118 Lord Rayleigh. On the [Dec. 13, 



occur in the immediate neighbourhood of the points of applica- 

 tion.* 



The work done upon the cylinder by the forces F during the 

 hypothetical displacement indicated by A 5 , &c., will be by (36) 



F2s (aSAj'+^SB,') (l + cosw), 

 so that the equations of equilibrium are 



" ==b dv -o 



~ ~ 



TT-T = (1 + cos STT) saF, -^-7 = (1 + cos STT) 



Thus for all values of s, 



A, = B = 0; 



and for odd values of s, 



A; = B/ = o. 



But when s is even, 



and the displacement M; at any point (2, 0) is given by 



w = 2 (A 2 'a + B 3 '^) cos 20 + 4 (A/a + B 4 Y) cos 40+ ............ (42), 



where A 2 ', B 3 ', A 4 ', .... are determined by (40), (41). 



If the cylinder be moderately long in proportion to its diameter, 

 the second term in the left hand member of (41) may be neglected, so 

 that 



P B/ = A/ 

 3a 3 Z L ~ ' a ' 



In this case (42) may be written 



...}... (43), 



* Whatever the curvature of the surface, an area up an it may be taken so small 

 as to behave like a plane, and therefore bend, in violation of Gauss's condition, when 

 subjected to a force which is so nearly discontinuous that it varies sensibly within 

 the area. 



