ON THE MECHANICAL EQUIVALENT OF HEAT, WITH SUBSIDIARY RESEARCHES ON THE VARIATION OF THE MERCURIAL FROM THE AIR THERMOMETER, AND ON THE VARIATION OF THE SPECIFIC HEAT OF WATER. BY HENRY A. ROWLAND, PROFESSOR OF PHYSIOS IN THE JOHNS HOPKINS UNIVERSITY. Presented June llth, 1879. [REPRINTED FROM THE PROCEEDINGS OF THE AMERICAN ACADEMY OF ARTS AND SCIENCES.] CAMBRIDGE: UNIVERSITY PRESS: JOHN WILSON & SON. 1880. OP ARTS AND SCIENCES. INVESTIGATIONS ON LIGHT AND HEAT, made and published wholly or in part with appropriation from the KUMFOUD FUND. Geol. 75 Lib. QC V. ON THE MECHANICAL EQUIVALENT OF HEAT, WITH SUB- SIDIARY RESEARCHES ON THE VARIATION OF THE MERCURIAL FROM THE AIR THERMOMETER, AND ON THE VARIATION OF THE SPECIFIC HEAT OF WATER. BY HENRY A. ROWLAND,* Professor of Physics in the Johns Hopkins University. Presented June llth, 1879. CONTENTS. I. Introductory Remarks ... . Thermometry .... (a.) General View of Thermome- try (6.) The Mercurial Thermometer 78 (c.) Relation of the Mercurial and Air Thermometers . . 83 1. General and Historical Remarks . . 88 2. Description of Apparatus 90 3. Results of Comparison . 97 (d.) Reduction to the Absolute Scale .... 112 Appendix to Thermometry . . 116 III. Calorimetry ..... 119 (a.) Specific Heat of Water . . 119 (6.) Heat Capacity of the Calo- rimeter .... 131 IV. Determination of Equivalent . . 137 (a.) Historical Remarks . . 137 1. General Review of Meth- ods .... 137 2. Results of Best Deter- minations . . .140 (6.) Description of Apparatus . 165 1. Preliminary Remarks . 155 2. General Description . 157 3. Details . . . .168 (c. ) Theory of the Experiment . 168 1. Estimation of Work done 163 2. Radiation . . . .168 3. Corrections to Thermom- eters, etc. ... in (d.) Results 173 1. Constant Data . . 173 2. Experimental Data and Tables of Results . .174 V. Concluding Remarks, and Criticism of Results and Methods . . 197 I. INTRODUCTORY REMARKS. AMONG the more important constants of nature, the ratio of the heat unit to the unit of mechanical work stands forth prominent, and is used almost daily by the physicist. Yet, when we come to consider * This research was originally to have been performed in connection with Professor Pickering, but the plan was frustrated by the great distance between our residences. An appropriation for this experiment was made by the Ameri- can Academy of Arts and Sciences at Boston, from the fund which was insti- tuted by Count Rumford, and liberal aid was also given by the Trustees of the Johns Hopkins University, who are desirous, as far as they can, to promote original scientific investigation. 76 PROCEEDINGS OF THE AMERICAN ACADEMY the history of the subject carefully, we find that the only experimenter tho has made the determination with anything like the accuracy demanded by modern science, and by a method capable of giving good results, is Joule, whose determination of thirty years ago, con- firmed by some recent results, to-day stands almost, if not quite, alone among accurate results on the subject. But Joule experimented on water of one temperature only, ai did not reduce his results to the air thermometer; so that we are still left in doubt, even to the extent of one per cent, as to the value equivalent on the air thermometer. The reduction of the mercurial to the air thermometer, and thence to the absolute scale, has generally been neglected between and 100 by most physicists, though it is known that they differ several tenths of a degree at the 45 point. In calorimetric researches this may produce an error of over one, and even approaching two per cent, especially when a Geissler thermometer is used, which is the worst in this respect of any that I have experimented on ; and small intervals on the mercurial thermometers differ among themselves more than one per cent from the difference of the glass used in them. Again, as water is necessarily the liquid used in calorimeters, its variation of specific heat with the temperature is a very important factor in the determination of the equivalent. Strange as it may appear, we may be said to know almost nothing about the variation of the specific heat of water with the temperature between and 100 C. Regnault experimented only above 100 C. The experiments of / Him, and of Jamin and Amaury, are absurd, from the amount of variation which they give. Pfaundler and Plattner confined them- selves to points between and 13. Munchausen seems to have made the best experiments, but they must be rejected because he did not reduce to the air thermometer. In the present series of researches, I have sought, firstly, a method of measuring temperatures on the perfect gas thermometer with an accuracy scarcely hitherto attempted, and to this end have made an extended study of the deviation of ordinary thermometers from the air thermometer ; and, secondly, I have sought a method of determin- ing the mechanical equivalent of heat so accurate, and of so extended a range, that the variation of the specific heat of water should follow from the experiments alone. As to whether or not these have been accomplished, the following pages will show. The curious result that the specific heat of water OP ARTS AND SCIENCES. 77 on the air thermometer decreases from to about 30 or 35, after which it increases, seems to be an entirely unique fact in nature, seeing that there is apparently no other substance hitherto experimented upon whose specific heat decreases on rise of temperature without change of state. From a thermodynamic point of view, however, it is of the Bame nature as the decrease of specific heat which takes place after the vaporization of a liquid. The close agreement of my result at 15.7 C. with the old result of Joule, after approximately reducing his to the air thermometer and latitude of Baltimore, and correcting the specific heat of copper, is very satisfactory to us both, as the difference is not greater than 1 in 400, and is probably less. I hope at some future time to make a comparison with Joule's thermometers, when the difference can be accurately stated. II. THERMOMETRY. (a.) General View. The science of thermometry, as ordinarily studied, is based upon the changes produced in bodies by heat. Among these we may men- tion change in volume, pressure, state of aggregation, dissociation, amount and color of light reflected, transmitted, or emitted, hardness, pyro-electric and thermo-electric properties, electric conductivity or specific induction capacity, magnetic properties, thermo-dynamic prop- erties, &c. ; and on each of these may be based a system of ther- mometry, each one of which is perfect in itself, but which differs from all the others widely. Indeed, each method may be applied to nearly all the bodies in nature, and hundreds or thousands of thermometric scales may be produced, which may be made to agree at two fixed points, such as the freezing and boiling points of water, but which will in general differ at nearly, if not all, other points. But from the way in which the science has advanced, it has come to pass that all methods of thermometry in general use to the present time have been reduced to two or three, based respectively on the apparent expansion of mercury in glass and on the absolute expan- sion of some gas, and more lately on the second law of thermo- dynamics. Each of these systems is perfectly correct in itself, and we have no right to designate either of them as incorrect. We must decide a priori on some system, and then express all our results in that system : the accuracy of science demands that there should be no 78 PROCEEDINGS OF THE AMERICAN ACADEMY ambiguity on that subject. In deciding among the three systems, we should be guided by the following rules : 1st. The system should be perfectly definite, so that the same temperature should be indicated, whatever the thermometer. 2d. The system should lead to the most simple laws in nature. ^ Sir William Thomson's absolute system of thermometry, coinciding with that based on the expansion of a perfect gas, satisfies these most nearly. The mercurial thermometer is not definite unless the kind of glass is given, and even then it may vary according to the way the bulb is blown. The gas thermometer, unless the kind of gas is given, is not definite. And, further, if the temperature. as given by either of these thermometers was introduced into the equations of thermo- dynamics, the simplest of them would immediately become compli- cated. Throughout a small range of temperature, these systems agree more or less Completely, and it is the habit even with many eminent physicists to regard them as coincident between the freezing and boil- ing points of water. We shall see, however, that the difference between them is of the highest importance in thermometry, especially where differences of temperature are to be used. For these reasons I have reduced all my measures to the absolute system. The relation between the absolute system and the system based on the expansion of gases has been determined by Joule and Thomson in their experiments on the flow of gases through porous plugs (Philosophical Transactions for 1862, p. 579). Air was one of the most important substances they experimented upon. To measure temperature on the absolute scale, we have thus only to determine the temperature on the air thermometer, and then reduce to the absolute scale. But as the air thermometer is very inconvenient to use, it is generally more convenient to use a mercurial thermometer which has been compared with the air thermometer. Also, for small changes of temperature the air thermometer is not sufficiently sensi- tive, and a mercurial thermometer is necessary for interpolation. 1 shall occupy myself first with a careful study of the mercurial thermometer. (6.) The Mercurial Thermometer. Of the two kinds of mercurial thermometers, the weight ther- mometer is of little importance to our subject. I shall therefore con- fine myself principally to that form having a graduated stem. For OF ARTS AND SCIENCES. 79 convenience in use and in calibration, the principal bulb should be elongated, and another small bulb should be blown at the top. This latter is also of the utmost importance to the accuracy of the instru- ment, and is placed there by nearly all makers of standards.* It is used to place some of the mercury in while calibrating, as well as when a high temperature is to be measured ; also, the mercury in the larger bulb can be made free from air-bubbles by its means. Most standard thermometers are graduated to degrees; but Reg- nault preferred to have his thermometers graduated to parts of equal capacity whose value was arbitrary, and others have used a single millimeter division. 'As thermometers change with age, the last two methods are the best ; and of the two I prefer the latter where the highest accuracy is desired, seeing that it leaves less to the maker and more to the scientist. The cross-section of the tube changes continu- ously from point to point, and therefore the distribution of marks on the tube should be continuous, which would involve a change of the dividing engine for each division. But as the maker divides his tube, he only changes the length of his divisions every now and then, so as to average his errors. This gives a sufficiently exact graduation for large ranges of temperature ; but for small, great errors may be intro- duced. Where there is an arbitrary scale of millimeters, I believe it is possible to calibrate the tube so that the errors shall be less than can be seen with the naked eye, and that the table found shall repre- sent very exactly the gradual variation of the tube. In the calibration of my thermometers with the millimetric scale, I have used several methods, all of which are based upon some graphical method. The first, which gives all the irregularities of the tube with great exactness, is as follows. A portion of the mercury having been put in the upper bulb, so as to leave the tube free, a column about l;) mm - long is separated off. This is moved from point to point of the tube, and its length carefully measured on the dividing engine. It is not generally necessary to move the column its own length every time, but it may be moved 20"""- or 25 nim- , a record of the position of its centre being kept. To eliminate any errors of division or of the dividing engine, readings were then taken on the scale, and the "lengths reduced to their value in scale divisions. The area of the tube at every point is inversely as the length of the column. We shall thus have a series of figures nearly equal to each other, if the tube is good. By subtracting the * Geissler and Casella omit it, which should condemn their thermometers. 80 PROCEEDINGS OP THE AMERICAN ACADEMY smallest from each of the others, and plotting the results as ordinates, with the thermometer scale as abscissas, and drawing a curve through the points so found, we have means of finding the area at any point. The curve should not he drawn exactly through the points, but rather around them, seeing they are the average areas for some distance each side of the point. With good judgment, the curve can be drawn with great accuracy. I then draw ordinates every 10 mra> , and estimate the average area of the tube for that distance, which I set down in a table. As the lengths are uniform, the volume of the tube to any point is found by adding up the areas to that point. But it would be unwise to trust such a method for very long tubes, seeing the mercury column is so short, and the columns are not end to end. Hence I use it only as supplementary to one where the column is about 50 ram - long, and is always moved its own length. This estab- lishes the volumes to a series of points about 50"' m- apart, and the other table is only used to interpolate in this one. There seems to be no practical object in using columns longer than this. Having finally constructed the arbitrary table of volumes, I then test it by reading with the eye the length of a long mercury column. No certain error was thus found at any point of any of the ther- mometers which I have used in these experiments. While measuring the column, great care must be taken to preserve all parts of the tube at a uniform temperature, and only the extreme ends must be touched with the hands, which should be covered with cloth. If V is the volume on this arbitrary scale, the temperature on the mercurial thermometer is found from the formula T =.O V < , where C and f. Q are constants to be determined. If the thermometer contains the and 100 points, we have simply G _ 100 - i o v ~ V Otherwise C is found by comparison with some other thermometer, which must be of the same kind of glass. It is to be carefully noted that the temperature on the mercurial thermometer, as I have defined it, is proportional to the apparent expansion of mercury as measured on the stem. By defining it as proportional to the true volume of mercury in the stem, we have to introduce a correction to ordinary thermometers, as Poggendorf has shown. As I ouly use the mercurial thermometer to compare with the air thermometer, and as either definition is equally correct, I will OF ARTS AND SCIENCES. 81 not further discuss the matter, but will use the first definition, as being the simplest. In the above formula I have implicitly assumed that the apparent expansion is only a function of the temperature ; but in solid bodies like glass there seems to be a progressive change in the volume as time advances, and especially after it has been heated. And hence in mercurial and alcohol thermometers, and probably in general in all thermometers which depend more or less on the expansion of solid bodies, we firyi that the reading of the thermometer depends, not only on its present temperature, but also on that to which it has been sub- jected within a short time ; so that, on heating a thermometer up to a certain temperature, it does not stand at the same point as if it had been cooled from a higher temperature to the given temperature. As these effects are without doubt due to the glass envelope, we might greatly diminish them by using thermometers filled with liquids which expand more than* mercury : there are many of these which expand six or eight times as much, and so the irregularity might be dimin- ished in this ratio. But in this case we should find that the correction for that part of the stem which was outside the vessel whose tem- perature we were determining would be increased in the same propor- tion ; and besides, as all the liquids are quite volatile, or at least wet the glass, there would be an irregularity introduced on that account. A thermometer with liquid in the bulb and mercury in the stem would obviate these inconveniences ; but even in this case the stem would have to be calibrated before the thermometer was made. By a com- parison with the air-thermometer, a proper formula could be obtained for finding the temperature. But I hardly believe that any thermometer superior to the mer- curial can at present be made, that is, any thermometer within the same compass as a mercurial thermometer, and I think that the best result for small ranges of temperature can be obtained with it by studying and avoiding all its sources of error. To judge somewhat of the laws of the change of zero within the limits of temperature which I wished to use, I took thermometer No. G1G3, which had lain in its case during four months at an average temperature of about 20 or 25 C., and observed the zero point, after heating to various temperatures, with the following result. The time of heating was only a few minutes, and the zero point was taken immediately after ; some fifteen minutes, however, being necessary for the thermometer to entirely cool. VOL. xv. (N. s. vn.) 6 82 PROCEEDINGS OF THE AMERICAN ACADEMY TABLE I. SHOWING CHANGE OF ZERO POINT. Temperature of Bulb before finding the Point. Change of Point. Temperature of Bulb before finding the Point. Change of Point. 22.5 700 .115 300 .016 81.0 .170 405 .033 90.0 .281 ' 51.0 .0:59 100.0 .313 CO.O .105 100.0 .347 The second 100 reading was taken after boiling for some time. It is seen that the zero point is always lower after heating, and that in the limits of the table the lowering of the zero is about propor- tional to the square of the increase of temperature above 25 C. This law is not true much above 100, and above a certain tempera- ture the phenomenon is reversed, and the zero point is higher after heating ; but for the given range it seems quite exact. It is not my purpose to make a complete study of this phenomenon with a view to correcting the thermometer, although this has been undertaken by others. But we see from the table that the error can- not exceed certain limits. The range of temperature which I have used in each experiment is from 20 to 30 C., and the temperature rarely rose above 40 C. The change of zero in this range only amounts to 0.03 C. The exact distribution of the error from this cause throughout the scale has never been determined, and it affects my results so little that I have not considered it worth investigating. It seems probable, how- ever, that the error is distributed throughout the scale. IE it were uniformly distributed, the value of each division would be less than before by the ratio of the lowering at zero to the temperature to which the thermometer was heated. The maximum errors produced in my thermometers by this cause would thus amount to 1 in 1300 nearly for the 40 thermometer, and to about 1 in 2000 for the others. Rather than allow for this, it is better to allow time for the thermometer to resume its original state. Only a few observations were made upon the rapidity with which the zero returned to its original position. After heating to 81, the* zero returned from OM70 to 0.148 in two hours and a half. After heating to 100, the zero returned from 0.347 to 110 in nine days, and to -0.022 in one month. Reasoning from this, I OF ARTS AND SCIENCES. 83 should say that in one week thermometers which had not been heated above 40 should be ready for use again, the error being then supposed to be less than 1 in 4000, and this would be partially eliminated by comparing with the air thermometer at the same intervals as the ther- mometer is used, or at least heating to 40 one week before comparing with the air thermometer. As stated before, when a thermometer is heated to a very high point, its zero point is raised instead of lowered, and it seems probable that at some higher point the direction of change is reversed again ; for, after the instrument comes from the maker, the zero point con- stantly rises until it may be 0.6 above the mark on the tube. This gradual change is of no importance in my experiments, as I only use differences of temperature, and also as it Was almost inappreciable in my thermometers. Another source of error in thermometers is that due to the pressure on the bulb. In determining the freezing point, large errors may be made, amounting to several hundredths of a degree, by the pressure of pieces of -ice. In my experiments, the zero point was determined in ice, and then the thermometer was immersed in the water of the com- parator at a depth of about G0 cm . The pressure of this water affected the thermometer to the extent of about 0.01, and a correction was accordingly made. As differences of temperature were only needed, no correction was made for variation in pressure of the air. It does 'not seem to me well to use thermometers with too small a stem, as I have no doubt that they are subject to much greater irreg- ularities than those with a coarse bore. For the capillary action always exerts a pressure on the bulb. Hence, when the mercury rises, the pressure is due to a rising meniscus which causes greater pressure than the falling meniscus. Hence, an apparent friction of the mercu- rial column. Also, the capillary constant of mercury seems to depend on the electric potential of its surface, which may not be constant, and would thus cause an irregularity. My own thermometers did not show any apparent action of this kind, but Pfaundler and Plattner mention such an'action, though they give another reason for it. (c.) Relation of the Mercurial and Air Thermometers. 