$\newcommand{\al}{\alpha}$
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][1] 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][2]. 

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.  





  [1]: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=2ahUKEwiRq5uEicrkAhUCXK0KHU-2AKcQFjABegQIBRAB&url=https%3A%2F%2Froyalsocietypublishing.org%2Fdoi%2Fpdf%2F10.1098%2Frspa.1929.0151&usg=AOvVaw0HxTCZr2nQT84iaFgNC0me
  [2]: https://works.bepress.com/iosif-pinelis/18/download/