342 ON THE APPLICATION OF ANALYSIS TO THE DISCOVERY 



If we prolong either of the two ordinates (GE) below the ab- 

 scisscB, by a quantity (EL) equal to the first ordinate, the extre- 

 mity of the ordinate, thus prolonged, will always be situated in a 

 curve, similar and equal to the given curve, which is also given 

 by position. 



The position of the given curve is parallel to the original 

 curve. The deduction of these porisms is so obvious, from 

 the property of the class of curves, that I consider any farther 

 explanation of them unnecessary. We shall, however, by as- 

 siofning particular values to some of the functions, find some 



very simple results. Let us suppose « x =: v/o^ — a;% then 

 the family of curves will be contained in the equation, 



C9 {x) 



fx-^f (v/a^ — X*) ' 



Of these curves, the following property may be stated : 



Any curve of this family being given, a straight line (HK) 

 may be found through any point, of which, if a line (HD) be 

 drawn at right angles to the axis of the abscissce, it will cut off 

 an abscissa (AD), and part of it will form the ordinate (CD), if 

 an abscissa (AE) be found, whose ordinate (BE) is equal to that 

 part of the line drawn, which is intercepted between the line found 

 and the curve ; and if both the two ordinates thus found, be pro- 

 longed below the axis, until the part of each below is equal to the 

 abscissa belonging to the other ordinate, the two points to which 

 these lines are prolonged, are situated in the circumference of a 

 circle given by position. See Fig. 4. 



Let « X = — oc, then the curves included in the family 



y — ^'''f possess the following property : 



fX-\-f{ — x) " 



Any 



