388 L Mr. J. H. Cotterill on the Equilibrium of 



which equations determine the stress on every section. From 

 them it appears that %w'r is the horizontal thrust nearly ; also 

 that M, which for small values of cj> is negative, increases to a 

 maximum negative vaiue when cos<£=§, then diminishes to 

 zero and changes sign when cos<£ = J, afterwards increasing 

 positively to the springing when its value is |-w/r 2 . 



5. By aid of the general expressions given in (2), the coeffi- 

 cients may be determined for a rib of any form (such that p is 

 a simple function of </>) loaded in any manner, though the pro- 

 cesses are of great complexity, which, however, is probably un- 

 avoidable by any method. In cases in which the thrust is very 

 great, it is moreover necessary to take the work done by it 

 into account, whereby a large number of fresh terms will be 

 introduced into the coefficients, but the process is rather com- 

 plex than difficult. When the rise of the rib is small compared 

 with its span, and it has a vertical load, it is simpler to use 

 rectangular equations of equilibrium ; but it is unnecessary to 

 enter into details, as the process is similar to the one already 

 employed. The results are the same as those Professor Ran- 

 kine has obtained in his work on Civil Engineering. Care 

 must be taken to include every part of the structure in which 

 work is done ; thus if abutments yield, the work done in over- 

 coming their resistance must be estimated and added to U. 



In the case of stone and brick arches, not only is work done 

 in the arch ring, but also in the mass of material resting on it. 

 The same law of Least Action, however, governs the distribu- 

 tion of the work ; and if it were possible to estimate its whole 

 amount, the problem of the arch might, by application of that 

 law, be completely solved. 



In my former article I endeavoured to show that if X, Y, Z be 

 the components of one of the forces acting on a perfectly elastic 

 body, u, v, w the displacements of its point of application pro- 

 duced by the action of the forces on the body, then 



dV _ d\J _ dU_ 

 dX~ U '' dY~ V ' dZ ~ W ' 



But the reasoning is not so conclusive as the following. 

 Since 



2U = 2{Xw + Yt> + Za;} 



2.SU=2{XSM + YS.i; + Z.SM;}+2{tt.SX + f;.8Y + tt;.SZ}; 



but %yX.8u + Y ,8v + Z .8w^ is the increment of energy ex- 

 pended, which, by the law of conservation of energy, is equal to 



