120 



APPENDIX. 



made by the engines is stated to have been 21, the speed of the vessel 11*7 statute miles per 

 hour, and the greatest immersion of the floats 2 ft. 6 in., or exactly their own depth. L L 

 (Fig. 2) is the water line, C the centre of the shaft, D D D the circumference of the rolling 

 circle, A B the float, and C A the radius, at the commencement of a revolution of the wheel ; 

 C C 1 is the line described by the centre during one revolution, which is equal to the distance 

 traversed by the vessel during the same time; D 1 D 1 D 1 is the position of the rolling circle, 

 and A" B" that of the float at the end of the revolution. The curtate cycloid A A 1 A 11 is the 

 line described by the outer edge of the float, and B B 1 B 11 that described by its inner edge; 

 and the various positions of the float are shown at intervals of one sixteenth of a revolution. 

 In order to render the motion of the float through the water more distinct, the node has been 

 drawn on an enlarged scale in Fig. 3, where Aa 1 A 11 is a part of the curve described by the 

 outer edge of the float, and B6 1 B 11 a part of that described by its inner edge, L L the water 

 line, a b the position of the float at the instant of entering the water, a b its position soon 

 after the immersion of the lower edge, being the end of one of the intervals mentioned above, 

 a 1 b 1 at the end of the next interval, a 1 6 1 in the middle of the stroke, and so on, a 11 6" being 

 its position at the instant of leaving the water. Figs. 4 and 5 relate to the performance of 

 the ' Salamander' at the mile trial, when the greatest immersion of the floats is stated to have 

 been 5ft. 6 in., the number of revolutions 15, and the speed 8'15 statute miles per hour. 



It has already been observed, that every point of a common paddle wheel situated on the 

 circumference of the rolling circle describes a common cycloid : let C A in the annexed 

 diagram represent a radius of the wheel, intersecting the circumference of the rolling circle at 

 the point B, and let D E be the vertical diameter of the rolling circle. On C D as a diameter 

 describe a circle intersecting the radius C A at the point F. The arc F D of this circle is 

 equal to the arc B D of the rolling circle, being the measure of twice the angle measured by 

 the latter in a circle of half the radius ; therefore if the two circles E B D, C F D, roll simul- 

 taneously along the horizontal straight line H K, so as to preserve the same relative positions, 

 the points B and F will coincide at some point G on that line, and will describe two cycloidal 



arcs, to the latter of which the radius C A, or its production on the opposite side of the 

 centre, will always be tangent at its intersection F with the circumference C F D ; this point 



