74 Mr. J. Larmor on the Influence of Flaws and 



These equations show that w is the velocity-potential of the 

 absolute motion in space of liquid contained in a box rotating 

 with angular velocity co. 



The tractions exerted across the section of the shaft are 

 (with unit rigidity) 



v dw A7 dw 



X=m/+ —, Y~—coxi- — . 

 ay dx 



These must vanish when integrated over the area of the 

 section ; therefore the box containing liquid must be supposed 

 to rotate round an axis through the centre of gravity of the 

 section. 



The couple transmitted across the section is 



G=-J(Xy-Y^)^S 



it is therefore less than the couple due to simple torsion by 

 the absolute angular momentum of the liquid. 



Also X, Y are the component velocities of liquid circula- 

 ting in a fixed box with vorticity w ; its resultant velocity 

 represents the shear at each point, and its angular momentum 

 represents the couple transmitted. Its linear momentum is 

 null. 



The analysis of this well-known result has been here indi- 

 cated in full, partly in order to point out that in the first form 

 of the analogy in which the box is made to rotate, the velocity 

 of the liquid relative to the box represents the shear whatever 

 be the axis of rotation, but the angular momentum of the 

 liquid represents the correction to the rigidity only when 

 taken about that axis for which its value is least, viz. the axis 

 through the centre of gravity of the section. If the motion 

 is referred to any other axis, as in the case of a rectangle 

 bounded by two concentric arcs and tw 7 o radii (Thomson and 

 Tait, § 707), then from the angular momentum round that 

 axis must be subtracted the moment of the linear momentum 

 of the whole mass of fluid supposed collected at its centre of 

 gravity. 



Suppose, now, a cylindrical tunnel of small circular section 

 bored down the shaft at a place where the velocity of the 

 rotational fluid motion is V ; the stream function near it will 



a 2 

 be changed from the form -v/^ = \y to the form ^ 2 = V?/ — V-^y, 



