Theory of Compound Curves in Railroad 



Hnoineerin^. 



BV ARNOLD EMCH. 



1. It is the purpose of this note to treat the problem of com- 

 pound curves as it occurs in railroad engineering from a general 

 geometrical stand point which enables us to discuss in an easy man- 

 ner all the essential parts of the problem. It will be seen that the 

 theory of compound curv^es is identical with the theory of two pro- 

 jective special pencils of circles. In Vol. Ill, No. 5, of the 

 Aiiicricaii Mathciitatical Moiitlily, the author has treated of projective 

 pencils of circles in connection with a special complex of lines of 

 the second degree*. 



The theorem has been established: 



TIic locus of the points of iangcncy of both taii^cnt-circlcs of two 

 pencils of circles is a bi-circnlar ci/rve of the fourth order. The same 

 curve is also produced b\ one of the pencils and the projective conjugate 

 pencil of the other pencil. 

 ■ This curve, of course, passes through the four fundamental 

 points of the pencils of circles. Now we may take the special 

 case where the two fundamental points of each pencil of circles 

 coincide, or where all the circles of the pencil are tangent to a fixed 

 line at a fixed point. This, however, represents precisely the case 

 of compound curves in railroad engineering. Evidently the bi- 

 circular curve of the fourth order, having also two finite double 

 points, must degenerate into ttco circles. 



2. In order to apply the previous result we will verify it directly. 

 First we will write the equations of the two special pencils of cir- 

 cles in the form 



U — 2AV=o, 



(I) 

 U'— 2A'V'=o, 



and assume as the double points (coinciding fundamental points) 

 of these pencils the points (0,0) and (a,b) fig. i. 



*A fJpecial Complex of the Second Degree and its Kelation with the Pencils of 

 Circles, 



(99) KAN- Umv. <^UAR., VOU V, NQ 2, OCTOBER, 1890. 



