1866.] 



Mr. Tarn on the Stability of Domes. 



183 



This method is adopted by Venturoli in treating on the equilibrium of 

 domes. 



Fig. 1 represents a rib of the form above described cut out of the dome, 

 and the corresponding part of the " drum" which sustains it. The arch 

 will have a tendency to fall in at the crown C D, and open outwards at the 

 haunches E F, causing the joint C D to open at D, and that at E F to open 

 at F. 



We may therefore suppose that N, the thrust of the corresponding rib 

 on the opposite side, acts at C. 



We shall now find the effect of the force N upon a given joint EF. Let 

 P be the weight of the portion of rib above EF ; x the perpendicular dis- 

 tance from E of a vertical from the centre of gravity of P ; y the vertical 

 distance of CN from E. Then the moments of these forces about E are 

 Vx and Ny ; and in order that there may be equilibrium, we must have N 



equal to the greatest value of P . -. We have therefore to express P - in 



y y 



terms of Q, the angle which FE makes with the vertical, and find what value 

 of Q makes P - a maximum. 



y 



Let r be the internal and R the external radius of the sphere, I the 

 weight of a cubic foot of the material of which it is composed. Then the 

 volume of any portion of the rib {p qrs CD) is 



V=fjJr»sin0.</0.«fr.t/0; . (A) 



so that P=a.V=a.0(l-cos0)5^I^', taking limits from 0=0. 



VOL. XV. R 



