H. S. Uhler — Deviation Produced by Prisms. 421 



transmission of light, the point characterizing the prism, (ft, c), 

 must lie within the angle AOP, since the equation of the 

 straight line OA is ft — 2c = 0. For points along this line 

 only one ray can be transmitted and the small-circle arcs C'AC 

 and C'BC of fig. 1 are tangent to each other. Again, by 

 hypothesis, the relative index of refraction, n, must exceed 

 unity, in other words, the critical angle must be less than \ir. 

 The equation of the straight line PH is 2 c — it = 0, hence, 

 for transmission of plane waves, the characterizing point must 

 fall inside the right triangle OQP, where Q denotes the point 

 of intersection of the lines OA and PH. The coordinates of 

 Q are ft = it and c = \tt. 



It has been shown that the necessary and sufficient condition 

 for equality between the deviations at grazing incidence in a 

 principal plane and at simultaneous grazing incidence and 

 emergence is 



cos f /3 — cot c sin ip = 0. [See (49) and (50).] 



A portion of the curve representing this equation is shown as 

 the line OY in fig. 8. [The reason for not plotting more of 

 the curve OY is that the arc OIH alone has physical impor- 

 tance, and by drawing this arc on as large a scale as possible 

 the practical value of the diagram is increased.] Hence, if the 

 characterizing point (j3, c) falls on this curve the relation 

 D 1 = D 2 will hold. The straight line UL has the equation 

 3 ft — 7r = 0, and it is seen that so much of the locus OY as 

 lies within the domain of physical reality falls to the left of 

 UL. This shows very clearly the truth of the theorem that in 

 order to have D x =D 2 it is necessary for the prism angle to 

 have a value less than ^ir. The function 



cos ■§/? — cot c sin i/3 



is positive on the left side of the partition OY and negative on 

 the right. Therefore, the domain bounded by OIHPO con- 

 tains the characterizing points of all prisms having the devia- 

 tion at grazing incidence in a principal plane greater than the 

 deviation at simultaneous grazing incidence and emergence. 

 On the other hand, the region having the periphery O AQHIO 

 includes all cases where D^D^. Since this area includes all 

 points for which ft = ^tt it follows that the 60° prisms usually 

 employed in spectroscopes always have D 1 < D 2 . 



The course of the curve OY will now be considered more in 

 detail. In the first place 



dc_ _1 + 2 cos ft — 2 cos 2 /3 

 dp ~ 2(1 — 2cos/3 + 2 cos 3 p) ' 



( l G 



Hence, when ft = 0, — -= J, so that the line OA is tangent to 



