544 PROCEEDINGS OP THE AMERICAN ACADEMY. 



In selecting the best values for /x aud a, the index of refraction of the 

 glass and the angle of the prism, one has to bear in mind that the angle 

 6, the angle of incidence of the reflected light on the under surface of 

 the glass plate, must not be so great as to cut down seriously the intensity 

 of the reflected beam. If 6 has a given value, which is equivalent to 

 fi cos 2 a having a given value, then x increases with fx, for if /x is 

 increased the perpendicular distance between the two rays is increased ; 

 but if we increase a we must increase /x if /x cos 2 a is to remain a con- 

 stant, so that /x should be as large as possible. The glass having been 

 selected a should be chosen as large as is consistent with the proper value 

 of 6. The angle a was taken 30°, making 6 — 58° and x about 2.75, 

 /x being 1.6936. 



The definition of the reflected image will be affected by the dispersion 

 of the lisht in entering the under surface of the plate. The magnitude 

 of this effect is obviously calculable from the above equation. In the 

 particular glass employed, both the plates and the prisms being of heavy 

 flint glass, the indices of the refraction were fVu = 1.6936 and /x^ = 1.7032. 

 This gave for the values of x corresponding to the lines D and E the 

 values 2.757 and 2.807. The variation in the value of x thus produced 

 was about 1.8 per cent, corresponding to a difference in the value of d 

 of about .001 mm., d being .05 mm. Thus the dispersion does not 

 seriously injure the definition of the reflected image, and the latter, though 

 somewhat weaker than the direct image, is of excellent definition. 



The object of using two prisms G and H and thus an image reflected 

 both in front of and behind the fiducial line is to correct for any error in 

 horizontality of the plate E, and the axis of the observing telescope. In 

 order to examine the effect of this let f^ be the angle between the axis 

 of the telescope and the horizontal, and let y be the angle between the 

 horizontal and the intersection of the surface of the plate with a vertical 

 • ))lane parallel to the plane of the figure. The values of x aud Xy are 

 given by the formulae : 



_ 2 sin [2 g + (y — /3)] tan [ si n^^ { ^ cos 2 a + (y — /?) } + fS] 

 ~ cos /3 + sin y8 tan [sin^^ {/x cos 2 a + (y — Z^)} -f (3] 



_ 2 sin [2 a— (y — (3)] tan [sin-^{/,c cos 2 a — (y — /3)} — /3] 

 ^^ "" cos (3 — sin ft tan [siu"^ | /x cos 2 a — (y — ft)} — ft] 



It is immediately evident, that small variations from the horizontal, 

 changing x in one direction, will cliange Xj to about the same extent in 

 the opposite direction, so that the mean magnifying power of the two 



