266 



Mr. Tarn on the Stability of Domes. [Nov. 22, 



II. ^^On the Stability of Domes."— Part II. By E. Wyndham 

 Tarn^ M.A.;, Merab. Eoy. lost. Brit. Architects. Communicated 

 by G. Godwin, Esq. Received October 23, 1866. 



(iVbstract.) 



Iq a former paper on this subject which the author presented to the 

 Royal Society, and which is pubhshed in the * Proceedings ' (vol. xv. 

 p. 182), he obtained formulje for calculating the thrust of a spherical 

 dome of uniform thickness, by supposing it to consist of a number of thin 

 ribs, each of which is formed by two vertical planes intersecting at the axis 

 of the dome, and making a small angle with each other ; and then treating 

 each rib as forming, with the corresponding one of the opposite side, a 

 complete arch. 



In the present paper the author applies the same method to domes of 

 other forms than the spherical. The following are the kinds for which the 

 formulae are investigated : — 



A. The Gothic dome, of which that of the cathedral at Florence is a 

 splendid example. This kind has for its section a pointed arch formed by 

 two segments of circles, the centre from which each is struck being on the 

 springing line, but not in the axis of the dome. The author denotes by 

 a the angle between the vertical and a line drawn from this centre to the 

 point, near the crown of the dome, where its axis is intersected by a circle 

 drawn around that centre with a radius equal to the mean of the radii of 

 the circles which generate the outer and inner surfaces of the dome, and 

 reduces his results to numbers for three different values of the angle a. 



B. A dome whose inner surface is a paraboloid of revolution, the thick- 

 ness of the shell being uniform throughout. 



C. The dome whose surface is formed by the revolution of an ellipse 

 about its major axis. In the investigation, the two surfaces are supposed 

 to be generated by the revolution of two concentric elliptic quadrants, 

 whose major axes differ from one another by the same quantity as the 

 minor axes, so that the thickness is very nearly uniform throughout. 



D. A form of dome commonly used in eastern countries, sometimes 

 called the ogival dome, the surface of which is generated by the revolu- 

 tion of a curve which has a point of contrary flexure. In the investiga- 

 tion, the author takes for this curve the curve of sines, the equation of 



which is y'=r sin—, where x', y are the vertical and horizontal coordi- 

 r 



nates of a point in the generating curve, and supposes the outer and inner 

 surfaces to be generated by the revolution of two such curves, differing only 

 by the value of r, the origin being the same. 



The following Table exhibits the principal results obtained from the 

 author's investigations. All the dom.es, except the last, are supposed to be 

 of the uniform thickness throughout of 1 foot. In the last the thickness is 

 1 foot at the springing, but gets rather less towards the top. The domes are 



