UC-NRLF 
 
 7DM 
 
UNIVERSITY OF CALIFORNIA. 
 
 FROM THE LIBRARY OF 
 
 DR. JOSEPH LECONTE. 
 
 GIFT OF MRS. LECONTE. 
 No. 
 
, 
 
LECTURE-NOTES 
 
 PHYSICS. 
 
 BY ALFKED M. MAYEK, P H I>., 
 
 PROFESSOK OF PHYSICS IN THE LEHIGH UNIVERSITY, BETHLEHEM, PA. 
 
 PART I. Containing 
 
 \ I. DEFINITIONS AND INTRODUCTION TO THE INDUCTIVE METHOD. 
 
 II. INSTRUMENTS USED IN PRECISE MEASUREMENTS. 
 
 $ III. METHODS OF PRECISION. 
 
 IV. MANNERS OF EXPRESSING A LA.W LAW EVOLVED FROM THE NUMERICAL 
 
 RESULTS OF OBSERVATIONS AND EXPERIMENTS. 
 
 \ V. THE GENERAL PROPERTIES OF MATTER THE CONSTITUTION OF MATTER 
 
 ACCORDING TO THE MOLECULAR HYPOTHESIS. 
 
 2 VI. CAPILLARY ATTRACTION. 
 
 PHILADELPHIA: 
 FROM THE JOURNAL OF THE FRANKLIN INSTITUTE. 
 
 1868. 
 
"Who hath measured the waters in the hollow of His hand, and meted out heaven 
 with the span, and comprehended the dust of the earth in a measure, and weighed 
 the mountains in scales, and the hills in a balance?" (!SAIAH xl. 12.) 
 
 " Thou hast ordered all things in measure and number and weight." (BooK OF 
 THE WISDOM OF SOLOMON, chap. xi. 20.) 
 
 " It ought to be eternally resolved and settled that the understanding cannot decide 
 otherwise than by induction, and by a legitimate form of it." 
 
 " Francis of Verulam thought thus, and such is the method which he determined 
 within himself, and which he thought it concerned the living and posterity to 
 know." 
 
 " Let no one enter here who is ignorant of geometry." (PLATO.) 
 
LECTURE-NOTES ON PHYSICS. 
 
 I. Definitions and Introduction to the Inductive Method. 
 
 Physical science is the knowledge of the laws (c) of the phenomena 
 (b) of matter (a). 
 
 (a.) Matter is that which affects the senses. It always presents 
 the three dimensions, or, in other words, occupies space. 
 
 It exists throughout all known space as a highly elastic and rare 
 medium called ether (proof of this given in optics), and in this medium 
 circulate dense bodies of spheroid forms, separated from each other by 
 distances which are immense when compared with the size of these 
 bodies; such are the planets and asteroids of celestial space, and the 
 earth on which we live. 
 
 Shooting-stars or Aerolites are celestial bodies of smaller size, 
 which at certain periods (November 13th and August 10th) cross 
 the orbit of the earth. 
 
 Comets are highly rarified nebulous bodies of various aud chang- 
 ing forms, circulating in orbits which are so eccentric, that they are 
 not visible to us in their remote situations; while at other times they 
 are much nearer to the sun than any of the planets. The comet of 
 1680, when nearest to the sun, was only one-sixth of the sun's dia- 
 meter from his surface. 
 
 The motion of comets retarded by the resistance of the ether? 
 
 The above is a statement of all known matter, and physical science 
 is that branch of knowledge which considers all the various pheno- 
 mena which this matter presents. 
 
 From the investigations of chemists on terrestrial matter, and from 
 the spectroscopic examination of celestial bodies, all matter can be 
 resolved into (at present) sixty -three elements. 
 
 According to the atomic theory see (V.) all matter consists of 
 exceedingly minute, absolutely jiard and unchangeable atoms, sepa- 
 rated from each other by distances which are very great when com- 
 pared with the dimensions of these atoms, 
 
 As in interplanetary space exists the rare ether, so interatomic 
 
 (3) 
 
 145274 
 
4 Lecture- Notes on Physics. 
 
 space is occupied by the same elastic medium. (Proofs of above given 
 under Undulatory Theory of Light and Heat.) Matter exists in the 
 three states of solids, liquids, and gases. 
 
 Each organ of our senses is so constructed that it can only take cog- 
 nizance of those effects which it was specially designed to receive. 
 Thus, through the eye we perceive light, but not sound ; and the ear 
 does not take cognizance of light, nor of flavor, nor of odor. 
 
 All the senses are modifications of touch. 
 
 In touch, taste, and smell the organs of these senses come in con- 
 tact with the matter that we touch, taste, and smell. In the cases of 
 sound, of light, and of heat, these effects are propagated through an 
 intervening elastic medium whose particles, vibrating in unison with 
 the particles of the sounding, luminous, or heated body, cause the 
 nerves of the ear, of the eye, or of the skin to pulsate; and thus we 
 have special sensations which we severally interpret as sound, light, 
 and heat. 
 
 Experiments showing the transmission of sound vibrations 
 through the air, through a liquid and through a long rod to the 
 ear. The human body can be the intervening vibrating medium to 
 transmit the sound. 
 
 Experiment. Sound not propagated through a vacuum. Ether is 
 the intervening vibrating medium in the case of light and heat. 
 
 A shadow is not matter, because it does not affect the senses. The 
 light reflected from the surface, contiguous to the shadow, affects the 
 eye; but from where the shadow is, no effect emanates, and the con- 
 sciousness of the absence of an effect on that part of the eye where the 
 shadow is projected gives us the idea of darkness. 
 
 According to Helmholtz and Du Bois Raymond an effect is propa- 
 gated along the nerves with a velocity of 93 feet in one second. 
 
 (b.) Phenomena. A collection of associated facts. 
 
 Example. Compression of air. Here the associated facts are two. 
 (1) The volumes of air corresponding to (2) different pressures. The 
 fall of a stone, associated fa-cts. (1) Spaces fallen through, and (2) 
 the times occupied in falling through the spaces. 
 
 Every physical phenomenon is either motion or the ^ result of 
 motion. 
 
 Examples. In the phenomena of astronomy, mechanics, acoustics, 
 light, heat, electricity, chemistry, botany, zoology. 
 
 The idea of motion necessarily embraces two others viz: that of 
 space and that of time; for a motion must take place in space, and 
 

 ~ 
 
 Lecture- Notes on Physics. 5 
 
 must happen with more or less rapidity, whence our idea of time. (See 
 Instruments used to Measure Time, under II.) 
 
 (c.) Law. The expression of the general relation which pervades 
 a class of facts, or the rule according to which a cause acts. 
 
 Examples. The law of the compression of gases; volumes of gases 
 are inversely as the pressures to which they are subjected. Gravita- 
 tion; the gravitating effect is directly as the masses, and inversely 
 as the square of the distance between the gravitating bodies. Sound, 
 light, heat, electric and magnetic attraction diminish in intensity in- 
 versely as the squares of the distances from their centres of origin. 
 Eeflection and refraction of light. 
 
 Universality of these general relations throughout the universe. 
 Importance of these general expressions in assisting the memory. A 
 law embraces in itself all the myriads of facts of which it is the gene- 
 ralization. 
 
 The use of a law is shown in its application to the practical pur- 
 poses of life. 
 
 Examples. Application of the law of the compression of gases ; of 
 the reflection and refraction of light ; of the pressure of steam at dif- 
 ferent temperatures ; of the laws of dynamic electricity. 
 
 Without the knowledge of the law of a class of facts, we are obliged 
 to make experiments to solve each individual problem, while if we 
 have the knowledge of the law, the solution of the problem is a mere 
 deduction from the law to which it belongs Examples drawn from, 
 above laws. 
 
 The money value of a law to the engineer, and to the industrial 
 and commercial world. 
 
 The discovery of laws is the object of science. 
 
 The logical arrangement of all the known physical laws consti- 
 tutes physical science. 
 
 " Classification of the Physical Sciences. 
 
 I. INORGANIC. II. ORGANIC, 
 
 a. Celestial Phenomena. 
 
 1. Astronomy. 1. Botany. 
 
 2. Meteorology (part of ) -2, Zoology." 
 
 b. Terrestrial Phenomena. 
 
 1. Physics. 
 
 2. Chemistry. 
 
 3. Geology, including Mineralogy* 
 
 PROF. J. HENRY. 
 
6 Lecture- Notes on Physics. 
 
 Physics considers the laws of the general phenomena of matter, or 
 is the study of the laws of those phenomena which do not bring about 
 a permanent change in the constitution of bodies. 
 
 Examples. Fall of a stone, compression of gases, expansion of 
 bodies bj heat, reflection and refraction of light, &c. 
 
 Chemistry considers the peculiar or individual phenomena of bodies, 
 or is that science which studies those phenomena which bring about a 
 permanent change in the nature of bodies. 
 
 Examples. Burning of a candle, rusting of iron, union of sulphur 
 and iron, union of oxygen and hydrogen. 
 
 The fundamental principle on which we repose in all our scientific 
 reasoning is, that the laws of nature are constant, or, in other words, 
 that like causes will always produce like effects. 
 
 "The mode of reasoning, in physical science, which is the most 
 generally to be adopted, depends on this axiom which has always been 
 essentially concerned in every improvement of natural philosophy, but 
 which has been more and more employed ever since the revival of 
 letters, under the name of induction, and which has been sufficiently 
 discussed by modern metaphysicians. That like causes produce like 
 effects, or that in similar circumstances similar consequences ensue, is 
 the most general and most important law of nature ; it is the foundation 
 of all analogical reasoning, and is collected from constant experience 
 by an indispensable and unavoidable propensity of the human mind." 
 (Dr. Thomas Young's Lectures on Natural Philosophy, Lecture I., 
 page 11. London, 1845). 
 
 "Most of the phenomena of nature are presented to us as the com- 
 plex results of the operation of a number of laws. 1 ' 
 
 Examples^ In astronomy, in souncl^ in light, in heat, and in elec- 
 tricity. 
 
 "" We are said to explain or give the cause of a simple fact when we 
 refer it to the law of the phenomena to which it belongs, or to a more 
 general fact ; and a compound one when we analyze it and refer its 
 several parts to their respective laws. 
 
 u The indefinite use of the term cause has led to much confusion and 
 'error. We distinguish two kinds of causes, intelligent and physical- 
 
 " By an intelligent cause is meant the volition of an intelligent and 
 efficient being producing a definite result. 
 
 " By a physical cause, scientifically speaking, nothing more is un- 
 stood than the law to which a phenomenon can be referred. 
 
 41 Thus we give the physical cause of the fall of a stone or of the ele- 
 
Lecture- Notes on Physics. 7 
 
 vation of the tides, when we refer these phenomena to the law of gra- 
 vitation ; and the intelligent cause (sometimes also called the moral 
 or efficient cause), when we refer this law to the volition of the Deity." 
 
 [It is generally easy to know to what physical cause we should refer 
 a class of physical phenomena ; but what is often difficult, is to de- 
 termine the nature and manner of acting of that cause, and to show 
 how the phenomena and their laws are consequences of its properties. 
 This is the nicest point in physical science.] 
 
 " In the investigation of the order of nature, two general methods 
 have been proposed I. The d priori method, and II. The inductive 
 method. 
 
 " I. The d priori method consists in reasoning downwards from the 
 original cognitions, which, according to the d priori philosophy, exist 
 in the mind relative to the nature of things, to the laws and the phe- 
 nomena of the material universe. " 
 
 [Examples given of ancient and medieval d priori reasoning in their 
 explanations of the motions of the heavenly bodies, and of the law 
 of falling bodies.] 
 
 " II. The inductive method, which is the inverse of the other, is 
 founded on the principle that all our knowledge of nature must be 
 derived from experience. It therefore commences with the study of 
 phenomena, and ascends from these by what is called the inductive 
 process, to a knowledge of the laws of nature. It is by this method 
 that the great system of modern physical science has been established. 
 It was used in a limited degree by the ancients, and especially by 
 Aristotle, but its importance was never placed in a conspicuous light 
 until the publication of the Novum Organum of Bacon in 1620." 
 
 "In the application of the inductive method to the discovery of 
 the laws of nature, four processes are usually employed. 
 
 " 1. Observation, which consists in the accumulation of facts, by 
 watching the operations of nature as they spontaneously present them- 
 selves to our view. 
 
 " This is a slow process, but it is almost the only one which can be 
 employed in some branches of science, for example, in astronomy. 
 
 "2. Experiment, which is another method of observation, in which 
 we bring about, as it were, a new process of nature by placing matter 
 in some new condition." 
 
 [In making an experiment we should not only evolve the phenomena 
 whose laws we wish to determine, but should a] so so produce these 
 phenomena that we can measure their related parts.] 
 
8 Lecture-Notes on Physics. 
 
 " This is a much more expeditious process than that of simple ob- 
 servation, and has been aptly styled the method of cross- questioning 
 or interrogating nature. 
 
 "The term experience is often used to denote either observation, 
 or experiment, or both. 
 
 [" A most important remark, due to Herschel, regards what are 
 called residual phenomena. When, in an experiment, all known 
 causes being allowed for, there remain certain unexplained effects 
 (excessively slight it may be), these must be carefully investigated, 
 and every conceivable variation of arrangement of apparatus etc. 
 tried; until,. if possible, we manage so to exaggerate the residual phe- 
 nomenon as to be able to detect its cause. It is here, perhaps, that in 
 the present state of science we may most reasonably look for exten- 
 sions of our knowledge ; at all events, we are warranted by the recent 
 history of Natural Philosophy in so doing. Thus, for example, the 
 slight anomalies observed in the motion of Uranus led Adams and 
 Le Yerrier to the discovery of a new planet ; and the fact that a mag- 
 netized needle comes to rest sooner when vibrating above a copper 
 plate than when the latter is removed, led Arago to what was once 
 called magnetism of rotation, but has since been explained, immensely 
 extended, and applied to most important purposes. In fact, this acci- 
 dental remark about the oscillation of a needle led to facts from 
 which, in Faraday's hands, was evolved the grand discovery of the 
 Induction of Electrical Currents by magnets or by other currents." 
 Natural Philosophy, by Sir William Thomson and P. G. Tait. Ox- 
 ford, 1867, p. 308.] 
 
 "3. The Inductive Process, or that by which a general law is in- 
 ferred from particular facts. This consists generally in making a 
 number of suppositions or guesses as to the nature of the law to be 
 discovered, and adopting the one which agrees with the facts. The 
 law thus adopted is usually further verified by making deductions from 
 it and testing these by experiment. If the result is not what was 
 anticipated, the expression of the law is modified, perhaps many times 
 in succession, until all the inferences from it are found in accordance 
 with the facts of experience. " 
 
 [Examples of induction in the discovery, by Newton, of the law of 
 gravitation; of the composition of light; of Wells's theory of dew; of 
 Ampere's laws of electro-dynamics.] 
 
 "4. Deduction, which is the inverse of induction, consists in rea- 
 soning downwards from a law, which has been established by indue- 
 
Lecture- Notes on Physics. 9 
 
 tion, to a system of new facts. In this process the strict logic of mathe- 
 matics is employed, the laws furnished by induction standing in the 
 place of axioms. Thus all the facts relative to the movements of the 
 heavenly bodies have been derived, by mathematical reasoning, from 
 the laws of motion and universal .gravitation." (Prof. J. Henry.) 
 
 [Examples of deduction. The discovery of the planet Neptune by 
 Le Verrier and Adams ; of conical refraction by Hamilton and Lloyd. 
 Laplace's work Mechanique Celeste^] 
 
 " Bacon's greatest merit cannot consist, as we are so often told 
 that it did, in exploding the vicious method pursued by the ancients 
 of flying to the highest generalizations for it, and deducing the middle 
 principles from them ; since this is neither a vicious nor an exploded 
 method, but the universally accredited method of modern science, and 
 that to which it owes its greatest triumphs. The error of ancient 
 speculation did not consist in making the largest generalizations first, 
 but in making them without the aid or warrant of rigorous inductive 
 methods, and applying them deductively without the needful use of 
 that important part of the deductive method termed verification," 
 (J. Mills's System of Logic, vol. ii., page 524-5.) 
 
 The agreement of the deductions which we make from a law, with 
 the facts of subsequent observations and experiments, is the only test 
 of our having truly arrived at a law. 
 
 Bacon well remarked, that the test of true science was its utility, 
 and Franklin was well aware of this when a friend, asking him the 
 use of electricity, he replied, characteristically, "What is the use of a 
 baby?" 
 
 "When one system of facts is similar to another, and when, there- 
 fore, we infer that the law of the one is similar to the law of the other, 
 we are said to reason from analogy. 
 
 " This kind of reasoning is of constant use in the process of induc- 
 tion, and is founded on our conviction of the uniformity of the laws of 
 nature. 
 
 "In the process of the discovery of a law, the supposition which we 
 make as to its nature must be founded on a physical analogy between 
 the facts under investigation and some other facts of which the law is 
 known. One successful induction is thus the key to another. " 
 
 [Examples. Young's discovery of the interference of light ; Frank- 
 lin's of the identity of lightning and electricity.] 
 
 "A supposition or guess thus made from analogy as to the nature 
 of the law of a class of facts, is usually called an hypothesis and some- 
 times the antecedent probability. 
 
10 Lecture- Notes on Physics. 
 
 "When an hypothesis of this kind has been extended and verified, 
 or, in other words, when it has become an exact expression of the law 
 of a class of facts, it ig then called a theory. 
 
 [Flourens, in his Eloge sur Buff on, says : " An hypothesis is the 
 explanation of facts by possible causes ; a theory is the explanation 
 of facts by real causes. "] 
 
 " Physical theories are of two kinds, which are sometimes called 
 pure and hypothetical. The one being simply the expression of a 
 law, which is the result of a wide induction, resting on experiment and 
 observation. Such is the theory of universal gravitation ; the theory 
 of sound. 
 
 " The other consists of an hypothesis combined with the facts of ex- 
 perience Of this kind is the theory of electricity which attributes a 
 large class of phenomena to the operations of an hypothetical fluid 
 endowed with properties, so imagined as to render the theory an ex- 
 pression of the law of the facts. " (Prof. J. Henry.) 
 
 A theory is therefore an assemblage of the facts and of the laws, 
 with the consequences which may therefrom be deduced, which be- 
 long to one and the same physical cause. Thus we have the theory 
 of gravitation ; of light ; of heat ; of sound, &c. 
 
 A theory, to be true and complete, should explain to their minut- 
 est details all the phenomena produced by the cause, and that, 
 naturally, without its being necessary to modify the given facts and 
 their laws. Also the numerical results deduced by mathematical 
 calculation should agree with those furnished by direct observation. 
 If, moreover, the logical deductions which we obtain enable us to pre- 
 dict new phenomena which experience afterwards confirms, that 
 theory carries with it the most convincing proof of its reality. The 
 theory of gravitation and the undulatory theory of light afford us 
 beautiful and instructive examples. 
 
 In order to establish a theory and follow logically the consequences 
 of the principle from which we have started, we are obliged to com- 
 pare among themselves quantities which are often bound together by 
 very complicated relations. The unaided attention is not equal to 
 such un effort, and hence we continually call in the aid of mathema- 
 tical analysis to assist us in our investigations in physics. 
 
 " Strictly speaking, no theory in the present state of science can be 
 considered as an actual expression of the truth. 
 
 44 It may, indeed, be the exact expression of the laws of a limited 
 class of facts, but in the advance of science it is liable to be merged 
 in a higher generalization or the expression of a wider law. 
 
Lecture- Notes on Physics. 11 
 
 " It should be recollected that the laws of nature are contingent 
 truths, or such as might be different from what they are, for anything 
 we know that they can only be established by induction from the 
 facts of experience that they admit of no other proof than the d 
 posteriori one of the exact agreement of all the deductions from them 
 with the actual phenomena of nature, and that no other cause can 
 be assigned for their existence than the will of the Creator. 
 
 " The ultimate tendency of the study of the physical sciences is the 
 improvement of the intellectual, moral, and physical condition of our 
 species. It habituates the mind to the contemplation and discovery 
 of truth. It unfolds the magnificence, the order, and the beauty of 
 the material universe, and affords striking proofs of the beneficence, 
 the wisdom, and power of the Creator. It enables man to control the 
 operations of nature, and to subject them to his use." (Prof. J. 
 Henry.)* 
 
 II. Instruments used in Precise Measurements. 
 
 A law of physics, as we have seen, consists in the general rela- 
 tion which exists between the numbers which are the final results 
 of our observations and experiments; and therefore, the first sub- 
 ject to be discussed in the exposition of that process by which we 
 arrive at a law, is the description of those instruments which have 
 given such minute precision in the numerical determinations of 
 modern physicists. 
 
 The instruments of precision used in making the measures required 
 by the physicist maybe arranged under the five following divisions: 
 
 1. Instruments used to measure . . Lengths. 
 
 2. " " . Angles. 
 
 3. " " . Volumes. 
 
 4. " " . Weight. 
 
 5. " . Time. 
 
 1. Instruments used to measure Lengths Dividing ^Engines, 
 
 Standards of Length. Before commencing a measurement of length, 
 the error of the unit we use should be accurately determined by com- 
 parison with a government standard, or with a well authenticated 
 
 copy. 
 
 * In \ I. we have made several long quotations from an admirable "Syllabus of 
 a Course of Lectures on Physics," by Prof. Joseph Henry; for it was not in our 
 power to have expressed these important fundamental truths so well, and we do not 
 know of any work in which they are so concisely stated. 
 
12 Lecture- Notes on Physics. 
 
 Two units of length are used in this country : the yard (subdivided 
 into feet, inches, and tenths of inches), and the metre (subdivided into 
 tenths, hundredths, thousandths, and so on decimally). 
 
 The yard, originally derived, about 1120, from the length of King 
 Henry the First's arm, was accurately fixed by Captain Kater, in 
 1818, when he obtained the ratio of the length of the standard Eng- 
 lish yard to the length of a pendulum beating seconds (i> e* making 
 86,400 vibrations in one mean solar day) in vacuo at the level of the 
 sea, in the latitude of Greenwich. He found that the length of such 
 a pendulum, from the point of suspension to the centre of oscillation, 
 was 39-13929 inches of Bird's standard of 1760, at 62 Fahr. 
 
 The actual standard of length of the United States is a brass scale 
 of 82 inches in length, prepared for the U. S. Coast Survey, by 
 Troughton of London; meant to be identical with the English Impe- 
 rial Standard, and deposited in the Office of Weights and Measures 
 at Washington. The temperature at which it is a standard is 62 
 Fahr., and the yard measure is the length between the 27th and the 
 63d inches of the scale, (See " Report on the Construction and Dis- 
 tribution of Weights and Measures," by Dr. A. D. Bache. 1857.) 
 
 Two copies of the new British standard, viz : a bronze standard, 
 No. 11, and a malleable iron standard, No. 57, have been presented 
 by the British Government to the United States. A series of care- 
 ful comparisons, made in 1856, by Mr. Saxton, under the direction 
 of Dr. Bache, of the British bronze standard, No. 11, with the 
 Troughton scale &f 82 inches, showed that the British bronze standard 
 yard is shorter than the American yard by 0*00087 inch. So that, in 
 very exact measures with the yard-unit, it is necessary to state 
 whether the standard is of England or of the United States, since 
 10,000 American feet = 10,000-5803 English feet. 
 
 " The Metre is a standard bar of platina, made by Lenoir in Paris, 
 which has its normal length at the temperature of zero centigrade, 
 or the freezing point. Its length is intended to make it a natural 
 standard, and to represent the ten millionth part of the terrestrial arc 
 comprised between the equator and the pole, or of a quarter of the 
 meridian. The length of this arc, given by the measurement, ordered 
 for the purpose by the Assemblee Nationale, of the arc of the me- 
 ridian between Barcelona, through France to Dunkirk (about 9J 
 degrees), combined with the measurements previously made in Peru 
 and Lapland, gave for the distance of the equator from the pole 
 5,130,740 toises. The standard toise was made in 1735, in Paris, by 
 
Lecture-Notes on Physics. 13 
 
 Langlois, tinder the direction of Godin. It is a bar of iron which has its 
 standard of length at the temperature of 1 3 Reaumur. It is known as 
 the Toise du Purou, because it was used by the French academicians 
 Bouguer and La Condamine in their measurement of an arc of the 
 meridian of Peru. The toise of Peru = 2-13145 English standard 
 yards. They found an ellipticity of 3^4, and the length of the metre 
 44:3-29596 lines of the toise du Perou, assumed to be 443-296 lines, 
 or 3 feet 11*296 lines. This last quantity was declared in 1799 to 
 be the length of the legal metre, and vrai et definitif, and is the length 
 of Lenoir's platina standard. Later and more extensive measure- 
 ments in various parts of the globe, however, seem to indicate that 
 this quantity is somewhat too small. The latest and most exact 
 results we now possess, combined and computed by Bessel, would 
 make the quarter of the meridian 10,000,856 metres, and the metre 
 = 443-29979 Paris lines. Schmidt's computation would make it 
 443*29977 lines, and both numbers are confirmed by Airy's results. 
 The legal metre is thus, in fact, as Dove remarks, a legalized part of 
 the toise du Perou, and this last remains the primitive standard. But 
 it must be added that a natural standard, in the absolute sense of the 
 word, is a Utopian one, which ever-changing nature never will give 
 us. The metre is, for all practical purposes, what it was intended 
 to be, a natural standard ; though it must be confessed that, in prac- 
 tice, the question is not whether and how far a standard is a natural or 
 a conventional one, but how readily and accurately it can be obtained, 
 or recovered when lost." (See A. Guyot's Tables, Meteorological 
 and Physical, p. 111.) 
 
 Captain Kater in 1818 determined with great care the value of the 
 metre at a temperature of 32 Fahr., in inches of Shuckburgh's copy 
 (made by Troughton) of Bird's standard yard at 62 Fahr., and found 
 one metre at 32 Fahr.; =39-37079 inches at 62 Fahr .= 39-36850535 
 United States standard inches at 62. Therefore the quadrant of the 
 French meridian contains 393,707,900 English standard inches. 
 
 It has recently been shown by M. Schubert that the equator of 
 the earth is not a great circle, but an ellipse ; having its major axis 
 = 41,851,800 feet, and its minor axis = 41, 850, 007 feet, giving an 
 ellipticity of 5 5 g<jths From this it necessarily follows that differ- 
 ent quadrants of the earth will differ, being longer when passing 
 near the major axis than when passing in the neighborhood of the 
 minor axis. Consequently, the metre is only the Toutnjssi^h P art 
 of the quadrant passing through Dunkirk. Sir John Herschel com- 
 
14 Lecture- Notes on Phycics. 
 
 putes 4008 feet for the excess of the true quadrant over that assumed 
 as the basis of the metrical system ; which makes the French stand- 
 ard ^Jgth of an inch too short. 
 
 Sir John Herschel proposes for the British standard an aliquot 
 part of the polar axis of the earth, as a natural unit, and shows that 
 by increasing the existing British standard yard (and with it, of 
 course, its subdivisions) by exactly the y^ijth part of its present 
 length, the polar axis will contain 500, 500, 000 of inches each y^^th 
 longer than the present standard inch. Herschel also shows, that by 
 adopting this new value of a foot, that one cubic foot of distilled 
 water at 62 Fahr. will contain 1000 ounces if we increase the present 
 ounce only the y^th of a grain ; and thus there will exist a simple 
 relation between the measures of length and of weight. (See Familiar 
 Lectures on Scientific Subjects, Article, " The Yard, Pendulum, and 
 Metre," by Sir John Herschel. A. Strahan : London and New York, 
 1866.) 
 
 In 1866, Congress passed an act directing that 15 grammes of the 
 metric system of weights shall be deemed, and be taken for U. S. 
 postal purposes, as the equivalent of one-half ounce avoirdupois. 
 
 The new U. S. five-cent piece is made 2 centimetres in diameter 
 and weighs 5 grammes. 
 
 See plate I. for the French measure of one decimetre, divided into 
 centimetres and millimetres ; and the English measure of four inches 
 divided into halves and tenths. 
 
 (The standards of one yard and of one metre exhibited.) 
 
 Measurement of lines of considerable length. There are two modes 
 of directly measuring lines of considerable length. The first consists 
 in placing in contact the ends of rods having an exact standard length, 
 and thus applying these units to the line whose length is required. 
 The second consists in bringing the beginning of one unit to coincide 
 with the end of another, by bisecting a fine X mark near the end of 
 one of the rods by the cross-threads in a microscope attached to the 
 end of the other rod. The distance on a rod between the cross-hairs 
 in the microscope at one end and the X mark at the other being equal 
 to a standard length. 
 
 With the precise apparatus of the U. S. Coast Survey, a base of 
 ten miles can be measured so accurately that the error made is only 
 about y^th of an inch. (See U. S. Coast Survey Eeport for 1854, 
 p. 103, et seq.j for a description of this apparatus for measuring "base- 
 lines.) 
 
Jonrnul TraiiMm Institute . 
 
 ONE DECIMETRE 
 
 O .005 01 .02 .03 .04 .05 .06 .07 .08 J09 .1 
 
 One Cubic Centimetre of Distilled Water at 4 Cr= 1 Gramme 
 
 FOUR INCHES 
 
 One Cubic Inch of Water at 62 P- 252. 456 Grains. 
 
 JV- IT Cmissa Lot 3* icDoA-'y.'. !V,3) 
 
Lecture-Notes on Pliysics. 15 
 
 The Vernier is a device for subdividing still further the lowest 
 divisions of a scale without dividing directly those divisions by equi- 
 distant lines. It consists of a sliding scale calle$ the vernier, which 
 glides along the length of the main scale, and has on it divisions 
 which differ from those of the main scale by a known fraction of 
 the smallest subdivision on the main scale. Thus, suppose we have 
 a main scale of inches divided into tenths, and we wish to divide the 
 tenths of inches into ten parts by a vernier, thus giving us the hun- 
 dredths of inches : we take a length of nine-tenths from the main scale, 
 and place it on the vernier, and we divide this length into ten equal parts, 
 which necessarily gives divisions on the vernier, each of which is T J^th 
 of an inch less than the smallest division, of tenths, on the main scale ; 
 therefore, by seeing where a line of the vernier corresponds with a 
 line on the main scale, the next following lines of the two scales are 
 distant from each other T J ^th of an inch ; the next following lines 
 differ yj-^ths, and so on; and thus we can subdivide any y^th of 
 an inch of the main scale into hundredths of an inch. 
 
 In general, to read to the nib. part of a scale division, n divisions 
 of the vernier must equal n -f 1 orn 1 divisions on the main scale, 
 according as these run in opposite or in similar directions. 
 
 Large models exhibited of vernier applied to straight line and to 
 arcs of circles. 
 
 Rule for finding the smallest reading by means of the vernier. 
 Divide the value of the smallest divisions on the main scale by the num- 
 ber of divisions on the vernier. 
 
 This beautiful and very useful invention is due to Pierre Yernier, 
 of France, who first described it in a work entitled "La construction, 
 I 7 usage et les proprietes du cadran nouveau." Bruxelles, 1631. 
 
