1 38 On the general principles of the Resistance of Fluids. 



fore, that act on the increment of the surface of the solid, is as the 



area of the increment, that is, as 2py(dx^ -{-dy^)", and their force 

 in the direction of the motion, as already said, is as the square of the 

 sine of the inclination of the increment to the direction of the mo- 



tion, that is, as j~7TT7' Therefore, the resistance to the incre- 



2pydy^ 

 ment is as — ==^ — — i and the resistance to the whole surface as 

 \/ax'^ -{-dy^ 



^PJ — — :• The resistance to the circle generated by the 



x/dx^+dy- 



revolution of the ordinate y, is as the area of that circle, that is, as 



py^. Therefore, F, the resistance to the circle whose radius is y, 



y^ ydy'-' 



is to/, the resistance to the solid, as tt : / — -, — — — • 

 •^ 2 ^^dx'^+dy^ 



y-dy' 



3 



r^ — (r^ —y'^Y 



and when y = r as 



U LIU 



If the solid be a sphere, then dx'^ = ^ _ / Therefore, F :/: : 



or 



r2 r^ 2 



•g ; g" .' ; 1 : o' That is, the resistance to a sphere is equal to two 



thirds of the resistance to one of its great circles. By the old theory, 

 the resistance is only one half of the resistance to the great circle. 



If the solid be a cone, dx^ =a'^dy'^. Therefore, F »f'.'.^ J 



/• ydy^ y' y^ 



/ , . -^— : : -^ .* ! '.\/a'^ -\-i : l; that is, resistance 



'' Va^-dy^+dy-" 2 2s/a"^l ^ ^ ' 



on the base is to that on the convex surface, as slant height to radius 



of base. The old theory gives the ratio, square of slant height to 



square of radius. 



If the solid be generated by the revolution of a parabola about its 



axis, then dx^J^-^'^^ and F :/: :^ :/ — --M==;-^-^ ♦ 



a^ ^ 



■ 7 ; and so on for other solids. 



To find the resistance to a curve moving in the direction of its 

 axis. The increment of the curve is Vdx^-\-dy'^^ and the square 