1. GENERAL AND HISTORICAL REMARKS. Since the time of Dulong and Petit, many experiments have been made on the difference between the mercurial and the air thermometer, 84 PROCEEDINGS OF THE AMERICAN ACADEMY but unfortunately most of them have been at high temperatures. As weight thermometers have been used by some of the best experi- menters, I shall commence by proving that the weight thermometer and stem thermometer give the same temperature ; at the same time, however, obtaining a convenient formula for the comparison of the air thermometer with the mercurial. For the expansion of mercury and of glass the following formulae must hold : For mercury, V = V (1 + a t -f J * 2 + &c.) ; glass, V = V' Q (1 + a t 4- (3 1 2 4- &c.). In both the weight and stem thermometers we must have V '= V. where V and F are the volumes of the glass and of the mercury reduced to zero, and t is the temperature on the air thermometer. The temperature by the weight thermometer is P where P , P t , &c. are the weights of mercury in the bulb at C., t C., &c. Now these weights are directly as the volumes of the mercury at 0. seeing that V is constant. .-. T 100 In the stem thermometers we have F , the volume of mercury at 0, constant, and the volume of the glass that the mercury fills, reduced to 0, variable. As the volume of the glass V' n is the volume reduced to 0, it will be proportional to the volume of bulb plus the volume of the tube as read off on the scale which should be on the tube. OF ARTS AND SCIENCES. 85 At .-. r=ioo which is the same as for the weight thermometer. If the fixed points are and t* instead of and 100, we can write T _ t ,At + Bfl^-Cfl + & c . r=<{i + (<-<<) [f + fl' + fc+o] +&<>} As T and t are nearly equal, and as we shall determine the con- stants experimentally, we may write t = T a t (t 1 t) (b t) + &c., where t is the temperature on the air thermometer, and T that on the mercurial thermometer, and a and b are constants to be determined for each thermometer. The formula might be expanded still further, but I think there are few cases which it will not represent as it is. Considering b as equal to 0, a formula is obtained which has been used by others, and from which some very wrong conclusions have been drawn. In some kinds of glass there are three points which coincide with the air thermome- ter, and it requires at least an equation of the third degree to repre- sent this. The three points in which the two thermometers coincide are given by the roots of the equation t(*-t) (6-0 = 0, and are, therefore, In the following discussion of the historical results, I shall take and 100 as the fixed points. Hence, t' = 100. To obtain a and b, two observations are needed at some points at a distance from and 100. That we may get some idea of the values of the constants in the formula for different kinds of glass, I will discuss some of the experimental results of Regnault and others with this in view. 86 PROCEEDINGS OF THE AMERICAN ACADEMY Renault's results are embodied, for the most part, in tables given on p 239 of the first volume of his Relation des Esper^ces. The Lres given there are obtained from curves drawn to represent he mean of his experiments, and do not contain any theorefcal results. The direct application of my formula to his experiments could hardly be made without immense labor in finding the most probable value the constants. But the following seem to satisfy the experiments quite ' Cristal de Choisy-le-Roi i = 0, = -000 000 82. Verre Ordinaire 6-245, = .000 000 34. VerreVert * = 270, a = .000 000 095. Verrede Suede i = +10, a = .000 000 14. From these values I have calculated the following : TABLE II. REGNAULT'S RESULTS COMPARED WITH THE FORMULA. 1 Choisy-le-Roi. Verre Ordinaire. Verre Vert. Verre de Suede. 1 | 1 1 1 1 1 1 *5 i 1 | 1 I 3 1 1 5 1 1 5 | .2 1 o 5 i 5 100 o o o 120 140 160 180 200 120.12 140 29 160.52 180.80 201 25 120.09 14025 KiO.49 18083 201 .28 +.03 +.04 +.03 -.03 -.03 119.95 139. W 5 159.74 179.63 199.70 119.90 139 80 159.72 179.68 19!l.fi9 +.05 +.05 +.02 .05 + .01 12007 140.21 16(1.40 180.60 20080 120.09 .01 140 22 ! .01 160.391 +.01 180.62 i .02 200.-S9 .09 120.04 120.04 140.11 ! 140.10 160 20 i 160.21 1MI.33 18034 20 ( 50 ! 200 53 +.01 .01 .01 .03 220 221 KJ 221.86 .04 2l9.HOl219.78 + .02 221.20 221.23 .03 220 75 ! 220 78 .03 240 242.55 242.50 239.90! 239 96 -.06 241.60 241.63 -.03 241.16 2*1.08 + .08 200 263 44 203.4H .02 260.20 2C021 -.01 262.15 262.09 +.07 WO 2H4.48 2S452 -.04 280.58 2SO.O>l 02 282.85 262.63 +.22 300 3?0 3(15.72 ::27.25 3115.70 327.20 -.04 -.05 301.08! 301. 12 321.80! 321.80 .04 00 340 349.30 348.88 +.42 434.00 34204 +.36 The formula, as we see from the table, represents all Regnault's curves with great accuracy, and if we turn to his experimental results we shall find that the deviation is far within the limits of the experi- mental errors. The greatest deviation happens at 340, and may be accounted for by an error in drawing the curve, as there are few ex- perimental results so high as this, and the formula seems to agree with them almost as well as Regnault's own curve. The object of comparing the formula with Regnault's results at temperatures so much higher than I need, is simply to test the formula through as great a range of temperatures, and for as many kinds of Corrected from 280.52 in Regnault's table. OF ARTS AND SCIENCES. 87 glass, as possible. If it agrees reasonably well throughout a great range, it will probably be very accurate for a small range, provided we obtain the constants to represent that small range the best. Having obtained a formula to represent any 'series of experiments, we can hardly expect it to hold for points outside our series, or even for interpolating between experiments too far apart, as, very often, a small change in one of the constants may affect the part we have not experimented on in a very marked manner. ' Thus in applying the formula to points between and 100 the value of b will affect the result very much. In the case of the glass Choisy-le-Roi many values of b will satisfy the observations besides 6 = 0. For the ordinary glass, however, b is well determined, and the formula is of more value between and 100. The following table gives the results of the calculation. TABLE III. REGNAOI/T'S RESULTS COMPARED WITH THE FORMULA. Calculated Calculated Calculated . ft = .00000032 a = .00000031 Observed. = .00000044 Thermome- ft = 0. b = 245. A 6 = 260. J ter. Clioisy-le-Roi. Verre Ordinaire. Verre Ordinaire. Verre Ordinaire. o 10 10.00 10.07 .... 10.10 20 19.99 20.12 .... 20 17 . . . 30 29.98 30.15 30.12 +.03 30.21 +.09 40 39.97, 40.17 40.23 -.06 40.23 60 49.96 60.17 50.23 .06 60.23 60 5:>.'.>5 60.15 60.24 .09 60.21 .03 70 09.95 70.12 70.22 .10 70.18 .04 80 79.96 80.09 80.10 .01 80.11 +.01 90 89.97 90.05 90.07 100 100 100 100 ' .... 100 Regnault does not seem to have published any experiments on Choisy-le-Roi glass between and 100, but in the table between pp. 226, 227, there are some results for ordinary glass. The separate observations do not seem to have been very good, but by combining the total number of observations I have found the results given above. The numbers in the fourth column are found by taking the mean of Regnault's results for points as near the given temperature as possible. The agreement is only fair, but we must remember that the same specimens of glass were not used in this experiment as in the others, and that for these specimens the agreement is also poor above 100. The values a = .000 000 44 and b = 260 are much better PROCEEDINGS OP THE AMERICAN ACADEMY above 100 for the given specimens. The table seems to show that between and 100 a thermometer of Choisy-le-Koi almost exactly agrees with the air, thermometer But this is not at all conclusive. Regnault, however, remarks * that be- tween and 100 thermometers of this glass agree more nearly with the air thermometer than those of ordinary glass, though he stat the difference to amount to .1 to .2 of a degree, the mercurial t mometer standing below the air thermometer. With the exception of this remark of Regnault's, no experiments have ever been publi. in which the direction of the deviation was similar to tins. All ( percenters have found the mercurial thermometer to stand above the air thermometer between and 100, and my own expenments agree with this. However, no general rule for all kinds of glass can b< laid down. Boscha has given an excellent study of Regnault's results subject, though I cannot agree with all his conclusions on this subject. In discussing the difference between and 100 he uses a formula of the form and deduces from it the erroneous conclusion that the difference is greatest at 50 C., instead of between 40 and 50. His results for T t at 50 are Choisy-le-Roi ...... .22 Verre Ordinaire ...... -J-.25 VerreVert ....... +.14 Verre de Suede ...... -f- 56 and these are probably somewhat nearly correct, except the negative value for Choisy-le-Roi. With the exception of Regnault, very few observers have taken up this subject. Among these, however, we may mention Recknagel, who has made the determination for common glass between and 100. I have found approximately the constants for my formula in this case, and have calculated the values in the fourth column of the following table. * Comptes Rendus, Ixix. OF ARTS AND SCIENCES. 89 TABLE IV. RECKNAGEL'S RESULTS COMPARED WITH THE FORMULA. Air Thermometer. Mercurial Thermometer. Difference. Observed. Calculated. 10 10.08 10.08 20 20.14 20.14 30 30.18 30.18 40 40.20 40.20 60 50.20 50.20 60 60.18 60.18 70 70.14 70.15 +.01 80 80.10 80.11 +.01 90 90.05 90.06 +.01 100 100.00 b = 290 a = .000 000 33 (100 t)(b t) It will be seen that the values of the constants are not very different from those which satisfy Regnault's experiments. There seems to be no doubt, from all the experiments we have now discussed, that the point of maximum difference is not at 50, but at some less temperature, as 40 to 45, and this agrees with my own experiments, and a recent statement by Ellis in the Philosophical Magazine. And I think the discussion has proved beyond doubt that the formula is sufficiently accurate to express the difference of the mercurial and air thermometers throughout at least a range of 200, and hence is probably very accurate for the range of only 100 between and 100. Hence it is only necessary to find the constants for my thermom- eters. But before doing this it will be well to see how exact the comparison must be. As the thermometers are to be used in a calorimetric research in which differences of temperature enter, the error of the mercurial compared with the air thermometer will be which for the constants used in Recknagel's table becomes Error = ^ 1 = .000 000 33 j 29000. 780 1 -f- 3 1* I . This amounts to nearly one per cent at 0, and thence decreases to 45, after which it increases again. As only 0.2 at the 40 point 90 PROCEEDINGS OP THE AMERICAN ACADEMY produces this large error at 0, it follows that an error of only 0.02 at 40 will produce an error of TT ^ T at 0. At other points the errors will be less. Hence extreme care must be taken in the comparison and the most accurate apparatus must be constructed for the purpose. 2. DESCRIPTION or APPARATUS. The Air Thermometer. In designing the apparatus, I have have had in view the production of a uniform temperature combined with ease of reading the ther- mometers, which must be totally immersed in the water. The uni- formity, however, needed only to apply to the air thermometer and to the bulbs of the mercurial thermometer, as a slight variation in the temperature of the stems is of no consequence. A uniform tempera- ture for the air thermometer is important, because it must take time for a mass of air to heat up to a given temperature within U.01 or less. Fig. 1 gives a section of the apparatus. This consists of a large copper vessel, nickel-plated on the outside, with double walls an inch apart, and made in two parts, so that it could be put together Avater- tight along the line a b. As seen from the dimensions, it required about 28 kilogi'ammes v of water to fill it. Inside of this was the vessel m d efg h k I n, which could be separated along the line c/. In the upper part of this vessel, a piston, q, worked, and could draw the water from the vessel. The top was closed by a loose piece of metal, o p, which fell down and acted as a valve. The bottom of this inner vessel had a false bottom, c /, above which was a row of large holes ; above these was a perforated diaphragm, . The bulb of the air thermometer was at <, with the bulbs of the mercurial thermometers almost touching it. The air thermometer bulb was very much elon- gated, being about 18 cm - long and 3 to 5 cra - in diameter. Although the bulbs of the thermometers were in the inner vessel, the stems were in the outer one, and the reading was accomplished through the thick glass window u v. The change of the temperature was effected by means of a Bunsen burner under the vessel w. The working of the apparatus was as follows. The temperature having been raised to the required point, the piston q was worked to stir up the water ; this it did by drawing the water through the holes at cl and the perforated diaphragm s, and thence up through the OP ARTS AND SCIENCES. 91 apparatus to return on the outside^ When the whole of the water is at a nearly uniform temperature the stirring is stopped, the valve op falls into place, and the connection of the water in the outer and inner vessels is practically closed as far as currents are concerned, and Fig.2. before the water inside can cool a little the outer water must have cooled considerably. So effective was this arrangement that, although some of the ther- mometers read to 0.007 C., yet they would remain perfectly station- ary for several minutes, even when at 40 C. At very high tempera- tures, such as 80 or 90 C., the burner was kept under the vessel w all the time, and supplied the loss of the outer vessel by radiation. The inner vessel would under these circumstances remain at a very 92 PROCEEDINGS OP THE AMERICAN ACADEMY constant temperature. The water in the outer vessel never.differed by more than a small fraction of a degree from that in the inner one. To get the and 100 points the upper parts of the vessel above the line a b were removed, and ice placed around the bulb of the air thermometer, and left for several hours, until no further lowering took place. For the 100 point the copper vessel shown in Fig. 3 was used. The portion y of this vessel fitted directly over the bulb of the air thermometer. On boiling water in x, the steam passed through the tube to the air thermometer. It is with considerable difficulty that the 100 point is accurately reached, and, unless care be taken, the bulb will be at a slightly lower temperature. Not only must the bulb be in the steam, but the walls of the cavity must also be at 100. To accomplish this in this case, a large mass of cloth was heaped over the instrument, and then the water in x vigorously boiled for an hour or so. After fifteen minutes there was generally no perceptible in- crease of temperature, though an hour was allowed so as to make certain. The external appearance of the apparatus is seen in Fig. 2. The method of measuring the pressure was in some respects similar to that used in the air thermometer of Jolly, except that the reading was taken by a cathetometer rather than by a scale on a mirror. .The capillary stem of the air thermometer leaves the water vessel at a, and passes to the tube b, which is joined to the three-way cock c. The lower part of the cock is joined by a rubber tube to another glass tube at d, which can be raised and lowered to any extent, and has also a fine adjustment. These tubes were about 1.5 cm - diameter on the inside, so that there should be little or no error from capillarity. Both tubes were exactly of the same size, and for a similar reason. The three-way cock is used to fill the apparatus with dry air, and also to determine the capacity of the tube above a given mark. In filling the bulb, the air was pumped out about twenty times, and allowed to enter through tubes containing chloride of calcium, sul- phuric acid, and caustic soda, so as to absorb the water and the car- bonic acid. The Cathetometer. The cathetometer was one made by Meyerstein, and was selected because of the form of slide used. The support was round, and the telescope was attached to a sleeve which exactly fitted the support. The greatest error of cathetometers arises from the upright support not being exactly true, so that the telescope will not remain in level OP ARTS AND SCIENCES. 93 at all heights. It is true that the level should be constantly adjusted, but it is also true that an instrument can be made where such an ad- justment is not necessary. And where time is an element in the accuracy, such an instrument should be used. In the present case it was absolutely necessary to read as quickly as possible, so as not to leave time for' the column to change. In the first place the round column, when made, was turned in a lathe to nearly its final dimen- sions. The line joining the centres of the sections must then have been very accurately straight. In the subsequent fitting some slight irregularities must have been introduced, but they could not have been great with good workmanship.* The upright column was fixed, and the telescope moved around it by a sleeve on the other sleeve. Where the objects to be measured are not situated at a very wide angle from each other, this is a good arrangement, and has the advantage that any side of the column can be turned toward the object, and so, even if it were crooked, we could yet turn it into such a position as to nearly eliminate error. It was used at a distance of about HO"" 1 - from the object, and no difficulty was found after practice in setting it on the column to fo mm - at least. The cross hairs made an angle of 45 with the horizontal, as this was found to be the most sensitive arrangement. The scale was carefully calibrated, and the relative errors f for the * The change of level along the portion generally used did not amount to more than .1 of a division, or about .Ol mm - at the mercury column, as this is about the smallest quantity which could be observed on the level. t These amounted to less than .016 mm - at any part. 94 PROCEEDINGS OF THE AMERICAN ACADEMY denoted by the subfix. Then approximat As the height of the barometer varies only very slightly during an experiment, the value of this expression is very nearly which does not depend on the absolute value of the scale divisions But the best manner of testing a cathetometer is to take readings upon an accurate scale placed near the mercury columns to b measured. I tried this with my instrument, and found that it agree< with the scale to within two or three one-hundredths of a millimeter, which was as near as I could read on such an object. In conclusion, every care was taken to eliminate the errors of this instrument, as the possibility of such errors was constantly present in my mind; and it is supposed that the instrumental errors did not amount to more than one or two one-hundredths of a millimeter on the mercury column. The proof of this will be shown in the results obtained. The Barometer. This was of the form designed by Fortin, and was made by James Green of New York. The tube was 2.0 cm diameter nearly on the outside, and about 1.7 cm - on the inside. The correction for capillarity is therefore almost inappreciable, especially as, when it remains constant, it is exactly eliminated from the equation. The depression for this diameter is about .08 mm -, but depends upon the height of the meniscus. The height of the meniscus was generally about l.S 1 ""' ; but according as it was a rising or falling meniscus, it varied from 1.4 to 1.2 mm -. These are the practical values of the variation, and would have been greater if the barometer had not been attached to the wall a little loosely, so as to have a slight motion when handled. Also in use the instrument was slightly tapped before read- OF ARTS AND SCIENCES. 