for Rejected and Refracted Light, 11 



36. In some of these investigations, reference has been made 

 to the mechanical doctrine of the impact of elastic bodies, to 

 which the communication of vibratory motion in aether presents, 

 so striking an analogy. The well-known equations expressing 

 the law of impact are adopted directly as the basis of FresneFs 

 proof in the case of vibrations perpendicular to the plane of in- 

 cidence, giving rise to his formulas (H). 



The very same equation is deduced by MaccuUagh from his 

 abstract dynamical principles, without reference to the above- 

 mentioned analogy; but applied on the same hypothesis as to 

 the vibrations, and on his assumption as to the density, it pro- 

 duces his formulas (K). 



37. This deduction, which is independent of any particular 

 supposition as to densities or direction ofvibrations, is as follows: — 



Combining equations (L) and (M), we have 



h^^j^i _m^ _ {h-Vh!){h'-h!) _ h-Ji 

 hf ""m ~ {h^lif ~A+A'' 

 whence, 



^^ = fi. . . . . . . . . . . (N) 



whence again, 



■ (*+*'^=2-=*"»'^f-'=^; ' • • • (0) 



Deduction of Expressions for the Amplitudes. 



38. (I.) On the hypothesis of equal densities, we have directly, 

 on substituting in the equation (N) the values of the masses and 

 dividing by k, supposed = 1, 



m=sin22, my=sin2r, 



' y— si^^^—si^^^ I. __ 2 sin 2i 



~" sin2i + sin2/*' '"" sin2i + sin2r* ' ' ^ ' 



These are MaccuUagh's formulas deduced in the same way, and 

 here (as before) A;' changes sign at i-{-r = 9(f. 



39. It is also evident that these values fulfil the equations of 

 vis viva (M), viz. 



(yt2~F) = A:2-^^=sin22sin2r, . 



^ ' ' yL6COS2 



as well as that of equivalence of vibrations (L), 

 A; + ^' = ^^=2sin2i; 



which last shows that they belong necessarily to vibrations jser- 

 pendicular to the plane of incidence. 



40. For vibrations joar«//ei to the plane of incidence : — since we 



