( 649 ) 



(ouches circle {M) in I'\ so /- is an eciuihiteml l)y[)erbola passing' 

 , tlirougii / witli directions of asymptotes I li and I A and touching 

 circle (J/) in P. 



Tlie centre C' of X- can be determined in tlie following manner. 



We suppose X' to be constructed bj means of tiie projective pencils 

 formed by rays parallel to IX and IT. If the two points united in 

 the point of contact P were separated then the two pairs of parallel 

 rays through these ])oints would determine two points A^ and .4 on 

 IX and two points B^ and B^ on / V and the centre would be the 

 point of intersection of A^ B^ and .4, B^. It is true .4^ and A coin- 

 cide in A, and B^ and B^ in B, but from the preceding follows 

 that the centre C lies on AB. If in P we draw the tangent to X- 

 perpendicular to the normal PI, then a point of each asymptote lies 

 at equal distance from P. So we measure Pl\ = P2\ and we draw 

 T.a/lX, T,C//IV; C' would be the centre of /Mfc' were situated 

 on AB. However, out of the figure is evident that C' lies on a right line 

 symmetric to 1\1\ with respect to PA, and therefore perpendicular 

 to AB. So the centre 6' of ^, is the footpoint of the normal let 

 down out of P on AB. 



If we consider diflferent positions of AB and if we construct the 

 successive positions of the point C, then the locus is an astroid on 

 the axes lA and IB. The hyperbola /' keeps touching the inva- 

 riable circle with fP as radius; so the astroid is the locus of the 

 centres of the equilateral hyperbolae with asj mptote directions I A 

 and IB passing through I and touching the last-mentioned circle. 



The (wo diameters I A and IB of circle / form with the right 

 line at infinity a polar triangle of the circle; so the points C have 

 the si^nitication of poles of one of the sides of that polar triangle. 



