R. S. Uhler— Deviation Produced by Prisms. 397 



In plotting the loci of constant deviation it is necessary to 

 know the coordinates of the points of intersection with the 

 Xt7/ boundary locus. Since 



y l = ^7r or y 3 = — \ir, 



and yV — 1 "= cot c sec rj^ 



elimination of the azimuths from equations (1), (2), (5), and 

 (6) leads to 



cot c sin ft = 2 cos r] l sin |i?sin {ft + \E). 

 Combining this result with relation (3), it will be found that 



2 esc ft sin 2 \D .. 



cos Vl = : , H (26) 



cot c cos ft ± v 4 sin 2 \D — cot 2 c sin 2 ft 



Although this equation pertains directly to the >y 1 ri 1 D surface, 

 it is also appropriate in this place because formula (4) gives rj/ 

 unambiguously as soon as n and rj 1 are known. Equation (26) 

 indicates that, under suitable conditions, the loci of constant 

 deviation will not intersect the Xr)/ boundary loci or will meet 

 each of them in two coincident points, or will intersect each in 

 two discrete points, according as sin \D is less than, or equal 

 to, or greater than -Jcot c sin /3 respectively. Reference is 

 made here to either half of the small-circle arcs C'AC and 

 C'BC since everything is symmetrical with respect to the prin- 

 cipal plane BLA. Moreover, since r) 1 > \tc and cos^ > 1, it 

 does not follow that both roots of equation (26) always have 

 physical significance. For instance, writing the radical as 



± ycoi' c cos 2 ft — (cot 2 c — 4 sin 2 \D) 



it is seen that the negative sign is inadmissible for 



cot c = 2 sin \B, 



The question of what constitutes the " suitable conditions " is 

 both important and interesting, and hence it will be discussed 

 later. Suffice it to say, in this connection, that the case of 

 equal roots suggests minima along the loci of greatest devia- 

 tion, that is, along the Xrj/ and y 1 r} l boundary curves. As far 

 as the coordinates of the points of intersection are concerned, 

 after having calculated 77/, (indirectly), from (26) the abscissae X 

 may be obtained from (9) and (10) or from (11). In the special 

 case of equal roots 



sin \D — |cotc sin ft, (27) 



cos rj l = |cot c tan ft. (28) 



