Desai and Lieber 



V = Vq + Vj + V^ + V3 + 



P = Po + Pi + P2 "^ P3 "^ 



= ^Q + 0j + ./;2 + 03 + 



(1.29) 



where u, v, p, and ^ correspond to the solution S, while u^,, v^, p^, and 0^ 

 correspond to the function S^,. - ,—'-•- - : ' 



Let us substitute Eqs. (1.29) in Eqs. (1.12), (1.13), (1.14), and (1.15). As- 

 suming that a rearrangement and grouping of terms in these equations as made 

 below is permissible, we get the following equations: 



1 "d^^o 2 ^^0 



J_( !!^ + 1?^ - — + 



du. 



'"■ '^"^'"'W— °,u. * 



— + (u„) + \ — 



1^ _ ?^ 



1 <^Pi _ J_/ ^2^ 1 __i _ ^ + Jl 1_^ 2 -1 



3u, 



-^ (Uo+-i)— + 



Vq + Vj\ Bu2 /^Uq 3uj 



Br 3r 



Ht ^ (^o + "i) - 7(^o + Vi)lv, 



1 _^ 



2 dr 



(1.30) 



1 / "^"2 1 ^"2 "_2 J^ ^_^2 2 ^^2 



Re \ 3r2 ^ 3^. 



Bu, 



i^"i ^1' 



' ^"1^ 'T^ 



0" M ' "2 



J3 Buj /v„+ V, + v,\ du, 



- + (U0 + U1+U2) — + 



3r 



Ju, du, \ / 1 3 2 \ 



1 ^Pa 1 P'"3 1^"3 



2 B r Re \ B r 



2 ^^3 



Bu2 v^Buj v^^ 



; 



532 



