Royal Irish Academy. 389 



in two ways : by measuring the angles 0', 0", and taking their sum 

 for 2 ; also by measuring the angles y', y", and taking their sum 

 for the same quantity. Now each of these methods gives a true 

 value of 0, because by the preceding formulae we have 2 = 0' + ' 

 =y'+ y" ; and this accounts for the agreement, shown by the tables 

 of M. de Senarmont, between the values* of 2 obtained by these 

 different methods. But the values of /3 were deduced from the angles 

 y', y", by simply making their difference equal to 2 j3 ; and we see 

 by the second of formulae (N.) that, when the plate is not of the 

 proper thickness, this value of 2 /3 is erroneous by the whole amount 

 of the angle 0'— 0", the difference between fi' and /3 being supposed 

 so small that it may be neglected. As M. de Senarmont proceeded 

 on the common assumption that when the thickness of the plate has 

 been adjusted to that part of the spectrum to which the observations 

 are intended to refer, it may afterwards, through the whole series of 

 experiments, be regarded as exact, he necessarily conceived 0' and 0" 

 to be the same angle ; and it was on the principle of taking an ave- 

 rage between two measures of the same quantity, that he made the 

 supposition 2 = 0'+ 0", which happened to be correct. When 

 therefore he found 0' and 0" to be different, he of course looked upon 

 the difference as merely an error of observation, which it would be 

 superfluous to tabulate. Not having the values of this difference, 

 therefore, we have not the means of immediately correcting the values 

 of 2 j3. But as observations were made for several azimuths at each 

 angle of incidence, we may use the values of to determine those of 

 /3 ; for when at any incidence (except that of maximum polarization, 

 where 0=0 for all azimuths) the values of are known for two given 

 values of a, we can deduce the corresponding values of j3, without 

 any other theory than that of the composition of vibrations. The 

 values of /3 so deduced must indeed be expected to be very inaccu- 

 rate, partly because of errors in the observed values of 0, partly be- 

 cause the observations in different azimuths do not answer to the 

 same ray of the spectrum ; but they will be accurate enough to show 

 the great amount of the error committed by neglecting the difference 

 0' — 0'. For example, putting O and /3 for the values of and /3 

 when a = 45°, M. de Senarmont gives, at the incidence of 60° 

 upon steel, 2 O = 64° 15' (taking the mean of his two determina- 

 tions), and for the azimuths 55°, 30°, 25°, he gives 2 equal to 

 88° 5', 37° 2', and 29° 36' respectively. Combining these values of 

 20 in succession with that of 20 o , we get for 2/3 the series of values 

 32° 38', 33° 28', 34° 30' ; the differences between which are to be 

 attributed to the causes above stated. The mean value of 2 /3 thus 

 found is 33° 32' ; while its value, as given by M. de Senarmont, is 

 only 28° 41'. The difference 4° 51' is the value of 0' — 0", which, 



* Or rather the values of 180°+ 2 & ; because the angle a, the double of 

 which appears in the tables of M. de Senarmont, is equal to 90°+ 0. The 

 angles which he calls yi and y 2 are equal to 90°+ y" and 90° + y' respect- 

 ively. It therefore comes to the same thing, whether the one set of angles 

 or the other is supposed to be measured. The letter /3 has the same signi- 

 fication in both notation?. 