 The Cathetometer (from Gr. KMttos vertical height, and pit poi/ a meas- 
 ure), consists of a vertical rod, which rotates round its axis, and carries 
 a telescope, provided with a spirit-level, whose line of collimation is 
 at right-angles to the axis of the rod. The rod rests on a base fur- 
 nished with levelling screws and spirit-levels, so that it can be brought 
 into a truly vertical position. The telescope is attached to a vernier 
 plate, which slides along the length of the rod which is divided into 
 fractions of the metre or of the foot. The instrument is used to 
 measure the vertical distance between two points, whether on the 
 same vertical line or not. It is of constant use in making all meas- 
 ures of vertical distances. 
 
 (Instrument exhibited, and method of using it shown.) 
 
16 Lecture- Notes on Physics. 
 
 This instrument was invented by Dulong and Petit, and subse- 
 quently improved by Pouillet. 
 
 Micrometer /Screw;. (from Gr. jutxpo? small, and uttpov a measure) con- 
 sists of a screw with a large circular head, whose circumference is 
 divided into a certain number of equal parts. Suppose the screw 
 has 50 threads to the inch, and that its circular plate, which rotates 
 with it, is divided into 200 parts; then if the screw is turned a 
 whole revolution, it will advance, in the block in which it turns, 
 g'oth of an inch ; but if the head is revolved only through 2 J O th of 
 a revolution, the screw will advance the y^^th of an inch. With 
 a good micrometer screw we can measure accurately the T O ' s O th of 
 a millimetre, or about the y^^^th of an inch. 
 
 (Large model exhibited, also the instrument itself in various 
 forms.) 
 
 Splierometer. As its name indicates, it is used to measure spheres ; 
 or, more concisely, to determine the radius of any spherical surface ; 
 as, for instance, the radius of the surface of a glass lens. It was 
 invented by the optician de Laroue, for the latter determination. 
 It consists in a micrometer screw, supported vertically by a tripod. 
 The points of the feet of the tripod are equally distant from each 
 other ; and they, as well as the screw, terminate in fine nicely rounded 
 extremities. The point of the screw and the points of the tripod 
 are brought into the same plane by placing the instrument on the truly 
 plane surface of a slightly roughened glass plate, and bringing the 
 screw's point to bear on the plane with the same degree of pressure as 
 the points of the tripod. This is attained by means of a jointed lever, 
 which moves with the screw, and its shorter arm bearing on the 
 glass, its longer arm is always brought into the same position. (This 
 can only be clearly understood by examining the instrument.) The 
 instrument is now removed to the lens, the radius of whose curva- 
 ture we desire, and the point of the screw brought down upon the 
 glass till it bears with the same degree of pressure as in the previous 
 instance. The difference of readings on the head of the micrometer 
 screw gives the perpendicular distance of the point of the screw above 
 the plane passing through the points of the feet of the tripod ; or, in 
 other words, the height of a segment of a sphere, having for radius 
 the radius of curvature of the lens, and for base the circumscribed 
 circle of the equilateral triangle made by the points of the three feet 
 of the spherometer. 
 
 Calling this height A, the radius of the circle passing through the 
 
Lecture- Notes on Physics. 17 
 
 feet of the tripod r, and the radius of the spherical surface of the 
 lens K, we have 
 
 2 R h : r : : r : h ; whence 
 
 2 R /i h 2 = r> therefore 
 
 half the value of 2 K being of coarse the radius of curvature of the 
 lens. 
 
 The spherometer will give accurate measures to the TT) Vfit;h of a 
 millimetre, and is so minute in its measurements, that if the finger 
 be momentarily placed on the plate of glass under the point of the 
 screw, and we again bring down "the screw to this portion of the 
 glass, we find that the heat from the finger has swelled the glass at 
 the part touched into a protuberance ! 
 
 It is easy to see how this instrument is also used for measuring 
 the thickness of plates. 
 
 Comparator. Model of instrument exhibited and described. In- 
 vented by Lenoir in 1800, to compare standards of length. 
 
 In this instrument, as well as in the following one, we measure a 
 multiple of the quantity we wish to estimate. 
 
 Saxton's Reflecting Comparator and Pyrometer. Model of this 
 instrument exhibited, and mode of using it shown. 
 
 It is so delicate that it will measure with precision the TaD^uu^ 
 of an inch. 
 
 " This very ingenious instrument has been applied to comparing 
 the yard of the kind called end-measure, in which the distance be- 
 tween the two ends of the bar is the length of the standard yard. 
 One end of the bar to be compared abuts against a fixed support^ 
 the other end is free to move, the whole bar being supported hori- 
 zontally on rollers. The free end presses against a small horizontal 
 bar or slide, connected by a chain, with a vertical axis, carrying a 
 small mirror. As the free end of the bar moves, the slide moves 
 also, and turns the mirror. A scale placed horizontally at any con- 
 venient distance from the bar and in the same general direction, is 
 reflected in the mirror, the image being viewed by a telescope placed 
 at the same distance as the scale, and directly over it. The distance 
 of the reflected image of the scale being twice that of the scale from 
 the mirror, the motion of the mirror is shown as on a divided circle 
 of that radius, and the smallest movement of the mirror is measurable." 
 
 (See " Keport by Dr. A. D. Bache, to the Secretary of the Trea- 
 3 
 
18 Lecture-Notes on Physics. 
 
 sury, on the Construction and Distribution of Weights and Measures," 
 Washington, 1857, p. 15, et seq.) 
 
 The illustrious Gauss, of Gottingen, in 1827, first used this method 
 
 of reflection in his measures of the forces of terrestrial magnetism. 
 
 It also can be applied to the galvanometer, by attaching its needles 
 
 to a very light silvered mirror ; and thus the smallest motions of the 
 
 needles can be shown to the largest audience. 
 
 Dividing Engines. To divide the straight line into fractional 
 parts, and the circumference of the circle into degrees, minutes, and 
 seconds, machines are used which, when made with care and used 
 with skill, are capable of working with great precision. 
 
 To divide the straight line. The simplest apparatus for dividing 
 the straight line is that described by Bunsen in his "Gasometry" 
 (London, 1857). The rod, tube, &c., to be divided is placed in a 
 line with a standard scale, and with long, rigid beam -dividers, hav- 
 ing one of its points on a division of the scale we describe, with the 
 other point a line on the rod &c. to be divided; we then move the 
 point of the dividers to the next division of the scale, describe with 
 the other point another line on the rod, and so on. 
 
 If the weight of the dividers is partly supported by a cord, pass- 
 ing over a pulley overhead, and attached to a weight, and an assist- 
 ant, with a glass, carefully places the point on a division of the scale 
 while the operator describes the line, this method will give an accu- 
 rate copy of the scale from which we transfer the divisions. 
 
 The most accurate machine for dividing the straight line consists 
 of a plate placed on A shaped guides, and moved in the direction of 
 its length by a screw working in a nut attached to its lower side. 
 The rod &c. to be divided is clamped to the upper surface of the 
 plate, and a cutting tool, which can only move in bne and the same 
 plane perpendicular to the length of the plate, cuts the divisions on 
 the rod after each rotation of the screw. If the screw contains ten 
 threads to the centimetre, then at each whole revolution of the 
 screw the plate will advance one millimetre ; and any fraction of a 
 millimetre can be cut by rotating the screw that same fraction of a 
 whole revolution. 
 
 So minute is the accuracy of which this machine is susceptible, 
 that it is able to cut 75,000 equidistant and parallel lines in the 
 breadth of an inch, and each division is distinctly visible under the 
 highest power of the microscope. A still greater number, probably 
 over 100,000 lines, can be cut in the length of an inch, but they can- 
 
Lecture- Notes on Physics. 19 
 
 not be discerned even by the highest powers of the best microscopes, 
 furnished with the most perfect means of illumination. In this case 
 our mechanical exceeds our optical power. 
 
 A graduated series of these lines, increasing in the number to the 
 inch, are used under the name of Nobert's test lines, to determine 
 the denning power of the objectives of microscopes. 
 
 Another example of fine engine work is given in " Barton's but- 
 tons," which are gold buttons stamped with a hard steel die on which 
 Barton cut hexagonal groups of fine equidistant lines with a diamond. 
 On account of the interference produced in the light reflected from 
 the surface of these buttons, they flash with the brilliancy and the 
 colors of gems. 
 
 From the above description of the performance of the dividing 
 engine, it is easy to see how, having fine lines cut at known inter- 
 vals apart on plates of glass, and placed in the ocular or on the 
 stage of a microscope, we have the means of measuring distances 
 which surpass in minuteness the determinations of the micrometer- 
 screw. 
 
 To divide the circumference of the circle. Two methods are used 
 in the graduation of the circumferences of the circles of astronomi- 
 cal and other instruments. 
 
 In the first, and probably the most accurate method, the circle to 
 be divided is centered and clamped upon a large accurately gradu- 
 ated circular plate, which revolves on a vertical axis, and under a 
 cutting tool which moves only in one and the same vertical plane 
 passing through the centre of the circle. The divisions on the 
 periphery of the large and originally divided circle are successively 
 bisected by the cross-wires in the ocular of a fixed microscope, by 
 revolving the plate under the microscope by means of a tangent 
 screw ; and at each bisection the tool cuts a line in the border of 
 the circle to be graduated. 
 
 In the other method a tangent screw works in the notched edge 
 of a circular plate; and by one whole revolution of the screw the 
 plate is carried round 10'. After each revolution the tool cuts the 
 division. Thus the circle is divided into arcs of 10' and into smaller 
 fractions of a degree if desired. 
 
 The first method is used in Germany ; the second, invented by 
 Ramsden, and subsequently improved by Troughton & Simms, and 
 by Gam bey, of Paris, is practiced in England and in this country- 
 At the U. S. Coast Survey Office in "Washington, can be seen a very 
 
20 Lecture-Notes on Physics. 
 
 superior dividing engine, made by. Troughton & Simms, of London. 
 
 In both methods the circle of the engine has originally to be 
 divided by the principle of bisection. To bisect a straight line, one 
 point of a beam-compass is placed at one extremity of the line, while 
 with the other point we strike a fine line nearly bisecting the line 
 we wish to divide. From the other extremity of the line an arc is 
 Struck with the same radius, and the minute space included between 
 the lines thus formed is bisected with a fine needle-point under a mi- 
 croscope. A scale of equal parts is thus formed with the most extreme 
 care ; and having attached to it an accurate vernier, we take in the 
 beam-compass from this scale a length which exactly equals the 
 chord of an arc of 85 20' of the circle to be graduated, and with 
 this chord we strike on the circle an arc of 85 20'. By continued 
 bisections this arc is subdivided into equal parts of 5' each. The 
 60 division can be obtained by a chord equal to the radius of the 
 circle, and this arc of 60 bisected gives 30, which, added to pre- 
 vious arc of 60, gives an arc of 90. 
 
 The reader is referred to an article on " Graduation," by the cele- 
 brated artist Troughton, in the Edinburgh Encyclopaedia, for the 
 best account extant of this, the most delicate and difficult problem 
 of mechanical art ; which has tested to the utmost the combined 
 skill of the first mechanicians, and of the ablest astronomers of 
 modern times. 
 
 2. Instruments used in the Measurement of Angles. 
 
 The circumference of the circle is divided into 360 parts, one of 
 which is called a degree the unit of angle measures. The degree 
 is divided into 60 minutes ; the minutes into 60 seconds ; the sub- 
 divisions of the second are decimal. The sign of the degree is () ; 
 of the minute (') ; of the second ("). 
 
 Theodolite, Meridian Circle, Sextant, <#c., described from the instru- 
 ments, and from diagrams, 
 
 The Micrometer consists of two spider-lines, stretched parallel to 
 each other, on two frames, each moved by a micrometer screw, so 
 that the threads can be made to approach to, or recede from, each 
 other. The spider-lines are exactly in the focus of the object glass 
 of an achromatic telescope, so that the image of any object formed 
 by the object-glass, and the lines, are in the same plane, and there- 
 fore both equally distinct, when viewed through the ocular of the 
 
Lee lure- Notes on Physics. 21 
 
 telescope. The value of one turn of the micrometer screw may be 
 estimated in angle units, by causing the spider-lines to embrace the 
 vertical diameter of the sun, when it is on or near the meridian ; 
 and then, by obtaining its exact diameter from the Nautical Al- 
 manac, we can, after allowance made on our measure for refraction, 
 reduce a turn of the screw to its value in angle. 
 
 The invention of the micrometer is due to William Gascoigne, 
 1640, who also first applied the telescope (invented in 1609, by 
 Galileo) to a graduated circle. Gascoigne was killed, at the early 
 ao-e of 23, while fighting for Charles I., at Marston-Moor, 2d July, 
 1644* 
 
 The Reading Microscope is a similar contrivance to the Microme- 
 ter ; the difference being, that the micrometer spider-lines, which 
 cross (x) each other, and are moved by one screw, are placed in the 
 focus of a microscope, instead of in the focus of a telescope. The 
 Eeading Microscope is placed over the divisions of a graduated cir- 
 cle, and is so adjusted that five turns of the screw will carry the point 
 of bisection of the cross (X) lines, the length of the smallest division 
 on the circle, which generally equals 5' of arc. The head of the 
 screw of the reading microscope is divided into 60 parts, So that when 
 the screw is revolved through one of these parts, the cross (X) lines 
 are carried over the circle, a distance equal to one second of arc. In 
 those circles used in the most accurate determinations of astronomy, 
 the microscopes read to y 1 ^ of a second of arc ; and this minute ac- 
 curacy is absolutely required in many problems now in process of 
 solution by Astronomers (e. g. the parallax of the fixed stars). 
 
 The difficulty of measuring with accuracy to a fraction of a sec- 
 ond, will be readily appreciated, when we know that one second of 
 arc occupies, on a circle of 6 feet in diameter, a length of only '0001745 
 of an inch. 
 
 For an excellent method of stretching spider-lines in the micro- 
 meter, devised by Lieut. M. F. Maury, see Washington Astronomi- 
 cal Observations, 1845. 
 
 Reflecting Goniometer. Used in the measurement of the angles 
 of crystals. Invented by Dr. Wollaston, in 1809. u This instrument 
 will give the inclination of planes of crystals, whose area is less than 
 iuiuo tn P art f a square inch, to less than one minute of a degree." 
 (Ency. Metrop. Art. Crystallography.) 
 
 * See Letters of Scientific Men of the Seventeenth Century, Oxford, 1841, vol. i., 
 p. 33, et seq. Letters XIX. and XX., Will. Gascoigne toOughtred, Dec. 2, 1640. 
 
22 Lecture-Notes on Physics. 
 
 Instrument exhibited, and method of using explained, by meas- 
 uring the angle of a crystal. 
 
 The Spirit Level is formed of a slightly curved glass tube, nearly 
 filled with ether. A bubble of air occupies the highest portion of 
 the convex side of the tube. 
 
 Spirit levels have been made of such extreme delicacy, that an 
 inclination of one second of arc, in the plane on which the level 
 rests, would cause the bubble to move three millimetres ; the curva- 
 ture of the tube of the level, in this case, was 619 metres. M. Biot 
 used, in his measurement of the arc of the meridian passing through 
 Dunkirk to Formentera, a level whose bubble moved one millime- 
 tre for an inclination of 1 /A *79, 
 
 By the aid of the levels of the best mechanicians we can safely 
 estimate tenths of seconds of arc. 
 
 The level appears to have been known to the ancient astronomers 
 of India. 
 
 3. Instruments used in the Measurements of Volumes. 
 
 Volumes of liquids and of gases are measured in graduated cyl- 
 indrical tubes. 
 
 A tube or vessel of any form may be graduated into cubic centi- 
 metres, by marking the levels reached by successive equal portions 
 of mercury poured into the tube or vessel ; each portion being 
 <equal to the exact weight of one cubic centimetre of mercury, of 
 the temperature at which the mercury is when the weighing is 
 performed. It is better, however, first to divide the tube into equal 
 divisions of length,- say millimetres, on a dividing engine, and then, 
 with a short tube, whose aperture is closed with a ground glass plate, 
 <and whose capacity is exactly one cubic centimetre, to measure off 
 sand pour into the tiibe, cubic centimetres of mercury. After each 
 addition, the number of millimetres to which the mercury rises is 
 noted in a table, and thence the value in capacity of each division 
 of the tube is deduced. 
 
 For further information on this subject, see Bunsen's Gasometry, 
 London, 1857 ; and the papers of Eegnault, published in the Trans- 
 actions of the Institute of France. 
 
 Volumes of solids, as well as of liquids, may be determined by the 
 estimation of their weight, at a known temperature, and under a 
 known atmospheric pressure. In this method we have also to know 
 
Lecture- Notes on Physics. 23 
 
 the weight of a cubic inch, or of a cubic centimetre of the solid or 
 liquid. The advantage of the French gramme weight is shown in 
 this determination of volumes, by weight; for the specific gravity 
 of a body is equal to the weight of one cubic centimetre of the 
 substance, in grammes ; for one cubic centimetre of water, at 4 C., 
 weighs one gramme. 
 
 4. Instruments used to Measure Weights. 
 
 Two units of weight are used in this country : the grain, fixed 
 by the weight of one cubic inch of distilled water, at 62 F. and 
 30 inches of the barometer, one of such cubic inches of water weigh- 
 ing 252-456 grains. The other is the French unit, the gramme, 
 which is derived from the weight, in vacuo, of one cubic centimetre 
 of distilled water, at 4 C. See Plate 1. 
 
 The Balance. Saxton's U. S. Standard Balance exhibited and 
 described. Becker's Analytical Chemist's Balance. This balance 
 will weigh to the T '<jth of a milligramme, with each pan carrying 
 a weight of 50 grammes; or, in other words, its beam will turn with 
 sni/rttnyth o f that weight, the index being deflected jd of a division 
 of the scale from the position of equilibrium. 
 
 See a paper by "W. Crookes, in the Chemical News for 19th April, 
 1867, On the Correct Adjustment of Chemical Weights; and the in- 
 vestigations of Prof. W. H. Miller, of Cambridge, in the R. S. Philo- 
 sophical Transactions for 1856. 
 
 Method of Double Weighing. Invented by Borda of Paris. Con- 
 sists in counterpoising a mass, whose weight we desire, by shot, sand, 
 thin copper foil, or pieces of fine wire, and then replacing the mass 
 by weights, until equilibrium is again established. The weights 
 are thus placed in exactly the same conditions as the mass, and there- 
 fore this process eliminates all errors arising from unequal length 
 of the arms of the balance, etc. This process is far more accurate 
 than single weighing. 
 
 The weights of very small portions of matter, such, for example, as 
 chemical traces, can be estimated by the deflection of a very fine fila- 
 ment of glass, one end of which is cemented to the edge of a 
 block, and the other end has hanging to it, a still finer filament, 
 supporting a disk, about ^^ggth inch thick, made of elder pith, on 
 which is placed the substance, whose weight we determine by the 
 amount of deflection it causes in the glass filament. With this ap- 
 
24 Lecture-Notes on Physics. 
 
 paratus (the invention of the author), can be estimated the deflection 
 produced by the y^^th of a millimetre. 
 
 See Silliman's Journal of Science, vol. xxv., page 39, 1858, for a 
 description of this instrument. 
 
 5. Instruments used in the Measurement of Time. 
 
 Definition of time, from Laplace's Systeme du Monde, ch. Ill : 
 "Time is for us the impression which is left in the memory by a 
 series of events, whose existence we are certain has been successive. 
 Motion can serve to measure it ; for a body not being able to be in 
 several places at the same time, can only go from one point to 
 another, by passing successively through all the intermediate points. 
 If at each point of the line which it describes, it is animated with 
 the same force, its motion is uniform, and the parts of that line can 
 measure the time employed to run over them. When a pendulum 
 at the end of each oscillation is found in exactly the same circum- 
 stances, the durations of the oscillations are the same, and time can 
 be measured by their number. We can also employ for that meas- 
 ure, the revolutions of the celestial sphere, which are always equal 
 in duration ; but all have unanimously agreed to use for that object 
 the motion of the sun, whose returns to the meridian, and to the 
 same equinox, or to the same solstice, form the days and the years." 
 
 The duration of one revolution of the celestial sphere constitutes 
 a sidereal day while the time of one mean revolution of the sun 
 equals a mean solar day. The sidereal day, which is used by as- 
 tronomers, is 3m. 56'5s. less than the solar day. 
 
 Both days are divided into 24 hours ; each hour into 60 minutes ; 
 and each minute into 60 seconds ; the seconds are sub-divided deci- 
 mally. The hours are designated by the sign h ; the minutes by 
 m ; the seconds by s. Both days are divided into 86,400 seconds. 
 
 The real unit of time must be the sidereal day, being the duration 
 of one revolution of the earth on its axis. From recorded ancient 
 eclipses, it has been computed that the time of this revolution has 
 not altered by 7-3 Wffuss tn f i ts length, from B. C., 720. From the 
 sidereal day is readily deduced the value of the mean solar day. 
 
 The Astronomical Clock A clock consists of a train of wheel- 
 work (moved by the descent of a weight) which maintains and 
 indicates the vibrations of a seconds pendulum. The Astronomical 
 clock is made with every refinement of workmanship ; it has gen- 
 
Lecture- Notes on Physics. 25 
 
 erally a " dead-beat " escapement, and has always a pendulum which 
 is so compensated that a change of temperature does not alter the 
 length from the point of suspension to the centre of oscillation of 
 the pendulum. 
 
 The Chronometer is a large watch, very accurately constructed, 
 having a " chronometer escapement " and a carefully compensated bal- 
 ance-wheel. It beats half-seconds. 
 
 It is not required that a clock or chronometer should keep the 
 exact time, but it is required that the daily rate of variation from the 
 true time should be constant. 
 
 The pendulum clock was invented by Huyghens, of Holland, in 
 1658 ; and in the same year Hooke of England applied the spring- 
 balance to time-keepers. 
 
 Seconds Stop- Watch exhibited and described. 
 
 A Uniformly Revolving Cylinder was first proposed by Dr. Thomas 
 Young, for the measurement of intervals of time. With this appa- 
 ratus we can, in certain cases, measure the y^^th of a second. 
 "By means of this instrument we may measure, without difficulty, 
 the frequency of the vibrations of sounding bodies, by connecting 
 with them a point, which will describe an undulated path on the re- 
 volving cylinder. These vibrations may also serve, in a very simple 
 manner, for the measurement of the minutest intervals of time ; for 
 if a body, of which the vibrations are of a certain degree of fre- 
 quency, be caused to vibrate during the revolution of an axis, and 
 to mark its vibrations on a roller, the traces will serve as a correct 
 index of the time occupied by any part of a revolution; and the 
 motion of any other body may be very accurately compared with 
 the number of alternations marked in the same time, by the vibrating 
 body." From Dr. Young's Lectures on Natural Philosophy and the 
 Mechanical Arts (a work to be thoroughly studied by all students 
 of physics), 2d Edit., London, 1845, page 147. See, also, an article 
 on "Apparatus and Experiments," devised by the author, in the 
 Franklin Institute Journal, for November, 1867. 
 
 The revolving cylinder has recently been applied, in connection 
 with the electric clock, to the registration of transit observations. 
 The apparatus described, with diagrams. 
 
 Rater's Method of determining small intervals of time, by weigh- 
 ing the mercury which flows from a small orifice, under a constant 
 pressure. 
 
 4 
 
26 Lecture- Notes on Physics. 
 
 Revolving Mirror. Invented by Prof. J. Wheatstone, of London, 
 who, in 1834, measured with it the velocity of electricity. Improved 
 by Arago, and proposed by .him to the Institute of France, in De- 
 cember, 1838, to determine the velocity of light. In 1850, Foucault 
 of Paris, so perfected its construction and arrangement, that he 
 measured with it, accurately, the TsWiiuuvth of one second of time. 
 
 The principle of the apparatus can be rendered clear in a few 
 words. Suppose a ray of light, entering a dark room by a fine slit, 
 falls upon a rapidly revolving mirror. This ray is reflected from 
 the revolving mirror to a fixed mirror, distant from it say 20 feet. 
 The ray reflected from the fixed mirror, falls again upon the revolv- 
 ing mirror. Now, suppose that in the time the light takes to go 
 from the revolving mirror to the fixed mirror, and to return to the 
 revolving mirror, that by its rotation, the face of the revolving mir- 
 ror changes its position : then this causes a deviation in the ray 
 reflected on its return to the revolving mirror, compared with its 
 direction when it left the revolving mirror to go to the fixed mir- 
 ror. Therefore, knowing the time it takes for the mirror to make one 
 revolution, we can compute, from the deviation produced, the time 
 required by the light to go over double the distance from the re- 
 volving mirror to the fixed mirror. This distance, in Foucault's 
 experiment of 1850, was only 4 metres (13*12 feet); and the mirror, 
 making 800 turns in a second, produced a deviation in the return 
 ray of -f^ millimetre. 
 
 In 1862 Foucault repeated his measurements with a far more pre- 
 cise instrument, and increased the distance of the mirrors from 4 to 
 20 metres (65 feet 7*4 inches). The result of his measures gives a 
 velocity of light equal to 185,177 miles per second, differing slightly 
 from the measure obtained by astronomic observations, when we use 
 for their reduction the solar parallax, with the increase in amount 
 which it appears from recent observation we should adopt. 
 
 The most important result obtained with this apparatus, was the 
 fact that light moves slower in water than in air, the velocities in the 
 two media being as 3 is to 4. In short, the index of refraction ex- 
 presses the ratio existing between the velocities of light in the two 
 media. 
 
 " An admirable experiment in physics, in which, by the power of 
 intellect and manual skill, we have succeeded not only in rendering 
 sensible, but even measurable, the time employed by light to run 
 over a path of 20 metres, although this time barely equals the 
 
Lecture ^ Notes on Physics. 27 
 
 jth of a second! and which, if we repeat so as to vary its 
 elements, and thus make evident the constant causes of error which 
 affect the result, appears capable of giving a determination of the 
 velocity of light altogether as precise as that which is deduced from 
 astronomical phenomena !" For further information of this method, 
 see "Essay on the Telocity of Light," by M. Delaunay of the In- 
 stitute of France; translated for the Smithsonian Institution by Prof. 
 Alfred M. Mayer, 1864. 
 
 Calculating Engines. 
 
 The great importance of such machines in giving us faultless loga- 
 rithmic, astronomical, nautical, and other tables. Impossible to cal- 
 culate and print any extensive table without making errors. 
 
 For a description of the Arithmetical Machine of Pascal, see Oevres 
 de Pascal, vol. ii., page 368, et. seq. Hachette, Paris, 1860. 
 
 For information in reference to Charles Babbage's celebrated 
 engines, see Edinburgh Review, 1834; Taylor's Scientific Me- 
 moirs, vol. iii., 1843, London, in which is a description by General 
 Menabrea, of Babbage's Difference Engine, and of his more recent and 
 far superior Analytical Engine, translated, with extensive notes, by 
 the accomplished Lady Lovelace, the only daughter of Lord Byron. 
 
 See also, Passages from the Life of a Philosopher, by Charles Bab- 
 bage, London, 1867. 
 
 Exhibit the " Tables calculated, stereomoulded, and printed by 
 machinery, London, Longmans & Co." 1857. The engine by which 
 these tables were calculated was made by George and Edward 
 Scheutz, of Stockholm, and is now the property of the Dudley 
 Observatory, at Albany, N. Y. 
 
 TABLES FOE THE COMPARISON OF THE FRENCH AND ENGLISH SYSTEMS 
 OF MEASURES AND WEIGHTS. 
 
 Measures of Length. 
 
 Denomination and Value. Equivalent in English Standard. 
 
 Myriametre 10,000 metres 6-2137 miles. 
 
 Kilometre 1 1,000 
 
 Hectometre 100 
 
 Decametre... 10 
 
 METRE ......................... 1 me re 39-37079 
 
 Decimetre 
 
 Centimetre 
 Millimetre 
 
 0-62137 mile = 3,280 feet 10 inches, 
 328 feet 1 inch. 
 393-707 inches. 
 
 3-937 
 0-3937 
 
 ' 8937 
 
28 
 
 Lecture-Notes on Physics. 
 
 Measures of Surface. 
 Denomination and Value. Equivalent in English Measure. 
 
 Hectare 10,0^0 square metres 2-471 acres. 
 
 ARE 100 " 119-6 square yards. 
 
 Centare 1 " metre 1,550 " inches. 
 
 Measures of Capacity. 
 
 Denomination and Value. Equivalent in English (Wine) Measure. 
 
 Kilolitre or Stere 1000 litres = 1 cubic metre 26417 gallons. 
 
 Hectolitre 10J ' = T V U " 26-417 " 
 
 Decalitre 10 = 10 " decimetres 2-6417 " 
 
 LITRE 1 litre = 1 cubic decimetre 1-0567 quart. 
 
 Decilitre & ' = T V " 0-845 gill. 
 
 Centilitre T 4 7 ' = lu cub. centim. 0-338 fluid oz. 
 
 Millilitre ^ ' =1" " 0-27 " drm. 
 
 Weights. 
 
 Denomination and Value. Equivalents in English Avd. Weight. 
 
 Millier or tonneau = l ) 0r,0,000grms.= 1 cub. met. of W.@4 C. = 2204-6 Ibs. 
 
 Quintal =-- 100,OOJ ' = 1 hectolitre " = 22046 
 
 Myriagramme == 10,000 ' = 10 litres " " = 22-046 < 
 
 Kilogramme or kilo ;- 1000 ' = 1 litre " = 2-2046" 
 
 Hectogramme = 100 ' = 1 decilitre " = 3-5274 oz. 
 
 Decagramme 10 ' = 10 cubic centim. = 0-3527 " 
 
 GRAMME =r 1 = 1 " = 15-43235 gs 
 
 Decigramme = T V " ^ " " = 1'5432 u 
 
 Centigramme = ^ " = 10 cub. millim. = 0-1543 u 
 
 Milligramme = ^o " = l " " " ' = 154 " 
 
 Inch = 25-39954 millimetres. 
 
 Foot = 3-047945 decimetres. 
 
 Mile =1609315 metres. 
 
 Square Inch = 6-451367 square centimetres. 
 '< Foot= 9-28997 " decimetres. 
 *' Yard = 8360971 
 " Acre=: -4046711 of a hectare. 
 " Mile= 258-9895 hectares. 
 Cubic Inch = 16-38618 cubic centimetres. 
 
 " Foot = 28-315312 " decimetres, or litres. 
 Gallon = 4-54346 litres. 
 
 " = 277-274 cubic inches. 
 
 Grain = 64 79896 milligrammes. 
 
 Pound Avd. = 453-5927 grammes. 
 
 III. Methods of Precision. 
 
 In II we described various instruments used in precise meas- 
 urements, and we will now explain certain methods used, both in 
 making a series of measures on a quantity and in subsequently com- 
 bining them, so that the final result has the greatest possible accu- 
 racy. These methods are designated as methods of precision. They 
 can be classed under three heads : 
 
 1. Method of means, or of averages. 
 
 2. Method of multiplication. 
 
 3. Method of successive corrections. 
 
Lecture- Notes on Physics. 29 
 
 1. Method of Means. If a series of measurements be taken with 
 an instrument of precision on one and the same quantity, it will be 
 found, when all the measures are made in the same circumstances 
 and with equal care, that they will differ from each other by & small 
 amount. 
 
 The mean value of all the measures is taken when the probability 
 is even that each measure is more or less than the true measure by a 
 small quantity. The mean result in this case is the same as if we 
 added the mean to itself as often as there are separate measures, and 
 divided by the number of measures. 
 
