144 THEORY OF HEAT. [CHAP. III. 



which the reader may supply, we remark that the fixed value 

 which is continually approached is JTT, if the value attributed 

 to y is included between and J-TT, but that it is Jvr, if y is 

 included between \TT and |TT ; for, in this second interval, each 

 term of the series changes in sign. In general the limit of the 

 series is alternately positive and negative ; in other respects, the 

 convergence is not sufficiently rapid to produce an easy approxima 

 tion, but it suffices for the truth of the equation. 



178. The equation 



, 3 cos ox + - cos ox * cos 7% + &c. 



O O / 



belongs to a line which, having x for abscissa and y for ordinate, is 

 composed of separated straight lines, each of which is parallel to 

 the axis, and equal to the circumference. These parallels are 

 situated alternately above and below the axis, at the distance JTT, 

 and joined by perpendiculars which themselves make part of the 

 line. To form an exact idea of the nature of this line, it must be 

 supposed that the number of terms of the function 



cos x 7. cos 3x + - cos 5x &c. 

 3 5 



has first a definite value. In the latter case the equation 



y = cos x - cos 3x + - cos ox &c. 

 o 5 



belongs to a curved line which passes alternately above and below 

 the axis, cutting it every time that the abscissa x becomes equal 

 to one of the quantities 



185 



0, g 7T, + 2 7T, g 7T, &C. 



According as the number of terms of the equation increases, the 

 curve in question tends more and more to coincidence with the 

 preceding line, composed of parallel straight lines and of perpen 

 dicular lines ; so that this line is the limit of the different curves 

 which would be obtained by increasing successively the number of 

 terms. 



