188 Lord Kelvin on the Reflexion 



This gives 



h ^ (p-p) S ini (33) 



r/ v r V, 



which is a very small numeric. Hence J 7 is very small in 

 comparison with I ; and 



I' ^ p, cot i-p cot ij ^ (M . 



1 * pi cot i + p cot i t 



§14. If the rigidities of the two mediums are equal, we 

 have p / \p = sin 2 i \ sin 2 i n and (34) becomes 



V _ sin 2i— sin 2i / _ tan (i—ij) ,„~. 



I ~~ sin 2i + sin 2i y ~ tan (i + i y ) ^ v' 



which is Fresnel's " tangent-formula." On the other hand, 

 if the densities are equal, (34) becomes 



I'_ —gin ft'— t,) (9C] 



I~ sin(t + t,) •.•••• WW. 



which is FresnePs " sine-formula " ; a very surprising and 

 interesting result. It has long been known that for vibrations 

 perpendicular to the plane of the incident, reflected, and 

 refracted rays, unequal densities with equal rigidities of the 

 two mediums, whether compressible or incompressible, gives 

 Fresnel's sine-law : and unequal rigidities, with equal 

 densities, gives his tangent-law. But for vibrations in the 

 plane of the three rays, and both mediums incompressible, 

 unequal rigidities with equal densities give, as was shown by 

 Rayleigh in 1871*, a complicated formula for the reflected 

 ray, vanishing for two different angles of incidence, if the 

 motive forces in the waves are according to the law of 

 the elasticity of an ordinary solid. Now we find for vibrations 

 in the plane of the rays, Fresnel's sine-law, with its continual 

 increase of reflected ray with increasing angles of incidence 

 up to 90°, if the restitutional forces follow the law of 

 dependence on rotation which I have suggested f for ether, 

 and if the waves of condensation and rarefaction travel at 

 velocities small in comparison with those of waves of dis- 

 tortion. 



§15. Interesting, however, as this may be in respect to an 

 ideal problem of dynamics, it seems quite unimportant in the 

 wave-theory of light ; because Stokes J has given, as I 



* Phil. Mag. 1871, 2nd half year. 



t " On the Reflexion and Refraction of light," Phil. Mag. vol. xxvi. 



1888. 



% "' Dynamical Theory of Diffraction." See footnote §5. 



