594 PROCEEDINGS OF SECTION H. 



farther from the point A will, after a given distance, decrease, so 

 that a limit to its length is imposed. When curves in opposite 

 directions are set out sufficient length of straight should be pro- 

 vided between them to allow for the two halves (not necessarilj- 

 equal) of the transition curves required by the circular arcs. The 

 writer is unable to suggest a method by which two reverse curves 

 with a common tangent can be united by transition curves. 



The cubic parabola can be readily ranged with the theodolite by 

 tangential angles. The instiument is set up at A. The angle for 

 the whole length up to C is J a (Fig. 3) and dividing the whole 

 length into n jjarts we get for 



1st angle ^, 



« X 2- 



2nd 

 3rd 



a X 3- 

 3 n"' 



Last 



3 n' "' 3 



the angles varying as the sqxwre of the distances from A. It will 

 be noted that the offsets these angles will give for their respective 

 distances are the same as those given by formula for ordinates. 



The curve may also be set out with the chain by offsets from the 

 chord (gee Fig. 4) : — 



a = (i) chord '^ ) «? . o ., . 



-^^^^ — = > offset from tangent, 



n R ) 



h = (1) chord ^ ^ 



«R 

 c = (2) chord 



n R 

 d = (3) chord "- ^Deflection off-sets. 



e = (4) chord " 

 ?i R 



R being radius in ft. and « = number of points. This method 

 may, however, lead to serious cumulative errors. 



An attempt has been made to ease the curves of existing lines at 

 their tangent points, and introduce the cubic parabola, but the 

 length from the start to the osculation with the curve is so great 

 (including an angle of OA'er 34° on each side of the tangent) as to 

 preclude its use, there being also a point on it where the least 

 radius of curvature is only 86 per cent, of that of the existing 

 curve. If an auxiliary curve of less radius inside the other be 

 introduced (see P^'ig. 5) the difficulty is partly overcome much in 



