The expansion in a power series of the Lambert W function up to first order is just the identity [2] and hence, line. Using trigonometric identity sec 2 $^{α}$− tan 2 α = 1 and Eq. (B1) we obtain that Where we used relation (B1) in order to obtain the last Using this result in the expression of ˜ r we obtain the desired result, Eq. (15). 7 ˜ r ( θ$_{m}$ ) = − 1 ε 2 sin θ $_{m}$W $^{(}$− ε $^{2}$tan θ$_{m}$f ( α )) ≈ 1 ε 2 sin θ $^{$_{m}$(}$− ε $^{2}$tan θ$_{m}$f ( α )) . = sin α cos α cos θ m . = 2 sin θ cos 2 α cos 2 θ $_{m}$. (B2) cos 2 θ$_{m}$ cos 2 α = 1 + 3 sin 2 θ$_{m}$. (B3) [2] R.M. Corless, G.H. Gonnet, G.H. Hare, D.E.G. Jeffrey, and D.E. Knuth, “On the Lambert W function”, Adv. in Comp. Mathematics 5 , 329-359 (1996). [1] See for instance scholar.google.com . [3] T.C. Scott, A.L¨ uchow, D. Bressanini, and J.D. Morgan III, “The Nodal surfaces of Helium atom eigenfunctions”, Phys. Rev. A, 75 . 060101-060104 (2007). [4] R.D.H. Warburton and J. Wang, “Analysis of asymp- totic motion with air resistance using the Lambert W function”, Am. J. Phys. 72 , 1404-1407 (2004). [5] E. Packel and D. Yuen, “Projectile motion with resis- tance and the Lambert function”, Coll. Math. J. 35 (5), 337-350 (2004). [6] J.L. Fern´ andez-Chapou, A.L. Salas-Brito, and C.A. Var- gas, “An elliptic property of parabolic trajectories”, Am. J. Phys. 72 ,1109-1109 (2004). [7] W.D. MacMillan, Theoretical Mechanics: Static and the Dynamics of a Particle (McGraw-Hill, New York and London, 1927). Reprinted in (Dover, New York, 1958), pp. 249-254. [8] G.B. Thomas, M.B. Weir, J. Hass, F.R. Giordano. Cal- culus . 11th Ed. (Addisson-Wesley, 2004). [9] G.B. Arfken. Mathematical Methods for physicist . 5th Ed. [10] E. Kreyzig. “Principal, Normal, Osculating Cir- cle” in Diffential Geometry . (Dover,N.Y., 1991). E.W. Wiesstein “Curvature” From MathWorld. http://mathworld.wolfram.com/Curvature.html . [11] S.M. Steward, “A little introductory and intermediate physics with the Lambert function”, Proc. of the 16th Biennial Congress of the Australian Institute of Physics. M. Colla ed. Vol 2. pp 194-197. Australian Institute of Physics. Parville, VIC (2005). [12] S.M. Steward, “Characteristics of the trajectory of a pro- jectile in a linear resisting medium and the Lambert W function”, Australian Inst. of Physics. 17th. National Congress 2006. Paper No. WC0035.(2006). [13] In Ref. 12, the author comment that for ε = 1 we obtain the special value α skew = sin − $^{1}$(1 /φ ) with φ = $_{(1 +}$√$_{5)}$$_{/}$$_{2, the golden ratio. However, the solution that}$ appears in the article is not longer valid for ε = 1. If the solution corresponds to another real root of the equation this special value corresponds to the case when the initial speed is equal to the limit speed b/g . This makes much more intriguing this fact. (Academic Press, 2000).