Railway Curves. 131 



The explicit directions for setting out (page 137) with the simple 

 example given on page 139, will make every thing else sufficiently- 

 comprehensible, and it is believed that it will all be found easy after 

 a little practice in the field, in the ordinary position of the parts 

 certainly, but it has been deemed most desirable to attempt to lay 

 down one general plan to provide for all possible cases. 



Ogee Curves of Common Radius. 



Investigation of formulae for finding the radius that shall be common 

 to the two arcs of an $ curve, together with the length and 

 position of the chords. 



It will be proved that 



C( . s sin. a + sin. 8 

 I. Sin. S = l. 



(and only so when the radius 

 is common) 



r. Chord DA = 2 r cos. a _ZL 



2 



deflected at an [_ = 90°— 



from the tangent. 



cos. a + cos. j3 + 2 cos. 8 



iv. Chord AB = 2 r cos. 

 at 90° -ill 



It has only to be remembered, as far as these formulae are con- 

 cerned, that, as before observed, the proper station to consider an 

 angle is the angular point (as B or D) and that the angle a or 8 is 

 then to be estimated in determining the sign (plus or minus) by the 

 " quadrant it is in ;" supposing the angle to be the degrees passed 

 over by the radii OD, PB, severally and turning out of the same 

 line DB on the same hand, right or left. The usual conventional 

 direction is to the right. Thus, then, an angle in the 3rd or 4th 

 quadrant has its sine minus, and one in the 2nd or 3rd quadrant 

 has its cosine minus. 



The numerical values of these ratios and their degrees are to be 

 taken simply as those of the angles designated a and 8 in the figures. 

 Thus, in the first place, in fig. R, a is in the 2nd quadrant, and (3 in 

 the 4th quadrant. 



. •. The sine of a is (+) ; its cosine ( — ) 

 Sine of 8 is ( — ) ; cosine (+) 



so that in formulae I. the sign (+) of sin. /3 is reversed, and becomes 

 ( — ) ; and in formulas II. cos. a becomes ( — ); whilst, secondly, as re- 

 gards their value in degrees, in formulas III. and IV. a, 8 and 8 merely 

 represent the angular quantities so designated in the figures, so that 



the angular quantity """*" and "' are necessarily less than a right 



2 2 



angle, because they are each half of two angles of a triangle. 



