56 Colorado College Studies. 



it subtends in the first of these circles will envelope a third 

 circle which will pass through the points of intersection of the 

 two former, or have in common with them the same secants, 

 whether real or ideal." 



By an "ideal secant" Poncelet means a real line, whose 

 points of intersection with the curve are imaginary. The 

 description " a common real or ideal secant " applies therefore 

 to the radical axis of two circles, which in all cases fulfils the 

 analytical condition for passing through the common finite 

 points of the two curves; but, in the case of circles which do 

 not meet in real i)oints, can be called their common secant only 

 in an "ideal" sense. Jacobi appears to prefer the designation 

 for this line which was introduced by Steiuer, viz., " the locus 

 of equal tangents," — a name given in reference to the well 

 known fact that whether two circles meet iu real points or not, 

 the tangents drawn to both from any point of the radical axis 

 which is exterior to them, are equal in length. 



The proof given by Poncelet for the theorem depends 

 upon the properties of conies having double contact with one 

 another. 



The property of the radical axis, to which reference has just 

 been made, furnishes an easy geometrical solution of the follow- 

 ing problem : 



Having given a circle, a chord of the same, and a line as 

 radical axis, to determine the circle which has, iu common with 

 the given circle, the given line as a radical axis, and which 

 touches the given chord. ( See Fig. 2j. 



From C the center of the given circle, draw a line CD per- 

 pendicular to the given radical axis. On this line, extended 

 indefinitely, lie the centers of the whole system — or "pencil,'' — 

 of circles which have with the given circle the given radical axis. 

 Produce, if necessary, the given chord PM to meet the radical 

 axis in Q. Determine in the usual manner, a point R on the 

 given circle so that QR would be a tangent. With Q as center 

 and QR as radius strike an arc RTG meeting the given chord 



