TABLES TO FACILITATE LOCATION OF THE CUBIC PARABOLA. 283 



when P equals 5 /^Ig, there are two equal and positive, and one 

 negative root. Under these conditions the following method of 

 solution is available. 



If we put ra = 2vj 



Cos 3 a = - M- \ Lo S n = 006246935 



therefore n 



Cos 3a= -p [04146520] 6 



then 



Sin <f> = n cos (120° - a) 7 



will be the root required. The quantity within brackets being a 

 logarithm. 



From a table of trigonometrical ratios, the angle cf> may now be 

 obtained, and from equation (2) the coefficient m may be deter- 

 mined. In the computation of the remaining quantities of the 

 table, the following formulae 1 have been used and require no 

 explanation. 



a?' = Sin cf> 



y c = mx c 3 



h = y c — versin </> 



S = x c (1 +Yb- tan 2 (/> - - 7 V tan 4 (/>+ 2-^g- tan 6 <£ etc.). 



The final column contains the circular measure of 2<£. 



In the tabulation of the several quantities, simplicity is gained 

 by removing the factor R from the formulae. The table therefore 

 involves this quantity and contains the constants for the cubic 

 parabola when R equals unity. For any other value of R the 

 quantities must be multiplied by the radius of the circular curve 

 to be used, with the exception of the angle <£ and log m. A 

 practical example will shew how the table is to be used. 



1 The notation here used is that adopted in a paper to t>e found in 

 Journal of the Royal Society of N. S. Wales, 1895, Vol. xxix., p. 51. See 

 also plate 10 of the same volume, x' - Distance along the axis x between 

 the point x c and a point at right angles to the tangent point of the circular 

 curve. y c = Value of the co-ordinate, to the point of contact of the tran- 

 sition and circular curve, measured along the axis X at the distance x c from 

 the origin, h = Distance between the parallel tangents, s = Length of 

 the arc of the cubic parabola. 



