level of the mercury in the short tube. When this point, 35′′, is reached, the mercury in the short tube will be found to stand at 5". The air in the short tube has thus been subjected to an additional pressure of 30" of mercury, i.e., to an additional pressure of one atmosphere; therefore, its pressure has been doubled. Before applying this pressure it occupied 10" of the tube; hence we see that its volume has been reduced by one-half by doubling the pressure on it, in accordance with the law just stated. It is important that the student should not overlook the fact, that this law is true, only when the temperature remains constant.

Since the pressure of an enclosed perfect gas kept at a constant temperature varies inversely as its volume, and since the density or weight per unit volume of the same, varies inversely as its volume, it follows that the pressure varies directly as the density. This law is not perfectly fulfilled by any actual gas, but very nearly so by those gases which cannot be condensed into liquids, such as air. When a gas is about to pass by condensation into a liquid (e.g., steam on the point of being transformed into water), then the density increases more rapidly than the

pressure.

Watt, however, assumed that Boyle's law held good in the case of steam, and he applied it in a most ingenious manner to prove the economy of the expansive working of steam in a cylinder, and to show that he could get a greater amount of work from the steam by cutting it off early in the stroke, and thus allowing it to force the piston forward during the remainder of the stroke, merely by expansion.

Watt's Diagram of Work.-Although, as we shall see later on, steam does not expand in strict accordance with Boyle's law (for the temperature of the steam falls the more it is expanded, unless external heat is applied to it, to make up for the loss due to the work got out of it), yet we shall gain a great insight into the action of steam in an engine cylinder, by first discussing "Watt's Diagram of Work done during Expansion," and then applying the corrections that have since been found necessary, in order to truthfully represent the actual state of matters.

The following figure will illustrate to the student the method adopted by Watt. The horizontal line, or abscissa, A B, indicates the length of the stroke, and is divided into 10 equal parts; the vertical line, or ordinate, A C, represents the pressure of steam used by Watt, say one atmosphere, and is also divided into 10, or decimal parts of an atmosphere of pressure. When the piston has travelled the distance, CD, i.e., or of the stroke, the steam is cut off, and the remainder of the stroke is effected by

the expansive action of the steam. The gradually falling curve, DE, marked "curve of pressures," is found by drawing verticals from each of the divisions of the stroke, 3, 4, 10, and marking them off in height corresponding to the pressures, p, at these points by the following formula, and joining their upper ends by a curved line:

Where v =

the volume swept out by the piston at the several points, and is, therefore, represented by the different distances, . . 10, from the commencement of the stroke.

2, 3,

For example

At point of cut off

2, p

1

End of stroke,

10, p =

==

0.2

Dividing by the Number of Parts, viz.,
We get roughly a Mean Pressure

⚫This mean pressure is less than the true mean as explained at pp. 82 and 137

10 4.85

=

485

=

By adding the several pressures, and dividing them by the number of divisions taken-viz., 10-we get the average pressure throughout the stroke, 485 of an atmosphere, or nearly half an atmosphere. The economy of cutting off the steam before the end of the stroke will, therefore, be at once apparent, for we have obtained an average pressure equal to nearly half that which would have been obtained by carrying full steam pressure throughout the whole stroke, and have only used of the quantity of steam.

Since work done is measured by force or pressure, multiplied by the distance through which the force or pressure acts, the area of the rectangle, A D (see upper part of Fig., p. 81), being equal to the pressure, A O, if reckoned in lbs., multiplied by the distance, A 2, or, O D in feet, measures to scale the work done upon the piston by the steam up to the point of cut-off in footpounds or units of work. In the same way, the area of the rest of the figure-viz., D E B 2, measures to scale the work done upon the piston by the steam while expanding in the cylinder, also in foot-pounds; for this area is equal to the mean pressure in lbs. between the points, D and E, multiplied by the distance, 2 B, in feet. Consequently, the area of the whole figure, ACDE B, measures to scale the whole work done by the steam in one stroke in foot-pounds. This area is equal to the calculated mean pressure throughout the stroke, multiplied by the whole stroke, A B, and expresses the result of Watt's diagram of work. Watt, in calculating the mean pressure throughout the stroke, assumed that the pressure at each of the points into which he divided the stroke commencing with number 1, remained constant until it arrived at the next in order, by which method he obtained a less value than the true mean, because he omitted to take into account the ordinate of pressure at the point, A, or the very commencement of the stroke. we now take into account the first ordinate at A, as well as the last one at 10, we have the following eleven pressures:-1, 1, 1, ·6, ·5, ·4, ·3, ·29, ·25, 2, and 2, giving a total sum of 5.86, which sum being divided by the number of ordinates, viz., 11, gives us a mean of 532 of an atmosphere instead of 485, or nearly 8 lbs. pressure on the square inch, which is a nearer approximation to the true mean.

