H. S. Uhler — Deviation Produced by Prisms. 395 



[Obviously, formulae (9) and (10) may be obtained directly 

 from the relations cos c — cos 7/ cos 77/ and cos c = cos 7 2 cos 1//.] 

 The equation of the \rj/ boundary locus may also be written 



sin X = ± [cos ip*/ sin (c + 17/) sin (c — 77/) — sin J/3 cos c] sec 17/, 



the upper or lower sign to be taken according as points on 

 C'AC or C'BC are desired respectively. When referred to 

 the rectangular coordinate frame X77/I}, the four acute angles 

 which these boundary loci make with the 77 /-axis are each 

 numerically equal to 



tan-'[sec c cot |/3Vsin (c 4- iP) sin (c — -J/3)] ( 1 2) 



On the other hand, the X-axis is normal to the small-circle 

 boundaries. The coordinates of the points A and B are 

 respectively 



X= +(c-i/9), 77/ = 0,} (13) 



and \= -(c- iP), yj,' =0. ) 



At C and C A , 



and 



77/= ± cos -1 (cose see -J/3). 



For simultaneous grazing incidence and emergence, formula 

 (8) reduces to 



sin \D = cot c sin -J/3. (15) 



For grazing incidence or grazing emergence in a principal 

 plane equation (8) leads to 



sinJ^Z) = [<\/cos-J/3cos(c— J/3) — Vsm J/3 sin(c — J/3)]a/J csc csm /? 



(16) 



Equation (16) is equivalent to the following formulae, which 

 are suitable for logarithmic computation, namely 



( . 04) 



cos (D -f- P) = esc c sin (c — /3), 



^YB)\ (17) 



sin -J(Z> + /3) = V esc c sin J/3 cos (c— J/3) 

 Finally, for the absolute minimum of deviation, e , (8) produces 



sin -Je n = [cos iP — ^/ sin (c + J-/3) sin (c— J73)]csc c sin J/3, (18) 

 which is equivalent to the familiar formula 



n = rin *(<. + (19) 



sin -J/3 v ' 



Of the various plane sections of the Xtj/Z* surface which 

 may be imagined, the curves of constant deviation seem best 



