280 



LAMBERT— ON THE SOLUTION OF 



[April 20, 



which is a solution of Fourier's equation for all values of <f>(x) for 

 which V either contains a finite number of terms or is an infinite 

 series uniformly convergent both in .r and in t. 



The following table shows several values of <^(.v) and the cor- 

 responding solutions of Fourier's equation. 



I (i) <f>(x) = A, 



(2) <f>{x) = A.V, 



(3) <^W = ^-^^ 



(4) (f){x) = A sin {nx), 



(5) <^('t') = ^ cos {71X), 



(6) 4>{x) = Ae^% 



(7) <^{x) = Ae-\ 



(8) 4>{x) = Ae"' sin {?ix). 



V= A, 



V= Ax, 



F= A{x^ + 2 A?), 



]^ = Ae-''-'^' sin {nx). 



V=Ae- 



{nx), 



V= Ae-"'+"'''', 



V= Ae""" sin (//.r -(- 211^ Kt). 



It will be noticed that in these solutions ^(.r) is the value of V 

 when t=o, that is F^</)(.r) is the initial heat distribution. 



It will also be noticed that in all these results x may be replaced 

 by X 4" 0" This statement is true of the results in the several fol- 

 lowing tables. 



If Fourier's differential equation is replaced by 



Ct OX' 



and the assumption made that 



makes this equation an identity, this identity arranged in ascending 

 powers of 5" is 



K 



dx^ 



+ K 



dx^ 



dV 

 '^ 



dt 



S + K ^^2 



dx' 



dt 



_dV, 

 dt 



S' + ■•' =0. 



Equating to zero the coefficients of the powers of 5" in this 

 identity, 



