191 1.] LINEAR DIFFERENTIAL EQUATIONS. 283 



(8) <i>{t) = ^ log /, V=A ^v log / + -^ 



3 



I x' 



+ 



Y 7 ! J 



K-t^ 5 ! ^ K^i 

 It will be noticed that in this set of solutions [' = when .r^o. 



Let u^^f^(xj), U2 = f2{y>^)> ^h = fz{^>^) represent solutions 

 of the three one-dimensional Fourier's equations, 



dV _ d-V (^V _ d^-V dV d'V 



~dt^^d7^' a7^ ^' "a7~ a^ 



respectively. It is readily proved that 



y=zn^u.-. and l^ -^ u-^\i.,u, 

 are solutions respectively of the two-dimensional Fourier's equation 



dV (d-V d-V\ 



and the three-dimensional Fourier's equation 



dV (d'V d^V d''V\ 



~df ^ ^\~dx^ ~^ ~dy '^ ~dz' )' 



This shows how solutions of the two- and three-dimensional 

 Fourier's equations can be obtained from the solutions of the one- 

 dimensional equation. 



For example, from the one-dimensional solutions 



Ae' 



4Kt 



V= -j^-r- and F= Ae''''-'" sin (nx) 

 the three-dimensional solutions 

 Ae~^* 



(2) F= /^^-(«-+P=+Y-)*''< sin {ax) sin {Qy) sin (7^), 



respectively, are obtained. 



