﻿in a Rectangular Bar by means of Polarized Liglit. 17 



7T + 21 



or less than the positive quantity 4 sin 3 — ^ — - . The critical 

 value is given by 



^' 2 = 4:sin 3 ( 7, ^)/sin2i. . . . (17) 



For this value the curve passes from the loop to the fork 

 type, and we have a node or cross in the sector DOB. 



In the sector DOH we have again u negative and a loop. 

 The general character of the curve is the same as in the 

 sector EOF. 



Case 77., q<0. For real values qb' 2 sin 21 < u. Now in the 

 sector EOF we have a fork or loop according as 



— 9i 



qb" 2 sin 2i > -4 sin 3 (^-^ \ 



In the sector DOE we have always a loop, qb' 2 sin 2i being 

 always negative, and therefore necessarily < u. This " loop " 

 meets radii from C at one point, and is of the type shown in 

 figs. 8, 9. 



In the sector DC EI we have a fork or loop according as 



2 i 



qb' 2 sin 21 ^ — 4 sin 3 ^ . 

 o 



If we write i r = ^ —i, we have a fork or loop in ECF 

 according as 



2# 2 / 3 siny — 4sm 3 j J> — 4sin 3 ^-, 



or, remembering that 0<i<^- and therefore 2i'/3 and 2i/3 

 are positive acute angles, according as 



3^' 2 >4sin 2 |-'(^' 2 -l), 

 that is, q (and therefore qb' 2 —l) being negative, according as 



sm 3^4#=iy (us) 



Similarly we have a fork or loop in DCH according as 



sm 3-^4p^T) (K) 



PA«. i%. S. G. Vol. 23. No. 133. Jan. 1912. G 



