320 PROCEEDINGS OF THE AMERICAN ACADEMY. 



equation. On the other hand, if & and 0" vary from the truth in oppo- 

 site directions, p quickly changes, a fact which is evident from an inspec- 

 tion of the expression for tan 2 p. The table shows an error of 1° 28' 

 in p when 6' is 20' too large and 6" 20' too small ; and again an error 

 of +2° 2' in p for corresponding errors of +30' and —30' in 6' and 6". 



If the rotation of the vertical circle had been in the opposite direction 

 through the same angle, so as to make C" = 47° 15', the errors for cases 

 1, 2, 3, 8, 9, and 10 of the foregoing table would have been considerably 

 less ; those for cases 4, 5, 6, and 7, on the contrary, somewhat greater. 

 The curves of Plates I., II., and III. show, however, that the extinction 

 angles for each of the different amphiboles in sections cut at respective 

 angles of 47" 15' and 62° 15' to the plane of symmetry would be nearly 

 the same, and that the variation in the extinction-angle at 47° 15' in pass- 

 ing from one amphibole to another would be much slower than that 

 peculiar to the 77° 15' position.* Hence I have chosen the latter as the 

 more useful ; in Table I., in the columns headed " C = 77° 15'," will be 

 found the values of extinction angles characteristic of the same amphi- 

 boles whose extinctions on cleavage pieces have already been calculated. 

 By the use of the whole table, a first approximation to the value of the 

 extinction-angle on (010) can be rapidly made without the necessity of 

 going through the rather tedious application of equation (12). 



Analogous results characterize the introduction of errors into the 6' 

 and 6" of pyroxenes. I have chosen an Ala diopside with C = 43° 33'? 

 2 K=: 59°, and p = 36°. Then, revolving the cleavage-face (110) 15° 

 away from (010), we have C" = 58° 33'. 6' was calculated to be 31° 16', 

 and 6" = 26° 8'. Introducing arbitrary errors in 6' and 6", we obtain 

 the results of the accompanying Table IV. 



Generalizing from the two error tables, supplemented by the inspection 

 of equation (12), we can reach certain conclusions regarding the in- 

 fluence of instrumental errors. Equation (12) is least sensitive for 

 errors in 6' and 6" when these are either both plus or both minus and 

 equal or nearly equal. When equal, p can be more accurately deter- 

 mined than by direct measurement on a section in the plane of sym- 

 metry (given the use of the same microscope in both cases, as well as 

 equal thicknesses, absorption, etc. for the cleavage piece and cut section). 



* It is, of course, evident that both readings (at 47^° and 77|^°) can be taken on 

 the same cleavage piece ; and also that amounts of rotation other than 15° may be 

 advantageously employed. Experiment shows that the cleavage piece will not 

 slide on the object-glass even at an angle of 20°, and thus 6 may be determined for 

 the 421° and 82^° positions. 



