98 THE BEST ARRANGEMENT FOR PRODUCING 



Let p. be the index of refraction of the prism, a its angle, <j>, and fa the 

 angles of incidence and emergence, 0, and 0, the angles of the ray within the 

 prism with the normals to the first and second surfaces, 8 the difference of these 

 angles then by geometry 



and by the law of refraction, 



.sin^, = fxsin 0,, sin fa = p. sin S , 



fa is constant, being the angle of incidence for all kinds of light, but the other 

 angles vary with p., so that 



i i 6, sin 0, d0, _ sin 0, dfa sin a 



lip. p. COS 0, ' dp. p. COS 0^ ' <//X COS 0, COS , ' 



The last expression gives the dispersion, or breadth of the spectrum, and 

 shews that it increases as the base of the prism is turned from the light. 



As the slit is parallel to the edge of the prism, we have only to consider 

 the primary foci of the pencils when we wish to render the image distinct. 



Let r, be the distance of the focus of incident light from the prism, 

 that of the emergent light, and u that within the prism, all measured to the 

 right, then by the ordinary formula, 



v, u u v, 



cos* fa ~ p, cos 2 0, ' p, cos 8 0, ~~ cos* fa ' 



r, cos 5 0, cos* fa = r, cos' 0, cos* fa t 

 Taking the differential coefficient of the logarithms of these quantities, 



1. <h\ _ 2sm 0, d0, _ 2sin fa dfa _ !_ c/v, _ 2sin0., d0 t 

 v, dp - 0, dp. cos fa dp, v, dp. cos 0, dp. ' 



1 dvi _^ 1 dv s _ 2 sin fa sin a 2 sin' 0, 2 sin 0, sin 0, 



/', (//X Vj rf/t COS 0, COS* fa p. COS* 0, /H. COS 0, COS 0, ' 



Substituting for these angles their values in terms of a and S we find 



1 di\ 1 dv, _ 4 sin a (/r- l) sin a sinSjl +/TCOS (a 8)} 

 v, dp. v t dp. (cos a + cos 8)' ' 1 ?>/r { 1 cos (a 8)} 



The quantity on the right of this equation is always positive, unless the 

 value of 8 exceeds that given by the equation 



(/i* 1 ) sin a = sin 8 {1 + /*' cos (a 8)}. 



