402 H. S. Uhler — Deviation Produced oy Prisms. 



formula (25) was derived, and the manner of using it for 

 numerical calculations was explained. For the same reason 

 equations (34) and (35) are not convenient for calculating the 

 constant deviation sections of the y^^D surface. Furthermore, 

 it is not desirable to attempt to transform (21) or (25) from the 

 Xrj/ coordinates to the 7^ system, since the equations of trans- 

 formation suggest that a very complicated function of <y 1 in 

 terms of 77, and D would result. For, the equations of trans 

 formation may be written, 



sin X = [cos ^fi sin y x cos rj 1 — sin £/3 y'eot 2 c + cos 9 y 1 cos 2 77,] X 



(cot 2 c 4- cos 2 r}^)-y* 

 and 



sin 77/ = sin c sin 77^ 



A much simpler plan is to assume values for D and rj^ and 

 then calculate J^and 77/ by means of (3) and (4) respectively. 

 Next evaluate X and hence 7/ from (25), and finally get y 1 

 from (5). 



Although it can be established by analysis that each locus of 

 constant deviation has an algebraic maximum, (r) 1 positive), and 

 a like minimum, (rj 1 negative), at points on the 7^ plane of 

 projection which correspond to X ~ 0, (7/ = -J/3), nevertheless 

 this property seems to be self-evident, and therefore, the 

 formal demonstration will be omitted. The equation of the 

 projection on the 7^ plane of the locus of these stationary 

 points may be written 



sin 2 y x cos 2 rj 1 — sin 2 |/3 cos 8 rj 1 = cot 2 e sin 2 %ft (44) 



Hence, the coordinates of points on this curve may be readily 

 obtained by assuming values for y 1 and calculating rj 1 from 



cos rj l = cot c sin £/J ycsc (y 1 + -J/3) esc (y 1 — $(3) (45) 

 Along the projected locus, 



— - = sec c tan c esc ^/8 cos y 1 cos -q l cot rj l cos 77/ 



hence it intersects both the 7 r coordinate axis and the right 

 boundary locus normally. The absolute minimum, e , falls at 

 the point 



y l = sin -1 (esc c sin ^/3), ) 



Vl = 0. J 



For X = formula (8) may be transformed into 



(46) 



sin \D = { V (cot 2 c + cos 2 77^ cos 2 -J/3 



- V(cot 2 c + cos 2 77J cos 2 ^ -cot 2 c } sin -J/3* (47) 

 * H. S. Uhler, this Journal, vol. xxvii, p. 227, 1909. 



