hersey: derivatives of physical quantities 625 



in any other product. The rule for keeping n c constant will 

 then be: Vary Q c simultaneously in such a manner as to com- 

 pensate the changes due to Q. 



If Q enters n c to the n th power and Q c enters it to the first, 

 the derivatives in (15) and elsewhere are subject to one or more 

 conditions of the type Q c «= Q~ n . For such a derivative let us 



adopt from now on the notation (^tt) _„• There are two 



experimentally independent methods for getting its numerical 

 value: First, by directly observing the change in Q Q with Q 

 while simultaneously changing Q c in the prescribed manner; 

 second, by calculating it from separate observations on the 

 change in Qo with Q at constant Q c , and the change in Q with 

 Qc at constant Q. Expanding the conditioned derivative into 



the form ( --£ ) + ( ~ ) ,° and taking account of the fixed 

 \oQ/Qc \oQc/Q uQ 



relation between Q c and Q leads to the working formula 



/5QA /&Qo\ _ w Q.7aQ.\ (25) 



VaQA^-n \?>Q/qc Q\^Qc/q 



for the second method. In the most general case where there 

 are i — 2 arguments to be kept constant, the second term on the 



right of (25) will be replaced by — ^ times the summation of 



i — 2 terms of the type nQ, 



( 



dQ \ 

 ZQc/q' 



While the procedure outlined in this section is always possible 

 and sufficient, it is not always necessary or even desirable. For 

 example: if the number of quantities, N, does not exceed the 

 number of fundamental units, k, by more than 2, there will be 

 no other arguments than n and II; again, if the remaining i — 2 

 arguments do not involve Q (i.e., Qi or Qo), their constancy will 

 not be disturbed at all by the fact that Qi and Q 2 do vary. 

 Further expedients for simplifying the work will suggest them- 

 selves upon examining each particular case by itself. 



Some illustrative examples. For reference in solving problems 

 it is convenient to rewrite (5) in the form 



