## Bertrand Curves Associated with a Pair of Curves

**Abstract**

John F. Burke 1. INTRODUCTION. A theorem due to C. Bioche (1) states that if there exists a one-to-one correspondence between the points of the curves C and C such that at corresponding points P on C and P on C 1 1 2 2 (a) The curvature K of C is constant 1 1 (b) The torsion r of C is constant 2 2 (c) The unit tangent vector T of C is parallel to the unit tangent 1 1 vector T of C , 2 2 then the curve C generated by the point P that divides the segment P P 1 2 in the ratio m: 1 is a Bertrand curve. It is the purpose of this paper to prove that if condition (c) is modified so that the binormals B and B at P and P are parallel, then the curve 1 1 2 C is a Bertrand curve; and if condition (c) is modified so that the tangent T at P is parallel to the binormal B at P then the curve C is again a 1 1 2 2 Bertrand curve. Gibbs methods and notation of vector analysis will be used in this