270 



THEORY OF HEAT. 



[CHAP. V. 



gives a known quotient X, and afterwards take n = -^ . It is 



JL 



evident that there are an infinity of such arcs, which have a given 

 ratio to their tangent ; so that the equation of condition 



nX - I _ XT 



, -vr- -L m.\. 



tan nX 

 has an infinite number of real roots. 



285. Graphical constructions are very suitable for exhibiting 

 the nature of this equation. Let u = tan e (fig. 12), be the equation 



Fig. 12. 



to a curve, of which the arc e is the abscissa, and u the ordinate ; 

 and let u = - be the equation to a straight line, whose co-ordinates 



A 



are also denoted by e and u. If we eliminate u from these two 

 equations, we have the proposed equation - = tan e. The un- 



A 



known e is therefore the abscissa of the point of intersection of 

 the curve and the straight line. This curved line is composed of 

 an infinity of arcs ; all the ordinates corresponding to abscissae 



1357 



2 71 &quot; 2 71 &quot; 2 71 &quot; 2 71 &quot; 



are infinite, and all those which correspond to the points 0, TT, 

 27T, STT, &c. are nothing. To trace the straight line whose 



. 6 



equation is u - = j-^ f we form the square oi coi, and 



A, 1 ilJL 



measuring the quantity hX from co to h, join the point h with 

 the origin 0. The curve non whose equation is utsm e has for 



