THEORY OF COMPOUND CURVES IN FIELD ENGINEERING 



139 



and the projective conjugate pencil 0] the other pencil, and is therefore, in 

 general, a bicircular quartic. 



In the special case where the two fundamental points of each pencil 

 of circles coincide, or where all circles are tangent to a fixed line at a fixed 

 point in each pencil, the bicircular quartic has also two finite real double 

 points, and degenerates therefore into two circles. This, however, repre- 

 sents precisely the case of compound curves in railroad engineering. 



Fig. i 



III. THEORY OF COMPOUND CURVES 



Assume the equations of the two special pencils of circles in the form 

 U l — 2 \JJ 2 = o, 

 U 3 — 2\'U 3 = o, (16) 



and as the coincident fundamental points of these pencils the points 

 (o } 0) and (a, b), Fig. 2. For the sake of convenience we replace the 

 parameters of the pencils (8) and (9) by — 2 X and — 2 X'. 



A verification of the results obtained in the preceding sections will 

 lead to furmulas applicable to actual field problems. 



