384 Mr. J. H. Cotterill on the Equilibrium of 



would oe the potential, by differentiation of which the values of 

 the constants could be determined ; but inasmuch as this func- 

 tion is of great complexity, it is necessary to proceed otherwise, 

 and calculate the several differential coefficients separately. By 

 differentiation, 



so that if we merge the factor EI in U, 



-tpT = \ mp\ p . cos <p . dcp d(p ; 



Mp \ p . sin $ . d<f> . d<f>, 



Jo 



In calculating these coefficients for a given form of beam, it is 

 convenient to observe that they consist of two parts— one due to 

 the arbitrary constants, and the same for every load, and the 

 other due to the mode of distribution of the load. If the beam 

 be acted on by detached forces, then the portion due to the con- 

 stants must be estimated afresh for every part of the rib lying 

 between those detached forces ; but the portion of the coefficients 

 due to the distributed load may be taken throughout the rib, 

 without reference to any detached forces which may act upon it*, 



3. I proceed to calculate the values of the parts of the coeffi- 

 cients due to the arbitrary constants in the case of a circular rib 

 of radius r. Here p = r, and consequently the coefficients are 



so also 



dU 

 dE 



dV 



d¥, 



-=r 2 l M.sin0.«ty; ~ = r 2 l M(l -cos <£).<% 

 o Jo «n Jo 



Md<j>; 



o 



dV 



dM f 



and so far as the arbitrary constants are concerned, 

 M = M + F . r . sin 0-h H r(l — cos0). 



On substitution and integration, we find 



* It is of course supposed that the change of form of the rib is incon- 

 siderable ; otherwise a preliminary question would have to be solved — 

 namely 3 to find p as a function of <p by means of the calculus of variations. 



