53 MR RUSSELL'S RESEARCHES IN HYDRODYNAMICS. 



2<)r 



we deduce in this case of diminished section by substitution, 

 of which the successive differential equations in regard to v are, 



dv X^'' 2g\2g • ■ ■ ■ U-) 



d^ \ 9 hg ' ' ' ' ^ ' 



dv^ -\ g hg ■ • • • ^^-^ 



From eq. (1.) if we make 



dv X-" 2gi2g "' 



we obtain, in the case of a maximum or minimum, 



2_g = and ,= 5^'. 

 By substituting this vahie in eq. (2.) we get 



~dv t" 2j2sr' 



being a negative quantity, whence it follows, that 



R' is a maximum, when v= ~\ 



s' = 0, when v = 2g. 



These expressions may be converted into the following laws. 



Laws of Dynamical Emersion and Diminished Resistance. 



1. If a floating body be put in motion wth a given velocity, the pressm-e which 

 it exerts doAvnwards upon the fluid in virtue of gi'avity, is diminished by a 

 quantity equal to the pressm-e of a column of the fluid having the height 

 due to the velocity of the motion. 



'1. The Section of Dynamical Immersion is less than the Section of Dynamical 

 Emersion, in the same proportion in which the difference between the velo- 

 city of the motion and the height due to it is less than the velocity of the 

 floating body. 



3. The Resistance being taken in the ratio of the square of the velocity upon 

 that part of the section only which remains immersed, the aggregate resist- 

 ance will increase in the ratio of the squares of the velocities, very nearly 



