CHAP. VII.] 



GEOMETRICAL CONSTRUCTION. 



31 



contained in the prism is so distributed, that it would vanish 

 entirely, if the end A were maintained at the temperature 0. 



328. We may add several remarks to the preceding solution. 

 1st, it is easy to see the nature of the equation e tan e = -j- ; we 



need only suppose (see fig. 15) that we have constructed the curve 

 u = e tan e, the arc e being taken for abscissa, and u for ordinate. 

 The curve consists of asymptotic branches. 



Fig. 15. 



The abscissa? which correspond to the asymptotes are ^TT, 



357 



o 71 &quot; o 77 &quot; 9 71 &quot; &c -&amp;gt; those which correspond to points of intersec 

 tion are TT, 2?r, 3?r, &c. If now we raise at the origin an ordinate 

 equal to the known quantity ~r , and through its extremity draw 



K. 



a parallel to the axis of abscissa?, the points of intersection will 

 give the roots of the proposed equation e tan e = -j- . The con 

 struction indicates the limits between which each root lies. We 

 shall not stop to indicate the process of calculation which must be 

 employed to determine the values of the roots. Researches of 

 this kind present no difficulty. 



329. 2nd. We easily conclude from the general equation (E) 

 that the greater the value of x becomes, the greater that term of 



the value of v becomes, in which we find the fraction jT &quot; 1 *&quot;* * l % 

 with respect to each of the following terms. In fact, n l9 n z , w 3 , 

 &c. being increasing positive quantities, the fraction e~ rx 2nr is 



