CUBIC PARABOLA APPLIED AS A TRANSITION CURVE. LIX. 
Further, it is not desirable that the radius of curvature at any 
point on the transition should be less than the radius of the 
circular curve. It is for this reason that the application of the 
cubic parabola, as a transition curve is limited. If this were not 
so, then it would be admissible to eliminate the circular arc in all 
cases. This could be done by taking the angle w/2 and a value of 
p; hence obtaining x, and the coefficient m, but it will be found, 
if a certain limit is exceeded, that the radius of curvature will be 
less at some point on the parabola than at 2, y.. 
This limit is reached when “H” has a maximum value and may 
be obtained thus 
EF ef tf —- R+ Roos ¢......... b. 
Eliminating mz’, also eae F equal to unity and reducing, we 
have 
H = 2sin® ¢ cos ¢ + cos¢ — 1 c 
from this equation it will be observed that H has a maximum 
1 ie” 
value when pS, : 
The angle ¢ corresponding to this cosine is 24° 5’ 41”-45 and the 
congruous value of a, being 0-6804...p. It is therefore advisable 
not to exceed this value of the ses ¢, or the objection mentioned 
will become evident. 
Remarks.— When alininntivig mzx* from equation (5) it will be 
noticed that a very useful form of the original equation is obtained 
namely 
= 3 Rsin® > COs g.....+.-+4+- d 
we have also at the same time 
x = 2R sin dp cos® P........000e 5 
By this method of procedure we may fix a point ¢ on the circular 
curve, and hence the angle ¢ becomes known, which should not 
exceed the limit previously mentioned. The solution of the 
equations (d) and (5) will give the rectangular cartesian co-ordin- 
ates of the point c relative to the axes of the parabola. The 
remaining qnantities required being obtained in a similar manner 
as explained in the paper previously referred to. 
