624 hersey: derivatives of physical quantities 



B "Sffe'-O a+sw (21) 



and 



C = 





-V (22) 



1/3/ 



Thus the curvature with respect to Qi can be calculated from 

 the slope and the curvature with respect to Q 2 . 



Integral form of the relation. 6 Integrating (14) at the point 



(Qo = go, Qi = <7i, Q2 = Qi) over an interval so short that — ^° 



may be treated as constant, and denoting its value by the symbol 



- , gives for the primitive equation of an element of the surface 

 oq 2 



in which 



On 

 _ a fq 2 bq 



(23) 



h = - l^^ + Po -«• (24) 



\q dq 2 / 



The use of (23) would permit a direct comparison of any new 

 results obtained by the present method with empirical results 

 previously published in one-term, constant-exponent formulas. 



Discussion of the constant-product restriction. Let IT C denote 

 any one of the i — 2 arguments which we have agreed to hold 

 constant, and let Q stand for either Qi or Q 2 . Then, unless n c 

 can be so chosen that it does not contain Q, it must be so chosen 

 that it will contain some additional quantity Q c not occurring 



6 If instead of an isolated value of ^—^ we were furnished with the entire curve 



OQ, 



Qo = f-i (Qi), the direct use of the n-theorem would be preferable, and would give 

 the whole curve Qo = /i (Qi)- If successively furnished with additional curves, 

 Qo= JiiQz) and so on, we could gradually build up generalized cross sections 

 of the surface (1) until, when N-k independent curves had been given, we should 

 have the whole of it. The problem of developing empirical equations syntheti- 

 cally has not been treated in the available papers. That problem is a general 

 one, of which the problem of the present paper is a special case; this situation is 

 illustrated by the fact that our final result (23) applies only to an infinitesimal 

 piece of the curve Qo = fi(Qi)- 



