Vol. 6, 1920 
MATHEMATICS: J. K. WHITTEMORE 
185 
tion of the pulleys on their axles must be eliminated or else the previous 
results cannot be applied. 
If it is found possible so to run the engine of a ship as to deliver constant 
effective thrusting power at the screw we may give a similar discussion of 
the motion of the ship, where now / = c/v. Even if it is not possible to 
produce constant effective thrusting power in starting a ship from rest it 
may be found possible to do so in increasing speed and the following dis- 
cussion would be applicable. We have 
— a = - — r{v), a = - - R{v) = F{v). 
g V V 
Since F(X) = 0, C = XF(X), and 
. \R{\) - vR{v) ... 
V 
L = — = (X - v)vdv 
~ J Vo ^{v) ~ J ^^'o \R{\) - iR(v)' 
If 'R{v) = Kiv^^ the conditions imposed on F{v) are satisfied as before. 
If in particular Vo = 0 and w = 2 we have L = 0.247/i^i, a value inde- 
pendent of X and C. The assumption R{v) = Kfo^ gives C = Ki\^, a 
formula connecting power and speed which is often used but which is 
certainly not correct. ^ 
It is not possible to represent the resistance offered by the water to the 
motion of a model or a ship by an expression of the form R{v) = Kiif. 
It is suggested that the law of resistance may be studied in two ways: 
more complicated laws of resistance may be assumed and tested by com- 
paring the measured values of L with the values given by equation (3) 
under the assumed law. Either constant force or constant power may 
be used. Secondly, it may appear from experiment that the distance 
L for a model started from rest and brought to full speed X, by a constant 
force, for example, can be expressed as a function of X. Then the func- 
tion R{v) determining the law of resistance might be studied from the 
integral equation 
Jo R{\) - R{v) J o R{\) - R{-\z) 
^ The complete discussion is to be published in the Annals of Mathematics, probably 
in the number for June, 1920. 
2 See, for example, the "Admiralty Coefficient Formula," H.P = A'^^V^/K, given by 
C. W. Dyson, Practical Marine Engineering, 7th edition, 1918, p. 614. 
