154 THEORY OF HEAT. [CHAP. III. 



TT 1 1 1 - I 



= cos x ; r cos 3x -f -z cos 5# = cos / # + g cos 9u; &c. 



A simpler method of obtaining this equation is as follows : 



If the sum of two arcs is equal to JTT, a quarter of the 

 circumference, the product of their tangent is 1; we have there 

 fore in general 



i - TT arc tan u -f arc tan - 



a 



the symbol arc tan u denotes the length of the arc whose tangent 

 is u, and the series which gives the value of that arc is well 



known ; whence we have the following result : 



If now we write e^&quot; 1 instead of u in equation (c), and in equa 

 tion (d), we shall have 



I / 



- TT = arc tan e x ^~ L + arc tan e~ x ^ ~ l j 



and j TT = cos x = cos ox + -- cos ox ^ cos 7x -}- -r- cos 9*i &c. 

 4 o o / 9 



The series of equation (d) is always divergent, and that of 

 equation (b) (Art. 180) is always convergent; its value is JTT 

 or ITT. 



SECTION IV. 



General solution. 



190. We can now form the complete solution of the problem 

 which we have proposed ; .for the coefficients of equation (b) 

 (Art. 1G9) being determined, nothing remains but to substitute 

 them, and we have 



^ .= e~ x cos y - -- e~&quot; x cos 3y 4- - e~ Bx cos 5y - ^ e&quot; 7 r cos 7.y + &c....(a). 



