THEORY OF COMPOUND CURVES IN FIELD ENGINEERING 1 47 



(P). In a similar manner it is proved that the diameter of the circle 

 (Q) is equal to TM + TO. To sum up, we may say: 



The locus of points 0} compound curves 0} all compound curves between 

 two tangents, TM and TO, and two tangent points, M and O, consists of 

 two circles which pass through the points M and O, and whose centers lie 

 on the hi- sectors 0} the tangents TM and TO. 



By this condition, the centers P and Q of these circles and, therefore, 

 the circles themselves are perfectly determined. In all compound curves 

 the radial lines through the points of compound curves belonging to this 

 system are all tangent to either of two fixed circles having as their centers 

 the points P or Q and for their radii the values 



TM — TO TM + TO' 



or , 



2 2 



The points, P, Q, O, M, T, in Fig. 3, all lie on the same circle, having 

 the line PQ as a diameter. 



Among the great number of special cases we will consider the problem 

 where the two tangents are parallel. The general theorem and con- 

 struction still hold, so that the solution is simply a matter of reduction 

 for special values. To find the equations of the locus we have to put 

 k = 00 in formulas (23) and (24). Observing that for an infinitely 

 large value of k 



lim (— bk -f b I 1 + kA =0 

 k = 00 



lim (ok — aV 1 + & 2 ) = o 



k = 00 



these equations become respectively 



bx —ay = o (28) 



and 



x 2 + y 2 — ax — by = o, 

 or 



s a 2 + b 2 



(x — a/ 2 ) 2 + (y — b/ 2 ) 2 = . (29) 



1 In practical treatises on this subject the conception of compound curves is not given under this general 

 point of view. Thus in Mr. W. H. Searles's treatise on Field Engineering the following restriction is made: 



"A compound curve consists of two or more consecutive circular arcs of different radii, having their 

 centers on the same side of the curve; but any two consecutive arcs must have a common tangent at their meeting 

 point, or their radii at this point must coincide in Dosition." 



