so that 

 dU 



Again, 

 dV 



Mr. J. H. Cotterill on Elliptic Ribs. 



Scos 2 /3-l "\ 

 4 J' 



31 



J;?a 4 |psin 2 /3 + 



s„=r M ^= a r MA ^' 



= Jjo« 3 sin 



in *a 



sin' 



fiA'dfi. 



Thus the parts of the coefficients due to constant fluid pres- 

 sure, so far as they arise from M, are 



dJl 



~ =l^ 4 cos/3{(4sin 2 ^-l)E + cos 8 /9}, 



o 

 dJJ 

 dE 



dV 

 dM, 



o *- 



Psin 2 /3 + 



Scos 2 /3- 



? }. 



(B) 



ip« 3 sin 2 /3P, 



P, R, S being the same quantities as in the first section, and whose 

 values were there given for /3=60°. Equations (A) and (B) 

 give the complete coefficients for constant fluid pressure when /3 



is not small, quantities of the order —^ being neglected. 



4. I proceed to apply these values to find the stress on any 

 section of an elliptic tube exposed to uniform fluid pressure ; the 

 tube being supposed of great length so as to eliminate the 

 effect of the ends. Here 



H =paA + H cos <£ — F sin <f>. 



But it is plain that H=pa when <£ = 0, and pb when <£ = 90°, 

 the initial section being at the extremity of the axis major ; 

 therefore the equations for the constants are 



H =0, F = 0, -g-=0; 

 .-. from (A) and (B), 



M^ + £;>«* sin 2 /8P = 0, 



M =-iKsin 2 /3_P. 



p 



For example, let /3 = 45°, then ^- =-542, and 



M = |^ 2 |sin 2 ^--542}, 