 The determination of means is of such constant occurrence in 
 physics, astronomy and chemistry, that the discussion of their de- 
 gree of precision is very important. It is founded on the principle 
 of the theory of probabilities; and we will here give a few of its 
 more important results. 
 
 Omitting from consideration gross errors, which can always be 
 avoided by operating with care, the causes of the errors of meas- 
 urements can be classified in two groups, 1. Constant or regular 
 causes; 2. Irregular or accidental causes; whence we have two 
 kind of errors, 1. Constant or regular errors, which are reproduced 
 when we repeat the observations in the same circumstances ; and 
 2. Irregular or accidental errors of which we are not able to get en- 
 tirely rid; but which, not being subject to any law connecting" them 
 with the circumstances of the measurements, occur indifferently to 
 increase or to diminish the true measure. 
 
 Examples. A constant cause of error would bean error in the length 
 of the unit used in measuring the base-line of a trigonometric survey. 
 This error will be regular and constant, and being made every time the 
 unit is applied to the length to be measured, it will be in proportion 
 to this length. It is evident- that this error can be allowed for, when 
 we know its amount, and thus our result is the same as if we ope- 
 rated with an accurate unit. As examples of irregular sources of 
 error, we may instance the measurement or sighting of angles in a 
 survey ; and the bisection of a line by the reading microscope, or by 
 the telescope of a-catheometer. 
 
 It is evident that constant errors are not eliminated by increasing 
 the number of observations; but the accidental errors tend to dis- 
 appear from the mean as we obtain it from a greater number of meas- 
 ures ; for this class of errors are as likely to be in one direction as 
 in another in successive trials, and therefore the mean of an infinite 
 
30 Lecture-Notes on Physics. 
 
 number of measurements on the same quantity would give the great- 
 est attainable precision, for the quantities would then exactly balance 
 each other in excess and defect. 
 
 To find the degree of precision of a mean we divide the whole 
 series of observations into two or three groups, selected at random, 
 and we calculate for each its mean. If these means differ very little 
 we can regard the observations contained in each group as being 
 sufficiently accurate. To proceed thus, we must have a considerable 
 number of observations, and they must not contain very discordant 
 measures. If the partial means do not thus agree, the precision of 
 the general mean is very doubtful. 
 
 Example. In the observation on the temperature of Providence, 
 K. I., by Prof. A. Caswell, during twenty years, the mean tempera- 
 ture of the first ten years is 48'l F. 
 
 " " second " " " . . 48-3 
 
 Difference, . . 0'2 
 
 The mean of the whole twenty years is . . . 48*2 
 
 This shows that one decade is sufficient to give the mean tempe- 
 rature to within about one-tenth of a degree. 
 
 It often happens that we find the measures grouped together in 
 series, depending on the periods or on the various circumstances 
 when they were made. It is then indispensable to calculate the 
 partial means of these series, and if there has existed, during the 
 time of the observations, a constant cause of errors, but of variable 
 intensity, it may manifest itself by taking, as above, the means of 
 several groups. 
 
 Errors of Observation, called also residual errors, are the differences 
 between each particular measure, and the mean of all the measures. 
 These errors are always very small when we operate with care. 
 
 The absolute value of the quantity sought remaining always un- 
 known, we substitute for it the mean, in the calculation of errors, 
 without which it would be impossible to appreciate them. 
 
 "When the errors are purely accidental it is found that they can 
 be arranged according to a most remarkable law. If they are 
 grouped by their signs and according to their magnitude, we find 
 that the positive errors and the negative errors are equal in num- 
 ber and in aggregate value, and that they diminish rapidly as we go 
 from the mean, according to a regular law; in other words, the 
 smallest errors are the most frequent and are principally accumu- 
 
Jounutl Franklin, htstitutt 
 
 \ 
 
 X 
 
 
 
 THE PROBABILITY CURVE 
 
Lecture-Notes on Physics. 31 
 
 lated around their mean value. This remarkable fact is shown in 
 the figure, where, on the left of the axis o Y are placed the differ- 
 ences or errors of observation less than the mean, or ' ; and on 
 the right those greater than the mean, or -f . 
 
 Theoretically, this curve, called the curve of probability, should 
 be perfectly symmetric and regular. If it departs much from this 
 form it shows that the observations or experiments have not been 
 made with sufficient care, or that their number is insufficient, or 
 that there exists a permanent cause of error arising either from the 
 observer himself (personal equation), or from the manner of oper- 
 ating. 
 
 Suppose we have now found a series of measures which, being 
 divided into groups, give as many means which differ by little ; that 
 the difference of each mean from the general mean is very small ; 
 and, finally, that these errors are distributed about the mean accord- 
 ing to the curve of probability ; then the cause of the errors is purely 
 accidental, they tend to diappear from the mean, and the theory of 
 probabilities gives us the numerical value of the degree of precision. 
 
 The first principle of this theory is, that the precision in the mean 
 increases as the square root of the number of observations from 
 which it is derived. Thus, the circumstances in both series of ob 
 servations remaining the same, from sixteen observations we have 
 in favor of the precision of the mean, a probability twice as great 
 as for four observations. 
 
 "We must now consider another element, which is, the error of 
 each partial measure, or its departure from the mean. 
 
 We call the mean error not the arithmetical mean of the errors, 
 but the square root of the mean of the squares of the errors. Let *, 
 
 *', *", &c., be the errors of observation, their mean 
 
 error, e, will be 
 
 e= 
 
 n being the number of observations. 
 
 By means of this mean error we can calculate what is called the 
 probable error, either of the mean or of a single observation. 
 
 The probable error is that quantity which has such a magnitude 
 that there is the same probability of the error, in the quantity de- 
 termined, being greater as there is of its being less by this quantity. 
 
32 Lecture- Notes on Physics. 
 
 It is demonstrated that the probable error of a single observa- 
 tion is 
 
 2 
 
 r- 
 
 The probable error of the mean result is 
 
 E=- 
 
 V 
 
 n 
 
 We thus see that it diminishes in the inverse ratio of the square 
 root of the number of observations. 
 
 We will now show from examples, taken from the appendix to 
 Gerhardt's Traite Elementaire de Ohimie, how the foregoing princi- 
 ples and formula can serve to indicate the exactness of a series of 
 measures. 
 
 M. Dumas made experiments to determine the composition of 
 water, and to verify the theoretical law of Prout, that all the chemi- 
 cal equivalents are exact multiples of that of hydrogen. The equi- 
 valent of oxygen being represented by 100, that of hydrogen should 
 be exactly 12'5. But a series of nineteen experiments gave, after 
 the corrections were made, the following numbers, which we arrange 
 according to their magnitude. 
 
 12-472 12-508 
 
 480 -522 
 
 480 -533 
 
 489 -546 
 
 490 -547 
 
 490 -550 
 
 490 -550 
 
 491 -551 
 
 496 -551 
 
 562 
 
 The mean of the numbers is 12'515 ; the sum of the squares of 
 the errors is 0'0173; the mean error e = 0'030; the probable error 
 of a single observation is 0'020, and the probable error of the mean 
 result E = 0'0046. It would appear from this that an error of 0'015 
 is impossible, and that the equivalent of hydrogen ought to be above 
 12-500. 
 
 But if we observe attentively the table, we will see that the num- 
 bers, instead of being accumulated, according to the law of proba- 
 bility, around the mean, tend rather towards the extremes. The 
 
Lecture-Notes on Physics. 33 
 
 means of the two columns are 12486 and 12-542, which differ too 
 much to accept the hundredths when we unite them. It therefore 
 follows that the method of experiment by which were obtained the 
 numbers, is too gross to decide that the equivalent of hydrogen departs 
 from 12'5, and we cannot therefrom decide against the law of Prout. 
 
 We see from this example, taken from The Theory of Chances, 
 by M. Cournot, how important it is, before drawing our conclusions, 
 to examine with care the individual observations whence we deduce 
 the mean, so that we may satisfy ourselves of their exactitude. 
 
 We often exaggerate the accuracy attained, which is necessarily 
 limited. A quantity cannot be estimated to an indefinite precision 
 even when the observations are indefinitely extended. In the meas- 
 ure of a length, for example, whatever care we may take we can 
 seldom surely rely on more than five figures. Thus, one of the 
 base-lines in the triangulation of France was found to be 6075'9 
 toises ; we find that there can easily occur in this number an error 
 of 0*'!. In the estimation of weights there is also a limit of preci- 
 sion which we cannot surpass, whatever maybe the accuracy of the 
 balance and the skill of the operator. Here also we cannot rely on 
 more than five significant figures. 
 
 Still greater reason is there for care in assuming a certain preci- 
 sion of result when the measure is the result of several different physi- 
 cal operations, each of which brings with it its own, error. In the 
 determination of the number # = 9 m -809, which represents the inten- 
 sity of gravity at Paris, (or the velocity acquired by a body after 
 falling one second,) the four figures which we have given are those 
 only on which we can rely, and it would be altogether illusory to 
 extend the decimal. If we compare the numbers found by ablo ob- 
 servers, we find that they differ even in this fourth figure. 
 
 Intensity of gravity at Paris. Length of second's pendulum. 
 
 Borda, 9'-SOS9 O m -99385 
 
 Biot, 9 -8091 -99387 
 
 Bes*el, 9 -8094 -99390 
 
 For the length of the second's pendulum we should take O m "9939, 
 but remembering that we can barely rely upon the result to 1 1 ff th of 
 a millimetre. 
 
 Physicists and chemists have often to estimate the exactness it is 
 possible to attain in the determination of densities and equivalents. 
 By repeating these operations several times in succession, by vary- 
 ing the methods, by taking different specimens of the same body, 
 5 
 
34 Lecture Notes on Physics. 
 
 and finally, by submitting the results to the numerical tests which 
 we have indicated, we arrive at the degree of precision of the results. 
 We thus find that the number of certain figures in this class of 
 determinations is less than we would d priori have supposed, and 
 that no more reliance is to be placed in several of the decimals that 
 are usually retained, than if we wrote them down at random. "We 
 can often discover this influence of chance in a series of measures. 
 and distinguish those decimal figures which we ought to suppress 
 as being altogether arbitrary, since they are less than the errors of 
 observation. Suppose that we estimate a length which can only be 
 appreciated to a millimetre ; if we retain the tenths of millimetre, 
 they will be entirely of an arbitrary value. The figures written in 
 the tenths will be irregular in the successive measures, absolutely, 
 as if we obtained them by drawing at random, out of our pocket, mar- 
 bles having pasted on them the figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. But, as 
 the mean of these ten figures is 4'5, it follows that if we have a great 
 many results, the mean of the figures of tenths of millimetre will be 
 exactly 4*5. And vice versa, if in a long series of numbers, given by 
 observation, we find that the mean of all the decimal figures of the 
 last order is 4*5, we can omit them. If it is the same with the fig- 
 ures of the next higher order, we suppress them also and so on. 
 
 M. Schrotter determined the equivalent of phosphorus, by weigh- 
 ing the phosphoric acid produced in the combustion of a known 
 weight of that element. These experiments were made upon amor- 
 phous phosphorus, previously dried at 150 C., in an atmosphere 
 of carbonic acid or of hydrogen. The transformation of the phos- 
 phorus into phosphoric acid, was effected in a combustion tube, 
 through which passed a current of perfectly dry oxygen. After 
 the combustion, the phosphoric acid was sublimed in an atmosphere 
 of oxygen, so as to oxydise any traces of phosphorus acid which 
 might have been produced. 
 
Phosphoric acid produced by one 
 part of phosphorus. 
 
 Lecture- Notes on Plnjsics. 35 
 
 The following are the results : 
 
 2-28909 
 
 -28783 
 29300 
 
 28831 
 29040 
 -28788 
 28848 
 28856 
 '28959 
 28872 
 
 Mean, , 2-28919 
 
 "We find that the sum of the figures of the last column on the 
 Tight is 46, the mean of which is 4*6, since there are ten results. In 
 the same manner the mean of the next column, to the left, is 4-4. We 
 therefore retain only the first three decimal figures, the remaining 
 two being solely due to hazard. This done, if we calculate the error 
 of each observation, or the departure from the mean, we will find 
 that the sum of the squares of these errors is 0'000022 ; the mean 
 error e = 0'0015, the probable error of a skigle observation is O'OOl, 
 and the probable error of the mean is E 0'0003. 
 
 We can therefore admit that one part of phosphorus gives 2-289 
 of phosphoric acid, and that the error of this result certainly does 
 cot exceed six times the probable error, that is to say, 0'002. The 
 -corresponding equivalent of phosphorus is 31*027, and its probable 
 ^error = 0'004, which is expressed thus 
 
 3 1-027 0-004. 
 
 But that supposes that the analyses are only effected with accidental 
 errors, while it may readily happen that they are subject to a con- 
 .stant error. It would then follow that the figure of the hundredths 
 could not be regarded as certain ; therefore M. Schrotter adopts sim- 
 ply 31 as the final result of his researches. 
 
 It is from the constant errors existing that we may err in the 
 conclusions that we prematurely draw from the calculation of the 
 mean 6f the probable errors. .And it often happens that subse- 
 quent observations show that the mean adopted at first is affected 
 with an error far exceeding the probable error which we attributed 
 to ^it. 
 
36 Lecture-Notes on Physics. 
 
 It is a characteristic of the arithmetical mean, that it makes the 
 sum of the squares of the residual errors a minimum. 
 
 To illustrate 'this principle geometrically, suppose that several 
 observations, made to determine the position in space of a point, A, 
 give several positions around the true point, A. Connect the points 
 given by observation by straight lines ; then the centre of figure or 
 centre of gravity of the polygon, which is formed by joining these 
 points, is evidently the most probable position of A. Draw lines 
 from the angles of the polygon to its centre of figure ; then these 
 lines will represent the several errors of the observations. Now, it 
 is a property of the centre of gravity of such a polygon, that the sum 
 of the squares of these lines is less than the sum of the squares of 
 similar lines drawn to any other point within the polygon. There- 
 fore, if the sum of the squares of the errors is the least possible, we have 
 the nearest mean we can obtain from the observations. 
 
 This is the fundamental principle of the celebrated theorem of 
 Least Squares, of Gauss and of Legendre, which has rendered such 
 inestimable service to physics and astronomy. 
 
 Suppose that, instead of having to determine, as in the above ex- 
 amples, a single unknown quantity which is the mean of the several 
 observations, we propose to determine several unknown quantities 
 entering as functions in equations. As observations or experiments 
 give the known quantities of these equations, we will have as 
 many equations as there are observations. Having, for example, 
 two quantities to be determined, x and y, suppose that we have made 
 three observations, giving three equations. The values of x and y 
 deduced from two of these equations, will not generally satisfy the 
 third. The difficulty consists in using all the equations at once, 
 so that we can deduce from them a series of values which will 
 satisfy, in the most accurate manner possible, all the equations. The 
 theory of the method of least squares shows, in order that this may 
 happen, that the sum of the squares of the errors must be a minimum. 
 
 In a series of observations or experiments, let us suppose that 
 the errors committed are denoted by e, e', e fl ', &c., and suppose that 
 by means of the observations, we have deduced the equations of 
 condition 
 
 e f = h' -f a! x -f V y -f c f z + &c. 
 e" = h" + a" x -f b" y + c"z + be. [ (1.) 
 <?"' == h'" + a'" x + b f " y + c" f z + be. I 
 &c. &c. &c. 
 
Lecture-Notes on Physics, 37 
 
 Let it be required to find such values of cc, y, z, &c., .that the errors 
 e, e', e", e'", &c., with reference to all the observations shall be the 
 least possible. 
 
 If we square both members of each equation, in group (1), and 
 add them together, member to member, we shall have 
 
 + 2 x (ah+a'h' + &c.) + a (& ?/ + c 2 + &c.) 
 
 4- a! (b' y + c' 2? 4- &c.) -f &c. 1 4- &o., 
 
 an equation which may be written 
 
 e 2 j_ <. <5 ,/ &(..=- w=s ^052 2 x + ^ &c. 
 
 Now, in order that e 2 + e f2 -f- &c., or w, may be a minimum, it is 
 necessary that its partial differential co-efficients, taken with re- 
 spect to each variable, in succession, should be separately equal to 0. 
 Hence, 
 
 ^ = o, or x (a 2 
 
 = = 2 
 
 X 
 
 -f a li -f u h f -f &c. + a (b y + c z -f &c.) 
 
 -f a' (i 7 y + c' a + &o.) + &c. = ; 
 
 and similar equations for each of the other variables. Hence, we 
 deduce the principle, that, in order to form an equation of condition 
 for the minimum, with respect to one of the unknown quantities, as 
 x for example, we have simply to multiply the second members of 
 each of the equations of condition by the co-efficient of the unknown 
 quantity in that equation, then take the sum of the products, and 
 place the result equal to 0. Proceed in this manner for each of the 
 unknown quantities, and there will result as many equations as there 
 are unknown quantities, from which the required values of the un- 
 known quantities may be found by the ordinary rules for solving 
 equations. 
 
 Let it be required, for example, to find from the equations 
 
 3 x 4- y %%= ' 
 
 5 3x 27/45* = 
 21__4<r y z = 
 14 -f x By 3z= 
 
 such values of x t y, and a, as will most nearly satisfy all of the 
 equations. 
 
 [It is to be remarked, that we must necessarily admit that these 
 
38 Lecture- Notes on Physics. 
 
 equations already approximate closely in value, that is to say, that 
 the values of x, y, and 2 deduced from the three first will satisfy 
 approximately the third ; without this, the problem is absurd, and 
 the observations which have given these equations are unworthy of 
 confidence.] 
 
 "Following the rule, and multiplying the first member of each 
 equation by the co-efficient of x in that equation, we get the pro- 
 ducts 
 
 3 + x y + 22 
 15+ 9x + 6y 152 
 
 84 + 16z +4y +162 
 14+ x By 32, 
 
 and placing their algebraic sums equal to 0, we have 
 
 27a+6y 88=0, .... (3.) 
 
 Proceeding in like manner with respect to the unknown quantities 
 y and 2, we obtain the equations 
 
 6cc + 15?/ + 2 70=0, . . . (4.) 
 y + 54 s 107 = 0, . . . (5.) 
 
 Combining equations (3), (4), and (5), we find 
 
 x = 2-4702, 2/=3-5507, and 2 = 1-9157, 
 
 which most nearly satisfy all of the equations in group (2). 
 
 To show the practical application of this principle, we will sup- 
 pose that it is required to investigate the values of a constant in an 
 equation, by means of several independent experiments. 
 
 It is demonstrated from theory that the length of the pendulum 
 which beats seconds, in any latitude, is given by the formula 
 
 L = x + y sin. 2 Z, .... (6,) 
 
 in which L denotes the length of the pendulum ; I the latitude of the 
 place on the surface of the earth, and x and y are constants to be 
 determined. In consequence of errors incident to observation, the 
 values of x and y cannot be accurately determined by means of a 
 single observation ; taking the metre (denoted by ra) as the unit of 
 measure, suppose that the length of the second's pendulum has been 
 measured in six different places, whose latitudes are known, and 
 that the following equations have been deduced : 
 
Lecture-Notes on Physics. 39 
 
 = x + y X O m .3903417 O m .9929750 
 = x + y X O m .4972122 O m .9934620 
 = SB + y X O m .5667721 0- .9938784 
 =x + y X O m .4932370 O m .9934740 
 = + y X O m .5136117 O m .9935967 
 = x + y X0 m . 6045628 O m , 9940932 ^ 
 
 Applying the rule already deduced, to these questions, we find 
 the equations 
 
 6 x + y X 3 m . 0657375 5 m . 9614793 = 0, . . (8), 
 x X m .0657375 + y X l m .5933894 3 .0461977 = 0, (9.) 
 
 Combining equations (8) and (9), we obtain 
 x = m .9908755, y = m .0052942. 
 
 Substituting these in equation (6), it becomes 
 
 L = O m .9908755 + O m .0052942 sin. 2 Z, . . (10.) 
 
 By means of formula (10), the length of the second's pendulum 
 may be found, by computation, at any place whose latitude is 
 known. In like manner, the method of least squares may be ap- 
 plied to a multitude of similar cases." (See art. Least /Squares, Diet, 
 of Mathematics, by Davies and Peck, N. Y., 1865.) 
 
 It is impossible, in lectures of this character, to give more than 
 an idea of this fertile principle of Least Squares. The reader is re- 
 ferred to the following works for a full discussion of the method. 
 Methode des moindres carrees Memoir es sur la combinaison des obser- 
 vations, par Oh. Fr. Gauss, traduit par J. Bertrand, Paris, 1855 ; 
 also Method of Least Squares, by Prof. William Chauvenet, in the 
 appendix to his Spherical and Practical Astronomy, Phila., 1863. 
 From this work the figure of the probability curve is taken. 
 
 2. Method of Multiplication. 
 
 The method of multiplication consists in measuring a known 
 multiple, by n, of the quantity to be determined, and then dividing 
 the result by n. By this method the error, committed in the direct 
 measure, is divided by n. 
 
 Example 1. To find the diameter of a fine metallic wire, we wrap 
 a certain number of turns, say 200, of the wire on a metal cylinder, 
 taking care to press the coils until all are in contact. Measure the 
 length of the 200 coils, and dividing this length by 200, we have 
 supposing all the coils are formed of wire of the same diameter the 
 
40 Lecture-Notes on Physics. 
 
 diameter of the wire to the 2 ^th of the error committed in the 
 measure of the length of the 200 coils. Suppose 2 for example, we 
 have made an error of ^ inch in measuring the length of 200 coils; 
 then the error in the determination of the diameter of the wire is 
 
 2tjo of 20 or the W<Jo tn of an incl1 - 
 
 Example 2. By a similar operation we determine, with great ac- 
 curacy, the distance between the contiguous threads (called the pitch) 
 of a micrometer screw. 
 
 Example 3. Saxton's Beflecting Comparator is a beautiful illus- 
 tration of the principle of multiplication, in which the direct mea- 
 sure is a very large multiple of the quantity whose value we desire. 
 
 Example 4. We may obtain a very accurate determination of the 
 weight of a body, with a balance with equal arms, by equilibrating 
 the body with shot, &c., placed in the other pan ; then placing the 
 body in the same pan with the shot, &c., we obtain in the other pan, 
 in shot, twice the weight of the body. "We now add the shot of twice 
 the weight of the body, to the pan containing the body, and on again 
 equilibrating, we have in the pan opposite the body, a weight of 
 shot equal to four times the weight of the body, and so on. We 
 finally have in a pan a known multiple, by n, of the weight of the 
 body, which, 'divided by n, gives the weight we seek. 
 
 Example 5. The Method of Repetition. Describe the method with 
 the aid of diagrams. It is a very ingenious application of the me- 
 thod of multiplication to the measurement of angles, by means of 
 the repeating circle, invented by Tobias Mayer, of Grottingen, in 1762, 
 and subsequently improved by Borda, of Paris. 
 
 3. Method of Successive Corrections. 
 
 This method, used frequently in astronomy, is also often em- 
 ployed in researches in Physics. We will explain it by an exam- 
 ple (from Daguin's Traite de Physique, vol. i., page 30). Let it be 
 proposed to determine the capacity of a ground glass-stoppered 
 flask. If we knew the weight of the water at 4 G., which the 
 flask holds, then just as many grammes as there are in this 
 weight, are there cubic centimetres in the flask ; for a gramme is 
 the weight of one cubic centimetre of pure water at 4 C. It is, 
 therefore, required to determine the weight of the flask-full of 
 water at 4 C., which we suppose is the temperature when the 
 experiment is made. We first weigh the flask, full of air, then 
 full of water at 4 C. Let p and p be the two weights, respectively. 
 
Lecture- Notes on Physics. 41 
 
 P p will represent the weight of the water, if the flask, during the 
 first weighing, had contained no air, whose weight is added to that 
 of the flask. The value P p is therefore too little by the weight of 
 that quantity of air. Suppose that the water at 4 C., weighs n times 
 as much as an equal volume of air, taken at the same atmospheric 
 pressure and temperature as in the experiment. We will see fur- 
 ther on how this number n is obtained. The weight of the air 
 
 which fills the flask will then be (p p) -, if we suppose for the 
 
 moment that P p is the exact weight of the water. Then the 
 weight of the water which fills the flask will be 
 
 (p-p) + (r-p) ^=(P-P) ( 1 + 1), -. . . (1.) 
 
 This expression, however, does not represent the exact weight 
 of the water, because the term (p p), and consequently the term 
 
 (p p) -, are too small. But the error which exists in value (1) 
 
 is less than that which affects the value P p. If we now employ 
 the value (1) to calculate the weight of the air, this weight will be 
 
 ' '" 
 
 . 
 
 and the weight of the water will become 
 
 /. ' . (2;) 
 
 a value which is a still greater approximation to the true value, 
 and which, multiplied by 1 : n, will give the weight of the air with 
 still greater precision. 
 
 By adding this weight to P p, we will have for the weight of 
 the water 
 
 n n 
 
 nearer than any of the preceding values. By continuing in this 
 manner, we can carry the approximation as far as we desire. 
 
 IY. Manners of Expressing a Law Law evolved from the nume- 
 rical results of Observations and Experiments. 
 
 The first step in that investigation, which has for its object the dis- 
 covery of the law of a class of facts, is the minute examination and des- 
 cription of the phenomena and of the circumstances which accompany 
 6 
 
42 Lecture- Notes on Physics. 
 
 them, and the determination of those conditions necessary for their 
 production. This having been accomplished, we observe that there 
 always exists between the different circumstances of an associated 
 class of facts, relations or dependencies which bind them together 
 in such a manner, that if we change one of the circumstances of the 
 phenomenon, the others experience determinate modifications. 
 
 For example. In the compression of air, we have seen that the 
 smaller the volume into which the air is forced, the greater the 
 force of compression required ; and on further examination with 
 measurements taken of (1) the volumes occupied (2) under different 
 pressures, we find that the relation which binds (1) and (2) is that 
 the volumes of the air are inversely as the pressures to which it is 
 
 Such an expression of the relation existing between the different 
 circumstances of a class of associated facts is, as we have seen, 
 ( I.) a physical law. 
 
 11 These laws rule all phenomena, and are their most complete 
 representation. Their existence did not escape the acute minds of 
 the philosophers of antiquity. Plato, . questioned concerning the 
 occupations of the Deity, replied that He geometrized without ceas- 
 ing ; wishing thereby to express according to Montucla, that the 
 universe is governed by geometrical laws" (DAGUIN'S TRAITE DE 
 PHYSIQUE, Yol. I., p. 8). 
 
 If the law is the expression of a general quantitative relation 
 existing between the associated facts of the phenomenon, it can be 
 replaced by a line referred to coordinate axes, as in the method of 
 Analytical Geometry. 
 
 Example 1. Thus, a Parabola represents the law of falling 
 bodies, because the origin of rectangular coordinates being at the 
 principal vertex, the abscissas are to each other as the squares of 
 their corresponding ordinates ; or, 
 
 x' :x" '.:y f *'.y"*; 
 
 and the law of falling bodies is that the spaces fallen through are as 
 the squares of the times ; therefore, if any ordinate represent in units 
 of length the units (seconds) of time occupied in the fall, the units 
 of length contained in its corresponding abscissa will give the units 
 of length fallen through ; one unit of length in this case being a 
 function of the intensity of gravity is, for the latitude of New 
 York, equal to 16 feet 1 inch. 
 
MAT JUN. JUL. AUG. SEP OCT. NOV DEC. JAN. FEB. MAR. APR 
 1 16 1 15 1 16 1 15 1 14 1 14 1 13 1 13 1 16 1 15 1 17 1 15 
 
 
 
 
 
 
 
 
 , 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 , 
 
 e 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / f 
 
 t 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ~M 
 
 
 
 
 A 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 / 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 ^ 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 00" 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 g 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 I 
 
 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 % 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^^ 
 
 fc 
 
 ^ 
 
 
 
 
 
 
 2 
 
 *>M 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 THERMAL CURVE OF BALTIMORE WITH CORRESPONDING SINUSOID. 
 
 L 1 
 CO 
 
 20 10 18 17 IB 15 14 13. 12 11 1O 9 8 7 fi 54 * '2 L 
 
Lecture- Notes on Physics. 43 
 
 The above is directly shown in MORIN'S MACHINE, where the 
 body describes the parabola by falling parallel to the axis of a uni- 
 formly revolving cylinder, against which a pencil, attached to the 
 falling body, gently presses. The curve thus formed is found on 
 spreading out the paper, which covers the cylinder, and measuring 
 its abscissas and ordinates, to be a parabola. Thus, in this beauti- 
 ful experiment, the body itself writes on the paper the law of its 
 motion. 
 
 Example 2. The law of the compression of gases is given by a 
 curved line (see Fig. 1, Plate III. Curve L I/), whose abscissas rep- 
 resenting pressures and ordinates volumes, has for its equation 
 
 This curve is an equilateral hyperbola, and therefore the axes of 
 X and of Fare asymptotes to the curve ; and this should be so, for 
 the gas, supposed to be nbn-liquinable, has under an infinitely great 
 pressure, still a definite volume, which is expressed by the ordinates 
 corresponding to that pressure. If, however, the gas is not perma- 
 nent (like carbonic acid), the curve will cut the axis of X at a point 
 corresponding to the pressure producing liquifaction. If all those 
 gases which are susceptible of liquifaction though the pressures 
 required for this result for certain gases might be beyond the limit 
 of practicable experiments had curves whose deflections towards 
 the axis of X all followed the same law, we could, by projecting 
 the curve of any gas, as far as the limit of experiment, determine 
 at what point it would cut the axis of JT, and thus determine the 
 pressure of liquifaction. 
 
 See Fig. 1, Plate III. The curve L I/ represents Mariotte's law, 
 that the volumes of gases are the reciprocals of the pressures ; while 
 L C0 2 shows the fraction of a unit of volume, measured on the axis 
 o Y, occupied by carbonic acid gas under pressures, from 1 up to 20 
 atmospheres, measured on the axis o x. If the curve of hydrogen 
 were projected on this diagram, it would sweep above L I/, departing 
 very slightly from this line on the side opposite the curve, L CO 2 . 
 
 The advantage of the graphic method of expressing a law is, that 
 it represents to the eye the continuous relation between the associated 
 facts of the phenomenon, and therefore gives, by direct measurement, 
 the quantity answering to any measure taken of either of the related 
 magnitudes embraced in the projected curve. Also, by the mathe- 
 
44: Lecture- Notes on Physics. 
 
 matical discussion of the curve, new relations which otherwise might 
 pass unknown, may be evolved. 
 
 Sometimes the relation existing between the associated facts is too 
 complicated to be susceptible of a concise quantitative statement. 
 In that case, we lay off on an axis of rectangular coordinates lengths 
 of abscissas equal to various values of one of the quantities, and on 
 their corresponding ordinates lengths equal to the corresponding 
 values of the other quantity. We then draw tibera manu, through 
 the several points thus determined, a curved line, and this line will 
 be the continuous expression of the relation existing between the two 
 quantities considered. 
 
