SECT. I.] ROOTS OF EQUATION OF CONDITION. 271 



tangent at the origin a line which divides the right angle into two 

 equal parts, since the ultimate ratio of the arc to the tangent is 1. 

 We conclude from this that if X or 1TiX is a quantity less than 

 unity, the straight line mom passes from the origin above the 

 curve non, and there is a point of intersection of the straight line 

 with the first branch. It is equally clear that the same straight 

 line cuts all the further branches mrn, H^TTH, &c. Hence the 



equation = X has an infinite number of real roots. The 



tan e 



first is included between and ^, the second between TT and 



, the third between STT and -^- , and so on. These roots 

 2t *2* 



approach very near to their upper limits when they are of a very 

 advanced order. 



286. If we wish to calculate the value of one of the roots, 

 for example, of the first, we may employ the following rule : write 



down the two equations e = arc tan u and u = - , arc tan u de* 



A&amp;lt; 



noting the length of the arc whose tangent is u. Then taking 

 any number for u, deduce from the first equation the value of e ; 

 substitute this value in the second equation, and deduce another 

 value of u ; substitute the second value of u in the first equation ; 

 thence we deduce a value of 6, which, by means of the second 

 equation, gives a third value of u. Substituting it in the first 

 equation we have a new value of e. Continue thus to determine 

 u by the second equation, and e by the first. The operation gives 

 values more and more nearly approaching to the unknown e, as is 

 evident from the following construction. 



In fact, if the point u correspond (see fig. 13) to the arbitrary 

 value which is assigned to the ordinate u ; and if we substitute 

 this value in the first equation e = arc tan u, the point e will 

 correspond to the abscissa which we have calculated by means 

 of this equation. If this abscissa e be substituted in the second 



equation u = - , we shall find an ordinate u which corresponds 



to the point u. Substituting u in the first equation, we find an 

 abscissa e which corresponds to the point e ; this abscissa being 



