The time dependence of the event rates in the Homestake chlorine solar neutrino experiment are reexamined using new Ar-37 production data covering the period from late 1986 to mid-1989. The data span almost two complete solar cycles. A careful statistical analysis using nonparametric rank-order statistics is used to calculate quantitative significance levels that do not depend on experimental errors. The results show that the Ar-37 production rate in the experiment is anticorrelated with solar activity for approximately 1977-1989. The shape of the Ar-37 production rate is different from the inverted sunspot activity curve. The Ar-37 production rate is better descrbed by a skewed sawtooth function than by the sunspot number. The best-fitting sawtooth function with sunspot period has a slow rise and a rapid decline. The Ar-37 maximum occurs about 12.5 yr after the solar sunspot minimum, while minimum Ar-37 production is more nearly simultaneous with the sunspot maximum.