198 MACKENZIE — EQUATIONS OF HEAT PROPAGATION. [April 4, 



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(27) 



We could arrive at the solution for this case by using Fourier's 

 integrals, as we did for equation (23), giving /(^) the value F„ 

 from X = — ooto:!t:==0 and the value zero from ;r = to ^ = oo. 

 We get at once equation (2T) again. 



This latter method gives the exact solution for the problem and 

 yet it gives the same result as the former method, from which one 

 might expect naturally enough an approximate solution, since we 

 get it by integrating solutions that were approximate. This is the 

 point to which attention was called in applying our results to this 

 case ; we have the integration of approximate solutions an exact 

 solution. The first explanation offered of this unexpected result is 

 apt to be that the approximation used is the more exact as the dis- 

 tance .T -(- ^ is the greater ; but we have seen earlier that just the 

 contrary is true and that at great distances (20) ceases to be 

 properly called a solution unless the time is taken very great. The 

 real explanation is simply that the operations of summation and 

 integration are not always the same, and this is a case in point. 

 Nothing is commoner in applying mathematics to physics than to 

 use mathematical processes with laxity and to test the legitimacy of 

 the application by the results. It is so uncommon to have a sum- 

 mation made improperly by integration that we lose sight of the 

 mathematical fact that the operations are not equivalent. We take 

 similarly the first two terms of a Taylor's series expansion as a 

 sufficiently close approximation in almost any piece of analysis, 

 without questioning whether the function under consideration can 

 be so expanded and without reference to the value of the terms 

 disregarded ; we take differential coefficients without asking 

 whether they can have a meaning, etc. The good excuse offered 

 is that the chances are overwhelmingly in our favor, and that if we 

 have made a mistake we shall quickly find it out from the results. 

 Had we actually made a summation in the above problem we should 

 have got an approximate result, but by integrating we get the limit 

 toward which the summation tends as ^^ tends towards zero, and it 

 happens in this case that this is the exact solution. In finding an 

 area we take a series of strips of area of y^x and however infinites- 

 imally small dx is, so long as it is something and not zero, the sum 



