486 X. STATICS OF DEFORMABLE BODIES. 



apply stresses varying in this manner, we make use, in this and the 

 other cases, of de Saint -Tenant's principle of equipollent loads, viz: 



If the cross- section is small with respect to the length, stresses 

 applied to a body near the ends produce approximately the same 

 effect if their statical resultants and moments are the same. 



Consequently we may apply to the end faces or to the convex 

 surface near them the forces and couples, 



X = I I X,dS = iifi I JydS = 0, 

 Y = / / F, dS = - [ift I i xdS = 0, 



Z= I J Z 2 dS = 0, 



114) L =( V Z > ~ * Y ') dS = 



M= C f(zX z - xZ z }dS = pp C iyzdS = 0, 







The twisting couple divided by the rotation per unit of length, 

 115) ^ 



(o z 



is proportional to the fourth power of the radius. This is the law 

 announced by Coulomb, in his work on the torsion -balance. This 

 factor of the applied couple multiplying the twist per unit length is 

 called the torsional rigidity of the prism. Thus the shear -modulus p 

 may be determined by experiments on torsion. 



For other contours than the circle it is convenient to introduce 

 into the problem instead of V its so-called conjugate function ^, 

 defined by the equations, 



3V r OW cV dW 



116) - = -jr-J H = -7f-> 



ex cy cy ex 



