726 PEOFESSOE MACQUOEN EANKINE ON THE DECOMPOSITION OF 



and make 



h _ f/Vz.dydz f/Vy.dydz m . 



ffz* .dydz > C ~ fftf .dydz ' ' V')- 



The denominators of these expressions are the geometrical factors of the 

 moments of inertia of the cross-section of the prism round y and z respectively. 

 It is obviously immaterial which end of the prism is chosen for the compu- 

 tations. 



(11.) Example of Homalostrephic Pressures ; Uniform Twisting Sti^ess in an 

 Elliptic Cylinder. 



Let O x be the longitudinal axis of an Elliptic Cylinder, O y and O z parallel 

 to its greatest and least diameters, and let 2 p and 2q be those diameters. 

 Then for the ends of the cylinder, 



n x = ± 1 ; n y = 0;n z = 0; 



and for the elliptic surface 



ym zm 



n x = u ; n y = —2 ; n z = — % ', 



when 



1 



m = 



\v* + 



i p* 



2 ' 



I 



Let there be two Homalostrephic systems of Tangential Stresses, thus re- 

 presented 



T y = by ; T z = cz . 



Then the external pressures constituting a pair of Homalostrephic systems com- 

 bined, will be as follows : — 

 On the ends, 



for n x = ± 1 ; P = ; Q = ± cz ; E = ± by, 

 on the elliptic surface 



P = n y T 2 + n t T y = myz : . { p- + -^ j ; Q = ; R = . 



Now let the external pressures be subject to the condition that the pressure on 

 the elliptic surface shall be everywhere null ; then we must have 



v 

 Consequently, let 



c b 



