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

 the plate, we find (2 = 2 ( CDVdx = lACBx/r.; and this value 







of A reduces (22) to the form (21). Hence the general form of 

 equation (21), which is approximate for a plate of actual thickness 

 Ax, is exact for the infinitely hot plane. We shall revert to this 

 important fact later. 



If we want the exact equation for the plate of thickness Ax we 

 can get it by the use of a Fourier integral. Making the obvious 

 changes in (17) to suit it to the case of linear flow, and giving/(:x:) 



the value ^o from x =^ -^to^==-^ and the value for all 



other values of x, we find 



^=77f e dy (23) 



2i/ kt 



Putting this in an approximate form, we have 



2{'Kkt) 





V,^x ^ ^^^ p^ , kt - (Ax] 



(24) 



the first term of which is equation (20). The forms of the curves for 

 (20) are exhibited on Plates XXIII and XXIV. With values of x as 



abscissae curves A'^ to A* are for values of the time -^, -^, -jr and 



-^ respectively ; with values of 4/CV as abscissa curves B^ to B'^ are 



for values of the distance 0, ^, ^, | and 1 respectively. The second 

 term of (24) enables us to tell approximately the degree of closeness 

 of (20) to the exact equation (23). Taking Jx as the unit of 



25 



length, if kt<.--^ the error will be everywhere greater than 1 % 



except in the neighborhood of x --= y^-lkt where it is practically 



25 

 zero. If kt = -^^ the error is not more than 1% from :x: = to 



X =z 2.9, being Ifo too high at x = 0, zero at jc = 2, and ifo too 

 low Sit X ^= 2.9. If /&/ = 25 the error is ^3-% too high at :v = 0, 

 zero at 7, and 1% too low at 26. This is then a nearer approxima- 

 tion than the one discussed for the case of a hot particle, as was to 



