first described by J. G. Ulrich, a compensated balance adjusted to keep correct time at a given low and high temperature will be a few seconds per day fast at intermediate temperatures. The reason is that the moment of inertia of the balance varies as the square of the radius of the compensation arms, and thus of the temperature. But the elasticity of the spring varies linearly with temperature. To mitigate this problem, chronometer makers adopted various 'auxiliary compensation' schemes, which reduced error below 1 second per day. Such schemes consisted for example of small bimetallic arms attached to the inside of the balance wheel. Such compensators could only bend in one direction toward the center of the balance wheel, but bending outward would be blocked by the wheel itself. The blocked movement causes a non-linear temperature response that could slightly better compensate the elasticity changes in the spring. Most of the chronometers that came in first in the annual Greenwich Observatory trials between 1850 and 1914 were auxiliary compensation designs. Auxiliary compensation was never used in watches because of its complexity. Better materials The bimetallic compensated balance wheel was made obsolete in the early 20th century by advances in metallurgy. Charles Édouard Guillaume won a Nobel prize for the 1896 invention of Invar, a nickel steel alloy with very low thermal expansion, and Elinvar (from Elasticité invariable) an alloy whose elasticity is unchanged over a wide temperature range, for balance springs. A solid Invar balance with a spring of Elinvar was largely unaffected by temperature, so it replaced the difficult-to-adjust bimetallic balance. This led to a series of improved low temperature coefficient alloys for balances and springs. Before developing Elinvar, Guillaume also invented an alloy to compensate for middle temperature error in bimetallic balances by endowing it with a negative quadratic temperature coefficient. This alloy, named anibal, is a slight variation of invar. It almost completely negated the temperature effect of the steel hairspring, but still required a bimetal compensated balance wheel, known as a Guillaume balance wheel. This design later fell out of use in favor of single metal Invar balances with Elinvar springs. The quadratic coefficient is defined by its place in the equation of expansion of a material; where: is the length of the sample at some reference temperature is the temperature above the reference is the length of the sample at temperature is the linear