230 Wave-Surface and Rotation of Polarization Plane. 



Let the actual position be obtained from the standard 

 position by the simplest kind of torsion round the axis 

 (St.-Venant' , s torsion-problem for a circular cylinder). Use 

 columnar coordinates R, cf>, z for the actual position of matter, 

 the axis being* the axis of torsion. Let i, j, k (i and j being 

 functions of the position of a point) be unit vectors in the 

 directions of rfR, dcj>, and dz respectively. Thus we put 



p f = eg kl2h pe -zlcj2k i (13) 



so that the torsion is a radian per length h along the axis. 

 For brevity put 



e*V 2h = r, (14) 



and note that rkr~ 1 =Ic. Thus 



dp'=rdpr~ l + 2Y . Ydrr~ l . p' 



= r{dp + h- 1 dzYkp}r- 1 . 



Remembering that dz = -SMp we see that (co being an arbi- 

 trary vector), 



X(o = rxocor-\ (15) 



where 



X co = co-h- l VkpScoJc (16) 



Now assuming that 



c = c , ii—fiQ t (17) 



where c and /jl are constant scalars, we see by eq. (9) § 9 of 

 " M.T.E.," that 



c ' = c oXX'> ^ = i"oX%', .... (18) 



since by physical considerations it is obvious that m = l. 

 Now 



XX f(0=r XoXo r ~ }oir ' r ~ l 



= r{r- i cor-h- l YkpSkr- ] (or-h- i k^kpr- ] cor-h- 2 YJcp^kpr- 1 cor\r 

 = co - h- l Ykp'Ska> - h- l k&kp'a - h- 2 Vkp'&kp'w, 

 or putting Vfy/=R/, 



^'G)==©~A- , R(yS/«+^S>)-A- 2 R 2 >S>. . (19) 



This gives XX?*=h (20) 



and if tan^ + A-'R tan0— 1 = 0, .... (21) 



or tan20=2/iR- 1 ; (22) 



XX ,(0 = m C0 ^Q where g> ==/ cos 6 j- k sin 0. . (23) 





