The Transition Curve Or Curve of Adjustment Based On the French of M Nord

Norman Benjamin Kellogg

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15. As indicated on page 20, we now proceed to verify certain equations (3-11, 61-691.
We have from the calculus: rda = dL, L = =Pr-i, (77) differentiating Pi* 2 ffr ^ ; ( 8o ) P P P( 1 1 \ a " ~ a ' = ~2^r ~ W = -2 V7~5 ~ Tjr ( 8l ) From (62), L, L, - ; whence L L -L-P- ~ (K*\ L n -L, - p\ r ^ - ;, - - - \- v - - ;, (83) substituting in value for a n a, we have // - *, = 4- (~- ^ -^) - ^> - PI. * &; (84) M^ + j/^ being used when curves are in the same direction and the sign when the ...curves are in a reversed direction. (If the sign is used the second member of (84) will still equal a. , a/, in which a ;/ is on the opposite side of the axis of X from a, or the curves are reversed, L = L lt + L, a. , = /3 y/ and a, = 0, ) in which - ftu + f*i (8 5 ) is the angle subtended by the arc L of the spiral as well as the sum of the circular arcs A, B, + A, B, or of a circular 2 arc whose length is = L and radius r = 1 1 .

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