sepcberte test sits cess secece= Sorspersstt read 3s enero Bitereitit ¥: ete eip ters Srittest Thee: Sf sites3 poe eOw reins phage te $3 3 peraeteseetetiee sesittess sa See sete setiet yeas peste hy peeth tenaa sree tel ine Peehs -r rereens oretee eet seas one twee oer ize? v2 ptaesserstias ate wisi Hit weslosacenece>: ussrieyresetessseh ss Teste tte: $3 Sed pretest RSE pe tirersrsccesesrrescs Tpshebes ter stre SCIENTIFIC MEMOIRS, Pa <} | ‘SELECTED FROM THE TRANSACTIONS OF FOREIGN ACADEMIES OF SCIENCE AND LEARNED SOCIETIES, AND FROM ¢ FOREIGN JOURNALS. EDITED BY RICHARD TAYLOR, F.S.A., FELLOW OF THE LINNZ AN, GEOLOGICAL, ASTRONOMICAL, ASIATIC, STATISTICAL, AND GEOGRAPHICAL SOCIETIES OF LONDON ; HONORARY MEMBER OF THE NATURAL HISTORY SOCIETY OF MOSCOW, UNDER SECRETARY OF THE LINNZAN SOCIETY. VOUT LONDON: PRINTED BY RICHARD AND JOHN E. TAYLOR, RED LION COURT, FLEET STREET. : “sou BY LONGMAN, ORME, BROWN, GREEN, AND LONGMANS; CADELL; RIDGWAY AND SONS; SHERWOOD, GILBERT, AND PIPER; SIMPKIN AND MARSHALL; B. ; FELLOWES; S. HIGHLEY; WHITTAKER AND CO.; AND J. B. BAILLIERE, LONDON: ia —AND BY A. AND C. BLACK, AND THOMAS CLARK, EDINBURGH; SMITH AND SON, GLASGOW :—MILLIKEN AND SON, AND HODGES AND M’ ARTHUR sDUBLIN : h — DOBSON, PHILADELPHIA :—-AND GOODHUGH, NEW YORK. 1841. oy. oh ior ti “Every translator ought to regard himself as a broker in the great intellectual traftic of the world, and to consider it his business to promote the barter of the pro- duce of mind. For, whatever people may say of the inadequacy of translation, it is, and must ever be, one of the most important and meritorious occupations in the great commerce of the human race.””—Goethe, Kunst und Alterthum. PREFACE TO THE SECOND VOLUME. eee IN the Advertisement to the Seventh Part of the Scientific Memoirs the Editor has already acknowledged the assistance ' afforded to the work by the British Association for the Ad- vancement of Science, concurring, as it has done most efficiently, with the other public bodies and individuals by whom the suc- cess of the undertaking had been promoted. It is now his pleasing duty to state, that the support thus given having af- forded an opportunity for the plan and objects of the work to become more generally known, the sale has been so far increased as to give an improved prospect of its permanence; and that a portion of the Third Volume is already in the press. Of the Memoirs contained in Part VIII. the following have been received from the Committee of the British Association for procuring the translation and publication of Foreign Scientific Memoirs, viz. :— The Galvanic Circuit investigated Mathematically. By Dr. G.S. Oum. Continuation. Besse. on the Barometrical Measurement of Heights. RupBeErG on the Expansion of Dry Air. WeseER on a Transportable Magnetometer. With a Plate. Wesex on the Magnetic Term-Observations for 1839 of the German Magnetic Association. Extract. With a Plate. Goutpscumipr on the Observations 6f Magnetic Declina- tion at Gottingen. | The continuation of the translation of Ohm’s Memoir has iv PREFACE. been paid for out of the grant at the disposal of the Committee ; as have the Plates for the two Memoirs of Professor Weber. The translation of Rudberg’s experiments has been presented to the Committee by Professor W. H. Miller, of Cambridge ; and the translation of the Memoirs of Bessel, Weber, and Goldschmidt, by Major Sabine; and by the Committee to the Editor. The Editor has also to acknowledge the valuable assistance which he has received from Professors Miller and Wheatstone in the revision of the translation of Ohm’s Memoirs, and of Professor Graham and Richard Phillips, Esq., for similar services with regard to the Chemical Memoirs. To the friendly and zealous cooperation of Major Sabine he is also most espe- cially indebted. Red Lion Court, Fleet Street, Feb. 20, 1841. CONTENTS OF THE SECOND VOLUME. ite AF sult, PART V. Page Art. I.—Electro-Magnetic Experiments, forming a Sequel to the Memoir on the Application of Electro-Magnetism to the Movement of Machines; presented to the Royal Academy of Sciences of St. Petersburg. By M. H. Jacos1, Doctor of Science, and Professor at the University of Dorpat....... I Art. Il —Results of the Observations made by the Magnetic Association in the year 1836. Gottingen: Edited by Cari Frizepricu Gauss, and WILHELM WEBER. ........ seit PD Pemaioductton's-by Prot. GAussts.veresc0csonsscssesctercevncsdsdosnsocessete 20 2. Remarks on the Arrangement of Magnetical Observatories, and Description of the Instruments to be placed in them; by Prof. Weber ....... Sone Sse tree maeuniadodentneneene aay racseaassesenecetcabenese 25 3. Method to be pursued during the Terms of Observation; by Prof. Mela Seirteeaceaeecseesarsceenesatsdevdadusteedclos ee Rydeasnaadesel seas oSemas 42 4. Extract from the Daily Observations of Magnetic Dailisinsion during three years at Gottingen; by Prof. Gauss..............see0e0e 54 5. Description of a smal] portable Apparatus for Measuring the Abso- lute Intensity of Terrestrial Magnetism ; by Prof. Weber ......... 65 6. On the Graphical Representations, and Table of Results; by Prof. (27 TES daowoande asain OS aaPa EE Ac sao -eeebsaSackcodd Meee edn hase sah 87 Arr. J11.—On the Combinations of Ammonia with Carbonic Acid. By Hernricu Roser, Professor of Chemistry in the Bere ew EE LICINRTY OSes Ue he OS rs Se Pe SOA ASE 98 PART VI. Art. [V.—Memoir on the Polarization of Heat. By Mace- SINDEN LONE 2S 8 225 clas SS mundi wtomon lta he 2a, esse aL Arr. V.—General Theory of Terrestrial Magnetism. By Cary Frrepricu Gauss, Professor in the University of Gottingen 184 Arr. WE the Changes in the Intensity of the Horizontal Portion of the Terrestrial Magnetic Force. By Cari Frieprich Gauss 252 vi CONTENTS. Art. VII.—Observations on the Arrangement and Use of the Bifilar Magnetometer. By Wirnetm WEBER .......... 268 Art. VIII.—Contributions to our Knowledge of Phytogenesis. By Dr. M. J. ScHEEIDEN £2... 1. 00. 2-0 ee ole oe Cee 281 PART VII. Art. IX.—Supplement to the Treatise entitled “ General Theory of Terrestrial Magnetism.” By Cari Frirpricn Gauss, of the University of (Gottingen. . ....\.. ... ness =enee 313 Arr. X.—On the Method of Least Squares. By J. F. Encke, Director of the Astronomical Observatory at Berlin ...... Si Arr. XI.—On the Theory of the Formation of Aither. By Heryricu Rose, Professor of Chemistry in the University of BROTURD eo d wispy oon og ont 6 3 wegen art ge 370 Art. XII.—Determination of the Axes of the Elliptic Spheroid of Revolution which most nearly corresponds with the ex- isting Measurements of Arcs of the Meridian. By F. W. BESSEL ©. 0. c:-} 871°35 10 871.0 | 871-60 | 20 8724 | 871-95) 30 872.9 The second column contains the several notations ; the third, the partial results ; 870°80 is the mean between the first and third notation, and therefore corresponds to 10" 19’ 40", and so forth. It is pleasing to perceive in this example, chosen from a time of rapid change in the declination, how a practised observer can recognize with certainty the changes occurring in 10 seconds. Observation on the 25th March, 1837, at 0» 5'. By Dr. Goldschmidt. Om 4 3Q" 847°3 39 847-2 ‘ 46. | S478) |. 8ta oe 53 848-7 847.95 | 5 0 848 9 £2? \847-91 7 848-1 847°85 14 847-0 cae) 21 346.9 | 84770 28 847°3 The first partial result in this case is obtained from the com- bination of the first and fourth notations ; the second from that of the second and fifth, &c. In this example the submultiple of the approximate time of vibration is an integer number; where this is not the case, the time must be divided into unequal parts, which has, however, no disadvantage, provided such an arrangement is made, that the notations to be combined shall always have for the in- terval to which they correspond the same approximate value of the time of vibration, and that the time, and also its portions, shall be registered. Thus, for instance, the observations in the astronomical observatory, with a bar of 25 pounds in weight, having a time of vibration of 43%-14, must be arranged according : | GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. 49 ‘ to the following scheme,—taking the approximate value at 438, ;| dividing it into four parts, and deriving the final result from | five partial results. o> 4/17! | 28 39 0 4! 38"-5> | 49 49 5 5 0 5 0°5 $0 5! OMI 1] 105 22 21 +5 32 43 The first column contains the times of notation; the second the times to which the partial results severally correspond: it is obviously unimportant that the final result, if accurately taken, falls at 0" 5! 0""1. If the final result is based on six partial re- sults, then the following scheme is adopted: 0» 4! 19" 22 33 0» 4! 33-55 44 43 “5 | 55 aS ae Aes Bon eh 16 16 5 27 26 +5 38 48 The advantage of this modification in the mode of observing is most evident, when it is desired to follow the course of the mag~ netic declination more closely than at intervals of 5 minutes. These intervals, sufficient for the ordinary progress of the changes of declination, are in fact too large for the examination of the greater and more rapid changes; and it was in this view, and because shorter intervals could scarcely be generally adopted through- out the terms of 24 hours, that subordinate terms were added, each of two hours’ duration, in which the observations were to be made at intervals of 3 minutes. As, however, the sub- ordinate terms occasioned some difficulties, and, as they have hitherto brought to light but few phenomena of correspond- ing importance, it has been decided to discontinue them. The Same object can be attained even more effectually in another manner. The rule of observing at every 5 minutes is retained ; but if at any time rapid changes of declination occur, the obser- VOL. II. PART V. D 50 GAUSS AND WEBER ON TERRESTRIAL MAGNETISM, vations are made at every 24 minutes, as long as it may appear desirable to do so. An example is added : 10" 22' 0” | 875-0 10 8748 | 875-50) 20 8760 | 875:95 | 30 8771 | 876-40 + 876-27 for 10" 22! 30” 40 8768 | 87660 | 50 8761 | 876-90) 23 0 877°] Observers in general are requested to pursue the course here pointed out whenever occasion may require it; and, in such case, it cannot be doubted that, whenever changes of such magnitude occur, a body of corresponding observations in close detail will be collected, and will furnish interesting conclusions respecting these remarkable phzenomena. If observers, instead of a clock beating seconds, are furnished with time-pieces marking other divisions of time, they must ar- range their observations in an analogous manner, corresponding to the beats of the time-piece. The observations with a chrono- meter are more difficult than with a clock, particularly if the second hand is not truly centred, as is sometimes the case. It may be well to add some general precautions for unprac- tised observers. It is of the first importance that the movement of the needle should be perfectly free. Spiders sometimes get into the box, and attach their web to the needle. This may be so fine as pos- — sibly to escape observation with the eye. Previously to each term, therefore, the finger should be passed carefully round the needle on every side. Any impediment which may exist to free motion will diminish the time of vibration of the needle. The most minute spider’s thread has a very considerable effect in this respect, of which a curious example will be related in its _place. In night observation it is necessary to illuminate the scale, which, at Gottingen, during the term-observations, is done by means of two Argand lamps. There is always an upward cur- rent of heated air above the flame, and, therefore, if one of the lamps is placed near and below the telescope, such a current passing before the object-glass will impair the distinctness of vision, and cause the divisions of the scale to appear tremu- lous and undulating. This inconvenience frequently occurred GAUSS AND WEBER ON TERRESTRIAL MAGNETISM, 5 at Gottingen in the first observations ; but has completely ceased since each lamp has been provided with a copper chimney, di- rected to the side. As in the term-observations several observers are required, there may be a considerable difference in the distance at which distinct vision is obtained by the several individuals. Ifa short- sighted person comes to the telescope adjusted for a long- sighted person, some alteration will be required for distinct vision. The use ofa concave glass would be inconvenient and un- advisable, on account of the considerable loss of light. The mere sliding of the eye-tube is not sufficient, as, although the image of the scale might thereby be rendered distinct, the cross threads would remain indistinct, and would have a parallax in respect to the image of the object. It would be necessary, therefore, (with the construction which the telescopes employed in these observations usually have) that the cell containing the cross threads should be moveable in the eye-tube, and that it should be brought nearer to the lens in the eye-piece ; but this requires a practised hand, takes time, and for other reasons is not to be recommended for the present case. The difficulty may, however, be got over in a very simple manner, if the following plan be adopted. The eye-tube in the telescope, and the cross threads in the same, are to be so adjusted previous to the observations, that the most short-sighted among the observers can see perfectly distinct both the image of the scale and the cross threads ; when _ a longer-sighted person arises in turn, he has merely, without displacing the eye-tube or the cross threads, to draw out the glass nearest the eye so far that he can define perfectly well the cross threads, and with this a completely distinct vision of _ the image of the scale is necessarily connected. A short-sighted person coming in turn has merely to make an adjustment in the contrary way. For the purpose of proving the undisturbed state of the tele- scope, a mark is employed, which is placed at such a distance i; that it may be seen distinctly with the same position of the _ eye-piece as is required for the distinct vision of the image of the scale ; this consists, in the Géttingen observatory, of a small vertical line on the northern wall*. Previously to the commence- * With respect to this arrangement, I may here observe that a mark for the verification alluded to must be considered as indispensable. Previously to the building of the present Gottingen magnetic observatory, it was doubted D 2 52 GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. ment of the observations, the telescope must be directed towards the mark, and this examination must be repeated from time to time ; and if a deviation is indicated in the optical axis, it must be again brought back to its original vertical plane. If the precau- tion is taken to note two other divisions on the wall, one on either side of the mark, they will furnish the means of estima- ting the amount of the requisite correction. But it should be remembered that these divisions, though they may be made to correspond exactly with the divisions of the scale, will not have exactly the same value in seconds. If no such auxiliary marks have been made, the amount of the correction must be judged of by the eye, in parts of the divisions of the scale itself. The observations are made at the vertical thread; the hori- zontal thread serving merely to indicate nearly the middle of the former. In order that it should make no difference whether the parts of the scale appear somewhat higher or lower in the field of view, the cross threads must have such a position, that a fixed object, seen on their crossing, remains accurately on the vertical thread, when the telescope is moved somewhat up and down. The mark also serves for this verification, which, how- ever, need not be frequently repeated when the position is left unchanged. The plumb-line suspended from the centre of the object-glass must be so near the scale that the image of both may appear with the same distinctness in the telescope, and that thus the division covered by the line may be precisely determined. The scale must be so placed that its zero must correspond with the plumb- line, or the division which does so correspond must be taken as an arbitrary zero. The verifying the undisturbed state of the scale should be repeated from time to time in the course of the whether it was not better to place this mark on an insulated pedestal in the interior of the room, than on an exterior wall exposed to the weather. The latter was decided on, as otherwise either the distance of the observer from the needle must have been diminished,—or the advantage of seeing distinctly the mark and the scale with the same position of the eye-piece be given up,— or the room must have been made of a greater length, which was not possible in the place fixed on. ‘lo have a separate foundation for a mark was regarded for many reasons as objectionable. Moreover, the fear that the place of the mark might be perceptibly altered by the influence of the weather on the wall, was regarded as of little importance, considering the solid construction of the building, and the small height of the mark above the foundation; and espe- cially as it was in our power to repeat, as frequently as desired, the measure- ment of the angle between the mark and a church spire seen through the northern window. The experience of three years justifies the propriety of this arrangement. GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. 53 observations ; it is, however, not requisite, when a small change is found, to bring back the scale to its former position; it is sufficient to note down in the registry the point of division cor- responding to the plumb-line. It may probably not be superfluous to draw attention to one or two points of comparatively minor importance. It has been supposed, that the magnetometer and telescope are so arranged that the mean position of the magnetic declination corresponds to about the centre of the scale. However, at times of consider- able variation, this centre frequently gets entirely out of the field of view, and then the above method of verification will no longer answer. If at such a time the verification appears necessary, the quieting bar must be made to perform an exactly oppo- site office to that which it generally serves; namely, to give the magnetometer a vibration of sufficient extent to reach, and even to go rather beyond, the spot required, and thus to allow the plumb-line to appear in the middle of the field, at that part of the vibration where the motion is slow, and where consequently the corresponding division of the scale can be determined with accu- racy. It is obvious that if such cases occur in the course of a periodical series, the magnetometer must be again quieted in time for the next observation, and, consequently, skill in the use of the quieting bar is of great moment. When the declination falls very nearly in the centre of the scale, unpractised observers must be on their guard not to confound the plumb-line with the vertical line of the telescope. In our apparatus both resemble one another so much, that with a very quiet state of the needle, a mistake is very possible, and did, indeed, once occur. When there is danger of such a mistake, it may be expedient temporarily to remove the plumb-line. With respect to the form of communication, some persons are accustomed to send in the observations im full, others the partial and final results only, and several merely the latter. The last may be sufficient, if the calculations have been revised, and the communicated numbers collated ; but the observations them- selves should be preserved, in case a reference should be wished ; and when unusually great changes occur, communication, in full detail, is most desirable. Besides the results of the observations, it is always proper to notice, in connection, the value of the parts of the scale (or the measurements on which the determination is founded), the time of vibration, the correction and rate of the 54. GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. clock, the name of the observer, and remarks on such observa- tions as may be somewhat doubtful. An early communication is always greatly to be desired. Gauss. Hil. Extract from the daily Observations of Magnetic Declination during three years at Gottingen. To discriminate the regular changes of declination, amidst those incessant changes of greater or less amount, which we call irregular, i so far as their occurrence seems unconnected with any periodical rules, requires a great number of observations on a fixed plan, persevered in for a length of time, in order to deduce, by suitable combinations, mean values, freed as far as possible from the influence of those anomalies by which the individual declinations are affected. In general, in this part of the globe, the declination increases during the forenoon, but the increase is unequal on different days ; it even sometimes happens, though rarely, that at the usual hour of maximum, the declination is less than it was during the earlier part of the same day. The cause of the morning increase may be in operation every day; but its influence is sometimes increased, sometimes diminished, and sometimes entirely masked, by other irregular intervening forces. Observations on a single day, or continued for a few days only, cannot therefore determine either the amount of the effect due to the regular cause, or its inequalities at different seasons. For this, mean values, taken from a great number of days, are required. The same is the case with those progress- ive changes which take place in one direction for a very long time ; these we call secular, because they require a long series of years to amount to many degrees. Single observations, repeated after an interval of only a few years, even though performed on the same day, in the same month, and at the same hour, can afford us no certain knowledge respecting them; but mean numbers, obtained by continued observations, allow us to an- ticipate, at the end of very few years, what it would otherwise take many tens of years to fix with any considerable degree of approximation. With this view, from the very commencement of the observa- tions to be performed at our Magnetic Observatory, I have in- - cluded among them the daily determination of the absolute de- GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. 55 clination at the same hour. In order to be able to calculate more easily on the possibility of a long and continuous perseve- rance, by which alone labours of this kind can be of value, I have at first rather chosen a limited plan than attempted to combine too much at once. On this account only two observa- tions are made daily; at eight in the forenoon, and one in the afternoon, according to mean time. These hours, which were most easily compatible with other duties, are also suitable ones, because in the regular course of the magnetic movements the position of the needle at 1, p.m. is never far from the maxi- mum of declination, and during the greatest part of the year, the hour of minimum is not far from 8, A.M. Observations at fixed hours of apparent solar time would, it is true, have been more in accordance with nature; but the much greater facility of an ar- rangement made according to mean time, renders it deserving of preference in this case, where the chief point is to secure a per- severing continuance in one and the same principle. A regular register was commenced on the Ist of January, 1834 ; but the first two months and a half have been omitted in the following extract, because during that time it was frequently necessary to wind up the suspension-thread, whereby changes were produced in the torsion which were at first not sufficiently attended to. From the 17th of March a stronger suspension- thread was employed, consisting of 200 fibres, of which the point of no-torsion had been previously accurately determined ; when- ever changes were subsequently made in respect to the thread, or to any other circumstance connected with the elements of reduction, the necessary corrections, or modifications of those elements, have each time been applied. During the first months various sufficiently practised observers took part with me in the observations ; but since the Ist of October, 1834, they have been regularly made by Dr. Goldschmidt, his place having been only occasionally supplied, when necessary, by other expert observers. I have already communicated in the Géttingen Gelehrten Anzeigen, 1834, p. 1269, and 1835, p. 345, the monthly means deduced from these determinations up to January, 1835: they are now given for three entire years. iL 56 GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. Mean value of the Westerly Magnetic Declination at Gottingen. 8, A.M. 1, P.M. ° ° i / 1834. March, secondhalf.| 18 38 16:0 | 18 46 404 prilee yeti 5 etree 36 69 47 38 WER Sedo aatigd 2) 36 28:2 47 15°4 Wane wes scons eee 37 40:7 47 59-5 Vily ..asaessecesee 37 57°5 48 19:0 INapuIsti re cine etemtels 38 48:1 49 11:0 September ........ 36 58:4 46 32:3 Octdtter: csc.cc sas 37 184 44 47:2 November ........ 37 38:4 43 43 December .......- 37 54:8 41 32-7 1S8e. anusnye. setae 37 51°5 42 14:4 February.....5.5.: of. 68'D 42 29:4 March sViiisanects 34 47°5 44 55:2 Aprils gine tren stet eiete 32 57:7 46 31°6 Ma Vics aererseesteais 32 13:4 45 17°71 Junie: Peer eeacees 32 56:4 44 41°3 Ditaliyiacteneaetiaiet acta 34 8:0 44 42:8 AUBUStesee 00056 ne 34 12:4 46 56:8 September ........ 33 21:2 44 27-6 OctObEE ie: ties: 33 23:0 43 5:3 November .......-. 36 15°3 43 49°5 December ........ 35 25:9 40 19:1 1836, January ........-- 35 24 40 346 February...... sees 33 26-7 41 15:2 Wharchisepstetnsenute 31 1:4 43 16:4 April’. .csc- cers 26 32:9 43 42°6 May csc sreescinss 28 0:8 44 37:2 JUNE: aie siastectenetas 27 351 42 52:4 July ...cceccconeee 26 54-2 42 26:0 August ........00. 25 42-4 41 45:0 September .....+ «+» 26 146 40 59:6 October .....-00-- 27 34-0 40 32:8 November .......- 29 21:0 36 54:3 December ....«-.- 29 13-7 35 46:8 1837. January ....++--«- 27 35:3 37 46:2 February ec es ceessee 27 356 36 28:3 IMarchieti. siekaseree 25 44.2 39 42 Some combinations of these observations may now be noticed. The difference between the declination of the morning and afternoon has one sign all through in the monthly means; the dependence of its magnitude on the season of the year will be perceived in the following tabular view: GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. 57 1834—1835. | 1835—1836. | 1836—1837. | Mean. i / i U i“ I u el ....... 10 569 13 339 17 97 13 53°5 Waves ices. 3's 10 47:2 13 37 16 36-4 13 29:1 DUNG: 0:50:00 10 18°38 11 44-9 15 17°3 12 27:0 ini SSeenee 10 21°5 10 34:8 15 31°8 12 9-4 August...... 10 22:9 12 44-4 16 26 13 3:3 September... 9 33:9 V1: 6:4 14 45:0 11 48-4 October...... 7 28°38 9 42:3 12 588 10 3:3 November .. 5 25:9 7 34:2 Loe 6 51-1 December.... 3 37:9 4 53:2 6 33:1 5 1-4 January ..... 4 22:9 5 32:2 10 10-9 6 42-0 February .... 5 25:9 7 48:5 8 52: 7 22:4 Marchese ccs| 10° 7-7 12 15:0 13 200 ll 542 Mean.... 8 142 10 28 12 54:3 10 23:8 It will be perceived that, not only in the mean values, but also in each of the separate years, the difference has been smallest in December ; and this is what we might expect, as those changes which vary according to the time of the day must necessarily be ascribed to the action of the sun, although as yet we know not how this action is effected. It may at first appear surprising, on the other hand, that the differences are not greatest at the time of the summer solstice, but appear smaller in June and July than in April, May, and August, especially as the coinci- dence of all three years in this circumstance affords a presump- tion that it is not accidental. It must not, however, be over- looked, that in the months immediately following the solstice, the time of the minimum of the declination is earlier, and there- fore the whole increase would be sensibly greater than the change reckoned from 8 o’clock. It is further observable that in each month the differences are greater in the second year than in the first; and again, in the third year greater than in the second. But these differences are by far too great in amount to admit of our considering them as parts of a secular increase, and it is rather to be expected that by continuing the observations for several years we shall not fail to discover a fluctuation. But, in any case, we hereby learn that one year may differ from another in respect to the effect of the sun on the earth’s magnetism, somewhat in the same way that one summer or one winter differs from another in temperature. On this account also we shall only arrive at an accurate determination of the mean values by observations con- tinued for several years. 58 GAUSS AND WEBER ON TERRESTRIAL MAGNETISM, It has been already stated that exceptions sometimes occur on single days, when the difference between the forenoon and after- noon declinations may have the opposite sign. But such ex- ceptions are rare; during the three years’ observations only fourteen cases of the kind have occurred; or, on an average, one in 79 days. I give them in this place, together with the amount by which, on each occasion, the declination at 8, a.m. exceeded that at 1, P.M. U a“ i i 1834. Aug. 15 6 80 1835. Nov. 8 3) 42-2 Dec. 24 3 43-0 Dec. 8 18 35°6 Dec. 25 0 38-2 1836. Jan. 20 0 46:3 Dec. 26 2 20:3 July 20 5 88 1835. Jan. 30 0 23:8 Nov. 9 ll 95 Feb. 7 0 32-5 1837. Feb. 13 4 10 Oct. 4 0 43-1 Mar. 14 1 22-6 Of these fourteen exceptions, twelve, as might be expected, occur in the winter months, and only two in the summer months ; the small regular action of the sun in the former being more easily exceeded by an anomalous movement than could be the case in regard to the far greater regular action in the summer months. To try how far the secular variation might be recognised in the present observations, the monthly means of the first year have been compared with the corresponding ones of the second, and these with those of the third year. Among the forty-eight comparisons thus obtained (for the incomplete month of March, 1834, has been excluded from this as well as from all the other combinations), forty-seven give a decrease, and only one an increase, which is therefore characterised in the following table by the sign —. Yearly Decrease of the Declination. First Year. Second Year. Mean 8, A.M 1, P.M 8, A.M 1, P.M i] 1 April...... 3 92 | 0322 | 6248 | 2490 | 3138 May .... 4 14:8 1 58:3 4 126 0 39:9 2 46-4 SHINE voces: 4 44:3 3 18:2 5 21:3 1 48:9 3 48:1 DULY sae. 3 49°5 3 36:2 7 138 2 16:8 4 141] August 4 35°7 2 14:2 8 30:0 5 11°8 fa September . 3 37:2 Dwi Ad 7 366 3 28:0 4 4] October....| 3 55:4 1 41:9 5 49:0 232°5 3 296 November . 1 23:1 |—0 45:2 6 54:3 6 55:2 3 36:8 December..| 2 28:9 1 136 6 12-2 4 32:3 3 36:7 January... 2 49:1 1 39:8 7 27:1 2 A8-4 3 41:1 February ..| 3 36:8 ] 14:2 5 55°] 4 46:9 3 52:2 March :...| $8 46:) 1 38:8 Tle 4 12:2 3 46:6 | Mean... | 3 308 1 42:2 6 21°7 oO mune 3 462 | |

jo! 47° 42! Uy — Us = 71° 49) 69° 21! 46° 12! 990 947! R, = 450 R, = 350 - millimetres ft =6"6/ = = H es Sew _~ Hou i il = 101°0 sre bahay millimetres. p 142000 milligrammes. From these may next be calculated, > = + (23° 9! + 290 27!) = 11° 24"-00 v, = ¢ (47° 42' + 46° 12!) = 23° 28'50 v, = + (71° 48! + 69° 21/) = 35° 17/25 If now we take the second and the millimetres as the funda- mental units of time and space in our calculation, we may deduce from the ascertained values of Ro, R,, Ras Vp, Vy» Va» the following values of A, A’, B, B', B", viz. tang 11° 24’ tang 23° 2e''5 |, tang 35° 197"25 _385°54 S | A =—~7508 3508 3008 =jou > Zs _tang 11° 24! tang 23° 28'5 = tang 35° 17'"25 __384'86 450° 350° 300° 10% 3 Ppa 1 _ 20362 = 750° * 350° * 300° 10 ° 1 1 1 20977 | ee je =e B' = 350° + 350° + 300° — 10 ° 1 1 yrs 1 20855 450 * 350% + 3900 = 0% * From these 7 is calculated: _; 385°54 + 2°0855 — 384°'86 + 2: 0277 10° ~ 2° 9:0362 + 20855 — (20277)? or 7 = 87650000. GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. 83 Finally, from this value of 7, and from that of ¢, determined by observation, may be deduced the value: By ctciey _ 5:0641 tVr ~~ 667° 8765000 =—:10” * This number suffices for the comparison of all intensities mea- sured with the same instrument, however the magnetic condition of the apparatus may have varied. Further, the number 7, which expresses in absolute measure the resulting intensity of the earth’s magnetism, may be ascer- tained by deducing from the observations the value of C, and multiplying the former number by its square root. C is calcu- lated from the observed values of a, 6, and p, the mass of the mil- ligramme being taken as the unity of mass: C = 9-8696 + LOL + 17°5° = 0°1227 10 = 9° coger pert + 142000 = 0°12272 + 10 “= whence 7 is deduced T = 5:0641 . 0712272 = 1°774. 5. Examination of the result. This number 1°774, expressing the intensity of terrestrial magnetism on the 18th of January, 1837, possesses, as an absolute measure, the advantage of being directly comparable with the results obtained in 1834 with the magnetometer of the Gottingen magnetic observatory, published in the Géttingen gelehrten Anzeigen of that year. They will be found in part 128, (with the account of the newly-constructed building, and of the instruments, as well as of the first experiments performed there). They are as follow: Pyenpoicd es Se Heres a. ek. ots — 20 . : : ° : 17740 — 271 . ; : : z 1°7761 Two apparatus destined for the same purpose can hardly be more dissimilar than the small apparatus above described, and the magnetometer. It results from the comparison, that the intensity of the terrestrial magnetism in Gottingen has under- gone hardly any alteration from 1834 to 1837. We have also a direct comparison of this number obtained for Gottingen with the result of observations with a third ap- paratus, differing widely from both the others made at Munich, April Ist, 1836, viz. 1-905, and with the number found for F2 84 GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. Milan, with the magnetometer of that place, in October, 1836, viz. 2°61839. To gain a clear idea of the import of these numbers, the de- termination and application of which have been hitherto under consideration, imagine a number of small steel bars, perfectly alike, and each weighing about 23 grammes, or 4 of an ounce. Imagine further a balance, of which the length of the arms bears to 1 metre the same proportion that 1 metre bears to the space of descent in 1 second (204 millimetres nearly) ; sup- pose one of these steel bars to be attached in a parallel direc- tion to the horizontal beam of the balance, in such manner that the equilibrium is not thereby disturbed. Then render all the steel bars (including the one attached to the balance) equally magnetic, and to such a degree that when another of their number is placed vertically beneath the scale at the distance of 1 metre from the attached magnet bar, ;15,th of a milligramme must be placed in the scale to preserve equilibrium. When the magnetism of all the bars has been regulated in this man- ner, place one of the bars horizontally, and at right angles to a small compass needle, 1 metre from the centre of the needle beneath, taking care that as the compass needle is deflected from the magnetic meridian, the bar be also turned so that they may preserve their rectangular position. Lastly, calculate how many such bars are required that their united force may deflect the compass needle 90°; the number of bars gives the terrestrial magnetism in thousandths of its absolute measure. We may conceive in like manner the number which repre- sents the absolute measure of the terrestrial magnetism to repre- sent the number of these bars reckoned in thousands, the forces of which must be united to cause, at a distance of a metre, a deviation of 90°. This would require at Gottingen the force of . . . 1775 bars MICA. ee Nal) CT Brae ese) coe ee ot ee 6. On the Advantages of the Dimensions selected for the small Measuring Apparatus. Before concluding this article, we have to discuss the accuracy of which the absolute measurement of intensity with the appa- ratus described is susceptible, and on what it is founded. It has been already remarked, that the absolute intensity can be GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. 85 measured with the accuracy it deserves only with the magneto- meter. It is therefore unnecessary to state that such extreme accuracy cannot be attained with the small apparatus. And in order to obtain with it a good approximation, it must combine all the advantages of which it is susceptible. The difficulty of an accurate measurement of intensity, with other instruments than the magnetometer, is thus stated in the memoir “ On Terrestrial Magnetism and the Magnetometer +” “ In all cases, if the elimination is to be satisfactory, the ex- periments must not be performed at too small distances ; conse- quently the effects are always comparatively small, and the means previously in use are inadequate to measure them with the necessary precision. It is this difficulty which has called for, and has given rise to the construction of a new apparatus, which may with propriety receive the name of magnetometer, since it serves to execute, with an accuracy equaling that of the most delicate astronomical determinations, all measurements— both of the force of magnetic needles, and of the intensity of the earth’s magnetism (at least its horizontal portion). The (hori- zontal) direction of the earth’s magnetic force is determined ac- curately with it to within one or two seconds of arc; the com- mencement and termination of a vibration is observed with it to within a few hundredths of a second of time, and consequently more accurately than the passage of stars behind the wires of a transit.” There are two circumstances, chiefly, on which the accuracy of an absolute measurement of intensity depends; first, the magnitude of the deflection produced; secondly, the delicacy of the instrument in measuring this deflection. In constructing an apparatus for this purpose we may therefore follow two different paths: we may either make the amount of deflection the main object, and pay only as much attention to the means of mea- surement as may be consistent therewith ;—-or we may attend chiefly to accuracy in the means of measurement, and let the amount of the deflection be the second object. The latter plan leads to much greater accuracy than the former, for this rea- son: the amount of deflection soon attains a limit, on account of the necessary condition of a considerable distance between the deflecting bar and the needle, so that the deflection produced 86 GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. must always be small. If, however, all pretensions to great accuracy of measurement are relinquished at the outset, by making the magnetic needle play on a pivot, instead of sus- pending it by a silk thread, the friction of the point renders fineness of measurement quite illusory, and the former much less advantageous plan is the only one that remains open; the endeavour must then be to adopt the arrangements and pro- portions best suited to produce the greatest possible deflection. This is the express object of the small size of the apparatus de- scribed, and not merely to render it light and convenient of transport. That the small size of the apparatus does actually allow of a great amount of deflection is evident by the result ; for in the experiments above mentioned all the measured angles exceeded 20°: it is easy to explain the reason. 1. The distance of the deflecting bar from the needle must be relatively great, but need not be absolutely so: it must at least be three or four times greater than the length of the deflecting bar, or of the magnetic needle. 2. By diminishing in proportion all the linear dimensions of the apparatus (viz. the dimensions of the magnets, and their distance apart), the angular magnitudes, of which the deflection is one, remain unchanged ; therefore such proportional reduc- tion in the size of the apparatus, causes no loss in the amount of the deflection to be measured. 3. But if instead of diminishing in equal proportion all the li- near dimensions of the apparatus, we diminish only the length of the magnets and their distance apart, the breadth and thickness of the deflecting bar being little or not at all diminished, then we even gain an increase in the angular magnitudes, and it only remains to know how far this increase may be carried. The limit depends on a single circumstance, viz. on the breadth and thickness of the deflecting bar, with a given length. Experience has shown, that neither the breadth nor the thick- ness of the bar ought to exceed the eighth part of its length. It follows that the greatest deflection may be produced by a magnet bar, of which the breadth and the thickness are equal, and of which the length is eight times greater than either, and acting upon a magnetic needle, placed at a distance equal to three or four times the length of the bar; the length of the needle must not exceed that of the bar. GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. 87 From this rule then we obtain the most advantageous di- mensions of such an apparatus, by knowing the limit in re- spect to thickness, which is determined by the nature of the steel. The thickness of the bar must not amount to much more than 12} millimetres, as otherwise the steel cannot be properly hardened and magnetized throughout. We thence obtain the following dimensions of the deflecting bar, as those which com- bine the greatest advantages, namely, for its breadth and thick- ness 121 millimetres, and for its length 100 millimetres. We have also the length of the magnetic needle 100 millimetres, and the smallest admissible distance between them, 300 milli- metres. By following these rules we obtain an apparatus, with which, in mean latitudes, the smallest deflections to be measured ex- ceed 22°, as in the experiments related. At greater distances from the magnetic poles of the earth, this deflection becomes somewhat smaller; nearer to the magnetic poles it is much larger. Therefore, if these deflections can be accurately mea- sured to within a tenth part of a degree, a final result can be obtained to within the 200th part of the force itself; sinceall other measurements required in the determination of the absolute in- tensity can be made with greater accuracy. This result, it is true, is far inferior to that which can be obtained with the mag- netometer ; but such results may still be of great utility in the absence of more accurate determinations. WEBER. V. Explanations of the graphical representations, and of the table of results. In Plates [V.—-IX. are given the graphical representations of the changes of declination during six terms, amounting, in all, to forty-six curves, from fourteen stations, viz. Berlin, Breda, Breslau, Catania, Freiberg, Gottingen, the Hague, Leipzig, Milan, Marburg, Messina, Munich, Palermo, and Upsala. The graphical representations begin with the November term of 1835, when the Association was strengthened by the accession of seve- ral new and zealous cooperators. The representations of two terms of the year 1836 have been omitted, viz. those of March and May, as the changes they present are comparatively less in- teresting than those of the two terms of January and July, be- 88 GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. tween which they occur; and the number of six terms, fixed on as the rule for the annual publication, is completed by the addi- tion of an extra term in August. Of the apparatus employed, three are exactly similar to those at Géttingen, but of smaller dimensions; namely, that of Dr. Wenkebach, first used at the Hague, and subsequently at Breda; the travelling apparatus with which M. Sartorius of Waltershausen, and Dr. Listing, observed in Palermo, Catania, and Messina; and the apparatus already mentioned at p. 22, in the Berlin Magnetic Observatory, which latter, however, will be shortly replaced by a larger one, of Meyerstein’s. The other apparatus in Breslau, Freiberg, Géttingen, Leipzig, Milan, Mar- burg, Munich, and Upsala, are all alike. The participators in the observations represented in the six terms, as far as the names have come to our knowledge, were as follows : In Berlin, besides Prof. Encke, MM. Bremiker, Galle, Madler, and Wolfers. In Breslau, besides Prof. V. Boguslawski and his sons, MM. Bratke, Brier, Dittrich, Héniger, Jacobi, Isaac, Klingenberg, Koch, Kérber, Kiintzel, Maywald, Miiller, Dr. Pappenheim, Reichelt, Reisern, Ribbeck, Riemann, Roedsch, Wiedemann, and Wilde. In Catania, Dr. Listing, MM. Sartorius von Waltershausen, and Zobel. In Freiberg, besides Prof. Reich, MM. Felgner, Neubert, and Walther. In Géttingen, MM. Briss, Lieut. Engelhard, Dr. Goldschmidt, Meyerstein, Schréter, Dr. Stern, Lieut. von Stolzenberg, Prof. Ulrich, Dr. Wappius, Dr. E. Weber, and Prof. W. Weber. At the Hague, (in the September term,) besides Dr. Wenke- bach, MM. von Cranenburgh, Rueb, and Simons. In Leipzig, besides Prof. Mébius, MM. Brandes, Faber, Hiilse, Kiihne, Michaelis, Netsch, and Zunck. In Milan, besides M. Kreil, MM. Capelli, Stambucchi, and Della Vedova. In Marburg, besides Prof. Gerling, MM. Beck, Deahna, Eich- ler, Fliedner, Hartert, Hartmann, Ise, Kutsch, Landgrebe, Lotz, and Oppermann. In Messina, Dr. Listing, MM. Sartorius von Waltershausen, and Tardy. GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. 89 In Munich, besides Prof. Steinheil, MM. Hierl, Lamont, Lip- -polt, Meggenhofen, Mielach, Pauli, Pohrt, Recht, Schleicher, Schréder, Siber, and Zuccarini. Other observations of some of these six terms have also come to our hands, but too late for insertion in the plates; this is the more to be regretted, as, for the most part, they accord with the others in a very interesting manner. The results of the obser- vations made at Upsala, in the September term, 1836, which are of this kind, are printed in the sequel. The Milan observations of November, 1835, which were also received after the curves for the six other stations had been drawn on stone, were inserted below them; but for this circumstance, their place would have been between the Munich and Palermo observations. The Gét- tingen observations have required no process of reduction, being drawn in accordance with the divisions of the scale as indicated in the margin, the height of each square being taken as two divi- sions of the scale in all the terms, with the single exception of that of January, 1836. The changes during that term are the greatest which have been hitherto observed, and rendered it necessary, in order not to increase the height of the page too much, to allow three divisions of the scale for each square. Increasing numbers always denote an advance of the needle from right to left,—in other words, diminishing westerly variations. The observa- tions at Breslau, Freiberg, the Hague, and Leipzig, where the divisions of the scale are nearly of the same magnitude as in Gottingen, have been drawn according to the same proportion. The distance between the curves is an arbitrary quantity in each case, determined solely by its fulfilling the one object of keeping them at a convenient distance apart. For those stations where the value of the divisions of the scale differs considerably from that at Gottingen, the original numeri- cal results were multiplied in each case by a common factor, ex- pressing, as nearly as possible, in convenient numbers, the pro- portion to the Gottingen scale. Thus, the various curves in each term are represented very nearly according to a common scale. In the January term alone the scale of representation is somewhat more unequal, the cause of which does not merit any mention in this place, as it suffices to know the scale for each curve. In the three first terms the height of each square corresponds to the following values of arc, viz. : 90 GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. November, | January, July, 1835. 1836. 1836. “a “i “ [a op ee opcepre 42°01 63°01 42°01 Gottingen. ...| 42-25 63°38 42:25 IErIU 2 cis oe s _- — 42:24 Breblau''::25-. 4 a -- 42:40 Leipzig ..... 41°34 63°01 41°34 Marburg ....| 42°20 60°28 42:20 Munich... 41°86 55°82 41°86 IMinlerte cfs: che he 40:27 60°40 41°33 Balernos. 6 oi 42:07 — — (CaLanity 9. <7s-0.s — 41°56 — Messina..... — — 43°06 For the three last terms, the value of the divisions of the scale, and the proportion, according to which they have been inserted in the plates, are stated in the table of numerical results. The curves are all drawn according to Géttingen mean time, (indicated at the top of each plate,) or at least very nearly so, and therefore contemporaneous movements appear all in one vertical line. The order in which the several curves are arranged in each plate was principally regulated by convenience as to the curves fitting into each other. The following remarks may be added in regard to particular terms : On the 28th of November, 1835, and during the following night, the observations at Palermo were much disturbed by an exceedingly violent Sirocco-wind, so that at one time they had even to be suspended for an hour and a half; and at other times only partial and uncertain determinations could be obtained. It is probable, therefore, that many of the apparent movements were not real magnetic changes. Nevertheless, we Wetermined not to exclude this curve; as the latter part of it, from the morning of the 29th November, when the storm had nearly passed over, offers a sufficiently satisfactory accordance with the stations to the north. I take this opportunity of mentioning that, according to all our experience hitherto, the most violent storms of wind appear to be wholly without influence on the magneto- meter, provided only the instrument is effectually protected from any effect of their direct mechanical action. Very frequently, either an extremely quiescent state of the needle, or a very regu- lar and uniform progress, has been remarked in the Magnetic GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. 91 Observatory of Gdéttingen, during the prevalence of the most violent storm. If any one, however, were inclined to infer from such experience, that storms in the atmosphere, on the other hand, counteract or enfeeble the magnetic forces, such an idea would be dispelled by what took place during the term of Janu- ary 1836. During this term a very violent storm prevailed at Gottingen, and at many other stations ; and several observers in other places accompanied the results which they communicated, by the expression of a fear that from this circumstance the un- usually large movements shown by the magnetometer might offer but little accordance. Nevertheless, the harmony of the curves from the various stations was so complete (see the repre- sentations in Plate V.) that it might have been termed wonder- ful, if the same thing had not been manifested before by so many experiments. As with wind storms, so it is with ¢hunder storms, which, even when close at hand, exercise (as attested by several cases which have occurred here and at other places) no percep- tible influence on the magnetic needle*. A letter from M. von Humboldt, received in August, 1836, con- tained the information that, from the 10th to the 18th of August, the magnetic changes would be observed uninterruptedly every quarter of an hour at Reikiavik, in Iceland, by a practised French astronomer, M. Lottin, with Gambey’s apparatus, and expressed the wish that corresponding observations might be made on one or on some of those days with magnetometers. In consequence an unusual term was fixed for the 17th and 18th of August, and as far as the shortness of the time allowed, seve- ral members of our Association at other stations were invited to take part in it. This unusual term was observed in Upsala, the Hague, Gottingen, Berlin, Leipzig, and Munich, in exactly the same way as the usual terms; and if the graphically re- presented observations in Plate VII. exhibit exceedingly in- teresting changes, we have only to regret that the place reserved at the top of the plate for the Iceland observations is vacant, as we have not been able to obtain the slightest information re- specting the result of the French Icelandic observations. The September term presents a case which may be noticed somewhat in detail, as it confirms, in a very instructive man- * There is, of course, no question here of experiments in which the atmo- spheric electricity is conducted to the earth by means of a conducting wire passing through a multiplier surrounding the needle. 92 GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. ner, what has been stated at p. 50. In the register of the Marburg observations, which on that occasion were made in the absence of Prof. Gerling, and at the hours of 120™, 12h5m, and 12" 10", there appeared an unusual irregularity, which ex- cited the suspicion, that about 12"5™ a spider had prevented the free motion of the needle by attaching a thread; and this sus- picion was increased by the circumstance, that from 12 10™ to the end, the changes of the needle were exactly similar to those which resulted from observations at other stations, but appeared proportionally much smaller than could have been expected from the experience of other terms. Prof. Gerling was requested, on his return to Marburg, to examine the apparatus carefully, and the result is contained in a letter from the Professor. The examination took place on the 5th of November, up to which time no one had entered the room of observation since the September term. In the first place the position of the needle was determined and found as follows : at 34 33m . . . 445°63 Bide 15. Toe ASS OF fh alt xe Ah Upon this the needle was set in moderate vibration by means of the moderating bar, and hence a time of vibration of 17 seconds was found, being nine seconds less than the usual duration: the lidof the case was then carefullyremoved, and avery minute living spider was noticed on its under surface ; a very small, and nearly imperceptible, thread was thought to be observed hanging to it: further, a number of small, black point-like bodies were found in the box, which, under the microscope, proved to be the dead bodies of gnats ; and finally, in one corner of the box, a regular undisturbed web, of such fine texture, that without the reflexion of the light it would hardly have been perceptible. From all these circumstances it may be supposed that the spider had been some time in the box. When the finger had been passed round the magnet bar in all directions, new observations of the time of vibration gave again the former value of 26 seconds. The position was also found to correspond to much lower numbers on the scale, namely, gh 45m. AG1°45 Aba he wt er ASd 4G Eh sata Setar: ad 1 GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. 93 Of course, however, these observations could not furnish an accurate determination of the amount of error introduced, as the declination may have. altered during the interval, which amount- ed to more than an hour. In the graphic representation the second half of the Marburg curve has been drawn on a reduced scale, the reduced divisions representing 28 on the Marburg scale. I may here mention a second case of a similar kind. The time of vibration of the magnet bar in Breslau, which, in March 1836, amounted to nearly 3°2 seconds, had from that period to November gradually increased, making altogether an increase of about 0-4. This is no unusual circumstance, as all magnetic bars in the course of time lose some part of their force, though in very various degrees dependent on the unequal tempering of the steel and other circumstances. But, from November 1836, to January 1837*, a decrease in the time of vibration of 1°27 took place. Prof. von Boguslawski, who informed me of this remark- able circumstance, seemed inclined to attribute it in part to an increased intensity of the terrestrial magnetism. I did not doubt, however, that the cause must be sought in the immediate neigh- bourhood of the magnet bar, probably in some impediment to its free motion, and this supposition was verified by the follow- ing letter of M. Boguslawski :— ** You were right in your supposition as to the cause of the al- teration in the time of vibration. By a slight accidental dis- placement of the box, the edge of the small aperture through which the suspension thread passes, had been brought near the thread, though by no means into contact with it. However, some of the finer fibres of the silk must have been touched thereby, for when it was again made to pass quite through the centre of the aperture, the time of vibration was found almost identical with that formerly observed.” This is perhaps the place for some remarks on the movements themselves, which are here represented during six terms. In the three summer terms, (Plates VI. VII. and VIII.) not- withstanding all the great anomalies, the regular diurnal move- ment is clearly seen in the curves, ascending during the hours * Probably during the interval no determinations of the time of vibration had been made. 94 GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. after noon, and descending in the following hours of the fore- noon; there is, on the other hand, scarcely a trace of this in the three winter terms, Plates IV. V. and IX. All our experience shows that partial or even total obliteration of the regular move- ments by the irregular is a verycommon occurrence. In theyears 1834 and 1835 some terms occurred in which the regular course was not at all obscured by any considerable anomalies, although there was no want of smaller ones. But what renders the ano- malous oscillations so remarkable, is their extraordinary coinci- dence, generally even in the smaller instances, at different sta- tions; nay, commonly at all the stations, only in dissimilar pro- portions of magnitude. It is quite unnecessary to demonstrate this agreement in individual instances : a view of the representa- tions of the six terms will speak sufficiently for itself. We cannot at present decipher these enigmatical hierogly- phics of nature: we must first endeavour to procure from the most diversified sources, authentic, numerous, and minutely faith- ful copies, in the confident hope, that when these rich materials are accumulated, the key to their hidden meaning will not be long wanting. In the mean time I may be allowed to add a few remarks, which may assist in the formation of a more correct judgement concerning them. First, it must not be forgotten that these anomalies are but comparatively small modifications of some of the effects of the great terrestrial magnetic force; that we must distinguish between the force itself and these supervening alterations ; and that nothing in the present state of our knowledge obliges us to ascribe both to the same or to similar causes. There- fore those who think it probable that these anomalies are the effects of electric currents, or of action, perhaps far beyond our atmosphere, (which view we leave entirely to its own merits) may continue to do so, without having to relinquish on that account the old view, of a force, residing in the solid portions of the earth, or rather being the collective action of all its mag- netized particles. If, according to the opinions of some phi- losophers, the interior of the earth be supposed still in a fluid state, the constantly advancing solidification, and the conse- quent thickening of the solid crust of the earth, would offer the most natural explanation of the secular variations of the mag- netic force. But we willingly leave the uncertain ground of hypothesis, : GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. 95 and return to facts. By far the greater number of the ano- malies are found to be smaller at the southern stations, and larger at the northern. For instance, the remarkable ascent of the curve, on the 30th January, 1836, between 94 25™, and 9% 40™, amounted in Catania to 6! (reduced to parts of arc) ; in Milan to 12’; in Munich to 134’; in Leipzig to 16’; in Marburg to 20'; in Gottingen to 26’; and at the Hague to 29’. Some- thing, it is true, must be deducted from this inequality, due to the circumstance that, at the northern points (where the horizontal portion of the terrestrial magnetic force has a weaker intensity than at the more southern ones,) similar disturbing forces must produce greater effects; but the difference of the horizontal intensities at the Hague and Catania is very small in comparison with the inequalities observed; and it is there- fore certain that the energy of the disturbing force was weaker the further we follow its action towards the south. With all the uncertainty under which we labour with respect to the nature of such disturbing forces, we cannot doubt that they have some definite source in space; and, as we must necessarily suppose those which produced the above-mentioned phzenomena to have their seat to the north or to the north-west of the places of obser- vation, (without venturing to define more precisely from so few data,) the northern districts, as far as we may venture to draw any such conclusions from experiments which embrace but a comparatively small portion of the earth’s surface, appear to be the great focus, from whence proceed the greatest and most powerful actions. A closer inspection of the data hitherto collected leads us to recognise, in the different successive movements, considerable variations in respect to their proportional magnitudes at dif- ferent places, even when the similarity in other respects is un- equivocal: thus, for instance, at one place, the first of two move- ments, following one shortly after the other, is the largest; at another place the reverse happens. We are therefore compelled to admit that, on the same day, and in the same hour, various forces are contemporaneously in action, which are probably quite independent of one another, and have very different sources ; and the effects of these various forces are intermixed, in very dissi- milar proportions, at various places of observation, relatively to the position and distance of these latter; or these effects may pass one into the other, one beginning to act before the other has 96 GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. ceased. The disentanglement of the complications which thus occur in the phenomena at every individual station, will un- doubtedly prove very difficult ; nevertheless, we may confidently hope that these difficulties will not always remain insuperable, when the simultaneous observations shall be much more widely extended. It will be a triumph of science, should we at some future time succeed in arranging the manifold intricacies of the phznomena,—in separating the individual forces of which they are the compound result,—and in assigning the source and mea- sure of each. Now and then we find at some places a small change, without any apparent counterpart at any of the other sta- tions. Such occurrences ought not to be at once looked upon as evidences of local magnetic action. In so great a mass of numbers an error may sometimes take place. Cases have fre- quently occurred to us where a revision of the original observa- tions, when these were in’our hands, has shewn an error of cal- culation in the reduction, or an evidently accidental error in the writing. In other cases, in which we had received only an ex- tract of the observations, a reference to our correspondent has led to a similar conclusion. As, however, it is impracticable to discuss all such cases by correspondence, those observers who do not communicate the original observations are requested, when they discover such cases in the curves representing them (as, for instance, at Leipzig, on the-26th of November, 1836, for 64 15™ Géttingen mean time), to refer to the original re- gister. If errors are thus discovered, they can be corrected in a following number. Even when the original papers do not decidedly indicate any error, yet we cannot have perfect assu- rance with respect to cases which rest only on a single set of ob- servations : it may happen, even to a practised observer, to write down in the same set repeated erroneous decimals. By such a conjecture, (somewhat hazarded it is true,) the above-men- tioned number 11°69 would be reduced to 6°69, and thus corre- spond with the others. But supposing the case of such an insulated movement to be established beyond all doubt, it does not follow that it is to be considered as local in the most limited sense. As the source of every anomaly must have its seat somewhere, it may be that the disturbing force is in the neighbourhood of the station it- self. If feeble, its action may still be perceptible at that station, on account of its proximity, and may disappear (2. e. be no longer Lda oe pmpet antes GAUSS AND WEBER ON TERRESTRIAL MAGNETISM. 97 perceptible) at all the other places of observations, because they are too remote. It appears, therefore, at least for the present, that there is no reason for admitting among the anomalies other than quantitative differences. Connected with this, it may be very useful, in many cases, to have two or more stations situated within a moderate distance of each other. It would have been desirable, for example, to have had obser- vations during the September term of 1836, at Augsburg, where the simultaneous observations are now regularly made. We should, in that case, have been able to form a decided opinion on the subject of the movement at 2 10™, everywhere sensible indeed, but which, at Munich, appears to have been of re- markable magnitude. Note. In the original work the observations made at the different sta- tions in the several terms are printed in tables, and graphical representations of them are contained in six Plates. Much care has been taken to make the plates which are annexed to this translation faithful copies of the originals. It has not been thought necessary to republish the tables ——Enir. VOL, II, PART V. G 98 HEINRICH ROSE ON THE COMBINATIONS OF ArRTIcLE III. On the Combinations of Ammonia with Carbonic Acid. By Hernricu Ross, Professor of Chemistry in the University of Berlin.* [From Poggendorff’s Annalen, vol. xlvi., part 3.] AN accurate examination of the combinations of ammonia with carbonic acid appeared to me to be important in several respects. Since the ultimate component parts of these combinations are exactly the same as those of animal substances, it was reason- able to suppose that they might easily combine in other propor- tions, and form new or already known combinations. It also appeared to me of importance to become acquainted with the properties of the anhydrous carbonate of ammonia, so as to be able to compare it with the other anhydrous salts of ammonia, The examination, however, of these combinations has not af- forded such results as I had expected. Carbonic acid and am- monia seem to belong to the last combinations into which sub- stances containing oxygen, hydrogen, carbon, and nitrogen, become converted ; and if therefore such bodies produce during their decomposition, by means of increased temperature, carbo- nate of ammonia, it is because the atoms of the elements in the carbonate of ammonia produced, are united in such a manner as to form combinations which are less easily decomposed, and, as it were, more stable, than the combinations consisting of these simple bodies generally aret. Neither did I find that the anhy- drous carbonate of ammonia possessed any remarkable properties analogous to those by which the anhydrous is distinguished from the hydrous sulphate ofammonia. However, I discovered in my experiments on the combinations of ammonia with carbonic acid a fact which to me appeared worthy of attention; for, although these combinations are less decomposable than other bodies which consist of the same elements, yet carbonic acid and am- monia have little affinity for one another, and this is the rea- * Translated by Mr. William Francis. + Something similar occurs with grape sugar. A great number of organic substances are convertible by the action of very dilute acids into grape sugar, a substance scarcely decomposable, at least by weak acids, ore a AMMONIA WITH CARBONIC ACID. 99 son why they can both combine in the most varied proportions. The number of these combinations is in fact surprising. I have prepared several of them, the existence of which was previously unknown. It would, however, have been easy for me to have greatly increased their number by further examination, but I have contented myself with indicating the possibility of the ex- istence of a great number of such combinations, since their pre- paration and examination would occasion more trouble than the subject appeared worthy of. The reason of the great number of these combinations arises less from the weak affinity which carbonic acid has for ammonia, than from the circumstance that the various combinations have a great tendency to form double salts with each other. I have attempted to consider several salts which carbonic acid forms with ammonia as double salts combined in certain proportions, by which the number of the more simple combinations is limited. Hitherto we were acquainted with only the following com- binations of carbonic acid with ammonia in a solid state: 1. the anhydrous neutral carbonate of ammonia, NH® + @; 2. the sesquicarbonate of ammonia, 2 NH? + 3C + 2H, or rather the sesquicarbonate of the oxide of ammonium, 2 NH‘ + 3 G ; and 3. the bicarbonate of ammonia, NH? + 2C + 2 H, or NH‘ 426 4 HL a ~ With respect to the analysis of the combinations of the car- bonic acid with ammonia, the proportions of the ammonia and carbonic acid were determined directly, the water by the loss. The determination of the ammonia may be effected with the greatest accuracy. The carbonate of ammonia was placed in a vessel which could be closed with a stopper, and a mixture of equal parts of muriatic acid and alcohol added; after the com- plete disengagement of all the carbonic acid, the solution was diluted by the addition of very strong alcohol (90 — 95 p..c:). Upon this an excess of a solution of chloride of platina, and then zether to nearly the amount of one fourth the volume of the alcohol, was added. The ammonio-chloride of platina is quite insoluble in a mixture of strong alcohol and zther, and may be collected without loss. I let it completely settle at the bottom of the stoppered bottle for twelve hours, and washed it out witha mixture of alcohol and zther. After desiccation it was cautiously exposed to ignition in a platina crucible. The salt was placed with the filtering paper in the crucible, and not as is usually done G2 100 HEINRICH ROSE ON THE COMBINATIONS OF with other precipitates, which are to be heated, taken out of the filter; the crucible was then closely covered with the lid, and ex- posed for a long time to a moderate heat, which was gradually raised to redness. This was continued until all the muriate of am- monia had evaporated. The crucible was then left to cool, the lid taken partly off, and the coal of the paper reduced in the usual way to perfect ash. If this precaution is not taken, and if the salt is heated too much at first, some undecomposed salt and metallic platina may be mechanically carried away with the vapours of the muriate of ammonia. From the weight of the metallic platina the ammonia in the salt is deduced. The determination of the ammonia may be effected in this way with great accuracy, much more so indeed than that of the carbonic acid. The determination of the carbonic acid may be effected by two methods. The most accurate is to convert it into carbonate of barytes, and to determine from the weight of this salt that of the carbonic acid. The carbonate of ammonia was dissolved in cold water in a bottle, which could be closed air-tight, and a so- lution of chloride of barium was added to it; if the combination underexamination does not consist of the neutral carbonate of am- monia, some pure liquid ammonia is added, after which the flask ‘is closed, and left to stand at least for twelve hours, or longer. Care must be taken not to use too little water for the solution, and especially to test the ammoniacal fluid by a solution of the chloride of barium, to see whether it is free from carbonic acid. I have usually distilled the ammonia previous to the experiment over quick lime. The liquid, after the twelve hours’ repose, was then passed through a filter, during which communication with the air was avoided as much as possible, and boiling water being poured upon the carbonate of barytes, it was filtered with the exclusion of air. This was frequently, and for a long time, washed with boiling water, but not until the water that passed through was no longer rendered milky by sulphuric acid, the residue not being quite insoluble in water. Repeated trials can alone deter- mine when it is time to leave off the washing. The carbonate of barytes must not be filtered for some hours after precipitation, and until it has entirely settled. Ifit be filtered sooner, carbonate of barytes is deposited from the clear filtered liquid, even when no communication with the air has taken place. When dry, the carbonate of barytes is heated. There is no need to fear, that by burning the filter any of the carbonic acid AMMONIA WITH CARBONIC ACID. 101 of the carbonate of barytes is expelled ; the strongest heat that an alcohol lamp, with double current of air, is capable of producing, may be applied without occasioning any loss. A solution of chloride of calcium cannot be employed with the same advantage as one of chloride of barium for precipitating the carbonic acid. The carbonate of lime, it is true, does not form so bulky a precipitate as the carbonate of barytes; but a portion of the precipitate adheres so firmly to the sides of the vessel, that it is impossible to separate it completely by mechanical means. The heating of the carbonate of lime has also its disadvantages, as it then loses a portion of the carbonic acid. In determining the carbonic acid in the neutral combinations, it was precipitated by chloride of barium without any addition of ammonia. In this case also the whole must be left to stand for some time after precipitation before the carbonate of barytes is filtered. When the solution of the neutral carbonate of am- monia is very weak, no precipitate is produced for some time by the chloride of barium, which is characteristic of the neutral salt. The liquid separated from the carbonate of barytes is then satu- rated with ammonia, in order to see whether any small precipi- tate would follow, which in general was the case; it was occa- sioned by the impossibility of obtaining the carbonate of ammo- nia always perfectly neutral. This precipitate, although filtered perfectly without exposure to the air, was nevertheless always more considerable than it should have been, and the amount of carbonic acid in the salt thus appeared to be greater than it really was. The second method of determining the carbonic acid was by measuring it in the state of gas. A weighed quantity of the salt was decomposed in a graduated cylinder under mercury by means of muriatic acid, in which shortly previous to the ex- periment some carbonate of ammonia had been dissolved, in order to saturate it with carbonic acid. When the salt could only be employed in the form of powder, it was wrapped up in bibulous paper. This method, however, gave, even when all. circumstances had been most carefully taken into consideration, less accurate results than by means of the carbonate of barytes. In general I obtained somewhat less carbonic acid than I ought. As, however, it is more quickly and easily performed, I have chiefly made use of it to ascertain to what known combinations of ammonia and carbonic acid any salt might belong. 102 HEINRICH ROSE ON THE COMBINATIONS OF 1. The Neutral Anhydrous Carbonate of Ammonia. It is well known that the neutral carbonate of ammonia is ob- tained by mixing dry ammoniacal gas with carbonic acid gas, and that both gases combine slowly, and only, (whichsoever of the two may be present in excess) as Gay-Lussac* first discovered, in the proportion of one volume of carbonic acid gas to two of ammoniacal gas. The properties of the anhydrous carbonate of ammonia obtained in this way are nevertheless almost unknown. Dr. John Davy +, who last experimented on the combinations of ammonia with carbonic acid, confirmed the previous experiments of Gay-Lussac, without however subjecting the combination to a more accurate examination. He states that it possessed the property of being decomposed without effervescence by a neutral solution of chloride of calcium, and formed with it a neutral fluid. I have only repeated the experiments of Gay-Lussac with the intention of learning whether, with an excess of ammoniacal gas, the two gases combined in the proportions above mentioned. I conveyed the carbonic gas into an excess of ammoniacal gas, and obtained the following results : 1, 29°7 vol. carbonic acid gas, combined with 61 vol. of ammoniacal gas 2. 24:9 a : - : : : 49°75 3. 20°1 : : ; : 38°15 The small differences are easily explained, by what I have on another occasion mentioned respecting the mixture of two gases which combine to form a solid body{. The volume of the absorbed carbonic acid gas in the first experiment is evidently smaller on this account than it should be, because the carbonic acid gas was mixed with too great an excess of ammoniacal gas ; in the second this was less, and in the third experiment still less. Since the combination of ammonia with the carbonic acid gas is formed very slowly, no vapour is observed when a glass rod moistened with ammonia is held over a carbonated alkali, from which the carbonic acid is disengaged by sulphuric acid, as is always the case when volatile acids, such as muriatic acid, sul- phurous acid, nitric acid, acetic acid, &c. are disengaged from a fluid by sulphuric acid. * Mémoires de la Société d’ Arcueil, tom. ii. p. 211, + Edinburgh New Philosophical Journal, vol. xvi. p. 245, ¢ Poggendorft’s Annalen, vol. xlii. p. 417. AMMONIA WITH CARBONIC ACID. 103 To obtain the combination in great quantities, considerable portions of the dried gases were brought into contact, in large vessels, which had been filled with dry air. The combination adheres so firmly to the sides of the vessels, especially when they have been artificially cooled externally, that it is frequently possible to obtain it in no other way than by breaking them. It is only when no external refrigeration has been applied to the vessels that a portion of the combination can be obtained in a pulverulent state. I therefore subsequently caused the two dried gases to pass through several glass tubes, which were kept cool at their outer surface, in order to obtain larger quantities of the neutral salt. These tubes were then cut, and the salt depo- sited in them taken quickly out. On preparing the salt in this way, it was observed that on the combination of the two gases a very considerable increase in temperature takes place. If in the preparation of this combination the greatest care is not taken to avoid every trace of moisture, which, with great quantities, it is very frequently difficult to effect, small admix- tures of the hydrous combinations of ammonia occur with the anhydrous carbonate. 1:3425 gramme of the neutral anhydrous carbonate, dissolved in water, gave with a solution of chloride of barium 3:321 grm. of carbonate of barytes. The liquid filtered from it, treated with ammonia, gave a slight precipitate, the weight of which was not determined. 1°444 grm. of the combination, treated in the manner previously described with muriatic acid, alcohol, chloride of platina and zther, gave 3°461 grm. of heated metallic platina. The carbonate of barytes obtained corresponds to 55°45 per cent. carbonic acid in the combination, and the quantity of pla- tina to 41°69 per cent. ammonia. Ifwe consider that a very small quantity of the hydrous bicarbonate of ammonia was contained in the combination, as would seem from a precipitate, although an inconsiderable one, being produced in the solution precipi- tated by chloride of barium and filtered, the compound formed agrees with the calculated formula NH® + C, according to which 56°31 per cent. carbonic acid is combined in the salt with 43-69 per cent. ammonia. It results from this examination, that in the solution of the anhydrous carbonate of ammonia, its constituents can be quan- titatively separated by the same re-agents as in the case of the solutions of the hydrous combinations of ammonia with carbonic 104 HEINRICH ROSE ON THE COMBINATIONS OF acid. The anhydrous carbonate of ammonia is therefore differ- ently circumstanced in this respect from the anhydrous sulphate of ammonia, the constituent parts of which cannot be separated ‘by the same re-agents as those producing this effect with the corresponding hydrous salts. Neither does the anhydrous car- bonate of ammonia in solution differ in its action with all the other re-agents from the other salts of the carbonate of ammonia, only that (which arises from its composition) the carbonic acid of the anhydrous neutral salt is entirely precipitated by solutions of the chloride of barium, and of the chloride of calcium, whilst this takes place in the solution of the other known combinations of carbonic acid and ammonia only after the addition of ammonia. The anhydrous carbonate of ammonia is very easily soluble in water. In the solid state it smells like free ammonia. This is peculiar to all the combinations of carbonic acid with ammonia ; but the greater the quantity of carbonic acid they contain, the weaker is the ammoniacal odour. It is not perceptible at first in the recently prepared combinations with excess of carbonic acid, and not till they have been preserved in a vessel for some time unexposed to the air. In the combinations, with more car- bonic acid than contained in the neutral salt, this peculiarity may be ascribed to the circumstance that they do not volatilize undecomposed ; in the anhydrous neutral salt, however, this is not the case, for it may be sublimed without changing its com- position. The neutral carbonate of ammonia is exceedingly volatile, and probably the most so of all the combinations of ammonia with carbonic acid. If exposed to the air, it disappears entirely in a short time. When sublimed, a very powerful ammoniacal odour is diffused, which, however, entirely arises from the volatili- zation of the undecomposed salt. 0°569 grm. of the sublimed salt, treated with chloride of platina, gave, after heating the pla- tina salt obtained, 1°4656 grm. of metallic platina. 0°930 grm. of a second quantity of the sublimed salt gave, with a solution of chloride of barium, 2°267 grm. carbonate of barytes. Ammonia still producedaslight precipitate of 0-046 erm. carbonate of barytes in the filtered liquid, corresponding to 1°11 per cent. of carbonic acid. The quantity of the platina answers to 44°79 per cent. of ammonia, and that of the carbonate of barytes to 54°64, or, rather, 55°75 per cent. of carbonic acid, whence it evidently results that the salt had undergone no change from sublimation. AMMONIA WITH CARBONIC ACID. 105 Since the salt undergoes no change in its composition by sub- limation, and volatilizes at a low temperature, it was easy to de- termine the specific gravity of its vapour. This was performed according to the well-known method of Dumas*. ‘Two expe- riments gave the following results : Weight of | weight of edad) testis | Tae" | Temoae-| Seer] “atthe | there | Stet mubepheric siete melting. theair. |harometer. globe. neglected. calculated. Grammes, | Grammes. Lg eee cab. Cub. 1. | 62-408 | 62-099 |176-25c| 15°C. | 750° | 602-73 | 0-75" | o-9048 II.| 61-6765 | 61:383 |140 8°75 7531 597 6:5 08936 But one vol. carbonic acid = 1°52400 Two vols. ammonia. . = 1°18240 2°70640 The calculated specific gravity of a volume of the vapour of the neutral carbonate of ammonia is consequently 0°90213, which coincides particularly well with the result of the first experiment, which was performed with great accuracy, and it agrees also pretty fairly with the second. The gaseous constituents in the vapour of the carbonate of ammonia are consequently combined without condensation. For the first experiment a salt was taken, as it had been ob- tained by the method above mentioned. For the second expe- riment, on the contrary, a sublimed salt was employed. It is hence evident, that, as already proved by the analysis of the sub- limed salt, its composition does not undergo any change by being sublimed once or even twice. These experiments, however, disagree with those of M. Bi- neaut. who alleges that he had observed that the gaseous product obtained by exposing the salt to heat retains its gaseous property at a temperature which is lower than that at which it is formed. But his statement of the specific gravity of the vapour coincides with the results obtained by me, although he determined it in an * The calculation was made according to Poggendorff’s formula (Annalen, vol. xli. p. 449), having regard to the cireumstance, that the gas of carbonic acid gas is lighter than the atmospheric air, the P of the formula, consequently the entire last member (p. 453) was taken negatively. 4 Annales de Chimie et de Physique, vol. \xvii. p. 240. 106 HEINRICH ROSE ON THE COMBINATIONS OF apparently very uncertain way, by keeping the salt for a long time in contact with a measured volume of atmospheric air and leaving it to evaporate in it; he then treated the few cubic cen- timetres of the mixture alternately with dry oxalic acid and with potash, and thus obtained the volume of the ammonia and of the carbonic acid*. Although the solution of the anhydrous carbonate of ammonia does not act differently towards the re-agents from the hydrous combinations of ammonia with carbonic acid, yet the combina- tion, in its solid state, is distinguished on account of the absence of water, by its action upon several substances, from the sesqui- carbonate of ammonia. If dry muriatic gas is passed over the anhydrous carbonate of ammonia, no action is perceptible in the cold, even when the gas is left for a long time in contact with the salt. But if, during the passing, the ammoniacal carbonate is heated at one spot only for a moment, it decomposes there, and the heat gradually diffuses itself through the whole combination ; and when it ceases, there is no longer any carbonic acid combined with the ammonia, and it has changed, naturally, without any disengagement of water, into hydrochlorate of ammonia. The common sesquicar- bonate of ammonia is decomposed, even in the cold, by muriatic gas, with disengagement of heat. Water is formed by a slow disengagement of the carbonic acid at the upper surface of the glass sphere in which the mixture is contained. The anhydrous carbonate of ammonia is at first not at all af- fected by gaseous chlorine; after an action of several days only does it gradually change, without any formation of water, into muriate of ammonia, in which case carbonic acid and nitrogen gas must necessarily escape. No formation of chloride of nitro- gen takes place. The common sesquicarbonate of ammonia gra- dually changes, with perceptible disengagement of water, into the muriate of ammonia. If the salt is employed in pieces, they de- compose very slowly, and, when taken out of the apparatus, ef- fervesce with acids. Even in this case no production of chloride of nitrogen could be observed ; however, the experiment was not continued until the salt had completely decomposed. The ex- ternal portion of the salt, and the fine powder, had become per- fectly converted into muriate of ammonia, without having indi- * Annales de Chimie et de Physique, vol. Ixviii. p. 434. AMMONIA WITH CARBONIC ACID. 107 cated any trace of chloride of nitrogen. When the anhydrous carbonate of ammonia is treated with dry sulphurous acid, it assumes, even in the cold, a pale yellowish colour. If it is heated in an atmosphere of sulphurous acid gas, it changes en- tirely into an orange sublimate of anhydrous sulphite of am- monia. The solution acts with acids, solution of the nitrate of silver, and solution of chloride of mercury, &c., quite in the same way as the solution of the anhydrous sulphite of ammonia, which has been directly prepared from dry ammoniacal gas and dry sulphurous gas*. When the common sesquicarbonate is treated with dry sulphurous gas, no change is perceptible in the cold. But if the salt be slightly heated in the sulphurous gas, a yellow sublimate of anhydrous sulphite of ammonia is pro- duced on the first action of caloric; but if the heat is continued, a white sublimate of the usual hydrous sulphite of ammonia is formed. If the whole is left to cool, and then suddenly heated anew, the same phenomena occur, and this may be repeated three or four times in the same way. But at last only white sublimate of anhydrous sulphite of ammonia is apparent. This decomposition of the salt into anhydrous and hydrous sulphite of ammonia is very easily explained if we regard the sesquicar- bonate as being composed of anhydrous carbonate and of hydrous bicarbonate of ammonia. When the anhydrous carbonate of ammonia is treated in the cold with dry sulphuretted hydrogen gas, no effect is produced. On the application of heat sulphuret of ammonia is formed with- out any evolution of water. The sesquicarbonate is likewise not affected by sulphuretted hydrogen gas in the cold, and even when heated it changes with difficulty, and partially only, with production of water, into sulphuret of ammonia; the greatest portion of the salt, however, may be sublimed in sulphuretted hydrogen gas. An important difference between the anhydrous neutral and the hydrous sesquicarbonate of ammonia, is manifested in their respective actions with anhydrous sulphuric acid. When the vapours of this acid are passed over some powdered sesquicar- bonate, it is decomposed, even when kept cold by a refrigerating mixture, with effervescence and evolution of carbonic acid, and the common hydrous sulphate of ammonia is formed. The neu- * Poggendorft’s Annalen, vol. xxxiii. p. 235. 