95 ing. The variation of the height of the meniscus from 1.2 to 1.4 mm - would affect the reading only to the extent of .01 to ,02 mm -. The only case where any correction for capillarity is needed is in finding the temperatures of the steam at the 100 point, and will then affect that temperature only to the extent of about 0.005. The scale of the instrument was very nearly standard at C., and was on brass. At the centre of the brass tube which surrounded the barometer, a thermometer was fixed, the bulb being surrounded by brass, and there- fore indicating the temperature of the brass tube. In order that it should also indicate the temperature of the barome- ter, the whole tube and thermometer were wrapped in cloth until a thickness of about 5 or 6 cm ' was laid over the tube, a portion being displaced to read the thermometers. This wrapping of the barometer was very important, and only poor results were obtained before its use; and this'is seen from the fact that 1 on the thermometer indi- cates a correction of ,12 mm- on the barometer, and hence makes a difference of 0.04 on the air thermometer. As this is one of the most important sources of error, I have now devised means of almost entirely eliminating it, and making continual reading of the barometer unnecessary. This I intend doing by an artificial atmosphere, consisting of a large vessel of air in ice, and attached to the open tube of the manometer of the air thermometer. The Thermometers. The standard thermometers used in my experiments are given in the following table. 96 PROCEEDINGS OP THE AMERICAN ACADEMY fl >> ffj 1 -flftrfi Owner or Lender. ( Physical Labora ( Johns Hopkins Uni Prof. Barkei Univ. of Pennsy Chemical Labor Johns Hopkins Un n ^o sS ^-s if I -siSilill ^ >> "^ ! I co 2 : 1_ i CO "3 CC & Q g a J ^ xj JS .S a ^ = - 5 .2 1 5 1 ' 1 : ^ S o 4*. 0> CO CO CN ^-i CD (N O5 OS T uation. | P^ |-i . ri d b b 2 3 a, b "2 "3 ^ C3 '0 ^ 3 CT 1 8 1 1 co o o Tli^ 2 s i H rH i-H i-H I-H C^ rH I-H i i 8> > ^ 2 + O O O CN rH . CO ^0*^ S= 2s o 1 1 1 \ \ ^_ T I ld ~-*~* a-| 49 l|| 58 58 J3 5S 5 CO CO 'o S CO CD O CO CO M* 1 S 3 o W CO CO CSJ CO CO CO 00 g ^ ^ OF ARTS AND SCIENCES. 97 The calibration of the first four thermometers has been described. The calibration of the Kew standard was almost perfect, and no cor- rection was thought necessary. The scale divided on the tube was to half-degrees Fahrenheit; but as the 32 and 212 points were not correct, it was in practice used as a thermometer with arbitrary divisions. The interval between the and 100 points, as Welsh found it, was 180.12, using barometer at 30 inches, or 180.05 as cor- rected to 760 mm - of mercury.* At the present time it is 179.68,f showing a change of 1 part in 486 in twenty-five years. This fact shows that the ordinary method of correcting for change of zero is not correct, and that the coefficient of expansion of glass changes with time4 I have not been able to find any reference to the kind of glass used in this thermometer. But in a report by Mr. Welsh we find a com- parison, made on March 19, 1852, of some of his thermometers with two other thermometers, one by Pastre", examined and approved by Regnault, and the other by Troughton and Simms. The thermometer which I used was made a little more than a year after this ; and it is reasonable to suppose that the glass was from the same source as the standards Nos. 4 and 14 there used. We also know that Regnault was consulted as to the methods, and that the apparatus for calibration was obtained under his direction. I reproduce the table here with some alterations, the principal one of which is the correction of the Troughton and Simms thermometers, so as to read correctly at 32 and 212, the calibration being assumed correct, but the divisions arbitrary. * Boiling point, Welsh, Aug. 17, 1853, 212.17 ; barometer 30* . Freezing point, " " " 82.05. Boiling point, Rowland, June 22, 1878, 2 12.46; barometer 760 mm .. Freezing point, " " " 32.78. The freezing point was taken before the boiling point in either case, t 179.70, as determined again in January, 1879. t The increase shown here is 1 in 80 nearly ! It is evidently connected with the change of zero; for when glass has been heated to 100, the mean coefficient of expansion between and 100 often changes as much, as 1 in 50. Hence it is not strange that it should change 1 in 80 in twenty-five years. I believe this fact has been noticed in the case of standards of length. VOL. xv. (N. s. vn.) 98 PROCEEDINGS OP THE AMERICAN ACADEMY TABLE VI. COMPARISON BY WELSH, 1852. Mean of Kew Standards Nos. 4 and 14. Fastre231, Kegnault. A Kew. Troughton and Simms (Royal Society). A Kew. 32.00 32.00 32.00 38!? 1 45.04 38.72 45.03 +.01 -.01 38.70 45.03 .01 .01 49.96 49.96 .00 49.96 .00 55.34 55.37 +.03 55.34 .00 60.07 60.05 .02 60.06 .01 65.39 65.41 +.02 65.36 .03 69.93 69.95 +.02 69.93 .00 74.69 74.69 .00 74.72 +.03 80.05 80.06 +.01 80.14 +.09 85.30 85.33 +.03 85.44 +.14 90.60 90.51 +.01 90.56 +.06 95.26 101.77 95.24 101.77 .02 .00 95.40 101.94 4-.14 +.15 . 109.16 109.15 .01 109.25 +.08 212.00 212.00 .00 212.00 .00 It is seeruhat the Kew standards and the Fastre agree perfectly, but that the Troughton and Simms standard stands above the Kew ther- mometers at 100 F. The Geissler standard was made by Geissler of Bonn, and its scale was on a piece of milk glass, enclosed in a tube with the stem. The calibration was fair, the greatest error being about 0.015 C., at 50 C. ; but no correction for calibration was made, as the instrument was only used as a check for the other thermometers. 3. RESULTS OF COMPARISON. Calculation of Air Thermometer. This has already been described, and it only remains to discuss the formula and constants, and the accuracy with which the different quantities must be known. The well-known formula for the air thermometer is H-h + T= l + at> _ Solving with reference to T, and placing in a more convenient form, we have H-h' nearly, OF ARTS AND SCIENCES. 99 where and For the first bulb, For the second bulb, = .0058. To discuss the error of T due to errors in the constants, we must replace a by its experimental value, seeing that it was determined with the same apparatus as that by which T was found. As it does not change very much, we may write approximately T= 100 H h V l + yt From this formula we can obtain by differentiation the error in each of the quantities, which would make an error of one tenth of one per cent in T. The values are for T = 40 nearly ; t = 20 ; ff lw h = 270""" ; and h = 750 mm -. If x is the variable, A x = dx dT ' *JL _L_ - ru d JL d T looo - u * d T TABLE VII. ERRORS PRODUCING AN ERROR IN T OF 1 IN 1000 AT 40 C. H H 100 orA. V T ha a - constant, a b a ^5 constant. 1,90 k^/con*. b ino -b a -^ constant. Absolute ralue, Ax n mm. 27 mm. .005 .00074 .00087 .0047 .00087 Relative value, Az X ... ... 0.9 .10 .12 .62 ... From this table it would seem that there should be no difficulty in determining the 40 point on the air thermometer to at least 1 in 2000 ; and experience has justified this result. The principal difficulty is in the determination of H, seeing that this includes errors in reading the barometer as well as the cathetometer. For this reason, as men- tioned before, I have designed another instrument for future use, in which the barometer is nearly dispensed with by use of an artificial atmosphere of constant pressure. The value of ^ does not seem to affect the result to any great extent; and if it was omitted altogether, the error would be only 100 PROCEEDINGS OF THE AMERICAN ACADEMY about 1 in 1,000, assuming that the temperature t was the same at the determination of the zero point, the 40 point, and the 100 point. It seldom varied much. The coefficient of expansion of the glass influences the result very slightly, especially if we know the difference of the mean coefficients between and 100, and say 10 and -f 10. This difference I at first determined from Regnault's tables, but afterwards made a deter- mination of it, and have applied the correction.* The table given by Regnault is for one specimen of glass only ; and I sought to better it by taking the expansion at 100 from the mean of the five specimens given by Regnault on p. 231 of the first volume of his Relation des Experiences, and reducing ' the numbers on page 237 in the same proportion. I thus found the values given in the second column of the following table. TABLE VIII. COEFFICIENT OF EXPANSION OF THE GLASS OF THE AIR THERMOMETER, ACCORDING TO THE AIR THERMOMETER. Tempera- ture ac- cording to Air Ther- mometer. Values of 6 used for a first Calculation b from Regnault's Table, Glass No. 5. Experimental Results. Apparent Coefficient of Expansion of Mercury. b, using Regnault's Value for Mercury .t 6, using Recknagel'g Value for Mercury 4 6, using Wullnel's Value for Mercury . 20 40 60 80 100 .0000252 .0000253 .0000256 .0000259 .0000262 .0000264 .0000263 .0000264 .0000267 .0000270 .0000273 .0000276 .00015410 .00015395 .00015391 .00015381 .0000254 .0000258 .0000261 .0000277 .0000264 .0000266 .0000267 .0000277 .0000273 .0000276 .0000278 .0000287 The second column contains the values which I have used, and one of the last three columns contains my experimental results, the last being probably the best. The errors by the use of the second column compared with the last are as follows : TTT ^ from using b m 6 40 = .0000008 instead of .0000011 ; TuW from using b m = .0000264 instead of .0000287 ; or, ffa-Q for both together. * This was determined by means of a large weight thermometer in which the mercury had been carefully boiled. The glass was from the same tube as that of the air thermometer, and they were cut from it within a few inches of each other. t Relation des Experiences, i. 328. t Pogg. Ann., cxxiii. 135. Experimental Physik, Wullner, i. 67. OP ARTS AND SCIENCES. 101 As the error is so small, I have not thought it worth while to entirely recalculate the tables, but have calculated a table of corrections as follows, and have so corrected them : TABLE IX. TABLE OP CORRECTIONS. Tf T Calculated Corrected Correction. Temperature. Temperature. 8 10 9.9971 .0029 20 19.9946 .0054 30 29.9924 0076 40 39.9907 .0093 50 49.9894 .0106 60 69.9865 .0135 80 79.9880 .0120 100 100. T = T' {1 + 373 (b' m - b m ) - (273 + T) (V - &)}. T T 1 {1 .000858 -f (273 -f T'} (b b')}. T = .99975 T' approximately between and 40. This last is true within less than T ^ 7 of a degree. The two bulbs of the air thermometer used were from the same piece of glass tubing, and consequently had nearly, if not quite, the same coefficient of expansion. In the reduction of the barometer and other mercurial columns to zero, the coefficient .000162 was used, seeing that all the scales were of brass. In the tables the readings of the thermometers are reduced to volumes of the tube from the tables of calibration, and they are cor- rected for the pressure of water, which increased their reading, except at 0, by about 0.01 C. The order of the readings was as follows in each observation : 1st, barometer; 2d, cathetometer; 3d, thermometers forward and backward ; 4th, cathetometer ; 5th, barometer, &c., repeating the same once or twice at each temperature. In the later observations, two series like the above were taken, and the water stirred between them. The following results were obtained at various times for the value of a with the first bulb : 102 PROCEEDINGS OP THE AMERICAN ACADEMY Mean .0036664 .0036670 .0036658 .0036664 .0036676 .00366664 obtained by using the coefficient of expansion of glass .0000264 at 100 or a = 0036698, using the coefficient .0000287. The thermometers Nos. 6163, 6165, 6166, were always taken out of the bath when the temperature of 40 was reached, except on November 14, when they remained in throughout the whole experi- The thermometer readings are reduced to volumes by the tables of calibration. TABLE X. IST SERIES, Nov. 14, 1877. Relative Weight. Air Thermometer. V 6163. V 6166. 7 6167. Temperature by 6167. A 4 o 115.33 21.25 6.147 4 17.1425 422.84 255.80 15.685 17.661 .236 4 23.793 634.71 341.05 19.157 24.089 .296 5 30.582 653.49 431.71 22.833 80.896 .814 2 38.569 793.18 27.175 38.935 .366 2 61.040 33.864 61.320 .280 4 59.137 38.256 69.452 .816 The first four series, Tables X. to XIII., were made with one bulb to the air thermometer. A new bulb was now made, whose capacity was 192.0 ccm -, that of the old being 201.98 c - cm -. The value of -p for the new bulb was .0058. The values of h' and a were obtained as follows : a hf June 8th .00366790 753.876 June 22d .00366977 753.805 June 25th .00366779 753.837 Mean .0036685 753.84 This value of a is calculated with the old coefficient for glass, new would have given .0036717. The OF ARTS AND SCIENCES. 103 W 53 5 gg 8 Jf fc-|7 '"*'*'*^-* 1 <1 ;i *'*'*T>50Ou3o Jih ooioo s^ i as ^ s s ^ s 104 ' PROCEEDINGS OF THE AMERICAN ACADEMY fi- J ^ iii s s s s T* t-- t 1 * CO OS 00 -H rH g 35 3 f S S5 a? s CO CO ^tl 00 g O W N c$ t- as i-i S 8 oo ec oj to to to S S OF ARTS AND SCIENCES. 105 *> it- Difference Column reduced ~ 8 S i s s s 10 cd ca t^ S S & & If 5 S 2 -N S S 3 g 8 S 5! c *Q CO CC ^ CO l~ t^ O 00 O r-! 2 3 $ co rh co S S 58 !o lo S 11 ft !J S at 11 P CO cc cc -^ 10 106 PROCEEDINGS OF THE AMERICAN ACADEMY 5 q q i + I o o 5 So o co 3" e S 3 TP eo to & S S 5 ll S S r- O CO t^- Nil- Ili ll 2 8 S I .8 I -f S ^ S S? x 8 9 S 2 5! S S S 8 8 o> oo 10 -j( O CD 5 *-l (N (N (M OP ARTS AND SCIENCES. 107 + + + + + + I" I t-< (N t-. ^ S So ? S S co Oi I-* I-H o i>- < 100 T*t-COiOiOOOcO S S ^ S S g S I S 88 I S 3 I S g s ji fe o' i |U I J_I_ f: S S 5 S

(1 +34.8 m). Hence m = ~^ . We thus obtain the following results : Date. Weight. Value of m. Nov. 14 1 .000236 Nov. 20 2 .000218 Jan. 25 3 .000217 Feb. 11 4 .000197 June 8 3 .000215 June 22 2 .000216 Mean .000213 OF ARTS AND SCIENCES. Ill The results for m are then as follows : From direct comparison of No. 6167 with the air thermometer .000187 From " " " No. 6163 " " .000154 From comparison of No. 6163 with No. 6167 .000213 The first and last are undoubtedly the most exact numerically, but they apply to No. 6167, and are also, especially the first, derived from somewhat higher temperatures than the 20 point, where the correction is the most important. The value of m, as determined in either of these ways, depends upon the determination of a difference of temperature amounting to 0.30, and hence should be quite exact. The value of m, as obtained from the direct comparison of No. 6163 with the air thermometer, depends upon the determination of a differ- ence of about 0.05 between the mercurial and the air thermometer. At the same time, the comparison is direct, the temperatures are the same as we wish to use, and the glass is the same. I have combined the results as follows : m from No. 6167 .000200 m " 6163 .000154 Mean .00018* It now remains to deduce from the tables the ratios of the constants for the different thermometers. The proper method of forming the equations of condition are as follows, applying the method to the first series: Weight. 4 21.25 C tll = 115.33 C, v 4 255.80 C UI = 422.84 C, v 4 341.05 C in = 534.71 G t v 5 431.71 C UI = 653.49 C, v where G UI is the constant for No. 6166, C, is that for No. 6163, and v is a constant to be eliminated. Dividing by (7/, the equations S~i can be solved for --. The following table gives the results. * See Appendix to Thermometry, where it is finally thought best to reject the value from No. 6167 altogether. 112 PROCEEDINGS OP THE AMERICAN ACADEMY TABLE XVI. RATIOS OF CONSTANTS. 6163 6166 6166 6165 6165 Date. Weight. 6167 6167 6163 6163 6166 Nov. 14 1 .031009 .040058 1.3111 Nov. 20 2 .030985 .040670 1.3128 Jan. 25 3 .... .... 1.3122 Feb. 11 4 1.3115 8.6588 6.1449 June 8 3 .... 1.3108 8.0605 6.1469 June 22 2 1.3122 8.0588 6.1428 Mean j .030993 '.00005 .040666 .000003 1.31175 .0004 8.0594 .0002 6.1451 .0004 From these we have the following, as the final most probable results : ' . C n =8.0601 G t , . C UI = 1.31175 (7,, C, = .031003 (7 iT , C,, .24991 C iv , C UI = .040661 <7 iv , of which the last three are only used to calculate the temperatures on the mercurial thermometer, and hence are of little importance in the remainder of this paper. The value of C" which we have found for the old value of the coefficient of expansion of glass was 6^ = . 056976; and hence, corrected to the new coefficient, it is, as I have shown, C, =.056962. Hence, <7 ;/ = .45912, where it is to be noted that _ , is nearly a constant for all values of 0" T a according to Newton's law of cooling. 134 PROCEEDINGS OF THE AMERICAN ACADEMY The temperature reaches a maximum nearly at the time and if O m is the maximum temperature, we have the value of 6" as follows : 6" = T" = 6 m + (t m + c * ); and this is the final temperature provided there was no loss of heat. When the final temperature of the water is nearly equal to that of the air, G will be small, but the time t m of reaching the maximum will be great. If a is a constant, we can put G = a (0" T a ), and O(t m _j- c t ) will be a minimum, when That is, the temperature of the air must be lower than the tempera- ture of the water, so that T a = 0" as nearly as possible ; but the for- mula shows that this method makes the corrections greater than if we make T a = 6', the reason being that the maximum temperature is not reached until after an infinite time. It will in practice, however, be found best to make the temperature of the water at the beginning about that of the air. It is by far the best and easiest method to make all the corrections graphically, and I have constructed the fol- lowing graphical method from the formulae. First make a series of measurements of the temperature of the water of the calorimeter, before and after the basket is dipped, together with the times. Then plot them on a piece of paper as in Fig. 5, making the scale sufficiently large to insure accuracy. Five or ten centimeters to a degree are sufficient. n abed is the plot of the. temperature of the water of the calo- rimeter, the time being indicated by the horizontal line. Continue the line dc until it meets the line la. Draw a horizontal line through the point /. At any point, b, of the curve, draw a tangent and also a vertical line b g ; the distance e g will be nearly the value of the constant c in the formulas. Lay off //equal to c, and draw the line fh k through the point h, which indicates the temperature of the atmosphere or of the vessel surrounding the calorimeter. Draw a vertical line, j k, through the point k. From the point of maximum, OF ARTS AND SCIENCES. 135 c, draw a line, j c, parallel to dm, and where it meets kj will be the required point, and will give the value of d". Hence, the rise of tem- perature, corrected for all errors, will be kj. This method, of course, only applies to cases where the final tem- perature of the calorimeter is^ greater than that of the air ; otherwise there will be no maximum. In practice, the line dm is not straight, but becomes more and more nearly parallel to the base line. This is partly due to the constant decrease of the difference of temperature between the calorimeter and the air, but is too great for that to account for it. I have traced it to the thin metal jacket surrounding the calorimeter, and I must con- demn, in the strongest possible manner, all such arrangements of calo- rimeters as have such a thin metal jacket around them. The jacket is of an uncertain temperature, between that of the calorimeter and the air. When the calorimeter changes in temperature, the jacket follows it, but only after some time ; hence, the heat lost in radiation is uncer- tain. The true method is to have a water jacket of constant tempera- ture, and then the rate of decrease of temperature will be nearly constant for a long time. The following results have been obtained by Mr. Jacques, Fellow of the University, though the first was obtained by myself. Correc- tions' were, of course, made for the amount of thermometer stem in the air. Temperature. Mean Specific Heat. 24 to 100 .0915 26 " 100 .0915 25 " 100 .0896 ,-' 13 " 39 .0895 14 " 38 .0885 9 " 41 .0910 136 PROCEEDINGS OP THE AMERICAN ACADEMY To reduce these to the mean temperature of to 40, I have used the rate of increase foundry B^de for copper. They then become., . for the mean from to 40, .0897 .0897 . .0878 .0893 .0883 .0906 Mean .0892 dc .00027 As the capacity of the calorimeter is about four per cent of that of the total capacity, including the water, this probable error is about J fo T of the total capacity, and may thus be considered as satisfactory. I have also computed the mean specific heat as follows, from other observers : Copper between 20 and 100 nearly. .0949 Dulong. .0935 ) ^ >0952 }Regnault. .0933 Be-de. .0930 Kopp. ' .0940 This reduced to between and 40 by Bede's formula gives .0922. Hence we have the following for the calorimeter: * Per cent. Specific Heat between and 40 0. Copper 91.4 .0922 Zinc .7 .0896 Tin 3.6 .0550 Lead 2.7 .0310 Steel 1.6 .1110 Mean .0895 The close agreement of this number with the experimental result can only be accidental, as the reduction to the air thermometer would decrease it somewhat, and so make it even lower than mine. How- * The cast brass was composed of 28 parts of copper, 2 of tin, 1 of zinc, and 1 of lead. The rolled brass was assumed to have the same composition. The solder was assumed to be made of equal parts of tin and lead. OF ARTS AND SCIENCES. 