414 H. S. JJhler — Deviation Produced by Prisms. 



sin %D = cot c sin ^/3, 

 cos rj 1 = cot c tan ^(3. 



Elimination of the parameter /3 leads to 



sin \D — 



cos rj 1 



(58) 



Vl + tan 2 c cos 2 rj l 



In general, the rationalized equation of the curve t may be 

 written 



tan 2 r} 1 — cot 2 %D + tan 2 c = (59) 



dD — 2 sin ^ 



Since 



dr} > (1 + tan 2 c cos 2 ^) V sin 2 ^ + tan 2 c cos 2 ^ 



it follows that the locns t makes the same angles with the coor- 

 dinate axes as the locus m. Therefore these curves have 

 common tangents for rj^ ±j7r. Although it can be shown 

 analytically that the curves m and t have points of inflection 

 for I)=0, it is easier to obtain this result from inspection of the 

 irrational equations of these loci, (56) and (58). The straight 



lines D = Q 7 r)^\ir, and 



V, 



— _! 



\iz are axes of symmetry. 



Fig. 7 bears the same relation to the general <y 1 j) x D surface 

 that figures 2, 3, 4, and 5 bear to the X 77/ D surface. As 

 before, n=l'6. In order to show the most complicated forms 

 of the curves of constant deviation, and also to have negative 

 values of the azimuth y x , (/3<c), the comparatively small value 

 20° has been taken for the prism angle j3. The values of D 

 along four constant deviation loci, together with certain other 

 theoretical data, are given in Table III. 



Table III. 



D 



Intercepts, 

 /i-axis. 



Intercepts, 

 ?7i-axis. 



Maxima and minima, 

 A=0. 



14° 31' 7" 



+ 52° 49' 12" 

 -18 18 5 



±48° 2' 12" 



Yl = 23 n 32' 1" 



?i=±57 19 7 



18 22 4.5 



+ 64 11 30 

 -25 49 26 



±60 18 1 



n- 42 21 4 



^ = ±72 38 47 



21 



+ 69 47 28 



-28 47 28 



±63 51 3 



y,= 70 16 32 

 7i = ±77 53 12 



22 13 2 



+ 72 1 49 

 -29 48 47 



±64 46 2 



y 1 = 87 31 29 



7/ 1= ±78 37 9 



Boundary 



+ 90 

 -33 52 10 



±65 59 16 



y x = 90 

 7i=±78 37 49 



