62 M. de St.-Venant on the Work or Potential of Torsion. 



pendicular, respectively, to the coordinates oc y y, z ; "d x , Q , "d z the 

 dilatations, that is to say, the relative elongations of the sides of 

 the elementary parallelopiped parallel to oc,y,z ; and g , g zx , g xy 

 the relative slidings, one over the other and per unit of their dis- 

 tance, of the opposite sides ; in other words, the cosines of the 

 three slightly acute angles formed, after the deformation, by the 

 three pairs of adjacent sides of the parallelopiped*. 



On substituting the values of the six components p, each of 

 which, as is known, is expressible as a linear function of the six 

 elementary and very small deformations d, g> an expression for 

 <I>, of the second degree, is obtained which contains twenty-one 

 terms, involving the squares and products of these quantities. 

 When, as in the case of the simple torsion of a prism around a 

 longitudinal axis parallel to that of x, the slidings g xyy g xz alone 

 enter into consideration, this expression reduces itself to 



*=i(G<4+§G'£L)> (2) 



where Gr and G' are the coefficients of elasticity of sliding in the 

 transverse directions of y and z. It may moreover be demon- 

 strated that each of the terms of the last expression is equal to 

 the sum of the quantities of work corresponding to a dilatation 

 \g or z9 xz i an d to an equal contraction in the directions, inclined 

 to each other at an angle of 45°, of the bisectors of the right 

 angles enclosed by the axes of x, y and those of x, zf. 



The potential of torsion for the volume corresponding to the 

 unit of length of the prism is equal to the integral of the above 

 quantity, previously multiplied by dy dz, extended over the sur- 

 face of a transverse section. 



The moment of torsion M, however — that is to say, the external 

 transverse force capable, when acting at the extremity of a lever 

 whose length is unity, of maintaining around the longitudinal 

 axis a certain acquired torsion 6 which shall develope on the sec- 

 tion the tangential components of pressure p xy ,p xz — has mani- 

 festly the value 



M=^dydz{p xg y--p xy z)\ .... (3) 



and if the force thus applied increases from to the value M, 

 or until the torsion, per unit of length of the prism, has attained 

 the magnitude 6, which latter is that of the angle described by 



* See Comptes Rendus, 1861, vol. liii. p. 1108, or the formula 200 in 

 the complementary appendix to the new (1864) annotated edition of the 

 Lecons of Navier. 



f See the annotated Leqons of Navier, second note of No. 42 of the his- 

 torical part. 