137 ever, the difference could not at most amount to more than 0.5 per cent, which is very satisfactory. The total capacity of the calorimeter is reckoned as follows : Weight of calorimeter 3.8712 kilogrammes. " screws ' .0016 " " part of suspending wires .0052 " Total weight 3.8780 " Capacity = 3.878 X -0892 = .3459 kilogrammes. To this must be added the capacity of the thermometer bulb and sev- eral inches of the stem, and of a tube used as a safety valve, and we must subtract the capacity of a part of the shaft which was joined to the shaft turning the paddles. Hence, .3459 + .0011 " -f .0010 - .0010 Jtyr Capacity = .3470 As this is only about four per cent of the total capacity, it is not necessary to consider the variation of this quantity with the tempera- ture through the range from to 40 which I have used. IV. DETERMINATION OF EQUIVALENT. (a.) Historical Remarks. The history of the determination of the mechanical equivalent of heat is that of thermodynamics, and as such it is impossible to give it at length here. I shall simply refer to the few experiments which a priori seem to possess the greatest value, and which have been made rather for the determination of the quantity than for the illustration of a method, and shall criticise them to the best of my ability, to find, if possible, the cause of the great discrepancies. 1. GEXERAL REVIEW OF METHODS. Whenever heat and mechanical energy are converted the one into the other, we are able by measuring the amounts of each to obtain the ratio. Every equation of thermodynamics proper is an equation 138 PROCEEDINGS OF THE AMERICAN ACADEMY between mechanical energy and heat, and so should be able to give us the mechanical equivalent. Besides this, we are able to measure a certain amount of electrical energy in both mechanical and heat units, and thus to also get the ratio. Chemical energy can be measured in heat units, and can also be made to produce an electric current of known mechanical energy. Indeed, we may sum up as follows the different kinds of energy whose conversion into one another may fur- nish us with the mechanical equivalent of heat. And the problem in general would be the ratio by which each kind of energy may be con- verted into each of the others, or into mechanical or absolute units. a. Mechanical energy. b. Heat. c. Electrical energy. d. Magnetic " e. Gravitation u /. Radiant " g. Chemical " h. Capillary " Of these different kinds of energy, only the first five can be meas- ured other than by their conversion into other forms of energy, although Sir William Thomson, by the introduction of such terms as " cubic mile of sunlight," has made some progress in the case of radia- tion. Hence for these five only can the ratio be known. Mechanical energy is measured by the force multiplied by the dis- tance through which the force acts, and also by the mass of a body multiplied by half the square of its velocity. Heat is usually referred to the quantity required to raise a certain amount of water so many degrees, though hitherto the temperature of the water and the reduc- tion to the air thermometer have been almost neglected. The energy of electricity at rest is the quantity multiplied by half the potential ; or of a current, it is the strength of current multiplied by the electro-motive force, and by the time; or for all attractive forces varying inversely as the square of the distance, Sir William Thomson has given the expression where It is the resultant force at any point in space, and the integral is taken throughout space. These last three kinds of energy are already measured in absolute OP ARTS AND SCIENCES. 139 measure, and hence their ratios are accurately known. The only ratio, then, that remains is that of heat to one of the others, and this must be determined by experiment alone. But although we cannot measure /, g, h in general, yet we can often measure off equal amounts of energy of these kinds. Thus, although we cannot predict what quantities of heat are produced when two atoms of different substances unite, yet, when the same quantities of the same substances unite to produce the same compound, we are safe in assuming that the same quantity of chemical energy comes into play. According to these principles, I have divided the methods into direct and indirect. Direct methods are those where b is converted directly or indirectly into a, c, d, or e, or vice versa. Indirect methods are those where some kind of energy, as g, is con- verted into 6, and also into a, c, d, or e. In this classification I have made the arrangement with respect to the kinds of energy which are measured, and not to the intermediate steps. Thus Joule's method with the magneto-electric machine would be classed as mechanical energy into heat, although it is first con- verted into electrical energy. The table does not pretend to be com- plete, but gives, as it were, a bird's-eye view of the subject. It could be extended by including more complicated transformations ; and, in- deed, the symmetrical form in which it is placed suggests many other transformations. As it stands, however, it includes all methods so far used, besides many more. In the table of indirect methods, the kind of energy mentioned first is to be eliminated from the result by measuring it both in terms of heat and one of the other kinds of energy, whose value is known in absolute or mechanical units. It is to be noted that, although it is theoretically possible to measure magnetic energy in absolute units, yet it cannot be done practically with any great accuracy, and is thus useless in the determination of the equivalent. It could be thus left out from the direct methods without harm, as also out of the next to last term in the indirect methods. 140 PROCEEDINGS OF THE AMERICAN ACADEMY TABLE XXV. SYNOPSIS OP METHODS FOB OBTAINING THE MECHANICAL ' EQUIVALENT OF HEAT. re. Expansion or compression ac- cording to adiabatic curve. b. Expansion or compression ac cording to isothermal curve. 1. Reversible Process c. Expansion or compression ac- cording to any curve with re- generator. f-Heat| M -So a n E " rgy - d. Electro-magnetic engine driven by thermo-electric pile in a cir- cuit of no resistance. ( n. Friction, percussion, etc. * 2. Irreversible Process \ b. Heat from magneto-electric cur- ( rents, or electric machine. f a Thermo-electric currents {1. Reversible Process b, Pyro-electric phenomena (prob- 2 ably). a. Heating of wire by current, or <4 2. Irreversible Process heat produced by discharge of electric battery. I y. Heat, Magnetic Energy 1. Reversible Process o. Thermo-electric current magnet- izing a magnet in a circuit of {a. Mechanical Energy. b. Electrical " M * c. Magnetic d. Gravitation Crooke's radiometer. Thermo-electric pile. Thermo-electric pile with electro- magnet in circuit. a Mechanical Energy ( 1. Cannon. < 2. Electro-magnetic machine run by ( galv. battery.. . Chemical Energy, Heat . . (Combustion, etc.) b. Electrical " c. Magnetic Current from battery. ( Electro-magnet, magnetized by a I battery current. d. Gravitation " a. Mechanical Energy. Movement of liquid by capillarity. y. Capillary energy, Heat . . . 6. Electrical {Electrical currents from capillary action at surface of mercury. (Heat produced when a liq- c Magnetic " uid is absorbed by a po- it. Gravitation " Raising of liquid by capillarity. rous solid.) (a. Mechanical Energy {Magneto-electric or electro-magnetic machine. Electric attraction. S. Electrical energy, Heat . . ( (Heat generated in a wire by 6. Magnetic " c. Gravitation " Electro-magnet. an electrical current.) ia. Mechanical Energy {Armature attracted by a permanent magnet. t Magnetic Energy, Heat . . (Heat generated on demag- 6. Electrical " {Induced current on demagnetizing a magnet. netizing a magnet.) c. Gravitation " (. Gravitation Energy, Heat . (Heat generated by a fall- a. Mechanical Energy. 6. Electrical c. Magnetic " Velocity imparted to a falling body. ing.body.) 2. RESULTS OF BEST DETERMINATIONS. On the basis of this table of methods I have arranged the following table, showing the principal results so far obtained. In giving the indirect results, many persons have only measured one of the transformations required ; and as it would lengthen out the OP ARTS AND SCIENCES. 141 TABLE XXVL-H,.*,., OF Exr ERIMENTAL RE8ULm Me 0< thod in neral. Method in Particular. . Observe Date Result. Compression of air Expansion " . ! . Joule 2 184 443.8 Theory of gases (see below) . Joule 2 184 437.8 ' vapors (see below) . Experiments on steam-engine Him 7 185 413.0 Him 7 186 420-432 Expansion and contraction of meta Edlund 8 1865 443.6 4301 428.3 Boring of cannon .... Friction of water in tubea . in calorimeter in calorimeter . . in calorimeter Junction of mercury in calonmete plates of iron . . . Rumforc Joule 3 Joule* Joule 5 Joule 6 Joule 6 Joule 6 1798 1843 1845 1847 1850 1850 1850 940ft.lbs. 424.6 488.3 428.9 423.9 424.7 4252 metals metals in mercury calo Him 7 Favre 9 1857 1858 371.6 413.2 metals .... Joring of metals Vater in balance a frottement . '. Flow of liquids under strong pressur Crushing of lead . Him 7 Him 7 Him? Him 7 TT- _T 1858 1858 1860 1860- 400-450 425.0 4320 432.0 Friction of metals . -TUttl' 1860- 425.0 Water in calorimeter .... Joule 1876 1878 426.6 423.9 a Heating by magneto electric cur- ) rents . .' t Joule 8 1843 460.0 Heat generated in a disc between ) the poles of a magnet . . . j vww. 1870 435~.2 434.9 435.8 437.4 A ft Heat developed in wire of known ( absolute resistance . . . . " Quintus 1857 399.7 ( so Weber D . do. do. J . current produces work . . ) Joule 3 1843 99.0 Do. do. do. Favre 15 1858 43.0 Heat due to electrical current, ] electro-chemical equivalent of Weber, water = .009379, absolute re- 1 sistance electro-motive force of f Daniell cell, heat developed by action of zinc on sul. of copper I Boscha, ivre, and jermann 857 32.1 eat developed in Daniell cell . . Electro-motive force of Daniell cell Joule oscha 12 859 19.5 142 PROCEEDINGS OF THE AMERICAN ACADEMY table very much to give the complete calculation of the equivalent from these selected two by two, I have sometimes given tables of these parts. As the labor of looking up and reducing these is very great, it is very possible that there have been some omissions. I have taken the table published by the Physical Society of Berlin, 1 as the basis "down to 1857, though many changes have been made even within this limit. I shall now lake up some of the principal methods, and discuss them somewhat in detail. Method from Theory of Gases. As the different constants used in this method have been obtained by many observers, I shall first give their results. TABLE XXVII. SPECIFIC HEAT OF GASES. Limit to Temperature. Approximate Temperature of Water Temperature reduced to Specific Heat. Air | Mercurial 1 2669 I Delaroche and 20 to 210 " I *14.2 \ ( Thermometer Air Thermometer r i | .23751" Be"rard. Regnault. 20 to 100 20 { Mercurial Thermometer I .2389" E.Wiedemann. Hydrogen | Mercurial I 3 2936 \ Delaroche and 15 to 200 ( 12.2 | Thermometer Air Thermometer j 3.4090' 6 Be'rard. Regnault. 21 to 100 21 j Mercurial Thermometer | 3.410" E.Wiedemann. TABLE XXVIII. COEFFICIENT OF EXPANSION OF Am UNDER CONSTANT VOLUME. Taking Expansion of Mercury according to Regnault. Taking Expansion of Mercury according to Wiillner's Re- calculation of Ragnault's Ex- periments. Regnault ..... Magnus .... Jolly Rowland .0036655 .0036678 .0036695 .0036676 .0036687 .0036710 .0036727 .0036707 Mean .0036676 .0036708 * Taking mean of results on page 101 of Rel. des Exp., torn. ii. OP ARTS AND SCIENCES. 143 TABLE XXIX. RATIO OF SPECIFIC HEATS OF AIB. Method. Observer. Date. Ratio of Specific Heats. Method of Clement & De'- ) sormes, globe 20 litres . . ) Never fully published . . . Method of Clement & De'- I sormes J Using Breguet thermometer . Cle'ment & Desormes, globe ) Cle'ment & D^sormes' 8 ) Gay-Lussac& Welter 19 . Delaroche & Be'rard*' . Favre & Silbermann 23 . Masson 20 1812 Publishe'd in 1819 1853 1858 ( 1.354 1.3748 1.249 1.421 1.4196 Clement & De'sormes .... Clement & Desormes, globe ) Weisbach.2i Him 22 1859 1861 1.4025 1.3845 Passage of gas from one ves- ) Cazin 2 * 1862 141 litres ) Pressure in globe changed by ) aspirator, globe 25 litres . J Heating of gas by electric \ Dupre' 25 Jamin & Richard 28 . . 1863 1864 1.41 Clement & De'sormes . . . Barometer under air-pump / receiver of 6 litres . . . ) Compression and expansion ) Tresca et Laboulaye 29 . Kohlrausch 26 .... 1864 1869 1871 < 1.302 Results lost of gas by piston . . . . J Clement & Desormes with ) Rontgen 27 . . , 1878 of Paris. 14053 70 litres ) Compression of gas by piston . Amagat 30 1874 1.397 References. (Tables XXVI. to XXX.) 1 Physical Society of Berlin, Fort, der Phys., 1858. 2 Joule, Phil. Mag., ser. 3, vol. xxvi. See also Mec. Warmeaquivalent, Gesammelte Abhandlungen von J. P. Joule, Braunschweig, 1872. 3 Joule, Phil. Mag., ser. 3, vol. xxiii. See also 2 above. * " " " " " xxvi. 6 xxvji. 6 " " " xxxi. 7 Him, The"orie Me'c. de la Chaleur, ser. 1, 3 me ed. 8 Edlund, Pogg. Ann., cxiv. 1, 1865. 9 Favre, Comptes Rend., Feb. 15, 1858; also Phil. Mag., xv. 406. i Violle, Ann. de Chim., ser. 4, xxii. 64. 11 Quintus Icilius, Pogg. Ann., ci. 69. 12 Boscha, Pogg. Ann., cviii. 162. 13 Joule, Report of the Committee on Electrical Standards of the B. A., Lon- don, 1873, p. 175. i* H. F. Weber, Phil. Mag., ser. 5, v. 30. 15 Favre, Comptes Rend., xlvii. 599. 16 Regnault, Rel. des Expe'riences, torn. ii. E Wiedemann, Pogg. Ann., clvii. 1. 144 PROCEEDINGS OF THE AMERICAN ACADEMY *!.,- CO t- CO NOD o ; .' o : !JB! S S CO CO CO CO [ _' CO Si : ff!| | j CO CO ; g S ^2 ea> -< 1 Velocity reduced to 0and Ordinary Air. a s OJ t- s 2 . 2-2 d : lO fc |* o S g + & c3 t. O Tempera Observ 6 to 7 o^ 2 00 * o? ^ 2 P o -^-1 2 5 2 " 2 D *t 5 < > | J3.2 IH "- J s 5 ^ cw M s y O s iS 1 1 11 .2 .2 'O 13 c c 8 -S 1 S eq I III s PC, < K a P* ' < % 1 1 S i g B >-> og a - S 'o u > PH . S S =a ; bo s *" og 1 -a 3 ~- c (S 1 3 2 "0 S B co S " (2 ( B 1 M 1 ^-v-w N-*^-^X a rH (M CO ^ CD t- QO C> S OP ARTS AND SCIENCES. 145 Estimating th< follows: 3 weight No. 1 rather arbitrarily, Velocity at 0. C. Dry Air. 332.6 I have combined them as Estimated Weight of Observation. 2 2 332.7 2 3 330.9 2 4 330.8 4 5 332.5 3 6 332.8 7 7 332.0 1 8 331.8 1 9 332.4 4 10 330.7 10 Mean 331.75 Or, corrected for the normal carbonic acid in the atmosphere, it be- comes 331.78 meters per second in dry pure air at C. 18 Clement et Desdrmes, Journal de Physique, Ixxxix. 333, 1819. is Laplace, Mec. Celeste, v. 125. 20 Masson, Ann. de China, et de Phys., ser. 3, torn. liii. 21 Weisbach, Der Civilingenieur, Neue Folge, Bd. v., 1859. 22 Him, The'orie Mec. de la Chaleur, i. 111. 23 Favre et Silbermann, Ann. de Chim., ser. 3, xxxvii. 1851. z * Cazin, Ann. de Chim., ser. 3, torn. Ixvi. 25 Dupre, Ann. de Chim., 3 me ser., Ixvii. 359, 1863. 26 Kohlrausch, Pogg. Ann., cxxxvi. 618. Rontgen, Pogg. Ann., cxlviii. 603. 28 Jamin and Richard, Comptes Rend., Ixxi. 336. 29 Tresca and Laboulaye, Comptes Rend., Iviii. 358. Ann. du Conserv. des Arts et Me'tiers, vi. 365. 80 Amagat, Comptes Rend., Ixxvii. 1326. Me'm. de 1'Acad. des Sei., 1738, p. 128. 82 Benzenberg, Gilbert's Annalen, xlii. 1. 38 Goldingham, Phil. Trans., 1823, p. 96. 8 * Ann. de Chim., 1822, xx. 210 ; also, CEuvres de Arago, Me'm. Sci., ii. 1. 85 Stampfer and Von Myrbach, Pogg. Ann., v. 496. S6 Moll and Van Beek, Phil. Trans., 1824, p. 424. See also Shroder van der Kolk, Phil. Mag., 1865. 37 Parry and Foster, Journal of the Third Voyage, 1824-5, Appendix, p. 86. Phil. Trans., 1828, p. 97. 38 Savart, Ann. de Chim., ser. 2, Ixxi. 20. Recalculated. 89 Bravais and Martins, Ann. de Chim., ser. 3, xiii. 6. Regnault, Rel. des Exp., iii. 533. 41 Delaroche and Berard, Ann. de Chim., Ixxxv. 72 and 113. 42 Puluj, Pogg. Ann., clvii. 656. VOL. xv. (N. 8. vn.) 10 146 PROCEEDINGS OP THE AMERICAN ACADEMY From Regnault's experiments on the velocity in pipes I find by graphical means 331. 4 m - in free air, which is very similar to the above. Calculation from Properties of Gases. K= specific heat of gas at constant pressure. k = " " " volume. p = pressure in absolute units of a unit of mass. v = volume " " " " p = absolute temperature. J = Joule's equivalent in absolute measure. y ~- 7 ~~k' General formula for all bodies : 1 7 = JK \dft) v \dn) p . J= L _ JL y K * /dvy Also, J= ^- \ dfl ' p K d - \d P V 2 Application to gases; Rankine's formula is, _ If a, is the coefficient of expansion between and 100, then Mo = (l-|- .00635m); r PH , . / 7 > ^ F a " a "(^i, OF ARTS AND SCIENCES. 147 where a' p and a' p are the true coefficients of expansion at the given temperature ; /= According to Thomson and Joule's experiments m = 0.33 C. for air and about 2.0 for CO 2 . Hence /i - 272.99. The equations should be applied to the observations directly at the given temperature, but it will generally be sufficient to use them after reduction to C. Using K= .2375 according to Regnault for air, we have for the latitude of Baltimore, From Rb'ntgen's value y = 1.4053 " Amagat's " 1.397 9 " velocity of sound 33 US- per sec. = 429.6. Using Wiedemann's value for K, .2389, these become = 427.8 ; = 434.0 ; L = 427.1. 9 y 9 As Wiedemann, however, used the mercurial thermometer, and as the reduction to the air thermometer would increase these figures from .2 to .8 per cent., it is evident that Regnault's value for K is the more nearly correct. I take the weights rather arbitrarily as follows : Weight. J. Rontgen 3 430.3 Amagat 1 436.6 Velocity of sound 4 429.6 Mean 430.7 And this is of course the value referred to water at 1 4 C. and in the latitude of Baltimore. My value at this point is 427.7. * Rontgen gives the value 428.1 for the latitude of Paris as calculated by a formula of Shroder v. d. Kolk, and 427.3 from the formula for a perfect gas, and these both agree more nearly with my result than that calculated from my own formula. 148 PROCEEDINGS OP THE AMERICAN ACADEMY This determination of the mechanical equivalent from the proper- ties of air is at most very imperfect, as a very slight change in either y or the velocity of sound- will produce a great change in the mechan- ical equivalent. From Theory of Vapors. Another important method of calculating the mechanical equivalent of heat is from the equation for a body at its change of state, as for instance in vaporization. Let v be the volume of the vapor, and v 1 the volume of the liquid, and H the heat required to vaporize a unit of mass of the water ; also let p be the pressure in absolute units, and ^ the absolute temperature. Then JH The quantity Hand, the relation of p to p. have been determined with considerable accuracy by Regnault. To determine J it is only required to measure the volume of saturated steam from a given weight of water ; and the principal difficulty of the process lies in this determination, though the other quantities are also difficult of determination. This volume can be calculated from the density of the vapor, but this is generally taken in the superheated -state. The experiments of Fairbairn and Tate * are probably the best direct experiments on the density of saturated vapor, but even those do not pretend to a greater accuracy than about 1 in 100. With Regnault's values of the other quantities, they give about Joule's value for the equivalent, namely 425. Him, Herwig, and others have also made the determination, but the results do not agree very well. Herwig even used a Giessler standard thermometer, which I have shown to depart very much from the air thermometer. Indeed, the experiments on this subject are so uncertain, that physicists have about concluded to use this method rather for the de- termination of the volume of saturated vapors than for the mechanical equivalent of heat. From the Steam-Engine and Expansion of Metals. The experiments of Him on the steam-engine and of Edlund on the expansion and contraction of metals, are very excellent as illustrat- * Phil. Mag., ser. 4, xxi. 230. OP ARTS AND SCIENCES. 