50 APPENDIX. 



tageous : the amount of vertical resistance or lost power is here increased to a considerable 

 amount, and the shock by the paddles striking the water at so great an angle is highly 

 injurious. 



EXPLANATION OF THE MANNER IN WHICH THE POWER OF THE ENGINE IS 

 EXPENDED IN THE TWO WHEELS. 



As the preceding formulae enable us to calculate the degree of resistance on the paddles at 

 any angle of inclination, we may readily find the whole power exerted by the engine, either 

 by obtaining the sum of all the resistances of one paddle in passing through the water, and 

 multiplying by the number ; or by finding the mean resistance of one paddle while in action, 

 and multiplying by the number acting at the same time, and the circumference passed through 

 by the centre of pressure. This latter mode is preferable, being arrived at by a less com- 

 plicated calculation, and as it represents more truly the real action of the resisting medium, 

 which, from there being several paddles in the water at the same time, is in fact equivalent 

 to that of an uniform weight acting at the distance and velocity of the centre of pressure. 



Not having the dip of immersion in all the experiments in the table, a separate calculation 

 is not given for each, but a dip of 3ft. '6 in,, equivalent to an entrance angle for the centre 

 of pressure of 44, is assumed as the average in the first class, which, from the experiments 

 being generally made after the engines were fitted, and before the vessel had taken in her 

 cargo of coals, must be very nearly the truth. 



The general expression for the tangential resistance with the COMMON WHEEL is (V 

 v cos. if) 1 ; the mean resistance will therefore be the integral of (V v cos. <p)* d <p divided by 

 p ; or it may be obtained sufficiently near, arithmetically, by calculating the resistance at 

 equal intervals or angles, and dividing their sum by the number, for the mean. 



Assuming V = 4 and v = 3, which is very nearly the mean ratio of the velocity of the 

 common wheel and vessel, we find the mean resistance of the paddle passing through the 

 whole arc to be to the resistance of the vertical one as 1'75 to 1. Now, as the whole circum- 

 ference contains sixteen paddles, and the arc passed through is 88, three paddles and a half 

 may be considered to be acting : this will make the whole resistance to the engine to be 6-12 

 times that opposed by the vertical paddle, or the proportion of the power of the engine 

 exerted on the vertical paddle to be = '163, while the mean obtained from the experiments 

 is -151. 



In the second class, the paddle wheels, though smaller, being proportionally immersed, will 

 enter the water at the same angle of inclination, so that the same mean resistance will result 

 from it, viz. 1-75. The number of paddles, however, being less in the small wheels, there are 

 not more than three of them effective ; which gives the proportion of the power of the engine 

 exerted on the lower or vertical paddle -190, the mean obtained from the experiments being 

 193. We are thus able to account in the radiating wheel for a power exerted on the paddles 

 equal to the whole nominal power of the engine, which not only speaks strongly as to the 

 accuracy of the principles adopted in the preceding calculations, but also that the supposed 

 loss from back water cannot be much. 



It now remains to account for the power of the engine in the NEW WHEEL, where we have 

 found the horizontal resistance to the paddle to be (V cos. <p v) 1 . The power of the engine 



