On the general principles of the Resistance of Fluids. 139 



of sine of inclination is , 21 / 2 ' therefore, the resistance to the 



curve is as / — — The resistance to the ordinate y is as its 



-^ '^dx'+dy^ ^ 



length, therefore F, the resistance to the ordinate, is to/, the resist- 



r ^y' 

 ance to the curve, as y \J — ==^=F=^* If the curve be a semicircle, 



\/dx--\-ay'^ 



/., / Vv/r^ — y2 4-^sin.~'y 



then F :/: '.y \jdyVr" —y- '. '.y \ ^-^ j and when 



pr"^ 

 y=r, as r : — j and putting radius unity, as 1 : .7854. From this 



it is also evident, that if a cylinder move in a direction perpendicularly 

 to its axis, the resistance to a section through the axis is to the resist- 

 ance to the convex surface as 1 : .7854. The old theory gives it as 

 3 : 2. 



To find the best angle for the sails of a mill, a rudder. Sic. at the 

 beginning of their motion. The number of particles being the same 

 at all inclinations, the best angle will be that at which the force of the 

 panicles in a direction perpendicular to their motion is a maximum. 

 This force, as already said, is as the product of sine and cosine of 

 angle of inclination. Let a;=sine of the angle, then ^x'^—x* will 

 be a maximum, 2xdx — Ax^dx — Q, and a:=v J=sine of 45°, which 

 is the maximum angle. The old theory gives 54° 44'. 



The following are two somewhat remarkable results of the new 

 theory. The resistance to a square is the same, whether it be moved 

 in a direction perpendicular to one of its sides, or in the direction of 

 its diagonal. In the latter case, the force of the particles in the direc- 

 tion of the motion is as square of sine of inclination, which is 45°. 

 The square of sine 45° is J, radius being unity. Their force, in the 

 latter case, is then half the force in the former. But in the latter 

 case two sides are presented to the action of the fluid, and the num- 

 ber of particles is doubled ; consequently, the resistance is the same 

 in both cases. The resistance to a cube is the same, whether it be 

 moved in the direction perpendicular to one of its sides, or in the di- 

 rection of its diagonal. In the latter case, the square of sine of in- 

 clination is ^, radius being unity. The force of the panicles is there- 

 fore only one third of their force in the former case. But, three sides 

 being presented to the fluid, their number is trebled ; consequently, 

 the resistance is the same in both cases. 



