724 PROFESSOR MACQUORN RANKINE ON THE DECOMPOSITION OF 



ordinate perpendicular to that direction, measured from a surface passing 

 through the body's centre of gravity. That is to say, the centre of gravity 

 being the origin, let Ox be any direction in which there is a normal stress N x , 

 let b, c, be two constants, and let 



N, = by + cz . . . (11), 



which fulfils of itself the differential equations (7) of internal equilibrium. Then 

 by the equations (4) — 



P = n x N* = n x (by' + czf) . . (12), 



will represent a system of Homalocamptic Pressures. 



Homalostrephic Pressures, or Pressures of Uniform Twisting. — A system of 

 pressures corresponding to a system of tangential internal stresses uniform as 

 to the pair of internal directions in which they act, and whose intensity is a 

 linear function of an ordinate perpendicular to those directions, and measured 

 from a plane passing through the body's centre of gravity ; that is to say, for 

 example, let 



T x = ax . . . (13), 



which fulfils of itself the equations (7). Then by the equations (4) — 



P = ; Q = n z T x = a of n z ; ) 



R= n y T x =ax'n y \ ' ( 14 '' 



A system of Homalostrephic Pressures is equivalent to a pair of systems of 

 Homalocamptic Pressures, making angles of 45° with the directions of the 

 Homalostrephic Pressures. For let Oy 1} Oz iy be a pair of axes in the plane 

 yz, inclined at 45° to y and z, so that + y lies between 4- y l and z x . Then is 

 T x equivalent to a pair of normal stresses, 



■N VI = -T x = -ax) 

 N„ = + T^ = + a x . 



Prop. V. — Theorem. Every Homalocamptic System of Pressures is Ar- 

 rhopic. 



For the following are the Rhopimetric co-efficients derived from equa- 

 tion (12)— 



B = 0; C = 0; D = 0; 



A = fjfPx' d 2 s = bffx' y . dy dz + cffz' of . dydz ; 

 E =f/Vz' d 2 s = bffy' z' .dydz + cffz"" .dydz; 

 F = f/Vy' d*s = bffy' 2 . dy dz + cfjy' z . dydz; 



(dy dz being as usual considered as + w or- ve , according to the sign of n x ). 



