Deriving the expression for wind velocity around low pressure area

Hi
Can somebody tell me the steps and assumptions made in deriving the equation to calculate the wind velocity around a low pressure area ? (see the attachment fro Kleppner & Kolenkow).
I tried to derive like this :
Coordinate system located on earth's surface with i direction towards E, j direction towards N and k direction upwards.
Initial wind velocity V = V_x i + V_y j
Ω = Ω sin λ j + Ω cos λ k where λ is the latitude of the place considered.
Position vector of parcel of air r = r_x i + r_yj

Eqn for the rotating system is
ma_rot = ma_inertial - 2m Ω * V - Ω * (Ω * r) ----- (1)

ma_inertial = -(ΔP)S j where S = cross sectional area of parcel of air

As the flow is assumed to be steady, LHS of equation 1 i.e. ma_rot = 0

RHS evaluated vectorially.

However, I cannot get the expression shown in the attachment.

Please help out. Is there any other text which shows this derivation ?

TIA

Attached Files

K&K.pdf (84.4 KB)