for Reflected and Refracted Light. 17 



or dividing, m, 



' m ' 



w —A. • tJ I — /I/ 1 , 



' m ' 



whence, calculating the values of A', h^ and A/, k^ for the several 

 incidences, we have the intensities directly. 

 But from the equation of vis viva we have 



^ m ' 



{^-n=^kr- 



rrif 

 m 



Thus, knowing only the values h! and A:', we obtain at once those 

 of the refracted intensities, instead of having to calculate h^ and 

 kif which is a troublesome process ; thus we take simply 



1^=1 -A'2^ J^=l-A;'2. 



And this, it will be observed, is independent of any hypothesis 

 as to densities. 



57. If we calculate I^ and Jy directly as above, the values of 

 the amplitudes will differ according to the hypotheses of the 

 density : but then we have to observe, as before, that (denoting 

 FresnePs values by roman letters) we have (14) and (19) 



z. I. 7 1 my m, 1 



and thus 



m ' m ' m ' m ' 



Thus, either way, the calculated intensities will be the same on 

 either hypothesis. 



58. To trace the intensities at different incidences : — 



(1 .) At the perpendicular incidence i = and r = 0. Thus the 



formulas become A' =^, ^' = 7:' But the actual values may be 

 found, since at perpendicular incidences we have (16), 



For equal densities, — ^ = - ; 

 ^ m fjb' 



m 

 For increased density, — = /jl. 



In either case the formula h' = '- will ffive 



- UTTW 



Phil. Mag. S. 4. Vol. 13. No. 76. July 1856. C 



