is called the rolling circle, and the resistance due to the difference of velocity of the rolling circle and centre of pressure is that which operates in the propulsion of the vessel.  The resistance upon any part of the float, therefore, will vary as the square of its distance from the rolling circle, supposing the float to be totally immerged; but, taking into account the greater length of time during which the extremity of the paddle acts, whereby the resistance will be made greater, we shall not err far in estimating the resistance upon any point at the third power of its distance from the rolling circle in the case of light immersions, and the 2.5 power in the case of deep immersions.

554. Q.—­How is the position of the centre of pressure to be determined?

A.—­With the foregoing assumption, which accords sufficiently with experiment to justify its acceptation, the position of the centre of pressure may be found by the following rule:—­from the radius of the wheel substract the radius of the rolling circle; to the remainder add the depth of the paddle board, and divide the fourth power of the sum by four times the depth; from the cube root of the quotient subtract the difference between the radii of the wheel and rolling circle, and the remainder will be the distance of the centre of pressure from the upper edge of the paddle.

555. Q.—­How do you find the diameter of the rolling circle?

A.—­The diameter of the rolling circle is very easily found, for we have only to divide 5,280 times the number of miles per hour, by 60 times the number of strokes per minute, to get an expression for the circumference of the rolling circle, or the following rule may be adopted:—­divide 88 times the speed of the vessel in statute miles per hour, by 3.1416 times the number of strokes per minute; the quotient will be the diameter in feet of the rolling circle.  The diameter of the circle in which the centre of pressure moves or the effective diameter of the wheel being known, and also the diameter of the rolling circle, we at once find the excess of the velocity of the wheel over the vessel.

556. Q.—­Will you illustrate these rules by an example?

A.—­A steam vessel of moderately good shape, and with engines of 200 horses power, realises, with 22 strokes per minute, a speed of 10.62 miles per hour.  To find the diameter of the rolling circle, we have 88 times 10.62, equal to 934.66, and 22 times 3.1416, equal to 69.1152; then 934.66 divided by 69.1152 is equal to 13.52 feet, which is the diameter of the rolling circle.  The diameter of the wheel is 19 ft. 4 in., so that the diameter of the rolling circle is about 2/3ds of the diameter of the wheel, and this is a frequent proportion.  The depth of the paddle board is 2 feet, and the difference between the diameters of the wheel and rolling circle will be 5.8133, which will make the difference of their