l8 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 



sarily, in order to have all its terms of the same dimensions, be 

 reducible to the form 



F(n„n„n3, /,/',... .)=o. (4) 



In addition to the dimensionless ratios r' , r", etc., there now appear 

 in the equation only three instead of the original six variables, so that 

 the labor of determining by experiment the form of the unknown 

 function is much less than if we had to deal with all six variables. 



The More Specific Form of the Equation of Fluid Resistance 



The dimensions of the quantities on the familiar mass, length, time, 

 or [m, I, t] system are : 



[R] = [mlt-^], [p-] = [ml-% 



and we see by inspection that the expressions 



are dimensionless products of the required form (2), and that they 

 are independent. Accordingly, we know that equation (i) must be 

 reducible to the form 



This equation is fundamental to the experimental study of the 

 hydrodynamic or aerodynamic forces acting on totally immersed 

 bodies. 



Solving for H^ we now have 



R ^IDS S 



in which the form of the unknown function </> remains to be found, 

 if it needs to be found at all, by experiment or by other than dimen- 

 sional reasoning. 



Simplification When Compressibility May be Disregarded 



A simplification is possible when the motion is not rapid enough to 

 cause any sensible compression in the fluid. In this event it is imma- 



terial what the compressibility is, so that ^ may be omitted from con- 

 sideration and equation (6) reduces to 



R ^/DS , . \ , , 



