850 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



With these relations we are now in a position to derive the differential 

 equation for U. 



This mechanical equation is obtained by first writing the obvious equality 



Then partial differentiation of the last term gives 



Sx SI Sx N 51 N^ St 



and, when the resultant time derivatives are replaced by (77) and (78) 



jjiU SU _j-SU 2 SK P dP , . 



the substitution of NU for F then gives the final differential equation for 

 ilj which may be written in the form 



It is the analogue of equation (2). 



The two sets of equations are now assembled and written in a form suit- 

 able for comparison. The equations for the idealized stream are 



ox Ot 6 



18U'8U „ . ,.. 



2 dx 8t 



and the analogous equations for the physical stream are 



6/6/6 



16^ 

 2 6:*; 



-it[^-m-'^'-'^ 



(82) 



