Resistance of Fluids. 391 



For AB being the direction of the plane, and BD that of the Fig. 240. 

 motion, ABD the angle whose sine is i ; the number of particles 

 or quantity of the fluid which strikes the plane will be diminish- 

 ed in the ratio of 1 to i ; and the force of each particle will 

 likewise be diminished in the same ratio ; so that on both these 

 accounts the resistance will be diminished in the ratio of 1 to i 2 ; 

 that is, in the duplicate ratio of radius to the sine of ABD. But fur- 

 ther, it must be considered that this whole resistance is exerted 

 in the direction BE perpendicular to the plane ; and any force in 

 a direction BE is to its effect in a direction AE, parallel to BD, as AE 

 to BE, or as 1 to i. Consequently, on all these accounts, the resist- 

 ance in the direction of the motion is diminished in the ratio of 

 1 to i 3 . And if this be compared with the result of the preced- 

 ing step, we shall have for the whole resistance, or the moving 

 force, on the plane, 



m = 



s v 



2 5 



(5.) If w represent the weight of the body whose plane sur- 

 face tf is resisted by the absolute force m, then the retarding 

 force 



f _ m __ as v 2 i 3 



J ~~ w 



(6.) And if the body be a cylinder whose surface or end is 

 d, and diameter D, or radius R, moving in the direction of its 

 axis; then, because i= 1, and tf = n R 2 i n D 2 , where 

 TT = 3,141593, the resisting force m will be 



71 S D 2 V 2 71 S R 2 V 2 ^ 



~~~ 



and the retarding force 



f n s o 2 v 2 

 / 



(7.) This is the value of the resistance when the end of the 

 cylinder is a plane perpendicular to its axis, or to the direction 

 of the motion. But were its face a conical surface, or an elliptic 

 section, or any other figure every where equally inclined to the 

 axis, the sine of the inclination being i ; then the number of parti- 

 cles of the fluid striking the surface being still the same, but the 

 force of each, opposed to the direction of the motion, diminished 



