48 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OP DIFFRACTION. 



Accordingly I have determined the index error of the analyzer in a way which will be 

 most easily explained by an example. Suppose the values of a to have been determined by 

 experiment corresponding to the following values of bt, - 15°, 0°, +.15°, ... + 75°, + 90°, + 105°. 

 The value of a for -ur = 0°, and the mean of the values for ■ar = — 15° and -ar = + 15°, furnish 

 two values of e ; and the value of a for sr = + 90°, and the mean of the values for •& = + 75° 

 and sr m + 105°, furnish two values of e + 90°. The mean of the four values of e thus deter- 

 mined is likely to be more nearly correct than any of them. In some few experiments no two 

 values of w were symmetrically taken with respect to the standard points. In such cases I 

 have considered it sufficient to take proportional parts for a small interval. Thus if a r , a 2 be 

 the readings of the analyzer for sr = - 10°, ■& = + 5°, assuming eti = e - 10° - 2 a?, a 2 = e + 5° 

 + x, we get 3x = a 2 — a, — 15°, whence e, which is equal to a 2 — 5° — x, is known. The index 

 error of the analyzer having been thus determined, it remains to get the most probable value 

 of m from a series of equations of the form (48). For facility of numerical calculation it is 

 better to put this equation under the form 



log m = log tan a - log tan w, (49) 



where it is to be understood that the signs of a and •& are to be changed if these angles 

 should lie between and — 90°, or their supplements taken if they should lie between + 90° and 

 + 180°. Now the mean of the values of log m determined by the several observations belonging 

 to one experiment is not at all the most probable value. For the error in log tan a produced 

 by a small given error in a increases indefinitely as a approaches indefinitely to 0° or Q0°, so 

 that in this way of combining the observations an infinite weight would be attributed to 

 those which were taken infinitely close to the standard points, although such observations are 

 of no use for the direct determination of log m, their use being to determine e. Let a + A a 

 be the true angle of which a is the approximate value, a being deduced from the observed 

 angle a corrected for the assumed index error e. Then, neglecting (A a') 2 , we get for the 

 true equation which ought to replace (49), 



, 2i!/Aa' , 



log m = log tan a H — ; ; log tan tst, 



sin 2 a 



M being the modulus of the common system of logarithms. Since the effect of the error 

 A a' is increased by the division by sin 2 a', a quantity which may become very small, in com- 

 bining the equations such as (49) I have first multiplied the several equations by sin 2 a, 

 or the sine of 2 (a — e) taken positively, and then added together the equations so formed, 

 and determined log m from the resulting equation. Perhaps it would have been better to 

 have used for multiplier sin 8 2 a, which is what would have been given by the rule of least 

 squares, if the several observations be supposed equally liable to error ; but on the other hand 

 the use of sin 2a' for multiplier instead of sin 2 2a' has the effect of diminishing the comparative 

 weight of the observations taken about the octants, where, in consequence of the defect of 

 polarization, the observations were more uncertain. 



The following table contains the result of the reduction of the experiments in the way just 

 explained. The value of e used in the reduction, and the resulting value of log m, are written 

 down in each case. The first column belonging to each experiment gives the value of a — "sr 



