EXTINCTION ANGLES. 139 
The difference in intensity between the two fields is accordingly 
/i 7j = sin 46 sin 28 sin 2 
A 
If 5 = 0, both sides of the field appear equally illuminated with the relative 
intensity 
7 = sin 2 8 
The ratio of the least perceptible intensity difference between the two 
halves when a crystal plate intervenes to the intensity of the field of the 
quartz plate alone determines the setting of the plate at total extinction. 
/i / K . sin 46 . sin 28 . 
= = 2K . sin 48 . cot 8 
/o sin 2 8 
The more sensitive the conditions, the smaller the angle 8 ; if, therefore, the 
reciprocal of 6 be taken as a measure of the sensibility, then for small angles 
of 6 we have 
i 8/0 . K 
-=- - .cot5 
e /i-/ 2 
This relation indicates that the sensibility increases with the value of K 
and also with that of cot 8. The most sensitive conditions are obtained 
accordingly when K is large and 8 small. 
THE QUARTZ HALF-SHADE PLATE OP S. NAKAMURA. 
In a recent paper,* S. Nakamura has discussed the problem of the sensi- 
tiveness of the half-shade system and arrived at practically the same con- 
clusions as those noted above. He suggests the use of a double quartz 
plate of 0.04 mm. thickness instead of 3.5 mm. or 7 mm. thick as in the 
Bertrand ocular, and by actual tests finds the theoretical deductions valid 
and the plate useful. The thickness of 0.04 mm. is equivalent to an angle 
(90^) of about 0.87 on each side of the junction line. Under certain con- 
ditions of strong illumination this angle is undoubtedly the best, and with 
the plate the accuracy of the measurements thereby attained equal to that 
of any of the other measuring devices. 
BI-QUARTZ WEDGE PLATE.f 
It is possible, however, to construct a combination wedge of quartz plates 
of such a character that any angle of rotation from o to any other value, 
positive or negative, can be had on insertion of the wedge, thus adapting 
to wedge form the advantage of the rotating bi-nicol ocular. This has been 
accomplished by combining two plates of quartz cut normal to an axis and 
of specified thickness, the one of right-handed, the other of left-handed 
quartz, each with a wedge of quartz of the opposite sign of rotary polari- 
zation, as indicated in fig. 82. 
The effect of this combination is to produce zero rotation in each half 
wedge where plate and wedge have the same thickness; as the wedge is 
*Centralblatt f. Mineralogie, 267-279. 1905. Compare also J. Mac* de L^pinay. Jour, de Phys. (2). 4, 
367, 1885; (3). 9, 585. 1900; Zeischrift Inst. Kundc, 21, 90. 1901. P. G. Nutting. Bull. U. S. Bureau of 
Standards, 2, 249-260, 1906. 
tF. E. Wright. Amer. Jour. Sci. (4) 26, 377-378. 1908. 
