by Rayleigh (1899) does not differ basically from that of Buckingham 

 (1914) but the latter method is preferred because, according to Langhaar 

 (1951), it does not involve the construction of an infinite series as in 

 the Rayleigh method. If n variables are connected by an unknown dimen- 

 sionally homogeneous equation, Buckingham's theorem, generally known as 

 the i: (pi) theorem, shows that the equation can be expressed in the 

 form of a relationship among n - k dimensionless products, where k is 

 the number of fundamental dimensions in the problem and n - k is the 

 number of products in a complete set of dimensionless products (tt terms) 

 of the variables, and that each ir term will have k + 1 variables of 

 which one must be changed from term to term. Thus, given a set of n 

 variables A^, A2, . • . An in which A^ depends on only the independ- 

 ent variables A2, A3, . . . An, the general function can be written in 

 the form 



which can be written 



Aj = f(A2, A3,. . .\) (2-20a) 



f'(Ai,A2, A3,...Aj,) = ; (2-20b) 



also, 



f"(7ri,7r2, 7r3,...7rj^_^) = . (2-20c) 



The variables of each it term must appear in this exponential form to 

 make each it term dimensionless. Buckingham's contention that there 

 will be n - k dimensionless products is now known to be a good general 

 rule, but it is not always true (Langhaar, 1951; van Driest, 1946). Van 

 Driest has shown that, to be generally true, the statement should be: 



"The number of dimensionless products in a complete set is equal 

 to the total number of variables minus the maximum number of 

 these variables that will not form a dimensionless product." 



Langhaar, using the algebra of determinants to study this problem, 

 obtained the following necessary condition, which is equivalent to that 

 of van Driest: 



"The number of dimensionless products in a complete set is equal 

 to the total number of variables minus the rank of their dimen- 

 sional matrix." 



Numerous examples showing the application of dimensional analysis to 

 engineering problems, involving different types of electrical, mechanical, 

 and fluid-flow phenomena, are presented in the literature by different 

 authors. The example selected in this instance is the general case of 

 fluid motion. According to Rouse (1938), the only variables that can 

 influence fluid motion are (a) those linear dimensions necessary to 

 define the geometrical boundary conditions (a, b, c, d, etc.); (b) the 

 kinematic and dynamic characteristics of flow (such as a characteristic 

 velocity V or a discharge Q, a time t, or an acceleration dv/dt, 

 a pressure increment Ap, and a pressure gradient dp/dx, or a resisting 



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