Photographic Impressions of Polarized Rings. 109 



along a given ring. Hence, when heterogeneous light is used, 

 the circumstances which determine the rings are so different in 

 the two cases that it is no wonder that the character of the rings 

 seen on a photograph should differ in some respects from that of 

 the rings seen directly. 



But not only is a difference of character indicated as likely to 

 take place ; a more detailed consideration of the actual mode of 

 superposition will serve to explain some of the leading features 

 of the abnormal rings as observed by Mr. Crookes. Let us take 

 for example calcareous spar, and suppose the transmitted rays 

 to be all of the same refrangibility. In this case the intensity 

 along a given radius vector, drawn from the centre of the cross, 

 varies as the square of the sine of half the retardation of phase 

 of the ordinary relatively to the extraordinary pencil (see Airy's 

 Tract). If t be the angle of incidence, the retardation varies 

 nearly as sin 2 i; and if sin 2 i=r, we may take, as representing 

 the variations of intensity I, 



I=sin 2 (mr 2 )=|(l-cos2mr 2 ) (1) 



In this expression m is a constant depending upon the refrangi- 

 bility of the rays. In the case of calcareous spar the tints of 

 the rings follow Newton's scale, and m is very nearly propor- 

 tional to the reciprocal of the wave-length. 



Suppose now that rays of two different degrees of refran- 

 gibility pass through the crystal together, and that the photo- 

 graphic intensities of the two kinds are equal. Suppose also 

 that the aggregate effect which the two systems produce together 

 on the plate is the sum of the effects which they are capable of 

 producing separately. The latter supposition, if not strictly 

 true, will no doubt be approximately true if the plate be not too 

 long exposed. Then, if m! be what the parameter m becomes 

 for the second system, we may represent the variation of inten- 

 sity along a given radius vector by 

 I = sin 2 (mr 2 ) + sin 2 (m'r 2 ) = 1 — cos (m—m!r q ) cos (m 4- m'r 2 ) . (2) 



Suppose the refrangibilities of the two systems to be mode- 

 rately different; then the difference between the two para- 

 meters m } m f will be small, but not extremely small, compared 

 with either of them. Hence of the two factors in the expression 

 for I the second will fluctuate a good deal more rapidly than the 

 first, and will be that which mainly determines the radii, &c. of 

 the rings. If the first factor were constant and equal to 1, its 

 value when r=0, the expression (2) would be of precisely the 

 same form as (1), the parameter being the mean of the two, m, m'. 

 However, the first factor is not constant, but decreases as r in- 

 creases, and presently vanishes, and then changes sign. Hence 



