ON PADDLE WHEELS. 



127 



) 2 3 rap 2ra? ap*cos. /3 j sin. ft j . . (2) 



The mode of applying these formulae will be illustrated by the following example : 

 The wheels of Her Majesty's steam frigate 'Salamander' are 21 ft. in diameter, the paddle- 

 boards 8 ft. 9 in. broad, and 2 ft. 6 in. deep, and there are sixteen on each wheel. Thus 

 R = 10-5, r = 8, b = 8'75, and m = 16. When tried at Woolwich, the greatest immersion 

 of the floats was 5 ft. 6 in., the number of strokes made by the engines was 15, and the speed 

 8'15 statute miles per hour. We have therefore a = the height of the axis of the wheel 



88 x 8-15 

 above the surface of the water = 5, n = 15, and p = 



2 TT n 



. The angles a. and 



TT n 



j8 and their sines are easily found in tables of logarithms, knowing that cos. a = -=- and 

 cos. /3 = 



a 



r 



Log. TT=O- 4971499 

 Log. n = 1 1 760913 

 Log. b =0-9420081 

 Log. m= 1-2041 200 



We will first apply the formula (1) to find the total power transmitted to the wheels, which 

 is most conveniently done by means of logarithms. We find, 



Log. R= 1-0211893 



Log. r = 0- 9030900 



Log. 0=0-6989700 



Log. p =0-8813691 



We have assumed w= 64- 125, which is the weight of a cubic foot of sea water, and 2 g, which 

 is the velocity of a falling body at the end of the first second, is equal to 64-38. We find 

 therefore, 



Log. w =1-8070274, 

 Log. 2^=1-8087510, 



Log. cos. = Log. a Log. R= 1-6777807, and the corresponding angle given in the table 

 of sines and cosines is 61 33'47""2, and the logarithm of the sine of this angle is found to be 

 equal to 1-9441579. The length of the arc a is equal to 1-074479, radius being taken as 

 unity, and its logarithm is consequently 0-0311979. In the same manner we find by sub- 

 tracting log. r from log. a, cos. /3= 1-7958800, whence =5119' 4"- 14 = 0-8956649, and 

 log. /3= 1-9521455, also log. sin. /9= 1-8924424. We shall first find the value of the sum 

 of the terms included between the brackets, and then perform the multiplication, by means of 

 logarithms. 



Log. 



3 = 



Ctr 



0-47712 

 4-08476 

 0-03120 



Log. 



a = 



3-93883 

 0-03120 



3fl 4 a 

 3 



= 4-59308 3B 4 a 



= 39181 



6aV = 3-97003 



6 a 2 p 2 



=9333 



,2 



0-47712 

 2-04238 

 1-76274 

 0-03120 



8 

 R* 



P 

 sin. a 



= 0-90309 

 = 3-06357 

 = 0-88137 

 = -1-94416 



= 4-31344 



=20580 



8 .R 3 p sin. a = 4-79219 8.R 3 psin. a =61970 



