60 APPENDIX. 



Let the power, or, which amounts to 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 a and the resulting velocity v from the equations, 



(V - r) 2 Fa : (m V - v')* m V a : : 1 : m, 



and 



( V - )* a : (m V ')* a : : v* : v' 2 



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

 paddles as v 2 to v' 2 . 



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 the 

 velocities Fand v will be changed. Let the former become p V, and the latter n v; our 

 equations are therefore, 



(F - )* V a : (p V - nvf p Va : : 1 : m, 

 (V - v) 2 a : (p V - n v) s a : : 1 : w 2 , 



which reduced, give p = n, and each equal to the ^/^ ; that is, the velocity of the 

 vessel will be increased in the ratio of the cube root of the powers expended. 



We see, therefore, that when an engine is not able to perform its whole duty, the diameter 

 of the wheel ought to be reduced, and not, as is usually done, the area of the paddle; for in 

 the former case the velocity is increased in the ratio of the cube roots of the number of 

 strokes, while in the former it remains the same as when the less power was developed. 



To find the change in the diameter required to produce this effect, we know the circum- 

 ferential velocities are V ': V ^^m, or as 1 : &/ m ; we know also that these velocities 

 are as the number of strokes multiplied by the radii of the wheels ; putting therefore r and r' 

 for the two radii, the velocities are as r : m r', or r : m r' : : 1 : v*/ m , whence 



.' r 



mi' 

 the required radius of the paddle. 



In the case of the Salamander, from the great immersion of the paddles, the engine could 

 only make 15 strokes instead of 20, its full duty. 



We may now find what increase of speed would have been given to the vessel by reducing 

 the wheel so as to allow the engine to perform its whole duty. 



We have m = 1-33, whence r' = -8264 r ; and n v = I'lOO v ; if therefore the diameter 

 of the wheel of the Salamander had been reduced in the ratio of 1 to -8264, the speed of 

 the vessel would have been increased in the ratio of 1 to I'lOO; that is, by reefing each 

 paddle about 19 inches, the speed of the vessel would have been increased about |ths 

 of a mile. 



In these calculations a similar action of the paddles has been assumed, with every variation 

 of diameter, which in reality is not strictly true, as every change of the position of the floats 

 will vary the angle at which the centre of pressure enters the water. This variation will 



