Reflection and Refraction of Light. 155 



*-_ a (ax + by+cz-<at)\f~l . j^ ^-ax+by+cz-ut)'/ ^l "\ 



^j>'£az + by+cz—wt)>/~l,j^(-az+by+cz—at')'/~l I ._. 



y p (ax+by+cz— (ot)<S-i ,r\ (—ax+by+cz— ut'^^-i. J 



whence, comparing these values with equations (2), 

 B„=C„=0*. 



Near the interface, and for values of x less than the small 

 quantity e, the differential equations change form by the 

 addition of terms whose coefficients are functions of x, which 

 vanish when x exceeds the small quantity e. These additional 

 terms may be reduced to linear functions of f , rj, f, and their 

 differential coefficients with respect to x], since the equations 

 will still be satisfied by taking the displacements proportional 



to the same exponential 



(by+cz-o>t)</ — 1 



We require now to determine the values of f , 97, £ which 

 satisfy these altered equations. 



Cauchy's method of doing this depends, as v. Ettings- 

 hausen| has pointed out, on the method of the variation of 

 parameters : by this method the constants A, A„ . . . are 

 treated as functions of x : and a first condition imposed upon 



them is that -j-> -t-j —^ must remain unaltered in form, so that 



ctx 



dx dx dx 



'_ _ S A (ax + by+cz-o>t)V-i * <-ax+by+cz-o>t)J~l 



+ B ll a*e (f,t * +b! ' +e '- a ° V ~ 1 -f C^V-^^^^V-l, 



drj 



dx 



d l 



dx 



_ i g (az+6y+©e-«rf)V— 1 g (~ax+by+cz-u>t)</ -i 



_ Sn ap (ax+by+cz-<ot)J'-i r-i (-ax+by+cz-wt)*/ —I 



>t)V-l 



W-l, 



K±) 



t)*/-i\ 



+ B / /" I+ ^ +cz - w ° v - 1 -C // a / /- v+ ^ K - w<)V - 1 (v / - L J 



Consider now any one of the parameters, say B /y ; its value 

 deduced from equations (2) and (4) is of the form 



B„= (*+„ + *+,« + ,g + .|)e-— ^. 



Differentiating this equation with respect to x, and substitu- 



t Tom. cit. p. 461. 



* C. R. viii. p. 440. 

 X Pogg. Ann. 1. p. 409 



M2 



