Astronomical and JVautlcal Collections. 401 



a' representing the velocities belonging to the two systems of 

 unduhitions, 2 ^r the circumference of the radius unity, c the 

 difference of the paths described, and X the length of the un- 

 dulation: and for the intensity of the homogeneous light in 

 the extraordinary image, we find 



F' [cos*z sin *(i-s) + sin *i cos ^ (i — — 2 sin £ 

 cos , sin (i - s) cos (i - s) sin 2 or — ] = P [{ — cos i 

 sin {i—s) + sin i cos (i— «) } '+ 2 sin i cos i sin (i— *) cos 

 (t — s) (1 — cos 2 9r ""T" )] = F* (sin ^s + sin 2 i sin 2 (i— s) 



Sm TT — ). 



Making a similar calculation for the two constituent pen- 

 cils of the ordinary image, and observing that the two 

 expressions, F cos i cos (i— &), and F sin i sin (i—s)^ must 

 have the same sign, in consequence of tlie direction of the 

 motion of their planes of polarisation, we find, for the inten- 

 sity of the light in the ordinary image : 



F* (cos *5— sin2i sin 2 (i — s) sin '^ 9r -— ). 



These are the general formulas which give the intensity of 

 each kind of homogeneous light in the ordinary and extraor- 

 dinary images, in terms of the length of the undulation and 

 the difference o—e of the paths described by the light in 

 Its passage. Knowing the thickness of the plate, and the 

 velocity of the ordinary and extraordinary rays in the 

 Crystal, it will be easy to determine o— -e. For the sulfate of 

 lime, for rock crystal, and for the greater number of the other 

 crystals possessing the property of double refraction, o—e 

 undergoes but very slight variations for the different kinds 

 of luminous rays, so that it may be considered as a constant 

 quantity, at least for the crystals here considered, in which 

 the dispersion belonging to the double refraction is very 

 small in comparison with the general separation of the two 

 kinds of refraction. If, after having calculated the difference 

 of the paths o — e, we divided each successively by the mean 

 length of the undulation of each of the "seven"" principal 

 kinds of coloured rays, and if we substitute successively 

 these different quotients in the above expressions, we shall 

 have the intensities of each kind of coloured rays in the 