149 ing the theory of the subject, but cannot have ajiy weight as accurate determinations of the equivalent. From Friction Experiments. Experiments of this nature, that is, irreversible processes for con- verting mechanical energy into heat, give by far the best methods for the determination of the equivalent. Rumford's experiment of 1798 is only valuable from an historical point of view. Joule's results since .1843 undoubtedly give the best data we yet have for the determination of the equivalent. The mean of all his friction experiments of 1847 and 1850 which are given in the table is 425.8, though he prefers the smallest number, 423.9, of 1850. This last number is at present accepted throughout the civil- ized world, though there is at present a tendency to consider the number too small. But this value and his recent result of 1878 have undoubtedly as much weight as all other results put together. As sources of error in these determinations I would suggest, first, the use of the mercurial instead of the air thermometer. Joule com- pared his thermometers with one made by Fastre. In the Appendix to Thermometry I give the comparison of two thermometers made by Fastre in 1850, with the air thermometer, as well as of a large number of others. From this it seems that all thermometers as far as measured stand above the air thermometer between and 1 00, and that the aver- age for the Fastre* at 40 is about 0.l C. Using the formula given in Thermometry this would produce an error of about 3 parts in 1,000 at 15 C., the temperature Joule used. The specific heat of copper which Joule uses, namely, .09515, is undoubtedly too large. Using the value deduced from more recent experiments in calculating the capacity of my calorimeter, .0922, Joule's number would again be increased 13 parts in 10,000, so that we have, Joule's value 423.9, water at 15.7 C. Reduction to air thermometer . . -{-1.3 Correction for specific heat of copper -f- .5 " to latitude of Baltimore -\- .5 426.2 It does not seem improbable that this should be still further in- creased, seeing that the reduction to the air thermometer is the small- est admissible, as most other thermometers which I have measured give greater correction, and some even more than three times as great 150 PROCEEDINGS OF THE AMERICAN ACADEMY as the one here used, .and would thus bring the value even as high as 429. One very serious defect in Joule's experiments is the small range of temperature used, this being only about half a degree Fahrenheit, or about six divisions on, his thermometer. It would seem almost im- possible to calibrate a thermometer so accurately that six divisions should be accurate to one per cent, and it would certainly need a very skilful observer to read to that degree of accuracy. Further, the same thermometer " A " was used throughout the whole experiment with water, and so the error of calibration was hardly eliminated, the temperature of the water being nearly the same. In the experiment on quicksilver another thermometer was used, and he then finds a higher result, 424.7, which, reduced as above, gives 427.0 at Baltimore. The experiments on the friction of iron should be probably re- jected on account of the large and uncertain correction for the energy given out in sound. The recent experiments of 1878 give a value of 772.55, which re- duced gives at Baltimore 426.2, the same as the other experiment. The agreement of these reduced values with my value at the same temperature, namely 427.3, is certainly very remarkable, and shows what an accurate experimenter Joule must be to get with his simple apparatus results so near those from my elaborate apparatus, which almost grinds out accurate results without labor except in re- duction. Indeed, the quantity is the same as I find at about 20 C. The experiments of Him of 1860-61 seem to point to a value of the equivalent higher than that found by Joule, but the details of the experiment do not seem to have been published, and they certainly were not reduced to the air thermometer. The method used by Violle in 1870 does not seem capable of accuracy, seeing that the heat lost by a disc in rapid rotation, and while carried to the calorimeter, must have been uncertain. The experiments of Him are of much interest from the methods used, but can hardly have weight as accurate determinations. Some of the methods will be again referred to when I come to the descrip- tion of apparatus. Method by Heat generated by Electric Current. The old experiments of Quintus Icilius or Lenz do not have any sxcept historical value, seeing that Weber's measure of absolute resistance was certainly incorrect, and we now have no means of find- ing its error. OP ARTS AND SCIENCES. 151 The theory of the process is as follows. The energy of electricity being the product of the potential by the quantity, the energy ex- pended by forcing the quantity of electricity, Q, along a wire of re- sistance, R, in a second of time, must be Q*R, and as this must equal the mechanical equivalent of the heat generated, we must have JH= Q 2 Rt, where H is the heat generated and t is the time the current Q flows. The principal difficulty about the determination by this method seems to be that of finding R in absolute measure. A table of the values of the ohm as obtained by different observers, was published by me in my paper on the " Absolute Unit of Electrical Resistance," in the American Journal of Science, Vol. XV., and I here give it with some changes. TABLE XXXI. Date. Observer. Value of Ohm. REMARKS. 1849 Kirehoff .88 to .90 Approximately. 1851 Weber .95 to .97 Approximately. 1862 Weber j 1.088 | 1.075 From Thomson's unit. From Weber's value of Siemens unit. 1863-4 B. A. Committee ( 1.0000 \ .993 Mean of all results. Corrected by Rowland to zero velocity of coil. 1870 Kohlrausch 1.0193 1873 Lorenz .975 Approximately. 1876 Rowland .9911* From a preliminary comparison with the B. A. unit. 1878 H. F. Weber 1.0014 Using ratio of Siemens unit to ohm, .9536. The ratio of the Siemens unit to the ohm is now generally taken at .9536, though previous to 1864 there seems to have been some doubt as to the value of the Siemens unit. Since 1863-4, when units of resistance first began to be made with great accuracy, two determinations of fche heat generated have been made. The first by Joule with the ohm, and the second by H. F. Weber, of Zurich, with the Siemens unit. Each determination of resistance with each of these experiments gives one value of the mechanical equivalent. As Lorenz's result was only in illustration of a method, I have not included it among the exact determinations. The result found by Joule was /= 25187 in absolute measure -* Given .9912 by mistake in the other tables. 152 PROCEEDINGS OF THE AMERICAN ACADEMY using feet and degrees F., which becomes 429.9 in degrees C. on a mercurial thermometer and in the latitude of Baltimore, compared with water at 18. 6 C. TABLE XXXII. EXPERIMENTS OF JOULE. Observer. Value of B. A. Unit. Mechanical Equivalent from Joule's Exp. Mechanical Equivalent reduced to Air Ther- mometer and cor- rected for Sp. Ht. of Copper. B. A. Committee Ditto corrected by Rowland 1.0000 .993 1.0193 429.9 426.9 438.2 431.4 428.4 439.7 .9911 426.1 427.6 H. F. Weber 1.0014 430.5 432.0 The experiments of H. F. Weber* gave 428.15 in the latitude of Zurich and for 1 C. on the air thermometer and at a temperature of 18 C. This reduced to the latitude of Baltimore gives 428.45. TABLE XXXIII. EXPERIMENTS o F H. F WEBEB Mean of Joule and Weber, giving Joule twice the Weight of Weber. Observer. Value of B. A. Unit. Mechanical Equivalent of Heat from Weber's Experiments. Mean Equivalent re- duced toAirThennom- eter In the Latitude of Baltimore. B. A. Committee Ditto corrected by Rowland Kohlrausch Rowland 1.000 .993 1.0193 9911 427.9 424.9 436.2 424 1 430.2 427.2 439.1 426 4 II . F. Weber. . . 1 0014 428 5 401 A My own value at this temperatuj-e is 426.8, which agrees almost exactly with the fourth value from my own determination of the ab- solute unit.f There can be no doubt that Joule's result is most exact, and hence I have given his results twice the weight of Weber's. Weber used a wire of about 14 ohms' resistance, and a small calorimeter holding only 250 grammes of water. This wire was apparently placed in the water without any insulating coating, and yet current enough was sent through * Phil. Mag, 1878, 5th ser, v. 135. t The value of the ohm found by reversing the calculation would be .992, almost exactly my value. OP ARTS AND SCIENCES. 153 it to heat the water 15 during the experiment. No precaution seems to have been taken as to the current passing into the water, which Joule accurately investigated. Again, the water does not seem to have been continuously stirred, which Joule found necessary. And further, Newton's law of cooling does not apply to so great a range as 15, though the error from this source was probably small. Further- more, I know of no platinum which Jias an increase of coefficient of .001054 for 1 C., but it is usually given at about .003. There can be no doubt that experiments depending on the heating of a wire give too small value of the equivalent, seeing that the temperature of the wire during the heating must always be higher than that of the water surrounding it, and hence more heat will be generated than there should be. Hence the numbers should be slightly increased. Joule used wire of platinum-silver alloy, and Weber platinum wire, which may account for Weber's finding a smaller value than Joule, and Weber's value would be more in error than Joule's. Undoubtedly this is a serious source of error, and I am about to repeat an experiment of this kind in which it is entirely avoided. Considering this source of error, these experiments confirm both my value of the ohm and of the mechanical equivalent, and unquestionably show a large error in Kohlrausch's absolute value of the Siemens unit or ohm. The experiments of Joule and Favre, where the heat generated by a current, both when it does mechanical work and when it does not, are very interesting, but can hardly have any weight in an estimation of the true value of the equivalent. The method of calculating the equivalent from the chemical action in a battery, or the electro-motive force required to decompose any substance, such as water, is as follows. Let E be such electro-motive force and c be the quantity of chemi- cal substance formed in battery or decomposed in voltameter per second. Then total energy of current of energy per second is E Q, where Q is the current, or cQ ffj, where ffis the heat generated by unit of c, or required to decompose unit of c. Hence, if the process is entirely reversible, we must have in either case CHJ E. But the process is not always reversible, seeing that it requires more electro-motive force to decompose water than is given by a gas battery. This is probably due to the formation at first of some un- stable compound like ozone. The process with a battery seems to be 154 PROCEEDINGS OP THE AMERICAN ACADEMY best, and we can thus apply it to the Daniell cell. The following quantities are mostly taken from Kohlrausch. The quantity c has been found by various observers, and Kohl- rausch* gives the mean value as .009421 for water according to his units (mg., mm., second system). Therefore for hydrogen it is .001047. The quantity H can be observed directly by short-circuiting the battery, or can be found from experiments like those of Favre and Silbermann. The electro-motive force E can be made- to depend either upon the absolute measure of resistance, or can be determined, as Thomson has done, in electro-static units. In electro-magnetic units it is Absolute Measure Siemens. Ohms. according to my Determination. After Waltenhofen 11.43 10.90 10.80 X 10 10 Kohlrausch f H-71 11.17 11.07 X 10 10 After Favre, 1 equivalent of. zinc develops in the Daniell cell 23993 heat units ; On the mg., mm., second system, we have E = 10.935 X 10 10 , c = .001047, H= 23993, g = 9800.5 at Baltimore. .-. y = 444160 mm - = 444.2 meters. Using Kohlrausch's value for absolute resistance, he finds 456.5, which is much more in error than that from my determination. I do not give the calculation from the Grove battery, because the Grove battery is not reversible, and action takes place in it even when no current flows. Thomson finds the difference of potential between the poles of a Daniell cell in electro-static measure to be .OQ374 on the cm., grm., second system. $ Using the ratio 29,900 000 000 cm> per second, as I have recently found, but not yet published, we have 111 800 000 011 the electro-magnetic system or 11.18 X 10 10 on the mm., mg., second system. This gives = 474.3 meters. * Pogg. Ann., cxlix. 179. t Given by Kohlrausch, Pogg. Ann., cxlix. 182. t Thomson, Papers on Electrostatics and Magnetism, p. 246. OF ARTS AND SCIENCES. 155 General Criticism. All the results so far obtained, except those of Joule, seem to be of the crudest description ; and even when care was apparently taken in the experiment, the method seems to be defective, or the determination is made to rest upon the determination of some other constant whose value is not accurately known. Again, only one or two observers have compared their thermometers with the air thermometer, although I have shown in " Thermometry " that an error of more than one per cent may be made by this method. The range of temperatures is also small as a general rule and the specific heat of water is assumed constant. Hence a new determination, avoiding these sources of error, seems to be imperatively demanded. (b.) Description of Apparatus. 1. PRELIMINARY REMARKS. As we have seen in the historical portion, the only experiments of a high degree of accuracy to the present time are those of Joule. Looked at from a general point of view, the principal defects of his method were the use of the mercurial instead of the air thermom- eter, and the small rate at which the temperature of his calorimeter rose. , In devising a new method a great rise of temperature in a short time was considered to be the great point, combined, of course, with an accurate measurement of the work done. For a great rise of tem- perature great work must be done, which necessitates the use of a steam-engine or other motive power. For the measurement of the work done, there is only one principle in use at present, which is, that the work transmitted by any shaft in a given time is equal to 2 w times the product of the moment of the force by the number of revo- lutions of the shaft in that time. In mechanics it is common to measure the amount of the force twisting the shaft by breaking it at the given point, and attaching the two ends together by some arrangement of springs whose stretching gives the moment. Morin's dynamometer is an example. Hirn* gives a method which he seems to consider new, but which is immedi- ately recognized as Huyghens's arrangement for winding clocks with- * Exposition de la Theorie Mecanique de la Chaleur, 3 me ed., p. 18. 156 PROCEEDINGS OF THE AMERICAN ACADEMY out stopping them. As cords and pulleys are used which may slip on each other, it cannot possess much accuracy. I have devised a method by cog-wheels which is more accurate, but which is better adapted for use in the machine-shop than for scientific experimentation. But the most accurate method known to engineers for measuring the work of an engine is that of White's friction brake, and on this I have based my apparatus. Him was the first to use this principle in determining the mechanical equivalent of heat. In his experiment a horizontal axis was turned by a steam-engine. On the axis was a pulley with a flat surface, on which rested a piece of bronze which was to be heated by the friction. The moment of the force with which the friction tended to turn the piece of bronze was measured, together with the velocity of revolution. This experiment, which Him calls a balance de frottement, was first constructed by him to test the quality of oils used' in the industrial arts. He experimented by passing a current of water through the apparatus and observing the tempera- ture of the water before and after passing through. He thus ob- tained a rough approximation to Joule's equivalent. He afterwards constructed an apparatus consisting of two cylinders about 30- in diameter and lOO 01 "- long, turning one within the other, the annular space between which could be filled with water, or through which a stream of water could be made to flow whose temperature could be measured before and after. The work was measured by the same method as before. But in neither of these methods does Hirn seem to have recognized the principle of the work transmitted by a shaft being equal to the moment of the force multiplied by the angle of rotation of the shaft. In designing his apparatus, he evidently had in view the reproduction in circular motion of the case of friction between two planes in linear motion. Since I designed my apparatus, Puluj * has designed an instrument to be worked by hand, and based on the principle used by Hirn. He places the revolving axis vertical, and the friction part consists of two cones rubbing together. But no new principle is involved in his apparatus further than in that used by Hirn.f * Fogg. Ann., clvii. 437. t Joule's latest results were published after this was written, and I was not aware that he had made this improvement until lately. The result of his experiment, however, reached me soon after, and I have referred to it in the paper, but I did not see the complete paper until much later. OF AETS AND SCIENCES. 157 In my apparatus one of the new features has. been the introduction of the Joule calorimeter in the place of the friction cylinders of Him or the cones of Puluj. At first sight the currents and whirlpools in such a calorimeter might be supposed to have some effect ; but when the motion is steady, it is readily seen that the torsion of the calorim- eter is equal to that of the shaft, and hence the principle must apply. This change, together with the other new features in the experi- ments and apparatus, has at once made the method one of extreme accuracy, surpassing all others very many fold. 2. GENERAL DESCRIPTION. The apparatus was situated in a small building, entirely separate from the other University buildings, and where it was free from dis- turbances. Fig. 6 gives a general view of the apparatus. To a movable axis, a b, a calorimeter similar to Joule's is attached, and the whole is suspended by a torsion wire, c. The shaft of the calorimeter comes out from the bottom, and is attached to a shaft, e f, which receives a uniform motion from the engine by means of the bevel wheels g and h. To the axis, a b, an accurately turned wheel, k I, was attached, and the moment of the force tending to turn the calorimeter was measured by the weights o and p, attached to silk tapes passing around the circum- ference of this wheel in combination with the torsion of the suspend- ing wire. To this axis was also attached a long arm, having two sliding weights, q and r, by which the moment of inertia could be varied or determined. The number of revolutions was determined by a chronograph, which received motion by a screw on the shaft ef, and which made one revolution for 102 of the shaft. On this chronograph was recorded the transit of the mercury over the divisions of the thermometer. Around the calorimeter a water jacket, t M, made in halves, was placed, so that the radiation could be estimated. A wooden box sur- rounded the whole, to shield the observer from the calorimeter. The action of the apparatus is in general as follows. As the inner paddles revolve, the water strikes against the outer paddles, and so tends to turn the calorimeter. When this force is balanced by the weights o p, the whole will be in equilibrium, which is rendered stable by the torsion of the wire c d. Should any slight change take place in the velocity, the calorimeter will revolve in one direction or the other until the torsion brings it into equilibrium again. The amount 158 PROCEEDINGS OF THE AMERICAN ACADEMY of torsion read off on a scale on the edge of k I gives the correction to be added to or. subtracted from the weights op. One observer constantly reads the circle k I, and the other con- stantly records the transits of the mercury over the divisions of the thermometer. A series extending over from one half to a whole hour, and record- ing a rise of 15 C. to perhaps 25 C., and in which a record was made for perhaps each tenth of a degree, would thus contain several hundred observations, from any two of which the equivalent of heat could b,e determined, though they would not all be independent. Such a series would evidently have immense weight ; and, in fact, I estimate that, neglecting constant errors, a single series has more weight than all of Joule's experiments of 1849, on water, put together.* The correction for radiation is inversely proportional to the ratio of the rate of work generated to the rate at which the heat is lost ; and this for equal ranges of temperature is only -fa as great in my measures as in Joule's ; for Joule's rate of increase was about 0.62 C. per hour, while mine is about 35 C. in the same time, and can be increased to over 45 C. per hour. 3. DETAILS, The Calorimeter. Joule's calorimeter was made in a very simple manner, with few paddles, and without reference to the production of currents to mix up the water. Hence the paddles were made without solder, and were screwed together. Indeed, there was no solder about the apparatus. But, for my purpose, the number of paddles must be multiplied, so that there shall be no jerk in the motion, and that the resistance may be great: they must be stronger, to resist the force from the engine, and they must be light, so as not to add an uncertain quantity to the calorific capacity. Besides this, the shape must be such as to cause the whole of the water to run in a constant stream past the thermometer, and to cause constant exchange between the water at the top and at the bottom. l^? 