144 APPENDIX. 



t 



is rather a disadvantage than otherwise in the float not being radial. The effective pressure in 

 the common wheel is proportional to [ V v cos. 0] 2 cos. <f>, and in the other to [ V v cos. $] 2 

 cos. a? cos. <f>. Thus the resistance overcome, and power required to overcome it with the 

 oblique floats, are to the resistance and power required with the common wheel (the dimensions 

 of the wheels and velocities being the same, except that the quantity of float in the former is 

 to that in the latter as 1 to cos. a) nearly in the ratio of cos. a 2 to 1. But if it be required to 

 determine what circumferential velocity would be necessary to give the same vessel the same 

 speed as with the common wheel and the circumferential velocity V, let V be the required 

 velocity ; then we must have 



[F" - V COS. <] 2 COS. a 2 = \VV COS. <] a , 



or [ V' v cos. $] cos. a = V v cos. <f>, 



..-, V V COS. <& (1 COS. a) 



whence V = e - i 



cos. a 



It will be at once observed that this is not a rigorous calculation, and that the above result 

 is not yet determined, as it contains the variable quantity cos. <f>, for which we ought to 

 substitute its average value. Assuming this to be -fc, a probable value, and cos. a = -f-, which 

 makes the obliquity less than that preferred by Mr. Hall, viz., 45, the value of V becomes 



, V ' ~ & . 50 V ~ 9 v 

 4- 40 



Assuming v to be equal to f of the mean value of V, we find by substitution, 



V' = ttV, 



and, as the resistance overcome is now equal to that overcome with the common wheel at the 

 velocity V, it follows that the former requires about -fa more power than the latter to produce 

 the same effect. 



We repeat that the above calculation is not rigorous, but it is obvious that whatever 

 probable values are substituted for a, <, and v, V' will always be found greater than V, which 

 is a sufficient indication of the inferiority of this wheel. 



It is superfluous to add, that the same effect would be produced, without altering the velocity 

 of the wheels, by increasing their breadth in the ratio of cos. a? to 1, but that would nearly 

 double the weight of the wheels under the circumstances assumed above. 



It is evident that, whichever method be adopted, the shock cannot be very much reduced, 

 and, the loss of power from oblique action being greater than with the common wheel, the 

 oblique floats are not likely to supersede the ordinary ones. 



We shall now briefly notice a modification of this wheel invented by Mr. Jacob Perkins, 

 which we will therefore call Perkins's paddle wheel. 



4. Perkins's Paddle Wheel. 



Mr. Perkins took out a patent for this wheel in the year 1829: it differs materially from 

 all others, although the floats are fixed, as in the preceding, at an angle with the shaft ; but 

 that angle is essentially 45, and the shafts, instead of traversing the vessel in the usual 

 manner, are carried in a sloping direction towards the stern, and meet in the plane of the keel, 

 making with it an angle of 45, and with each other a right angle. On the extremities of the 



