Mr. J. H. Cotterill on Elliptic Ribs. 33 



Here the equations for the constants are 



H -o dV -0 dU -0 



H °- ' d¥ -°' dM -°> 



the initial section being at the springing; that is, using the 

 values given in (A) and (B), 



r o «cos / 8P + M R + ijo« 2 {(4sin 2 / S-l)R+cos 3 /3}=0, 

 F a cos /3R + M^ + ipa 2 sin 2 j3P = 0. 



For example, suppose the rise of the rib one-fourth the span, 

 so that ft =60°, then these equations become 



351F « + 854M +228^ 2 =0, 



427F a + 1211M + 263p« 2 =0. 



The roughly approximate values of M , H from these equations 

 are M = '08pa 2 , ¥ =-'85pa, 



.-. M. = ±pa 2 sin 2 ft sin 2 <£' + F a cos j3 sin <£' + M 



= ipa 2 {$sin* (/>'-'85sin f + '16} 



= '0Spa 2 when <£' = 90°, that is, at the crown. 



M is a maximum when -j-j-. =0, that is, when 



fsin(//=-85, 

 or 



sin <£'=-56 ((/>' = 34°); 



substituting which value of sin 0', we find 



M=--16p« 2 . 



The points of contrary flexure will be found, on solving the 

 equation M = 0, to be given by <£' = 15° and <£' = 61° nearly. 

 Also 



H =pa& + 'S5pa sin $ 



=%pa + -85pa when = 90° 



= l'S5pa at the crown. 



This solution serves to determine approximately the greatest 

 admissible fluctuation of the surface of water whose weight, 

 when the surface is at its normal height, is sustained by a linear 

 arch ; for a semielliptic arch approaches closely to an hydro- 

 static arch which has its radii of curvature at crown and springing 

 in the same ratio as the corresponding radii of the semiellipse. 



Phil. Mag. S. 4. Vol. 30. No. 200. July 1865. D 