1 "? expertments > with an average rise of temperature of 0.56 F., equal ' -31 C, gives a total rise of 12.4 C., which is only about two thirds the erage of one of my experiments. As my work is measured with equal accu- racy, and my radiation with greater, the statement seems to be correct. OP ARTS AND SCIENCES. 159 160 PROCEEDINGS OP THE AMERICAN ACADEMY Fig. 7 shows a section of the calorimeter, and Fig. 8 a per- spective view of the revolving paddles removed from the appa- ratus, and with the exterior paddles removed from around it; which could not, however, be accomplished physically without destroying them. To the axis e b, Fig. 7, which was of steel, and 6 mm - in diame- ter, a copper cylinder, a d, was attached, by means of four stout wires at e, and four more at/. To this cylinder four rings, g, //, i,j, were attached, which supported the paddles. Each one had eight paddles, but each ring was displaced through a small angle with reference to Fig. 7. Fig. 8. the one below it, so that no one paddle came over another. This was to make the resistance continuous, and not periodical. The lower row of paddles were turned backwards, so that they had a tendency to throw the water outwards and make the circulation, as I shall show afterwards. Around these movable paddles were the stationary paddles, consist- ing of five rows of ten each. These were attached to the movable paddles by bearings, at the points c and k, of the shaft, and were removed with the latter w^ien this was taken from the calorimeter. "When the whole was placed in the calorimeter, these outer paddles were attached to it by means of four screws, I and m, so as to be immovable. The cover of the calorimeter was attached to a brass ring, which was nicely ground to another brass ring on the calorimeter, and which OF ARTS AND SCIENCES. 161 could be made perfectly tight by means of a little white-lead paint. The shaft passed through a stuffing-box at the bottom, which was entirely within the outer surface of, the calorimeter, so that the heat generated should all go to the water. The upper end of the shaft rested in a bearing in a piece of brass attached to the cover. In the cover there were two openings, one for the thermometer, and the other for filling the calorimeter with water. From the opening for the thermometer, a tube of copper, perforated with large holes, descended nearly to the centre of the calorimeter. The thermometer was in this sieve-like tube at only a short distance from the centre of the calorimeter, with the revolving paddles outside of it, and in the stream of water, which circulated as shown by the arrows. This circulation of water took place as follows. The lower paddles threw the water violently outwards, while the upper paddles were Fig. 9. prevented from doing so by a cylinder surrounding the fixed paddles. The consequence was, that the water flowed up in the space between the outer shell and the fixed paddles, and down through the central tube of the revolving paddles. As there was always a little air at the top to allow for expansion, it would also aid in the same direction. These currents, which were very violent, could be observed through the openings. The calorimeter was attached to a wheel, fixed to the shaft a b, by the method shown in Fig. 9. At the edge of the wheel, which was of the exact diameter of the calorimeter, two screws were attached, from which wires descended to a single screw in the edge of the calorimeter. Through the wheel, a screw armed with^ a vulcanite point pressed upon the calorimeter, and held it firmly. Three of these arrange- ments, at distances of 120, were used. To centre the calorimeter, a piece of vulcanite at the centre was used. By this method of suspension very little heat could escape, and the amount could be allowed for by the radiation experiments. VOL. xv. (N. s. vii.) 11 162 PROCEEDINGS OP THE AMERICAN ACADEMY The Torsion System. The torsion wire was of such strength that one millimeter on the scale at the edge of the wheel signified 11.8 grammes, or about 7 ^ of the weights o p generally used. There were stops on the wheel, so that it could not move through more than a small angle. The weights were suspended by very flexible silk tapes, 6 mm or 8 mm broad and 0.3 mra - thick. They varied from 4.5 k - to 8.5 k - taken together. The shaft, a b, was of uniform size throughout, so that the wire c sus- pended the whole system, and no "weight rested on the bearings. The pulleys, m, n, Fig. 6, were very exactly turned and balanced, and the whole suspended system was so free as to vibrate for a con- siderable time. However, as will be shown hereafter, its freedom is of little consequence. The Water Jacket. Around the calorimeter, a watef jacket, t u, was placed, so that the radiation should be perfectly definite. During the preliminary experiments a simple tin jacket was used, whose temperature was determined by two thermometers, one above and the other below, inserted in tubes attached to the jacket. The Driving Gear. The cog-wheels, g, h, were made by Messrs-. Brown and Sharpe, of Providence, and were so well cut that the motion transmitted to the calorimeter must have been very uniform. The Chronograph. The cylinder of the chronograph was turned by a screw on the shaft ef, and received one revolution for 102 of the paddles ; 155 revolutions of the cylinder, or 15,810 of the paddles, could be recorded, though, when necessary, the paper could be changed without stopping, and the experiment thus continued without interruption. The Frame and Foundation. The frame was very massive and strong, so as to prevent oscillation; and the whole instrument weighed about 500 pounds as nearly as could be estimated. It was placed on a solid brick pier, with a firm foundation in the ground. The trembling was barely perceptible to the hand when running the fasteot. OP ARTS AND SCIENCES. 163 The Engine. The driving power was a petroleum engine, which was very efficient in driving the apparatus with uniformity. The Balance. For weighing the calorimeter, a balance capable of showing the presence of less than ^ gramme with 15,000 grammes was used. The weights, however, by Schickert, of Dresden, were accurate among themselves to at least o"* for the larger weights, and in proportion for the smaller. A more accurate balance would have been useless, as will be seen further on. Adjustments. There are few adjustments, and they were principally made in the construction. In the first place, the shafts a b and ef must be on line. Secondly, the wheels m n must be so adjusted that their planes are vertical, and that the tapes shall pass over them symmetrically, and that their edges shall be in the plane of the wheel k I. Deviation from these adjustments only produced small error. (c.) Theory of the Experiment. 1. ESTIMATION OF WORK DONE. The calorimeter is constantly receiving heat from the friction, and is giving out heat by radiation and conduction. Now, at any given instant of time, the temperature of the whole of the calorimeter is not the same. Owing to the violent stirring, the water is undoubtedly at a very uniform temperature throughout. But the solid parts of the calorimeter cannot be so. The greatest difference of temperature is evidently soon after the commencement of the operation. But after some time the apparatus reaches a stationary state, in which, but for the radiation, the rise of temperature at all points would be the same. This steady state will be theoretically reached only after an infinite time ; but as most of the metal is copper, and quite thin, and as the whole capacity of the metal work is only about four per cent of the total capacity, I have thought that one or two minutes was enough to allow, though, if others do not think this time sufficient, they can readily reject the first few observations of each series. When there is radiation, the stationary state will never be reached theoretically, 164 PROCEEDINGS OF THE AMERICAN ACADEMY though practically there is little difference from the case where there is no radiation. The measurement of the work done can be computed as follows. Let M be the moment of the force tending to turn the calorimeter, and d 9 the angle moved by tHe shaft. The work done in the time i, will be f M d 6. If the moment of the force is constant, the integral is simply M 9 ; but it is impossible to obtain an engine which runs with perfect steadiness, and although we may be able to calculate the integral, as far as long periods are concerned, by observation of the torsion circle, yet we are not thus able to allow for the irregularity during one revolution of the engine. Hence I have devised the follow- ing theory. I have found, by experiments with the instrument, that the moment of the force 1 is very nearly, for high velocities at least, proportional to the square of the velocity. For rapid changes of the velocity, this is not exactly true, but as the paddles are very numerous in the calorimeter, it is probably very nearly true. We have then where C is a constant. Hence the work done becomes Yrfi, As we allow for irregularities of long period by readings of the torsion circle, we can assume in this investigation that the mean velocity is constant, and equal to v . The form of the variation of the velocity must be assumed, and I shall put, without further dis- cussion, ' d / . 2 IT t\ -T = v ( 1 4- c cos ) . V a / We then find, on integrating from a to 0, w= Cv s a (1 -|- f c 8 ), which is the work on the calorimeter during one revolution of the engine. ^The equation of the motion of the calorimeter, supposing it to be nearly stationary, and neglecting the change of torsion of the sus- pending wire, is where m is the moment of inertia of the calorimeter and its attach- ments, f is the angular position of the calorimeter, W is the sum of OF AETS AND SCIENCES. 165 the torsion weights, and D is the diameter of the torsion wheel. Hence, When W D = 2 C v 2 (1 -f- c 2 ), the calorimeter will merely oscil- late around a given position, and will reach its maximum at the times t = 0, | a, a, &c. The total amplitude of each oscillation will be very nearly , _ ,/ __ C v^ga^c _ TF > gr a z c V Y ~ * m 2ir* m ' If a; is the amplitude of each oscillation, as measured in millimeters, on the edge of the wheel of diameter D, we have \jr i// = -^. where n is the number of revolutions of the engine per second. Having found c in this way, the work will be, during any time, W = TT WD N(l +c 2 ), where N is the total number of revolutions of the paddles. A variation of the velocity of ten per cent from the- mean, or twenty per cent total, would thus only cause an error of one per cent in the equivalent. Hence, although the engine was only single acting, yet it ran easily, had great excess of power, and was very, constant as far as long periods were concerned. The engine ran very fast, making from 200 to 250 revolutions per minute. The fly-wheel weighed about 220 pounds, and had a radius of lj feet. At four turns per second, this gives an energy of about 3400 foot pounds stored in the wheel. The calorimeter required about one-half horse-power to drive it ; and, assuming the same for the engine friction, we have about 140 foot pounds of work required per revolution. Taking the most unfavorable case, where all the power is given to the engine at one point, the velocity changes during the revolution about four per cent, or c would nearly equal .02, causing an error of 1 part in 2500 nearly. By means of the shaking of the calorimeter, I have estimated c as follows, the value of m being changed by changing the weight on the inertia bar, or taking it off altogether. The estimate of the shaking was made by two persons independently. 166 PROCEEDINGS OP THE AMERICAN ACADEMY m x observed. calculated. 2,200,000 grins, cm. 2 .6 mm. .016 3,100,000 " -36 " -013 11,800,000 " .13 " jO!7 Mean, c = .015 causing a correction of 1 part in 5000. Another method of estimating the irregularity of running is to put on or take off weights until the calorimeter rests so firmly against the stops that the vibration ceases. Estimated in this way, I have found a little larger value of c, namely, about .017. But as one cannot be too careful about such sources of error, I have experimented on the equivalent with different velocities and with very different ways of running the engine, by which c was greatly changed, and so have satisfied myself that the correction from this source is inappreciable in the present state of the science of heat. Hence I shall simply put for the work w = -trNWD, in gravitation measure at Baltimore. To reduce to absolute measure, we must multiply by the force of gravity given by the formula g = 9.78009 -f- .0508 sin 2 <, which gives 9.8005 meters per second at Baltimore. If the calo- rimeter moved without friction, no work would be required to cause it to vibrate back and forth, as I have described; but when it moves with friction, some work is required. When I designed the apparatus, I thus had an idea that it would be best to make it as immovable as possible by adding to its moment of inertia by means of the inertia bar and weights. But on considering the subject further, I see that only the excess of energy represented by c z wN W D can be used in this way. For, when the calorimeter is rendered nearly immovable by its great moment of inertia, the work done on it is, as we have seen, TT N W D (1 -\- c 2 ) ; but if it had no inertia, it is evident that the work would be only TrNWD. If, therefore, the calorimeter is made partially stationary, either by its moment of inertia or by fric- tion, the work will be somewhere between these two, and the work spent in friction will be only so much taken from the error. Hence in the latter experiments the inertia bar was taken off, and then the calorimeter constantly vibrated through about half a millimeter on the torsion scale. Besides this quick vibration, the calorimeter is constantly moving to OF ARTS AND SCIENCES. 167 the extent of a few millimeters back and forth, according to the vary- ing velocity of the engine. As frequent readings were taken, these changes were eliminated. In very rare cases the weights had to be changed during the experiment ; but this was very seldom. The vibration and irregular motion of the calorimeter back and forth served a very useful purpose, inasmuch as it caused the friction of the torsion apparatus to act first in one direction and then in the other, so that it was finally eliminated. The torsion apparatus moved very freely when the calorimeter was not in position, and would keep vibrating for some minutes by itself, but with the calorimeter there was necessarily some binding. But the vibration made it so free that it would return quickly to its exact 'position of equilibrium when drawn aside, and would also quickly show any small addition to the weights. This was tried in each experiment. To measure the heat generated, we require to know the calorific capacity of the whole calorimeter, and the rise of temperature which would have taken place provided no heat had been lost by radiation. The capacity of the calorimeter alone I have discussed elsewhere, finding the total amount equal to .347 k ' of water at ordinary tempera- tures. The total capacity of the calorimeter is then A -f- -347, where A is the weight of water. Hence Joule's equivalent in absolute measure is ~~ (4 + .347) (t t>) y \ where n is the number of revolutions of the chronograph, it making one revolution to 102 of the paddles. The corrections needed are as follows : 1st. Correction for weighing in air. This must be made to JF, the cast-iron weights, and to A -j- .347, the water and copper of the calo- rimeter. If A is the density of the air under the given conditions, the correction is .835 A. 2d. For the weight of the tape by which the weights are hung. This is '-rp; 3d. For the expansion of torsion wheel, D' being the diameter at 20 C. This is .000018 (t" 20). Hence, nnofi (1 -j_ .000018 (f '-20) 4- jp .835 A)/^.a where t t 1 is the rise of the temperature corrected for radia- tion. . Oo . 168 PROCEEDINGS OP THE AMERICAN ACADEMY 2. RADIATION. The correction for radiation varies, of course, with the difference of temperature between the calorimeter and jacket; but, owing to the rapid generation of heat, the correction is generally small in propor- tion. The temperature , generated was generally about 0.6 per minute. The loss of temperature per minute by radiation was approxi- mately .0014 6 per minute, where 6 is the difference of the tempera- ture. This is one per cent for 10.7, and four per cent for 14.2. Generally, the calorimeter was cooler than the jacket to start with, and so a rise of about 20 could be accomplished without a rate of correction at any point of more than fpur per cent, and an average correction of less than two per cenl. An error of ten per cent is thus required in the estimation of the radiation to produce an average error of 1 in 500, or 1 in 250 at a single point. The coefficients never differ from the mean more than about two per cent. The observations on the equivalent, being at a great variety of tempera- tures, check each other as to any error in the radiation. The losses of heat which I place under the head of radiation include conduction and convection as well. I divide the losses of heat into the following parts: 1st. Conduction down the shaft; 2d. Conduction by means of the suspending wires or vulcanite points to the wheel above ; 3d. True radiation ; 4th, Convection by the air. To get some idea of the relative amounts lost in this way, we can calculate the loss by conduction from the known coefficients of conduction, and we can get some idea of the relative loss from a polished surface from the experi- ments of Mr. Nichol. In this way I suppose the total coefficient of radiation to be made up approximately as follows : Conduction along shaft . . .00011 ' " suspending wires .00006 True radiation 00017 Convection .00106 Total . . . .00140 The conduction through the vulcanite only amounts to .0000002. From this it would seem that three fourths of the loss is due to radiation and convection combined. The last two 'losses depend upon the difference of temperature between the calorimeter and the jacket, but the first two upon the difference between the calorimeter and frame of the machine and the wheel respectively. The frame was always of very nearly the same OF ARTS AND SCIENCES. 169 temperature as the water jacket, but the wheel was usually slightly above it. At first its temperature was noted by a thermometer, and the loss to it computed separately; but it was found to be unnecessary, and finally the whole was assumed to be a function of the tempera- ture of the calorimeter and of the jacket only. At first sight it might seem that there was a source of error in having a journal so near the bottom of the calorimeter, and joined to it by a shaft. But if we consider it a moment, we shall see that the error is inappreciable ; for even if there was friction enough in the journal to heat it as fast as the calorimeter, it would decrease the radiation only seven per cent, or make an average error in the experi- ment of only 1 in 700. But, in fact, the journal was very perfectly made, and there was no strain on it to produce friction ; besides which, it was connected to a large mass of cast-iron which was attached to the base. Hence, as a matter of fact, the journal was not appreciably warmer after running than before, although tested by a thermometer. The difference could not have been more than a degree or so at most. The warming of the wheel by conduction and of the journal by friction would tend to neutralize each other, as the wheel would be warmer and the journal cooler during the radiation experiment than the friction experiment. The usual method of obtaining the coefficient of radiation would be to stop the engine while the calorimeter was hot, and observe the cooling, stirring the water occasionally when the temperature was read. This method I used at first, reading the temperature at inter- vals of about a half to a whole hour. But on thinking the matter over, it became apparent that the coefficient found in this way would be too small, especially at small differences of temperature ; for the layer next to the outside would be cooled lower than the mean tem- perature, and the heat could only get to the outside by conduction through the water or by convection currents. Hence I arranged the engine so as to run the paddles very slowly, so as to stir the water constantly, taking account of the number of the revolutions and the torsion, so as to compute the work. As I had foreseen, the results in this case were higher than by the other method. At low temperatures the error of the first method was fifteen per cent ; but at high, it did not amount to more than about three to five per cent, and probably at very high temperatures it would almost vanish. I do not consider it necessary to give all the details of the radiation experiments, but will merely remark that, as the calorimeter was 170 PROCEEDINGS OP THE AMERICAN ACADEMY nickel-plated, and as seventy-five per cent of the so-called radiation is due to convection by the air, the coefficients of radiation were found to be very constant under similar conditions, even after long intervals of time. The experiments were divided into two groups ; one when the temperature of the jacket was about 5 C., and the other when it averaged about 20 C. The results were then plotted, and the mean curve drawn through them, from which the following coefficients were obtained. These coefficients are the loss of temperature per minute, and per degree difference of temperature. TABLE XXXV. COEFFICIENTS OF RADIATION. Difference between Jacket Jacket 5. Jacket 20. and Calorimeter. -5 .00138 .00134 .00135 .00130 +5 .00137 .00132 10 .00142 .00138 15 .00148 .00144 20 .00154 .00150 25 .00158 .00154 As the quantity of water in the calorimeter sometimes varied slightly, the numbers should be modified to suit, they being true when the total capacity of the calorimeter was 8.75 kil. The total surface of the calorimeter was about 2350 sq. cm., and the unit of time one minute. To compare my results with those of McFarlane and of Nichol given in the Proc. R. S. and Proc. R. S. E., I will reduce my results so that they can be compared with the tables given by Professor Everett in his " Illustrations of the Centimeter-Gramme- Second System of Units," pp. 50, 51. The reducing factor is .0621, and hence the last results for the jacket at 20 C. become : TABLE XXXVI. Difference of Temperature. Coefficient of Radiation on the C. G. S. System. McFarlane's Value. K&tio. 3 .000081 .000168 2.07 6 .000082 . .000178 2.17 10 .OOOOSff .000186 2.16 15 .000089 .000193 2.17 20 .000093 .000201 2.16 25 .000096 .000207 2.16 OF ARTS AND SCIENCES. 