Closed and Open Tubes. 523 



The condition that there should be no stress at the surface 

 of the rod is now, in terms of yfr, 



yjr — ^T(.r + y 2 )=constant, .... (8) 

 this equation being trim all along a boundary of the section. 

 If the rod is a tube, so that the boundary of the section 

 consists of two closed curves, the constant in (8) is a 

 different constant along each closed curve. 

 The equations of internal equilibrium give 



n d equations (4) and (5) give also 



h + w =0 (10) 



It follows from (4) and (5), which depend upon (9), that 

 ic + *>=/(* + ?», (11) 



where i denotes V — 1- 



It is coiwenient to introduce a function f such that 



£=+- M^+J^ (12) 



so that the boundary condition makes 



f= constant (13) 



along each closed boundary of a section. 

 The torque on a section is 



Q^jJOi'Sx-ySo)^^, .... (14) 



the double integral extending over the whole area of the 

 section. 



Now S > = - n (st- T ' r ) 



Therefore 



(16) 



3? 



a// 



2 M 2 



