particular crank giving the piston velocity corresponding to that particular crank position.

It is simple to prove, as in the position curve, that the curve consists of two circles, but on OB and OB' now as diameters, instead of OA and OA'.

to

The piston velocity V, then varies from

Zero at A and A'

V, at B and B',

and is found for any intermediate crank position OP by measuring OQ on the particular velocity scale previously described.

The inverse proposition, to find the crank position for a given piston velocity, is equally simple. For take OQ on the velocity scale to represent the given velocity, and then describe a circle with centre O, radius OQ, which will cut the curve in four points, either of which joined to O gives a crank position fulfilling the required condition.

The curve just drawn gives us crank positions, and if piston positions be either data or required results, we should have first to find the crank position corresponding to the given datum, and then deduce the velocity or piston position required.

This process is simple enough, but we can render it even simpler by drawing a curve, once for all, which shall give velocities corresponding to given piston positions, and vice versa.

To do this we must proceed thus:—

Taking any piston position N (Fig. 126), set up NP perpendicular to AA', and along NP mark off the piston velocity corresponding to the piston position N. But since OP is the corresponding crank, this piston velocity is Vo sin 0, i.e. NP itself; and thus the curve drawn through the points so obtained is the crank circle itself. We have then two velocity curves :

One in which intercepts are measured from a point O, or a pole; hence called the Polar curve.

And one in which intercepts are measured from a base line; hence called a Linear curve.

2d. Let V, be not constant. — We cannot now represent V, by a line of constant length. But the work by which we obtained the velocity ratio did not depend on the constancy or otherwise of Vo, and hence we still have

Velocity ratio of piston to crank=sin 0,

and if we take such a scale that OP represents unity, OR (Fig. 126) will represent sin 0.

We can then still draw the curve as usual, but OQ now does not give the velocity of the piston, but only the ratio of its velocity to that of the crank on the scale just explained.

The preceding is not of very great practical importance, since V, is in most cases so nearly uniform that we may without sensible error assume it to be exactly so.

Mean Piston Velocity.-Referring to chap. i. (page 22) we have for this simply

[Stroke meaning either the backward or forward movement, but not the whole.]

But Vo, besides being the crank speed, is also the maximum piston speed,

... Ratio of maximum to mean piston speed=π : 2.

Actual Motion with Obliquity.—We must now see what results we obtain in the actual motion, taking account of the obliquity, and we can also see how they compare with our preceding simple ones.

To obtain the relation between Vp and V。, we cannot proceed quite so simply as we did before.

Drawing the usual figure, we have C moving towards O with velocity Vp, and P perpendicular to OP with velocity Vo.

Now there is a connection between V, and Vo, because P and C are connected together, and the relation which exists between them depends on the particular

B

a

A

Fig. 127.

method of connection. We must then examine into the connection, and when we do so, we see that its essential characteristic is that C and P cannot approach or recede from each other, the distance CP being constant.

From the preceding it follows that if, at any instant, C have a velocity in the direction CP, P must have the same velocity in that direction. Because if it were not

so, then P would be either approaching to or receding from C; and this is not affected by either C or P, or both, having in addition velocities at right angles to CP, because no finite velocity at right angles to CP can be at the instant altering the length of CP. There may be a little difficulty experienced here, one cause perhaps

being that we speak of velocity along CP, while CP is a rod which is continually changing its direction; this difficulty can be removed if we are careful to remember that the whole question relates to an instantaneous state of affairs, and that the direction CP does not mean the solid rod, because of course neither C nor P can move along the rod, but it means the line along which the rod at the instant lies, and it is a perfectly definite direction, in spite of the fact that, at the next instant, the rod will be along some other line. Some aid to understanding the point can perhaps be got from splitting the instantaneous motion of the rod into two parts; the first a bodily translation along CP, in which plainly C and P have the same velocity; and the second a turning of the rod, so as to keep C up to the line of stroke, and P down on the crank circle, this latter consisting only of motions of C and P at right angles to CP. We have then

The total velocity of C is Vp along CO, which

resolves into

and

Vp cos along CP,

V, sin perpendicular to CP.

The total velocity of P is V, at right angles to OP, which resolves into

V, cos a along CP produced,

V, sin a perpendicular to CP (Fig. 127).

We have seen that the two first components must be

equal,

.. Vp cos =V, cos a.

Let now CP produced cut OB in T. Then

cos sin OTP,

cos asin TPO,

.. V, V. sin TPO: sin OTP,

=OT: OP,

which gives us the velocity ratio, and if V, be constant

Fig. 128.

and we take OP as before to represent it, then OT represents Vp.

We can now proceed as in the preceding to draw a curve of velocity, or of velocity ratio, by taking OQ along OP, equal to OT, and doing this for a large number of cranks, draw a curve finally through all the Q's. The detailed construction is shown in Fig. 128. We explained very fully in the preceding case, and so we need not again do so. The crank circle is divided in I, 2, etc.; then II, 22, etc. are the connecting rods, which being produced cut OB in T1, T2, etc. Then we make OIOT1, OIIOT,, etc., and finally draw the curve through I, II, etc., the lower half being symmetrical with the upper. The