If

Let us take another example of Watt's diagram of work, taking the first as well as the last pressure ordinate into account, in order to get a nearer approximation to the true mean. Suppose we have an engine using steam of 100 lbs. pressure per square inch, and cutting off at of the stroke, to find the curve of expansion by Boyle's Law and the mean pressure.

Dividing by the Number of Points, viz.,

We get an approximate Mean Pressure

11 | 657-2

= 59.7 lbs.

There are several rules for obtaining approximately the mean pressure from a diagram of work such as we have been discussing. The plan most commonly adopted by engineers (as we shall see at pp. 127 and 137) in finding the mean pressure from actual indicator diagrams is, to measure by a suitable scale or rule the length of each of the ten ordinates, taken at the centre of each of the ten spaces into which the diagram is divided, add them together, and divide by their number.

For instance, applying this rule to the last example, we should measure the length of the vertical lines midway between the points 0 and 1, 1 and 2, 2

and 3,

9 and 10, add these ten pressure ordinates

together, and divide the sum by 10, to get the mean pressure; and doing so (or calculating these pressures by pv constant), we find them to be respectively, 100, 100, 100, 71·43, 55·5, 45.45, 38-46, 33-3, 29-41, and 26-31 lbs., giving a mean of 59-9 lbs., or slightly greater than that found above.

Simpson's Rule is as follows:-Divide the length of the figure into n equal parts, n being an even number, and draw ordinates through the points of division to touch the boundary lines. Add together the first and the last ordinates, call the sum A; add together the even ordinates 2, 4, 6, &c., call the sum B; add together the odd ordinates 3, 5, 7, &c., except the A+4 B+2 C first and the last, and call the sum C; then = mean ordinate 3 n or pressure. This quantity multiplied by the length, L, of the figure gives the area of the figure, or what we would call the area of work in this case.

Methods of Constructing the Curve of Pressures and Volumes by Boyle's Law.-We shall now show how to construct the curve for the relation between pressure and volume of a perfect gas expanding according to Boyle's law. This curve may be constructed in two different ways:

1. By making use of the formula expressing Boyle's law-viz., pv= = a constant, and thus calculating the pressure at various points during the expansion.

2. Or, we may adopt a purely graphical method for determining a series of points on the curve. The curve of expansion can then be drawn freehand or by aid of French curves, or by bending a thin flexible strip of wood until its lower edge passes through the several points. These two methods will be clearly understood from the solution of the following example.

EXAMPLE I.-Steam is admitted into the cylinder of an engine at a pressure of 30 lbs. by gauge, and is cut off at of the stroke. Draw to scale the diagram of work done during admission and expansion, assuming that the steam expands according to Boyle's law. From the diagram thus constructed, find the pressures at,, and of the stroke respectively.

ANSWER. First Method, by Calculation.

--

Draw two axes O P, O V, at right angles to each other. Along O P, measure off a distance O A, to represent the initial absolute pressure of the steam.*

The initial pressure as given by the question is 30 lbs. by gauge. The pressure as indicated by a steam gauge on a boiler or cylinder of an engine, has for its starting (or zero) point, the pressure of the atmosphere-viz., about 15 lbs. per square inch. We cannot, therefore, base our calculations respecting a law of nature on such an arbitrary and variable starting-point as this. Consequently, we must refer all our pressures to the absolute zero or perfect vacuum line before applying Boyle's law. The absolute zero is