14 PHILOSOPHICAL TRANSACTIONS. (^ANNO 1735. 



express the force of gravity, and the centrifugal force at the middle circle. 

 Since the spaces described in the same time, by the action of two forces, are 



as those forces, s : 5 :: g : c, and — := c; then substituting in this expression, -— 

 instead of s, we have - — = c; and putting -^r + t'" instead of its equal m, 

 -. — ■—: = c. So that the ratio of gravity to the centrifugal force, at the 



middle circle, is that of g to '^ , or that of 1 to —^ ; which 



(R + r) X s' (a + r) X s ' 



being multiplied by the number of the revolving circles u — r, gives for the 

 pressure of the column of air r — r, proceeding from gravity, r — r, and the 



pressure proceeding from the centrifical forces ■ ■— , where r — r being 



a factor common to both, may be thrown out of the expression; and since the 

 velocities produced from different pressures, are as the square roots of the 

 pressures, the velocity gravity would give from the natural weight or pressure 

 of R — r, will be to the velocity the same column would have from the pres- 

 sure occasioned by the centrifugal force, as ^Z J , or 1 , to t/- r . 



•' ° '(R + r)xs 



Lastly, since the velocity proceeding from the action of gravity, on a column 

 = R — r, is always a known quantity, it may be called = a, equal in this case 

 to 15,38 feet per second, and consequently the velocity proceeding from the 



centrifugal force will be a X /: c , or av X \^ , — :— r , or ,■ , - =^; 



^ ' (R + r) X s- (r + r) X s' v(r -|. r) x s 



1 5-38 X 26 71 



that is, in this machine, . .'. — = 40.67 feet per second. And if we add 



to this, the velocity of the outer circle in the tangent of which the air escapes, 

 which, in the supposition we made of two revolutions in a second, is 44 feet 

 per second, we shall have = 93.67 feet per second. 



Note. — This calculation supposes the bore of the sucking-pipe sufficient to 

 furnish as much air as would escape, according to this velocity ; but in this 

 machine the sucking-pipe being no larger than the ajutage or blowing pipe, 

 the velocity proceeding from the pressure occasioned by the centrifugal force, 

 and from the velocity in the tangent, which may be represented by a column 

 of air of sufficient height to give the velocity of 93.67 feet, which is 145. 882 

 feet, must be divided into two equal parts, one half employed in sucking, and 

 the other in blowing; therefore the half of 145.882 feet, which is 72-941 feet, 

 will represent the height of a column of air, that would occasion the same 

 pressure with which the centrifugal force and the circular motion act in this 

 machine; and a column of this height producing a velocity of 68.53 feet per 

 second. This number will express the velocity with which the air is sucked 

 into the wheel; and the same number will also express the velocity of the air 