171 The variation which I find is almost exactly that given by McFar- lane, as is shown by the constancy of the column of ratios. But my coefficients are less than half those of McFarlane. This may possibly be due to the fact that the walls of McFarlane's enclosure were blackened, and to his surface being of polished copper and mine of polished nickel : his surface may also have been better adapted by its form to the loss of heat by convection. The results of Nichol are also much lower than those of McFarlane. The fact that the coefficients of radiation are less with increased temperature of jacket is just contrary to what Dulong and Petit found for radiation. But as I have shown that convection is the principal factor, I am at a loss to check my result with any other observer. Dulong and Petit make the loss from convection dependent only upon the difference of temperature, and approximately upon the square root of the pressure of the gas. Theoretically it would seem that the loss should be less as the mean temperature rises, seeing that the air be- comes less dense and its viscosity increases./ Should we substitute density for pressure in Dulong's law, we should have the loss by con- vection inversely as the square root of the mean absolute tempera- ture, or approximately the absolute temperature of the jacket. This would give a decrease of one per cent in the radiation for about 6, which is not far from what I have found. To estimate the accuracy with which the radiation has been obtained is a very difficult matter, for the circumstances in the experiment are not the same as when the radiation was obtained. In the first place, although the water is stirred during the radiation, yet it is not stirred so violently as during the experiment. Further, the wheel above the calorimeter is warmer during radiation than during the experi- ment. Both these sources of error tend to give too small coefficients of radiation, and this is confirmed by looking over the final tables. But I have not felt at liberty to make any corrections based on the final results, as that would destroy the independence of the observa- tions. But we are able thus to get the limits of the error 'produced. During the preliminary experiments a water jacket was not used, but only a tin case, whose temperature was noted by a thermometer above and below. The radiation under these circumstances was larger, as the case was not entirely closed at the bottom, and so per- mitted more circulation of air. 3. CORRECTIONS TO THERMOMETERS, ETC. Among the other corrections to the temperature as read off from the thermometers, the correction for the stem at the temperature of 172 PEOCEEDINGS OP THE AMEEICAN ACADEMY the air is the greatest. The ordinary formula for the correction is .000156 n (t ") But, in applying this correction, it is difficult to estimate n, the number of degrees of thermometer outside the calo- rimeter and at the temperature of the air, seeing that part of the stem is heated by conduction. The uncertainty vanishes as the thermometer becomes longer and longer, or rather as it is more and more sensitive. But even then some of the uncertainty remains. I have sought to avoid this uncertainty by placing a short tube filled with water about the lower part of the thermometer as it comes out of the calorimeter. The temperature of this was indicated by a thermometer, by aid of which also the heat lost to the water by conduction through the ther- mometer stem could be computed ; this, however, was very minute compared with the whole heat generated, say 1 in 10,000. The water being very nearly at the temperature of the air, the stem above it could be assumed to be at the temperature of the air indicated by a thermometer hung within an inch or two of it. The correction for stem would thus have to be divided into two parts, and calculated separately. Calculated in this way, I suppose the correction is per- fectly certain to much less than one hundredth of a degree : the total amount was seldom over one tenth of a degree. Among the uncertain errors to which the measurement of temper- ature is subjected, I may mention the following : 1. Pressure on bulb. A pressure of 60 cm - of water produced a change of about 0.01 in the thermometers. When the calorimeter was entirely closed there was soon some pressure generated. Hence the introduction of the safety-tube, a tube of thin glass about lO 6 - long, extending through a cprk in the top of the calorimeter. The top of the safety-tube was nearly closed by a cork to prevent . evaporation. Had the tube been shorter, water would have been forced out, as well as air. 2. Conduction along stem from outside to thermometer bulb. To avoid this, not only was the bulb immersed, but also quite a length of stem. As this portion of the stem, as also the bulb, was surrounded by water in violent motion, there could have been no large error from this source. The immersed stem to the top of the bulb was generally about 5 cm - or more, and the stem only about .8- in diameter. 3. The thermometer is never at the temperature of the water, be- cause the latter is constantly rising ; but we do not assume that it is so in the experiment. We only assume that it lags behind the water to the same amount at all parts of the experiment, -and this is doubt- less true. OF ARTS AND SCIENCES. 173 To see if the amount was appreciable, I suddenly threw the appa- ratus out of gear, thus stopping it. The temperature was observed to continue rising about 0.02 C. Allowing 0.01 for the rise due to motion after the word " Stop " was given, we have about 0.01 C. as the amount the thermometer lagged behind the water. 4. Evaporation. A possible source of error exists in the cooling of the calorimeter by evaporation of water leaking out from it. The water was always weighed before and after the experiment in a balance giving ^ gramme with accuracy. The normal amount of loss from removal of thermometer, wet corks, &c. was about 1 gramme. The calorimeter was perfectly tight, and had no leakage at any point in its normal state. Once or twice the screws of the stuffing- box worked loose, but these experiments were rejected. The evaporation of 1 gramme of water requires about 600 heat units, which is sufficient to depress the temperature of the calorimeter about 0.07 C. As the only point at which evaporation could take place was through a hole less than I" 1 - diameter in the safety-tube, I think it is reasonable to assume that the error from this source is in- appreciable. But to be doubly certain, I observed the time which drops of water of known weight and area, placed on the warm calo- rimeter, took to dry. From these experiments it was evident that it would require a considerable area of wet surface to produce an ap- preciable effect. This wet surface never existed unless the calo- rimeter was wet by dew deposited on the cool surface. To guard against this error, the calorimeter was never cooled so low that dew formed ; it was carefully rubbed with a towel, and placed in the appa- ratus half an hour to an hour before the experiment, exposed freely to the air. The surface being polished, the slightest deposit of dew was readily visible. The greatest care was taken to guard against this source of error, and I think the experiment is free from it. (d.) Results. 1. CONSTANT DATA. Joule's equivalent in gravitation measure is of the dimensions of length only, being the height which water would have to fall to be heated one degree. Or let water flow downward with uniform velocity through a capillary tube impervious to heat ; assuming the viscosity constant, the rate of variation of height with temperature will be Joule's equivalent. Hence, besides the force of gravity the only thing required in ab- 174 PROCEEDINGS OP THE AMERICAN ACADEMY solute measure is some length. The length that enters the equation is the diameter of the torsion wheel. This was determined under a microscope comparator by comparison with a standard meter belong- ing to Professor Rogers of Harvard Observatory, which had been compared at Washington with the Coast Survey standards, as well as by comparison with one of our own meter scales which had also been so compared. The result was .26908 meter at 20 C. To this must be added the thickness of the silk tape suspending the weights. This thickness was carefully determined by a micrometer screw while the tape was stretched, the screw having a flat end. The result was .0003 l m -. So that, finally, J7 = .26939 meter at 20 C. Separating the constant from the variable parts, the formula now becomes f= g = 9.8005 at Baltimore. It is unnecessary to have the weights exact to standard, provided they are relatively correct, or to make double weighings, provided the same scale of the balance is always used. For both numerator and denominator of the fraction contain a weight. 2. EXPERIMENTAL DATA AND TABLES OF RESULTS. In exhibiting the results of the experiments, it is much more sat- isfactory to compute at once from the observations the work neces- sary to raise 1 UL of the water from the first temperature observed to each succeeding temperature. By interpolation in such a table we can then reduce to even degrees. To compare the different results I have then added to each table such a quantity as to bring the result at 20 about equal to 10,000 kilogramme meters. The process for each experiment may be described as follows. The calorimeter was first filled with distilled water a little cooler than the atmosphere, but not so cool as to cause a deposit of dew. It was then placed in the machine and adjusted to its position, though the outer half of the jacket was left off for some time, so that the calorimeter should become perfectly dry ; to aid which the calo- rimeter was polished with a cloth. The thermometer and safety- tube were also inserted at this time. After half an hour or so, the chronograph was adjusted, the outer half of the jacket put in place, the wooden screen fixed in position, and all was ready to start. The engine, which had been running OP ARTS AND SCIENCES. 175 quietly for some time, was now attached, and the experiment com- menced. First the weights had to be adjusted so as to produce equi- librium as nearly as possible. The observers then took their positions. One observer constantly recorded the. transit of the mercury over the divisions of the ther- mometer, making other suitable marks, so that* the divisions could be afterwards recognized. He also read the thermometers giving the temperatures of the air, the bottom of the calorimeter thermometer, and of the wheel just above the calorimeter ; and sometimes another, giving that of the cast-iron frame of the instrument. The other observer read the torsion wheel once every revolution of the chronograph cylinder, recording the time by his watch. He also recorded on the chronograph every five minutes by his watch, and likewise stirred the water in the jacket at intervals, and read its tem- perature. The recording of the time was for the purpose of giving the con- necting link between the readings of the torsion circle and of the ther- mometer. This, however, as the readings were quite constant, had only to be done roughly, say to half a minute of time, though the records of time on the chronograph were true to about a second. The thermometers to read the temperature of the water in the jacket were graduated to 0.2 C., but were generally read to 0.l C., 'and had been compared with the standards. There was no object in using more delicate thermometers. After the experiment had continued long enough, the engine was stopped and a radiation experiment begun. The last operation was to weigh the calorimeter again, after removing the thermometer and safety-tube, and also the weights which had been used. The chronograph sheet, having then been removed from the cylin- der, had the time records identified and marked, as well as the ther- mometer records. Each line of the chronograph record was then numbered arbitrarily, and a table made indicating the stand of the thermometer and the number of the revolutions and fractions of a revolution as recorded on the chronograph sheet. The times at which these temperatures were reached was also found by interpolation, and recorded in another column. From the column of times the readings of the torsion circle could be identified, and so all the necessary data would be at hand for cal- culating the work required to raise the temperature of one kilo- gramme of the water from the first recorded temperature to any succeeding temperature. ' 176 PROCEEDINGS OP THE AMERICAN ACADEMY As these temperatures usually contained fractions, the amount of work necessary to raise one kilogramme of the water to the even degrees could then be found from this table by interpolation. Joule's equivalent at any point would then be merely the difference of any two succeeding numbers ; or, better, one tenth the difference of two numbers situated 10 apart, or, in general, the difference of the num- bers divided by the difference of the temperatures. It would be a perfectly simple matter to make the record of the torsion circle entirely automatic, and I think- I shall modify the apparatus in that manner in the future. It would take too much space to give the details of each experiment ; but, to show the process of calculation, I will give the experiment of Dec. 17, 1878 as a specimen. The chronograph sheet, of course, I cannot give. The computation is at first in gravitation measure, but afterwards reduced to absolute measure. The calorimeter before the experiment weighed 12.2733 kil. " " after " " " 12.2716 " Mean 12.2720 Weight of calorimeter alone 3.8721 " .-. Water alone 'weighed 8.3999 " .3470 Total capacity 8.7469 " The correction for weighing in air was .835 X = .00106. The total term containing the correction is therefore .99878. log 86.324 =1.9361316 log .99878 = 1.9994698 1.9*356014 log 8.7469" = .9418542 log const, factor = .9937472 = log 9.85706. Hence the work per kilogramme is 9.85706 2JFn in gravitation measure, the term 2 W n being used to denote the sum of products similar to Wn as obtained by simultaneous readings of torsion circle and records on chronograph sheet. Zero of torsion wheel, 79.3 mm -. Value of l mm - on torsion wheel .0118 kQ -. The following were the records of time on the chronograph sheet : OP ARTS AND SCIENCES. 177 Time observed. Revolutions of Chronograph. Time calculated. 15 8.74 15.2 20 25.32 20.1 25 42.10 25.0 30 59.05 30.0 35 76.00 35.0 40 93.03 40.0 45 109.97 45.0 50 126.92 50.0 55 144.14 55.0 The times were calculated by the formula Time = .294 X Revolutions -f 12.66, which assumes that the engine moves with uniform velocity. As the principal error in using an incorrect interpolation formula comes from the calculation of the radiation, and as this formula is correct within a few seconds for all the higher temperatures, we can use it in the calculation of the times. The records of the transits of the mercury over the divisions of the thermometer were nearly always made for each division, but it is useless to calculate for each. I usually select the even centimeters, and take the mean of the records for several divisions on each side. While the mercury was rising l cm on No. 6163, there would be about seven revolutions of the chronograph, and consequently seven readings of the torsion circle, each one of which was the average for a little time as estimated by the eye. I have obtained more than thirty series of results, but have thus far reduced only fourteen, five of which are preliminary, or were made with the simple jacket instead of the water jacket, the radiation to which was much greater, as there was a hole at the bottom which allowed more circulation of the air. The mean of the preliminary results agrees so closely with the mean of the final results, that I have in the end given them equal weight. On March 24th, the same thermometer was used for a second ex- periment directly after the first, seeing that the chronograph failed to work in the first experiment until 8 was reached. The error from this cause was small, as the first experiment only reached to 26 C., and hence there could have been no change of zero, as this is very nearly the temperature at which the thermometer was generally kept. Having thus calculated the work in conjunction with the tempera- ture, I have next interpolated so as to obtain the -work at the even VOL. xv. (N. s. vn.) 12 178 PROCEEDINGS OF THE AMERICAN ACADEMY degrees. The tables so formed I have combined in two ways : first, I have added to the column of* work in each table an arbitrary number, such as to make the work at 20 about 10,000, and have then combined them as seen in . Table LI. ; and, secondly, I have subtracted each number from the one 1 farther down the table, and divided the num- bers so found by 10, thus obtaining the mechanical equivalent of heat. In these tables four thermometers have been used, and yet they were so accurate that little difference can be observed in" the experi- ments which can be traced to an error of the thermometer, although the Kew standard has some local irregularities. The greatest difference between any column of Table LI. and the general mean is only 10 kilogramme-meters, or 0.023 degree, and this includes all errors of calibration of thermometers, radiation, &c. This seems to me to be a very remarkable result, and demonstrates the surpassing accuracy of the method. Indeed, the limit of accuracy in thermometry is the only limit which we can at present give to this method of experiment. Hence the large proportional time spent on that subject. The accuracy of the radiation is demonstrated, to some extent, by the agreement of the results obtained even with different temperatures of the jacket. But on close observation it seems apparent that the coefficients of radiation should be further increased as there is a ten- dency of the end figures in each series to become too high. This is exactly what we should suppose, as we have seen that nearly all sources of error tend in the direction of making the radiation too small. For instance, an error came from not stirring the water dur- ing the radiation, and there must be a small residual error from not stirring so fast during radiation as during the experiment. Besides this, some parts around the calorimeter were warm during the radiation which were 'cool during the experiment. And both of these make the correction for radiation too small. However, the error from this source is small, and cannot possibly affect the general conclusions. In each column of Tables LI. and LII. a dash is placed at the tem- perature of the jacket, and for fifteen degrees below this point the error in the radiation must produce only an inappreciable error in the equivalent: taking the observations within this limit as the standards, and rejecting the others, we should still arrive at very nearly the same conclusions as if we accepted the whole. Most of the experiments are made with a weight of about 7.3 kiL as everything seemed to work best with this weight. But for the sake of a test I have run the weight up to 8.6 and down to 4.4 kil - by which the mte of generation of the heat was changed nearly three times. OF ARTS AND SCIENCES. 