The paper deals with an experimental investigation of the similarity of motion in fluids, of widely different viscosities and densities, in motion relative to geometrically similar surfaces, the existence of which has been predicted from considerations of dynamical similarity by Stokes, Helmholtz, Osborne Reynolds, and Lord Rayleigh. The theory in its most general form may be expressed by the relation given by Lord Rayleigh R = ρ v 2 f ( v L/ v ), where R is the resistance per unit area of the surface, ρ the density of the fluid, v the velocity, L a linear dimension of the surface, v the kinematical coefficient of viscosity of the fluid, and the assumptions made in the derivation of the expression are that R depends solely on ρ , L, v , and v .

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 P oiseuille, Darcy and Osborne R eynolds on the friction of water flowing in pipes, th a t of W illiam F roude 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 R eynolds 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 H elmholtz in 1873,t by Osborne R eynolds in 1882,J by Lord R ayleigh §|| in 1899 and 1909, and as has been pointed out by Sir George Greenhill, were foreshadowed by N ewton in Proposition 32, Book II., of the ' Principia. ' The object of the present paper is to furnish evidence confirming the existence,

It has long been realised that experiments in the laboratory on small models afford in certain cases a means of obtaining, cheaply and rapidly, information on the behaviour of the actual structure under the conditions in which it is to be used. This plan has been adopted in many branches of science, and in none to a larger extent than in naval and aeronautical architecture. The problems to be investigated are usually those connected with the motion of bodies through a fluid, but if we accept the principle of relative motion we have an alternative method open to us for our model experiments. We may, if we wish, reverse the conditions and set the fluid in motion, the model remaining stationary. Generally speaking, one plan has been adopted in naval architecture and the other in aeronautics. In the former case models are towed through water in a ship tank, while in the latter they are placed in a tube through which air is passed.

The resistance offered by bodies to motion through fluids has for many years been a problem of the first importance, and one which has received a correspondingly large degree of attention both from scientific investigators and those commercially interested in the subject. A considerable amount of mathematical investigation has been carried out; but a detailed study of fluid motion has only proved possible in the case of a perfect fluid, i.e., one in which viscosity is not present. Assumptions have been made as to the quantities upon which the motion of the fluid and the body depends, and formulæ have been prepared expressing the laws connecting force and speed for dynamically similar bodies.In the present note attention will be directed chiefly to the case where the only properties of the fluid upon which the motion depends are the viscosity and density, though brief mention may be made of instances where the acceleration due to gravity is also important.