132 Railway Curves. 



With this caution, these fonnulse are universally applicable, or, as 

 it is technically called, " general." 



Investigation of Formula No. I. 



Let PS be drawn parallel to BD. Then, in Fig. Q, 

 OS = OF+EP, but OS^OP sin. 8+2 r sin. 8 



and OF+EP = OD sin. a+PB sin. fi=r (sin. a+sin. /3) 



o • 9 / • n \ a- * sin. a+sin. 8 



.: 2 r sin. 8 = r (sm. a x sin. /3) .*. Sin. B= ^ 



2 



N.B. — By the trigonometrical law of signs, this formula is true for 

 all values of a and j3, which angles are in this figure acute angles, 

 and have all functions (Sin., Cos., Tangt., &c.) positive. 



And it is easily shown that what is said above of Fig. R as to the 

 alteration of sign ( =+= ) in the fourth quadrant, makes sine /3 negative 

 in Fig. R. 



Where OS = OP sin. 8 = 2 r sin. 8=OF-EP=r (wa.a-wa.fi) 



so that the equation becomes sin. 8 = ~ ' " in Fig R. 



2 



Investigation of Formula No. II. 

 Fig. Q. 



BD = r cos. a + 2 r cos. 8 + r cos. /3 



cos. a +2 cos. <5 + cos. ft 



N.B. — This is true also for all values of a and /?, which are in 

 this figure acute angles, and have all functions positive, and in Fig. R 

 it is easily shown that the sign ( =L ) of cos. a is reversed from its 

 being in the second quadrant, for BD =r cos. (3+2 r cos. 8 — r cos. a 



BD 



.-. The equation becomes, ^ = cos . ^ +2 cos. 8-cos. a ^ Fi °" R > 



As to Formulae III. and IV., it is only required to show that — the 

 triangles DO A and BPM being isosceles, their equal sides (rot 

 itroo-KeXoi) being the equal radii, and their equal angles, at the bases 

 DA, BA, being severally a mean between a and 8 and (3 and 8 — if 

 then each be bi-sected by the perpr. from apex upon base, into two 

 right-angled triangles, of which half the chord is the adjacent side to 

 the I at the base, and of which, radius is the hypothenuse ; then 



. •. aa J- 5 - cos. . \ cos. (L at base) = '— = J^ 



hyp. 



.-. chd. = 2 r cos. (|_ at base.) 

 (i.e.) chd. AD and AB = 2 r cos. a ~*~ and 2 r cos .LZ_, severally. 



