1866.] 



Mr. Tarn on the Stability of Domes. 



185 



thickness of the pier, we proceed to take their moments, together with 

 those of the part of the rib below EF, and the pier itself, about the outer 

 bottom edge S of the pier. 



If we call i/S=b,ihe perpendicular distance from S of the direction of N, 

 then 



5 = H + r.cos0, . (2) 



where H is the height RS of the pier or '*drum." 



Let Se—tty the distance from S of a vertical dropped from E; F=:the 

 weight of the portion of the rib below EF ; c=the perpendicular distance 

 from S of a vertical from the centre of gravity of the lower portion of the 

 rib whose weight is F; Q=the weight of the portion of the "drum" on 

 which the rib stands, and q=t]ie distance from S of a vertical from its 

 centre of gravity ; we have then (in equilibrium) the equation 



N.5=P.« + Q.^ + F.c, (3) 



which I will call the Equation for Equilibrium. From this equation we 

 can find the thickness (t) of the pier necessary to produce equilibrium. 



In order, however, that we may have stability in the structure, we must 

 multiply N6 by the coefficient of stability , which we may take as 2. The 

 equation from which to find the value of t then becomes 



2N.6 = P.« + Q.(? + F.c, (4) 



which I will call the Equation of Stability. 



I now proceed to find the values of P . a, Q . and F . c. 

 The value of P was previously found to be 



■R3_*.3 



P = a.0 \-co?>Q)- — ^; 



o 



and if we substitute for ^ and Q their values, 



P=-007656 a(R3-r3). 

 Also o!=Se=^ + r (1 — sin 0) = ^ + '06031 r ; so that we find for the value 

 of P . «, 



P«=-007656a(R3~r3)^+-0004618a.(R^-r^) (5) 



The value of F is found by means of the integral (A), taking the limits 



from 0=0 to 0=- ; which gives us 

 2 



F=a.^ .cos0^^^ 



3 



= -00398a(R=^-r3). 



Let be the distance from O on OA of the perpendicular dropped from 

 the centre of gravity of the portion of the rib whose weight is F. Then 

 the integral (B) gives us, by taking the same limits of Q as for F, 



R 2 



