Vol. 6, 1920 
MATHEMATICS: J. K. WHITTEMORE 
F{v) = {\ — v)(p{v), \>V^Vo, 
and that the function (p{v) is not less than a positive number m for such 
values of v and is finite ior v = \. It may then be proved 
(X - Vo)e~'^' >\-v>0, (1) 
}.t-x = L- (2) 
L = p ^, U^-v.)e--% P ^ >0. (3) 
These results may be interpreted as follows: the velocity v increases 
from Vo and approaches X indefinitely as t increases without ever reaching 
that limit. But this "limiting velocity" X is, for practical purposes, at- 
tained and is hereafter called "full speed." To interpret equation (2) 
we consider a particle P' describing the same path as P, starting from 
coincidence with P at the time t = 0, and moving uniformly with velocity 
X; then evidently PP' = \t — x. Equation (2) shows that PP' increases 
from zero approaching the finite limit L. We call L the "lost distance" 
or the distance lost in attaining full speed. It is clear that L is practically 
the distance that P is behind P' when P has reached full speed. If P 
is, for example, a ship attaining and then proceeding at full speed, L could 
be found from observation, for this lost distance is the difference of the 
distance run in any time r at full speed and the distance run in the time 
r from the start, provided that full speed is attained in the time r from the 
start. We introduce the time T lost in attaining full speed, defined by 
L = \T. This lost time T would, in the case of a ship or of a towed 
model, perhaps be more easily measured than L, for T is the difference 
of time in running any distance 6 at full speed and in running the distance 
5 from the start, provided that full speed is attained in the distance 5 
from the start. Then if T is measured L is given by L = \T. Now 
if the acceleration F(v) is known L may be calculated from equation (3). 
If F{v) is expressed by a formula containing an unknown constant and 
if L is found by observation the constant may be calculated from (3). 
These remarks lead to the suggestions of applications to the motion of a 
ship starting from rest or increasing speed under its own power or to the 
motion of a towed model. 
For a towed model or for a ship increasing speed under the action of 
the thrust of the screw the component of accelerating force in the direc- 
tion of the tangent to the path has the form, / — r, where / is the tension 
of the towing line in the case of a towed model or the effective thrust of 
the screw in the case of a ship, and r is the resistance of the water. No 
general expression for r is known, but it may be assumed that for any 
particular model or ship f is a function of v alone. We consider two forms 
of /.• first / is constant, f = k; second / is the force exerted by a constant 
power, / = c/v. 
