PRBSIDKNT's address — SECTION A. 33 



It is easy to see, however, that in general the solution of 

 these equations must be exceedingly complicated — that it is 

 only in special cases that they can be solved at all. For 

 example, our sources must be spheres or ellipsoids, or plates, 

 or other simple surfaces. Our body must not be very 

 irregular ; if its conductivity varies in different parts and in 

 different directions it must vary in some simple manner; some 

 spherical portion, perhaps, may be of different conductivity to 

 the rest. 



But though we cannot, except in a few simple cases, solve 

 the problem completely, we can obtain some general results 

 which are of great importance, and practically almost as use- 

 ful as a complete solution would be. 



If we follow from its source the course of some small 

 quantity of heat, our path is called a line of flow. By 

 imagining these lines of flow to be drawn in all parts of the 

 body we can form a complete mental picture of what is going 

 on. If all the lines of flow which pass through the contour 

 of some small area be supposed drawn, there is formed a tube 

 along which heat flows as water in a pipe. No heat ever 

 crosses the walls of the tube, so that if a section of the tube be 

 made at any place, the area of the section multiplied by the 

 rate of flow at right angles to the section is constant all along 

 the tube. This constant is called the " flux " along the tube. 

 Each tube must then start at some positive source and end at 

 some negative source, or else lose its contents in the sands of 

 infinity. If the conductivity of the body change suddenly 

 at any point of the tube of flow, the tube will be bent more 

 or less -at the surface of separation, but the flux along the 

 tube must always remain the same. Further, the size of each 

 tube may be so adjusted that the flow of heat across any cross 

 section is at the rate of one unit of heat per second. 



By drawing tubes of flow in this manner it is possible to 

 exhibit diagramatically the results of the solution of the 

 equations in any particular case ; and even when we cannot 

 solve the equations we can often make a very good guess as 

 to how the tubes will go, and so obtain a result near enough 

 for practical purposes. 



To put the matter in a mathematical and terser form, let us 

 use the symbol V to denote the temperature at any point, c 

 the conductivity at any point, and C the flow of heat across a 

 unit area round aiiy point. All these quantities are in the 

 general case functions of the co-ordinates x, y, ^ ; C and c 

 are also functions of direction, but in most practical cjises c is 