 It may happen that the inspection of the curve leads to the dis- 
 covery of a law, which we could not have made from the direct com- 
 parison of the numbers given by experiment or observation. For 
 example, if the curve was one of those lines studied by geometers, 
 and whose properties are well known, the relation which exists 
 between the abscissas and ordinates of the curve, or, in other words, 
 its equation (referred to the axes X and ]T), will express the law 
 which we seek. If, therefore, we recognize by mere inspection, that 
 the curve resembles a known line, we must proceed in the follow- 
 ing manner to the verification of that supposition. "We write the 
 general equation of the curve, referred to the axes X and Y, giving 
 it indeterminate co- efficients ; we then successively substitute in that 
 equation as many values of x and of their corresponding ordinates 
 y., as there are co-efficients; which gives equations of condition, by 
 means of which we calculate the values of these co-efficients, which 
 we then substitute in the general equation. We then successively 
 substitute for x and y other numerical values given by the experi- 
 ments or observations, and we examine if these numbers satisfy the 
 equation. If this is the case, we conclude that the curve is really 
 the one we suspected, and its equation expresses the mathematical 
 law of the phenomena. 
 
 For a gimple example, take the curve described by Morin's Machine 
 for showing the law of falling bodies. If, from preliminary meas- 
 urements, we suspect the curve to be a parabola, we have, for the 
 general equation of that curve 
 
Lecture-Notes on Physics. 45 
 
 in which, placing x r and y r from measured coordinates of the curve, 
 we have 
 
 substituting this value of the parameter in the general equation, and 
 making x and y successively x" y" , and x" r , y' n ', &c., obtained from 
 other measurements on the curve, we find that 
 
 whence we conclude that the -curve is the quadratic parabola, and 
 therefore 
 
 ,/ . ,// . . ./2 . ,// . 
 
 x . .x . . y . y j 
 
 and since in the curve described by this machine, x 1 and x" are equal 
 to the spaces fallen through by the body in the corresponding times 
 y f and y", we have the law that the spaces fallen through are as the 
 squares of the times. 
 
 Example, in which more complex relations exist 
 
 "We have (in a paper preparing for publication), projected the 
 thermal -curves, of the variation of temperature from day to day 
 throughout the year, of fifteen places, differing in latitude, longi- 
 tude, elevation above and distance from the sea, and in other topo- 
 graphical features ; and on discussing these curves, we find that 
 they can all be replaced by sinusoids, differing from each other only 
 in the amplitude of their flexures, which corresponds to half the annual 
 range of temperature of the different stations. 
 
 Lengths of abscissas standing for days and lengths of ordinates rep- 
 resenting degrees of temperature {see Plate III., Thermal Curve of 
 Baltimore, with corresponding sinusoid ; where the horizontal red 
 line marks the mean annual temperature, while the curved red line 
 is the annual thermal variation, and the curved black line its cor- 
 responding sinusoid), we obtain a .series of points, by laying off on 
 ordinates erected from abscissas corresponding to certain days, the 
 mean temperatures of those days, and then drawing a curved line 
 through tKese points, we have the annual thermal curve. This 
 curve has its origin at the point of the axis of X which corres- 
 ponds, in the average, to the date of the 21st of April; the mean 
 temperature of that day being the mean temperature of the year. 
 
 (Diagrams of thermal curves, with corresponding sinusoids of 
 fifteen stations exhibited.) 
 
46 Lecture-Notes on Physics. 
 
 Now projecting on the line of mean annual temperature of each 
 curve, as the base, a sinusoid the length of whose base (=360) 
 equals the length of the year ; whose point of origin is on the base 
 at the point corresponding to the 21st of April, and whose ampli- 
 tude equals the half yearly range of temperature, we have, in every 
 instance, so close a coincidence between the thermal curve and the 
 sinusoid, that we can confidently say, that the variation of tempera- 
 ture throughout the year follows the variation of the sinusoid curve. 
 
 The equation of the sinusoid is 
 
 y==sin. x; 
 
 and as the sines of the first and second quadrants are -f and of the 
 third and fourth , it follows that this is a recurring curve, and up 
 to sin. (x = 180) the flexure is above the base line, while from 
 sin. (x = 180) up to sin. (# = 360) the curve lies below the line, 
 and so on; each succeeding 180 being on the opposite side of the 
 line of mean temperature from the preceding. 
 
 Therefore, expressing any date in days reckoned from April 21st, 
 and converting these days into lengths of arc x at the rate of one 
 
 onrvo * 
 
 day = , = 59' I", and making the sin. 90 or E = the mean 
 
 yearly range of temperature, we can readily determine the mean 
 temperature of any date of the year. 
 
 1. Problem. What is the mean temperature of the 10th of June, 
 at Paris ? 
 
 The mean annual temp, from 46 years observations, = 51' 3 Fah. 
 Half the yearly range, v. .... =15* 47 
 Days from 21st April to 10th June, . ' V ' " =50 days. 
 The above number of days in arc, V . . = 49 10' 50" 
 Natural sine of above arc, . " . . .- V ='756773 
 Eadius in this example, . .">.:* : . ; -=15'47 
 Then the temperature of 15th June equals 
 
 756773 X 15-47+51-3 v' . "V . =-62-90 
 Mean temperature of 15th June, from observa- 
 tions of 46 years, . '.'.-' . ';"" . = 62-99 
 Difference, ?Jg3 . = 0'09 
 
 2. Problem. What is the mean temperature of the 13th of Dec. 
 at Baltimore? The temperature from observations = 35'6. Data 
 for solution. From 35 years observations, the mean annual tempe- 
 rature 54-4. The yearly range 44-4, 
 
Lecture-Notes on Physics. 47 
 
 3. Problem. Calculate the mean temperature at Baltimore, of 
 of the 14th October. The observed mean temperature of this day is 
 equal to 56'5. 
 
 4>. Problem. Determine which is the warmest and which is the 
 coldest day in the year at St. Petersburg. Data for the solution. 
 Mean temperature, 38*75. Annual range, 50. 
 
 From observations of 30 years, the warmest day at St. Petersburg 
 is on the average, the 25th July ; the coldest, the 19th January. 
 
 We can therefore determine the mean temperature of any day of 
 the year for any station (not situate in the tropics), whose mean 
 temperature and yearly range of temperature are known. The mean 
 temperature of any place can be found by taking the temperature 
 once a month during one year, of any spring which issues from a 
 depth of a few feet below the surface of the ground. Thus, the 
 temperature of a spring in Baltimore, from the mean of one obser- 
 vation a month, during one year, was 54 0< 25, only 0*15 below the 
 mean given by 35 years observations of the thermometer in the air. 
 
 If a law could be established connecting different parallels of lati- 
 tude with the yearly range of temperature, we could arrive at the 
 other datum, and then the curve of annual variation could be pro- 
 jected from a knowledge of the latitude and longitude, with the 
 determination of the mean temperature of a spring. 
 
 This method of representing to the eye, by means of curved lines, 
 the nature of the laws concealed within a mass of numbers, presents 
 the advantage of representing the continuous and periodical nature 
 of cosmical action. It also presents to the eye a form, easy to seize 
 and remember, instead of an abstract statement referring to relations 
 of quantities, difficult to be conceived and sometimes impossible other- 
 wise to discover. Thus the thermal curve of Berlin is the graphic 
 expression of the relation of the 365 days of the year to observa- 
 tions made three times a day for 110 years, or, to 120,450 separate 
 numerical quantities. 
 
 The curves show by their deformities, either that our data are not 
 the results of sufficiently extended observations to give, by their pro- 
 jection, the expression of a law, or, that another cause is acting which 
 increases or diminishes the effect due to the cause whose rule of action 
 is expressed, in the main, by the curve. 
 
 Thus, in the thermal curves of Greenwich, of Paris and of Eome, 
 an upward deflection exists, from the middle of January to the 
 middle of April, causing a departure of about 2 Fahr. from the 
 
48 Lecture- Notes on Physics. 
 
 sinusoidal curves. This departure, at first sight, would appear as 
 opposed to the assumption that the annual variation of temperature 
 follows the variation of the sinusoid ; but, in reality, it is " an excep- 
 tion which proves the law ;" for it is found that during those months 
 the prevailing winds for those stations are W. S. W., and Dove has 
 conclusively shown that in the northern hemisphere, the thermometer 
 stands highest, on an average, with a S. W. wind. This deflection, 
 however, does not occur in the curve of Berlin, and there a S. 17 
 W. wind prevals during those months. But this is easily accounted 
 for r and the explanation will at the same time give a sufficient reason 
 for the upward deflection of 2 in the thermal curves of Greenwich, 
 Paris and Rome. 
 
 The prevailing wind of Greenwich from the middle of January 
 to the middle of April, reaches that station after having traversed 
 the Atlantic ocean, whose surface water has, during the above 
 months, even in the latitude of the English Channel, an average tem- 
 perature of 52 Fahr. Now, one cubic foot of water in cooling 1 
 Fahr. will give out heat sufficient to raise 3080 cubic feet of air 1 
 Fahr., and the wind after traversing this surface of a liquid, of high 
 specific heat, 13*5 above the mean temperature of the air during 
 February, passes over only about 85 miles of lowlands before reach- 
 ing Greenwich. The same prevailing wind reaches Paris after hav- 
 ing also passed over the Atlantic, of even a slightly higher tem- 
 perature than in the case of the Greenwich wind, and traversed 
 about 250 miles of land-surface not likely to affect its temperature, 
 since this very land has received its thermal condition mainly from 
 the supernatant air; but the S. 17 W. wind reaches Berlin after 
 having crossed the snow covered summits of the Appenines and 
 Alps, and 750 miles of land. 
 
 This general graphic method of expressing the relation which per- 
 vades a class of facts, evidently gives. a ready means for interpola- 
 tion, and even where deformities exist, a mean or true curve can be 
 obtained by sweeping a line through the mean position of the irregu- 
 larly placed points given by the observations or experiments. By 
 this last method, Sir John Herschel determined the orbits of several 
 of the double stars (Transactions of the Royal Astronomical Society, 
 Vol. Y., 1832), and M. Regnault (Memoirs de TInstitut, t. 21, 1847, 
 page 316, et seq., and page 574, et seq.), from his measures of the 
 tension of vapor of water corresponding to different temperatures, 
 succeeded in ascertaining the. extreme oscillations of the errors 
 
Lecture- Notes on Physics. 49 
 
 of those measures, by drawing a mean curve through the sphere of 
 the points given by his experiments. He then substituted this 
 curve for the numbers directly determined by experiment, because 
 this curve expressed the relations of those numbers practically 
 corrected of their errors. 
 
 It should be remembered that each number at which we finally 
 arrive, before we place it down as a point to serve in the projection of 
 the curve, is the most probable mean which we can obtain from the 
 discussion of the numbers from which it is derived, (see III.), and 
 that each of these points so determined is arrived at independend- 
 ently of any other point, and therefore the graphic method is espe- 
 cially adapted to combine all of these independent determination of 
 points in one curve, and to make them mutually correct each others 
 errors. 
 
 The curve thus obtained can be rendered more serviceable in 
 practice, by obtaining from it, if possible, an equation which will 
 express it ; but if we find that it cannot be expressed by a single 
 equation, we can, by the aid of formulae of interpolation, find the 
 value of each ordinate corresponding to as small an increase in each 
 successive abscissa as we desire. We should not, however, apply 
 a formula of interpolation to the numbers before we have obtained 
 from them a mean curve. The interpolation is merely used to give 
 a ready means of obtaining the values of any related magnitudes 
 expressed by a curve, which shows the relations of the numbers 
 corrected of their errors ; for, as M. Eegnault remarks, " the graphic 
 method when it is properly executed, is preferable to all methods 
 of interpolation by calculation ; it permits us to distinguish, imme- 
 diately, the variations due to the accidental errors of the observa- 
 tions, and the constant errors which depend on the diversity of the 
 methods which we have employed." 
 
 Y. The General Properties of Matter. The Constitution of Matter 
 according to the Molecular Hypothesis. 
 
 MATTER is that which affects our senses. (See I). 
 
 A body is a portion of matter limited in all directions. 
 
 In the different manners in which bodies affect our senses, con- 
 sist the properties of bodies. These properties are either general, 
 that is, belonging to all bodies, whether solid, liquid or gaseous, 
 and therefore come under the head of physics, or they are special, 
 1 
 
50 Lecture-Notes on Physics. 
 
 and therefore belong to the province of chemistry. (See definitions 
 of Physics and Chemistry, in I). 
 
 Matter possesses several general properties, of which two are 
 called essential properties ; for without them we cannot conceive of 
 the existence of matter, and therefore they can serve to define it. 
 These two properties are extension and impenetrability. 
 
 The following is a list of the general properties of bodies : 
 
 1. Extension. ) 
 
 2. Impenetrability, j Necessar 7 to our perception of matter. 
 
 3. Figure. 
 
 4. Divisibility. 
 
 5. Compressibility. 
 
 6. Dilatability. 
 
 7. Porosity. 
 
 8. Mobility. ] 
 
 9. Inertia. j Ultimate properties according to the mole- 
 
 10. Attraction and j cular hypothesis. 
 
 11. Eepulsion. J 
 
 12. Polarity. 
 
 13. Elasticity. 
 
 1. Extension. 
 
 Extension is the property which every body possesses of occupy- 
 ing a portion of space, which we call its volume. A body always 
 presents the three dimensions length, breadth, and thickness ; and 
 it is by abstraction only that in geometry we consider surfaces, which 
 are the boundaries of bodies, and have only length and breadth; 
 and lines, which are the boundaries of surfaces, and have only 
 length; and points, which are the terminations of lines, and have 
 alone position. 
 
 For the measurements of extension, see II. 
 
 2. Impenetrability. 
 
 The property which every body has of excluding every other 
 body from the space which it occupies. 
 
 In ordinary language, we say that one body is penetrated by 
 another, but in all these cases it is found that the particles of the 
 one body are merely displaced by the other. 
 
 Examples. A needle penetrating (displacing) mercury contained 
 in a fine glass tube. The mercury rises in the tube, as the needle 
 
Lecture- Notes on Physics. 51 
 
 descends, to an amount exactly equal to what would be produced 
 by pouring into the tube a quantity- of mercury equal to the volume 
 of that portion of the needle below the surface of the mercury. 
 
 A liquid cannot be poured into a vessel unless the air, which it 
 contains, goes out as it enters. 
 
 Consider the images formed in the foci of converging lenses and 
 mirrors ; and also a shadow. 
 
 We can now define matter as all that which has both extension 
 and impenetrability, and therefore void space or vacuum is space 
 without impenetrability, being penetrable. 
 
 3. Figure. 
 
 The bounding surfaces of a body give it its figure or form. 
 That department of science which discusses and classifies the 
 forms of nature, is called Morphology. 
 
 The forms of bodies may be arranged under two divisions. 
 
 I. Regular Forms. II. Irregular or Amorphous Forms. 
 Regular forms can be embraced in two classes (a and b). 
 
 a. Forms of matter produced by the action of forces not directed by 
 the vital principle. 
 
 1. Forms of the heavenly bodies. Considered in Astronomy. 
 Forms of liquids in motion; as "the liquid vein;" (see researches 
 
 of Poncelet, Savart and Magnus), the rain drop ; the forms of waves. 
 
 Forms of liquid at rest ; (not contained in vessels) as the dew-drop ; 
 the soap-bubble. 
 
 Forms of liquids not subject to the action of gravity ; as the forms 
 assumed by oil suspended in a mixture of water and alcohol of 
 the same density as itself when subjected to various conditions of 
 molecular action. (See Plateau's experimental and theoretical researches 
 on the figures of equilibrium of a liquid mass withdrawn from the action 
 of gravity. Translated in Smithsonian Reports, 1863, et seq. 
 
 2. Forms of crystals. The laws ruling these forms considered 
 in the science of Crystallography. 
 
 b. Forms of matter produced by the action of forces directed by. the 
 vital principle. 
 
 1. Forms of plants. Considered in botany. 
 
 2. Forms of animals. Considered in zoology. 
 
 All the forms of matter not contained in the foregoing classes are 
 irregular or amorphous, and belong to class II. 
 
 Diagrams illustrating above classification of forms, 
 
52 Lecture-Notes on Physics. 
 
 4. Divisibility. - 
 
 Every body can be divided into many parts, and these parts can 
 be further subdivided until the parts become so minute as to escape 
 observation, even when aided by the most powerful microscopes. 
 
 That matter was susceptible of very minute division, was known 
 to the philosophers of ancient Greece and of India, and they dis- 
 cussed the -question, whether matter was infinitely divisible, or 
 divisible only to a certain minuteness, when the parts were sup- 
 posed to be unalterable by any means, whether mechanical or 
 chemical. Mathematically speaking, the division has no limit, for 
 however small the residual particles may be, nevertheles's, since 
 they have extension they have divisibility. But we demand not 
 what may be conceived, but what really exists ; and in fact, the dis- 
 cussions of the ancients are about the infinite divisibility of space, 
 and prove nothing as to the actual divisibility of matter. 
 
 Anaxagoras and Aristotle admitted its infinite divisibility ; Leu- 
 cippus maintained a contrary opinion, and Democritus, siding with 
 Leucippus, gave the name of atoms to the ultimate unalterable parts 
 of which he supposed all bodies to be composed. Epicurus attempted 
 to develope the ideas of Democritus, that found in the seventeenth 
 century a zealous defender in Grassendi, whilst Descartes upheld 
 the opinion of Aristotle. 
 
 But we have in the phenomena of molecular physics, in the laws 
 of combination in chemistry, .and in the facts of crystallography 
 strong evidence that all bodies are composed of ultimate excessively 
 small parts, which are invisible, even with the aid of the most pow- 
 erful microscopes, but still of finite dimensions, of infinite hardness, 
 unalterable by any means, and separated from each other by spaces 
 which are very great when compared with their size. These parts 
 which are the limits of the possible division of matter, are called 
 atoms (Gr. d-ro/toj ; privative and *e>v, to cut.) 
 
 According to this hypothesis, a union or grouping of two or more 
 atoms forms a molecule; a combination of molecules a compound 
 molecule, and a union of the latter a particle. 
 
 The sensible division of matter can be carried to a very minute 
 limit. 
 
 Examples. Gold can be beaten into leaves ^^(jTjth of a milli- 
 metre thick. 
 
 Silver wire, gilded with ? } v th of its diameter of gold, can be 
 drawn so fine that one metre weighs only eight milligrammes. The 
 
Lecture- Notes on Physics. 53 
 
 gold film on this wire is now only g^ss^ of a millimetre thick. 
 By placing a short piece of this wire in nitric acid, the silver core 
 is dissolved out, leaving a tube of gold, having a wall only the 
 gWfiuoth millimetre thick. Now, under the best microscopes, we 
 can discern a surface of ^^th f a millimetre in diameter ; therefore 
 we can divide gold into particles ? ^^th millimetre in diameter, and 
 BWfffiff tn millimetre thick ; yet each of these parts has all the phy- 
 sical and chemical properties of a large mass, as can be determined 
 by testing it under a microscope. 
 
 Dr. Wollaston drew platinum wire so fine that its diameter was 
 only r sVflth millimetre (3^5^ t>th inch) ; and although platinum is the 
 heaviest of the metals, yet it took 200 metres of this wire to equal 
 one centigramme in weight ; or, in other words, one mile of this 
 wire weighed about one grain. Dr. Wollaston accomplished this 
 by wire-drawing a cylinder of silver Jth of an inch in diameter, 
 having in its axis a platinum wire ,-J^th inch diameter, and after 
 having obtained a very fine wire having in its interior a platinum 
 wire of still greater tenuity he dissolved with nitric acid the silver 
 coating, and thus obtained an almost invisible platinum wire. 
 
 The thickness of a soap-bubble, in the dark spot which is formed 
 on it just before it bursts, is rWfftfs tn millimetre. 
 
 Divisibility of matter in solution. One grain of carmine tinges 
 ten pounds of water, which, we can divide into about 617,000 drops. 
 If we suppose that 100 particles of carmine are requisite to produce 
 a uniform tint in each drop, it follows that the one grain of carmine 
 has been divided into 62,000,000 parts. 
 
 Metallic solutions and chemical tests. 
 
 Illustrations from organized bodies. 
 
 The thread of a spider is composed of more than 1000 separate 
 threads. 
 
 The diameter of the red disks contained in human blood is ? ^ O th 
 inch ; while the diameter of the blood-disks of the Java musk-deer is 
 only the r 1 25 incn, so that a drop of this deer's blood, such as would 
 adhere to the point of a fine needle, would contain 150,000,000 disks. 
 
 It has been calculated that some of the siliceous plates which cover 
 and give rigidity to the minute vegetable cell-plants, called diato- 
 maceae, weigh only jfluoiuoffijti 1 of a grain ; yet the surfaces of these 
 plates are covered with the most exquisite tracery of siliceous stria 
 or bars, often not more than ^g^^th inch in width and thickness. 
 Now we can discern a surface of th inch in the best micro- 
 
54 Lecture- Notes on Physics. 
 
 scopes, and a portion of one of these siliceous disks of that area 
 would weigh only about w ^^^th of a grain. 
 
 Divisibility of odorous substances. 
 
 A portion of musk will give off a powerful odor during a year, 
 and yet its diminution in weight has not been sufficient to be detected 
 by the most delicate balance. 
 
 "In order to offer an inexact idea of the minuteness of the par- 
 ticles of musk which are still capable of imparting some odor, we 
 state, after a well known experimenting physiologist, that a certain 
 liquid, containing as much of an extract of spirit of musk as 
 ssimffVfiffUfirti part of its whole weight, was at this time still dis- 
 tinctly odorous. A grain's weight of a liquid of which s^^J^Tj^th 
 part was of that extract, spread an intensely penetrating odor. Next 
 after musk are to be mentioned certain flower ethers, especially the 
 oil of roses, a little drop of which is sufficient to fill with odor an 
 immense atmosphere. The same physiologist states that a certain 
 space filled with air, of which, at the highest, only fTj^Jo"OTi tn P art 
 was vapor of oil of roses, still diffused a distinct odor of roses." 
 See article " On the Senses," in Smithsonian Eeport, 1865. 
 
 5. Compressibility. 
 
 All bodies when subjected to exterior pressure are reduced in 
 volume, and since all bodies can thus be indefinitely compressed, 
 it follows that the matter of which bodies are composed does not 
 fill the space which is contained within their bounding surfaces ; or, 
 as it is sometimes stated, a body does not form a plenum of matter. 
 
 The compressibility of solids is seen in all structures and machines 
 where matter experiences great pressure ; e. g. The stone pillars 
 which support the dome of the Pantheon, at Paris, sensibly dimin- 
 ished in height as the weight which they bear was placed upon 
 them, and a visible depression takes places in the arches of massive 
 bridges when the "centres" are knocked away. 
 
 The relief of coins and medals is produced by subjecting disks 
 of metal, placed between hard steel dies, to an intense and sudden 
 pressure, which is so great that the metal disk is not only changed in 
 form, but its volume after being struck is appreciably less than before. 
 
 For a long time, liquids were regarded as incompressible, and 
 this opinion seemed to be confirmed by many experiments which 
 the Academy of Florence made during the end of the seventeenth 
 century. Their most celebrated experiment, which, however, had 
 
Lecture-Notes on Physics. 55 
 
 previously been made by Lord Bacon, consisted in filling a hollow 
 silver sphere with water, and after stopping the orifice with a screw- 
 plug, subjecting it to the powerful pressure of a screw-press. It is 
 evident that if the sphere is flattened, its capacity is diminished, 
 and the water compressed. They were surprised, however, to see 
 the water appear on the exterior surface, as a fine dew, and they 
 thence concluded that water was incompressible, and silver porous 
 to this liquid. 
 
 John Canton, in Trans. E. S. 1762, first showed that liquids were 
 compressible, even with the pressure of one atmosphere. His appa- 
 ratus consisted of a large thermometer which contained the liquid, 
 and the bulb of which was inclosed in an exhausted receiver. The 
 liquid was thus entirely relieved of atmospheric pressure. The 
 height of the liquid was now marked upon the stem, and the end 
 of the sealed thermometer tube being broken, the air was allowed 
 to enter the stem and press upon the surface of the liquid, and to 
 enter the receiver and press upon the outside of the bulb. The 
 liquid instantly fell in the tube, thus clearly showing a compression 
 produced by the pressure of one atmosphere. 
 
 Canton thus found that water was compressed nj^Vuiiu^ 8 f i ts 
 volume by the pressure ef one atmosphere, and this determination 
 is quite exact. 
 
 In 1826, Jacob Perkins described in the Trans. R. S. an apppa- 
 ratus which renders the compressibility of liquids very evident. 
 Into a hollow cylinder passed a rod through a stuffing-box, and 
 around this rod tightly fitted a leather washer, which rested on the 
 top of the cylinder. This apparatus being filled with water, or 
 other liquid, was placed in a closed cannon, into which water was 
 forced by a strong pump, until a certain pressure was attained, 
 which was measured by the lifting of a safety-valve and the escape 
 of water. 
 
 On taking out the cylinder from the cannon, the leather washer 
 was found on the rod several inches above the place it occupied 
 before the experiment, thus showing that the rod had entered by 
 that quantity into the vessel, as the pressure which it exerted com- 
 pressed the liquid. 
 
 Describe and use (Ersted's piezometer, and exhibit table of Reg- 
 nault and Grassi's results on the compressibility of liquids. 
 
 Gases, of all bodies, are the most easily compressed. This fact is 
 readily shown by enclosing a gas in a cylinder, into which glides 
 
56 Lecture-Notes on Physics. 
 
 an air-tight piston. By pressing on the piston, it is forced into the 
 cylinder, and the gas is reduced in volume. On relieving it of 
 pressure, the gas expands to the volume it had before the experi- 
 ment, and will do so no matter into how small a volume it may 
 have been compressed, or how often the experiment may have been 
 made. In this respect (of perfect elasticity) gases and liquids differ 
 from solids. This is explained by the fact, that the ultimate parts 
 of solids have polarity. 
 
 It is found that by pressure, gases are reduced to volumes, which 
 are the reciprocals of the pressures. 
 
 6. Dilatability. 
 
 When any body is subjected to exterior pressure on all its sur- 
 faces, it is forced into a smaller volume, but on the pressure being 
 removed, it expands to the dimension it had before the pressure 
 was applied. Also, the volumes of all bodies are increased by a 
 rise in their temperature; and Cagnard Latour has shown that 
 wires and rods when stretched, have their volumes increased. 
 
 This general property of increase in volume from these causes is 
 called dilat ability. 
 
 The dilation of solids and of liquids is shown by their expan- 
 sion, when relieved of hydrostatic and atmospheric pressures ; and 
 any gas expands when the atmospheric pressure is removed. 
 
 Experiments. The expansion of solids shown when relieved of 
 powerful pressure. The experiment of Canton reversed, shows the 
 expansion of a liquid as the pressure of the atmosphere is removed. 
 
 The expansion by heat, shown by Saxton's pyrometer, of liquids 
 and of gases by large liquid and air thermometers. 
 
 Experiments of stretching wires and India-rubber, with Cagnard 
 Latour's apparatus. 
 
 7. Porosity. 
 
 The spaces which exist between the ultimate parts of bodies 
 that is, between their atoms and between their molecules are 
 called pores ; by virtue of which all bodies have porosity. 
 
 This property is a natural deduction from the properties of com- 
 pressibility and dilatability ; for all bodies can be reduced in vol- 
 ume by pressure, and this reduction of volume is only limited by 
 the pressures which our machines can produce, and all bodies expand 
 
Lecture-Notes on Physics. 57 
 
 in volume when relieved of pressure ; we therefore conclude that the 
 ultimate parts of bodies do not touch, but are separated by intervals 
 that we call pores. 
 
 This property of porosity, first indicated and concisely defined by 
 Gassendi, can, however, be directly proved by numerous experiments. 
 
 Porosity of solids. Water and oil have, by enormous pressures, 
 been forced through the walls of vessels of gold and of iron; iron 
 converted into steel by absorbing carbon at a high temperature ; 
 gold and the densest bodies are transparent when reduced to suffi- 
 ciently thin leaves ; lead is so porous to mercury, that Prof. Henry 
 made a syphon of a solid bar of lead draw mercury over the side 
 of a vessel containing it ; marble and all rocks are porous to gases, 
 allowing them to "diffuse" through them. We can ascertain how 
 a stone will be affected by frost, by allowing it first to absorb a solu- 
 tion of sulphate of soda, and then ascertaining the effects produced 
 by its subsequent crystallization. It is better, however, to deter- 
 mine directly the effect of the freezing of absorbed water. See 
 Smithsonian Report, 1857, p. 305. (It is probable that the two pre- 
 ceding instances are examples of the porosity existing between the 
 integral crystals of the stone and not between the ultimate parts, 
 and therefore is not porosity in the sense in which we have defined 
 it.) Hydrophane, a species of agate, becomes translucent by the 
 absorption of water, when immersed in that liquid, while the dis- 
 placed air issues from the hydrophane in minute bubbles. 
 
 M. M. Deville and Troost have recently shown that platinum and 
 other dense metals and cast iron, when heated to redness, are porous 
 to gases. The injury to health produced by heating dwellings by 
 means of cast iron stoves, is caused by the vitiation of the atmos- 
 phere by the gases of combustion transmitted through the hot iron. 
 (See Comptes Rendus, Jan. 20, 1868. Report of commission on the 
 observations of Dr. Garret, on the deleterious effects of cast iron 
 stoves.) 
 
 Porosity of Liquids. If measured volumes of water and of alco- 
 hol are mixed, it is found that their combined volume is less than 
 the sum of their volumes before mixing. The greatest contrac- 
 tion takes place when 100 volumes of water are combined with 116 
 volumes of anhydrous alcohol, the diminution in volume being 
 equal to 3*7 per cent. 
 
 This experiment, due to Reaumur, is made as follows : to a bottle 
 is adapted a deeply fitting cork, perforated by a glass tube. The. 
 8 
 
58 Lecture- Notes on Physics. 
 
 requisite proportion of water is placed in the bottle, and on this, 
 colored alcohol is gently poured until it completely fills the bottle. 
 The stopper is now forced in, and the alcohol rises in the glass tube. 
 The height of the liquid is marked, and the bottle being so inclined 
 that the mixture of the liquids takes place, the fluids contract, and 
 the alcohol descends in the tube. 
 
 Contraction also takes place when water is mixed with sul- 
 phuric acid, with salt and with other substances. No experiment 
 could more clearly show that the atoms of water and alcohol do not 
 really fill the space which the liquids appear to occupy. 
 
 "Water and other liquids dissolve gases; for example, 1 volume 
 of water at 60 Fah. will dissolve 720 volumes of ammonia, 
 and increase in its volume only one-half, and in its weight one- 
 third. 
 
 Porosity of gases. Six parts by weight of carbon will unite with 
 eight parts by weight of oxygen, and yet the resulting carbonic 
 acid gas has the volume of the oxygen before combination. 
 
 Air and vapor are porous to each other, and between the atoms 
 of air there may exist the ultimate parts of other gases. 
 
 It is to be remarked that this property of porosity is not opposed 
 to the property of impenetrability, for it is the atoms which are 
 impenetrable, while it is the inter molecular spaces of one body 
 which are penetrated by the atoms of another. 
 
 We must be careful not to confound the true meaning of porosity, 
 as given above, with what we designate as organic, or structural 
 porosity. Organic pores are interstices which are visible with the 
 aid of the microscope, and often even with the naked eye, while the 
 intermolecular spaces are invisible by any means. 
 
