32 president's address — section a. 



made upon the analogy, and can be avoided once and for all 

 by mending the imperfection. Moreover, the student of the 

 theories of electricity and magnetism can hardly avoid the 

 use of some sort of analogy, for these theories deal with the 

 quantitative relations between quantities of whose real nature 

 we are completely ignorant, and most minds cannot for long 

 consider these relations in mere symbols, but must finally 

 give them some sort of form. 



Of all these theories, then, each is the analogue of any 

 other. Each is founded on laws which have the same 

 mathematical expression ; each must be capable of similar 

 development. The solutions of particular problems in any 

 one theory must, mutatis mutandis, be solutions of correspond- 

 ing problems in all the others. Not until by-laws peculiar to 

 any one theory are introduced will the analogy break down. 



Let me begin by stating briefly the fundamental laws and 

 the mode of development of some one of these theories ; 

 it will be convenient to take as our first example that of the 

 uniform motion of heat in a solid, as it is easy to form a 

 mental picture of it. Then passing on to the other theories 

 I will try to make clear the nature of the analogy, and the 

 extent to which it holds good. 



Imagine, then, a solid body in which heat is in motion in 

 any uniform manner — uniform, in that the nature of the 

 motion does not alter with time. The conductivity of the 

 body may, in the most general case, vary from point to 

 point, or even be different in different directions at the same 

 point. There may be various places in the body where heat 

 is being continually suj^plied, others where it is being con- 

 tinually abstracted ; in other words, there may be sources 

 positive or negative. All boundaries, internal or external, of 

 the body may with advantage be considered as sources. 



Suppose that we are supplied with information as to the 

 conductivity of the body in all parts, as to the position and 

 extent of its boundaries or sources. Suppose that we are 

 also told either the temperature at each unit of area of 

 boundary, or the rate at which heat is flowing across it ; if, 

 however, over any bounding surface closed in itself the tem- 

 perature is known to be constant, it will be sufficient to know 

 this temperature or the total flow across the surface. We 

 can then solve the problem ; that is to say, we can write 

 down equations whose solution wouhi tell us the temperature 

 at every point of the body, or, if we like to express it so, the 

 rate and direction of the flow of heat at every point. 



