By De Volson Wood, Professor of Engineering in Stevens Institute of Technology.

It is clearly intimated by Mr. Hanssen, in his determination of the mechanical equivalent of heat, published in the Scientific American Supplement, No. 642, April 21, 1888, that his object is to determine the absolute value of this constant. With his data he finds it to be 771.89 foot pounds. But the determination by direct experiment gives a larger value. Thus, the most reliable experiments - those of Joule and Rowland - give values exceeding by several units that found by Hanssen. A committee of the British Association, appointed for this purpose, reported in 1876 that sixty of the most reliable of Joule's experiments gave the mean value 774.1. The experiments were made with water at a temperature of about 60° F., according to the mercurial thermometer, and reduced to its value at the temperature of melting ice, according to the formula given by Regnault for the variation of the specific heat of water at varying temperature under the constant pressure of one atmosphere. According to this formula the specific heat of water increases with the temperature above the melting point of ice, so that the equivalent would be somewhat less at 32° F. than at 60° F. It will be found in Regnault's Relation des Experiences that he experimented on water at high temperatures, but more recently Professor Rowland has found that the specific heat of water is greater at 40° F. than at 60° F., thus reversing between these limits the law given by Regnault; the increase, as given by the most probable values, being, roughly, about 1/ of its value at 60° F. The proper correction due to this cause would make the equivalent over 777 foot pounds, instead of 774.1. Professor Rowland's experiments, when reduced to the same thermometer, same temperature, and same latitude as Joule's, agreed very nearly with those of the latter, being about 1/ part larger; so that the chief difference in the ultimate values consists in the reductions for temperature and latitude. The force of gravity being less for the lower latitudes, the number representing the mechanical equivalent will be greater for the latter, since the unit pound mass must fall through a greater number of feet to equal the same work; so that the equivalent will be greater at Paris than at Manchester. Professor Rowland also found that the degrees on the air thermometer from 40° F. upward to above 60° F. exceeded those on the mercurial thermometer throughout the corresponding range, and that from 40° to 41° the degree was between 1/ and 1/ of a degree larger on the air thermometer than on the mercurial.

Although this fraction is too small to be observed by ordinary means, yet, if it exists, it cannot be ignored if absolute values are sought. Regnault employed the air thermometer in his experiments, while Joule used the mercurial thermometer, and if Joule's value 774.1 be increased by 1/ of itself in order to reduce it from the equivalent of the degree on the mercurial thermometer to that on the air thermometer, we get 778 foot pounds, nearly. Rowland found from his experiments that when reduced to the air thermometer and to the latitude of Baltimore, the equivalent was nearly 783, subject to small residual errors.

Nearly all writers upon this subject - except Rankine - have considered that the mechanical equivalent of heat, in British units, was the energy necessary to raise the temperature of one pound of water from 32° F. to 33° F., but Rankine defines it as the heat necessary to increase the temperature of one pound of water one degree Fahrenheit from that of maximum density, or from 39° F. to 40° F. For ordinary practice it is immaterial which of these definitions is used, for the errors resulting therefrom are much less than those resulting from ordinary observations. But when the value is to be determined by direct experiment at the standard temperature, Rankine's limits are much to be preferred; for it is so very difficult to determine exact values by observation when the substance is near the state bordering on a change of state of aggregation, as that of changing from water to ice. Observations made at about 60° F. were reduced by means of Regnault's law for the specific heat of water, as has been stated, which is expressed by the formula

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

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

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.

The method above called Hanssen's is really that of Dr. Mayer (the German professor), who in 1842 used it for determining the mechanical equivalent; but on account of erroneous data, the value found by him was much too small.