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



the equation just obtained, and the position of the centre of 



M 



pressure of the section will change by the quantity ^ nearly 



(W being supposed small), and thus it can be found whether 

 the arch -ring is of sufficient depth for the stability of the arch 

 under the load W. If, for instance, an arch sustain the pres- 

 sure of an indefinite mass of earth of such frictional tenacity 

 that the linear arch, which is in equilibrium under it, has its 

 axes in the ratio 1 : 2, the thrust at the crown produced by a 



pressure of earth p is ~' 9 and the bending moment produced 



at the crown by a weight W hung from it was shown to be 

 yWa ; the centre of pressure at the crown therefore moves upwards 



4W 

 through a space -^— approximately. It is here supposed that 

 \)p 



no work is done in the mass of earth, that is, that the spreading 

 of the arch caused by W does not cause any additional pressure 

 between the arch and the earth resting on it. And on the 

 other hand, it is also supposed that the change of form pro- 

 duced in the arch by W does not sensibly change the stress 

 produced by the forces originally acting. The first condition is 

 not realized in practice, nor if the arch be originally linear can 

 the second either; but the errors so produced counteract each 

 other more or less. 



Again, suppose a semielliptical rib placed with its major axis 

 horizontal, and let it be fixed at the springing and loaded with 

 a weight W at the crown; the initial section being now at the 

 springing, the equations for the constants will be 



W. ^U 'rfU 



n °~ 2 ' d¥ ~ V > dM Q - V > 



and having determined the values of F and M , the stress on 

 any section will be known ; but, on account of the rigidity of the 

 backing, the actual deviation of the centre of pressure at the 

 crown of a semielliptical arch would probably be very different 

 from that obtained from this solution in the manner indicated 

 above. It is therefore unnecessary to pursue the subject. 



3. I shall next calculate the parts of the coefficients due to the 

 action of uniform fluid pressure on an elliptic quadrant, which 

 do not contain the constants. 



The general value of the function K is 



