Manchester Memoirs, Vol xlii. (1898), No. 0. 13 



where x is a function of x and r satisfying the equation 

 (V- + ^-)X = 0, or 



^~^^+^r^'^;^^ + ^^X = o . . (49;- 



If we now assume that, as regards dependence on x, 

 all our functions vary as ^"""^j then, writing 



so that r denotes the radial displacement, we have 



^'^^^ /Z.0 ox \ dl dx , , 



a = - yj + (^- - ;;r)x, r =^ - /^, 7r+^'Vr ' ^5o). 



The normal stress across any plane perpendicular to 

 ;' is then given by 

 dr 



= ( X +2^- 2-p- le + 



2^ ^3 

 ^V ~dr 



-./,,«(^^-,«^)x-^| . . (5x); 



and the shear parallel to x and r by 



^a d- 2im dl ^^^ dy 



In the reduction of (51) use has been made of the 

 differential equations (47) and (49). 



The pressure in the fluid is subject to the same 

 equation (17) as before, and we find, as the condition to 

 be satisfied at the inner surface (r=a) of the cylindrical 

 tube, 



p„^-.fi^,^.-§^;^. . .(S3).* 



There is a similar condition to be satisfied at the 

 external surface, if the influence of the surrounding fluid 

 is to be taken into account ; see § 2, above. 



* The change of sign, from (23), is due to the fact that p denotes a 

 pressure, and/,.r a tension. 



