II. S. Uhler — Deviation Produced by Prisms. 417 



a cos 6 7j 1 + « 2 cos 4 rj 1 + a 4 cos 2 rj l + a 6 — (60) 



where 



a = 16 sin 4 c sin 2 /5 sin 2 £ D cos 3 y x cos 2 (y l — /?), 



« 2 = 16 sin 2 c sin 2 /3 sin 2 £ Z> cos y a [cos 2 c sin /3 sin (y 1 — /3) 



— sin 2 c sin 2 ^D cos y, cos 2 (y 1 — /?)] — [cos 2 c sin 2 /3 



— 4 sin 2 c sin 2 £ 2> -f 4 sin 2 c cos (3 sin 2 ^Z? sin y x sin (y a — /?)] 2 , 

 a 4 = 8 sin 2 c sin 4 \D [2 cos 2 c sin 2 (3 sin 2 (y t — /?) — cos 2 c sin 2 /3 



+ 4 sin 2 c sin 2 ^Z> — 4 sin 2 c cos /? sin 2 %D sin y x sin (y t — (3)], 

 « 6 = — 16 sin 4 c sin 8 £Z>. 



Obviously, equation (60) is altogether unmanageable. For this 

 reason the conditions under which stationary values of y t may 

 arise willnot be investigated. Suffice it to say that the con- 

 stant deviation loci of fig. 7 are quite consistent with the above 

 sextic in cos^. There are several ways in which the cubic can 

 give two and only two values of rj l which pertain to the y 1 r) 1 

 diagram. One way is for the equation to have three real roots 

 in cos 2 rj 1 , two of which are positive and less than unity, while 

 the third root is either negative or else positive but greater 

 than unity. Another way is for equation (60) to degenerate. 

 Thus for 7 x =|-7r, a = and a biquadratic results. In order to 

 test the correctness of the coefficients of the sextic, y 1 was put 

 equal to \ir and an attempt was made to factor the quartic 

 which was obtained. The factors were found to be 



(cos 2 c — 4 sin 2 c sin 2 ^D) sin 2 (3 cos 2 r) x 



=F sin 2c sin 2/3 sin 2 \D cos rj 1 + 4 sin 2 c sin 4 %D = 0. 



The quadratic with the upper sign is the equation which, when 

 solved for cos 7) 1 , gives formula (26). [The sextic was also 

 found to be satisfied by cos r] x = cot c tan (3 when 7 X = and 

 sin \D was taken from (42).] 



Inspection of the constant deviation loci of ^. 7 throws 

 some light on the most general form of intersection which a 

 plane given by y 1 = constant can have with the y l n 1 D sur- 

 face. [A discussion of equation (60), in this connection, is 

 entirely out of the question.] In other words, a generalized 

 constant-/? curve of fig. 6, (for which 7 1 =i'7r), can have the 

 following course. The deviation will have a minimum value 

 for ^j = 0, then as rj 1 increases so also will D increase until a 

 maximum value is reached. After this the deviation will de- 

 crease to a minimum and then increase up to, and beyond, the 

 intersection with the boundary locus. The coordinates of some 

 points of the rj l D curve corresj^onding to 7 X = 75° are given in 

 Table IY. These points are not presented graphically because 

 the deviation is roughly constant so that the wavy form of the 

 curve would not be sufficiently marked on a small diagram. 



Am. Jour. Sci.— Fourth Series, Vol. XXXV, No. 208.— April, 1913. 

 29 



