﻿716 Prof. S. R. Milner on 



which twisting occurs in a corresponding tube. In a twisted 

 tube .the bounding lines will not be parallel to the tube axis 

 but will tend to run in spiral curves around it. Fig. 2 

 (p. 712) shows the effect of a twist on the A'-tube. Let 



xM^-, xM^- be the angles Y'0%, Z'0% through 

 OX O'*' 



which O'Y', O'Z' have been rotated from OY, OZ in the 



plane yz, while Y'YA, Z'ZB are the angles yM^ , zM^ 



oy oz 

 through which the ^-lines through Y and Z have been 

 rotated in the planes zx, wy respectively from the #-line 

 through 0. We see at once by expressing AY', BZ' in 

 terms of these that 



"60 ex _ pfa"' 6 X y _ 0<j>zy f , . „. 



oy ox ' oz '' o* ' ' ' { } 



When Y'O'Yj, Z'O'Zx are not equal there is distortion as 

 well as twist ; the two may be separated in the usual way 

 by writing 



^yz=~f^= —i(<l>zy'-<l>zy") 



- ,,,■• • ( 14 ) 



— Qzv hJ 



o4> 

 .J^ 3 measures the pure twist, or the rate at which the 



rectangle y^z x is undergoing rotation in its own plane in 

 the direction from y to z as we proceed along the .t'-tube^ 



while -Tf* expresses the rate at which it undergoes dis- 

 tortion. These considerations are applicable to any tube. 

 For the x- and Z-tubcs erected on the same base y\Z\, 



y yg 9 j;f g represent the respective twists. The y- and 



2-tubes, based on x-J,^ are twisted in the xl plane ; let 



_ x , ^f x be their respective twists. We then have from 

 oy o^ 



(13) and (14), and corresponding formulae for the other 



tubes 



pOlz . ~d@yl _ , 9 Bfez O^zl , oftrg _ • q "d<l>xl 



oy oz ~ +w QZ ' 3* + ol ~ + ~ oz ' 



oOzx , "60 xy _ _ 2 ofyyz o@lv , o^® yx _ _ -9 Bf; 

 oy 0^ ' o% ' o^ o^ " oy ' 



