INTO THE FORM OF THE WAVE-SURFACE OF QUARTZ. 
319 
values of a—b and c. We may therefore use (11) as an equation to find a—b, 
inserting the observed value of i for each of the rings measured. If the value of a—b 
does not come out the same for each ring, then MacCullagh’s theory is incon¬ 
sistent with the facts, unless indeed the discrepancies may be attributed to errors of 
observations. 
This mode of presenting the results of the measurements possesses the great 
advantage that it shows at once how far the observed wave-surface departs from the 
theoretical one. For instance, if for a certain value of <f> a — 6 comes out “nmutli 
part too large, this indicates that the real separation of the two sheets is very 
approximately Toooth part greater than that given by the theory. This is true 
throughout, except for the first two small rings. 
The results from the large rings in Plate 2 are exhibited in Table III. The first 
line gives the value of n. The next four lines give the observed diameters, two sets 
of observations having been taken in plane A and two in plane B. The oblique 
strokes indicate the position of the principal plane of the analysing Nicol. To each 
set is attached the mean temperature at which it was taken. The sixth line gives 
the mean observed diameters. The seventh the values of a — b deduced from Mac¬ 
Cullagh’s theory. The eighth the change in a — b, which would be produced by an 
error of V in the measured diameter; in this, to save space, the first five ciphers after 
the decimal point are omitted. The ninth line gives the value of <f>. 
Table III. (Plate 2). 
Temp. 
Value of n. 
22. 
32. 
52. 
77. 
102. 
127. 
152. 
21f J 
Plane A analyser \ . 
47° 54' 
58° 111' 
75° 13' 
93° 26' 
110° 231' 
127° 501' 
148° 50' 
22f u 
/ • • 
47° 531' 
58° 111' 
75° 14' 
93° 25' 
110° 23' 
127° 50' 
148° 51' 
221° 
Plane B analyser \ . 
47° 53' 
58° 10i' 
75° 121' 
93° 25' 
110° 231' 
127° 50' 
148° 521' 
23° 
/ • • 
47° 53' 
58° 10' 
75° 13' 
93° 241' 
110° 24' 
127° 50' 
148° 511' 
22-1' 
Mean diameter .... 
47° 531' 
58° IP 
75° 13' 
93° 25' 
110° 231' 
127° 501' 
148 51i' 
a — b MacCullagh . . . 
■0037884 
•0037899 
•0037913 
0037922 
•0037931 
■0037925 
•0037931 
Change in a — b per 1' in 
diameter X 10 5 . 
■247 
T97 
•142 
T02 
•075 
•053 
033 
Approximate 0 . 
15° 14' 
18° 21' 
23° 161' 
28° 71' 
32° 7' 
35° 34' 
38° 36' 
a — b Saerau. 
•0037916 
■0037922 
•0037927 
0037931 
•0037939 
•0037932 
•0037936 
The eighth line is an important one since the accuracy with which a — b is deter¬ 
mined varies so much. Thus, while an error of 1' in the diameter of the 150th ring 
gives a change of one part in eleven thousand, the same error in the 20th ring gives a 
change of one part in fifteen hundred. To bring up the numbers in the seventh line 
to one constant value we shall have to suppose errors of 2', \\ r , 1^', and 1' for n— 22, 
32, 52, and 77 respectively. But it is very improbable that the mean observed value 
of the diameter of any ring, except, perhaps, the 150th, is more than l' in error so 
MacCullagh’s theory must be condemned. 
Now, the theory that explained the observations on the small rings most satis- 
