52 C. J. MERFIELD. 
applied to the circular arcs. The advantage of such a system 
cannot be overestimated. It is however not the object of this 
paper to discuss these matters. 
Mr. Shellshear in his paper read before this Society, has given 
nearly all that can be desired about this particuiar curve, and his 
method of location is carried to a fair degree of approximation. 
For transitions of from one to two chains in length, his procedure 
is practically exact, provided that the radius of curvature be large 
at the point of contact with the circular arc. Increasing the 
transition to say four chains, the error becomes noticeable, and 
seems to require a more exact method of location. 
In the following pages the necessary equations are deduced, - 
which upon solution, will give data for the location of the cubi¢ 
parabola to any circular arc, so that the radius of curvature will 
be equal to that of the circular are at the point of contact. Table 
No. 1 gives the offsets, and other quantities, according to this 
exact method, in several useful cases. 
This degree of accuracy may not always be required, it depends 
upon the solution of an awkward equation, which would perhaps 
deter some engineers from employing the formule. 
During the author’s duties in the Railway Construction Branch 
Public Works, N.S.W., he was called upon to prepare tables of 
the offsets to this curve, for several cases, to be used by the 
surveyors of the department. 
The method adopted in their preparation, was to comput? 
the offset, at the point of contact, from the equation Yor 
3 { R- / R?— ) . \ the intermediate offsets being varied as the 
cube of the distance along the axis of X. This would represent 
a cubic parabola, but the radius of curvature is in error, de 
small extent, at the point of contact with the circular are. This 
error is however not sufficiently large to injuriously affect the 
usefulness of this approximate method. 
To many the method of locating curves by a system of offsets 
is tedious. Tables of the angles, contained between the radius 
