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



the equations for determining M and H would be 



dM ~ v ' dH > 



W 



F being replaced by — — • 



4. I shall next calculate the parts of the coefficients due to a 

 continuous load on a circular rib. 



First, for the parts due to the weight of the rib, let w be the 

 weight of the rib per unit of length of the axis, so that the weight 

 of a frustum cut off by planes very near to each other is w . ds; 

 then we have 



p = w . cos cf) ; q = w . sin <f>, 



K== { 1 + ^}" l {^-^w} 

 =wr \ 1+ W 2 S p-***) 



= — wr . <fi . cos^>, 

 M = 1 K/^<£ = — wr 2 I <p . cos (j> . d(j> 



= —wr 



i(f> . sinc£ + cos^ — lj- 



On substitution of this value of M in the general values of the 

 coefficients formerly given, the parts of those coefficients due to 

 the weight of the rib are found to be 



^H_ = -i^T 4 {2^(^-sin20)~3.cos20 + 8.cos^-5}, 



~ = -±wr 4 {2$ {cos 2$ -4>.cos(j) -6) + 24 sin ^-3 .sin 20}, 

 dV 



dM 



wr 



4 {<£(cos0 + l)— 2.sin$}. 



If the rib be supposed loaded with a vertical load of uniform 

 horizontal intensity w', then 



p = w ! . cos 2 cf) ; q = w'. &incj) . cos 0. 



And by a similar process we find for the parts of the coefficients 



* The consideration of the small amount of work done in compressing 

 the central bar is omitted. The question has been inadvertently treated as 

 if W were a thrust instead of a pull on the chain ; to obtain results correct 

 in sign, W must be changed into — W wherever it occurs. 



