Of the earlier observations in the preceding table, I need not say much. Those of Mr. Watt, which he suffered to lie by him for forty years, and, in the caustic phrase of Tredgold, only produced when they had become unnecessary, he was himself dissatisfied with; but as appears upon comparison, with more modesty than reason. I have specially calculated but two or three of his temperatures; and, of the whole sixty-two experiments, have inserted but twenty-two; among which, however, both the limits are to be found. Of his friend Robison's, I have had to calculate none specially; but all happened to find a place in the table. Of Mr. Betancourt's numerous observations which were reported originally in degrees of Reaumur and French inches, I have inserted only those which have been reduced to English scales by Sir D. Brewster for the Edinburgh Encyclopædia.

The experiments of Mr. Southern are, in fact, the supplement of those of Mr. Watt; having been made and reported at the desire of the latter. The numbers will be found to differ, somewhat, from those generally found in professed treatises on the Steam-engine; they are in fact the mean of the actual observations; while those usually given, have been selected now from one and now from the other set, and reduced (by himself) to what they might have been, had the pressure at 212° been thirty inches. For the present purpose, it seemed to me proper to state the real, not the possible, result.

Mr. Dalton's experiments were distinct; and are therefore given in distinct columns. The numbers in the earlier column, marked with an asterisk, were not from actual experiment; but by interpolation, according to the method he has himself explained. I have inserted them opposite to experimental numbers in the adjoining column, for the sake of comparison and the benefit of the inference which may flow from the variations. The numbers in the later column were not, in every case, given by his own experiments; but they were accepted by him as authentic, and the most reliable he knew. It is more complimentary to his reputation than to their own research, that compilers of Chemical manuals, even down to the present time, retain among their tables his ancient results whose inaccuracy he himself has recognized. All of his experiments, of Southern's, and of Dr. Ure's, are in the table. To the originals of Mr. Arzberger, I have not had access; but I have found these quoted, in so many authorities and so uniformly accordant that I have not hesitated in recording them. Of the extensive table of Mr. Taylor, whose remarkable accord with the results from the formula I may be allowed to call attention to, I have taken only those epochs of temperature which were already in my table.

The experiments of the French Academy have been already signalized. It is enough to establish their claim to distinction, to say that they were executed by Dulong and Arago; names that have been long since inscribed in the very highest rank of physical philosophers. The numbers found in the appropriate column, are, agreeably to what I have already mentioned as governing me throughout, quantities actually observed. The temperatures and pressures generally quoted in the text-books on Steam, as of the French Academy, are not, in fact, what they observed; but what they deduced (in part, by a formula of their own, and in part, by Tredgold's) from the present experimental series. The pressure 29 92 inches corresponding to the temperature 2120, is marked with an asterisk, because it is not expressly declared to have been observed. It is the height which is constantly taken in France for the barometric standard, as thirty inches are in England: in the latter assumption, the temperature is rated at 60° F.,-in the former, at 32° F.; and the difference of heights is nearly identical with the difference of expansion at the respective temperatures.

The pressure in this series corresponding to the temperature 368-51, is also noted with a dagger; it may be presumed to be erroneous, not only because it differs so much from the result by my formula, but because it varies so much and so suddenly from the rate accused by the pressure on either side of it. Nor does it correspond at all with their own formula; calculated by that, the pressure will be 335 87 inches. The error is not, in this instance, SECOND SERIES, Vol. VI, No. 17.—Sept., 1848.

29

of the press; since it makes its appearance in both ways of reckoning, by atmospheres and by metres.

I do not know how to account for another discrepant pressure, corresponding to the temperature 405°-04; which has been indicated by a note of interrogation. On both sides, above and below, the observed pressures are higher than the calculated one; in this instance, it is suddenly lower. It agrees, to be sure, with an independent calculation by the formula of Dulong and Arago, at the temperature; but very manifestly breaks the uniformity or any regular progression of the series. What adds to the difficulty, is that the same observation is given again in another part of the Memoir of the Academicians; but the ciphers do not agree. have neither altered nor omitted either of these instances; it is obvious that they are not to be used in comparison with the present formula.

The temperatures of the Franklin Institute, which were taken for the composition of the table, come from the second series of their reported experiments. Pressures have been also taken from both the other series, when their temperatures were already in the table; and, adopting this method as a uniform system, I did not allow myself to exclude the anomaly which shews itself between the different series at the temperature of 300°.

Of the experiments of Mr. Regnault, I have already spoken sufficiently.

It is apparent, upon a slight examination of the table, that the calculated pressures do not differ more from the average of the experimental ones, than these experimental ones do among themselves; which is about as much as could be desired to shew the validity of the formula, and the reasonableness of its application, instead of others which are in general merely means of interpolating a particular experimental series. But in order to establish this more clearly, it will be necessary to ascertain more distinctly what the difference is between the results of the formula and those of observation. This difference is, of course, best expressed in the arithmetical scale of temperatures; as I have tabulated it here, upon the maximum deviation in each instance.

The mean of the sum of these differences, is +10-09 Fahrenheit; which is the maximum error of the formula, compared with these six series.

It will be observed that I have left out of this comparison, the last observation of the Academy; because it was the very utmost point which the apparatus could carry, and because it might therefore be expected to be affected by the untrustworthiness which forbade the series from being extended farther. I have also neglected the last observation of Arzberger; which, compared with the Academy's, is in error more than 100,-a deviation sufficient to discredit it entirely. Ure's experiments I have not compared at all; because, if we admit the series just now tabulated, his results are altogether too high. He may, however, be compared with himself, in the two results he has recorded for his last observation. These two different pressures accuse a corresponding difference of temperature of 0.63 F.; a possible error, not so materially less than what we have found as the maximum that can attach to the formula. The Franklin Institute experiments, which correspond closely with Ure's, I have omitted for a similar reason; they do not profess, even, to read nearer than 0.25 F. They may, however, for illustration be compared with those of the Academy, as under:

Academy pressure, 145 15 inches; temp. observed, 3050-39; temp. calculated, 3070-51. Institute 154-28 66

305 .50;
00.11

64

311 73.

40.22

Discounting the observed difference from the calculated one, we have left 40.11 F., as the error of one or the other series; an amount nearly four times that of the formula.

It is manifest that the comparative error of the formula is only approximate; because it is based, in each case, upon only one observation instead of upon the combined mean of all the observations or, rather, the mean of the differences at every epoch observed. Also, it can only be called an error, upon the assumption of the mean of all the experiments resulting in absolute accuracy; an assumption by no means to be made; for, in general, the utmost that can be done for any experimental series, is to determine the limits of its necessary or accidental errors. Such a research and determination, I have thought the present formula of sufficient interest to warrant. The account, which is in fact the promised and proper conclusion of the present paper, will appear in a future Number of this Journal.