136 UNIVERSITY OF COLORADO STUDIES 



There are as many terms in the summation as there are n divisions 

 in the fibre. If n becomes a differential the summation becomes an 

 integration. 



For a fibre at unit's distance from the neutral surface, z is equal 

 to unity, hence: — 



l^,nV'=0. (2) 



From the principles of the common theory of flexure: — 



M is the bending moment at any section in the rib, I the mo- 

 ment of inertia of that section with respect to an axis at right angles 

 to the neutral curve and in the neutral surface of the rib, and E the 

 coefficient of elasticity of the material of the rib. 



Therefore:— 2^,/iP' =2^,— =0. (4) 



If the lengths n, measured on the neutral curve, are selected so 



n 

 that— is a constant for all parts into which the rib is divided and if 



n 

 at the same time E is constant, then — is constant, and: — 



Jill 



Therefore:— 2y,M=0. (6) 



n 

 It follows, therefore, when — is constant that the sum of the 



bending moments throughout the entire length of the rib must be 

 zero. When the cross section of a rib varies, n must vary. For 

 ribs of constant cross section n becomes constant. 



Again since the points V and V ' , see plate I, are fixed, while the 

 rib may deflect and distort in any manner between these points, the 



