REFLECTED ON THE SURFACE OF A CRYSTAL. 61 



or 2cota{mn' cot(b m' n cot (b}=2nsin6 cot (b 2 n sin 6 cot $, 



2m cos # cot (b' + 2 wz cos & cot < 

 and 2 cot a{m' cos 0' cot <b' + ' sin cot (b + *' cos & cot + n' sin cot 0,} 



= 4 sin 6 cos 0' cot </>, 4 sin cos 6' cot ^>' 

 or cot a cot (j) (mri m'n) nsin 6 (cot cot <,) + w cos ^ (cot (f) cot </)') . . (A) 



and cot a (z' cos 0' cot + cot <' + ' sin . cot< + cot< / ) = 2sin0cos0'(cot<|>, cot<') (B) 



Eliminating cot a between these two equations, there results 



2 sin 6 cos & (cot 0, cot (b') cot (mri m! n) = 

 (m' cos & cot + cot (b r 4- w' sin cot + cot 0,) x 



(M sin 6 cot (b cot </>, + m cos #" cot cot (b'} . . (11) 

 This is the equation which determines the polarizing angle. 



We can reduce it to a much more simple form by expunging one of the fac- 

 tors : thus, 



L j. i. w./ sind). <b' 

 cot d> cot <p = . T ., : 

 sin <p, sin 9' 



but </>, $' is a small quantity, depending on the differences of the squares of the 

 refractive indices : call it of the first order. 



n n ( sin d> d>' cos (b + d)' 

 n'-m' n = COS 6 sin & sin c OS in ^ sin - sin (f) + fr cos $=fc-u si 



sin d) + (b' cos (b 





_ sin </) /S in ^ cos frl 

 sin J 



sin (b' sin (j. 



Now u is a very small quantity: hence (mn'-m'ri) (cot (b,- cot $') is of the 

 second order in small quantities, and may in an approximation be neglected. 

 We must therefore equate to zero the first factor of equation (11). 



This gives m' cos 6' sin (b + $' + M , sin sin + ^ = Q 



sin <p sin <b' sin (b sin <b t 



or cos 6 cos ^ sin (p (b / cos (b + (p / sin (b + d)' 



sin sin ^ sin (> cos(b + <'sin (b + () 



n -L r sin (b d). M cos cos ^ s j n ^ 



(j cos <p + (b -. , ~ . , . - J- ~~0 (13) 



em m 4- m Sin (D + Q). cin ^ ^ ' 



T I ' olll v 



From the form of this equation, we are inclined to think that it is the correct so- 

 lution of the problem. 



We must remember that we owe the equation altogether to our neglect of 

 quantities of the second order. We ought, therefore, at once to reject from it all 

 such quantities as interfere with the simplicity of its form. 



VOL. xv. PART i. R 



