What Are the Two Free Parameters in Nondimensionalization?

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Please check the image, I got it from wikipedia which is similar to the problem i am trying to solve, just a quick question, there's a line on it that says "Since there are two free parameters, at most only two coefficients can be made to equal unity.", i want to know what are the two free parameters, and how do they determine it? This is something that's mentioned in my textbook as well which I am confused of. Thanks
 

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The free parameters are x_c, t_c that were introduced by defining

x = x_c \chi ,~ t = t_c \tau.

The idea is that you can simplify the form of the equation by choosing x_c, t_c in terms of a,b,c,A.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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