272 



THEORY OF HEAT. 



[CHAP. V. 



then substituted in the second equation gives rise to an ordinate 

 w , which when substituted in the first, gives rise to a third 

 abscissa e&quot;, and so on to infinity. That is to say, in order to 

 represent the continued alternate employment of the two pre- 



Fig. 13. 



Fig. 14. 



ceding equations, we must draw through the point u a horizontal 

 line up to the curve, and through e the point of intersection draw 

 a vertical as far as the straight line, through the point of inter 

 section u draw a horizontal up to the curve, through the point of 

 intersection e draw a vertical as far as the straight line, and so on 

 to infinity, descending more and more towards the point sought. 



287. The foregoing figure (13) represents the case in which 

 the ordinate arbitrarily chosen for u is greater than that which 

 corresponds to the point of intersection. If, on the other hand, we 

 chose for the initial value of u a smaller quantity, and employed 



in the same manner the two equations e = arc tan u, u - , we 



A 



should again arrive at values successively closer to the unknown 

 value. Figure 14 shews that in this case we rise continually 

 towards the point of intersection by passing through the points 

 ueu e u&quot; e&quot;, &c. which terminate the horizontal and vertical lines. 

 Starting from a value of u which is too small, we obtain quantities 

 e e e&quot; e &quot;, &c. which converge towards the unknown value, and are 

 smaller than it ; and starting from a value of u which is too great, 

 we obtain quantities which also converge to the unknown value, 

 and each of which is greater than it. We therefore ascertain 



