FOECES EXTERNALLY APPLIED TO AN ELASTIC SOLID. 725 



Now each term of A is null, because the origin is the centre of gravity of 

 the body ; and each term of E and F is null, because for each positive 

 element dy dz of the projection of the body's surface there is an equal negative 

 element. 



. " . A = ; E = ; F = ; 



and the system of pressures is Arrhopic. Q. E . D. 



Corollary 1. Every Homalostrephic System of Pressures is Arrhopic. 



Corollary 2. The subtraction from any Arrhopic system of pressures, of 

 a Homalocamptic or Homalostrephic system, leaves an Arrhopic residual 

 system. 



(10.) Example. — Homalocamptic Pressures, Uniform Bending Stress in a 

 Prism. — The consideration of Homalocamptic and Homalostrephic Pressures 

 does not, like that of Homalotatic and Antibarytic Pressures, form an essential 

 part of the solution of every problem of the internal equilibrium of an Elastic 

 Solid ; but is to be employed only when it evidently tends to simplify the 

 problem. 



The most generally useful example of a single system of Homalocamptic 

 Pressures is the following : — 



Let the axis of x be that of a prismatic pillar, traversing its centre of 

 gravity. Then for the ends of the prism respectively, 



n x = ± 1 ; n y = ; n, = j 



and for the sides, n x = 0. Let b z = c y be the equation of any plane passing 

 through the centre of the prism, and let each element of the ends of the prism 

 be acted on by normal pressures, proportional to the distance from that plane, 

 tensile towards + y, and compressive towards — y. Then for 



n x = ±1 ,P = ±(by + cz) ; Q = ; It =-- ;-\ 



and for > . (15), 



% = , P = 0, Q = 0, R = ; ) 



and the internal stresses are 



N„ = by + cz ; N„ = N z = T, = T y = T z = . (16). 



When a system of normal pressures is distributed in any manner on the 

 two ends of a prism, the system of Homalocamptic Pressures which approxi- 

 mates most nearly to the actual system is found by computing the moments of 

 the pressures on one end relatively to the planes xy and zx, viz., 



(fVz dy dz and —JJ'Ty . dy dz , 