 Examples of organic porosity. Mercury forced through wood. 
 Circulation of sap in plants. 
 
 Examples of structural porosity. Porosity of stones, caused by 
 the minute interstices produced by the crystalline structure of the 
 stone. Paper used to filter liquids. 
 
 8. Mobility. 
 
 Everywhere in the universe, we see masses of matter changing 
 their positions with reference to each other ; while others appear to 
 be at rest. 
 
 We only know that a body changes its position by referring it 
 
Lecture- Notes on Physics. 59 
 
 to other bodies, supposed to be at rest ; and therefore Descartes 
 defined motion as a rectilinear change of distance between two 
 points. From this definition, if only one point existed in the uni- 
 verse, its conditions, whether of rest or of motion, could not be 
 determined. 
 
 If the points to which we refer the moving body be really at 
 rest, then the successive varying measures give us the absolute motion, 
 but if this point be only apparently at rest, we obtain the relative 
 motion. Give illustrations. 
 
 Rest is however only apparent ; we know of no point in the uni- 
 verse which is absolutely at rest ; therefore all the motions which 
 we have determined are relative. 
 
 All bodies on the surface of the earth have the relative positions 
 of their atoms changed by every change of temperature, by every 
 vibration which they transmit; they are in motion with the earth 
 on its axis and in its orbit, while the earth with the planets and 
 sun are being translated in an unknown path in space. The stars, 
 improperly called fixed, have also motions of their own, which, in 
 some cases, are evidently of great magnitude. 
 
 Kind of motions. A motion may be one of translation, of rotation 
 round an axis, or of a combination of these motions ; it may be rec- 
 tilinear or curvilinear, continued or reciprocating. Give illustrations. 
 
 The line a point of a body describes in space, is called its trajectory. 
 
 The velocity of motion of a moving body, or the ratio of the space 
 described to the time of describing it, may be uniform, uniformly 
 accelerated, uniformly retarded, or irregular. 
 
 Uniform motion is that in which equal spaces are passed over in 
 equal, small portions of time. 
 
 Formulae of uniform motion. 
 
 L; s s 
 
 (1) s = VT; T = S- ; V==- 
 
 in which s = space described ; v = velocity of moving body, or the 
 space described in unit of time ; T= time occupied in going over space s. 
 
 Uniformly varied motion is that in which the change in velocity 
 at the end of a certain time, is proportional to this time. 
 
 Let u be the initial velocity of the body, that is to say, the velocity 
 at a given instant from which we commence to reckon the time ; a 
 the acceleration, or the constant quantity by which the velocity varies 
 
60 Lecture- Notes on Physics. 
 
 in a unit of time ; v the velocity at the end of t seconds ; we have 
 from definition of uniformly varied motion. 
 
 (2) v = u a t 
 
 The sign -f- when the motion is accelerated ; the sign when it is 
 retarded. In the last case, the velocity will become o when u=at, 
 that is to say, at the end of a number of seconds represented by u : a. 
 
 If, in formula (2), we make u = o, then the moving body starts 
 from a state of rest, and we have at the end of time t 
 
 (3) v=at; 
 
 that is, the velocity acquired at the end of a certain time is proportional 
 to this time. 
 
 In uniformly accelerated motion, the spaces described ~by a body start- 
 ing from a state of rest, are proportional to the squares of the times 
 employed to describe them; or 
 
 (4) s = JcU 2 
 
 which shows that the space gone over in uniformly accelerated 
 motion, by a body which starts from a state of rest, is equal to the 
 space which it would go over with a uniform motion with the velocity 
 J v or J a t. 
 
 It is often required to know the velocity acquired in function of 
 the space gone over. We obtain this by eliminating t in the two 
 equations v = a t, s = J at 2 , which gives 
 
 (5) v = 
 
 A body falling vertically, in vacuo, by the action of gravity, from 
 a height which is exceedingly small compared with the radius of the 
 earth, may be regarded as having a uniformly accelerated motion. 
 In this case a is equal -to the velocity acquired by the body at the 
 end of the first second of its fall. This velocity which is the measure 
 of the intensity of gravity, and which is equal, in the latitude of New 
 York, to 32 feet 2 inches, is always indicated by the letter g. The 
 formula (5) then becomes 
 
 Problem. "What velocity will a body acquire by falling 772 feet, 
 in the latitude of New York? Ans, 222 feet, 10-28 inches. 
 
Lecture- Notes on Physics. 61 
 
 When the moving body possesses an initial velocity w, at the 
 moment from which we count the time, the above formulae become 
 v = u a t\ s u t + J a t 2 , v = yV +~2a s 
 
 To Galileo is due the 'discovery of the laws of uniformly varied 
 motion. Give his classical demonstration of these laws. 
 
 Co-existence of separate motions. When a body in motion is acted' 
 on by a force, the same effect in motion is produced as if that force 
 moved the body from a state of rest. Thus, a cannon ball pro- 
 jected horizontally from an elevation, will reach the ground by 
 the action of gravity, in the same time (assuming that no air resists 
 its motion). as if it dropped vertically from the elevation through 
 the same height. 
 
 "A body describes the diagonal of a parallelogram by two forces 
 acting conjointly, in the same time in which it would describe its 
 sides by the same forces acting separately." Newton. 
 
 The discovery of the law of the co-existence of motions, is also 
 generally ascribed to Galileo (Dial. 4, Prop. 2), but Aristotle clearly 
 announced it in his Mechanical Problems, c. 24. 
 
 The various relations of space and time can readily be repre- 
 sented geometrically, by making lengths of ordinates stand for 
 velocities acquired at the end of times represented by lengths of 
 abscissas. Thus, uniformly varied motion is represented by a right 
 angled triangle, whose base equals in units of length the units of 
 time during which the motion took place, its altitude, the velocity 
 acquired at the end of this time, while the units of area of the 
 triangle represents the distance gone through by the moving body. 
 
 The consideration of motions, irrespective of force and of the 
 properties of matter, constitutes the part of science called Kinematics. 
 (Gr. Kerens, motion.) The ablest discussion of this subject is found 
 in first chapter of the Natural Philosophy, by Profs. Thomson and 
 Tait, Oxford, 1867. 
 
 9. Inertia Force. 
 
 The property of matter by which it tends to retain its state, 
 whether of rest or of motion, is called inertia. 
 
 By saying that a body has inertia, we merely understand that a 
 body cannot of itself modify its condition, whether of rest or of 
 motion ; and that whenever a body begins to move, or to change 
 the velocity or the direction of its motion, these changes in its con- 
 dition are to be referred to some extraneous cause. 
 
62 Lecture-Notes on Physics. 
 
 When a body is set in motion and abandoned entirely to itself 
 when it is conceived as being alone in space it will move in a 
 straight line, which is the direction of its first motion, and with its 
 first velocity forever. This truth, called the law of inertia, is the 
 result of an extended induction, and was not recognized before the 
 time of Kepler. Descartes made it the ' foundation of his princi- 
 ples of mechanics. 
 
 Give illustrations of above principle, from observations of the 
 motions of the heavenly bodies, and from experiments on the 
 motions of bodies on the surface of the earth. The rotation of the 
 earth on its axis. A pendulum will vibrate two days in a vacuum, 
 when the friction of the point of suspension is reduced to its 
 minimum. 
 
 The apparent departures from the law of inertia, can all be re- 
 ferred to the action of forces or of resistances exterior to the body, 
 and which are opposed to its uniform motion in a straight line. 
 
 Force. 
 
 All the phenomena, or changes which we observe in the condi- 
 tion of matter, are motions or the results of motions of either masses 
 or of their ultimate parts or atoms; and that which produces these 
 changes in the condition of matter is denominated force. 
 
 To Dr. Julius Robert Mayer, of Heilbronn, Germany, we owe the 
 first successful attempt to give as clear conceptions in reference to 
 force, as previously existed in relation to matter. In 184:2, he pub- 
 lished in Liebig'sAnnalen, a short paper of eight pages, entitled Bemer- 
 kungen uber die Krdfte der unbelebeten Natur, which from the funda- 
 mental importance of the truths which it unfolds, and from the re- 
 sults which have been deduced from them, is to be considered as 
 one of the most important additions to knowledge produced in this 
 century. 
 
 Mayer reasons thus: " Forces are causes; accordingly, we may 
 in relation to them, make full application of the principle, causa, 
 &quat effectum. If the cause c has the effect e, then c = e; if, in its 
 turn, e is the cause of a second effect, /, we have e /, and so on ; 
 c = e =f . . . . = c. In a chain of causes and effects, a term or a 
 part of a term can never, as plainly appears from the nature of an 
 equation, become equal to nothing. This first property of all causes 
 we call their indestructibility. 
 
 " If the given cause, c, has produced an effect, e, equal to itself, it 
 
Lecture- Notes on Physics. 63 
 
 has in that very act ceased to be; c has become e; if, after the pro- 
 duction of e, c still remained in whole or in part, there must be 
 still further effects corresponding to this remaining cause ; the total 
 effect of c would thus be >e, which would be contrary to the sup- 
 position c = e. Accordingly, since c becomes e, and e becomes /, 
 &c., we must regard these various magnitudes as different forms 
 under which one and the same object makes its appearance. This 
 capability of assuming various forms, is the second essential pro- 
 perty of all causes. Taking both properties together, we rnay 
 say causes are (quantitatively) indestructible and (qualitatively) con- 
 vertible objects." 
 
 In another important paper, " Bemerkungen ilber das mechanische 
 Aequivalent der Warme, 1851," Mayer says: "Force is something 
 which is expended in producing motion ; and this something which 
 is expended is to be looked upon as a cause equivalent to the effect, 
 namely, to the motion produced." 
 
 Now, in these motions or effects, there are evidently two things to 
 be considered, (1) the mass of matter moved, and (2) the space 
 through which it is moved; and we have therefore force = mass X 
 space gone through ; but as we can measure and compare forces only 
 by measuring and comparing their effects, and as bodies in free 
 motion will move in the same right line, and with uniform velocity 
 forever, we must place the moving bodies in such circumstances 
 that their motions are destroyed, and we have remaining in their 
 stead, equivalent effects, which we can measure; then the compari- 
 son of these measures will give us the relative intensities of the 
 forces. 
 
 These effects, either directly or indirectly obtained from the mov- 
 ing body, are as various as there are kind of forces and resistances 
 existing; thus, we may oppose to a body, moving vertically up- 
 ward, the resistance of gravity (which we may regard as constant, 
 if the upward flight of the body is a very minute fraction of the 
 radius of the earth) ; or, we may oppose the constant resistance, 
 which a body of homogeneous structure offers when it is penetrated 
 by another, as, for example, when a cannon ball penetrates earth, 
 clay, or pine wood ; or, again, we may have, for the effect of the 
 destroyed motion, heat, which makes its appearance whenever a 
 moving body is brought to rest either by friction, percussion, com- 
 pression, &c., with some other body, or, as in Foucault's experi- 
 ment, where a copper disk being forced to revolve between the 
 
64 Lecture-Notes on Physics. 
 
 poles of a powerful electro-magnet, the motion of the wheel being 
 opposed by the reaction existing between the electric currents flow- 
 ing in it and in the magnet, the motion (force) lost by the wheel 
 appears as heat (force) in its substance. 
 
 The heat which is produced by any of these means can readily be 
 caused to evolve, as it disappears, dynamic electricity, light, and 
 chemical action. Thus, in an experiment which the author devised 
 for his classes, about four years ago, the heat developed by the "fall- 
 ing force " of a weight striking the terminals of a compound thermal- 
 battery, formed of pieces of iron wire and German silver wire twisted 
 together at alternate ends, caused a current of electricity through 
 the wires, which being conducted through a helix magnetized a 
 a needle (which then attracted fine iron particles), caused light to 
 appear in a portion of the circuit formed of Wollaston's fine wire, 
 decomposed iodide of potassium, and finally moved the needles of a 
 galvanometer. 
 
 Let us now try to arrive at comparable measures of force by first 
 opposing to the moving body the constant resistance of gravity, 
 and see if the measures thus given for different velocities compare 
 with measures given by other resistances overcome, and for differ- 
 ent quantities of heat, electricity, &c., developed by the disappear- 
 ance of different velocities in the moving mass. 
 
 A body, shot vertically upward, with a velocity v t comes to rest, 
 by the opposing resistance of gravity, after having reached a certain 
 height, which we will call /?. Giving the body twice the initial 
 velocity, 2v, it reaches 4h before it begins to return to the earth ; 
 with the initial velocity 3?;, it reaches the height 9A; while 4v gives 
 16A, and so on; in other words, the heights reached are as the squares 
 of the initial velocities. 
 
 Wherefore, as force = mass X space gone through, it follows that 
 the measure of force is mass X v 2 ; v being the velocity of the mass; 
 or, force = mass X distance gone through in overcQming the constant 
 resistance =mass X v 2 . 
 
 If this measure be true, the same ratio of the square of the 
 velocity, will exist when other resistances are opposed to the mov- 
 ing body. Take the resistance offered by an earth or clay bank to 
 the penetration of cannon balls having different velocities. It is 
 found by artillerists, that a ball striking with the velocity 2t>, will 
 penetrate four times as deep as the same ball with velocity v; while 
 a velocity of 3v will give a penetration of nine times the depth and 
 
Lecture-Notes on Physics. 65 
 
 so on ; the penetration of the same ball being as the squares of its 
 velocities. (See Dr. Wollaston's Bakerian Lecture on the Force of 
 Percussion, Phil. Trans. 1806; and Benton's Ordnance and Gun- 
 nery, K Y., 1862, p. 476, et seq.) 
 
 Having found this measure of force true for these two cases, 
 where motion disappears, let us determine, as we only can, experi- 
 mentally and inductively, whether the relative quantities of heat 
 evolved as different velocities of the moving mass disappear, also 
 preserve the ratio of the squares of those velocities. 
 
 In the year 1850, 'there appeared in the Phil. Trans. E. S. Lond., 
 a paper "On the Mechanical Equivalent of Heat" by Dr. James 
 Preseott Joule, of Manchester, England. This memoir contains 
 one of the most important physical constants ever determined. 
 
 In this investigation was first obtained, by direct experiment, the 
 exact quantity of heat developed by the falling of a given weight 
 through a known height. His experiments on this subject began 
 in 1843, and were continued during six years, until 1849. Dr. 
 Joule, during this long experimental experience, gradually per- 
 fected his apparatus, and learned to eliminate various sources of 
 error, until finally his measures arrived at by different processes 
 gave, within small limits, almost identical values for " the mechani- 
 cal equivalent of heat."* 
 
 His apparatus consisted of an upright copper cylinder, which 
 contained either water, oil, or mercury ; in the lid of this vessel 
 were two apertures, one for a vertical axis, to revolve in without 
 touching the lid, the other for the introduction of a thermometer. 
 The vertical axis, which was perfectly fitted into the bottom of the 
 vessel, carried eight revolving arms or paddles, which, as they went 
 round, passed between openings in four stationary vanes, so that the 
 water could not acquire a motion of rotation and move with the 
 arms; and resistance was thus made to their motion. Two weights 
 were attached to fine flexible cords, which passed over pulleys, and 
 were wound round a roller on the vertical axis, armed with the 
 paddles. These weights in falling caused the paddles to revolve, 
 and by the resistance which it opposed to their motion, the liquid 
 was heated by the equivalent in motion expended. The height of 
 
 * It is to be remarked that Mayer, in 1842, used the expression " mechanical 
 equivalent of heat," and from the difference in the specific heats of the same weight 
 of air under constant pressure, and under constant volume, theoretically deduced a 
 value for this equivalent, which, corrected with the exact data of the above quan- 
 tities as furnished by Kegnault, gives a result nearly identical with Joule's. 
 
66 
 
 Lecture-Notes on Physics. 
 
 the fall was about sixty-three feet (to which we may say that practi- 
 cally the radius of the earth was infinite), and was measured by 
 vertical scales, along which the weights descended. 
 
 The mode of experimenting was as follows : the temperature of 
 the liquid having been ascertained by a thermometer, which was 
 capable of indicating a variation of temperature as small as 2 Jo tn 
 of a degree Fah., and the weights wound up by detaching the roller 
 from the vertical paddle-axis, the precise height of the weights were 
 ascertained after keying the roller; the weights then descended until 
 they reached the floor. The roller was again detached from the 
 paddle-axis, the weights wound up, and the agitation of the liquid 
 renewed. This was repeated twenty times, and then the tempera- 
 ture of the liquid was again observed. The mean temperature of 
 the room was derived from observations made at the beginning, 
 middle, and end of each experiment. 
 
 Corrections were now made for the effects of radiation and con- 
 duction ; and, in the experiments with water, for the quantities of 
 heat absorbed by the copper vessel and by the paddle wheel. In 
 the experiments on the heat produced by the agitation of mercury, 
 and in the heat given out by the rubbing of cast iron plates, the 
 heat capacity of the entire apparatus was ascertained by observ- 
 ing the heating which it produced on a known weight of water in 
 which it was immersed. In all the experiments, corrections were 
 also made for the velocity with which the weights came to the 
 ground, and for the rigidity of the strings. The force expended in 
 friction of the apparatus was diminished as far as possible by the 
 use of friction wheels, and its amount was determined by connect- 
 ing the pulleys without connection with, the paddle-axis, and ascer- 
 taining the weight necessary to give them a uniform motion. 
 
 The following table gives the results of Joule; the second col- 
 umn, as they were obtained in air, the third, the same corrected, as 
 though the weights had descended in vacuo. 
 
 MATERIALS EMPLOYED. 
 
 MEAN EQUIV. 
 IN AIR. 
 
 MEAN EQUIV. 
 IN VAC. 
 
 NO. OF EXP'S 
 FROM WHICH 
 DERIVED. 
 
 Water 
 
 773-640 
 
 772-602 
 
 40 
 
 
 775-032 
 
 774-083 
 
 50 
 
 Cast Iron 
 
 775-938 
 
 774-987 
 
 20 
 
 
 
 
 
Lecture-Notes on Physics. 67 
 
 In the experiments of producing heat by the rotation of one 
 cast iron plate on another, the friction produced considerable vibra- 
 tion in the frame work of the apparatus, and a loud sound ; allow- 
 ance was therefore made for the quantity of force expended in pro- 
 ducing these effects. 
 
 The number 772'692, obtained as the mean of forty experiments 
 on the friction of water, Joule considered the most trustworthy ; 
 but this he reduced to 772, because, even in the friction of liquids, 
 he found it impossible entirely to avoid vibration and sound. The 
 deductions of Joule from these experiments are : 
 
 1. That the quantity of heat produced by the friction of bodies, 
 whether solid or liquid, is always proportional to the force ex- 
 pended. 
 
 2. That the quantity of heat capable of increasing the temperature 
 of one pound of water (weighed in vacuo, and between 55 and 60) 
 by 1 Fah., requires for its evolution the expenditure of a mechanical 
 force represented bj the fall of 1 pound through 772 feet, or 772 foot- 
 
 This is the " Mechanical Equivalent of Heat" or the unit of heat, 
 generally called in honor of the illustrious physicist, "Joule's 
 Unit." 
 
 In French measures, the above heat-unit is thus stated : The heat 
 capable of increasing the temperature of 1 kilogramme of water 1 C., 
 is equivalent to a force represented by the fall of 423'55 kilogrammes, 
 through the space of 1 metre. The descent or ascent of 1 pound 
 through 1 foot is called afoot-pound, while the descent or ascent of 
 1 kilogramme through 1 metre is denominated a kilo gramme-metre. 
 By the adoption of these terms, the expressions of the above truths 
 can be mere concisely enunciated; thus, using French measures, 
 we say, 423 kilogramme-metres is equivalent to 1 kilogramme-degree 
 centigrade. 
 
 Thus, Joule showed that the heat developed was in proportion to 
 the mass X the distance fallen through ; or, what is the same, equiva- 
 lent to the mass X square of the velocity. 
 
 In a remarkably interesting paper, " On the Production of Thermo^ 
 Electric Currents by Percussion," Prof. O. K. Eood, of Columbia 
 College, N. Y., shows directly by careful and skilful experiments,, 
 that the heat and its equivalent in dynamic electricity (which latter 
 gave the measure of the heat), produced by a weight falling from 
 different heights on a compound plate of German silver and iron,j 
 
68 
 
 Lecture- Notes on Physics. 
 
 was in proportion to the height of the fall of the weight, or, what 
 is the same, to the square of the velocity of impact. 
 
 For the details of these experiments, and the precautions taken 
 to avoid the action of extraneous causes mingling themselves with 
 the main effect, the reader is referred to Prof Eood's paper in "The 
 American Journal of Science, July, 1866." 
 
 We here only give some of the results, which speak for them- 
 selves. 
 
 TABLE 2. WITH Two SKINS ABOVE AND BELOW JUNCTION OF METALS. 
 
 
 1 in. 
 
 2 ins. 
 
 2-7 
 
 3 ins. 
 3-55 
 
 4 ins. 
 5-2 
 
 Deflection of Galvanometer-needles 
 Average of eight Observations 
 
 TABLE 3. WITH FOUR LAYERS OF PLAIN SILK ABOVE AND BELOW THE 
 JUNCTURE. 
 
 Distances Fallen 
 
 lin. 
 l-5 
 
 2 ins. 
 3-0 
 
 3 ins. 
 40.5 
 
 4 ins. 
 6-0 
 
 Average Deviation of eight exp'ts. 
 
 TABLE 4. WITH FOUR LAYERS OF WAX-COATED WOVEN SILK. 
 
 Distances Fallen 
 
 1 in. 
 1-07 
 
 2 ins. 
 2-l 
 
 3 ins. 
 3 -28 
 
 4 ins. 
 4 -3 
 
 5 ins. 
 6-0 
 
 Keduced Average Deviation of Five 
 
 
 For a proper interpretation of the above results, it is to be re- 
 marked that the deflection of the galvanometer needles up to 6 
 was in proportion to the intensity of the current, which is itself, 
 within these limits, proportional to the heat at the juncture which 
 developed it. 
 
 It can be further shown that the measure of force, or energy (a 
 term now more generally used, and which we owe to Dr. Thomas 
 Young), is proportional to the square of the velocity of a moving 
 
Lecture-Notes on Physics 
 
 mass, by obtaining the equivalents of effects other than those of 
 resistance offered by gravity, by penetrable bodies, and of the heat 
 developed by friction and by impact. Joule, in 1843, showed that 
 the same relation existed between the heat evolved by the electric 
 current of an electro-magnetic engine, and the mechanical energy 
 expended in producing it, and in 1844 he showed that the heat ab- 
 sorbed and evolved by the rarefaction and condensation of air, is 
 proportional to the amount of mechanical energy evolved and ab- 
 sorbed in these operations. 
 
 In the following table, taken from Yerdet's " Expose de la Theorie 
 MecJianique de la Chaleur^ Paris, 1863, are given the most reliable 
 determinations of the mechanical equivalent of heat. The numbers 
 represent the number of kilogramme-metres, which is equivalent 
 to one kilogramme-degree centigrade of water. 
 
 
 >-' ^ ' 
 
 *3 
 
 
 
 M U 2 
 
 gS 
 
 
 
 ^ H W 
 
 ^ 
 
 EH 
 
 
 ; w H 
 
 W pq 
 
 i 
 
 
 < pj 3 
 
 5 
 
 pa 
 
 
 W O H 
 
 W ^ 
 
 jj 
 
 NATURE OP THE PHENOMENON 
 
 see 
 
 K S 
 
 u 
 
 WHENCE THE DETERMINA- 
 
 a ^ 
 
 ^ H 
 
 & 
 
 TION IS DRAWN. 
 
 K 
 
 02 
 
 H 
 
 
 | H H 
 
 P5 *j 
 
 ri 
 
 
 SgS^ 
 
 r . 
 
 
 
 
 o 2 o o 
 
 
 ^ 
 
 
 hr ^ S 
 
 1/2 W "^ 
 
 5 
 
 
 r s ^ 
 
 p2 
 
 1 
 
 General Properties of Air 
 
 J Mayer 
 
 Y. Eegnault, 1 
 Moll&Van [ 
 
 426 
 
 
 1 Clausius 
 
 Beek J 
 
 
 
 
 Joule 
 
 424 
 
 
 Joule 
 
 Favre 
 
 413 
 
 "Work done by the Steam Engine 
 
 Clausius 
 
 Him 
 
 413 
 
 Heat evolved by Induced Currents 
 
 Joule 
 
 Joule 
 
 452 
 
 Heat evolved by an Electro- ~| 
 
 
 
 
 Magnetic Engine at Rest and > 
 
 Favre 
 
 Favre 
 
 443 
 
 in Motion J 
 
 
 
 
 Total Heat evolved in the cir- ) 
 cuit of a Daniell's Battery... / 
 
 Bosscha 
 
 fw. Weber | 
 j Joule J 
 
 420 
 
 Heat evolved in a metallic wire, ") 
 
 
 
 
 through which an electric I 
 
 Clausius 
 
 Quintus Icilius 
 
 400 
 
 current is passing J 
 
 
 
 
 
 
 
 
 From the above discussion, we conclude that the true measure of 
 force is mass X v 2 . 
 
70 Lecture-Notes on Physics. 
 
 The Theory of Energy considers its Conservation, its Transfor- 
 mation and its Dissipation. 
 
 The Conservation of Energy. The total amount of energy in the 
 universe, or in any limited system which does not receive energy 
 from without, or part with it to external matter, is invariable. 
 Energy is, in other words, as indestructible as matter, and is neither 
 created nor destroyed, but merely changes its form. 
 
 The Transformation of Energy. By an extended induction, we 
 find that any one form of energy may be transformed, wholly or 
 partially, to an equivalent amount in another form. These trans- 
 formations are, however, subject to limitations contained in the 
 principle of 
 
 The Dissipation of Energy. "No known natural process is 
 exactly reversible, and whenever an attempt is made to transform 
 or retransform energy by an imperfect process, part of the energy 
 is necessarily transformed into heat and dissipated, so as to be 
 incapable of further useful transformation. It therefore follows 
 that as energy is constantly in a state of transformation, there is a 
 constant degradation of energy to the final unavailable form of 
 uniformly diffused heat ; and this will go on as long as transforma- 
 tions occur, until the whole energy of the universe has taken this 
 final form." See N. Brit. Rev., May, 1864. 
 
 " There is consequently," says Prof. Thomson, " so far as we under- 
 stand the present condition of the universe, a tendency towards a 
 state in which all physical energy will be in the state of heat, and 
 that heat so diffused that all matter will be at the same temperature ; 
 so that there will be an end of all physical phenomena." 
 
 Yast as this speculation may seem, it appears to be soundly 
 based on experimental data, and to represent truly the present con- 
 dition of the universe, so far as we know it. See Prof. Thomson 
 U 0n a Universal Tendency in Nature to the Dissipation of Mechanical 
 Energy." Proc. K. S. Edinb. and Phil. Mag., 1852. 
 
 The energy of a moving body is the work which it is capable of 
 performing against a resistance before being brought to rest, and is 
 equal to the force which must act on the body to move it from a 
 state of rest to the given velocity. This force is measured by the 
 product of the mass of the body into the height from which it 
 must fall in order to acquire the given velocity ; which is expressed 
 thus: 
 
 M V 2 
 
Lecture-Notes on Physics. 71 
 
 g representing as usual, the velocity acquired by a body at the end 
 of the first second of its fall. 
 
 Energy may be of two kinds (1), Kinetic energy, or energy of 
 motion, and (2) Potential energy, or energy of position or of condi- 
 tion. Thus, the energy of a ball shot vertically upward, is entirely 
 kinetic at the moment of its discharge, while its energy is all 
 potential, or one of position, when it has reached the summit of 
 its flight and begins to descend. It is evident that the ball in 
 descending, will gradually lose potential energy and gain kinetic 
 energy (the sum of the two energies always remaining constant), 
 and when it has reached the level from which it was discharged 
 upward, its energy is again all kinetic, and equal to what it was 
 when it began its upward flight. An oscillating pendulum is an 
 instance where the energy is alternately kinetic and potential. 
 
 All the various forms of energy may be brought under two 
 classes. 
 
 I. Visible Energy, or energy of visible motions' and positions. 
 II. Molecular Energy. 
 
 Under class I. we have 
 
 A. Visible kinetic energy. 
 
 B. Potential energy of visible arrangement, as for examples, 
 
 a head of water ; a coiled spring ; a raised weight. 
 Under class II. 
 
 C. The energy of electricity in motion. 
 
 D. The energy of radiant heat and light. 
 E- The kinetic energy of absorbed heat. 
 F. Molecular potential energy. 
 
 Gr. Potential energy caused by electrical separation. 
 H. Potential energy caused by chemical separation. 
 Now with regard to these various forms of energy, the principle 
 of the conservation of energy asserts that for a body left to itself, 
 or for the entire material universe, we must have 
 
 A-+- B + C -f D + &c., =a constant quantity. 
 
 On the other hand, the various terms of the left hand member of 
 this equation must be considered as variable quantities, subject, 
 however, to the above limitation, but capable of being trans- 
 formed into one another according to certain laws. 
 
 Laws of the transmutations of energy. The following are among 
 
72 Lecture-Notes on Physics. 
 
 the most important cases of transmutation of these energies into 
 one another. 
 
 A into B, when a weight is projected upwards ; into C, when a 
 conductor revolves between the poles of a magnet. A is not trans- 
 muted directly into D ; it is into E, F and Gr. A is not directly con- 
 verted into H. 
 
 B can be converted into A, and through it into other forms of 
 energy. 
 
 C can be transmuted into A, into E, into F, and into H. 
 
 D can be transmuted into E, into F, and into H. 
 
 E and F is converted into A and into B, in the action of any 
 heat-engine ; into C, into D ; into Gr when tourmalines are heated ; 
 and into H. 
 
 Gr can be converted into A and into C. 
 
 H can be transmuted into C ; into E, into F, and into G. (See 
 Elementary Treatise on Heat, by Balfour Stewart, Oxford, 1866.) 
 
 Sources of Energy. The energy available for the production of 
 mechanical work is almost entirely potential, and consists of 
 
 Potential forms of Energy. 
 
 1. Energy of Fuel. 
 
 2. Energy of Food. 
 
 3. Energy of Ordinary Water-Power. 
 
 4. Energy of Tidal Water-Power. 
 
 5. Energy of Chemical separation implied in native sulphur, 
 
 native metals, free oxygen, &c. 
 
 . 
 
 Of Kinetic forms of Energy. 
 
 6. Energy of Winds and Ocean Currents. 
 
 7. Energy of Direct Eays of the Sun. 
 
 8. Energy of Volcanoes, Hot Springs and Internal Heat of the 
 Earth. 
 
 "The immediate sources of these supplies of energy are four : 
 
 I. Primordial Potential Energy of Chemical Affinity, which 
 
 probably still exists in native metals, native sulphur, &c., 
 
 but whose amount, at all events near the surface of the 
 
 earth, is now very small. 
 
