| c = | 1 | + | 4 — 105 | t | + | 9 — 107 | t2 |
in which t denotes the temperature according to the Centigrade scale. According to this law, the mechanical equivalent would not be 0.2 of a foot pound greater at 5° C. (41° F.) than at 0° C. (32° F.); hence, if this law were correct, it would make no practical difference whether the temperature were at 0° C. or 5° C. This law makes the computed value at 32° F. about 0.95 of a foot pound less than that determined by experiment at 60° F.; whereas Rowland's experiments make it greater at 40° F. by more than four foot pounds, for the air thermometer. In determining a fixed value to be used for scientific purposes, it is necessary to fix the place, the thermometer, and the particular degree on the thermometer. The place may be known by its latitude if reduced to the level of the sea. The air thermometer agrees most nearly with that of the ideally perfect gas thermometer, while the mercurial thermometer differs very much from it in some cases. Thus, Regnault found that when the air thermometer indicated 630° F. above the melting point of ice (or 662° F.), the mercurial thermometer indicated 651.9° above the same point (683.9° F.), a difference of 22° F. It is apparent that the air thermometer furnishes the best standard. As for the particular degree on the scale to be used for the standard, it is apparent, from the observations above made, that the temperature corresponding to that at or near the maximum density of water is more desirable than that at the melting point of ice. The fact, also, that the specific heats at constant pressure and at constant volume are the same at the point of maximum density, as shown by theory, is an additional argument in favor of selecting this point for the standard. It thus appears that the solution of this problem, which appears simple and very definite by Mr. Hanssen's method, becomes intricate and, to a limited degree, indeterminate when subjected to the refinements of direct experiment. If the constants used by Hanssen are absolutely correct, then his result must be unquestioned; but since physical constants are subject to certain residual errors, one would as soon think of finding the specific heat of air at constant volume, by using the value of the mechanical equivalent as one of the elements, and trusting the result, as he would to trust to the computed value of the mechanical equivalent without subjecting it to the test of a direct experiment. We will, therefore, examine the constants used to see if they are the exact values of the quantities they represent.
He says they are universally accepted as correct; and this may be true, when used for general purposes, and yet not be scientifically exact. He uses 0.2377 as the specific heat of air. This is the value, to four decimals, found by Regnault. Thus, Regnault gives for the mean value of the specific heat of air
| Between —30° C | and + | 10° C. | 0.23771 |
| " 0° C | " | 100° C. | 0.23741 |
| " 0° C | " | 200° C. | 0.23751 |
And we know of no reason why one of these values should be used rather than another, except that the mean of a large range of temperatures may be more nearly correct than that of any other; and if this reason determines our choice, the number 0.2375 would be used instead of 0.2377. Although this difference is small, yet the former value would have reduced his result about 0.7 of a foot pound.
Again, he uses 0.1686 for the specific heat of air at constant volume. The value of this constant has never been found to any degree of accuracy by direct experiment, and we are still dependent upon the method established by La Place and Poisson, according to which the constant ratio of the specific heat of a gas at constant pressure to that at constant volume is found by means of the velocity of sound in the gas. The value of the ratio for air, as found in the days of La Place, was 1.41, and we have 0.2377 ÷ 1.41 = 0.1686, the value used by Clausius, Hanssen, and many others. But this ratio is not definitely known. Rankine in his later writings used 1.408, and Tait in a recent work gives 1.404, while some experiments give less than 1.4, and others more than 1.41.
An error of one foot in a thousand in determining the velocity of sound will affect the third decimal figure one or two units. A small difference in the assumed weight of a cubic foot of air also affects the result. M. Hanssen gives 0.080743 pound as the weight at 32° F. under the pressure of one atmosphere; while Rankine gives 0.080728 pound. In my own computations I use 1.406 as a more probable value of the constant sought. This will give for the specific heat of air at constant pressure
0.2375 ÷ 1.406 = 0.1689
This is only 0.0003 of a unit greater than the value used by Hanssen, but it would have given him nearly 775, instead of 771.89.
Again, he uses 491.4° F. for the absolute temperature of melting ice. The exact value of this constant is unknown; but the mean value as determined by Joule and Thomson, in their celebrated experiments with porous plugs, was 492.66° F. This value would slightly change his result. It will be seen from the above that a small change in the constants used may affect by several units the computed value of the mechanical equivalent. I have computed it, using 1.406 for the ratio of the specific heat of air at constant pressure to that at constant volume, 491.13° F. as the temperature of melting ice above the zero of the air thermometer, 26,214 feet for the height of a homogeneous atmosphere, and 0.2375 for the specific heat of air, and I find, by means of these constants, 778. If computed from the zero of the absolute scale, 492.66° F., I find 777 to the nearest integer. Recently I have used 778. If the value given by Rowland, about 783 according to the air thermometer at 39° F., should prove to be correct, it seems probable that the constant 1.406 used above would be reduced to about 1.403, or that the other constants must be changed by a small amount. The height of the homogeneous atmosphere used above, 26,214 feet, is the value used by Rankine as deduced from Regnault's figures, and only one foot less than the value used by Sir William Thomson; but the figures used by Mr. Hanssen give 26,210½ feet.