400 ^Astronomical and Nautical Collections. 



cos i F„ sin i Fe 



P.O. P.E'. 



The small letters, placed below F, are not intended to 

 alter its magnitude, they merely indicate the length of the 

 paths which have been described at the same instant by the 

 ordinary and extraordinary rays, after their emersion from 

 the plate, and determine by their difference, o — e, the inter- 

 val separating the corresponding points of the two systems 

 of undulations. The capital letters P.O. and P.E'. show 

 the successive situations of the planes of polarisation of each 

 pencil, to facilitate the application of the rule which has 

 been laid down. 



Each of the two systems of undulations will be divided 

 into two others, by the action of the rhomboid of calcarious 

 spar, which will afford us in the whole the four following 

 pencils, of which the first two are produced by the first 

 system of undulations, and the latter two by the second. 



cos i cos {i — s) F„^„<, cos i sin (^ — s) F^^,, 

 P. O. S. P. O.T. 



sin i sin [i-s) Y,^,^ sin i cos (i-s) F,^^, 

 P. E^ S. P. E^ V, 



The first and third of these compose the ordinary image, 

 the second and fourth the extraordinary. 



In the case of the latter, we see, from the progress of the 

 planes of polarisation indicated by the capital letters, that 

 the paths of the rays, reduced to a common plane, must be 

 supposed to differ half an undulation, independently of the 

 difference o — e of the paths described; we must, therefore, 

 add half an undulation to o — e, or change the signs of one 

 of the expressions representing the intensity of the common 

 factor of the velocities. The problem is thus reduced to 

 that of finding the result of the combination of two systems 

 of undulations, for which the difference of the paths is 

 o e, and the velocities of oscillation are respectively ex- 

 pressed by F cos I sin {i—s') and — F sin i cos (i — s). 



Applying to this case, the general formula given in the 

 abstract of my Memoir on Diffraction, {Ann. Ch.y vol. xi., 



p. 258,) we have A^ = a^ + a ^ + 2 a a\ cos 2 tt f - J ; a and 



