45 



skill is not able to effect, is solved in the arrangement of the fibres of 

 the heart of every person in this room. 



I shall now apply the principle of least action to the case of ellipsoidal 

 muscles. We call an ellipsoidal muscle a muscle that surrounds a 

 cavity — a muscular bag surrounding a cavity which generally contains 

 fluids. In attempting the solution of the problem of an ellipsoidal 

 muscle, I found myself brought into contact with a problem in architec- 

 ture which has baffled architects for many years : I mean the problem 

 of the equilibrium of an elliptical dome. Every portion of a curved 

 ellipsoidal muscle forms a portion of a small flat dome ; and to deter- 

 mine the equilibrium of tensions and strains amongst the muscular 

 fibres of such an animal structure, is the same thing as solving the pro- 

 blem in architecture of what are the strains in various directions in an 

 elliptical dome. I believe I have succeeded completely in solving the 

 problem ; and I have done so by an application of pure geometry, in 

 which I have not used a single letter of analysis. The difficulty of con- 

 structing equilibrated domes may be illustrated when I tell you that, 

 with the exception of the Pantheon in Paris (fortunately saved from de- 

 struction), there is not a truly equilibrated dome in existence. The 

 dome of St. Paul's, in our own city, is braced up with double chains 

 of iron, and other chains of timber and lead put on to cover the defects 

 in the original structure of the dome. Even Sir Christopher Wren was 

 not able, from his want of knowledge of the solution of this problem, to 

 apply the principles of architecture to make the dome of St. Paul's 

 stand by its own intrinsic strength without support. In the great dome 

 of St. Peter's, at Rome, many hoops of iron are employed which were 

 never intended by the great mind of Michael Angelo, who conceived 

 it. They are confessedly failures. Brunelleschi's octagonal dome at 

 Florence is perfectly equilibrated ; but then it is octagonal. No case exists, 

 I believe, of a self-supporting perfectly equilibrated spherical dome but 

 that of the Pantheon at Paris. An attempt has been made in the construc- 

 tion of the roof of the Albert Hall, to make an elliptical dome, but 

 whether that construction has been successfully carried out on the prin- 

 ciple of least action, I cannot say. The principle of least action applied 

 to the building of a dome would require that not a single pound of 

 material more than was sufficient was used in any part of it. The solu- 

 tion of the problem is so simple that I will venture to give you the 

 result of it. Here is an ellipse that represents a section of the elliptical 

 dome of the Hall of Albert the Good, the father of our future king. I 

 draw a line in any direction from the centre, and I require a construc- 

 tion which shall give me the strain which the structure must be capable 

 of bearing in that direction. The problem requires me to draw a line 

 in every possible direction, radiating from the top of the dome, and to 

 assign what amount of strength I must give the materials in that direc- 



