the Colours of Mixed Plates. 691 



be readily calculated and shown to be 



COS <fi — COS (j)' 



¥{ 



v 



2yl<i(cOS <j) — COS(f) ') 



cosc/>-cos</>' ) _g 



where the angles <£, <£', </>" are connected by the relations 

 <£ + <£' = 24>", 



tan- y - = 



2 yu./ sin <£ 



wjiere k 2 = 1 — At 2 (l — I 2 cos 2 <£), 



/' = A; cos </) + /xZ sin 2 </>. 



The deviation D of the scattered light is given by the 

 relation 



cos D = 1^2 + ft 2 n 2 , 



where / 1? o, n x and h, m 2 , n 2 are the direction- cosines of the 

 incident and the final emergent ray respectively, and 



I J = 1 -fi*(l -P cos 2 2<t>\ 



With the help of the foregoing expressions, the phase- 

 difference of the interfering rays and their deviations may 

 be readily calculated. The results show that the path- 

 differences increase with the obliquity at which the plate 

 is held and for the same deviation depend on the plane in 

 which the emergent rays lie, and are different also on the 

 two sides of the regularly-transmitted pencil in the plane of 

 incidence. Fig. 5 shows the decrease of the path-difference 

 of the interfering rays with increasing deviation in the three 

 cases indicated in figs. 2, 3, and 4, the angle of incidence 

 being 27° # 5. & has not been taken into account in drawing 

 the graphs. It will be noticed that the path-difference 

 in fig. 2 falls off throughout much more rapidly than in 

 fig. 3 or fig. 4, and, as regards the two latter, their graphs 

 intersect at a deviation of about 35°. 



From these graphs the position of four points on each of 

 the dark rings in the halo may be readily found, and hence 

 the general shape of the rings may be ascertained. This 

 has been done in fig. 6 for the first few rings for a case in 

 which (p- l)t=10\. The oval asymmetrical shape of 



