$\newcommand{\al}{\alpha}$
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]<sup>1</sup> 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.  




<sup>1</sup><cite authors="Fisher, R. A.">_Fisher, R. A._, [**Tests of significance in harmonic analysis.**](http://dx.doi.org/10.1098/rspa.1929.0151), Proceedings Royal Soc. London (A) 125, 54-59 (1929). [ZBL55.0950.16](https://zbmath.org/?q=an:55.0950.16), [MR2079](https://mathscinet.ams.org/mathscinet-getitem?mr=2079).</cite>

  [1]: https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1929.0151
  [2]: https://works.bepress.com/iosif-pinelis/18/download/