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



the curves approach each other more and more for greater and 

 greater values of the time, no matter what the distance. This is 

 seen more distinctly from an examination of Plate XXVI. There it 

 will be seen also that the approximate curves are slower in reaching 

 their maximum values, as well as that they have different maxima. 

 For distances less than unity the approximate curves start at go , 

 while the exact curves start at F= F^; for the distance unity the 



exact curve starts abruptly at-^, while the approximate curve starts 



at then gradually rises and has a maximum value less than 



-^. For distances greater than unity both curves start at the origin. 



From an inspection of the second term of (19) we can foretell 

 the approximate accuracy of (16). Taking 7? as the unit of length, 



if /('/< 15 the error in the value of — ^^ will be everywhere greater 



than 1% except in the immediate neighborhood o{r=i/^kf, at which 

 point the error is practically zero. For instance, for k^ = ^ (curves 

 IV and IV^) the approximate curve is 33% too high at r = 0, 22 fc 

 ■21 r = 1, correct at about 1.8, and 38% too low at 3. If kt =^ 15, 

 the error is not more than 1% from r = to r = 13.4. If ^/ ==: 25 

 the error is not more than 1% from r = to r = ?0. In general, 

 for any value of kt the error is not more than \% from r = to 



r = yfSkt + ^{kt)\ and from r = to r= |/6^ the error 



15 * 



decreases gradually from -j7% to zero, and after that increases 



again. If we want results accurate to .01%, kt must be at least 

 1500, and in general for any value of ki greater than this the error 

 is not more than .01% from r =-- to r = y(Skt -J- Yiu ('^0^ ^^^ 

 from r = to r = |/6/C'/ the error decreases gradually from -rrfo 

 to zero, and after that increases again. 



From equation (15) we can build up by summation the equation 

 for the case of a body of any shape or size initially at Vq cooling in 

 an infinite medium initially zero. In order to bring out a very 

 interesting difference between summation and integration we shall 

 apply equation (15) to the case of an infinite space, one-half of 

 which is initially at V^ and the other half at zero, the two parts 

 being separated by an infinite plane surface. We shall first have to 

 find the solution for a plane lamina. Take the central plane of the 

 lamina as the plane of ^'2,''and the origin where a perpendicular 



