202 THEORY OF HEAT. [CHAP. III. 



We might change the limits of the integrals and write 



/2/r T+T r-n 



I or I instead of I ; 



J J _7T JO 



but in each of these two cases it would be necessary to substitute 

 in the first member TT&amp;lt; (x) for JTT&amp;lt; (x). 



233. The function &amp;lt; (x) developed in cosines of multiple arcs, 

 is represented by a line formed of two equal arcs placed sym- 



Fig. 11. 



metrically on each side of the axis of y, in the interval from 

 TT to +TT (see fig. 11) ; this condition is expressed thus, 



The line which represents the function i|r (x) is, on the contrary, 

 formed in the same interval of two opposed arcs, which is what is 

 expressed by the equation 



Any function whatever F(x\ represented by a line traced 

 arbitrarily inTEe interval from TT to + TT, may always be divided 

 into two functions such as &amp;lt; (V) and ^H[g) I n fact, if the line 

 F F mFF represents the function F(x} } and we raise at the point 

 o the ordinate om, we can draw through the point m to the right 

 of the axis om the arc mff similar to the arc mF F of the given 

 curve, and to the left of the same axis we may trace the arc mff 

 similar to the arc mFF ; we must then draw through the point m 

 a line &amp;lt;^&amp;lt;^ m^ which shall divide into two equal parts the differ 

 ence between each ordinate ooF or x f and the corresponding 



