II. S. Uhler — Deviation Produced by Prisms. 401 



now be stated: "At the discrete minima on the boundary 

 loci of the <y 1 ?7 1 Z> surface the deviation of the projected ray is 

 the supplement of twice the prism angle" In calculating the 

 coordinates of points on the left boundary of the yji^ surface, 

 it is convenient to assume 7, and to derive. 77 x from the follow- 

 ing modified form of (36), namely : 



cos r), = \ cot c sin (3 esc i(Jir— /8'+y x ) sec ^(|tt— (3— y,) (37) 



The right boundary is simply y x = %ir. 



The following special cases will be found useful. For graz- 

 ing incidence or grazing emergence formula (35) reduces to 



sin \D = \ sin (3 V cos Vi \ Vcos 77, + cot c cot %(3 



— Vcos 77j — cot c tan \ ft \ (38) 



For t) 1 = formula (38) reduces to (16). For grazing 

 emergence in a principal plane (36) gives 



sin y 1 = esc c sin ((3 — c) (39) 



For simultaneous grazing incidence and emergence (36) leads to 



cos 77, == cot c tan 4/3 ) 



ll , [ (40) 



or tan ^77, = ± yese (c + -J-/J) sin (c — i(3), ) 



and (38) becomes (15). Anywhere along the left boundary 



dy\ 



= tan c esc (3 cos y x cos 77 x cot 77,, 



hence this locus crosses the 7 r axis, and also intersects the 

 right boundary, at right angles. The intercepts on the Ti^axis 

 of the left boundary may be derived at once from (36). They 

 are given by 



cos r) i = cot c tan (3 ) 



or tan iy y = ± y'ese (c + (3) sin (c — (3)A ^ ' 



Then (35) simplies to 



sin |_Z) = cot c tan /? sin |(tj7r — /?) (42) 



In order that these intersections may be real and discrete, and 

 that 7, may have negative values, the necessary and sufficient 

 condition is /3<<?. 



For normal incidence or emergence formulae (8) and (34) 

 each lead to 



sin (D -f- (3) = esc c sin (3 (43) 



In the case of the \n,'D surface, equation (8) was not used 

 to calculate the coordinates of the points of the constant devia- 

 tion loci because of the involved nature of (8). Instead, 



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



28 



