If a = 0, or p=bv", as in most approximate formulas for expansion curves, the result simplifies, by cancelling out b from numerator

These formulas, which are all practically useful, give the work done during expansion in terms of the ratios between the initial and final volumes, and of the initial and final pressures; also in terms of the initial product pv and of the final product pv. The latter formula is most useful in the case of air and gas compression pumps where the initial and known volume and pressure are v2P2. The "admission" part of the indicator diagram has an area P11, and this has to be added to the above, giving

These calculations do not take account of the back pressure deduction from the area of the card.

The "

mean pressure,” measured from zero pressure, is the last

value of W divided by v2, or

From this the back pressure must be subtracted to obtain the "effective" mean pressure.

In the case of isothermal gas expansion, n integration for work done during expansion is

1 or pvb, and the

P1

V2

P2

V1

111. Graphic Construction for Indicator Diagrams.-In fig. 25 the upper curve is a common hyperbola or curve of reciprocals,

and is the gas isothermal. The lower is drawn to the formula p=bv-12. The product pv is the same at all points of the upper curve, and, therefore, at all points equals p,v,. Therefore for the

point 2 on the lower curve, the horizontal strip of area rafined over equals (P-P22); and this divided by n-1='2, i.e., multiplied by 5, equals the work done under the lower curve during the expansion from 1 to 2.

The mean pressure, including the admission period, therefore, equals 5 times the height of the strip rafined over plus the height to the upper edge of the same strip.

The gradient of the curve p-bu-" is negative, and equals

=

sign which only indicates that the forward slope is downwards.

But if T be the subtangent, then p'. Therefore we find

v

1

T

=

T

n = and T' n-1 v T cards taken from engines or compressing pumps, at each point of the expansion curve at which a fair tangent can be accurately drawn, the value of the index n can be found by measuring the ratio of v to T. Also in finding the mean pressure by adding to the height of the upper edge of the rafined strip of fig. 25 the depth of this strip divided by (n-1), this division can be performed very easily by an evident graphic construction, since

Thus in investigating actual indicator

=

T

n-1 v-T

T

=

n

Conversely, in constructing theoretical indicator diagrams, when a few points of the curve have been calculated, it much assists in the fair drawing in of the curve to draw the tangents at these points, which can easily be done by setting off for each point If an oblique line be drawn at a tangent of inclination n to the vertical axis (it is drawn dotted in fig. 25), then at each v the height of this line will give the corresponding T. In fact, by this construction the whole curve may be accurately drawn out from point to point by drawing a connected chain of short tangents whose direction is at each point obtained in this way; the accuracy of the construction being very considerable if care be taken that each short tangent length shall stretch equally behind and in front of the point at which its direction is found by plotting T. By this construction the labour of logarithmic calculation of the heights of a series of points is rendered unnecessary.

112. Sin-1 and (2x2)--In § 74 it was found that the anglegradient of a sine.is the cosine, and that of the cosine minus the

E

sine. That is, if a be the angle and s its sine, or sin as; then since cos2a = 1-s2, we have

When the angle is measured by its sine it is symbolically ex pressed as sin1s="the angle whose sine is s." Using this notation, and taking the reciprocal of the above; i.e., taking the cogradient or the "sine-gradient of the angle," we have

The corresponding integrations are, when x instead of s is used to indicate the variable,

The two angles having the same fraction for sine and cosine respectively are complementary; so that these two forms of the

πα

integral only differ in the integration constants (C'-C=T2) and

in the sign of the variable parts.

The sine of an angle cannot be greater than +1 nor less than – 1. These integration formule would have, therefore, no meaning in cases in which x>r or x< -r. These limits correspond with those within which (2 – x2) remains real, because the square root of a negative quantity is "impossible" or "imaginary." If (2-2) arises from any actual physical problem, such a problem can never throughout the whole actual range of make

x>r.

113. (1-2) integrated or Area of Circular Zone.-In § 59 it was shown that the area of a sector of a circle equals r2a. The

* See Classified List, III. B. 6 and 5.