THEORY OF COMPOUND CURVES IN FIELD ENGINEERING 1 43 



which both pass through the origin and through the point (a, b), the two 

 finite double points. Thus, as established before, in this case the 

 bicircular cubic degenerates into two circles. 



The co-ordinates of the center of the first circle are 



a — bk — bl'i + k 2 



and of the second 



(25) 



(26) 



The radii of these circles are V m 2 + n 2 and V y m' 2 -f- n' 2 . It is 

 easily verified that 



(m — m') 2 + (n — n') 2 = m 2 + w 2 + m' 2 + n' 2 . 



This, however, is the condition that two circles are normal to each 

 other. Hence : 



The two circles forming the locus intersect each other at right angles. 



From this follows that the points P, Q, O, M, T in Fig. 3, all lie on 

 the same circle with the line PQ as a diameter. 



The normal pencil of circles of the pencil (18) is obtained by con- 



OC ' — OL 



sidering the normal to the straight line (17), -, = k, which is 



xk -\- y — ak — 6 = 0, 



and the zero-circle at the point (a, b) as two circles of the pencil. The 

 required normal pencil is therefore given by the equation 



(x — a) 2 + (y — b) 2 — 2 X" (xk + y — ak—b) = o. 



For a fixed value of V the co-ordinates of the center of the correspond- 

 ing circle are 



