TWENTY-SIXTH ANNUAL MEETING. 19 



The length S of the curve may be represented by the definite integral, 



/tan -^ / a* cos' k — b* sin^ k \ \ 

 \a' cos' k — b' sin' k) ^^''■ 



The whole area inclosed by the Curve is given by the value of a & + {a' — b-) 

 tan ~. The entire area of the lemniscate of Bernouilli is found by putting 

 a = 6 in the preceding expression, which gives the value a-. The inverse of the 

 curve from the center is the hyperbola 



o- X- — b- if- = 1, (20') 



which has the same eccentricity as the quartic. The polar reciprocal of this 

 hyperbola is 



--J7 = U (21') 



a' b- 



ehowing that the inverse of an hyperbola from the center is the pedal of its polar 



reciprocal with respect to the center, as in the corresponding case of the ellipse. 



The curv^e may be described mechanically by means of its pedal properties, 

 very much as in the preceding case of the ellipse. 



Many of the inverse theorems relating to the ellipse and oval are equally true 

 when applied to the hyperbola and this curve, which may be called a lemniscate. 

 Following are a few additional theorems relating more especially to the hyper- 

 bola and the lemniscate : 



1. The difference of the two lines drawn fi-om any point of an hyperbola ta 

 the foci is equal to the transverse axis. 



2. The difference of the two lines drawn from any point on a lemniscate to the 

 foci bears a constant ratio to the radius vector of the point. 



1. PN is the ordinate of a point P on an hyperbola, PO is the normal meet- 

 ing the axis in O; if NP be produced to meet the asymptote in Q, then is QG 

 at right angles to the asymptote. 



2. PN is a " central " circle through the point P on a lemniscate and whose 

 "central diameter" is the axis of abscissae, PG is a "central" circle cutting 

 the curve orthogonally at P and meeting the axis of abscissae at G; if the " cen- 

 tral" circle NP cut the tangent at the origin at Q, then is the "central" circle 

 QR cut orthogonally by this tangent. 



The lemniscate may be said to have a "conjugate" lemniscate, just as the 

 hyperbola has a conjugate hyperbola. 



1. The equation of an hyperbola is a-x- — b-y"- = 1. The equation of its con- 

 jugate is arxr — b-i)'' = — 1. The equation of its asymptote is a'x'' — b'y- = 0. 



&- + a' ^^ 

 The equation of an hyperbola referred to its asymptotes is xy = — — . That 



li' + a' ^""^ 



of its conjugate is xy ^= — 777~- 



4a-b'' 



2. The equation of a lemniscate is a-x- — b''y^ = (x^ + y-) . The equation of its 

 conjugate is a'x- — b-y^ = — {x^ + y'') . The equation of tangents at the origin is 

 a^x^ — b'y^ = 0. The equation of a lemniscate referred to tangents at the origin 



^2 _r_ 52 J . (^2 _|_ 52 ^ 



is xy = ■ (.r^ + y') . That of its conjugate is xy = — — - {x^ + y-). 



4a'b^ 4Ci 0- 



1. The two lines joining the points in which any two tangents to an hyperbola 

 meet the asymptotes are parallel to the chord of contact of the tangents and are 

 equidistant from it. 



2. The two " central " circles joining the points in which any two " central " 



