156 Mr. J. Walker on Cauchy's Theory of 



(pt ^ v d 2 K 

 ting for -j\j -=-^ -r-| from the changed differential equations, 



OlX U.X cix 



all the terms will cancel out except those which depend on 

 the change of form; and we shall have 



where L, M, . . . vanish for finite values of x. 



Now the values of f , rj, £ will differ but slightly from those 

 given by (3), so that this last expression may be written 



dB 



dx 



-jf = (AL + BM + . . .y-*.>^-* + (A,L + B,M + . . .) e ~<*+*»»<-* 



whence the variable part of B /; is 

 B n = ( (AL + BM + . . .)e^- a ^~ l dai 



+ \ (A,L + B,M + . . .)e-( a+a -^ v - l dx. 



Similar values are obtained for the parts of A, A /? . . . which 

 depend on x. NowL, M, . . . vanish for finite values of x ; 



if l Jjdx, I 

 Jo Jo 



so that if 1 L dx, 1 M dx, . . . are very small relatively to 



Jo Jo 



\, ft, v, . . .*, the variable part of B /y may be neglected if 

 — (a-\-a tl )\/ — 1, (a — a lf )\/ — 1 have no real positive part ; so 

 that those among the coefficients A, A n . . . will remain un- 

 altered, when the change in the medium near the interface is 

 taken into account, which have the coefficient of x in their 

 exponential factor with a real part not less than that of a\/ — 1. 

 In the present case this will be so for all the parameters 

 except B /y ; and hence, calling f, rj, ? the corrected values 

 of f, 7], f, we have 



£ __ A (ax + by+cz—o)t)V~l , a (— ax+by+cz-wt)</~L . t> (a„x+by+cz-<at)</~l 



~__"D (ax+by+ez—at)V^l , t> (-ax + by+cz-<*t)<S^l , T> 7 (a n x+by+cz~ <at)V~i 

 7 __ r\ (ax+bjl cz-(at)^^\ _^_ ri (—ax+by+cz-u>t)V~l , t> (a^+by+cz-at^V ^i 



* This necessitates, first, that the coefficients of the added terms in 

 the altered differential equations are all finite, and their product by e 

 very small ; secondly, that the thickness of the modified layer is small 

 compared with the wave-length (Comptes JRendus, viii. p. 439; ix. p. 5). 



