BUCKINGHAM: PHYSICALLY SIMILAR SYSTEMS 351 



6. If equation (8) is solved for any one of the tt's, e.g. tti it may 

 be written in the form 



Pi = QIQ\ ■ ■ ■ Ql f (X2, 7r3, • • • 7r„_„ r', r", . . • ) (11) 

 in which 



a = — ai, 6 = — /3i, etc. 



If it is desired to obtain an equation of the form (11) with a par- 

 ticular quantity Pi appearing separately and in the first member 

 only, it is evident that this quantity must, from the start, be 

 excluded from the list of quantities to be used as the [Q]'s in 

 equation (9) . It will then act as a [P], will appear in only a single 

 X, and will be separable. 



Equation (8) may also, of course, be solved for any one of the 

 r 's, such as r' , and put into the form 



r' = ^ (xi, TTs, • • • Tn-k, r", r'", ■ ■ etc.) (12) 



which is sometimes useful. 



7. Since equation (11) contains an unknown function (p, the 

 form of which can not be found by dimensional reasoning, the 

 equation does not give us any definite information in the general 

 case when all the quantities involved in the second member vary 

 arbitrarily. If, however, all the r's are held constant; and if the 

 variations of the Q's and of Po,....Pn-k are not entirely arbitrary 

 but subjected to the (n—k — 1) conditions that 7r2,....x„_A; shall 

 remain constant; then we do have a definite statement of the 

 dependence of Pi on the Q's. For under these circumstances, 

 although its general form remains unknown, the function cp de- 

 generates into a dimensionless constant N, because its arguments 

 are all constant. Hence equation (11) assumes the definite form 



Pi = NQl Ql ■ ■ ■ Ql (13) 



A single measurement of simultaneous values of Pi and the Q's 

 suffices to determine the numerical value of N; and by equation 

 (13) with this value of N, the value of Pi may be computed for any 

 values of the Q's without further experiment. 



