^j^ MR RUSSELL'S RESEARCHES IN HYDRODYNAMICS. 



stant representing the modified resistance derived from the form of the anterior 

 part of the solid, and p the density of the fluid, to which might also have been 

 added a constant quantity for adhesion, but we have omitted it, for the sake of 

 simplifying the expression. 



If, however, we include the element of diminished immersion, we shall have 

 by (Sec. I.) 



W=^^.p.ms(\ — l^) .... (2.) 



If we now include the element of change in the position of the body, and of in- 

 creased anterior immersion when behind the wave, we shall have, by using & for 

 the angle of elevation of the axis of the solid, ^ for the difference of the anterior 

 section or height of the wave forming on the sohd, modified by the constant n, 

 for the form of that part of the solid, and measured at a given unit of velocity 

 in relation w the velocity due to the wave, we shall have 



R" = |.p{,«.(l-f^).a+s>ne)+;^J • • • (3-) 



When the velocity is less than that of the wave, the quantity is posi- 

 tive, w being greater than v, and the effect of the wave is then to increase the 

 resistance ; as v increases, qv — v diminishes, still remaining positive. If the sides 

 of the channel and of the vessel were infinitely high, and the increments of force 

 uniform and very slow, the phenomena would give the case represented, when 



n'iv niv 



00 > 



w — V 



the resistance being infinitely great ; and when the velocity v becomes greater than 



that of the wave w, sin 6 being = 0, the expression becomes negative, its 



denominator having become negative, and the expression is reduced to 



R"' = f>.{,».(l-f^)-.^} . . . (4.) 



the expression of the case when the velocity is greater than that of the wave. 



The line of resistance AP corresponding to Eq. (1.), is a parabola, AX being 

 the axis of the parabola, and AY the tangent of the vertex, the velocities being 

 represented by the ordinates parallel to AY, and the resistances being represented 

 by the abscissae parallel to AX ; A being the origin.— (See Fig. 9.) 



The line of resistance AM2E corresponding to Eq. (2.), has all its abscissae 



