MATHEMATICAL THEORY OF STREAM-LINES. 
297 
by the sine of the angle of obliquity of the wave ; that is, 
V sin «= X /V 2 -W 2 ; (74) 
and if this be correct, the resistance arising from the dispersion of energy by a set of 
waves of a given speed may be expressed as follows : — Let R' be the propelling force which 
would be required in order to produce the disturbance constituting the wave-motion, if 
the whole of the energy of that motion were dispersed ; then the actual propelling force 
required in order to restore the energy dispersed by those waves will be 
R/ sin a=R'. \J (75) 
The total wave-resistance of a ship, according to this hypothesis, is the sum 'of a set 
of terms similar to the above expression, each term belonging to a different set of waves 
and containing its proper values of IT and of W'. 
Each value of R' is probably proportional to the square of the speed of the ship, and 
to some function of her dimensions and of the position of that part of her where the 
set of waves in question originates, and may therefore be expressed in units of weight 
by where a is such a function, and % the density of the water. Hence the total 
wave-resistance may be expressed as follows : 
S.R , sin a =^.^y / (76) 
For waves of the first class the value of W is that given by equation (72 d), the period 
Tj being expressed as follows, 
T 1= AdAi. (77) 
where l x and L are the lengths of the fore body and after body respectively, and/, and/) 
two coefficients, depending on the forms of those bodies. From the practical results of 
the rules given by Mr. Scott Russell, there seems to be reason to believe that those 
coefficients are sensibly equal to, or not very different from, the coefficients of fineness, 
found by dividing the displacement of the fore body and after body respectively by the 
area of midship section. The speed of waves of the first class is thus given by the fol- 
lowing formula, 
W ^(/l6+// 2 ) f>7Q\ 
1— 2ttV ’ 
and in order that such waves may not disperse energy by their divergence, it is necessary 
that W, should be equal to or greater than V ; that is to say, that 
fA+fA = OT >~ (™) 
It appears further, from results of practice, that it is advisable that the two terms of 
the left-hand member of this equation should be equal to each other ; that is to say, 
ttY 2 
ff=fif=0Y> — ; 
(80) 
