SECT. I.] SOLUTION OF THE LINEAR EQUATION. 355 



b being any constant whatever and the limits of the integral the 

 same as before. From the equation 



we conclude, by making 6 2 = kt 



hence the preceding value of u or e~* e kt is equivalent to 



we might also suppose u equal to the function 



a and w being any two constants ; and we should find in the same 

 way that this function is equivalent to 



We can therefore in general take as the value of u the sum of an 

 infinite number of such values, and we shall have 



+ &c.) 



The constants a lt a 2 , a 3 , &c., and n v n z , n s&amp;gt; &c. being undetermined, 

 the series represents any function whatever of x 4- Zg_&amp;gt;Jkt ; we have 



therefore u= ldqe~ qi ^&amp;gt; (x + fyjkfy The integral should be taken 

 from 2 r = cotog ss+x, and the value of u will necessarily satisfy 

 the equation -j- = k -y-j . This integral which contains one arbi 



trary function was not known when we had undertaken our re 

 searches on the theory of heat, which were transmitted to the 

 Institute of France in the month of December, 1807: it has been 



232 



