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



when pushed towards the limit of strength of the materials. 

 In the most important examples we shall be concerned only 

 with shears. 



It will appear on consideration that a small spherical cavity 

 in a column or other mass under tension or compression 

 cannot seriously affect its strength. For its strength could 

 be reduced only by an increase of shear in the neighbourhood 

 of the cavity. Now this shear must act all round it in the 

 planes containing that diameter which lies along the direction 

 of the stress, and at the free surface of the cavity it must be 

 zero, in the absence of surface-tractions there ; hence the 

 shear is diminished in the neighbourhood of the cavity. The 

 compression may be slightly increased in a ratio depending on 

 that of the area of the section of the cavity to the area of the 

 section of the shaft. The same argument applies of course to 

 any symmetrical form of cavity, and generally to any cavity 

 of regular shape. 



The case is different, however, when the cavity exists in a 

 shaft which transmits a couple. If we suppose the cavity to 

 consist of a narrow tunnel bored down the length of the shaft, 

 we may make use of the result of St. Yenant's torsion problem. 

 The distribution of the shear across the section of the shaft is 

 simply and succinctly expressed by hydrodynamical analogy*. 

 If a cylindrical shell of the same form of cross section as the 

 shaft is filled with frictionless fluid and is set in rotation, the 

 velocity of the fluid relative to the shell will at each point 

 represent the shear, in direction and magnitude ; and the 

 momentum of the fluid relative to the shell, which must 

 necessarily have no linear component, will be proportional to 

 the torsional rigidity of the shaft, For the present purpose 

 it is convenient to state the proposition in a form less practi- 

 cally realizable : suppose the shell fixed and the fluid circula- 

 ting inside it with uniform vorticity, the velocity at each point 

 will represent the shear, and its resultant momentum (angular) 

 will be proportional to the rigidity of the shaft. 



Now the result of boring a small tunnel will be to modify 

 the velocity system in the neighbourhood in the same way as 

 a solid cylinder changes the velocities in a stream flowing- 

 past it. The velocities in front and rear are reduced to zero, 

 while those at the sides are doubled. A tunnel of this kind 

 therefore halves the strength of the portion of the shaft in 

 which it is situated ; and the same statement practically 

 applies to any cavity of elongated form and circular section 

 which lies parallel to the axis of the shaft. The possibility of 



* Thomson and Tait's 'Natural Philosophy/ § 705. 



