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



—ka'^t 



V= e cos /5(a — x), where a and /5 are arbitrary constants, 



—ka^t —ka-t 



for it is made up of V = Ae cos ax and F= Be sin ax^ 



both of which are solutions of (3) as shown above. This equation 

 denotes a distribution of temperatures which has maxima and 

 minima values, the latter being at certain fixed points given by the 



equation x = a — (2n -\- 1) -^. In general it is very similar to 



the distribution represented by (4) already studied. V^ = V (f{a) 

 is also a solution, where the temperatures are as before except that 

 they are increased by multiplying every one by ^(a), an arbitrary 

 constant function of a. Another solution is got, as described 

 before, by superposing all the heatings formed on reducing the 

 temperatures in V^ by multiplying each by the very small quantity 

 da, and giving a all values from — oo to + °^> ^^^ then taking the 

 limiting case where da tends to zero. Call this new solution V^ ; 



then v^ — J ^ cos /5(a — x)<p{a)da. Repeat this last operation 



with regard to /5; that is, take the distribution of temperatures 

 represented by V^ and reduce the numerical value of each by 

 multiplying by ^/5, then superpose all such heatings formed by 

 giving /5 every value from to oo, and finally take the limiting case 

 where d^ tends to zero. Call this new solution V^ ; then 



//* — Ka-t 



di3 J e cos iS(a — x')(p{fi)da. Still another solution 



— cc 



is got by reducing every temperature in ^ in the ratio of - to 1. 



cc 4- cc 



if f ~^'^"^ 



Call this solution F^; then ^ = -J d^ J e cos /5(a — x)<p{a)da; 



^ — oo 



it has the special importance and peculiarity, as was first shown by 

 Fourier, that at time zero the distribution of temperature it 

 represents is the same function of x, (fi^x), that we took originally 

 of a. Similarly every Fourier integral may be interpreted. 



Returning now to equation (6) and the curves drawn for it, we 

 can find new solutions by addition ; at each point r let us add the 

 temperature for that point and all other points farther from the 

 centre, even to infinity, but first reduced in absolute value by 

 multiplying each by the small quantity dr, which we make ulti- 

 mately tend to zero. We have but to add on Plate XXIV for any 



