MATHEMATICS: J. K. WHITTEMORE Proc. N. A. S. 
Suppose f = k. Then if W is the weight of P, 
^la = k - r{v), a = K - Riv) = F(v). 
g 
Since F(X) = 0, K = R{\), and 
Fiv) = RW - R{v) = (X - vMv), 
~ ^ ^{v) ~ J R{\) - 
If we assume the resistance of the water proportional to some positive 
power of the velocity, R{v) = KiV^, the conditions imposed on F{v) are 
satisfied, and 
^ 1 fx (X - v)dv 
Ki 
If in particular = 0 and if n = 2, 
, 1 1 o 0.693 
L = — log, 2 = — - . 
Ki Ki 
We have the curious result for this particular case that L is independent 
of X and consequently of the force k. If L is obtained from observation 
Ki is found from the preceding equations and the force required to develop 
speed X is determined. Comparison of Ki for different models would serve 
to determine the relative merits of their designs. 
An apparatus might be simply constructed to give the effect of towing 
models with a constant force. Suppose, to outline such an apparatus 
in its simplest form, a model to be towed by a perfectly flexible horizontal 
cord passing over a pulley without weight or friction on its axis and at- 
tached to a descending weight. The tension of the cord would, under the 
assumptions, be the same on both sides of the pulley, but would vary with 
the velocity. The equations of motion for the model, weight W, and the 
descending weight W are 
—a=T- r{v), —a = W ' - T, 
g g 
where T is the tension. Adding the equations we have 
a - ^ [W - r{v)l 
W 
from which it appe?.rs that the motion is the same as if the towing force 
were constant, the force of resistance however being modified by a constant 
factor. The preceding discussion would be applicable. In a more prac- 
tical but less simple apparatus consisting of several cords and pulleys, 
the former not perfectly flexible, the latter neither weightless nor per- 
fectly smooth, the results appear still to be applicable, account being taken 
of the stiffness of the cords and of the weight of the pulleys, but the fric- 
