124 Railway Curves. 



consideration of the capabilities of the ground — what curves 

 are possible, first for the evasion of obstructions, and next 

 for the recovery of the direct course, from point to point, pre- 

 determined to be passed through. 



I have treated the subject analytically, and have aimed at 

 a tangible set of simple deductions from the data, to be fixed 

 in the mind, and as applying to all cases, so as in a 

 great degree to supersede theorising in the field, and to have 

 the mind as free as possible for viewing leisurely and coolly 

 the practical and physical difficulties of execution. 



The track of a railway may be considered as a system of 

 curves and their tangents, each curve to the right or left 

 necessitating an early reverse curve ; for the further a 

 divergence is continued, the more there is to rectify by the 

 return curve. It has least to do when the two reciprocating 

 arcs are contiguous. 



It is proposed to prove that the locus of the point of con- 

 tact, or of osculation of these two arcs, and the locus also for 

 a compound curve, is the arc of a circle, which may be called 

 the " arc of contact," and that where both are possible the 

 locus of both is the same arc. 



The formulas for finding the radius and position of the 

 tangents of the arc of contact are simple, so that the arc 

 may be set out very readily ; and it will be seen that it at 

 once affords a key to the capabilities of the ground. I pro- 

 pose to consider other formulas connected with or in dependent 

 of the arc of contact. 



The Arc of Contact for Simple and Compound Curves. 



Investigation No. I. (of the " arc of contact," or locus of the 

 osculating point (A) of every possible $ or compound curve, 

 from a given railway tangent at B to a railway tangent at D. 



It is first to be shown that the locus of the osculating point is the 

 arc of a circle, in either case. 



Figure M shows the usual position of the parts in an $ curve, the 

 angles a and /3 opening each to the right out of BD, to an observer 

 stationed on the angular points B and D respectively. Figures N 

 and O show a possible arrangement of the parts, the angles a and /3 

 opening right and left respectively out of DB, so that, considered as 

 opening to the right, /3 in each figure (N and O) is, evidently, as 

 determining the sign ( = ) of its sine, cosine, &c, to be placed or taken 



be, in the fourth quadrant. Figure P will be sufficient for the 



