M. de St.-Venant on the Work or Potential of Torsion. 63 

 one edge of the base of the prism, the work performed will be 



BtJ. .' . (4) 



In order that this expression may be equal to the one obtained 

 by substituting the value of <f>, given in (2), in the integral 



^Qdydz, 



the following condition must be satisfied identically : 



provided we limit ourselves to the sufficiently general hypothesis 

 of three planes of symmetry of contexture perpendicular to the 

 axes of x, y, z } in which case 



JW=%** />« = ( ty«- ...... (6) 



Now, since for the torsion we have 



du n du A ,„. 



on denoting by u the longitudinal displacement parallel to x of 

 the point (y, z) of the section, the equation (5) becomes, by sub- 

 stitution and reduction, 



^[^e-^sSt*)]^ • 



(8) 



If we integrate, partially, the first and second terms according 

 to y and z respectively, we shall detach the simple integrals 



a) 



where the indices 0, 1 indicate that the difference between the 

 two values of the /"must be taken relative to the two intersec- 

 tions of the contour of the section by a parallel to y, and by a 

 parallel to z. At these points we have 



dz= +ds cos (n, y), dy= +ds cos {n, z), 



where ds denotes the arc-element of the contour and n the direc- 

 tion of its externally drawn normal; so that the two differences (8#) 

 become sums of the form (ds . . . , extended over the whole con- 



