/<w ~ir°> ■ ■ ■ ■ w> 



the Method of Dimensions. 715 



the familiar resistance equation for a body of fixed shape 

 moving, in a given attitude, through a quiescent liquid 

 or into still air at a constant speed which is not high enough 

 for the compression of the medium to have an appreciable effect 

 on the resistance. 



The initial equation for this problem is 



F (R, D, S, P , fi) = 0, .... (47) 



where R= resistance, D = a linear dimension of the body, 

 S = the constant speed of advance, p = density, and yu, = 

 viscosity; and the dimensional equations are 



[BQ =[«*-«]! [D] = [Z] ; [S] •=[«->];) 



Since n=5 and k = 3, n—~k = 2, and the II theorem reduces 

 equation (47) to the form 



R Dp_S\ 



or 



?w -.♦(*£> <50 > 



which is the well-known air-resistance equation for aeroplane 

 speeds or lower, and is known to agree with the facts within 

 the rather wide limits of experimental error *. 



Now let us ignore JSfawton's second law of motion and 

 use an independent arbitrary unit of force f, so that instead 

 of (48) we have 



M = [/]; [B] = [Zj; [S] = [iri];! 



l P l = [mi-*]., m = [fi-ni ] ■ • ( 51 > 



The number of fundamental units has increased from 3 

 to 4, while the number of quantities remains at 5, so 

 that there is now only a single II ; and by using the 

 II theorem we get the equation 



R = NDS/a, (52) 



where N, as usual, represents an unknown pure number. 



* It is sometimes overlooked, that in passing from the behaviour of a 

 model in a wind-channel to that of the full-sized aeroplane in free flight, 

 the very wide extrapolation along- an empirical curve for 0(DS v) is not 

 the only possible source of error. Another is the difference between 

 turbulent and quiet air, of which little is known except that it nun 

 sometimes be very important. 



3 B 2 



