412 Prof. Ketteler on the Boundary-Conditions of Reflection 



in which p denotes the vibration-excursions, c the velocities of 

 vibration, and the suffixes specify whether they refer to the inci- 

 dent, reflected, or transmitted undulation. Both equations hold 

 for the particles of the boundary-surface, consequently for # = 0, 

 when, namely, through it is placed a system of coordinates, 

 moving with it, such that the #-axis coincides with the perpen- 

 dicular, and, perhaps, the 2-axis is perpendicular to the piaue of 

 incidence. 



The first is identical with Cauehy's well-known equation of 

 continuity ; the second is a generalization of Fresnel and Neu- 

 mann's, containing instead of the maximum velocities of oscil- 

 lation their variable values. 



If we write them thus, 



Pe + Pe = Pv 1 



d pE + d pR = d Pp f * = 0, 

 dt dt dt J 



and substitute in them the known cosine-form, A denoting the 

 amplitude, T the time of an oscillation, taking into account the 

 translation of the coordinate-system by which T R and T D are 

 determined (in correspondence with Doppler's principle), they 

 reduce to 



Ae + Ar = Ap, 



^e Ar__ Ap 

 T E + T R ~ T D * 



A E +A R =A D ,/( 



Ce + C-r = Cj))J 



when the amplitudes A accompany the maximal velocities C 

 of oscillation. The two equations are sufficient by themselves 

 for the deduction of the intensity-formulse, which, for isotropic 

 media in a state of rest, accord with those in Appendix H of my 

 Astronomical Theory of Undulation*. Certainly, applied imme- 

 diately to the same, the two would become identical. 



For the second case we have the combination of the principle 

 of the conservation of vis viva with the principle of the equiva- 

 lence of the quantities of motion in the direction perpendicular 

 to the effective dividing surface ; for if by //, E , fi R , fjbjy be under- 

 stood the reduced masses of the aether, and by a E , or, CLb the 

 angles of incidence, reflection, and refraction reckoned from the 



* Astronomische Undulationstheorie, oder die Lekre von der Aberration 

 des Lichtes. Bonn, 1873. 



or, abbreviated, 



