1902.] MACKENZIE— EQUATIONS OF HEAT PROPAGATION. 193 



r—R 



2\/ki 



This then is the exact equation for a sphere of any size of initial 

 temperature Fg put into an infinite medium of the same material as 

 the sphere of initial temperature zero and left to cool there. The 

 forms of the curves given by this equation are exhibited on Plates 

 XXV and XXVI, along with those of the approximate equation 

 (16). Curves I to IV correspond to V to IV^ and curves 1 to 5 

 correspond to 1* to 5\ 



We can get an approximate form from equation (18) by expand- 

 ing it in": erms of J? ; we find 



Tkir- 4^-6 



6i/^ W^ L 40 i^ J 



(19) 



The first term of this is the same as equation (16), found otherwise. 

 Equation (19) gives us a second approximation, and the second 

 term within the bracket will enable us to determine the closeness of 

 (16) as an approximation. In a similar problem, Fourier (Free. 

 man's translation, p. 380) gives a limit to the time when the 

 approximation may be used, but he does not give any means of 

 telling how great the error is in general, and it was for the purpose 

 of bringing this out distinctly that equation (19) and the curves on 

 Plates XXV and XXVI were produced. From Plate XXV we see that 

 the approximate curves are at first steeper and afterward flatter than 

 the exact curves ; they make the temperatures too high for points 

 nearer the origin than a certain distance, and too low for points 

 farther away. Indeed curves I and I^ are very little alike for any value 

 of/'. As the value of the time for which the curve is drawn is taken 

 greater and greater the curves approach each other more and more 

 nearly, even for points less distant than unity (which are inside the 

 little sphere), for which we might have expected little agreement. 

 This makes evident the fact to which Fourier calls attention at the 

 place just cited ; one is very apt to assume that the curves would 

 approach each other more and more as r is taken greater and 

 greater, no matter what the value of /; bat just the reverse is true, 



PROG. AMER. PHILOS. SOC. XLI. 169. M. PRINTED JULY 28, 1902. 



