It is possible to employ the same apparatus at constant volume as well as at constant pressure, but the manipulation is not quite so simple, in consequence of the change of pressure. Instead of removing mercury from the overflow bulb M in connection with the thermometric bulb, mercury is introduced from a higher level into the standard bulb S so as to raise its pressure to equality with that of T at constant volume. The equations of this method are precisely the same as those already given, except that w now sig nifies the "inflow" weight introduced into the bulb S, instead of the overflow weight from M. It is necessary, however, to take account of the pressure-coefficient of the bulb T, and it is much more important to have the masses of gas on the two sides of the apparatus equal than in the other case. The thermometric scale obtained in this method differs slightly from the scale of the manometric method, on account of the deviation of the gas com pressed at o° C from Boyle's law, but it is easy to take account of this with certainty.
Another use to which the same apparatus may be put is the accurate comparison of the scales of two different gases at con stant volume by a differential method. It is usual to effect this comparison indirectly, by comparing the gas thermometers sep arately with a mercury thermometer, or other secondary standard. But by using a pair of bulbs like M and S simultaneously in the same bath, and measuring the small difference of pressure with an oil-gauge, a higher order of accuracy may be attained in the measurement of the small differences than by the method of indi rect comparison.
Expansion Correction.—In the use of the mercury ther mometer we are content to overlook the modification of the scale due to the expansion of the envelope, which is known as Poggendorff's correction, or rather to include it in the scale cor rection. In the case of the gas thermometer it is necessary to determine the expansion correction separately, as our object is to arrive at the closest approximation possible to the absolute scale. It is a common mistake to imagine that if the rate of ex pansion of the bulb were uniform, the scale of the apparent ex pansion of the gas would be the same as the scale of the real expansion—in other words, that the correction for the expansion of the bulb would affect the value of the coefficient of expansion only, and would be without effect on the value of the tem perature t deduced. A result of this kind would be produced by a constant error in the initial pressure on the manometric method, or by a constant error in the initial volume on the volumetric method, or by a constant error in the fundamental interval on any method, but not by a constant error in the coefficient of expansion of the bulb, which would produce a modification of the scale exactly analogous to Poggendorff's correction. The correction to be
applied to the value of t in any case to allow for any systematic error or variation in the data is easily found by differentiating the formula for t with respect to the variable considered. Another method, which is in some respects more instructive, is the following:— Let T be the function of the temperature which is taken as the basis of the scale considered, then we have the value of t given by the general formula (r), already quoted in § 3. Let dT be the correction to be added to the observed value of T to allow for any systematic change or error in the measurement of any of the data on which the value of T depends, and let dt be the cone sponding correction produced in the value of t, then substituting in formula (r) we have, from which, provided that the variations considered are small, we obtain the following general expression for the correction to t, dt=(dT—dT0)— — dT.) oo. (ro) It is frequently simpler to estimate the correction in this manner, rather than by differentiating the general formula.
In the special case of the gas thermometer the value of T is given by the formula where p is the observed pressure at any temperature t, V the volume of the thermometric bulb, and M the mass of gas remain ing in the bulb. The quantity M cannot be directly observed, but is deduced by subtracting from the whole mass of gas M. con• tained in the apparatus the mass M2 which is contained in the dead space and overflow bulb. In applying these formulae to deduce the effect of the expansion of the bulb, we observe that if dV is the expansion from o° C, and V. the volume at o° C, we may write whence we obtain approximately dT (12) If the coefficient of expansion (3' the bulb is constant and equal to the fundamental coefficient f (the mean coefficient between o° and ioo° C), we have simply dV/V.= ft; and if we substitute this value in the general expression (14) for dt, we obtain dt= (T— TO ft= ft(t— loo). (i3) Provided that the correction can be expressed as a rational in tegral function of t, it is evident that it must contain the factors t and since by hypothesis the scale must be correct at the fixed points o° and ioo° C, and the correction must vanish at these points. It is clear from the above that the scale of the gas thermometer is not independent of the expansion of the bulb even in the simple case where the coefficient is constant. The correction is by no means unimportant. In the case of an average glass or platinum reservoir, for which f may be taken as 0-000025 nearly, the correction amounts to —0-0625° at 50° C, to 3.83° at 445° C, and to at 1,000° C.