 II. Solar Kadiation. III. The Earth's rotation about its axis. 
 IV. The Internal Heat of the Earth. 
 
Lecture -Notes on Physics. 73 
 
 Thus, as regards (1) our supplies of fuel for heat-engines are, as 
 was long ago remarked by Herschel and Stephenson, mainly due to 
 solar radiation. Our coal is merely the result of transformation 
 in vegetables, of solar energy into potential energy of chemical 
 affinity. So, on a small scale, are diamond, amber and other com- 
 bustible products of primeval vegetation. As Prof. Thomson 
 remarks, wood fires give us heat and light which have been got 
 from the sun a few years ago. Our coal fires and gas lamps bring 
 out, for our present comfort, heat and light of a primeval sun, 
 which have lain dormant as a potential energy, beneath seas and 
 mountains for countless ages. 
 
 Though (II) thus accounts for the greater part of our store of 
 energy, (I) must also be admitted, though to a very subordinate place. 
 
 As to (2), the food of all animals is vegetable or animal, and 
 therefore ultimately vegetable. This energy, then, depends almost 
 entirely on (II). This, also, was stated long ago by Herschel. 
 
 Ordinary water-power (3) is the result of evaporation, the diffu- 
 sion and convection of vapor, and its subsequent condensation at a 
 higher level. It also is mainly due to (II). 
 
 Tidal water-power (4), although not yet much used, is capable, if 
 properly applied, of giving valuable supplies of energy. As the 
 water is lifted by the attraction of the sun and moon, it may be 
 secured by proper contrivances at its higher level, and then becomes 
 an available supply of energy when the tide has fallen again. Any 
 such supply is, however, abstracted from the energy of the earth's 
 rotation (III). This was recognized by Kant; Mayer also and J. 
 Thomson showed that the ebb and flow of the tides being due to 
 the earth's revolving on her axis under the moon's attraction, the 
 energy of the tides is really taken from the energy of the earth's 
 revolution ; part of which is thus ultimately dissipated in the heat 
 of friction caused by the tides. The general tendency of tides on 
 the surface of a planet is to retard its rotation till it turns always 
 the same face to the tide-producing body ; and it is probable that 
 the remarkable fact that satellites generally turn the same face to 
 their primary, is to be accounted for by tides produced by the pri- 
 mary in the satellite while it was yet in a molten state. 
 
 "Winds and ocean currents (6), both employed in navigation, and 
 the former in driving machinery, are, like (3), direct transformations 
 of solar radiation (II). 
 
 10 
 
74 Lecture- Notes on Physics. 
 
 As to (8), which is due to (IV), no application to useful mechanical 
 purposes has yet been attempted. 
 
 We must next very briefly consider the origin of these causes, 
 with the exception of (I), which is of course primary. Laplace, 
 Mayer and Helmholtz come to our assistance, and suggest as the 
 initial form of the energy of the universe, the potential energy of 
 gravitation of matter irregularly diffused through infinite space. 
 By simple calculations it is easy to see that, if the matter in the 
 solar system had been originally spread through a space enclosing 
 the orbit of Neptune, the falling together of its parts into separate 
 agglomerations, such as the sun and planets, would far more than 
 account for all the energy they now possess in the forms of heat 
 and orbital and axial revolutions. 
 
 The sun still retains so much potential energy among its parts, 
 that the mere contraction by cooling must be sufficient (on ac- 
 count of the diminution of potential energy) to maintain the 
 present rate of radiation for ages to come. Moreover, the capacity 
 of the sun's mass for heat, on account especially, of the enormous 
 pressure to which it is exposed, is so great that (at the least and 
 most favorable assumption) from 7,000 to 8,000 years must elapse, 
 at the present rate of expenditure, before the temperature of the 
 whole is lowered one degree centigrade, although the amount of 
 solar heat received by the earth in one year is so enormous that it 
 would liquify a layer of ice 100 feet thick, covering the whole 
 surface of the earth, and if we bear in mind that the solar heat 
 which reaches the earth in any time is only ssouuouotj^j of the heat 
 which leaves the sun, we may obtain, some idea of the immense 
 heating power of the radiation from our luminary.* 
 
 It thus appears that if we except tidal-power, the sun's rays are 
 the ultimate source of the available forms of energy with which we 
 are surrounded.f 
 
 We see, from the above exposition, that " Philosophers have 
 extended their ideas of quantity from matter to energy, and thus 
 has arisen the new science of Energetics, or the quantitative study 
 of the transformations of energy (as chemistry is the quantitative 
 study of the transformations of matter), comprehending and uniting 
 all the different branches of physical science." 
 
 * If the entire solar radiation were employed in dissolving a layer of ice, enclos- 
 ing the sun, it would dissolve a stratum 10J miles thick in a day. 
 
 f The sketch we here give of the Sources of Energy, is taken almost entirely 
 from an article on " Energy,' in the N. Brit. Rev., May, 1864. 
 
Lecture- Notes on Physics. 75 
 
 Efficiency of Heat- Engines. After Joule had determined the 
 mechanical equivalent of heat, engineers had the means of testing 
 the actual efficiency of heat-engines. 
 
 If the number of thermal units produced by the combustion of 
 one pound of a given kind of fuel, be multiplied by Joule's unit, 
 772 foot-pounds T the result is the total heat of combustion of the 
 given fuel expressed in foot-pounds. This quantity ranges between 
 5,000,000 and 12,000,000 foot-pounds. But in the best existing 
 steam-engines, it is found that on an average only about J- of the 
 mechanical value of the heat produced by the fuel burning in the 
 furnace, is obtained as useful mechanical effect, the remaining f 
 being wholly lost. 
 
 To understand the cause of this great loss, it is to be remembered 
 that in every heat-engine the heat of the expansible fluid which 
 is the medium by which the heat of the fuel is transformed into the 
 motion of the engine disappears as heat by the exact equivalent, 
 expressed in Joule's units, of the motion produced. Therefore the 
 greater the fall in heat in the vapor, which, in expanding, cools and 
 gives up its heat as motion, so will be the efficacy of the engine. 
 Just as in a head of water, where the greater the difference between 
 the higher and lower level, the greater the power obtained. But in 
 the steam-engine we are obliged to obtain our power from the fall 
 in temperature of the steam, which takes place in its expansion, so 
 that in a steam-engine the power obtained is measured by the differ- 
 ence of temperature between the boiler and condenser, and not by 
 the difference between the temperature of the furnace and con- 
 denser. Now, in the furnace the temperature is about 3,000 degrees 
 above that of the atmosphere, while the temperature of the boiler 
 is only about 200 degrees in excess of that of the condenser ; there- 
 fore it is evident that the larger fall in temperature taking place 
 between the furnace and the steam, the heat is lost or at least not 
 utilized. 
 
 In a perfect engine the steam would enter the cylinder at the tem- 
 perature of the furnace, and expand down until it had given up all 
 its heat as motion to the piston, and would then enter the condenser 
 at the temperature of the atmosphere ; indeed, such an engine would 
 require no condenser, for the steam would condense itself as the 
 heat disappeared in its transmutation into motion. 
 
 We will conclude this sketch on the subject of Energy, with a 
 concise statement in reference to the efficiency of heat-engines 
 
76 Lecture-Notes on Physics. 
 
 taken from Prof. Eankine ; referring the reader who desires further 
 information on this important and interesting subject to the works 
 given below. 
 
 " The total heat produced in the furnace is expended, in any 
 given engine, in producing the following effects, whose sum is equal 
 to the heat so expended : 
 
 1. The waste heat of the furnace, being from 0*1 to 0*6 of the 
 total heat, according to the construction of the furnace and the skill 
 with which the combustion is regulated. 
 
 2. The necessarily -rejected heat of the engine, being the excess of 
 the whole heat communicated to the working fluid by each pound 
 of fuel burned, above the 7 portion of that heat which permanently 
 disappears, being replaced by mechanical energy. 
 
 3. The heat wasted ly the engine, whether by conduction or by 
 non-fulfilment of the conditions of maximum efficiency. 
 
 4. The useless work of the engine, employed in overcoming friction 
 and other prejudicial resistances. 
 
 5. The useful work. The efficiency of a heat-engine is improved 
 by diminishing as far as possible, the first four of those effects, so 
 as to increase the fifth. 
 
 " It appears, then, that the efficiency of a heat-engine is the pro- 
 duct of three factors, viz : I. The efficiency of the furnace, being the 
 ratio which the heat transferred to the working fluid bears to the 
 total heat of combustion ; II. The efficiency of the fluid, being the 
 fraction of the heat received by it, which is transformed into 
 mechanical energy ; and III., The efficiency of the mechanism, being 
 the fraction of that energy which is available for driving machines." 
 
 From the above discussion, we see immediately that by super- 
 heating the steam before it reaches the cylinder, we obtain a greater 
 range of temperature for the steam to fall through in expanding, 
 and thus render efficacious yet more of the heat of the furnace. 
 
 List of Works on the Conservation of Force and Thermodynamics. 
 
 The Correlation and Conservation of Forces ; a collection of the 
 papers of Mayer, Helmholtz, Faraday, Grove, Liebig and Carpenter. 
 Edited by Dr. Edward L. Youmans, N. Y., 1865. 
 
 Joule. On the Calorific Effects of Magneto-Electricity, and on 
 the Mechanical Yalue of Heat. Phil. Mag., Vol. XXIII., 1843. 
 
 On the Changes of Temperature produced by the Earefaction and 
 Condensation of Air. Phil. Mag., May, 1845. 
 
Lecture-Notes on Physics. 77 
 
 On the Mechanical Equivalent of Heat. Phil. Trans., 1850. 
 
 On some Thermodynamic Properties of Solids. Phil. Trans., 
 1859. 
 
 On the Thermal Effects of Compressing Fluids. Phil. Trans., 
 1859. 
 
 Clausius. The Mechanical Theory of Heat, with its applications 
 to the Steam Engine and to the Physical Properties of Bodies. 
 Edited by T. A. Hirst, with an Introduction by Prof. Tyndall. 
 London, 1867. 
 
 Thomson (William). An Account of Carnot's Theory of the 
 Motive Power of Heat. Trans. K. S., Edinb., 1849. 
 
 On the Dynamical Theory of Heat. Trans. E. S., Edinb., 1852. 
 
 Thomson and Joule. On the Thermal Effects of Fluids in Motion. 
 Phil. Trans., 1853. 
 
 On the Changes of Temperature experienced by Bodies moving 
 through Air. Phil. Trans., 1860. 
 
 Rankine. The Steam Engine and other Prime Movers. London, 
 1859. 
 
 Verdel. Expose de la Thdorie Mechanique de la Chaleur. Paris, 
 1863. 
 
 Him. Exposition Analytique et Experimentale de la Theorie 
 Mechanique de la Chaleur. Paris, 1865. 
 
 Saint Robert. Principes de Thermodynamique. Turin, 1865. 
 
 Bacon. Novum Organum, De Forma Calidi, book 2, aph. 20. 
 
 " Now from this our first vintage it follows, that the form or true definition of 
 heat (considered relatively to the universe and not to the sense) is briefly thus : 
 Heat is a motion, expansive, restrained, and acting in its strife upon the smaller 
 particles of bodies. But the expansion is thus modified : while it expands all ways, 
 it has at the same time an inclination upwards. And the struggle in the particles 
 is modified also; it is not sluggish, but hurried and with violence." 
 
 Locke "Heat is a very brisk agitation of the insensible parts of the object, 
 which produce in us that sensation from whence we denominate the object hot; so 
 what in our sensation is heat, in the object is nothing but motion." 
 
 Rumford. Trans. R. S. Lond., 1798. Kumford placed a cannon in a water- 
 tight box, so that it could rotate against a blunt borer firmly pressed against the 
 bottom of its chamber. The box was filled with water of a temperature of 60 F., 
 and the cannon set in rotation by the power of horses. 
 
 " The result of this beautiful experiment was very striking, and the pleasure it 
 afforded me amply repaid me for all the trouble I had had in contriving and 
 arranging the complicated machinery used in making it. The cylinder had been 
 in motion but a short time, when I perceived, by putting my hand into the water, 
 and touching the outside of the cylinder, that heat was generated. 
 
 "At the end of an hour the fluid, which weighed 18-77 Ibs., or 2^ gallons, had 
 its temperature raised 47 degrees, being now 107 degrees. 
 
78 Lecture-Notes on Physics. 
 
 " In thirty minutes more, or one hour and thirty minutes after the machinery 
 had been set in motion, the heat of the water was 142 degrees. 
 
 "At the end of two hours from the beginning, the temperature was 178 degrees. 
 
 "At two hours and twenty minutes it was 200 degrees, and at two hours and 
 thirty minutes it actually boiled. 
 
 " From the results of my computations, it appears that the quantity of heat 
 produced equably, or in a continuous stream, if I may use the expression, by the 
 friction of the blunt steel borer against the bottom of the hollow metallic cylinder, 
 was greater than that produced in the combustion of nine wax candles, each f inch 
 in diameter, all burning together with clear bright flames. 
 
 "One horse would have been equal to the work performed, though two were 
 actually employed. Heat may thus be produced merely by the strength of a horse, 
 and, in case of necessity, this heat might be used in cooking victuals. But no 
 circumstances could be imagined in which this method of procuring heat would be 
 advantageous ; for more heat might be obtained by using the fodder, necessary for 
 the support of a horse, as fuel. 
 
 * * * * * " It is hardly necessary to add, that anything which any insulated 
 body or system of bodies can continue to furnish without limitation, cannot possibly 
 be a material substance; and it appears to me to be extremely difficult, if not 
 quite impossible, to form any distinct idea of anything capable of being excited 
 and communicated in those experiments, except motion" 
 
 2)avy. First scientific memoir, entitled "On Heat, Light, and 
 the Combinations of Light." "Works Vol. II. 
 
 " Experiment. I procured two parallelopipedons of ice, of the temperature of 
 29, six inches long, two wide, and two-thirds of an inch thick ; they were fastened 
 by wires, to two bars of iron. By a peculiar mechanism, their surfaces were 
 placed in contact, and kept in a continued and most violent friction for some 
 minutes. They were almost entirely converted into water, which water was 
 collected, and its temperature ascertained to be 35, after remaining in an atmos- 
 phere of a lower temperature for some minutes. The fusion took place only at the 
 plane of contact of the two pieces of ice, and no bodies were in friction but ice. 
 
 " From this experiment it is evident that ice by friction is converted into water 
 and according to the supposition, its capacity is diminished ; but it is a well-known 
 fact that the capacity of water for heat is much greater than that of ice ; and ice 
 must have an absolute quantity of heat added to it before it can be converted into 
 water. Friction consequently does not diminish the capacity of bodies for heat. 
 
 * ****** 
 
 " Now a motion or vibration of the corpuscules of bodies must be necessarily 
 generated by friction and percussion. Therefore we may reasonably conclude that 
 this motion or vibration is heat, or the repulsive power. 
 
 Davy in his Chemical Philosophy, p. 95, says : 
 
 < * * * * The immediate cause of the phenomena heat, then, is motion, and 
 the laws of its communication are precisely the same as the laws of the communi- 
 cation of motion." 
 
 A process similar to the above classical experiment of Davy, has from time 
 immemorial been used by savage nations in obtaining fire by means of friction: 
 not by rubbing together sticks, as usually stated, for no one could thus produce 
 ignition, but by whirling rapidly, a pointed rod against a wooden block, by means 
 of an arrangement similar to the watchmaker's bow and drill. It is worthy of 
 remark that this apparatus has everywhere the same form, whether used by the 
 
Lecture-Notes on Physics. 79 
 
 islanders of the Pacific or by the aborigines of our own country. One of these 
 instruments can be seen in the State Cabinet of Natural History of New York. 
 
 Henry (Dr. Joseph). Meteorology Patent Office Keport for 
 1857. 
 
 On Acoustics applied to Public Buildings Smithsonian Eeport, 
 1856, p. 228. 
 
 "The tuning-fork was next placed upon a cube of India rubber, and this upon 
 the marble slab. The sound emitted by this arrangement was scarcely greater 
 than in the case of the tuning-fork suspended from the cambric thread, and from 
 the analogy of the previous experiments, we might at first thought suppose the 
 time of duration would be great; but this was not the case. The vibrations con- 
 tinued only about forty seconds. The question may here be asked, what became 
 of the impulses lost by the tuning-fork? They were neither transmitted through 
 the India rubber nor given off to the air in form of sound, but were probably 
 expended in producing a change in the matter of the India rubber, or were con- 
 verted into heat, or both. Though the inquiry did not fall strictly within the 
 line of this series of investigations, yet it was of so interesting a character in a 
 physical point of view to determine whether heat was actually produced, that the 
 following experiment was made. 
 
 * * * And the point of a compound wire, formed of copper and iron, was 
 thrust into the substance of the rubber, while the other ends of the wire were 
 connected with a delicate galvanometer. The needle was suffered to come to rest, 
 the tuning-fork was then vibrated, and its impulses transmitted to the rubber. A 
 very perceptible increase of temperature was the result. The needle moved through 
 an arc of from one to two and a half degrees. The experiment was varied, and 
 many times repeated; the motions of the needle were always in the same direc- 
 tion, namely, in that which was produced when the point of the compound wire 
 was heated by momentary contact with the fingers. The amount of heat generated 
 in this way is, however, small, and indeed, in all cases in which it is generated by 
 mechanical means, the amount evolved appears very small in comparison with the 
 labor expended in producing it." 
 
 Leibnitz. " The force of a moving body is proportional to the square of its 
 velocity, or to the height to which it would rise against gravity." 
 
 Wollaston. Bakerian Lecture on the Force of Percussion. Phil. 
 Trans., Yol. XCYL, 1806. 
 
 " In short, whether we are considering the sources of extended exertion or of accu- 
 mulated energy, whether we compare the accumulated forces themselves by their 
 gradual or by their sudden effects, the idea of mechanic force in practice is always 
 the same, and is proportional to the space through which any moving force is 
 exerted or overcome, or to the square of the velocity of a body in which such force 
 is accumulated." 
 
 Tynddll. Heat considered as a' Mode of Motion. New York, 
 1863. 
 
 I need hardly refer the reader to this charming exposition of 
 Prof. Tyndall ; who, by his enthusiasm and vividness of illustration, 
 has rendered his subject so popular that his work is, in this country, 
 in the library of nearly every man of culture. 
 
80 Lecture-Notes on Physics. 
 
 10. Attraction, 11. Repulsion. 
 
 Attraction exists between the minute parts or atoms of bodies, 
 and they would continually approach, until contact supervened, if 
 they were not kept apart by an equal and opposing repulsion. 
 These molecular attractions and repulsions are often designated as 
 the molecular forces. 
 
 The phenomena of traction, of compression, of porosity, and of 
 the transmission of vibrations through all kind of matter, prove 
 indirectly that bodies are formed of minute parts which do not 
 touch, but are kept at certain distances depending on the intensity 
 of the attractions and repulsions subsisting between them; while 
 there are many direct proofs of the above statement which will be 
 given further on. 
 
 All masses of matter mutually attract each other with an inten- 
 sity directly proportional to their masses, and inversely as the 
 squares of their distances. This tendency is called gravitation, and 
 is common to all matter. This is shown in the celebrated experi- 
 ment devised by the Eev. John Michell, and generally known as 
 the Cavendish experiment for determining the density of the earth. 
 Describe this apparatus. 
 
 Electric and magnetic attractions and repulsions also follow the 
 law of the inverse squares of the distances. 
 
 The different manifestations of attraction and repulsion may be 
 thus arranged : 
 
 GRAVITATION, which act at scnsible distances . L e , 
 
 the **W* of an inch, 
 
 MAGNETISM, 
 
 COHESION, 
 
 ADHESION, I Which act at insensible distances ; i. e. 
 
 CAPILLARITY, within the ^oo 111 of an incl1 - 
 
 CHEMICAL AFFINITY, j 
 
 Molecular Attraction. 
 
 Cohesion designates the attraction existing between the minute 
 parts of the same body ; adhesion the attraction between the parts 
 of dissimilar bodies. Sometimes designated respectively as homo- 
 geneous and heterogeneous attraction. 
 
 Experiments on Cohesion of solids. Two leaden planes pressed 
 together, cohere with a force of forty pounds to the square inch. 
 Two plates of glass cohere even in vacuo, which shows that the 
 
Lecture- Notes on Physics. 81 
 
 phenomenon is not due to the atmospheric pressure. The intensity 
 of the cohesion in this experiment is proportional to the surface, 
 and increases with the time of contact. In plate glass manufacto- 
 ries, mirror glasses sometimes cohere with such force, from having 
 been placed on each other without intervening paper, that it is 
 impossible to separate them. It is to be remarked that in the above 
 instances only a comparatively few points of the cohering surfaces 
 are in contact. 
 
 " There is a precious experiment by Mr. Huyghens in No. 86 of 
 the Philosophical Transactions. A piece of mirror glass being laid 
 on the table, and another, to which a handle was cemented on one 
 surface, being gently pressed on it, with a little of a sliding motion, 
 the two adhered, and the one lifted the o^her. Lest this should have 
 been produced by the pressure of the atmosphere, Mr. Huyghens 
 repeated the experiment in an exhausted receiver, with the same 
 success. 
 
 u He found that one plate carried the other, although they were 
 not in mathematical contact, but had a very sensible distance between 
 them. He found this by wrapping round one of the plates a single 
 fibre of silk drawn off from the cocoon. The adhesion was vastly 
 weaker than before, but still sufficient for carrying the lower plate. 
 
 " Here, then, is a most evident and and incontrovertible example 
 of a mutual attraction acting at a distance. Mr. Huyghens found 
 that if, in wrapping the fibre round the glass, he made it cross a fibre 
 already wrapped round it, there was no sensible attraction. In this 
 case, the glasses were separated by a distance equal to twice the 
 diameter of a fibre of silk. 
 
 " I said that this experiment showed that it was not the attrac- 
 tion of gravitation that produced the cohesion. I have repeated 
 the experiment with the most scrupulous care, measuring the dis- 
 ance of the glasses (the diameter of a silk fibre), and the weight 
 supported. I find this, in all cases, to be nearly 14 J times the action 
 of gravity. The calculation is obvious and easy. I tried it in dis- 
 tances considerably different, according to the diameter oT the fibre. 
 I must inform the person who would derive his information from 
 his own experiments, that there are many circumstances to be at- 
 tended to which are not obvious, and which materially affect the 
 result. The silk fibres are not round, but very flat, one diameter 
 being almost double of the other. The 2400th part of an inch may 
 be considered as the average smaller diameter of a fibre. A mag- 
 11 
 
82 Lecture-Notes on Physics. 
 
 nif ying glass must be used, and great patience in wrapping the fibre 
 round the glass so that it may not be twisted. A flaxen fibre is 
 much preferable, when gotten single, and fine enough, for it is a 
 perfect cylinder. I must also -inform him, that no regularity will 
 be had in experiments with bits of ordinary mirror; these are 
 neither flat enough, nor well enough polished. We must employ 
 the square pieces which are made and finished by a very few Lon- 
 don artists for the specula of the best Hadley's [Godfrey's] quadrants. 
 These must be most carefully cleaned of all dust or damp. Yet this 
 must not be done by wiping them with a clean cloth ; this infal- 
 libly deranges everything by rendering the plate electric. I suc- 
 ceeded best by keeping them in a glass jar, in which a piece of 
 moist cloth was lying, but jaot touching the glasses. "When wanted, 
 the glasses are taken out with a pair of tongs and held a little while 
 before the fire, which dissipates the damp which had adhered to 
 them, and which prevented all electricity. With these precautions, 
 and a careful measurement of the diameter of the silk fibre, the ex- 
 periments will rarely differ among themselves one part in ten. 
 
 * * * * * * If the plates have been hard 
 pressed, with a sliding or grinding motion, the adhesion is then 
 either very strong, or nothing at all ; when they do adhere, it seems 
 to be another stage or alternation of the force, as will be explained 
 by and by. But they rarely adhere, owing to fragments torn off 
 by the grinding. The glasses will be scratched by it. 
 
 " I thought this capital experiment worthy of a very minute de- 
 scription, it being that which gives us the means of mathematical 
 and dynamical treatment in the greatest perfection." "A System of 
 Mechanical Philosophy, by John Eobison, L. L. D. Edited by Sir 
 David Brewster, Edinburg: printed for John Murray, London, 1822." 
 Yol. I. page 240, et seq. 
 
 Cohesion in solids is measured by the force in pounds avoirdu- 
 pois required to tear apart, by a direct pull, a rod of one square 
 inch area of section. This measure is called the tenacity of a body. 
 
 TABLE OF TENACITIES FROM "KANKINE'S APPLIED MECHANICS." 
 
 Steel -. 116.COO 
 
 Iron, wire 95,000 
 
 " wire ropes 90,000 
 
 u wrought bars and bolts 65,000 
 
 cast 10,500 
 
 Copper, wire 60,000 
 
Lecture-Notes on Physics. 83 
 
 Copper, cast 19,fOO 
 
 Brass, wire 49.0')0 
 
 " cast 18,0>0 
 
 Gun-metal (copper 8, tin 1) 36,000 
 
 Zinc 7,50J 
 
 Tin, cast , 4,600 
 
 Lead, sheet 3,300 
 
 Teak 18,00) 
 
 Ash 17,000 
 
 Mahogany 16,20) 
 
 Locust 16,(H 
 
 Oak, European 14,900 
 
 " American red 10,250 
 
 Fir red pine '.? 13.* 00 
 
 " spruce 12,400 
 
 " larch 9,500 
 
 Chestnut 12,0.0 
 
 Beech 11,5(0 
 
 Maple 10,600 
 
 Hempen cables 5,600 
 
 Slate 11,2-10 
 
 Glass 9,400 
 
 Brick 290 
 
 Mortar 50 
 
 Cohesion of liquids is shown in the force required to separate a 
 disk of wood from a liquid which wets it ; this force varies with 
 the liquid, requiring 52 grains per square inch to separate the disk 
 from water; 31 grains for oil of turpentine; 28 grains for alcohol. 
 These experiments, however, as will be seen below, give only the 
 relative cohesion. 
 
 Prof. Joseph Henry, in a valuable contribution to molecular 
 physics, published in the Proceedings of the American Philoso- 
 phical Society for April, 1844, showed that the molecular at- 
 traction of water for water, instead of being only about fifty-two 
 grains to the square inch, is really several hundred pounds, and is 
 probably equal to that of the attraction of ice for ice. The follow- 
 ing are extracts from Dr. Henry's paper : 
 
 " The passage of a body from a solid to a liquid state is generally 
 attributed to the neutralization of the attraction of cohesion by the 
 repulsion of the increased quantity of heat; the liquid being sup- 
 posed to retain a small portion of its original attraction, which is 
 shown by the force necessary to separate a surface of water from 
 water, in the well known experiment of a plate suspended from a 
 scale beam over a vessel of the liquid. It is, however, more in 
 accordance with all the phenomena of cohesion to suppose, instead 
 
84 Lecture-Notes on Physics. 
 
 of the attraction of the liquid being neutralized by the heat, that 
 the effect of this agent is merely to neutralize the polarity of the 
 molecules so as to give them perfect freedom of motion around 
 every imaginable axis. The small amount of cohesion (52 grains 
 to the square inch), exhibited in the foregoing experiment, is due, 
 according to the theory of capillarity of Young and Poisson, to the 
 tension of the exterior film of the surface of water drawn up by 
 the elevation of the plate. This film gives way first, and the strain 
 is thrown on an inner film, which, in turn, is ruptured ; and so on 
 until the plate is entirely separated ; the whole effect being similar 
 to tearing the water apart atom by atom. 
 
 " Keflecting on the subject, the author has thought that a more 
 correct idea of the magnitude of the molecular attraction might be 
 obtained by studying the tenacity of a more viscid liquid than water. 
 For this purpose, he had recourse to soap water, and attempted to 
 measure the tenacity of this liquid by means of weighing the 
 quantity of water which adhered to a bubble of this substance just 
 before it burst, and by determining the thickness of the film from 
 an observation of the color it exhibited in comparison with New- 
 ton's scale of thin plates. Although experiments of this kind 
 could only furnish approximate results, yet they showed that the 
 molecular attraction of water for water, instead of being only about 
 52 grains to the square inch, is really several hundred pounds, and 
 is probably equal to that of the attraction of ice for ice. The effect 
 of dissolving the soap in the water, is not, as might at first appear, 
 to increase the molecular attraction, but to diminish the mobility 
 of the molecules, and thus render the liquid more viscid. 
 
 " According to the theory of Young and Poisson, many of the 
 phenomena of liquid cohesion, and all those of capillarity, are due 
 to a contractile force existing at the free surface of the liquid, and 
 which tends in all cases to urge the liquid in the direction of the 
 radius of curvature towards the centre, with a force inversely as 
 this radius. [The explanation of the existence of this contractile 
 force will be given in the next section of the Notes, which con- 
 siders Capillarity.] 
 
 " According to this theory, the spherical form of a dew-drop is 
 not the effect of the attraction of each molecule of the water on 
 every other, as in the action of gravitation in producing the globu- 
 lar form of the planets (since the attraction of cohesion only extends 
 to an appreciable distance), but it is due to the contractile force 
 
Lecture-Notes on Physics. 85 
 
 which tends constantly to enclose the given quantity of water within 
 the smallest surface, namely, that of a sphere. The author finds a 
 contractile force similar to that assumed by this theory, in the sur- 
 face of the soap-bubble ; indeed, the bubble may be considered a 
 drop of water with the internal liquid removed, and its place sup- 
 plied by air. The spherical form in the two cases is produced by 
 the operations of the same cause. The contractile force in the sur- 
 face of the bubble is easily shown by blowing a large bubble on 
 the end of a wide tube, say an inch in diameter ; as soon as the 
 rnouth is removed, the bubble will be seen to diminish rapidly, and 
 at the same time quite a forcible current of air will be blown 
 through the tube against the face. This effect is not due to the 
 ascent of the heated air from the lungs, with which the bubble was 
 inflated, for the same effect is produced by inflating with cold air, 
 and also when the bubble is held perpendicularly above the face, so 
 that the current is downwards. 
 
 " Many experiments were made to determine the amount of this 
 force, by blowing a bubble on the larger end of a glass tube in the 
 form of a letter U, and partially filled with water ; the contractile 
 force of the bubble, transmitted through the enclosed air, forced 
 down the water in the larger leg of the tube, and caused it to rise 
 in the smaller. The difference of level observed by means of a 
 microscope, gave the force in grains per square inch, derived from 
 the known pressure of a given height of water. The thickness of 
 the film of soap-water which formed the envelope of the bubble, 
 was estimated as before, by the color exhibited just before burst- 
 ing. The results of these experiments agree with those of weigh- 
 ing the bubble, in giving a great intensity to the molecular attrac- 
 tion of tne liquid ; equal at least to several hundred pounds to the 
 square inch. Several other methods were employed to measure the 
 tenacity of the film, the general results of which were the same ; 
 the numerical details of these are reserved, however, until the ex- 
 periments can be repeated with a more delicate balance. 
 
 " The comparative cohesion of pure water and soap- water was 
 determined by the weight necessary to detach the same plate from 
 each ; and in all cases the pure water was found to exhibit nearly 
 double the tenacity of the soap-water. The want of permanency 
 in the bubble of pure water is therefore not due to feeble attrac- 
 tion, but to the perfect mobility of the molecules, which causes the 
 equilibrium, as in the case of the arch, without friction of parts, to 
 be destroyed by the slightest extraneous force." 
 
86 Lecture-Notes on Physics. 
 
 The above investigation of Dr. Henry will be referred to again 
 under the head of Capillarity. 
 
