66 C. S. Hastings — Double Refraction in Iceland Spar. 



In the case of the prism angles each was repeated three 

 times, whence, since they were all quite close to 60°, not only 

 were all errors of graduation eliminated, but the absolute values 

 of the instrumental arcs 120° and 240° determined with great 

 accuracy. The influence of temperature on the magnitudes of 

 the angles becomes evident even in comparatively rude observa- 

 tions. Table I gives all the measures of these angles. Of 

 course the angles given are the supplements of those directly 

 observed ; they are also corrected for circle errors. Following 

 the column containing the observed angle is given the temper- 

 ature of the prism, and then the value reduced to a tempera- 

 ture of 20° C. The method by which the last column was cal- 

 culated will be given farther on. 



The observation of PR enclosed in brackets is rejected. 

 Two or three others might have been rejected without chang- 

 ing the results except to give them smaller probable errors. 



In order to find the values of the angles a standard tempera- 

 ture (20° C.) was chosen as the standard, observation equations 

 of the form 



M = m-\-n (t— 20), 



whence normal equations of the form 



2a\ m+2a(l. n-2a. M=0, 

 2»p. m+2f. n-20. M = 0, 



gave the means of finding m and n. The values of the coeffi- 

 cients of the normal equations are as follows : 



2a 2 2a/3 2/3 2 2aM 2/?. M 



PQ 24 13-7 72-67 34-514 22-647 



PR 16 7-9 64-6 —37-570 -15-588 



QR 15 2-4 66-42 14-324 — 3'980 



The observed values of a from these equations are : 



For PQ 60° 1'. 412±0'-006+0'-0454 («— 20°) 

 PR 59 57'. 628±0'-009-i-0'-0489 (t— 20°) 



QR 60 0'. 970±0'-008 — 0'-0950(Z — 20°) 



The probable errors of a single observation of an angle were 

 found to # be 0'-028, 0'035 and 0'-032, respectively, and the 

 probable errors of the coefficients of the terms containing the 

 temperature 0'-0035, 0''0045 and 0'-0039, respectively. The 

 probable error of 2" for a single observation seems large, con- 

 sidering the refinement of the method used, and indeed it 

 would be for a glass prism ; but regarding the enormous change 

 from temperature and the extreme difficulty of determining 

 that of the prism, it must, I think, be regarded as satisfactory. 



These constants derived directly from observation are subject 

 to certain geometrical conditions which will modify them 

 very slightly and reduce the probable errors. As it was 



