38 ECKART [chap. 2 



which were neglected for the zeroth approximation and are now to be calcu- 

 lated using Uo = 0, p=po, P = po, etc. In this way, the equations 



-— i + vo Wpi + vi Vpo + Q X ui = 0, (38) 



ot 



^ + ui.Vr,o = ^(G^o-V.ho), (39) 



^ + ui-V^o= -voV-so, (40) 



ot 



%i + ui-V^o + /)oCo2V.Ui = [(yo-l)Mo][G^o-V.ho-i^proV-So], (41) 



ot 



are obtained. 



In these equations, ui, 171, pi and Si will be considered as the unknowns, 

 while the quantity vi will be eliminated by using the first of equations (2), 

 which is essentially 



pi = -XoVi+Yorji + KoSi; 



it then follows from (6) and (7) that 



vi Vpo = - pogvi Vx = [piiglpoco^) - ^^'(r?i + HpvoSildo)] Vx- (42) 



If qi is defined by the equation (analogous to (37)) 



N^qi = -eA'im + HpvoSildo), (43) 



it will have the dimension of length ; its interpretation will appear in Section 8. 

 Equation (42) may then be written 



vi Vpo = [piiglpoco^) + N^i] Vx, (44) 



while (38) becomes 



^ + voVpi + [?)i(g'/poCo2) + iV2gi]Vx + ftxui = 0. (45) 



ot 



It is convenient, now, to replace tji as an unknown by qi ; differentiating (43) 

 with respect to t, and using (39) and (40), it becomes 



m^ = dA'ui-{V-qo + Hj,ro'^Soldo)-{voldo){Go-W-ho-Hp,oV-So), 

 ot 



which (37) converts into 



%_ui.Vx = -{dA'lpoeoN^){Go-V-ho-Hp,oV-so). (46) 



ot 



The equations (45) and (46), together with (40) and (41), are the most con- 

 venient independent set with which to work. They can be simplified by two 

 additional definitions. 



