.! N.I 7. Y llC OEOMl.lHY 



;<. xi. 



and, therefore, the sum >f tho acute angles which a pair nt" s U j. pigmentary 



chords of th juilaU-ral hyiH>rhola make with its transverse axis is 90 



(cf. Art. 171' (ft)). 



174. Equations representing an hyperbola, but involving only one 

 variable. 



(a) Eccentric angle. In th theory of the hyperbola, tin- auxiliary 

 s described upon the axes of the curve as diameters are not as 

 useful as the corresponding circles for the ellipse, since the ordinate for 

 a point on the hyperbola does not cut the ^-auxiliary circle, and, there- 

 fore, there is no simple construction for the eccentric angle. It is, how- 

 ever, sometimes desirable to express by means of a single variable the 

 condition that a point shall be on an hyperbola; anl I'm- thi> purpos*. 

 the equations 



z = asec<k y = fttan4, . . . [7IJ 



similar to equations [<X)], may be used; for these evidently satisfy the 

 equation of the hyperbola 



The angle <f> may be defined as tin- eccentric angle for tin- hyperbola, 

 and the corresponding point of the curve may be constructed as follows : 



Draw the auxiliary circles, and any Z A OQ = <f>. At the points R and Q, 

 \\ li-Tf the terminal side of <f> cuts the circles, draw tangents cutting the 

 transverse axis in the points M' aud M, respectively. Erect at M an 



