Understanding Critical Flow and Minimum M in Fluid Mechanics

[fonseh]

Homework Statement

I am not sure how to get the Mmin . Why δA will become B (refer to the pink circled part)

Homework Equations

The Attempt at a Solution

Is there anything wrong with the author's steps ? I know the author's intention is to show that the Mmin occur when the Fr = (Q^2)B/ g(A^3) = 1 , but we have (Q^2)δA / g(A^3) ... How to make the B=δA ?

 

[haruspex]

I'm afraid I cannot make sense of the development over those two pages.
On the left page, the parameter appears to be distance along the stream. At any such distance, there is an A, a $\bar y$ etc. The height, y, is not a parameter here.
On the right page we are concerned with the minimum M has a function of height, and yet the equations appear to be derived from those on the left page. The step "dividing by ∂y" is not adequately justified by that description.
I see no explanation of what B represents.