Help with Poisson Brackets (original paper)

[gibsonphysics]

Here I have a translation from French to English of the original paper by Poisson about his brackets. I cannot understand why the function a=f(q,u,t) doesn't have a second order derivative (in q or u). The problem is on the top of the third page (second .JPG) after he took the time derivative. Can somebody help me?

 

[gibsonphysics]

Also, a=f(q,u,t) is a constant of motion. Is there any restriction about second order derivatives for q or u or (p) for a constant of motion?

 

[gibsonphysics]

Here is a quotation that I found on Wolfram website:

"A first integral associated with the independent variable t exist if f is independent of t and does not contain any second or higher derivatives of the coordinates."

Since we have a=f(q,u,t) as a firt integral, it will not have a second derivative of any canonical variables.

What I can't understand and ;also, I didn't find anywhere is why a first integral of the motion can't have a second order derivative.

Does anybody know?