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



the only new integral required is 



( 



cos (/> A d<f> ! =8 suppose ; 



and after some reductions it is readily found that 



~_ sin/3 + cos 2 /31og{tan/3 + sec/3J- 



~ 2 sin £ 



Again, 



re ir 



and substituting, as before, the value of M, we find 



dV 

 dM 



- = F a 2 cos/3 1 sin<£'A'#' + H « 2 1 (1 - cos <£') A'df</>' 

 o Jo Jo 



7T 



+M «r 2 A'#'. 



These integrals are the same which have already been consi- 

 dered, and we consequently have, as the parts of the coefficients 

 due to the constants, 



4S- = Vcos 2 y 8P + H « 3 cos y 8(ll-Q)+M ^cos/3R, 

 dh 



^=F « 3 co S/ e(R-Q) + H « 3 (2E 1 -P-2S) + M « 2 (E 1 -S), 



-^- = F « 2 cos /3 R + H a 2 (E 1 - S) .+ M aE v 



The numerical values of P, Q, E, S to three places of decimals 

 when /3 = 60°, are written below for use in examples which 

 follow : — 



P = -702, R = -854, E^l-211. 



Q=-389, S = -690, 



2. The equations (A) are sufficient for the solution of all 

 questions concerning elliptical or semielliptical ribs acted on 

 by forces at the extremities of its axes. For example, take the 

 following case. 



A semielliptic rib is fixed at the springing, and a weight (W) 

 is placed at the extremity of its major axis, which is supposed 

 vertical, to find the stress on any section. Here the equations 



(A) 



