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



tour. By this well-known mode of transformation, first employed 

 by Lagrange, the equation (8) to be verified becomes 



(u ds \^(~ - 6z\ cos {n, y) + G' f~ + 6y\ cos [n, *)] "] 



^$#4*$ + e£]^ j' (9) 



Now the squares of the second, and of the first parenthesis, 

 each equated to zero, give precisely and respectively the indefi- 

 nite differential equation applicable to all points of a section, and 

 the definite differential equation having reference to points on 

 the contour; which equations I established in 1847 and in 1853, 

 and presented* as containing, implicitly, the whole theory of the 

 torsion of prisms having any base whatever and composed of 

 matter whose contexture was doubly symmetrical, as in the case 

 to which, for simplicity, we have limited ourselves in the above 

 considerations. Moreover, in the several cases of circular, ellip- 

 tic, triangular, equilateral, &c. sections, I obtained by calculation 



the same value for the potential of torsion from both its expres- 

 n 



sions, ff<& dy dz and M-. 



It will be seen that the consideration of the potential or work 

 of elasticity completely verifies the general and special results to 

 which I was led, in a different manner, on establishing the theory 

 of torsion. This consideration, moreover, is evidently connected 

 with the methods of the Mecanique Analytique employed by 

 Navier in 1821, and recently verified again by several mathe- 

 matical physicists, amongst whom may be mentioned the re- 

 gretted Clapeyron; for his theorem, after supplying the for- 

 gotten factor ^, resolves itself to a particular case of the equa- 

 tion (1). 



The calculation of the potential of torsion has also, in itself, a 

 practical value; for the helical springs frequently opposed to 

 shocks of different kinds, work almost wholly by the torsion of 

 their threads, as I showed in 1843, and as was also remarked 

 by Binet in 1814, and M. Giulio in 1840, and recently by rail- 

 way engineers. 



* Memoires des Savants Etrangers, vol. xiv. ; or note to the No. 156 of 

 the Leqnns of Navier. 



