Supplement to the article Thermometer. Immediately connected with the construction and improvement of thermometers are M. Biot's "Researches into the laws of the dilatations of liquids at all temperatures;" of which, therefore, I shall present, in this place, a translation with slight abridgment. The knowledge of the laws observed in the dilatation of liquids is necessary in an infinity of chemical and physical enquiries. We require the dilatations of water, in order to reduce the specific gravities observed in that liquid to comparable terms. We require those of alcohol to determine its density at different temperatures, or to observe the thermometers in which that substance is employed. Or, if we would attempt theoretically to compare the dilatability of different liquids respectively, and to connect their more or less rapid progress, with their tendency more or less near to ebullition and to solidification, we cannot accomplish it generally, or obtain any precise ideas. on this point, without expressing the dilatations by formulæ which shall represent them at all temperatures, and at the same time render evident the particularities of each of the liquids it is wished to examine. Such is the object M. Biot proposes to himself. He shows that for all liquids whose dilatations have hitherto been observed, the general progress of each respective dilatation may be represented at all temperatures by an expression of this form: d=at + b2+ct3, t in which t denotes the temperature in degrees of the mercurial thermometer, and a, b, c, constant coefficients which depend on the nature of the liquid. He here supposes that is the true dilatation for the unit of volume reckoned from the temperature of thawing ice; but it is easy to conclude, hence, that the apparent dilatation follows similar laws: for, representing this latter byt and denoting by K the cubic dilatation of the matter of the vessel that contains the observed liquid †, we have A=-Kt; neglecting here the square of the co-efficient K, which is almost The paper that contains these researches was read to the Society of Arcueil, August 8th, 1813. It is printed in France; but the work in which it is to appear had not been published in March 1815. I have to acknowledge my obligations to M. Biot, for transmitting these investigations to me, through the medium of M. Hachette, (Professor of Mathematics in the Polytechnic school,) for the purpose of insertion in this work. †M. Biot means by cubic dilatation the triple of the linear dilatation. always allowable, since the dilatation of solid bodies is extremely small. Let it be supposed that the primitive volume of the liquid being 1 when t=0°, occupies at +t degrees a number of divisions x in the vessel whose cubic dilatation is K. This number of divisions will indicate a greater capacity than when t was nothing. It will answer to the capacity x (1+kt+÷K22) limiting the expression to the square of K; and as by supposition it is equal to 1+, since & is the true dilatation for the unit of volume, we shall have the equation x(1+kt+÷K22) = 1+8;; which gives The first term of this expression is the primitive volume at 0°; the second is the apparent dilatation A,: we have, therefore, The term affected by K2 is absolutely insensible in the most exact observations on the dilatations of liquids made with glass vessels, between the temperatures of 15° and +100° (centi 1 + kt a grade). Neglecting it, therefore, we have simply 4=value which, by neglecting the square of K and the product of K byd, reduces to A-Kt, as we have assumed above. Now, to establish the preceding law, and determine the coefficients a, b, c, relatively to different liquids, M. Biot employed the results of a series of experiments, made, with much care, by De Luc*, on the dilatation of nine different liquids with which he had constructed thermometers, regulating them from the temperatures of melting ice, and boiling water; marking 0 at the first point, 80° at the second, and dividing the interval into 80 equal parts. These liquids were, 1. Mercury. 2. Oil of olives. 3. Essential oil of camomile. 4. Essential oil of thyme. 5. Water saturated with muriat of soda. 6. Alcohol highly rectified. 7. Alcohol and water in equal parts. 8. Alcohol one part, water three. 9. Water. • Researches on the Modifications of the Atmosphere, vol. i. T If we express by D, the number of degrees indicated by each of these thermometers, on its own scale, when T is the number indicated by the mercurial thermometer divided into 80 parts, all the experiments of De Luc may be represented by the general formula DAT+BT2+CT3; A, B, C, being arbitrary constant quantities, differing in the different liquids: and of which the absolute values, as inferred by M. Biot, from De Luc's experiments, are presented in the following table, Values of the co-efficients. Nature of the liquids. A B C Mercury +1·000000 +0·0000000 + 0·000000000 +0-820006+0-0020275+0000002775 muriat of soda Alcohol highly rect. +0784000 +0·0020800+0·000007750 1 Alcohol, 1 water+0-705333 +0.0027500 +0·000011667 1 Alcohol, 3 water +0 010333+00155277-0·000089444 Pure water −0·160000+0·0185000 −0·000050000 To prove the correspondence of these results with the observations, M. Biot has computed the values of D by the formula for each of these liquids for every 10°, and compared them with the numbers given by De Luc's observations. 1. Thus, computing the thermometer of oil of olives from the formula D=0;950667T+0.00075 T2-0'000001667T3, we have the following comparative table. Here the greatest variation between the observation and the computation is less than one fourth of a degree; and nearly all the rest are exceedingly minute. Mr. De Luc put the oil of olive thermometer several times in a refrigerating mixture which caused the mercurial thermometer to descend to 14°; and he relates that the oil thermometer remained nearly at that degree so long as the oil was not congealed. This result agrees with the preceding formula; for if we suppose т = — 14°, the formula gives D = - 13°.21. But when the oil began to congeal, the olive oil thermometer fell, all at once, much lower than the mercurial thermometer; the oil, indeed, retired entirely within the ball. It will be seen that it was the congelation which produced this sudden depression: for, when it has taken place, if the temperature be raised, the mercurial thermometer will rise immediately, but the oil thermometer will remain at its extreme point of depression, during an interval sometimes considerable, being, without doubt, that which the oil would require to become uncongealed. But having once resumed its liquid state, it will soon regain its relative sition as to the mercurial thermometer, and manifest its accustomed progress. De Luc supposed that it was the privation of air which enabled the oil to undergo, without congealing, a degree of cold which would have caused its congelation in the open air. But it appears from the experiments of Sir Charles Blagden, that neither the exclusion of the air, nor rest, are absolutely necessary to the production of that effect, though they may contribute to it. po We may learn from these phenomena, 1. That oil of olives may, in certain circumstances, as well as water, be depressed to a temperature far lower than its ordinary degree of congelation, without ceasing to be liquid. 2. That it contracts in congealing as mercury does, which is evident of itself, as the parts congealed retire to the bottom of the vessel. S. That down to the very moment in which it becomes solid, it retains exactly or very nearly the same law of dilatation: this appears also to obtain with respect to mercury, as we conclude from the discussion of Mr. Cavendish relative to the experiments of Hutchins at Hudson's Bay. Hence we see that olive oil in cooling to any degree whatever, cannot, like water, have an apparent maximum of condensation, at least in glass tubes. This, also, is shewn by our formula for this maximum would answer to the case in which we should have |