16 



KANSAS ACADEMY OF SCIENCE. 



straight lines coincident with the axis of abscisste, if e is made unity by b 

 assuming a value that is infinite compared with a. 



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



iS 



=/ 



+ — r ck, 



which becomes, after values are substituted 



ra'' cos-k-\-h^sWk 

 Q L d^ cos- k + 6- sin^ k 

 The area. A, of the curve may be found by the formula 





\A 





r'- dk, 



(18; 



dk. 



(19) 



which gives for the whole area inclosed by the curve the value, — (a^ + b'^). 

 The inverse of the given curve with respect to the center is, 



o^ x'^-\- V- y- = 7, 

 an ellipse whose eccentricity is identical with that of the given curve, 

 reciprocal of this ellipse is. 



a^ b-' 



(20) 

 The polar 



(21) 



If circles be described upon the semi-diameters of this latter ellipse, they will en- 

 velop the quartic under discussion. Hence, in general, the inverse of an ellipse 

 from the center is the pedal curve of its polar reciprocal with respect to the 

 center. 



The above relations furnish a convenient method of investigating many prop- 

 erties of the curve given at the beginning of this paper. 



The curve may be described mechanically by means of its pedal property, 

 thus: 



Let R be an elliptical board with semi-axis a and 6, a 

 "T" square is held by a pivot at O working in a slot, 

 while the arm AB slides against the board; a pencil or 

 crayon at -Pwill describe the required curve Q,. 



Following is a partial list of theorems relating to the 

 curve. They were obtained by inverting those proper- 

 lies only of the ellipse which in some way were depend- 

 ;)t upon the center of the curve. The first paragraph 

 ontains the original theorem in the conic; the second 

 paragraph, that relating to the quartic. 



From the fact that straight lines invert into circles 

 passing through the origin, it is convenient to call all 

 such circles "central" circles. The quartic will be 

 called an oval. 



1. Two tangents can be drawn to an ellipse from any point, which will be 

 real, coincident, or imaginary, according as the point is outside, upon or within 

 the curve. 



2. Two "central" tangent circles can be drawn to an oval from any point, 

 which will be real, coincident, or imaginary, according as the point is within, 

 upon, or outside the curve. 



1. If the polar of a point P with respect to an ellipse pass through the point 

 Q, then will the polar of Q pass through P. 



