122 APPENDIX. 



vessel, gives p x B E for the propelling effect. The sum of these two quantities is p [A E + B E] 

 = p x A B, whiclj is the product of the total pressure on the given point by its circum- 

 ferential velocity, as was to be proved. 



This is true only for wheels with radiating floats : in all other cases the velocity perpendicular 

 to the surface of the float which the given point would have if the axis of the wheel were 

 stationary, must be substituted for the circumferential velocity, and, as in the common wheel 

 the motion of each float is perpendicular to its surface when the axis is at rest, these two 

 velocities are identical. 



In order to be able to calculate the absolute amount of power required to produce a given 

 effect, it is necessary to be acquainted with the laws which govern the resistance of fluids to the 

 motion of solid bodies in them, which are generally admitted to be based on the following 

 theorem. Prop. 1. If a plane surface move at a given velocity through a fluid at rest in a 

 direction perpendicular to itself, the resistance is proportional to the density of the fluid and 

 to the square of the velocity of the plane, or it is equal to the weight of a column of the fluid, 

 whose base is equal to the area of the surface, and altitude equal to the height through which 

 a body must fall by the force of gravity to acquire the given velocity ; which, if v denote the 

 A'elocity of the surface in feet per second, a its area in square feet, and w the weight of a cubic 



v 2 v 2 



foot of the fluid, is equal to a w , the altitude due to the velocity v being. 



2/ *2ff 



It is assumed that the resistance to a plane moving in a fluid at rest is equal to the pressure 

 of the fluid on the plane at rest, the fluid moving at the same velocity and in the contrary 

 direction to that of the plane in the former case; on which hypothesis the ratio of the square of 

 the velocity is explained in two very different ways. The first is, that " the resistance must 

 " vary as the number of particles which strike the plane in a given time, multiplied into the 

 " force of each against the plane ; but both the number and the force are as the velocity, and 

 " consequently the resistance is as the square of the velocity." The second explanation is, that 

 " the force of the fluid in motion must be equal to the weight or pressure which generates 

 " that motion, which, it is known, is equal to the weight of a column of the fluid, whose base 

 " is equal to the area of the surface, and altitude the height through which a body must fall 

 " to acquire the given velocity." 



These explanations are extracted from Dr. Gregory's Treatise on Mechanics, in which he 

 states that they are founded on the hypothesis that " the particles of the fluid move freely 

 " without disturbing each other's motions, and that it flows in behind as fast as a plane body 

 " moves forward, so that the pressure on every part of the body is the same as if the body 

 " were at rest," which is well known to be incompatible with the nature of any fluid with 

 which we are acquainted. 



In what precedes, only one plane has been mentioned, namely, that which strikes the fluid, 

 the after surface of a body being supposed, by the hypothesis, to have no influence on the 

 resistance ; but we know that the pressure on the after surface must decrease as the velocity 

 increases, for if the body were suddenly abstracted, so as to leave a void, the water which had 

 been pressing on the after surface could only move to fill it with the velocity due to its depth 

 below the surface of the fluid ; so that, if the body moved forward with a greater velocity, the 

 fluid would cease to be in contact with it, and could not therefore exert any pressure upon it, 



