the method. The absolute value of the specific heat deduced necessarily
depends on the absolute values of the electrical standards
employed in the investigation. But for the determination of relative
values of specific heats in terms of a standard liquid, or of the variations
of specific heat of a liquid, the method depends only on the
constancy of the standards, which can be readily and accurately
tested. The absolute value of the E.M.F. of the Clark cells employed
was determined with a special form of electrodynamometer
(Callendar, *Phil. Trans.* A. 313, p. 81), and found to be 1.4334 volts,
assuming the ohm to be correct. Assuming this value, the result
found by this method for the specific heat of water at 20° C. agrees
with that of Rowland within the probable limits of error.

§ 15. *Variation of Specific Heat of Water.*—The question of the
variation of the specific heat of water has a peculiar interest and
importance in connexion with the choice of a thermal unit. Many
of the uncertainties in the reduction of older experiments, such as
those of Regnault, arise from uncertainty in regard to the unit in
terms of which they are expressed, which again depends on the scale
of the particular thermometer employed in the investigation. The
first experiments of any value were those of Regnault in 1847 on the
specific heat of water between 110° C. and 192° C. They were conducted
on a very large scale by the method of mixture, but showed
discrepancies of the order of 0.5%, and the calculated results in many
cases do not agree with the data. This may be due merely to deficient
explanation of details of tabulation. We may probably take
the tabulated values as showing correctly the rate of variation
between 110° and 190° C., but the values in terms of any particular
thermal unit must remain uncertain to at least 0.5% owing to the
uncertainties of the thermometry. Regnault himself adopted the
formula,

s = 1 + 0.00004t + 0.0000009t^{2} (Regnault), (3)

for the specific heat s at any temperature t C. in terms of the specific heat at 0° C. taken as the standard. This formula has since been very generally applied over the whole range 0° to 200° C., but the experiments could not in reality give any information with regard to the specific heat at temperatures below 100° C. The linear formula proposed by J. Bosscha from an independent reduction of Regnault's experiments is probably within the limits of accuracy between 100° and 200° C., so far as the mean rate of variation is concerned, but the absolute values require reduction. It may be written—

s = S_{100} + .00023(t − 100) (Bosscha-Regnault) (4).

The work of L. Pfaundler and H. Platter, of G. A. Hirn, of J. C.
Jamin and Amaury, and of many other experimentalists who succeeded
Regnault, appeared to indicate much larger rates of increase
than he had found, but there can be little doubt that the
discrepancies of their results, which often exceeded 5%, were due
to lack of appreciation of the difficulties of calorimetric measurements.
The work of Rowland by the mechanical method was the
first in which due attention was paid to the thermometry and to
the reduction of the results to the absolute scale of temperature.
The agreement of his corrected results with those of Griffiths by
a very different method, left very little doubt with regard to the
rate of diminution of the specific heat of water at 20° C. The work
of A. Bartoli and E. Stracciati by the method of mixture between
0° and 30° C., though their curve is otherwise similar to Rowland's,
had appeared to indicate a minimum at 20° C., followed by a rapid
rise. This lowering of the minimum was probably due to some
constant errors inherent in their method of experiment. The more
recent work of Lüdin, 1895, under the direction of Prof. J. Pernet,
extended from 0° to 100° C., and appears to have attained as high
a degree of excellence as it is possible to reach by the employment of
mercury thermometers in conjunction with the method of mixture.
His results, exhibited in fig. 6, show a minimum at 25° C., and a
maximum at 87° C., the values being .9935 and 1.0075 respectively
in terms of the mean specific heat between 0° and 100° C. He paid
great attention to the thermometry, and the discrepancies of individual
measurements at any one point nowhere exceed 0.3%, but
he did not vary the conditions of the experiments materially, and it
does not appear that the well-known constant errors of the method
could have been completely eliminated by the devices which he
adopted. The rapid rise from 25° to 75° may be due to radiation
error from the hot water supply, and the subsequent fall of the
curve to the inevitable loss of heat by evaporation of the boiling
water on its way to the calorimeter. It must be observed, however,
that there is another grave difficulty in the accurate determination
of the specific heat of water near 100° C. by this method, namely, that
the quantity actually observed is not the specific heat *at* the higher
temperature t, but the *mean specific heat* over the range 18° to t.
The specific heat itself can be deduced only by differentiating the
curve of observation, which greatly increases the uncertainty. The
peculiar advantage of the electric method of Callendar and Barnes,
already referred to, is that the specific heat itself is determined over
a range of 8° to 10° at each point, by adding accurately measured
quantities of heat to the water at the desired temperature in an
isothermal enclosure, under perfectly steady conditions, without
any possibility of evaporation or loss of heat in transference. These
experiments, which have been extended by Barnes over the whole
range 0° to 100°, agree very well with Rowland and Griffiths in the
rate of variation at 20° C., but show a rather flat minimum of specific
heat in the neighbourhood of 38° to 40° C. At higher points the rate
of variation is very similar to that of Regnault's curve, but taking the
specific heat at 20° as the standard of reference, the actual values
are nearly 0.56% less than Regnault's. It appears probable that
his values for higher temperatures may be adopted with this reduction,
which is further confirmed by the results of Reynolds and
Moorby, and by those of Lüdin. According to the electric method,
the whole range of variation of the specific heat between 10° and
80° is only 0.5%. Comparatively simple formulae, therefore, suffice
for its expression to 1 in 10,000, which is beyond the limits of accuracy
of the observations. It is more convenient in practice to use a few
simple formulae, than to attempt to represent the whole range by a
single complicated expression:—

