The Ar-37 production rate in the Homestake solar neutrino experiment is anticorrelated (significance level of parts in 10(5)) with solar activity (as measured by sunspot number) in the second two-thirds of the data, approximately 1977-1989; no significant correlation is substantiated in the first third of the data, 1970-1977. The shape of the Ar-37 production rate is different from the inverted sunspot activity curve. The Ar-37 production rate is better described by a skewed sawtooth function than by the sunspot number; the best-fitting sawtooth function with the sunspot period has a slow rise and a rapid decline. The Ar-37 maximum occurs approximately 1.5 yr after the solar sunspot minimum, while minimum Ar-37 production is more nearly simultaneous with the sunspot maximum. The Fourier spectrum of the Ar-37 data has no feature that is more significant than 50%, between 0.3 and 3 cycles per year. A sinusoidal modulation of 25% would probably have been detected, while a sinusoidal modulation of 35% would exceed the observed signal 95% of the time. The average event rates in the first third and last two-thirds of the data are statistically compatible, 2.0 +/- 0.35 SNU and 2.1 +/- 0.2 SNU, respectively (1-sigma-errors). The two segments differ significantly in their best-estimate constant component. However, a small (0.1 SNU) constant component, which fits best the second two-thirds, is marginally allowable for the first one-third at the extreme of its 95% confidence interval. Fitting the second two-thirds of the data to a model of a constant plus a time-variable sawtooth function yields an underlying best-estimate constant signal of 0.1 SNU, and a 95% upper limit of 0.9 SNU. Fitting all the data to a pure time-dependent sawtooth function gives a peak rate of 3.9 SNU. The variable event rate could in principle be due to slow drifts in the experimental sensitivity or to an unrecognized background in the chlorine solar neutrino experiment, one that is modulated by the solar cycle. We are unable to identify any plausible process to account for either possibility. If the correct explanation of the variability does involve unidentified drifts or backgrounds, then other solar neutrino experiments, including Kamiokande II, the gallium experiments, and the Sudbury Neutrino Observatory, will not exhibit the same modulation as observed in the Homestake chlorine experiment, since each experiment has a unique environment and detection scheme.
