Crystalline Reflexion and Refraction. 151 



values of the differential coefficients obtained from (d} and (d'}. 

 Thus we get 



-f cos + L cos /3' + cos P" cos y 



7 cos + cos 



+ ( -r-7 COS 7 + -r- r COS 7' + -:-, COS y" } COS 3' 



\dy dy dy J 



+ ( -j cos 7 + -j-7 cos 7' + 7 cos 7" ) cos /3" ; 

 \^3 dz dz J 



and when we subtract these equations, attending to the formulas 

 in Lemma I., we find 



rfn 



+ --? - --7 COS O 



or simply, 



X = Z' COS a + Y" cos a x + Z' cos a'', 



which is the first of formulae (D) . And in like manner the others 

 may be proved. 



The same things will obviously hold with respect to quanti- 

 ties derived from X, Y, Z in the same way that these are derived 

 from , rj, . That is, if we put 



T - - d Y - d - d 7 - d - 

 ' ~ Hz ~ ~dy' ' ~ dz~ ~dz' ' ~ dy ~ ~dx ' 



and then suppose the axes of co-ordinates to be changed, the 