108 HEINRICH ROSE ON THE COMBINATIONS OF tral anhydrous carbonate of ammonia, on the contrary, loses, by the action of the vapour of the anhydrous sulphuric acid, its carbonic acid, without any effervescence, and is converted into anhydrous sulphate of ammonia. - The neutral carbonate of ammonia may be prepared from the common sesquicarbonate in various ways, but not in a dry state. There is no way of obtaining it crystallized from asolution. The solutions of all the combinations of ammonia with carbonic acid, which contain more carbonic acid than the neutral salt, lose, when heated, carbonic acid, and are converted into the neu- tral combination ; while the solution of this latter, evaporated at the common temperature, (in vacuum either over sulphuric acid or hydrate of potash,) loses ammonia, and changes into super- carbonates. When the solution of the sesqui- or bi-carbonate of ammonia is boiled for a short time, it acquires the property of being thrown down entirely by an excess of a solution of the chloride of ba- rium, or the chloride of calcium ; so that pure ammonia produces no precipitate in the liquid filtered from the carbonated earth, nor even an opalescence. In the solution, therefore, there is a neutral combination of carbonate of ammonia. If the boiling is continued, the salt volatilizes entirely from the solution. M. Hiinefeldt* has shown, that when solid sesquicarbonate is subjected to distillation along with alcohol, on the boiling of the alcohol the carbonic acid escapes as gas; a portion of the alcohol then passes over, upon which a sublimate of a solid salt volatilizes with the remainder of the alcohol, at first adhering to the neck of the retort, and finally passing into the receiver with the vapours of the alcohol: this salt is the neutral carbonate of ammonia. I have frequently repeated this experiment in various ways, and convinced myself of the correctness of the fact. If the sublimed salt is dissolved in water, the solution is com- pletely precipitated by a solution of the chloride of barium or the chloride of calcium in excess, and in such manner that no milkiness is produced by an addition of ammonia to the solution filtered from the carbonated earth. It is, however, impossible to dry the neutral salt moistened with alcohol without its changing in its composition and losing some ammonia. When I dried it as quickly as possible by * Journal fiir praktische Chemie, vol. vii. p. 25 AMMONIA WITH CARBONIC ACID. 109 means of bibulous paper, and then precipitated the solution of the dried salt by chloride of barium, I obtained from 1:042 grm. of the salt only 1°714 grm. of carbonate of barytes, which only answers to 36°87 per cent. carbonic acid in the salt. When, however, some ammonia was added to the filtered liquid, and the precipitate formed protected from the action of the air, I ob- tained 0°688 grm. carbonate of barytes, corresponding to 14°80 per cent. of carbonic acid in the salt. I then attempted to dry the neutral salt, by placing it immedi- ately in vacuo over sulphuric acid. The salt, it is true, became dry, but was no longer perfectly neutral, for its solution gave, after precipitation by the chloride of barium, a precipitate with ammonia. It is, nevertheless, the best method of drying the salt without its composition being considerably affected. When I attempted to desiccate the salt moistened by alcohol over a considerable quantity of the hydrate of potash in vacuo, it remained moist although I kept it for more than a week under the air pump. The hydrate of potash became, it is true, carbo- nate at its surface, but a great quantity of ammonia was evolved in the gaseous form during the pumping. The moist salt was then placed in a basin filled with fused chloride of calcium, and this again put imto a larger basin containing hydrate of potash, and the whole then quickly placed inavacuum. The chloride of calcium became covered with car- bonate of ammonia; the remaining portion of the salt was dry, but after desiccation was no longer neutral. I obtained a similar result when I employed quick lime in- stead of the hydrate of potash. When I brought this, as was the case with chloride of calcium, warm into the vacuum with the moist salt, the greater portion of it volatilized and deposited itself on the chloride of calcium; the small quantity of the salt remaining was not neutral. I then placed the moist salt with another combination of chlo- rine of easy solubility in alcohol in vacuo. I chose for this purpose pulverized bichloride of mercury. The carbonate of ammonia remained moist, but the chloride of mercury attracted some of it, and did not dissolve entirely in water, the solution being opalescent. I obtained a remarkable result when I placed ‘the moist salt under the air pump with quick lime and the ace- tate of lead. The carbonate of ammonia volatilized sooner than the alcohol with which it was moistened; it combined with the ace- 110 HEINRICH ROSE ON THE COMBINATIONS OF tate of lead, forming a tumid, white pasty mass, which effervesced with acids, and dissolved in water, leaving carbonate of lead be- hind. The alcohol was left in the fluid state, and contained some, although very little, ammonia. For this, and most of the other experiments, the sesquicarbonate was distilled with anhy- drous alcohol. It appears to result from these experiments that the carbonate of ammonia combines with some salts, and that it has towards these, even when they are soluble in alcohol, a greater affinity than alcohol towards them. However, this affinity does not seem to be very considerable, and probably occurs only under peculiar circumstances, perhaps not without the presence of a trace of water or alcohol, or at the common pressure of the atmosphere. For when I placed some anhydrous fused chloride of calcium, and some fused acetate of soda, in bottles which con- tained anhydrous neutral carbonate of ammonia, which had been prepared from a mixture of the carbonic and the ammoniacal gases, none of it was absorbed by the fused salts, not even when they had been moistened with some alcohol or water. The same is the case with fused chloride of calcium, which absorbs none of the usual sesquicarbonate of ammonia, when both are placed together in vessels. If, therefore, under certain conditions, the carbonate of ammonia appears to combine with some salts, this affinity cannot be compared to that which pure ammonia exhibits towards a great number of salts. The experiments above mentioned, of drying the neutral car- bonate of ammonia moistened with alcohol, were modified in various ways, but I never succeeded in obtaining a dry, unde- composed salt. The result was either that the salt remained moist or volatilized previous to desiccation, or that when it did become dry the salt was no longer neutral. If the sesquicarbonate is distilled in a similar manner with ther, the phenomena are nearly the same: a considerable evo- lution of carbonic acid gas takes place during the distillation of the ether, but a far smaller quantity of carbonate of ammonia escapes with the ether than with the alcohol. The sublimed mass is the same neutral salt as that obtained with alcohol, and like it cannot be obtained pure in a dry state. The only method by which I succeeded in obtaining a dry neutral carbonate of ammonia, besides that of preparing it from a mixture of carbonic acid gas with ammoniacal gas, AMMONIA WITH CARBONIC ACID. 111 was by the sublimation of a mixture of anhydrous sulphate of ammonia and carbonate of soda. If every trace of moisture is avoided, a product is obtained as pure as by the mixture of the gases. The impossibility of combining the anhydrous neutral carbo- nate in any way with the quantity of water which is requisite to convert the ammonia into the oxide of ammonium is in so far a very remarkable circumstance, as the carbonate of ammonia dissolved in water exhibits quite the identical properties which the carbonate of the oxide of ammonium would present, and, moreover, does not differ essentially in its other relations from other combinations of carbonic acid with ammonia, in which the latter may be regarded as the oxide of ammonium. Berze- lius’s view of considering the ammoniacal salts, on account of their water, as salts of the oxide of ammonium, is so plausible, and has justly been adopted by so many chemists, that the composition and properties of the anhydrous carbonate of am- monia do not suffice to render this view less probable. It must, therefore, be regarded as a body of a peculiar kind, belonging, with respect to its composition, to a class with the anhydrous combinations of ammonia with sulphuric acid and sulphurous acid, which latter, however, essentially differ in their properties from the carbonate of ammonia, in so far as these ammoniacal salts vary considerably in their action upon re-agents from the corresponding salts of the oxide of ammonium, and indicate in the most evident manner the distinction between combinations of ammonia and those of the oxide of ammonium. The most important distinction which exists between the anhydrous neu- tral carbonate and the hydrous combinations of ammonia with carbonic acid, which contain more carbonic acid, is that the former may be sublimed undecomposed, which is not the case with the latter. It must be here mentioned that I have also prepared some anhydrous combinations of ammonia with oxy-acids, which, dissolved in water, did not differ, in their properties, from their corresponding salts of the oxide of ammonium, and in this respect are analogous to the carbonate of ammonia. Il. The Neutral Hydrous Carbonate of Ammonia. The experiments mentioned in the preceding section show that it is not possible to combine the neutral anhydrous carbonate 112 HEINRICH ROSE ON THE COMBINATIONS OF of ammonia with the quantity of water which exactly suffices to change the ammonia into the oxide of ammonium. I was much surprised at obtaining a hydrous neutral carbo- nate of ammonia in an unexpected manner. For if the sesqui- carbonate of ammonia of commerce is exposed in a retort to a very gentle heat, and if the neck of the retort is connected with a longish glass tube, the other end of which is immersed in mer- cury, a disengagement of pure carbonic acid gas is first perceived, and in that part of the glass tube furthest from the heated retort a crystalline salt is deposited, the solution of which, in water, is so entirely precipitated by a solution of the chloride of barium, or the chloride of calcium, that ammonia produces no opacity, or at least only a very slight one, in the liquid separated from the carbonate of the earth. This salt is the most volatile of the solid products, which are produced during the distillation of the ses- quicarbonate; if a gentle heat is applied for some time to the retort, the salt melts, and other combinations are formed and sublimed, which will subsequently come under our notice. If the sesquicarbonate is exposed to a stronger heat, but little of the neutral salt is produced. It is therefore necessary to ap- ply a very gentle heat, and only to employ for examination the products which are deposited in the part furthest from the heated portion of the retort. When this is not carefully attended to, a mixture of other combinations is obtained. A mixture of sal-ammoniac and carbonate of soda gives, when exposed to heat under similar circumstances, the same salt. With this distillation, at first only ammoniacal gas escapes, as will be subsequently shown. 1:609 erm. of the sublimate, treated after having been dis- solved with the chloride of barium, gave 3596 grm. of carbonate of barytes: and 0°860 grm. of the sublimate, prepared in the same way, gave, when treated in the manner above mentioned, with alcohol, «ther, muriatic acid, and chloride of platina, 1:942 germ. of metallic platina. This corresponds to the fol- lowing composition : Carbonic acti eles os S09 Ammoniae ee wertecis ee Yo ee DT W ate? Hite Wesaihiyl ravilen =) 2": LORS 100°00 The composition of this salt is yery remarkable ; only half the AMMONIA WITH CARBONIC ACID. 113 quantity of water necessary to convert the ammonia into the oxide of ammonium is present. A composition calculated ac- cording to the chemical formula C + NH?* + 2 H, gives in the hundred, Garbonicacid iw 2b. o Uso. 50°52 AMTTOMIAKEe RL... UA tue oO oe2O WERE rma? st ce oe EP LO:28 100°00 On repeating the experiment I obtained from 1°420 grm., 3°288 erm. of platina, and from 0°390 grm., 0°837 grm. of car- bonate of barytes; and after an addition of ammonia, also 0:059 grm. This answers to the following composition : @Wanbonic- acid. 26 2s Se bg AITO Ia tee hy ea eee ee | 426 Water ptt BEML BE 14) Sy dt REAM «C0 100°00 It sometimes happens that it is difficult to obtain the salt per- fectly pure. That its solution is not entirely precipitated by a solution of chloride of barium, but that, after the precipitation, a slight precipitate is still produced by ammonia, is almost al- ways the case even with the solution of the anhydrous neutral salt. The hydrous neutral carbonate of ammonia can, without changing very essentially in its composition, be again sublimed. 1°552 grm. of the twice sublimed salt gave, treated in the man- ner above mentioned, 3°692 grm. of metallic platina; and 0°446 grm. by means of chloride of barium, 1°009 grm. of carbonate of barytes ; a precipitate of 0°077 grm. was nevertheless produced by ammonia. This answers to 41°37 per cent. ammonia, 50°71 per cent. carbonic acid, and also 3°87 per cent. carbonic acid in the precipitate caused by ammonia. We see clearly, that this salt, by the double sublimation, had changed in a small degree into a combination containing more carbonic acid, though it remained doubtful whether this was in consequence of the renewed action of heat, or on account of the attraction of moisture. I then sublimed the first sublimate which had been obtained from two pounds of the sesquicarbonate, not less than five times, in order to see whether, by this means, it might entirely lose its water, and change into an anhydrous salt. The renewed sublimations VOL. II, PART V. H 114 HEINRICH ROSE ON THE COMBINATIONS OF ? were effected in such manner that only the most volatile sublimate of each operation was employed for the following sublimation : 0:619 grm. of the obtained product gave 1°441 grm. of metallic platina, and from 0°552 grm. 1:288 grm. of carbonate of barytes were obtained by the chloride of barium ; the liquid filtered from it gave, with ammonia, 0°068 grm. more. This corresponds to 40°48 per cent. ammonia, and 52°30 per cent. carbonic acid; and the last precipitate obtained 2°76 per cent. carbonic acid. The salt then does not become anhydrous by frequent sublimation. If we admit that the composition first obtained is the correct one, and that the other salts contained a slight mixture of a combination, with a larger proportion of carbonic acid, then in fact this composition must appear a very remarkable one, for it is not favourable to the ingenious hypothesis proposed by Berze- lius, that ammonia is changed into the oxide of ammonium by the reception of 1 atom of water, and is thus converted into a base. I shall, however, subsequently endeavour to show that the neu- tral anhydrous carbonate of ammonia has great tendency to form double salts, especially with the bicarbonate of the oxide of ammonium. This tendency it appears to evince also towards the simple carbonate of the oxide of ammonium, which does not seem to exist independently in a solid state. The most probable view which we may therefore take of the composition of the neutral hydrous carbonate of ammonia is, that we should look upon it as a combination of the carbonate of ammonia with the carbonate of the oxide of ammonium, (G + NH’) + (C NH%). If the anhydrous neutral salt, obtained by the mixture of the two gases, is not well preserved and protected from moisture, it appears to change into the hydrous neutral combination. On analysing such a salt, which had been sublimed, I obtained from 1°259 grm., 2°929 grm. of metallic platina, and from 0°784 grm. 1°844 of carbonate of barytes. This answers to 40°46 per cent. ammonia, and 52°72 per cent. carbonic acid. It is surprising that the formation of the neutral carbonate of ammonia, during the distillation of the common sesquicarbonate, or a mixture of sal-ammoniac and dry carbonate of soda, has escaped the attention of chemists. I must, however, remark, that John Davy mentions in his paper* that his brother had obtained, on exposing the sesquicarbonate to heat, a salt which * Edinburgh New Philosophical Journal, vol. xvi. p. 257. AMMONIA WITH CARBONIC ACID. 115 possessed a decided ammoniacal odour, deliquesced when ex- posed to the air, and, as he believed, contained more ammonia than the known combinations. John Davy confirmed this ex- periment, and adds that it is more volatile than the last, and that probably it is hydrous carbonate of ammonia. I did not find that the hydrous neutral salt deliquesced in the air ; but the salt, it is true, becomes moist, and remains so if the distillation is continued for any length of time, and water passes over. III. The Sesquicarbonate of Ammonia. This is the salt which occurs in commerce. I have analysed it several times, and found that, if it had not effloresced at its surface from the action of the atmosphere, and had not changed into the bicarbonate, it generally had, but not always, the compo- sition which R. Phillips has assigned to it. The analyses were performed with quantities which had been obtained from various manufactories. 2-143 orm. of salt gave 3°530 grm. of metallic platina; 1:113 grm., however, of another quantity, 1-965 grm. of platina. The first quantity answers to 28°66 per cent., and the last to 30°70 per cent. of ammonia. The quantities of carbonic acid, which were determined in the gaseous form by means of muriatic acid over mercury, varied quite as much. 0°607 grm. gave 155 cub. centim.; 1°480 grm., 399°44 cub. centim.; and 1°419 grm. of the salt, 403 cub. centim. carbonic acid gas. This answers to 50, 55, 53, 40, and 56°23 per cent. carbonic acid in the salt. These differences are explained by the modes of preparing the salt. When it has been prepared directly by sublimation from carbonate of lime and sal-ammoniac, or from sulphate of ammonia, then it is sesquicarbonate of ammonia. When, however, it has been once more sublimed in the manufactory, probably in order to purify it, it has changed into $-carbonate of ammonia, of which we shall speak hereafter. The calculated composition of the sesquicarbonate, according to the formula 3 CG +2NH? + 2H, is Ammonia! 3) 2 219 Jo 109 28°92 Carbonic-acid sv 2) 2" 2) 55°91 Waters Oy) sat. ogrreaed 6°17 Se 100°00 116 HEINRICH ROSE ON THE COMBINATIONS OF We tind that there is sometimes in commerce a salt that con- tains about 31 per cent. ammonia, 51 per cent. carbonic acid. This is 2 of carbonate of ammonia. The composition of the sesquicarbonate of ammonia is such that it may be conceived as a combination of anhydrous neutral salt, and hydrous bicarbonate of the oxide of ammonium (C + NH°) + (2C + NH‘ + H); or if it is thought that the anhy- drous neutral salt cannot exist in combination with hydrous salts of the oxide of ammonium, we might consider the for- mula (© + NH‘) + (2 C + NH‘) to be the more correct. Perhaps the preference might be given to the first formula, partly because the bicarbonate of the oxide of ammonium can- not be prepared alone, but is mixed with water, and at least with 1 atom of water; and partly because, as will be shown here- after, anhydrous neutral carbonate of ammonia is volatilized when exposed to the air from the sesquicarbonate, and leaves behind hydrous bicarbonate of the oxide of ammonium. This view is, in a great measure, confirmed by some recent experiments of Scanlan, and some earlier ones of Dalton*. They found that if the sesquicarbonate of ammonia is treated at the usual temperature for several times with less water than is ne- cessary to dissolve it completely, the first saturated solutions had a greater specific weight than the last. In the same degree that the specific gravity of the solutions decreased, they lost their ammoniacal odour ; the last solution gave crystals of the bicar- bonate. They hence concluded, that either the sesquicarbonate is a mixture of two salts, or that the water exerts an action upon | the salt similar to that it is usually imagined to have on some salts of bismuth, and that it decomposes it into two salts of two dissimilar degrees of saturation. Should, however, the last action take place, the salt of more difficult solution would remain in the form of a powder, which is not the case, for it is left as a skeleton. The crystalline structure of the salt evidently shows that it is not a mere mixture, but is composed according to fixed propor- tions, which is also confirmed by analysis. But the experiments above mentioned prove that it is a double salt composed of 1 atom of neutral, and 1 atom of the bicarbonate of ammonia, both which constituents may be separated by water, according to their solu- bility in it. _ This separation, from the two salts being perfectly *The Atheneum, 1838, No. 565, p. 596. AMMONIA WITH CARBONIC ACID. 117 soluble, never more than approximates. When I, in the manner already mentioned, poured a little water upon the sesquicarbonate, I could not manage to obtain pure carbonate without a small mixture of dissolved bicarbonate ; for, as I precipitated the solu- tion with a solution of the chloride of barium, the filtrated liquid was rendered opalescent by ammonia. The affinity between the two constituents in a double salt varies. The carbonate of ammonia is combined so feebly with the bicarbonate in the sesquicarbonate of ammonia, that water alone may cause a separation of both constituents. We find something similar in several double salts which are composed of a salt difficult, and of one easy of solution. Of the Bro- gniarti (Glauberit), a crystalline double salt of sulphate of lime and of sulphate of soda, the latter dissolves in water and leaves the sulphate of lime undissolved. In the same manner, according to Stromeyer, sulphate of potash and sulphate of mag- nesia is dissolved from the polyhallit of Ischl, by water, whilst sul- phate of lime is left. According to Bauer, from the artificially prepared combination of carbonate of potash and carbonate of lime, water dissolves the first salt and leaves the last undis- solved*; whilst, according to Boussingault, the Gaylussite, oc- curring in nature, which is similarly composed, withstands the action of the water, and is only easily decomposed by it when it has lost its water by being heated}. Most of the other double salts, likewise composed of a salt easy and of one difficult of solution, are not at all decomposed by water. Common alum dissolves equally in water, without the readily soluble sulphate of alumina being separated by it from the sulphate of potash, which is of more difficult solution. The bisulphate of potash, which must be considered as a double salt, consisting of sulphate of potash and hydrate of sulphuric acid, acts in a similar way towards water; it also dissolves in water without decomposition. But between the two examples of double salts there is this difference, that, from the last salt alco- hol separates the insoluble sulphate of potash, and dissolves the hydrate of sulphuric acid, whilst the alum resists the decompo- sition by aqueous alcohol, though the sulphate of alumina is soluble, and the sulphate of potash insoluble, in it. The carbonate in the sesquicarbonate of the ammonia can * Poggendorff’s Annalen, vol. xxiv. p. 367. t Ibid. vol. vii. p. 99. 118 HEINRICH ROSE ON THE COMBINATIONS OF also be separated from the bicarbonate, not only by water, but also by being preserved in vessels from which the air is not en- tirely excluded. The more volatile carbonate gradually disap- pears entirely, and the less volatile bicarbonate is left quite free from the carbonate. This succeeds especially well if the sesqui- carbonate is employed pulverized in the way above mentioned, and if the atmosphere in which the vessel is situated be not too moist. The remaining bicarbonate of the oxide of ammonium contains 1 atom of water; the volatile carbonate of ammonia is consequently anhydrous, and contains no oxide of ammonium, on which account, as remarked above, the latter can hardly be con- sidered to exist in the common sesquicarbonate. The double salts, which the carbonate of ammonia forms with the bicarbonate, are, however, in so far of an uncommon kind, that whereas in general the simple salts which form the constituents in other double salts are of one and the same degree of satura- tion, this is not the case here. We must, however, certainly distinguish two kinds of double salts. In the double salts of one kind, which form the majority, the simple salts are of the same degree of saturation ; in them, generally half, or another defi- nite portion of one base is replaced by an equivalent of another base, and the one salt consequently cannot act in them the part of an acid or a base towards the other, which was formerly the view taken with regard to the composition of these combinations. In the second kind of the double salts, on the contrary, both the combinations of which they consist are not of the same degree of saturation ; in these double combinations one constituent part may be considered as the acid, the other as the base. Certain combinations of carbonic acid, of silicic acid, and of other weak acids with bases, belong to this class; and also the property of boracic acid to dissolve, when melted, all substances of acid and basic properties, depends on the tendency to form double salts of this second class. In the combinations of the carbonate and of the bicarbonate of ammonia, which also belong to this class of double combina- tions, the carbonate is naturally the base, and the bicarbonate the part which replaces the acid. The tendency which the carbonate has to form a double salt with the bicarbonate, when sal-ammo- niac or sulphate of ammonia is subjected with the carbonate of 1 me oradry carbonated alkali to distillation, rests in part on this circumstance; that the carbonate of the oxide ofammonium, C + AMMONIA WITH CARBONIC ACID. 119 NH+4, which ought here to be formed, does not seem to exist in a solid state of itself, as has already been remarked. On this account, at the beginning of the heating, ammonia is disengaged, and this escapes, in common, with so much water as would be ne- cessary to convert it into the oxide of ammonium, whilst the ses- quicarbonate of ammonia is formed. From 3 atoms of carbonate of oxide of ammonium, which ought to evolve from the mixttre when heated, 1 atom of carbonate of ammonia is formed, and 1 atom of hydrous bicarbonate, which two form the double salt, and it disengages 1 atom of ammonia and 1 of water. 3 C + 3NH? + 3H =(C + NH®) + (2C + NH* + H) + NH? +H. Ifthe products of this operation are received in the order in which they are produced, over mercury, pure ammoniacal gas is first obtained, which is wholly absorbed by muriatic acid; and af- terwards come the products, which appear during the sublima- tion of the common sesquicarbonate, of which we shall speak further on. As the sesquicarbonate can be evaporated only with the disengagement of carbonic acid gas, this gas is found amongst the products of the sublimation ; there is, however a de- finite interval between the disengagement of the ammoniacal gas and of the carbonic acid gas. The latter first begins to escapewhen the evolution of the ammonia has entirely ceased, and when the glass cylinder, in which the gaseous products are received, begins to be covered with a thin incrustation of the carbonate of am- monia, and at the same time water passes over. When all the gaseous products are received together in one glass cylinder, over mercury, the ammoniacal gas which first goes over gradually combines with the carbonic acid gas which subsequently passes over. IV. Sesguicarbonate of Ammonia with a larger proportion of Water. If the common sesquicarbonate is exposed for some time to a very gentle heat, in a retort, the neck of which is connected with a long glass tube, the following appearances occur : at the very beginning carbonic acid gas is disengaged, and then the hydrous neutral carbonate of ammonia sublimes, which, as the most volatile of the solid products of sublimation, consolidates in that part of the glass tube furthest from the retort. The nearer to the retort the sublimate adheres, the more the solution is precipitated 120 HEINRICH ROSE ON THE COMBINATIONS OF by ammonia, after having been treated with chloride of barium, and the precipitated mass filtered. The salt in the retort continually becomes moister, whilst the sublimate in the neck of the retort increases, and begins to be deposited in the body of the retort. At last a clear liquid only is left in the retort, from which, when the heat is over, a salt crystallizes, in the form of tables, in great quantity. The bulb of the retort must be broken, in order that the crystals may be well separated and obtained pure from the original mass. If the mass is preserved for a long time in closed vessels, a quantity of tables of the same salt is deposited from it, of more beautiful and distinct crystalline structure. This deposition of crystals continues for some weeks. When it ceases, the mass contains only neutral carbonate of ammonia in solution ; by means of a solution of chloride of barium it is thrown down so com- pletely, that ammonia produces no precipitate in the filtered li- quid. The salt sublimed in the neck and in the body of the retort, as well as that crystallized from the solution, are two combina- tions hitherto unknown. This sublimed salt will be treated of in the following section. The crystals of the salt from the solution have the form of thin six-sided plates. On account of their thinness and rapid efflorescence the angles could not be measured. No cleavage could be observed. Since this salt may be obtained in distinct crystals, it is con- sequently free from foreign mixtures; and the various analyses agree better with one another than is the case with those of the sublimed and non-crystalline combinations of carbonic acid with ammonia, and are more in unison with the calculated result. 1°904 germ. gave 2°594 erm. of metallic platina ; 1°816 grm., witha solution of chloride of barium, treated with an addition of am- monia, gave 3°674 grm. of carbonate of barytes. This corresponds to the following composition : IATMINOUIAN Sf » € 2 158194 » ie A 111 472 » a ] 3 101 56* The planes are in general very smooth and bright, and are well adapted for accurate measurements, the plane ¢ alone is generally somewhat rounded. The crystals are perfectly cleavable, parallel to the planes of the vertical prism g ; no other surfaces of cleavage besides this were observed. The planes of the vertical prism are, in the various deposits, sometimes large, sometimes small; fre- quently so small that the planes of the superior and inferior ex- tremity are in contact. These crystals have undoubtedly been obtained before now ; I produced them many years ago of considerable size, from a solution of a great quantity of the sesquicarbonate, for which * Only the inclinations of g to g=112° 9!, and g to d=115° 5’, are the direct results of measurement ; the other angles mentioned are calculated from these, but these also were measured, and the measured and calculated angles were mostly found to differ only by a few minutes. AMMONIA WITH CARBONIC ACID. 133 hot water was employed. It seems to me that they have always been looked on as the common bicarbonate, from which they differ, not only by the form of the crystals, but also, as will be seen immediately, in their composition. It is, however, also possible that the bicarbonate, with 2 atoms of water, which was treated of in the preceding chapter, may never have been ob- tained crystallized ; since, notwithstanding all my endeavours, I only succeeded once in producing distinct crystals, at other times only powder and crusts ; and, that all the crystallized bi- carbonate formerly obtained contained more water. The imper- fect descriptions of the crystals given by Schrader* and others seems at least to render this view probable, although, on the other hand, all the analyses with which I am acquainted, give 55 to 56 per cent. carbonic acid in the bicarbonate, which agrees with the salt described in the preceding chapter. 2-722 grm. of the salt gave 3179 grm. metallic platina, and 3°406 grm. of the same preparation, treated with muriatic acid, 890°66 cub. centim. carbonic acid gas. 1-802 grm. of the salt produced 2°043 grm. of metallic platina, and 1-116 grm., with a solution of the chloride of barium and ammonia, 2°630 grm. of carbonate of barytes. 0°764 germ. of the same salt afforded, with muriatic acid, 185°11 cub. centim. car- bonic acid gas. These answer to the following compositions : iZ II. Ammonia ...::).., 20°31 19:72 Carbonic acid . 52°82 52°96 — 52°05 BECP. is ks tend HO OL ag hs 100°00 100:00 which corresponds to the chemical formula 4G + 2 N NH? + 5 H, or rather 4 C + 2 NH* + 3 H; calculated according to this, i it would be in the hundred, PAM IMONIA ie a)» hiaint oi necwateh, cto 20°45 BASB ROIS SIM. oi oo bo ey wat MOET RM a th. Bites haiti te ni | BORE 100°00 The salt therefore differs from the other bicarbonate only by its containing one half of an atom more water. * Neues allgem, Journal der Chemie, vol. ii. p. 582. 134 HEINRICH ROSE ON THE COMBINATIONS OF X. Bicarbonate of Ammonia, with greatest quantity of Water. By the distillation of the } carbonate of ammonia, with 5 atoms of water, which was prepared by distilling the 4 car- bonate, with 4 atoms of water, I obtained a sublimate, which, on examination, proved to be a bicarbonate, with a still larger portion of water. 1°351 grm. of it gave 1'408 grm. of metallic platina ; 0°681 grm., with muriatic acid, 174°3 cub. centim., and 0°512; in a second experiment 131°55 cub. centim. carbonic acid. This corresponds to the following composition: _ JAMMONIA:2-0.0. tebe eee Carbonic acid . . . 50°67—50°86 Watert. Soka eoee 100°00 which would be represented by 2.C + NH? + 3 H, or rather 2C + NH*+2 Hi; and the salt, calculated according to this, would be composed of Ammonia’) °c ee os eon Carbonie acid) 2) 8s 22°95 2 eo0s Waters <0 a ee ae Ore 100°00 As this salt was procured only in small quantity, its preparation should, by right, have been repeated in larger quantities. I have, however, thought it better to mention these experiments in this place, to show that the bicarbonate, like all the other com- binations of carbonic acid and ammonia, is capable of combi- ning with very different quantities of water. XI. Seven-four Carbonate of Ammonia. This salt was obtained by distilling the bicarbonate with a greater quantity of water, 4C + 2 NH‘ + 3H, during which process a disengagement of carbonic acid gas took place, and, in fact, phenomena similar to those which occurred with the sub- limation of the sesquicarbonate. 0-923 grm. of the sublimed salt gave 1°030 grm. metallic platina; 0°602 grm. produced 148°7 cub. centim., and 0°545 grm., 131°32 grm. of carbonic acid gas, giving the following composition : IATATMONIANS oh vse oci.t Satta sed ee Carbonie acidy,..... . . . ARR = 46°90 Vere PRET uel ih Ah SBBBO 100°00 AMMONIA WITH CARBONIC ACID. 135 which correspond best with the chemical formula 7 © + 4 NH® + 12H, or rather, 7C + 4 NH?! +8 H. When the composi- tion is calculated according to o this we obtain in the hundred, PASI 2 5. ne oy oye. HOE Wamppnic acid, ., «. .. sien of 4ogt EMER SG) sts, a) ae es pes GA eee 100°00 The preparation of this salt, like that of the bicarbonate with the greatest quantity of water, should be repeated ; for the experi- ments with these two salts were the last which I made on the combinations of ammonia with carbonic acid, and, moreover, at a time when I had already resolved to discontinue these examina- tions. : I have, however, made especial mention of this salt, be- cause, if it be regarded as a double salt, in which the anhy- drous neutral carbonate forms the one constituent, we are then compelled to consider the other constituent as a combination of 4 atoms of carbonic acid, and 1 atom of ammonia or the oxide of ammonium,—a combination which has never yet been pre- pared isolated. According to this view the composition of the salt would be expressed by the formula 3 (C + NH?) + (40 + NH? + 12H), or rather, 3 (C + NH?*) + (4€ + NH* + 11H), or 3(C + NH*) + (4C + NH* + 8H). XII. Nine-four Carbonate of Ammonia. When a solution of the common sesquicarbonate is evaporated over sulphuric acid in vacuo, and too much pumping is avoided, so that the solution may not boil, small crystals are obtained, which, immediately on their formation, must be withdrawn from the influence of the sulphuric acid, otherwise they effloresce and pass into the common bicarbonate. 1507 grm. of these crystals gave 1°657 of platina, and 1°176 grm. of crystals from the same preparation gave, treated with chloride of barium and ammonia, 2°929 grm. of carbonate of ba- rytes, which gives the ares composition : Ammonia. . . EIS: hs A EIQ e'D Carbonic acid . .. . . . 5583 WUAtER Sab Se A as ip A er BO5 100°00 136 HEINRICH ROSE ON THE COMBINATIONS OF This answers to the chemical formula 9 C + 4 NH* + 10 H MH or, 9C + 4NH* + 6H. Calculated accordingly, the compo- sition would be, WANITIEIONIIA.© se ct onc" 0+ on weteitinoulee @arbomie acid. .. .-.- «+! « oones WWiatercs cs co eres) tet Jee eee 100°00 0°866 grm. of beautiful crystals prepared on another occasion gave 0°920 grm. metallic platina, which corresponds to 18°47 per cent. ammonia in the salt. It is necessary to determine, in the analysis, the quantities of ammonia and carbonic acid from the same portion of crystals. I examined some transparent crystalline incrustation of a differ- ent portion, only for carbonic acid, as I obtained but a small quantity of them in a pure state: 0°664 grm. gave 192°29 cub. centim. carbonic acid gas by means of muriatic acid, which answers to 57°33 per cent. carbonic acid. Perhaps this crust might have been the ¢% carbonate of ammonia with a different quantity of water ; for if we only suppose 9 atoms of water, in- stead of 10, in the salt, which, in fact, is a more probable amount of water, then it would contain, Ammonia ©. 2 elo eee, Sees Carbonic. acid; << 5) » «Jats 57 oe Water... .c..) .20) CUA Enc eye 100°00 The 3 carbonate of ammonia is only produced by the ac- tion of sulphuric acid on a solution of the sesquicarbonate. Whilst it changes into the bicarbonate by the loss of neutral ‘carbonate of ammonia, the ammonia alone of this last is absorbed by the sulphuric acid, the surface of which becomes covered with a strong efflorescence of sulphate oxide of ammonium, and the carbonic acid gradually combines with the bicarbonate. If the pumping is performed rather quickly bicarbonate only is formed, under the above-mentioned circumstances, from the ses- quicarbonate, because the carbonic acid is then too rapidly re- moved. Also, on employing hydrate of potash, lime, or even chloride of calcium, instead of sulphuric acid, bicarbonate only is produced ; the chloride of calcium absorbs the carbonate of AMMONIA WITH CARBONIC ACID. 137 ammonia, and is not capable, even when employed in great quantity, of decomposing it, although pure ammonia combines easily, and in great quantity, with the chloride of calcium. The 2 carbonate of ammonia only feebly retains that portion of carbonic which it contains, more than is requisite to form the bicarbonate. It is only when the salt has separated in crystals that it resists speedy decomposition. But even in the solid state it loses the last portion of carbonic acid, and easily effloresces into the bicarbonate: on which account the crystals, when once formed, under the air-pump, must not be allowed to remain there. But the salt not only effloresces and passes into the bicarbonate, when exposed to the air, but even in the closed vessels in which it is intended to preserve it. The preparation of this salt, moreover, is very uncertain, and I have succeeded but very rarely. It seems to depend on the concentration of the solution, and on the evaporation under the air-pump, which must neither be too quick nor too slow. Ifa saturated solution of the sesquicarbonate is evaporated over sul- phuric acid, without being placed in vacuo, the whole of the carbon disappears with a slower evaporation of the water, and nothing is left behind. If we regard the % carbonate of ammonia as a double salt, we must admit the presence of the carbonate, together with the bicarbonate; and the quadricarbonate contained in the 3 car- bonate of ammonia, and the very compound formula would be (C + NH®) + (4€ +2NH*) + (4 C + NH*) + 7H, or 6 H, if 9 atoms of water are contained in the salt. It must be evident to every one that the number of combina- tions of carbonic acid with ammonia might easily have been in- creased had I continued further my experiments on this subject ; for the sublimation of the various kinds of bicarbonate, of +, and of the various kinds of the 3 carbonate of ammonia, would cer- tainly have produced new double salts, which may be imagined as formed of the carbonate, united with bicarbonate and the quadricarbonate. I thought it best, however, after having con- tented myself with indicating the possibility of the existence of this great number of double salts, to discontinue the examination. Among the combinations of carbonic acid with ammonia, it 138 COMBINATIONS OF AMMONIA WITH CARBONIC ACID. is only the carbonate which loses a portion of the carbonic acid in its solid state; the other combinations, on the contrary, undergo at the ordinary temperature quite a reverse decomposi- tion ; they have a tendency to evolve partly ammonia, partly an- hydrous carbonate, which have just the same odour as pure am- monia. When, therefore, the recently-prepared acid salts do not smell of ammonia, it nevertheless arises when they have been kept for some time in a well-closed vessel, if this be opened. Even the bicarbonate forms no exception to this ; and this perhaps seems to indicate that it has a tendency to pass into the $ car- bonate. The solution of the carbonates of ammonia have a ten- dency, at the common temperature, to smell of ammonia. With an increased temperature, the solutions of the carbo- nates, as well as they themselves, the carbonate naturally ex- cepted, evolve on the contrary carbonic acid. The solutions of all the carbonates are converted by boiling into neutral salt, and the solid salts lose a portion of the acid, and produce partly neutral, and partly less acid salts. I think it will be advantageous to enumerate, in a table at the end of this memoir, the chemical formule of the combinations examined according to the various views which may be enter- tained with regard to their composition, so that they may easily be compared one with another. H +HOS+ sHNOP H +HO + HINO | H@ + H +HOZ+ vHNOZ | He + HO =e sHINO H + uet+Hoe+ sHINOF wttt+ He+Ho + sHINOZ | HP + Ho +eHNOtVHNO [H + Hit+Ho + sHINOP | HIL+ HO os sHNOP HP + HO +cHNO++HNo8 | He + cHNO+tHNO “plow o1uoqaeg Jo ajerpAP 94} YIM pure ‘eruoMOLy joaqwuoqieg Yt ‘untuouUTYy JO APIXO A}BUOgILD [BAyNEN 943 JO HL + sHNO++HNcO@+tHNO vHNcO sHNz0@ +HNeO HINVO+ rHNcO+ pHIN,O+ rHNeO+ vHINcO+ vHNGO+ rHNO+ eo ee tHNOS sHNO HNO cHNOS cHNOE cHNOE eHNO *‘ayeuoqieolIpend pur ayeuoqivorge YIM fayeuogieD snoxpAyuy Jo sz[es a[{nop sv papavdar suoeUIQUIOD BUT | prow oruogae, oy, SUOT}LUIQUIOD SB papavdad 8}[Bg OUT, H9++sHNF+60 He+tHN +:0 He+sHNo++0 H ++HN +20 H8++HNb+29 He+rHNe+eo tHN@+e0 H8++HNt+s0 H +tHN?+s0 sHN+s0 vHNO+SHNO “pIuoWULy JO OPIXO 94} YIM poutq =u109 oq 0} pasoddns HOI+sHN}+60 Hg +cHN +30 Sic diet H@ +cHN +20 H2tt+eHN¢+20 HS +cHNZ+s0 HZ +sHNz2 +29 Hett+cHN¢+s0 HS +sHNP+s0 Hr +cHNP+s9 H +cHN@+c0 cHN +0 “BIUOMIUTY YITA\ pourq -u109 aq 0} pasoddns Ploy o1uogueD ay, seneeesereeseee a | BIMOUULY JO ayeuoqrey & | “IIX tee seeeeees JOIBAN JO JUNOWL 489}8913 ay} IM ‘eMoMUy jo syeUoqIvolg | “1X ssesacesessseeneeeserseergq@ MA JO FUNOUIE 1078913 YILM ‘eIUOWIUY JO ByeUOqIBOTg | *X seeeeeeeeeeeees BIUOULULY jo ayeuoqieorg ‘XI PTTTT TTT eIUOWULy jo oyvUuogiey f teeeoeverees JOIBAA JO JUNOW J0}vOIT WIA ‘euouury jo syeuoqreombseg sseeeeeeeeesprmourUTy Jo ayeuoqivombsag | “TA seseeeeeeess FOIBAA JO JUNOUIL 4S9}B91T ay YIM ‘euouUry Jo ayeuoqieg F | "A seesaesaesersesseeveseeezgq@ AA JO JUNOTTR Ja}vo13 YIM ‘eIUOMIUTY Jo ayBuog.tey) El say seece sesveeees BIMOUUY jo ayeuoqieg ft ‘TIT senceccensnecssccscceseussssoveseees BITIOUL -wy jo oyeuoqieg snompAH [eayNeN | “IT ceeccecoececenevsecsecesseoeoesseee® BITIOUL -ury Jo oyvuoqieg snorpAquy [eyneN | “T Nn ——————— 140 COMBINATIONS OF AMMONIA WITH CARBONIC ACID. I may be allowed to add a few additional remarks on the che- mical formule in the last column. If the various salts of car- bonic acid with ammonia be regarded as combinations of the carbonate oxide of ammonium, with carbonate of ammonia and the hydrate of carbonic acid, some of the salts will still contain superfluous water. This is the case with the 5th, 7th, 8th, 10th, 11th, and 12th; yet it is possible, as was previously noticed, that the latter salt may be prepared with 1 atom less of water, and then it would not belong to this section. I am inclined to con- sider this water as real water of crystallization ; I have, however, not performed any experiments to ascertain whether it might be removed without any change in the composition. With respect to the view, that carbonate oxide of ammonium is partly com- bined with carbonate of ammonia, partly with the hydrate of carbonic acid, and partly with both together in the salts de- scribed, this is founded on a hypothesis, upon which I lay but little stress, and which needs more confirmatory facts before it can be adopted. In a memoir communicated many years ago to the Annalen*, I endeavoured to show that in numerous salts ammonia acted quite the same part as the water of crystalliza- tion, and that it might, as it were, replace it. It may, therefore, be possible, that water, even when not existing as water of cry- stallization and ammonia, both combined with carbonic acid, might form bodies which might equally be replaced. This at least appears to be the case in those combinations which these bodies form with the carbonate oxide of ammonium. If we admit this view, several of the ammoniacal salts described would have, as was already noticed with respect to the hydrous neutral carbonate and the bicarbonate, an identical composition. From a subsequent communication from M. Bauer, the arti- ficially-prepared combination of carbonate of soda and carbonate of lime stand in the same relation to water as the Gay-Lussite which occurs in nature. * Poggendorff’s Annalen, vol. xx. p. 163. SCIENTIFIC MEMOIRS. VOL. II.—PART VI. ARTICLE IV. Memoir on the Polarization of Heat ; by MaceDoINE MELLoni. Second Part*. [From the Annales de Chimie et de Physique, vol. Ixv., May, 1837.] In the first part of this Memoir it was shown, that calorific rays, transmitted by a pair of tourmalines which completely polarize light, undergo every degree of polarization. Certain species of heat, in sensibly equal quantities, traverse the two plates, what- ever be the position, parallel or perpendicular, of their axes of crystallization : others pass, in different proportions, in these two directions of the axes ; and, lastly, others only traverse the system when the axes of crystallization are parallel. In the examination of the method according to which the polarization of light becomes sensible by means of tourmalines, it was shown that, notwithstanding the great differences of effect presented by the various species of heat, it was not necessary to suppose the existence of a different aptitude for polarization in each ; but, on the contrary, that all might undergo an equal and complete polarization in the interior of the tourmalines, and yet appear more or less polarized at their emergence. These effects are sufficiently accounted for by supposing that the tourmalines refract doubly every sort of radiant heat, and that, in each par- ticular case, one of the two pencils of rays proceeding from this double refraction is more or less absorbed during its passage. The two refracted pencils being of the same intensity, polarized * (The first part of this Memoir will be found in Screntiric Memorrs, vol. i. p. 325.] VOL, IT. PART VI. K 142 M. MELLONI ON THE POLARIZATION OF HEAT. at right angles, and sensibly superposed, it might be expected that if they have undergone the same degree of absorption, no index of polarization would be presented by them when issuing from the plates; but if one of them has lost, in its passage, a greater portion of its intensity, the other will necessarily give signs of polarization on its emergence ; and the appearances of this phenomenon will become exactly similar to those presented by light when one of the two refracted pencils is entirely absorbed in the interior of the plates. According to this manner of considering the subject, the more or less absorbent action of the tourmalines upon one of the doubly refracted pencils of heat would enter, so to speak, into the class of facts that have been observed in our examina- tion of simple calorific transmission by solid and liquid bodies, and all the rays of heat, like light of every colour, would be sus- ceptible of complete polarization by the forces which produce reflection and refraction. This latter conclusion will appear, with the clearest evidence, from the numerous experiments that we proceed to describe. We know that a pencil of common light, traversing a series of parallel plates of glass, or other diaphanous substance, at a certain inclination, becomes polarized perpendicularly to the plane of refraction; so that if a second series of plates be pre- sented to the emergent rays at the same inclination, the light either passes through, or is in great part intercepted, accordingly as the second plane of refraction is disposed in a direction parallel or perpendicular to the first. In order to see whether analogous effects are produced rela- tively to radiant heat, we have only to submit these two oblique piles of glass to trial by the thermomultiplier, disposing the planes of refraction successively in a parallel or in a perpendicular direction. But if the plates be sufficiently numerous, the quan- tity of emergent heat is very small, and scarcely appreciable by the most delicate instruments, especially for sources at a low temperature, the rays of which undergo an almost complete absorption in penetrating the first vitreous layers. Rock salt may be substituted for glass with the greatest success ; but it is difficult to procure several plates of that substance sufficiently large and pure. To obviate, in a great degree, these various in- conveniences, Mr. Forbes suggested the employment, for the polarization of heat, of mica reduced into very thin lamine, M. MELLONI ON THE POLARIZATION OF HEAT. 143 which then, like all other liquid and solid bodies in a very atte- nuated state, transmits notable portions of radiant heat emitted from any source whatever.* We have noticed some of the results at which Mr. Forbes arrived by the employment of piles of mica. The calorific polarization obtained by means of two piles of ten plates each was far from complete, for it was always less than half, while for light it appeared about ;%,. But the circumstance particu- larly worthy the attention of physicists, is the great difference which he observed in the proportion of heat polarized, according to the nature of the source: for in similar circumstances the same mica piles gave 29, of polarization for the heat of an Ar- 100 gand lamp, 524; for a Locatelli lamp, 3%, for a flame of alcohol, d ; é 349, for incandescent platinum, 74% for copper at a temperature of 390 to 400°, =1%, for an iron crucible heated by mercury at 280°, and +4, for a vessel filled with boiling water. These numbers, compared with the indices of polarization, which vary so greatly with the nature of the calorific rays given by the same system of tourmalines, might at first sight induce an opinion that the different species of heat are more or less susceptible of polarization. But by examining attentively the manner in which Mr. Forbes obtained his results, it may easily be seen that the numbers which have just been stated do not in the least represent the quantities of heat polarized. In fact, to measure and compare together these quantities of heat, Mr. Forbes has had recourse to the same method which I employed for the purpose of putting beyond doubt the constancy or vari- ability of the calorific transmission of various diathermanous substances, in passing from one calorific source to the other, * This fact results from a great number of experiments which I have per- formed upon glass, rock crystal, sulphate of lime, mica, water, alcohol, &c. It is intimately connected with the pheenomenon of decreasing transmission pre- sented in general by a given lamina successively exposed to radiant heat from sources at less and less elevated temperatures. It is also in strict relation with the greatly varied proportion of heat which passes through the same body when submitted to emergent calorific rays from different substances. In investigating the analogous effect in relation to light, we find, as I have shown elsewhere, (vol. lv. p. 361 of these Annals), [Scrent. Mem. vol. i. p. 53], that all diapha- nous substances, excepting rock salt, produce precisely analogous effects upon radiant heat to those of coloured media upon light; and in fact, the coloration which diminishes the transparency of bodies exposed to luminous rays of several qualities is entirely removed by reducing them into very thin laminae, so that these tenuous layers then become equally permeable to all kinds of coloured rays. gE 2 144 M. MELLONI ON THE POLARIZATION OF HEAT. viz. he varied the distance between the source and the thermo- scope, in order to render nearly constant the quantity of heat radiated upon the instrument. Now it may easily be seen that the variation in the distance of the calorific focus cannot have any injurious influence upon the measures of transmission, be- cause the diathermanous body placed against the opening of the intermediate screen being of very small dimensions, and the sources of heat being always at considerable distances, the most eccentric rays are never deflected more than a few degrees from the perpendicular, which produces no sensible alteration in the quantities of heat reflected and absorbed by the body submitted to radiation, as may be proved by direct experiment, by placing the same calorific source at different distances, and taking each time the transmission of a given lamina, which will be found constant, if everything be well arranged for observations of this nature. But the result is different in experiments of polariza- tion by means of piles; for the proportion of polarized heat varies generally with the slightest variation of incidence of the calorific rays; and in the experiments under examination, the alteration in the inclination of these rays upon the piles would necessarily amount to several degrees, considering the proximity of the source to the thermoscope, the extent of the polarizing surfaces, and the absence of any intermediate diaphragm. Besides, Mr. Forbes neglected to secure his thermomultiplier from the influence of the heat absorbed by the mica lamine’*, * A single series of observations will suffice to show the small distance at which Mr. Forbes placed his sources of heat; and the very appreciable influ- ence of the heating of the piles upon the results. Source of heat, copper heated to 400° by an alcohol flame. Distance of the thermoscopic body, five inches and a half. Deviations of the galvanometer. The plane of refraction of one pile at 0°, the other at 0° 64° “J 90 . 5s 1 180 7 : 270 6 OWala: ” (Trans. of the R. S. of Edin. vol. xiii. part i. p. 150.) London and Edin- burgh Philosophical Magazine, vol. vi. p. 212. The two piles were placed at the same inclination, in the interior of two gra- duated tubes, turning one within the other; the first was fixed upon the cylin- drical envelope of the thermomultiplier, the second was free, and could move so as to direct the zero of the division successively into the positions indicated by the table. If the galvanometrical deviations corresponding to each of these positions represented the effect of the radiation transmitted immediately through the piles only, the values of the first, third, and fifth observations would be evidently equal to each other, and the case would be similar with the second M. MELLONI ON THE POLARIZATION OF HEAT. 145 so that the effects observed represent the sum of the actions exerted upon the thermoscopic instrument by the two portions of heat which always co-exist in the phenomena dependent upon the passage of calorific rays by diathermanous substances, viz. the immediate transmission and the conductivity. The latter, in altering the absolute value of the index of polarization, would have allowed the equality in the proportion of heat polarized by sources of every description to subsist, could it have operated in these different cases with the same intensity ; forall the calorific rays being equally polarizable, it is evident that the continuance of the action of heating cannot disturb the continuance of the effect due to the polarization. But the diathermancy of mica being analogous to that of glass, the quantity of heat which it absorbs, and consequently its proper heating, varies with the temperature of the source, and thus alters, by this variation of the perturbing cause, the constant effect produced by the prin- cipal cause. In a new series of experiments which has appeared in the last volume of the Edinburgh Philosophical Transactions, Mr. Forbes has partly avoided the different incidence of the calorific rays upon the piles by placing the sources at a distance, always the same, but about three times greater than that which he had adopted in his previous researches. The results obtained have a nearer approach to equality. In fact, the index of polarization of the same system of piles inclined about 34° upon the axis of radiation was 772, to 74, for the Argand lamp, 22, for incandes- 100 100 1 cent platinum, ;63, for copper heated to 400°, 48 for the iron crucible filled with mercury at 280°, and ;44, for the vessel full of boiling water*. But the perturbation due to the heating of the piles still re- mained, and the existence of this cause of error, which Mr. Forbes allowed in the arrangement of his apparatus, is alone amply sufficient to account for the differences observed without its being necessary to suppose that different species of heat un- dergo, in a parity of circumstances, degrees of polarization so various. It might even be demonstrated that the influence of and fourth observation. But instead of the two equalities, we find increasing quantities, which prove with the clearest evidence the progressive action of the heating of the piles. * Trans. of the R. S. of Edin., vol. xiii. part ii., Researches on Heat, second series, p. 14. London and Edinburgh Philosophical Magazine, vol. xii. p. 551. 146 M. MELLONI ON THE POLARIZATION OF HEAT. the heat acquired by the mica laminz should operate in the direction indicated by Mr. Forbes’ experiments; viz. that the action due to the heating of the piles should diminish the appa- rent index of polarization in proportion as the temperature of the source whence the radiation emanates is lower, as the follow- ing will show. The heated piles transmit their own heat to the thermoscope ; and if this secondary radiation be sensible, it always alters, as has just been remarked, the effect due to the heat polarized. But does the alteration produced tend to augment the real index of calorific polarization, or does it render it less apparent? In order to ascertain, I took some paper well blackened upon each side, which, in this state, is absolutely athermanous, but which, as is known, absorbs a large quantity of heat, and also radiates it in abundance. I substituted a rectangle of this paper at the pile nearest the source, and concentrated upon it a large quantity of heat by means of a lens of rock salt. The virtual plane of refraction of the blackened paper was parallel to the plane of refraction of the posterior pile. The heat absorbed by the paper, and after- wards radiated upon the pile, heated the leaves of mica, and they threw the heat acquired upon the thermomultiplier placed at a little distance; the needle of the galvanometer gradually removed from zero in proportion as the mica became heated ; but as the source was at a constant temperature, after five or six minutes, the quantity of heat acquired by the pile became equal to that lost by radiation and contact with the air, and the needle then had a permanent deviation, which in the circumstances under which I experimented was from 25 to 26°*. This being * We know that spiders’ threads do not burn when exposed to the solar rays concentrated by the action of the strongest lenses. From this isolated fact, some physicists have inferred that the heat acquired by thin plates, (corps minces) under the action of a constant calorific radiation, is in the inverse ratio of their thickness, and that it becomes null or insensible when they are of an extreme tenuity. ‘This proposition cannot be true in all its generality, and is even completely false in several circumstances ; for in the experiment described above, the effect of the heat of the blackened paper upon the thermomultiplier, instead of diminishing, constantly increased in proportion as the paper employ- ed was thinner. It is needless to say that I had previously ascertained that this increase was not produced by an immediate transmission ; which had no appre- ciable value in any of the sheets under experiment. Thus, in cases of this sort, the fact is directly contrary to the opinion I have alluded to; paper, and ather- manous substances in general, when exposed to a constant source of heat, beco- ming more heated in proportion to their thinness : at least, when once they have attained their state of mobile calorific equilibrium, they radiate by their pos- M. MELLONI ON THE POLARIZATION OF HEAT. 147 so, I turned the plane of refraction of the mica leaves perpendi- cularly to the virtual plane of refraction of the blackened paper, without on that account altering the common inclination of the laminz upon the axis of calorific radiation: no difference was produced in the permanent deviation of the galvanometrical index, which, after a few minutes’ oscillation, again stopped be- tween 25° and 26°. The action due to the heating of the piles in experiments of polarization is therefore equal for the two directions, parallel and perpendicular, that are given to their plane of refraction *. Now the index of polarization of a pair of piles at a given in- clination, being only the difference between the two quantities immediately transmitted in the parallel and perpendicular posi- tions of the planes of refraction, referred to the greater of them, it might be inferred that the action of the proper heating of the mica piles would diminish this index, by adding the same quan- tity to the two terms of relation. But mica becomes more terior surface a quantity of heat which increases according to the diminution of their thickness. But is the fact the same with regard to diathermanous bodies ? If the impossibility of measuring the elevation of temperature of thin laminz prevents the solution of this question by direct experiment, the properties ac- tually known of immediate transmission furnish us with a satisfactory reply. In fact, glass, water, alum, and diaphanous substances the most refractory to the passage of calorific rays, admit the passage of notable quantities of heat thrown off by sources of all kinds, when reduced into thin laminz ; and as their faculties of transmission increase in proportion as the thickness diminishes, it is clear that in this case the quantity of heat retained will follow the contrary propor- tion, that is, the heating of the lamina will be in the direct ratio of its thick- ness. But this latter law requires invariability in the radiating source. Jt eannot always take place in cases in which the lamin are submitted to rays of different origins, for these rays pass in various proportions by tle same lamina, and, consequently, heat it in a degree proportionate to their intransmissibility. Certain sorts of heat that traverse in abundance a thin Jamina, may therefore communicate to it a slight elevation of temperature, while others will heat it considerably by virtue of their feeble transmission through the substance of which it is composed ; and if two lamine, of the same substance, but of different thickness, be exposed to equal quantities of heat, thrown off by different sources, the thick lamina will become less heated than the thin one, if it receives the heat of the source whose rays are more transmissible. According te all analogy, the substance which forms the spider’s threads is extremely permeable to calorific radiation ; on the other hand, heat emanating from the sun passes with greater facility throngh diathermanous bodies in general, than heat emitted from any other source. These two causes combined appear to be sufficient to account for the phenomenon of incombustibility pre- sented by spiders’ threads placed in the focus of lenses under the action of the solar rays. * Mr. Forbes arrived at the same conclusion, by substituting for the anterior pile the sloping side of a metallic vessel containing hot water. London and Edinburgh Philosophical Magazine, March 1836, p. 248. 148 M. MELLONI ON THE POLARIZATION OF HEAT. heated as the temperature of the source is reduced, since, like glass, it transmits immediately quantities of heat decreasing with this same temperature. If, therefore, the proper radiation of the mica piles exerts an appreciable action, the index of po- larization will, in appearance, undergo a greater diminution for the sources at low temperatures than for those at elevated ones. By the same principle of the secondary radiation of the piles another experiment of Mr. Forbes’s may be explained, which, according to him, proves the unequally polarizable nature of calorific rays. The radiant heat of copper raised to a temperature of 400°, by means of an alcohol lamp, gave him, as we have seen above, 63, of polarization for the action of a certain system of mica piles. By interposing a lamina of glass between the same source and the same system of piles, the proportion of heat polarized increased ten hundredths, that is to say, when the heat traversed the glass lamina before being submitted to the polarizing action of the piles, seventy-three rays in a hundred, instead of ;63,, disappeared in consequence of the intersection of the planes of refraction. The heat of incandescent platina ha- ving given him 7/4, of polarization, without the interposition of the lamina, Mr. Forbes concludes from it that “ the heat from a dark source, after transmission through glass, became as polari- zable as that from incandescent platinum*.” But it is easy to see that the lamina of glass interposed between the source and the piles of mica itself absorbed the greater part of the rays which previously heated these piles in the experiment of direct heat; so that the perturbing cause being considerably enfeebled, the apparent effect of polarization was increased to the point of becoming sensibly equal to that given by the rays of incandes- cent platinum, the passage of which through the mica excites in it avery slight elevation of temperature, on account of their great transmissibility through that substance. The experiment shows that the radiant heat of incandescent platinum, and that of flame, traverses the thin mica leaves in nearly equal proportionst. This equality of heat transmitted being accompanied by an equality of heat absorbed, the piles must necessarily exert the same perturbation upon the imme- * Researches on Heat, second series, by J. D. Forbes, p. 14. London and Edinburgh Philosophical Magazine, vol. xii. p. 551. : + Annales de Chimie et de Physique, vol. lv. p. 346. M. MELLONI ON THE POLARIZATION OF HEAT. 149 diately transmitted rays of each source, which is the reason that Mr. Forbes found, in both cases, the same proportion of polarized heat. Thus the action derived from the proper heating of the piles, an action varied by the nature of the source, or the interposition of a glass lamina, is of itself sufficient for the explanation of all the alterations observed by Mr. Forbes in the index of calorific polarization ; and it has been already observed that the greater or less obliquity with which the various rays fall upon the po- larizing laminz may also produce analogous variations. In order to obtain exact and comparable results, it is therefore necessary to avoid these two causes of error ; for which purpose T have successfully employed the means which we proceed to examine. But it must first be seen how the piles of mica in- tended for experiments of polarization are prepared. There are several different methods, but the following appears to be preferable: First, carefully determine, by any one of the known optical methods, the directions of the aves or neutral sections of luminous polarization for a natural sheet of mica, one or two millimeters in thickness, and cut, according to these two perpendicular directions, a rectangle eight or ten centimeters long, by four or five in width. Then take another rectangle of very thin card, a little larger than the piece of mica which has been cut, and remove all the inner part in a direction parallel to the sides, so as to form a rectangular frame, of which the open- ing will be six or eight millimeters smaller each way than the mica ; then separate from the rectangle of mica, by means of a lancet, the thinnest lamina possible; fix it with a little gum upon the frame of card, carefully keeping its sides exactly paral- lel to those of the opening; and after having fastened upon those portions of the longer sides that rest upon the frame two narrow bands of gummed paper, detach a second lamina of mica, and superpose it exactly upon the first; cover the sides in the same manner with gummed paper, and proceed thus with all the lamin subsequently separated from the rectangle of mica. When the pile is finished, place a second frame of thin card ‘equal to the first on the top of it, apply some glue between the free parts of the two cards, and fasten together their exterior edges by bands of glued paper, in such a manner that no move- ment may take place in the sheets of mica, and that their sides may remain perfectly parallel or perpendicular to the sides of 150 M. MELLONI ON THE POLARIZATION OF HEAT. the frame and to the neutral sections, one of which should always be found in the plane of refraction of the radiation, an indispensable condition, as is known, to render the polarizing action of these sorts of piles independent of their crystalline state, and consequently, similar to the action of piles of glass or any other amorphous substance. I procured thus four pairs of mica piles, composed of three, five, ten, and twenty lamine. The next step was to give them the necessary arrangement for experiments of polarization. The apparatus which appeared to me most suitable is exactly similar to that described in M. Biot’s Traité de Physique, vol. iv. p. 255, with the exception of a few slight modifications, which render it still more simple, and more especially adapted to experiments of polarization by re- fraction. It consists of a horizontal tube, to the two ends of which are adjusted two drums, open at one end, which by hard friction may be turned round, like the ordinary covers of cylindrical boxes ; each of them is divided at the edge of friction into 360 degrees ; from two opposite points of their free circumference proceed two arms parallel to the axis; they are pierced at a cer- tain distance with two small holes, in which are fastened the pivots of a rectangular frame intended to receive the two piles of mica: these pivots, placed in a contrary direction upon the transverse line which passes through the centre of the frame, allow of its being more or less inclined in relation to the axis of the tube; they may be fixed in a determinate position by a clamp. The measure of the angle is furnished by an arc of the circle fixed upon one of the two salient arms of each drum. Thus, when the piles are attached to the apparatus, they may be directly placed, by means of their moveable supports, at any inclination whatever in relation to the axis of the tube, and by afterwards turning the drums any possible position around this axis may be given them ; that is to say, they may be made to travel over, in succession, all the imaginable angular positions around the calorific pencil; for we shall presently see that the rays of heat always enter the tube in the direction of the axis. The circular divisions of the two drums mutually correspond by means of a line, parallel to the axis, traced at the upper part of the tube, and prolonged to the graduated edges in form of an index. The exterior arms being placed symmetrically upon the two sides of the tube, the reciprocal directions of the planes of refraction M. MELLONI ON THE POLARIZATION OF HEAT. 151 of each pile may afterwards be known by means of the degrees marked by the two extremities of this line. Thus, when the drums both mark 0°, or 360°, these planes are parallel, and always preserve the same direction, if any number of degrees whatever upon each division be passed before the index. But if one drum be left at 0°, and all the degrees of its circumference be successively marked at the other, the plane of refraction of the second pile inclines more and more upon that of the first, becomes perpendicular at 90°, still advances towards the primi- tive direction, and reaches it at 180°. The same successive changes of inclination are afterwards produced further ; that is to say, the planes of refraction are gradually separated, and take, at 270°, a perpendicular direction, in order to approach and re- sume the initial position of 0°, or 360°. To avoid the diverse incidence of the rays upon the piles, I placed the source of heat in the focus of a lens of rock salt, suf- ficiently distant from the tube, and in the prolongation of its axis. A horizontal pencil of concentrated heat is thus obtained, which traverses the piles of mica in a direction parallel to the axis, and is propagated beyond, still preserving its cylindrical form, and a considerable portion of its primitive energy, which allows of the removal of the thermoscopic instrument intended to investigate the properties of calorific radiation in the different positions of the piles, to such a distance that the effect of their proper heating becomes perfectly insensible. The employment of the salt lens has, therefore, two great advantages: 1. The giving of intense and sensibly parallel rays ; 2. The possibility of completely securing the thermoscope from the influence of the heat absorbed by the mica lamin. As to the heating of the apparatus which supports the piles, that may be easily avoided by covering every part of it with a double or triple metallic screen, having an opening of an equal or smaller diameter than the smallest dimension of the laminz. We will now recapitulate, fixing our ideas by a particular example. Let the source of heat be a Locatelli lamp: the lumi- nous and calorific rays emanating from it are received at a proper focal distance upon a lens of rock salt ; they issue from it sen- sibly parallel and horizontal, travel over a free space of forty or fifty centimeters, reach the metallic screen which covers the polarizing apparatus, enter by its central opening, fall only upon 152 M. MELLONI ON THE POLARIZATION OF HEAT. the piles, and traverse in a greater or smaller proportion the sheets of mica. We will suppose, for the sake of perspicuity, that each pile is composed of five laminz:, and that the planes of these laminz are all parallel, vertical, and inclined 45° upon the axis of radiation. After emergence, the pencil of heat passes over another interval of from twenty to thirty centimeters, pene- trates the envelope of the thermomultiplier, and, lastly, arrives at the anterior surface of the thermoscopic pile which transmits the impression received to the galvanometer. The indicating needle of the instrument commences its movement, and describes a certain angle, say of 35°92. Before proceeding to experiments of polarization, we must ascertain, 1. That the heat absorbed by the sheets of mica has no sensible influence upon the thermoscope; 2. That the effect observed is independent of the vertical direction of the planes of the two piles during their parallelism. We may easily satisfy ourselves that these conditions are really fulfilled in the circumstances of the experiment, by first removing the thermoscopic body out of the space occupied by the pencil of transmitted heat, without increasing its distance from the last laminz of mica, and still keeping the opening of its envelope directed towards them; and then reinstating the thermoscope in the direction of the calorific pencil, and turning, by means of the drums, the two piles of mica quite round the axis of the tube, without altering either their parallelism or their inclination. In effect, in the first case the needle of the galvanometer returns exactly to the zero of the dial*; and in the second, it gives constantly 35°-92 of deviation. The heating of the mica lamine and the assumed vertical direction of their parallel planes have therefore no influence upon the results ; and the deviation observed in any case of parallelism is produced e In the supposed arrangement of the apparatus, the planes of the mica laminz are vertical: the axis of the thermoscopic pile, which at first formed an angle of 45° with these planes, may, therefore, become perpendicular to them during its lateral movement. The thermoscope then receives anteriorly this same action, caused by the heating of the laminz, which was previously ex- erted obliquely, and yet the needle of the galvanometer always returns to 0°. Therefore, the proper radiation of the piles of mica does not produce any ap- preciable effect. : It is evident that this demonstration should be repeated each time that the source of heat is changed, or its position relatively to the piles and the thermo- scope altered. M. MELLONI ON THE POLARIZATION OF HEAT. 153 solely by the heat directly transmitted by the piles inclined 45° upon the axis of radiation, whatever, in other respects, may be the particular position which they affect around it. Now leave one of the drums at 0°, and place the other at 90°, or 270°: the common inclination of the piles upon the axis undergoes no alteration, but the planes of refraction deviate from their parallelism, and take a perpendicular direction; so that one of them, for example, being horizontal, the other ne- cessarily becomes vertical. Now, upon transmitting the invari- able radiation of the lamp through our ten laminz thus dis- posed, we shall no longer have, as before, 35°92 of deviation, but only 28°54. There is then a very distinct diminution in the quantity of heat that reaches the thermoscope. According to the two preliminary experiments just described, this diminu- tion can only be attributed to an effect of polarization. The arcs of 35°92, and 29°°54*, described by virtue of the primitive impulsions of the galvanometrical index, correspond to forces of 32°10; and 24°95. Dividing the difference of these two quantities by 32°10, and multiplying the quotient by 100, we have 22°06, a number which evidently represents the quan- tity of heat polarized by the /pair of five lamine, expressed in hundredths of the quantity transmitted when the two planes of refraction are parallel. But this result was obtained at an incidence of 45°. In what direction is the variation of the polarizing action of the lamine, when the angle which they form with the calorific rays is dimi- nished? Is the proportion of heat polarized notably increased with the number of the laminze? What degree of polarization may be reached by the concurrence of these two elements ? I have made several series of experiments, in order to resolve these different questions. Their results are laid down in eight tables, which we proceed to notice, first endeavouring clearly to explain the circumstances under which they were made. I combined successively my eight piles, singly, two and two, and three and three; I thus formed of them eight pairs, com- posed of 3, 5, 10, 15, 20, 25, 30, and 35 lamine. Each pair was then raised upon the apparatus, and exposed to the calo- rifie flux of the lamp, in the parallel and perpendicular direc- tions of the planes of refraction, and at different inclinations. [* So in the original. ] 154 M. MELLONI ON THE POLARIZATION OF HEAT. The quantity of heat which reaches the thermoscope, at a given inclination of the piles, diminishes in proportion as the number of the lamine increases. In order to operate as much as possible in the same circumstances, it appeared desirable to render the largest galvanometrical deviation produced in each of the eight series nearly constant. To effect this, 1 employed a small spherical metallic mirror, making the centre of curvature coincide with the middle of the flame ; the concavity was turned towards the lens of rock salt. In this situation the calorific rays thrown off in the direction opposite the lens, were reflected upon themselves; and, being mixed with the heat thrown di- rectly upon the lens by the flame, increased the intensity of the pencil parallel to the axis. I commenced each series by blacken- ing the whole surface of the mirror by the flame of a resinous taper; then by partly removing the lamp-black with a linen cloth I gradually restored the metallic lustre upon a portion of its surface, increasing its extent until the intensity of the heat which reached the thermoscope at that inclination of the piles at which the maximum effect was obtained, had nearly attained the value of the largest galvanometrical deviation adopted, which was from 35° to 37°. It is almost superfluous to add, that I afterwards left the apparatus in the same state during the whole series of experiments having relation to the same pair of piles, so that all the quantities contained in each table may be com- pared together. The titles inscribed at the head of each column sufficiently denote the objects to which the series of numbers which they contain relate. The first gives the angle under which the pair of piles meets the calorific pencil, which is measured from the surface. The second and fourth indicate the ares, reckoned from 0°, described by the index of the galvanometer at the ini- tial effect, when, in establishing the radiating communication with the source, the heat arrives upon the thermoscope, through the piles, in the two directions, parallel and perpendicular, of their planes of refraction : each of the numbers they contain has been established from a series of ten observations. The third and fifth columns contain the intensities of the forces corre- sponding to the arcs of the second and fourth. The last column comprehends the quantity of heat polarized in 100 transmitted rays when the planes of refraction are parallel; which quantity M. MELLONI ON THE POLARIZATION OF HEAT. 155 is easily obtained, as has been seen above, by multiplying the difference of the two forces corresponding to the parallel and perpendicular positions by 100, and dividing the product by the first of those two numbers. This polarized heat, or, in other terms, the heat which dis- appears in the act of the intersection of the two planes of refrac- tion, is neither destroyed nor absorbed, but simply reflected, as occurs in the polarization of light. This may be proved by taking two of our bundles, composed of twenty or thirty lamine, which are to be inclined from 30° to 40° upon the axis of radia- tion, and to receive at first a parallel and vertical direction. Af- terwards withdraw the thermoscopic body from its place, and dispose it laterally at the same distance from the posterior pile, keeping it still turned towards it, but in such a manner that the axis of its cylindrical envelope forms, with the anterior sheet, an angle equal to that formed on the other side of the normal by the incident calorific pencil. The effect of the reflection, which should take place evidently in the direction of the ther- moscopic body, is then extremely feeble, and the index of the galvanometer scarcely departs a few degrees from its natural position of equilibrium; for the heat transmitted by the first pile arrives upon the second, and continues to be transmitted by the remaining laminz, in consequence of the parallelism of the planes of refraction. But if the anterior pile be turned in such a manner as to place its plane of refraction perpendicular to that of the posterior pile, leaving all the rest in their previous state, a considerable deviation is immediately manifested in the indi- cating needle of the galvanometer, which proves a very abundant reflection of heat upon the surface of the second pile: now, in experiments on polarization, it is precisely when the two planes of refraction are thus disposed, that a great portion of the heat no longer reaches the thermoscope. The following are the eight tables. 156 M. MELLONI ON THE POLARIZATION OF HEAT. TABLE I. iF F Calorific transmissions when the planes of refraction are Quantity of heat Eos a polarized in a 236 PARALLEL. PERPENDICULAR. ea eae, SbEa | aresor Ares of Spee Es E> s Teinilton. Forces. Impulsion, Forces. parallel. fy oO ° ° 45 35-29 31-68 32-01 29°12 8-08 43 34-99 31-52 30°77 27°78 11:87 41 34-24 21:12 29°55 2618 15°87 a | 39 33-58 30°55 28:13 24:49 19°84 BOS, 32-84 29:81 26:22 22:70 23:85 | 35 31-78 28:88 24:23 20°86 27-77 33 30°71 27°70 21-98 18-87 31-87 2:4 3) 29°44 26°04 19°40 16°73 35°76 = | 29 27°41 23°81 16°53 14°35 39-73 S197 24:57 31-18 13°63 11-90 43°81 @ | 25 21-24 18°25 10-94 9-54 47°73 i | 23 17°31 15-01 8:27 7:22 5189 21 13-31 11°63 5:88 515 55°72 19 9-22 8:02 3-71 3:24 59°60 (17 502 4:39 1:83 1:60 63:55 TABLE II. fe) ie} fe} 45 35-92 32-01 28:54 24-95 22-06 43 35°69 31-89 27-01 23-45 26°46 141 35-42 31-75 25°16 21-73 31-56 & | 39 35-21 31-64 23-47 20°15 36°31 a 37 34:33 31-17 21°39 18-30 41-03 @ | 35 33-30 30-26 19°73 16-46 4561 > | 83 31-64 28°74 16°39 14-23 50-49 & 731 29°71 26°38 13°80 12-03 54:39 ‘3 | 29 27°38 23-79 11-29 9-85 58:59 2 | 27 23-70 20°36 8-72 7-61 62-62 i | 25 20-04 17-23 6-60 5°77 66°51 23 16-01 13°91 4-74 414 70-24 21 11-71 10-24 3-06 2:68 73°83 19 7-58 6-63 1-71 1:50 77°37 17 3-42 2:99 0-66 0:58 80-60 TABLE III. 45 28-82 26-53 17°21 14-93 43-73 _ | 48 31-41 28-49 16-48 14:31 49-77 8 | 41 33-29 30-24 15°36 13:32 55-95 e | 39 35°19 31-63 13-95 16°16 61:56 Ss | 37 36-46 32:50 12°31 10:77 66-86 a J 35 36°86 32°88 10-638 9-26 71:84 £2 ) 33 36-72 32°75 8-90 7-75 76°34 ‘S| 31 33-79 30°76 6:92 6-05 80°33 3 | 29 30-94 28-00 5-25 4-59 83-61 es 27-89 24-25 3-72 3-25 8660 25 23-19 19-89 2-44 214 89:24 23 17°60 15:26 1:55 1:36 91-09 a? M. MELLONI ON THE POLARIZATION OF HEAT. 157 TABLE IV. o23 Calorific transmissions when the planes of refraction are ow ——————— > =| oe PARALLEL. PERPENDICULAR. fas meaos Ee | = pcuiee Forces, tae: Forces, fo} fo} (eo) 45 24-12 20°75 9-30 8-09 g | 43 27-08 23°51 8-95 7:79 4141 29:59 26-23 8-16 7-13 E139] 31-66 28-76 7-23 6-32 Pay 33°79 30:77 6°15 5°38 stipe 35°58 31-83 4:99 4:36 = | 33 35°44 31-76 3-90 3-40 ‘ | 31 32:13 29-22 2-90 2-54 > | 29 29-04 25°52 2-14 1:87 3 | 27 24-4) 21-03 1°55 1:36 Pa | 95 18:23 15°78 1:07 0-94 23 12-05 10°54 0-68 0-60 TABLE V. ° ° ° _ (45 21-23 18-24 6:56 5-74 @ | 43 24-60 21-23 6-51 5-69 41 28-08 24-44 6-22 5-44 = | 39 30-66 27°63 5-68 4-97 e | 37 33-55 30°52 5-00 4:37 EJ 35 36-21 32-25 4-24 3-70 2 | 33 36-18 32-22 3-41 2-98 % | 31 34-60 29:50 2:52 2-21 © | 29 27-63 24-01 1-68 1-47 S | 27 21-52 18-49 1-13 0-99 B15 14-41 12°53 0-73 0-64 93 8°31 7:26 0-41 0:36 TABLE VI. 8 [45 18°57 16-05 4:17 3-64 2 | 43 22-78 19-53 4:19 3-66 3/41 26-51 22-97 4-00 3-49 2: | 39 29-71 26-39 3-71 3-24 & | 37 32-45 29-48 3-28 2-84 BJ 35 35-42 31-75 2-61 2-39 5 | 33 35-56 31-82 2-20 1:93 — [31 31-75 28-85 1-73 152 % | 29 27-20 23-62 1:33 1:17 2 | 27 20°51 17-63 0:99 0:87 i | 25 13-13 11-48 0-65 0:57 23 6-90 6-03 0-34 0:30 VOL. II. PART VI. Quantity of heat polarized in a hundred trans. mitted rays when the planes of refraction are parallel. 61-01 66°87 72°82 78°03 82°51 86°30 89-29 91°31 92°67 93°53 94:04 94:3] 68°53 73°20 7774 82-01 85-01 88°53 90°75 92°51 93°88 94°64 94°89 95°04 77°32 81-26 84-8] 87:72 90-33 92:47 93°93 94-73 95-05 95-06 95-03 95-02 158 M. MELLONI ON THE POLARIZATION OF HEAT. TABLE VII. S22 Calorific transmissions when the planes of refraction are Quantity of heat sou —$—_—A.—___—. —, | polarized in a aot PARALLEL. PERPENDICULAR. hundred trans. BUsc mitted rays, Ba as a f Arcs of pepe tern gs =F Fa e Sapabion: Forces. Impulsion, Forces. parallel. 45 16-92 14:68 2-73 2:59 83-72 g | 43 21-50 18°47 2-74 2-40 87-01 gq | 41 25-84 22-18 2-52 2:21 90-04 | 39 29:36 25-93 2:30 2-01 92-25 pay 32-38 29-43 2-12 1:86 93-68 BJ 35 35:96 32:03 1:90 1:67 94:79 a 33 36-53 32°56 1:83 1:60 95:09 % | 31 31:90 29-01 1-62 1:42 95°11 nm | 29 27-11 23°54 1:30 1:14 95°16 = | 27 19°89 17:13 0-94 0-83 95-15 ate 12:33 10°79 0°59 0-52 95-18 23 8:81 8-08 0-28 0:25 95:08 TABLE VIII. a (4 14-69 12°75 V71 150 88:24 A | 43 19°35 16°69 1:72 151 90-95 =} 41 23:86 20°51 1:63 1-43 93-03 = | 89 27:99 24°34 1:56 1:37 94-35 & | 37 30°83 27°85 1:60 1:40 94:97 3.2 35 33°88 30°86 1-74 1:52 95:07 = | 33 34:93 31:49 1:76 1:54 95°11 = | 31 30°89 27:93 1:57 1:38 95-06 S | 29 25-67 22-19 1-24 1-09 95-09 2 | 27 18-23 15:78 0:88 0-77 95°12 i | 25 10-92 9-52 0:53 0:47 95-06 23 4-34 3:79 0:22 0:19 94-99 From the various numerical results contained in these tables, we deduce the following consequences : I. The proportion of heat polarized by the piles increases, as the angle at which the rays meet their surfaces is diminished. II. In piles containing a sufficient number of elements, the calorific polarization attains a maximum effect, at a certain angle of inclination which it afterwards preserves for all the smaller inclinations that the ray may successively form with the laminez. III. The inclination which is always reckoned from the surface, at which the manifestation of the invariable effect commences, increases with the number of laminz of which the piles are com- posed. As to the value of this limit of polarization, it is nearly con- M. MELLONI ON THE POLARIZATION OF HEAT. 159 stant in all the series, and is not much less than complete polar- ization, or 199. It would, without doubt, arrive at it if the optical axes of the different mica lamine which compose each of the two piles were exactly in the direction required for render- ing the proper action of the crystallization totally inappreciable, and if the rays introduced into the system were all exactly par- allel, conditions which it is extremely difficult, not to say im- possible, rigorously to fulfil. By substituting my eye in the place of the thermoscopic body, when the two planes of refrac- tion intersected, I constantly perceived, through the system, traces more or less decided of coloration. These colours, due to the action of the central lamina, showed definitively that the light itself did not undergo a complete polarization under the influence of the mica piles: and I have little doubt that, had it been possible to measure with precision their degree of lumi- nous polarization in the oblique positions, at which they gave their maximum effect, the value would have been found to be nearly -25., as for the greatest effect of calorific polarization. M. Biot had previously remarked that the proportion of light polarized by refraction, is increased indefinitely with the angle of incidence, so that the maximum effect occurs at the greatest degree of obliquity at which the rays of light can penetrate the substance of which the refracting laminz are formed. Sir D. Brewster has, besides, enunciated that the light of a wax candle, at a distance of from ten to twelve feet, is completely polarized, at an inclination of 10° 49', by eight plates of crown glass; at 32° 50’, by twenty-seven plates; and at 48° 19’, by forty-seven plates; so that setting out from perpendicular inci- dence, the angle limit, at which complete polarization com- mences, is so much nearer the normal in proportion to the am- plitude of the number of the polarizing lamine. The laws, therefore, of polarization by refraction, in reference both to heat and light, are exactly similar. A very simple observation upon the numbers contained in the second or third column of the last six tables, will show that the calorific rays are also polarized by reflection; that in this case there is a given incidence at which the polarization takes place in the highest degree; and that the planes of the two polariza- tions, produced upon the radiant heat by the action of the forces of refraction and reflection, are respectively perpendicular. If we look at an object through a lamina of glass, or any other L 2 160 M. MELLONI ON THE POLARIZATION OF HEAT. diaphanous substance, in a direction more and more inclined upon the plane of the lamina, it will be found gradually to dimi- nish in intensity, in proportion to the increasing obliquity. It may readily be conceived that such would be the case ; because the rays that fall obliquely upon the lamina traverse a greater thickness of glass than those which arrive in a direction more nearly approaching the perpendicular, and suffer consequently a greater absorption. But even if the matter of which the lamma is composed were perfectly limpid, and admitted the passage of all the light which penetrates into the interior, at any incidence whatever, the decrease of intensity corresponding to the increa- sed inclination would still be observable, because the luminous rays undergo a partial reflection at the two surfaces of the lamina, which is at first feeble and sensibly constant, for angles of from 30° 40° around the normal, but which is rapidly augmented at in- creased inclinations, so that the pencil transmitted in a very ob- lique direction to the surfaces of the lamina, loses a very great portion of its intensity, solely on account of the reflection. The same results are produced with two or several successive lamine ; but when the number is increased to about thirty, and beyond that number, the effects produced are very different. If, for instance, a pile composed of forty or fifty plates of glass superposed, be held, first perpendicularly to the incident rays, and afterwards gradually inclined upon them, the feeble light transmitted under the normal incidence, instead of being dimi- nished by an increase of obliquity, becomes, on the contrary, more and more vivid and brilliant, up to a certain inclination ; it then loses by degrees the intensity acquired, and, lastly, becomes extinguished, when the rays by an excess of obliquity can no longer penetrate into the vitreous matter. Now, the angle at_ which the pencil transmitted attains its maximum intensity, is precisely that at which light is completely polarized by reflection. This singular deviation from the ordinary laws of transmission, is attributable, therefore, to a pheenomenon of polarization. Sup- pose, first, for example, the pile inclined 35°-25/, the value of the angle at which light is completely polarized by reflection upon glass: the refracted rays at this inclination will be strongly po- larized at a certain depth of the pile; for we have seen that light, as well as heat, is susceptible of complete polarization by refraction, at any angle whatever, if the laminz traversed be suf= ficiently numerous. We also know that the plane of polariza- M. MELLONI ON THE POLARIZATION OF HEAT. 161 tion of refracted light is perpendicular to the plane of refraction, or of reflection, at which reflected light is polarized. On the other hand, the rays polarized perpendicularly, to the plane of reflection, are no longer capable of being reflected from the la- mine of glass at 35° 25’, but penetrate into its substance without undergoing any diminution of intensity. Therefore, the refracted light in the interior of the pile, being completely polarized in a plane perpendicular to that of the refraction, after traversing a certain number of laminz, and arriving upon the surfaces of the suc- ceeding laminz at an angle of 35° 25’, will experience the same negative effect ; viz. it will traverse them all without suffering any loss by reflection. But this entire transmission cannot be effectuated at any other inclination, because, in that case, the luminous rays which are penetrated at a certain depth of the pile, and which become polarized by refraction, then undergo only an effect of incomplete polarization, by the action of the reflecting surfaces of the remaining laminz, which consequently resume a portion of their ordinary activity, which increases in proportion to its further removal, in either direction, from 35°25’. The losses of the luminous pencil will therefore follow the same progression, so that the maximum intensity, in transmitted light, will necessarily occur at the angle of complete polarization. Thus the known fact of luminous polarization by reflection and refraction, and the equally known fact of the perpendicula- rity of the planes of these two polarizations, necessarily conduct to the consequence, that light transmitted by a pile of numerous diaphanous laminz attains a maximum intensity at the angle of complete polarization, produced by reflection. Vice versd, starting from the observation of this maximum, in the quantity of light transmitted at different inclinations of the pile, we deduce from it the existence of the two polarizations, the angle at which polarization by reflection takes place completely, and the perpendicularity of the two planes at which the light is found polarized by virtue of the forces of reflection and refrac- tion. Now, this is precisely the case with the transmission of ra- diant caloric by piles of mica; for, by examining the series of numbers contained in the first columns of the last six tables, it will be seen that the transmission by the series of parallel la- mine increases with the inclination, up to an angle comprised between 33° and 35°, and decreases again beyond that limit. 