Below 20° C. s = 0.9982 + 0.0000045(t − 40)^{2} − 0.0000005(t − 20)^{3}.
From 20° to 60°, s = 0.9982 + 0.0000045(t − 40)^{2} (5).

Above 60° to 200°{ | s = 0.9944 + .00004t + 0.0000009t^{2} (Regnault corrd.) |

s = 1.000 + 0.00022(t − 60) (Bosscha corrd.) |

The addition of the cubic term below 20° is intended to represent the somewhat more rapid change near the freezing-point. This effect is probably due, as suggested by Rowland, to the presence of a certain proportion of ice molecules in the liquid, which is also no doubt the cause of the anomalous expansion. Above 60° C. Regnault's formula is adopted, the absolute values being simply diminished by a constant quantity 0.0056 to allow for the probable errors of his thermometry. Above 100° C., and for approximate work generally, the simpler formula of Bosscha, similarly corrected, is probably adequate.

The following table of values, calculated from these formulae,
is taken from the *Brit. Assoc. Report,* 1899, with a slight modification
to allow for the increase in the specific heat below 20° C. This was
estimated in 1899 as being equivalent to the addition of the constant
quantity 0.20 to the values of the total heat h of the liquid as
reckoned by the parabolic formula (5). This quantity is now, as the
result of further experiments, added to the values of h, and also represented
in the formula for the specific heat itself by the cubic term.

Specific Heat of Water in Terms of Unit at 20° C. 4.180 Joules

t° C. | Joules. | s. | h | Rowland. |

0° | 4.208 | 1.0094 | 0 | 0 |

5° | 4.202 | 1.0054 | 5.037 | 5.037 |

10° | 4.191 | 1.0027 | 10.056 | 10.058 |

15° | 4.184 | 1.0011 | 15.065 | 15.068 |

20° | 4.180 | 1.0000 | 20.068 | 20.071 |

25° | 4.177 | 0.9992 | 25.065 | 25.067 |

30° | 4.175 | 0.9987 | 30.060 | 30.057 |

35° | 4.173 | 0.9983 | 35.052 | 35.053 |

40° | 4.173 | 0.9982 | 40.044 | |

50° | 4.175 | 0.9987 | 50.028 | |

60° | 4.180 | 1.0000 | 60.020 | |

70° | 4.187 | 1.0016 | 70.028 | |

80° | 4.194 | 1.0033 | 80.052 | |

90° | 4.202 | 1.0053 | 90.095 | Shaw |

100° | 4.211 | 1.0074 | 100.158 | Regnault |

120° | 4.231 | 1.0121 | 120.35 | 120.73 |

140° | 4.254 | 1.0176 | 140.65 | 140.88 |

160° | 4.280 | 1.0238 | 161.07 | 161.20 |

180° | 4.309 | 1.0308 | 181.62 | 182.14 |

200° | 4.341 | 1.0384 | 202.33 | |

220° | 4.376 | 1.0467 | 223.20 |

The unit of comparison in the following table is taken as the specific heat of water at 20° C. for the reasons given below. This unit is taken as being 4.180 joules per gramme-degree-centigrade on the scale of the platinum thermometer, corrected to the absolute scale as explained in the article Thermometry, which has been shown to be practically equivalent to the hydrogen scale. The value 4.180 joules at 20° C. is the mean between Rowland's corrected result 4.181 and the value 4.179, deduced from the experiments of Reynolds and Moorby on the assumption that the ratio of the mean specific heat 0° to 100° to that at 20° is 1.0016, as given by the formulae representing the results of Callendar and Barnes. This would indicate that Rowland's corrected values should, if anything, be lowered. In any case the value of the mechanical equivalent is uncertain to at least 1 in 2000.

The mean specific heat, over any range of temperature, may be
obtained by integrating the formulae between the limits required,
or by taking the difference of the corresponding values of the total
heat *h*, and dividing by the range of temperature. The quantity
actually observed by Rowland was the total heat. It may be remarked
that starting from the same value at 5°, for the sake of
comparison, Rowland's values of the total heat agree to 1 in 5000
with those calculated from the formulae. The values of the total
heat observed by Regnault, as reduced by Shaw, also show a very
fair agreement, considering the uncertainty of the units. It must
be admitted that it is desirable to redetermine the variation of the
specific heat above 100° C. This is very difficult on account of the

steam-pressure, and could not easily be accomplished by the electrical