X 2 = p/180 where p is the rotation, measured in degrees, due 

 to a plate of quartz of unit thickness. Brock has found the rota- 

 tion for a millimetre to be 21-67°. He experimented on several 

 different pieces of quartz, and found the mean differences to be 

 about a tenth of a degree. 



I have used the values of 1/a and 1/b given by Rudberg 

 1-54418 and 1-55328. 



In different specimens of quartz they apparently vary about 

 •0001. Thus a — b might vary about 1/60 of itself which would 

 make a difference of about 8' in the tenth ring. The values I have 

 taken are confirmed by the agreement between the observed and 

 the calculated radii in the larger rings. 



In the tenth ring the absence of G would only make a differ- 

 ence of 12', while it would almost double the first ring. The 

 observed and the calculated values of the radii would be reconciled 

 in the case of the second ring by a change in G of l/10th of its 

 value and in the case of the third ring of 1/7 th ; while a much 

 smaller change would suffice in the case of the first ring. 



The final formula from which the values of <p' were calculated 

 was 



sin 4 f = -00004207 n 2 cos 2 <f> - -0003487 (9). 



An approximate value of <j> is sufficient in the second term. 



