304 THEORY OF HEAT. [CHAP. VI. 



having nul values. Similarly to determine the second coefficient 

 a a , we multiply both terms of the equation 



&amp;lt;f&amp;gt; (x) = a z u^ + 2 w 2 + o 3 u B -f &c. 



by another factor &amp;lt;r 2 dx, and integrate from x = to x - X. The 

 factor &amp;lt;r 2 must be such that all the integrals of the second member 

 vanish, except one, namely that which is affected by the coefficient 

 * a 2 . In general, we employ a series of functions of x denoted by 

 &quot;i&amp;gt; &quot;2 s ^ a wn i cn correspond to the functions u iy u# u s , &c. ; 

 each of the factors cr has the property of making all the terms 

 which contain definite integrals disappear in integration except 

 one ; in this manner we obtain the value of each of the coefficients 

 a,, GL, a a , &c. We must now examine what functions enjoy the 



1 2 3 .^..I^IMB^B^^^^^^^ :.., . J -... 



property in question. 



316. Each of the terms of the second member of the equation 

 is a definite integral of the form a I audx u being a function of x 

 which satisfies the equation 



m d?u 1 du 



_ 



~j~ U ~T&quot; ~7 n *l ~7~ ^ 



A; da? x dx 



we have therefore alcrudx = -a |(--7^-fo -T~). 

 J m]\xdx dx J 



Developing, by the method of integration by parts, the terms 

 du , d*u , 



, /V du , ~ (T C , /&amp;lt;r\ 



we have \- -^-dx = C + u |wa- 



Jicau; x ) \xj 



, f c 2 w , -p. &amp;lt;&* cZcr T c?V 7 



and I &amp;lt;7 7 o dx V } -^- a u-^ h |w -7, a#. 



J ad? a,^ dx J dx^~ 



The integrals must be taken between the limits x = and 

 x = X, by this condition we determine the quantities which enter 

 into the development, and are not under the integral signs. To in 

 dicate that we suppose x = in any expression in x, we shall affect 

 that expression with the suffix a; and we shall give it the suffix 

 co to indicate the value which the function of x takes, when we 

 give to the variable x its last value X. 



