ON THE RELATION BETWEEN THE DIAMETER, &c. 59 



ON THE RELATION BETWEEN THE DIAMETER OF THE WHEEL, AREA OF THE PADDLE, 



AND THE VELOCITY OF THE VESSEL. 



When the area of the float of a paddle-wheel is so adjusted to any given diameter that the 

 engine is capable of performing its whole duty, it is evident that the same duty might also 

 be performed with a less paddle and larger wheel, or with a smaller wheel and larger 

 paddle, but the velocity of the vessel will not be the same in the two cases ; and the 

 question therefore is, to determine what change must be made in the area of the paddle, and 

 what change would take place in the speed of the vessel, with a given change in the 

 diameter of the wheel, so that the engine in both cases may perform its whole duty. 



Let d = diameter of the first wheel, 



V = its circumferential velocity, 



a = the area of paddle, 



v = the velocity of vessel, 

 r d = the diameter of the second wheel, 

 r V the circumferential velocity, 



a' = the required area of paddle, 



v' = the new resulting velocity of the vessel, 



all of which quantities are given except ' and v', which may be determined from the 

 following considerations, viz. 



1st, That the whole duty of the engine is exerted in both cases, consequently, 



(V-v) 2 Va = (r V i/) 2 r Va'. 



2nd, That the resistance on the paddle in each case is equal to that of the vessel, and 

 therefore proportional to the squares of the two velocities, v' and v, that is, 



( V - i') 2 a : (r Vv')* a : : * : i'' 2 . 

 From these two equations we find 



, v , (V vY 

 v = . and a = x a. 



From the first it appears that the two velocities are to each other inversely as the square 

 roots of the radii. And by the second, the new area of paddle will be found to increase and 

 decrease so rapidly, that generally little practical advantage can be taken of the condition of 

 the first equation. 



It appears from the above, that there are two different diameters of wheel, with dependent 

 area of paddles, that will allow the full power of the engine to be developed. And when 

 from circumstances of loading, &c., the whole power of the engine cannot develope itself- 

 there are two ways in which this effect can be insured ; the one by reducing the paddle 

 and the other by reducing the diameter of the wheel : by the former it will be seen that 

 the speed of the vessel will remain the same, but by the latter it will be increased as the 

 cube root of the power developed in the two cases. 



We have seen that (Vv) *V a expresses the whole amount of the power exerted, 

 which, in the case we are now supposing, is less than the engine is capable of exerting. 



