a

} =

2P,{(1-cos.) (1-2 sin..), sin., 2U,+

a

a

4

1

)—

Wp, sin. 4, F2W,(1-cos.) (1— sin. 9.)

in which equation 2U, represents the work done in the descending or ascending arcs of the imaginary arm, according as the ambiguous sign is taken positively or negatively. Taking, therefore, the sum of the two values of the equation. given by the ambiguous sign, and representing by 4U, the whole work done in the descending and ascending arcs, during those portions of each complete revolution when both of the arms are not on the same side of the centre, we have

a

Фа

4P, {(1-cos.) (1-2 sin. 9. )—p, sin. ç, } =

Φι

2P, { a(√2—1)—a( √2—1) A sin. 9. — 2 P, sin. 9, }

4U,+Wap, sin. ..

=

Adding this equation to equation (331), and representing by U, the entire work yielded during a complete revolution of the imaginary arm,

2P, { a √2 — a( √2 — 1) o; sin. 9. — (2p, sin.q,+p, sin.e,)}

But if U, represent the whole work done by the driving pressures at each revolution of the imaginary arm, then 4. P, U,. Since 2

described by the extremity of the arm during the ascending and descending strokes respectively, therefore 2P,= Substituting this value for 2P,,

a √2

which is the modulus of the double crank, the directions of the driving pressure and the resistance being both supposed vertical; or if the friction resulting from the weight of the crank be neglected, and W be therefore assumed 0, then does the above equation represent the modulus of the double crank, whatever may be the direction of the driving pressure, provided that the direction of the resistance be parallel to it. Dividing by the coefficient of U1, and neglecting terms of more than one dimension in sin., and

sin. Pa

264. In some of the most important applications of the steam engine, the crank is made to receive its continuous rotatory motion, from the alternating rectilinear motion of the piston rod, directly through the connecting rod of the crank, without the intervention of the beam or parallel motion; the connecting rod being in this case jointed at one extremity, to the extremity of the piston rod, and the oblique pressure upon it which results from this connexion being sustained by the intervention of a cross piece fixed upon it, and moving between lateral guides.*

* This contrivance is that well known as applied to the locomotive carriage.

Let the length CD of the connecting rod be represented by b, and that BD of the crank arm by a, and let P, and P, in the above figure be taken respectively, to represent the pressure upon the piston rod of the engine and the connecting rod of the crank, and RS to represent the direction of the resistance of the guide in the state bordering upon motion by the excess of the driving pressure P. Then is RS inclined to a perpendicular to the direction of the guides, or of the motion of the piston rod, at an angle equal to the limiting angle of resistance (Art. 141) of the surfaces of contact of the guides.

Since, moreover, P,, P., R are pressures in equilibrium,

Let BCD=0; limiting angle of resistance of guide =; therefore, P,CS= -9, P ̧CS=2+9—8;

T

T

2

Let BDa, CD=b, and DBC=4,, and assume P, to remain constant, P, being made to vary according to the conditions of the state bordering upon motion;

..4U,=P1.4AC=—P1.4BC=—P, . 4 (a cos. 8, +b cos. 4)=

1

P, sec. cos. (q) (a sin. 6, A8, +b sin. 0^0);

2

AU,=-P,(ABC) cos. 8-P, (a sin. 8,48, +b sin. eae) cos. ;

2

π

..U,P,sec. {asin. 4, cos.(—)dê, +b sin. cos.(ê —q)d3}.

si

U‚=P, {a / sin. §, cos. êdê, +b sin. è cos. êdê}

The second integral in each of these

equations vanishes

a

between the prescribed limits; also sin. sin. 4,; there

=sin.
b

:: U‚=P‚a ƒ sin. § ̧ cos. êd3 ̧=P,af (1 — % sin. %, )' sin. 0,d3 ̧ =

4,

U1=P,a sec. q fsin. 6, cos. (-4)de,P,af sin. §, cos. êdî, +

265. The angular velocity of the fly-wheel.

Let P, be taken to represent a constant pressure applied through the connecting rod to the arm of the crank of a

Church's Diff. and Int. Cal. Art. 199.

steam engine; suppose the direction of this pressure to remain always parallel to itself, and let P, represent a constant resistance opposed to the revolution of the axis which carries the fly-wheel, by the useful work done and the prejudicial resistances interposed between the axis of the Hy-wheel and the working points of the machine.

Let the angle ACB=8, CB=a, CP,=a,.

Now the projection, upon the direction of P,, of the path of its point of application B to the crank arm, whilst that arm describes the angle ACB, is AM, therefore (Art. 52.), the work done by P, upon the crank, whilst this angle is described, is represented by P,. AM, or by P, a vers. . And whilst the crank arm revolves through the angle, the resistance P, is overcome through the arc of a circle subtended by the same angle 4, but whose radius is a,, or through a space represented by a,. So that, neglecting the friction of the crank itself, the work expended upon the resistances opposed to its motion is represented by P,a,, and the excess of the work done upon it through the angle ACB by the moving power, over that expended during the same period upon the resistances, is represented by

Now 2aP, represents the work done by the moving pressure P, during each effective stroke of the piston, and 2a,P, the work expended upon the resistance during each revolution of the fly-wheel; so that if m represent the number of strokes made by the piston whilst the fly-wheel makes one