162 M. MELLONI ON THE POLARIZATION OF HEAT. The precise value of the angle at which the complete polariza- tion of heat is effected, is not so easily obtainable as at first sight itappears to be. Indeed all the calorificrays do not traverse themica laminz in the form of the phenomena of polarization that we have just described ; this may be proved by placing the lamine perpen- dicularly to the incident pencil, for with this arrangement a sensi- ble effect of heat is still obtained through the system. Now, we know that the polarizing action is null under the perpendicular incidence ; there is, therefore, a portion of heat which passes in- dependently of polarization ; and though it cannot well be de- monstrated excepting when the rays fall perpendicularly upon the laminz, yet it is not the less certain that it must exist under every other incidence. If this portion of heat, transmitted inde- pendently of polarization, had the same value, whatever were the obliquity of the rays upon the laminz, the angle under which the greatest calorific effect would occur would be always that of complete polarization by reflection. But this value varies with the incidence, according to a progression that differs entirely from the law observed by that portion which passes by virtue of the polarizing forces; for we have seen that instead of first in- creasing, as this does, until it reach the angle of complete polari- zation, it constantly decreases from 90° of incidence to 0°. If the influence of non-polarized heat, upon the transmission of polarized heat, be sensible, it must produce a displacement in the position of the maximum, and bring it evidently nearer the perpendicular incidence. To obtain security from this cause of error, it may be observed, that the absolute quantity of non-polarized heat which traverses the lamine diminishes as their number increases. The probable error, in the determination of the angle of polarization, will therefore follow the same decreasing progression, and become null for a series composed of a sufficient number of laminz. In accordance with this principle, I procured a supplementary pile of forty-four elements, which, added to the other piles, formed a series of a hundred and twenty lamine. Here the quantity of non-polarized heat could have no appreciable influ- ence upon the calorific rays that traverse the system by virtue of the polarizing forces ; for the transmission was sensibly null un- der the perpendicular incidence. Thus, the maximum trans- mission, in oblique incidences, would give exactly the angle of complete polarization. M. MELLONI ON THE POLARIZATION OF HEAT, 163 I therefore mounted my hundred and twenty laminz on one frame, which was provided with two pivots upon the transversal section, and an alhidade indicating the inclination of the polar- izing surfaces to the calorific rays, upon a vertical circle ten inches in diameter. The transmissions observed at each half degree comprised between 33° and 35°, are subjoined: and to them are added two series of similar observations made upon two piles consisting of a smaller number of laminz, one of twenty, the other of sixty elements. The quantity of incident heat varies from one series to another, and consequently the transmissions given under the same inclinations by the three systems of laminz, cannot be compared together. The fulfilment of this condition of comparison was neglected, in order to render the transmissions from the series consisting of numerous laminz more sensible; nor was it of utility for the end actually proposed. Calorific transmissions by Inclination of the Laminz a to the Rays. 20 Lamina. | 60 Lamine. | 120 Lamine. 35:00 37°34 35°97 31-86 34°30 37°42 36°48 32-71 34-00 37°46 36°87 33°07 33°30 37°39 37:10 33°29 33°00 37:09 36°82 33°02 It will be perceived from a glance at this table, that the greatest calorific transmission occurs at an incidence of 34° in the first series, at 33° 30! in the second, and that it maintains the same obliquity in the third. The influence of the non- polarized heat upon the value of the angle of polarization, is therefore sensible only when the pile is composed of a small number of lamin. By comparing, in each of the last two series, the number which represents the greatest transmission, with the numbers that immediately precede and follow it, it will easily be seen that the maximum cannot differ much from 33° 30’, and that if there be a deviation of a few seconds, it is rather in the direction of the 34th degree than in the opposite one. According to the law discovered by Sir D. Brewster, the tan- gent of the angle of polarization, for light, is given by the num- ber which represents the index of refraction of the body employed as a reflector. Now, we know that mica has an index of refrac- 164 M. MELLONI ON THE POLARIZATION OF HEAT. tion equal to 1°5*; the angle corresponding to this quantity, taken as a tangent, is 56°19’, or 33° 41/ reckoning from the surface. Thus, the angle of complete polarization, by reflection, is very nearly the same for both heat and lighty. Now take a pair of piles, each of twenty lamine, and after mounting them properly upon the apparatus of polarization, place a lamina of alum, amber, or black glass, or a layer of water, oil, or some other diathermanous substance, against the opening of the screen which covers the apparatus. The emer- gent rays of the layer added to the system, will then pass through the two packets of mica, which are to receive in succession the two directions adapted for measuring the quantity of heat polar- ized by refraction. Now, in effectuating this experiment it will be found that the index of polarization does not alter in the smallest degree with the nature of the substance interposed, and that its value coincides precisely with the proportion of heat po- larized under the actual incidence of the two piles, when the opening of the screen is free. To exhibit this fact with facility, and in a very evident manner, I employ a means which to me appears capable of carrying con- viction even to the most prejudiced mind. I choose two sub- stances endowed with contrary diathermanciest, that is to say, two bodies which, when exposed to the same calorific flux, admit * Biot, Traité de Physique, vol. iv. p. 80. + From what precedes, it will easily be conceived that to polarize heat or light by means of refraction, it is nearly always requisite to give the piles a great degree of obliquity upon the incident rays. When the lamine are suffi- ciently numerous we may stop at the inclination at which complete polarization commences, which, in certain cases, allows of placing the piles at incidences nearly approaching the perpendicular. However, when the two series of plates consist of very many elements, it is often useful to dispose them, in preference, at the angle of complete polarization by reflection, in order to have an abun- dant transmission of luminous or calorific rays. + Experiments have just been commenced at Geneva, upon the quantity of heat radiated by bodies under a serene sky, at different hours of the day. An .— account of them may be found in the number for April, 1837, of the Bibliotheque Universelle, in which one of the learned editors of that excellent repertory has discussed the results of those observations under the title of Diathermansie de l' Atmosphere. The word diathermancy, as I have defined it in my second memoir upon Transmission, (vol. lv., p. 378 of these Annals) signifies the pro- perty possessed by nearly all diathermanous bodies, of admitting the passage only of certain species of calorific rays. When we wish to denote the quantity of heat transmitted independently of the quality, the term diathermaneity is perhaps preferable, in order to preserve the same termination as the word diaphaneity, indicating the analogous property in relation to light. M. MELLONI ON THE POLARIZATION OF HEAT. 165 _ the passage of rays of so distinct a nature, that the emergent heat of the first can scarcely pass by the second, and vice versa ; of them I form laminz of such thicknesses that the quantities of heat transmitted through each of them and the whole of the two piles may be equal; I then place one of these lamine against the opening of the screen, and observe the calorific actions produced by the rays which reach the thermoscopic body in the two principal directions of the planes of refraction of the mica piles. I repeat the same observations with another lamina, and obtain exactly the same deviations upon the galvanometer. If we take in the table of reduction the forces corresponding to the two galvanometrical deflections observed in either case, and calculate the index of polarization from these data, we shall have a value equal to that indicated by Table V., viz. for example, Z ** or "according as the calorific rays traverse the piles under the obliquities of 41°, 35°, or 29°; and that whatever be the nature of the lamina placed against the opening. The substances best adapted for these experiments of com- parison are opake black glass, or green glass which is imper- meable to red rays, on the one hand; and water, citric acid, or alum, on the other. It may be remembered that the heat transmitted by this latter class of bodies undergoes, under the influence of tourmalines, a polarization which reaches -%°,, while the emergent heat of the antagonistic substances, green or black glasses, submitted to the same polarizing system, give scarcely any sensible trace of this phenomenon, the apparent index of polarization being, in certain cases, scarcely elevated to one or two hundredths. And then these indices, determined by the system of the two piles, no longer present any appreciable difference. Thus the calorific fluxes transmitted by bodies of different natures, and which fluxes are of a constitution so different, are all equally polarizable by refraction ; which proves that the polarization pro- duced by the refractive forces of the media is independent of the quality of the calorific rays. Though this consequence is rigorously established by the ex- periments that have been just related, it appeared to be not altoge- ther useless to verify it also upon calorific rays emanating from dif- ferent sources. To this end I replaced the Locatelli lamp by a spiral of platina maintained in a state of incandescence, by means of the flame of alcohol. The indices of polarization were again equal to those indicated by our eight tables. The same thing 166 M. MELLONI ON THE POLARIZATION OF HEAT. happens when, instead of incandescent platina, a metallic lamina heated to 400° is employed, or simply a vessel filled with boiling water. But the heat of these two latter sources being very slightly transmissible by the mica, and consequently unable to traverse piles composed of a great number of laminz, notwithstanding the action of the salt lens by which their parallelism is established, I receive the parallel rays emerging from the apparatus of polar- ization upon a second lens of rock salt, which collects them all, and concentrates them upon the thermoscopic body. The divergent rays, which proceed from the heating of the posterior pile, must be weakened until they become perfectly insensible, either by re- moving the collecting lens to a suitable distance, or by bringing it very near. In the first case these rays are more and more dis- persed by their natural divergence, and arrive upon the collector- lens with too little intensity to give an appreciable effect. In the second case, the central parts of the last sheets of mica being within the principal focal distance, the greater part of their pro- per rays of heat, instead of being concentrated and consequently mixed with the direct heat, are, on the contrary, dispersed by the lens still more rapidly than is effected by their natural diver- gence, and have no action whatever at a very short distance. Whatever be the means adopted, care should always be taken, after the collector is added, to ensure that the condition of the insensibility of the thermoscope to the heating of the piles is exactly fulfilled. For this purpose the anterior pile is to be re- moved from its frame, and in its place is to be substituted, as in the experiment at page 146, a sheet of paper blackened upon each side, which becomes as much heated as mica, and even more, because it does not immediately transmit radiant heat. If every- thing be well arranged, no appreciable calorific action is obtained. In the apparatus which I possess, the use of the collector about doubles the intensity of the effects, while preserving, according to the method just indicated, the direct rays completely pure, and without mixture with the secondary heat of the piles. *Experiments of polarization may be thus carried, with the * It is evident that more might be gained with piles of mica, and a lens of larger dimensions. ‘To attain this amplification of the thermoscopie effects, Mr. Forbes, in his second series of experiments upon polarization, employed the conical reflector of brass that M. Gourjon generally uses, in addition to my apparatus of transmission; but this reflector collects at the same time the direct heat of the source, and that proceeding from the heating of the piles, M. MELLONI ON THE POLARIZATION OF HEAT. 167 obscure heat of copper at 400°, to the limit of ;%5,, already ob- tained by means of the direct heat of flame. It is impossible to attain this limit with the heat of a vessel filled with boiling water, because the mica exerts upon it an action too strongly absorbent as we have shown when considering the results of his observations. Mr. Forbes appears to attribute the application of the reflector to the thermo-electrical piles to M. Nobili. Another physicist, M. Despretz, says, in the last edition of his Traité de Physique, that the thermomultiplier which [ employ is due entirely to M. Nobili, and that I have only rendered its indications regular. Perhaps I may be here allowed to state the real facts. The first idea of measuring temperatures by thermo-electrical currents is due to M. Becquerel. His object being to estimate high degrees of heat, he con- structed his electrical thermometer of wires of platina and palladium, which he put in communication with a multiplying galvanometer, made according to the principles of Poggendorf. A few years later M. Nobili proposed the employ- ment of thermo-electricity, for the production of a thermoscope of contact supe- rior in sensibility to that of the late M. Fourier, which consists of a common thermometer, around which is tied a small bag of flexible skin, filled with mer- cury. For this purpose he made use of bismuth and antimony, which develope the maximum thermo-electrical effect; of these substances he formed a pile, which he immersed almost entirely in a wooden cylindrical box, containing li- quid mastic, so as to leave exposed only the superior alternate contacts, which were polished and reduced into the same plane; two bars of copper passing _through the sides served to establish the communications with the two ends of an astatic galvanometer: The box was held in the hand, and the bodies, whose differences of temperature were sought, were touched with the uncovered face of the pile. The elements of this pile were twelve in number, (six pairs) folded over rectangularly, and in a contrary direction at the two extremities, in order to prevent the contact of the intermediate parts when they were sol- dered together. Their section was from forty to fifty square millimetres, and the diameter of the box from two to three inches. Upon the instrument thus constructed, I commenced my labours to convert it into a thermoscope of radi- ation. Having observed in some preliminary trials, that the action upon the multiplier depended much more upon the number than upon the bulk of the elements, and likewise, that the thermo-electrical currents never acquired, within certain limits, the tension necessary for traversing non-metallic bodies; I gave the elements the form of small flat bars, from thirty to forty times lighter than M. Nobili’s, and kept them insulated in their whole length, except at the extremities where the solder was placed, by small bands of paper ; I in- creased the number of them considerably, and fixed them by the middle upon an operculum adapted to a transversal ring seven or eight lines in diameter, and low enough for the two extremities and a great part of the rest to be perfectly free. I then covered all the salient parts of the pile with lamp-black, and sur- rounded them with cylindrical tubes, or conical reflectors, according as my ob- ject was to appreciate the action of a small pencil of parallel rays, or to collect the divergent heat proceeding from the walls of a room, or any other large distant surface. Lastly, I gave it the form and the proportions of the thermo- multipliers so skilfully constructed by M. Gourjon, and which are now to be found in the principal collections of philosophical instruments, both at Paris and in foreign countries. The advantage obtained by diminishing considerably the transversal section of the elements, is not only that of being able to introduce a larger number into a very smal! space, and thus to increase the tension of the electric current which is to traverse the long wire of the galvanometer, but it is specially useful in preventing the formation of the returning currents which took place in the 168 M. MELLONI ON THE POLARIZATION OF HEAT. to allow it to traverse a considerable number of laminz, while — maintaining a sufficient intensity ; but, happily, this experiment is not necessary to prove that calorific rays proceeding from dif- ferent sources have an equal aptitude for polarization. It is suf- ficient to show that, under the action of a given number of la- minz, placed at a determined inclination, every species of heat, when rendered parallel by means of a rock-salt lens, and separated from the rays produced by the variable heating of the polarizing piles, gives indices of polarization sensibly equal. Piles composed of a small number of elements which furnish a sufficiently abun- dant transmission of heat from any source whatever, may be very advantageously employed for this purpose. The indices of polarization are easily calculated from the data of observation, by means of the table which furnishes the ratios between the forces, and the deviations of the magnetic needles of the galvanometer; but if we would be independent of this table, and show by the simple inspection of the movements of the galvanometrical index, the equal polarization of radiant heat thrown off by sources of different temperatures, incandes- cent platina, and copper at 400°, for example, an artifice must be employed analogous to that recently described when speak- ing of the calorific rays transmitted by different species of bodies exposed to the radiation of flame. After observing the greatest calorific effect obtainable by means of the heat derived from copper at 400°, we must again take incandescent platina, and interpose one or more plates of interior of M. Nobili’s pile, and destroyed a part of the effect produced. The mastic which covered one of the faces of his pile was also a great inconveni- ence, for it hindered the exterior thermometrical variations from communicating with equal rapidity with all the metallic parts, the consequence of which was, that deviations of from 30° to 40° were produced, during whole hours, solely by the difference of temperature between the mastic and the ambient air. At last, by substituting polished metal for wood in the construction of the envelopes, the instrument was secured from the exterior calorific radiations, and the ob- server thereby enabled to approach it, without the apprehension that the heat of his own body would injure his experiments. The greater part of the alterations that I made in the thermo-electrical pile, are described in a note published by M. Nobili himself, who so far recognised their importance as to say, ‘“ In future I shall combine a second pile of this kind with my first thermomultiplier.” (Bibliotheque Universelle, vol. xliv., p- 233.) But from that time the original pile of contact was really of little importance, which was the reason that M. Nobili thought it just and proper te add my name to his when the electrical thermometer actually in use, that is, the thermomultiplier for the measurement of radiant heat, was presented to the Institut, September 5th, 1831. M. MELLONI ON THE POLARIZATION OF HEAT, 169 glass in the course of the rays transmitted by the piles with par- allel planes of refraction, in order considerably to diminish the more intense energy of the calorific radiation, and render it equal to that from the feeble source, when the planes of refrac- tion of the piles are also parallel. We afterwards dispose these planes perpendicularly, and the index of the galvanometer de- scends precisely the same quantity in both cases. Sir D. Brewster found that to arrive at the limit of the obli- quity at which the polarization of light becomes complete, by means of refraction, the number of laminz requisite diminishes as their refractive power increases. The refrangibility of each coloured ray that enters into the composition of white light di- minishes gradually from the violet to the red; therefore, for a certain series of laminz disposed at a determinate inclination, inferior to the angle which is the limit of complete polarization, the quantity of light polarized will be greater for the violet rays than for the blue, greater for the blue than the green, &c. Analogy induces us to believe that similar phenomena occur with respect to the different species of calorific rays which we have frequently compared to light of various colours. But, as we have just seen, these variations entirely escape the existing resources of calorimetry. Nor can this circumstance occasion much surprise if we consider, I. that in the case of light the differences between the quantities polarized by glass or mica, acting at a given incidence upon violet and red, which are the rays of the greatest and least refrangibility, do not much exceed the hundredth part of the entire quantity, even in the most favourable circumstances; II. that these small variations would probably not have been discovered and measured, if the criterion of coloration, which enables the eye immediately to distinguish luminous rays of different refrangibility, had been wanting in light as well as in heat; III. that the differences of refraction of the divers rays of heat proceeding from terrestrial sources are very small, and only exceed the amount of the analogous varia- tions of light by a scarcely sensible quantity*; IV. that we can never operate alone upon one sort of calorific rays, since every direct flux of heat contains several species, which pass, in groups more or less complex, through the piles of mica and other la- minz interposed, and consequently give a species of interme- * Vol. lv. p. 368, of the Annales de Chim. et de Phys. [or Screnr. Mem. vol. i. p. 56. Epir.] 170 M. MELLONI ON THE POLARIZATION OF HEAT. diate index between the extremes, the values of which already approach so nearly. The variable calorific transmission of a series of numerous parallel laminz, presented under increasing inclinations to the radiation of flame, has recently led us to the inference that heat like light is polarized by reflection; that is to say, that this species of polarization takes place in a plane perpendicular to that of heat polarized by virtue of the refracting forces, and that the angle at which it is completely effectuated differs by a scarcely appreciable quantity from that given by the complete polarization of light. It may here be added that this angle does not undergo any sensible variation if the nature of the calorific radiation be altered, either by the interposition of laminz of different diathermancy, or by substituting other sources of heat for flame. The emergent rays of opake black glass, trans- mitted by a pile of seventy lamin, give actions upon my ap- paratus, which, at an inclination of 33° 30’, the moment of the maximum effect, push the index to more than 30°, and allow it to descend rapidly towards zero when the laminz are inclined in either direction. The direct rays of copper heated to 400° pro- duce the same relations of intensity at different inclinations, but upon a much smaller scale. I shall here observe, once for all, that in the greater number of experiments on calorific polarization, in which rays of heat unmixed with light are required, the obscure heat of bodies below incandescence may be very advantageously replaced by the emergent heat of perfectly opake black glass exposed to the calorific fluxes of flame or incandescent platina. For this sort of heat is certainly perfectly obscure, and, in addition, is endowed with a diathermancy very analogous to that of mica ; it consequently presents all the conditions desirable for the ve- rification, upon heat alone, of the facts corresponding to those observable in luminous polarization. The invariability manifested in the angle of the complete po- larization of heat by reflection, notwithstanding the differences of the mean indices of the refraction of the various incident pencils, may be conceived relatively to the limits of precision furnished by our actual calorimetrical instruments, from reasons exactly analogous to those that have just been alleged when treating of polarization by refraction. And even should we at some future time succeed in insulating M. MELLONI ON THE POLARIZATION OF HEAT. 171 the different calorific rays, and in measuring their indices of polarization for a given incidence with the greatest exactitude, we shall only add a new element to the science of radiant heat, which will separate, by a few fractions of a degree, the inclina- tions actually known, at which the different species of rays give the same quantity of polarization. But all these species being susceptible of complete polarization, will still, notwithstanding these little distinctions, be of the same polarizable nature. Heat then is polarized absolutely, like light, by refraction and reflection; a conclusion which fully confirms the theory deve- loped in the first part of this memoir, to show how the pheno- mena of polarization may take place in the interior of tourma- lines without the possibility of perceiving them outwardly*. Indeed, it may be remembered that certain species of tourmalines give a calorific polarization, which is either total, incomplete, or null in appearance, according to the quality of the heat em- ployed. But we have just seen that all the calorific rays are equally polarizable: there exists, therefore, in the tourmalines, a cause which sometimes conceals, and sometimes exhibits the polarizing action. This cause can only be double refraction, which always produces two superposed pencils, equally intense, but differently polarized, in plates divided parallel to the axis of crystallization. When the action of the tourmalines manifests itself, one of these pencils is completely absorbed, and the other remains alone and exhibits its proper direction of polarization ; in the opposite case, the two pencils undergo an equal absorp- tion, and issue together completely neutralized with regard to polarization. Now if in the latter case the emergent heat re- sembles ordinary heat, the second pencil, which was previously absorbed, must necessarily be polarized at right angles to the other; its polarization must also be complete, for it is in that state that the first pencil of heat is separately exhibited. The production of two calorific pencils in bi-refractive media, and their rectangular polarization, is also inferred from another experiment exactly analogous to those performed in optics to show the action which bodies possessing double refrac- tion exert upon polarized light. If a ray of light, reflected by a mirror of black glass, at an angle of 30° 25’, traverse perpendicularly a lamina of sulphate * Vol. Ixi. p. 408, of the Annales de Chim. et de Phys. [or Scrent. Mem. vol. i. p. 345. Enir.] 172 M. MELLONI ON THE POLARIZATION OF HEAT. of lime or mica, and afterwards fall upon a second surface of glass at an equal inclination of 30° 25’, the latter reflects the in- cident light in a larger or smaller quantity, according to the positions which the principal section of the crystallized lamina and the plane of the second reflection affect, in relation to the plane of primitive reflection, in which the ray of light is first polarized. Let us consider the two mirrors independently of the crystal- lized lamina. If we first make their planes of reflection to coincide, and afterwards place them perpendicularly, in the first position we shall obtain the maximum of reflected light, in the second the minimum. The effect is unaltered if the doubly refracting lamina be interposed between the two mirrors, after tracing upon its edges the direction of the principal section, and bring- ing it parallel or perpendicular to the primitive plane of reflec- tion; the proportions of light reflected by the second mirror remain the same in each case; hence the denomination of neutral axes, given to these two directions of the lamina. But if the principal section or its perpendicular be inclined in such a manner that one of them forms an angle of 45° with the plane of primitive polarization, there is a very considerable alteration in the reflection of the second mirror; the maximum of reflected light is diminished, the minimum increased ; and the diminu- tion of intensity produced in the first case, when the planes of reflection are parallel, is found to be precisely equal to the aug- mentation which occurs in the second case, when the planes of reflection are perpendicular. These variations of intensity, caused by the particular posi- tion of the principal section of the bi-refracting crystal, in rela- tion to the plane of primitive polarization, require for their pro- duction a certain thickness according to its nature, but always extremely small, of the lamina interposed: they are besides ac- companied by a brilliant coloration, which ceases also at certain limits of thickness, equally dependent upon the quality of the crystal interposed. We here lay aside the subject of colours, and shall only consider the intensity, which always follows the law enunciated whether the colours be perceived or not, the re- flected light in the latter case appearing perfectly white, as hap- pens with plates of sulphate of lime rather more than a demi- millimetre in thickness, and with plates of mica nearly twice as thick. M. MELLONI ON THE POLARIZATION OF HEAT. 173 It would be superfluous, for the end proposed, to enter into all the theoretical details relative to the different modifications that the lamina interposed impresses upon the luminous pencil in proportion as its principal section turns around the plane of primitive polarization ; they may be found in all optical trea- tises. Let it suffice that we call to mind that the equality of the two variations of which we have recently treated, is a necessary consequence of the double refraction and the complete and rect- angular polarization that the luminous pencil undergoes in the interior of the bi-refracting crystal. The light polarized by the first mirror, when traversing this thin crystal, is either sub- divided into two parts sensibly superposed, or preserves its unity, according as any one of the neutral axes is inclined, or parallel to the plane of primitive polarization. When the sub- division takes place, there results from it, at the particular incli- nation of 45°, two pencils of equal intensity, ordinary and ex- traordinary, which, in the two cases under examination, always have their planes of polarization so turned that one of them is found precisely comprised in the plane of reflection of the second mirror, and the other in the perpendicular direction: it is the first only which can undergo the second reflection and reach the eye. Now one of these two pencils is sometimes added to the light reflected by the second mirror, and sometimes subtracted, which is the reason that the augmentation produced, when the _ planes of reflection are perpendicular, is equal to the diminution which takes place when these planes are parallel. The results just related do not absolutely require the employ- ment of two mirrors, but may be also obtained with a pair of tour- malines, whose axes are rendered successively parallel or perpen- dicular. They may also be observed by means of two series of parallel laminz of glass, properly inclined to the incident rays, and so disposed as that their planes of refraction are sometimes parallel, at others perpendicular. Now, if the same phzenomena can be produced upon calorific rays, we may conclude that heat is refracted and polarized, like light, in bodies possessed of double refraction. This experiment was tried by Mr. Forbes with two of his mica piles, giving from twenty to thirty hundredths of sensible polarization, be- tween which he interposed a large vertical lamina of mica, which was provided with two contiguous bases, and inclined to each at an angle of 135°. The principal section having a direc- VOL. II, PART VI. M 174 M. MELLONI ON THE POLARIZATION OF HEAT. tion perpendicular to one of the two sides of which this double basis was formed, and the plane of primitive refraction remain- ing always vertical, by resting the lamina first upon one, then upon the other basis, and disposing the second plane of refrac- tion, first vertically, then horizontally, the actions indicated in the following table were obtained. Variation, in degrees of the thermomultiplier, observed when the principal section of the interposed lamina of mica passes from the vertical to 45° of obliquity, whilst Sources of Heat. the plane of refraction of the second pile is 4 Vertical. | Horizontal. Mercury at 280°.......+. — 0°23 + 0°26 Copper at 400° ......... — 0°°:517 + 0°°545 , Incandescent Platina... — 2°18 + 2°32 Argand Lamp.......++++- — 19-43 + 19:37 By comparing each positive variation with the corresponding negative variation, we see that the first is constantly superior to the second, for the two sources of heat possessing light, and in- candescent platina; and the contrary for the radiation of the lamp; the difference, which is five or six hundredths for the three latter cases, amounts to nearly twelve-hundredths for the first. But, from the nature of the experiments, says Mr. Forbes, the table tends generally to show the coincidence of the two va- riations*. I know not whether this tendency will appear suffi- cient to physicists in general. It is really alarming when we see that the effects produced scarcely amount to a few fractions of a degree for obscure heat, the subject in which we are chiefly interested; for it is very difficult to estimate smaller quantities than a quarter of a degree upon the circle of the thermomultiplier, which is scarcely more than five centimetres in diameter: on the other hand, circum- stances, in appearance the most unimportant, may produce varia- tions equal at least in extent to the deviations observed in the first two cases. It is true Mr. Forbes has adapted a microme- trical system to his galvanometer, by means of which he can, with greater ease, appreciate, as he thinks, even tenths of a de- gree; he also endeavours to secure himself from perturbing causes * The following are the author’s own expressions: “ The table generally points to a coincidence, and that as close as by the nature of the experiments we should perhaps be warranted in expecting.’”—Trans. of the R. S. of E. vol. xiii. p. 162., [or Lond. and Edinb. Phil. Mag. vol. vi. p. 366. Eprr.] M. MELLONI ON THE POLARIZATION OF HEAT. 175 by taking the mean of several observations. But these expedients were not sufficient in the case under consideration, as is even evident from the nature of the results obtained upon the rays of obscure heat, which, though giving a tolerably considerable dif- ference, and always in the same direction, would yet be far from proving the equality of the two actions, if it were not deducible from the analogous case of light, in which this equality is esta- blished from inductions that cannot admit of the least doubt. To render the experiment conclusive of itself, it must be per- formed upon an obscure calorific flux, very intense and very trans- missible through mica, in order to be able to polarize it almost completely by piles of numerous laminz, while still preserving a notable portion of its energy, and to render it, thus strongly polarized, more sensible to the doubly refracting action of the interposed laminz. It must also be secured from the heating of the mica system, which always tends to diminish the apparent effects of polarization. Nothing is more effectual for satisfying these conditions than our calorific rays rendered parallel by the rock-salt lens, and completely separated from light, and from the greater portion of the heat absorbable by the mica, by their pre- vious transmission through opake black glass. I therefore caused a pencil of this obscure heat to fall upon my two piles of twenty laminz inclined 33° 30’ upon the axis of radiation; I placed between them the perpendicular lamina of mica, and after ascertaining, by the means indicated above, that the proper heat radiated by the last pile upon the thermoscopic body was insensible, I proceeded to the measurement of the two variations, which were then very considerable, as may be seen from the following table. Variation, in degrees of the thermomultiplier, observed when the neutral axes of the interposed lamina pass, re- latively to the plane of refraction of the first pile, from Origin of the obscure rays parallel and perpendicular directions to an inclination of transmitted by opake 45°, while the plane of refraction of the second pile in re- black glass. lation to it is Parallel. Perpendicular, Argand Lamp...........- — 299-32 + 299-27 Locatelli Lamp ......... — 27°51 + 27°°56 Incandescent Platina... — 31°19 + 31°15 Each of the three sources of heat was placed at the centre of a spherical reflector; the calorific pencil of parallel rays, after having traversed the black glass and the system of mica laminz, M 2 176 M. MELLONI ON THE POLARIZATION OF HEAT, arrived upon the thermoscopic body, without being there con- densed by the collector, which was not in the least required, in consequence of the intensity of the effects produced. The mica lamina, interposed between the two piles, was of a circular form, and in thickness equal to 0™™-2489; it could only revolve in its own plane around the centre, which consequently remained im- mobile during this rotatory motion. The equality of the negative, and the corresponding positive variations, is here established with all the requisite exactitude, for their differences are less than ;4,, sometimes in one, and sometimes in the other direction. Yet each number contained in this table is the result of only ten observations. It is true that these observations were made with the greatest care, and that the differences between the maximum and minimum of each series scarcely exceeded half a degree. Now, suppose that a horizontal pencil of obscure heat issuing from black glass be thrown upon a vertical surface of glass or mica, at the angle of complete polarization; that the reflected rays be afterwards transmitted perpendicularly through the cir- cular lamina of mica; and that the emergent heat be received upon another surface of glass or mica, disposed parallel to the first ; it will there undergo a second reflection, and return in a direction parallel to the primitive direction, but always removing further from the source. If the thermoscopic pile be placed at a certain distance from the two reflectors, so that it may receive the impression of the pencil of heat which has undergone the two reflections and the intermediate transmission of the mica disc, by turning this disc in its proper plane, a much less ener- getic action is observable when the principal section is inclined 45° upon the horizon, than when it is horizontal or vertical. The effects obtained are nearly as sensible as the differences re- corded in the preceding table ; for the index of the galvanometer, in passing from one position of the principal section to the other, travels over arcs of from 20° to 25°. This experiment, which is perfectly analogous to the preceding ones, is very interesting, as it enables us completely to insulate, as to their mode of manifestation, the polarizing forces developed in the act of reflection, from the similar forces developed during simple refraction. Indeed, until now, it has been necessary to have recourse to the second forces of polarization to render the first sensible. The rays, in the experiment under considera- ' M. MELLONI ON THE POLARIZATION OF HEAT. 177 tion here do not undergo any ordinary refraction, but simply two successive reflections; and the lamina interposed perpen- dicularly to the pencil of obscure heat which passes from one mirror to the other, only reveals, so to speak, its state of po- larization produced by reflection alone. Indeed this species of calorific polarization may be separately developed by more direct means, exactly similar to those employed to exhibit the ana- logous phenomenon of light; but to do this would hazard displacing the source or the thermoscope in giving the perpen- dicular direction to the two planes of reflection, for it may be ob- jected that the calorific rays do not present themselves at the opening of the thermoscopic tube, with the same directions that they affect when the two planes of reflection are parallel ; or, that the intensity of the source, or its position in relation to the mirrors, has been altered during the necessary movement. But let us return to the piles. When the planes of refraction are perpendicular, the interposition of the circular disc of mica between the two series of lamina, increases the calorific trans- mission if its principal section be inclined 45° upon the first plane of refraction, and leaves it in nearly its natural state if the disc present its principal section parallel to that plane. According to the denominations adopted in England, Mr. Forbes calls the relation of the quantities of heat transmitted through the system, in these two positions of the lamina, the effect of de- polarization. When endeavouring to determine a similar rela- tion for heat proceeding from different sources, Mr. Forbes found that it varies even when employing the same depolarizing lamina, and the same system of piles arranged wnder a constant inclination. Thus, in certain circumstances, the heat of copper at 400° gave him, as the mean of several observations, 100:118; and the heat of incandescent platina 100:134. He thence concludes that calorific rays are more or less depolarizable*, according to their proper nature. _ If the tenor of the reasoning with which this second part of the memoir commenced has been well understood, it will be easily seen that Mr. Forbes’s conclusion is inadmissible. Indeed we have seen that in the conditions of distance which he adopted, the heat proceeding from the whole of the system of mica was mixed in a sensible manner with the direct rays of the source * Trans. of the R.S. of Edin. vol. xiii., part i. p. 155, [or Lond. and Edinb, Phil. Mag. vol. vi. p. 286. Eprr.] 178 M.MELLONI ON THE POLARIZATION OF HEAT. which immediately traversed the laminze. In each of the sources employed, the heating of the piles, and, consequently, the quan- tity of proper heat which they radiate upon the thermoscopic body, does not alter in the two positions which are successively given to the principal section of the interposed lamina. How- ever, the calorific absorption of the mica, whence this heating is derived, varies with the quality of the incident rays, and becomes strong in proportion to the intransmissibility by the system, of the heat supplied from the source. Besides, we shall see that all the calorific rays undergo the same effect of depolarization, and consequently give the same difference between the two por- tions of heat that immediately traverse the system, when the principal section is parallel, and afterwards inclined the same [angular] quantity upon the plane of primitive polarization. But it is evident that by adding a given number to two different quantities, they must necessarily approach to equality, and that in a proportion corresponding to the largeness of the number added. Wherefore the heat from sources at low temperatures, that is, heat from sources whose rays are not very transmissible by mica, undergoing a greater absorption, must have produced, in Mr. Forbes’s experiments, a depolarization smaller in appear- ance than the heat from sources at elevated temperatures, whose rays communicate less heat to the system. I demonstrate the equality of the depolarization of heat of every kind by means perfectly analogous to those which I employ to prove the equality of their polarization. If the subject under investigation be heterogeneous calorific fluxes transmitted by different bodies submitted to the radiation of flame, I take those endowed with the most opposite diather- mancy, which, combined separately with the system of depolar- ization, transmit equal quantities of heat, when the principal section of my circular lamina is parallel or perpendicular to the plane of primitive polarization, and in each case I incline the principal section 45° upon this plane; the progress of the gal- vanometrical index is precisely the same for both experiments. If we desire to verify this equality relatively to the heat emit- ted by different sources, the maximum transmission obtained with the source at a low temperature is to be first observed, and then glass laminz, more or less thick, are to be interposed upon the exterior passage of the radiation from the source at an ele- vated temperature, until the effect of the minimum transmission M. MELLONI ON THE POLARIZATION OF HEAT. 179 be equal to that observed upon the preceding source. We then pass to the augmentations produced in both cases, by inclining the principal section 45° upon the plane of refraction of the an- terior pile ; these two augmentations are again respectively equal. In all these experiments the indicating needle of the galvano- meter moves over a considerable space, for we have recently seen that it sometimes describes arcs which exceed 30°. The smaller angle described by virtue of the alteration of the direction of the principal section is due to the action of the heat thrown off by copper at 400°, which scarcely propels the needle beyond 7°; but as, by means of the artifice just indicated, the heat of flame may be employed to a sufficient extent to give precisely the same movement, the equality of the depolarization in these two extreme cases is proved in the most evident manner. The two pencils of light produced by the plate of mica or sul- phate of lime, in positions in which the principal section is in- clined 45° upon the primitive plane of polarization interfere when they are reflected together by the second mirror, or transmitted by the second pile, and thus develope the beautiful colours treated of above. Is there an analogous interference of the calorific rays ? Coloration being here the criterion of the interference, I at first thought that I should easily succeed in verifying the exist- ence of this phenomena in heat by experiments of diathermancy. I will endeavour to explain my ideas more clearly. We know that the two coloured images obtained by the inter- position of the plate of mica or sulphate of lime, having the prin- cipal section inclined 45° upon the plane of primitive polarization, whilst the second plane of polarization is rendered alternatively parallel or perpendicular to the first, have always complementary tints. We will suppose that these tints are the red and the green. If we view these two images produced thus successively through a glass of a very pure red colour, the first will be seen and not the second. If, instead of red, white or some other kind of coloured glass be employed, the two images will be seen, sometimes in their natural state, and sometimes altered; the red image more than the green, or the green more than the red, according to the nature of the screen glass interposed. Would not these different alternations, produced in the relative energy of the two images, by the interposition ofa given screen, be equally sensible to us, if our eyes lost the faculty of distin- guishing colours, only retaining a perception of luminous in- 180 M. MELLONI ON THE POLARIZATION OF HEAT. tensity ? Now the sense of vision, reduced to this state of sim- plicity, would become, as to light, what our thermometers are as to heat. Wherefore, if the two complementary pencils of obscure heat were transmitted by a substance possessing a high degree of diathermancy, it is very possible that they might not be equally absorbed, in which case we should have an indirect proof of the interference of the two calorific pencils. I have tried the experiment with several sorts of plates, and have al- ways obtained the same relation of transmission in the two cases. These results do not decide the question negatively. It is very possible, I will even say probable, reasoning from analogy, that the calorific rays interfere; but, in my opinion, we have not yet a single fact whence any experimental proof whatever, direct or indirect, of these interferences, may be de- duced. As to the polarization of heat, its existence and its general laws appear to me to be fully proved by the numerous facts recorded in this memoir. I have endeavoured to describe the fundamental experiments as clearly as possible, in order that all who are interested in the progress of physics may study them with facility. I may add, they are neither difficult nor uncertain; I have repeated them very many times, and in the presence of several physicists, and always with perfect success. At the commencement of these researches we proposed to explain the contradictions presented in the results obtained by different experimenters upon calorific polarization; but this task becomes needless after the long investigation, into which we found it necessary to enter in relation to Mr. Forbes’s expe- riments. All the differences observed in the polarization of heat de- veloped by the forces of reflection and refraction, are attribu- table to the MORE OR LESS SENSIBLE HEATING of the appa- ratus of polarization ; excepting in the case of the tourmalines, which render the phenomena of polarization sensible or not, ac- cording to the quality of those minerals. The portion of heat regularly reflected by the mirrors, and re- fracted or transmitted immediately by the piles, is very small, relatively to the quantity of heat absorbed by the mirrors or Jaminz. If the thermoscopic body be placed so as to be simul- taneously affected by these two species of heat, the difference existing between the feeble reflected or refracted rays in the M. MELLONI ON THE POLARIZATION OF HEAT, 181 two positions, parallel or perpendicular, of the planes of polari- zation, is concealed by the enormous quantity of heat which the polarizers radiate upon the thermoscope equally in both cases. The manifestation of this difference commences, if the action of the secondary radiation of the mirrors or piles upon the thermo- scope be comparatively feeble to that of the calorific pencil, which undergoes immediate reflection or transmission. Lastly, it attains its normal state, when, by a suitable arrangement of the apparatus, the thermoscope is completely secured from the sensible effect of this radiation, whilst left exposed to the action alone of reflected or refracted heat. If we take a general survey of the whole of the facts which, at the present day, compose the science of radiant heat, it will be seen that this agent is propagated, reflected, refracted, and polarized, in absolutely the same manner as light. If these pro- perties often remain unperceived, it is to be attributed to a de- fect of diathermaneity in the greater number of bodies, or to the particular manner, according to which their absorption is manifested upon the radiation of heat. ‘Some media, such as air and rock-salt, transmit equally all sorts of calorific or luminous rays ; but others act in a different manner upon the rays of the two agents, extinguishing some- times more light than heat, at others more heat than light. We have thus the singular spectacle of bodies which completely ab- sorb the luminous rays, and admit the passage of certain calo- rific rays; and of substances permeable to light, completely ar- resting every species of heat. Analogous differences are produced in the diffuse reflection which the two radiations experience at the surface of opake and athermanous bodies ; for perfectly white substances reflect or absorb extremely diverse proportions of heat, according to the quality of the calorific rays; and yet the same white sur- faces absorb all the rays of light in equal proportions. It is deducible even, with the clearest evidence, from the absence of any coloration whatever, which would not fail to appear when these surfaces were exposed to ordinary light, if, by a difference of absorption, the coloured rays, which enter into the composi- tion of light irregularly reflected, had not between them exactly the same relations of intensity of the incident rays. Other inequalities, also deriving their origin from absorption, are manifested in the phenomena of polarization presented by 182 M. MELLONI ON THE POLARIZATION OF HEAT. tourmalines. In these phenomena the two pencils into which a ray of light is divided, in penetrating into the interior of the plates, are so modified in their progress, that the ordinary pencil is completely absorbed during its passage, and the ex- traordinary pencil presents itself alone completely polarized at the emergence ; and that whatever be the colour of the incident light. The case is different with radiant heat, the two pencils of which produced at the entrance of the same polarizing plates, undergo absorptions sometimes extremely diverse, sometimes perfectly equal, which occasions great variations in the appear- ances of polarization, according to the quality of the calorific rays. Polarization becomes equal for radiations of every kind, if it be produced by the forces of refraction and reflection, which are perfectly independent of the absorption of the media. It is similar when this latter force has no longer any influ- ence upon the phenomenon of reflection. Indeed, we have just seen above, that diffuse reflection, in which absorption acts a part so important, varies considerably from one ray of heat to the other; but the portion of incident radiation, which is re- flected in a regular manner at the polished surface of rock salt, and other diaphanous substances, is equal for every species of heat and light. All bodies exposed to radiant heat become hot, and, when withdrawn from the action of radiation, preserve for some time the heat acquired ; but very few substances, after exposure to light, retain it so as to become luminous in darkness : in general the light disappears even at the moment of absorption. In short, the heat absorbed is found, so to speak, to have changed its nature. It then forms a homogeneous flux, and the mode of its transmission acquires characters quite oppo- site to those effected by calorific or luminous radiation. This absorbed heat makes its way, in the body, in every direction, is propagated in it slowly, like heat communicated by contact, and its propagation is considerably modified by the displace- ment of the different parts, of which the body is composed. Light, and radiant heat, on the contrary, are composed of he- terogeneous fluxes, they move only in a rectilinear direction, travelling over any interval whatever in an imperceptible space of time, and do not receive any influence from the agitation, whether more or less violent, of the media which transmit them, M. MELLONI ON THE POLARIZATION OF HEAT. 183 In conclusion, these two great agents of nature, and the mo- difications which they undergo from the action of ponderable matter, are governed by similar laws, while their rays move freely. Numerous differences are manifested as soon as the progress of the two radiations suffers any interruption what- ever, either at the surface, or in the interior of bodies. 184 C. F. GAUSS ON THE GENERAL THEORY OF ARTICLE V. General Theory of Terrestrial Magnetism. By PRoFEssor Caru Friepricu Gauss, of the University of Gottingen. [Translated by Mrs. Sabine, and revised by Sir John Herschel, Bart.] Tue unwearied zeal with which, in recent times, endeavours have been made to examine the direction and intensity of the mag- netic force of the earth, at all parts of its surface, is the more worthy of admiration, as it has been prompted by the pure love of science. Great as is the importance to navigation of the most complete attainable knowledge of the lines of declination, more than this is scarcely required for its purposes. Whilst science delights to render such useful services, her own requisitions have a wider scope, and make it necessary that equal efforts should be devoted to the examination of all the magnetic elements. It has been customary to represent the results of magnetic observations by three systems of lines, usually termed Iso- gonic, Isoclinal, and Isodynamic lines. In course of time these lines undergo considerable alterations both in position and in figure, so that a drawing of them represents the phenomena correctly only for the epoch to which it corresponds. Hal- ley’s Chart of Declination for 1700 is very different from that of Barlow for 1833; and already Hansteen’s Dip Chart for 1780 differs greatly from the present position of the Isoclinal lines. Doubtless, in course of time, similar alterations in the lines of intensity will be manifested ; but observations of this nature are altogether too recent to furnish such indications at present. In all these maps there exist spaces either blank, or in which the lines are but indifferently supported by observation. The inaccessibility of parts of the earth’s surface renders perfection in this respect impossible; but a rapid progress towards it may be confidently hoped for. Viewed from the higher grounds of science even a complete representation of the phenomena after this manner is not itself the final object sought. It is rather analogous to what the astro- nomer has accomplished, when, for example, he has observed the apparent path of a comet in the heavens. Until the complicated phznomena have been. brought in subjection to a common prin- ciple, we have only building-stones, not an edifice. TERRESTRIAL MAGNETISM. 185 The astronomer, after the comet has disappeared from his view, begins his chief employment, and resting on the laws of gravita- tion, calculates from the observations the elements of its true path, and is thus enabled to predict its future course. And in like manner the magnetician proposes to himself as the object of his research, as far as the different and in some respects less favourable circumstances permit,—the study of the fundamental causes which produce the phznomena, their magnitude and their mode of operation,—the subjection of the observations, as far as they extend, to those elementary principles,—and the anticipa- tion, with some approximation at least, of their effects, in those regions where observation has not yet penetrated. It is at least well to keep in view this higher object, and to endeavour to pre- pare the way for it, even though the great imperfection of the data may render its attainment impossible at present. It is not my purpose here to notice the earlier fruitless attempts: to explain the enigma of these phenomena by hypotheses ha- ving no physical foundation. A physical foundation can only be allowed to such attempts as have considered the earth as a real magnet, and have employed in the’ calculation only the demon- strated mode of action of a magnet operating at a distance. All attempts of this nature hithert omade have this in common ;— that instead of first examining what the conditions, whether simple or complex, of this great magnet must be to satisfy the pheno- mena, certain determinate and simple conditions were presup- posed, and the subject of inquiry has been the accordance or non- accordance of the phenomena with these presupposed condi- tions. We see here a repetition of what has often occurred in the early history of astronomy and of other sciences. The simplest hypothesis of this kind is that which supposes a very small magnet in the centre of the earth; or rather (as it is not likely that any one ever believed in the actual existence of such a magnet) supposes magnetism to be so distributed in the earth, that its collective action at and beyond the surface is equi- yalent to the action of an imaginary infinitely small magnet ; much as gravitation towards a homogeneous sphere is equivalent to the attraction of a sphere of equal mass condensed in its central point. In the supposed case, the magnetic poles are the two points where the prolonged axis of the little central magnet intersects the earth’s surface; where the magnetic needle is vertical and the intensity is also greatest. In the great circle midway be- 186 C. F. GAUSS ON THE GENERAL THEORY OF tween these two poles called the magnetic equator, the dip is =o and the intensity is half as great as at the poles; between the magnetic equator and either pole, both the dip and the in- tensity depend on the distance from the said equator (which distance is termed the magnetic latitude) in such manner, that the tangent of the dip is equal to twice the tangent of the mag- netic latitude. Lastly, the direction of the horizontal needle must everywhere coincide with the direction of a great circle drawn through the northern magnetic pole. There is in nature only a rude approximation to all these ne- cessary consequences of the above hypothesis. In reality the line of no dip is not a great circle, but a line of double flexure ; equal intensities do not correspond to equal dips; the directions of the horizontal needle are far from all converging to one point ; and soon. A very slight consideration is sufficient therefore to show the inadmissibility of this hypothesis. One of the above propositions is however still employed as an approximation in deducing the line of no dip from observations of dips of small amount made at some little distance from it. About eighty years ago, Tobias Mayer used a similar hypo- thesis, but with this modification; that instead of supposing the infinitely small magnet at the centre of the earth, he placed it at about the seventh part of the earth’s radius from the centre ; at the same time (probably in order to avoid greater complica- tion in the calculations) he retained the wholly arbitrary suppo- sition, of the plane perpendicular to the axis of the magnet pass- ing through the centre of the earth. In this manner, on a com- parison of the observed variations and dips, at a very small num- ber of places it is true, he found them agree very well with his calculation. A more extended comparison would have shown that this hypothesis did not afford a much better representa- tion than the first-mentioned one, of the whole phenomena of the dip and declination. No observations of the intensity had been at that time made, at least as far as we know. Hansteen went a step further, by the endeavour to represent the phenomena on the hypothesis of two infinitely small eccen- tric magnets of unequal strength. The decisive test of an hypo- thesis must always be the comparison of its results with those of experiment. Hansteen compared his with observations at forty-eight different places, amongst which however there were only twelve at which the intensity had been determined, and TERRESTRIAL MAGNETISM. 187 only six complete in the three elements. In these comparisons we find in the dip differences of 13° between calculation and ob- servation *, If these differences are greater than are admissible in a satis- factory theory, one cannot avoid drawing the conclusion, that the magnetic conditions of the earth are not such as to admit of re- presentation by means of a concentration in either one or two infinitely small magnets. It is not denied that with a greater number of such fictitious magnets, a sufficient agreement might be ultimately attainable; but how far such a mode of solving the problem might be advisable is quite a different question. The calculations are extremely laborious even with two magnets; with an increased number they would probably present insuper- able difficulties. It will be best to abandon entirely this mode of proceeding, which reminds one involuntarily of the attempts to explain the planetary motions by continued accumulation of epi- cycles. In the present treatise it is my purpose to develope the gene- ral theory of terrestrial magnetism independently of all particu- lar hypotheses as to the distribution of the magnetic fluids in the body of the earth; and to communicate the results which I have obtained from the first application of the method. Imper- fect as these results must be, they give an idea of what may be hoped for in future, when trustworthy and complete observations from all parts of the earth shall be obtained, and employed in renewed and more refined attempts. 1. The force which at each part of the earth imparts a certain direction to a magnetic needle suspended by its centre of gravity, (supposing it free from all extraneous influence, such, for ex- ample, as that of another artificial magnet, or the conductor of a galvanic current,) is termed the earth’s magnetic force, in so far as the source whence it is derived is to be sought for in the earth itself. It may indeed be doubted, whether the seat of the proxi- mate causes of the regular and irregular changes which are hourly taking place in this force, may not be regarded as exter- * Tn the declination there is even a difference in one instance of 29 degrees ; but it is proper to estimate the error of the calculation, not by the number of degrees of declination, but by the true angular difference between the calcu- lated and observed directions, which in the case in question is 11} degrees. 188 Cc. F. GAUSS ON THE GENERAL THEORY OF nal in reference to the earth. We may hope, that from the ge- neral attention now directed to these phenomena, much light may shortly be thrown upon their causes. But it should not be forgotten that these changes are comparatively very small, and that there must therefore exist a much more powerful and con- stantly acting principal force, of which we assume the seat to be in the earth itself. A consequence which follows from this con- sideration is, that the facts which are to serve as the foundation on which the study of the principal force must be based, ought properly themselves to be first freed from the effects of the ano- malous changes. This can only be done by mean values, drawn from numerous and continued observations; and until we shall possess such purified results, from a great number of stations di- stributed over the whole surface of the globe, the utmost that can be looked for is an approximation, in which there must still re- main differences of the order of these anomalies. 2. The foundation of our researches is the assumption, that the terrestrial magnetic force is the collective action of all the mag- netized particles of the earth’s mass. We represent to ourselves magnetization as a separation of the magnetic fluids. Admitting this representation, the mode of action of the fluids (repulsion- of similar and attraction of dissimilar particles inversely as the square of the distance) belongs to the number of established physical truths. No alteration in the results would be caused by changing this mode of representation for that of Ampére, whereby, instead of magnetic fluids, magnetism is held to con- sist in constant galvanic currents in the minutest particles of bodies. Nor would it occasion a difference if the terrestrial magnetism were ascribed to a mixed origin, as proceeding partly from the separation of the magnetic fluids in the earth, and partly from galvanic currents in the same; inasmuch as it is known, that for each galvanic current, may be substituted such a given distribution of the magnetic fluids in a surface bounded by the current, as would exercise in each point of external space pre- cisely the same magnetic action as would be produced by the galvanic current itself. 3. For the measurement of the magnetic fluids we take, as in TERRESTRIAL MAGNETISM. 189 the Intensitas Vis Magnetice, &c., for our positive fundamental unity, that quantity of northern fluid which at the unit of di- stance exercises on an equal quantity of the same fluid a moving force equivalent to what we assume as unity. When we speak of the magnetic force which in any point of space is produced by the action of the magnetic fluid elsewhere, we always mean to speak of the moving force which is there ex- ercised on the unity of the positive magnetic fluid; therefore in this sense the supposed magnetic fluid ~ concentrated in a point exercises at the distance p the magnetic force 3 , of either repul- sion or attraction in the direction p, according as yp is positive or negative. Representing by a, b, c, the co-ordinates of # in relation to three rectangular axes, and by 2, y, and z, the co-or- dinates of the point where the force is exercised, so that p= Vv (lv—a) + (y— 4)? + —e])5 and resolving the force in parallels to the co-ordinate axes, the components are w(e—a) wly—8) wle—0) 3 2 3 2 3 2 P p p which, as is easily seen, are equal to the partial differential co- efficients of — ; relatively to x, y, and z. If besides yz, there are also in operation other portions of the magnetic fluids p!, w!’, !", &c., concentrated in points, of which the distances from the spot where the force is exercised are p’, p', pl”, &c., then the components of the whole resulting mag- netic force, parallel to the co-ordinate axes, are equal to the partial differential co-efficients of ! " 7 (F454 r+ by + &e.), relatively to x, y, and z. 4. Hence may easily be shown what magnetic force is exercised in each point of space by the earth, however the magnetic fluids may be distributed therein. Imagine the whole volume of the earth, as far as it contains free magnetism (that is to say, sepa- rated magnetic fluids), to be divided into infinitely small ele- ments; designate generally the quantity of free magnetic fluid VOL, Il. PART VI. N 190 Cc. F. GAUSS ON THE GENERAL THEORY OF contained in each of these elements by d yz, in which the southern fluid is always considered as negative ; call p the distance of d uw from a point in space, the rectangular co-ordinates of which may be 2, y, 2; lastly, let V denote the aggregate of ch compre- hending with reversed signs the whole of the magnetic particles of the earth: or say dw V= — J —. Hi p Thus V has in each point of space a determinate value, or it is a function of x, y, z, or of any other three variable magnitudes, whereby we may define points in space. We then obtain, by the following formule, the magnetic force ~ in every point of space, and the components of yf, parallel to the co-ordinate axes, which we shall call &, », &, dV dV dV b= Get mage ee 5s I shall first develope some general propositions which are in- dependent of the form of the function V, and are worthy of at- tention from their simplicity and elegance. The complete differential of V becomes rr eee ge TN CNy sty Rt a v dy z =Edxa+ndy+Cdz. If we designate by ds the distance between the two points to which V and V + d V belong, and by @ the angle which the di- rection of the magnetic force yy makes with ds, we have : dV=vywcos 6.ds, because as 5, rm ah are the cosines of the angles which the di- dx dy dz rection of y makes with the co-ordinate axes, so aah es are the cosines of the angles between ds and the same axes. Therefore ae is equal to the force resolved in the direction of ds; the same follows from the equation ote & if we bear in mind that the co-ordinate axes may be arbitrarily chosen. TERRESTRIAL MAGNETISM. 191 6. If two points in space P°, P', be connected by an arbitrary line, of which ds represent an indeterminate element, and if, as before, @ signify the angle between d s and the direction of the magnetic force there existing, and 7 its intensity, then Sy 00s @.ds=V’'—V°® if we extend the integration through the whole line, and desig- nate by V°, V’, the values of V at the extremities. The following corollaries of this fruitful proposition deserve especial notice :— I. The integral aye a cos. 0. ds preserves the same value by whatever path we proceed from P® to P’. II. The integral fy cos @.ds, extended through the whole length of any re-entering curve, is always = 0. Ill. In a re-entering curve, if @ is not throughout = 90°, a part of the values of @ must be greater and a part must be less than 90°. ¥- Those points of space in which V has a value greater than V°, are divided from those in which the value of V is less than V°, by a surface in all the points of which V has one determinate value= V%, It follows from the proposition in Art. V., that in each point of this surface the magnetic force has a direction perpendicular to the surface, and towards the side where the higher values of V are found. Let ds be an infinitely small line perpendicular to the surface, and V° + d V° the value of V at its other extre- mity; then the intensity of the magnetic force will be = ae The series of points for which V = V° + d V°, form a second surface infinitely near to the first, and at different points in the whole intervening space the intensity of the magnetic force is in the inverse ratio of the distance apart of the two surfaces. * If the function V could have any arbitrarily chosen form, then in parti- eular cases a maximum or a minimum value of V might correspond to an in- sulated point, or to an insulated line, around which only greater or only less values might be found, or it might correspond to a surface on both sides of which there might be greater or on both less values. But the conditions to which the function V is subjected do not allow the occurrence of such excepted cases. A full development of this subject, as it is unnecessary for our present object, must be reserved for another occasion. N 2 192 C. F. GAUSS ON THE GENERAL THEORY OF Let V alter by infinitely small but equal steps. A system of surfaces will be produced, dividing space into infinitely thin strata, and the inverse ratio of the thickness of the strata to the intensity of the magnetic force will then hold good not only for different points in one and the same stratum, but also for dif- ferent strata, 8. We will now take into consideration the values of V on the surface of the earth. At a point P of the earth’s surface let y be the intensity ; P M the direction of the whole magnetic force ; w the intensity, and P N the direction of the force projected on the horizontal plane, or P N the direction of the magnetic meridian, meaning thereby the direction indicated by the north pole of the mag- netic needle ; 7 the angle between P M and P N, or the dip; 9, ¢, the angles formed by the elementary portion ds of a line on the surface of the earth and the directions PM, PN. Lastly, V and V + d V correspond to the two extremities of ds. We have consequently cos 8. = cosi cos t, a = cos i. And the equation in Art. V. becomes dV=acost.ds If two points on the earth’s surface P° and P’, at which V has the value of V° and V', are connected by a line traced on the surface of the earth of which d s is an indeterminate element, then Sfocost.ds= ia si if the integration be extended through the whole line; and it is plain that three corollaries hold good similar to those in Art. VI., namely, I. That the integral Sf wcost.ds keeps the same value by whatever path you proceed on the surface of the earth from P° id. II. The integral f cos ¢. ds throughout the whole length of a closed line on the surface of the earth is always = 0. III. In such a closed line, unless throughout its course ¢ = 90°, a part of the values of ¢ must necessarily be acute and a part obtuse. ; Propositions I. and II. of the foregoing article (which, pro- TERRESTRIAL MAGNETISM. 193 perly speaking, are only different modes of expressing the same thing) may be tested, at least approximately, by a reference to observation. Let P°, P’, P!....P® be a polygon on the surface of the earth, the sides of which are the shortest lines that can be drawn between their respective extremities, and are therefore portions of great circles, the earth being here considered simply as a sphere. Let o°, ’, w', &c. be the intensities of the horizontal magnetic force at the points P°, P', P", &c.; further, let 5°, &, 6, &c. be the declinations reckoned in the usual manner, west of north as positive, east of north as negative; lastly, let (01) be the azimuth of the line P® P! at P®°, reckoned in the customary manner, from the south by the west; in like manner (10) the azimuth of the same line taken backwards at P’, and so on. Let it be observed that ¢ alters continuously in each of the sides of the polygon, but suddenly at the corners, where there- fore ithas two different values ; for example, at P, ¢ has the value (10) + é', in consideration that P! is the end of the line P° P’; and the value of 18° + (12) + &, in regard that it is the begin- ning of P! P", We may consider the approximate value of the integral va w cos ¢.ds, extended through P°® P’, to be 4 (w° cos 7° + w’ cos dt’). P® P’, where 7° and ?’ signify the values of ¢ at P® as the beginning, and at P! as the end of P® P’, This approximation is all that can be obtained, because we have the values of @ and ¢ only at the extremities P° P', and is deserving of confidence in propor- tion to the shortness of the line. The given expression is, in our notation, = 4 (a! cos ((10)) + 8’) — w° cos ((01)) + 6° ]). PP’. In like manner, the approximate value of the integral, extended through P! P", is = 1 (0! cos ((21) + 8") — ow’ cos ((12) + 8']). P' P", and so on through the whole polygon. Therefore, for a triangle our proposition gives the approxima- tively correct equation w° (P° P' cos ((01) +6°) — P® P" cos ((02) + 8°]) + o! (P’ P'"'cos ((12) + 8’) — P® P’ cos ((10) + 8) + o! (P° P" cos ((20) + 8") — P’ P" cos ((21) + 8"]) =0. 194 Cc. F. GAUSS ON THE GENERAL THEORY OF It is obvious that in this equation the units of intensity and of distance are arbitrary. 10. As an example, we will apply the formula to the magnetic ele- ments of Gottingen | 6° = 18° 38! | ® = 67° 56! ap? = 1:357 Milan § = 18 33 | # =63 49 | W = 1-294 Paris S' = 22 04 | = 67 24 | al = 1:348 whence it follows that w° = 0°50980 wo! = 0°57094 ow! = 0°51804. Taking the geographical position of Gottingen | 51° 32! latitude | 9° 58' longitude from Greenwich Milan 45 28 9 O09 Paris 48 52 2 21 and performing the calculation for a spherical surface only, we find (01) = 5° 11! a Rai a se (10) = 184 35 35 fio he (12) = 198 47 31 - ee | (a1) = 30d agit Jota eae OS (20) = 238 20 20 ee: - (02) = 64 10 12 oe oo eee Substituting these values in our equation, and those given above for 5°, d!, 6", we have O = 17556 w° + 2774 w! — 20377 o", or, o! = 0°86158 w° + 0:13613 a’. Hence we deduce from the observed horizontal intensities at Gottingen and Milan, that at Paris w!! = 0°51696, agreeing al- most exactly with the observed value 0°51804. It is easily seen that if we permit ourselves to take the di- stances P°, P", &c. instead of their sines, the above formula can be expressed immediately by the geographical longitudes and latitudes of the places. Viale The line on the earth’s surface, in all points of which V has the same value = V°, divides generally speaking the parts of the surface in which the value of V is greater than V°, from those in TERRESTRIAL MAGNETISM. 195 which it is less. The direction of the horizontal magnetic force in each point of this line is obviously perpendicular to it, and towards the side where the greater values of Vare found. Ifds be an infinitely small line in this direction, and V°+dV° the 0 value of V at the other extremity of this line, then ~_ is the in- tensity of the horizontal magnetic force at this place. As here also the series of points corresponding to the value of V= V° + dV°® forms a second line situated infinitely near to the first, | and thus marks out on the surface of the earth a zone, within which the values of V are between V° and V° + d V°, and where the horizontal intensity is in an inverse ratio to the breadth of the zone; so by making V vary by infinitely small but equal steps from the lowest value on the surface of the earth to the highest, the whole surface of the globe becomes divided into an infinite number of infinitely narrow zones, the direction of the horizontal magnetic force being everywhere perpendicular to the dividing lines, and its intensity being in an inverse ratio to the breadth of the zone at the place in question. The two extreme values of V correspond in this point of view to two points, in- closed by the zones, at which the horizontal force is = 0, and where therefore the whole magnetic force can only be vertical : these points are termed the magnetic poles of the earth. The lines dividing the zones are no other than the intersections of the surfaces considered in Article VII. with the surface of the earth, whilst it is only at the poles that they are in contact with it. 12. The form of the system of lines described in the above article is strictly but the simplest type, which might be subject to many exceptions were we to take into account every possible distribu- tion of magnetism in the earth. We shall not, however, exhaust this subject here, but shall only add a few elucidatory remarks as to the cases of exception. The magnetic condition of the earth, no doubt is such, that the form of the system of lines on its surface corresponds to the description. At least there are cer- tainly no exceptions on the great scale, though probably there may occur local ones. Some philosophers have considered the earth as having two north and two south magnetic poles, but it does not appear that an essential condition was previously ful- 196 Cc. F. GAUSS ON THE GENERAL THEORY OF filled, by a precise definition being given of what should be understood by a magnetic pole. We intend to apply this de- nomination to each point of the earth’s surface where the hori- zontal intensity = 0: where therefcre, speaking generally, the dip = 90; but including the singular case, did it exist, where the total intensity=0. If we were to give the name of magnetic poles to those places where the total intensity is a maximum (2. e. greater than anywhere in the surrounding vicinity), it must not be forgotten that this is something quite different from the above definition ; that neither the situation nor the number of these last-named points have any necessary connexion with those of the points first spoken of; and that it tends to confusion when dissimilar things are called by the same name. If we look away from the actual condition of the earth and take the ques- tion in its generality, there may certainly exist more than two magnetic poles; but it does not appear to have been noticed that if, for example, two north poles exist, there must necessarily be between them yet a third point, which is likewise a magnetic pole, but is properly neither a north nor a south pole, or is both if that expression be preferred. A consideration of our system of lines will best serve to elucidate this subject. If the function V have at a point of the earth’s surface P* a maximum value V*, and all around smaller values, then a series of progressively de- creasing values will correspond to a system of rings, each of which will inclose all the preceding ones, together with the point P*, and on each of these rings the direction of the horizontal magnetic force, or that of the north pole of the magnetic needle, will be inwards t. This is the characteristic mark of a magnetic northt pole. It is clear that the rings may be made so small, or the cor- responding values of the function V may differ so little from V*, that any other point may be excluded. We will designate by S the space included by all the points on the surface of the earth at which the value of V is greater + These rings, themselves assumed as infinitely small, are not necessarily cir- cular, but generally speaking oval, so that the normal direction of the mag- netic needle in reference to them only coincides with the direction towards P* at four points of each ring. Great error may be involved, therefore, if with- out further precaution, the intersection of the prolongations of two compass di- rections at considerable distances is assumed to be P*. { We conform here to the mode of speaking in common usage, according to ~ which the point established by Captain James Ross is so designated, although pro- perly speaking it is a south pole, when the earth itself is considered as a magnet. TERRESTRIAL MAGNETISM. 197 than a given value WV. It is clear that S may either be one con- nected surface or several detached spaces, and that V = W, on the bounding lines or lines which separate S from other parts where V is less than W; by increasing or diminishing WV, we en- large or contract the space 8. Now let us assume P** to be a second point of similar pro- perties to P* so that at it also V may have a maximum value = V**, As according to what has been before noticed, W may have a value less than V*, and differing from it by so small an amount that P** shall fall outside that part of S in which P* is situated ; then if we arrange (as we may do) that V** shall not be less than V’*, it will be greater than W, and P** will neces- sarily also belong to a part of S. Thus P* and P** will both be situated in S, but in separate portions of it. On the other hand, it is evident that W may be taken so small that P* and P** shall both be situated in one connected part of S; for by only taking W small enough, S may be made to embrace the whole surface of the earth. If then W be made to pass progressively through all the values between the first and the second values spoken of, there must be amongst them one which we will call = V***, characterised by being the lowest at which P* and P** are still situated in sepa- rate portions of S, which separate portions will unite whenever W is diminished further. If this union occur at a point P***, the bounding line on which V = V*** will have the form of an 8, crossing at that point ; where also we may easily satisfy ourselves that the hori- zontal intensity must = 0. In fact, the crossing either does or does not take place under an angle of sensible amount. In the first case, the horizontal force, if it be not=0, must be directed in the normal to the two different tangents, which is absurd; in the second case, in which the two halves of the 8 touch each other at P***, or would have the same tangent, the force normal to this tangent could only be directed towards the interior of one half surface of the 8, which involves a contradic- tion, as the value of V increases towards both sides; therefore P*** is a true magnetic pole according to our definition, but must be considered as a south pole as regards the points nearest to it inside the two openings of the 8, and as a north pole as re- gards the points which lie outside. Figure 1. illustrates this form of the system of lines. 198 Cc. F. GAUSS ON THE GENERAL THEORY OF If the junction take place at two different points, what has been demonstrated for one point would hold good for the two ; and one may easily see that inside the space inclosing P* and p** an insular space would be formed, which would gradually contract itself as W was diminished, and would necessarily at length resolve itself into a true south pole. The case is similar when the junction takes place at three or more separate points; but if it take place at once on a whole line, then the horizontal force must disappear on all the points of that line. It is evident that the assumption of two south poles would in like manner necessitate the existence of a third polar point, which would be neither a south pole nor a north pole, or rather would be both at once. 13. From what has been developed in the foregoing article, its ap- — plication to many conceivable exceptions from the simplest type of our system of lines will be readily understood. The whole > of the points to which a certain value of V corresponds, may be a line consisting of several portions, of which each returns” back into itself, but which are quite separate from each other 5 it may be aline crossing itself; lastly, it may be a line having on > both sides spaces where V is greater than on the line, or where |} it is less. a We may assert that on the earth there are, on the great scale, no deviations of such a nature from the simplest type. TERRESTRIAL MAGNETISM. 199 Local deviations, indeed, may well be supposed to exist. Mag- netic masses near the surface, though producing no sensible effect at any considerable distance, may obscure and even obli- terate the regular progress of the terrestrial magnetic force in their immediate vicinity. In the simplest case the system of lines in such a district might take the form represented in Figure 2. eee 14. After this geometrical representation of the relations of the horizontal magnetic force, we proceed to develope the mode of submitting them to calculation. On the surface of the earth V becomes a simple function of two variable magnitudes, for which we will take the geographical longitude reckoned eastward from an arbitrary first meridian,—and the distance from the north pole of the earth ; we will designate the first of these, or the lon- gitude, by 4, and the second, or the complement of the geogra- phical latitude, by uv. Considering the earth as a spheroid of re- volution, of which the greater semi-axis = R, and the lesser semi- axis = (1—e) R, an element of the meridian is »,. (l=e)*R.du y ~ (1—(2 e—e*) cos wu?) *’ and an element of the parallel is 2 Rsinu.dxr ~ WW (1—(2 e—e*)cos u?) 200 C. F. GAUSS ON THE GENERAL THEORY OF Resolving the horizontal magnetic force into two portions, one of which, X, acts in the direction of the geographical meridian, and the other, Y, perpendicularly to that meridian,—and considering X as positive when directed towards the north, and Y as posi- tive when directed towards the west,—then x= mu — (2e—e*) cosu)3. dV (1—e)? Rdu dV a = — & 2) ne Seen s te =). Rsinu. dr The total horizontal force is then = / (X°+ Y*), and the tangent of the declination cs Ere. Neglecting the square of the ellipticity, e, the expressions become £ dV X= — (1 + (2 — 3 cosw’)e). Raa : dV Y= — (1 — €cos uw’) Wana Gor or, setting the ellipticity quite aside, Sts dV Ae Rdu dV ama Rsinu.dd* The data furnished by the observations which we possess are much too scanty, and most of them much too rude, to make it advisable at present to take into account the spheroidal form of the earth. It would not be difficult to do so; but it would complicate the calculations without affording any corre- sponding advahtage. We will therefore adhere to the last- mentioned formula, in which the earth is considered as a sphere, whose semi-diameter = R. ils. If X be expressed by a given function of w and A, Y can be be deduced from it @ priori. Let the integral f° X du=T, considering » as constant in the integration: it is then clear that if we differentiate in a similar . TERRESTRIAL MAGNETISM. 201 d(V+ RT) _ du 7 value independent of wu, or, what is the same thing, constant in all the points of a meridian,—it must hence also be absolutely constant, because all meridians converge and meet at the poles. If we call the value of V at the north pole = V*, then [i T= V: 3 E and hence manner according to w, 0; V+ RT having a is dT ~ sinu.dr This result may also be expressed as follows: Sa u dX Y= aT. mit: 16. This remarkable proposition, that, if the component of the ho- rizontal magnetic force directed towards the north be given for the whole surface of the earth, then the component directed towards the west (or towards the east) follows of itself, is true, conversely, only with a certain modification. If Y be expressed by a given function of uw and i, and if Urepresent the indeterminate integral sinwu. Yd, u being assumed constant in the integration, d(V+RU) then an = 0, or V+ RU has a value independent of X, and is, generally speaking, a function of w. Thus gi el) — U) dU = —— — Xis such a function; that is to say, the formula wal du du gives an imperfect expression for X, a part of it containing uw only remaining undetermined. This want would be sup- plied if, besides the expression for Y, we had also that for X, for some one given meridian, or to speak generally, for some line extending from the north to the south pole. We see there- fore that, if we know the component of the horizontal magnetic Sorce in the direction towards the west for the whole of the earth’s surface, and the component in the direction towards the north for all points of some one line extending from the north pole to the south pole, the latter component, for the whole of the earth’s surface, follows of itself. 202 Cc. F. GAUSS ON THE GENERAL THEORY OF LF. The foregoing investigations apply only to the horizontal por- tion of the earth’s magnetic force. In order to embrace the vertical force also, we must consider the problem in all its gene- rality ; therefore V must be regarded as a function of three vari- able magnitudes, expressing the position in space of an undeter- mined point O. We select for the purpose the distance r from the centre of the earth, the angle u which 7 makes with the northern part of the earth’s axis, and the angle , which a plane passing through r and the axis of the earth makes with a first meridian, counted as positive towards the east. Let the function V be expanded into a series, decreasing ac- cording to the powers of 7, and to which we give the following form : R?2 P°® R3 P' R4 Pll R pl V= re 4- + oo + pe +P mre ae &e. The co-efficients P®°, P', P","&c. are here functions of u and; in order to see how they are connected with the distribution of the magnetic fluid in the earth, let dy be an element of the earth’s magnetism, p its distance from O, and let r®, u®, r°, signify for d w the same as 7, wu, X for O. We have then V=— 18 = extended so as to include every dw; further p= W (r?—27r7?) cos ucosu® + sinwsin uw cos (A—A®) + 797°, and if : be developed in the series, Ltd 7 Pr Se ple Pip eee &e. ) then R? Po — — fT du, R? pl = — [Tr ap, “tyne at — for 7° 79 dw, &c. As T° = 1, and as according to the fundamental supposition with which we set out, the quantities of positive and of negative fluid are equal in every measureable particle in which they exist, and therefore are equal in the whole earth; that is to say, fe p = 0, it follows that i 0, or the first number of our series for V goes out. TERRESTRIAL MAGNETISM. 203 We see further that P’ has the form R? P'=acosu + BsinucosrA + ysin uw sin A, where a = - f cos CT a, B= = f sin w cos rA° red pz, ¥ =— / sinusind®r°dy. Therefore, according to the expla- nation laid down in page 13 of the Intensitas Vis Magnetice, —a,—P,—+, are the moments of the earth’s magnetism, in re- lation to three rectangular axes, of which the first is the axis of the earth, and the second and the third are the equatorial radii for longitudes 0 and 90°. The general formule for all co-efficients of the series for a may p be assumed as known; it is merely necessary for our purpose to remark, that in relation to uw, d, the co-efficients are rational integral functions of cos uw. sin w cos X, and sin w sin X, and of 7" of the second order, 7" of the third, &c. It is the same as to the co-efficients P", P'’", &c. The series for E and for V, converge, So long as 7 is not less than R, or rather, not less than the half diameter of a sphere, which includes all the magnetic particles of the earth. 