1902.] MACKENZIE — EQUATIONS OF HEAT PROPAGATION. 197 



be expected. In general, for any value of kt the error is not more 



than \% from ^ = to x = y^Uf -f- ^{kty, and for any value of kt 

 greater than ^-^ the error is not more than M% from jc = to 



2^V -f 23^(^0' y from :\: = to a,- = V 2k/ the error decreases 



25 

 gradually from t^^^ to zero, and after that increases again. 



The correspondingly approximate equation for the current or 

 flow of heat in this case is 



J = — A ^= — ^= ^e = -^^ -3 e ....(25) 



The forms of these curves are given on Plates XXVII and XXVIII. 

 With values of a: as abscissae curves C^ and C/, C^ and Q^, and Q^ 



are for values of the time zrwr, -^ and ^-t- respectively ; with values 



of 4^/ as abscissae curves Z>^ and Z?/, D " and Z>i^, and Z>i' are for 

 value of the distance i, J and 1 respectively. 



The exact equation for the flow, found from (23), is 



r^rz I— 4:kt Akt — i 



/^-ALlF, _, n (26) 



2(:7/^/)^L J 



the curves for which have not been drawn. 



By adding up the effects of an infinite number of such plates we 

 can get the temperature due to one-half of space initially at a uniform 

 temperature V^ and the other half at zero temperature. Take the 

 point F, at which the temperature is desired, in the cold half and 

 at a distance x from the surface of separation, and take the origin 

 in that surface at the foot of the perpendicular from F. Let one 

 of the plates making up the other half of the medium be distant I 

 from the origin. Then the x of equation (20) becomes x -\- ^, 

 and Ax becomes J| ; hence the temperature at F due to a series of 

 such plates extending from c =: to 1^ =: oc , as found by inte- 

 gration, is 



00 TT-, -r, — ft2 



V /• ■^^^ V 





2V kt 



