ON THE LEAD OF THE SLIDE.

When a slide has lap and lead on both the steam and exhausting sides.

Let ab and ac represent the double lap on the steam side, af and ag the same on the exhausting side, be and cd the steam ports, and the line ed both the travel of the slide and the stroke of the piston; then, supposing ch to represent the lead of the slide, ai will be the position of the eccentric when that of the crank is a e, the piston being at the top of its downward stroke.

When the eccentric reaches the point k, the port c d will be fully closed and the piston will have descended to 1, the arc em being equal to the arc ik. Again, when the eccentric

arrives at n, exhaustion commences from above the piston, which has descended to o, the arc emp being equal to the arc ikn. When the eccentric arrives at q, the port be begins to open for the admission of steam beneath the piston, which has descended to r, the arc ems being equal to the ark ikq. When the eccentric has reached the point

i, opposite to i, the port be will be open to the extent of the lead b'h' equal to ch, and the piston will have completed its descent.

Steam continues to enter the port be during the ascent of the piston until the eccentric reaches the point k', when the port be will be reclosed, the direction of the slide's motion being downward and the piston having ascended to l'. Exhaustion ceases from above the piston when the eccentric reaches the point t, the piston being then at u. When the eccentric reaches the point n', opposite to n, exhaustion commences below the piston. Finally, when the eccentric reaches the point q', and the crank the point s', opposite to s, steam begins to enter the port cd, for the return stroke, at the commencement of which the port cd will be open to the extent of the lead ch, the crank and eccentric occupying their original positions a e and a i.

It is here shown that four distinct circumstances result from the use of a slide having lap on both sides of the port, with lead, during a single stroke of the piston. These are

1. The cutting off the steam for the purpose of expansion.

2. The cessation of the exhaustion on the exhaustion side.

3. The commencement of exhaustion on the steam side.

4. The readmission of steam for the return stroke.

With regard to the first of the results, we found the steam port cd closed when the crank and eccentric had described the equal arcs em and idk. Now cd, the steam port, is the versed sine of dk; and hd, the steam port minus the lead, is the versed sine of id.

Now, to reduce the versed sine of an arc from any given radius to radius unity, we must divide the versed sine to the arc by the radius of the arc, since the versed sines of arcs are as the radii of these arcs. Hence we have the first rule at page 53.

Exhaustion was shown to cease during the ascent of the

L

piston, when the eccentric had reached the point t, and the crank the point x, the crank having described the arc dkx equal to i dt described by the eccentric.

Now, ie is equal to arc the second, and et is equal to 90° minus tt, or the arc whose versed sine is ef; and ef is half the slide's travel minus the lap on the exhaust side. Hence the rule at the bottom of page 53, observing always that the versed sines are proportional to their radii. Exhaustion was shown to commence from above the piston, when the crank and eccentric had described the equal arcs ek'p and idn.

Now, idn is equal to 180° minus ni, ni' is equal to n'i, and n'd is equal to arc the third. Hence the first rule at page 54. Steam was found to be readmitted for the return stroke when the piston had reached the point r in its descent, the crank and eccentric having described the equal arcs ek's and idq. Now, idq is equal to 180° minus q, being diametrically opposed to i. And qi is equal to iq', the difference between arcs the first and second. Hence the second rule at page 54.

ON PADDLE WHEELS*.

The difference of the curves during the lower part of the motion, amounts nearly to what is due to an arc described with a radius equal to the difference of the extreme radius of the wheel, and that of the circle of equal velocity with the ship.

I have considered from this cause, that the resistance, on any part of the float, varies nearly as the square of its distance from the rolling circle; and having at the same time taken into consideration the greater length of time of the action of the extremity than of the inner edge of the paddle, I find, from the examination of several experiments, that in the case of slight immersions the as

* From Mr. Barlow's Paper in the Philosophical Transactions, vol. cxxiv. p. 315.

sumption of the resistance on any point varying as the cube of the distance from the rolling circle, and in deep immersions as the 25 power, will be a sufficiently near approximation for the present purpose.

Having thus assumed the ratio of resistance with respect to the radius, we readily find the position of the centre of pressure by the following equation.

Let pAr be the difference of the radius of the rolling circle and that of the wheel, n the power of the resistance in relation to the radius, AB=b_the depth of the paddle, Ah=x any variable distance from its upper edge, hk = dx; AC = y being the distance of the mean centre of pressure from the upper edge of the paddle: then

f(r + x)". dx

will be the sum of all the resistances, and

(r + y)". b

the expression to which it is to be equal. We have therefore, when

A

A striking difference is observable between the ratio of the resistance of the paddle in a vertical position to the power of the engine in the common wheels and in the new ones; the former being (in the Table) 157 and 193

with the large and small boats, and the latter 546. This difference arises from the nature of their action. In the new wheels the vertical position is the most effective in propelling the vessel, and in the common wheels it is least so, as may be proved in the following manner :

Let AB be the position of the paddle-rod of a vessel in motion, V being the velocity of the wheel, and v that of the ship, and the angle of inclination of the paddle-rod with a vertical line; let CD represent the velocity V at right angles to the paddle, and ĒC that of the vessel in a

horizontal direction. Then it is evident that CF, which is the resultant of these velocities, will represent the velocity and direction of motion of the paddle with respect to still water.

Resolve FC into the two velocities FG, CG, one at right angles to, and the other in the direction of the paddle, of which the latter is lost, while the former will represent the velocity with which the paddle meets the water in a direction at right angles to its face; then CG or HF EF — EH = V — v. cos. Consequently v. cos.)2, will represent the whole resistance which the paddle opposes to the engine at any angle 4.

(V