18. The function V being composed of — S; —/ satisfies the fol- lowing partial differential equation : pera V d? V coe dV 1 d2V Sy taal (7 Fisinw 2 adhe? which is only transformation of the well-known equation os d*V a2 V d? V = de tay + de | where 2, y, z signify the rectangular co-ordinates of O. If we substitute the value of V, 3 pl 4 " 5 i V AP BAP. R ms ete ye + ee) rf it is clear that for the several Sa P', P!, P",i&es there will likewise be partial differential equations, of which the pedal expression is d? P®) d P™ i d2 P®) o= 1) P™ ee lard el Bia + 1) ts dus Gabi du sin wu? drn= 204 C. F. GAUSS ON THE GENERAL THEORY OF From this equation, combined with the remark in the pre- ceding article, we obtain the general form of P™. If we repre- sent by PP” the following function of w, (cos w fe @ ao) om he 2 (2n—1) (n—m) (n—m—1) (n—m—2) (n—m—3) n—m—4 ) om $4 (Qn=1), @a=s a then P™ has the form of an aggregate of 2 n + 1 parts, P® = 9 pm? + (9%! cos d+ AM sin r) PD! + (g™? cos 2+ h™ sin 22) P™ +, &e. + (g™” cosnd + h™ sin nd) P™", where g”®, g”', n™1, g™, &c. are determinate numerical co-effi- cients. oe If the magnetic force at the point O be resolved into three forces perpendicular to each other, X, Y, and Z, of which Z is directed towards the centre of the earth, and X and Y are tan- gential to a spherical surface concentric with the earth, passing through O, X directed northwards in a plane passing through O and the axis of the earth, and Y directed westwards in a plane parallel to the equator of the earth, then dV dV dv <= — 747 * > penne ee consequently, RB dP GP aie ane State ate dw te qe &) ee c.) rsnu\drn ‘rr dx ieee R® SHRP! See Z= = (oP! + +3 , &e.) On the surface of the earth X and Y are the same horizontal components which we have designated above by those letters ; Z is the vertical component, which is positive when directed downwards. The expressions for these forces on the surface of the earth are, then, : Pg Pl Pl aes ‘du* du +t du +, &e. ) ; TERRESTRIAL MAGNETISM. 205 ! iT] mt Bia) Go GAR! Wb" /n'd P 4,80.) sin u Pott dx ee dx Z=2P!+3P"44P"+,&e. 20. If we combine, then, with these propositions, the known theo- rem, that every function of X and w, which, for all values of 2. from 0 to 360°, and of uw, from 0 to 180°, has a determinate finite value, may be developed into a series of the form P° + P)+P" 4 Pl" +, &e. _the general member of which, P” satisfies the above partial dif- ferential equation,—that such a developement is only possible in one determinate manner,—and that this series always converges,— we obtain the following remarkable propositions. I. The knowledge of the value of V at all points of the earth’s surface is sufficient to enable us to deduce the general expres- sion of V for all external space, and thus to determine the forces X, Y, Z, not only on the surface of the earth, but also for all ex- ternal space. It is clearly only necessary for this purpose to develope into a series according to the above-mentioned theorem. In the sequel, therefore, unless it is expressly stated otherwise, the symbol V is always to be taken as limited to the surface of the earth, or as that function of X and uw which follows from the general expression, when 7 is made = R: thus V=R(P! + P" + Pl" 4, &e.) Il. The knowledge of the value of X at all points of the earth’s surface is sufficient to obtain all that has been referred to in Prop. I. In fact, according to Art. 15, the integral f™ Xdu = i ) ll V ° signifying the value of V at the north pole, and the developement of if “ X duinto a series of the form referred 0 to must necessarily be identical with Vo — Pl_ Pl _ Pl, &, Ill. In like manner, and under the considerations in Art. 16, it is clear that the knowledge of Y on the whole earth, combined with the knowledge of X at all points of a line run- VOL. Il, PART VI, Oo 206 C. F. GAUSS ON THE GENERAL THEORY OF ning from one pole of the earth to the other, is sufficient for the foundation of the complete theory of the magnetism of the earth. IV. Finally, it is clear that the complete theory is also de- ducible from the simple knowledge of the value of Z on the whole surface of the earth. In fact, if Z be developed into a series, Z= M+ Q+ QU + Q" 4, &e. so that the general member satisfies the often-mentioned partial differential equation; Q° must necessarily = 0, and P!=1Q,P" =1Q, Pl!" =1Q", &e. Ole On account of the simple nature of the dependence of the several forces X, Y, Z, on a single function V, and the simple relation which they bear to each other, they are far better caleu- lated to serve as a foundation for the theory, than the usual ex- pression of the magnetic force by the three elements, total in- tensity, inclination, and declination. Or rather, the latter mode, natural as it appears in itself when the question is solely that of comprehending the facts, cannot directly serve for the founda- tion of the theory (at least not for the first foundation) until it has been translated into the other form. In this view it would be very desirable that a general graphical representation of the horizontal intensity should be made ; partly because it would be more immediately useful for theory than the total intensity ; partly because, in far the greater number of cases, the horizontal intensity was originally that which was actually observed, the total intensity having been subsequently deduced from it by means of the dip. It is the more advisable to keep. the elements of the horizontal force unmixed, as they can be de- termined with extreme accuracy with the present instrumental means; at any rate, the observed horizontal intensity should never be suppressed when publishing the deduced total intensity, without at least giving the dip employed in the calculation; so that a person wishing to employ the horizontal intensity for the theory may either have, or be enabled to reproduce, the original observed numbers. Interesting as it would be to found the theory of terrestrial magnetism on observations of the horizontal needle only, and thus to anticipate the vertical part, or the inclination, it is at present TERRESTRIAL MAGNETISM. 207 much too soon to do so: the scantiness of the data which we now possess does not allow of our dispensing with the assistance of the vertical part. It is a confirmation of the theory, if we can show the agreement of the different elements when reduced to one principle. 22. Although we are a priori certain that the series for V, X, Y, Z, converge, nothing can be determined beforehand as to the de- gree of convergence. If the seats of the magnetic forces be limited to a moderate space around the centre of the earth, or if ere were such a distribution of the magnetic fluids in the earth as to be equivalent thereto, the series would converge very ra- pidly ; on the other hand, the further the seats of the magnetic forces extend towards the surface, and the more irregular the distribution, the slower we must be prepared to find the con- vergence. In the practical application, absolute exactness is unattainable ; we have to desire only a degree of approximation commensurate with the circumstances. The slower the convergence, the greater will be the number of members which must be taken into account to attain a certain degree of accuracy. ; Now, P! contains three members, and requires, therefore, the knowledge of three co-efficients g!°, g'!, h’1; P" requires five co-efficients; P’”’ seven; PV nine, &c. As we consider P', P", P'", &c. as magnitudes of the first, second, and third order, and so on, it is clear that if the calculation is to be pushed to magni- tudes of the order inclusive, the values of n® + 2 n co-efficients must be determined ; therefore, for example, 24 coeflicients, if we would go as far as the fourth order. Every given value of X, Y, or Z, for given values of u and 2), furnishes an equation between the co-efficients, whilst for each place where the complete elements of the terrestrial magnetic | force are known, three equations are given. If we could venture to assume that the members have a sensible influence only as far as the fourth order, complete observations from eight points would be sufficient, theoretically considered, for the determina- tion of all the co-efficients. But such a supposition can hardly be | ventured upon, and the accidental errors which beset all obser- vations, together with the neglected members of higher orders, o 2 208 Cc. F. GAUSS ON THE GENERAL THEORY OF might have a very injurious effect on the results of the elimi- nation*. To diminish the unfavourable effect of these circumstances, the number of series of observations from stations well distri- buted over the whole globe ought to be much greater than that of the unknown values, and these should be derived from the observations by the method of least squares. As all the equa- tions are only linear, the process would, it is true, be uniform ; but the extent of the labour, arising from the great number of unknown values and equations, would be such as might well deter the most courageous calculator from undertaking it in this form, especially as the result might be wholly vitiated by the in- ~ troduction either of defective observations or of accidental errors of calculation. 23. There is another mode of proceeding, which, as it is free from a part of these difficulties, appears better adapted for a first trial. We shall develope it in this place without omitting to notice objections to which its application may be liable in the pre- sent state of the inquiry. This method supposes the knowledge of all three elements at points so grouped on a sufficient number of parallels as to divide them into a sufficient number of equal portions. The numerical values of X, Y, and Z, are to be first deduced from the given elements of the usual form. The values of X, Y, Z, are then brought by the known method in each parallel to the form X=k+k' cost + K' sind + k" cos2X + K" sin22r + kK" cos 3X + K" sin 3X 4+, &e. Y=1+U cosX+ Ll snd + 2! cos2X+4 L" sin2nr +1 cos3X + Ll" sin3 Xr 4+, &c. Z=m-+m cosr+ M'sind + m'cos2X + M"sin 2X +m!" cos3 + M" sin3 r+, &e. We then obtain as many values for each of the co-efficients k, i, m, k', &c., as there are parallels of latitude under consideration. Theory would give in each parallel 7 = 0 ; therefore the values of 7 which result from the calculation furnish a kind of measure * In such a mode of determination, the effect of these circumstances would be least injurious if the eight points were distributed symmetrically on the sur- face of the earth; that is to say, if they coincided, or nearly so, with the corners of a cube inscribed in the globe. TERRESTRIAL MAGNETISM. 209 of the degree of uncertainty which still attaches to the funda- mental members. From the equations 1-0 2-0 3-0 pangs Pg SEO Ea m = 2g Pl0 + 3 y20 P20 4 4 g30 p30 4, &e., the total number of which is double the number of the par- allels, we have to obtain, by the method of least squares, (after 10 , &e., and in P'°, &c. the corresponding nu- . «A a substituting in Fi merical values of w,) as many of the co-efficients g', g*°, g*°°, &c. as require to be taken into account. In like manner the equations ll — Bg SEO 4g ge Ey ot 4, be. P i pi ll 21 371 ae sna’? Sines) ai Fo +, Be. m! = 2 gi! Pl + 3g?) P21 4 4 93) P31 4, &e., the number of which is three times as great as the number of parallels, serve to determine the co-efficients g'', g*', g®}, &c. And the following, Vl 21 31 pr pl ps See Ae eee a1 3-1 mite sing * _ aaa sin u rhs ie. oan ges PPS Bel pel 2 4 Rel Pol 1. &e, determine the coefficients h!", h*', h*1, &c. Further, the equa- tions d P?? — fll — 22 lied du piges SEE pigs Ee ee du du = — 6030 3 = — 0178 Wi —= + 47°794 fA = + 4127 WW’ = + 64112 LAS = 32175 These numbers, which may be considered as the elements of 212 Cc. F. GAUSS ON THE GENERAL THEORY OF the theory of terrestrial magnetism, are used both here and in the formation of the table to be described in the sequel, just as they were given by calculation, without omitting decimals. To any one conversant with calculation it is superfluous to remark, that these fractional parts have in themselves no value, as we are still far from being able to eliminate with certainty even the integers. But it is important that the observations should be closely compared with one and the same definite system of ele- ments ; and, as by leaving out decimals nothing would be gained in point of convenience in computing, there was no reason for altering in any respect the elements given by calculation. 27. The expression for V, developed according to the above num- bers, is as follows: for the sake of brevity e stands for cos wv, and J for sin uw. t= —1:977 + 937°103 e +71'245 e — 18°868 e — 108°855 e* + (64:437 — 79°518 e + 122°936 e? + 152°589 e?) f cos x 4 (— 188°303 — 33°507e + 47°794 & + 64112 &) fsind + (7:035 — 73193 e — 45°791 e) f? cos 20 + (—45:092 — 22°766 e — 42°573 e*) f* sn 2X + (1396 + 19°774 e) f3 cos 3% + (— 18°750 — 0178 e) f? sn 3A + 4127 f4cos 4nr + 3°175 f? sin 4X. We may here add the completely developed expressions for the three components of the magnetic force. X = (937°103 + 142:490 e — 56°603 & — 435°420 e°) f +(— 79°518 + 181°435 e — 298°732 e? — 368°808 e* + 610°357 e*) cosX + (— 33°507 + 283°892 e + 259°349 e® — 143°383 & —256°448 e+) sinX + (— 73193 - 105°652 e + 219°579 e® + 183-164 &) fcos 2 + (—22°766 + 175°330 e + 68-098 e° — 170°292 e°) f sin 2X + (19°774 — 4°188.e — 79096 e”) f* cos 3X + ( — 01178 + 56°250e + 0°716e*) f? sin 3X — 16508 ef? cos 4X — 12°70lef*? sin 42 Y = (188303 + 33°507 e — 47°794 e? — 64'112 e*) cosr TERRESTRIAL MAGNETISM. 213 + (64437 — 79°518e + 122°936 e? — 152°589 e?) snr + (907184 + 45°532 e — 185°46 e*) fcos 2X + (14070 — 146°386 e — 91°582e?) fsin 2X + (56°250 + 0°534e) f? cos3 Xr + (4188 + 59°322e) f? sin3 — 12°701 f? cos 42 + 16508 £3 sin 40 Z= — 24:593 + 1896.847 e + 400°343 e? — 75°471 & — 544:275 e+ + (79°700 — 107°763 e + 491°744 ce? — 762°946 e%) fcosr + (—395°724 — 155:473 e + 191°176 e? + 320°560 e?) f sin A + (34°187 — 292°772 e — 228°955 e*) f? cos 2X + (— 147:439 — 91:064e + 212°865e?) f? sn 2X + (5°584 + 98°870e) f2 cos 3X + ( — 75°000 — 0:890 e) f3 sin 3X + 20°635 f4 cos 4X + 15°876 f* sin 42. After these components have been calculated for a given place, we obtain in the following manner the several parts of the de- termination of the magnetic force, according to the customary form. Let 6 be the declination, 2 the inclination, the total, and w the horizontal intensity. Determine first 6 and by means of the formule X =o cos 6, Y = w sin 6, and then i and yy by means of the following formule : o®=wcosi, Z= sini. 28. As the formule for X, Y, Z, contain 71 members, their immediate calculation is a considerable labour. Its repetition for a great number of places appears the more alarming, con- sidering that we could hardly hope to be secure from the pos- sibility of mistake without going twice over the whole. But little would be gained by suppressing all those members of which the co-efficients are less than an integer, or even less than 10 integers, for the remaining members would still amount to 65. But as the whole value of the work would remain uncer- tain if not tested by a considerable number of actual observations, I have encountered the labour of calculating a table, by the assistance of which the work will be in the highest degree 214 Cc. F. GAUSS ON THE GENERAL THEORY OF abridged and facilitated, and at the same time the important object of security against errors of calculation will be materially promoted. For the construction of the table the values of the coefficients were brought into the following form : X = @& + a’ cos (A + 4’) + a cos (2X + A") + a!" cos (3% + A") + al cos (4X + A) Y = J cos () + B’) + b" cos (2X + B") + b" cos (3% + Bl) + bY cos (4 + BY) Z= © +c cos (x + C’) + ec! cos (2X + C") + ce cos (3% + Cl") + cl’ cos (4X + CTY). The first table contains those parts of X and Z which are in- dependent of X. In the four next tables are found the values of the auxiliary angles 4’, A", &c., and the logarithms of a’, a", &c., all for the several degrees of latitude ¢ = 90° — u. The table is placed at the end of the memoir. The calculation for Gottingen is given as an example. For latitude 51° 32! we find from the tables : a = + 500°8 c° = + 1465°2 log a! = 228980 | log J = 2-18900 | loge = 220204 log a! = 1°79403 | log J’ = 2°03220 | log ce" = 212777 log a!” = 1:32522 log db! = 1°46845 log ec” = 1°43199 log atV= 059391 log b'%= 0°70016 log clY= 0°59091 A! = 249° 30 B! = 358° 24! C' = 105° 44’ a Bll Ag Bil = 64 50 C" 22 1G, a Al = 234 10 B= 318 13 Ci'= 42 22 AN= 142 26 BW = 232 26 Civ — 322.26 And for longitude 9° 56/3, the parts of X, ¥, Z, are found as follows : Z 500°8 + 14652 Sota | + 15289 — 68:99 + 54°76 + 9°92 — 13367 — 22) + 28°77 + 8:27 — 3-92 + O19 +o B90 X = + 513°72 Ft (0077, Z= + 127471 TERRESTRIAL MAGNETISM. 215 The farther calculation then gives : 6 = + 20° 28! .lop.» = 2°73907. += + 66 43 ay = 1387-6, or, in the unity commonly employed, ar = 1°3876. 29. The following table contains the comparison of our formule, with observations at 91 stations in all parts of the earth. As the three maps from which we have taken the data for our calcu- lation are intended to represent the phenomena for the most recent epoch, we have included in our comparison only very re- cent observations, and we have taken, by preference, observations at those stations where all the three magnetic elements were ob- served. We are not at present in a condition to demand that the observations should be strictly cotemporaneous, unless we would see our stock reduced to a very small number. 216 Cc. F. GAUSS ON THE GENERAL THEORY OF Declination. Latitude. | Longitude. | computed. | Observed. | Difference. Oo t] oO ! oO i 0 i] o 1 | Spitzbergen........ +79 50} 11 40/496 31 [495 12 |+ 19 2 | Hammerfest........ 70 40 | 23 46 |+12 23 |+10 50 |4+ 1 33 3 | Mag. Pole of Ross. | 70 5 | 263 14 |\—22 23 4 | Reikiavik .......... 64 8! 3388 5 +40 12 |4+43 14 |— 3 2 Bi dakittsk.. 25 sccc6.s. 62 1| 129 45/+ 0 5 /+ 5 50 |— 5 45 6 | Porotowsk ......--| 62 1) 181 50|+ 0 4 /+ 4 46 |— 4 42 7 | Nochinsk........++ 61 57 | 1384 57|/— 0 3 |+ 211 |— 214 8 Tschernoljes SAVES 61 31 | 136 23 0 0O}+ 3 30 |— 3 30 9 | Petersburg ...-.... 59 56| 3019/+ 6 47 |+ 6 44/4 0 3 10 | Christiania ........ 59 54 10 44 +19 55 |+19 50/4 0 5 HI Ochotsk? 7.2.20... 59 21 | 143 11 |— 0 18 |+ 2 18 2 36 12) obolsk: 2cisare se 58 11] 68 16 |— 7 19 |—10 29 |+ 3 10 13 | Tigil River ........ 58 1] 158 15 ||— 4 20 |— 4 6 |— 0 14 HORE scree ot fee ees 57 3 | 224 35 |—28 45 |—28 19 |— 0 26 5H diaravesi. 2. Vee 56 54) 74 4 |\— 7 44 |— 9 36 |+ 1 52 16 | Catharinenburg....| 56 51 | 60 34 |— 5 20 |— 6 18 |+ 0 58 VAIPROMGE:. ¢ scar coe cee 56 30} 85 9 |— 7 21 |— 8 34/4 113 18 | Nishny Novogorod..| 56 19} 43 57 |+ 110 — 0 27 |+ 1 37 19 Krasnojarsk sede A 56 1 92 57 |i— 5 49 |— 6 40 |+ O 51 DOM KARAM e225, cereals 55 48} 49 7|—1 7 |— 2 22\/4 115 21 | Moscow .......... 55 46 | 37 37 + 4 26|4+ 3 2/4+ 1 24 22 | Konigsberg ........, 54 43 | 20 30 /+14 15 |+13 22 |+ 0 53 23 | Barnaul . ........| 53 20| 83 56|/— 7 0|— 7 25/4 0 25 24 | Uststretensk ...... 53 20 | 121 51 |/+ 1 29 |4+ 4 21 |— 2 52 25 | Gorbizkoi.......... 53 6/119 9|+ 1 5/|+ 2 54 |— 1 49 26 | Petropaulowsk ....| 53 0 | 158 40 |— 3 34 |— 4 6 |+ 0 32 27 \ublriupina....\>» ~=dGaa0 Inclination. | Exman, 1828,.. «.- 4s; <<) “7a Von Humboldt, 1829, . . 70 56 Wires Deore ar Sa a et Fedor, 1898, 5! < (eo. | Cpe 16. Catharinenburg. Declination. Hansteen, 1828,. . . — 6°27! Erman, 1828, «. . jvc 4s -— 3 3 Remke, 1836;3° .. “2 “) ~—_ es Inclination. Erman, 1828,. . .. . 69 24 Von Humboldt, 1829,. . 69 6 Fuss, 1830, 3? 3S. 40) .4ee Fedor, 1832, ....~.%s.n.- seine 17. Tomsk. Declination. Hansteen,1828,. . . — 8°39! Erman, 1829,. . . «98236 Inclination. Erman, 1829,. . . . . 70 59 Wuisy-1820, 55 52 2 ten FOyal to bo —_ TERRESTRIAL MAGNETISM. 18. Nishny Novogorod. Declination. Erman, 1828,. . . . — 0°46! PsA Sy wilh gies) eA, 8 19. Krasnojarsk. Declination. Hansteen, 1829,. . . — 6° 43! Brman,.1829,.055 ou. = 6. BT Bedor. LSS55-54- s. ae -3’ — % 26 Inchnation. Erman, 1829,. . . . . 7O 53 Bedor. USS5y) oso. sce at 88 20. Kasan. Inclination. Erman, 1828,. . . . . 68°21! Von Humboldt, 1829,. . 68 27 PUES I S50, ate he) ee 68 26 21. Moscow. Declination. Hansteen, 1828,. . . + 3° 3! Brmanjis28yeur. 94)... Be] Inclination. Erman, 1828,. . . . . 68 58 Von Humboldt, 1829,. . 68 57 30. Irkutsk. Declination. Hansteen, 1829,. . . — 1939! Erman, 1829,. . . . — 1 52 Puss, 1830069 maser, + —) 125 Inclination. Erman, 1829,. . .. . 68 7 Wass. 0830, ti. ws Se 6816 Huss WSs er eles ESS ae 36. Orenburg. Inclination. Von Humboldt, 1829,. . 64° 41! Fedor, 1832,. . .. . 64 47 44. Troizkosawsk. Declination. Hansteen, 1829,. . + 0° 5! Manian, TEI97! ies ts oe O33 Fuss, 1830, . e . © ay O 1 Inclination. Erman, 1829, .. . . 66 14 EUS LEO pees tes i's 6 24 Most of the determinations in the southern hemisphere are VOL, Il. PART VI. P 222 Cc. F. GAUSS ON THE GENERAL THEORY OF supplied by Captains King and Fitz Roy, and are taken from a little work by Sabine, (Magnetic Observations made during the Voyages of H. B. M.’s Ships Adventure and Beagle, 1826-1836.) The determinations for the several other stations are taken partly from the above-named sources, and partly from the following : 1. Spitzbergen. Observer, Sabine, 1823. (From his Account of Experiments to determine the Figure of the Earth.) 2. Hammerfest. The declination and inclination are the means of the determinations of Sabine, 1823 (Pendulum Expe- riments) ; and of Parry, 1827. (Narrative of an Attempt to reach the North Pole.) 3. Magnetic Pole, from Captain James Ross, 1831. (Phil. Trans. 1834.) 4. Reikiavik, from observations by Lottin, 1836, (Voyage en Islande.) 28. Berlin, from Encke, 1836. (Astronomisches Jahrbuch, 1839.) 38. Gottingen. The declination is for October 1, 1835 (Re- sultate fiir 1836, page 39); the inclination is reduced to the same epoch by interpolation between von Humboldt’s observation in 1826, and Forbes’ in 1837. 39. London, from observations communicated in manuscript. The declination, by Captain James Ross, for the mean epoch, April, 1838 ; and the inclination by Phillips, Fox, Ross, Johnson, and Sabine, for the mean epoch of May, 1838. 48. Paris, for 1835, from the Annuaire for 1836. 54. Milan, 1837, by Kreil. Communicated by him in ma- nuscript. 58. Naples, from observations by Sartorius and Listing. The intensity, which was determined according to absolute mea- sure, has been reduced to the common unity, by the application of the factor given in Article 31. 64. Madras, 1837, from observations by Taylor, taken from the Journal of the Asiatic Society of Bengal, May, 1837. 30. In judging of the differences between calculation and observa- tion, as shown in the foregoing tabular comparison, it must be remembered, on the one hand, that almost all the observations are charged both with the errors of observation, and with the influ- TERRESTRIAL MAGNETISM. 293 ence of the accidental anomalies of the magnetic force itself, and that they do not correspond to the same year* ; and, on the other hand, that our formule do not include members beyond the fourth order, whereas those of the following order may still be very sensible. When due weight is allowed to these circum- stances, the agreement between calculation and experiment ap- pears to be as satisfactory as we are entitled to expect from a first attempt. As our expression for x may therefore be safely regarded as coming near the truth, at least in its more important members, it has appeared worth while to form a graphical representation of the course of the numerical values of this function. This has been done in a map drawn by Dr. Goldschmidt, in three parts, the first on Mercator’s projection, passing round the globe, and including all the parallels between 70° north, and 70° south lat.; the other two being polar projections, extend- ing to lat. 65°. The corrections and additions which will arise from a fresh calculation resting on more perfect data, may, doubt- less, cause material alterations of position in these lines, parti- cularly in the high southern Jatitudes ; but no important change in the whole form of the system of lines can be supposed without such alterations in the expression for 4 as would destroy the agreement with existing observations. We are thus led to the important result, that the system of lines of equal values of V, on the surface of the earth, is actually comprehended by the simplest type described in Art. 13, and that consequently there are on the earth only two magnetic poles, apart from the possible case of local exception spoken of in Art. 13. * The last article presents instances of discordances between different ob- servers at one and the same place; I will notice some others, which are much greater than can with any degree of probability be attributed to yearly changes. The dip at Valparaiso was, in 1829, according to King, 40° 11’; in 1835, ac- cording to Fitz Roy, 38° 3'. In Mauritius the intensity was 1-096 in 1818, ac- cording to Freycinet, and 1/192 in 1836, according to Fitz Roy. The difference is still greater at Otaheite, where Erman’s intensity = 1:172 in 1830, and Fitz Roy’s, in 1835, = 1-017. Otaheite is a station of the highest importance for the future improvement of the elements: the difference between the two determinations made there by different observers, considerably exceeds the greatest difference between the computed and observed intensities in our eighty- SIX comparisons. Pp 2 224 Cc. F. GAUSS ON THE GENERAL THEORY OF The exact computation of the places of these two poles, ac- cording to our elements, gives them as follows : 1. In 73° 35’ north lat., 264° 21’ long. east from Greenwich, the value of the total intensity being = 1°701 in the unity in common use. 2. In 72°35! south lat., 152°30! long., the total intensity = 2°253, At the first of these points + hasits greatest value, = + 895°86 ; at the second its smallest value, = — 1030°24, According to Captain James Ross’s observation the north mag- netic pole falls 3° 30! to the south of its position according to our calculation, which gives at that place a direction of the magnetic force, differing 1° 12’ from observation, as may be seen in the table of comparisons. We must expect a considerably greater displacement of the position of the southern pole. At Hobart Town, which is the nearest station to this pole, calculation gives too low a dip by 3° 38’, as far as the observation can be de- pended upon. It seems probable, therefore, that the actual south magnetic pole is considerably north of the position given by our calculation, and that it may be looked for in about 66° lat., and 146° long. 31. The two points on the earth’s surface where the horizontal force vanishes, and which are called magnetic poles, may, it is true, be allowed a certain significancy on account of their relation to the form of the phenomena of the horizontal force all over the earth ; but we must be careful not to give them undue consider- ation. The chord which unites these two points has no signi- ficancy, and it would be a gross mistake to call it the magnetic axis of the earth. The only mode of giving a generally valid signification to the idea of the magnetic axis of a body is laid down in the 5th Article of the Intensitas Vis Magnetice, where it is understood to mean the straight line in which the moment of the free magnetism contained in the body is a maximum. In order to determine both the position of the magnetic axis of the earth in this sense, and the moment of the earth’s magnetism in relation to this same axis, we only require, as noticed in Art. 17, a knowledge of the members of the first order of V. According to our elements, Art. 26, P! = + 925°782 cosu + 89°024 sin u TERRESTRIAL MAGNETISM. 9295 cos X — 178°744 sin uw sin A, and — 925°782 R®, — 89°024 R’, 4+ 178°744 R3 are the moments of terrestrial magnetism with respect to the axis of the earth, and to the two radii for longi- tudes 0 and 90. In speaking of the earth’s axis, the direction towards the north pole is to be understood, and the negative sien of the corresponding moment shows that the magnetic axis makes with it an obtuse angle, or that its magnetic north pole is turned towards the south. The direction hence found for the magnetic axis is parallel to that diameter of the earth which is from 77° 50’ north lat., and 296° 29’ lon., to 77° 50’ south lat., 116° 29’ lon.; and the mag- netic moment in relation to this axis is = 947°08 R®. It must be remembered that in our elements the unity of intensity em- ployed is a thousandth part of the unity in common use. In order to obtain the reduction to the absolute unity establish- ed in the Intensitas Vis Magnetice, we must remark that in that work the horizontal intensity at Géttingen for the 19th of July, 1834, was found = 1:7748, which, combined with the dip 68° 1’, gives the total intensity = 4:7414. The total intensity, according to the unity employed above, was 1357. Thus the reducing factor is = 0°0034941, and the magnetic moment of the earth, expressed according to the absolute unity, = 3°3092 R?. As the millimetre is the unit of length employed in the above absolute unity for the earth’s magnetic force, R must.also be given in millimetres ; and, as the ellipticity of the earth need not be taken into account, it will be sufficient to consider R as the radius of a circle 40000 millions of millimetres in circum- ference. Hence the above magnetic moment will be expressed by a number of which the logarithm = 29,93136, or by 853800 quadrillions. By experiments made in the year 1832 (Jntensitas, Art. 21) the magnetic moment of a magnet bar, of a pound weight, was found to be, according to the same absolute unity, = 100877000. The magnetic moment of the earth is there- fore 8464 trillion times greater. Thus 8464 trillions of such magnet bars, with parallel magnetic axes, would be required to replace in external space the magnetic influence of the earth. Supposing the magnetism of the earth to be uniformly distri- buted throughout its volume, it would hence be equal to eight such bars (more exactly 7:831) for every cubic metre. This re- sult thus enounced preserves its meaning even, if instead of 9296 C. F. GAUSS ON THE GENERAL THEORY OF considering the earth as an actual magnet, we should prefer to ascribe terrestrial magnetism simply to constant galvanic cur- rents in the earth. But if we consider the earth as an actual magnet, we are obliged to ascribe to each of its portions, of the size of the eighth of a cubic metre, on an average, at least* as great a force of magnetism as that contained in one of the above- mentioned bars. Such a result will be an unexpected one to phi- losophers. 32. The manner of the actual distribution of the magnetic fluids in the earth necessarily remains undetermined. In fact, accord- ing toa general theorem which has been already mentioned in the 2nd article of the Intensitas, and will be treated of in greater detail at a future opportunity, we may substitute for any supposed distribution of the magnetic fluids in the interior of a body oc- cupying space, a distribution on the surface of the same space, which shall leave the effect on every point of external space pre- cisely the same. It may be easily concluded from hence, that one and the same action on all external space may be deduced from an infinite number of different distributions of the magnetic fluids in the interior. We are enabled to assign on the other hand that fictitious distribution on the surface of the earth, which shall be perfectly equivalent to the actual distribution in the interior, as regards the external resultant of the forces ; and the spherical form of the earth allows us to do so in a very simple manner. We may express the density of the magnetic fluid in each point of the earth’s surface, i.e. the quantum of the fluid which corresponds to the unit of surface, by the formula 1 /V a, (z aA ): or by — zB P'+ 5 P+ 7 Pl, + 9 PN, Be, The result of this formula will be hereafter exhibited by a graphical representation. We shall only notice here that it is negative in the northern and positive in the southern parts of the earth, but in such manner that the dividing line cuts the * Tn as far as we are not prepared to assume the magnetic axes of all the magnetized particles of the earth to be everywhere parallel to each other,—the more imperfect this parallelism, the greater must be the average force of mag- netism in the parts to produce the same total magnetic moment, TERRESTRIAL MAGNETISM. 297 equator twice (in longitudes 6° and 186°); its points most di- stant from the equator being in about 15° north and 15° south latitude: and further that in the northern hemisphere there are two minima, but in the southern hemisphere but one maximum. According to a cursory computation, these minima and this maximum are — 209'1 in 55° N, lat., 263° lon. — 200°0 in 71° N. lat., 116° lon. + 277°7 in 70° S. lat., 154° lon. These values are founded on the unity of our elements, and must therefore be multiplied by 0°0034941 to obtain their ex- pression in absolute measure. 33. It has been already said that our elements are to be regarded only as a first approximation. So considered, their agreement with the observations in Art. 29 is sufficiently satisfactory. It cannot be doubted that a much greater agreement would be ob- tained by an improved calculation, even with these observations. The only difficulty of such a calculation is its length, which would be still alarmingly great, even supposing it abridged by the introduction of such skilful methods as have been employed by astronomers in correcting the elements of the planetary and cometary paths. Although this difficulty might be easily sur- mounted by dividing the work amongst a number of computers, it does not appear advisable to undertake such an amended calcu- lation at present, when there is still so little certainty in the data from a great number of places which it would be important to employ. It will be preferable in the first place to pursue further the comparison of the elements with observations, whence the means will be afforded of giving much greater certainty to the general maps, than has been hitherto possible by the exclusively empirical mode. We may be allowed to give a few glances at the future pro- gress of the theory, the perfect realization of which may indeed be far distant. 34. For the satisfactory refinement and completion of the ele- ments, it will be requisite to make much higher demands than have been hitherto complied with, as to the data furnished by observation. Their accuracy at all the points employed ought 228 C. F. GAUSS ON THE GENERAL THEORY OF to be such as has hitherto been obtained at a very few only ; they should be cleared from the effect of irregular changes ; they should be all for the same epoch. It will probably be long before such demands are satisfied. Next to this the chief desideratum is to obtain complete ob- servations (i. e. including all three elements) from points in those large parts of the earth’s surface where such observations are still wholly wanting. Every new station will have for the gene- ral theory an importance proportionate in great measure to its distance from those we already possess. After a sufficient interval of time shall have elapsed, the ele- ments may be determined afresh for a second epoch, and their secular changes may be thence deduced. Manifestly it will be essential for this purpose to reject altogether the present measure of the intensities, and to substitute for it an absolute measure. In the course of the present century these alterations will no longer appear uniform, and the examination of the course and progress of the elements will offer to men of science inexhaust- ible materials for research. 35. Conclusions as to interesting points of theory may also be ex- pected in future. In our theory it is assumed that every determinate magnet- ized particle of the earth contains precisely equal quantities of positive and negative fluid. Supposing the magnetic fluids to have no reality, but to be merely a fictitious substitute for gal- vanic currents in the smallest particles of the earth, this equality is necessarily part of the substitution; but if we attribute to the magnetic fluids an actual existence, there might without ab- surdity be a doubt as to the perfect equality of the quantities of the two fluids. In regard to detached magnetic bodies (natural or artificial magnets), the question as to whether they do or do not contain a sensible excess of either magnetic fluid might easily be decided by very exact and delicate experiments. In case of the existence of any such excess in a body of this nature, a plumb-line to which it should be attached should de- viate from the true vertical position in the direction of the mag- netic meridian. If experiments of this kind, made with a great number of TERRESTRIAL MAGNETISM. 929 artificial magnets and in a locality sufficiently distant from iron, never showed the slightest deviation, (which we should rather expect,) the equality of the two fluids might with the highest degree of probability be inferred for the whole earth; though without wholly excluding the possibility of some equality. The only difference which the existence of such an inequality would occasion in our theory would be, that P® (Art. 17) would no longer be = 0. The consequence of this would be, that for all external space it would be necessary to add to the expression 2 po for Z the member = ; so that on the surface of the earth the (constant) member P° must be added, but X and Y would be in no respect affected. When there shall exist in future times a much more extensive collection of accurate observations than we at present possess, it may be examined whether a vanishing value of P° is or is not required for their accurate representation. With our present data such an undertaking would be wholly useless. 36. Another part of our theory on which there may exist a doubt is, the supposition that the agents of the terrestrial magnetic force are situated exclusively in the interior of the earth. If we seek for their immediate causes, partly or wholly, without the earth, and confine ourselves to known scientific grounds, we can only think of galvanic currents. But the atmosphere is no conductor of such currents, neither is vacant space; thus, in seeking in the upper regions for a vehicle of galvanic currents we go be- yond our knowledge. -But our ignorance gives us no right ab- solutely to deny the possibility of such currents; we are for- bidden to do so by the enigmatical phenomena of the Aurora Borealis, in which there is every appearance that electricity in motion performs a principal part. It will therefore still be in- teresting to examine what form magnetic action arising from such currents would assume on the surface of the earth. 37. Let us, then, assume the existence of constant galvanic cur- rents in a concave sphere, S, surrounding the earth, and call S’ all the space included by S, and S" all the space external to S. Whatever may be configuration of the galvanic currents, we can always substitute for them a fictitious distribution of the 230 Cc. F. GAUSS ON THE GENERAL THEORY OF magnetic fluids in the space S, the magnetic action of which, in all other spaces S' and S"’, will be exactly similar to that of the currents. This important proposition, which has been already men- tioned (Art. 3.), rests on the following grounds: first, that these currents may be resolved into an infinite number of elementary currents (i.e. such as may be considered linear) ; secondly, the well-known theorem, first demonstrated, I believe, by Am- péere, that in place of each linear current bounding an arbitrary surface, we may substitute a distribution of the magnetic fluids on both sides of this surface, at immeasurably small distances from it, with the same action; thirdly, the evident possibility of assigning for every re-entering line inside S, a surface bounded by it and situated wholly inside S. If we designate by —v the aggregate of all the quotients pro- duced by dividing all the elements of the imaginary magnetic fluid by the distance of an indeterminate point, O in S! or S"; of course it is understood that the elements of the southern fluid are to be considered as negative. Then will the partial differen- tial quotients of v, (just like those of V in our theory) express the components of the magnetic force which the galvanic cur- rents produce at O. 38. Although we must defer to another opportunity the detailed developement of the theory from which the proposition employed in the last article is taken, yet there is an important point re- lating to it which deserves to be noticed here. If we construct two different surfaces, F and F’, each bounded by the same linear current G,—and (taking the simplest case for the sake of brevity) having no other point in common,—they will include a portion of space. Now, if O be situated without this space, we obtain for that constant portion of » which belongs to G, one and the same value, whether we assign the magnetic fluids to For F’; and this value is equal to the product of the intensity of the galvanic current G (measured by an appropriate unity) multi- plied by the solid angle, the summit of which is at O, and which is included by straight lines, drawn from O to the points of G; or, which is the same thing, multiplied by that portion of the spherical surface described with radius 1 round O, which is the common projection of both F and F’. TERRESTRIAL MAGNETISM. 231 If, on the other hand, O be situated inside the space included by F and F’, it is true that the two values of the part of v in ques- tion will not be the same, whether we assign the magnetic fluids ‘to F or to F’, because different parts of the spherical surface alluded to correspond to them,—which parts, taken together, make up the whole spherical surface. But as the galvanic cur- rent has opposite directions towards F and F’, opposite signs must be applied in the two cases to the intensity of the current, in the multiplication into the parts of the spherical surface. The consequence is, that the algebraic difference between the values of the part of v in question is equal to the product of the intensity of the current multiplied by the whole spherical sur- face, or by 4 7. Hence it may easily be deduced, that if O is situated in 8", the value of v remains independent of the choice of the con- necting surface; that if, on the other hand, O is situated in S', the absolute value of v does indeed depend on that choice, but the differential of v does not. The highly fruitful theorem here touched upon,—according to which, in relation to the magnetic action of a linear galvanic current, the product of the intensity of that current, into the portion of spherical surface which is bounded by the line of current from O outwards, has the same import in regard to at- tracting or repelling forces, as the parts of the mass divided by the distance from O,—still requires in its generality many fuller explanations, which must be reserved for a detailed treatment of this subject. 39. The value of v, which in general is a function of r, wu, and 2d, passes on the surface of the earth into a function of uw and X, and dv dv ~ Rdw Rsinudx are the horizontal components of the magnetic force proceeding from the galvanic currents, directed respectively towards the north andwest. It is manifest that the remarkable propositions mentioned in Art. 15. and 16. hold good likewise in this case. But as to the third component, the vertical magnetic force, the ease will be somewhat different, if the agents are situated above, from what it would be supposing them to be situated in the in- terior. To eliminate the vertical magnetic force resulting from 932 Cc. F. GAUSS ON THE GENERAL THEORY OF the former supposition, v must first be considered as a function of — both 7, wv, and 2; it must be differentiated according to 7, and then r = R must be substituted. But for the inner space S’, to which the surface of the earth belongs, v can only be developed in a series according to ascend- ing powers of r. If we make 7 V F liad g=P + y ot se Bl + aye BM Bec. p® is a constant magnitude, namely, the value of + at the centre of the earth; p', p", p'", &e., on the other hand, are functions of u and X, which satisfy the same partial differential equations as p's p's pls above. Hence it follows, in a similar manner to Art. 20, that the knowledge of the value of v at every point of the earth’s surface is sufficient to enable us to deduce therefrom the general expres- sion for the space S'; that we may arrive at the knowledge of this value with the exception of a constant part,—or, which is the same thing, at the knowledge of the co-efficients p!, p"’, p!", &c.,— by the knowledge of the horizontal forces on the surface of the earth ; but that the value of the vertical force on the surface of the earth is not = 2p! + 3p" + 4p" +, &e. as it would be if the forces acted outwards from the interior of the earth, but is = — pf — 2p! — 3p!" —, &e. Now, as our numerical elements (Art. 26.), determined under the supposition of the first formula, give a very satisfactory re- presentation of the phenomena generally, whereas, the pheno- mena would be wholly incompatible with the second formula, the fallacy of the hypothesis, which places the causes of terrestrial magnetism in space external to the earth, may be looked upon as proved. 40. At the same time, this must not be looked upon as decidedly disproving the possibility of a part, though comparatively a very small part, of the terrestrial magnetic force proceeding from the upper regions: a far more full and far more accurate knowledge of the phenomena may in future throw light on this important point of theory. If, under the supposition of TERRESTRIAL MAGNETISM. 233 mixed causes, we attach the same meaning as before to the signs V, P°, P', P", &c., v, p°, p', p, applying the former to the causes acting from within, and the latter to the causes acting from without ; and if we further put V + v = W, P® + p® = IT, P' + p' = II’, P" + p" = II", &c., then on the surface of the earth, 7 = Te + 1 + I", &e, where II) satisfies the same partial differential equation as P™ (Art. 18.) ; and the two components of the horizontal magnetic force there existing are expressed by nae? Renae The propositions mentioned in Articles 15. and 16. retain there- fore their validity in this case, and we can determine the magni- tudes II’, II”, II’, &c. simply from the knowledge of the hori- zontal forces, but without being able in any degree to conclude from hence only as to the existence of mixed causes. But if we consider the vertical force by itself, and bring it into the form Per OOO. be. ee. so that Q™) satisfies the above-mentioned partial differences, then Q° = P° C= ar — yf QSaP iiss Dip" Q” = 4P" — 3p", &e.; and, consequently, ae i +, an = oil ~ 266 Cc. F. GAUSS ON A NEW INSTRUMENT FOR OBSERVING sists in entirely destroying, in avery short time, vibratory motions, which would otherwise be continued for several hours. The damper constructed at first for the magnetometer in the magnetic observatory produced this effect in a very high degree, so that the greatest vibratory motions disappeared entirely in a few minutes. A similar arrangement can be applied to any vibrating needle, to the magnetometer, and to the new apparatus here treated of ; and will certainly form an essential part of every ap- paratus which is to be employed for telegraphing by the method described above. A more complete explanation of this apparatus would, however, lead us too far from our present subject. No particular name has been as yet given to the new appa- ratus. From its chief application it might be termed an Jn- tensitometer. But as it is applicable to as many and as ac- curate magnetic measurements as the magnetometer, it has per- haps an equal claim to the name. The essential difference is, that the new apparatus is suspended by éwo threads, by which a new directive force is obtained with which the magnetic force is commensurable. The other differences, viz. in the mode of attaching the mirror, and in the means of measuring the relative amount of rotation of the several parts of the instrument, are conditions necessarily arising from the objects to be obtained. The new apparatus may therefore be termed a bifilar or bipensil- magnetometer, to distinguish it from the older mstrument, the simple or wnifilar magnetometer. I may express my conviction, that its more extended use, and especially its employment, con- jointly with the declination magnetometer, in the term observa- tions, at stations widely remote from each other, will be soon followed by an important progress in our knowledge of the disturbances of the earth’s magnetism. Gauss. [The graphical representations of the changes in the direction and intensity of the horizontal force at Géttingen in the terms of July 29-30, Aug. 31—Sept. 1, and Nov. 13-14, 1837, referred to in the preceding memoir, page 259, are contained in the Resultate aus den Beob. for 1837. As our purpose is rather that of illus- tration than of record, it has appeared sufficient to give one of these plates ; and we have selected for the purpose that of the term of November 13-14, appointed expressly on those days, on account of the great number of falling stars which had been — THE INTENSITY OF TERRESTRIAL MAGNETISM. 267 observed in them in preceding years. Part 1, Plate XIII., repre- sents the changes of the Intensity in the upper line, and of the Declination in the lower. The justice of the remark, in p. 259, will be at once recognised, namely, that, when considerable changes take place in the one element, they ar usually accom- panied by considerable changes in the other. Part 2 represents the changes of both elements united in one curve, and affords an illustrative delineation of the variation of the horizontal por- tion of the earth’s magnetic force. To avoid the confusion arising from too repeated involutions of the curve, it is divided into three separate portions, and in each of these half the curve is drawn in an unbroken line, and half in a dotted line. M. Gauss remarks, that “ the observations during this term of Noy. 13-14 do not present greater disturbances than had been noticed in many of the terms at other seasons of the year. On the preceding and following evenings, very great and rapidly- varying changes took place in the declination; but these are _known to be the general accompaniments of the Aurora Borealis, which was extremely brilliant on those two nights.”—Epir. | 263 ArtTIcLE VII. Observations on the Arrangement and Use of the Bifilar Magnetometer. By WitHELM WEBER*. [From the Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1837.—Herausgegeben von Carl Friedrich Gauss und Wilhelm Weber. Got- tingen, 1838. ] AFTER the full development in the preceding article of the prin- ciple of the Bifilar Magnetometer, and of all that is essentially necessary for its construction and application, an exact drawing of the instrument will be particularly interesting. The drawing (Plate XIV.) is so accurate that any skilful artist can work from it. The following observations are added with a view of rendering the drawing still more intelligible, and of facilitating the adjust- ment of the instrument by other observers, as far as may be done by such directions. 1. General Observations. The height and other dimensions of the Gottingen astrono- mical observatory, where the instrument figured has been esta- blished, allowed of large size in the instrument, and therefore a 25lb. bar very powerfully magnetized was employed. At other places it will perhaps be necessary to employ smaller dimensions, and we shall notice at the conclusion the difference in the cost produced by diminishing the size. Large dimensions are gene- rally, however, more to be recommended for the bifilar than for the unifilar, for the following reasons: Ist, because no pro- portionate increase in the price is occasioned, as the principal expense arises from the fine division of the circle and from the mirror; and since the latter is not attached to the end of the mag- net bar, it is not requisite that its size should be increased with that of the bar; 2nd, because the enlargement of the instrument does not require any considerable enlargement of the room; which would be the case with the unifilar, on account of the experiments of deflection in the measurements of the absolute intensity; 3rd, because the magnet bar need very rarely be re- moved from the stirrup; and therefore the size of the bar pro- * Translated by Mr. William Francis, and revised by Professor Lloyd and Major Sabine. ON THE BIFILAR MAGNETOMETER. 269 duces no inconvenience in its use, which would be the case to a certain extent with the other magnetometer. It does not fol- low that a bar of exactly five-and-twenty-pounds’ weight, such as we have used, must be employed; one of ten pounds will suffice for the most delicate measurements, and even one of four pounds might answer. The small bars have only one advantage over larger ones, in the greater facility of imparting to them a strong degree of magnetism; and this is only of importance where powerful means of exciting magnetism by friction are wanting. With respect to a suitable locality, a room similar to that em- ployed for the unifilar is all that is requisite, even if a bar of 25lbs. is employed. The breadth of the room may even be less, and its length may form any angle with the magnetic meridian, because the mirror in this case is not attached to the extremity of the mag- netic bar, but to the stirrup at the centre of the bar, and it can be turned in any direction. A considerable height is requisite, so that the interval of the two threads or wires to which the in- strument is suspended may be sufficient for convenient measure- ment without rendering the directive force too great. As it is rare that a room is sufficiently high, it is advisable to pierce the ceiling, and to carry the wires as high as the roof will allow. In regard to the height, it is of little consequence whether a heavy or light bar be placed in the stirrup, supposing only both bars to be proportionally magnetized, and both to be much heavier than the stirrup. It is not necessary to construct a separate building free from iron for the bifilar, as is done for the other magnetometer; it may be placed, as is the case at GGttingen, in the middle of a room in a building from which iron has not been excluded: it is sufficient to remove all iron from the immediate neighbourhood of the instrument : it is best, however, to place it in the magnetic observatory which contains the other magnetometer, if the room is large enough and adapt- ed for the purpose. If, for instance, the changes of the declina- _ tion and of the intensity are to be observed simultaneously du- | ring the terms, a double number of observers is necessary if : | the apparatus are in different buildings. But if both are in one large room, and so arranged that, whilst the magnetometers are at a sufficient distance asunder, the theodolites with which the observations are made are situated near one another, one clock may serve both observers, and one practised observer may ob- serve alternately with both instruments, allowing an ‘interval 270 W. WEBER ON THE ARRANGEMENT AND of two minutes. The two magnetometers may be so placed re- latively to each other in a large room, that the mean declination may remain unaltered, and the changes of the declination and of the intensity be only so far affected, that the determina- tion of the value of the divisions of the scale is somewhat dif- ferent from what it would otherwise have been. This is the case when the pillar supporting the theodolites forms with the two magnetometers a triangle, of which one side (viz. that be- tween the pillar and the declination-magnetometer) is situated in the magnetic meridian, while the other side, viz. the line which connects the central points of the two magnetometers, forms an angle of 35° 15’ 52’ with the magnetic meridian*. The * Prof. Gauss has given, in a very simple geometrical construction, the com- plete solution of the problem of the reciprocal action of two magnets at a great distance, in any given position relatively to each other. It is as follows: Let 4 be the centre of a small magnet, ns; 4 B the prolonga- tion of ns; C a particle of free magnetism of theotherbar; ACB aright angle; 4D=%4 4B; then C D is the direction of the force which acts upon C, when Cis a north magnetic particle; (when C == is a south magnetic particle, the nAs D B direction of the force is, on the contrary, in the prolongation of D C beyond C) 3 : Hees is the magnitude of the force, M designating the magnetism of n s, and m the magnetism at C. This simple proposition, which is useful in numberless cases, is especially applicable to this case, in which the most advantageous reciprocal position of the mag- netometers to be placed in the same room is required ; i. e. that position in which they will least disturb each other, and in which, whatever slight disturbance may be produced can easily be brought into calculation as a correction. The ap- plication of Gauss’s proposition to our case shows that in the position above described, Ist, the mean declination remains unchanged; 2nd, the value of the divisions of the scale, not only for the variations of the declination, but also of the intensity, are only altered in so far as the directive force of the two apparatus undergoes a change ; for the value of the divisions of the scale changes with the directive force, and in the same proportion. This may all be seen from the © geometrical construction of the reciprocal action of two magnets at a great di- stance, without its being necessary to give a detailed development of the theory of the two magnetometers. The first assertion is evident from the consideration of the above figure, where 4 is the central point of the intensity-bar x s, C the central point of the declination-bar situated in the line CD, C D the magnetic meridian, and where the straight line 4 C, which connects the centres of the two bars, forms the angle 4 C D = 35° 15! 52" with the magnetic meridian C D,—or, more accu- rately, forms such an angle, 4 C D, that sinks) -4/ = cotan 4d CD= 4/2 cosec AC D= 4/3 According to the above proposition, CD is the direction of the force which t | USE OF THE BIFILAR MAGNETOMETER. 271 height of suspension, which is of such great importance for the objects of this instrument, renders it very desirable that access to the points of suspension should be rarely or never required. acts on the declination-bar C, for if 4 CB=90°, 4D=1AB. This latter case is the actual one, because C D is perpendicular to 4 B (the magnetic axis of the declination-bar must be situated in the magnetic meridian, and the mag- netic axis 4 B, of the intensity-bar must be placed perpendicularly to it); then A Cheing the half diameter, 4 Dis the sine of 4d CD, A B the secant of B A C, or the cosecant of 4 C D; consequently, AD:AB=sin ACD: coseceACD= V7}: /3=1:38. The direction of the force with which the intensity-bar acts on the declination- _ bar is therefore that of the magnetic meridian C D: it may consequently have - some influence on the time of vibration of the declination-bar, the directive _ force of which is somewhat changed by it; but it will exert no influence on its direction, so long as this direction coincides with the assumed mean meridian CD: the deviations from it will, however, be somewhat diminished or increased by this force, according as it acts conjointly with, or in opposition to, the ter- _ restrial magnetic force ; but even this is provided for if we alter the value in are of the divisions of the scale, in which the deviations from the mean meri- V2 Mm AC3" Tm, -cot dC D dian are expressed proportionately to the force of direction, i. e. by Mm AC3 designates the magnitude of the directive force produced by the where, according to the above proposition, =e Ap es cD Mm oad. 4 CY intensity-bar, and Z m designates the directive force of the earth. The /atter assertion, in so far as it relates to the changes of the intensity, is tees by letting fall a perpendicular on C 4 at C, which intersects the pro- ongation of the line C Dat Z. It then results from the similarity of the tri- angles d CD, ABC, EAD, ECA, thatE D=1 EC, because 4 D was equal to 4 4 B; North Cc E South consequently, if C D is bisected in F, CF =+ CE. Now, as all that has been said of 4, 4 B, C, A BC, A D, and CD, istrue also of C, C E, A, CAE, CF, and 4 F, it results that 4 F is the direction, and — : ae is the magnitude of the force with which the declination-bar acts on the intensity-bar. If now 272 W. WEBER ON THE ARRANGEMENT AND Even in the construction of the Unifilar Magnetometer it was noticed that it would be convenient that the torsion circle, of which frequent use is made, should be fixed to the stirrup of the magnetometer instead of to the ceiling. The same object has been considered in the construction of the bifilar, where, on account of the greater height of suspension, it was of much more consequence. To make it quite unnecessary to go to the ceiling in the case of the bifilar, several other arrangements are requisite at the stirrup. The screws, for instance, which serve to lengthen and shorten the wires, have to be fixed to the stirrup instead of the ceiling. They are very clearly marked in Pl. XIV., figs. 1, 2, 3, where it is seen how they are connected with the circle on which the stirrup is placed, and arranged in the same way as the elevating screw of the unifilar, so that the wires may be lengthened or shortened without any lateral displacement. It is also necessary to be able to bring the two wires closer to, or further from each other, at the stirrup, so as to increase or diminish their directive force at pleasure. Although it is most simple that the two wires which support the instrument should be always equi-distant above and below, and that whenever it we resolve this force into a force acting in its magnetic axis, by multiplying AD : . : FF? and into a force perpendicular to it (to- wards the magnetic meridian) by multiplying the entire force by the fraction the entire force by the fraction _ we obtain for the first the value AD AF Mm_AD Mm_Mm_ AF CF’ 40" OF AG— ac Vi and for the latter the value DF AF Mm DF Mm Mm AP” CH AC CE AG a sincee2CF=CD=AD WA/2, or 42 =4/ 2, and CF=DF. M ‘ oes : : The force Ta a/2 directly changes the directive force of the intensity-bar miei he : M : ie in its transversal position. This force Fr, . would alter its position if the effect were not counteracted by asuitable'change in the suspension, sothat the bar should remain unmoved in its transversal position. In the latter case it is true that Mm : - : 5 Fos 2° longer comes into consideration ; but the changed suspension has certainly some influence on the directive force, and consequently on the value of the divisions of the scale. This, however, does not require any separate calculation, being included in the calculation of the directive force from the given suspension,—a problem which belongs to the Theory of the Bifilar Mag- netometer, which will subsequently be developed. the force USE OF THE BIFILAR MAGNETOMETER. 273 is desired to increase or diminish the directive force, they should be moved through an equal quantity at both extremities, it is by no means necessary. The change in the interval of the wires may be effected below only, but in such case to a greater degree. The apparatus figured is, in fact, so arranged, that, with a mean distance of the upper ends, every necessary in- crease or diminution of the directive force can be produced by a displacement of the suspension screws at the stirrup; how- ever, for the sake of completeness, the apparatus is provided with an. arrangement at top for an equal displacement of the two eylinders, over which the wire is conducted, and by which its two vertical suspended ends are kept separate from each other ; so that, if it is desired, the upper distance may always be ren- dered equal to the lower. In case it is not desired to retain the power of making this upper displacement, these two cylinders may be united into a roller of a suitable diameter, and the axis of this roller, like that of a friction wheel, may be allowed to run on wheels, so as to diminish the friction, and cause the two wires to have an equal tension ;—a point which is of great importance in absolute determinations. 2. On the separate parts of the Bifilar. The description of these is reduced almost wholly to a descrip- tion of the stirrup, because it unites nearly all the parts which in the unifilar are distributed among the stirrup, the ceiling, and the extremity of the bar. It is also unnecessary to speak of the theodolite and its stand, the clock, the scale, or the mark, as all these have been treated of in the account of the former instru- ment. But as so many arrangements are united in the stirrup, its construction requires to be particularly explained. Plate XIV. gives three different views of the instrument, of the natural size, and as arranged for the 25lb. bar ; the small and compound parts have been represented in a separate section, so as to exhibit their interior mechanism. It requires an attentive consideration on account of the many impor'ant parts compressed into so small a space at the stirrup : a cieay comprehension of its mechanism will be obtained when we know the various concentric rotations which are performed at the stirrup,—the mode of checking and mea- suring these,—and their objects. The rotations are the following. 1. Of the mirror on its pivot ; ;—the whole of the other por- tions of the instrument remaining unchanged. 274 W. WEBER ON THE ARRANGEMENT AND 2. Of the mirror, with its pivot and alidade, on the circle to which the suspension-screws of the wires are fixed, and on which the stirrup and its alidade rest. 3. Of the stirrup with its alidade, on the circle on which it rests. In order to complete the view of all the rotations, we may here add, 4. That of the two upper extremities of the wire around one another, z. e. around the same axis as that on which the other ro- tations take place. The first rotation will be sufficiently intelligible from figs. 1 and 3, Pl. XIV. The arrangement is simple, because its amount does not require to be measured. Its object is merely to allow of perfect freedom in fixing the theodolite; the axis of the mirror can always be made to revolve, to suit the position of the telescope and the scale, wherever they may be placed. The image of the scale which appears in the mirror serves itself to regulate the ro- tation, and no further arrangement for measuring it is required. A screw, as exhibited in the figs. 1 and 3, fixes the mirror in its position. For the second rotation, the three pieces, the mirror, the pivot, and its alidade, are firmly connected as one piece, and revolve together in the cavity of the circle ; they are represented, together . with the latter, in a cross section, at fig.4. The mirror is placed on the upper end of the pivot B at 4; C is the alidade of the pivot ; D is the circle. The only essential difference between the second rotation and the first is that in the second the angular amount can be measured. As the revolving alidade of the pivot, situated beneath the circle, embraces at its two extremities the edge of the circle, it forms on its upper and graduated surface two verniers, the inner margins of which lie close to the outer margin of the divided circle. A clamp, by which the alidade of the pivot can be pressed firmly against the circle, is seen in the section at E, fig. 4. The second rotation alone would be sufficient if there were at no time an impediment to its use. The verniers on the alidade of the pivot come in certain cases beneath, and are hidden by the alidade of the stirrup. In the instrument represented in Pl. XIV. much care has been employed to restrict this within very narrow limits, as will be plainly perceived in fig. 2; but, in order to meet the rare cases in which it does still occur, without having to alter the position of the theodolite, both rotations may be em- USE OF THE BIFILAR MAGNETOMETER. 275 ployed at the same time, so as to free the indices without turning the mirror from the scale. The third rotation is that of the stirrup with its alidade, on the circle upon which it rests. The directive force of the wires acts immediately on the circle to which the suspension screws are fixed: the directive force of magnetism acts immediately on the stirrup in which the magnet-bar is placed. When, there- fore, these two directive forces form an angle with each other, the two parts upon which they act will have a tendency to move in opposite directions. That no such displacement of the parts may occur, they are made to slide on each other with so much friction, that the two directive forces, when forming a large angle with each other, may not be able to overcome it. For a similar reason it was provided in the unifilar that the alidade of the stirrup should be placed on the outermost margin of the circle, so that the friction produced by its pressure might act with the greatest leverage. The same has been done with the bifilar, where this provision is much more essential and im- portant, the forces which tend to displace the two parts being much more powerful. Further, we must be able to measure with great exactness this rotation, on which depends the angle which the two directive forces form with each other. The sim- plicity of construction of the bifilar consists chiefly in this cir- cumstance, that the same circle and graduation serve for mea- suring both the second and the third rotation. For this reason the alidade of the stirrup is also furnished with two noniuses. The instrument consists, therefore, of a circle with two alidades, which may be used independently of each other. In order that this independent use may never cause the two alidades to inter- fere, the one is situated beneath, and the other above the circle. But since each alidade is provided with two noniuses, and all four are to move on the divided limb of the circle, which is its upper surface, the inferior alidade embraces the margin of the circle and forms noniuses which abut at the outer margin, whilst those of the upper alidade, in order not to come in conflict with those of the lower, abut on the inner margin. The noniuses of the upper alidade can thus pass by those of the lower one, and even an interval may exist between them, which, however, must be smaller than the length of the divisions on the circle. Thus the graduation of the circle serves two purposes, the one not in- terfering with the other, only it cannot serve both purposes at the 276 W. WEBER ON THE ARRANGEMENT AND same time, as the figures must be covered either by the noniuses of the one or of the other alidade, according as they are inside or outside of the graduation. For this reason the figures are placed alternately on the inner and on the outer side, as exhi- bited in Pl. XIV. fig. 2. The fourth rotation is that of the two upper ends of the wires. No mechanical arrangement is required for this rotation ; but the bearer on the ceiling, by which the wires are carried and ad- justed, is turned by the hand. As the bearer must be fixed to the ceiling, no use is ordinarily made of this rotation ; but it is so arranged in the first instance as to be in the most convenient position for all purposes. That position may be regarded as most convenient in which the lower ends of the wire interfere least with the mirror which is situated between them. It will be evident that, in the various uses to which this instrument is applied, if the bearer is not moved, the lower ends of the wire are brought into various positions, while the mirror retains its position between them nearly unchanged, being always directed towards the scale. The two wires, for instance, will sometimes be in one vertical plane throughout their whole length ; some- times they perform part of a revolution round each other, and a vertical plane drawn through them will form with the former one an angle, which is, however, always less than 90 degrees. If it be now so arranged that in the first case the plane of the wires coincides with the vertical plane of the optical axis of the telescope, the one wire passes just as far from the mirror in front as the other does behind, and both wires are as far as possible from the mirror. If the instrument is then arranged for the other use, the wires are brought nearer to the mirror, but not so as to touch it, even if the mirror were larger than the inter- vening space, because the rotation does not amount to 90°. It is always less than 90°, because the directive force arising from the suspension must be greater than the magnetic directive force ; hence the moments of rotation arising from the two forces will only equilibrate when the wires undergo a smaller rotation than the magnetic axis; and since the latter, in the transverse, must. be 90° from its natural position, it follows that the rotation of the wires must be less than 90°. 3. On the use of the Bifilar Magnetometer. I shall in conclusion briefly notice the series of experiments USE OF THE BIFILAR MAGNETOMETER. 277 which must be performed in fixing and adjusting the appa- ratus. 1. The clock, the theodolite, and the scale are fixed, and a plumb-line dropped from the centre of the object-glass across the scale. The theodolite is to be leveled. 2. The telescope is directed to the opposite wall, on which there is a mark, serving to designate the terminal point of the optical axis. The scale is placed perpendicularly to the vertical plane of that axis. 3. A place is sought in the vertical plane of the optical axis for the mirror, the distances of which from the centre of the ob- ject-glass, and from the division of the scale across which the plumb-line is suspended, are, together, equal to the distance of the mark from the centre of the object-glass. The horizontal plane of this point must bisect the plumb-line from the centre of the object-glass. A plumb-line is let fall from the ceiling through this point. _ 4, The bearer is either fixed to the ceiling, or perpendicularly above a hole made through the ceiling, from 80 to 100 milli- metres wide ; so that the ends of a thread passed over it, and ex- tended by small weights, pass freely through the aperture, and are both situated in the vertical plane of the optical axis of the telescope. 5. One end of a steel wire, sufficiently strong to carry half the weight of the instrument without danger of breaking, is fast- ened to one end of the thread, and drawn up to the bearer by drawing the other end of the thread down (care being taken that the wire and the thread should always be extended in a straight line); it is passed over the two cylinders of the bearer, and drawn down; the thread is then removed, and the two ends of the wire, weighted, are left to hang freely until they have assumed their natural position. 6. The two ends of the wire are cut off about 100 or 150 mil- limetres below the place where the magnetometer is to be sus- pended, and are fixed to the suspension screws. The stirrup thus carried is then, with the help of the screws, wound up into its proper position. 7. A box, sufficiently large to contain the magnet-bar, is placed underneath to protect the instrument in case the wires should break, and to prevent currents of air. This box is closed on all sides. Its lid consists of two halves, which fit close, and leave only one round aperture, through the centre of which the pivot VOL. Il. PART VI. 7 278 W. WEBER ON THE ARRANGEMENT AND passes; the upper end of the pivot carries the mirror, which must be above the box. The two wires having the mirror be- tween them pass through the same aperture. This circular aper- ture is usually closed by two semicircular flaps, in which there are small slits for the pivot and the wires. 8. Before the magnet bar is laid in the stirrup, a weight of the same size, but unmagnetic, is placed therein, and the wires are suffered to arrange themselves in their natural position, in which both are in one vertical plane throughout their whole length. The alidade of the stirrup is then brought as exactly as possible into the mean magnetic meridian from which the changes of variation are to be measured. The other alidade on the pivot of the mirror should be so fixed as to form a right angle with the alidade of the stirrup, in order that the noniuses may be far apart. The weight in the stirrup is moved until the mirror is situated exactly between the two wires, when the axis of the mirror should be very nearly horizontal. Employ the first ro- tation to direct the mirror towards the scale, without disturbing the alidade. If the scale does not appear in the telescope, it will be seen by the naked eye a little above or beneath, and may be brought into the field by the help of a light running weight placed on the stirrup. The first observation is then performed, and the position of the scale determined. 9. The time of vibration for determining the directive force of the wires may be observed before the magnet-bar is inserted, and again with a known increase of the moment of inertia. It is better, however, to perform this experiment somewhat later, when the distance of the wires from each other has been accu- rately adjusted, in case this distance has not been previously determined by calculation, and regulated accordingly. 10. The magnet-bar is then placed in the reverse position, (north towards the south) and the position of the scale again ob- served: this ought to agree with the observation (8.) If the two readings do not coincide, agreement must be attained by merely turning the stirrup with its alidade. The coincidence of the two readings proves that the magnetic axis of the bar is situated in the magnetic meridian. The less the directive force arising from the mode of suspension exceeds the magnetic di- rective force, (see 8.), the more delicate is this test, so that it may be impossible to obtain a perfect coincidence of the two readings ; a difference of a few divisions of the scale may then be considered as unimportant. The influence of the hourly va- USE OF THE BIFILAR MAGNETOMETER. 279 riations must be attended to, by making continued observations with a second apparatus of the same kind, or by making con- tinued observations of the time of vibration with a common mag- netometer. 11. The time of vibration, ¢, is observed in this reverse position. 12. The magnet-bar is then placed in its natural position, (north towards the north,) by turning the stirrup with its alidade exactly 180 degrees; the time of vibration, 7, is again observed. Then the magnetic directive force, M, is to the directive force arising from the mode of oe S, in the ratio M:S=—7:f4 7%. When this proportion deviates in from unity, the wires must be brought nearer to or moved further from each other, until the altered directive force of the wires exceeds but little the magnetic directive force; for instance, by about the tenth part of the latter. This is the case in the Gottingen magnetometer. 13. Seek the angle z, the sine of which is {2 — 7? turn the alidade of the stirrup (say in the direction of the daily motion of the sun) 90° — z, and turn the alidade of the pivot of the mirror in the opposite direction through the angle z. The equilibrium is then disturbed : the wires can ne longer remain in their natural position, but must turn the circle to which they are fixed (and thus the whole instrument) exactly through the angle z, in the direction of the daily revolution of the sun. In this new position the equilibrium may be re-established, since the bar makes with its former position an angle (90° — z) + z= 90°, while the wires have only been turned through the angle z at their lower ends. It follows, thence, that if the wires were pre- viously in their natural position, and if the magnetic axis of the bar was situated in the magnetic meridian, the opposite moments of rotation arising from the: two forces M and S are to each other in the proportion M sin 90°: S sin z. But as M:S=#—-—7:2%+47? ; i aint 2 = oe sin 90° = 1 280: W. WEBER ON THE BIFILAR MAGNETOMETER. The equality of these opposite moments of rotation, or the equilibrium of the instrument in this position, is the result. Whether the true position of equilibrium coincide with the cal- culated one or not, is proved immediately by an observation of the scale, which ought to be the same as before. For the mirror has been turned (together with the whole apparatus) the angle z in the direction of the daily motion of the sun; but having been turned by its independent motion the same angle z in the opposite direction, it consequently retains its first posi- tion, and the point of the scale is unchanged. 14. If, however, the observation shows an alteration of the scale, it follows that the supposition in the first experiment— of the magnetic axis of the bar being in the magnetic me- ridian—was not accurately fulfilled. The amount of the error can be calculated, and the experiments repeated. This calcula- tion will be still more accurate and certain, if a corresponding experiment has been previously made, proceeding precisely as described in (13.), only making all the rotations in the contrary direction. 15. When the required coincidence has been obtained, the magnetometer remains in its transverse position. Its time of vibration is then, according to a simple theorem, the geo- metrical mean between the times of vibration ¢ and 7, and the observations of changes of intensity can be arranged lke those of the changes of declination. The changes of inten- sity are obtained in divisions of the scale. If we desire to con- vert them into fractions of the entire intensity, these are ob- tained by multiplying the are value of the scalar divisions (ex- pressed in parts of the radius) by ae for the value in arc of the parts of the scale, expressed in parts of the radius, gives immediately the changes of intensity in parts} of the directive force, which, under the prescribed conditions, is S cos z. If we divide this directive force by the whole intensity. i.e., by S sin z, we obtain by multiplying the value of the arc by the quotient, cot z—the changes of intensity in fractions of — the whole intensity. ArTicLeE VIII. Contributions to our Knowledge of Phytogenesis ; by Dr. M. J. ScHLe1pEN*. [From Miiller’s Archiv fiir Anatomie und Physiologie, Part I1., 1838.] THE universal fundamental law of human reason, its undevia- ting tendency to unity in its acquirements, has from the first been evinced in the department of organized bodies as in all branches of science, and various attempts have been made to establish the analogies between the two great divisions, the animal and vegetable kingdoms. But although so many emi- nent men have devoted their attention to this subject, it can- not be denied, that all attempts hitherto made with this view must be considered as entirely unsuccessful. If, indeed, the fact has been of late generally admitted, still the reason of this circumstance has not always been correctly conceived and stated in its full clearness and precision. The cause, however, lies in this; that the idea of individual in the sense in which it occurs in the animal nature, cannot in the least be applied to the vegetable creation. At the most we can speak of an in- dividual in its true sense only in some of the lowest orders of plants, in some Algze and Fungi, which consist only of a single cell. But every plant developed to a somewhat higher degree, is an aggregate of fully individualized independent beings, even the very cells. Each cell leads a double life: an entirely independent one, belonging to its own development alone; and an incidental one, in so far as it has become the constituent part of a plant. But it is easy to perceive that, as regards vegetable physiology as well as comparative anatomy in general, the vital process of the single cells must form the very first, absolutely indispensable fundamental base; and, therefore, at the very outset this question especially presents itself: how does this peculiar small organism, the cell, originate ? The great importance of this subject may, perhaps, be a sufficient excuse for my venturing at present to publish the fol- lowing remarks, feeling as I do only too well that more extended _ * Translated and communicated by Mr. William Francis.—The Editor is indebted to J. J. Bennett, Esq. for his assistance in revising the Translation. 282 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. observations can alone impart to them their proper scientific value. Perhaps, however, I may succeed by these remarks in drawing attention to this highly important subject. Since no real advance in the acquisition of knowledge results from the attempt to explain processes in nature hypothetically, and least of all, where all requisites for a tenable hypothesis, namely all guiding facts, are wanting, I may omit all historical introduction ; for as far as I am acquainted, no direct experi- ments exist at present on the origin of the cells of plants. That Sprengel’s supposed primitive cells are solid granules of amy- lum, has long since been demonstrated. To enter into Raspail’s observations seems to me to be inconsistent with the dignity of the science. He who feels any desire to do so may turn to the work itself. The only researches connected with this subject, the highly important ones of De Mirbel, I shall have occasion to advert to subsequently, since even he does not make any mention of the progress of the formation of cells. It is to be regretted that Meyen, who, perhaps, has studied vegetable anatomy more extensively than any one up to the present day, has confined himself almost solely to the examination of developed forms, and has not yet brought the formative process itself in any degree into the field of his inquiries. I have still many doubts, the solution of which I had hoped to have found in his Vege- ~ table Physiology, but found them not. It was Robert Brown, who, with his natural genius and comprehensive power of mind, first conceived the importance of a phenomenon which, although observed previously by others, yet had been left totally unregarded. He found at first in the Orchidee, in a great portion of their cells, chiefly in the epi- dermis, an opake spot designated by him areola or nucleus of the cell. He subsequently pursued this phenomenon in the earlier stages of the pollinic cells, in the young ovulum, in the stigmatic tissue, not merely in the Orchidee, but also in many other Monocotyledons, and even in some Dicotyledons. It was natural that the constant presence of this areola in the cells of the very young embryo and in the newly originated albumen should strike me in my extensive researches respecting the development of the embryo; and thus, from the consideration of the various modes of its occurrence, the thought very naturally arose, that this nucleus of the cell must stand in a close ye- DR. M. J. SCHLEIDEN ON PHYTOGENESIS,. 283 lation to the origin of the cell itself. I consequently directed my attention especially to this point, and was so fortunate as to see my endeavours crowned with success. Before, however, proceeding to the communication of these observations, it is necessary that I should describe somewhat more at length the nucleus of the cell. As I have to treat of an entirely peculiar, and, as it appears to me, of an universal elementary organ of vegetables, I do not deem it necessary to excuse myself for applying to this body a definite name, and shall term it Cytoblast («vtos BXactos) with reference to its function, which will be subsequently described. This formation varies in its outline from oval to circular, according as its solid form seems to pass from that of the lens into the perfect sphere. The oval and flat ones I have found most frequently in monocotyledonous plants, in the albumen, and in the pollen ; the globular chiefly in the Dicotyledons, and in the leaf, stem, articulated hairs, and similar formations ; how- ever, no exclusive rule can be asserted in this respect. The colour of the cytoblast is in general yellowish, yet some- times passing almost into a silver white. I observed it to be most transparent in the albumen of some water plants, in the unripe pollen, in some Orchidee, and also in the rudiments of the leaf in Crassula portulacea. It is scarcely to be distinguished, on account of its excessive transparency, in the sporidia of some Helvelloids. It is coloured by iodine, according to its various modifications, from a pale yellow to the deepest brown. Its size varies considerably. It is in general largest in Mono- cotyledons, and in the albumen; smallest in Dicotyledons, in the leaf, stem, and their metamorphosed parts. The largest that I have seen was 00022 Prussian inch* in diameter (in Fritillaria pyrenaica) ; the smallest in the embryonal extremity of the pol- linic cellule of Linum pallescens, from 0°00009 to 0:0001 Prus- sian inch. In the albumen of Adies excelsa I found it, on the average of several admeasurements of individuals of appa- rently equal size, from 0°00034—0:00059—0-00079. In the young leaves of Crassula portulacea=0-0003, and in the albu- men of Pimelea drupacea=0-00095 —0:001055. However, little importance can be attached on the whole to these measurements, as they increase and diminish; and it cannot be determined in what period of its life the cytoblast is examined. Its internal * The Prussian inch is to the English as 1-03 to 1.—Ep. 284 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. strugture is mostly granular, without however the granules of which it consists being clearly distinct from one another. Its consistence is very various, from such a softness that it almost dissolves in water, to that degree of firmness that it bears even the pressure of the compressorium without losing its form. The nearer it is to its origin the softer it is, and also where its ex- istence is merely transitory. It is denser and more sharply de- fined where it goes through the whole vital process of the plant as a permanent tissue, as in the Orchidee. These peculiarities have been more or less completely de- scribed by R. Brown (Organs and Mode of Fecundation in Or- chidee and Asclepiadee; Linn. Trans. 1833*, p.710) and recently by Meyen (Physiology, &c., vol. i. p. 207). But a circumstance has escaped these two most acute observers, which I nevertheless am inclined to place amongst the most essential. In very large beautifully developed cytoblasts, for instance in the recently ori- ginated albumen of Phormium tenax and Chamedora Schiedeana (Plate+ XV. fig. 5), there is observed (whether sunk in the in- terior or on its surface was not evident to me) a small, sharply defined body, which, judging from the shadows, appears to repre- sent a thick ring or a thick-walled hollow globule. In less deve- loped individuals only the outer sharply defined circle of this ring can be observed, and in its centre a dark point, for instance in the stipes of the embryo of Limnanthes Douglasii, Orchis latifolia (Pl. XV. fig. 21), Pimelea drupacea (fig. 14,15). In still smaller cytoblasts it appears merely as a sharply circum- scribed spot; this is most frequently the case in the pollen of Richardia ethiopica, in the young embryo of Linum pallescens, and in almost all Orchidee (fig. 16). Or lastly, there is observed only a remarkable small dark point. In the very smallest and most transitory cytoblasts (for instance in the leaves of Dicoty- ledons) I have hitherto not been able to discover it. In very rare cases, and those probably mere exceptions, and always only where the majority exhibited the simple nucleus, I have also found two; for instance in Chamedora Schiedeana (fig. 6, 7), Secale cereale, Pimelea drupacea (fig. 14) :—in the two latter I have found sometimes even three (fig. 15). From my obser- vations on all plants which admitted of a complete watching of * Read at the Linnean Society Nov. 1, 1831. + The Editor has been favoured, through the kindness of Dr. Schleiden, with impressions from the copper-plates engraved under his superintendence. Having been printed abroad, the numbers have not been placed on the plates, but they are referred to in this work as Plates XV. and XVI, DR. M. J. SCHLEIDEN ON PHYTOGENESIS. 