upon motion remain the same as before; that is, the same

α

whose length is and whose √2'

position coincides with CF. Now, referring to equation (324), it is apparent that this condition will be satisfied if, in that equation, the ambiguous sign of (P2+W) be suppressed, and the value of P, in the second member, which is multiplied by p1 sin. 1, be assumed 0; by which assumption the term p1 sin. 1 will be made to disappear from the left-hand member of equation (325), and the ambiguous signs which affect the first and second terms of the right-hand member will become positive. Now, these substitutions being made, and the equation being then integrated, first, between the limits 0 and

3π

4

and then between the limits and , the symbol U, in 4 it will evidently represent the work done during each of those portions of a semi-revolution of the imaginary arm in which the two real arms of the crank are not on the same side of the centre. Moreover, the integral of that equation between the limits 0 and, is evidently the same with its

Зп

integral between the limits 37 and 7. Taking, therefore,

4

twice the former integral, we have

a

π

2P,4,{(1-cos.),sin. p,} = {4, + p, sin. f, } 2U,

COS.

Φι

Dividing this equation by (a2+ p1 sin. 41), or or by a (1+sin.), and neglecting terms above the first di

mension in sin. 4, and sin. 21

+Wp, sin. 4, F2W, {(1-cos.) (1-sinf)-, sin. 4, };

1

a

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

π

2P, {a(√2—1)—a (√2−1)o¦ sin. ¢,—, sin. 4.}=4U, +Wwp, sin. ? ̧•

2

Adding this equation to equation (331), and representing by U2 the entire work yielded during a complete revolution of the imaginary arm,

2

Π

2P, {a√/2—a(√2—1) sin. c,— (2, sin. ç2+p, sin.,)}=U2+Wπp, sin. §,— 1

But if U, represent the whole work done by the driving pressures at each revolution of the imaginary arm, then

a

a

4a1⁄2 P1=U1. Since 2 is the projection of the space

√2

√2

described by the extremity of the arm during the ascending and descending strokes respectively, therefore 2P1=

Substituting this value for 2P,,

√2-101

v,{1– √2

U1

a√2

sin.in.+sin.)}=U2+Wzp, sin., . . · · (332),

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. O̟, and sin. P2

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.*

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 P2 in the following 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 P1. Then is

E

R

RS inclined to a perpendicular to the direction of the guides,

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

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, P1, P2, R are pressures in equilibrium,

P1,

Let BCD=0; limiting angle of resistance of guide

π

=; therefore, P1CS=3—4, P2CS=¦¿ +¢—0;

sin. (3-)

P.

cos.

2

(334).

Let BD=a, CD=b, and DBC=0,, and assume P2 to remain constant, P being made to vary according to the conditions of the state bordering upon motion;

· · AU1=P1 . AAT=-P1. ABC=-P1. A(a cos. 01+b cos. 0)=Pg sec. cos. (0—4)(a sin. 6, A01 +b sin. (A6); AU2=—P2(AAC) cos. =P2(a sin. A1 + b sin.§A¤) cos. ;

0

cos.(0-4)do}.

... U1--P, sec. 4 (a/sin. 6, cos. (0-4)de,+bsin. cos. (0

=P2{assin. 0, cos. 9d9, +bfsin. 9 cos. 9dê}.

گولا

0

0

The second integral in each of these equations vanishes

0=

between the prescribed limits; also sin. = sin. 01; there

U‚=P ̧a sec. ¿f sin. 0, cos. (0—4)d',=P ̧af sin. 6, cos. ed', +Patan. sin. O sin. 0,d9,

265. The angular velocity of the fly-wheel.

Let P1 be taken to represent a constant pressure applied

1

2

through the connecting rod to the arm of the crank of a 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 fly-wheel and the working points of the machine.