22 Mr. J. H. Cotterill on Elliptic Ribs. 



F=K + F cos<£ + H sin</>, 

 K=pp+ -tt- + H cos<£— F sin<£, 



M= i Kpfy + YA /ocos0^+H o i /3sin^^ + M , 



Jo *^o Jo 



and that the three constants F , H , M 0J which depend on the 

 physical constitution of the rib, may be approximately deter- 

 mined by making 



TT f/M 2 H 2 \, 



a minimum. 



It was further remarked that when, as in the present case, the 

 angle subtended by the rib was considerable, and the distribution 

 of the load different from that necessary for the equilibrium of a 



linear arch of like form, then it was sufficient to use 





for U. In fact in the present case, whatever be the load, the 

 error in the parts of the coefficients containing the constants is 



of the order -%, where k is the radius of gyration of the transverse 



W- 

 section, and a the semi axis major ; and -%, even in such an ex- 



treme case as the link of a chain, is always very small*. The 

 same thing is true for the parts of the coefficients which do not 

 contain the constants, provided that the load is not distributed 

 as for a linear arch. But the problems in which this is the case, 

 though common in practice, it is probably not going too far to say, 

 cannot at present be satisfactorily treated, the question being a 

 complex case of crushing by bending, and involving other difficul- 

 ties besides. Thus for practical purposes it is generally useless to 

 calculate any terms in the coefficients arising from H when the 

 rib subtends a large angle, though there may be cases in which 

 it is advisable to consider such of them as do not involve the 

 constants. 



I proceed therefore to calculate the differential coefficients 



-^ri -f=- i -ST- from the simpler value of U with reference to an 

 ar aii dm r 



elliptic quadrant. 



* The effect of curvature on the strength of beams appears to me im- 

 portant. I hope to attempt to estimate it at a future time ; pending which 

 investigation the question of the stress on links of a chain is omitted, 

 though naturally belonging to this paper. 



