In the figure, draw L K parallel to G V, and passing through centre of first branch. Then MK= (R+r) X cos. B. NL (R + r) X cos. C. NL - MK= D. A = (90° — C) — (90° — B)=B — C. Example. log. 3.352568 log. 9.916859 3.269427 log. 3.290925 log. 3.352568 . TO PASS A CURVE THROUGH A FIXED POINT, Given the intersection angle, A, of two tangents, to find the radius, R, of a curve which shall pass through a point, C; the position of said point, with reference to the tangents or the point of intersection, being known. 1. By what data soever point C is located, they may be commuted by simple processes to the form shown in the figure; namely, the ordinate BC and the distance IC to apex. Call the angle BIC a, and complete the triangle ICO. '180 - A In this triangle, x = - a= 90° = (A + a). 2 Also, CO:10:: sin. X: sin. 2. R R ..R: :: sin. X: sin. 2. cos. | A cos. LA The triangle ICO may then be cos. * A (180,- A) Example. Then BC • IB = 32 - 80 0.4 nat. tan. 21° 49'; and IC=BC : nat. sin. 21° 49' = 32 • .372 = 86 feet. Also, x= 90° (1 A +a)= 90° (41° 49') 48° 11'. Next, sin. X, 48° 11' log. 9.872321 Divided by cos. $ A, 200 log. 9.972986 = sin. 2, 127° 31' log. 9.899335 Or, since the sine of any angle is equal to the sine of its supplement, the supplement in this case, 52° 29', may be taken directly from the logarithmic table, from which supplement deducting x, or 48° 11', the remainder is the angle y = 4° 18'. Finally, IC= 86 log. 1.934498 Multiplied by sin. X, 48° 11' log. 9.872321 CD log. 1.806819 And C D divided by sin. y, 4° 18' log. 8.874938 log. 2.931881 2. In the case of a rectangular intersection, the solution is more simple. It is quite plain, from the figure, that – . . . . . R?= (R — a)2 + (R — b)?, from which equation, R=a +6+12 ab. 3. Cases of this kind are disposed of with great ease in the field by means of the curve-protractor. TO FIND THE RADIUS OF A TURNOUT CURVE, THE FROG ANGLES, AND THE DISTANCES FROM THE TOE OF SWITCH TO THE FROG POINTS. 1. Draw the figure as in margin, C being the centre of the turnout curve, CK parallel to main track, and OK, IE, LM, perpendicular to it. Call the angle of the frogs at 0, F; that of the intermediate frog at I, 2 F'; the throw of the switch-rail for single turnout, D; its angle with main track, S; the gauge of the track, G; and radius of outer rail, R. 2. Usually the length and throw of switch-rail and the angles of the frogs at O are given. In that case, to find R, F, 3. The angle HNW, between the line of the switch-rail prolonged and a tangent to turnout curve at frog point 0, NOP NII W = F – S. The angle NOL or NLO, between chord and tangent, - half the intersection angle HNW (F-S). The angle NOB=NOL+LOB. But NOL is seen to be - $ (F S), and NOB = F; then LOB= NOB - NOL=F Ž (F – S)=} (F+S). The distance LO, from toe of switch to point of main frog, = LB : sin. LOB=(G-) = sin. 1 (F +S). 4. Again: the angle LCY N LÔ = (F– S); LY=; LO } (G D) • sin. $ (F – S). LY • sin. LCY LC; i.e., [1 (G - D) • sin. $ (F + S)] • sin. $ (F – S) R. OK OC cos. KOC=R cos. F; LM=LC cos. CLM= R cos. S; LM – OK=LB; i.e., R (cos. S — cos. F)=(G – D). Hence R= (G - D) = (nat. cos. S — nat. cos. F). 6. If R be known, to find F. This equation gives nat. cos. F = nat. cos. S - |(GD) R). 7. To find the angle, 2 F', of the middle frog at I. IE IP + PE or OK; i.e., R cos. F'=fG+ cos. F. Hence nat. cos. F' = nat. cos. F + (1 G + R). 8. The angle LIV, by similar reasoning to that used in relation to LOB, is found to be : } (F' +S). The distance LI, from toe of switch to point of middle frog, =LV • sin. LIV (I G – D) - sin. (F' +S). RULES FOR FROGS AND SWITCHES. 9. To find the Angle of Switch-Rail with Main Track. Divide its throw, in decimals, by its length: the quotient will be the natural sine of the angle sought. 10. To find the Distance from Toe of Switch to Point of Main Frog. Subtract the throw of switch-rail from the gauge of track, both in decimals; call the remainder a. Add together the angle of switch-rail with main track and the angle of the main frog; find the natural sine of half this sum, and call it b. Divide a by b: the quotient will be the distance |