This Paper describes the determination of the coefficient of viscosity of water by a method in which water is forced along glass capillary tubes of about 1.5 to 2.0 mm. bore at rates of flow varying from 1 litre in two years to 1 litre in 24 years. The liquid fills a closed tubular circuit which for purposes of description may be said to be rectangular in shape. Two of the tubes of the rectangle are horizontal, and two vertical. The circulation is caused by a difference of density between the contents of the two vertical tubes. The difference is obtained by having a weak solution of uranine in one vertical tube, and pure water in the other. In Apparatus I. the two vertical tubes are at a distance apart, and an allowance has to be made for the difference of temperature. In Apparatus II. a modification of the rectangular circuit is employed, the vertical tubes are close together, and no such corection is necessary. Applying the temperature correction to the results recorded in this Paper, the value of the co-efficient of viscosity at 18°C. is 0.010624 ± 0.000229 (assuming observations of equal weight) and 0.010569 gm./cm. sec. from Hosking's formula. Thus there is no experimental evidence that at the extremely low rates of shear (0.0017 to 0.0233 radians/sec.) the viscosity of air-free water in glass capillary tubes differs from its value at normal rates of shear (5,000 to 10,000 radians/sec.).