400 H. S. TJhler — Deviation Produced by Prisms. 



From (3), 



cos D = sin 2 rj 1 + cos 2 rj 1 cos E, 



so that substitution in (33) with due attention to (4), gives 

 cos D = sin 2 77j + cos rj 1 sin (y 1 —/3) [cos /3 sin y l cos rj 1 



— sin ft ^ cot 2 6 , + cog 2 ^ cos . ^j + cog ^ cog (y i _ / g) x 



|/ cos 2 ^ — [cos /3 sin y x cos ^ — sin y8 Vcot 2 c + cos a y 1 cos 2 ^] 2 (34) 



This is the general formula for the y^^D surface, it being under- 

 stood that D is always positive. The last equation may also 

 be written 



sin \D = 



i j Vcos^[l +sin(y l — •/?)] j [1— sin (y x — /^Jcos^-f i^sin/?} 



— V^os^^l — sin(y 1 — ^)]{[l +sin (y 1 — /3)] cos^ — usinfi] f ( 35 ) 

 where 



U — Vcot 2 C + COS* y i COS 2 r; 1 —COS y x COS 77 x . 



Since y^^Tr, the expression under the first long radical is 

 always positive. The second long radical will become imagi- 

 nary when 



(1 -f-cos (3 sin y x ) cos r) 1 < sin (3 ycot 2 c + cos 2 y l cos 2 rj 1 , 



therefore the equation of the left boundary of the y^^D 

 surface may be written 



sin y x cos rj 1 + cos /3 cos r) } — cot c sin ft (36) 



[The unrationalized form of (36) can be obtained directly by 

 combining the condition j 2 f = —^tt with (1), (5), and (6). 

 Also, along the left boundary locus formulae (33) and (35) 

 reduce to E=i7r + y l — /3 and sin \D — sin J( \ ir 4- <y 1 — j3) cos rj v 

 as they should do, conformably to (2) and (3).] Combining 

 the last equation with (28) it will be found that 



and therefore 



E = 7T - 2/3. 



Assuming, for the time being, (since it will be formally 

 demonstrated later), that the case of equal roots, characterized 

 by (27) and (28), corresponds to discrete minima on the bound- 

 ary loci, the following theorem, which seems to be new, may 



