Mr. Ivory's Notes relating to the Theory of Barker's Mill, 425 



Notes relating to the Theory of Barker's Mill, By 

 J. Ivory, Esq, M.A. F.Il.S. 



I. On the Pressure caused by the Centrifugal Force. 



The centrifugal pressure may be investigated in the manner 

 following. — The whole pressure at the orifice is the sum of 

 the variable pressures of all the molecules, or infinitely small 

 portions of the fluid, in the length of the arm. Let r be the 

 length of the arm between the orifice and the axis; g = 16^ 

 feet ; v 9 the absolute velocity of the orifice in feet : then if d x 

 (or x) be a small portion of the fluid in the arm at the di- 

 stance x from the axis, the centrifugal pressure of dx will be 



sfiv* dx v^.xdx 



X 



** 2xg 2gr* ' 



and the pressure of the prism of the fluid, of which the length 



V 1 x^ 



is x, will be = - — j- ; and the whole pressure of all the fluid 

 in the length r 9 will be = 4~ r = tt- 



The same thing may be more shortly stated, thus: The 

 centrifugal force of every small portion of the fluid in the arm 

 being as the distance from the axis, we may assume that every 

 portion is acted upon by the force which takes place at the 

 middle of the arm, or at the distance J r from the axis. Now 



this force is m >j+* ; which multiplied by r, the sum of the 



portions of the fluid, gives -7— x r = ^- 5 the same as be- 

 fore. ier W 



And since we have, 



v2 — / v V* 



4g " 4g \ r / 9 



-- being the angular velocity of the machine, or the angle 



through which it turns in a second ; it follows that the angular 

 velocity being the same, the centrifugal pressure varies as the 

 square of the length of the arm. 



Some authors, and in particular Bossut (Hydrod, torn. i. 

 § 429), reckon the centrifugal pressure, not in the whole 

 length r, but in the distance between the upright tube and the 

 orifice in the horizontal arm. Let r 1 be the radius of the up- 



New Series, Vol. 3. No. 18. June 1828. 3 I right 



