408 PROFESSOR KELLAND ON FRESNEL'S FORMULAE FOR THE 



The values thus substituted will, of course, have to be replaced by others, when 

 the particle under consideration is near the common surface of the media. And 



Sa = {Sl-8R)smip + 8l, 



but if we adopt the particular values of I^ R^ &c. which we may do since all 

 values have the same/orw?, we have the following results : 



81 = acos(ex+f^ + ct + e8x+/8t/)—acosex+/i/ + ct 



i(e X +/i/ + c t)2 sin- (^-^J—^\ -asm{ex +fy + cf)sm(e d x +/8 y) 



= — acos( 



= -2l.sm2-_- + - — smKii' 

 2 e ax 



if we denote eSa; +/8p'bjlix' instead of k^. 



Let us in like manner assume 



e 8 x—f8i/ = Kx" 



e,8 X +f8y = K x^ _ 



then 5R=-2Rsin'-^-+-^-sinK^' 



Z e ax 



T = - 2 T sin^ .— -^ + - -5— sm K x, 

 2 e, ax 



8l,=ke-^''-^'^^ coi(Jy + ct + n+f8y)- ke-^'^ cos/y + ct + n 



= A«-'"| (e-"''-^ cos/8 y-l) cos/y + ct + n-e-'"^'' sin/5^ sin/y + ct+,} 

 = _ I,(l_,-"»S-cos/5y)+j,lI' e-'"^' sm/8y 



ST- _ T,(l-,-'"'^^cos/a./)+j.^' e-^"'''sm/8y 



11. Now for a particle at a distance from the common surface 8 L and 8 R, va- 

 nish ; in such cases the values of 8 a and 8 /3 are 



f „T . „K-^ Idl . ,, „„ . Kx" If/R . ,, . 



Oa=sm d> < —21 sm^— „— H t- sin K j/ + 2 R sm^ — >, 3 — smKar S 



f^ I 2 e dx 2 e dx J 



8fi=cos(j) \ -2Isin*-„-+-^sinKy-2Rsm--2— + --J— smK.r"|- 



By substituting these values in the equations in art., 10, and at once omitting 

 terms of the form iM8x8i/,we obtain 



^"=z2 {^'•+~Sa'»cos-^ + 6V'sin'(/)} X {sin0( -2 Ism= ~)} 

 + 2 [(pr + '!^-8a/"'eos'(p + 8i/'^sm'4>] {sm^2Rsm=-^ } 



