476 PHILOSOPHICAL TRANSACTIONS. [aNNO l7Qd. 



of water, the axis mn must be produced up above k, and the string applied to 

 that part ; the machine must be immersed in a large reservoir of water, leaving 

 the part of the axis to which the string is applied above the surface. Before we 

 proceed to the application, we must investigate a point called the centre of 

 resistance. 



Def. If a plane body revolve in a resisting medium about an axis by means of 

 a weight acting from it, that point into which if the whole plane were collected 

 it would suffer the same resistance, I call the centre of resistance. 



Let a be the area of the plane, and a the fluxion of the area at any variable 

 distance x from the centre of the axis, and d the distance of the centre of resist- 

 ance from that of the axis. Now the effect of the resistance of a to oppose the 

 weight is, from the property of the lever, as the resistance multiplied into its dis- 

 tance from the axis, or as x'a; but the resistance is supposed to vary as the square 

 of the velocity (which is found by experiment to be true under certain limitations), 

 or as the square (x') of its distance from the axis ; hence the effect of the resist- 

 ance of a to oppose the weight, is as x^'a; therefore the whole effect is as the 

 fluent of a^a. For the same reason the effect of the resistance of the whole 

 plane a at the distance d is as d^a; hence d^a = fluent x^'a, consequently 



, , flu x^i 



If the plane be a parallelogram, two of whose sides are parallel to the arms, 

 and m and ?i the least and greatest distances of the other two sides from the axis, 

 , , n* —m* («' + »«') X (n + m) 



then d =. iJ — = v' ; • 



Now to find the resistance of the planes striking the fluid perpendicularly, first 

 set them parallel to the horizon, so that tiiey may move edgeways, or in their 

 own plane, and let 2 equal weights be put, one into each scale, such as to give 

 the arms a uniform velocity, and then these weights together {iv) will be just 

 equivalent to the friction of the axis and the resistance of the arms. Then place 

 the planes perpendicular to the horizon by a plumb-line, and put in 2 more equal 

 weights, one into each scale, making together w, so as to give the planes the 

 same uniform velocity as before. Then, from what has been already observed, 

 there is no additional friction, and therefore this weight w must be equivalent to 

 the resistance of the planes. But this equivalent weight w acts only at the dis- 

 tance of the radius r of the axis from the centre of motion, whereas the resist- 

 ance is to be considered as acting at the distance d of the centre of resistance from 



the centre of motion; hence d : x :: ^ : -j X w the weight acting at the distance 

 d, which is equivalent to the resistance acting at the same distance, and con- 

 sequently it must be equal to the absolute resistance against all the planes. And 

 to find the velocity, let c feet be the circumference described by the centre of 



