7L2 



3Ir. E. Buckingham : Notes 



Since the dimensions of p are ml' 3 , we may set [»»Z~ 3 ] 

 = "1" or ' ~m~ = ~_l z ~_ ; and the dimensions o£ the five quan- 

 tities which appear in equation (41) are 



m 



G- = ~ml~' 2 t~ 



- 2 ] 



= ~7r 2 ~ 



D] = [Z] 





r 7~i 



S~_ = Ir 1 ] 





= P'" 1 ] 



' ' p" ~ i m ^~ z ] 





= [1], 



v = :mi- i t- 



-: _ 



^ "/-r 1 





Since p is now treated as a dimensionless number it may 

 be omitted from the initial equation, which becomes 



F G-. D, S. fx) = : 

 whence by the Id theorem 



'DS^ 



°-w?> 



(43) 



(**; 



Upon comparing (44) with [19) it is seen that the new 

 equation is merely the special form into which the more 

 general one degenerates when p is a constant, and that 

 under the present restrictions it is equivalent to the earlier 

 result. Since there are now only two fundamental units, 

 the algebra of the solution is a trifle simpler than before 

 after the new dimensional equations have been obtained; but 

 practically it is easier to keep to the more familiar m, 1, t 

 system. 



To carry the process a step farther, let the viscosity also 

 be constant and treat it as a dimensionless number, so that 



- i2 > W -[«-] = [1] and 



'ti=TP 



we now nave 



[p] = » x ; [d ] = ra s [G] = [s] = x 1 : ■ («) 



The initial equation is F (G, D. S) = 0. and the solution 

 by the II theorem is 



Gt=^*(DS) do 



which is the form into which (19) degenerates when both p 

 and /J, are constant. 



Bv taking stdl another step and letting D be constant. 

 vre arrive at the result 



q= s-c/>;s) =^(S 



which is. of course, obvious beforehand. 



