Polarizing Prisms. 



359 



mean of the values, irrespective 

 of sign, of the amplitudes of the 

 vibrations normal to the plane 

 OAX. 



Let Q be the direction of 

 vibration for the wave-normal 

 N, and Q R perpendicular on 

 the plane AX. Then the dis- 

 placement normal to the plane 

 AOX is p sin QR ; and, with the 

 same notation as before, the ave- 

 rage displacement 



Fig. 4. 



= <M sin/3sin<£cot0d0 (9) 



Also sin 6d0= sin a sin /3 sin <j> dcp 



Hence average displacement 

 _ P 



-[ 'cos 6 dQ (10) 



27rsin 



We must consider now the limits between which the inte- 

 gral is to be taken. Fig. 5. 



Describe a circle (fig. 5) pass- 

 ing through N with X as centre, 



and cutting A X in L and M, ^/a. 



and if possible let JSTj be a point 

 on this circle such that 



Nl A=f. 



Suppose we treat displacements 

 to the right of AX as positive. 

 When N coincides with L, 



If N lies between L and Ni, Q is to the right of AX, as in 

 figure, and the displacement is positive. We must therefore 



IT 



integrate for 6 from a — 13 to x . But if N lie between N x and 



M, Q will be to the left of AX, and the displacement will be 

 negative. Thus, to get the whole average displacement, irre- 

 spective of sign, we have to subtract the value of the integral 



from - to a +/3. If, however, a +/3 is < ^, no position such 



as Ni can be found, and we have to integrate straight from 



