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

 meanings. Without thinking of the special form of the differential 



equation, we can find the meaning of -_~ as follows : Let a solution, 



F, be/(r,t); then another, F^, is -^/(r,/), where Ar is a small 



constant; and another, F^, is — ^/(^O- Superpose on the medium 



these two states of temperature, Fi and F^, after first displacing F^ 

 bodily to the positive side of the origin by an amount Ar. When 

 Ar is indefinitely decreased the limiting state of temperature is that 



represented by -^, or — 1— . That is, -^ represents a heating due 



to a kind of doublet. We must next find out whether such a state 



of temperature as that represented by-^ is a solution of (1). We 



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saw that -y- was a limiting case, and hence it is not a solution in 



the limit (except by some unusual accident) unless it is so just 

 before the limit is reached. While Ar is still finite, but as small as 

 we please, the superposed heatings do not satisfy the same differen- 

 tial equation; for F^ satisfies the equation -j^ ^^^ = j ^^p-^ 



\ , while F, satisfies the equation -r-^^ — -^ = — ^^^ — - -{- 



-, and on account of the variable coefficient these are not 



5/-2 



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 the same equation. Hence -j- is not a solution of (1), and is only a 



solution of an equation in Fwhen that equation has constant coeffi- 

 cients, that is, coefficients not containing r. Equation (2) is of that 



kind, and hence knowing a solution of it, Fr, we can say that -~ 



is also a solution. Call this new solution F^r, then F^ is a solution 



of (1). Since ^^^= kH- r^-, and since \ — ^ is a solution of 



^ ^ dr dt ^ r dr 



(1), we have h .- a solution of (1) ; this is what we have just 



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 called F^. Now F satisfies (1), but we have just seen that -^ 



V 

 does not, and it can easily be seen that doesnot in general; so we 



have the interesting fact that the solution F^ is the sum of two func- 

 tions of K (itself a solution) neither of which is a solution. We 

 can at least give a physical interpretation to the method of finding 



