The laws of the surface friction of fluids have formed the subject of many important investigations during the last 100 years, among which may be mentioned the work of Poiseuille, Darcy and Osborne Reynolds on the friction of water flowing in pipes, that of William Froude on the resistance of thin plates towed in water, and the corresponding experiments of Zahm on flat plates in a current of air. Researches in this field have also been carried out by Brix, Stockalper, Mallock, Coker, Gebers, Brightmore, Grindley and Gibson, and others. As a result, the effect on the resistance, of the dimensions of the body over whose surface the fluid moves, and of the velocity of flow, are tolerably well known for the particular fluid and character of motion observed. In the case of the surface friction of water in pipes, the researches of Osborne Reynolds have demonstrated the existence of similar motions in pipes of different dimensions, hut, as far as the authors are aware, no systematic series of experiments appears to have been made for the purpose of establishing a general relation which would be applicable to all fluids and conditions of flow, although the existence of such relationships for different aspects of the problem were predicted as a consequence of the laws of motion by Stokes in 1850, by Helmholtz in 1873, by Osborne Reynolds in 1882, by Lord Rayleigh in 1899 and 1909, and as has been pointed out by Sir George Greenhill, were foreshadowed by Newton in Proposition 32, Book II., of the ‘Principia.’