and Loci of Apollonms, ^c. 79 



of the polygon is even or odd. Now the points and magni- 

 tudes being as above indicated, it is e-\adent tliat when tlie 

 number of sides is odd, the locus of a is a circle having as 

 centre the fixed point in which the tangent at -a cuts ABC, 

 and a radius equal this tangent, Avhose magnitude is evidently 

 constant. And w^hen the number of sides of the polygon is 

 even, the locus, under the prescribed necessary conditions for 

 inscriptable polygons, is not restricted in the plane. 



Again, returning to the case in which the points are not in 

 a straight line, avc have seen that the circle O is fixed, and 

 that the locus of a — if a is at all capable of innumerable con- 

 tinuous positions — must be the circumference of circle O. And 

 here, in order to facilitate the investigation of the relations, 

 let us first suppose a polygon of three sides, and see whether 

 it is possible to have three points A, B, C not in a straight line, 

 so related that if through any one of them C we draw any chord 

 rs to a fixed circle, aiul Ar to cut the circle again in t, and 

 Xs to cut the circle again in v, then will st and rv pass 

 through B. 



Now, from the properties of poles and polars, it is easy to 

 see that when each of the three points is the pole of the line 

 joining the other two, these conditions will be fulfilled; more- 

 over, since if any two A, B of the points so determined be 

 fixed, the third C is in the intersection of the perpendiculars 

 from A on OB and from B on OA, it follows that the given 

 points must be restricted to the number three when they are 

 not all in a straight line. 



I need scarcely mention that similar investigations may be 

 made when instead of points in a plane, &c., we are proposed 

 points on the surface of a sphere, and the products of the tan- 

 gents of the halves of the segments made on the sides of the 

 inscriptable polygon by the fixed points through wliicli they 

 pass; or, when we arc proposed points in space, and a figuie 

 inscriptable in a sphere, and constant rectangles under the 

 distances from the fixed points to its angles situated on the 

 sides passing through these fixed points. 



PORISMATIC RESEARCHES. 



(The figures to be supplied by the reader.) 



Given three straight lines MM, NN, and LL, in position, 

 and given the point P in the first line MM, and the point Q, i?i 



