622 hersey: derivatives of. physical quantities 



in which the n's are all the independent dimensionless products 

 which can be built up by combining in any way the N physical 

 quantities involved. Further, the total number of such prod- 

 ucts, or dimensionless arguments, will always be the same, no 

 matter how the quantities are grouped. This number will be 



i = N -k (6) 



if k is the number of fundamental units needed for measuring 

 the N quantities. 4 



Let LT and LT designate any two of the i products in (5) which 

 contain between them the three quantities Q , Qi, and Q 2 in which 

 we are interested. Let Q appear to the first power in n and 

 not at all in any other product. This can always be done, for 

 Buckingham has shown that a certain standard arrangement is 

 possible in which each product contains to the first power some 

 one quantity of type P which occurs nowhere else. 5 We shall 

 then have 



n„ = #•<??' ••<£' -Go (7) 



and 



n = Q? Qi ■ ■ Qt ■ Q k+l (8) 



The exponents are abstract numbers fixed by the dimensions of 

 the N quantities; in any particular problem some of them may 

 be zero. If we now agree to keep the remaining i — 2 products 

 constant, (5) becomes 



n = </>(n) (9) 



in which the form of </> is unknown. The restriction to constant 

 products can always be fulfilled in theory, but it may lead to 

 difficulties in practice; it will be discussed in a later section. 

 Differentiating (9) and then (8) gives in succession 



dllo __ dcf> dn _ dct> all ,-^v 



From (7) 



dlTo _ dQ lip apllo .--. 



dQi = = dQi Q Qi 



4 The question of the number of fundamental units needed has been discussed 

 by Riabouchinsky, Rayleigh, and Buckingham; see Nature, 93: 396-397. 1915. 

 • 5 Trans. Am. Soc. Mech. Engs., 37: 291-292; note eq. (11) and its discussion. 



