•184 MACKENZIE — EQUATIONS OF HEAT PROPAGATION. [April 4, 



instead of doing so we ought rather to be able to say that this opera- 

 tion means such and such and foretell the distribution of tempera- 

 ture it will give. This illustrates what was meant above when 

 saying that we ought if possible to give the physical interpretations 

 of mathematical processes. What is the meaning of the operation 

 involved in (5)? Perhaps some light can be had on it from the 

 following consideration : We are to take a series of distributions 

 of temperature like that given by (4) and described above, where 

 the constant a (determining the thickness of the shells) has the 



successive values, 0, da, ^da, a, and superpose them on the 



medium after first reducing every temperature by multiplying it by 

 da. We are then to take da indefinitely smaller and smaller, and 

 finally to make a indefinitely greater and greater. We have thus 

 the difficulty of a double limit entering, and if we wish to seek the 

 initial condition it becomes a triple limit. This is sufficient to 

 prevent any rash prediction in this problem as to the exact nature 

 of the solution to be obtained ; and this case serves as an excellent 

 example of the difficulties to be overcome in any such efforts at 

 physical interpretation. Before the limit is reached the state of 

 temperatures is given by 



p —kt{daY- —Akt{da)"- -I 



Vr = d(j\ \ -\- e cos rda + e cos 'irda + etc. 



The limiting value of this series, which is equation (5), is not very 

 evident without considerable study, but on account of the dying- 

 out factor in each term the series is convergent, and the more 

 rapidly convergent the greater the value of t, and its value could 

 be found for any given / and da. Another way of finding this 

 value at any time and distance required is to take an axis along 



which a's are measured and draw the logarithmic curve e and 



the curve cos ra, then form the curve whose ordinate at each 

 point is the product of the ordinates of these two curves at the 

 point, and the area between this new curve and the axis gives the 

 numerical value of Vr. Since this area is formed of pieces alter- 

 nately above and below the axis of a and of decreasing numerical 

 value, we see that Vr is always of the same sign and that, for any 

 finite value of ;-, it begins by increasing in value and finally falls 

 off to zero, and by inference that it is zero at time / = ; but 

 that at the origin it has initially a value greater than zero. The 



