I91I.] 



LINEAR DIFFERENTIAL EQUATIONS. 



279 



Fourier's partial differential equation for the linear flow of 

 heat is 



= IC 



dt 

 Replace Fourier's equation by 



dx 



dV _ d^V 



and assume that 



makes the latter equation an identity. 



When arranged in ascending powers of S this identity is 



^0 



a/ 



+ 



dt 



5 + 



dt 





K 







- K 



^2 



S^ + ... = 0. 



Equating to zero the coefiicient of the powers of 5" in this identity, 

 there result the following partial differential equations for the deter- 

 mination of Fq, F^, Vo, Fg, • • •, 



= o, 





^0 



dt 



dj-^'dx' = ''^ 



These partial differential equations solved in regular order give 

 V, = c^{x), l\ = 4>%r){Kt), F, = c/>-(-r)^\ 



f^3 = </> ('^)-Yr' •••• 



Substituting these values of F^, F^, F„, Fg, ... in the assumed 

 value of F and finally making 5" unity, there results 



{A) F= 4>{x) + r{-r){Kt) + <t>'\x) ^' + r\^) ^' + • • . , 



