A manual of the principles and practice of road-making: comprising the location, construction, and improvement of roads, (common, macadam, paved, plank, etc.) and rail-roads. By W. M. Gillespie ...

APPENDIX C. 417
PROBLEM II.
It is required to make the difference of the two radii a minimum.
t - t'' =t. tan. a 2cos.'r =t' tan. a -- - = qy 90 - a
PROBLEM III.
When one radius is given to find the other.
By construction. Draw perpendiculars at A and z. Set off the
given radius on each of them, from A to some point, o, and from z
to some point, P. Join o P, and bisect it by a perpendicular. This
perpendicular meets the perpendicular from z in o'. o and o' are
the desired centres, and the two curves will unite on the line
through these points.
Analytically.* Let A z, the angles I A z and I Z A, and the radius
A o = r, be given, to find the second radius r'.
From A run a curve with the first Fig. 16.
radius to D, where the tangent, D E, IDo
becomes parallel to zI. The line, E
z D, prolonged will meet the curve at
the common tangent point, c.
ZD
r i,r+r` A
2 sin. C zI'
When the angle at z is greater than
the angle at A, the formula becomes,
ZD\ 0
0'
~ 2 sin. c zI - /
In the field the point, D, may be found by laying off the
angle, I A D = j (I A Z + I Z A), and measuring the distance, A D
= 2r. sin. (IAz + IZA).
REVERSED CURVES.
If the two branches of the curve, instead of both lying on the same
* From Henck's Field Book.
18*