285 the entire process of formation, it follows that these small bodies are formed earlier than the cytoblast (Plate XV. figs. 1 and 2), and I would almost suggest the conjecture that they are not en- tirely foreign to the nuclei shown by Fritzsche to exist in starch, and probably even identical with them. The size of this cor- puscle also varies considerably, from the extent of half the dia- meter of the cytoblast to the most minute point, whose size did not allow of measurement, because it was even much exceeded by the thickness of the thread of the diaphragm of the micro- scope. In the albumen of Abies excelsa I found it to be on an average from 0:000045—0:000095 Prussian inch ; in Pimelea drupacea, from 0°00029—0:0003. Sometimes it appears darker, sometimes brighter than the remaining mass of the cytoblast. In general it has more consistency than the latter, and still con- tinues sharply defined when this has been changed by pressure into an amorphous mucus, as for instance in Pimelea dru- pacea. A second point, on which I must make a few observations so as to be able to express myself hereafter more briefly without being unintelligible, relates to the various inorganic substances which occur during the vital process of plants, and belong to the series of amylum and of the woody fibre. I do not at all pretend fully to enumerate in this place all substances chemi- eally distinct ; as little do I require that chemists should ap- prove and adopt all my terms and characteristics (perfection independent of this would at present be an impracticable task) : it is my intention merely to notice in a few words the most im- portant modifications, their consequence and purport in the progress of the development of vegetable organization, in order to spare repetitions in future. Starch, in the plant, appears to take the place of animal fat. It is superfluous nutritive substance, which is deposited for future use, and we consequently find it abundant in places where after a short repose a new formative process is to commence, or where a too luxuriant life has originated a superabundance of nutritive matter. It has of late been the subject of such deep research that it is not necessary to enter more fully upon this head: I shall merely refer the reader to the most recent and best summary of the results, in Meyen’s Vegetable Physiology, vol. i. p- 190, &c. Frequently the place of the starch is occupied by a semi-granular substance, for instance in pollen, in the albu- 286 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. men of some plants, and frequently in the cells of the leaf as containers of the chlorophylle. It is chiefly distinguished by its occurrence in all kinds of granular forms without any inte- rior structure, and from its being coloured by tincture of iodine, brownish-yellow, or brown. This substance, which I shall call mucus, is probably identical with that of which the cytoblast consists, and with the small granules in gum which I shall pre- sently mention; the first conjecture Meyen (Vegetable Physio- logy, vol. i. p. 208) has noticed as being very probable. Now when the starch is to be employed in new formations, it dissolves, in a manner as yet totally unknown to chemistry, into sugar or into gum, the latter appearing at times to pass into the former, or vice versd. The sugar appears in the form of a per- fectly transparent fluid, almost as clear as water, is not rendered turbid by alcohol, and takes from the tincture of iodine only a colour in proportion to the strength or weakness of the solution of the agent. The gum appears as a somewhat yellow, more consistent, less transparent fluid, which is coagulated granularly by the tinc- ture of iodine with a pale yellow permanent colour. In the further progress of organization, in which the gum is always the last immediately precedent fluid, a quantity of exceedingly minute granules appear in it, most of which, on account of their minuteness, appear merely as black points. The fluid then seems to take from iodine a somewhat darker yellow, but the granules, when their size enables their colour to be distinguished, seem to become by this process of a dark brownish-yellow. This is always the mass in which organization takes place, and the newly formed parts consist again principally of this se- parate transparent substance, which, on being subjected to pres- sure, presents to view an homogeneous colourless mass; when dried it imbibes water and swells; it is not at all affected by iodine, nor does it even imbibe it, but appears after pressure colourless as before, and so completely transparent, that, if not surrounded by coloured or opake bodies, it is totally invisible. This substance is of frequent occurrence in plants (for in- stance, in great quantity, together with a little starch, in peculiar large cells in the tubers of Orchis) ; I shall call it for shortness sake vegetable gelatin, and am inclined to enumerate under this head, as mere slight modifications, pectine, the basis of gum DR. M. J. SCHLEIDEN ON PHYTOGENESIS. 287 tragacanth and many of those substances commonly arranged under vegetable mucus. It is this gelatin which is ultimately converted by new chemical changes into the actual cellular membrane, or its thickening layers, and into vegetable fibre. I now return to the subject itself. There are two places in plants where the formation of new organization may be ob- served most easily and clearly, from their being cayities closed by a simple membrane, viz. in the large cell which subsequently contains the albumen of the seed, the embryonal sac, and at the end of the pollen tube, from which the embryo itself is developed. They are chiefly distinguished from each other by the embryonal sac, never originally containing starch, but pro- bably, in general, the saccharine solution, (whence arises the sweet taste of unripe pod fruits and cerealia,) or gum. The pollen on the contrary constantly contains starch, or the above-mentioned granular mucus representing it, as an essential constituent part. The so-called vegetable Spermatozoa will pro- bably on more accurate examination be generally reduced to one of these substances. These substances, however, are soon dissolved, and change either into sugar or into gum; both, at times, even before the pollen grain has commenced sending forth tubes on the stigma, frequently in the progress of the descent of the tube through the style to the ovule ; so that in some cases eyen unaltered starch is still found in the embryonal end. At both these places the above-mentioned minute mucous granules very soon originate in the gum, upon which the solution of gum, hitherto homogeneous, becomes opalescent, or, through the presence of a larger mass of granules, even opake. Single, larger, more sharply defined granules (fig. 2 above) now become apparent in this mass; and very soon afterwards the cytoblasts occur (fig. 2 below,) appearing as it were like granular coagula- tions around the granules. The cytoblasts, however, in this free state grow very considerably ; and I have observed, for instance in Fritillaria pyrenaica, a gradual expansion from 0:00084 to 0°001 Prussian inch. As soon as the cytoblasts have attained their full size, a deli- cate, transparent vesicle rises upon their surface: this is the young cell, which at first represents a very flat sezment of a sphere, whose plane side is formed by the cytoblast, and the convex side by the young cell, which is situated on it somewhat like a watch glass on a watch. In its natural medium, it is di- 288 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. stinguished almost by this circumstance alone, that the space between its convexity and the cytoblast is perfectly clear and transparent, and probably filled with an aqueous fluid, and is bounded by the surrounding mucous granules, pressed back by its expansion, as I have endeavoured to represent it in Plate XV. figs. 4,5, and 6. But if these young cells are isolated, the mucous granules may almost entirely be removed by shaking the stage. They can however not be observed for any length of time, for they dissolve entirely after some minutes in distilled water, and only leave the cytoblasts behind. The vesicle gra- dually extends, and becomes more consistent (fig. 1 6.), and the covering now consists, with the exception of the cytoblast, which always forms one portion of the wall, of gelatine. The entire cell now gradually increases beyond the margin of the cytoblast, and quickly becomes so large, that at last the latter merely ap- pears like a small body inclosed in one of the side walls. At the same time the young cell frequently exhibits highly irregular indentations (fig. 1 ¢.), a proof that the expansion does by no means proceed uniformly from one point. After further pro- gressive growth of the cell, and evidently arising from the pres- sure of the neighbouring objects, the form becomes more regular, and then also frequently passes into the form, so beautifully de- termined @ priori by Kieser, of the rhomboidal dodecahedron (compare fig. 1. from 6—e and fig. 8.). The cytoblast is still found to be inclosed in the wall of the cell, at which place it passes through the whole vital process together with the cell formed by _ it, if it be not in cells destined to higher development, either reabsorbed at its place, or, after having been cast off as a use- less member, dissolved in the cavity of the cell, and there reab- sorbed. It is only after its absorption, that the formation of se- condary depositions, as far as I was able to observe, commences on the inner surface of the sides of the cell. (fig. 9.) In general it is rare that the cytoblast accompanies the cell which it produced through its entire vital process: nevertheless eats. 1. Characteristic of the families of Orchidee and Cactee that in them a portion of their cellular tissue remains during the whole vital period in a lower stage of development ; 2. It sometimes occurs in various plants, that the cellular tissue, which is merely of transitory import, is not perfectly developed, but retains the cytoblast, and is subsequently reab- | DR. M. J. SCHLEIDEN ON PHYTOGENESIS. 289 sorbed cotemporaneously with it. Yet I have also observed that the latter in the middle period of its existence lost much of its distinctness and sharpness of outline, which however reap- peared when the reabsorption had commenced ; for instance, in the nucleus of the ovules of Abies excelsa, Tulipa sylvestris, and Daphne alpina. It is inconceivable how some physiologists have been able to deny reabsorption in plants, since even very considerable portions of cellular tissue, for instance of the nu- cleus of the ovule, become wholly fluid again, and are received into the common mass of sap. This indeed only takes place so long as the cell still consists of the simple original membrane, and whilst it is not so far advanced in its individual development that its wall is thickened by secondary deposits. 3. In some rare cases the cytoblasts also remain persistent in the pollen granules; this is the case in some, perhaps all the Abietine. The lenticular cytoblast has already been observed by Fritzsche in Larix europea, but its nature not understood. 4. Many hairs, especially the articulated and such as exhibit circulation of the sap in their cells, retain the cytoblasts (c, /. fig. 25.). Itis remarkable, and moreover a proof of the close relation in which the cytoblast stands to the whole vital activity of the cell, that the small currents frequently covering the entire wall reticularly, always proceed from it and return to it, and that in statu integro, it is never situated without the current (fig. 25.). The above described development of the cells I have observed in their whole course in the albumen of Chamedorea Schiedeana, Phormium tenax, Fritillaria pyrenaica, Tulipa sylvestris, Elymus arenarius, Secale cereale, Leucgji spec., Abies excelsa, Larix eu- ropea, Euphorbia pallida, Ricinus leucocarpa, Momordica elate- rium, and in the embryonal end of the pollen cell of Linum pal- lescens, Cinothera crassipes, and a number of other plants. It was only in the summer of 1837, after this memoir had been written, that I took up the examination of the Leguminose, and found to my surprise that in these plants, so frequently examined and everywhere employed as examples in the history of vege- table development, this process, overlooked by all observers, might most beautifully and easily be studied. But, indeed, the saccharine fluid contained in the embryonal sac had not been considered worth examining. Without exactly following up the whole course of the forma- tion of the cells, I found the cellular nuclei previous to the oc- 290 DR. M. J. SCHLEIDEN ON PHYTOGENSIS. currence of cells floating loose in a great many plants. Finally, not a single example has occurred to me of newly originated cellular tissue, cambium excepted, in which the cytoblasts were wanting. I therefore think I am justified in supposing the pro- cess above described to be the general law of formation of the vegetable cellular tissue in Phanerogamia. j My observations are much more limited with regard to the Cryptogamia; nevertheless I found the cytoblasts in the sporidia of the Helvelloids, where however, on account of their great transparency, they are only perceptible with very high magni- fying powers and with a considerable darkening of the field. In the large yellowish cells in the interior of the so-called anthers in Chara vulgaris I have observed them. In the sporules of Mar- chantia polymorpha I also noticed their development into cells, one of which, pressing the original parietes of the sporule for- wards, forms the long capillary root (Plate XV. fig. 18—20.). It is evident from the preceding that the cytoblast can never lie free in the interior of the cell, but is always inclosed in the cellular wall; and in fact, as far as observation will allow us to draw conclusions from such ticklish examinations, in such a man-— ner that the wall of the cell splits into two lamina, one of which passes interiorly, the other exteriorly, over the cytoblast. That on the inner side is in general the most delicate, and frequently only gelatinous, and is also reabsorbed at the same time with the cytoblast (fig. 8, 16, 21.). Sometimes, in preparing sections, they are ruptured and scattered over the glass, which might lead — one to suppose that they were free. And probably they are subsequently, on incipient reabsorption, disengaged from their connexion with the cellular wall, and then a slight touch may suffice to disturb them from their position. The wall of the cell is frequently thickened in their neighbourhood, especially where | they are rather globular, for instance in the pollen tube of species of Orchis which has become cellular (fig. 16 and 20). Meyen, who should always be consulted in relation to anato- mical subjects, has endeavoured in his Physiologie, vol. i. p. 45, &c., employing in a very ingenious way his beautiful observa= tions on the relations of structure in developed cells, to establish - the opinion that the cell is formed of spiral fibres intimately superposed. My direct observation, which may easily be re- peated by every one, gives, it is true, quite a different mode of formation ; nevertheless I must bring the facts related by Meyen DR. M. J. SCHLEIDEN ON PHYTOGENESIS. 291 into connexion with my discovery, in order not to leave an appa- rent contradiction unresolved. Meyen himself correctly observes, when treating of those spi- ral tubes whose very narrow fibres lie close upon one another, that an enveloping membrane could indeed not be observed, but that this by no means justified our concluding on its absence ; for if the thickenings of the cellular walls, which are formed in most, perhaps in all cases, in spiral lines, in those places where they make their appearance early, even when the original cellular wall itself is in statu nascentie and soft, are connected firmly with this latter, and at the same time the simple coils of the spiral fibre lie perfectly close on one another, so that with our present micro- scopes no space between them remains perceptible,—it naturally follows that on rupturing the entire membrane (the so-called rolling of the fibres) the rupture in the direction of the coils of the fibre must be so sharply defined that our instru- ments would not possibly be able to show the unevenesses. At the same time it should be well remembered, that the original ellular membrane, especially in long cells of hairs, frequently jandergoes so great an expansion, that at last it would be infinitely, ‘hin, so that even the thinnest and apparently most simple sellular wall would not exclude the possibility of its being com- »0sed of the original membrane, and of the secondary deposit. Now if we set out from the spiral cells and vessels, the distant oils of which admit of no doubt as to the existence of an exte- ‘ior enveloping membrane, and if we follow up the presence of his membrane through all the forms of the constantly approxi- nating coils of the fibre until only the feebleness of our optical ‘neans prevents further direct observation, the law of sound ana- logy would require us to admit even here the presence of a simi- ‘armembrane. But there is yet a more direct mode of proof, vamely the observation of the history of the development. It $ quite an essential law that each cell (laying aside for the pre- ent the cambium) must occur in the form of a minute vesicle, vradually expanding to the size in which we find it in the deve- oped state. Moreover it is the constant result of an extensive \xamination of this formative process that a cell never evinces trace of a spiral formation, either in its appearance or on upture, previous to its complete growth, 7. e. before it has reab- orbed the cytoblast. In all spiral cells, cells which exhibit se- ; | | | 292 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. parate fibres, we find the full-grown cells in the commencement still perfectly simple in their walls. Thus, for instance, I have ob- served this to be the case in all aérial roots in their outer parch- ment-like layer*. Meyen discovered the spiral fibres in Oncidium altissimum, Acropera Loddigesii, Vanda teretifolia, hort. bot. Be- rol. (rectius Brassavola cordata), Cyrtopodium speciosissimum, Aerides odorata, Epidendron elongatum, Cattleya Forbesii, Colax Harrisonii and Pothos crassinervia. This is still more evident in the true cortical layer of these aérial roots, where I discovered in Colax, Cyrtopodium, and Acropera the far more beautifully de- veloped and much broader spiral fibres. In quite young aérial roots not a trace of them can be found, and their formation belongs: decidedly to a process of lignification. We may further be convinced of the subsequent period of the occurrence of spiral fibre in the pericarp of the Casuarine, the cells of which previous to or shortly after impregnation evince not a trace of spiral formation. Meyen has treated these fibrous cells in the envelopes of many seeds in a somewhat stepfatherly way in his Physiologie, which is the more to be regretted, as these inter- esting and often highly beautiful formations promise many con- clusions respecting the physiology of the life of the cell, espe- cially if we should take occasion to investigate accurately the individual development of several of them. I may be permitted to make a few observations on this head. Their occurrence is more extensive than is generally supposed. They occur in the hairs of the pericarp in some Composite, where they were found by Lessing in Perdiciwm taraxact and Senecio flaccidus, and by myself in Trichocline humilis and hete- rophylla. They occur in the epidermis in many Labdiate, for instance in Ziziphora, Ocymum, in most Salvie, e. g. limbata, hispanica, Spielmanni, &c., and lastly in Horminum pyrenaicum. My uncle * Meyen called this, in his Phytotomie, p. 163, an outer cortical layer, which was situated on the true epidermis of the aérial roots. In recent times some doubts have been raised as to the correctness of this view. It may however be almost incontrovertibly proved that the cellular layer termed epidermis by Meyen possesses actual stomata, which, from their being covered, usually indeed occur only in a rudimental state, frequently manifest a complicated structure, although deviating only in appearance, as in Aérides odorata, but often likewise occur quite in the ordinary form and distinct, as in Pothos crassinervia. More- over it was not Dutrochet, as it would seem from Meyen’s Physiologie, p. 48, but Link who first drew attention to this layer. DR. M. J. SCHLEIDEN ON PHYTOGENESIS. 293 Horkel discovered them in all these many years ago; Baxter only noticed and published their occurrence in Salvia verbenacea. I can add to these Dracocephalum Moldavica. R. Brown discovered them in the parenchyma of the pericarp in the Casuarine; and I have met with them in the spongy in- flated cellular aaeae of Picridium vulgare, occurring generally in a reticular form, and presenting an exquisitely beautiful ap- pearance. Horkel also discovered them in the epidermis of the seed itself in the Polemoniacee long before Lindley made known their pre- sence in Collomia linearis. They occur in Collomia, Gilia, Ipom- opsis, Polemonium, Cantua, Caldasia, and perhaps in the entire family, with the exception of Plox, to which genus Leptosiphon, in which are the first indications of them, is closely allied. Horkel had also studied them on the seeds of Hydrocharis, where they occur in the highest state of development, long before Nees von Esenbeck published this fact. Rob. Brown makes mention of them in the Orchidee, which statement I find con- firmed as to most of our native species of Orchis. Moreover I have discovered very beautiful spiral fibrous cells in the epidermis of the seed of Momordica elaterium, and a more reticular fibrous formation in Linaria vulgaris, Datura Stramonium, in Salvie and in several other Labiate; probably it is common to the whole family. Lastly, they occur, according to Horkel’s discovery, in the pa- renchyma of the seminal integuments in Cassyta and Punica. - Whether these formations be studied in their individual development in a single species, or in their progressive stages in a series of allied plants, highly interesting general results will be found in both ways. The general and essential fact at which we first arrive is, that the fibres are never formed free, but in the interior of the cells; and that the walls of these cells in the young state are simple, and generally very delicate. M. Corda’s statement respecting spiral cells without enveloping membrane (Ueber Spiralfaserzellen, &c., p. 7 and 8) is founded merely on inaccurate observation. ‘These cells in the commencement are usually filled with starch, rarely with mucus or gum. The starch always passes into the latter state, in the progress of development; and this is con- verted into gelatin, and, as it seems, gradually from the exterior towards the interior. This gelatin finally passes at its outer VOL. II. PART VI. U 994 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. surface into vegetable fibre, following the direction of a spiral line, the coils of which are sometimes narrower, sometimes wider. If these forms be observed in their successive stages of development and in their various conditions, the idea in- voluntarily forces itself upon one that the spiral formation arises from a spiral movement of a fluid on the walls of the cells be- tween them and the central gelatin. Horkel has once actually observed, in Hydrocharis, the motion of small globules between the coils of the fibre whilst in the act of forming. The highly varied appearance of the fibres seems to depend chiefly upon the time of their origin, and on modifications in the chemical changes of the formative substance. It probably depends solely upon the first circumstance, whether the spiral fibre lies free in the cell when it is formed very late, or whether it is ad- hering to the membrane of the cell, if its origin happens at a period when the cellular membrane itself is still very soft and gelatinous, and consequently can glue itself to the fibre, likewise still in a gelatinous state. This is the case in Camarina, Cassy- tha, Hydrocharis, Trichocline, Orchis, &c., but in general the wall of the cell is too far advanced to unite with the fibre, and it then lies loose in the interior of the cell. In this case the ma- terial is rarely consumed entirely in the formation of the fibre (although it always is when the fibre coheres with the wall) e. g. in Salvia Spielmanni, Mormordica elaterium. I have reason to suppose that this complete consumption almost always takes place, especially in spiral vessels, and is the cause of their conveying only air. More frequently, however, one or more fibres are formed; but then a great portion of the gelatine has still remained unconsumed, which, on moistening the cell with water, oozes out in a vermicular form, and in swelling expands itself over the fibres, thus appearing to surround them; this is the case in most Salvie and Polemoniacee, in Senecio flaccidus, Ocymum polystachyum and polycladum (Lumnitzera, Jacq.). There is an intermediate form between this and the former when the gelatine itself forms a broad spirally wound band, which ap- pears to be composed at its surface of innumerable delicate fibres ; their occurrence in this state is very beautifully seen in Perdi- cium Taraxaci and Ziziphora. A much less advanced formation exhibits merely a thread or a cone of gelatine in the interior of the cell, the surface of which, however, is covered with delicate spiral lines. This occurs in some Salvie, for instance in S. verticillata, DR. M. J. SCHLEIDEN ON PHYTOGENESIS. 295 and in Leptosiphon androsaceum. Lastly, the lowest stage of de- velopment is that where the gelatinous thread, which is furnished with spiral striz, has a hollow cavity in its interior, which still contains undecomposed starch; this instructive appearance is found in Dracocephalum Moldavica, Ocymum basilicum, and some allied species. In illustration of the above, consult Plate XVI. figs. 26-35, with their explanations. Before I quit the spiral fibre, I will merely add, what indeed has been of late admitted by every good observer, that the only difference between spiral cell and spiral vessel consists in the di- mensions ; although constant transitions between them may be observed quite as well as between the liber and parenchymatous cells; and consequently, as regards the doctrine of this sub- ject at least, there is no longer any place for natural philoso- phical phantasies of rigid images of higher types, and such like empty words. That which forms a cell of the liber out of a round cell, the preponderating expansion of an organ lengthwise, is also that which converts the spiral cells (the vermicular body) into spiral vessels. But the function of the spiral fibre is, as every honest vegetable physiologist will certainly admit, entirely un- known to us at the present day. It is certain that spiral vessels and spiral cells occur in the living plant quite as frequently filled with sap (in the young vegetating portions) as with air (in the older organs which have attained their full dimensions) ; and it is this which has given rise to the conflicting views of authors. But the same also occurs in all cells under certain circum- stances; and the influence of spiral fibre remains totally in the dark and unexplained. Perhaps it may seem probable from the preceding that the spiral is everywhere only a secondary variation in form in the product of the vital principle (the fibrous sub- stance) produced by a different tendency of the vital activity of the cell, as soon as this is forced, at a certain stage of its de- velopment, to give up its independent individuality, and to enter as an integral portion into the complexity of the entire plant. Moreover I believe we may venture in conclusion to deduce from the data above enumerated, that this indication of spiral formation is the surest sign that we have no longer anything to do with the simple cellular membrane. I now return, after this somewhat lengthy digression, to my subject. The process of the formation of cells, which I have en- deavoured to explain at full, is in effect that which I have ob- u 2 296 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. served in most of the plants which I have examined. There are however several modifications of this process which add in many parts to the difficulty of observation, nay sometimes render it quite impossible, although notwithstanding this the law remains in general indisputably valid, because the analogy requires it, and moreover we can sufficiently account for the reasons of the impossibility of direct observation. The difficulties which I here notice arise especially from the physical and chemical properties of the substances preceding the formation of cells. The above enumerated ingredients are to be considered as scarcely anything else than some few points, which for the purpose of giving a general view, and to render the clas- sification more easy, I have intentionally selected from the or- ganic chemical processes of vegetable life, which are constantly in operation, and with which as yet we are entirely unacquainted. Almost all those substances exist constantly together in the living plant, and only their greater or less preponderance author- izes the expression, that the cell contains amylum or gum, and so forth. Towards the termination only of the individual life of the single cells do we find them filled with a less number of dif- ferent substances; with one only, probably in those cells alone which contain volatile oil. If now we suppose that the cell is entirely filled with a limpid solution of sugar in which gum is rapidly generated, but only just so much as is necessary to form, by as quick a conversion into gelatine, a delicate cellular membrane, whose existence, in consequence of a similar refracting power of the wall, of the con- tents, and of the surrounding medium, we are not able to distin- guish with the microscope ;—then it becomes highly probable, that a number of such formative processes may be going forward which escape our observation, and become known to us only by their re- sults, when we find, after the reabsorption of the primitive cell, two new ones suddenly in its place. If on the contrary our attention has been previously directed to this process, we have, it is true, by employing reagents, especially iodine, which is quite indispen- sable to the physiological botanist, several means at hand of ren- dering it visible where such a formative process is suspected. Gra- dual transitions to the perfectly invisible processes will be easily found by extensive examinations: I will as an example just mention one of the most difficult cases I have met with. This occurs in the germination of the spores of Marchantia poly- DR. M. J. SCHLEIDEN ON PEYTOGENESIS. 297 morpha. Of the cellular nuclei evident in the spores only few, in general only from 2 to 4, serve for the formation of the cells; the others are quickly enveloped in chlorophylle, and thus with- drawn from the vital process. The transparent liquid in which these cytoblasts float, passes through the ether stages of the me- tamorphosis into cellular membrane just at the boundary of this latter, and so rapidly, that the excessively delicate. young cells are distinguishable by nothing else than a fine, in general more or less uninterrupted circle of infinitely small, black granules, and by a scarcely perceptible greater transparency of the con- tents of the newly formed cells in comparison with that of the primitive cell ; and, finally, under the most favourable circum- stances by the place where the newly originated cells come in contact, and when this juncture is still covered by the mem- brane of the primitive cell. (Pl. XV., fig. 18—20.) In the Cryp- togamia, and especially in water plants, this may perhaps be general; and probably Mohl’s division of the cells of Confervze might be thus explained. If we consider, however, that there are undoubtedly many plants, among which should probably be reckoned more espe- cially the Fungi and Infusorial Algz, in which we are totally unacquainted, as yet at least, with the cytoblast, on account of its absolute minuteness and transparency ; if further we bear in mind that the nucleus in the cell-germ, even in larger cytoblasts, appears frequently immeasurably small, or even with the highest ‘magnifying power, still entirely escapes the eye ; and, lastly, if we deduce from what has been previously stated, that nevertheless this granule, which can no longer be rendered perceptible, pro- bably affords in the proper medium a sufficient cause for the for- mation of cytoblasts with which the whole formative process of the cells originates; then indeed we are forced to confess that imagination here obtains ample space to explain in every case the origin of infusorial vegetable forms even without the aid of a deus ex machina (the generatio spontanea). But it is my inten- tion to communicate only facts and their immediate conse- quences, and not to dream ; and I will therefore rather add a few more observations on the growth of the plant. What is to grow? is a question which every child quickly answers in the expression, “ When I am as big as father.” There is truth in this answer, but this little will not satisfy science. Words have no yalue of themselves, but are like coin, only signs 298 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. of avalue not exhibited in specie, in order to facilitate commerce. And to carry the comparison further, there follows an insecurity in this intellectual property, and frequently bankruptcy, if this coinage has not its unchangeable, accurately determined stand- ard: in a word, the utility of a scientific expression depends on the accurate definition of the idea upon which it is based. Un- fortunately the perversity of our social relations has made us entirely forget the original meaning of money; the sign has be- come to us the thing itself: may some good genius preserve us in our intellectual life from similar mistakes! We must here guard against two dangerous rocks; first, when words are trans- ferred from one science to another without accurately testing whether they fit in their new place as to all their accompanying meanings also; and secondly, when we lose sight of the signi- fication of a word consecrated by the spirit of the language and its historical development, and employ it without any further ceremony in compounds, where perhaps, at most, only an unes- sential part of its signification suits. Thus, for instance, E. Meyer (Linnea, vol. vii. p. 454.), after repeating the well-known experiments of Duhamel, lays down this position: “the law of the longitudinal growth of the inter- nodes is, to grow inter se, or from above downwards.” ‘This po- sition he requires for his theory, and consequently he must de- fend it in all ways, although he himself confesses that this re- verse growth must appear to every one of his readers contrary to good sense. He would never have arrived at this position if he had more accurately analysed the word “ grow,” (to which he was accustomed in animal physiology,) in reference to its appli- cability to the plant: he would soon have found that the origin of new cells, and consequently the actual growth of the plant, constantly takes place in its outer portions upwards, and that his very comparison of the building up of a voltaic pile is ex- ceedingly well adapted to refute himself. Nothing further would result from the experiments of Duhamel and Meyer, than that the inferior, 7. e. precisely the first originated, older cells of the internode possess a greater power of extending in the longitudi- nal direction, and retain this capability longer than the younger cells. With respect to the second point, we find an admirable exam- ple in the position frequently expressed of late, that the stem of the plant is formed of the cohering petioles. The word “cohere” — DR. M. J. SCHLEIDEN ON PHYTOGENESIS. 299 (verwachsen, to grow together) has possessed however from time immemorial, both in common life and in science, the signification that two or several originally and naturally separate parts have be- come by the process of growth either abnormally, or, under cer- tain circumstances regularly, united. If, therefore, we apply the word “ cohere” (verwachsen) to the stem of the plant, an organ, which, in every period of its existence, under all forms of its ap- pearance, is a simple and undivided one, and at the origin of the plant even constantly makes its appearance earlier than the leaves with their petioles, there certainly is in this a monstrous misuse of language, and science itself can gain nothing by it, and even loses in the eye of the intelligent layman who sees through such a play upon words. What would the zoologist say were we to regard the trunk as a cohesion of the extremities ? But I come back to my question: What isTo grow? An old twaddler says, To grow signifies increase of the mass of an indi- vidual, and takes place in the inorganic world by juxtaposition, in the organic by intus-susception. Have we gained anything by this for vegetable physiology? I think not. If the plant is to grow by intus-susception, then I say the plant consists of an ag- gregate of single, independent, organic molecules, the cells ; it increases its mass by new cells being deposited on those already existing ; consequently by juxtaposition. But the single cells in their expansion, frequently to an enormous bulk in compari- son with their original size (I need merely call to mind the pol- len tubes), also increase in substance in the interior of their membrane, and in this way also the mass of the whole plant is increased; it consequently grows by intus-susception also. Lastly, the cell deposits after a certain time new organic matter in layers upon its primitive membrane, therefore a juxtaposition again, which still however belongs to the cycle of the life of the plant. It is hence easily apparent that the idea “ grow” still requires for the purposes of scientific botany a new foundation in order to be capable of being applied with certainty. Of the three above-named cases, the second and third belong more to the individual life of the cells, and are of secondary importance only, as concerns the idea of the whole plant, re- garded as an organism composed of a certain (1 to ©) num- ber of cells. The plant considered in its totality increases its mass, that is, the number of the cells composing it, in the first way only. 300 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. We must therefore discriminate here three processes essen- tially distinct from each other, which accurately considered scarcely find their analogue in the other kingdoms of nature. 1. The plant grows, i.e. it forms the number of cells it is to have. 2. The plant unfolds itself by the expansion and development of the cells that are formed.—It is this phenomenon especially, altogether peculiar to plants, which, because it results from their composition of cells, can never in any form, not even a re- mote one, occur in crystals or in animals. 3. The walls of the full-grown cells are thickened by fresh- deposited layers ;—a process which, according to the old rule, a potiori fit denominatio, may be most properly termed the lignifi- cation of the plant. ' If, with regard to the growth of the plant, we keep at present to the meaning of the word given under No. 1, then this ques- tion will arise—Where are the new cells formed? Three cases here comprise all possible answers: namely, the new cells are either formed outside on the surface of the entire previous mass ; or in its interior ; and then again either in the intercellular pass- ages or in the cells themselves; guartum non datur. Mirbel has,in two excellent and profound memoirs on the Mar- chantia polymorpha, which he presented to the French Academy in 1831 and 1832 (p.32), proposed the idea, that all the three cases just mentioned as possible do actually occur in plants. Without meaning here to anticipate what follows, I must yet remark, that only one case (the formation of new cells in old ones) appears to be proved by his direct observations. The se- cond case is merely a conclusion assumed ; and lastly, the germi- nation of the spores of the Marchantie, which was to explain the third case, has been observed by me to be quite different, as I have already represented above. Lastly, however, we must still examine whether the differ- ence of organs establishes a physiological difference of growth which deserves our attention. We may distinguish here four cases. We observe: 1. The development of the plant upwards in puncto vegetationis, C. Fr. Wolff). 2. The elongation downwards. We thus comprise the formation of the neces- sary organs of the plant, of the stem, of the leaves, with their metamorphoses, and of the root. 3. We have to keep in view the production of the accidental organs, e. g. bulbs, &c. And 4. DR. M. J. SCHLEIDEN ON PHYTOGENESIS. 301 We find an annual thickening of the axile formations, the de- velopment of the woody stem. Let us now see which of the three possibilities of the forma- tion of new cells, in each of the cases just enumerated, is actu- ally realized. I have shown how the new cells are developed in the embryonal sac, and consequently in a large cell. A similar process is evident in the embryonal end of the pollen tube, consequently in a highly elongated cell; and I shall now proceed to delineate the further development of the embryo. After the first cells, generally few in number, have formed, they rapidly expand so much that they fill the pollen tube, which is then very soon no longer recognisable as the old enveloping membrane. But at the same time several cytoblasts again originate in the interior of each of these cells, and produce new cells, on the rapid expansion of which, the mother-cells also cease to be ap- parent and are reabsorbed. The same process is repeated indefi- nitely. But since the newly originated cells have continually less room to expand, and therefore constantly become smaller, the previous transparency is soon destroyed by the cytoblasts which are constantly being produced anew in the interior, and by the tissue becoming more and more compressed; and from this stage to the perfect completion of the embryo we are conducted by the clearly logical inference that the process thus introduced con- tinues the same, since no new force comes into action, which might determine us to admit a sudden variation of the vital ac- tion, more especially as we very soon meet with the same indi- cation of the vegetative power again. The seed, meanwhile, germinates, and the embryo becomes a plant ; and then indeed the question may arise, Does the process of life continue the same thenceforward, in the internodes and foliaceous organs? Now we are here very soon convinced of the negative,—that an origin of new cells on the surface of already existing organs does not take place. The surface is always smooth, and generally provided in a very early state with akind of epidermis, the outer layer being more transparent and almost as clear as water; and never do we find even an indication of a newly formed cell on the surface. But if the embryo is the image of the whole plant, and this latter does not present anything that is not a repetition of its organs, if we have found in the embryo that its growth only consists in the formation of cells within cells, we may expect to 302 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. find the same result also in the process of the growth of the entire plant. It is principally a foliaceous organ, the anther, which has hitherto been studied and followed in its development by many celebrated men (particularly well by Mirbel) ; and here it is quite decided that the increase of cells takes place within the old ones. And in this case the formative process certainly coincides with that above described. R. Brown and Meyen have enumerated many cases where they had observed the cyto- blast in very young cells of the pollen. In Pinus, Abies, Podo- stemon, Lupinus and others, I have followed up completely the development of pollen after Mirbel; in Adzes I have decidedly observed the cellular nuclei and their development into new cells within one another; and I have never missed the cytoblast in young cells. Now if the pollen grains are nothing more than converted leaf- parenchyma, if the anther is merely a metamorphosis of the leaf, we may undoubtedly infer inversely, that the process which we have observed in it, and which characterized the formation of the embryo and cotyledons (as prototypes of the leaf), will be again found in all foliaceous organs. For the same reason which was stated in reference to the later stages of the development of the embryo, actual observation is infinitely difficult in this case. With a view to this Ihave nevertheless examined a large number of buds, and have convinced myself in the most decided way of the iden- tity of the process both in the constantly elongating apex of the axis and in the leaves, which always originate somewhat beneath it. The best adapted for this purpose are the succulent plants, the Aloinee and Crassulacee. Crassula portulacea seemed to me mostadvantageous, and in this I first succeeded in separating from their connexion some cells, in whose interior young cells were already developed, without however entirely filling the original cell. But having once become familiar with the subject, I was subsequently able to detect in all other plants these individualities from among the apparently merely semi-organized chaos. An- other additional circumstance here indeed presents itself, which renders the subject much more difficult than with the embryo. For, setting aside the smallness of the cells, their walls, in the new-forming vegetable parts, still consist only of gelatine, and are so delicate, that it is exceedingly difficult to separate the ji parts intended for examination without destroying the organiza- _ tion altogether. (Compare Plate XV. figs. 22—24.) DR. M. J. SCHLEIDEN ON PHYTOGENESIS. 303 This process is more easily discernible in articulated hairs, and such as have a head consisting of several cells, where the same appearances, which I have so frequently observed in the young embryo, and such as Mirbel has so beautifully described in the development of the gemmez in the cups of Marchantia, may be easily and beautifully seen, for instance in the common potatoe. Meyen also has published similar observations, although he still expresses himself with some doubt. (Wiegmann’s Archiv, 1837, vol. ii. p. 22.) It is not until after as many cells are formed as the organ re- quires for its completion that the walls of the cells become firmer ; and then commences the development of the organ by the mere expansion of the cells already formed. But I must here enter somewhat more into detail, in order to explain the probable origin of the vascular bundles, and of the epidermis. At an early period a stripe of more transparent cells is defined in the axis of the leaf which is in the act of forming, in which no more new cells are developed, and these cells soon considerably exceed in size the cells of the remaining mass, which are constantly becoming smaller by continual divi- sion. These cells are the foundation of the future vascular bun- dle which forms the midrib of the leaf. For while the paren- chymatous cells subsequently expand on all sides, these cells are only developed in their longitudinal dimension, and are thus able, although fewer in number, to follow the expansion of the other cells in the longitudinal direction of the leaf. Itis not till a later period that these cells, by a difference of the internal depositions, separate themselves into spiral vessels and cells of the liber. The spiral vessels begin to be visible in the newly-formed parts, and also in the entire bud, always in the immediate vicinity of old, already formed spiral vessels; and they proceed in this manner away from the stem into the new parts. I do not under- stand therefore what is meant when the fibres of the stem are regarded as proceeding from the buds; one might quite as well consider the river as running from the ocean to its source. A similar process takes place in the development of the side nerves of leaves. The formation of new cells generally ceases quite early in the outermost layer of cells. The cells are soon filled with a limpid fluid, and naturally become, on the expansion of the subjacent parenchyma, superficial, flat, and expanded. The cells of the vascular bundle and of the epidermis appear in this way to be less potentialized [minder potenzirt|—are as 304 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. it were cells of lower dignity than the parenchymatous cells ; and perhaps this physiological peculiarity is connected with the fact, that they more rarely secrete peculiar chemical substances, but are mostly thickened only by depositions within their walls of new vegetable fibrous substance, or more correctly, membranous sub- stance. I cannot omit here venturing to throw out some hints, which perhaps are less intimately connected with the purpose of this Memoir, but which may probably at some future time be of importance, for the understanding of the entire plant. Let us once more pass under review the process of growth of the plant just depicted. A simple cell, the pollen-tube, is its first foundation. In this originate cells; in them are formed new cells, and so forth through the entire life. But here, the mode just mentioned of the origin of the vascular bundles and of the epidermis in relation to the parenchyma, would point to this fact ; that the lower the dignity of the cell, 1. the greater power does it possess of expanding and of extending in length, and 2. the less capacity does it possess of forming peculiar fine sub- stances in its interior. If now the potentialization of the cells goes on throughout the entire growth of the plant, there follows from thence a constantly closer approximation of organs otherwise kept asunder, and a constantly higher ennobling of the substances developed in the cells. Consequently, the lower parts of the internodes will appear to be more elongated than the upper ; the leaves and young shoots (summitates herbarum, Pharmacol.) con- tain nobler saps than the stem; the members are shortened as they approach nearer to the upper terminal point of the plant, the leaves come closer together, and the result of this internal higher potentialization of the cell, of the constantly diminished expansion in length, of the constantly nearer approximation of the lateral organs, of the constantly more nobly developed sub- stances, is, last of all, the flower, in its exclusive individuality, with its splendour of colour, its perfume, and its secret capacity of determining, by means of its juices, a single cell which is to develope itself anew into an independent plant and pass anew through the same cycle. I return after this digression to my subject. Hitherto I believe I have demonstrated conclusively enough and in accord- ance with nature, that in the whole growth of the plant* cells are constantly formed only within cells. Let us now proceed to the * T would observe, that in the whole Memoir in general only phenogamous plants are intended. ——— DR. M. J. SCHLEIDEN ON PHYTOGENESIS. 305 root. Here I can contribute but little to the explanation of the subject; for as yet I have not succeeded, in the rather limited researches that I have instituted, in coming to any satisfactory result ; for I found it quite impossible to decide the question, whether there is secreted at the extremity of the radicle a liquid in which new cells are formed. On the other hand, it is certain that there exists in the extremity of the root a concavo-convex group (a meniscus) of cellular tissue in which the process of the formation of cells takes place in the same way as in the ascend- ing parts of the plant. A main cause of the elongation of the root consequently consists in this,—that on the convex side of that cellular mass new cells are constantly formed in the interior of those already present, while on the concave side the cells already formed expand cotemporaneously, and generally indeed most predominantly in the longitudinal direction, and thus constantly push the extremity of the root before them. The third case, the formation of the accidental organs of the plant, I must here entirely pass over, as I am wholly unfurnished with any personal observations on the subject. Probably, how- ever, the process here is the same as in former cases, for Meyen (Physiology, Vol. i. p. 209,) observed the cellular nuclei in the ' germinating tubers of Orchidee. Moreover, analogy leads to the same result, since all these parts are nothing more than morphological modifications of organs which have already been previously treated of. It still remains, however, for me to men- tion a fourth point, namely, the increase in thickness of plants forming woody stems (Dicotyledons). The origin and import of cambium is the nut upon which so many young phytologists have already broken their milk-teeth, the Gordian knot which so many botanical Alexanders have cut instead of untying, and the enigma in the solution of which almost all the Coryphezi of our Science have laboured with more or less success. My inquiries respecting this layer of distinct origin between bark and wood are by no means concluded. Before I proceed, however, to the communication of my ob- servations on this subject, it is requisite once more to take up the question of the individuality of plants.—I have above ob- served, that in the strictest sense of the word, only the simple cell deserves to be called an individual. If we go a step further, we might regard each axis with its lateral organs as simple beings. If, however, we disregard this composition of the plant of cells 306 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. and similar axes, and conceive as an individual in the organic world, that body which cannot, without losing its idea of totality, be divided into two or several, and whose vital process has a fixed point of beginning and ending in definite periodicity, it thence follows that only the herbaceous (planta annua) and the true biennial plants, which flower in the second year and then die off entirely, can be considered as individuals in the vege- table kingdom. The idea of individual life necessarily requires for a character individual death as a condition of the organiza- tion itself. But where such a death does not exist as a final termination from internal necessity, as an internal precondi- tioned cessation of the organizing force, there individuality must be out of the question. But this is only the case in the above-mentioned plants; and, consequently, from them solely must we set out, as from the prototype, in all inquiries regard- ing the nature and life of the vegetable organism. To prepare for a transition to what follows, I shall turn to the exposition of the two different modes of propagation. It either takes place by a process which has hitherto been termed in plants impregnation, and to which has been ascribed a sexual difference, (Wiegmann’s Archiv, 1837, Vol. i. p. 200, &c.) or by division, the plant, for instance, developing on itself a per- fectly similar individual, and then at a certain time dismissing it. This latter, the formation of so-called bulbilli, &c. occurs to- gether with the former only in a small number of plants. We must however make ourselves better acquainted with it. This creation, for instance, does not take place always in such a way as that the mother plant separates itself entirely from them, and scatters them singly ; but it forms most frequently, before its in- dividual death, a peculiar organ, which places the offspring in a peculiar vital connexion with one another, and at the same time serves as a reservoir for a certain quantity of nutritive sub- stance, by which the first development of the young individuals is facilitated. But in general this organ is merely a metamor- phosis of some other single well-known one, the stem or the root, or, as in the potatoe, the axillary buds; and consequently, in this case, no one has ever hesitated to speak of these things as of mere parts of a plant, which continue to live as connecting members between the younger individuals after the death of the parent. A different course on the contrary has been taken, when stem and root cotemporaneously, and therefore nearly DR. M. J. SCHLEIDEN ON PHYTOGENESIS. 307 the entire totality of the plant, take part in this formation; and although the result in this case may probably be that there can be no question of an heteromorphy of a known part of a plant, yet the physiological identity in signification of this and the former case has not been steadily maintained, and the view has thus been obscured. Most botanical writers set out quite at their ease, as if it were self-evident, from the tree as the perfect plant, and I believe it is not difficult to demonstrate that where Vegetable Physiology lies very deep in error this very misconception is solely to blame. Two quite distinct ideas have here been confounded, viz. the highest stage of development to which vegetable life can in fact raise itself, and the type upon which the idea of individual must be based. Now if the first of these ideas may be truly main- tained with regard to the tree, yet the application of the second to it is in every respect totally false, as has been very correctly asserted before by M. Meyer (Linnea, vil. p. 424). It necessa- rily belongs to the idea of a plant that it produces on its stem fo- liaceous organs ; yet there is no tree that has leaves. Paradoxical as this may sound, yet it is not the less true. It is a fact that certainly no botanist is ignorant of, that no lignified part of a plant, even though only in its second year, is capable of producing a leaf; but the direct consequence is by no means so generally acknowledged, that for that very reason the woody stem can- not come under the idea of plant. From the error of regard- ing the tree as a single plant much confusion has arisen in our physiology, the definitions of the ideas of root, stem, bud, &c. have become very unsettled, and bitter controversies have been carried on respecting the functions of these parts, which could have no result, because the one party spoke of this, the other of that, this one of stalk, the other of stem, this of root-fibrils, that of ligneous root-substance. But the so-called lignified root is just as little a root as the lignified stem is a stalk ; but both together are, according to the idea, inseparable, and they form, moreover, altogether a purely accidental organ in the plant, which the annual individual has secreted on its surface, in order to bring into connexion, by means of a single organized membrane, the whole sum of new and young individuals. The tree corresponds entirely to the polypidom, and it appears to me not more sound to set out from it as the type in plants, than were the zoologist to set up a Gorgonia as the idea 308 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. of animal individuality. -And this analogy is not in the least weakened by the circumstance, that exactly in the highest deve- loped plants we meet with this woody stem most frequently ; but it is on the contrary natural, that, if the animal kingdom receives in a certain measure its vegetative side from the vegetable king- dom, it should connect itself through the lowest stage of animals to the highest plants, while this vegetative half of the vital phz- nomena in the higher animals is in like manner illustrated and ennobled by the constantly more surely and more obviously in- dependent individuality. With this explanation of the woody stem (the root included), it will appear henceforward by no means remarkable that this organ (as if it were a mere organized groundwork) can produce upon every part of its surface young vegetable individuals, 2. e. buds, as soon as it is in a condition to convey nutritive substance to these buds from any part, whether it correspond apparently to the former root or to the stem ; while this purified idea of the plant leads to the law, that in the regular course of vegetation, neither root nor internode, but only the axilla of the leaf, is ca- pable of generating a bud, é. e. a new axis with lateral organs. But the following remarks, which in nature (who never, like a bad artist without a plan, fluctuates between the most oppo- site methods,) would be in the usual way of treating it an inex- plicable contradiction and an absolute miracle, will serve for the decided establishment of this view. We miss quite suddenly, for instance, upon the secretion of this organized mass, the wood, the influence of the law of forma- tion, which, till then, had without exception, presided over the growth of the entire plant in allits parts. There are here formed, so far as we are yet acquainted with the subject, no cells within cells; there occurs here no expansion on all sides of the primi- tively minute vesicle ; there is here no cytoblast from which the young cell might be developed—but under the outermost layers of cells which are comprised in the term bark, an organizable fluid pours itself, as it were, into a single large intercellular space, which fluid, as it appears, very suddenly consolidates in its whole extent into a new, peculiarly formed tissue of cells de- posited on one another, the so-called prosenchyma. Here, moreover, decidedly no vascular bundles are formed from cells of lower dignity; for all the cells are cotemporaneous, and ori- ginate at their full size; and what has been called “spiral DR. M. J. SCHLEIDEN ON PHYTOGENESIS. 309 vessels of the wood” is something immensely different from the spiral vessels of herbaceous plants, both with respect to their origin and probably also with respect to their physiological destination. In the controversies carried on, sometimes with great warmth, respecting the function of spiral vessels, no result has been obtained, nor could any be obtained, because each person meant, quite ad libitum, the spiral vessels of her- baceous plants, or of the wood, completely shutting their eyes to the possibility that the two might be exceedingly different things. If, for instance, we consider the cambium in the ear- liest period in which it begins to acquire organization, we find that it consists throughout of entirely similar prosenchymatous cells still in a gelatinous state. A short time afterwards some longitudinal series of these cells appear to have increased in breadth, by which alone they are distinguishable from the adja- cent mass. On a further development we observe that some dark spots appear on the walls of some of these expanded cells, which we soon recognise to be small flat air-bubbles that have formed between the walls of this and of the neighbouring cell. Gradually all the expanded cells which are superposed one upon the other are changed in this way ; the air-bubble gradually ap- pears more circularly or ovately bounded, and there appears in its centre a smaller circle which constantly becomes more dis- tinct, and which originates in the following manner :—on the deposition of new masses upon the inner wall of the cell, the parts corresponding to the outer air-bubble remain free from this deposition, thus forming a small canal which traverses the newly deposited mass. We now distinguish the fully developed porous vessel, the septa between each two superposed cells appear- ing at the same time to be more or less reabsorbed. This history of the formation of the porous vessels, which may easily be ob- served on limes and willows, greatly contradicts the general no- tion that the porous canals serve to facilitate the communication of the saps. As the air-bubble is first formed on the outer sur- face of the wall, it renders the passage of the sap impossible at this spot, and for this reason the origin of the porous canal might probably be most easily and naturally explained as a lo- cal atrophy of the cellular wall. At the same time it is evident from hence that the distinction between wood in general [laub- holz) and fir-wood, as to its anatomical structure, cannot be of such vast physiological importance; for, with like elements and VOL. Il. PART VI. 7 310 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. - like development, the distinction depends in fact on the larger or smaller number of cells that are converted into porous ves- sels. There are, however, a vast number of gaps still to fill up: and, more especially,-the origin of the medullary rays and their rela- tion to the wood, the formation of the new bark, and lastly, the origin of the buds in the wood, are so many questions for ex- tensive researches, to the execution of which, however, we may look forward at no distant time, considering the ardent and gra- tifying zeal which has been awakened and cherished, especially among our contemporaries, in behalf of the sound and scientific study of the anatomy and physiology of plants. I have, as far as lay in my power, attempted in this Memoir to solve many interesting questions in vegetable physiology ; or, by more accurate definitions of the question, to advance nearer to a future solution. May these observations meet with a friendly reception, and be speedily improved upon and extended among the vegetable physiologists of Germany. EXPLANATION OF PLATES XV. AND XVI. Fig. 1. Cellular tissue of the albumen from the embryo-sac of Chamedorea Schiedeana in the act of formation. a. The inner mass consisting of gum with intermixed mucous granules and cytoblasts. 6. New cells, still soluble in distilled water. c—e. Further development of the cells, which by a slight pressure still form into an amorphous gelatinous mass, with the exception of the cytoblasts. Fig. 2. The formative substance from fig. 1. a. more highly magnified, gum, mucous granules, nuclei of the cytoblasts and cytoblasts. Fig.3. A single, still free cytoblast, still more highly magnified. Fig. 4. A cytoblast with the cell forming on it. Fig. 5. The same, more highly magnified. Fig. 6. The same. The cytoblast here exhibits two nuclei and is represented in Fig. 7. in an isolated state after the destruc- tion of the cell by pressure. Fig. 8. The same cellular tissue still further advanced in de- velopment than in Fig. 1. e. The walls of the cells in contact already cohere. In a. their horizontal section, it may be distinctly perceived that the cytoblast is inclosed in the cellular wall. Fig. 9. Cells of the nearly mature albumen in a thin cross section. Fig. 10. Common septum between two cells from Fig. 9, under DR. M. J. SCHLEIDEN ON PHYTOGENESIS. 311 a higher power. The strata-like depositions (near 5.) upon the inner wall, and the porous canals (near a.) produced by their local failure are apparent. I could distinctly count nine to - twelve layers which had originated within fourteen days. Fig. 11. A spore from Rhizina levigata, Fries, with the cyto- blasts. = Figs. 12—14. Several cytoblasts from the embryo sac of Pi- melea drupacea before the appearance of cells. Fig. 15. Young cells with their cytoblasts from the same. The latter here unquestionably present three nuclei. Fig. 16. A portion of the embryonal end of the pollen-tube projecting from the ovulum in Orchis Morio, in which towards the upper part cells have already developed. Below, the original pollen-tube is still distinguishable. The almost globular cyto- blasts in this case are distinctly included in the cellular wall. Fig. 17. Embryonal end of the pollen-tube from Linum pal- lescens, together with the appended lobule of the embryo sac (a.). The process of the formation of cells is in its beginning. Above, a young cell with its cytoblasts is already perceptible ; beneath this are seen several cellular nuclei floating in a free state. Fig. 18—20. Commencement of the germination in the spores of Marchantia polymorpha. Compare the text, p. 297. Fig. 21. Portions of the pollen-tube become cellular in Orchis latifolia in the highest stage of development. The covering de- rived from the pollen-tube is no longer perceptible. The cyto- blast is exactly as in Fig. 16. included in the wall of the cell. Figs. 22 and 23. Two isolated cells from the terminal shoot (punctum vegetationis, Wolff.) of Gasteria racemosa; in 22, two free cytoblasts are seen ; in 23, two newly formed cells in the original cell. Fig. 24. A very young leaf of Crassula portulacea, the five cells solely composing it are still surrounded by an original cell. Fig. 25. Three cells from an articulated hair of a potatoe, with a quantity of currents of mucus at the sides, giving them a re- ticulate appearance. In the middle cell the direction of the cur- rents is partly indicated by arrows. Wherever hitherto I have observed in Phanerogamia these movements in the cells, I have constantly found that the moving part consisted of a yellowish gelatinous fluid, perfectly insoluble in distilled water, and mixed with a quantity of minute black granules, differing entirely from the other aqueous cellular sap ; and even where the currents were so minute that they appeared merely as excessively minute delicate lines of black points, yet I succeeded with higher magnifying powers in distinguishing the yellowish gelatinous fluid, especially with the favourable cir- cumstance, which frequently occurs, of the current being ar- rested by some preventive, thus causing a somewhat large quan- 312 DR. M. J. SCHLEIDEN ON PHYTOGENESIS. tity of the moving water to be aggregated, and upon this followed in general either a change of direction or a division of the cur- rent. Fig. 26. Cells from the epidermis of the pericarp of Ocymum basilicum, moistened with water, so that the globule of mucus has expanded, ahd has torn the outer cellular wall (a) from the side walls (0). Fig. 27. Cells from the epidermis of the pericarp of Ziziphora dasyantha. Fig. 28. Cells from the epidermis of the pericarp of Salvia verticillata. Fig. 29. Cells from the epidermis of the pericarp of Salvia Horminum. Fig. 30. Cells from the epidermis of the pericarp of Salvia Spielmanni. Fig. 27—30 a. shew the remains of the side walls of the rup- tured cells. Fig. 31. A portion of the epidermis (a) and of the integument (0) of the ovule of Collomia coccinea. The epidermis cells only contain granules of starch. Fig. 32. The cells of the epidermis of the half-ripe seed of the same plant, containing mostly gum, near a. some still undecom- posed starch. Fig. 33. The same cells in the nearly ripe seed. Beautiful spiral fibres have been formed from the entirely consumed con- tents. Fig. 34. Cells of the epidermis of the seed of Leptosiphon an- drosaceum, moistened with water, so that the globule of gelatine has come out. (a) remains of the cellular walls. Fig. 35. Cells from the epidermis of the seed of Hydrocharis morsus rane. In the lower part of the cells, where they are connected with one another, the spiral coils take a direction dif- ferent from that in the upper free part of the cells. For Figs. 26—35 compare the text p. 293 to p. 295. SCIENTIFIC MEMOIRS. VOL. II.—PART VII. ARTICLE IX. Supplement to the Treatise entitled “ General Theory of Terres- trial Maynetism+.” By Professor C. F. Gauss, of the Uni- versity of Gottingen. [Translated from the Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1838. The references in the translation of the ‘‘ Supplement ” are made to the corresponding pages of the translation of the General Theory in the Scientific Memoirs. ] AFTER the table of comparisons, pp. 216—219, was printed, two slight inaccuracies were remarked in it :—the one at Callao arose from a wrong longitude in the work referred to in page 222 ; that at St. Helena arose from an error of calculation. I have sub- joined the corrected result at those two places, and have availed myself of this opportunity to give the comparison of the theory with observations at eight other stations which have since come to my knowledge. } Declination. Station. Latitude. | Longitude. oi als Computed. Observed. | Difference. o i 0 i 0 i 0) Ul 0 / 8* Port Etches ...... +60 21) 213 19 || —28 33) —31 38] +3 5 8** | Lerwick............ +60 9] 358 53 || +27 10/+27 16| —O0 6 11. Stockholm......... +59 20) 18 4 || +15 22}+14 57] +0 25 34* Valencia ......... +51 56] 349 43 |} +30 2) +28 43] +1 19 40* | Brussels............ +50 52 4 50 || +23 23)+22 19} +1 4 54* | Montreal ......... +45 27) 286 30 || + 5 23'+ 7 30; —2 7 62* DAD Wises oescsesecs.- +21 17| 202 0 || —12 19, —10 40; —1 39 64* Panama.........0s+ + 8 37] 280 31 ||— 6 44'— 7 37] +0 53 68 Callaot si soi. 0500% —12 4] 282 52 ||— 9 32 —10 0/| +0 28 71 St. Helena......... —15 55] 354 17 || +19 27 +18 0] +1 27 + Translated in the Scientific Memoirs, vol. ii., Art. V. VOL, Il. PART VII. = 314 Cc. F. GAUSS ON THE GENERAL THEORY OF Inclination. Intensity. Computed. Observed. Difference. |] Computed. | Observed. | Difference, 0 Oo / g* | + 76 25| + 76 +622 | 1-678 | 1-75 | — 0-072 8** | + 73 46]}+ 7345] +0 1 1:469 1°42] + 0-048 Ui? + 70 52| + 71 40] + 0 48 1:45] 1:382 | + 0:069 34* + 71 25| + 70 52) + 0 33 1448 1:409 + 0-039 40* + 67 29| + 68 49 | — 1 20 1:393 1:369 | + 0°024 54* 77 24) + 7619); +1 5 1-713 1-805 — 0:092 62* + 37 36] + 41 35] — 3 59 1°125 114 — 0015 64* + 34 40] + 31 55] +4 2 45 1-238 1:19 + 0-048 68 — 439) — 614] 4 1 35 1-003 0:97 + 0:033 71 — 1452};-—18 1] +3 9 0811 0:836 | — 0:025 The observations at Stockholm were made by Rudberg ; those of intensity and inclination in 1832, and those of declination in 1833: Poggendorff’?s Annals, vol. xxxvii. The observations at Brussels are for 1832; those of declination and inclination are by Quetelet, (Bulletin de ? Académie de Bruzelles, tome yi.,) those of intensity by Rudberg, (Sabine, Report on the Variations of the Magnetic Intensity). 1 am indebted to the obliging com- munication of Major Sabine for the determinations at the other six new stations, as well as for the intensity at Callao, and for a more recent determination of the dip at that place. The obser- vations at Lerwick and Valencia were made by Captain James Ross in 1838; those at Port Etches, Panama and Oahu, by Captain Belcher in 1837; and those at Callao by the same officer in 1838: the inclination and intensity at Montreal were observed by Major Estcourt in 1838; the declination at that station is for 1834, but the observer is not named. There are two other trifling corrections to be made in the table of comparisons. By an error of the press, the longitude of Naples is made 10! too small, although the true longitude 14° 16’, was employed in the calculation. The declination observed by FitzRoy at Otaheite is printed in one part of the Magnetic Observations made during the Voyages | of H. B. M.’s ships Adventure and Beagle as 7° 34! E., in an- other part of the same work 7° 54! EK, Of these two numbers, the one employed in the table of comparisons was the erroneous one; the difference between calculation and observation at that station is therefore + 2° 0°9!. The following errors of the press are also to be corrected :— Page 186, line the last, for twelve read fourteen. TERRESTRIAL MAGNETISM. 315 Page 239, in p = 45°, log. a’, for 2°29724 read 2°29796. — 251,in d= — 13°, log. cl’, for 1°27047 read 1°37047. The public is indebted to M. Weber for the map (Pl. XVII.) containing the values of the declination as computed from the Ele- ments of the Theory of Terrestrial Magnetism (Scien. Mem. vol. ii. page 211). In order to give a clearer view of the intricate form of the system of lines of equal declination, the points at which the declination has a maximum value, as well as those points where two lines of equal declination intersect each other, (or where one such line crosses itself,) have been computed with especial care. There are two points of the first kind, and four of the second kind. The common character of such points consists in this, that the first differential of the declination in every direction disappears. It is almost superfluous to remark, that in those regions where the declinations alter very slowly on all sides, as in Southern and South-Eastern Asia, small alterations in the values of the declination may produce very great changes in the form of the system of lines. The same remark applies to the maps of the Total Intensity (Pl. XVIII. and XIX.) computed by Dr. Goldschmidt from the tables, pages 236—251. These maps show, in the northern hemisphere, two points of maximum intensity, and one point of intersection of lines of equal intensity ; in the southern hemisphere, one point of maxi- mum ; and in the middle zone, two points of minimum inten- sity, and two points of intersection. Similar maps, grounded on the theory, are in preparation for the inclination, for the horizontal intensity, for the three com- ponents of the earth’s magnetic force (2. e. the values of X, Y, and Z), and for that distribution of the magnetic fluids on the surface of the earth, which may be taken as the representative of the actual distribution in the interior. We hope to publish these maps in the Resultate for 1839. Note by the Translator. The maps of the Declination and of the Total Intensity, com- puted according to M. Gauss’s theory, are given in the present number of the Scientific Memoirs. By the kind permission of MM. Gauss and Weber, the translator is also enabled to give in y2 316 C. F. GAUSS ON TERRESTRIAL MAGNETISM. the present number the maps of the Inclination and of the Hori- zontal Intensity (two of each), Pl. XX., XXI., XXII., and XXIII., computed also according to M. Gauss’s theory, and not yet published in Germany. It is requested that the following corrections, which have been kindly pointed out by M. Gauss, may be made in the Translation of the “ General Theory” (Sc. Mem., vol. 11. Part vi.) Page 196, line 35, for the space included by read the space com- prehending. — 196, note, line 1, for themselves read even if. — 202, line 21, for V (r?—2rr°) cos ucosu® + sin u sin u° cos (A—A®) +79 7° read y/ (r?—2rr° (cos u cos u?+sin w sin vu? cos (A—A®)) + 7° 7°), — 204, line 4. In the second term of the factor of sin wu” for (n —m) (n—m +1) (n—m) (n—m—1) a aCe read aay — 210, line 35, for on seven parallels read on each of seven parallels. — 228, line 16,/or the present century read future centuries. — 236, line 38, for eliminate read obtain. . It is also requested that the following corrections may be made in the magnetic papers in the Sc. Mem. vol. ii. part 5 :— Page 57, line 17 from bottom, for immediately following read nearest. — 80, lines 11 and 12, for B?—B'?, read B?—B B". — $2, last line but two, in the value of r, for + read x throughout. — 83, line 13, in the value of C, for + read x throughout. REFERENCE TO THE PLATES. XVII. Map of the Lines of Declination. XVIII. Map of the Lines of Total Intensity: Part 1. XIX. Ditto: Parts 2 and 3. XX. Map of the Lines of Inclination: Part 1. XXI. Ditto: Parts 2 and 3. XXII. Map of the Lines of Horizontal Intensity : Part 1. XXIII. Ditto: Parts 2 and 3, | 317 ARTICLE X. On the Method of Least Squares. By J. F. Encxe, Director of the Astronomical Observatory at Berlin. [From the Astronomisches Jahrbuch for 1834. ] THE frequent application of the method of least squares, or of the calculus of probabilities, to the results of observations, in- duces me to hope that a useful service may be rendered, by giving as brief and elementary a view as is possible of the pro- positions on which this method is founded,—adding thereto certain rules which I know from much experience to be most convenient in practical application. With this design I have drawn the present paper from the following sources: Gauss, Theoria motus corporum celestium, lib. i. sect. 3; Disquisitio de elementis ellipticis Palladis. Com. Gott. recen. vol. i. 1808-1811 ; LinpDENAU and BoHNENBERGER, Zeitschrift fiir Astronomie und verwandte Wissenschaften, bd. 1. pp. 185, et seg.; Theoria combinationis observationum erroribus minimis obnoxie. Com. Gétt. recen. 1821 and 1823, Parts I. and II. ; combined with re- marks by Brssex in the Fundamenta Astronomia, pp. 18 and 116, and in his treatise on the Comet of Olbers. No proposition of any importance is here put forward which is not taken from the above-mentioned sources ; but the form of the demonstration has occasionally been altered with the view of rendering it more easy of comprehension. I have not thought it necessary to refer to the particular places where the several propositions are to be found. The classical labours of other mathematicians, especially those of Laplace and Poisson, agree perfectly with those here given, as respects the results: the form of representation and the mode of deduction are different, chiefly because Laplace confined himself to a strictly theoretical view of the subject, and appears to have viewed but one amongst the many ap- plications of the calculus of probabilities. For the present ob- ject, it has been thought preferable to follow the path pursued by the two above-named astronomers, who combine the strictest theory with the happiest practical application of theoretical truths; 318 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. a rare combination, but of high importance in the cultivation of modern astronomy. Experience shows that in the simplest kind of observations, and with the utmost care to avoid all circumstances which may occasion error, continued repetitions of the same ob- servations always give results differing somewhat from each other. The causes of these differences are unknown to us; or, if we choose to ascribe them to the imperfection of our instru- ments, and to the uncertainty of all the perceptions of sense, at least their action cannot be subjected to calculation. We may however assume, that in a certain kind of observation, both the number of the sources of error, and the number of combinations of which they are susceptible, remain the same ; and also that the same combination, whenever it occurs, will pro- duce the same error, If we knew the number of all the possible combinations of the sources of error, and if we knew how often those combinations which produce equal errors are contained in this number, we should be enabled, by the calculus of proba- bilities, to compute a priori how often a certain error ought to appear in a given number of observations, and we could calcu- late the probability that it would not appear more or less fre- quently than a certain number of times. The causes being un- known, we may, on the other hand, apply the calculation of pro- babilities to the results of experiments; or, from the number of times that an error has actually appeared in a number of ob- servations, we may infer how often it should have appeared ac- cording to rule, and how often it would appear in future repeti- tions. This application only supposes that the continued re- petition does not bring in any new source of error. The num- ber of the sources of error, and of their combinations, remains wholly undetermined. By the probability of a certain combination, or of all the combinations which produce an error of a certain amount, we understand the proportion which the number of such com- binations bears to the number expressing all possible com- binations. On this proportion the probability of an error A will depend. If this probability (which is necessarily a func- tion of A, and of one or more constants having reference to the ; ( : q J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 319 kind of observation) be designated generally by ¢ A, then among m observed errors there will be, according to proba- bility, mA errors of the value A; and this determination will be so much nearer the truth as m is greater; so that if m be indefinitely increased, there can be no assignable dif- ference between the value of m ¢ A, and the true number of the errors A. Even with this indeterminate. designation some of the pro- perties of the function ¢ A can be shown. We know that in each kind of observation the errors can in no case go beyond a certain, though not precisely definable limit; consequently, if a denote the value of this limit, for A > a (abstracting signs) 6A becomes impossible, or=0. In like manner, on the supposi- tion of the greatest possible care in the observations, and with the assumption, which is the only warrant of certainty in ex- perimental science, that a greater number of observations gives hope of a more exact result,—it is implied that @ A is a maxi- mum for A = 0, and is equal for equal positive and negative va- lues. If indeed this were not the case in a continued repetition of the observations, the erroneous values of the quantity to be determined would prevail so much on either the positive or the negative side, that we should find ourselves in the impossibility of attaining the truth, and should be in danger, even with an in- finite number of observations, of taking an erroneous value for the most probable one.. We have then as the most probable value resulting from our observations, that for which ¢ A is a maximum with A = 0, and which is besides a direct function of A; and as we have no other means than the observations of determining the true value, this value must be to us the true one. In these assumptions, however, the distinction between con- stant and irregular errors requires consideration. By constant errors, are generally understood those of which the sources are not general, but belong to the particular observations, some- times to a particular instrument, or to the individuality of the observer. Irregular errors, on the other hand, are those which oceur under all circumstances, and which are therefore properly subject to the calculus of probabilities. The causes of the smaller constant errors are in themselves analogous to those which produce the irregular errors, and the total avoidance of them may even be regarded as impossible. Our aim should be 320 J. ¥F. ENCKE ON THE METHOD OF LEAST SQUARES. to avoid wholly the greater constant errors,—or to lessen them as much as possible,—or to bring their influence so far within the power of computation, that the remaining constant errors in one mode of making the observations may appertain to those sources of error which in other modes of observing can exist only in a different degree. In this case, it is as important to multiply the methods themselves as the observations in each; and by making as many repetitions as possible, and by varying the methods themselves as much as possible, the nearest approach is made to the truth. This distinction between constant and ir- regular errors does not influence the application of the calculus of probabilities, so long as we do not know whether any and what constant errors exist. Their existence may be ascertained, if, on comparing together the results of different methods, we find that a greater difference exists between them, than the treatment of the observations by each method separately would justify us in expecting. For the most part, the multiplication of the observations according to one method is easier to obtain, and is more frequently met with, than the multiplication of the methods themselves. On this account the result deduced as most proba- ble is usually a partial one; and, in order to come as near as possible to the pure truth, the chief object of attention should be to avoid every possible constant error. In the sequel this distinction will be disregarded ; it only causes the estima- tion of the exactness of such a partial result to be always somewhat faulty,—a circumstance so much the less influential on the general consideration, as the estimation itself lays no claim to absolute certainty. If now, with the following conditions, ¢ A a maximum for A =0, $A an even function of A, and ¢ A= 0 for A > a,— we combine the remark drawn from experience, that in general smaller errors are more frequent than greater ones,—that in approaching a, the extreme limit, the number of errors de- creases with great rapidity,—and that between A =O and A = a there is in general no value of A for which ¢ A is im- possible, or that all errors from 0 to a may exist,—then the march of the function may be assigned @ priori. A geome- tric consideration may be here employed to facilitate the con- ception of it. If the values of A be taken as abscissa, and the ¢ A belonging tv them as rectangular ordinates, the curve of probabilities on both sides of the axis of ordinates will be J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 321 symmetrical. A maximum maximorum will be found at A = 0, From this point forward, according to the law, the curve will be drawn continuously, so that in the neighbourhood of A = a, it will approach the axis of abscisse very rapidly. Hence follows another circumstance of great importance in the se- quel. The absolute limit @ can never be strictly determined: but as in the neighbourhood of a the ordinates ¢ A decrease very rapidly, we may without any sensible error assume the limits — © and + o, instead of the values of a, provided the function, which within the values 0 and a@ should agree with the march of the curve, has the property of decreasing constantly as A increases. For in the rapid approach to the axis of the abscissee, so soon as A approaches a, each func- tion which beyond a decreases still more, and was before ap- proaching rapidly, will give for its values between + @ and+ only insensible magnitudes. The definition of ¢ A implies, that when the number of ob- servations is so great that all errors will occur, each in due pro- portion of frequency, mopQ+moA'+mgA". . . . =m, +o or = (@ A) =1. — © Hence we perceive that if the number of A be infinite, when all the gradations from A = 0 to A =a are taken into account, the function ¢ A will be infinitely small for any given error A. We may express this condition more conveniently, in the lan- guage of analysis, by not considering the probability of one determinate error only, but the probability of all the errors lying between the infinitely near limits A and A + dA. Within these infinitely near limits, the value of ¢ A may be regarded as constant. Hence the probability of the errors between A and A+dAis $AdA;; and the probability of the errors between the limits a and 4 is equal to the sum of these elements within the given limits, or j ae if pada. (1.) a For the limits — » and+ ©, which include all errors, it becomes +a rf pAdd=1, (2.) equal to certainty. 322 J. ¥F. ENCKE ON THE METHOD OF LEAST SQUARES, The last integral gives the area of the curve of probability taken from the axis of the abscissz to the curve. It represents the number of observations which are possible, and embrace all errors. Each element of surface 6 Ad A compared with the whole surface, shows the proportion which the number of observations giving errors between A and A + dA bears to the total number of observations; or it gives the probable num- ber of observations charged with these errors, the whole number being = 1. The object of every observation is the deduction of one or more quantities, by which the observed phenomenon is pro- duced. In the places of the planets, for example, these magni- tudes may be the elements of the paths of the planets and of the earth. The manner of combining thé elements so as to obtain the observed value must be supposed known, if we wish to determine the value of the elements from observation ; therefore every observed quantity M will give an equation M=/ (a, y, 2;.--) where the function f is known, and x, y, z are to be determined according to their most probable values. The equality will be more or less presented according to the values assumed for x,y,z. If we suppose 7 =p, y = q, z = 1, and if V=S(p,%7); then M —V would be'the error of the observations in case the values p, g, 7 were the true ones. If several observations of the same kind have been made, in which all the same elements p, g, 7 determine the observed value, then, in similar manner, by the assumption of x=p, y=q, 2=7, the errors M!— V', M"— V"", M!'— V"' will be ob- tained. By another assumption, v=p', y= q', z=7", substituted in the same manner in all the equations, other values of V, and consequently also other values of M—V will be obtained, so that to every hypothesis as to the value of v, y, z, appertains a deter- minate system of errors A, A’, A", which depend on the hypo- thesis. In order to determine from hence the most probable values of 2, y, 2, we need two propositions from the calculus of probabilities, one of which gives the probability of a connected system of errors when the probability of each single one is known ; the other teaches how to determine the probability of the hypothesis from the probability of the system of errors belonging to it. J. F. ENCKE ON THE METHOD OF LEAST SQUARES, 323 For the first proposition the calculus of probabilities gives the following expression. i. If ¢ A is the probability of the error A, $ A’ that of A’, and so on, then the probability of the concurrence of the errors A, A’, A”, &c. is = oA.¢A'.gA"... We may convince ourselves of the truth of this in the fol- lowing manner. Let us assume, for instance, that in three observations the error A be found twice, and the error A’ once; further, let 6A =f ¢g A! = 1, Let the three observa- n n tions be regarded as belonging to a series of observations, m, so extensive that in it all errors shall occur according to their pro- P n to A’, will occur in it. Let the number of the remainder be s, in which it is here indifferent how many equal or unequal there are among them. Apart from s, the number of all possible ar- rangements of the errors in the m observations will be LS Opa ae id bability ; consequently, m errors equal to A, rs m errors equal PON: ae ee ee eae n nN As three places are taken up by the two A and one A’, there are left for the remaining m — 3 observations, Se (m —3) 1.2.3..(2m—2).1.2.3.0..(% m—1) yh n possible mutations. Consequently the probability that in any three observations two A and one A! should be found Pm —1) sem . In ee n. n ~ (m— 2). (m—1).m (Et) 24 OS Ol ee EM 1— =) .Q-5)a m m The assumption on which we have proceeded is, however, strictly true only for m = , or the probability of a single com- ] or 324 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. bination of two A and one A’ in any otherwise arbitrary ar- rangement = (pA) PA, whence the above formula is deducible. To obtain the second proposition, let us consider the case in which any observation has given the value of M. Now com- pare together two hypotheses as to x, y, z. Let Before M is observed, we have no measure of the relative probabilities of these two hypotheses, or of any others ; there- fore, before the observation they must be regarded as equally probable. But after M has been found, Hyp. I. will give the error A, with the probability ¢A, and Hyp. II. will give the error A’, with the probability ¢ A’. If we denote by m the number of cases in which, assuming Hyp. I., M will pro- ceed from it, and by » the number of cases in which, by the same supposition, M will not be obtained, then will m ee m+n Let m! and m! have the same signification in Hyp. I., then yb ea se m+n But besides these two suppositions, of either Hyp. I. or Hyp. II. being the true one, there are also cases in which neither are true, and amongst these there may be some which, in certain cases, give M. Let the signification of m! and m" for all other hypotheses be the same as above, then the number of all possible cases will be = m + 2 + m' + n’ + m' + n"; therefore the pro- bability of Hyp. I., before the observation is made, oa m+n ~ m+n+mi +n + m+ A!? and that of Hyp. II., before the observation is made, ~ m+n +m +n! +m! +n? these two values must be considered equal, whence it follows that m+n=m +n. But after M has actually been found, the cases where it does J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 325 not result are excluded; consequently, in reference to the ob- served value M, the relative probability of Hyp. I. als m ae gare ge and that of Hyp. II. m! = in ea? or they are to each other as m:m’, and in consequence of ‘ m m+n' m+n @A: dA’. Hence follows the proposition : II. The probabilities of two hypotheses, which are equally probable before the observation is made, and which exclude each other, are directly proportional to the probability of the errors, or system of errors, proceeding from them. Consequently, if the magnitudes M are found by a kind of observation of which it is by other means known what errors may occur in it, and in what proportion, or for which the law of the probability of the errors ¢A is known, (which is in- dependent of the use to be afterwards made of these observa- tions for determining one or more unknown values,) then the probability of each hypothesis as to 2, y, z, is proportional to the product the equation m + n= m' + mn’, as -» OY as PGA.GA'. GA". GA"....