302 THEORY OF HEAT, [CHAP. VI. 



7)1 X* 



If we make = , and consider u as a function of x, we have 



K u 



du Q d*u 



u + d~e + de^- 



The following value 



_i a 2 J* J* 4 _ & 

 u 1 u + a ^2 02 ~r 2 2 3 2 4 2 



satisfies the equation in u and 0, We therefore assume the value 

 of u in terms of x to be 



- mo? m* a? m 3 x 3 



~ I 2*&quot; + F 2 2 .1 2 ~ ,77& * :c 



the sum of this series is 



the integral being taken from r = to r = TT. This value of v in 

 terms of x and m satisfies the differential equation, and retains a 



finite value when x is nothing. Further, the equation hu + -j- =0 



must be satisfied when x = X the radius of the cylinder. This 

 condition would not hold if we assigned to the quantity m any 

 value whatever ; we must necessarily have the equation 







2 &quot;1-2-3-4- 5- &c. 



i&amp;gt; . Vj m X* 



in which denotes -j- -^ . 



This definite equation, which is equivalent to the following, 



l fi^ * &amp;gt; * \ fi ^V ^ Xr 



+ 2 ~ 2 ~&quot; 2 + ~ *~ + 2 &quot;&quot; &quot; 



gives to 6 an infinity of real values denoted by V Z , 3 , &c. ; the 

 corresponding values of m are 



2 3 



V2 &amp;gt; Y 2 JT 2 &amp;lt; &quot; * 



thus a particular value of v is expressed by 



_2 2 Atf?i f / x i- 



Trv-e ~x*~ I cos f 2 -y, v^ sin 



