Crystalline Reflexion and Refraction. 149 



one of these equations may have the same sign, as well as the 

 same magnitude. 



If the last two right lines be perpendicular to each other, 

 we have sin = 1, and the formulae become 



cos a =. cos |3' cos 7" - cos ft" cos 7', 



cos /3 = cos 7' cos a" - cos y" cos a', (B) 



cos 7 = cos a' cos |3 r/ - cos a" cos )3' ; 



but in this case the three right lines are perpendicular to each 

 other, and therefore we have, in like manner, 



cos a' = cos |3" cos 7 - cos j3 cos 7", 



cos ]3' = cos 7" cos a - cos 7 cos a", (B') 



cos 7' = cos a" cos/3 - cos a cosjS"; 

 and also 



cos a" = cos /3 cos 7' - cos j3' cos 7, 



cos ]3" = cos 7 cos a' - cos 7' cos a, (B") 



cos 7" = cos a cos ]3' - cos a' cos /3. 



The last three groups of formulas will still be true, if we 

 suppose the first right line to make with the axes the angles 

 a, a', a", the second the angles ]3, /3', /3", and the third the 

 angles 7, 7', 7" 



Lemma II. Let , 17, denote, as before, the displacements 

 of a particle whose initial co-ordinates are #, y, 2 ; and after 

 putting 



rfi, 4i M 



~te~dy> -Tx~te> ~ dy dx* 



suppose the axes of co-ordinates, still remaining rectangular, to 

 have their directions changed in space, whereby the quantities 

 X, F, Z will be changed into X', Y', Z' ', answering to the new 

 co-ordinates x' 9 /, a', and to the new displacements ', ?', 2' ; then 

 will the quantities X', Y', Z' be connected with X, F, Z by 



