The hydrometer, by some called areometer, is an instrument commonly made of glass, as represented by Figure 1, consisting of a stem AB, graduated by small beads of glass of different colours, stuck on the outside, a larger Ball, B, quite empty as well as the stem, and a small ball, C, filled with quicksilver before the end A, was hermetically sealed, in such manner as to make the hydrometer sink in rain water as deep as
m
, the middle of the stem.
Quadratic functionals and small ball probabilities for the $m$-fold integrated Brownian motion
