Is something arbitrary if it is a function of something else?

Question

I'm working through Gauss' proof of the Normal Distribution (pg 106, pdf pg 10). At this point we have:

\begin{equation*} f(M_1-\bar{M}) + f(M_2-\bar{M}) + \cdots + f(M_n-\bar{M}) = 0 \end{equation*}

where

\begin{equation*} f(x) = \frac{\phi'(x)}{\phi(x)} \end{equation*}

and $\phi(x)$ ends up being the Normal Distribution. The author says that since $M_i$ can assume arbitrary values (it is some observation that occurs with probability determined by $\phi$) we can use $N$ and $M$ as arbitrary reals where:

$$M_1 = M, M_2=M_3=\dots=M_n = M - Nn$$

and thus the mean (which is the usual mean)

$$ \bar{M} = \frac{M_1+M_2+\dots+M_n}{n}=M-(n-1)N$$

By substitution into the first equation we find

$$f((n-1)N)=(n-1)f(N)$$

(Note that f(-x)=f(x))

and thus by homogeneity combined with the continuity of $f$ we can write:

$$f(x)=\frac{\phi'(x)}{\phi(x)}=kx$$

and integrate our way to glory.

I have some questions. Although $M$ and $N$ are arbitrary:

1. We have set the first element, $M_1$ to something arbitrary. Fine.
2. We have set every other item $M_2$ to $M_n$ equal to the same thing. Doesn't this create an assumption?
3. The thing we set those other items to is a function of both $M$ and $n$. Does that not also create an assumption?