IN THE INTERIOR OP TRANSPARENT BODIES. 431 



§ 38. We may find the extent of this dispersion as follows : 



Let m' be the greatest value of f{<p't), and let 0, and t t be the values 

 of 0' and / which satisfy the equations 



sin -/(ft*,) sin ft, /<#*,) = m„ 



then 0, is an angle at which light emerges at the time t; also, it is the 

 least angle ; for suppose 0, to be diminished by any quantity, then sin <p t 

 is diminished, and f(<p,t) is not increased, since it is the greatest value 

 of /(fttf); therefore f{<p,t,) sin ft is made less than sin 0, and consequently, 

 no value of 0' less than ft will satisfy the equation (1) at the time t. 

 Hence if we put ft in the equation (2) and find the corresponding value of 

 >\f (ft suppose), ft will be the least angle at which light emerges at the 

 time t t , and it may be considered the least angle at any other time, 

 since, as we have seen, the variation of t does not produce any percep- 

 tible change in the values of ft which satisfy (1). 



Hence the equations 



. . . . sin sin 



sin = m, sin , -. — 2_ = _ — r, — 

 Y ' r '' sin ft sin {i - ft) ' 



will give us ft, the least angle at which the light emerges from the prism. 



In the same manner it may be proved that if ^ be the least value 

 of f{<p,t), and ft be obtained from the equations 



sin sin 2 



sin = Ms sin ft, -. — f- = . y ^ , 



r r sin \//a sin (a - 2 ) 



ft will be the greatest angle at which the light emerges from the prism. 



Thus we may obtain ft - ft, which will be the whole dispersion 

 produced by the prism in a homogeneous ray. As hi — m 2 is small, we 

 easily obtain the following value of 2 — t by the common method, viz. 



* rh Sln j ( V 



^-^-cosftcosftk 1 "" 2) - 



By this formula if we knew the value of /*, — ^ we might find the 

 dispersion 0, — 2 of a homogeneous ray ; and vice versa, if we determine 

 ft — 0a by experiment, we shall then know /u, - m 8 , i.e. the difference 



3c2 



