424 THEORY OF HEAT. [CHAP. IX. 



Integrating this linear equation, and determining the arbitrary 

 constants, so that, when x is nothing, y may be 1, and 



dij fry d?i/ 

 tx&amp;gt; dx 2 d? 



may be nothing, we find, as the sum of the series, 



It would be useless to refer to the details of this investigation ; 

 it is sufficient to state the result, which gives, as the integral 

 sought, 



v - |cZa/(a) Idq q -jcos 2&amp;lt;? 2 (t a) (e^ + e~v x ) cos qx 

 - sin 2^ 2 (t - a) (&* - e~ qx ] sin gx } + W. ..... 



The term W is the second part of the integral; it is formed by 

 integrating the first part with respect to x, from x = to x = x, 

 and by changing / into F. Under this form the integral contains 

 two completely arbitrary functions f(t) and F (t). If, in the value 

 of v, we suppose x nothing, the term W becomes nothing by 

 hypothesis, and the first part u of the integral becomes f(t}. If 



we make the same substitution x = in the value of -r- it is 



ax 



evident that the first part -j- will become nothing, and that the 



dx 



dW 

 second, -j , which differs only from the first by the function 



F being substituted for f t will be reduced to F (t). Thus the 

 integral expressed by equation (00) satisfies all the conditions, 

 and represents the sum of the two series which form the second 

 member of the equation (X). 



This is the form of the integral which it is necessary to select 

 in several problems of the theory of heat 1 ; we see that it is very 

 different from that which is expressed by equation (/3), Art. 897. 



1 See the article by Sir W. Thomson, &quot;On the Linear Motion of Heat,&quot; Part II. 

 Art. 1. Camb. Math. Journal, Vol. III. pp. 2068. [A. F.] 



