28 



POLARISATION OF LIGHT. 



Tfi r* 



a =Tint produced separately at E by 



the axis O o 

 b =Tint produced separately at E by 



the axis A B 

 ^ =The angle of the parallelogram of 



forces 



*=The angle CEF 

 =The angle OEF 

 A=The archFO 

 D =The arch F E 

 = Half the difference of the angles 



of the base at the diagonal of 



the parallelogram of forces 

 We have then, by spherical trigonometry, 



Cos. 6 = Cos. Ax Cos. D 

 <p= goo _D 

 Tang. D 



Tang. 

 Tang. 



Tang. <p 

 2 A E O = ^ = 2 (180 ) =2 u. 

 . Now, since the tints produced by each 

 axis O o, A B at E, are as the squares 

 of the sines of inclination to the axis, or 

 as sin. 2 O E and sin. 2 A E> and as the 

 relative intensities of the axes are as 1 to 



we shall have a = sin. 2 O E, and 



si$l. 8 OP, 



n. 2 O P. 



Having thus found a and b the sides 

 of the parallelogram of forces, whose 

 angle is ^, ^the diagonal T of this paral- 

 lelogram will be thus obtained : 



Tang. C - 



g+^ = Greater angle at the base. 

 Hence, 



_ 

 * 



a sn. 



When a b, then 



T = 2 a (cos. *+*) 



When a = b, and the axes O o, A B, of 

 equal intensity, then * = *> and 



T = 2 a (cos. 2 r), or T = 2 a (cos. 2 ). 

 When twice the angle, formed by the 

 planes OE, A E, is 90, or ^-90, we 

 have _ 



T= V 2 + 62 



When ^ = 180 T= a - b 



When ^= 0or360 T= a +b 



Such is the method of determining the 

 tints and the form of the rings, in relation 

 to the real axes from which the forces 

 emanate; but in relation to the poles 

 P, P, the law may be expressed more 

 simply by the formula : 

 T = * sin. PEx sin. P E, where t is 

 the maximum tint, 



This result was deduced mathematically 

 by M. Biot from Dr. Brewster' s law ; and, 

 by independent observations, it was esta- 

 blished experimentally by Mr. Herschel, 

 who found, also, that the curves belonged 

 to the class called Lemniscates,whichhave 

 this property, that the rectangles under two 

 lines drawn from the poles P, P' to any 

 point in the periphery N, for example, is 

 invariable throughout the whole curve 

 that is, P N x P' N is a constant quantity, 

 If the axis O o, fig. 33, is exactly equal 

 to the axis A B in intensity, it is obvious 

 that the points of compensation P, P', 

 where the tints of each axis are equal and 

 opposite, and therefore destroy one ano- 

 ther, will be at C and D, the extremities 

 of an axis C D, at right angles to the two 

 axes O o and A B ; and as there cannot 

 be any other points of compensation, the 

 phenomena will now be related to one 

 axis C D, and this axis will be of an 

 opposite character to O o and A B that 

 is, it will be positive if they are negative, 

 and negative if they are positive. Dr. 

 Brewster has demonstrated that a single 

 system of rings will be seen by looking 

 along C D, and that all the phenomena 

 produced by the two equal axes will be 

 mathematically the same as in crystals 

 with a single axis. Hence he ascer- 

 tained that a single system of rings 

 did not necessarily indicate the action 

 of a single axis, but that certain phy- 

 sical circumstances might occur which 

 would determine that the system of 

 rings might be the result of two equal 

 axes, or even of three axes which are not 

 all equal. Such circumstances in the 

 condition of the rings have been disco- 

 vered by him ; and it is therefore an un- 

 doubted fact that crystals, with apparently 

 one axis, have in reality a greater number. 

 System of Rings produced by Common 

 Light. Hitherto we have considered the 

 system of rings as produced by polarised 

 light ; but under certain circumstances 

 they may be produced by common light, 

 and it was indeed by common light that 

 they were first discovered in topaz by 

 Dr. Brewster. If, for example, Jig. 3 1 , 

 A B, 1)6 a plate of topaz, and if common 

 light is incident in the direction D C of one 

 of the resultant axes, and is reflected from 

 the posterior surface of the plate at d so 

 as to reach the eye in the direction C' E' 

 of the other resultant axis, we shall see, 

 by the analysing plate, the system of rings 

 shown infg. 32 ; or if DC is polarised 

 light, the rings will be seen at E' without 

 an analysing plate. Other curious phe- 

 nomena are seen when the rings are 



