1902.] MACKENZIE — EQUATIONS OF HEAT PROPAGATION. 181 



ON SOME EQUATIONS PERTAINING TO THE PROPA- 

 GATION OF HEAT IN AN INFINITE MEDIUM. 



BY A. STANLEY MACKENZIE. 



(Plates XXIII-XXVIII.) 

 ( Read April ^, igo2. ) 



We may attack a problem in the theory of the conduction of 

 heat in two ways \ we may make use of a Fourier's series or inte- 

 gral, or, since the general differential equation is a partial linear 

 one, we may build up the required solution out of known solutions 

 for simpler cases. The former way is usually much the more 

 expeditious if the proper ''trick " can be hit upon, but the method 

 is a purely artificial one, throwing no light on the process 

 involved. The student or reader sees at once that this method pro- 

 duces the required result and that a limited number of very similar 

 problems might be treated in the same way, but he is apt to feel 

 instinctively at first that the mathematical tool he has employed is 

 one of which he has only a superficial knowledge and that will fail 

 him when he gets out of a certain set of problems ; he wonders what 

 a Fourier's integral means and why it has a special value in such 

 problems. The trouble here, as in many other departments of 

 physics, is that the physical interpretation of mathematical opera- 

 tions is usually avoided. There can be but one good reason for 

 this, since all must admit the desirability of such interpretations, 

 that it is at times exceedingly difficult, if not impossible, to give 

 the inherent physical meaning of a mathematical operation. Much 

 more, however, might be done than is done, and there is perhaps 

 no branch of mathematical physics more suited to the purpose of 

 introducing to those just beginning such studies the meanings and 

 the limitations of mathematical operations than heat conduction. 

 The second method of treating heat conduction problems, by 

 building up solutions from known solutions for other cases, is full of 

 suggestiveness, and brings into view the meaning of many of the 

 mathematical processes employed in any treatment of the conduc- 

 tion of heat, and the relationships of the equations involved. An 

 attempt is made in the following pages to point out the necessity 

 for effort along the lines indicated above, and among other things 

 to give careful drawings of some of the more important curves of 

 temperature and current. 



