278 Remarks on the theory of the Resistance of Fluids. 



Since the area of the plane is given, the number of particles in 

 action at any moment is givenj and consequently the force of each, 

 at any instant, is as the square of the velocity of the plane. 



We may now note a fundamental error in the received theory, 

 which assumes, usually without argument, that the force of each par- 

 ticle is as the velocity of the plane, instead of the square of the 

 velocity, as we have now shown it to be. 



Since the power of resistance is as the force into the distance 

 moved through, the power of resistance that will have been encoun- 

 tered at the end of any distance, will be as the square of the velocity 

 into that distance, or if that distance be given, it will be as the square 

 of the velocity simply. 



Since the fluent power of resistance, is as the product of the force 

 - into the velocity, it will be as the square of the velocity into the ve- 

 locity, or as the cube of the velocity. 



In order to bring these results together in a concise form, I will 

 recapitulate them. 



The force exerted by any particle of the fluid, is as the square of 



the velocity of the plane. 



The force felt by the whole plane at any instant, is as the square 



^ of the velocity of the plane. 



The power of resistance, or the amount of resistance encountered 

 by the plane, in moving a given distance, is as the square of the ve- 

 locity of the plane. 



The fluent power of resistance, or the amount of resistance en- 

 countered in a given time, is as the cube of the velocity of the 

 plane. 



Second. — ^Let the plane be oblique to the line of motion, and let 



Its velocity be given in the direction of that line. 



It is manifest that the plane will now strike the fluid with a force 

 corresponding to its velocity, estimated in a direction perpendicular 

 to the plane ; and that this velocity will be to that of the plane 

 radius to the sine of inclination. But it has been shown already, 

 that the force is as the square of the velocity ; therefore the force es- 

 timated in a direction perpendicular to the plane, is as the square o 



; as 



the sine of inclination. 



nd dis- 



The power of resistance, being as the product of the force and < 

 tance, is as the product of the square of the sine of inclination into 

 distance, estimated in a direction perpendicular to the plane. ^ But 

 this distance, when the distance in the other direction is given, 