= 0, (3.) where A, A’, A", A! are the errors which remain over in each hypothesis. The most probable hypothesis will be that in which © is a maximum, or in which, in differentiating, d 0 becomes = 0. On account of the mutual independence of the quantities x, y, z, this equation divides itself into the separate . &0, dQ dQ equations en 0, ae 0, wen Generally, each A=M~—V. If consequently, before the substitution of a numerical value for 2, Y, z, the functions M — V be designated by v, so that’ M — V =», M' — Vi=yv, M'! —V" =", &e.; and if, for the sake of easier differentiation, we make log. OX = log. A + log. f A’ + log. pA"... 3296 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. and designate the logarithmic differential by ¢' A, so that dpa _ 4 prada pes the equations of condition of the maximum become the following: dv dv! dv! dv"! Tat a qa?” ae ae gl vl + aa PO eine i dv dv! dv! dv a Bayh” an ee © g! x!!! =O} dv dv! dv dv It TT aU a, LE PNET a dsked se Wp Srl heh hatuf by), pee = St FY Poe Tce ee 0; whence the values of 2, y, z, which satisfy them, and which con- sequently are the most probable values, must be determined. These general propositions can, however, only be applied when the function ¢ is known in each separate case. Instead of making different hypotheses as to its most appropriate form, and then trying which of these corresponds best with experi- ence, we shall attain our object more directly, by considering in a converse manner the simplest case,—examining for it what values experience (apart from the general formulz of the cal- culus of probabilities) teaches us to prefer,—and then trying to determine from thence the form of ¢ by means of the general formule. Let us suppose any arbitrary number of observations, all made under equal circumstances, so that beforehand no pre- ference can be given to any one above the rest. Let us say that these observations are to be applied to the determination of the value of an unknown quantity, of which the true value would be given directly by each single observation, if there were no errors of observation. An examination of the differ- ence between two right lines may serve as an example. First, if one observation has been made, giving the value a, there is no choice but to put ea. If two observations have given the values a and }, and if neither of these is to be preferred to the other, then from these observations alone the value of 2 must be determined in such manner that the differences —a and2— 4 may come out equal. This gives x=} (a+), under the supposition that a positive and a negative deviation, J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 327 of equal absolute amount, are to be viewed as equal errors. If a fundamental principle is necessarily required, this supposi- tion appears to be the simplest of all. It rests on the conscious- ness of having exercised the greatest possible care, so that no reason exists for assuming that an error has been made, either in the positive or in the negative sense. But let it even be granted that an error tends to occur more frequently in one sense than in the other, still, so long as we do not know in which sense it occurs, the value } (a + 6) is the only one which in this uncertainty will make the error of the result the smallest ; or, at least, which will most securely avoid the danger of in- creasing the error. Now let three observations have been made. On account of the fully equal worth of the observations, the values found, a, 8, c, must be so combined that no one shall influence the result more than another, independently of their numerical values. Or it must be assumed that x =symmetric function (a, 4, c). But we may consider the subject in another point of view. If two of the observations alone be taken, we should have, ac- cording to which two were selected for that purpose, one of the three following results, which in each case would be the only result that could be chosen :— $ (+8), kato, $ (+0. To this the third observation adds c, 4, a. It is true, that we can no longer combine the two values in each arrange- ment symmetrically, because one rests on two observations, the other on one. But whatever may be the form for the combina- tion of both, it must unquestionably be that, which would pro- duce the result to which the preference is due as derived from the three observations; and this form, which may be arbitrarily designated by yy, must be the same for all three. Hence we have for z the three values 2n=vy (} (a+), ©), z (@+¢), d), =~ (2 (6 + ¢), a). If we introduce here the sum of a, 8, c, or if we say a+b+c=s, 328 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. then r= (3 (8 — 6), ¢) = os =v (3 (s — 4), 6) = ¥ (8, 4); =v (3 (s — 4), a) = (84). But these three formule, from what has been said above, should give a symmetrical form to 7 in reference to a, 6, c, which, as s is already in itself a symmetrical form, can only be if c, 4, a dis- appear in the development by the powers of s; consequently, in the same manner, from all three, v= (s). If now in a given case a = b = ¢, the only possible value of x would be 2 = a; consequently, a= (34), or the function sign Wy would signify the dividing by 3. Hence follows, 7, an b+¢ 3 for three observations. In like manner it follows generally, that if for n observations, the value to be chosen is ya cei aS then, if another observation p is added, for (n + 1) observations, _~G@+b4+C....4+n+p a | ought to be chosen; for the equality of the observations re- quires that if we put atb+c....4+¢n+p=s, av then 1 esa (6p); Ke, should be a symmetrical function of all the n+ 1 values. Now, as this form is geod for three values, it follows that it is so also for any number of observations, great or small. This proposition,—that in any number of equally good obser- vations of an unknown quantity, the arithmetical mean of all gives the value which is to be preferred, and which consequently must be regarded as the most probable value,—has been re- ceived as a fundamental proposition ever since combinations of J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 329 several observations have been made. Rightly understood, the confidence which we place in all quantities derived by expe- riment in any science, rests essentially on this proposition ; it may, therefore, safely be affirmed concerning it, that its truth has been confirmed by experience. The deduction here given shows somewhat more clearly than would be done by the simple statement of the proposition itself, the suppositions on which it is founded. If the observations are made under strict- ly equal circumstances, and if in two observations a positive and a negative deviation of equal amount are regarded as equal, the arithmetical mean is the only value which does not con- tradict these suppositions. Then it cannot well be denied that the same value ought to be obtained, whether the observa- tions are considered all together, or divided imto arbitrary groups, provided only that no arbitrary supposition is made in the combination of the results of these groups amongst them- selves. To deny this, would be to deny that there is any value which ought to be chosen in preference to others. It may perhaps serve to illustrate the importance of the supposition of the equality of the observations in reference to the arith- metical mean, if we refer to the example furnished by Lam- bert, in the Photométrie, §. 276, in which the arithmetical mean obviously does not give the greatest approximation to the truth. The periphery of a circle is always between the values of the perimeter of an inscribed and a circumscribed polygon of an equal number of sides. If, therefore, we consider- ed the perimeter of a polygon of n sides as an observation of the length of the periphery, and regarded the arithmetical mean be- tween the inscribed and circumscribed polygon of n sides as the most probable value of the periphery, we should be in error. We come nearer the truth if we add to the perimeter of the in- scribed polygon the third part of the difference between the two. Whether, therefore, we regard the principle of the arith- metical mean, in observations of equal worth, as a fundamental proposition which experience has confirmed,—or whether we prefer to take those propositions, on which the deduction here given is based, as more simple fundamental propositions re- quiring no proof,—in either case the founding of the application of the calculus of probabilities to observations on the principle of the arithmetical mean is, perhaps, of all the modes of demon- stration, that which is most useful to the practical mathemati- VOL, II. PART VII. Z 830 J. F. ENCKE ON THE METHOD OF LEAST SQUARES, cian. Therefore we give the following deduction, which is based on proposition II. Any number of equally good direct observations of an unknown magnitude being given, the arithmetical mean of all the observed values determines the most probable value of the unknown magni- tude, so far as it is determinable by these observations, without requiring or universally admitting any other condition. Let there be m equally good observations of the unknown magnitude 2, and let them have given for it the values M, M’, M", &c. According to the last proposition, if M+M'+M".... m the most probable value of x in every case, so far as it can be concluded from these m observations, will be the magnitude p. Consequently M—p, M’—p, M” —>p must be regarded as errors of observation; or the equation from which the most probable value of x proceeds according to the arithmetical mean, is p= M—2+ M’—2+M"—-2+4+M"—2...=0. (4) If we apply to the same case the general formulz of the cal- culus of probabilities, we have v=M—2, v= M!/—a, v!=M"—2z....; consequently the only equation of condition of the most pro- bable value is g! (M — 2) + 4’ (M’—2) + 4! (M"—2) + 9!(M"—2)...=0, to which the following form may also be given: ¢' (M=2) | yp _,) o” (M2) inked a ae PnP are +(M"—2) OE) 0. It follows immediately from this latter form, that the above equation deduced from the arithmetical mean, will universally satisfy this last equation only when ¢(M—2z) _ ¢'(M’—2) _ ¢ (M'—2) a Me ch eT Se at aa gi A is independent of the value of A, or when - tiie ny nee i.e. when x J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 33] ! is equal to a constant. Any function ae can only contain, besides this common constant, such members as vary with the value of (M — 2), or are a function of (M — 2). But whateverfunction may be assumed, asum of products of the form (M —2) f(M—z) will never in general equal=0, by virtue of the single equation (4.). For let it be granted that it might happen that for the values M, M’, M"...this sum might, with the equation (4.),= 0, still in all cases in which, with the un- changed sum M + M'+ M"...= mp, somewhat different values M—a, M/ + «, M"”—£, M'’+ 6, &c. have been found, a new equation would arise, which, if the arithmetical mean holds good, must be, together with the equation (4.), = 0. But from the infinite diversity which not only may, but, ac- cording to experience, will be found to exist in the amount of the changes of M, M’, M", as well as in their distribution, there ean be no function which shall fulfill all these conditions at once. Although the values of M—p, M'— p, M"—p are not ab- solutely independent of each other, because p depends on their sum, yet they must, in case the arithmetical mean holds uni- versally good, be considered as independent variables, because the only equation which expresses this dependence, with every number of observations, disappears in consequence of the in- finite diversity of the values which may still be found after this equation has been fulfilled. This equation, ? ¢' A _dlog. (9 A) _ k SSR AY, a yee where is an arbitrary constant, gives immediately the form of oA. Integrating, Const. + log. ¢ A = 43k A?; 4k Az, Ore yA "x e>" 35 in which formula the value of the constants remains still be to be determined. In regard to k, the above remark, that ¢ A must be a maxi- woum for A = 0, shows at once that & must always be nega- live. It may therefore be more convenient to write —h2 A? OUAN cast ee : Die 832 J. F. ENCKE ON THE METHOD OF LEAST SQUARES, The equation (2.) may then serve for the further determination of a constant. If we make h A = #, this integral becomes x He be aS Se ee ant dtiany —o where the limits remain the same as before. In order to obtain the value of this definite integral, let us exainine the double integral * +a V= i e “TM dedy, ek where x and y signify two variable magnitudes independent of each other, and the limits — to + o refer to the integration according to x as well as to that according to y. If we integrate first according to y, considering w as constant, and make the value + 0 2 e-"dy=L, +o V= Lf ed a —@ consequently, if we now integrate according to 2, Via: But we may also compare the expression for V with the ge- neral formula for the cubature of a solid. If we consider 2, y, z as three rectangular co-ordinates, and imagine the surface the equation of which then pe Pic ain i V will express the volume of the body bounded by this infinitely extended surface. But this surface would obviously have arisen by rotation round the axis of the z, because z comes alike to all points of the plane of the x, y, which are equally distant from the point of beginning. On this account the volume of the * According to the verbal communication to which I am indebted for this short and elegant mode of finding the value of the definite integral, M. Cauchy has given it thus in his lectures. : : | J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 333 body may also be expressed by a simple integral, if we imagine it decomposed into a series of infinitely thin cylindrical shells, all perpendicular to the plane of the vz, y. If we make Tr? = x + y’, the solid contents of every such cylindrical shell of infinitely small thickness will be found = Qrz2nx.dr, ; consequently, the volume of the body (for which, in relation to r, we must take the limits 0 to w); or oe ae: V= Irm7e dr, 0 for which the integral is immediately found, ie 2) NRE 0 or for the given limits, V = T. Hence, by virtue of the above, L= Jr; and consequently, by substituting this value in (5.), = vx = 1, or ily % S lanes The complete expression for ¢ A will be accordingly h — h2 A? ~A= Tag e ’ (6.) which not only contains in itself the principle of the arithme- tical mean, but depends so immediately upon it, that for all those magnitudes for which the arithmetical mean holds good in the simple cases in which it is principally applied, no other law of probability can be assumed than that which is expressed by this formula. It is therefore not limited to any special kind of ob- servation, but is altogether general. Equally general is the re- sult in regard to 2, which follows immediately from this form : namely that, for any arbitrary number of magnitudes to be deter- mined, the most probable values are those which make the sum ee WPA LA, seals « a minimum. x 334 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. Tt follows from this formula, that the probability that an error lies between A and A +d A, is hk _pzae Sid Wa dA; (7.) and the probability that it lies between the arbitrary limits a and 0, A=b h —h*® a ed dA. ee dad: £f (8.) Calling the number of the errors unity, this integral expresses also the number of errors which should occur between a and according to the law, and which will occur very approximately if the number is sufficiently great. If we make WK ='t, the integral takes the form t=bh ak} on We V0 =ah If we take for the limits an equal positive and negative value —ahto + ah, then on account of the even power of ¢ in the differential, we may write tan : — ex dt; VAs t=0 and we may thence, by means of a table giving this integral for successive values of a h, obtain a clear representation of the distribution of the errors, without regard to signs, but simply in respect of their magnitudes, proceeding from 0 to the extreme limits. Such a table is appended (Tab. I.). It is de- duced directly from the table for the integral fe" d ¢ in Bessel’s Fundamenta Astronomia. The calculation of such a table from the developement of the integral according to as- cending and descending powers of ¢, or according to a continued fraction, is found frequently given, as this remarkable function is applied in many ways in different researches. This table shows at the same time, how rapidly the number of errors included in equal intervals of the value of ¢ decreases in the higher values. It justifies, therefore, our assumption of the limits — © and + o in lieu of the actual limits, which must be narrower, although they are not susceptible of being J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 335 defined with precision. In a thousand observations there are between 5 =) Oi andes ¢=0°5 520 errors P= OD. 4%) = LO GLa. ves t=10 4 ¢=15 Tos fa OU a ke a) Do. and between this latter limit and ¢ = » there remain only five errors; a number so small that it is scarcely probable that any- thing will ever be experimentally decided in respect to this de- viation of the theory from the rule. Among the different values of ¢, there is one especially which may lead to a determinate view as to the proportionate exact- ness of different kinds of observations. It is that value of ¢ for which the integral has the value 0°5, or which, if we take a suf- ficiently large number of errors, and imagine them arranged in the order of their magnitudes without respect to signs, will di- vide them into two parts, each containing an equal number of errors. The number of errors is assumed to be large only in order that the law of probability may actually be fulfilled with sufficient approximation. From the table, it is found, by inter- polation, that the value of the integral 0°5 corresponds to the value of ¢ = 0°476936. Let this number, which holds good for all kinds of observations, be designated by e, on account of the frequent use to be made of it, so that ae a 2 aie = e = 0°476936 wa Sef § dt=%}. (9,) If we designate by 7 the error which belongs to the value t = ¢@ in each kind of observations, then e=hrorh= 2. (10.) German astronomers call the magnitude 7 the probable error of any particular class of observations*. It is that error below which there are as many errors less than itself as there are larger ones above it; so that there are as many cases in which the errors are less than 7, as there are cases in which the errors are greater. Therefore it is an equal chance, that the error of an * The French geometricians are in the habit of giving to this value of r the name of l’erreur moyenne: this is the more to be borne in mind, as the import yg given by German writers to the term mean error differs essentially rom r. 336 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. isolated observation does not exceed 7, supposing the value of r for the class of observation to be known. On account of its frequent use, the value of the integral 2 = : : : pes if edt has been also given in a second table, arranged according to an argument in which the value of 7 has been as- sumed as unity. This table gives for the argument 4 the value of eA so that it shows immediately how many errors will occur up to a determinate error A (always without reference to the sign), provided the proportion of the given A to the probable error be known. It further facilitates a view of the distribution of the errors according to their magnitude. If half the number of all the errors are less than an error =7, then among 1000 observed errors, there will be 823 less than 27, 957 less than 37, and 993 less than 47. There will not be more than one error greater than 57. By means of this view of the probable error, we may also ob- tain a clearer view of the signification of the constant hk. In different kinds or sets of observations the errors always follow the same law, which is expressed by ¢ A. The difference of any one kind or set from all others depends, therefore, solely on the value of the constants /, and these again afford the means of comparing together observations of different kinds in re- spect to exactness, and thus enabling them to be subsequently combined. With two kinds of observations, one of ia has the constant h and the other the constant 4’, the integral fe Ad A, taken up to any assigned limit, will have equal values, if the value of the limit, determined in both cases by the variable ¢, is the same. Or (as in one ¢ = A, and in the other ¢ = h’ A!, the errors of the second kind being designated by A’) there will be as many errors in proportion to the whole number in both kinds within the hmits A and A’, if we determine one value from the other by the equation nee =e! (11.) J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 337 the constants A are therefore in the inverse proportion of the equally probable errors of two kinds of observations. This is true for all errors, consequently for the probable errors of each kind, as already shown by the equation h = oe because g is here an absolute number. If, therefore, there is an even chance that an error falls in one kind within one limit, and in the other kind within the other limit, for which generally the pro- bable errors r and 7’ are chosen, we have also the reciprocal proportion of the constants / and h’, from the inverse propor- tion of the limits, or from the probable errors 7 and 7’. Hence may be derived a preliminary estimation of the proportion of 4 and #’. If in two measurements of angles there is reason to fear that an error of w” may have been made in one as easily as an error of mo! in the other, then, if 4 be taken for the latter, m h must be put for the former. On account of this constant proportion between the increase of exactness and the magnitude of h, Gauss calls this constant the measure of precision. The geometric representation of the curve of probability may be also applied to this consideration. Take any unity as the general measure of A, or of the abscissz ; then, by means of the equation : h —h? a? Va the whole curve could at once be drawn if the value of h were known. Consequently, if we only know an ordinate belonging to a determinate A, the whole curve will be fully given. If the ordinate for which A =0 be chosen, by comparing its value gA= with ae we have at once the value of h. If the ordinate Vv were chosen, which divides the superficial contents of the curve into two equal parts on either side of zero, we should obtain from the abscissa belonging to this ordinate, by means of the equation A = 4. If we even knew merely the relative propor- tion of two ordinates which correspond to any abscissa, A . . . —f2 a2 —h2 a2 and A!, then as this proportion is as e eee oR, oF’ as pee “), we should be able to determine / from hence. It is most convenient to choose for the one ordinate that which corresponds to the value A =0. Hence follows a 338 J. ¥F. ENCKE ON THE METHOD OF LEAST SQUARES, proposition which will be frequently applied in the sequel, viz. :— ie If the probability of an error = 0 is to the probability of an error = A,asl:e ? es then for this set of observations we must assume h= V p. Such a determination of 4 admits even of combining together observations relating to heterogeneous magnitudes, as for ex- ample angular and linear magnitudes, provided only it be pos- sible to deduce the relative values of A in reference to the re- spective unities. An actual exemplification taken from experience may perhaps serve to illustrate this subject further, by showing how very nearly the function ¢ A expresses the distribution of the errors in a sufficient number of observations. It is taken from the Fundamenta Astronomie, in which Bessel has given a memor- able example of the consecutive, strict, and elegant treatment of a series of observations. He determines the value of 7 by a direct observation of the difference of right ascension of the sun, and of one of the two stars, a Aquile and a Canis minoris, as de- rived from BrADLEY’s observations, to be r = 07-2637, and then compares the number of errors which ought, accord- ing to theory, to occur between the limits 0-0 and 01, 0""1 and 0:2, and so on (always increasing by 0°1), with the errors given by actual experience in 470 observations. Expressed in units of 7, the interval of 0-1 = 0°3792 r._ If, therefore, we seek in the second table the value of the integral for the different limits, we find for Ol . . . . 03792 the number 0°20186 Goma... Or7584 0°39102 Cf: «11376 ¥ 0°55705 Ga T5168 3 069372 OB endl ybzoutse960 i 0°79904 OG i e750 a 087511 Ory UO eis 3 092661 68>; SS osa6 f 095926 OF) Cina eas te 097866 WO Se wld eg oED % 0°98983 at) Or LOO ” 1:00000 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 339 If we now deduct every number from the following one, and multiply the remainder by the number of the observations = 470, we find According According to Between Number of errors. to theory. experience. 0-0 and O-1 0°20186 95 94 Ort, 70°2 0°18916 89 88 O27 ,, O03 0°16603 78 78 O3 5, O04 0°13667 64 58 04 ,, O°5 0°10532 50 51 0:5. .5,».0:6 0°07607 36 36 WG. 5 » Or7, 0°05150 24 26 oy, O8 0°03265 15 14 O08 ,, 0:9 0°01940 9 10 Os, 1:0 0°01117 5 7 Above 1:0 0°01017 5 8 In other examples also it is found, for the most part, that the larger errors occur somewhat more frequently in experience than according to theory, a proof that the assumption of the limits — © and + o has not misled us; for, if it had, the con- trary would have been the case. This deviation is easily ex- plained from the circumstance, that the larger errors suppose in the rule a very unusual combination of unfavourable influ- ences, and, indeed, are frequently occasioned by occurrences so insulated that no theory could subject them to calculation. The determinate value of one of the constants / or 7 in a set of observations can, however, only be deduced from actual ex- perience, or from a series of errors which have been found to occur in this set of observations. We must first learn how to proceed, in order to obtain in the given observations those errors which approximate most nearly to the true errors; and we must then see how the function ¢ A is to be determined numerically in all its parts from those errors. We may begin with the most simple case. It will afterwards be more easy to take a view of the rules for the more general and complicated cases, as the ge- neral fundamental propositions remain unaltered. For the value of an unknown magnitude z, let direct observa- tion, repeated m number of times, in the same manner and under completely equal circumstances, have given m values. n, nt, n', nil, &e. Each insulated observation will have given an approximate value by virtue of the equations of condition, 340 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. eo = 0, and each equation of condition is the general expression of the error of the observation in any hypothesis as to 2 Conse- quently, if the constant / belong to this set of observations, so that for it h — hh? A2 Be ae ig P then the general expression for the probability of one error in the first observation will be in every assumption as to 2, h —(e—n) vt and the joint probability of the concurrence of m errors in these observations will be h™ ie h? {(« — nj? +(a—n’')? + («— wre be This probability will be greatest when the sum of the squares of the remaining errors according to an adopted hypothesis is the least possible, and consequently according to Proposition 11.—That hypothesis as to x in which the sum of the squares of the remaining errors is an absolute minimum, is the most probable of all possible hypotheses. This minimum may be obtained either by the differential cal- culus, by which 2(¢—n) +2(~@—w) +2(v—n").....=0, or me +n! + nl +... m thus the arithmetical mean, as was before laid down, is the most probable value in equally good observations. But when 2 is left undetermined, we may give to the sum of the squares of the errors such a quadratic form, that both the most probable value of z, and the remaining minimum squares of the errors, may at once proceed therefrom. For the sake of brevity we will desig- nate the sum j nen! oa ee en si, 2, Dy [ae] (12.) nt eM eas. DY [ed r This mode of designation will be extended in the sequel to any | J; F. ENCKE ON THE METHOD OF LEAST. SQUARES. 341 ‘symmetrical function of any given magnitude. The compound probability, if every error is actually squared, will be : meen ‘ | 1 e—l? {mx? — 2[n] 2+ [n*]}, ™ ‘to which expression it is easy to give the form Be {wi- Bi +aG-2)} sm T Consequently the negative exponent will be the smallest for n= fel (13.) : and the minimum of the squares of the remaining errors is aipaey ce le = fey ER, (14.) - This form leads at once to the estimation of the exactness of this determination of x. If we take [x] m then the probability of this hypothesis becomes Rp” - { [n?] — a “= > mae But any other value of By has the probability : Jet Penta — bet + mar} = m © Consequently, a to Proposition II., the probability of the arithmetical mean being the true value, is to the probability of its being erroneous by the magnitude A’, as —h? m A” “ Fe 5 or, according to the above proposition (IV.), the value of H, which is deduced from m equal observations, and which belongs to this determination of 2, is H=h of Mm, (15.) so that the function ¢ A for this determination of x becomes pa atime wan, | In some cases, instead of expressing the relative exactness of 342 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. two determinations by the proportions of their respective values’ of h and r, it is more convenient to bring in the new idea of weight. By the weight of a given value we understand the num- ber of equally good observations of a determinate kind (of which the exactness is to be viewed as the unit of exactness), which are required to furnish, by their arithmetical mean, a determination of equal exactness to that of the given value. According to this, in the present case, if the weight of the single observation be regarded as unity, the weight of «= m;—if h be the mea- sure of the exactness of the single observations, the measure of the exactness of «= h m,—and if the probable error of an observation be designated by 7, the probable error of x Sy aN, COR r “Ho asm ff are to each other in the direct proportion of the squares of their respective measures of exactness, and in the inverse proportion of the squares of the probable errors. If we substitute in the equations of condition the most probable values of 2, then the differences, between the result calculated with this value, and the actual observation, are to be regarded as the errors of observation which approximate most nearly to the truth; therefore, so long as we have no means of determining the value of z more nearly, the errors thus obtained are to be regarded as the true ones. The sum of their squares must, according to the whole deduction hi- therto, be equal to the minimum just determined, or it must [n]° be = [n?] — ok In order to obtain, generally, a more con- The weights of two determinations venient expression for this sum, we introduce a new idea, that of the mean error. By mean error we understand the magnitude which is obtained, if the sum of the squares of the true errors of observation be divided by the number of observations, and the square root of the quotient be extracted. Consequently, in the present case, the mean error being desig- nated by ¢,, ray? g) es om v(e a ), m [m]° me,” = [n*] — ek, inasmuch as we are at present obliged to regard the errors re- sulting from the most probable hypothesis as the true ones. We or J: F. ENCKE ON THE METHOD OF LEAST SQUARES. 343 _ may also define the mean error thus: it is the error which, if it alone were assumed in all the observations indifferently, would give the same sum of the squares of the errors as that which actu- ally exists. According to this, the probability W, of the concur- rence of m true observation errors is, generally, in any hypothe- sis which can be made as to the constant h of the function ¢ A, A” sm —h? m £52 2". WwW = From this value we are now enabled to determine the most probable value of 4; for if the m observation errors, and conse- quently also ¢,, have actually been found, and cannot be further altered, then the maximum of this function W will depend only on the value of h. The most probable value of h will be that which makes this function W a maximum. We may first seek this maximum by the differential calculus. If we write the above expression thus: log. W = m log h — 4 m log x — h? me,?, then the condition of the maximum is = a 2mh,°; or 1 = 2h? «,”, whence coma. ie fy / 2° We may also develope generally the magnitude W as a func- tion of h, for altered values of h. Let the value W! belong to a value h + A, just as the value W belongs to h, we shall then have log W' = m log (hk + A) — 3 mloga — (h+ A)?me,2; if we write here for m log (h + A) the expression m log h + mlog (i + +) and develope the latter part into the known series, we have log W' = m logh — i mloga — h? m«,2 A Ae AS leg Pane Sale 7 —2meezh A — me? A?: A4 1 a 7 844 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. and combining with the expression of log W Ww" m m log (qr) = (G—-2maat)a—(4% + mee )ar + $55 A845 Att, Be. If the value of / is here to become the most probable (conse- ! quently if W is to be an absolute maximum, and log i is on that account always to have a negative value) the coefficient of A must be made =0. For the maximum of W there will be ; —~2mhe2= 0, or > ot 5 h/aemrtinney and if we substitute this most probable value in the remaining members, every other value of W, so far as it depends on another h, will be given by the formula W! = W. en mee {l= 52/2) 4 + 3 (a 2)?0? jal; } We may here make the series contained as factor in the ex- ponent = 1. Forif we introduce the value of the most probable h, it becomes Ae oes b= $3 Zot doe Sage ° . . 2 which series must still be multiplied by m 5 aut = is a small fraction the series will deviate little from unity, and, still more, the difference of the complete rigorous value from the ap-_ 2 A proximate e—”72 will be quite insensible. But if oe were to have a greater value, W' would be very small compared to W, and for that reason the whole accurate expression would have no material interest. Hence the probability that h = : 5 9 oF 2 W, is to the probability that the value of = aie + A,or W! i) = as . mem €g? AP eC l:e Or all ss ccauys J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 345 Consequently, according to Proposition IV., the measure of pre- 1 age of h = /2m or == vm; cision for the value and the probable error of this determination acer g 1 vm %V¥2° vm? or there is an even chance that the true value of / lies between aaa + $a} and é 54) — ay: (17.) hence, inasmuch as r=, it follows at once that the probable error of a single observation depends on the mean error by the equation r=eW 2. = 0674489 «, (18.) if the numerical value of e s/ 2 is substituted. The certainty of this determination is given by the limiting values of h. It is an even chance that r lies between Bae a ae arid) eeetees yy wb. & ts) a BS nied EE 1 cits instead of which, as absolute exactness is not contemplated, we may permit ourselves to make the limits of r = ag ( a ~—) and é,.@ “2 @ ie ~-). (19.) We neglect in this the higher powers than the first of the un- certainty of the probable error, considering the uncertainty as a small magnitude of the first order. There still remains a circumstance to be attended to. The mag- nitude <, and with it h also, ought properly to have been deter- mined from the true errors of observation, whereas it has been only deduced from the obtained minimum of the squares of the errors. It is clear that this mode of deduction is necessarily somewhat faulty, as every value of x which differs ever so little from the arithmetical mean must give a greater «, and a lesser h. In order to gain a clearer view of this, let the most probable value of 2, as far as it follows from m observations, = p, or ia at VOL, Il. PART VII. 2a 346 J. F. ENCKE ON THE METHOD OF LEAST SQUARES, but let the true value be p+ Ap. By substituting p in the equations of condition, we obtain as the errors of the observations the magnitudes p —n, p — n', p — vn! ...., which for the sake of brevity may be designated by «4, a, «. The substitution of the true value p + A p would have given p+ Ap—a, ptAp—v,p+Ap—n'...., and these latter magnitudes, which may be called 8, 2', 8”, would have been the true errors of observation. We have consequently the equations a+ Ap=d a+ Ap= of a’! + Ap=", &c. As [a]= 0, the sum of the squares taken on both sides will give [a] + mAp? = [e]. Thus if we assume [a*] as the true sum of the squares of the © errors, we shall always err by the positive magnitude m A p?. This representation gives at once the means of correcting the error as far as circumstances permit. If to the m observations a new one were added without our knowing determinately what value it had given, we should have to add to the [a*] the value e,2 as the mean value of such a square. The equation shows that m A p? must be added in every case ; and it follows from what has been said above, that p has the weight m, or that if a single observation has the mean error ¢,, the mean error of p will be equal to a Hence it follows, that we approach the truth as nearly as possible, if in this equation we take the mag- nitude of A p such as its proportion to the single observations gives it, or if we substitute the value A p = =) Tien Ym [a"] +e" = [*], and as it follows from the assumed definition that [27] = me,? the value of «, derived from the m errors remaining over after the substitution of the arithmetical mean is obtained by (m.— 1). ¢,” = [a*|. (20.) In order to obtain as nearly as possible the true mean errors of the observations, we must, with an unknown magnitude, regard the sum of the squares of the errors as if it belonged not to m, but to (m—~—1) errors. — — _- dee — —_ J. F, ENCKE ON THE METHOD OF LEAST SQUARES. 347 We may convince ourselves also of the general correctness of this rule in the following way; at least we may do so in a preliminary manner. If »% unknown magnitudes are to be found, # equations of condition independent of each other are in every cease requisite; and if no more than » such equations are given, we have no remaining standard for the estimation of the pos- sible error. We do not obtain this until we substitute in other equations of condition the values found instead of the unknown magnitudes, and compare the errors which result; so that with m observations treated in this manner there result m— p% errors, which allow us to form a judgement as to the exactness. Inasmuch as we do not regard the » determinate equations alone as the absolutely correct ones, and the deviations of all other results from those drawn from the » chosen results as errors, but as we give to all an equal share in determining the unknown magnitudes, we are certainly nearer the truth ; but we do not by this means get rid of the analytical necessity of always applying to the determination of » unknown magnitudes, if not » determinate equations, yet an equivalent to such w equations taken from all together. Consequently the functions of the re- maining errors thus obtained will always refer, not to a number of m errors, but to the number of m — p» errors, as has been shown for ~ = 1, and as will be shown in the sequel for any » taken at pleasure. A view of the rules for the heretofore considered simplest case, i. e. the case of equally good direct observations of an unknown quantity,—may be facilitated by applying them to BENZEN- BERG’s latest and most exact experiments on the fall of bodies, made in the Schlebuscher coal mines. The object of these ex- periments was to demonstrate directly the rotation of the earth round its axis, by showing that balls let fall from a consider- able height without initial velocity, deviate, in falling to the lower station, towards the east, more than a plumb-line sus- pended from the same upper point. The experiments, although divided into separate parts, were so made as all to have the same value. As they are only used as an example, I leave quite out of consideration the deviation (not agreeing with theory) of single balls towards the north and south, which moreover almost entirely disappeared in the mean of all the experiments. I also take as valid only those experiments which the observer himself selects,—Table, page 424, ‘ Versuche 2a2 348 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. iiber das Gesetz des Falles, &c., by J. F. BENZENBERG, Dort- mund, 1804,—although the reasons given for the exclusion of others are perhaps not quite convincing. Designating the easterly deviation from the perpendicular by +, and the westerly by —, the following deviations in Parisian lines were observed in a height of 262 Parisian feet. n n a a Experiment 1. — 3-0 Experiment 16. — 80 5; 2, +120 fi 17. + 80 i Soni) 3x0 3 18. +100 . 4. +13°0 3 19. + 7-0 ‘s 5. +200 i 20. + 75 4g 6.2 =o ¥ 21. + 60 53 7 +115 - 22, — 2-0 ‘5 8. — 4:0 5 23. +110 4 9. + 20 g 24, — 4-0 cs 10. + 20 i 25. — 9:0 i ll. +12°0 iy 26. —10-0 is 12. + 7:0 i: Q7, + 85 ie 13. +13°5 43 28. +10°0 “5 14. +110 is 29. + 5°5 f 15. + 9:0 If x designates the deviation sought, the simple form of the equations of condition is here z2—n=0; consequently, according to (13.) the most probable deviation is op = £.189°5 — 42-0 ue I. 35 = + 5!"-086, and the errors remaining over, arranged according to their abso- lute magnitudes, are, —,,, ia Experiment 29. —0°414 Experiment 7. —6°414 PtSi <2 Org1a as 2. —6914 uD TO 14 on eh Ses s 19. —1:914 ¥ 6. +7:086 if 3. + 2-086 » 22. +7-086 b--S oaGel! ups fi 4, —7-914 " 17. —2914 if 1. —8-086 3” 9, + 3'086 ” TS: oe 8414 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 349 “i Experiment 10. + 3°086 Experiment 8. + 9°086 & 27. —3'414 ‘3 24, + 9086 » 15. —3:914 iF 16. + 13°086 » 18. —4:914 hi 25. +14:086 + 98, —4:914 3 5. +14:914 ; 14. —5:914 i 26. +15°086 Y 23. —5:914 The sum of the squares of these errors will be found, either by immediately squaring each single error, or by means of formula (14.) to be = 1612°0; also m= 29; consequently, 1612°0 Be eit) 58 = 7''°588, whence g==7 8,06 014/ Zi 5-118, and hk = = = 0°093, 3 lwo the unit being the Parisian line. As m=29, and therefore 7 = =0'08846, it is aa equal chance that é,-. Will be between 67916 and 8'*260, Tess 4 4 665 , 5 ‘571, ee a“ 0 :085 ,, 0 ‘101, Lastly, the most probable deviation, in reference to a single one of these experiments, has the weight 29; consequently its probable error (and in like manner the H belonging to it and the mean error) r = Vv 29 = 0!"""950, the limits of certainty of which are given in the same manner from the limits of 7, and it is an equal chance that the true de- viation lies between 4136 and 6036. The value given by theory, 4’"6, is within these limits ; there- fore the experiments agree with it. They also agree sufficiently well for their small number with the value of 7, according to which half the errors should be less than 5-118. Of twenty- nine errors thirteen are less than this amount, and sixteen - 350 J. F, ENCKE ON THE METHOD OF LEAST SQUARES. exceed it. If there were no easterly deviation, there would be in # an error of 5!-086; but as this is more than five times the probable error of 2, the existence of an easterly deviation — borders closely on certainty. If it were desired to determine the absolute value within narrower limits, it would be necessary to make a considerably larger number of experiments of the same kind. About 2600 would be required in order to reduce probable error of x to O!'"1. It must not be overlooked, that the limit of error is clearly much too narrow, partly because with the absolute smallness of # a constant error in the kind of observation would have pro- portionally very great influence, partly because the exclusion of observations which deviate more than two inches (of which there were in all eleven in forty)-can hardly be perfectly justi- fied. Such an exclusion, moreover, if made after the result, is open to the danger of leading away from the pure truth, and it always produces an erroneous representation of the certainty of the result. The most troublesome part of the calculation in this simplest case being the determination of the sum of the squares of the errors, we may wish to attain in a simpler manner to the know- ledge of r and hf. This examination is besides useful, as the subject is considered in it from another quarter, and the de- termination of h, from the sum of the squares of the errors, is attained by another way. If the law of the errors were given generally by ~ A (with- out determinate assumption of the above function ¢ A), and if this function were completely known, then, in respect to m observations of any kind, we should be able, even before we knew their result, to form a conclusion as to the distribution of the errors and as to the magnitude of any of their functions 5 which conclusion would be so much the more confirmed after the observations were made, as m is greater. So for example, according to probability, there would be between A =a and A = 6a number of errors ~ nf WA) d A; also as m W (A) is the number of errors of the magnitude A, the magnitude m A” (A) will be the sum of the zth powers: of the errors of the magnitude A in m observations; and, con- sequently, J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 351 4=+o m A® y(A)dA=mk") will express generally the sum of the zth powers of all the errors which, according to the law of probabilities, should occur in m observations. The magnitude k”), in which the index depends on the power of A, or the integral taken between the widest limits, cannot be merely an absolute number, but will contain one or more constants, having respect to the class of observations. If, therefore, we knew truly the form of (A), but were still uncertain of the exact value of the con- stants contained in it, then any number of m observations, when the pure errors of observation are found thereby, would lead us to the knowledge of the constants. For, let the errors 4, 8, y; 5; be given immediately up to the number m, then the most probable value of k) will be found by Km”) peta p” ai y” ei OS [A”] a m mm Any other hypothesis as to the value of &") would not suppose the errors distributed according to the law} (A) ; consequently, _it would assume an error in one or several values of a”, 6”, y”» - &e. The value of &), which, in its conditions, involves no error, must be the most probable, according to these m observa- tions. This form gives also at the same time the limits of certainty of the determination thus obtained of k”)., With k\) the principle of the arithmetical mean holds rigorously good, by which, for each m, we find, from the results given by the observations singly, the most probable value of one and the same unknown magnitude. The magnitudes 2”, 6”, y”, come consequently into | the series of direct observations of the magnitude *™), and the | differences k™ — a”, K”) —", k™ —), are to be viewed as the errors of one such single determination. The above-deter- mined form ¢(A) holds good for them, apart from the original form (A) in every case. Hence the mean deviation of such a single determination “ y cee — a”)? + (KO) — B)2 4 (KO) —y")P +... is m instead of which, by substitution of 352 J. F, ENCKE ON THE METHOD OF LEAST SQUARES. LAY Sa 6" py? + Oem = mk”, i a | = an 4 Ben 4 2n so s22 = m k2", squaring, we may write, V {km — Ko) KO}, the probable deviation of a single datum is =p VW {2 (k2%)— 2) 2™)}, and consequently that of the arithmetical mean of m data 2 (k2")— £0) k@)) 6 consequently it is an equal chance that k” lies between la EVA (2 (k2”) as A 9 ese ; ie / e m z or that A”) (2 n) ‘ m My Iie Fore km) i where the bracket refers to the ae values, between which the probability = . In the application to the law ¢ A found above for (A), we require each time the value of / k™, Thus, if we designate enerall n n generally \/ Flees m and extract on both sides the nth root, neglecting the higher powers of the limiting values, then ps FACED) a/ H=0 REL VARVA (sage 73)} This formula requires besides only the determination of the values of k() for any n that may be taken. For the function $¢ A which here obtains, we have h said = ae af AG eV 4g AS or if, in order to be able to bring into calculation the uneven powers of the errors (which else must always destroy each other), we regard all errors as positive ae J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 353 2h ps Gotan BuO T Ave, KO) — 3, KO) = 2 2, Oat MaRS 354 J. ¥F. ENCKE ON THE METHOD OF LEAST SQUARES. If these values be substituted in the above formula, we have on the left side of the equation 2/ k P OARS Ve, BEL ase 8) or meven= =. yom ee a 1 aA: nf1.2.3...% (n—1) n odd = aie fee consequently, if we multiply both sides by eg, and then leave on the left side = = r standing alone, we then obtain the following . Fanl o-%F val 18 Fon} + al } oy ‘paagrlly Gabe tim 45 or in numbers,— 0°509584 y = 0'845347 . a4] Se eset | 0°476936 1 = 0674489 » & 41 +o =0°577190 « ¢, {1 ag QUERY values: pes 441 ae I+ —_ i ll = — /m 055071 r = 0°512502 . s, 41 + meson | 635508" r = 0°465553 41 a a m = 0°429497 « &% {1 ™ oreriit | where ¢ is the arithmetical mean of all the errors without re- garding their signs; ¢, is the square root of the arithmetical J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 355 mean of the squares of the errors; and, generally, ¢, is the nth root of the arithmetical mean of the nth powers, without regard to signs. © We see from the numerical part of the limiting values, that the determination by the sum of the squares is the most advan- tageous one. With an equal number of observations, we ob- tain by its means the narrowest limits within which there are equal chances that r lies. The number of observations neces- sary for-attaining equal limits, according as we employ «,, 5, €35 &c., will be to each other as zw—2:1: eka 3 ed : sa Nat age 36 3 1600 45 ’ or if with «, one hundred observations are required to attain certain limits, there are required for the same limits, with epee eat ot od 14 ODSeWahORS mee ere, 108 z. Cea e eae ss PED sate ty fe dell tos Sa ive DE ee eT cae, On account of the great convenience of ¢,, and the not very con- siderable difference in the narrowness of the limits, the employ- ment of <«, will be most frequently preferred if the sum of the squares of the errors is not already known. For the above example the sum of the absolute errors = 181°898 ; consequently, epi alas = 6-496, and thence pr = 5!l-499 within the limits 4"-972 and 6!"-012, a value which, if it differs from that above given, still leads, for the small number of observations, to a sufficient estimation of the accuracy of the result. We may employ besides for this determination the proposi- tion which has no direct reference to the magnitude of the single errors, but only declares that, according to the universal law of probability [without determinate assumption of ¢(A)], the idea of the probable error contains the condition that there occur as many errors less than 7, as there are greater. If, therefore, we arrange the errors, without regard to their 356 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. signs, according to their absolute magnitude, and begin to reckon from the smallest, then the error which belongs to the index 4 (m + 1), if m is an odd number,—or the arithmetical mean between the errors of which the indices are 3 m, and } m +1, if m is an even number,—will give an approximate value for 7. Inthe example given above, m being = 29, it would be the 15th, or we should find r= 5914, As in the sums of the powers a greater number of errors so greatly increases exactness in respect to the probable limits, the effect must be so much the greater in this mode of computing. As the necessary formula has been given without proof, by Gauss, in the Zeitschrift fiir Astronomie, vol. i. p. 195, the following elegant demonstration, for which I am indebted to my respected colleague, Professor Diricuuet, will have the more value, as the proposition has not yet been demonstrated elsewhere. Let us seek the probability that, with (2 2 + 1) observations, the distribution of the errors shall be such, that there shall be one error between ¢ and ¢ + dt, n errors between O and ¢, and n errors between ¢+d¢ and. Let the probability that there is one error less than ¢ be generally t =/[(V(A)dA=u; 0 then the probability of an error > ¢ + d¢ will be t l—pidt— fya dA =1-u—wtdt, as the probability of an error between ¢ and ¢ + dt = (?) dt. Hence the compound probability of an arrangement of errors in which » errors < ¢, one error between ¢ and ¢ + d?#, anda errors > ¢ + dt. = u" (l—u)”.v (2) dt, neglecting the members of the second order, as the result is of the first order. But there may be as many such cases or arrange- ments as there are possible transpositions of 2 » + 1 elements, if there occur among them x equal elements of one kind (of which the probability = u), and n equal elements of another kind (of which the probability =(1—z)). Consequently the pro- bability of all possible arrangements of this kind . 1263... (n+) Be eer wu" (l1—4u)" fp (t) dé =U. | J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 357 _ If we regard the magnitude d ¢ of the interval between ¢ and _ ¢+ dt as constant, there is a value of ¢, for which U is a maxi- mum. The equation obtained by differentiation for its deter- - mination is | ny (t) nv (t) / mu eae P= 0 where / ¢ has the same signification as 9! A above. du, or the increment of WY a Ww A dA, in reference to an infinitely small Siieration of the limit 7, is equal top (¢) dt. We may give to the last equation the form 1 il v't ET Pla RYE The last member will be so much the less as is greater, or as there are more observations given. With a sufficiently large number it may be neglected. Or as m increases, the value of ¢, for which the maximum takes place, approximates continually to the value which follows from the equation = 0 er whence or the value of 7, according to the definition given above. | If we take the integral of U between determinate limits, we _may obtain the probability that the error which is situated in the middle is contained in these limits. It will be for the limits | r—6andr +0 r+9 Ses /, (Lu) yi dt, | | = or because W (¢) dt =d u, if for the limits in relation to ¢ we | put fee? RVR (6 p (t) dt=u', then the probability that the middlemost value lies between r — 3 and r + 8 will be 2 Si cd(O nae oh u 358 J. F. ENCKE ON THE METHOD OF LEAST SQUARES, The greater the number of observations, the narrower will be the limits between which ¢ will lie with equal probability. Therefore, if the observations are sufficiently numerous, deve- loping wu! and w'! according to TAyLor’s theorem, we may be permitted to consider only the first power of @. Whence, da fyidt 84 (1) = 4-200); and similarly, = 4 + oy (x). This form, as well as the combination of w and 1 — w in the integral, shows that a still more convenient form may be ob- tained by bringing in another variable for w; and this may be best done by the equation eit: he nay ia =) a aT A Toe consequently, the limits in relation to s being found by 8 oy (7) = ayn According to this the integral becomes 23,/n r 1.2.8...@n+1) 1 si a el es AT 7 “ gr+l Vf n (1 a =) ds. —23/n¥ (7) or, because s in the differential contains only even powers, 23 ,/nv(r) IOs ty (2 7 1) 1 (he 23.15. 0)* 9" Jn G-4) ds. Now let 8/7 be a finite oe = y; thus the limit 3 decreasing with the increase of 4/ n, s remains finite within the | assumed limits, however much » may increase. But if n is large, we may, according to the development of logarithms in Eucer’s Introductio, put (1-<)’ me 3 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 359 and 1.2.3...2” 2°” (ie 2 ase a8)? abs fw Euter Cale. Diff. P. ii. Cap. vi. § 160-162, as the limiting value to which it continually approximates as m increases; so that the expression becomes In+1 23./ n¥(r) ef aes Nf 7 0 for which we need not scruple to write 2 frre e-* ds fm ; 0 as the expression of the probability that with numerous ob- servations, the middlemost error, all being arranged accord- ing to their magnitudes, lies between 7 —0 and 7 + 8 This probability consequently becomes 3, or the probable limits are given by 23 3 28/nv(r) =e, whence 1 e FTO OGG * For the law of the errors assumed above Ne Qh —h2 a2 ——__——= | ¥(A) =29(A) = —— the probable limits of 7 will be nyt eer W 1 — A th or if, instead of 2 + 1, we call the number of observations m, and if we employ the equation hr = @, {is veh. The numerical value of e is 1:2554176, whereby the expres- sion becomes 0°786716 Ua 41 alia Tah This mode of determination of 7 is consequently still more inexact 360 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. than any one of the former ones up to the sum of the 6th powers. Applied to the above example, 7 becomes r = 5!"-914 + 0!""864, or the limits 5-050 and 6!"-778. In the demonstrations hitherto given, it has frequently been necessary to conclude from the probability of one value to that of another value depending on that of the first in a simple man- ner. For the sequel it becomes necessary to solve the general problem. If we know the most probable values of certain inde- pendent magnitudes 2, 2’, 2, &c., and the different limits within which these most probable values will lie, if any deter- minate probability is to be ascribed to them, to determine the most probable value of any function of these variables, OP ti PY and also the limits within which X has the same determinate’ probability. As, when we know the value of r in a magnitude de- duced by observations, we can at once find A, ¢,, and all the other functions of the errors, as well as their complete law ¢ (A), the problem may be proposed in this way: for 2, a’, xv, the most probable values a, a', a’ having been found independently of each other with the probable errors 7, 7’, r...., it is required to determine the most probable value of X = f(z, 2’, 2...) and its probable error. To begin with the simplest case, let X be a linear function o one unknown quantity = a we In all the cases in which wv = a, X = aa, consequently this will also be the most probable value of X. So also the cases in which z lies between a—r and a+r are equal in number to the cases in which X lies between aa—ar, and aa+ar; or “X=saatar, where the last member denotes the probable error of X. :. Now, in the second place, let X be the simple linear function of two variables t X=-27- 2’. For the sake of more convenient expression, let us now intro- duce, in lieu of the probable error, the weight of the values a and a'. If an observation, of which the probable error is w, be taken J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 361 as a common standard, then the weight of a, by reason of its probable error 7, will be we j= Pr? and likewise for a! la a ber Hence, if h belongs to w, the probability of any value for x al h vp —h? p(x—a)? = AE rs é > and for 2! F = h Vv p' eh Pr (a’—a’)?, ft : ' thus the probability of the concurrence of two arbitrary values will be WV pp v pp e he {Pp (2—a? + pi (2’s-0'}?} T _and the probability of the concurrence of two values x and 2! which satisfy the equation v+a=X, in which X signifies an arbitrary but determinate value, is found by considering one of the magnitudes z or 2’! as a func- tion of the other, and of the magnitude X, and by substituting the value so obtained. Hence the probability that any value z, by its concurrence with the value 2/, should give the result X, is Ww = he J pp e—? {p (a—a)?+p’ (X—2—a’)?} us Then, if we take the sums of all possible values of W, or the : W dz within the limits in which a value of can exist, being here — » and + w, we shall have embraced all cases in which X can be obtained, or have determined the probability of X. In order to facilitate the integration, let us give to the exponents the following form : — 1 £ (pty) («PA tBeray p+p pp x ! a, +s —a—a Sarr Ph which is at once obtained if we combine in a quadratic form all VOL. Il. PART VII. 2B 362 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. the members which contain z. Now, for temporary abbrevia- tion, let pX +pa-pa pty X—a—d=X,; then —72 PP ys fiona FF Aue ) eo” pe false st eee eo) SeB Or dy the value of the factor which contains the integral will, accord- ing to (5), = 1; consequently the probability of X a h pp —h? PP (X—a—a’)? ars pt+p 2 € p+p’ is a maximum, if ed = Lo, X=a-4d, and the weight of this determination will be given immediately by the form py. pt p’ consequently the probable error w £3 ww rag + RAL. Papi a) aaa Sag? e), The simple proposition thus found is this: If @ and a’, the most probable values of x and z', independently of each other, toge- ther with the probable errors r and 7’, are given, then the most probable value of X = @ + a! =a+d, and the probable error of this value = V7 (7? +7”), Combining this with the preceding proposition, we obtain consequently for any linear function X=ar+fha'+yao'.... the most probable value =aat+Ba't+ya'.... = & (203) with the probable error = Vf (227? + Br? tor? ... )J J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 363 because, by virtue of the form for two unknown magnitudes, the form for any number is at once derived,—if there are three, by first combining two with each other, and then com- bining their result with the third,—if there are four, by first combining three with each other, and then their result with the fourth, and so on. The general problem might be solved in a similar manner if the integrations could be performed. For eee eee ae: Oa) We on Ran a) <',, (2S) the probability of the concurrence of arbitrary values of the pz variables will be a (pe ppl +s) 4-12 (p (ea) +p (a etp" (aa...) T If we are here to consider only the cases in which a determi- nate value for X is to be found, let us express one of the vari- ables x as a function of X and the remainder. If we substi- tute this value in the exponents, and take the sums or in- tegrals within all possible limits for x, 2’ ..., we shall obtain the probability of the value X, and we may thence determine the most probable value and its limits. But for this the know- ledge of the function is obviously needed; and, if this function is not linear, the complete integration will be impracticable in most cases. However, under the supposition that the limits for the several variables are already so narrow, that the higher powers of the probable error may be neglected, we may find an approximative value for X and its limits, which will be always _ sufficient in practice. Let us take for arbitrary values of 2, 2’, 2... the form @+Az,ad+ A2z',a'+ Ax"; then, if =e 2k Sea are 0.5 the general expression for X, neglecting those powers of A 2, A #, A x, which exceed the first, will be ow dV d . ers) *°* Se wy Aas (TS ra) A a. ey or X-V= Tn) Ae v4 (Tart (S VY Aa. and the cae of the concurrence of these LF ae will be fet — MY fp. p' sp eee —h? (pAaxtt p! az?+p"ax?,,.), me 2 2B2 364 J. F. ENCKE ON THE METHOD OF LEAST SQUARES. The most probable value of X—V and its limits are given im- mediately by the most probable value of X and its limits, and conversely, because the two magnitudes X—V and X differ only by a constant; so also the probable errors of Aw, Aa’, Aa", &c. will be the given magnitudes 7, 7,7”, &c. and the most pro- bable values of Av, A 2’, A x", &c. will be nothing by virtue of the equations 7 = a + A x, &c. Hence follows, according to (20.), the most probable value of X, X—V=0, and the probable error of X —V, ne d Ng nN? 6, av? , } ( F=/1(] Li +(F)" +(Gaoae J or the most probable value of X is V, and the probable error of this determination is equal to the above-determined F; a solu- tion which is rigorously true for linear functions, but only ap- proximately so for higher ones. It is a different case, supposing that we have found for one and the same unknown «, by different examinations, the values a, a’, a”. .., with the probable errors 7,7", 7” . . ., or the weights p> Pp’, p" ..., and that we seek to find from them all the most probable values. The definition of the idea of weight, according to which a, a’, a’ must be considered as respectively found by the number p, p', p", of equally good observations, gives here, by virtue of the arithmetical mean, the most probable value of z _ apt+a p! +a" p"+,&e. — ptp +p" +, &e. ” with the weight pt+p' tp" +, &e.; or, which is the same thing, the most probable value of a a’ a! 7 23.) Pr ar pat pie +; &e. PSS See et 1 1 1 with the probable error > + + (24) It 1 = eT * reek Wie == pe er aat.&e.) 0-18 0:19 0:20 0:24 0:25 0:29 0:30 0-32 0°33 0°39 0°40 0-41 0:46 0°48 0:49 0°50 TABLE I. 365 @ (2) | ¢ | e@ 0:00000 00 | 145 0|| 0:50 | 052049 99 8 78 01128 33 | 1138 33} sol 0-51 | s2924 37 | 82% 38 | bs 02256 44 | 1)57 gg | 45|| 0°52) 53789 87 | gx6 54/8 96 03384 10 | 119g gg | 67/| 0°53 | 54646 41 | of 27 | 9 03 04511 09 | 115g 99 |. 90|| 0°54] 55493 92 | B56 4) |9 10 05637 18 | 494 g7 |112| 0:55) 56332 33 ge9 04 (2 17 06762 15 | 1193 Gg |1 35] 0°56 | 97161 57 | gon 6) | 9 28 07885 77 | j199 94 |} 58] 0:57 | 57981 58 | o79 7) |9 30 09007 81 | 159 95 |1 79/058 | 58792 29 | On) 54/9 35 10128 06 | -°~~ ~? |2 o1|| 0:59 | 59593 65 9 40 1118 24 791 96 011246 30 2 24|| 0-60 | 060385 61 9 45 12362 30 ue ey 2 46| 061 | 61168 12 rhe a 9 49 13475 84 | 1419 g7 |2 67|| 0°62 61941 14 | 763 4g 9 58 14586 71 | 1197 99 |2 88] 063 | 62704 63 | 253 94 |9 55 15694 70 | 1194 go | 10/064) 63458 57 | 544 3. | 9 59 16799 59 1101 58 3 31]| 0°65 64202 92 734 73 9 62 17901 17 | y99g og |3 52|| 0:68 | 64937 65 | 555 19 [9 68 18999 23 | jo94 34 |3 72|| 0°67 | 65662 75 | 15 45 |9 65 20093 57 | i999 41 |3 93] 068 | 66378 20 | 595 79 |9 66 21183 98 4 14| 0-69 | 67083 99 | 4 9 68 1086 27 cava one 696 11 0:22270 25 > |4 34]|| 0°70 | 9°67 ] , |9 67 23352 18 ey rr 453| 0-71 | 68466 54 eb 76 9 68 24429 58 | 1975 gy |4 73|| 0°72 69143 30 | ge7 Og |9 68 25502 25 | ioe, 75 | 4 92| 0-73 69810 38 | G25 45 |9 66 26570 00 | too G3 |S 12|| 074 | 70467 80 | G45 ~6 |9 66 27632 63 | 957 34 |5 29] 0-75 71115 56 | gag Hy 9 65 28689 97 24 9* 15 49] 0:76] 71753 67 9 62 1051 85 : 628 49 29741 82 | so46 13 |5 87|| 0:77 | 72382 16 | rg gq |9 61 30788 00 1040 34 5 84|| 0:78 73001 04 609 31/2 97 31828 34 6 01|| 0-79 | 73610 35 9 56 1034 33 eh 599 75 032862 67 6 19|| 0-80 | 0°74210 10 9 52 33890 81 tie: aa 6 36|| 0:81 | 74800 33 49 = 9 48 34912 59 | 3915 5g |6 52|| 0°82 | 75381 08 | 57) 39 |9 45 35927 85 1008 59 6 67|| 0°83 75952 38 561 89 |2 4! 36936 44 1001 75 6 84]|| 0:84 76514 27 552 53. |2 36 37938 19 | “oo4 4 |6 98] 0°85 | 77066 80 | 545 95/9 31 38932 96 987 63 7 14|| 0-86 77610 02 533 96 9 26 39920 59 | 985 34 |7 29|| 0°87 | 78143 98 | 554 75 |9 21 40900 93 | 79 99 |7 42| 0°88 | 78668 73 515 59/216 41873 35 | °!* 7 55|| 089 | 79184 32 9 09 965 37 506 50 0:42839 22 ~~ wo |7 69] 0:90 | 0:79690 82 9 04 43796 90 oH Be 7 82|| 0-91 | 80188 28 id v4 8 97 44746 76 941 91 |7 95 0:92 80676 77 479 58 8 91 45688 67 933 84 8 07]| 0°93 81156 38 470 75 8 83 46622 51] 95° G7 |8 17| 0:94 | 81627 10 | 46) og |8 77 47548 18 917 37 |8 30] 0:95 | 82089 08 | 453 og |8 70 48465 55 | O08 g7 {8 40// 0:96 | 82542 36 | 444 G7 |8 61 49374 52 900 46 |8 53] 9°97 | 82987 03 | gag 19 |8 55 50274 98 391 85 |8 61 || 0:98 83423 15 427 66 8 46 51166 83 8 69] 0:99 | 83850 81 8 39 883 16 419 2 0-52049 99 8 78|| 1:00 | 084270 08 8 30 366 TABLE I. (continued). t wis a 0) ts) 7 nf t 6 (2) | t | 6 (t) 1:00 | 0:84270 08 s 30|| 1°50 | 0:96610 52 3 57 1-01 | 84681 05 | 449 97 |s 20|/ 1-51] 96727 68 | 114 Go |3 49 1-02 | 85083 80 | 397 73 |s 1s|| 1:52) 96841 35 | 115 of [3 40 1:03 | 85478 42) Soe 27 |8 05] 1:53 | 96951 62 | 196 on |3 32 1-04 | 85864 99) 3° 7 96|) 1:54] 97058 57 | j93 79 [3 25 1:05 86243 60 370 75 7 86|| 1°55 97162 27 100 54 3 16 1:06 | 86614 35 ae 5 7 77|| 1:56 | 97262 81 | “9, 4 |3 09 1-07 | 86977 32| 3 7 |7 68|| 1:57 | 97360 26 a (80 1:08 | 87332 61 | 255 291, gol! 158 | 97454 70 0 (2 94 1-09 | 87680 30| 347 69 |, 49] 1-59 | 97546 20| 91 59 lo g6 1:10 | 0:88020 50 Shak ie 60 | 0:97634 84 ik : . ~ 7 40]| l- 2 79 1-11 | 88353 30| 332 8) ly sil 1-61 | 97720 69 | 88 85 |e 73 1:12] 88678 79 | 379 53 |7 21]) 1-62 | 97803 81 83 12] 9 64 1:13 | 88997 07 | 3,7 Te |7 12|| 1:63 | 97884 29 80 48 | 5 59 1:14| 89308 23 Hs 617 oi} 1-64! 97962 18] 4&7 89 Jo 51 1:15] 9612 38} 39% 1516 oi 1-65 | 98037 56| 75 38 2 45 116] 89909 62| 297 24) gol 1-66] 98110 49| £2 93 |e ss 117| 90200 04] 299 42\¢ 7ol] 1-67] 98181 04} 29 55 Jo 31 118 | 90483 74| 783 70 |¢ 61] 1-68 | 98249 28 | 68 24 lo 26 1:19 | 90760 83| 277 9916 sol] 1-69| 98315 26| 65 98 |2 20 270 57 63 78 1:20 | 091031 40 | 264 15 |6 42]] 1°70 | 0:98379 04 212 1:21] 91295 55| 957 gq {6 31|| 1-71} 98440 70| 61 66/2 og 1:22 | 91553 39| 95) g2 |6 22|| 1-72| 98500 28| 99 98/2 o1 1:23 | 91805 01] 945 5; [6 11|| 1-73| 98557 85 | 57 37 |1 96 1:24 | 92050 52 | 539 4g [6 02|| 1-74| 98613 46| 55 61)1 90 1:25 | 92290 01 | 935 53|5 91] 1:75| 98667 17] 53 7111 85 1:26 | 92523 59) <° 5 81|} 1-76| 98719 03| 91 86|1 79 1:27| 92751 36| 227 77 |5 711 1-77] 98769 10| 50 07 |) 75 1:28 | 92973 42] 222 06 | 5 61] 1-78| 98817 42| 48 32 |1 68 1:29| 93189 87| 716 45|5 sol 1-79| 98864 06| 46 6411 65 210 93 44 99 1:30 | 0°93400 80 wh eae 1:80 | 0:98909 05 1 59 1:31 | 93606 32| 295 52/5 soll q-81| 98952 45| 434011 54 1:32| 93806 52| 200 20/5 901] 1-82] 98994 31] 41 8611 50 1:33 | 94001 50| 194 98/5 11|/ 1-83] 99034 67 | 40 36 |1 44 1:34| 94191 37] 189 87/5 o2|| 1:84] 99073 59 | 38 92}1 41 1:35 | 94376 22| 184 85 |4 93] 1-85 | 99111 10| 37 51/1 36 1:36 | 94556 14| 179 92/4 go|| 1-86 | 99147 25 | 3615 ]1 33 1:37 | 94731 24| 175 10|4 74] 1-87] 99182 07 | 34 82|1 27 1:38 | 94901 60| 170 36 |4 63] 1-88] 99215 62| 3355 |1 24 1:39 | 95067 33] 165 73/4 55]| 1-89.| 99247 93 | 32 31 |1 20 161 18 Ci Pall 1-40 | 095228 51 _. |4 45|| 1:90 | 099279 04 1 16 1-41 95385 24 | 156 73/4 35|| 1-91 | 99308 99| 29 95 }1 12 1:42] 95537 62| 152 38) 4 27/1 1:92 | 99337 82| 28 83 ]1 o8 1:43] 95685 73} 148 11/4 18] 1-93| 99365 57| 27 75 |1 06 1:44| 95829 66| 143 93/4 09|| 1:94] 99392 26} 26 6911 o1 1:45 | 95969 50} 139 84/3 99/1 1-95 | 99417 94} 2568] 99 1:46 | 96105 35| 135 85}|s3 91/1 1-96] 99442 63| 2469] 95 1:47 | 96237 29| 131 9413 g2|| 1-971 99466 37| 2374] 91 1:48 | 96365 41| 128 12/3 74] 1-98] 99489 20| 2283] 89 1:49 | 96489 79| 124 38/3 65|| 1-99 | 99511 14] 2194] 85 20 7: 2 1:50 | 096610 52| 179 73] 5 s7l| 2-00 | o-gn5s2 23} 7! | sal A r (2) 0-00000 00538 01076 01614 02152 02690 03228 03766 04303 04840 0:05378 05914 06451 069387 07523 08059 08594 09129 09663 10197 0:10731 11264 11796 12328 12860 13391 13921 14451 14980 15508 0°16035 16562 17088 17614 18138 18662 19185 19707 20229 20749 0:21268 538 538 538 538 538 538 538 537 537 538 536 537 536 536 536 535 535 534 534 bss 23S 532 532 532 531 530 530 529 528 527 527 526 526 524 524 523 522 522 520 519 TABLE II. Is ae AN 4) fart an 0(e4 E °(e+) 0:21268 21787 22304 22821 23336 23851 24364 24876 25388 25898 0:26407 26915 27421 27927 28431 28934 29436 29936 30435 30933 0°31430 31925 32419 32911 33402 33892 34380 34866 35352 35835 0°36317 36798 37277 37755 38231 38705 39178 39649 40118 40586 0:41052 °(¢?) 0:41052 41517 41979 42440 42899 43357 43813 44267 44719 45169 0°45618 46064 46509 46952 47393 47832 48270 48605 49139 49570 0-50000 50428 50853 51277 51699 52119 52537 52952 53366 53778 0:54188 54595 55001 55404 55806 56205 56602 56998 57391 57782 0°58171 367 465 462 461 459 458 456 454 452 450 449 446 445 443 441 439 438 435 484 431 430 428 425 424 422 420 418 415 414 412 410 407 406 403 402 399 397 396 393 391 389 368 TABLE IL. (continued). Aa 5 Dro uta A Lane ie Se *.) ¢ =0'4769360 A A A A A A tle | * |G) aie eee 1:20 | 058171 1:60 | 0°71949 2-00 | 0:82266 1:21 | 58558 Oe 161 | 72249 ~ 2-01 | 82481 ee 1:22 | 59942 | $84 || 1-62 | 72546 a 2.02 | 82695 | 215 1:23 | 59325 | 58° | 1-63 | 72841 | 39° | 2-03 | 82907 | 315 1-24 | 59705 | $50 | 1-64| 73134 | 59° | 2-04) 83117 | 35, 1:25 | 60083 | 87° || 1-65 | 73495 | 52) || 2-05 | 83324 | oo7 1:26 | 60460 | 377 || 1-66) 73714 *4 2-05 | $3530 | Soy 1.27 | 60833 | $75 || 1-67 | 74000 | 22° || 2-07 | 83734 | ooo 1:28 | 61205 | 37° || 1-68 | 74285 | 582 | 208 | 83936 | 30) 1:29 | 61575 | 27° | 1-69] 74567 2-09 | 84137 0 | 061942 ih 1-70 | 0-74847 ay 2:10 | 0:84335 0. 1:3 5 < a ¥ 2 ° D 131 | 62308 oi 71 | 75124 | 277 || 2-11 | 84531 rts 132 | 62671 | 5°" || 1.72) 75400 a 2-12] 84726 | 102 133 | 63032 | $6) 11-73 | 75674 | 57 || 213 | 84919 | 105 134 | 63391 | $29 | 1-74| 75945 | 375 || 214] 85109 | 75, 1:35 | 63747 | $70 || 1:75 | 76214 ao 2:15 | 85298 | 128 1:36 | 64102 | 372 || 1-76 | 76481 | 5°72 | 216 | 85486 | 135 137 | 64454 | 3°° | 1-77 | 76746 | 52> || 2:17 | 85671 | 133 138 64804 See || 1:78 | 77009 | Se? | 218 | 85854 | \o5 1:3 5152 1:79 | 77270 2:19 | 86036 sah Give coe 346 258 180 65 . || 1:80 | 077528 2-20 | 086216 | __ 1-41 | 65841 | 24° | 1-81 | 77785 | 227 || 221 | 6394 | 77° 1-42 | 66182 | $*) || 1:82] 78039 | 578 || 222) 86570 | 7 1:43 | 66521 | 2° || 1-83 | 78291 | 22° | 223 | 86745 | 770 1-44 | 66858 | S27 || 1-84| 78542 | 57° | 224 86917 | 177 1:45 | 67193 | 32° || 1:85 | 78790 | 46 || 2:25 | 87088 | 1, 1:46 | 67526 | 35° || 1-86 | 79036 | 27° || 226 | 87258 | 16 1-47 | 67856 | 35° | 1-87 | 79280 ae 2-27 | 97425 se 148 65184 soe || 1:88 | 79522 aa 2-28 | 87591 | 5 : : 1:89 | 79761 | 739 || 2:29 | $7755 1°50 | 0°68833 ae “Oi a . 1:90 | 0-79999 2:30 | 0:87918 1:51 | 69155 se 1:91} 80235 | 23° | 2-31 | 88078 a 1-52 | 69474 | $1° || 1-92 | sodeo | 25% || 2-32 | 88237 | 18 1:53 | 69791 | $1; | 1-93 | 80700 23) || 2:33 | 88395 | 3 + 1:54 | 70106 | 31° || 1-94 | 80930 aed 2-34 | 88550 | 12? 1:55 | 70419 | $15 | 1.95 | 1158 | 22° | 2-35 | 88705 | 120 1:56 | 70729 | $00 | 1:96] 81383 | $5? || 236 | 88857 | 15 1:57 | 71038 | $°° | 1-97| si607 | $34 || 237 | 89008 | 145 1:58 71344 o0¢ || 1-98 81828 | 22) | a-38 | 9157 | 12> . 1648 1:99 | 82048 2:39 | 39304 301 218 146 1:60 | 0:71949 2-00 | 0-82266 2-40 | 089450 TABLE Ii. (continued). A & 20 va e-P dt=O (¢ *) e = 0°4769360 A A A A A A + \e(e>) | Cal WA Gia, | al a Gre, 2-40 | 089450) |,. | 280| 094105) ,, | 320 | 096910] g41| 89595 | 145 | 2-81] 94195 | 20 | 321 | 96962] 4 g42| 89738 | 14° || 282| 94284| 2? | 322) 97013 | 9 243 | 99879 | }4) | 283| 94371| 871323) 97064 | 3 244| 90019 | 120 | 284] 94458) 27 | 324] 97114 | 20 245 | 90157 | 128 || 2-85| 94543) 8% | 325 | g7163| 4° 246 | 90293 | 96 | 286| 94627| 84/326] 97211| 4° 2-47 | 90428 | 125 || 287) 94711| 8% | 327] 97259| 48 248 | 90562 | 12% || 288| 94793) 8° 328 | 97306 | 47 249 | 90694 2-89 | 94874 329 | 97352 131 80 45 2:50 | 0-90825 | ,,, || 2-90 | 094954 | _,. || 330 | 097397] ,. 251 | 90954 | 392 || 291| 95033) 72 | 331 | 97442 | 4% 252| 91082 | 128 || 292) 95111| 78 | 332| 97486 | 4% 253 | 91208 | 3° || 293| 95187| 76/333] 97530| 44 254 | 91332 | 124 || 294) 95263| 76 | 3:34] 97573) 48 255| 91456 | 15s || 295| 95338) 72335 | 97615| 79 256| 91578 | 12° || 296| 95412) 73 | 336| 97657| 4 257 | 91698 | 170 || 297] 95485] 78 | 3:37| 97608| 4° 258| 91817 | 11° 298 95557 | 79 338 He is 2:59 | 91935 2:99 | 95628] 7 || 3 7 116 70 39 2-60 | 0:92051 | |, || 3:00 | 0-95698 | . || 3-40 | 0:97817 | ,.. 2-61| 92166 | 117 || 301 | 95767| Fe || 350) 98176 | $°2 2.62 | 92280 | 115 || 302] 95835] °° | 3.60 | 98482.| $00 263 | 92392 | 117 || 3:03 | 95902 | 67 || 3:70 | 98743 | 3° 264| 92503 | 11) || 304} 95968 | &° | 380) 98962 | 722 265 | 92613 | 1)? || 305 | 96033| 62 || 390] 99147 | 18? 266 | 92721 | }°° || 306| 96098 | 6° || 4-00 | 99302 | 13° 267 | 92828 | 1°" | 3-07 | 96161 | °2 | 410] 99431 | 100 2-68 | 92934 3-08 | 96224 4.20 | 99539 2-69 | 93038 | 1°* || 309 | 96286 | ©? | 4:30 | 99627| °° 103 60 73 270 | 093141 |... || 310 | 096346 | || 4-40 | 099700] ,, 271 | 93243 | 1°" | 311 | 96406 | §° || 4:50 | 99760| $2 272| 93344 ae 312) 96466) 7. | 460 | 99808] 45 273| 93443) °° || 313| 96524| $8 | 470| 99848] 39 274| 93541 | °° | 314] 96582] $8 | 480 | 99879) 36 275 | 92638 315 | 96638! 5° || 490} 99905 276| 93734| 2° 11316] 96694! °° || 500] 99996| 7! 277 | 93898| 97 || 317] 96749} 55 278 | 93922 23 318 | 96804 a 279| 94014 3:19 | 96857 1 69 9 53 i 2°80 | 0°94105 3°20 | 0:96910 ARTICLE XI. On the Theory of the Formation of Atther. By HEinricw Rose, Professor of Chemistry in the University of Berlin.* [From Poggendorff’s Annalen, vol. xlviii., part 11, November 1839. ] IT is well known that many salts of the oxide of bismuth, o the oxide of mercury, of antimony, and several other metallic oxides are decomposed by water. They are generally convert- ed by it into basic salts; but sometimes, by employing a suffi- cient quantity of water, the decomposition even goes to the separation of the pure oxide, as in the case of the nitrate of the oxide of mercury. The explanation usually given of these decompositions is, that the water resolves the neutral salt of a metallic oxide into an acid and a basic salt, in a similar manner as nitric acid converts the red superoxide of lead into protoxide o lead and the brown superoxide. But the existence of acid salts, which are said to be formed by the action of water on several neutral salts of metallic oxides, is far from being proved ; in most cases the water only deprives the salt of a part of the acid, and this dissolves a portion of the neutral salt, which, after the acid solution has been concentrated by evaporation, most fre- quently crystallizes as a neutral salt, and rarely as a double combination of neutral salt and acid hydrate. In many cases the quantity of the salt which dissolves in the liberated acid is exceedingly small, frequently none at all, and the entire quan- tity of the oxide forms an insoluble basic salt. The simplest explanation that can be given of such de- compositions produced by water, appears to me to be this, that water, acting the part of a base, separates the metallic oxide as a basic salt, or at times even in the pure state, and combines with the acid to form a hydrate. This explanation is the more admissible, as we have been long accustomed to regard the hydrates of acids as saline combinations in which water replaces a fixed base. It is well known what happy conclusions for the whole theory of chemistry, more espe- * Translated and communicated by Mr. William Francis. HEINRICH ROSE ON ATHERIFICATION. S71 cially Graham, Berzelius and Liebig, have drawn from this view. In fact it is particularly the salts of such metallic oxides as are not possessed of strong basic properties that are decomposed | by water. The salts of the powerful bases do not exhibit this | phenomenon. According to this view the decompositions in question are analogous to the conversion of the red oxide of lead into the brown superoxide and protoxide of lead, by nitric acid, only that they are of exactly the converse kind, the strong acid ex- pelling from a combination of the oxide of lead, with oxygen, the weaker electro-negative body, and combining with the basic. _ Water also occurs in other cases as a base, and sometimes | displaces other bases from their combinations. As it, how- ever, belongs to the weaker bases, and, at the same time, is volatile, these cases are not very frequent; but although vola- tile, it is nevertheless able to expel the more volatile oxide of ammonium from its combinations. Ifa solution of the sulphate of the oxide of ammonium be boiled for a long time, it becomes | acid; and if the boiling is effected in a retort, a liquid passes over into the recipient, which contains free ammonia. This result evidently arises from the water, as base, eliminating the oxide of ammonium (which cannot exist in a free state, and passes into water and ammonia) from its combination with the sulphuric acid, and combining with the same. The quan- tity of the sulphate of the oxide of ammonium decomposed in this way, is indeed but small; but it must also be remembered that the oxide of ammonium is one of the most powerful bases, and this result is chiefly to be ascribed to its greater volatility. If we apply the above explanation of the decomposition of many salts by water to the theory of the formation of ether, it will acquire great simplicity. Berzelius and Liebig have advanced the view that «ther may be regarded as a base; which has found such general assent, that it is almost universally adopted, at least in Germany. It is well known that the salts of the oxide of ethyl (the com- pound zthers) may be more or less easily decomposed by bases, water being present; the bases combine with the acid cf the compound, and separate the oxide of zethyl as a hydrate (alcohol). But water itself, which in this case acts evidently the part of 372 HEINRICH ROSE ON ZTHERIFICATION. a base, also causes the same decomposition. Some compounds of the oxide of zthyl are as easily decomposed by water as by many bases ; so, for instance, is oxalic zther, which is converte¢ by water into hydrate of oxalic acid and alcohol. A high tem: perature is not even requisite to produce this change ; for i takes place at the common temperature, and indeed in a very short time. But the acid sulphate of the oxide of zthyl,—or, rather the combination of the sulphate of the oxide of zthyl, with hydrated sulphuric acid (sulphovinic acid), also undergoes in its solution in water quite a similar decomposition. Even at the usual temperature alcohol and the hydrate of sulphuric acid are gradually formed; it proceeds more rapidly by boiling. This process may likewise be most easily explained by the suppositior that water acting as a base eliminates the oxide of ethyl from its combination with sulphuric acid, which, at the moment o its expulsion, takes up water and forms alcohol. The solutions of nearly all sulphovinates in water are de composed, especially on boiling, in a similar manner. Alcoho and water evaporate, and a so- eaied acid sulphate, z. e. a double compound of the neutral salt, which already pre-existed in the sulphovinate salt with the hydrate of sulphuric acid, is forme¢ in the solution. If sulphovinic acid is heated with merely a small quantity o water, no alcohol is obtained, but chiefly hydrated sulphuric acid, and pure oxide of zthyl or ether :—there is not sufficien water present to convert the liberated «ther into alcohol. . If alcohol is mixed with the hydrate of sulphuric acid, sulpho- vinic acid is formed, or a double compound of the neutral suk phate of the oxide of zthyl with the hydrated sulphuric acid. By the formation of the sulphate of the oxide of zthyl two atoms of water are set free, one from the hydrated sulphuric acid, the other from the alcohol. On heating the mixture, one of these liberated atoms of water eliminates oxide of zthyl from its com) bination with sulphuric acid, and combines with the acid, form- ing the hydrate of sulphuric acid. But why does not the zther at the moment of its expulsion} combine with water and form alcohol? There is sufficient | water present, for only one atom of water is requisite to ex ma the ether; and at the formation of sulphovinic acid, even when anhydrous alcohol is employed, two atoms are set free. 5 HEINRICH ROSE ON ZTHERIFICATION. 373 It is well known that sulphuric acid can take up more than h one atom of water to form a hydrate. Besides the common djhydrate, with one atom of water, we are acquainted with a second, which may be prepared in a crystalline state, and con- jtains two atoms of water. This combination corresponds to a nbasic sulphate salt. The disposition of the hydrate of sulphuric acid to take up ? more water is very great, and it is employed on this account for various purposes in our laboratories. It is this which pre- ixture is uninterruptedly boiled for some time, the hydrated isulphuric acid loses the acquired water, which may then be listilled over in company with the ether. The ether therefore hey are not the products of one, but of two chemical pro- esses, which are both active together in the boiling mixture. At the commencement of the operation but very little water )/passes over along with the zther and that alcohol contained in ithe mixture, which has not been converted into sulphovinic acid, so that the water remains dissolved in the distilled alcoholic zther, and does not separate: the quantity of water increases by further distillation, especially at a high temperature, when ze quantity of the second hydrate of sulphuric acid has augmented. Alcohol is scarcely ever employed anhydrous in the prepara- tion of zther, but generally hydrated. It is evident that in the Matter case the quantity of the second hydrate of sulphuric acid must be considerably increased. The experiments of Liebig, Magnus, and Marchand have shown that in the cold this |second hydrate cannot form sulphovinic acid with alcohol, but does so at a higher temperature, and therefore that such a mix- jture on boiling can give zther by distillation. But it is well known that on employing hydrated, or even anhydrous alcohol, there is always a portion of it which is not converted into sul- \phovinic acid, and this quantity may be distilled as alcohol from the mixture. A second portion of alcohol, which distils over in company with the ether, in the formation of ether, may, however, be produced in this way,—that «ther and water are cotemporaneously disengaged from the mixture, and com- 374 HEINRICH ROSE ON ATHERIFICATION. bine to form alcohol; for it is produced only in this way when a solution of pure sulphovinic acid is boiled with much water, or compound zthers decomposed by water or by the hydrates of bases. q When, however, from the tendency of the hydrate of sul- phuric acid to take up more water, zther has been evolved from a mixture of alcohol and sulphuric acid, it does not take up any water after being once separated: but water may be di- stilled over by heating the diluted sulphuric acid. We know that when ether is treated with water, or even dissolved in it, no alcohol is formed. When ether is once separated from 4 compound of oxide of ethyl, the former can in no way be con. verted by water into alcohol. Only when, as above observed the zther comes in contact with water at the moment of its ex- pulsion does it form alcohol with it. The cotemporaneous dis- engagement of ether and water, from a boiling mixture of al evidently that both owe their origin to two distinct processes. Moreover, it is by no means an anomalous phznomenon that a base, which is capable of forming a hydrate, does not combine with water when brought into contact with it in a pure state; @ great number of cases of this kind occur in inorganic chemistry, We need only compare zther with that numerous class of ig nited oxides in which so compact a state of cohesion is pro duced by heat, that they not only withstand the action of water, but even entirely or partially that of acids, to find abundant proof of such analogies. The ignited oxides with these pro- perties always belong to the weaker bases, under which zthet these oxides the more, as it like them combines directly with acids with difficulty. j But even among the stronger bases we find some whose relations to water resemble those of «ther. When oxide of tact with it at a higher, or at the common temperature. To find out at what period, in the preparation of zther by boiling a mixture of alcohol and sulphuric acid, water com-} mences to pass een M. Wittstock, at my request, instituted’ a HEINRICH ROSE ON £THERIFICATION. 375 series of experiments, which he had the kindness to communi- cate to me. Two pounds of the hydrate of sulphuric acid were mixed cold with two pounds of anhydrous alcohol, the mixture was made to boil with all possible haste in a retort, the distilled products, well cooled, were gradually received, and the distilla- tion continued until the contents of the retort boiled over. The weight and specific gravity of the products were deter- mined as they distilled over in succession. The results are as follow : First product : 3 drachms 50 grains; spec. gr. 0°776*; pro- duced before the boiling of the mixture. The following products passed over after its boiling : Second product: 3 ounces 6 drachms; spec. gr. 0°808. Third product: 3 ounces 6 drs.; spec. gr. 0°800. Fourth product: 3 ounces 6 drs. ; spec. gr. 0°786. Fifth product: 5 ounces 3 drs. 50 grs.; spec. gr. 0°776. Sixth product: 4 ounces 1 dr. 50 grs.; spec. gr. 0761. Seventh product: 1 ounce 7 drs. 10 grs.; spec. gr. 0°809. Eighth product: 1 ounce 2 drs. The first five products consisted of a single liquid; the sixth was the first in which a layer of water and of zther were erceptible. The quantity of separated water amounted to 3 drachms ; the ethereal liquid had the specific gravity mentioned bove. The seventh product consisted in volume of two parts water, and three parts of an ethereal fluid of the specific gravity stated ; the eighth consisted almost entirely of water, above which floated a very thin layer of zther, which was coloured yellow by oil of wine. The contents of the retort boiled over on the continued application of heat. The first five products consisted of zether mixed with alcohol, which last was contained in the retort as such, and not con- verted into sulphovinic acid, and evaporated from the mixture in company with the ether. The first product, which distilled over at the lowest temperature, contained, to judge from its specific gravity, much ether, and little alcohol, quite opposed to the general opinion that ether is only formed at the boiling-point of the mixture. The succeeding products gradually became, ac- cording to their specific gravity, constantly more ethereal, and * The specific gravities, both here as well as those to be mentioned subse-~ quently, were all determined at 14° Reaum. (63°5° Fahr.). 376 HEINRICH ROSE ON £THERIFICATION. contained less alcohol ; but only in the sixth product was there so much water that it separated, and the quantity increased in | proportion as the distillation was continued. The first six products smelt but slightly of oil of wine; bu the seventh contained a portion, and also smelt of sulphurous | acid. After the first seven products had been mixed together, | and the separated water removed, they had a specific gravity of ; 0°788. It is well known that ether is prepared, of late, in the most advantageous manner, by allowing a small stream of alcohol to flow constantly into a mixture of alcohol and the hydrate of sul- phuric acid, and distilling off ether in proportion as alcohol is added*. It has been denied that the presence of sulphovinic acid is of essential influence in the formation of zther, and as-_ serted that it is not necessary that the formation of this acid J should precede that of «ther, because in the method of preparing zether alluded to, the boiling mixture must be constantly at a temperature of 140° cent., at which sulphovinic acid could not | exist. But at the point where the current of cold alcohol flows” into the boiling mixture, the temperature is under 140°. The | sulphovinic acid formed is decomposed it is true, in a very_ short time, from its soon acquiring the temperature of the | boiling liquid. The preparation of zther, according to the above method, consists therefore in a constant formation, and } continual decomposition of sulphovinic acid. It is a pretty ge-| nerally entertained opinion that the production of zther from a mixture of alcohol and sulphuric acid, is solely effected by the boiling of the mixture, which takes place at a high temperature, about 140° cent. In many works on chemistry we meet with | the assertion that when a mixture of sulphuric acid and alcohol | are heated at a temperature, not high enough for it to boil, na ; ether, but merely anhydrous alcohol, is obtained. tion to the hypothesis I have advanced; for, according to that, it would be somewhat difficult to explain the circumstance why the oxide of zthyl is separated at a lower temperature, as a hy- drate, and at a higher one in an anhydrous state. But this common opinion is founded on an error, which to_ me is quite incomprehensible. Aither is obtained even from a mixture of the hydrate of sulphuric acid and anhydrous alcohol, * See Poggendorfi’s Annalen, vol. xx. p. 46]. 4 HEINRICH ROSE GN A THERIFICATION. S77 ' when distilled in a water-bath, at a temperature which need not | always amount to the boiling heat of water. It is not indeed requisite to employ anhydrous alcohol, but the hydrated, of 90 per cent. Tralles*, to obtain zther from a mixture at the above- mentioned temperature. M. Wittstock had the goodness, at my request, to institute a ) series of experiments on this point, and communicated the re- sults to me. _ I. Fifteen ounces of anhydrous alcohol were mixed in the cold, with an equal weight of the hydrate of sulphuric acid, and the mixture distilled at a temperature at which it could not boil strongly. The products, well cooled, were successively received, and the temperature at which they passed over accurately ob- served. First product: 1 dr. 10 grs., spec. er 0°817, 5 passed over at from . . ty $4) 08 16060807 R, Second product: 3 oz. 1 dr. 10 onl spec. gr. 0°792, passed over at from . . . . 90° ,, 93°,, Third product: 3 drs. 57 grs., spec. gr. 0°772, passed over at fiom SU it TR eBOP S Fourth product: 2 oz. 40 grs., spec. “ah 0749; passed over at from’): . 9. . .* 90° ,; 95°,; Fifth product: 5 drs. When the mixture had reached the temperature of 90° it Began to boil very slightly ; the boiling, however, subsequently eased at this temperature, but even then zther was disengaged from the mixture in bubbles, just as carbonic acid gas escapes at the common temperature from a liquid strongly saturated with it. | From these experiments it is evident that ether is formed jat far lower temperatures than is usually supposed. The first product smelt indeed strongly of zther; but chiefly con- sisted, which is also indicated by the specific gravity, of alcohol, which had not been converted, by mixing with sulphuric acid, into sulphovinic acid; zther could not be separated from it, either by water or even by chloride of calcium. The second, ird, and fourth products consisted, on the contrary, principally of zther, which could even be separated by mere washing with * That is, 90 per cent. absolute alcohol by volume; when in Germany it is reckoned by weight, Richter’s scale is employed. In Prussia alechulometers after Tralles are employed by the Excise. VOL. Il. PART VII. 2.6 378 HEINRICH ROSE ON ATHERIFICATION. water. The fifth was the first that contained free water, and — indeed, in volume, more than the half. The specific gravity of ~ the ethereal liquid floating above it was not determined. This last product distilled over very slowly, although at times the | temperature was raised to 100° R. It results from these experiments that ather which is pro- _ duced at lower temperatures than is requisite to boil the mix- — ture, is at the same time purer, and contains less alcohol and — water than ether which has been prepared by strong boiling. — A comparison of the specific gravities with those previously — mentioned, set this evidently beyond all doubt. At a low temperature the water especially escapes later, and therefore — only in the last product could separated water be observed, a— proof that it is not disengaged in company with the zther. II. A second series of experiments proved this in a still more _ decided manner, so that there can no longer remain any doubt — on the subject that ether can be evolved in abundant quantity § at the boiling-point of water. Seventeen ounces of anhydrous alcohol of specific gravity ; 0°792 were mixed cold with 18 ounces of the hydrate of sul- — phuric acid, and the mixture subjected to distillation in a water- bath whose temperature frequently did not even attain that of © boiling water. The quantities taken are in the proportion of | single equivalents of each of the substances employed; they | were taken in this proportion, partly because it approaches — that which otherwise is employed in the preparation of ather, when equal parts by weight of alcohol and sulphuric acid are | employed, and also in order to have no excess of sulphuric | acid. The results of the experiments are as follow: First product: 3 drs. Second product: 3 oz. 6 drs.; spec. gr. 0°755. Third product: 3 drs.; spec. gr. 0°745. Fourth product. Even the first product consisted of nearly pure ether; for a slowly in the water-bath, that several hours were necessary to | obtain a few drachms of it. From the specific gravities it will HEINRICH ROSE ON &THERIFICATION. 