172 BOOK OF MATHEMATICAL PROBLEMS, 



with respect to the former is 



I' m' m 

 I 7ii m 



__"?' ? 

 I n m 



or the reciprocal of one of these, according to the order in which 

 the points are taken : and these are also the anharmonic ratios of 

 the range formed by the four common tangents on any tangent to 

 the latter. 



X. Reciprocal Polars, Projections. 



If there be a system of points, and straight lines, lying in the 

 same plane, and we take the polars of the points and the poles of 

 the straight lines with respect to any conic in that plane, we obtain 

 a system reciprocal to the former j so that to a series of points 

 lying on any curve in the first system correspond a series of 

 straight lines touching a certain other curve in the second system, 

 and vice versa : and, in particular, to any number of points lying 

 on a straight line or a conic, correspond a number of straight 

 lines passing through a point or touching a conic. Thus, from 

 any general theorem of position may be deduced a reciprocal 

 theorem. It is in nearly all cases advisable to take a circle for 

 the auxiliary conic, with respect to which the system is recipro- 

 cated; the point (p) corresponding to any straight line being 

 then found by drawing from 0, the centre of the circle, OP per- 

 pendicular to the straight line, and taking on it a point p, such 

 that OP . Op = k*, k being the radius of the circle : and similarly 

 the straight line through p at right angles to OP is the straight 

 line corresponding to the point P. 



To draw the figure reciprocal to a triangle ABC, with respect 

 to a circle whose centre is 0, or, more shortly, with respect to the 

 point O y draw Oa perpendicular to C, and in it take any point a ; 

 through a, draw straight lines perpendicular to OC, CA, meeting 



