I III iv KljUAi 







\ 



r.. 



/)gtf*Sfion o/iif femffosj y* = r S) 



x.0,tben f is imaginary; if y 

 4; hence UM locus crosses the xas at the three 

 (* 0), and C (4,0), and 



y-aiw at all. 



Moreover, since jr U imaginary if 



negative, UM locus lies wholly 



e positive aide of UM y*xis. 



This locus is symmetrical with 



regard to the x-axU; it has no 



point nearer to the yntxis than 



i-otwecn A and B it consists 



of aclosed branch; and it has no 



real points between B and C, but 



is again real beyond C. The 



entire locus consists, then, of a 



closed oval, and of an open branch 



which recedes in.l.-tinitoly from 



both axes (see Fig 



) Difctution of ikt flotation y = tan x. This equation has already 

 been examined in (' >*ut in practice it may be much more iinjIv 



the following method: 



Describe a circle w ith unit radius; draw the diameter AOC, and the 

 lines OB V OB 9 OB V .-, meeting the tangent .1 / 

 in the points T |f T r T v -; then the tangent of 

 . the angle AOB l is Af A : ^i = A T i '- OA (Art. 

 * 14), and, sine* O.I = 1, its value is graphically rep. 

 resented by A T v So also 

 tan AOBi = 3/,/i, : O.\f t = A T t : OA = A T t : 1. 



and may be graphically represented b\ 

 the same way, AT T AT r AT. ... are the tangents 

 of the angles .4O/^ AOB V A OB. .... Again, 

 since angles at the center of a circle are propor- 

 tional to the arcs intercepted by their side*. I / 

 A T r ... may be said to be the tangents of the 

 arcs AB V ABy T, = tan A B v A T, = tan 





Therefore the coordinates of the points 

 P l3 ( :S(AB T ^T,),... satisfy the 



given equation, and if s nufficit-nt number of points, 

 whose coordinates are thus determined, be plotted, 

 they will all lie on a euro like that in Fig. 10. 



