150 APPENDIX. 



lower edge of the float, and 6 the inclination of the radius when it enters the water ; we shall 

 have 



The moment of resistance due to the float in any given position is found by substituting, 

 in its former value, the depth of the float /, for the immersion R cos. <f> a, which gives for 



its mean value 



6 

 Tr 2 Rn 3 w b f f r 



-27060^ J [#-?***. 



o 

 and, after integrating and substituting for cos. 6 its value, we get 



which, multiplied by 2 m and reduced to horse powers, becomes 



The moment of effective resistance is found in the same manner as above ; and if we call z 

 the distance of the centre of propelling effect from the lower edge of the float, and the 

 inclination of the radius when that point enters the water, its mean value is 







7i 3 an s wbff* rTt -11 j i 



-- ^ / \R cos. <f> a] 2 , ad), 

 27000 x2ffj 

 o 



which, after integrating and substituting for cos. f its value, --_, becomes 



This gives for the number of horse powers effective, or propelling effect, 



H.P.E.= 7r [P? + 2a*]{-[SRa-Rz]sm.{ ....(4) 



14850000 x 2ff I L / 



We regret not having the time to construct tables for these two points, although they 

 would be more interesting than useful, as Buchanan's wheel is not used ; we will, however, 

 apply the two first equations to a case that will serve in a measure as a comparison of this 

 with the common wheel. 



Let it be required to find the breadth of the floats, and the power of the engines necessary 

 to give the ' Salamander' the same speed under the same circumstances, as on the trial with 

 the common wheels alluded to in a former part of this paper : the extreme diameter of the 

 wheels, the depth of float, and the height of the shaft above the water, being the same ; the 

 radius of the rolling circle, however, being limited by the condition explained above, the 

 number of revolutions is thereby fixed. All the necessary data will be found at page 127. 



The condition that the same effect is to be produced, requires that the value of H. P. E. 

 should be equal to 88-23, as found, page 130, for the common wheel ; and, b being the only 

 quantity not yet determined, its value may be found by substituting the value of H. P. E. in 

 the equation (2), which we will now proceed to solve with the assistance of logarithms. 



