Air-cavities on the Strength of Materials. 75 



because the boundary of the tunnel, r = a, must become a 

 stream line, and therefore give a constant value to i^ 2 . The 



velocity along the tunnel is — -~, and is therefore 2V at 



the sides, as stated above. 



The angular momentum of the fluid is altered, owing to the 

 tunnel, by 



C dx 



that is 1 (^""^i) j~; Ground the boundary. For a circular 



boundary this is equal to Ya J cos 2 6 ds, or 7ra 2 V. The rigidity 

 of the surrounding parts is therefore diminished by the pre- 

 sence of the cavity, just as if the shearing over the material 

 which originally occupied its place were reversed in direction; 

 the loss of rigidity is due in equal proportion to the removal 

 of the matter and the release on the constraint of the sur- 

 rounding parts. 



The case when the section of the cavity is an elliptic cylinder 

 is of interest, as it illustrates the effect of making it more 

 and more flat until it is finally a mere crack for which the 

 strain is theoretically infinite at the edge. The corresponding 

 hydrodynamical problem has been solved by Prof. Lamb*: 

 his value of the stream function i/r, which may easily be 



verified, is 





V.* : 



where a, b are the sea mixes of the ellipse, V is the velocity of 

 the stream past it parallel to the axis b, and £, y are the con- 

 jugate functions given by 



x-\-iy — c sin (f + tv\) . 



The velocity at the end of the longer axis is the value of d\jr/cLv 

 when j/ = 0, that is, when % = ±7r, and is found to be 



K> 



Thus in the elastic problem the increase of shear produced by 

 the cavity is the original shear multiplied by ajb. 



* Quart. Journ. of Math. 1875: " Fluid Motion," p. 90. 



