358 Mr. R. T. Glazebrook on 



And if a be < j8, 

 Intensity 



=*&w{ Tr^T*-^} =V^{sec 2 |-sm^}. (8) 



7T 



In the first case the intensity is clearly least when a is j }9 its 

 value then being 



7T/3 2 sin/3(l— cos/3) 2 ; 



and in the second case it is least when a is 0, and its value is 



7rp 2 sin |3 cos 2 (3. 



This second minimum will be greater than the other if 



cos /3 is > 1 — cos /3, 



i. e. if cos /3 is > -J, 



i. e. if |9 is < 60°. 



If, then, a conical pencil whose semi- vertical angle is less 

 than 60° be passing through the spar, the pencil will be most 

 nearly plane-polarized if the axis of the pencil is at right 

 angles to that of the spar. 



Now if the axis of a conical pencil pass normally through a 

 prism cut as already described, it will be at right angles to the 

 optic axis; and hence the pencil, if its semi- vertical angle be 

 less than 60°, will be more nearly plane-polarized than it would 

 be if the axis occupied any other position. This constitutes a 

 second advantage in favour of the new prism. 



Again, suppose we have a parallel pencil of wave-normals 

 in direction ON, and that the axis round which the prism 

 rotates is OX (fig. 4). In our observations we suppose that 

 these two coincide, and work as if the plane of polarization of 

 the emergent light coincided with that of light travelling along 

 OX, thus introducing an error. The amount of this error will 

 depend of course partly on the angle NX (/3 say), and partly 

 on the angle NXA (</> say), OA being the optic axis. If we 

 know (3 and </> we can calculate the error, and could determine 

 the value to be given to XA or a to make it the least possible. 



But in practice </> may be anything between and 2ir, and 

 /3 anything between and a not very large angle /3 X ; and 

 the question arises, what value must we assign to a in order 

 that the error produced by any chance values of /3 and </> may 

 most probably be as small as possible ? To answer this we 

 require to determine, between these limits for /3 and <f>, the 



