184, 185] TORSION OF PRISMS. 489 



where I is the moment of inertia of the section about the 

 If we call 



122) ff( X |T - y fl) dS = (1 - q)ff(*> + f) dS 

 then 



123) N = 



and the moment per unit twist per unit length qpl is called the 

 torsional rigidity of the bar, and q is de Saint -Tenant's torsion -factor. 

 For the circle q = 1, for the ellipse q = 2 ^ fc2 2 , since I = ^-ab(a 2 + & 2 ). 

 If S = nab is the area of the ellipse, the rigidity may be written, 



ii S*. 



and by a generalization of this formula, de Saint -Venant writes 



Having solved the problem for a great variety of sections, he found 

 that, when the section is not very elongated, and has no reentrant 

 angles, K varies only between .0228 and .026, its value for the ellipse 

 being .02533. We may thus put in practice 



obtaining a most valuable engineering formula. Considering the 

 dimensions of S and I, we see that for similar cross -sections, the 

 rigidity varies as the square of the area of the section, as stated by 

 Coulomb, but for different sections the results differ much from those 

 of the old theory, in which q was supposed to be unity. 



. 185. Flexion. For the third case we put 

 III. ,+(), 



124) u = - y{ 2 -M0 2 -r% v 



125) Z z = E d ^ = Ea l x ) 

 The force on any section, 



Z= I i Z z dS = Ea I I 



since the origin is the center of mass of the section. In this case 

 the couples L and N vanish, while 