379 be perceived that the second, and especially the third product, consisted of zther far more pure than is obtained in other modes of preparing that substance. III. As the idea is so general, that «ther is formed from a mixture of alcohol and sulphuric acid only on boiling, and as in the usual mode of distilling, hydrated, and not anhydrous alcohol is employed, a new series of experiments were performed with the former. A pound of alcohol of 90° Tralles, such as is usually employed in the preparation of zther, was mixed in the cold with a pound of the hydrate of sulphuric acid, and the mixture subjected to }) distillation in a water-bath, as in the second series of experi- “ments. The results were: First product: 4 drs. 36 grs.3 spec. gr. 0°833. ~~ Second product : 2.02. 4 drs. 20 grs.; spec. gr. 0°787. Third product: 4 drs. 50 grs.; spec. gr. 0°789. Fourth product: 5 drs. 17 grs.3; spec. gr. 0°789. Fifth product. _ The first product consisted almost entirely of alcohol, as in- dicated by the specific gravity. The succeeding ones contained much ether, or consisted mostly of it. Free water also was evident in this case only in the fifth and last product, which consisted of 1 drachm of liquid, of which only one fourth was separated water. To distil this small quantity over, it was ne- ssary to heat for more than five hours. ' The ether obtained from a mixture of sulphuric acid and al- cohol, at the temperature of boiling water, is far more pure, as ‘may be anticipated, and is indicated by the specific gravities of the products, when anhydrous, instead of hydrated, alcohol is employed. The ether obtained from hydrated alcohol in this way contains more alcohol, because upon mixing hydrated aleohol with sulphuric acid, less is converted into sulphovinic acid, and more remains in a free state in the mixture, than when absolute alcohol is used. According, however, to the theory advanced in this memoir, only that portion of the coal- hol can produce zther which has been converted into sulpho- Vinie acid, and this ether distils over when heated, in com- ‘pany with the free alcohol. _ The fact that «ther is produced from a mixture of alcohol and sulphuric acid even at the boiling-point of water, is indeed highly important in the theory of the formation of zther, and 2c2 380. HEINRICH ROSE ON ATHERIFICATION. by this method the ather is also obtained more pure, especially — from water, and of a far lower specific gravity than when distilled — at a boiling heat ; but it is not convenient in the preparation of — zther, in so far as at this low temperature the zther, and par- ticularly the last products, pass over with great slowness. One fact, however, seems not to admit of being quite satis- factorily explained by the present theory. Seeing that water acts as a base upon the oxide of ethyl, and disengages it from its combinations, it must appear surprising that stronger bases than water do not effect this separation still more per- fectly. But solutions of the sulphovinate of potash and soda may be treated with an excess of potash without the oxide of zthyl being expelled; and even the salts of the alkaline earths can exist in contact with an excess of base. But there seems to be a difference in properties between the double compound of the hydrate of sulphuric acid with the sul- phate of the oxide of zthyl and the other sulphovinates. The former is far easier decomposed by water than the latter ; but this fact is by no means without analogy. Water is able to de- compose many salts of the oxide of antimony, and displace the latter from these combinations as a basic salt; but the combina- tions of the oxide of antimony with tartaric acid, and other non- volatile organic acids, are not decomposed by water. According to the earlier method in use, ether was obtained from a mixture of equal parts, by weight, of sulphuric acid and alcohol; here there is more alcohol at the commencement than is requisite. In the progress of the distillation, however, the quantity of sulphuric acid becomes constantly predominating, in proportion as the alcohol passes over as zther; and from the great excess of the hydrate of sulphuric acid, the liberated wether is itself decomposed by the boiling, which in this case takes place at a high temperature, and is then first converted into a double compound of the sulphate of the oxide of zthyl with sulphate of atherol (oleum vini); and lastly changed by the boiling into olefiant gas, from the presence of too great a quantity of the hydrate of sulphuric acid, and from too high a temperature. This change of ether into oil of wine and olefiant gas, by an excess of sulphuric acid and too high a temperature, is not the result of a mere deprivation of water, as might be concluded from | a comparison of the composition of these substances with that of cS HEINRICH ROSE ON 2 THERIFICATION. 381 ather ; for 4s soon as the slightest trace of oil of wine is evident in the formation of the «ther, a corresponding trace of sulphur- ous acid is disengaged, the quantity of which becomes more considerable if olefiant gas is formed. The production of sul- phurous acid stands therefore in definite connexion with that of the oil of wine and olefiant gas. Since the origin of these two bodies takes place only at a high temperature, especially that of the olefiant gas, these substances undoubtedly owe their origin to a similar action of sulphuric acid on ether, as this acid exerts on other bodies of organic origin at high temperatures. The sulphuric acid is coloured black by these, at the high tem- ') perature, with the evolution of sulphurous acid and separation _ of a carbonaceous substance; the same also takes place in the - distillation of «ther, when continued to the production of oil of ‘| wine and olefiant gas. The origin of this coally matter, which has recently been ) examined by Erdmann and Lose*, stands therefore in connexion with that of the sulphurous acid, oil of wine, and olefiant gas ; consequently the formation of this body is the result of another '| process, which very likely has nothing to do with the formation of the ether. When therefore ther is prepared from a mixture of sulphuric acid and alcohol at a very low temperature, it is perfectly free |) from oil of wine ; and, in fact, not a trace of that substance could | “De observed in the first products which were obtained by the 1) above distillations, not only in those that were performed in the ‘| water-bath, but also in those which were carried on at a gentle heat in the sand-bath. Even the last products appeared to be _ perfectly free from it ; but if a considerable quantity of the zthe- real liquid was evaporated on blotting-paper, a very slight smell of it might be discovered, a trace however so insignificant, that individuals not well acquainted with the odour of oil of wine could not perceive it. Morcover, when the distillation was at an end, the residuum in the retort was, it is true, of a dark colour, but not deep (foncé), so that it resembled a brownish vitriol, such as frequently occurs in commerce ; the residue smelt as slightly of sulphurous acid as the distilled ather did of oil of wine. Not a trace of carbonaceous substance was separated. ‘The process by which oil of wine is produced, commences, * Poggendorft’s Annalen, vol. xlvii. p. 619. 388 HEINRICH ROSE ON AATHERIFICATION. therefore, in the mixture prepared for the distillation of zether, even at the boiling-point of water, at least when this is long continued ; but even then the formation of this body at that temperature is quite trifling in amount. When ether is distilled from a mixture of sulphuric acid and alcohol in the water-bath, we obtain, as is evident from the above results, less «ther than we might expect from the quan- tity of alcohol employed, and the residue weighs more in pro- portion. In the last series of experiments described, in which ether was prepared in the water-bath, the residuum, on em- ploying 17 ounces of absolute alcohol and 18 ounces of sulphu- ric acid, weighed 27 ounces, and the distilled alcoholic ether 41 ounces ; the loss consisted partly in the water distilled, the quantity of which was not determined, in volatilized ether, which in this case volatilized the more, as it was nearly pure, and also in the loss which occurs by pouring out. On employ- ing 1 pound of hydrated alcohol and 1 pound of sulphuric acid, the residuum weighed 263 ounces, the products 4 ounces and some drachms ; the loss consisted partly in the water which passed over, the quantity of which was not accurately deter- mined. In both cases therefore, besides water, ather also re- mained with the sulphuric acid, undoubtedly as iszethionic acid, probably also in part as zthionic acid. It is very probable that the products which present themselves with zther in a distilla- tion when long continued and at high temperature, are produced, not by the direct decomposition of the ether, but by the de- composition of the isaethionic acid, occasioned by the excess of sulphuric acid and a high temperature ; such as the precipitated carbonaceous substance, the sulphurous acid, oil of wine, and lastly, the olefiant gas. It is well known that the formation of these products is generally avoided in the preparation of zther by the new and most profitable method, in which, as zther passes over, a like quantity of alcohol is allowed to flow into the boiling mixture. The action of an excess of sulphuric acid on the alcohol, or rather on the iszthionic acid, at a high temperature, is thus prevented. A When formerly the production of zther was sought to be ex- plained by the subtraction of the water from it, by means of sulphuric acid, it might with much justice be objected to the present explanation, that other bodies, which have, like sulphu- HEINRICH ROSE ON ATHERIFICATION. 383 ric acid, a great affinity to water, such as the hydrate of potash, chloride of calcium, &c., are not able to transform alcohol into zether ; but this objection now falls entirely to the ground, as we know that the ether is not formed by any subtraction of water, but by the decomposition of the sulphovinic acid. If «ther is regarded as a base, then all the theories on the formation of ether are not capable of satisfactorily explaining how a base is discharged from a strongly acid liquid, and by a powerful acid. It is only by the present explanation, and by the analogy which the separation of ether from sulphovinic acid bears to the decomposition of several inorganic salts by “means of water, and also by the above-mentioned analogy of ether with a series of oxides which do not, or to a very slight extent, combine with acids, that this phenomenon loses se “anomalous appearance. It seems to me highly desirable in organic chemistry, to illus- trate its processes always as much as possible by analogous processes in inorganic chemistry. The greatest advantages have accrued to organic chemistry by the endeavours of Berzelius, Liebig, and atest who have pursued this path, frequently starting, it is true, from very different views. It is certainly advantageous i in so imperfect a science as che- ‘mistry, and especially organic chemistry, to ascribe provisionally ‘to a common force all phenomena which stand isolated, for which no suitable analogues can be detected, and which on this ‘account appear wonderful, and thus openly to admit that in the present state of science it is better to avoid explaining a process altogether, than to explain it by some artifice or in a constrained er. The smaller the number of phenomena which we are compelled to refer to this class, the more perfect the science be- comes. Setting out from this point of view, I have ventured to ex- plain a process in organic chemistry, which has long, and parti- cularly of late years, engaged the attention of chemists, as being analogous to several processes in inorganic chemistry; and if the explanation should not give general satisfaction, the attempt to attain so important an object, will, I trust, meet with appro- ion. . e present theory is valid, it is true, only for the formation of thes from a mixture of alcohol and sulphuric acid; but | quite a similar one may undoubtedly be advanced for the forma- 384 HEINRICH ROSE ON 2 THERIFICATION. tion of ether from mixtures of phosphoric and arsenic acids with alcohol. For the present, however, I leave it undecided whe- ther the formation of «ther, by treating alcohol with fluo-— boracic gas, as also with the chloride of zine and other chlo- rides, is to be explained by a mere subtraction of water by these substances; or in this way, that they form with alcohol, at the common temperature, combinations analogous to sulpho- vinic acid, which are decomposed like it, at a high temperature, by the agency of water. The latter view I regard as being the most probable. PostTscriptT*. In the preceding Memoir I have compared the formation of wether from a mixture of sulphuric acid and alcohol, with the decomposition of several inorganic salts by means of water; I have endeavoured to show that it is the water which in these cases acts the part of a base, and separates the oxide of «ethyl or the metallic oxide, the latter generally as basic salt. The inorganic salts which I enumerated in this comparison as examples, were those of the oxide of bismuth, the oxide of mercury, and of antimony. These undergo the said decompo- sition by water even at the common temperature; zther, how- ever, is first separated from a mixture of sulphuric acid and al- cohol, or from sulphovinic acid, at a high temperature. There are, however, among the inorganic weak bases, a con- siderable number which are eliminated by water, from their combinations with acids only at a high temperature; and the decomposition of the salts of these bases, by means of water, is therefore still more fit to be compared to the formation of zther. To these bases belongs more especially the peroxide of iron, which is precipitated by water as basic salt from solutions of most of its neutral salts at a high temperature. The weaker the solution of the salt of peroxide of iron, the lower is the tem- perature which occasions precipitation, and the more completely * The present Postscript appeared in the following part of Poggendorff’s: Annalen, under the title ‘On the precipitation of some metallic oxides by water.” - HEINRICH ROSE ON ATHERIFICATION. 385 is the peroxide of iron thrown down, so that with a certain dilu- | tion, as M. Scheerer has shown*, scarcely a trace of the per- _ oxide of iron remains in solution, but the entire quantity is sepa- | rated as basic salt. As stronger bases are not precipitated by water on boiling, this property of the peroxide of iron has been employed to separate it from the oxides of cobalt, nickel, and other metalst. It may even be separated, by boiling the solu- tion, from alumina, which, although it has with regard to its properties much similarity to the peroxide of iron, is evidently a stronger base; this separation of alumina from the peroxide of iron by means of water at a high temperature, is of some importance to the arts, as in the fabrication of alum the per- _ oxide of iron contained in the mother-liquor is precipitated by cre boiling, and is thus more easy to separate from the alumina _ than the protoxide of iron, although the former, with sulphuric acid and an alkali, forms an alum which has quite an analogous composition with alumina-alum ; and, from being isomorphous with that alum, could crystallize with it in all proportions. Several other bases have the same property as the peroxide of iron, which like it belong to the class of weaker bases, and also several substances which act as bases towards strong acids, and also as acids towards strong bases, and which on that account are frequently classed among the acids. Among these are the _ oxide of zirconium, thorina, the peroxide of cerium, peroxide of _ tin, titanic acid, tellurous acid, columbic acid; also in certain _ respects molybdic acid, tungstic acid, and vanadic acid. Several ‘combinations of these oxides with acids are soluble in the cold “in water, and are precipitated from the solution, on boiling, as oxides or basic salts. _ Several of the oxides precipitated in this manner possess, after precipitation by boiling, properties which they do not evince before their solution in acids and precipitation; they are more indifferent than before, are partly of difficult solution in acids, partly insoluble, and do not combine after precipitation with them, even when these are employed in a concentrated state. Titanic acid, peroxide of tin, and many others may be classed here. This peculiarity is in a certain degree ana- ¥ ma Poggendorff’s Awnalen, vol. xliv. p. 453. [or Lond. and Edinb. Phi o vl 3 lepine Pp [ and Edinb, Phil. Mag., + Scheerer in Poggendorff’s Annalen, vol. xlii. p. 104. for Lond. ; i Phil. Mag., vol. xvi. p- 131.—Eorr. j F Seine 386 HEINRICH ROSE ON ATHERIFICATION. logous to that of «ther, which, when it has been once separate by boiling from a mixture containing sulphovinic acid, appears not to combine directly with acids. NOTE. Our readers will be able to judge how far the theory of ztherification, sup- ported by so much research in the foregoing Memoir, coincides with that previously announced by Professor Graham, in Part II. of his Elements of Chemistry published in 1838, by the following extract from the latter work. Under the head of “ Circumstances which affect the order of decomposition,” the alternate displacement of zether and water by each other, as bases, is announced and described by Mr. Graham in the following terms :— “‘ The remarkable decomposition of alcohol by sulphuric acid, which affords zther, is another similar illustration of decomposition depending upon volati- lity, and affected by changes in the nature of the atmosphere into which eva- poration takes place. Alcohol or the hydrate of «ther is added in a gradual manner to sulphuric acid somewhat diluted, and heated to 280°. In these cir- cumstances, the double sulphate of «ther and water is formed; water, which was previously combined as base to the acid, being displaced by wther, and evolved together with the water of the aleohol. ‘The first effect of the reaction therefore, is the disengagement of watery vapour, and the creation of an atmo- sphere of that substance which tends to check its farther evolution. But the existence of such an atmosphere offers a facility for the evaporation of ether, which accordingly escapes from combination with the acid and continues to be replaced by water, the affinity of sulphuric acid for water and for ether being nearly equal, till «ether forms such a proportion of the gaseous atmosphere as to check its own evolution, and to favour the evolution of watery vapour. Then again alcohol is decomposed, and more of the double sulphate of water and zther formed as at first; the sulphate of «ther of which comes in its turn to be decomposed as before, and zther evolved. Hence, both zether and water distil over in this process, the evolution of one of these bodies favouring the separa- tion and disengagement of the other. In this description, the evolution of water and ether are for the sake of perspicuity supposed to alternate, but it is evident that the result of such an action will be the simultaneous evolution of the two vapours in a certain constant relation to each other.” p. 188. 387 Articte XII. Determination of the Axes of the Elliptic Spheroid of Revolu- tion which most nearly corresponds with the existing Measure- _ ments of Arcs of the Meridian. By F. W. Besseu. [From the Astronomische Nachrichten, No. 333.] THE observed latitudes of points on the earth’s surtace, and the distances between the parallels on which those points are situated, have a relation which would be given by a knowledge of the figure of the earth. If the equation of the earth’s surface were known, we should be enabled to determine the constants, whereby the measured distances between the parallels and the observed latitudes of the parallels would be brought into accord within the limits of the errors of observation. But the figure of the earth is not known,—or, rather, we know that it is irregular. There is, however, an elliptical spheroid of revolution, the sur- face of which is not far removed from the surface of the earth ‘at any point; but whether at all points of the respective sur- faces this distance may be regarded as a small quantity, com- pared with the ellipticity of the spheroid, is a question yet to be decided by the combination of several measurements of arcs. In the mean time we may make progress in the inquiry by deter- mining the axes of the spheroid which would most nearly repre- sent the existing measurements. If we regard the deviations of the surface of the earth from the surface of the spheroid as following no definite law, their influence on the latitudes is com- bined with that of the errors of observation of the latitudes, and we must consider that spheroid to be the one sought for, which brings the measured distances between the parallels in corre- spondence with the latitudes, by correcting the observed Jati- tudes in accordance with the conditions of the method of least squares. _ Walbeck first commenced the investigation upon this correct view, but took into account only the most southern and the most northern points of each measured arc, omitting in his calculation the intervening astronomically determined points. Schmidt improved on the earlier calculation, not only by giving proper weight to all the observed latitudes, but also by taking 388 M. BESSEL ON THE ELLIPTIC SPHEROID OF REVOLUTION into consideration other measurements of degrees which had — been made known in the interim. I return again to the same — subject, partly because Schmidt employed several data which | appear to me incorrect, partly because I am enabled to avail , myself of three additional measurements of arcs. I am in- debted for the knowledge of the first to manuscript communi- cations from General von Tenner, who has executed an under- taking of his own of this kind, and has connected it with the northern extremity of von Struve’s arc, so that the two together give the measurement of an arc of the meridian of 8° 2! 29", I owe the second to manuscript communications of Schumacher, whose measurement includes 1° 31' 53”. The third, extending over 1° 30! 29", has been executed by myself, conjointly with Major Baeyer, in the district of Kénigsberg: as this is the first public notice of it, I may remark that its more immediate — object was to unite the arcs which have been measured in the South and West of Europe with those which have been and will be executed in the North and East; so that a connected chain of triangulation, comprehending the principal European obser- vatories, may extend from Formentera to Finland. The mea- surement of an arc was combined with this more immediate object, by comparing the latitude of the most southern and most northern points of the triangulation with the latitude of the observatory of Kénigsberg. Oy la I will first give the data on which the calculation is based, and the sources from whence they are taken. 1. Peruvian Are. Distance Latitude. Amplitude. | between the Parallels. o 74 “s ang Ute ss - = > — 3 4 32-068 T Cotchesqui..... +0 2 31:387 3 7 3-455 176875°5 4 These data rest on the new reductions of the observations by f Delambre and von Zach. Delambre, in the Base du Sj yt Meétr. IIL. p. 133, gives the latitudes — 3° 4! 31"-9 and + 0° 2! 31"-29, making the amplitude 3° 7/312. Von Zach finds the spill REPRESENTING MEASURED ARCS OF THE MERIDIAN. 389 tude 3° 7! 3-79 (Mon. Corresp. xxvi. p. 52). I have taken the mean of these as the amplitude, and have altered Delambre’s latitudes only so much as to bring them into accord with this mean. The distance between the parallels of the two points is found by Delambre = 176877 toises; by von Zach = 176874". The values employed by Schmidt differ considerably from the above ; the amplitude being greater by 5'*205, and the distance less by 9733. . 2. First East Indian Arc. Ti 44 52°590 13 19 49-018 Trivandeporum . AMONG) 5:0 s9's 0 \0 is} ae Tt 1 34 56-428 | 9813-01 The account of this measurement is given in the Asiatic Re- searches, vol. vill. p. 137. The distance is given by Lambton himself = 95721°32 fathoms. But Kater’s examination of the standard scale on which the measurement rests, shows that a correction of — 0°000018 must be applied, in order to reduce it to true English measure. The distance thus corrected is = 95719°60 fathoms, which gives the number of toises in the proportion of 1:06576542 to 1. 3. Second East Indian Are. fe} ‘ “4 PUTING N3 4h 0.6) 6! 8 9 31-132 enh z Ps T Putchapolliam ..}| 10 59 42:276 | 2 50 11-144 | 160944-20 Dodagoontah ...| 12 59 52-165 | 4 50 21-033 | 274694-:30 Namthabad ....| 15 5 53:562 | 6 56 22-430 | 393828-09 Daumeragidda..| 18 3 16:245 | 9 53 45-113 | 561690-06 Takal K’hera ..| 21 5 51:582 | 12 56 20-400 | 734570-43 Kullianpoor....| 24 7 11:860 | 15 57 40-728 | 906171-67 A part of this great undertaking is described in the Asiatic Researches, vols. x., xil., xill., and another part in Colonel Ever- est’s account of the measurement of an arc of the meridian, London 1830. It has appeared to me necessary to subject the observations with the zenith sector for determining the latitude to a fresh calculation, which I shall publish in a separate me- moir. The data as above are the results of this calculation. The original observations are to be found: for Punnae, Asiat. Res. xii. p. 68 ; for Putchapolliam, xii. p- 61; for Dodagoontah, X. p- 356; for Namthabad, xii. p. 339; for Daumeragidda, xiii. p- 83; and for Takal K’hera and Kullianpoor, in Everest’s Account, &c., pp. 287 and 306. Pages 112—114 of the latter 390 M. BESSEL ON THE ELLIPTIC SPHEROID OF REVOLUTION work contain the distances between the parallels of the astrono- mically determined points from whence I have deduced Punnae Putchapolliam . . . 17152876 fathoms. Dodagoontah, . . . 292759°68) ‘+, Namthabad . . . . 41972836 ,, Daumeragidda . . . 598629°84_ _,, Takal K’*hera . . . 4782879°76 os Kullianpoor®. «>. '. 965766°43. —” ,, The proportion of the toise to the fathom has been given above (2). 4. French Arc. ‘ “ Formentera ....| 38 39 56-11 My OL T Montjouy ..... 41 21 44:96 | 2 41 48-85 | 153605-77 Barcelona ..... 4) 22 47:90 2 42 51-79 | 1545489 Carcassonne....| 438 12 54°30 4 32 58°19 | 259104-8 Bivank, on) vexaiee's 46 10 42°54 7 30 46:43 | 427951:5 Pantheon. 0...) 48 50 49°37 | 10 10 53-26 | 580244-6 Dunkirk ...... 51 2 8:85 | 12 22 12-74 | 705189-4 The distances of the parallels of the different points from the most southern point are given, with the exception of Barcelona, in the Base du Syst. Métr. iii. p. 549. The spot in Barcelona where the astronomical observations were made is 943713 north of Montjouy (i. p. 565). The latitudes of Formentera, Car- cassonne, Eyaux, and the Pantheon, are to be found in pages 89 and 459, vol. iii.; in page 89 the latitude of Montjouy is also given. That of Barcelona is (ii. pp. 565 and 615) = 41° 21! 48"-37 + 5953. For Dunkirk I have taken the result in iii. p- 548. I have followed Delambre’s example in leaving out Perpignan, because the observations of the latitude at that place seem less certain than the others. The latitudes taken by Schmidt differ from the above; at Montjouy + 0'"49; at Barce- lona—O"-74 ; Carcassonne + 0""01 ; Eyvaux-—0'"35 ; the Pantheon — 043 ; Dunkirk — 0"-11. He places the parallel of Barcelona — 57-9 more to the north than is done here. . 5. English Are. te} ‘“ “ Dunnose .....- 50 387 76338 | 5, oy T Greenwich...-. 51 28 39-000 | 0 51 31367 | 49059-89 Blenheim ..... 51 50 27-652 | 1 13 19-999 | 69829-19 Arbury Hill....| 52 13 28-031 | 1 86 20398 | 9169639 a Clifton. $.,...2' +s. 53 27 31°130 | 2 50 28-497 | 162075-93 a REPRESENTING MEASURED ARCS OF THE MERIDIAN. 391 These latitudes differ from those given in the Phil. Trans. for 1803. They result from a new combination of the reductions made by General Mudge of his own observations. More par- ticulars on this point, and the reasons which have obliged me to introduce alterations in General Mudge’s own data, will be con- tained in a separate memoir. The distances of the different parallels from Dunnose, were originally given in the Phil. Trans. 1803, pp. 441 and 487, as follows: SareenWich «. 00% jc: «:-« -92282°67 fathoms, gt a Or Ngee 2 2 et: ee OUR Y TAN scc 5) adie 079772000. «sy ements: gS 3), SORE MESA I OSB. mie From Kater’s examination of the scale employed in the mea- surement, these distances require to be multiplied by 0°00007, and will be thereby respectively augmented 4 3°66 5 5°21; 6°84; 12°09. ‘The distances given above in toises correspond to the distances in fathoms thus augmented. 6. Hanoverian Arc. Gottingen ..... 31 47-85 TBO o 5\'o.0 <7 4551.0 32 45°27 OD) iv “ T 2 0 57-42 | 115163-725 Taken from Gauss’s Breitenunterscheid, &c., p. 71. 7. Danish Arc. On 6 “ 53 22 17-046 54 54 10352 Lauenburg..... Lysabbel .....- 1 31 53-306 T 87436-5385 These results have been communicated to me by Schumacher. They might be combined with those of the preceding measure- ment, if it were not that the latitudes of the two arcs rest on different stars, which would render the combination dependent on the determinations of the declinations of these stars. I think it right to avoid the danger of introducing error into undertak- ings of such distinguished exactness, by bringing in a foreign element, the more so as Lauenburg, which is 9860™46 south of Altona, is 21031751 to the east of the same, forming an angle of nearly 65° with the meridian. The distance of its parallel from that of Altona could not therefore be found with the exactness which is attainable when the inclination to the meridian is less. 392 M. BESSEL ON THE ELLIPTIC SPHEROID OF REVOLUTION 8. Prussian Arc. Tiare chase 54 13 11°466 T Kénigsberg ....| 54 42 50-500 | 6 39 39°034 | 28211-629 Memel ......- 55 43 40-446 | 1 30 28-980 | 86176-975 A memoir on this arc is now in the press. 9. Russian Are. Berlin coke 52 2 40-864 x T Wemescli 4. ss, 54.39 4-519 | 2 36 23-655 | 148811-418 Jacobstadt ..... 56 80 4-562 | 4 27 23-698 | 254543-454 Bristen. ... 5 sats 56 84 51-550 | 4 32 10-686 | 259110-085 Dorpati te cic stetel 58 22 47-280 | 6 20 6-416 | 361824-461 Hochland ..... 60 5 9771 | 8 2 28-907 | 4593863-008 These numbers have been communicated to me by General von Tenner. Such of them as refer to Struve’s arc—i. e., those for Jacobstadt, Dorpat, and Hochland—agree with the values given in the treatise on the measurement of an arc of latitude in the Baltic provinces of Russia, i. pp. 312 and 338. 10. Swedish Are. 65 31 30-265 67 8 49-830 WHEN a pees Oe one Pahtawara..... T 92777981 1 37 1s 565 These are Swanberg’s data, p. 157 of his work. Schmidt has taken the amplitude as 0'"785 greater, and the distance be- tween the parallels as 177251 less. These differences have been occasioned by two remarks made by Swanberg in re- ducing the observations; first as to what the latitudes would have been, if the density of the air, and with it the refraction, had been assumed as dependent, not, as in the usual manner, on the height of the thermometer, but in a more complicated relation to the same, which certain experiments of Prony’s appeared to indicate; and, secondly, as to what the distance would have been inferred from the measurement, if it had been assumed that the double metre, sent from Paris, had its true length at — the normal temperature of the toise, viz. 13° Reaumur. As the doubts which gave rise to these two remarks have been fully re- moved, they ought not to be attended to. §. 2. ; The theory with which the ten above-mentioned arcs should — REPRESENTING MEASURED ARCS OF THE MERIDIAN. 393 be compared is as follows. If the two semi-axes of an elliptic spheroid of revolution be designated by a and 4, and if web © then the length of the arc of the meridian between the equator and the latitude ¢ is the integral gia 0-8) f outa a Taga erent or, developed, s=a (1—n)? (l+n)N{o—asin 29+ ¢' sn 49 —iel!sn6o+...} _ wherein : N =1 ar, 3. 3.5\? 4 , i TDF 2.4 nN” + oy 3 Bb. 08. cami g, BISUe ; alt 5 4a” T o.4ace Baa" fo . Eee EL ae ; oS a OW a aM Se a5 On) ges 3 bes ggg? tea ea De ot and so forth. If we desire to make this expression dependent on the length of the mean degree of the meridian (g) instead of on the greater _ semiaxis, we must put ¢ = 180°, whence we obtain + 180g = a (1—n)? (1+ 7) Nz, and thus a = 809 {¢—asn2¢+¢sn4¢—lae!sn6$9+..}. Hence follows the expression for the distance between the pa- rallels corresponding to the latitudes $ and 4’, - ¢—s= ae {¢'—o—2asin ($¢’ — ¢) cos (¢' + 4) 42a! sin 2 (¢'— ¢$) cos 2 (9'+ 9). If for brevity we write 7 for the amplitude ¢'— 9, and 2 L for the sum of the latitudes 4! and 4, and understand by w the number a = 2062648, and if we express / in seconds, _ we thence obtain 3 : ; (s! —s) =1—2wasin/7 cos 2 L+4+ we! sin 2] cos 4 L —Zwe' sin 3lcos6L +... VOL. 11. PART VII. 2nD 394 M. BESSEL ON THE ELLIPTIC SPHEROID OF REVOLUTION The problem requires that the observed latitudes 9, 4, ¢”... should be brought in correspondence with the measured di- stances between the parallels by the application of alterations 2, a, w!..., of which the sum of the squares a? + a? + al? shall be a minimum: the values of g and a, a’, a” ..., which fulfil this condition, belong to the elliptic spheroid of revolution sought. If we write ¢ +2, and $!+ x', for ¢ and ¢', and if we neglect the influence of the alterations on L as well as the squares and products of z and z', the above expression becomes 3600 (s) —s) =J—2wasinl cos2L+woa' sn2/cos4L—... (7 — where g stands for 1—2.«cos/cos2L+22! cos 2lcos4L—...; we have thus 3600 a — eu ef (s' —s) —(J—2 we sin/ cos2 L es sin 2%cos4L...)}, and we must now so determine g and the compression as to fulfil the above-named condition. If we take gy, and «, as the approximate values of g and a, make a : Ure Rae and if we neglect the squares and products of i and k, the ex- pression for 2! — x becomes ae —s)—1)- em sin 7cos2 L—a,'sin 2/cos4L+...), a=a, (1+h); Sel eA 1 3600 da! +-.— i+ —(2a,sin/cos2 L—a "1 sin2]cos4L+.. k, eg dit Se i ) ! in which e¢’, and e, a expressed by «, are respectively 5 : iecthap ton eu bee als and 7% pre 317 Then, if we make m= — a3 -1\ na o4 2m, sin 7 cos 2L g i it ne es un isa) Phra Re aN REPRESENTING MEASURED ARCS OF THE MERIDIAN. 395 1 _ 3600 a (2 — 8), oo 2 fon, sind cos 2 L— (Fa? rant) sin 21c0s41 | we have a—x=m+ai+t bk, and a similar equation for the combination of the southernmost point of an arc with each of the points to the north of it. The sum of the squares of the alterations to be applied to all the latitudes of an arc is thus: a+(m+ai+ bk+a)?4+(m'4+adi+0k4+ 2), &c.; for other arcs the sums are Wa? + (m, 40,646, k+4+ 0)? + (mi, +a,i4+ 0k 4m), &e, D a2 + (mz + a,i+ bok + %)2+m',+a'git+ Uk + x,)*, &e., &c., &c., &e., each of these gives thus for the determination of its own value of # the equation 0=pert (m+ (ait (Dh in which » is the number of the observed latitudes, and (m) (a) and (4) denote as in the usual notation of Gauss. It furnishes also towards the determination of 7 and 4, which must be found- ed on all the existing measured arcs, the following contributions ; (am) + (a) v+ (a*)t+ (ab) k, ¥ (bm) + (b) 2 + (ab) i+ (Bk, which, eliminating x, become (am - 9). fy -O Obs 4 fay -O OV, the sums of the first as well as of the second of these contribu- tions, so furnished by all the existing measured arcs, being made = 0, give the two equations necessary for the determination of i and k. Bios I will now communicate the several equations of condition which I have deduced from each of the ten arcs on which this examination is founded. My view in so doing is to obtain the advantage of being able to avail myself of any subsequent 9 D2? 9) a) 896 M. BESSEL ON THE ELLIPTIC SPHEROID OF REVOLUTION alterations of the results assumed as observed, for the purpose of correcting my calculation without being obliged to repeat it throughout. In order to avoid unnecessary multiplication of figures, I will assume the unknown quantities sought, instead of i and k, to be 10000i=p and 10k=q. I set out from the as- sumption that 570087 1+k a= 1+¢ ? 400 1. Peruvian Arc. a} — a= +1966 + 1°1225 p + 5°6059. g. 2. First East Indian Arc. Xo! — L, = + 0""937 + 0°5697 p + 2°5835 .¢. 3. Second East Indian Arc. u ag! — %3= +0°455+4+1°0212p+ 4:8270.q, ago — = + 6681 +1°7428 p+ 8:1250.9, #3 — %3= + 1°745 + 2°4983 p + 11°4652 . 9g, X51 — &,= + 3°878 + 3°5624 p + 15°9264.g, 2° — = + 8'272 + 46585 p + 20°1840.9, a6 —a#5= +2°677 + 5°7458 p + 24-0262 .q. 4. French Arc. ; ul x,! — x, = — 0297 + 09709 p + 0°8601 . g, ve — L,= —3°641 + 0'9768 p + 08642. g, vg — 0, = — 4259 + 1°6374 p + 1:1889.9, a} —a,= —9°319 + 2°7037 p + 1°2671.9, v4 — 2, = — 3°092 + 3°6651 p + 0°8659 . g, 28 —xv,= +0°889 + 4:4533 p + 0°2051 . p. 5. English Are. u as! — = + 3°504 + 0°3095 p — 0'3178 .g, Ls — #,= + 4937 + 0°4405 p — 0°4658 . 9, a? —x@,= + 3°758 + 0°5784 p — 0°6308 .q, rs — #,= —0°892 + 1°0223 p — 1°2226.9. 6. Hanoverian Are. Xe —Xg= + 5679 + 0°7263 p — 0°9294 . g. 7. Danish Are. 2) — x. = —0!"369 + 0°5513 p — 0°8537 .g. REPRESENTING MEASURED ARCS OF THE MERIDIAN. 397 8. Prussian Arc. ul Lq1 — @, = —0°368 + 0°1779 p — 0°2852..q, Lg — , = + 3°790 4+ 0°5433 p — 0°9157 . g. 9. Russian Arc. ui Lo! — Ly = + 0'248 + 0°9384 p— 1°3293 .Q, Ly — %y= + 5°110 + 1°6049 p— 2°5184. gq, oo + 5°939 + 1°6337 p — 2°5741.4q, 24! — &y = + 2°909 + 2°2809 p — 3°9289.q, Ly — Ly = + 5°276 + 2°8953 p — 5°3824.q. 10. Swedish Arc. 2! 19 — £19 = O" 507 +. 0°5839 p —1°9711. 9. From these equations of condition I have obtained for each of _ the measured arcs the sums marked (m), (a), (6), (am), (aa), &e. (a) (2) (am) (aa) (ab) (bm) (bb) V4 19664 1-1225\4 5°6059\4+ 2-2068) 1:2600/4+ 6-2926|/4+ 11-0211) 31-4261 0:5697|4- 2-5835)+ 0°5338] 0:3246/+ 1:4718/4+ 2-4207 6°6745 19-2290'+ 84-5538) 4 84-1994) 77-7283) + 336°5465|4 369-5289, 1459-0687 14-4072|4 5-2513)— 43:3870, 45:1527/+ 11:1889/— 22-7680) 5:2976 2:3507 2-6370 + 4:5209) 1:6694;— 1:9183)— 4-6932 2°2105 0:7263 0-9294|+ 4:1247) 0:5275|— 0- 6750) — 5°2780) 0°8638 0°5513 0°38537|— 0:2034 0-3039 — 0°4706 ite 03150 0:7288 9°3532|— 15-7331/+ 40: 0469 19:7106)— 34:0396)— 68-3130) 59-1418 0:7212 fa 1:2009)+ 1:9986| 0-3268/— 0- 5482|— 33655 0:9198 0°5839|— 1:9711|\— 0 2960, 0°3409|— 1 1509)+ 0°9994 3°8852 “Aer the elimination of 7,72, #3, we have the data furnished by different arcs for the equations serving to determine p and gq. 9°5188 | — 17-2270 17-8868 05755 | + 0°4997 1-9426 9-6768 51302 01480 0-1705 _ | (am) (a a) (ay) (b m,) (b 5) a + 1:1034 0:-6300 | + 31463] + 5-5106 15-7131 2 | + 0-2669 01623 | + 07359 | + 1-2104 3°3373 3 + 19:0734 | 24-8940 | + 104-2771 | + 83-1572 | 437-7342 4 — 28019 | 155002 | + 03308 | — 7-9757 1-3582 5 — 0°7950 05642 | — 06785 | + 1-2701 90-8197 6 + 2-0624 0-2638 | — 0:3875 | — 2-6390 04319 7 — 01017 01519 | — 02853 | + 0-1575 0°3644 8 + 1:1710 0:15384 | — 0°2595 | — 1-9957 0-4391 9 + — 0 eae oe Sums| + 29-5073 47-6205 + 96° 58900. + 61: 9681. 4800273 Thus we have for the determination of p and qg the equations 0 = + 29°5073 + 47°6205 p + 96°8900 g, 0 = + 61-9681 + 96°8900 p + 480:0273 g, 398 M.BESSEL ON THE ELLIPTIC SPHEROID OF REVOLUTION from the solution of which we obtain p=—0°60574; Weight = 28-064 q = — 0'0068280; yo = 282°892, §. 4. If we compare the several observed latitudes with this deter- mination of p and g, we obtain the alterations required to make them agree with the elliptic spheroid of revolution, to which the found values of those quantities belong. 1 “= 0-624 «L, =— 1-980 ay! = + 0°624 a5! = + 1°338 ve = +2°793 Ly = —0:287 ae= + 1°432 cs =<} 0:987 oe = — 3°483 @, = — 1°640 x® = — 2623 ag! = — 1837 tg = + 2°623 2 = + 3°929 x;° = — 1487 X, = + 0°349 &5'= —0'029 x, = — 0°349 r= + 3°672 v8 = — 2608 2, = —0°998 a! = —1°472 ty = +4069 arg = + 2469 z,! = +3-178 v2 = —0'170 vy = —2°321 2,3 = —1:190 Lo = — 2'632 av = —6'897 ag? = + 1°834 ae = —1:249 v3 + 2646 aS = +2259 zt —0°766 to + 1°238 ayy = + 0°424 io! = —0°424 . The sum of the squares of these alterations is = 203°391, and the mean value of each of them 4 sp BOBO ae k = V 55g = £21695. | From this determination, and from the above given weights of the determinations of p and g, the mean errors of these quanti ties are = +0°5087 and = + 0°1602, or the mean errors of i = + 0:00005087, of k = + 0°01602; REPRESENTING MEASURED ARCS OF THE MERIDIAN. 399 thus we have M. error 57008 eS eee ee ns UP eet SyO1 {P45S Sp g- I = 7—0-000060574 7011453 . + 27-900 —=( 2 pe 2— 010006828 = 0:002498293 . -+ 000004002 400 Seeds I have further only to seek for the two axes of the elliptic spheroid of revolution, and to develop the numerical values be- longing to it of some formule which are of frequent application. The reversion of the series (§. 2) 34 45,34 5% 2 16 128 Ltn + nt te. gives the expression of m in terms of a, namely Te Tee Se 2 Ling 93 ies eg @ +—a@4...,, 9 486 and by substitution of the found values of a, n = “= — 0.0016655304. a+6 Hence we obtain the proportion between the axes of the elliptic spheroid of revolution, which corresponds most nearly to the ares which are under examination, hs a : ae — = = 300°7047 : 299°7047 ; mean error + 4°81. Further, we obtain the axes themselves according to the for- mulz given in §. 2. |] w]e i 180g ~ r(l—n) (1 +n)N 180g a(1 +n)? (1—n) N°? or, numerically expressed, a= 3271953°854 log a = 6. 5148071699 = 3261072:900 log 6 = 6 . 5133605073. The length of the quadrant of the meridian, which according to the original view, ought to be 10,000,000 metres, is according to this determination, 864 Se m+947 90 9 « 73-996 = 10000565™278. 400 M. BESSEL ON THE ELLIPTIC SPHEROID OF REVOLUTION. | Its mean uncertainty is = 508™-7, which is almost equal to its - difference from the round number. Hence we see how uncer- — tain the length of the metre would have been, even now that the — number of measured arcs has been considerably augmented, if its original definition as the 10,000,000th part of the quadrant of the meridian had been adhered to. Its uncertainty would | still amount to at least 0%0225, a quantity which could only be — deemed insignificant in very rough measurements. g The formule of which I have to give the numerical develop-_ ment are the following : j 1. The length of a degree of the meridian of which the mean | latitude = 9: | m= 570117453 —2847-851cos2 > + 07:593.cos4¢—O?001cos6¢. 2. The length of a degree of the parallel : p=571537'885 cos ¢6—477576 cos 3 ¢+07-059 cos 5 4, or if sin) = e sin $,... (log e = 89110835), then log p = 4°7566845'4 + log cos ¢ — log cos tp. 3. Let the radius of curvature in the meridian = 2’, in the. direction perpendicular to it = rv", in the azimuth «=r: — = 0"-06314600+4 000031552 cos 2 ¢ + 000000013 cos 4 ¢. or log i = 8'8025112 .9+3 log cos log a = 8°7996179 . 6 + log cos fp, pa =A+A!' cos 2a, wherein 4, Let the distance from the centre of the earth = ge, and a corrected latitude = ¢’: log . eg cos ¢! = log . cos $¢ — log cos, log . eg sin ¢' = log . sin ¢ — log cos f — 0:0028933 . 3. 7 on 4 401 ARTICLE XIII. The Galvanic Circuit investigated Mathematically. By Dr. G. S. Oum*. PREFACE. I HEREWITH present to the public a theory of galvanic elec- tricity, as a special part of electrical science in general, and shall successively, as time, inclination,and means permit, arrange more ‘such portions together into a whole, if this first essay shall in some degree repay the sacrifices it has cost me. The cir- cumstances in which I have hitherto been placed, have not been ‘ane either to encourage me in the pursuit of novelties, or to enable me to become acquainted with works relating to the “same department of literature throughout its whole extent. I hhave therefore chosen for my first attempt a portion in which ‘Thaye the least to apprehend competition. May the well-dis- posed reader receive the performance with the same love for the ‘object as that with which it is sent forth. Tue AvuTHOR. Berlin, May 1st, 1827. * INTRODUCTION. _ Tue design of this Memoir is to deduce strictly from a few oss obtained chiefly by experiment, the rationale of ‘those electrical phenomena which are produced by the mutual contact of two or more bodies, and which have been termed Jalvanic:—its aim is attained if by means of it the variety of facts be presented as unity to the mind. To begin with the most simple investigations, I have confined myself at the outset to those cases where the excited electricity propagates itself only in one dimension. They form, as it were, the scaffold toa greater structure, and contain precisely that portion, the more accurate knowledge of which may be gained from the elements of natural philosophy, and which, also, on account of its greater | aecessibility, may be given in a more strict form. To answer | ae Die Galvanische Kette mathematisch bearbeitet von Dr. G. S. Ohm: Berlin, 1827.” Translated from the German by Mr. William Francis, Student > in the University of Berlin. 402 OHM ON THE GALVANIC CIRCUIT. this especial purpose, and at the same time as an introduction to the subject itself, I give, as a forerunner of the compressed mathematical investigation, a more free, but not on that account less connected, general view of the process and its results. Three laws, of which the first expresses the mode of distribu- tion of the electricity within one and the same body, the second the mode of dispersion of the electricity in the surrounding atmosphere, and the third the mode of appearance of the elec- tricity at the place of contact of two heterogeneous bodies, form the basis of the entire Memoir, and at the same time con- tain everything that does not lay claim to being completely established. The two latter are purely experimental laws; but the first, from its nature, is, at least partly, theoretical. With regard to this first law, I have started from the suppo- sition that the communication of the electricity from one par- ticle takes place directly only to the one next to it, so that no immediate transition from that particle to any other situate at a greater distance occurs. The magnitude of the transition between two adjacent particles, under otherwise exactly si- milar circumstances, I have assumed as being proportional to the difference of the electric forces existing in the two parti- cles; just as, in the theory of heat, the transition of caloric be- tween two particles is regarded as proportional to the dif- ference of their temperatures. It will thus be seen that I have deviated from the hitherto usual mode of considering molecular actions introduced by Laplace; and I trust that the path I have struck into will recommend itself by its generality, sim- plicity, and clearness, as well as by the light which it throws upon the character of former methods. With respect to the dispersion of electricity in the atmosphere, I have retained the law deduced from experiments by Coulomb, according to which, the loss of electricity, in a body surrounded ~ by air, in a given time, is in proportion to the force of the — electricity, and to a coefficient dependent on the nature of the atmosphere. A simple comparison of the circumstances under which Coulomb performed his experiments, with those at pre- a sent known respecting the propagation of electricity, showed, — however, that in galvanic phenomena the influence of the atmo- _ sphere may almost always be disregarded. In Coulomb’s expe- riments, for instance, the electricity driven to the surface of the — be “tee OHM ON THE GALVANIC CIRCUIT. 403 body was engaged in its entire expanse in the process of di- spersion in the atmosphere; while in the galvanic circuit the electricity almost constantly passes through the interior of the bodies, and consequently only the smallest portion can enter into mutual action with the air; so that, in this case, the disper- sion can comparatively be but very inconsiderable. This con- sequence, deduced from the nature of the circumstances, is confirmed by experiment ; in it lies the reason why the second law seldom comes into consideration. The mode in which electricity makes its appearance at the place of contact of two different bodies, or the electrical tension _ of these bodies, I have thus expressed: when dissimilar bodies ni nas touch one another, they constantly maintain at the point of contact the same difference between their electroscopic forces. With the help of these three fundamental positions, the con- _ ditions to which the propagation of electricity in bodies of any kind and form is subjected may be stated. The form and treatment of the differential equations thus obtained are so similar to those given for the propagation of heat by Fourier and Poisson, that even if there existed no other reasons, we might with perfect justice draw the conclusion that there exists an intimate connexion between both natural phenomena ; and this _ relation of identity increases, the further we pursue it. These researches belong to the most difficult in mathematics, and on that account can only gradually obtain general admission ; it is therefore a fortunate chance, that in a not unimportant part of the propagation of electricity, in consequence of its peculiar nature, those difficulties almost entirely disappear. To place this portion before the public is the object of the present memoir, and therefore so many only of the complex cases have been admitted as seemed requisite to render the transition ap- parent. The nature and form commonly given to galvanic apparatus favours the propagation of the electricity only in one dimension ; and the velocity of its diffusion combined with the constantly acting source of galvanic electricity is the cause of the galvanic phenomena assuming, for the most part, a character which does not vary with time. These two conditions, to which most fre- quently galvanic phenomena are subjected, viz. change of the electric state in a single dimension, and its independency of time, 404 OHM ON THE GALVANIC CIRCUIT. are however precisely the reasons why the investigation is brought to a degree of simplicity which is not surpassed in any branch of natural philosophy, and is altogether adapted to secure incontrovertibly to mathematics the possession of a new field of physics, from which it had hitherto remained almost totally excluded. The chemical changes which so frequently occur in some, gene- rally fluid, portions of a galvanic circuit, greatly deprive the re- sult of its natural simplicity, and conceal, to a considerable extent, by the complications they produce, the peculiar progression of the phzenomenon; they are the cause of an unexampled change of the phenomenon, which gives rise to so many apparent ex- ceptions to the rule, frequently even to contradictions, in so far as the sense of this word is itself not in contradiction to nature. I have distinctly separated the consideration of such galvanic circuits in which no portion undergoes a chemical change, from those whose activity is disturbed by chemical action, and have devoted a separate part to the latter in the Appendix. This total separation of two parts forming a whole, and, as might appear, the less dignified position of the latter, will find in the following circumstance a sufficient explanation. A theory, which lays claim to the name of an enduring and fruitful one, must have all its consequences in accordance with observation and experiment. This, it seems to me, is sufficiently established with respect to the first of the parts above-mentioned, partly by the previous experiments of others, and partly by some performed by myself, which first made me acquainted with the theory here developed, and subsequently rendered me entirely devoted to it. Such is not the case with regard to the second part. A more accurate experimental eit Gabo is in this case almost entirely” wanting, to undertake which I need both the requisite time and means; and therefore I have merely placed it in a corner, from which, if worth the trouble, it may be drawn hereafter, and may then also be further matured under better nursing. # By means of the first and third fundamental positions we obtain a distinct insight into the primary galvanic phenomenon | in the following way. Imagine, for imstance, a ring every- where of equal thickness and homogeneous, having, at any one place, 1 in its whole thickness, one and the same electrical tension, i. e. inequality in the electrical state of two surfaces situated’ close to each other, from which causes, when they have come * OHM ON THE GALVANIC CIRCUIT. 405 into action, and the equilibrium is consequently disturbed, the electricity will, in its endeavour to re-establish itself, if its mobility be solely confined to the extent of the ring, flow off on both sides. If this tension were merely momentary, the equilibrium would very soon be re-established; but if the tension is permanent, the equilibrium can never be restored; but the electricity, by virtue of its expansive force, which is not sensibly restrained, produces in a space of time, the dura- tion of which almost always escapes our senses, a state which comes nearest to that of equilibrium, and consists in this; that by the constant transmission of the electricity, a percep- _ tible change in the electric condition of the parts of the body ~ through which the current passes is nowhere produced. The _ peculiarity of this state, also occurring frequently in the trans- “mission of light and heat, has its foundation in this; that each article of the body situated in the circle of action receives in each moment just so much of the transmitted electricity from the one side as it gives off to the other, and therefore constantly retains the same quantity. Now since by reason of the first fundamental position the electrical transition only takes place directly from the one particle to the other, and is, under other- wise similar circumstances, determined according to its energy _ by the electrical difference of the two particles, this state must _ evidently indicate itself on the ring, uniformly excited in its entire thickness, and similarly constituted in all its parts, by a mstant change of the electric condition, originating from the _ point of excitation, proceeding uniformly through the whole ring, and finally again returning to the place of excitation; whilst at this place itself, a sudden spring in the electric condition, con- ‘stituting the tension, is, as was previously stated, constantly per- ceptible. In this simple separation or division of the electricity lies the key to the most varied phenomena. The mode of separation of the electricity has been completely determined by the preceding observation ; but the absolute force of the electricity at the various parts of the ring still remains uncertain. This property may be best conceived, by imagining the ring, without its nature being altered, opened at the point of excitation and extended in a straight line, and representing the force of the electricity at each point by the length of a per- pendicular line erected upon it ; that directed upwards may re- present a positive electrical, but that downwards a negative 406 OHM ON THE GALVANIC CIRCUIT. electrical, state of the part. The line A B (Plate XXIV., fig. 1) _ may accordingly represent the ring extended in a straight line, © and the lines AF and BG perpendicular to A B may indicate ~ by their lengths the force of the positive electricities situated at the extremities A and B. If now the straight line F G be drawn from F to G, also FH parallel to A B, the position of F G will give the mode of separation of the electricity, and the quantities BG —AF or GH the tension occurring at the extremities of the ring; and the force of the electricity at any other place C, may easily be expressed by the length of C D drawn through C perpendicularly to AB. But, from the nature of the galvanic excitation, merely the quantity of the tension or the length of the line G H, therefore the difference of the lmes AF and BG, is determined, but not at all the absolute magnitudes of the lines AF and BG; consequently the mode of separation may be represented quite as well by any other line parallel to the former, e.g. by I K, for which the tension still constantly retains the same value expressed by K N, because the ordi- nates situated at present below A B assume a relation opposed to their former one. Which of the infinitely numerous lines parallel to F G would express the actual state of the ring can- not be stated in general, but must in each case be separately determined from the circumstances which occur. Moreover, it is easily conceived that, as the position of the line sought is given, it would be completely determined for one single part of the ring by the determination of any one of its points, or, in other words, by the knowledge of the electric force. If, for instance, the ring lost all its electricity by abduction at the place C, the line LM drawn through C parallel to F G would in this case express with perfect certainty the electrical state of the ring. This variability in the separation of the electricity is the source of the changeableness of the phenomenon peculiar to the galvanic | circuit. I may further add, that it is evidently quite indif- ferent whether the position of the line F G with respect to tha of AB be fixed; or whether the position of the lime FG re~ main constantly ae same, and the position of A B with respect to it be altered. The latter course is by far the more simple where the separation of the electricity assumes a more complex form. " The conclusions just arrived at, which hold for a ring ho- mogeneous throughout its whole extent, may easily be ex- : OHM ON THE GALVANIC CiRCUIT. 407 tended to a ring composed of any number of heterogeneous parts, if each part be of itself homogeneous and of the same thickness. I may here take as an example of this extension a ring composed of two heterogeneous parts. Let this ring be imagined as before open at one of its places of excitation and stretched out to form the right line A BC (fig. 2), so that AB and B C indicate the two heterogeneous parts of the ring. The perpendiculars AF, BG, will represent by their lengths the electrical forces present at the extremities of the part AB; on _ the other hand, B H and CI, those present at the extremities of the part BC; accordingly AF + CI or F K will represent the tension at the opened place of excitation, and G H the ten- ‘sion occurring at B at the point of contact. Now if we only bear in mind the permanent state of the circuit, the straight lines FG and HI will, from the reasons above mentioned, in- dicate by their position the mode of separation of the elec- tricity in the ring; but whether the line A C will keep its place, or must be advanced further up or down, remains uncertain, and can only be found out in each distinct case by other se- parate considerations. If, for instance, the point O of the cir- euit is touched abductively, and thus deprived of all electri- city, ON would disappear; and therefore the line LM drawn through N parallel with A C would in this case give the posi- tion of AC required. It is hence evident, how sometimes this, metimes another, position of the line AC in the figure 'G HI, representing the separation of the electricity, may be ‘the one suited to the circumstances; and herein we recognise the source of the variability of galvanic phenomena already mentioned. It is, however, essentially requisite, in order to be able to i judge thoroughly of the present case, to attend to a circum- || Stance the mention of which has hitherto been purposely avoided, that the various considerations might be separated as distinctly as possible. The distances FK and GH are indeed given by the tensions existing at the two places of ex- citation, but the figure FGHI is not yet wholly determined 7 by this alone. For instance, the points G and H might move | down towards G! and H’, so that G'H’ would equal GH, giving rise to the figure F G! H'I, which would indicate quite a different mode of separation of the electricity, although the | individual tensions in it still retain their former magnitude. 408 OHM ON THE GALVANIC CIRCUIT. Consequently if that which has been stated with respect to the circuit of two members is to acquire a sense no longer subject to any arbitrary explanation, this uncertainty must be removed. The first fundamental law effects this in the followmg way :— | For since the state of the ring alone, independent of the time, is regarded, each section must, as has already been stated, re- ceive in every moment the same quantity of electricity from one side as it gives off to the other. This condition occasions upon such portions of the ring as have perfectly the same con= stitution at their various points, the constant and uniform change in the separation which is represented in the first figure | by the straight line FG, and in the second by the straight lines FG and HI. But when the geometrical or the physica nature of the ring changes in passing from one of its compo- § nent parts to another, the reason of this constancy and uniform-_ ity no longer obtains; consequently the manner in which the several straight lines are combined into a complete figure must _ first be deduced from other considerations. To facilitate the difference of the single parts, each independently. Let us first suppose that every section of the part BC is m’ times smaller than in the part A B, while both parts are com=| posed of the same substance; the electric state of the ring, received on one side as is given off from the other, can evi- dently only exist under the condition that the electric trans- the portion BC is m times greater than in the portion AB; because it is only in this manner that the action in both parts can maintain equilibrium. But in order to produce} this m times greater transition of the electricity from element § to element, the electrical difference of element to element within mination is transferred to the figure, the line HI must sink m times more on equal portions, or have an m times greater “dip” than the ine FG. By the expression “ dip” (Gefdille), is to be understood the difference of such ordinates which be- long to two places distant one unit of length from each other. From this consideration results the following rule: The dips of —. OHM ON THE GALVANIC CIRCUIT. 409 the lines F G and 1 in the portions A B and BC, composed of like substance, will be inversely to each other as the areas of the sections of these parts. By this the figure F G H I is now fully determined. When the parts AB and BC of the ring have the same section but are composed of different substances, the transition of the electricity will then no longer be dependent solely on the _ progressive change of electricity in each part from element to element, but at the same time also on the peculiar nature of each substance. This difference in the distribution of the elec- tricity, caused solely by the material nature of the bodies, whe- ther it have its origin in the peculiar structure or in any other peculiar state of the bodies to electricity, establishes a distinc- tion in the electrical conductibility of the various bodies; and even the present case may afford some information respecting the actual existence of such a distinction and give rise to its more accurate determination. In fact, since the ring composed of the two parts A B and BC differs from the homogeneous one only in this respect, that the two parts are formed of two differ- ent substances, a difference in the dip of the two lines F G and HI will make known a difference in the conductibility of the two substances, and one may serve to determine the other. In this way we arrive at the following position, supplying the place of a definition: In a ring consisting of two parts AB and BC, of like sections but formed of different substances, the dips of the lines F G and HI are inversely as the conducting powers of the two parts. If we have once ascertained the conducting powers of the various substances, they may be employed to determine the dips of the lines F G and H I in every case that may occur. By this, then, the figure F GH I is entirely determined. The determination of the conductibility from the separation of the electricity is rendered very difficult from the weak intensity of galvanic electricity, and from the imperfection of the requisite apparatus ; subsequently we shall obtain a more easy means of ecting this purpose. From these two particular cases we may now ascend in the usual way to the general one, where the two prismatic parts of the ring neither possess the same section nor are constituted of the same substance. In this case the dips of the two paris must be in the inverse ratio of the products of the sections and powers of conduction. We are hereby enabled to deter- VOL. Il. PART VII. 2E 410 OHM ON THE GALVANIC CIRCUIT. mine completely the figure FG HI in every case, and also to distinguish perfectly the mode of electrical separation in the ring. All the peculiarities, hitherto considered separately, of the ring composed of two heterogeneous parts, may be summed up in the following manner: In a galvanic circuit consisting of two heterogeneous prismatic parts, there takes place in regard to its electrical state a sudden transition from the one part to the other at each point of excitation, forming the tension there oc- curring, and from one extremity of each point to the other a gradual and uniform transition ; and the dips of these two trans- itions are inversely proportional to the products of the conducti- bilities and sections of each part. Proceeding in this manner, we are able without much diffi- culty to inquire into the electrical state of a ring composed of three or more heterogeneous parts, and to arrive at the following general law: In a galvanic circuit consisting of any indefinite number of prismatic parts, there takes place in regard to its elec- trical state at each place of excitation a sudden transition, from one part to the other, forming the tension there prevailing, and within each part a gradual and uniform transition from the one extremity to the other; and the dips of the various transitions are inversely proportional to the products of the conductibilities and sections of each part. From this law may easily be deduced the entire figure of the separation for each particular case, as I will now show by an example. Let ABCD (fig. 3) be a ring composed of three heteroge- neous parts, open at one of its places of excitation, and extended in a straight line. The straight lines FG, HI, K L represent by their position the mode of separation of the electricity in each individual part of the ring, and the lines AF, BG, BH, Cl, CK, and DE drawn through A,B,C and D perpendicular to AD such quantities that GH, KI and LM or DL—AF show by their length the magnitude of the tensions occurring at the individual places of excitation. From the known magni- tude of these tensions, and from the given nature of the single parts AB, BC, and C D, the figure of the electrical separation has to be entirely determined. i If we draw straight lines parallel to A D, through the points F,H and K, meeting the line drawn through B, C and D per- pendicular to A D, in the points F’, H’, K’, then according to what has already been demonstrated, the lines G F’, IH! and OHM ON THE GALVANIC CIRCUIT. 411 LK’ are directly proportional to the lengths of the parts A B, BC and CD, and inversely proportional to the products of the conductibility and section of the same part, consequently the relations of the lines GF’, IH! and LK’ to each other are given. Further, that G’+1H' + LK’=GH—KI+(DL — AF = LM) is also known, as the tensions represented by GH, KI and DL — AF are given. From the given relations of the lines GF’, I H', LK’ and their known sum, these lines may now be found individually ; the figure F GH I K L is evi- dently then entirely determined. But the position of this figure with respect to the line AD remains from its very nature still undecided. __ If we recollect, that proceeding in the same direction AD, the tensions represented by GH and DL — AF or LM indicate a sudden sinking of the electric force at the respective places of excitation, that represented by I K on the contrary a sudden rise of the force ; and that tensions of the first kind are regarded and treated as positive quantities, while tensions of the latter kind are considered as negative quantities, we find the above ex- ample lead us to the following generally valid rule: If we divide the sum of all the tensions af the ring composed of several parts into the same number of portions which are directly proportional _ to the lengths of the parts and inversely proportional to the pro- | ducts of their conductibilities and their sections, these portions | will give in succession the amount of gradation which must be | assigned to the straight lines belonging to the single parts and i representing the separation of the electricity ; at the same time the positive sum of all the tensions indicates a general rise, on | the contrary the negative sum of all the tensions a general depres- sion of those lines. , I will now proceed to the determination of the electric force | at any given position in every galvanic circuit, and here again I shall lay down as basis fig. 3. For this purpose let a, a’, a!’ indicate the tensions existing at B,C, and between A and D, so that in this case also a and a! represent additive, a’ on the contrary a subtractive line, and a, A’, 2” any lines which are | directly as the lengths of the parts AB, BC, and CD, and in- versely as the products of the conductibilities and sections of the same parts; further, let | at@d+al=A 2n2 412 OHM ON THE GALVANIC CIRCUIT. and Ata 4+A°=L then according to the Jaw just ascertained GF’ is a fourth proportional to L, A and 4 IH’ a fourth proportional to L, A and LK’ a fourth proportional to L, A and a". Draw the line F M through F parallel to A D, regard this line as the axis of the abscisse, and erect at any given points X, X’, X" the ordinates XY, X!'Y’, X" Y", we obtain their respective values, thus: In the first place we have, since AB = F F" AB): Gi =i x: 2, whence follows: FX 1GF' XT See ; : ' A ehA or if we substitute for G F’ its value L 553: SUAS WT ERE ai a fone AB If now 2 represent a line such that A Bi Pe A ae then p. = +8 Secondly, since BC and F’X! are equal to the lines dra through I and Y' to GH parallel to AD BC:IH'=FX':FH—-XY"|’, whence ! I] / Bex yn others =’ Bh or, since FH = GH — GF’ ! lj ! xy tt FS ere nr If now for IH’ and GF’ we substitute their values =e » we obtain ! —-wy=t (a+ es ae) -4 and if by z! we represent a line such ne Ae L é "parts of the circuit and different in form from each other, may OHM ON THE GALVANIC CIRCUIT. 413 B@y: POX! =a ya; then A — XY =F (A+ 2’) — a. Thirdly, since CD = K K’ and F" X" is equal to the part of K K’ which extends from K to the line X" Y", we have Cis R= xe key — KF", whence ! W! WW Xryn — eee x + KF", or, since KF’ = KI +4+I1H'—IF’H and PH=GH—-GF, ! fi i] soy = EE PX yg are 4d). 3 7 4 If now for L K’, I H', G F’ we substitute their values A.a! A.al Awa ‘ Shit bliach sabes obtain wy on ed — Eat +7 a5) —(a+a); and if by 2 we represent a line such that CD: F’X" =a"; gi! we have xX" y" — = A+aA 4 2")—(a4+a’). These values of the ordinates, belonging to the three distinct be reduced as follows to a common expression. For if F is taken as the origin of the abscisse, F X will be the abscissa corresponding to the ordinate X Y which belongs to the ho- mogeneous part AB of the ring, and ~ will represent the length corresponding to this abscissa in the reduced proportion of AB:a. In like manner F X’ is the abscissa corresponding to the ordinate X’ Y’ which is composed of the parts F F’ and F’ X’ belonging to the homogeneous portions of the ring, and A, @ are the lengths reduced in the proportions of AB:a and BC: 2 corresponding to these parts. Lastly F X” is the ab- Scissa corresponding to the ordinate X” Y’’, which is composed of the parts F F’, F’ F’, F’ X” belonging to the homogeneous portions of the ring, and A, a’, 2” are the lengths reduced in the proportions of AB:a, BC:2,CD:a”. If in consequence of this consideration we call the values 7, A + a’, A + V+ a” 414 OHM ON THE GALVANIC CIRCUIT. } reduced abscisse and represent them generally by y, we obtain Sima ge A xy =F .y- (a+), and it is evident that L is the same in reference to the whole length AD or FM as y is to the lengths F X, FX’, F X", on account of which L is termed the entire reduced length of the circuit. Further, if we consider that for the abscissa corre- sponding to the ordinate X Y the tension has experienced no abrupt change, but that for the abscissa corresponding to the ordinate X! Y' the tension has experienced the abrupt changes a, a'; and if we represent generally by O the sum of all the abrupt changes of the tensions for the abscissa corresponding to the ordinate y, then all the values found for the various ordi- nates are contained in the following expression : - -y —O. But these ordinates express, when an arbitrary constant, cor- responding to the length AF, is added to them, the electric forces existing at the various parts of the ring. If therefore we represent the electric force at any place generally by u we obtain the following equation for its determination : ee = y—-O+¢, in which c represents an arbitrary constant. This equation is generally true, and may be thus expressed in words: The force of the electricity at any place of a galvanic circuit composed of several parts, is ascertained by finding the fourth proportional to the reduced length of the entire circuit, the reduced length of the part belonging to the abscissa, and the sum of all the tensions, and by increasing or diminishing the difference between this quan- tity and the sum of all the abrupt changes of tension for the given abscissa by an arbitrary quantity which is constant for all parts of the circuit. When the determination of the electric force at each place of the circuit has been effected, it only remains to determine the magnitude of the electric current. Now in a galvanic circuit of OHM ON THE GALVANIC CIRCUIT. ALS the kind hitherto mentioned, the quantity of electricity passing through a section of it in a given time is everywhere the same, because at all places and in each moment the same quantity in the section leaves it on the one side as enters it from the other, but in different circuits this quantity may be very different: therefore, in order to compare the actions of several galvanic circuits inter se, it is requisite to have an accurate determination of this quantity, by which the magnitude of the current in the circuit is measured. This determination may be deduced from figure 3 in the following manner. It has already been shown that the force of the electric transition in each instant from one element to the adjacent one is given by the electric differ- _ ence between the two existing at that time, and by a magnitude _ dependent upon the kind and form of the particles of the body, _ viz. the conductibility of the body. But the electrical difference of the elements in the part BC, for instance, reduced to a con- stant unit of distance, will be expressed by the dip of the line H I ! or by the quotient a ; if, therefore, we now indicate by x the magnitude of the conductibility of the part B C, x, «LH! BC _ will express the force of the transition from element to element, or the intensity of the current in the part BC; consequently if ‘w represent the magnitude of the section in the part BC, the quantity of electricity passing in each instant from one section to the adjacent one, or the magnitude of the current, will be ex- pressed by ne x.wo.1H! BC ? or if S represent this magnitude of the current, we have ga t8: IH’ Corman > 1 OR Nig ; ‘ : Das A and if we substitute for I H’ its value L A x.0.A — iv . eey) C . Hitherto the letters a, ’, A" have represented lines which are proportional to the quotients formed of the lengths of the parts AB, BC, CD, and the products of their conductibilities and their sections. If we restrict for the present this determination, 416 OHM ON THE GALVANIC CIRCUIT. which leaves the absolute magnitude of the lines a, a!, a! un- certain, so that the magnitudes a, a’, x" shall not be merely proportional to the said quotients, but shall be likewise equal to them, and henceforth vary this limitation in accordance with the meaning of the expression “reduced lengths,” the first of the two preceding equations becomes IH! Ke s= \ which gives the following generally: The magnitude of the cur- rent in any homogeneous portion of the circuit is equal to the quotient of the difference between the electrical forces present at the extremities of this portion divided by its reduced length. This expression for the forces of the current will be continued to be employed subsequently. The second of the former equa- tions passes, by the adopted change, into A L? which is generally true, and already reveals the equality of the force of the current at all parts of the circuit; in words it may be thus expressed: The force of the current in a galvanic circuit is directly as the sum of all the tensions, and inversely as the entire reduced length of the circuit, bearing in mind that at present by reduced length is understood the sum of all the quo- tients obtained by dividing the actual lengths corresponding to the homogeneous parts by the product of the corresponding con- ductibilities and sections. From the equation determining the force of the current in a galvanic circuit in conjunction with the one previously found, by which the electric force at each place of the circuit is given, may be deduced with ease and certainty all the phenomena be- longing to the galvanic circuit. The former I had already some time ago derived from manifoldly varied experiments* with an apparatus which allows of an accuracy and certainty of mea- surement not suspected in this department ; the latter expresses all the observations pertaining to it, which already exist in great number, with the greatest fidelity, which also continues where the equation leads to results no longer comprised in the circle of previously published experiments. Both proceed un- interruptedly hand in hand with nature, as I now hope to Ss * Schweigger’s Jahrbueh, 1826, part 2. OHM ON THE GALVANIC CIRCUIT. 417 demonstrate by a short statement of their consequences ; at the same time I consider it necessary to observe, that both equa- tions refer to all possible galvanic circuits whose state is per- manent, consequently they comprise the voltaic combination as a particular case, so that the theory of the pile needs no separate comment. In order to be distinct, I shall con- stantly, instead of employing the equation vu = ay —O6¢, only take the third figure, and therefore will merely remark here, once for all, that all the consequences drawn from it hold generally. In the next place, the circumstance that the separation of the electricity, diffusing itself over the galvanic circuit, maintains at the different places a permanent and unchangeable grada- tion, although the force of the electricity is variable at one and the same place, deserves a closer inspection. This is the reason of that magic mutability of the phanomena which admits of our predetermining at pleasure the action of a given place of the galvanic circuit on the electrometer, and enables us to produce it instantly. To explain this peculiarity I will return to figure 3. Since the figure of separation FGHIKL, is always wholly determined from the nature of any circuit; but its position with respect to the circuit A D, as | was seen, is fixed by no inherent cause, but can assume any change produced by a movement common to all its points in the direction of the ordinates, the electrical condition of each point of the circuit expressed by the mutual position of the two lines, may be varied constantly, and at will, by ex- ternal influences. When, for example, A D is at any time the position representing the actual state of the circuit, so that, therefore, the ordinate S Y” expresses by its length the force of the electricity at the place of the circuit to which that or- dinate belongs, then the electrical force corresponding to the point A, at the same time will be represented by the line A F, If now the point A be touched abductively, and thus be entirely deprived of all its force, the line A D will be brought into the position F M, and the force previously ex- isting in the point S will be expressed by the length X" Y"; this force, therefore, has suddenly undergone a change, corre- sponding to the length SX". The same change would have occurred if the circuit had been touched abductively at the point 418 OHM ON THE GALVANIC CIRCUIT. Z, because the ordinate Z W is equal to that of A F. If the cir- cuit were touched at the place where the two parts AB and BC join, but so that the contact was made within the part BC, we should have to imagine AD advanced to NO; the elec- trical force at the point S would in this case be increased to the force indicated by TY". But if the contact took place, still at the same point, viz. where the parts A B and BC touch each other, but within the part AB, the line A D would be moved to PQ, and the force belonging to the point S would sink to the negative force expressed by U Y". If, lastly, the pile had been touched abductively at the point D, we should have prescribed for the line AD the position RL, and the electrical force at the point S would have assumed the negative force indicated by V Y". The law of these changes is ob- vious, and may be expressed generally thus: each place of a galvanic circuit undergoes mediately, in regard to its outward- ly acting electrical force, the same change which is produced immediately at any other place of the circuit by external in- fluences. Since each place of a galvanic circuit undergoes, of itself, the same change to which a single place was compelled, the change in the quantity of electricity, extending over the whole circuit, is proportional, on the one hand, to the sum of all the places, 7. e. to the space over which the electricity is diffused in the circuit, and moreover, to the change in the electric force produced at one of these places. From this simple law result the follow- ing distinct phenomena. If we call 7 the space over which the electricity is diffused in the galvanic circuit, and imagine this circuit touched at any one place by a non-con- ducting body, and designate by u the electric force at this place before contact, by u that after contact, the change produced in the force at this place is u, — wu; consequently the change of the whole quantity of electricity in the circuit is (w,—w) r. If, now, we suppose that the electricity in the touched body is diffused over the space R, and is at all places of equal strength, and, at the same time, that at the place of contact itself the circuit and the body possess the same electric force, viz. u, it it is evident wR will be the quantity of electricity ee to the body, and (u, —u)r=uR, whence we obtain . _ was already shown, OHM ON THE GALVANIC CIRCUIT. 419 ur pes The intensity of the electricity received by the body will, there- Sore, be the more nearly equal to that which the circuit possessed at the place of contact before being touched, the smaller R is with respect to r; it will amount to the half when R = r, and become weaker, as R becomes greater in comparison with 7. Since these changes are merely dependent on the relative mag- nitude of the spaces r and R, and not at all on the qualitative nature of the circuit, they are merely determined by the dimen- sions of the circuit, nay, even by foreign masses brought into conducting connexion with the circuit. If we connect this fact with the theory of the condensor, we arrive at an explanation of all the relations of the galvanic circuit to the condensor, noticed by Jager*, which is perfectly surprising. I content my- self with regard to this point to refer to the memoir itself, to give room here for the insertion of some new peculiarities of the galvanic circuit. The mode of separation of the electricity, within a homo- geneous part of the circuit, is determined by the magnitudes of the dips of the lines FG, HI, KL, (fig. 3,) and there again GP jy ae DIKE by the magnitudes of the ratioss 5 Be CD: “= But, as GF=2.., 1H = 4, LK = +! hence it may be seen, without much trouble, that the magnitude of the dip of the line corresponding to any part of the circuit, and representing the separation of the electricity, is obtained by multiplying the value = by the ratio of the reduced to the actual length of the same part. If, therefore, (A) represent the reduced length of any homogeneous part of the circuit and (J) its actual length, the magnitude of the dip of the straight line belonging to this part, and representing the separation of the electricity, is A (A) DP)? which expression, if we designate by (x) the conductibility, * Gilbert’s Annalen, vol. xiii. 420 OHM ON THE GALVANIC CIRCUIT. and by (w) the section of the same part, may also be written thus: MARAT (eK) L * () (#) This expression leads to a more detailed knowledge of the separation of the electricity in a galvanic circuit. For since A and L designate values which remain identical for each part of the same circuit, it is evident that the dips in the sepa- rate homogeneous parts of a circuit are to one another in- versely as the products of the conductibility, and the section of the part. If consequently a part of the circuit surpasses all others from the circumstance, that the product of its conduc- tibility and its section is far smaller than in the others, it will be the most adapted to reveal, by the magnitude of its dip, the differences of the electric force at its various points. If its actual length is, at the same time, not inferior to those of the other parts, its reduced length will far surpass those of the other parts; and it is easily conceived that such a relation be- tween the various parts can be brought about, that even its re- duced length may remain far greater than the sum of the reduced lengths of all the other parts. But in this case the reduced length of this one part is nearly equal to the reduced lengths of the entire circuit, so that we may substitute, without committing any great error, “S for L, if (2) represent the actual length of the said part, («) its conductibility, and (w) its section ; but then the dip of this part changes nearly into A (2)’ whence it follows that the difference of the electrical forces at the extremities of this part is nearly equal to the sum of all the _ tensions existing in the circuit. All the tensions seem, as it were, to tend towards this one part, causing the electrical sepa- ration to appear in it with otherwise unusual energy, when all the tensions, or, at least, the greater part in number and magnitude, are of the same kind. In this way the scarcely perceptible gradation in the separation of the electricity, in a closed circuit, may be rendered distinctly evident, which, other- wise, would not be the case without a condensor, on account of the weak intensity of galvanic forces. This remarkable pro- OHM ON THE GALVANIC CIRCUIT. 42} perty of galvanic circuits, representing, as it were, their entire nature, had already been noticed long ago in various bad con- ducting bodies, and its origin sought for in their peculiar con- stitution * ; I have, however, enumerated in a letter to the editor of the Annalen der Physik}, the conditions under which this property of the galvanic circuit may be observed, even in the best conductors, the metals; and the necessary precautions, founded on experience, by which the success of the experiment is assured, described in it, are in perfect accordance with the present considerations. The expression os : () denoting the dip of any portion of Li _ the circuit, vanishes when L is indefinitely great, while A and an retain finite values. Consequently, if L assumes an in- definitely great value, while A remains finite, the dip of the straight lines representing the separation of the electricity, in all such parts of the circuit, whose reduced length has a finite ratio to the actual length, vanishes, or what comes to the same thing, the electricity is of equal force at all places of each such part. Now, since L represents the sum of the reduced lengths of all the parts of the circuit, and these reduced lengths evi- dently can only assume positive values, L becomes indefinite as soon as one of the reduced lengths assumes an infinite value. Further, since the reduced length of any part represents the quotient obtained by dividing the actual length by the product of the conductibility and the section of the same part, it becomes infinite when the conductibility of this part vanishes, 7. e. when this part is a non-conductor of electricity. When, therefore, a part of the circuit is a non-conductor, the electricity expands uniformly over each of the other parts, and its change from one part to the other is equal to the whole tension there situated. This separation of the electricity, relative to the open circuit, is far more simple than that in the closed circuit, which has hitherto formed the object of our consideration, as is geome- trically represented by the lines F G, HI, K L, (fig. 3) taking a position parallel to A D. It distinctly demonstrates that the difference between the electrical forces, occurring at any two * Gilbert’s Annalen, vol. viii. pp. 205, 207, and 456. Vol. x. p. 11. + Jahrgang, 1826. Part v. p. 117. 422 OHM ON THE GALVANIC CIRCUIT. places of the circuit, is equal to the sum of all the tensions situ- ated between these two places, and consequently increases or decreases exactly in the same proportion as this sum. When, therefore, one of these places is touched abductively, the sum of all the tensions, situated between the two, makes its appear- ance at the other place, at the same time the direction of the tensions must always be determined by an advance from the latter place. All the experiments on the open pile, with the help of the electroscope, instituted at such length by Ritter, Erman, and Jager, and described in Gilbert’s Annalen*, are expressed in this last law. All the electroscopic actions of a galvanic circuit of the kind, described at the outset, have been above stated; I therefore pass at present to the consideration of the current originating in the circuit, the nature of which, as explained above, is expressed at every place of the circuit by the equation A S=7: Both the form of this equation, as well as the mode by which we arrive at it, show directly that the magnitude of the current in such a galvanic circuit remains the same at all places of the circuit, and is solely dependent on the mode of separation of the electricity, so that it does not vary, even though the electric force at any place of the circuit be changed by abductive contact, or in any other way. 'This equality of the current at all places of the circuit has been proved by the experiments of Becque- relf, and its independency of the electric force at any de- terminate place of the circuit by those of G. Bischoff. An abduction or adduction does not alter the current of the gal- vanic circuit so long as they only act immediately on a single place of the circuit; but if two different places were acted upon contemporaneously, a second current would be formed, which would necessarily, according to circumstances, more or less change the first. The equation A S=— L shows that the current of a galvanic circuit is subjected to a * Vol. viil., xii., and xiii. + Bulletin universel. Physique. Mai, 1825. t Kastner’s Archiv, vol. iv. Part 1. | | | | OHM ON THE GALVANIC CIRCUIT. 423 change, by each variation originating either in the magnitude of atension or in the reduced length of a part, which latter is itself again determined, both by the actual length of the part, as well as by its conductibility and by its section. This variety of change may be limited, by supposing only one of the enu- merated elements to be variable, and all the remainder con- stant. We thus arrive at distinct forms of the general equa- tion corresponding to each particular instance of the general capability of change of a circuit. To render the meaning of this phrase evident by an example, I will suppose that in the circuit only the actual length of a single part is subjected to a continual change; but that all the other values denoting the “magnitude of the current remain constantly the same, and, consequently, also in its equation. If we designate by 2 this © variable length, and the conductibility corresponding to the same part by x, its section by w, and the sum of the reduced lengths of all the others by A, so that L = A + ee then the general expression for the current changes into the following: oes. sce A+ 5) av x. | or if we multiply both the numerator and denominator by x a, _ and substitute @ for x w A, and 0 for x A, into the following: a 6+2 where a and 6 represent two constant magnitudes, and 2 the variable length of a portion of the circuit fully determined with respect to its substance and its section. This form of the general equation, in which all the invariable elements have been reduced to the smallest number of constants, is that which I had practically deduced from experiments to which the theory here developed owes its origin*. The law which it expresses relative to the length of conductors, differs essentially from that which Davy formerly, and Becquerel more recently, were led to by experiments; it also differs very considerably from that advanced by Barlow, as well as from that which I had previously drawn from other experiments, although the two latter come much nearer to the truth. The first, in fact, Ss = * See Schweigger’s Jahrbuch, 1826. Part 2. 