628 hersey: derivatives of physical quantities 



p 



or 



t = — (l+BVgD s ) (33) 



gD 



in which A and 5 (both intrinsically positive) do not involve g 



at all, nor D except in a shape factor. The values of — — and 



t bg 



D <:t 



— — — found by differentiating (33) do satisfy (31). 



III. Without knowing the empirical equation let it be required 

 to predict the change in roll time due to any small change in 

 liquid density, such as would occur upon using the tube under 

 pressure, by reference to an observation on the effect of chang- 

 ing the ball density. Since an expression for — in terms of — 



dp b Po 



is sought, t, p, and p are selected for Q , Q h and Q 2 respec- 

 tively. If (30) were to be used as it stands there would be a 



restriction on the derivative — , which is hardly to be desired. 



Op 



An equivalent result in a more convenient form can evidently 

 be obtained by confining p to a smaller number of arguments. 

 This is done by replacing (30) by one of the alternative forms 

 provided by the n-theorem, such as 



\k * = funct. h, g -fJ^, shape) (34) 



~D \p p? / 



Comparing this with (26), « = 0, /? = 0, a = 1, (3 = - 1; hence 

 by (14) 



bt po/dl\ 



or by (25) 



= - £-° (^L ) (35) 



Op p \C)po/ M ccpo 



op 



1 / bt . bt\ , QA ; 



= --(po — + p— ) (36) 



p \ Opn bp/ 