 Gases. Between the molecules of the same gas repulsion exists, 
 but a slight attraction appears to prevail between the molecules of 
 different gases. 
 
 Adhesion. Of solids to solids. Placing of metals. Gold leaf stamped 
 on metals. " It is known that if two pieces of metal are scraped 
 very clean, a severe blow will make them to cohere so as to be in- 
 separable. It is thus that flowers of gold and silver are fixed on 
 steel and other metals. The steel is first scraped clean, and a thin 
 bit of gold or silver is laid on it, and then the die is applied by a 
 strong blow with a hammer. It is remarkable that they will not 
 adhere with such firmness, if they adhere at all, when the surfaces 
 have been polished in the usual way, with fine powders, &c. This 
 is always done with the help of greasy matters. Some of this pro- 
 bably remains, and prevents that specific action that is necessary. 
 I am disposed to think that the scraping of the surfaces' also ope- 
 rates in another way, viz : by filling the surface with scratches, 
 that is, ridges and furrows. These allow the air to escape as the 
 pieces come together by the blow. If the mere blow were sufficient, 
 a coin would adhere fast to the die. But, in coining, the flat face 
 of the die first closes with the piece of metal, and effectually con- 
 fines the air which fills the hollow that is to form the relief of the 
 coin. This air must be compressed to a prodigious degree, and in 
 this state, it is still between the die and the coin. "We may say that 
 the impression on the coin is really formed by this included air ; 
 for the metal in this part of the coin is never in contact with the 
 die. I know of two cases, which greatly confirm this conjecture. 
 The dies chanced to crack in the highest part of the relief, and after 
 this were thrown aside (although in one, for a common die, the crack 
 was quite insignificant), because the coin could seldom be parted 
 from them." [JRobison.~] 
 
 "When bladder is dried on glass, the adhesion is so great that it 
 cannot be torn off without bringing with it some of the glass. 
 
 Of solids to liquids. A rapidly issuing jet of water is deflected 
 from its course by touching a glass rod. Experiments quoted above, 
 on relative cohesion. 
 
 Of liquids to liquids. Oil and spirits of turpentine spread over 
 the surface of water. 
 
 Of gases to solids. Air and vapor of water adhere with consider- 
 
Lecture- Notes on Physics. 87 
 
 able force to the surface of glass. This shown by placing a beaker 
 of water under the receiver of an air pump, when bubbles of air, 
 previously coating as a film the surface of the glass, collect on its 
 surface. That vapor of water also coats with a film the glass, is 
 known from the increase in weight of a dry light glass vessel, when 
 exposed to a damp atmosphere. The action of clean platinum and 
 gold in condensing gases on their surfaces. Charcoal absorbs 98 
 times its volume of ammonia, and 14 volumes of carbonic acid gas. 
 As ammonia is condensed into a liquid by a pressure of seven at- 
 mospheres, at a temperature of 60 F., it follows that the absorbed 
 gas must exist in the liquid state in the interstices of the charcoal. 
 Gold leaf will not sink in water from the air condensed on its surface. 
 
 Of gases to liquids. Air and all gas absorbed by water. Oxygen 
 gas absorbed from the air by melted silver. 
 
 Of gases to gases. In the diffusion of gases, one gas acts as a 
 vacuum to another. Yapor diffuses in space containing a gas, until 
 the same tension is produced as would have been acquired by the 
 same vapor evaporating from its liquid in a vacuum. 
 
 Molecular Repulsion. 
 
 "When the convex surface of a plano-convex lens, of a radius of 
 curvature of 20 feet or more, is pressed upon a plate of glass, a 
 system of concentric colored rings are observed. These rings are 
 produced by the interference of certain of the rays of light reflected 
 from the under surface of the lens with those reflected from the 
 upper surface of the glass plate. By knowing the diameter of any 
 ring, and the radius of curvature of the lens, we can calculate the 
 distance between the convex surface and the glass plate correspond- 
 ing to the ring. Newton thus found that the distance at each ring 
 exceeded the distance of the ring immediately within it by the 
 
 ffvl* 11 of an incn - 
 
 Now, unless the lens be heavy, or pressed against the glass plate, 
 no colored spot appears in the centre, and it can be shown that the 
 glasses are, in this case, not in contact, but distant from each other 
 at least ^^th of an inch, and at this distance reposes the upper 
 glass, kept from the plate by a repulsion existing between them. 
 
 By forcing the glasses nearer together, we at length produce a 
 Hack spot at the centre of the ring system, and Prof. Kobison found 
 that u a very considerable force is necessary for producing the black 
 spot. A greater pressure makes it broader, and in all probability 
 
88 Lecture-Notes on Physics. 
 
 this is partly by the mutual yielding of the glasses. I found that 
 before a spot, whose surface is a square inch, can be produced, a 
 force exceeding 1,000 pounds must be employed. When the ex- 
 periment is made with thin glasses, they are often broken before 
 any black spot is produced. 
 
 " What is it that we properly, and without any figure of speech, 
 call a pressure ? It is something that we are informed of solely by 
 our sense of touch. What do we feel by means of this sense, when 
 the upper lens lies in our hand ? It is not the matter of this lens, 
 for we now see that there is some measurable distance between the 
 lens and the hand ; it is this repulsion. Give a blind man a strong 
 magnet in his hand, and let another person approach the north pole 
 of a similar magnet to its north pole. The blind man will think 
 that the other has pushed away the magnet he holds in his hand 
 with something that is soft. 
 
 ### #'.' "v* ^ #..-' "ii 
 
 "There is, therefore, an essential difference between mathematical 
 and physical contact ; between the absolute annihilation of distance 
 and the actual pressure of adjoining bodies. We must grant that 
 two pieces of glass are not in mathematical contact till they are ex- 
 erting a mutual pressure not less than 1,000 pounds per inch. For 
 we must not conclude that they are in contact till the black spot 
 appears ; and even then we dare not positively affirm it. My own 
 decided opinion is, that the glasses not only are not in mathemati- 
 cal contact in the black spot, but the distance between them is 
 vastly greater than the 89,000th part of an inch, the difference of 
 the distances at two successive rings. 
 
 *####*## 
 
 " While gravity produces sensible effects at the utmost boundary 
 of the solar system, these [attractions and repulsions] seem limited 
 in their exertion to a small fraction of an inch, perhaps not exceed- 
 ing ^(juth part in any instance ; and in this narrow bounds we 
 observe great diversity in the intensity, although we have not yet 
 been able to ascertain the law of variation. What is of peculiar 
 moment, we have seen that those corpuscular forces even change 
 their kind by a change of distance, producing at one distance, the 
 mutual approach, and at another distance the mutual separation of 
 the acting corpuscles, from being attractive, becoming repulsive. 
 # # * * # * # ' * 
 
 " Physical contact, or pressure, becomes sensible at the distance of 
 
Lecture- Notes on Physics. 89 
 
 the 5000th part of an inch nearly, and decreases much faster than in 
 the inverse duplicate ratio of the distances. I could infer this from 
 my experiments with the glasses with great confidence, although I 
 could not assign the precise law," JRofasorts Mechanical Philosophy, 
 Yol I. p. 250, et seq. 
 
 All bodies expand when relieved of pressure, and this expansion 
 is caused by the mutual repulsion of their atoms. 
 
 A dew drop does not touch the leaf above which it reposes, but 
 is held at a certain distance by repulsion. Certain insects walk on 
 water, which is repelled by their feet, so that each foot rests in a 
 pit. A needle floats on the surface of water, in which it forms a 
 trough in which it rests. 
 
 Prof. Baden Powell has shown (Phil Trans. 1834:, p. 485), that 
 the colored rings known as Newton's rings, change their breadth 
 and position in such a manner, when the glasses which produce 
 them are heated, that he inferred that the glasses repelled each 
 other. 
 
 " The distance at which the repulsive power can act, is shown by 
 these experiments to extend beyond that at which the most extreme 
 visible order of Newton's tints is formed. But I have also re- 
 peated the experiment successfully with the colors formed under 
 the base of a prism placed upon a lens of a very small convexity ; 
 and according to the analysis of these colors given by Sir John 
 Herschel, the distance is here about the 1100th of an inch." 
 
 Yery finely divided solids, such as elutriated silica and wood- 
 ashes will, when rendered incandescent, flow like liquids about the 
 capsule which contains them ; while it can be directly shown that 
 a sensible distance exists between a layer of these powders and a 
 heated plate on which it rests. 
 
 The Molecular Constitution of Matter. The Atomic Theory of 
 
 Boscovich. 
 
 Of the foregoing facts which we have brought together, concern- 
 ing the general properties of matter and the effects of attraction 
 and repulsion existing between the minute parts of bodies, we can 
 frame a hypothetical theory which, in a few lines, or postulates, will 
 embrace what we otherwise could not express in many pages. 
 
 This theory together with the doctrine of the Conservation of 
 Energy, are the two most important generalizations in Physics, and, 
 12 
 
90 Lecture-Notes on Physics. 
 
 in our opinion, the following generalization forms an absolutely 
 essential introduction to the proper study of this department of 
 science. 
 
 The ancient Greek philosopher, Democritus, propounded an hypo- 
 thesis of the constitution of matter, and gave the name of atoms to 
 the ultimate unalterable parts of which he imagined all bodies to 
 be constructed. In the 17th century, Grassendi revived this hypo- 
 thesis, and attempted to develope it, while Newton used it with 
 marked success in his reasonings on physical phenomena ; but the 
 first who formed a body of doctrine which would embrace all known 
 facts in the constitution of matter, was Eoger Joseph Boscovich, of 
 Italy, who published at Vienna, in 1759, a most important and inge- 
 nious work, styled Theoria Philosophise Naturalis ad unicam legem 
 virium, in Natura existentium redacta. This is one of the most pro- 
 found contributions ever made to science ; filled with curious and 
 important information, and is well worthy of the attentive perusal of 
 the modern student. In more recent days, the theory of Boscovich 
 has received further confirmation and extension in the researches of 
 Dalton, Joule, Thonison, Faraday, Tyndall and others. 
 
 We present here a generalization which, while giving the sub- 
 stance of the important postulates of Boscovich embraces others 
 made necessary by the progress of science since 1760. 
 
 1. Matter has trilineal extension. 
 
 2. Is impenetrable. 
 
 3. Does not form a plenum. 
 
 4. All matter consists of indefinitely small but finite parts; of ex- 
 treme hardness ; indivisible, and unalterable by either mechanical 
 or chemical means ; and endued with impenetrability and inertia. 
 These ultimate parts are called Atoms. 
 
 5. These atoms are not in mathematical contact, but are separated 
 from each other by distances which are great when compared with 
 the size of the atoms. 
 
 6. A union of atoms forms a molecule, and combinations of mole- 
 cules form particles of which all bodies are composed. 
 
 7. There exist between the atoms, attractions and repulsions: 
 when these tendencies are equal, the atoms preserve fixed positions 
 and the volume of the body is constant. These molecular forces 
 vary both in intensity and direction, by a change of distance, so 
 that at one distance two atoms attract each other, and at another 
 they repel ; there being, within the distance in which physical con 
 
VolLV, 3* Scries.Pl.W 
 
 Fig. 3. 
 
 H 
 
 Fig. 4 
 
 Fig. 5. 
 
 Fig. 6. 
 
 Fig. 8. 
 
 . 7. 
 
Lecture-Notes on Physics. 91 
 
 tact is observed (about 5^3 3 th inch), several alternations of attrac- 
 tion and repulsion. 
 
 8. The repulsion of two atoms generally diminishes more rapidly 
 than their attraction when the distance between them is increased ; 
 while their repulsion increases more rapidly than their attraction 
 when their distance is diminished. 
 
 9. The law of variation is the same in all atoms. It is therefore 
 mutual ; for the distance of a from b being the same as that of I 
 from a, if a attract or repel 6, b must attract or repel a with exactly 
 the same force. 
 
 10. At all sensible distances (i. e. beyond ^^th inch), this mutual 
 tendency is attraction, and varies inversely as the squares of the 
 distances. It is known as gravitation. 
 
 11. The last force which is exerted between two atoms as their 
 distance diminishes, is an insuperable repulsion, so that no force 
 however great can press two atoms into mathematical contact. 
 
 12. Between the molecules of gases continued repulsion seems 
 to exist, so that when relieved of exterior force, a gas expands 
 indefinitely. 
 
 Between the molecules of liquids exist attraction and repulsion, 
 which maintain them at determinate distances ; but they have no 
 fixed axial direction, so that a liquid molecule will rotate around 
 any imaginary axis on the action of the slightest force. Thus 
 liquids have fluidity, while at the same time they have a small 
 range of compressibility. 
 
 In solids the molecules have, besides their mutual attractions and 
 repulsions, polarity or fixity of position of their axes inter se, so that 
 when a molecule in any solid is turned around any axis, it will return 
 to its primitive position after a series of decreasing oscillations. 
 
 Those of the above postulates which refer to the mutual action 
 of atoms, can be geometrically expressed by means of an exponen- 
 tial curve. See Plate IV., Fig. 2. 
 
 Let an atom be at A, while another is anywhere on the line AX. 
 Suppose that when placed at i for example, that an attraction exists 
 between the atoms. The intensity of this attraction is represented 
 by the length of the line i 7, and we show that mutual attraction 
 exists by drawing the ordinate to the point i above the axis AX. If 
 the atom be supposed at m and repelled by A, then the repulsion 
 and its intensity are expressed by drawing the ordinate to the point 
 w, below the axis AX. This may be supposed to be done for every 
 
92 Lecture-Notes on Physics. 
 
 point of the length AG, which represents the distance between the 
 glass plates in Huyghens' experiment, or about the 3^00 inch, and 
 thus we will form an exponential curve. 
 
 As there are several alternations of attractions and repulsions, the 
 curve will consist of various inflections lying alternately above and 
 below AX. The, last inflection, most distant from A, viz : G' G" is 
 of such a form that the lengths of its ordinates being the recipro- 
 cals of the squares of their distances from A, it expresses the law 
 of gravitation ; the atom at G being at the point called the limit of 
 gravitation, or about 5^0 i nc h from A. AX will be an asymptote 
 to this curve, while the inflection c' Dn will have AY for asymptote; 
 for the ordinate expressing repulsion increases beyond all limit when 
 the distance from A is just vanishing. The intermediate branches 
 of the curve must be determined by means of the alternations of 
 attraction and repulsion, in the experiments already described and 
 by the aid of the various phenomena of capillarity and of molecular 
 physics. 
 
 If an atom, supposed at the point c', or c", or &c., have its dis- 
 tance increased from A, it will, being under the curve, be attracted 
 with an intensity represented by its ordinate. When set free it will 
 move with an increasing velocity towards its primitive position of 
 equilibrium, which it will surpass on account of its inertia, and, 
 coming into the sphere of repulsion, it will be repelled from A, and 
 thus oscillate about its point of equilibrium. The atom will there- 
 fore eventually return to c', or c", or &c. These positions are 
 called limits of cohesion, c' being designated as the last limit of 
 cohesion. 
 
 If an atom at D', or D", or &c., be moved ever so little from its 
 position, it will rush to an adjacent limit of cohesion, either to the 
 right or to the left, according as it was moved from or towards A. 
 These points, D', D" &c., are called limits of dissolution, and differ 
 from the limits of cohesion in being positions of unstable equilib- 
 rium, and therefore only a temporary molecule can be formed by an 
 atom placed at D', D", &c., with an atom at A ; while an atom at c', 
 c", &c., together with the atom A forms a permanent molecule, which 
 resists compression and dilatation^ and whose component atoms 
 return to their primitive positions when the extraneous force is 
 removed ; provided the compression or dilatation has not been too 
 great ; for, in that case, the atom G", for example, might be forced 
 beyond D' by a compression, or removed beyond D" by a dilatation, 
 
Lecture-Notes on Physics. 93 
 
 and would then rush to another position of permanent equilibrium, 
 either to c' or to G. The only molecule that cannot possibly be 
 changed by compression is AC'. "When, however, the amount of 
 compression or dilatation of a body formed of permanent molecules 
 is a very small fraction of its volume, the body regains the dimen- 
 sion it had before the compression or dilatation was applied, and it 
 is found that the compression or dilatation is proportional to the 
 force employed ; for, in this case, the small portion of the curve 
 which expresses the variation of repulsion or attraction may be 
 considered a straight line, and therefore its ordinates are as its 
 abscissas. 
 
 These logical consequences of the theory are confirmed by the 
 most extensive experience. " Mr. Coulomb was engaged (for a par- 
 ticular purpose), in a series of experiments on the oscillations of 
 springs, particularly of twisted wires. He suspended a nicely turned 
 ball or cylinder by a wire of a certain length, and fitted it with an 
 index, which pointed out the degrees of the torsion. He found that 
 when a wire of twenty inches long was twisted ten times, the index 
 returned to its primitive position, if repeated a thousand times, and 
 the oscillations were made in equal times, whether wide or narrow. 
 But if it was twisted eleven times, the index did not return to its 
 first place, but wanted nearly a whole turn of it. Here, then, the 
 parts of the wire had taken new relative positions, in which they 
 were again at rest But what was most remarkable in Coulomb's 
 experiments was this: He found that after the wire had taken this 
 set (as it is termed by the artizans), it exhibited the same elasticity 
 as before. It allowed a torsion of ten turns, and when let go, it 
 returned, and after its oscillations were finished, it rested in the 
 position from which it had been taken. I was much struck with 
 this experiment, and immediately repeated it on a great variety of 
 substances with the same result. The most inelastic substance that I 
 know is soft clay. I got a thread made of fine clay at a pottery, 
 by forcing it through a syringe. It was about T J 2 th of an inch in 
 diameter, and eleven feet long. While quite soft (and smeared with 
 olive oil, to prevent its stiffening by the evaporation of its moisture), 
 I fastened it to the ceiling, and fixed a small weight and an index 
 to its lower end. I found that it made 5| turns a hundred times 
 and more, without the smallest diminution of its elasticity, always 
 recovering its first position. But when I gave it 7 turns it re- 
 turned only 5J. Thus it took a set. In this new arrangement of 
 
94 Lecture- Notes on Physics. 
 
 its parts, I found that it again bore a twist of 5J turns without 
 taking any new set. And I repeated this several times. I then 
 gave it 10 turns, in the same direction with the first seven. It 
 returned 5J as before, and was again perfectly elastic within this 
 
 limit." 
 #&## * * ** 
 
 " Another appearance of tangible matter shows a most encour- 
 aging conformity to the theory. Where bodies are very moderately 
 compressed or dilated, the forces employed are proportional to the 
 change of distance between the particles. This appears most ex- 
 actly true in the experiments of Dr. Hooke, on which he founded 
 his theory of springs, expressed in the phrase ut tensio sic vis, and 
 his noble improvement of pocket watches by applying a spiral 
 spring to the axis of the balance, which, by its bending and unbend- 
 ing, produced a force proportional to the angle of the oscillations, 
 and therefore made them isochronous, whether wide or narrow. It 
 is also confirmed by the experiments of Coulomb on twisted wires, 
 and by the form of the elastic curve, as determined by Bemouilli, 
 on the supposition that the forces with which the particles attracted 
 and repelled each other, are proportional to their removal from their 
 natural quiescent positions. But it is found that when the com- 
 pression or dilatation is too much increased, the resistance does not 
 increase so fast ; that it comes to a maximum by still increasing 
 the strain, then decreases, and the body takes a great set or breaks. 
 All this is perfectly analogous to the forces expressed by the ordi- 
 nates of our exponential curve. In the immediate vicinity of the 
 limits of cohesion, the ordinates increase nearly in the ratio of the 
 abscissae, then they increase more slowly, come to a maximum, 
 decrease again, till we come to a limit of dissolution." 
 
 VI. Capillary Attraction. " 
 
 The phenomena of capillary attraction consist in the elevation 
 or depression of the surfaces of liquids along the line of contact with 
 the walls of the vessels which contain them ; in the ascent or depres- 
 sion of liquids between slightly separated plates, or in tubes of such 
 small internal diameters as to approach to the dimensions of a hair ; 
 whence the name of capillarity, from capillus, a hair. 
 
 These effects are due to the attractions of the molecules of the 
 
Lecture-Notes on Physics. 
 
 95 
 
 liquid for each other combined with the attractions existing between 
 them and the molecules of the solid. 
 
 Generalization of the Phenomena of Capillary Attraction. 
 
 1. The ascent or depression of the liquid is inversely as the 
 diameter of the tube ; provided that this diameter does not exceed 
 two millimetres. In tubes over twenty millimetres in diameter, 
 there is neither elevation or depression of liquids. 
 
 2. The phenomena are independent of the pressure to which the 
 apparatus is subjected ; being the same in vacuo as in compressed 
 air. 
 
 3. They do not depend on the thickness of the tube ; hence the 
 action of the tube is limited in its effects to insensible distances. 
 
 4. The phenomena vary with the material of the tube, and with 
 the nature of the liquid ; thus, in a tube of glass, water rises above 
 and mercury is depressed below the level of the outside liquid. The 
 following table of the experiments of M. Frankenheirn, gives the 
 heights in millimetres to which different liquids rise, at a tempera- 
 ture of C., in a glass tube of 1 millimetre in diameter. 
 
 LIQUIDS. 
 
 DENSITY. 
 
 ELEVATION. 
 
 Water 
 
 1-000 
 
 30-73 
 
 Formic Acid 
 
 1-105 
 
 20-40 
 
 Acetic Acid .. . 
 
 1-290 
 
 17-02 
 
 Sulphuric Acid 
 
 1-840 
 
 16 80 
 
 Solution of Potassa. . . 
 Petroleum 
 
 1-274 
 
 0-847 
 
 15-40 
 13-90 
 
 Spirits of Turpentine. . 
 
 0890 
 0-905 
 
 13-52 
 12-20 
 
 Alcohol . 
 
 0-821 
 
 12-10 
 
 Alcohol ... 
 
 0-967 
 
 14-54 
 
 Ether 
 
 737 
 
 10-80 
 
 Bisulphide of Carbon.. 
 
 1-290 
 
 10-20 
 
 5. When the liquid wets the tube, it rises above the level of the 
 
96 Lecture- Notes on Physics. 
 
 liquid outside the tube ; and in this case the surface of the elevated 
 liquid is concave. 
 
 Example. "Water in glass tube. 
 
 6. When the liquid does not wet the tube, it is depressed below the 
 surface of the exterior liquid ; and in this case, the surface of the 
 liquid in the tube is convex. 
 
 Example. Mercury in tube of glass. 
 
 7. When the liquid in the tube has a plane surface, there is neither 
 elevation or depression. 
 
 Example, Water in a tube of steel. 
 
 These facts are readily explained by the atomic theory,, of which 
 they are a beautiful illustration and a natural deduction. 
 
 (a.) An attraction exists between the neighboring molecules of a 
 liquid, and between the molecules of a liquid and of the contiguous 
 solid. 
 
 (b.) This force decreases very rapidly as the distance between the 
 molecules increases, and becomes null when that distance exceeds 
 the radius of sensible attraction. 
 
 (c.) The attraction existing between the molecules forming the 
 surface of a liquid, and those extending below the surface as far as 
 the radius of sensible attraction, produces a molecular pressure, or 
 tension, on this surface, whose effect has to be added to the pressures 
 produced by gravity and the atmosphere. 
 
 (d.) The molecular pressure is greater with a convex and less 
 with a concave than with a plane liquid surface. 
 
 The truth of the four preceding postulates, is made clear by what 
 follows : 
 
 Let s s', Fig, 3, be a liquid surface of any form. M is a molecule 
 on the surface ; M r is a molecule distant from the surface less than 
 the radius of sensible attraction ; and M" a molecule whose dis- 
 tance from the surface equals the radius of sensible attraction; 
 while all molecules between s s' and R R' are distant from the sur- 
 face less than the radius of sensible attraction. 
 
 The molecule, M, on the surface, is attracted downward by all 
 the molecules contained in the portion of sphere which has for its 
 radius M P, the radius of sensible attraction. The effect of all these 
 attractions on M will be a resultant in the direction M P, perpen- 
 dicular to the surface. 
 
 j The molecule, M', is attracted by all the molecules contained in 
 he spherical portion ABC, which we can divide into three parts by 
 
Lecture- Notes on Physics. 97 
 
 three equidistant planes, A B, p Q, A' B', paralleFto the surface, s s'. 
 The attraction produced by A B p Q, is destroyed by the attraction of 
 p Q A' B', and therefore the molecule M 7 is drawn downward as 
 though it were attracted only by the liquid contained in A' B' c, 
 which gives a resultant, p', also perpendicular to the surface, but 
 less than P. 
 
 The molecule, M", whose distance from the surface equals the 
 radius of sensible attraction, and all other molecules placed at 
 greater distances, are equally attracted on all sides, and therefore 
 they produce no tension in the surface-film of the liquid, which has 
 for its thickness the radius of sensible attraction. 
 
 The influence of the curvature of the liquid surface on the molecular 
 pressure. 
 
 Let M', Fig. 4, be a molecule at a distance M H from the surface 
 s s' of the liquid. With M as a centre, draw a sphere whose radius 
 M' P equals the radius of sensible attraction. 
 
 If the surface is a plane, AB, the attractions of the liquid in 
 A B P Q are destroyed by those produced by the symmetrical portion 
 below, A' B r p Q, and there remains for resultant only the action of 
 A' B' c. 
 
 Suppose the surface concave and D H E ; if we draw through H' 
 the symmetrical surface, D' H' E', it is evident that the attractions 
 of the molecules comprised between D H E P Q and of those contained 
 within D' H' E' P Q equal and oppose each other, and there remains 
 only the attraction of D' H' E' c on M', which is less than when the 
 surface was a plane. 
 
 If the surface is convex, and is represented by K H L, draw the 
 symmetrical surface, K' H' I/ ; then the efficient attracting portion 
 of the liquid will be increased and represented by K r G' I/, and con- 
 sequently the molecular pressure is greater with a convex than with 
 a plane surface. 
 
 We can now explain the rise and depression of liquids in capillary 
 tubes. 
 
 When the surface of the liquid in the tube is concave, the mole- 
 cular pressure on the liquid in the tube is less than the pressure on 
 the liquid outside the tube, and therefore the liquid rises in the tube 
 to a height which measures the diminution of pressure produced 
 by the concave surface. 
 
 When the surface of the liquid in the tube is plane, there is 
 13 
 
98 Lecture- Notes on Physics. 
 
 neither elevation or depression, for the pressures are the same on 
 the surfaces of the liquid inside and outside the tube. 
 
 When the surface of the liquid in the tube is convex, the mole- 
 cular pressure on the liquid in the tube is more than the pressure 
 on the liquid outside the tube, and therefore the liquid column is 
 depressed in the tube below the level of the outside liquid, and the 
 depth to which the column is forced below this level, is the measure 
 of the pressure produced by the convex surface. 
 
 As we have seen that the elevation or depression of liquids in 
 capillary tubes, is due to a diminution or increase of molecular pres- 
 sure, produced by a concave or convex surface, it remains, to ren- 
 der the explanation complete, to show the cause of the special figure 
 of each surface. 
 
 Cause of the (1) plane, (2) concave, and (3) convex surfaces of 
 liquids in capillary tubes. 
 
 Let D A in Figs. 5, 6 and 7, be the vertical surface of a solid 
 plunged in liquids, whose surfaces are M L. Let M be a molecule of 
 the surface of the liquid contiguous to the plate. This molecule is 
 attracted by all the molecules contained in the quarter-spheres 
 D M c and A M c, whose radii are equal to the distance of sensible 
 attraction ; giving as resultants M s and M s', while the resultant of 
 the attractions of the liquid on the molecule, M, will be M. p. 
 
 Three cases can present themselves. 
 
 1. If the resultant, M P, Fig. 5, is twice M s, or its equal, M s', the 
 effect of these three attractions on M will be the resultant, M K ; 
 which being perpendicular to the liquid surface, the fluid will 
 remain horizontal, for the surface of a liquid is always perpendicular 
 to the forces acting on it. 
 
 2. If the resultant, M. P, Fig. 6, is less than twice M s or M s', the 
 three attractions will result in M K, which will produce a concave 
 surface M I/, inclined against the plate. 
 
 3. If the resultant, M P, Fig. 7, is more than twice M s or M s', the 
 resultant of the three attractions on liquid contiguous to solid will 
 be M R, which will, for the reason given above, produce the surface 
 M r L', which will be convex. 
 
 The above results may be expressed concisely as follows : 
 
 I. On the free surface of every liquid there exists a molecular 
 pressure from without inward, which always adds its effect to that 
 produced by gravity and the pressure of the air. 
 
 II. The intensity of this molecular pressure varies with the form 
 
Lecture-Notes on Physics. 99 
 
 of the surface, being greater when the surface is convex and less 
 when concave, than when it is plane. 
 
 III. The form of a liquid surface in a tube, depends on the rela- 
 tive amounts of attraction existing between the molecules of the 
 liquid and the molecules of the solid and of the liquid. 
 
 1. When the attraction between the molecules of the liquid is 
 twice as great as the attraction between the molecules of the liquid 
 and those contained in an equal portion of the solid, the surface in 
 the capillary tube is horizontal. 
 
 2. When the attraction between the molecules of the liquid is 
 less than twice that existing between the molecules of the liquid and 
 solid, the surface in the tube is concave. 
 
 3. When the attraction between the molecules of the liquid is 
 more than twice that between the molecules of the liquid and solid 
 the surface in the tube is convex. 
 
 IV. When the surface of the liquid in the capillary tube is 
 (a) horizontal, it is in the same plane with the exterior liquid. 
 (/;) concave, it is above the plane of the exterior liquid, (c) convex, 
 it is below the plane of the exterior liquid. 
 
 V. The amount of elevation or of depression of the same liquid 
 in tubes of the same material, is inversely as the diameter of these 
 tubes. This is known as the law of Jurin, after the philosopher 
 who established it ; and with the aid of the table already given 
 we can by means of it readily calculate the heights to which different 
 liquids will rise in glass tubes of various dimensions, contained 
 within diameters of two millimetres to a few hundredths of a 
 millimetre. 
 
 The reason of this law is as follows. The force which elevates or 
 depresses the liquid columns in the tubes depends evidently, from 
 what has preceded, upon the number of the molecules on the sur- 
 face of the liquid contiguous to the sides of the tubes. Therefore, 
 the forces of elevation or of depression are as the interior circum- 
 ferences of the tubes, arid the forces are measured by the quantity 
 (or weight) of liquid elevated above or depressed below the level of 
 the liquid exterior to the capillary tube. Therefore, let h and h f 
 be the lengths of liquid columns elevated or depressed in tubes 
 whose interior diameters are respectively d and a" '. Their interior 
 circumferences are ft d and it d f . 6 being the specific gravity of the 
 liquid, the weights of the columns elevated or depressed will be 
 Jrtc? 2 /i5, and JrfcTVi'S. These weights are equal to the force* 
 
100 Lecture-Notes on Physics. 
 
 which produce the elevations or depressions of the liquid columns, 
 and these forces being to each other as the interior circumferences 
 of the tubes, we have 
 
 or 
 
 d:d'::h':h 
 which is the expression of the law given above. 
 
 Experiments. The apparatus with which Gay Lussac verified 
 the above law, explained and used. 
 
