114 JOURNAL OF THE 



z z 



^ = — , D° = — (20). 



D° ^ 



Line C gives the semi-chord of the arc of a 1° circular 

 curve, of which a subtends half that arc .*. C = 5729.65 

 sin a. For a D° curve, divide by D° expressed in degrees 

 and decimals. 



Line Q gives the product ^ D°, where ^ is the distance 

 CD (fig. i) and D° the degree of circular curve LA. We 

 find this product as follows: Call, for brevity, Ri the radius 



Rx 



of a 1° curve (5729.65 was used), then in fig. i, R = — 



D° 



and ^ = Kh -b O^ — OB = X + Kcos a — R. There- 

 fore, from (17), 



.2 NX R, R, 



g = 1 cos a 



D° D° D° 



.'. Q = qD"" = .2 NX -f R, t-^i ^ — R, (21). 



Hence, calling the ratio of q to x (the ordinate at L) F, 

 we have, 



./ ^D° Q 



F = — = = (22) 



X .2 NX .2 NX 



from these formulas, rows F and Q of the tables were com- 

 puted. It is seen that this ratio is 14^ or nearly so through- 

 out. 



From the above exac^ formulas, the table of "Equiva- 

 lents" given below is made out. They are identical with 

 the formulas given by Mr. Howard, though the latter were 

 deduced in a totally different way. 



20 N^/ 100 X 100 y 



Q X Y 



20 N^ 200 a 



D° D° 



