254 ON THE REFLEXION AND REFRACTION OF LIGHT. 



It now only remains to substitute for < 2 and < 2 (1) their values, 

 to effect the integrations by parts, and to equate separately to 

 zero the coefficients of the independent variations. Substituting 

 therefore for </> 2 its value ((7), we get 



1 1 1 dx dy dz 8</> 2 



A fff ^ ^ ^ (f^U dv dw\ fdSu dSv d$W\\ 



= - A HI dx dy dz ] (-j- + -j- + -j- -j + -j- + j- r 

 JJJ (\dx dy dzj\dx dy dz J} 



nfffj j j ((& u dv\fdSu d$v\ fdu dw\fdSu d$w\ (dv dw\fdSv dbu 



B\ \dxdy dz\ -T-+J-}( -j-+-j- -f -J-+-T- ~r- + ~r~ + T+-J- -y-+^r" 



JJJ IW^ dx)\dy dx) \dz dxj\dz dx J \dz dyj\dz dy 



o [ ' (^ ^ w ^ w d$v\ fdu d$w dw du\ fdu d$v dv d$u 

 \\fiy'~d* *"dz'~dy) + (dx'~dz ^ dz'~d^} + (dx'lty + dj/'~d^ 



ff 

 = - 

 Jj 



du dv dw\ -r^fdv dw 

 -j- + - r + - r }---2B (-j- + 

 dx dy dz) \dy 



u d z u d fdv 

 + _-_(_ 



d fdu dv dw\ d*v d*v d fdu dw 



, f A d fdu . dv dw\ J\d*w d*w d fdu dv\]} * 

 T j^^-. U=- + - -f - J + B\ -7-5 +-^-5 ^-.^- + -7- r^5 

 ( az \dx dy dz) [_dx* dy 1 dz \dx dy)]} 



seeing that we may neglect the double integrals at the limits 

 ^ = ^00 , 2/=co , 2 = +co; as the conditions imposed at these 

 limits cannot affect the motion of the system at any finite dis- 

 tance from the origin; and thus the double integrals belong 

 only to the surface of junction, of which the equation, in a state 

 of equilibrium, is 



= x. 



