Quetelet's Rings and other Allied Phenomena. 833 



decrease in the intensity of one of the interfering pencils 

 due to multiple reflexion is to some extent compensated by 

 the increase in the obliquity of the incidence at which the 

 reflexion occurs ; but this is much more so in the cases (^) 

 and Q-) than in the cases (2) and (3), and hence the former 

 systems are much more easily seen. It should be particu- 

 larly noticed that the scattering film used should be very 

 thin and transparent, in order that these systems of higher 

 order may be well seen. 



From the figures showing paths, we may readily write 

 down the formulae 



2t (cos r -2 cos 6) =±nX .... 5(i) 



2*(cosr— 3cos0) = +.7iX .... 5(£) 



2t (cos r- cos 6) = ±nX .... 5(1) 



2*(2cos>-cos0)=±7iX .... 5(2) 



2t (3 cos r-cos 6) =±n\ .... 5(3) 



The position of the achromatic centres for the different 

 systems are thus given by the relations 2 cos 6 = cos r ; 

 3 cos 6 = cos? 1 ; cos# = cosr; cos 6 = 2 cos?'; cos 6 = 3 cosr. 

 It will be seen that the two latter systems of rings cannot be 

 formed unless r>cos _1 (-|) and r>cos -1 (^) respectively. 

 The formulae also readily enable the width of rings in the 

 different systems to be calculated, and the results are in 

 agreement with observation. 



5. Quetelet's Rings in Crystalline Plates. 



No reference has hitherto been made to the interesting 

 effects that arise in the rings observed with the mica owing 

 to its doubly-refractive property. As is well known, mica is 

 a bi-axial crystal, the'planes of cleavage being practically 

 perpendicular to the bisectrix of the angle between the optic 

 axes. In muscovite mica the apparent (external) angle 

 between the axes is about 70°. That special effects must 

 arise from the doubly refractive property is clear in view of 

 the analogy with the Haidinger's rings observed by reflected 

 monochromatic light which have been recently very fully 

 studied by Chinmayanandam *. It appears that in the latt« r 

 case we have not one, but two sets of interference-curves 

 corresponding respectively to light polarized in and perpen- 

 dicular to the principal plane, which are approximately 



* Proc. Roy. Soc. ser. A. vol. xcv. (Jan, L939). 