 If two squares of plane glass, touching along two vertical edges, 
 are opened to an acute angle and placed in colored water, the liquid 
 will rise between the plates, forming an equilateral hyperbola, and 
 therefore the liquid at various points stands at heights inversely as 
 the distance of the plates at these points. 
 
 The relation which exists between the form of the surface which 
 terminates the capillary column and its vertical distance above the 
 plane of the exterior liquid, is beautifully shown by the following 
 experiment, which, with those above cited, can be readily thrown 
 on a screen by means of the lantern and erecting prism of Prof. 
 Morton (see Journal of Franklin Institute, Yol. LIIL, p. 406). A 
 large glass tube has connected with it a capillary tube, as shown in 
 Fig. 8. Water, colored with carmine, is poured into the larger 
 tube until its level reaches, say, A, and the liquid in the capillary 
 tube just attains the top, s, and in these circumstances, will there 
 form a concave surface. Now, on pouring into the large tube more 
 liquid, the concave surface becomes flatter and flatter as the liquid 
 rises in the tube A, until, when the surface rises to B on the same 
 level as s, the terminal surface at s is a plane. When liquid is 
 further added until the surface reaches c, at a higher level than s, 
 the capillary surface at s is convex. 
 
 When s is concave, the molecular pressure on this surface is less 
 than on A by the pressure of the column from the level A to s. 
 When s is plane, equality of pressure exists in both tubes, and 
 therefore the liquid surfaces are in the same plane. When s is 
 convex, more molecular pressure is on s than on c, by the column 
 from s to the level c, 
 
 Professor Plateau, of the University of Ghent, has made a series 
 of very important investigations in molecular physics, which are 
 contained in a series of papers entitled, " Experimental and Theo- 
 retical Researches on the Figures of Equilibrium, of a Liquid Mass 
 
Lecture-Notes on Physics. 101 
 
 withdrawn from the action of Gravity" translated and published by 
 the Smithsonian Institution, in the Reports of 1863, et seq. The 
 fifth series of these investigations (Smith. Rep. 1865), contains a 
 research on the molecular pressure exerted by liquid films, with 
 applications to capillary action ; and so interesting has this investi- 
 gation appeared to us, that we thought it proper to present a rather 
 full abstract from Prof. Plateau's paper. 
 
 Pressure exerted by a spherical film, on the air which it contains. 
 App lication. 
 
 11 The exterior surface of a laminar sphere being convex in every 
 direction, the pressure which corresponds to it is greater than that 
 of a plane surface, and consequently the resultant of the pressures 
 exerted in any point of the bubble by the two surfaces of the latter, 
 is directed towards the interior ; whence it results that the bubble 
 presses on the air which it encloses. It is, indeed, well known 
 that when a soap bubble has been inflated, and while it is still 
 attached to the tube, if the other extremity of this last be left open, 
 the bubble gradually collapses, expelling the air which it contained 
 through the tube. We see now what is the precise cause of this 
 expulsion. 
 
 " But we may go further, and determine according to what law it 
 is, that the pressure, exerted by such a bubble on the confined air, 
 depends on the diameter of that bubble. We can compute, more- 
 over, the exact value of the pressure in question for a bubble hav- 
 ing a given diameter, and formed of a given liquid. The pressure 
 corresponding to a point of a laminar figure, has for its expression 
 
 A( h , -). R and R', standing for the radii of curvature, P 
 
 \ R R / 
 
 being the pressure which a plane surface would occasion, and A a 
 constant which depends on the nature of the liquid. Now, in the 
 case of the spherical figure, we have R = R / = the radius of the 
 sphere. If, therefore, we designate by d the diameter of the bubble, 
 
 the value of the pressure will simply become- , always, be it un- 
 
 Cu 
 
 derstood, neglecting the slight thickness of the film ; whence it fol- 
 lows that the intensity of the pressure exerted by a laminar spherical 
 bubble on the air which.it confines, is in inverse ratio to the diameter 
 of that bubble. 
 
 "This first result established, let us recur to the general expres- 
 sion of the pressure corresponding to any point of a liquid surface, 
 
102 Lecture-Notes on Physics. 
 
 an expression which is p + -= I }--, ). For a surface of convex 
 
 A \ R R / 
 
 spherical curvature, if we designate by d the diameter of the sphere 
 to which this surface pertains, the above expression becomes 
 
 2 A 
 P H- -7- , and for a spherical surface of concave curvature pertaining 
 
 2 A 
 
 to a sphere of the same diameter, we shall have P -,-. Thus, in 
 
 ct 
 
 the case of the convex surface the total pressure is the sum of two 
 forces acting in the same direction force, of which one designated 
 by P is the pressure which a plane surface would exert, and the 
 
 2 A 
 
 other represented by -=- is the action which depends on the curva- 
 ture. On the contrary, in the case of the concave surface the total 
 pressure is the difference between two forces acting in opposite 
 directions, and which are again, one the action P of a plane surface, 
 
 2 A 
 and the other ^ which depends on the curvature. Whence it is 
 
 4 A 
 
 seen that the quantity r- , which represents the pressure exerted 
 
 by a spherical film on the air it encloses, is equal to double the 
 action which proceeds from the curvature of one or the other sur- 
 face of the film. 
 
 "Now, when a liquid rises in a capillary tube, and the diameter of 
 this is sufficiently small, we know that the surface which terminates 
 the column raised 'does not differ sensibly from a concave hemi- 
 sphere, whose diameter is consequently equal to that of the tube. 
 Let us recall, moreover, a part of the reasoning by which we arrive, 
 in the theory of capillary action, at the law which connects the 
 height of the column raised with the diameter of the tube. Let us 
 suppose a pipe, excessively slender, proceeding from the lowest point 
 of the hemispheric surface in question, descending vertically to the 
 lower orifice of the tube, then bending horizontally, and finally 
 rising again so as to terminate vertically at a point of the plane 
 surface of the liquid exterior to the tube. The pressures corres- 
 ponding to the two orifices of this little pipe will be, on the one 
 
 2 A 
 
 part, P, and on the other, p , if by 5 be designated the diame- 
 ter of the concave hemisphere, or, what amounts to the same thing, 
 v that of the tube. Now, the two forces P, mutually destroying one 
 
Lecture- Notes on Physics. 103 
 
 2 A 
 another, there remains only the force , which, having a sign 
 
 contrary to that of P, acts consequently from below upwards at the 
 lower point of the concave hemisphere, and it is this which sustains 
 the weight of the molecular thread contained in the first branch of 
 the little pipe between the point just mentioned and a point situ- 
 ated at the height of the exterior level. This premised, let us re- 
 
 2 A 
 mark that the quantity is the action which results from the 
 
 curvature of the concave surface. The double of this quantity or 
 
 4 A 
 
 -, will therefore express the pressure exerted on the enclosed 
 
 air by a laminar sphere or hollow bubble of the diameter 5, and 
 formed of the same liquid. It thence results that this pressure con- 
 stitutes a force capable of sustaining the liquid at a height double 
 that to which it rises in the capillary tube, and that, consequently, 
 it would form an equilibrium to the pressure of a column of the 
 same liquid having that double height. Let us suppose, for the 
 sake of precision, J equal to a millimetre, and designate by h the 
 height at which the liquid stops in a tube of that diameter. We 
 shall have this new result, that the pressure exerted on the enclosed 
 air by a hollow bubble formed of a given liquid and having a 
 diameter of 1 millimetre, would form an equilibrium to that exerted 
 by a column of this liquid of a height equal to 2 h. Now, the pres- 
 sure exerted by a bubble being in inverse ratio to the diameter 
 thereof, it follows that the liquid column which would form an 
 equilibrium to the pressure exerted by a bubble of any diameter 
 
 whatever, c?, will have a height equal to ^-. 
 
 "It would seem, at first, that this last expression ought to apply 
 equally well to liquids which sink in capillary txibes, h then desig- 
 nating this subsidence, the tube still being supposed 1 millimetre 
 in diameter ; but is not altogether so, for that would require, as is 
 readily seen by the reasonings which precede, that the surface which 
 terminates the depressed column in the capillary tube should be 
 sensibly a convex hemisphere ; now we know that in the case of 
 mercury this surface is less curved ; according to the observations 
 of M. Bede, its height is but about half of the radius of the tube ; 
 whence it follows that the valuation of the pressure yielded by our 
 formula would be too small in regard to such liquids. It may be 
 considered, however, as a first approximation. 
 
104 Lecture-Notes on Physics. 
 
 Let us take, as a measure of the pressure exerted by a bubble, 
 the height of the column of water to which it would form an equili- 
 brium. Then, if designates the density of the liquid of which the 
 bubble is formed, that of water being 1, the heights of the columns 
 of water and of the liquid in question which would form an 
 equilibrium to the same pressure will be to one another, in the in- 
 verse ratio of the densities, and, therefore, if the height of the 
 
 second is -^, that of the first will be --* . Hence, designating 
 by p the pressure exerted by a laminar sphere on the air which it 
 encloses, we obtain definitely p J ^-- , ^ being, as we have seen, 
 
 the density of the liquid which constitutes the film, li the height 
 to which this liquid rises in a capillary tube 1 millimetre in dia- 
 meter, and d the diameter of the bubble. If, for example, the bub- 
 ble be formed of pure water, we. have = 1, and, according to 
 the measurements taken by physicists, we have, very exactly, 
 ^ = 30m m ; the above formula, therefore, will give, in this case, 
 
 60 
 p = -^-. If we could form a bubble of pure water of one decimetre 
 
 or 100 mm , in diameter, the pressure which it would exert would 
 consequently be equal to O mm *6, or, in other terms, would form an 
 equilibrium to the pressure of a column of water O mm '6 in height; 
 the pressure exerted by a bubble of the same liquid one centimetre, 
 or 10 mm in diameter, would form an equilibrium to that of a column 
 of water 6 mm . As regards soap-bubbles, their pressures, if the 
 solution were as weak as possible, would differ very little from 
 those exerted by bubbles of the same diameters formed of pure 
 water. 
 
 "For mercury we have 13*59, and, according to M. Bede, h 
 about equal to 10 mm ; the formula would therefore give, for a bub- 
 
 971 '8 
 ble of mercury p = -^ , but, from the remark which closes the 
 
 last paragraph, this value is too weak, and can only be regarded as 
 a first approximation. It only instructs us that, with an equality 
 of diameter, the pressure of a bubble of mercury would exceed four 
 and a half times that of a bubble of pure water. For sulphuric 
 ether, we have 0'715, and conclude from measurements taken 
 from M. Frankenheim, h to be very closely to 10 Rim> 2 ; whence re- 
 
Lecture- Notes on Physics. 105 
 
 suits j9 -T-, and thus, with an equal diameter, the pressure of a 
 
 bubble of sulphuric ether would be but the fourth of that of a bub- . 
 ble of pure water. 
 
 "We know that the product h being the product of the capillary 
 height by the density, is proportional to the molecular attraction 
 of the liquid for itself, or, in other terms, to the cohesion of the 
 liquid ; (see research of Prof. Henry on Cohesion of Liquids, quoted 
 in Y.) it is, moreover, the result from a comparison of the values 
 
 j- and -j , which have been found to represent the pressure 
 
 exerted by a laminar sphere on the air which it contains ; hence 
 we deduce h $ = 2 A, and it will be remembered that A is the capil- 
 lary constant ; that is to say, a quantity proportional to the cohe- 
 
 2 hs 
 sion of the liquid. The formula p == -=- indicates, therefore, as 
 
 must be evident, that the pressure exerted by a laminary bubble 
 on the included air is in the direct ratio of the cohesion of the liquid 
 which constitutes the film and the inverse ratio of the diameter of 
 the bubble. 
 
 " As early as 1830, a learned American, Dr. Hough, had sought 
 to arrive at the measure of pressure exerted, whether on a bubble 
 of air contained in an indefinite liquid or on the air enclosed in a 
 bubble of soap. 
 
 (Inquiries into the principles of liquid attraction. Silliman's 
 Journal, 1st series, vol. xvii., page 86.) 
 
 " He conceives quite a just idea of the cause of these pressures 
 which he does not, however, distinguish from one another, and, in 
 order to appreciate them, sets out, as I have done, with a considera- 
 tion of the concave surface which terminates a column of the same 
 liquid raised in a capillary tube ; but, although an ingenious ob- 
 server, he was deficient in a knowledge of the theory of capillary 
 action, and hence arrives, by reasoning, of which the error is palpa- 
 ble, at values and a law which are necessarily false. 
 
 "Prof. Henry, in a very remarkable verbal communication on the 
 cohesion of liquids, made in 1844, to the American Philosophical 
 Society, (Philosophical Magazine, 1845, vol. xxvi., page 541), de- 
 scribed experiments by means of which he had sought to measure 
 the pressure exerted on the internal air by a bubble of soap of a 
 given diameter. According to the account rendered of this com- 
 14 
 
106 Lecture-Notes on Physics. 
 
 munication, the mode of operation adopted by Mr. Henry was essen- 
 tially as follows : lie availed himself of a glass tube of U form, of 
 . small interior diameter, one of whose branches was bell-shaped at 
 its extremity, and inflated a soap-bubble extending to the edge of 
 this widened portion ; he then introduced into the tube a certain 
 quantity of water, and the difference of level in the two branches 
 gave him the measure of the pressure. Unfortunately, the state- 
 ment given does not make known the numbers obtained, nor does 
 it appear that Mr. Henry has subsequently published them. This 
 physicist refers the phenomenon to its real cause, and states the law 
 which connects the pressure with the diameter of the bubble; the 
 account does not say whether the experiments verified it. But Mr. 
 Henry considers that a hollow bubble may be assimilated to a full 
 sphere reduced to its compressing surface; that is to say, he attri- 
 butes the phenomenon to the action of the exterior surface of the 
 bubble, without taking into account that of the interior surface. 
 Let us add that, in the same communication, Mr. Henry has men- 
 tioned several experiments which he had made on the films of soap 
 and water, and which, from the statement given would elucidate, 
 in a remarkable manner, the principles of the theory of capillary 
 action. It is much to be regretted that these experiments are 
 not described 
 
 " In a memoir presented to the Philomatic Society in 1856, and 
 printed in 1859 in the Comptes Rendus, (tome xlviii., pap^ "* 405.) 
 M. de Tessan maintains that if the vapor which forms olouc ,*.'. / 
 fogs were composed of vesicles, the air enclosed in a vesicle of 0*0- 
 millimetre diameter would be subjected, on the part of this vesicle, 
 to a pressure equivalent to \ of an atmosphere. M. de Tessan does 
 not say in what manner he obtained this valuation ; but it is easily 
 seen that he has fallen into an error analogous to that of Professor 
 Henry, in the sense that he pays no attention except to the exterior 
 surface of the liquid pellicle. According to the formula of the pre- 
 ceding paragraph, the pressure exerted on the interior air by a 
 bubble of water of 0*02 millimetre diameter would, in fact, be equi- 
 valent to that of a column of water 3 metres in height, which equals 
 nearly f of the atmospheric pressure ; M. de Tessan has found then 
 but half the real value, and we know that this half is the action due 
 to the curvature of one only of the surfaces of the film. 
 
 " After having obtained the general expression of the pressure 
 exerted by a laminar sphere on the air which it encloses, it r em aim d 
 
Lecture- Notes on Physics. 
 
 107 
 
 for me to submit my formula to the control of experiment. I have 
 employed, with that view, the process of Mr. Henry, which means 
 that the pressure was directly measured by the height of the column 
 of water to which it formed an equilibrium. 
 
 From our formula we deduce p d = 2 h $ ; for the same liquid, 
 and at the same temperature, the product of the pressure by the 
 diameter of the bubble must, therefore, be constant, since h and 
 are so. It is this constancy which I have first sought to verify for 
 bubbles of giyceric liquid (a solution of Marseilles soap and gly- 
 cerine in distilled water, with which Plateau made his bubbles) 
 of different diameters." 
 
 Plateau here describes his apparatus and the precautions to be 
 used in making the measures, which the reader will find detailed 
 in the Smithsonian Report for 1865. 
 
 " The following table contains the results of these experiments; I 
 have arranged them, not in the order in which they were obtained, 
 but in the ascending order of the diameters, and I have distributed 
 them into groups of analogous diameters. During the continuance 
 of the operations the temperature varied from 18*5 to 20 C. 
 
 * * * * As the first diameter is to those of the last group 
 very nearly as 1 to 6, these results suffice, I think, to establish dis- 
 tinctly the constancy of the product p d, and consequently the law 
 according to which the pressure is in the inverse ratio of the dia- 
 meter. 
 
 ********* 
 
 "As to the general mean 22'75 of the results of the table, its deci- 
 mal part is necessarily a little too high, on account of the excessive 
 
 Diameters, or 
 values of d. 
 
 Pressures, or 
 values of p. 
 
 Products, or 
 values of p d. 
 
 m m 
 
 mm 
 
 
 7-55 
 
 3-00 
 
 22-65 
 
 10-37 
 
 2-17 
 
 22-50 
 
 10-55 
 
 2-13 
 
 22-47 
 
 23-35 
 
 0-98 
 
 22-88 
 
 26-44 
 
 0-83 
 
 21-94 
 
 27-58 
 
 0-83 
 
 22-89 
 
 46-60 
 
 0-48 
 
 22-37 
 
 47-47 
 
 0-48 
 
 22-78 
 
 47-85 
 
 0-43 
 
 20-57 
 
 48-10 
 
 0-55 
 
 26-45 
 
108 Lecture- Notes on Physics. 
 
 value 26-45 of the last product. As this product, and that which 
 precedes it, are those which alone deviate materially from 22 in 
 this integral part, it will be admitted, I think, that a nearer approach 
 to the true value will be made by neglecting these two products and 
 taking the mean of the others, a mean which is 22*56, or more sim- 
 ply 22-6 ; we shall adopt, then, this last number for the value of 
 the product p d in regard to the gly eerie liquid. 
 
 It remained to be verified whether this value satisfied our for- 
 mula, according to which we have p d = 2 h S-, the quantities and 
 h being respectively, as we have seen, the density of the liquid and 
 the height which this liquid would attain in a capillary tube 1 
 millimetre in diameter. With this view, therefore, it was necessary 
 to seek the values of these two quantities in reference to the glyce- 
 ric liquid. The density was determined by means of the aerometer 
 of Fahrenheit, at the temperature of 17 C., a temperature little 
 inferior to that of the preceding experiments, and the result was 
 > = 1-1065. To determine the capillary height the process of Gay 
 Lussac was employed, that is, the measurement by the cathetometer, 
 all known precautions being taken to secure an exact result. The 
 experiment was made at the temperature of 19 C. **** 
 * * * rp]^ rea di n g o f the cathetometer gave, for the distance 
 from the lowest point of the concave meniscus to the exterior level, 
 27 mm -35. 
 
 This measurement having been taken, the tube was removed, 
 cut at the point reached by the capillary column, and its interior 
 diameter at that point measured by means of a microscope, furnished 
 with a micrometer, giving directly hundredths of a millimetre. It 
 was found that the interior section of the tube was slightly elliptical, 
 the greater diameter being O mm -374 and the smaller O mm '357; the 
 mean was adopted, namely, O mm '3655, to represent the interior dia- 
 meter of the tube assumed to be cylindrical. To have the true 
 height of the capillary column, it is necessary, we know, to add to 
 the height of the lowest point of the meniscus the sixth part of the 
 diameter of the tube, or, in the present case, O mm> 06 ; the true height 
 of our column is consequently 27 miu> 41. Now, to obtain the height 
 h to which the same liquid would rise in a tube having an interior 
 diameter of exactly a millimetre, it is sufficient, in virtue of the 
 known law, to multiply the above height by the diameter of the 
 tube, and thus we find definitively & = 10 min -018. 
 
 "I should here say for what reason I have chosen for the experi- 
 
Lecture-Notes on Physics. 109 
 
 ment a tube whose interior diameter is considerably less than a 
 millimetre. The reasoning by which I arrived at the formula sup- 
 poses that the surface which terminates the capillary column is 
 hemispherical ; now that is not strictly true, but in a tube so nar- 
 row as that which I have employed, the difference is wholly inap- 
 plicable, so that in afterwards calculating, by the law of the inverse 
 ratio of the elevation to the diameter, the height for a tube one mil- 
 limetre in diameter, we would have this height such as it would be 
 if the upper surface were exactly hemispherical. 
 
 u The values of and h being thus determined, we deduce there- 
 from 2/0 = 22-17, a number which differs but little from 22'56 
 obtained above as the value of the product p d. The formula 
 pd=2h may therefore be regarded as verified by experiment, 
 and the verification will appear still more complete if we consider 
 that the two results are respectively deduced from elements altogether 
 different. I hope hereafter to obtain new verifications with other 
 liquids. 
 
 Investigation of a very small limit below which is found, in the 
 gly eerie liquid, the value of the radius of sensible activity of the mole 
 cular attraction. 
 
 2 h& 
 " The exactness of the formula p = -v- supposes, as we are about 
 
 to show, that the film which constitutes the bubble has, at all points, 
 no thickness less than double the radius of sensible activity of the 
 molecular attraction. 
 
 "We have seen that the pressure exerted by a bubble on the air 
 which it encloses is the sum of the actions separately due to the 
 curvatures of its two faces. On the other hand, we know that, 
 in the case of a full liquid mass, the capillary pressure exerted by 
 the liquid on itself emanates from all points of a superficial stratum 
 having as its thickness the radius of activity in question. Now, if 
 the thickness of the film which constitutes a bubble is everywhere 
 superior or equal to double that radius, each of the two faces of the 
 film will have its superficial stratum unimpaired, and the pressure 
 exerted on the enclosed air will have the value indicated by our 
 formula. But if, at all its points, the film has a thickness inferior 
 to or double this radius, the two superficial strata have not their 
 complete thickness, and the number of molecules comprised in each 
 of them being thus lessened, these two strata must necessarily exert 
 
110 Lecture- Notes on Physics. 
 
 actions less strong, and consequently the sum of these, that is to 
 say, the pressure on the interior air, must be smaller than the for- 
 mula indicates it to be. Hence it follows that if, in the experiments 
 described above, the thickness of the films which formed the bub- 
 bles had, through the whole extent of these last, descended below 
 the limit in question, the results would have been too small, but in 
 this case we should have remarked progressive and continued 
 diminutions in the pressures, which, however, never happened, 
 although the color of the bubbles evinced great tenuity. But all 
 physicists admit that the radius of sensible activity of the molecular 
 attraction is excessively minute. 
 
 "But what precedes permits of our going further, and deducing 
 from experiment a datum on the value of the radius of sensible 
 activity, at least in the glyceric liquid. 
 
 #-3f*##### 
 
 "After the film has acquired a uniform thinness, if the pressure 
 exerted on the air within the bubble underwent a diminution, this 
 would be evinced by the manometer, and it would be seen to progress 
 in a continuous manner in proportion to the ulterior attenuation of 
 the film. In this case the thickness which the film had when the di- 
 minution of pressure commenced would be determined by the tinge 
 which the central space presented at that moment, and the half of 
 that thickness would be the value of the radius^ of sensible activity 
 of the molecular attraction. If, on the contrary, the pressure re- 
 mains constant until the disappearance of the bubble, we may infer 
 from the tint of the central space the final thickness of the film, and 
 the half of this thickness will constitute at least a limit, very little 
 below which is to be found the radius in question. 
 
 fc#-fc#-5f-fc4f# 
 
 * * * I deposited in the bottom of the dry jar, morsels of 
 caustic potash, and contrived by the application of a little lard 
 around the orifice of the jar and of the aperture through which 
 passed the copper tube, that after the introduction of the bubble, 
 
 the pasteboard disk should close the opening hermetically. * * 
 ##-*####* 
 
 "Now, under these conditions, the diminution of thickness of the 
 film was continuous, the bubble lasted for nearly three days, and 
 when it burst, it had arrived at the transition from the yellow to 
 the white of the first order ; it then presented a central space of a 
 pale yellow tint, surrounded by a white ring. 
 
Lecture- Notes on Physics. Ill 
 
 ******<*# 
 
 * * * if the pressure varied, it was in an irregular manner, 
 in both directions, and terminating not in a diminution, but an 
 augmentation, at least relative; we may, therefore, admit, I think, 
 that the final thickness of the film was still superior to double the 
 radius of sensible activity of the molecular attraction. 
 
 "Let us now see what we may deduce from this last experiment. 
 According to the table given by Newton, the thickness of a film 
 of pure water which reflects the yellow of the first order is, in mil- 
 lionths of an English inch, 5J, or 5'333, and for the white of the 
 same order 3 {, or 3*875. We may therefore take the mean, namely 
 4-064, as the closely approximative value of the thickness corres- 
 ponding, at least in the case of pure water, to the transition between 
 those colors, and the English inch being equal to 25-4 millimetres, 
 this thickness is equivalent to 55^4 of a millimetre. Now we know 
 that, for two different substances, the thickness of the films which 
 reflect the same tint is in the inverse ratio of the indices of refrac- 
 tion of those substances. In order, therefore, to obtain the real 
 thickness of our film of glyceric liquid, it suffices to multiply the 
 denominator of the preceding fraction by the ratio of the index of 
 the glyceric liquid to that of water. I have measured the former 
 flpproximatively by means of a hollow prism, and have found it 
 equal to 1-377. That of water being 1-336, there results, for the 
 thickness of the glyceric film g^T of a millimetre. The half of this 
 quantity, or 77-552 of a millimetre, constitutes, therefore, the limit 
 furnished by the experiment in question. Hence we arrive at the 
 very probable conclusion, that in the glyceric liquid the radius of 
 sensible activity of the molecular attraction is less than I^^-Q-Q of a 
 n.illimetre. 
 
 "I had proposed to continue this investigation with a view to 
 reach, if possible, the black tint, and to elucidate the variations of 
 the manometer ; but the cold season has intervened, diminishing 
 the persistence of the bubbles, and I have been forced to postpone 
 attempts to a more favorable period." 
 
 In the report of the transactions of the Society of Physics and 
 Natural History, of Geneva, 1862, we find the following : " Prof. 
 Wartmann, Jr., repeated before the Society the recent experiments 
 of M. Plateau on bubbles of soap, of varied forms as well as much 
 persistency, obtained by mixing with soap-suds a small quantity of 
 glycerine, and causing the bubbles to attach themselves to iron 
 
112 Lecture-Notes on Physics. 
 
 wires arranged in different manners. At a subsequent session M. 
 Wartmann exhibited an apparatus of the same kind, still more 
 varied, so as to produce more perfectly than by former processes 
 the phenomena of coloration in extremely thin surfaces of the liquid. 
 The dark part presents not more than y^Vsu of a millimetre, whence 
 we may conclude, says M. Wartmann, that the radius of the sensi- 
 ble activity of molecular attraction is below w?\w f a milli- 
 metre." 
 
ADDITIONS AND CORRECTIONS. 
 
 Page 3, line 11 from top, for such are the planets, &c., read such 
 are the stars, the sun, the planets and asteroids of, &c. 
 
 Page 3, after line 21, read: Nebulas are faintly luminous aggre 
 gations of matter so far removed from our solar system 
 as to have no sensible parallax. They are formed 
 either of clusters of stars, and are then resolvable, or 
 qonsist of self-luminous gaseous matter, and are then 
 unresolvable into star points ; while some seem formed 
 of stars surrounded by atmospheres of intensely heated 
 vapors, as is the case with our sun. These facts are the 
 results of the spectroscopic analysis of these bodies. If 
 the spectrum of a celestial body is formed only of iso- 
 lated bright lines, then that body is composed only of 
 luminous gas, as is the case with the great nebula in 
 Orion. When the spectrum is continuous, the light 
 comes from an incandescent solid or liquid mass; while 
 if the spectrum is continuous and crossed by fine dark 
 lines, we know that the light emanates from an incan- 
 descent solid or liquid body surrounded by vapors 
 which have, by their absorption, produced these dark 
 lines. 
 
 Of 60 nebulas examined by Mr. Huggins, the spectroscope 
 showed that 20 were gaseous, while 40 consisted of ag- 
 gregations of stars. 
 (See Results of Spectrum Analysis Applied to the Heavenly 
 
 Bodies. By Wm. Huggins. London: 1868.) 
 The Zodiacal light is a faint nebulous light, resem- 
 bling that of the tail of a comet. It is visible in 
 the W. about 1st March, after sunset, and in the 
 
E. about the middle of October, before sunrise. It is 
 supposed to be caused by a nebulous envelope which 
 surrounds the sun and extends beyond the orbit of the 
 earth, so that, if the sun were seen from one of the stars 
 exterior to our system, it would appear as a bright 
 point enveloped in a haze, just as some of the fixed 
 stars appear to us. 
 
 Page 4, line 3 from top, for solids, liquids and gases, read solid, 
 liquid and gaseous. 
 
 Page 4, line 8 from bottom, for stone, associated facts. (1) Spaces 
 read stone. Associated facts, (1) spaces. 
 
 Page 17, line 8 from top, for T J^ read O 1 IJU . 
 
 Page 18, line 7 from top, after audience, read, by the aid of a 
 lantern provided with a small slit and a lens. 
 
 Page 19, after line 3 from top, read: See papers on Robert's 
 Test-plates, by Mr. Stodder, in the American Naturalist 
 for April, 1868, and by Messrs. Sullivant and Wood- 
 ward in Amer. Jour. Science, Nov., 1868. 
 
 Page 20, after line 17 from top, read: An arc of 17 10', of which 
 the cord is J of radius, may be employed as a test of 
 the accuracy of the work. 
 
 Page 24, line 2 from top, for millimetre read milligramme. 
 
 Page 24, line 16 from bottom, for mean revolution, read mean 
 apparent revolution. 
 
 Page 25, line 1 from top, for dead-beat, read dead beat and gravity. 
 
 Page 25, after line 4 from bottom, read Dr. Locke, of Cincinnati, 
 in 1848, first recorded transits by means of an electric 
 clock and the ordinary Morse telegraph register; .and 
 Messrs. Saxton and Bond subsequently replaced the 
 telegraph fillet by a uniformly revolving cylinder* 
 This mode of registering transits is known as "the 
 American method." 
 
 Page 28, add to the table Litre = 0-035317 cubic feet, 
 
 " =61-02705 " inches. 
 
 Page 31 and 32. In this discussion we have assumed that the 
 arithmetical mean is the mean of an infinite number 
 of observations, and, therefore, that its error is an infi- 
 nitesimal ; but, as the number of observations is always 
 limited, better approximations to the values of the 
 
probable errors of a series of directly observed quanti- 
 ties are given by the formulas : 
 
 nl 
 Probable error of a single observation is 
 
 0-6745 e 
 Probable error of the mean result is 
 
 E = 0-674:5 - =0-6745 \J * 2 + * /2 + t//2 + 
 
 Vn n(n 1) 
 
 Page 35, line 11 from bottom, for effected read affected. 
 Page 66, line 8 from top, for height read heights. 
 Page 72, line 11 from top, for E and F is, read E and F are. 
 Page 81, line 22 from top, dele and. 
 
 Page 87, line 11 from bottom, for of the ring read at the ring. 
 Page 90, line 19 from top, for Tyndall and others read Tyndall, 
 Henry, Norton and others. 
 
Entered, according to the Act of Congress, in the year 1869, by 
 
 ALFRED M. MAYER, Ph.D., 
 
 In the Clerk's Office of the District Court of the United States in and for the 
 Eastern District of Pennsylvania. 
 
* 
 
419