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III. Discussion of Formula for Error.

The plus sign signifies that the effect of increasing a variable in (1) is to make P greater, and a minus sign means of course an opposite result. That is, if in observing & we have recorded its value too great by 4, then by using this too great value in (1) to compute the percentage of priming we obtain too large a percentage. This is evidently as it should be for by using a too large we charge the steam with more heat than it should possess and hence compute too great a value for the priming. On the other hand if we record W too large we credit the steam with having imparted more heat to the calorimeter than it actually delivers and thereby we compute too little priming; hence the sign of 4Pw is opposite to that of AP Similarly 4P1 and 4Pt2 are of opposite signs.

Since 4PH and 4P depend upon the same error of observation namely, that of the steam pressure we may put their sum equal to AP, p representing the pressure of steam. Now an inspection of a table of the properties of steam shows that for any given difference of steam pressure the corresponding variation of his about three times that of H. Hence, we may put 4h=34H and combine (10) and (11) to give

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To discuss the relative importance of these partial errors we must know something of the relative magnitude of the several variables. The latter are controlled by the equation

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Throughout the range of steam tables covering pressures from atmosphere to 210 pounds per square inch, H varies only from 1178 to 1230 thermal units. The maximum value of Q may be taken at 200 thermal units equivalent to 25% of priming, or about 400° Fahrenheit superheating; t is limited to temperatures not lower than 40°, as water at 32° quickly becomes 40° in being handled in a boiler room, tą + c should not be greater than 125°, as the losses by evaporation above this temperature are too great to be properly controlled.

If we assume (t2 + c) s2 to exceed tis1 by 5°, 10°, 30°, 60° and 85° respectively, we shall have, by (13), the possible values of

follows in round numbers:

W

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TABLE I.

W

SHOWING POSSIBLE VALUES OF FOR INITIAL TEMPERATURE 40° FAHRENHEIT.

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W

For any value of in Table I. we may have an infinite num

ber of values for either W or ∞ by assigning a value to W and determining to correspond.

Hence for any given steam pressure or state of superheating or priming, as for example for dry steam at 210 pounds pressure, we may have a range of values of W and ∞ like that in the following table:

TABLE II.

SHOWING POSSIBLE VALUES FOR W AND ∞ FOR DRY STEAM OF 210 LBS. PRESS.

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As the steam pressure is lower or the per cent. of priming is greater the extreme values of ∞ in the above table will be greater. Similarly for superheating the extreme values of o will be less. But the same range of intermediate values will occur for all pressures. We will therefore obtain an idea of the relative values of the several partial errors for all conditions of steam, by applying to formulæ (6) to (12) the range of values of ∞ in Table W II. for the greatest and least values of

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To this end let the values of the several errors of observation be as follows:

Ap=3 pounds per square inch, which, at steam pressures in the neighborhood of 100 pounds per square inch, makes

AH = about 1 thermal unit, and

Ah = about three thermal units.

4t1 = 0.1 degree Fahrenheit.

4t2 = 0.2 degree Fahrenheit, as this includes the errors of both

tą and c.

Aw=

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pound = 4W.

Then by substitution in formulæ (6) to (12) we have results as per Tables III. and IV.

24

TABLE III.

SHOWING VALUES OF PARTIAL ERRORS FOR CASE OF DRY STEAM AT 210 LBS. PRESSURE.

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SHOWING PARTIAL ERRORS FOR DRY STEAM AT 210 LBS. PRESSURE.

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For other conditions of pressure, superheating or priming, such as are covered in Table I., the figures for 4P, would increase or decrease by about 0.07 units and those for 4P by about 25 per cent. of their own value. The other partial errors would not sensibly alter.

We may therefore draw conclusions as follows from a study of Tables II., III. and IV.

1st. For constant errors of observations.

Equal to lbs. for weights.

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"3 lbs. per square inch, for steam pressure.

The discrepancy in percentage of priming calculations due to instrumental errors is less, as the range of temperature of the con

W

densing water is greater and the ratio of less.

W

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particular value of the error is less the greater w.

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And for any

2d. For the same conditions of constant error the total instrumental error of a calorimeter employing about 200 pounds of condensing water and condensing from 5 to 15 pounds of steam (case of the ordinary barrel calorimeter) is as follows for dry steam:

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If the total error is represented by + 1, the proportions of the several errors average as follows:

Error due weight of condensed steam is + 1.0

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weight of condensing water"

For excessive superheating to 400° Fahr., or

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The error due to weight of condensing water changes 25% of itself, and error due steam pressure makes an accompanying change of 25% itself, but the other errors do not sensibly vary.

It is evident from inspection of Tables III. and IV. that for weights of condensing water less than 50 pounds, such amounts of error of weights as have been assumed are inadmissible and quite

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