FORCES EXTERNALLY APPLIED TO AN ELASTIC SOLID. 



lp 2 = 



727 



cq 2 = a , 



and we shall have for the internal stresses 



y ~ p 2 ' 2 ~ q 2 



(18), 



and for the external pressures at the ends of the cylinders, which are wholly 

 tangential ; 



riw not 



(19). 



az -p, ay 



for % = ± 1 ; Q = =f -^ ; K = ± — g 



The resultant tangential pressure at any point (y, z) of one end of the cylinder, 

 or of one of its sections, 



Jq* + R 2 = _ ; being pro. 

 and its direction-cosines are 



Q 



R 



zm, 



= zb 



ym 



(20), 



showing it to be a tangent to an ellipse similar and concentric to the outline of 

 the end of the cylinder, and proportional to the diameter of that ellipse to 

 which it is parallel. 



The total moment of torsion M, that is, the moment about x of the forces 

 applied to one end of the cylinder, is as follows : — 



ff (Ry - Qz) dy dz = ajjr(^ + ^)dydz , 

 which, because 



lf_ z^ 



p 2 + q 2 - 1 , 



becomes 



M = a x area of elliptic base = irapq 



which equation serves to determine the constant a when the moment of torsion 

 is given, viz., 



M 



(21), 



a = 



irpq 



The tangential stresses at the extremities of the greatest and least diameters 

 of the ends are inversely as those diameters, viz., 



T = - = 



" p irp'q 



a M 



/7 J 1* — 77 — ~ 



(22). 



q irpq" 



These results agree with those obtained by Cauchy, but have the peculiarity 

 of being arrived at independently of the co-efficient of elasticity of the sub- 

 stance. 



VOL. XXVI. TART TV. 9 C 



