14 Mr. O'BRIEN ON THE REFLEXION AND REFRACTION OF LIGHT 



Observing 



that -Em- f'(r) 5y' ^x = - D, 2W - <p'(r') A j/= A jsr = i>', 



and putting -E and E' to represent ^m-f'(r)S^ and ^'m -^ ^'(r) As^ respectively. 



(10.) Hence it appears that, if a, /3, 7 be the displacements at any point (xyx) of the 

 lower medium, and a', /3', 7' those at any point (x'y'z) of the upper, and if we put a?'= .r, 

 y'=y, ;^'=sf=0, then the equations (1), (2), (3), and the equations, a = a', ^ = /3', 7 = 7', 

 will hold for all values of .1; and y. Now this being the case, we may differentiate these 



equations with respect to .1; or 2^ ; therefore "^ = "^ ' ""^ therefore (1) may be put in the 



form 



^^ fda dy\ ,^ ^,. Ida dy'\ 



and a similar alteration may be made in (2) and (3). 



Hence, if we put C + D = M, C + D' = M', C + E = N, C + E' = N', we have the follow- 

 ing equations : 



a = a', 13-=/^', 7 = 7' (-0)' 



W, 



«e^s)=-(f^f)l 



A^%^.M(^^.».i^^tt^.«•(l^: + 'f) (F,. 



dss \dw dyl dss \dx dy J 



These are the general equations of connection of the vibratory motion of the two media ; 

 they hold at all points of the plane of separation, i. e. they are true for all values of x and y, 

 z being put equal to zero. 



(11.) We shall now compare with the last of these equations the equation of connection 

 furnished by the common law of elasticity, in the case of two ordinary elastic fluids separated 

 t)y plane surface. 



Let p be the pressure at any point of the lower medium when at rest, considered as a 

 common elastic fluid; then the pressure when it is in a state of vibration, will (by the law of 

 elasticity) be (See Airy's Tracts, note, p. 278, 2nd Ed.) 



/ Sa>SySz 1" f (<tL.^+^]\ 



^\(S.v + Sa)(Sy+S(i){Sz + Sy)\' "'■''i ""[dx^dy dzlj' 



n being a constant nearly equal to unity, depending upon the alteration of temperature during 

 the vibration. 



Similarly, the pressure in the upper medium will be 



\ \dw dy dz ) ) 



Now these two pressures ought to be equal at the plane of separation ; also, by the con- 

 ditions of previous equilibrium, p = p'. 

 Hence, when x = 0, we have 



fda d(i dy\ _ , Ida d^' dy'\ 

 Kdcc dy da I \d(v dy dz j 



