Probability – Try to understand Fisher's proof

To the $ i = 1, points, n $, To let
begin {equation}
R_i: = frac {X_i} {X_1 + dots + X_n},
end {equation}

where the $ X_i $These are exponential standard random variables. To let
$$ R _ *: = max_ {1 lei le n} R_i.
$$

Fisher gave the formula
begin {equation}
P (R _ * x) = sum_ {j = 1} ^ n (-1) ^ {j-1} binom nj (1-jx) _ + ^ {n-1}
end {equation}

to the $ x in (0,1) $ (with some other notation) where $ u _ +: = max (0, u) $, I have proof of this result and a certain generalization of it.

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

  1. What does (the polynomial (?)) Mean $ f $ in the $ t $ (Introduced (?) On page 57 of Fisher's Paper) deals with the spline (?). $ text {P} $ in the $ g $;
  2. Why does $ f $ have to have different properties in a neighborhood of $ t = 1 $ Fisher says that $ f $ must have?
  3. How does Fisher leap from these properties? $ f $ to the (correct) final expression for $ text {P} $? Fisher seems to give absolutely no information.

I will be glad to receive any help to fill these gaps in my understanding.