of Physical Quantities to Directions in Space. 243 



ment be the planes XY, and the direction of displacement X. 

 The applied stress (tangential force per unit area) is 



MXT^ , 2 



XY x L ' 



The shear is XZ _1 . Hence the rigidity is of the dimensions 

 ^4- = (MX- 1 Y" 1 ZT- 2 ). 

 The velocity of propagation of a wave of distortion is 

 V=a/^, n— rigidity. Hence 



L J_ V M(XYZ)- 1 ~ T ' 



Thus, the applied stress acts over the face XY, and the dis- 

 turbance is propagated along Z. 



27. Torsion. — Let OABC be the section of a cylinder, axis Z 

 and radius Y, subjected to torsion; aABb the face of an elemen- 

 tary cube edge ~dr } bounded as in the figure; and the twist per 



Fit?. 2. 



CB A 



unit length. The shear experienced by the elementary cube is 

 ultimately ^ =r6, or dimensionally XZ"" 1 (for 6 is of the 



dimensions XY -1 ^ -1 , and ?' = Z). Let P = m*#bethe applied 

 stress (tangential force per unit area). Then [P] = MY _1 T~ 2 . 

 The tangential force over ring of which aABb forms part 

 = (27rrBr)m^ = 27rr 2 n^r, or dimensionally MY _1 T~ 2 (XY) = 

 MXT -2 , and the moment of this round = 27n*n0'dr, or 

 dimensionally MXYT -2 . The moment due to the whole face 



= 2™<^=p^)0, 



where R = OA, and this is of the same dimensions as Znnr^O'dr, 

 for the r 4 in the former expression is of the same dimensions 



