402 ^Astronomical and JVaiitical Collections, 



ordinary and extraordinary image, and we shall then he 

 able to determine the tints of these images by means of the 

 empirical formula which Newton has given for finding the 

 colour resulting from any mixture of different rays, of which 

 we know the relative intensities. We may, therefore, con- 

 sider the general formulas which give the intensity of each 

 species of homogeneous light, in terms of the length of 

 the undulation, as the direct expression for the tint pro- 

 duced by white light ; this, at least, is all that can at pre- 

 sent be deduced from theory, and for the rest, we must 

 have recourse to the empirical construction of Newton, which 

 agrees pretty well with experience, at least with regard to 

 the principal divisions of colours. 



These formulas may be considered independently of the 

 common factor F^, which may be called unity, and we have 



Ordinary image . . cos ^s — sin 2i sin 2 (i~s) sin ^ir 



o—e 



Extraord. image. . sin ^s + sin 2i sin 2 (i-s) sin ^tt 



We see by inspection that the two images must both be 



white when the term containing -^— ■ vanishes, since this is the 



only part which varies with the length of the undulation, 

 and which causes the intensity to differ for the different 

 coloured rays. Thus the images will become white when 

 sin 2i sin 2 (i—s) = 0, which will happen when either of 

 the angles i ov i—s vanishes, or becomes equal to 90°, 180°, 

 or 360°. 



We have, therefore, eight cases in which the images must 

 become white; that is to say, the principal section of the 

 crystallized plate must be parallel or perpendicular to the 

 primitive plane of polarisation, or to the principal section of 

 the rhomboid, which might easily be inferred from the 

 theory without this formula : for when the principal section 

 of the plate is parallel or perpendicular to the primitive 

 plane, the incident light undergoes but one kind of refrac- 

 tion within the crystal ; and when the principal section is 

 parallel or perpendicular to that of the rhomboid, each 

 image contains such rays only as have undergone the same 

 refraction within the plate : tlius in the one case, as well as 

 in the other, each image will contain one system of undula- 



