OF ARTS AND SCIENCES. 15 



45° to the jilane of polarization. The plane may also be determined, 

 more easily but less accurately, by removing the Nicol's prism, and 

 turning the tube until the two images are equally bright, and adding 

 45° to the reading. 



We next wish to determine the delicacy of this instrument in differ- 

 ent parts of its scale. If the Nicol's prism is set at an angle v', differing 

 slightly from its true value v, the brightness of the two images will be 

 A sin^ v' and B cos^ v' respectively. Now it is commonly assumed 

 that the difference in two such images will be perceptible, when the 

 difference in brightness, divided by the brightness of either, equals 



a certain fraction — , in which a equals about 80. Now — = 

 a a 



A sin2 y' — B cos^ v' sin^ v' cos^ v — cos^ v' sin^ v 4 sin {v + v') sin {v' — v) 



A sin2 v' sin- ?/ cos'- o sin''^ 2 v ' 



hence, since v is substantially equal to v', — = : — ^ = "■ — o~ > 



again, since p = cos 2 v, dp= — 2 sin 2 v dv, and dp =. — ^ — = — ^ — » 



from which the error in the result for any unobserved difference in 

 brightness of the two images is readily determined. 



If p = 0, dp = -=— , its greatest value, which diminishes as the polar- 



ization increases, becoming zero when p = 0. Hence the greater the 

 polarization, the more accurately it can be measured. If a = 80, dp 

 = Yio fo^ its greatest value ; hence the instrument should always give 

 results within two-thirds of 1 per cent. Observation, however, shows 

 that the error is much greater, a difference in brightness of -g\ being 

 by no means perceptible. To determine" this point, ten observations 

 of a bright unpolarized cloud were taken, and gave a probable error 

 of l°.l, which corresponds to 3.8 percent. The beam reflected from 

 a plate of glass was then observed in the same way, and the probable 

 error was found to be 0°.7, equal 1.0 per cent. As the polarization in 

 this case was 87 per cent, the probable error should have been 3.8 

 (1—p^) == 3 8 X 2-43 = 0.9, a result agreeing very closely with the 

 observed amount. As might be expected, the error varies also with the 

 intensity of the light, so that a sheet of unpolarized paper gave a prob- 

 able error of 5.2 per cent. To compare the absolute brightness in this 

 case with that of the cloud, the polarimeter was directed towards the 

 latter, and its aperture half covered by the paper. The prism was 

 then turned until the dark image of the sky just equalled the bright 

 image of the paper. The mean angle was then found to be 25°, 

 ■whence the relative brightness of the two images was found to be 



