282 C. J. MERFIELD. 



the circular curve completely, this is very simply done in all cases, 

 so long as the deflection angle of the tangents does not exceed 

 48° IT 22 -"9, methods for so doing are explained in the note above 

 referred to. When tables are accessible, then there is no more 

 difficulty in laying out a cubic parabola than the circular curve,, 

 and little, if any, additional time involved in the location. 



Method of preparing the tables. — The second column of the table 

 contains $> the angle between the axis of X and the tangent to the 

 transition and circular curve at the point of contact, where the 

 radius of curvature p of the parabola must be equal to the radius 

 R of the circular curve. It is therefore necessary to determine 

 the coefficient of the adopted equation 



y = mx 3 1. 



so that p equals R at the point of tangency. 



From equation (1) we have 



d y =3m- 2 =tan <\> 2 



ax 



d 2 y n 

 __* = o mx 

 ax 2 



substituting these values in the equation 



'- {'.♦(»■ }*-*3 ' 



it will be found after reduction, that 



x/2p = sin cf> cos 2 </> 4 



Making p equal to unity and adopting a suitable value of x Y 



denoted by x c , and which should not exceed # 68...p, then putting 



fS in place of x c /2 and writing [x for sin <f> the above equation (4) 



becomes 



^3_ /x + i g = 5 



from which ll can be found and hence sin <j>. 



The solution of either equation 4 or 5 may be facilitated in the 

 following way. Between the limits of the table this equation has 

 three real roots, two positive and one negative, and at the limit, 1 



1 It has been demonstrated that the practical application of the cubic 

 parabola as a transition is limited. The cosine of the angle <£ must not 

 exceed Vf, the congruous value of x c j p being equal to 5 V-^. See Journal 

 of the Koyal Society, N.S. Wales. 1897, Vol. xxxi., p. lix. 



