18 



Mr. W. H. M. Christie on the 



[Mar. 1, 



II. Half-prism diminishing. 



1. Magnifying-power . .ra •= — 



m 



2. Dispersion A, = — '. 



3. Purity ir;,=m,II/=: A> 



III. Isosceles prism. 



1 . Magnify ing-p o vver . . m ( ( ( = 1 . 



2. Dispersion A /;/ = 2A ( . 



3. Purity n„, = 2A ; . 



In order to fix the ideas, let us take the same flint, as in the case of the 

 simple prism, and for the crown glass p A = 1*510, ^i H = 1*530. Then 

 da 1 hu' 1 i $u <W 2 j 7 -i 



f = W f = 75' and = 75' and ^ 4 VCTy nearIy ' 



.*. A = — — I 7m tan ^ — 2 tan x | • 



The first term increases with \p', or the angle of the half-prism of flint, 

 which should therefore be as large as practicable, consistently with good 

 definition and moderate loss of light : it also increases with m ; but in that 

 case the second term increases too in so far as m depends on x . It will 

 therefore be desirable to find the rate of variation of m with x , supposing 

 \p and \p' to remain constant. Since 



COS y' COS \L' 



m = <> . i , 



cos x cos \p 



we have, taking logarithms of both sides and differentiating, 

 ^=tan x 5 x -tan x ' 3 X '; 



whence, since 



* , i * , j tan y' 1 cos y 



cos y d\ = a cos y cv , and ^- = — *. , 



tan x ^ cos x " 



hm=m tan v 1 1 — -L C0S „ ^ 1 <W 

 x l ^' 2 cos 2 x 'J x 



= m tan x . -£ — jL- . 3 X , 

 — sm x 



which increases rapidly with x , and 



3 . tan y = — 



* cos 2 X 



Taking as an example x =60° and m = 3, we have 

 5 



lm=2 • ^X = ^*^^X anc ^ ^ * ^ an X = / ^X* 

 The following Tables are arranged in the same form as those for the 

 simple prism, taking 



^ A =1*700, fi' A = 1*510, and therefore =1*130, 



^ A 



^ = 1-770, ^h = 1-530, ^=1-160. 



A 1 H 



