326 MR. P. W. BARLOW ON THE LAWS WHICH GOVERN 



Let d = the required area of paddle. 



v' = the new resulting velocity of the vessel. 

 All of which quantities are given except d 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-z;)2Va = (rV-i;')3rV«'. 

 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 - vf a : (r V - v'f d : v^ : v'K 

 From these two equations we find 



^'' = ;^, and « = p^^— -^ X «. 



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 de- 

 pendent 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 

 develop 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 inversely as the cube root of the power developed in the 

 two cases. 



We have seen that (V — v^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. 

 Let the amount of power, or, which is the same thing, the number of strokes made in 

 the two cases be as 1 to m. 



Now supposing, in the first place, the diameter to remain the same, the velocity V 

 will become m V ; and we may find d and the resulting velocity v' from the equations 



(V-vyya:{mY-v')^myd::l:m, 

 and 



(V - vy a : (w V - v'f d : : v^ : v'^ ; 



that is, by making the whole power in the two cases as 1 to m, and the resistances on 

 the paddles as v"^ to v'^. 



From these equations it appears that v' = v, or that no increase of velocity will be 

 given to the vessel by reducing the paddle, so as to bring out the full power of the 

 engine. 



But if the diameter of the wheel be changed, the paddle remaining the same, both 



