l6 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 



In order that the results of such observations shall be interpretable 

 as definite statements about the behavior of the full-sized original of 

 which the model is a copy, certain requirements must be satisfied, and 

 when they are satisfied the original and the geometrically similar 

 model are said to be dynamically similar. The conditions for 

 dynamical similarity are bound up with the general question of the 

 possible forms of equations which describe relations subsisting among 

 the physical quantities involved in physical phenomena. 



Nature of the Problem to Be Discussed 



Let us suppose that a solid body is moving, with the constant 

 velocity S, through a fluid which is itself sensibly at rest at points 

 far distant from the body ; and let us consider the forces exerted on 

 the body by the surrounding fluid. Since these forces are evidently 

 due to the relative motion, they would remain unchanged if the body 

 were held at rest and the fluid made to flow past it with the velocity 

 (—S). The boundaries of the fluid are supposed to be so distant 

 from the solid body that no sensible disturbance reaches them, and 

 their nature can then have no influence on the forces with which we 

 are concerned and need not be further referred to. If the fluid is a 

 liquid with a free surface, the foregoing condition requires that the 

 moving body be so deeply immersed as not to cause any surface dis- 

 turbances. 



Let R be any force exerted by the fluid on the body ; for example, 

 the component in any specified direction of the force on some par- 

 ticular part of the solid surface ; or, to make it more definite, let R 

 be the total head resistance in the direction of motion. Then R will 

 depend on and be completely determined by the relative speed, the 

 size, shape, and attitude of the body, and the mechanical properties 

 of the fluid ; and there must be a definite relation connecting these 

 various physical quantities, which can be described by an equation. 

 We wish to consider the nature of this equation in so far as it is fixed 

 by the natures of the separate quantities involved in it. 



The Physical Quantities Which Influence Fluid Resistance 



Let D be some linear dimension of the body, such as its greatest 

 length. The shape of the body and its attitude, i. e., its orientation 

 with regard to the direction of motion can be specified by stating the 

 ratios of a number of lengths to the particular length D. If these 

 ratios are denoted by /, /', /", . . . . , etc., the size, shape, and attitude 

 of the body are completely specified by the values of D, /, r", . . . . , etc. 



