112 Prof. Stokes on the Occurrence of Abnormal Figures in 



the formula (3), or from an inspection of the numbers in the 

 table. 



It will be seen that in the first three columns the numbers lying 

 between the horizontal lines beginning *20 and -60 correspond 

 to bright rings, and the remainder of each column, together with 

 the beginning of the next, corresponds to a dark ring. But the 

 dark ring, which would regularly follow the fourth bright ring, 



is converted, by the change of sign of the factor cos (^7nr j, into 



a bright ring, forming with the former one broad bright ring 

 having a minimum corresponding to a? = 3, where however the 

 intensity falls only down to its mean value unity. A similar 

 displacement occurs in the seventh column, but here the whole 

 variation of intensity is comparatively small. 



In the case of calcareous spar the character of the rings is the 

 same all the way round, but in the photographs of the rings of 

 nitre a new feature presents itself. Mr. Crookes's figure of the 

 abnormal rings of nitre is rather too small to be clear, but with 

 the assistance of his description there is no difficulty in ima- 

 gining what takes place. With reference to these photographs he 

 observes, "But here a remarkable dislocation presented itself; 

 each quadrant of the interior rings, instead of retaining its usual 

 regular figure, appeared as if broken in half, the halves being 

 alternately raised and depressed towards the neighbouring rings." 

 This effect admits of easy explanation as a result of the super- 

 position of systems of rings which separately are perfectly regular, 

 when we consider that the poles of the lemniscates of the several 

 elementary systems do not coincide, since in nitre the angle be- 

 tween the optic axes increases from the red to the blue. Now 

 the change of character which may be described as a displace- 

 ment of half an order is due to the circumstance that the smaller 

 rings corresponding to the more refrangible rays are, as it were, 

 overtaken by the larger rings corresponding to the less refran- 

 gible. It is plain that the variation of position of the poles of 

 the lemniscates would tend to retard this effect in directions 

 lying outside the optic axes, and to accelerate it in directions 

 lying between those axes. Hence what was a bright ring in one 

 part of its course would become a dark ring in another part, so 

 that each quadrant would exhibit a dislocation of half an order 

 in the rings. In order to show this dislocation to the greatest 

 advantage, a crystal of a certain thickness should be used. With 

 a very thin crystal there would be no dislocation of this nature, 

 but only a displacement like that which takes place with calca- 

 reous spar. With a very thick crystal the effect of the chro- 

 matic variation of position of the optic axes would be too much 

 exaggerated. 



