I 



?.oWk«'vvvI for Reflected and Refracted Light. -o^i Joi 269 



A' __ sin 22— sin 2r _ tan {i—r) rjljsv kof^iV^fiq aui* j<» 

 k "" sin 2i + sin 2r ~~ tan {i -f r) 



1 u -^ 



A;^ = (X: + A:') -2sin22— =4sinrcos2 

 aoiaaa-iqxa 9fio jr /^ /^ ^oe/iO iflod 8Uiil 



-fli io '^/ _ 4 sin r cos 2 __ /^ tan {i—r) Vcosi' "^nhd % 



k ~ sin %i + sin 2r ~ \ tan (i + r) / cos r >^«9t)io 



9. These amplitudes are portions of the sides of the triangle 

 taken in the ratio of the components ; and if in that ratiq W^F*^ 



given we substitute the value of -j-f we find -t M \ 



' ^ (#)v j^vgiij (i—r) cos {i + r) _ sin 22- sin 2r ''^' 



(A) sin (i + r) cos (z- r) sin 2z -f- sin 2r' 



which coincides with the value just deduced on the principle of 

 the vis viva. 



Thus for both cases we have FresneFs original formulas. 



10. It is also worthy of remark, that as the investigation of 

 case (a) is obviously equivalent to the direct application of the 

 formula for impact, viz. ^^^^ ■t,4,.«3d}oqTd siW ao iM 



„= ^:ra, j>,=-?5-; - •" '■ --• ■■"■'■- 



m + m^ ' m + ml 



so on the principle here adopted for case (/3), it is easily seen 

 that the same is true, if here, instead of m m^ we take (as 

 above (8)) u^ ,ix h ^^, -:. .^ 



, . msmz . ft. 1 / X m, smr . ^ 



(m) = -^ = sm 2i and (m,) = —k — — = sm 2r. 



' smr ' " smz 



The result is the same as would be obtained directly from the 

 equations 



W + W ' W + K) ' ' 



11. It is also remarkable that this result is the same as if in 

 case (a) we had supposed increased density, but in case (/3) equal 

 densities. 



It may also be observed, that in the investigations of Fresnel 

 and Maccullagh the analogy of impact is referred to only in the 

 case of vibrations perpendicular to the plane of incidence, where 

 the mechanical analogy is not apparent ; whereas in the present 

 investigation we see that that analogy applies also to the case of 

 vibrations parallel to the plane of incidence, which is the actual 

 case in ordinary mechanical impact. 



. "sS >t ms {^i — f) 



Erratum in first Paper. 

 § 37, line 2, dele " or direction of vibrations." 



