210 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The distance between two points (pi, 0]), (p 2 , #2) is, 

 d = pi — Pi + pi pi (02 — 0i). 



For the angle between OP2 and OPi is 02 — 0i and expressing the angle 

 in terms of the sides of a triangle, we have, 



e 2 - dl = pl + d - p2 

 — p\pi 



which is the relation above. 



The equation of a straight line is, 



Ap0 + Bp + C = 10, 



which is obtained by finding the locus of a point at a constant distance 

 from a fixed point. The angle which this line makes with / is, 



0= l 



A+ B 



Hence for lines through F, A = — C. The element of arc is, 



ds = dp + phld. 



A second and for most purposes more convenient system has two 

 points of reference. Algebraically this is very similar to cartesian 

 coordinates in euclidean geometry but geometrically like bipolar 

 coordinates. Let X and Y be the points of reference so chosen that 



3(*i,Vi) 



x(o,-n r(i,o) 



Figure 2. 



the line XY does not pass through F. The coordinates of a point A 

 is then the distances from X, Y to the point. The only indeterminate 

 points for this system are those of the line/. For convenience we will 

 choose the distance from X to Y to be unity. The coordinates of X 

 and Y are (0,-1), (1, 0). 



