182 



Mr. Tarn on the Stability of Domes, 



[May 31, 



of such an instrument as that ! But if, as seems probable, we can without 

 much difficulty increase the heating-power of the induction-discharge an 

 hundredfold, we shall have made a very great step, and by means which 

 are everywhere accessible. Inductoria are common ; there are few situa- 

 tions where a physicist cannot obtain access to several, and combine them 

 as Despretz did the voltaic batteries of Paris to make the experiments 

 which have thrown such splendour on his name. 



IV. On the Stability of Domes." By E. Wyndham Tarn, M.A. 

 Communicated by George Godwin, Esq. Received May 5, 

 1866. 



The few writers who have attempted to treat the subject of the Equili- 

 brium of Domes mathematically, have entirely failed to obtain results that 

 are of any practical use to the architect. Their failure has arisen from 

 taking a too theoretical view of the subject, and endeavouring by mathe- 

 matical reasoning to find the form which a dome ought to have in order 

 that it might stand safely. Such a question is of no practical utility, as 

 domes of various sizes and forms have been erected for centuries past, and 

 the question for the architect is, — given a dome of certain form and size, 

 what are the conditions which must obtain between it and the wall of the 

 building it is intended to cover, in order that the whole structure may be 

 in a condition of stability ? 



The object, therefore, of this paper is to find a solution to the following 

 problem : — 



Given a spherical dome, built of stone or brick, of any radius and thick- 

 ness, and standing on a " drum^^ or walls of any height; to find the thrust 

 of the dome on the " drum,'' and the thickness that must be given to the walls 

 in order to insure the stability of the structure. 



I take the case of the dome having a spherical section, as being the 

 form most commonly used ; but the same method of investigation will 

 apply to domes of any form. 



In this investigation I shall consider the dome as made up of a large 

 number of arched ribs, of which the bases resting on the top of the 

 *' drum" subtend a small angle (0) at the centre, and the vertices have no 

 thickness ; each rib having the form of a wedge cut out of the spherical 

 shell by two planes intersecting in a vertical line through the centre, and 

 making the small angle ^ with each other. I shall then consider that the 

 two wedges thus formed on opposite sides of the dome, thrust against each 

 other at the vertex, as in the case of an ordinary semicircular arch, and 

 by this means keep each other in equilibrium. Of course no arch of 

 this form if built alone could stand for a moment, as it would give way 

 laterally ; but in the dome this is prevented by the parts on each side 

 of the rib under consideration. 



OCT. 8. ISG6. 



