$\newcommand{\al}{\alpha}$ For $i=1,\dots,n$, let \begin{equation} R_i:=\frac{X_i}{X_1+\dots+X_n}, \end{equation} where the $X_i$'s are iid standard exponential random variables. Let $$R_*:=\max_{1\le i\le n}R_i. $$ Fisher gave the formula \begin{equation} P(R_*>x)=\sum_{j=1}^n(-1)^{j-1}\binom nj(1-jx)_+^{n-1} \end{equation} for $x\in(0,1)$ (using somewhat different notation), where $u_+:=\max(0,u)$. I have a proof of this result and a certain generalization of it.

My problem is that I understand almost nothing in Fisher's proof (on pages 57--58 of his paper). In particular, I don't understand the following:

1. What does (the polynomial (?)) $f$ in $t$ (introduced (?) on page 57 of Fisher's paper) have to do with the spline (?) $\text{P}$ in $g$;
2. Why does $f$ have to have the differential properties in a neighborhood of $t=1$ that Fisher says $f$ has to have?
3. How does Fisher make the jump from those properties of $f$ to the (correct) final expression for $\text{P}$? Fisher seems to provide absolutely no details on this.

I will appreciate any help in filling these huge gaps in my understanding.