Balance spring with two hairsprings and improved isochronism

A balance spring comprising:

• a first hairspring, a curve of which extends in a first plane and an inner coil of which includes a collet,
• a second hairspring, a curve of which extends in a second plane parallel to the first plane, and
• an attachment member securing an outer coil of the first hairspring to one end of the second hairspring so as to form a dual balance spring in series, wherein
• the curve of the first hairspring and the curve of the second hairspring each have a continuously variable pitch and are symmetrical relative to a straight line parallel to the first and second planes and pass through a median plane of projection of the attachment member and each curve being such that the following relation is zero:
• P
• →
• (
• n
• )
• =
• n
• +
• 1
• L
• n
• +
• 1
• ⁢
• ∫
• 0
• L
• ⁢
• ⅆ
• s
• ·
• s
• n
• ·
• x
• →
• ⁡
• (
• s
• )
• where:
• {right arrow over (P)}(n) is a moment of the balance spring of order n, n being an integer greater than or equal to zero;
• L is a length of the balance spring;
• sn represents a curvilinear abscissa along the balance spring to a power of n;
• {right arrow over (x)}(s) is a parameterization of the balance spring by the curvilinear abscissa thereof;
• so as to reduce displacements of a centre of mass thereof during contraction and expansion, and
• each hairspring further includes at least two counterweights so as to compensate for an unbalance formed by a mass of the attachment member and personalize an anisochronism slope of the balance spring.