the Method of Dimensions. 699 



long one, it is usually evident that under the actual circum- 

 stances a number of these quantities may safely be ignored ; 

 so we cross them off the list and thus pass from our most 

 general conception of the phenomenon to an ideal simpler 

 one, in which these quantities are not involved at all. 



To illustrate, let it be required to find out how the 

 resistance R of still air to the motion of a smooth sphere 

 depends on the speed S. We may be interested only in 

 the relation of R to S for a particular sphere and under 

 particular atmospheric conditions ; but if any information 

 is to be got from dimensional reasoning, other quantities 

 must also be taken into account. The resistance evidently 

 depends on the diameter of the sphere D, and on the 

 properties of the air. Of these, the density p, viscosity \x y 

 and compressibility 0, may obviously be of importance. 

 But since the density and viscosity are affected by tem- 

 perature and there is a dissipation of work into heat 

 about the sphere, it is evident that the specific heat Q p , 

 the thermal conductivity X, and the emissivity <r, will also 

 have some effect on what happens. Hence, to go no 

 farther, our list already contains the nine quantities 



R, S; D, /?, /£, C, Cp, X, a. 



But it seems safe to assume that C p , X, and a cannot 

 play any important role in the determination of air 

 resistance, because the dissipation is spread over so large 

 a mass of air that the temperature effects must be small. 

 We therefore strike these quantities off the list without 

 any hesitation ; and if the speed is to be so low that the 

 air is not sensibly compressed in front of the sphere, we 

 strike off C as well. Thus the problem has been idealized 

 and simplified for the sake of concentrating attention 

 on the things that may probably be of real importance ; 

 but no omission has been made which seems likely to 

 prevent the result of correct reasoning about the ideal 

 phenomenon from being very nearly correct for the real 

 one. To put it in another way, it seems likely that the 



e<1 " ati0n F(R,S,D,^)=0 



is sensibly complete for the problem in hand. 



In some such way as this, the list of quantities is ma do 

 out, and the initial equation corresponding to (1) is written 

 down. The basic assumption which underlies the sub- 

 sequent purely algebraic reasoning is that these quantities 

 are concerned in the real phenomenon and that no others 



3 A 2 



