Crystalline Reflexion and Refraction. 151 



values of the differential coefficients obtained from (d) and (). 

 Thus we get 



dn (dg dj ., d? D ,A 



~ = , cos /3 + -j-, cos j3 + ~ cos /3 cos 7 



dz \dx dx dx J 



cos j3 + -~ cos 8' + - cos i3" ) cos ?' 



,, 



= , COS 7 + -7-7 COS 7 + -7-7 COS 7 COS 3 



dy \dx dx dx 



+ -j-7 COS 7 + -T-? COS 7' + , COS 7" COS 8' 



y dy dy 



"- cos 7 + -^7 cos 7' + -3-7 cos 7" ] cos 3" ; 



3 <& fl&^ / 



and when we subtract these equations, attending to the formulae 

 in Lemma I., we find 



= I -7-7 - -7-7 ) cos a + I -7-7 - -r-7 ) cos a' 



3 dy \dz dy J \dx dz J 



'd% dn \ 



~T7 T~7 cos a > 



,C?^ ^/ 



or simply, 



X = X' COS a + y COS a' + /T 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, F, Z in the same way that these are derived 

 from > 17, 2- That is, if we put 



____ --~ -- 



' ~ Hz " ~dy' ' " dx ~ ~dz* ' ~ dy dx' 



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



