\ (X) 



20 



is positive in the case of refraction, and negative in that of reflexion. 

 The equations (V), when combined with the relations (H), determine 

 the six partial differential coefficients of V„ of the second order, toge- 

 ther with the three quantities 



3a 3a Ja 

 3jr ' 3y ' 3z ' 



since they give, for these three latter quantities, the conditions 



3a / hi iu 3« \ , 3m / 3a 3a 3a \ 



+ 3t( *^ SJ + ^* 37 + "^ fr) + 3t( ** 37 + ^' 37 + ''^ 3TJ' 



3a / 3w 3u , 3« \ , 3» / 3a 3a 3a \ 



+ 37(''^Sr + ^'37 + ^'37y + 37(*^ 37 + ^-37 + ^'3tJ = 



« = *^(3i3^+^3j3^)+^^ (W+^1^37; + ^'(Tz-+^3ir) 



+ 57(-'57 + ^^37 + ^'S-)+3f(«^ 37 +^'37+^^37; = 

 in which the trinomial 



(*'3i +^^37 + ^'57; 

 can be determined by the following relation : 



<> = --(-37^+^3^/+ ^Hl/+^3i;^i+'''(l/ + "5F-j + 



(3a , 3a , 3a \ / 3m . 3tt , Sm \ 

 -'3T + ^'37+^'3TJi''' 37 + ^^ 37 + ^' 37J- 



In a similar manner we can calculate the new values which are 

 given, by reflexion or refraction, to the partial differential coefficients 

 of V, of the third and higher orders ; and can thence deduce the cor- 



