72 



Geodesic Lines on the 8yntracikix oe' Revolution. 



E. L. Hancock. 



The syntractrix is defined as a curve formed by taking a constant 

 length, d upon the tangent c to tlie tractrix*. The surface formed by re- 

 volving tliis cur\'e about its asymptote is the one under consideration. We 

 shall call it S. 



Being a surface of revolution it is represented by the equations 

 x^u cos V 

 y = u sin v 



Z: 



= — v^d^ — uM- Jlog- 



2 d— v/cl2_u2 



Using the Gaussian notationf we find : 



Tj, u2(d2— 2cd) + c2d2 -p, r- 2 A u2^-cd_ 



E = — ^^ LJl F = o, G = u2, A = — — , — u cos v, B = — 



u2(d2 — u2) "^Vd^ — v? 



V? — cd ^ _ n2rfl2 2cd1-l-cd3 u(u2 — cd) 



usinvC = u, Di=li^i /ca)+ca j)/_ p/^— \ '_ 



Tl/dS — U2 ' ' U(d2 — U2)| ' ' l/d2 



1 DD" — D'2 (u2 — cd)[u2(d — 2c)-f cd2] 



K 



-RiR2~ EG — F2 — (d2 — u2)[u2(d — 2c) + c2d 

 In the particular surface given by d = 2c the Gaussian curvature be- 



comes o. o d^ 1 



d2 — U2 



Here d is positive, and since d > u, the denominator is always positive. 

 We get the character of the curvature of different parts of tlie surface by 

 considering the numerator. When u2:= d2 2, K = 0, i. e., the circle u^ d/2 



~7= is made up of points liaving zero-curvature. When u2 > d2/2, K> O, 



and when u2 < d2/2, K < O. 

 For tliis particular surface 



14 2u2-^d2 



E = - , F = o, G = u2, A = — >y— y u cos V, B = — 



4u2(d2 — U2) 2u^/(i2_ii2 



2u« — d2 Ai u(2u2 — d2 



o ,-^ .^usin vC = o, D= ^- , T>' = o, B'' = i^ 



2u/(i2— u2 2u(d2 — u2)|' 2/(12— u2 



To get the geodesic lines of the surface we make use of the metliod of 



the calculus of variations according Weierstrass§. This requires that we 



minimize the integral : 



■' Peacock, p. 175. 



t Bianchi, Differential Geometric, pp. 61, 87, 105. 



j Osgood, Annals of Mathematics, Vol. II (1901), p. 105. 



