306 THEORY OF HEAT. [CHAP. VI. 



m and n being constant coefficients, we have 



n m[ , fdu do- &amp;lt;r\ fdu do- o\ 



7 \ffudx =-7-0- u-j- +u-} -T-O- W-7- + M-) . 

 k J \dx dx x/v \dx dx x/ a 



Between u and a a very simple relation exists, which is dis- 



covered when in the equation 7;&quot; + :^ -- T~~ = we su PP ose 

 cr = xs ; as the result of this substitution we have the equation 



n d*s Ids _ 

 k S *~d a ? + xdx~&quot;&amp;gt; 



which shews that the function s depends on the function u given 

 by the equation 



m d*u 1 du f. 



T u + -7~2 + ~ T~ = 0- 



A; cZ^ 2 cc rfa? 

 To find s it is sufficient to change m into n in the value of u ; 



the value of u has been denoted by ^ (#A/ T; J , that of cr will 



therefore be xty (x A/ -^ J . 



We have then 

 cZ?^ do- a- 



-j- (7 U-j- + U- 



dx dx x 



= -Vf * ( Vf ) t ( VS - V^ K/l^ ( Vf ) 



the two last terms destroy each other, it follows that on making 

 x 0, which corresponds to the suffix a, the second member 

 vanishes completely. We conclude from this the following equa 

 tion 



n m m 