179 By this the correction for the radiation and the error due to the irregularity of the engine are changed, and yet scarcely an appreciable difference in the results can be observed. The tables explain themselves very well, but some remarks may be in order. Tables XXXVII. to L. inclusive are the results of fourteen experiments selected from the total of about thirty, the others not having been worked up yet, though I propose to do so at my leisure. Table LI. gives the collected results. At the top of each column the date of the experiment and number of the thermometer are given, together with Xhe .approximate torsion weight and the rate of rise of temperature per hour. The dash in each column gives approximately the temperature of the jacket, and hence of the air. There are four columns of mean values, but the last, produced from the combination of the table by parts, is the best. Table LII. gives the mechanical equivalent of heat as deduced from intervals of 10 on Table LI. The selection of intervals of 10 tends to screen the variation of the specific, heat of water from view, but a smaller interval gives too many local irregularities. In taking the mean I have given all the observations equal weight, but as the Kew standard was only graduated to ? F. it was impossible to calibrate.it BO accurately as to avoid irregularities of 0.02 C. which would affect the quantities 1 in 500. , Hence, in drawing a curve through the results, as given in the last column, I have almost neglected the Kew, and have otherwise sought to draw a regular curve without points of inflection. The figures in the last column I consider the best. Table LIII. takes the mean values as found in Tables LL and LII., and exhibits them with respect to the temperatures on the different thermometers, to the different parts of the earth, and also gives the reduction to the absolute scale. I am inclined to favor the absolute scale, using m = .00015, as given in the Appendix to Thermometry, rather than .00018, as ufeed throughout the paper. Table LIV. gives what I consider the final result of the experiment. It is based on the result m = .00015 for the thermometers, and is corrected for the irregularity of the engine by adding 1 in 4000. The minor irregularities are also corrected so that the results signify a smooth curve, without irregularity or points of contrary flexure. But the curve for the work does not differ more than three kilo- gramme-meters from the actual experiment at any point, and generally coincides with it to about one kilogramme-meter. These differences signify 0.007 C. and 0.002 C., respectively. The mechanical equiv- alent is for single degrees rather than for ten degrees, as in the other tables. 180 PROCEEDINGS OP THE AMERICAN ACADEMY TABLE XXXVII. FIRST SERIES. Preliminary. January 16, 1878. Jacket and Air about 14 C. ji Ja H 140 ' 160 180 203 220 240 259 289 | S Correction. Corrected Temperature. Revolutions of Chronograph Mean Weight W. Work per Kilogramme. I Work per Kilogramme. IWork per Kilogramme + 6380 1 1 52.0 '56.0 59.2 63.4 66.5 70.2 74.0 80.0 .005 .003 +.006 -j-.Oll +.020 +.028 +.045 .017 .022 .015 .001 +.027 +.067 +.161 9.185 11.412 13.650 16.230 18.137 20.392 22.538 25.943 5.485 18.023 30.652 45.329 56.241 69.158 81.484 101.214 7.509 7.478 7.442 7.394 7.364 7.354 7.292 951 1906 3010 3825 4786 5702 7156 o io n 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 348 775 1202 1629 2056 2484 2012 3340 3767 4193 4619 5048 5472 5899 6326 6753 7180 5728 6155 6582 7009 7436 7864 8292 8720 9147 9573 9999 10428 10852 11279 11706 12133 12560 TABLE XXXVIII. SECOND SERIES. Preliminary. March 7, 1878. Jacket 18.5 to 22.5. Air about 21 C. j Correction. $1 If i MS j J jj SI fijfj P.S o - ifc IS" 0^ &te 1 S& Je Ji | 1 1 n Is E * S 1 1 g ^ + 170 19.9 .016 12.537 6.03 13 198 7010 180 190 13.646 14 755 11.12 17 22 7.710 474 947 14 15 625 1052 7437 200 210 26.8 15.863 16.972 23.36 29.55 7.666 7.642 1421 1897 16 17 1480 1909 8292 8721 .010 .036 220 230 .... 18.085 19.196 35.70 41.90 7.630 2369 2845 18 19 2333 2761 9145 9573 24.0 250 260 270 280 290 33.8 +.003 .036 20.305 21.419 22.533 23.642 24.754 25.867 48.09 64.30 66.69 72.92 79.16 7.600 7.596 7.582 7.552 7.547 3319 3794 4740 5213 5687 20 21 22 23 24 25 3189 3615 4041 4467 4892 6318 10001 10427 10853 11279 11704 12130 *"* 40.8 +.020 .001 300 26.990 85.42 6164 26 6744 12556 * In the calculation of this column, more exact data were used than given in the other two columns, seeing that the original calculation was made every 6 mm. of the thermometer. Hence the last figure may not always agree with the rest of the data. OP ARTS AND SCIENCES. 181 TABLE XXXVIII. Continued. ji 1 Correction. Corrected Temperature. 1 Mean Weight TV. 8 1 Work per Kilogramme. 1 1 I 310 320 330 340 350 360 370 3.80 390 4V.8 61.4 +.044 +.073 28.119 29.253 30.393 31.540 32.689 33.842 34.998 36.158 37.321 91.67 97.98 104.28 110.67 117.12 123.54 130.04 136.56 143.08 7.611 7.604 7.611 7.617 7.602 7.592 7.576 7.550 7.550 6643 7125 7608. 8097 8590 9081 9576 10071 10567 28 29 30 31 33 34 35 36 37 6168 6593 7017 7441 7867 8294 8722 9149 9577 10004 10430 12980 13405 13829 14253 14679 15106 15534 15961 16389 16816 17242 65.0 58.7 +.072 +588 +.184 +.261 TABLE XXXIX. THIRD SERIES. Preliminary. March 12, 1878. Jacket 13.2 to 16.6. Air about!5 C. Is p g p Correction. Corrected Temperature. Revolutions of Chronograph 2n. i- & |l S3 g>^? p S w II Temperature. Work per Kilogramme. ijs *&* *a+ 00 I 205 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 28.0 28.6 29.9 31.1 32.4 33.6 34.9 36.2 37.4 38.7 39.9 41.2 42.5 43.7 45.0 46.3 47.6 48.9 50.1 51.4 52.7 64.0 55.3 +.663 +.002 +.6io 14.368 14.754 15.529 16.307 17.090 17.875 18.662 19.452 20.242 21.029 21.825 22.619 23.418 24.220 25.023 28.825 26.628 27.438 28.253 29.069 29.884 30.703 31.519 3.156 6.334 9.770 14.184 18.642 23.080 27.550 32.014 36.474 40.924 45.424 49.838 54.302 58.844 63.366 67.874 72.403 76.987 81.550 86.100 90.720 95.316 99.920 7.5167 7.5462 7.5668 7.5875 7.5763 I 7.5872 7.5801 164 495 827 1160 1495 1831 2167 2504 2840 3179 3514 3853 4194 4536 4876 5219 5565 5910 6255 6604 6951 7299 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 269 696 1122 1548 1975 2401 2828 3253 3676 4101 4526 4951 5378 5803 6226 6653 7078 7868 8295 8721 9147 9574 10000 10427 10852 11275 11700 12125 12550 12977 13402 13825 14252 14677 +.009 +.021 +.014 +.038 +.019 +.055 +.024 +.030 +.038 +.047 +.056 +.066 +.089 +.i20 +.159 +.202 +.251 +.304 * As this table was originally calculated for every 5 mm. on the thermometer, I have given the weights which were used to check the more exact calculation. 182 PROCEEDINGS OP THE AMERICAN ACADEMY TABLE XL. FOURTH SERIES. Preliminary* March 24, 1878. Jacket 5.4 to 8.2. Air about 6 C. ft Correction. 1| t - $ fg f JL S.ss I c If Rag I s Jafc 1 03 t Si* u 9* *ly ! II lit 130 140 150 160 27.4 29.2 31.0 32.9 +.002 8?071 9.204 10.340 11.480 42.364 48.898 55.438 62.066 7.471 7.446 7.442 485 968 1458 8 9 ]0 11 30 398 823 1252 4872 5300 5725 6154 +.010 +.019 170 180 34.7 36.6 +.017 +.050 12.620 13.763 68.669 75.330 7.390 1944 2433 12 13 1680 2107 6582 7009 190 200 38.4 403 +.025 +.093 14.908 16.054 81.973 88.597 7.431 2921 3410 14 15 2534 3960 7436 8862 210 220 42.2 44.2 +.034 +.150 17.202 18.350 95.264 101.941 7.437 3902 4395 16 17 3387 3815 8289 8717 230 46.1 +.046 +.222 19.604 108.588 4886 18 4245 9147 240 !. 19 4672 9574 250 7.4617 ''0 5098 10000 260 21 5524 10426 270 280 53.6 55.7 +.073 +.399 24.124 25.288 135.158 141.803 7.509 6855 7350 22 23 5950 6376 10852 11278 290 57.7 +.084 +.524 26.456 148.427 7844 24 6802 11704 .... 25 7228 12130 26 7651 12553 TABLE XLI. FIFTH SERIES. Preliminary. March 24, 1878. Jacket 5.4 to 8.4. Air about 6 C. 1 . Corre tion. 4 * M 1 4 1 dj p 1 I i l! Ei lf H Ir I* IS? I If f" 75 80 0.9 1.7 .003 L891 2451 3.154 6 118 8.1544 239 2 g 46 477 2296 2727 90 100 110 120 130 140 3.4 5.1 6.8 8.5 10.2 12,0 .002 b +.003 .012 .61 7 .612 3.569 4.690 5.810 6.936 8.060 9190 12.174 18.172 24.212 30.397 36.621 42 854 8.0900 8.0409 8.0074 7.9170 7.8973 7.8786 723 1200 1677 2161 2647 3132 4 6 6 7 8 g 906 1332 1759 2189 2621 3050 3156 3582 4009 4439 4871 5300 150 160 13.7 15.5 +.007 +.005 10.323 11 459 49.068 55 398 7.8512 7.8061 3614 10 3477 5727 170 180 17.2 190 +.015 +.032 12.600 13 742 61.707 68036 '7.7799 7.7622 4588 12 4333 6583 7009 190 200 210 20.8 22.6 243 +.024 +.028 +.068 +.092 14.882 16.025 17 170 74.358 80.716 87 064 7.7643 7.7807 7.8419 5558 6047 14 15 5183 5608 7433 7858 QOQO 220 230 26.1 27.9 +.039 +.150 18.316 19.467 93.402 99.677 7.8468 7.8579 7030 7518 17 18 6466 6895 8716 9146 * The first part of the experiments were lost, as the pen of the chronograph did not work. OF ARTS AND SCIENCES. TABLE XLI. Continued. 183 Thermometer No. 6163. , Correction. Corrected Temperature. Revolutions of Chronograph 2n. ^' Work per Kilogramme = 2 9.8816 Wn. Temperature. Work per Kilogramme. 1 I I 240 250 260 270 280 290 300 310 29.6 +.050 +.270 20615 105.950 7.8802 f 7.8980 7.9038 7.9091 7.8979 7.8974 8006 9482 9976 10474 10974 11481 19 20 21 22 24 25 26 27 7320 7745 8170 8597 9024 9451 9878 10305 10733 11160 9570 9995 10420 10847 11274 11701 11128 12555 12983 13410 34'.9 36.7 38.5 40.2 42.1 +.069 +.087 +.109 +.351 +.450 +.583 24.072 25.231 26.395 27.565 28.748 124.863 131.181 137.560 143.972 150.467 TABLE XLIL SIXTH SERIES. May 14, 1878. Jacket 12.l to 12.4. Air about 13 C. lo Correction. if ^ sg- %* e & i 5 1.1 !ii B* I 5 is Ii* 2 8 Jl 1 a 1 3J II i *g I *I if + 140 46.4 .002 9319 1.93 9 137 5296 150 47.9 10.178 7.07 7.2291 370 10 293 6726 160 49.4 ' '.000 .007 11.032 12.19 . 735 11 721 6154 170 50.9 11.886 17.37 } 7.1608 1102 12 1151 6584 180 62.5 +.002 .008 12.740 22.52 1467 13 1579 7012 190 54.0 13.596 27.70 7.1600 1835 14 2007 7440 200 55.5 +.006 .002 14.454 32.88 2201 15 2434 7867 210 57:0 15.314 38.07 7.1512 2568 16 2863 8296 220 68.5 +.6io +.6ii 16.174 43.29 2938 17 3290 8723 230" 60.0 17.037 48.50 7.1446 8306 18 3716 9149 240 61.6 +.015 +.031 17^093 63.70 3675 19 4142 9575 250 7.1536 20 4567 10000 260 21 4993 10426 270 280 66.2 67.7 +.024 +.075 20.500 21.362 69.27 74.50 7.1230 4778 5148 22 23 6420 5846 10853 11279 290 300 69.2 70.7 +.031 +.113 22220 23.076 79.69 84.84 7.1344 5514 5878 24 25 6271 6696 11704 12129 310 320 72.2 73.7 +.039 +158 23.928 24.774 89.97 95.05 7.1302 6240 6600 26 27 7121 7547 12554 12980 330 340 75.2 76.2 +.047 +.2i2 25.624 26.467 100.19 105.27 7.1117 6962 7319 28 29 7973 8400 13406 13833 350 360 78.2 79.7 +.056 +.272 27.309 28.147 110.39 115.44 7.0958 7680 8035 30 31 8829 9259 14262 14692 370 380 81.2 82.7 +.065 +.341 28.990 29.825 120.57 125.66 7.1076 8396 8754 32 33 9678 10096 15111 15529 390 84.2 +.076 +.417 30.663 130.78 7.1088 9115 .... 400 85.7 31.505 135.90 9475 .... 410 420 87.2 88.7 +.087 +.504 32.377 33.226 140.98 146.08 7.1064 9833 10192 '.'.'.'. 184 PROCEEDINGS OF THE AMERICAN ACADEMY TABLE XLIII. SEVENTH SERIES. May 16, 1878. Jacket 1P.8 to 12. Air about 12 C. Thermometer 1 No. 6163. 1 Correction. if 1! Revolutions of Chronograph Mean Weight W. 1 Temperature. Work per Kilogramme. 1 1 i 130 140 30.9 32.2 .004 8538 9.315 5.07 9.73 7.2350 335 9 199 6296 150 160 170 33'.6 350 36.3 .002 .006 10.094 10.875 11.654 14.36 18.98 23.56 7.3011 668 1003 1335 10 11 12 628 1056 1484 5725 6153 6581 .010 180 37.6 12.433 28.16 7.3165 1670 13 1913 7010 190 200 38.9 40.2 +.003 .008 13.209 13.984 32.74 37.31 7.3460 2003 2337 14 15 2344 2770 7441 7867 210 220 41.5 42.8 +.006 .000 14.758 15.536 41.84 46.38 7.3094 2667 2998 16 17 3196 3623 8293 8720 230 240 44^2 45.5 +.010 +.013 16.317 17.103 50.99 55.62 i 7.2846 3332 3667 18 19 4052 4478 9149 9575 250 260 46.9 48.3 +.014 +.032 17.891 18.682 60.29 } 7.2822 4005 20 21 4906 5324 10003 10421 270 280 49.6 50.9 +.619 +.056 19.475 20.269 69.63 74.34 7.2610 468i . 5021 22 23 6754 6179 10851 11276 290 300 52.3 53.6 +.025 +.090 21.079 21.866 79.01 83.71 7.2504 5358 5697 24 25 6603 7028 11700 12125 310 320 65.0 56.4 +.032 +.127 22.665 23.471 88.42 93.14 7.2893 6037 6379 26 27 7454 7883 12551 12980 330 67.8 +.039 +.172 24.281 97.88 6722 28 8307 13404 340 59.2 25.088 102.61 7.3047 7065 29 872'.) 13826 350 60.5 +.046 +.222 25.896 107.36 7410 30 9157 14254 360 61.9 .... 26.706 112.14 7.3389 7759 31 9582 14679 370 63.2 +.055 +.279 27.523 116.88 1 ' 8104 32 10009 15106 380 64.6 .... 28.346 121.62 \ 7.4109 8454 .... .... 390 66.0 +.065 +.345 29.172 126.34 1 . , 8801 400 67.4 29^996 13L12 7.4356 9155 410 68.8 +.075 +.419 30.827 135.90 9508 420 70.1 +.080 +.456 31.653 140.66 9861 .... TABLE XLIV. EIGHTH SERIES. May 23, 1878. Jacket 16.2 to 16.5. Air about 20 C. ii Ji 1 Correction. Corrected Temperature". Revolutions of Chronograph 2n. Mean Weight ,|* IgjW i 1 Work per Kilogramme. *9* & S 1 230 240 250 260 270 280 290 23.9 25.4 26.8 28.3 29.7 31.2 32.7 .007 16287 17.063 39.120 43.982 6.9137 > 6.9358 6.9007 6.9125 333 i338 1673 2010 17 18 19 20 21 22 306 735 1163 1592 2019 2446 8715 9144 9572 10001 10428 10866 .000 +.005 19.405 20.190 20.978 58.602 63503 68.428 .... OP ARTS AND SCIENCES. TABLE XLIV. Continued. 185 I Correction. . ** | . 5 ll 1!.- f !i| I la ig jM i 1 i l| r 1* ||| | : Iff H 3 EH PSO s n fcH P M N 300 34.2 21/765 73.351 6.8878 2346 23 2871 11280 310 320 35.6 37.1 +.008 +.040 22.554 23.350 78.283 83.245 6.8866 6.8594 2682 3020 24 25 3298 3722 11707 12131 330 38.6 24.151 88.314 6.8358 3363 26 4150 12559 340 350 40.1 41.6 +.6i7 +.085 24.952 25.751 93.294 98.275" 6.8748 6.9184 3702 4044 27 28 4574 4999 12983 14408 360 43.1 26.552 103.232 6.9444 4385 29 5423 13832 370 380 44.6 46.0 +.028 " " +.144 '*" 27.361 28.175 108.216 113.269 6.9291 6.9338 4727 5074 30 31 5851 6275 14260 14684 390 47.5 28.989 118.231 6.9385 5418 .... 400 410 49.0 50.6 +.039 +217 29.800 30.624 123.329 128.399 6.9444 6.9467 5766 6115 .... .... 420 52.1 +.047 +.281 31.445 133.480 6.9314 6464 .... TABLE XL V. NINTH SERIES. May 27, 1878. Jacket 19.6 to 20. Air about 23 C. I Correction. g* J ^ life 4 C 6 II lii t ll 3 M *l tii 1 1 fl (Sw i " S ll 1 Mr 200 380 .015 15890 6.33 16 47 8293 210 394 17:000 11.74 8.8108 473 17 473 8719 220 409 .011 .010 18.106 17.17 946 18 901 9147 230 49, 3 19.219 22.62 8.7341 1419 19 1326 9572 240 438 .005 .011 20.329 28.13 1895 20 1754 10000 250 45.3 21.442 33.68 2368 21 2180 10426 260 270 +.002 .004 22.552 23659 [ 8.4800 .... 22 23 2606 3031 10852 11277 280 290 49.8 51.3 +.009 +.012 24.771 25.885 60.55 56.25 ) 8.4399 3785 4263 24 25 3457 3883 11703 12129 300 310 52.9 544 +.019 +.037 27.006 28.133 61.93 67.63 8.4765 4737 5215 26 27 4312 4734 12558 12980 320 330 56.0 575 +.029 +.072 29.264 30.404 73.36 79.15 8.4552 5697 6182 28 29 5159 5584 13405 13830 340 350 59.1 60 fi +.042 +.118 31.552 32.702 84.97 90.85 } 8.4015 6669 7159 30 31 6010 6435 14256 14681 360 370 62.2 638 +.056 +.173 33.853 35.011 96.78 102.66 [ 8.4222 7652 8143 32 33 6860 7286 15106 15532 380 390 65.4 670 +.071 +.242 36.170 37.331 108.59 114.45 ) 8.4706 8638 9128 34 35 7714 8138 15960 16384 400 410 68.6 70.2 +.088 +.322 38.497 39.664 120.36 126.33 8.4316 9626 10126 36 37 8565 8988 16811 17234 420 71.8 +.105 +.419 40.833 132.26 10620 38 39 9414 9842 17660 18Q88 40 10268 18514 41 10691 18937 186 PROCEEDINGS OP THE AMERICAN ACADEMY TABLE XL VI. TEHTH SERIES. June 3, 1878. Jacket 18.l to 18.4. Air about 20 C. g b Correction. "Sji ^ |P S i if fl If" 1 ||1 5 g 11 Iff H I 1 1 k II |v *$% H *B 250 4.1 .007 17838 7.82 18 69 9145 260 7.0 18.617 4.3899 19 496 9572 270 9.9 .003 +.004 19.401 23.'l9 '667 20 925 10001 280 12.8 20.188 3095 4.3919 1005 21 1350 10426 290 15.7 +.003 +.020 20.978 38.70 1341 22 1778 10854 300 18.7 21.763 46.41 i 4.3912 1676 23 2204 11280 310 320 21.6 24.5 +.008 +.037 22.551 23.354 54.21 62.04 ) { 4.3907 2014 2354 24 25 2627 3054 11703 12130 330 340 27.5 30.5 +.014 +.078 24.162 24 970 69.92 77.92 J 4.3624 2696 3041 26 27 3479 3904 12555 12980 350 360 336 36.6 +.020 +.132 25.780 26.593 85.89 93.94 4.3542 3385 3731 28 29 4332 4852 13408 13828 370 396 +.028 +.198 27.415 102.05 4081* 30 5179 14255 380 42.7 28.246 110.34 4.3362 4437 31 5604 14680 390 45.8 +.036 +.281 29.079 118.49 4786 .... .... 400 48.9 29.911 126.66 4.3978 5141 410 52.0 +.044 +.377 30.754 134.89 5499 .... .... TABLE XLVII. ELEVENTH SERIES. June 19, 1878. Jacket 19.6 to 20. Air about 23 C. Thermometer No. 6163. 5 H Correction. g SI ReTolutions of Chronograph 2n. Mean Weight III O C OJ s w Temperature. Work per Kilogramme. *3 + 1 I 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 ... .002 +.002 +.006 +.029 +.063 -f.iis 2L450 22.562 24.789 25.007 27.032 28.168 29.307 30.456 31.612 32 774 8.933 16.087 30.281 37.439 44.655 61.848 69.098 66.390 73.724 81.163 88.462 95.734 103.093 110.560 118.121 126.693 133.250 '6.7572 6.7678 6.7749 6.7896 6.7973 | 6.8188 6.9165 6.7876 6.7808 476 i42J 1899 2379 286p 3344 3832 4323 4817 5311 5807 6307 6808 7311 7815 8321 21 22 23 24 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 192 235 602 1087 1511 1939 23*55 2789 3214 3638 4003 4488 4913 5337 6760 6187 6614 7040 7465 7891 8317 10428 10855 11282 11707 12131 12569 12985 13409 13834 14258 14683 15108 15533 15957 16380 10807 17234 17660 18085 18511 18937 +.010 +.019 +.031 +.043 +.177 +.058 +.072 +.257 +.351 33.939 35.110 36.280 37.456 38.637 39.821 41.010 +.087 +.106 +.463 +.596 OF ARTS AND SCIENCES. 187 0961 + tauiBjSonji J3d Jt.lO.ii. ~ ~ X ^1 1^ I-H i work pe ilogram CC W CO C3i * CC CO O * rti j co co ' co ao co a>Q5oa>Qajao0Q S qdBoSouoaqo jo SiggggSSSSScSiSSSg g S5 : : : (6 co cc t-^ t^ CD co to o so d o co t-^ t-^ co t^ 05 *i t^ r I I jo suonn[OA8H 1 1 1 ++++++++++++++++++ .I._I_._I_._L._L.O --^ -co - \^ . T^ . l^ . |7 . |^ .(?q CO '-^t* 'i ' "M 7* * .. 1 .. .. .. .. . '8919 J ffl JB CO O lO O O O 'CO -CD CO 'CO CO 'CO '"^ *O "lO '*O Ico Ico Ico led Ico" Ico Ico !cd I I I 8919 -ON *q J3J3UIUOIBOJO oanjK.iadiu.JX * r-! cs co * vo co r^ oJ o r cc 'S9T9 'ON g ; J3;3uiouijaqx oo to . 188 PROCEEDINGS OP THE AMERICAN ACADEMY TABLE XLIX. THIRTEENTH SERIES. Dee. 19, 1878. Jacket 3.2 to 3.5. Air 4.2 to 5.2 C. 55 Corrections. ^ ^ ,-i S |i if IL I f|* I 1 El 1 ; H I I || If G**" * 6 a! 1 1 H I 70 1248 1.72 Oj 106 1858 8.6610 485.0 80 .... ' 2.378 7.38 485.0 2 +323 2287 8.5571 485.1 90 -.003 3.500 13.11 970.1 3 754 2718 8.4325 482.2 100 4.626 18.89 1452.3 4 1184 3148 8.3688 481.1 110 +.001 +.003 5.751 24.70 1933.4 5 1612 3576 8.4155 487.1 120 6.881 30.55 2420.5 6 2041 4005 8.4189 485.6 130 +.005 +.019 8.013 36.38 2906.1 7 2472 4436 8.3953 489.2 140 9.148 42.27 3395.3 g 9001 .Q_ 8.4366 486.6 150 +.009 +.044 10.284 48.10 3881.9 9 3331 5295 8.4484 486.5 160 11.424 53.92 AnpQ A - -. Q7Pf\ _^ . 8.4189 490.6 &12A 170 +.016 +.080 12.569 59.81 4859.0 11 4187 6151 8.3988 491.1 180 13.713 65.72 5350.1 12 4615 6579 8.4153 487.1 190 +.023 +.126 14.859 71.57 5837.2 13 5045 7009 8.3811 491.7 200 16.005 77.50 6328.9 14 5472 7436 210 +.033 +.183 17.154 83.40 8.3835 489.4 6818.3 15 5898 7862 8.3976 490.2 220 18.300 89.30 7308.5 16 6327 8291 8.4035 493.0 230 +.044 +.251 19.452 95.23 7801.5 17 6753 8717 8.4460 496.4 240 20.604 101.17 8297.9 18 7180 9144 250 +.056 +.332 21.760 f- 8.4555 981.3 19 7608 9572 260 22.912 112.90 9279.2 20 8038 10002 270 +.069 +.424 24.065 118.81 8.4602 494.7 9773.9 21 8465 10429 280 25.221 124.70 8.4779 494.0 10267.9 22 8891 10855 23 9317 11281 25 9746 10173 11710 12137 OF ARTS AND SCIENCES. 189 TABLE L. FOURTEENTH SERIES. December 20, 1878. Jacket 1.5 to 1.9. Air about 3.4 C. Temperature by Kew Standard. 1 Corrections. Corrected Tem- perature, Abso- lute Scale. Revolution of Chronograph Mean Weight TV. Work per Kilogramme = 2 9.7832 Wn. I Work per Kilogramme. Work per Kilo- gramme -f 2210. Reduction to Absolute Scale. I 1 36.0 66.0 .00 1.82 8.03 7.3682 2 77 2287 38 5 584 3.23 1637 7.3458 '601 3 * 503 2713 41.0 .9 .01 .00 +.01 4.62 24.78 7.3705 1206 4 936 3146 43.5 3.3 .... .... 6.02 33.19 7.4012 1812 6 1370 3580 46.0 5.8 .02 +.01 +.04 7.43 41.48 7.4142 2412 6 1803 4013 485 8 2 884 4981 7 4177 3016 7 2226 4436 51.0 10.7 .03 +.02 +.09 10.26 68.18 7.4390 3624 8 2656 4866 535 13 2 11 68 66 56 7 4107 4234 9 3084 5294 56.0 15.6 .04 +.03 +.16 13.12 74.95 7.3493 4842 10 3513 5723 58.5 18? 14.56 83.56 7.3269 5461 11 3942 6152 61.0 20.7 .04 +.05 +.25 16.01 92.27 7.2335 6085 12 4369 6579 63.5 23.3 .... 17.46 100.99 7.1603 6703 13 4790 7000 66.0 25.9 .05 +.06 +.38 18.92 109.95 7.2076 7330 14 5220 7430 685 28 5 2039 118 84 7 1839 7957 15 5650 7860 71.0 31.2 .05 +.08 +.62 21.86 127.83 7.2122 8689 16 6081 8291 73.5 33.8 ..:. 23.34 136.75 7.2252 9218 17 6507 8717 76.0 36.5 .06 +.10 +.69 24.84 145.78 7.2134 9857 18 6935 9145 785 392 26 33 15480 10493 19 7364 9574 ?0 7791 10001 ftl 8219 10429 9,9: 8648 10858 TO 9074 11284 ?4 9499 11709 fln 9925 12135 26 10352 12562 190 PROCEEDINGS OP THE AMERICAN ACADEMY ifc h O M u II ERS AT BALTIMORE TO HEA GIVEN TEMPERATURE ON TH anoq aad 1!^ Si *q3 IS *qa jnoq aad 'III 9'8 jnoq jad ( R1 6'9 ?U a anoq aad 1PI 8'9 ? 9919 anoq aad IW I'i 9919 anoq jd '111 8'i 'R 3 '8919 '88U lt* i 'COTfOCOO^OCOCOOO TJI r~ Q o S S 51 r- -. 3 S 5 cp .c* S t- OO5^fc6c^COr-IOC5COCOO >6 3 5J | Si 3 $ 3 O r-4 Si Si Si q <=>. -i 8 cs co cq p >o i s 8 s s 1-4 co 10 t- Tt< a'- l ?DlMCO^OiiOC5 ^t*t^-O^iOt^-OC^iOt OCNtOr^-O^or-OC^Ot OT*i OJOSOOO' ; -'^SNCiOOOOOO .pppr-jr-;^-;^-.-;^ e4eo4igH COCOCOCOCOCOCOCO-tl-* gggggggo sssssssssss ^5cie