424 OHM ON THE GALVANIC CIRCUIT. is nothing more than a formula of interpolation, which is valid only for a relatively very short variable part of the entire cir- cuit, and, nevertheless, is still applicable in very different possible modes of conduction, which is already evident, from its merely admitting the variable portion of the circuit, and leaving out of consideration all the other part; but all partake in common of this evil, that they have admitted a foreign source of variability, produced by the chemical change of the fluid portion of the circuit, of which I shall speak more fully hereafter. I have already treated, in other places, more at length of the relations of the various forms of the law to one another. . From the numerous separate peculiarities of the galvanic circuit resulting from the general equation S= - I will here merely mention a few. It is immediately evident that a change in the arrangement of the parts has no influ- ence on the magnitude of the current if the sum of the ten- sions be not affected by it. Nor is the magnitude of the cur- rent altered, when the sum of the tensions, and the entire reduced length of the circuit, change in the same proportion ; conse- quently a circuit, the sum of whose tensions is very small in comparison to that of another circuit, may still produce a cur- rent, which, in energy, may be equal to that in the other cir- cuit, when merely that which it loses in force of tensions is replaced by a shortening of its reduced length. In this circumstance is the source of the peculiar difference between thermo- and hydro-circuits. In the former only metals occur as parts of the circuit; in the latter, besides the metals, aqueous fluids, whose power of conduction, in comparison to that of the metals, is exceedingly small; on which account the reduced lengths of the fluid surpass, beyond all proportion, those of the metallic parts, with in all respects equal dimen- sions, and even remain considerably greater when diminished by shortening their actual lengths, and increasing their sec- tions, so long, at least, as this diminution is not carried too far. And thence it is that the reduced length of the ther- mo-circuit is, in general, far smaller than that of the hydro- circuit, whence we may infer a tension smaller in the same pro-: portion in the former, although the magnitude of the current, OHM ON THE GALVANIC CIRCUIT. 425 in the thermo-circuit, cedes in nothing to that in the hydro- circuit. The great difference between a thermo- and hydro-cir- cuit, both of which produce a current of the same energy, is evident when the same change is made on both, as will be shown in the fol- lowing consideration. Let the reduced length of a thermo- circuit be L, and the sum of its tensions A, the reduced length of an hydro-circuit m L, and the sum of its tensions m A, then the magnitude of the current in the former is expressed by 4 in the latter by — an both circuits. But this equality of the current no longer exists if the same new part A of the reduced length be intro- duced into both, for then the magnitude of the current is in the first d is consequently the same in A i x in the second mA wig oh Tf we connect with this determination an evaluation, even if _ merely superficial, of the quantities m, L, and A, we shall readily be convinced that in cases where the simple hydro-circuit can still produce in the part A actions of heat or chemical decompo- sition, the simple thermo-circuit may not possess the hundredth, and in some cases not the thousandth part of the requisite force, ‘whence the absence of similar effects in it is easily to be under- stood. Weare also able to understand why a diminution of the reduced lengths of the thermo-circuit (by increasing, for instance, _ the section of the metals constituting it) cannot give rise to the _ production of those effects, although the magnitude of its ' eurrent may be increased by this means to a higher degree than in the hydro-circuit producing such efiects. This difference in the conductibility of metallic bodies and aqueous fluids, is the cause of a peculiarity noticed with respect to hydro-circuits, which it is here, perhaps, the proper place to mention. Under the usual circumstances, the reduced length of the fluid portion is so large, in comparison to that of the metallic portion, that the latter may be overlooked, and the former alone taken in- stead of the reduced length of the entire circuit; but then the | magnitude of the current in circuits which have the same ten- \sion is in the inverse ratio to the reduced length of the fluid VOL. Il. PART VII. 2F 426 OHM ON THE GALVANIC CIRCUIT. portion. Consequently, if merely such circuits are compared in which the fluid parts have the same actual lengths and the same conductibilities, then the magnitude of the current in these circuits is in direct ratio to the section of the fluid portion. However, it must not be overlooked, that a more complex defi- nition must take the place of this simple one when the reduced length of the metallic portion can no longer be regarded as evanescent towards that of the fluid, which case occurs when- ever the metallic portion is very long and thin, or the fluid portion is a good conductor, and with unusually large terminal surfaces. From the equation A cee we can easily perceive that, when a portion is taken from the galvanic circuit, and is replaced by another, and after this change the sum of the tensions as well as the energy of the current still remains perfectly the same, these two parts have the same reduced length, consequently their actual lengths are as the products of their conductibilities and sections. The actual lengths of such parts are therefore, when they have like sections, as their conductibilities, and when they have like conductibilities as their sections. By the first of these two relations we are enabled to determine the conductibilities of various bodies in a far more advantageous manner than by the previously men- tioned process, and it has already been employed by Bec- querel and myself for several metals*. The second relation may serve to demonstrate experimentally the independence of the effect on the form of the section, as has previously been done by Davy, and recently by myselft. In the voltaic pile, the sum of the tensions, and the reduced length of the simple circuit, is repeated as frequently as the number of elements of which it consists expresses. If, there-_ fore, we designate by A the sum of all the tensions in the simple circuit, by L its reduced length, and by » the number of ele- ments in the pile, the magnitude of the current in the closed pile is evidently * Bulletin universel. Physique, Mai 1825, and Schweigger’s Jahrbuch, 1826. Part 2. i + Gilbert's Annalen, nn. Folge. Vol. xi. p. 253, and Schweigger’s Jahrbuch, 1827. OHM ON THE GALVANIC CIRCUIT. 42 =i ia nL’? while in the simple closed circuit it is Es ) A ¥ 1 we now introduce into the simple circuit, as well as into the one and the same new part A of the reduced length, upon h the current is to act, the magnitude of the current thus red in the simple circuit will be 4 n Iti is hence evident that the current is constantly greater in a ic pile than in the simple circuit, but it is merely imper- bly greater so long as is very small in comparison with L ; contrary, this increase approximates the nearer to n times, eater A becomes to nL, and consequently the more so in ison with L. [Besides this mode of increasing the mag- : of the galvanic current, there is a second one, which con- in shortening the reduced lengths of the simple circuit, may be effected by increasing its section, or placing seve- ple circuits by the side of each other, and connecting them ich a way that together they only form one single simple iif we now retain the same signs, so that A . L+A ir | denotes the magnitude of the current in one element, theabove-mentioned combination of 7 elements into a @ circuit, the magnitude of the current is evidently 730 we he dicates a slight increase in the action of the new combi- when \ is very great in comparison with L; on the con- : very powerful one when J is very small in comparison with =, and consequently the more so in comparison with L. It hence 2m 2 428 OHM ON THE GALVANIC CIRCUIT. follows that the one combination is most active in those cases _ where the other ceases to be so, and vice versd. If therefore we are in possession of a certain number of simple circuits in- tended to act upon the portion whose reduced length is A, much depends on the way in which they are placed, in order to produce the greatest effect of current; whether all be side by side, or all in succession, or whether part be placed by the side of each other, and part in series. It may be mathe- matically shown that it is most advantageous to form them into a voltaic combination, of so many equal parts, that the square of this number be equal to the quotient When es is equal to, or smaller than A, they had best be arranged by the side of each other, and in succession when is equal to, or larger than the square of the number of all the elements. We see in this de- termination the reason why in most cases a simple circuit, or ai least a voltaic combination of only a few simple circuits, is sufficient to produce the greatest effect. If we bear im mind, that since the quantity of the current is the same at all places of the circuit, its intensity at the various places must be in inverse proportion to the magnitude of the section there situated, and if we grant that the magnetic and chemical effects, as well as the phaenomena of light and heat in the circuit, are direct indica- tions of the electrical current, and that their energy is deter- mined by that of the current itself, then a detailed analysis of the current, here indicated merely in outline, will lead to the perfect explanation of the numerous and partially enigmatical anoma- lies observed in the galvanic circuit, in as far as we are justified in considering the physical nature of the circuit as invariable*. Those great differences which are frequently met with in the statements of various observers, and which are not consequences of the dimensions of their different apparatus, have undoubtedly their origin in the double capability of change of the hydro-cir- ) ' cuits, and will therefore cease when this circumstance is taken | into consideration on a repetition of the experiments. The remarkable variability in the circle of action of one and the same multiplier in various circuits, and of different multipliers in the same circuit, is completely explained by the * See Schweigger’s Jahrbuch, 1826, Part 2, where I have given a somewhat more detailed explanation of the separate points. be ae Wen ae pit d OHM ON THE GALVANIC CIRCUIT. 429 . : preceding consideration. For if we denote by A the sum of the tensions, and by L the reduced length of any galvanic oo et L x Bi tresses the magnitude of its current. If we now imagine A A L+na dicates the magnitude of the current when the multiplier is _ brought into the circuit as an integral part. Moreover, if we grant, for the sake of simplicity, that each of the » convolutions exerts the same action on the magnetic needle, the action of the - multiplier on the magnetic needle is evidently - nA = Car yhen the action of an exactly similar coil of the circuit, without ultiplier on the needle, is taken as A L dence it follows directly that the action on the magnetic needle is augmented or weakened by the multiplier, according as nL s greater or smaller than L + nA, i. e., according as n times the reduced length of the circuit without the multiplier is greater or er than the reduced length of the circuit with the multiplier. ther, a mere glance at the expression by which the action multiplier on the needle has been determined, will show e greatest or smallest action occurs as soon as L may lected with reference to m A, and is expressed by A rey f we compare this extreme action of the multiplier with that bor the multiplier shows that it is proportional to the ten- 1 Of the circuit, and independent of its reduced length; conse- 430 OHM ON THE GALVANIC CIRCUIT. quently the extreme action of the same multiplier may serve not merely to determine the tensions in various circuits, but it also indicates that the extreme action may be also augmented to the same degree as the sum of the tensions is increased, which may be effected by forming a voltaic combination with several simple circuits. If we represent the actual length of a coil of the multiplier by J, its conductibility by x, and its section by l ; : w, so that A= a the expression for the extreme action of the multiplier is converted into A Mie ecare from which it will be seen that the extreme action of two mul- tipliers of different metals, constructed of wire of the same thickness, are in the same ratio to each other as the conductibi- lilies of these metals, and that the extreme actions of two multi- pliers formed of wire of the same metal, are in the same pro- portion to each other as the sections of the wires. All these various peculiarities of the multiplier I have shown to be founded on experience, partly on experiments performed by other persons, and partly on those by myself*. The most recent experiments made on this subject on thermo-circuits, have, in a different, and, in a certain sense, opposite manner, already afforded the conclusion deduced above from an equa- tion of the reduced lengths, that. the sum of the tensions in a thermo-circuit is far weaker than in the ordinary hydro-circuits ; and a preliminary comparison has convinced me, that, with re- spect to the heating effects, if they are to be predicted with certainty, a voltaic combination of some hundred well-chosen simple thermo-circuits is requisite, and for chemical effects of some energy a far greater apparatus. Experiments, which place this prediction beyond doubt, will afford a new and not un- important confirmation to the theory here propounded. The previous considerations are also sufficient to indicate the process which is carried on when the galvanic circuit is divided at any place into two or more branches. For this purpose 1 call attention to the circumstance, that at the time we found the equation 5 = = we also obtained the rule that the magni- tude of the current in any homogeneous part of the galvanic” * Schweigger’s Jahrbuch, 1826. Part 2; and 1827. . OHM ON THE GALVANIC CIRCUIT. 431 circuit is given by the quotient of the difference between the electrical forces existing at the extremities of the portion and its reduced length. It is true, this rule was only advanced above for the case in which the circuit nowhere divides into several branches; but a very simple consideration, analogous to the one then made, derived from the equality of the abducted and adducted quantity of electricity in all sections of each pris- matic part, is sufficient to prove that the same rule holds good for every single branch in case of a division of the circuit. Let us suppose that the circuit be divided, for instance, into three _ branches, whose reduced lengths are 4, A’, A”; and, moreover, that at each of these places the undivided circuit and the single branches possess equal electrical force, and consequently no ten- sion occurs there, and designate by « the difference between the electrical forces at these two places; then, according to the above rule, the magnitude of the current in each of the three branches is a gz a Repo “ANS whence it directly follows that the currents in the three branches are inversely as their reduced lengths ; so that each separate one may be found when the sum of all three together is known. But the sum of all three is evidently equal to ‘the magnitude of the current at any other place of the non-divided portion of the _ citeuit, for otherwise the permanent state of the circuit, which is ~ still constantly supposed, would not be maintained. If we connect with this the conclusion resulting from the above con- siderations. namely, that the magnitude of the current, and the nature of each homogeneous part of the circuit, give the dip of the corresponding straight line, representing the separation of the electricity, we are certain that the figure of the separation belonging to the non-divided portion of the circuit must remain the same so long as the current in it retains the same magni- tude, and vice versd; whence it follows that the variability of the current in the non-divided portion necessarily supposes that the difference between the electrical forces at the extremi- ties of this portion is constant. If we now imagine, instead of the Separate branches, a single conductor of the reduced length A brought into the circuit which does not at all alter the magni- tude of its current and its tensions, then, according to what has just been stated, the difference between the Besicibal forces 432 OHM ON THE GALVANIC CIRCUIT. at its extremities must still always remain ~, and consequently _ be . a a a Al =e th ir red or 1 1 1 1 SRE tava) vic itn ahs which equation serves to determine the value of A. But if this value is known, and we call A the sum of all the tensions in the _ circuit, and L the reduced length of the non-divided portion of | the circuit, we obtain, as is known, for the magnitude of the cur- — rent in the last-mentioned circuit A L+ A’ which is equal to the sum of the currents occurring in the separate t branches. Now since it has already been proved that the cur- rents in the separate branches are in inverse proportion to one another as the reduced lengths of these branches, we obtain for — the magnitude of the current in the branch whose reduced length is A, cid tel L+A* 2X’ in the branch whose reduced length is a’, mRedetal ieee Tyteedia.pal? and in the branch whose reduced length is 4", A A L+A‘ ar This remote, and hitherto but slightly noticed peculiarity of the © galvanic circuit, I have also found to be perfectly confirmed © by experiment*. I herewith conclude the consideration of such galvanic cir-_ cuits which have already attained the permanent state, and which neither suffer modifications by the influence of the surrounding atmosphere, nor by a gradual change in their che- mical composition. But from this point the simplicity of the subject decreases more and more, so that the previous element- ary treatment soon entirely disappears. With respect to those | * Schweigger’s Jahrbuch, 1827. OHM ON THE GALVANIC CIRCUIT. 433 circuits on which the atmosphere exercises some influence, and whose condition varies with time, without this change origin- ating in a progressive chemical transformation of the circuit, and is thus distinguished from all the others by the magnitude of its current being different at different places,—I have been content, with respect to each of these, always to treat of only the most simple case, as they but rarely occur in nature, and in general appear to be of less interest. I have adopted this plan the more willingly, as I intend to return to this subject at some future time. But with regard to that modification of galvanic circuits which is produced by a chemical change in the circuit, proceeding first from the current, and then again react- ing on it, I have devoted separate attention to it in the Ap- pendix. The course adopted is founded on a vast number of experiments performed on this subject, the communication of which, however, I omit, because they appear to be capable of being far more accurately determined than I was able to do at that time, from failing to attend to several elements in opera- tion; nevertheless, I consider it proper to mention the cir- cumstance in this place, in order that the careful manner with which I advance in the inquiry, and which I consider to be due to truth, may not operate more than is just against its reception. Ihave sought for the source of the chemical changes caused by the current, in the above-described peculiar mode of separa- tion of the electricity of the circuit; and, I can scarcely doubt, have at least found the main cause. It is immediately evident that each disk belonging to a section of a galvanic circuit, which obeys the electric attractions and repulsions and does not oppose their movement, must in the closed circuit be propelled always towards one side only, as these attractions and repul- sions, in consequence of the continually varying electric force, are different at the two sides; and it is mathematically de- monstrable that the force with which it is driven to the one side, is in the ratio compounded of the magnitude of the electric cur- rent and of the electric force in the disk. It is true, how- ever, that merely a change of position in space would be imme- diately produced by this. But if this disk be regarded as a com- pound body, the constituent parts of which, according to electro- chemical views, are distinguished by a difference in their elec- trical relation to one another, it thence directly follows that this 434 OHM ON THE GALVANIC CIRCUIT. one-sided pressure on the various constituent parts would in most cases act with unequal force, and sometimes even in con- trary direction, and must thus excite a tendency in them to se- parate from one another. From this consideration results a di- stinct activity of the circuit, tending to produce a chemical change in its parts, which I have termed its decomposing force, and I have endeavoured to determine its magnitude for each particu- lar case. This determination is independent of the mode, in which the electricity may be conceived to be asssociated with the atoms.* Granting, which seems to be most natural, that the electricity is diffused proportionately to the mass over the space which these bodies then occupy, a complete analysis will show that the decomposing force of the circuit is in direct propor- tion to the energy of the current, and, moreover, that it depends on a coefficient, to be derived from the nature of the constituent parts and their chemical equivalents. From the nature of this de- composing force of the circuit, which is of equalenergy at all places of an homogeneous portion, it directly follows, that when it is capable of overcoming, under all circumstances, the reciprocal connexion of the constituent parts, the separation and abduc- tion of the constituents to both sides of the circuit are limited solely by mechanical obstacles ; but if the connexion of the con- stituent parts inter se, either immediately at the commencement everywhere, or in the course of the action anywhere, overcome the decomposing force of the circuit, then from that time no further movement of the elements can take place. This general descrip- tion of the decomposing force is in accordance with the experi- ments of Davy and others. There is a peculiar state which seems to be produced in ~ most cases of the separation of the two elements of a chemically — compound liquid, which is especially worthy of attention, and which is caused in the following manner. When the decompo- — sition is confined solely to a limited portion of the circuit, and — the constituent parts of the one kind are propelled towards the be one side of this part, and the constituent parts of the other © kind to its opposite side, then, for this very reason, a na- tural limit is prescribed to the action; for the constituent — —_ * J shall shortly have occasion to speak of the peculiar import of this re- mark, when I shall attempt to reduce the actions of the parts of a galvanic cir- _ cuit on one another, as discovered by Ampére, to the usual electrical attrac-_ tions and repulsions. a 4 OHM ON THE GALVANIC CIRCUIT. 435 part preponderating on the one side of any disk, within the portion in the act of decomposition, will, by force of its inate repulsive power, constantly oppose the movement of a similar constituent to the same side, so that the decomposing force of the circuit has not merely to overcome the constant connexion of the two constituents infer se, but also this reaction of each constituent on itself. It is hence evident that a cessation in the chemical change must occur, if at any time there arises an equilibrium between the two forces. This state, founded on a peculiar chemical and permanent separation of the con- stituents of the portion of the circuit in the act of decom- position, is the very one from which I started, and whose nature I have endeavoured to determine as accurately as pos- sible in the Appendix. Even the mere description of the mode of origin of this highly remarkable phenomenon shows that at the extremities of the divided portion no natural equilibrium can occur, on which account the two constituents must be re- tained at these two places by a mechanical force, unless they pass over to the next parts of the circuit, or, where the other circumstances allow, separate entirely from the circuit. Who would not recognise in this plain statement all the chief circum- stances hitherto observed of the external phenomenon in che- mical decompositions by the circuit ? If the current, and, at the same time, the decomposing force, be suddenly interrupted, the separated constituents gradually return to their natural equilibrium ; but tend to re-assume im- mediately the relinquished state, if the current is re-established. During this process, both the conductibility, and the mode of excitation between the elements of the portion in the act of decomposition, obviously vary with their chemical nature; but this necessarily produces a constant change in the electrical separation, and in the magnitudes of the current in the gal- vanic circuit dependent. thereon, which only finds its natural . limits in the permanent state of the electrical separation. For the accurate determination of this last stage of the electric | current it is requisite to be acquainted with the law which governs the conductibility and force of excitation of the vari- able mixtures, formed of two different liquids. Experiment has hitherto afforded insufficient data for this purpose, I have therefore given the preference to a theoretical supposition, which will supply its place until the true law is discovered. 436 OHM ON THE GALVANIC CIRCUIT. With the help of this law, which is not altogether imaginary, I — now arrive at the equations which make known for each case all the individual circumstances constituting the permanent — state of the chemical separation in the galvanic circuit; I have, — however, neglected the further use of them, as the present state r of our experimental knowledge in this respect did not appear — to me to repay the requisite trouble. Nevertheless, in order to compare in their general features the results of this examination — with what has hitherto been supplied by experiments, I have fully carried out one particular case, and have found that the | formula represents very satisfactorily the kind of wave of the — force, as I have above described it*. Having thus given a slight outline of the contents of this © Memoir, I will now proceed to the fundamental investigation — of the individual points. * Schweigger’s Jahrbuch, 1826. Part 2. (To be continued. } SCIENTIFIC MEMOIRS. VOL. I1.—PART VIII. ArticLe XIII. continued. The Galvanic Circuit investigated Mathematically. By Dr. G. 8. Oum*. Tue Gatvanic Circuit. A. General observations on the diffusion of electricity. 1. A PROPERTY of bodies, called into activity under certain circumstances, and which we call electricity, manifests itself in _ Space, by the bodies which possess it, and which on that ac- count are termed electric, either attracting or repelling one : another. _ In order to investigate the changes which occur in the | electric condition of a body A in a perfectly definite manner, | this body is each time brought, under similar circumstances, into ‘contact with a second moveable body of invariable electrical con- dition, called the Electroscope, and the force with which the | electroscope is repelled or attracted by the body is determined. This force is termed the electroscopic force of the body A; and to distinguish whether it is attractive or repulsive we place be- fore the expression for its measure the sign + in the one case, and — in the other. The same body A may also serve to determine the elec- troscopic force in various parts of the same body. For this purpose we take the body A of very small dimensions, so that when we bring it into contact with the part to be tested of any third body, it may from its smallness be regarded as a sub- | futeces ee puicke Kette mathematisch bearbeitet von Dr. G. S. Ohm: | VOL. Il. PART VIII. 2G 438 OHM ON THE GALVANIC CIRCUIT. the way described, will, when it happens to be different at the — various places, make known the relative difference with regard — to electricity between these places. i The intention of the preceding explanations is to give a sim- — ple and determinate signification to the expression “lectro-_ scopic force”; it does not come within the limits of our plan to take notice either of the greater or less practicability of this ~ process, nor to compare inter se the various possible modes of | proceeding for the determination of the electroscopic force. 2. We perceive that the electroscopic force moves from one | place to another, and from one body to another, so that it does not merely vary at different places at the same time, but also at a single place at different times. In order to determine in what manner the electroscopic force is dependent upon the time when it is perceived, and on the place where it is elicited, we must set out from the fundamental laws to which the exchange of electroscopic force occurring between the elements of a body is subject. These fundamental laws are of two kinds, either borrowed from experiment, or, where this is wanting, assumed hypothe= tically. The admissibility of the former is beyond all doubt, and the justness of the latter is distinctly evident from the coinci+| dence of the results deduced from calculation with those whic actually occur; for since the phenomenon with all its modifica- tions is expressed in the most determinate manner by calculation it follows, since no new uncertainties arise and increase the earliel ones during the process, that an equally perfect observation of nature must in a decisive manner either confirm or refute its statements. This in fact is the chief merit of mathematical) analysis, that it calls forth, by its never-vacillating expressions, a generality of ideas, which continually excites to renewed eX= periments, and thus leads to a more profound knowledge of nature. Every theory of a class of natural phenomena founde upon facts, which will not admit of analytical investigation im the form of its exposition, is imperfect; and no reliance is to | placed upon a theory developed in ever so strict a form, whieh is not confirmed to a sufficient extent by observation. So lo ng therefore as not even one portion of the effects of a natural f ore’ has been observed with the greatest accuracy in all its grad tions, the calculation employed in its investigation only treat OHM ON THE GALVANIC CIRCUIT. 439 on uncertain ground, as there is no touchstone for its hypo- theses, and in fact it would be far better to wait a more fit time ; but when it goes to work with the proper authority, it enriches, at least in an indirect manner, the field it occupies with new natural phenomena, as universal experience shows. I have thought it necessary to premise these general remarks, as they not only serve to throw more light on what follows, but also because they explain the reason why the galvanic phenomena have not long since been mathematically treated with greater success, although, as we shall subsequently find, the requisite course has been already earlier pursued in another, apparently less prepared, branch of Physics. __ After these reflections we will now proceed to the establish- ment of the fundamental laws themselves. _ 3. When two electrical elements, E and E’, of equal magni- tude, of like form and similarly placed with respect to each other, but unequally powerful, are situated at the proper distance from each other, they exhibit a mutual tendency to attain electric equilibrium, which is apparent in both constantly and unin- terruptedly approaching nearer to the mean of their electric state, until they have actually attained it. That is to say, both elements reciprocally change their electric state so long s a difference continues to exist between their electroscopic forces; but this change ceases as soon as they have both at- © fained the same electroscopic force. Consequently this change of the electric difference of the elements is so dependent that the me disappears at the same time with the other. We now sup- ose that the change, effected in an extremely short instant of \time in both elements, is proportional to the difference of their cotemporaneous electroscopic force and the magnitude of the instant of time; and without yet attending to aay material di- jstinetions of the electricity, it is always to be understood hat the forces designated by + and — are to be treated exactly \is opposite magnitudes. That the change is effected accurately vecording to the difference of the forces, is a mathematical sup- yosition, the most natural because it is the most simple; all the fest is given by experiment. The motion of electricity is effected jm most bodies so rapidly that we are seldom able to determine jts changes at the various places, and on that account we are 10t in a condition to discover by observation the law according 9 which they act. The galvanic phenomena, in which such 2G2 : 440 OHM ON THE GALVANIC CIRCUIT. changes occur in a constant form, are therefore of the highest — importance for testing this assumption: for if the conclusions — drawn from the supposition are thoroughly confirmed by those phznomena, it is admissible, and may then be applied without — any further consideration to all analogous researches, at least — within the same limits of force. We have assumed, in accordance with the observations hitherto ; made, that when by any two exteriorly like constituted elements, — whether they be of the same or of different matter, a mutual — change in their electrical state is produced, the one loses just so much force as the other gains. Should it hereafter be shown by experiments that bodies exhibit a relation similar to that which in the theory of heat is termed the capacity of bodies, — the law we have established will have to undergo a slight altera- tion, which we shall point out in the proper place. 4, When the two elements E and E’ are not of,equal magni- | tude, it is still allowed to regard them as sums of equal parts. Granting that an element E consist of m perfectly equal parts, - and the other E! of m’ exactly similar parts, then, if we imagine the elements E and E’ exceedingly small in comparison with their mutual distance, so that the distances from each part of the one to each part of the other element are equal, the sum * of the actions of all the m! parts of the element E! on a part of E will be m’ times that which a single part exerts, and the sum of all the actions of the element E! on all the m parts of E will be mm! times that which a part of E’ exerts on a partof E. It is hence evident, that in order to ascertain the mutual actions of dissimilar elements on each other, they must be taken as pro- portional not merely to the difference of their electroscopic forces and their duration, but also to the product of their relative mag- nitudes. We shall in future term the sum of the electroscopice actions, referred to the magnitude of the elements—by which therefore we have to understand the force multiplied by the magnitude of the space over which it is diffused, in the case where the same force prevails at all places in this space—the quantity of electricity, without intending to determine anythir o thereby with respect to the material nature of electricity. le same observation is applicable to all figurative expressions intro- duced, without which, perhaps for good reasons, our language could not exist. ¥ In cases where the elements cannot be regarded as evanescer { OUM ON THE GALVANIC CIROUIT. 441 in comparison with their relative distances, a function, to be de- termined separately for each given case from their dimensions and their mean distance, must be substituted for the product of the magnitudes of the two elements, and which we will desig- nate where it is employed by F, 5. Hitherto we have taken no notice of the influence of the mutual distance of the elements between which an equalization of their electric state takes place, because as yet we have only considered such elements as always retained the same relative distance. But now the question arises, whether this exchange is directly effected only between adjacent elements, or if it ex- tends to others more distant, and how on the one or the other upposition is its magnitude modified by the distance? Fol- ing the example of Laplace, it is customary in cases where olecular actions at the least distance come into play, to em- y a particular mode of representation, according to which a direct mutual action between two elements separated by others, still occurs at finite distances, which action, however, - decreases so rapidly, that even at any perceptible distance, be i ever so minute, it has to be considered as perfectly eva- nescent. Laplace was led to this hypothesis, because the suppo- sition that the direct action only extended to the next element @ quantities*,—a non-uniformity which is opposed to the t of the differential calculus. This apparent unavoidable * Poisson, in his Mémoire sur la Distribution de la Chaleur, Journ. de cole Polytechn. cah. xix. expresses himself on this subject thus :— Ifa bar be divided, by sections perpendicular to the axis, into an infinite number of infinitely saat elements, and if we consider the mutual action three consecutive elements, that is to say, the quantity of heat that the intermediate clement at each instant communicates to and abstracts from the two others, in proportion to the positive or negative excess of its tempera- re over that of each of them, we may thence easily determine the augmenta- on of temperature of this element during an infinitely small instant; assu- ming therefore this quantity equal to the differential of its temperature taken with respect to the time, the equation of the propagation of heat according to he length of the bar is formed; but on examining the question more atten- tely, it is seen without difficulty that this equation would be founded on the parison of two infinitely small non-homogeneous quantities, or of different which would be contrary to the first principles of the differential calculus. lifficulty can only be made to disappear by supposing, as M. Laplace ' emarked, (Mémoires de la Ire classe de |’Institut, année 1809,) that he action of each element of the bar extends itself beyond the contact, and nn itself on all the elements contained within a finite space, as small ase,’ 442 QHM ON THE GALVANIC CIRCUIT. disproportion between the members of a differential equation, belonging nevertheless necessarily to one another, is too remark- able not to attract the attention of those to whom such inquiries are of any value; an attempt therefore to add something to the explanation of this znigma will be the more proper in this place, as we gain the advantage of rendering thereby the subsequent considerations more simple and concise. We shall merely take as an instance the propagation of electricity, and it will not be difficult to transfer the obtained results to any other similar subject, as we shall subsequently have occasion to demonstrate — in another example. 6. Above all, it is requisite that the term goodness of conduc- — tion be accurately defined. But we express the energy of con- — duction between two places by a magnitude which, under other- — wise similar circumstances, is proportional to the quantity car- — ried over in a certain time from one place to the other multi- — plied by the distance of the two places from each other. If- the two places are extended, then we have to understand by their distance the straight line connecting the centres of the dimensions of the two places. If we transfer this idea to two electric elements, E and E’, and call s the mutual distance of their centres, g the quantity of electricity, which under accu- rately determined and invariable circumstances is carried over from one element to the other, and x the conductibility between them, } = 0.8. We will now endeavour to determine more precisely the quan-_ tity of electricity denoted by g. According to § 4 the quantity of electricity, which is transferred in an exceedingly short time’ from one element to the other, is, the distance being invariable, in general proportional to the difference between the electro- scopic forces, the duration, and the size of each of the two ele- ments. If therefore we designate the electroscopic forces of the two elements E and E! by uw and w/, and the space they occupy by m and m', we obtain for the quantity of electricity carried over from E’ to E in the element of time d¢ the following expression : i amm! (u’ —u) dt, where « represents a coefficient depending in some way on the distance s. This quantity changes every moment if uw’ — wis yariable; but if we suppose that the forces u! and u remaix Seng: Fes OHM ON THE GALVANIC CIRCUIT. 443 constant at all times, it merely depends on the magnitude of the instant of time d7¢, we can consequently extend it to the unity of time; if we place the present constant difference of the forces u’ — u equal to the unity of force, it then becomes am mM. This quantity of electricity is for the two elements E and EH’ whose position is invariable, constant under the same circum- stances, on which account it may be employed in the determi- nation of the power of conduction just mentioned. For if we understand by g the quantity of electricity transferred from E! to E in the unity of time, with a constant difference of the electroscopic forces equal to the unity of force, we have e=amm, and then x=amm s. If we take from this last equation the value of « m m’ and sub- stitute it in the expression amm (u'—u) dt, we obtain for the variable quantity of electricity which passes over in the instant of time d ¢ from E’ to EK, the following : x (ul —u) dt (3) 8 which expression is not accompanied by the above-mentioned disproportion between the members of the differential equation, as will soon be perceived. _ 4. The course hitherto pursued was based upon the suppo- Sition that the action exerted by one element on the other is proportional to the product of the space occupied by the two elements, an assumption which, as was already observed in § 4, can no longer be allowed in cases where it is a question of the mutual action of elements situated indefinitely near each other, because it either establishes a relation between the magnitudes of the elements and their mutual distances, or prescribes to these elements a certain form. The previously found expres- sion (3) for the variable quantity of electricity passing from one element to the other, possesses therefore no slight advan- tage in being entirely independent of this supposition ; for what- ever may have to be placed in any determinate case instead of the product m m’, the expression (3) constantly remains the 444 OHM ON THE GALVANIC CIRCUIT. same, this peculiarity being solely referrible to the power of conduction x. If, for instance, F designate, as was stated in ~ § 4, the function, corresponding to such a case, of the dimen- sions and of the mean distance of both elements, the expression amm (u’—u) dt not merely changes apparently into F (wv —u) dt, but also the equation x=amm s into the other, oi ee (©) : so that if we take the value of F from this equation and place it in the above expression, we always obtain x (ul —u)dt. s Moreover, the circumstance of the expression (¢) still remain- — ing valid for corpuscles, whose dimensions are no longer inde- finitely small, is of some importance when the same electrosco- pic force only exists merely at all points of each such part. It is hence evident how intimately our considerations are allied to the spirit of the differential calculus ; for uniformity in all points — with reference to the property which enters into the calculation is precisely the distinctive characteristic required by the differ- ential calculus from that which it is to receive as an element. If we institute a more profound comparison between the pro- cess originating with Laplace and that here advanced, we shall — arrive at some interesting points of comparison. If for instance i we consider that for infinitely small masses at infinitely short distances all particular relations must necessarily have the same weight as for finite masses at finite distances, it is not directly evident how the method of the immortal Laplace—to whom we are indebted for so many valuable explanations respect- ing the nature of molecular actions,—according to which the elements must be constantly treated as if they were placed at finite distances from each other, could nevertheless still afford correct results; but we shall find on closer examination that it acts in fact otherwise than it expresses. Indeed, since Laplace, when determining the changes of an element by all surrounding it, makes the higher powers of the distance disappear compared with the lower, he therewith assumes, quite in the spirit of the t Z OHM ON THE GALVANIC CIRCUIT. 445 differential calculus, the difference of action itself to be infinitely small, but terms it finite, and treats it also as such; whence it is immediately apparent that he in fact treats that which is infi- nitely small at an infinitely short distance as finite. Disregard- ing however the great certainty and distinctness which accom- pany our manner of representation, there might still be some- thing more to say, and perhaps with some justice, against La- place’s mode of treatment in favour of ours, in this respect, that the former takes not the least account of the possible nature of the given elements of bodies, but merely has to do with imagi- nary elements of space, by which the physical nature of the bodies is almost entirely lost sight of. We may, to render our assertion intelligible by an example, undoubtedly imagine bo- dies in nature which consist only of homogeneous elements, but whose position to each other, taken in one direction, might be different than when in another direction; such bodies, as our mode of representation immediately shows, might conduct the electricity in one direction in a different manner than in another, notwithstanding that they might appear uniform and equally dense. In such a case, did it occur, we should have to take refuge, according to Laplace, in considerations which have remained entirely foreign to the general process. On the other hand, the mode im which bodies conduct affords us the means by which we are enabled to judge of their internal structure, which, from our almost total ignorance on the subject, cannot be immediately shown. Lastly, we may add, that this, our hi- therto developed view of molecular actions, unites in itself the two advanced by Laplace and by Fourier in his theory of heat, and reconciles them with each other. 8. We need now no longer hesitate about allowing the elec- trical action of an element not to extend beyond the adjacent surrounding elements, so that the action entirely disappears at every finite distance, however small. The extremely limited circle of action with the almost infinite velocity with which elec- tricity passes through many bodies might indeed appear sus- Picious ; but we did not overlook on its admission, that our com- parison in such cases is only effected by an imaginary relative standard, which is deceitful, and does therefore not justify us to vary a law so simple and independent until the conclusions drawn from it are in contradiction to nature, which in our sub- ject, however, does not seem to be the case. 446 OHM ON THE GALVANIC CIRCUIT. The sphere of action thus fixed by us, has, although it is infi- nitely small, precisely the same circumference as that introduced by Laplace, and called finite, where he lets the higher powers of the distance vanish compared with the lower, the reason of which may be found already in what has been stated above. The sup- position of a finite distance of action in our sense would corre- spond to the case where Laplace still retains higher powers of distance together with the lower. 9. The bodies on which we observe electric phznomena are in most cases surrounded by the atmosphere ; it is therefore re- quisite, in order to investigate profoundly the entire process, not to disregard the changes which may be produced by the adjacent air. According to the experiments left us by Coulomb on the diffusion of electricity in the surrounding atmosphere, the loss in force thus occasioned is (during a very short con- stant time), at least when the intensities are not very consider- able, on the one hand proportional to the energy of the electri- city, and on the other is dependent on a coefficient varying according to the cotemporaneous nature of the air, but other- wise invariable for the same air. The knowledge of this enables us to bring the influence of the atmosphere on galvanic phzeno- mena into calculation wherever it might be requisite. It must however not be overlooked here, that Coulomb’s experiments were made on electricity which had entered into equilibrium and was no longer in the process of excitation, with respect to which both observation and calculation have convinced us that — it is confined to the surface of bodies, or merely penetrates to a — very slight depth into their interior; for from thence may be — drawn the conclusion, of some importance with respect to our — subject, that all the electricity present in those experiments may have been directly concerned in the transference to the atmo- sphere. If we now connect with this observation the law just announced, according to which two elements, situated at any finite distance from each other, no longer exert any direct action on each other, we are justified in concluding, that where — the electricity is uniformly diffused throughout the entire mass — of a finite body, or at least so that proportionately but a small quantity resides in the vicinity of the surface, which case does not in general occur when it has entered into motion, the loss which is occasioned by the circumambient air can be but ex- tremely small in comparison to that which takes place when the OHM ON THE GALVANIC CIRCUIT. 447 entire force is situated immediately at the surface, which inva- riably happens when it has entered into equilibrium; and thence, therefore, it happens that the atmosphere exerts no perceptible influence on galvanic phenomena in the closed circuit when this is composed of good conductors, so that the changes produced by the presence of the atmosphere in phe- nomena of contact-electricity may be neglected in such cases. This conclusion, moreover, receives new support from the cir- cumstance, that in the same cases the contact-electricity only remains during an exceedingly short time in the conductors, and even on that account would only give up a very slight por- tion to the air, even if it were in immediate contact with it. _ Although, from what has been stated, it is placed beyond all doubt that the action of the atmosphere has no perceptible in- fluence on the magnitude of effect of the usual galvanic circuits, it by no means is intended to admit the reverse of the conclu- sion, viz. that the galvanic conductor exerts no perceptible in- fluence on the electric state of the atmosphere; for mathema- tical investigation teaches us that the electroscopic action of a body on another has no direct connexion with the quantity of electricity which is carried over from the one to the other. 10. We arrive at last at that position founded on experi- ment, and which is of the highest importance for the whole of natural philosophy, since it forms the basis of all the phzeno- mena to which we apply the name of galvanic: it may be ex- pressed thus: Different bodies, which touch each other, con- stantly preserve at the place of contact the same difference be- tween their electroscopic forces by virtue of a contrariety pro- ceeding from their nature, which we are accustomed to desig- nate by the expression electric tension, or difference of bodies. Thus enounced, the position stands, without losing any of its simplicity, in all the generality which belongs to it ; for we are nearly always referred to it by every single phenomenon. Moreover, the above expression is adopted in all its generality, either expressly or tacitly, by all philosophers in the explana- tion of the electroscopic phenomena of the voltaic pile. Ac- cording to our previously developed ideas respecting the mode in which elements act on one another, we must seek for the source of this phenomenon in the elements directly in contact, and con- sequently we must allow the abrupt transition to take place from one body to the other in an infinitely small extent of space. 448 OHM ON THE GALVANIC CIRCUIT. 11. This being established, we will now proceed to the sub- ject, and in the first place consider the motion of the electri- city in a homogeneous cylindric or prismatic body, in which all points throughout the whole extent of each section, per- pendicular to its axis, possess contemporaneously equal electro- scopic force, so that the motion of the electricity can. only take place in the direction of its axis. If we imagine this body divided by a number of such sections into disks of infinitely small thickness, and so that in the whole circumference of each disk the electroscopic force does not vary sensibly for each pair of such disks, the expression 3 given in § 6 can be applied to determine the quantity of electricity passing from one disk to the other; but by the limitation of the distance of action to only infinitely small distances mentioned in the preceding para- graph, its nature is so modified that it disappears as soon as the divisor ceases to be infinitely small. If we now choose one of the infinite number of sections invariably for the origin of the abscissz, and imagine any- where a second, whose distance from the first we denote by 2, then dz represents the thickness of the disk there situated, which we will designate by M. If we conceive this thickness of the disk to be of like magnitude at all places, and term uw the electroscopic force present at the time ¢ in the disk M, whose abscissa is 2, so that therefore w in general will be a function of t and x; if we further suppose w! and w, to be the values of u when « + da and « — d= are substituted respectively for 2, then w and u, evidently express the electroscopic forces of the disks situated next the two sides of the disk M, of which we will denote the one belonging to the abscissa 2 + dx by M’, and that belonging to the abscissa 2 + dx by M,; and it is clearly evident that the distance of the centre of each of the disks M’ and M, from the centre of the disk M isda. Con- sequently, by virtue of the expression (¢) given in § 6, if x re- presents the conducting power of the disk M! to M, x (ul —u)dt dx is the quantity of electricity which is transferred during the in- terval of time d¢ from the disk M! to the disk M, or from the latter to the former, according as wu! — wu is positive or negative. In the same manner, when we admit the same power of conduc- tion between M, and M, " OHM ON THE GALVANIC CIRCUIT. 449 x (u,—u) dt Whar i: is the quantity of electricity passing over from M, to M when the expression is positive, and from M to M, when it is nega- tive. The total change of the quantity of electricity which the disk M undergoes from the motion of the electricity in the in- terior of the body in the particle of time d ¢, is consequently x(ul +u,—2u) dt dx ‘ and an increase in the quantity of electricity is denoted when this value is positive, and when negative a diminution of the same. But according to Taylor’s theorem du d?u dx? a - — a a’ Ut tay 4# + 7 Qo teers and in the same way du d?u dx ae ar aad Riaboay geoaiignmeetas consequently ! au 1 u+tu=2ut = Ax. d x According to this the expression just found for the total change of the quantity of electricity present in the disk M is converted during the time d¢ into 2 — dx dt, _ where x represents the power of conduction which prevails from one disk to the adjacent one, which we suppose to be invariable throughout the length of the homogeneous body. It must here be observed, that this value x is, on account of the infinitely small distance of action, proportional to the section of the cylin- drie or prismatic body ; if therefore we denote the magnitude of this section by w, and separate this factor from the value x, always calling the remaining portion x, the former expression changes into the following : x w we dex dt, in which x now represents the conductibility of the body inde- pendent of the magnitude of the section, which we will term the absolute conductibility of the body in opposition to the for- mer, which may be called the relative. Henceforward wherever 450 OHM ON THE GALVANIC CIRCUIT. the word conductibility occurs without any closer definition, the absolute conductibility is always to be understood. Hitherto we have left out of consideration the change which the disk suffers from the adjacent atmosphere. This influence may easily be determined. If, for instance, we designate by ¢ the circumference of the disk belonging to the abscissa x, then c d x is the portion of its surface which is exposed to the air;. consequently, according to the experiments of Coulomb, men- tioned in § 9, beuduxdt is the change of the quantity of electricity which is occasioned in the disk M by the passing off of the electricity into the at- mosphere during the moment of time d¢, where 6 represents a coefficient dependent on the cotemporaneous nature of the atmosphere, but constant for the same atmosphere. It expresses a decrease when wu is positive, and an increase when » is nega- tive. But in accordance with our original supposition, this action cannot occasion an inequality of the electroscopic force in the same section of the body ; or at least, this inequality must be so slight that no perceptible alteration is produced in the other quantities 5 ; a circumstance which may nearly always be supposed in the galvanic circuit. Accordingly, the entire change which the quantity “3 elec- tricity in the disk M undergoes in the moment of time d ¢ is x w in which the portion is comprised which arises from the motion of the electricity in the interior of the body as well as that which is caused by the cireumambient atmosphere. But the entire change of the electroscopic force u in the disk M effected in the moment of time df is - Sas: consequently the total a in the quantity of electricity in the disk M during the time d¢ is where, however, it is supposed that under all circumstances similar changes in the electroscopic force correspond to similar changes in the quantity of electricity. If observation showed that different bodies of the same surface underwent a different RAG OHM ON THE GALVANIC CIRCUIT. 451 change in their electroscopic force by the same quantity of elec- tricity, then there would still remain to be added a coefficient y corresponding to this property of the various bodies. Experience has not yet decided respecting this supposition borrowed from the relation of heat to bodies. If we assume the two expressions just’ found for the entire change in the quantity of electricity in the disk M during the moment of time d¢ to be equal, and divide all the members of the equation by w dx dt, we obtain du a@u be Deng (iad atte inant (a) from which the electroscopic force u has to be determined as a function of x and ¢. 12. We have in the preceding paragraph found for the change in the quantity of electricity occurring between the disks M! and M during the time dt x (u'—u) dt Fae? and have seen that the direction of the passage is opposed to the course of the abscissze when the expression is positive ; on the contrary, it proceeds in the direction of the abscissee when it is negative. In the same way the magnitude of the transition between the disks M, and M, when we retain the same relation to its direction, is x (u,—u) dt Pe yiic If we substitute in these two expressions for w, and w! the trans- formations given in the same paragraph, and at the same time x for x, i.e. the absolute power of conduction for the relative, we obtain in both cases du a dt, whence it results that the same quantity of electricity which enters from the one side into the disk M during the element of time d?, is again in the same time expelled from it towards the other side. If we imagine this transmission of the electricity, occurring at the time ¢ in the disk belonging to the abscissa Bs of invariable energy reduced to the unity of time, call it the electric current, and designate the magnitude of this current by 8, then 452 OHM ON THE GALVANIC CIRCUIT. S= xo; () and in this equation positive values for S show that the current takes place opposed to the direction of the abscisse ; negative, that it occurs in the direction of the abscisse. 13. In the two preceding paragraphs we have constantly had in view a homogeneous prismatic body, and have inquired into the diffusion of the electricity in such a body, on the supposi- tion that throughout the whole extent of each section, perpen- dicular to its length or axis, the same electroscopic force exists at any time whatsoever. We will now take into consideration the case where two prismatic bodies A and B, of the same kind, but formed of different substances, are adjacent, and touch each other in a common surface. If we establish for both A and B the same origin of abscisse, and designate the elec- troscopic force of A by u, that of B by w', then both w and w! are determined by the equation (a) in paragraph 11, if x only retain the value each time corresponding to the peculiar substance of each body: but w represents a function of ¢ and 2, which holds only so long as the abscissa 2 corresponds to points in the body A; u on the other hand denotes a function of ¢ and x, which holds only when the abscissa x corresponds to the body B. But there are still some other conditions at this common surface, which we will now explain. If we denote for this purpose the separate values of w and w, which they first assume at the common surface, by enclosing the general ones between crotchets, we find according to the law advanced in § 10 the following equation between these separate values : (u) — Ww) =4, where a@ represents a constant magnitude otherwise dependent on the nature of the two bodies. Besides this condition, which relates to the electroscopic force, there is still a second, which has reference to the electric current. It consists in this, that the electric current in the common surface must in the first place possess equal magnitude and like direction in both bodies, — or, if we retain the common factor a, $ du yg (4H xo (2) = #e (22), where x represents the actual power of conduction of the als OHM ON THE GALVANIC CIRCUIT, 453 body A, x! that of the body B, and (7 au = ) the particular du du. : ; values of =p immediately belonging to them at the com- mon surface, and in which it was assumed that the origin of the abscissee was not taken on this common surface. The necessity of this last equation may easily be conceived; for were it otherwise, the two currents would not be of equal energy in the common surface, but there would be more con- yeyed from ‘the one body to this surface than would be abs- tracted from it by the other; and if this difference were a finite portion of the entire current, the electroscopic force would _ increase at that very place, and indeed, considering the sur- F ' prising fertility of the electric current, would arrive in the shortest time to an exceedingly high degree, as observation has long since demonstrated. Nor can a smaller quantity of elec- tricity be imparted from the one body to the common surface than it is deprived of by the other, as this circumstance would be evinced by an infinitely high degree of negative electricity. It is not absolutely requisite eo the validity of the preceding determinations, that the two bodies in contact have the same base. The section in the one prismatic body may be different in size and form to that in the other, if this does not render the electroscopic force sensibly different at the various points of 1e same section, which, considering the great energy with which the electricity tends to equilibrium, will not be the case when the bodies are good conductors, whose length far surpasses their other dimensions. In this case everything remains as before, only that the section of the body B must everywhere be distinguished from that of A; consequently the second condi- tional equation for the place where the two bodies are in contact changes into the following :— 1 (2¥ x w ot) =e Ps b Dire » still represents the section of A, but w! that of the body B, which at present differs from the former. There may even exist in the prolongation of the body A two prismatic bodies, B and C, separated from each other, which are both situated immediately on the one surface of A. If in this case x! w! uw! signifies for the body B, and x" w', w for the body VOL. II. PART VIII. 2n 454 OHM ON THE GALVANIC CIRCUIT. C what xu does for A, we obtain instead of the one conditional equation the two following :— (u) — kas =4, (u) — (ul) =a’, where a represents the electric tension between the bodies A and B, and a! that between A and C. In the same manner we now obtain instead of the second conditional equation the fol- lowing :— x w (=) Re) ) + x! a! (ai dz dx dx) It is immediately apparent how these equations must change when a greater number of bodies are combined. We shall not enter further into these complications, as what has been stated suffices to throw sufficient light upon the changes which have in such a case to be performed on the equations. 14. To avoid misconception, I will, at the close of these gene- ral observations, once more accurately define the circle of appli- cation within which our formulz have universal validity. Our whole inquiry is confined to the case where all the parts of the | same section possess equal electroscopic force, and the magni- tude of the section varies only from one body to the other. The nature of the subject, however, frequently gives rise to circum- stances which render one or the other of these conditions super-_ fluous, or at least diminishes their importance. Since the know-— ledge of such circumstances is not without use, I will here illus-— trate the most prominent by an example. A circuit of copper, zinc, and an aqueous fluid, will wholly come under the above formula when the copper and zine are prismatic and of equal section ; when, further, the fluid is like- wise prismatic and of the same or of smaller section, and its terminal surfaces everywhere in contact with the metals. Nay, when only these last conditions are fulfilled with respect to the fluid, the metals may possess equal sections or not, and touch one another with their full sections, or only at some points, and even their form may deviate considerably from the prisma- tic form, and nevertheless the circuit must constantly obey the laws deduced from our formulz ; for the motion of the elec- tricity produced with such ease in the metals, is obstructed to such a considerable extent by the non-conductive nature of the fluid, that it gains sufficient time to diffuse itself thoroughly OHM ON THE GALVANIC CIRCUIT. 455 with equal energy over the metals, and thus re-establishes in the fluid the conditions upon which our calculation is founded. But it is a very different matter when the prismatic fluid is only touched in disproportionately small portions of its surfaces by the metals, as the electricity arriving there can only advance slowly and with considerable loss of energy to the untouched _ surfaces of the fluid, whence currents of various kinds and di- rections result. The existence of such currents has been suffi- ciently demonstrated by Pohl’s manifoldly varied experiments, and nothing more now stands in the way of their determination _ by analysis, after the additions which it has received from the _ successful investigations respecting the theory of heat, than the ¥ ‘complexities of the expressions. Since their determination pe, exceeds the limits of this small work, which has for its object to __ inyestigate the current only in one dimension, we will defer them _ to a more fit occasion. We will now proceed to the application of the formulz ad- vanced, and divide, for the sake of a more easy and general survey, the whole into two sections, of which the one will treat of the electroscopic phenomena, and the other of the phzno- mena of the electric current. B. Electroscopic Phenomena. _ 15. In our preceding general determinations we have con- stantly confined our attention to prismatic bodies, whose axes, By on which the abscissz have been taken, formed a straight line. But all these considerations still retain their entire value, if we ima sine the conductor constantly curved in any way whatso- ‘ever, and take the abscisse on the present curved axis of the conductor. The above formule acquire their entire applica- bility from this observation, since galvanic circuits, from their very nature, can but seldom be extended in a straight line. ‘Having anticipated this point, we will immediately proceed to the most simple case, where the prismatic conductor is formed in its entire length of the same material, and is curved back- on itself, and conceive the seat of the electric tension to be where its two ends touch. Although no case in nature re- ables this imaginary one, it will nevertheless be of great service in the treatment of the other cases which do really occur : semb in nature. 2u2 : 456 OHM ON THE GALVANIC CIRCUIT. The electroscopic force, at any place of such a prismatic — body, may be deduced from the differential equation (a) found _ in §11. For this purpose we have only to integrate it, and to determine, in accordance with the other conditions of the pro- blem, the arbitrary functions or constants entering into the in- tegral. This matter is, however, generally very much facili- tated, with respect to our subject, by omitting one or even two members, according to the nature of the subject, from the equa- tion (a). Thus nearly all galvanic actions are such that the phzenomena are permanent and invariable immediately at their origin. In this case, therefore, the electroscopic force is inde- pendent of time, consequently the equation (a) passes into Orme x Ca ea u d x otis Moreover, the surrounding atmosphere has (as we have already — noticed in § 9.) in most cases no influence on the electric na- — ture of the galvanic circuit; then 6=0, by which the last equa- — tion is converted into : 1 da = FF But the integral of this last equation is u=fute, (c) where / and c represent any constants remaining to be deter-_ mined. The equation (c) consequently expresses the law of } electrical diffusion, in a homogeneous prismatic conductor, in all cases where the abduction by the air is insensible, and the action no longer varies with time. As these circumstances in reality most frequently accompany the galvanic circuit, we shall on that account dwell longest upon them. We are enabled to determine one of the constants by th tension occurring at the extremities of the conductor, which has to be regarded as invariable and given in each case. If, fot instance, we imagine the origin of the abscissee anywhere in the axis of the body, and designate the abscissa belonging to one of its ends by 2,, then the electroscopic force there ac is, according to the equation (c), Sx + C3 H i] in the same way we obtain for the electroscopic force of the other extremity, when we represent its abscissa by 2, g iD OHM ON THE GALVANIC CIRCUIT. 457 S %q + 6. If we now call the given tension or difference of the electro- scopic force a, we have a=4+f (2). | But x,—2, evidently represents the entire, positive or negative, length of the prismatic conductor ; if we designate this by J, we obtain accordingly a=+fl, whence the constant f may be determined. If we now intro- duce the value of the constant thus found into the equation (c), _ it is converted into “4 oo ae v2 ae C5 80 that only the constant c remains to be determined. We may _ consider the ambiguity of the sign + to be owing to the ten- ¥: sion a, by ascribing to it a positive value when the extremity of the conductor, Plnnene to the greater abscissa, possesses So the greatest electroscopic force, and when the contrary a negative. Under t)iis supposition is then generally : | u=Tate. (d) The constant ¢ remains in general wholly undetermined, _ which admits of our allowing the diffusion of the electricity in the conductor to vary arbitrarily, by external influences, in ch manner that it occupies the entire conductor everywhere uniformly. __ Among the various considerations respecting this constant, a is one of especial importance to the galvanic circuit, I - mean that which supposes the circuit to be connected at some one place with a perfect conductor, so that the electroscopic force has to be regarded as constantly destroyed at this place. _ If we call the abscissa belonging to this place a, then according to the en (d) % = + A+ c¢. _ By determining from this the constant c, and placing its value in the same equation (d), we obtain = a h . u= 7 (t—A), from which the electroscopic force of a galvanic circuit of the 458 OHM ON THE GALVANIC CIRCUIT. length /, and of the tension a, which is touched at any given place whose abscissa is A, may be found for every other place. If any constant and perfect adduction, from outwards to the galvanic circuit, were to be given instead of the permanent abduction outwards, so that the electroscopic force pertaining to the abscissa A were compelled to assume constantly a given energy, which we will designate by «, we should obtain for the determination of the constant ¢ the equation a 2 ee and for the determination of the electroscopic force of the cir- cuit at any other place the following: us + (@—2) 4 ae We have seen how the constant c may be determined when the electroscopic force is indicated at any place of the circuit by external circumstances; but now the question arises, what value are we to ascribe to the constant when the circuit is left entirely to itself, and this value can consequently no longer be deduced from outward circumstances? The answer to this question is found in the consideration, that each time both elec- tricities proceed contemporaneously, and in like quantity from a previously indifferent state. It may, therefore, be asserted, that a simple circuit of the present kind, which is formed in a perfectly neutral and isolated condition, would assume on each side of the place of contact an equal but opposite electric condition, whence it is self-evident that their centre would be indifferent. For the same reason, however, it is also apparent that when the circuit at the moment of its origin is compelled by any circumstance to deviate from this, its hone state, it would certainly assume the abnormal one until again caused tay change. ; The properties of a simple galvanic circuit, such as we have hitherto considered them to be, accordingly consist essen= tially in the following, as is directly evident from the equa tion (d) : a. The electroscopic force of such a circuit varies throughout the whole length of the conductor continually, and on like extents constantly to the same amount ; but where the two extremities are in contact, it changes suddenly, and, indeed, y OHM ON THE GALVANIC CIRCUIT. 459 from one extremity to the other, to the extent of an entire tension. b. When any place of the circuit is disposed by any circum- stance to change its electric state, all the other places of the circuit change theirs at the same time, and to the same amount. 16. We will now imagine a galvanic circuit, composed of two parts, P and P’, at whose two points of contact a different elec- tric tension occurs, which case comprises in it the thermal cir- cuit. If we call u the electroscopic force of the part P, and a/ _ that of the part P’, then, according to the preceding paragraph, __as here, the case there noticed is repeated twice, in consequence é of the equation (c), by =f 2 +0 for the part P, and w=fletic _ for the part P’, where f, c, f’, c! are any constant magnitudes to be deduced from the peculiar circumstances of our problem, and each equation is only valid so long as the abscisse refer to that part to which the equations belong. If we now place the _ origin of the abscissz at one of the places of contact of the part P, and suppose the direction of the abscissee in this part _ to proceed inwards; moreover, designate by / the length of the _ part P, and by /' that of P’; and, lastly, represent by w’, and u, e values of uw and w at the place of contact where # =o, and uz and u', the values of uw and w! at the place of contact where t= l, we then obtain “o w,=f' 1+?) +¢ ly =C Uz, =fl+e “= f'ls ec. Tf we now designate by a the tension which occurs at the place of contact where # = 0, and by a! that which occurs at the place of contact where # = /; and if we once for all assume, for the _ sake of uniformity, that the tension at each individual place of _ contact always expresses the value which is obtained when we deduct the electroscopic force of one extremity from the elec- troscopic force of that extremity belonging to the place in ques- tion, upon which the abscissa falls before the abrupt change takes Pplace—(it is not difficult to perceive that this general rule con- tains that advanced in the preceding paragraph, and which, in fact, expresses nothing more than that the tensions of such 460 OHM ON THE GALVANIC CIRCUIT. places of contact, by the springing over of which, in the direc- tion of the abscissa, we arrive from the greater to the smaller electroscopic force, are to be regarded as positive, in the con- trary case as negative, where, however, it must not be over- looked that every positive force has to be taken as greater than every negative, and the negative as greater than the actually smaller), we obtain ( a=fi(l+)+cd—e, al =fl—fil+e—d, whence directly results at+ad=fl+/f'l. But now at each of the places of contact when x and w repre- sent the power of conduction and the section of the part P, and x' and w! the same for P’, in accordance with the considerations developed in § 13, there ay the conditional equation (75) =* o*), where (35) and ie rE tu represent the values of = and —— at the place of contact. From the equations at the commence- ment of this paragraph for the determination of the electroscopice force in each single part of the circuit, we, however, obtain the value of z to be allowed to each, and 7 which converts the conditional equation in question into xo f= x ow! f', From this, and the equation a + a’ = f/1 + /'l! just deduced from the tensions, we now find the values of f and /' thus: wee (a+a’) x! w! ~ Xo l + xwl? a+a')xw f= aTpaeP and with the help of these values we find A (a+ a’) (x! col E— x0 I) 9 all+xol Hence the electroscopic force of the circuit in the part P is ex- pressed by the equation ec =c—a'+ =A Wie 2 tH OHM ON THE GALVANIC CIRCUIT. 461 (a +a’) x! we x al +xal! mz) and that in the part P’ by the equation SAE I ee eh es loll + xwl! —a+c. 1 If we substitute A and a’ for and —, the following more simple form may be given to these equations :— _a+ad ¢e 71 nk ROSS) oc gach | " a ¢_atd ee oF a Si int an rae x! a! xw ‘] From the form of these equations it will be immediately per- ceived, that when the conductibility, or the magnitude of the ‘section, is the same in both parts, the expressions for uw and w! undergo no other change than that the letter representing the conductibility or the section entirely disappears. 17. We will now proceed to the consideration of a galvanic circuit, composed of three distinct parts P, P’, and P’, which case comprises the hydro-circuit. If we represent by u, w, wv! respectively the electroscopic - forces of the parts P, P’, and P”, then, according to § 15, the case there mentioned being here thrice repeated, we have, in accordance with the equation (c) there found, with respect to e part P, .. u=furte, : with respect to the part P’, w=f'e+e, od with respect to the oe |i =fle+e', where fs J', f", ¢, , c' may represent any constant magnitudes a ‘remaining to be ap ecained from the nature of the problem, and each equation has only so long any meaning as the ab- ‘scissee refer to that part to which the equations appertain. If “We suppose the origin of the abscissee at that extremity of the part P, which is connected with the part P", and choose the direction of the abscissz so that they lead from the part P to that of P’, and from thence into P”; if we further respectively 462 OHM ON THE GALVANIC CIRCUIT. designate by /, 7, and Z! the lengths of the parts P,P’, P”; and — lastly, let w’, and uw, represent the values of w! and wu at the place of contact where x = 0, and w, and w, the values of w and u' at the place of contact where # = /, and w/, and wu", the values of w/ and w! at the place of contact where x = 1+ /, then we obtain ule=f" (+04 h')+el uw =e i ae ee a uy=f Loe wo =f' (lt l)+e ul = fl (L4U) +l. If we call a the tension which occurs at the place of contact where x = 0, a’ that at the place of contact where w= /, and a! — that at the place of contact where # =/-+ I’, we obtain, if we | pay due attention to the general rule stated in the preceding © aragraph 2h, onan i aa nae ae ad =fl—fli+e—d a! =f! (I+1) —f" (141) 4+e—e', atadt+al=fl+flu4 fl. But from the considerations developed in § 13, when x and — w represent the power of conduction and the section for the part P, x’ and w! the same for the part P’, and x" and w! for the _ part P", at the individual places of contact, the following condi- tional equations are obtained: x Ww (3) = x! a! (<*) = x!! g!! (= dx day, da ); ! M a where (5) 3 = ) : (*) represent the particular values” | du dw du dz da’ dz’ the equations stated at the commencement of the present para- | graph for the determination of the electroscopic force in the single parts of the circuit, we obtain for every admissible value of x, and hence of belonging to the places of contact. From du _ dul _ / pele ei hae ’ ide into | | . aster ( —(7+f).. # iz = ‘ae * it is easy to see, that these equations, with the omission of the letter x or w (both where they are explicit, as well as in the a 4, OHM ON THE GALVANIC CIRCUIT. 463 duced from the tensions, we now find, when 4, 4, x’ are respect- 1 7! qi ively substituted for ap taal pi at+ad+a" 1 TOA EAE A x ieee +at+a" 1 ~ ATA HAT Oe lo? ,» at+a+al! 1 P= NaN * al? and by the aid of these values we find further, atad+a'(l ) ' Bees Ags ixtal) oot ® Ee eee ae rere fan xe opal iat alae N tee act: Git By substituting these values, we obtain for the determination of the electroscopic force of the circuit in the parts P, P’, P" re- spectively, the following equations : _at+a+a! «x ig eid | By eto +o (+4) -ate (U) ea EA ANN xl ol | nw 7 Y t fl DEN EAT Hl el Jatz2)~@ ape eri expressions for A, 2’, 2"), are the true ones for the case x = x’, or = a = w!, : _ 18. These few cases suffice to demonstrate the law of progres- sion of the formule ascertained for the electroscopic force, and to comprise them all in a single general expression. To do this with the requisite brevity, for the sake of a more easy and ge- neral survey, we will call the quotients, formed by dividing the length of any homogeneous part of the circuit by its power of conduction and its section, the reduced length of this part; and when the entire circuit comes under consideration, or a portion of it, composed of several homogeneous parts, we understand by its reduced length the sum of the reduced lengths of all its parts. Having premised this, all the previously found expres- sions for the electroscopic force, which are given by the equa- 464 OHM ON THE GALVANIC CIRCUIT. tions (L) and (L!), may be comprised in the following general statement, which is true when the circuit consists of any num- ber of parts whatever. The electroscopic force of any place of a galvanic circuit, composed of any number of parts, is found by dividing the sum of all its tensions by its reduced length, multiplying this quo- tient by the reduced length of the part of the circuit comprised by the abscissa, and subtracting from this product the sum of all the tensions abruptly passed over by the abscissa ; lastly, by varying the value thus obtained by a constant magnitude to be determined elsewhere. If, therefore, we designate by A the sum of all the tensions of the circuit, by L its entire reduced length, by y the reduced length of the part which the abscissa passes through, and by O the sum of all the tensions to the points to which the abscissa corresponds, lastly, by u, the electroscopic force of any place in any part of the circuit, then u=fy-O+e, where c¢ represents a constant, but yet undetermined mag- nitude. Thus transformed, this exceedingly simple expression for the electroscopic force of any circuit will allow us hereafter to com- bine generality with conciseness, for which purpose we will, more- over, indicate by y the reduced abscissa. This form of the equa- tion has besides the peculiar advantage that, without anything further, it is even applicable when in any part of the circuit the tensions and conductibilities constantly vary; for in this case we should merely have to take, instead of the sums, the cor- responding integrals, and to define their limits according as the nature of the expression required. Since O does not change its value within the entire extent of the same homogeneous part of the circuit, and y constantly varies to the same amount on like portions of this extent, the following properties, already — demonstrated less generally with respect to the simple circuit, evidently apply to every galvanic circuit, and in these is ex- pressed the main character of galvanic circuits :— a. The electric force of each homogeneous portion of the circuit yaries throughout its entire length constantly, and on like extents always to the same amount; but where it ceases and another commences, it suddenly 7 é OHM ON THE GALVANIC CIRCUIT. 465 changes to the extent of the entire tension situated at that place. 8. If any single place of the circuit is induced by any circum- stance whatsoever to change its electric condition, all the other places of the circuit change theirs at the same time, and the same amount. The constant ¢ is in the rule determined by ascertaining the electroscopic force at any place of the circuit. If, for instance, we designate by w the electroscopic force at a place of the circuit, the reduced abscissa of which is y’, then, in accordance with the general equation above stated, , W=ty-O'+e where O! represents the sum of the tensions abruptly passed over by the abscissa y'. If we now subtract this equation, valid for a certain place of the circuit, from the previous one belong- ing in the same manner to all places, we obtain u—u =f y—y) —(0-0), in which nothing more remains to be determined. If the circuit, during its production, is exposed to no external deduction or adduction, the constant c must be sought for in the circumstance that the sum of all the electricity in the cir- _ cuit must be zero. This determination is founded on the fun- damental position, that, from a previously indifferent state, both electricities constantly originate at the same time and in like quantity. To illustrate, by an example, the mode in which the constant c is found in such a case, we will again consider the case treated of in § 16. In the portion P of that circuit, wu is generally = 3 y + c, where y = , and in the portion P’ we have constantly u = as y—a' +c, where y = +a. Since now the magnitude of the element, in the portion P, is wda or x” dy, but in the portion P! is w! dw or x! w? dy, we ob- tain for the quantity of electricity contained in an element of the first portion x w* ay (ty +c), and for the quantity contained in an element of the second portion 466 OHM ON THE GALVANIC CIRCUIT. x! w!? dy (Fy —a+t e). If we now integrate the first of the two preceding expressions from y = O to y = A, we then obtain for the whole quantity of electricity contained in the part P, fos Weare x wm lane te | 5 in the same manner we obtain, by integrating the second ex- pression from y =A to y=A-+A‘, for the entire quantity of electricity contained in the portion P! x! wl? [= (2+ 2anN)—adan + on]. But the sum of the two last found quantities must, in accordance with the above-advanced fundamental position, be zero. We thus obtain the equation required for the determination of the constant c, and it only remains to be observed that a and A’ are the reduced lengths corresponding to the portions P and P’. We have hitherto always tacitly supposed only positive ab- scisse. But it is easy to be convinced that negative abscisse may be introduced quite as well. For let — y represent such a negative reduced abscissa for any place of the circuit, then L—y is the positive reduced abscissa pertaining to the same place, for which the general equation found is valid; we ac- cordingly obtain u -+ (L—y)—O+e Be u=—ty-(-A)+e. But O —A evidently expresses, if regard be had to the general © rule expressed in § 16, the sum of all the tensions abruptly passed over by the negative abscissa, whence it is evident that — the equation still retains entire its former signification for ne-_ gative abscissa. . 19. If we imagine one of the parts of which the galvanic circuit is composed to be a non-conductor of electricity, 7. e.— a body whose capacity of conduction is zero, the reduced length of the entire circuit acquires an indefinitely great value. If we now make it a rule never to let the abscissze enter into the non-_ conducting part, in order that the reduced abscissa y may con- stantly retain a finite value, the general equation changes into — the following : a OHM ON THE GALVANIC CIRCUIT. 467 u=—-O+e, which indicates that the electroscopic force in the whole extent of each other homogeneous portion of the circuit is everywhere the same, and merely changes suddenly from one part to the other to the amount of the entire tension prevailing at its place of contact. To determine the constant ¢ in this equation, we will suppose the electroscopic force, at any one place of the circuit, to be given. If we call this w’, and the sum of the tensions there abruptly passed over by the abscissa O', we have u—u=—(O—-O). ‘The difference of the electroscopic forces of any two places of an open circuit, 7.e. a galvanic circuit interrupted by a non- ‘conductor, is consequently equal to the sum of all the tensions ‘situated between the two places, and the sign which has to be placed before this sum is always easily to be determined from mere inspection. 20. We will now notice another peculiarity of the galvanic circuit, which merits especial attention. To this end let us keep in view exclusively one of the homogeneous parts of the circuit, and imagine, for the sake of simplicity, the origin of the abscisse placed in one end of it, and the abscissz directed towards the other end. If we designate its reduced length by and the reduced length of the other portion of the circuit by then within the length a; the following form may also be given to this equation : Aa A+A “= -¥+C3 the extent is consequently similarly circumstanced to a simple Pe . AA homogeneous circuit, at whose ends the tension Aan oo i accordingly, A has a very sensible sae such as it can "acquire in the voltaic pile, and if the ratio approaches 8 A will likewise be still very per- AA to unity, then the tension aes 468 OHM ON THE GALVANIC CIRCUIT. ceptible ; consequently its various gradations in the extent of the portion A are very easily perceptible. This conclusion is of — importance, because it affords the means of presenting to the senses the law of electric distribution even on compound cir- cuits, when it is no longer possible on the simple circuit, on account of its extremely feeble force. It is, moreover, immedi- ately evident, that, with equal tensions, this phenomenon will be indicated with greater intensity, the greater A is in compari- son with A. 21. A phenomenon common to all galvanic circuits is the | sudden change to which its electroscopic force may incessantly, and arbitrarily, be subjected. This phenomenon has its source in the previously developed properties of such circuits. Since, as we have found, each place of a galvanic circuit undergoes the same alterations to which a single one is exposed, we have it in our power to give sometimes one, sometimes another value to the electroscopic force at any certain place. Among these changes those are the most remarkable which we are able to produce by deductive contact, i. e. by destroying the electro- scopic force sometimes at one, and sometimes at another place of the circuit; its magnitude, however, has its natural limits in the magnitude of the tensions. There is another class of phenomena which is immediately connected with these. If, for instance, we call r the space over which the electroscopic force is diffused in a given gal- vanic circuit, u the electroscopic force of the circuit at one of its points, which is immediately connected with an external body M, and w' the electroscopic force of the same circuit at the same place as it was previous to contact with the body M, uw! —u, is evidently the alteration in the electroscopic force pro= duced at this place; consequently, since this change likewise occurs uniformly at all the other places of the circuit, r (u! —u) is the quantity of electricity which the change produced over the entire circuit comprises, and accordingly that which has passed over into the body M. If now we suppose that in the state of equilibrium the electroscopic force is everywhere of equal intensity at all places of the body M in which it occurs, and represent by R the space over which it is diffused in the body : : sid saienne r (ul —u) aN M, then its electroscopic force is evidently — ae But this force is in the state of equilibrium equal to the w/, which the ra OHM ON THE GALVANIC CIRCUIT. 469 place of the circuit, brought into contact with M, has assumed when no new tension originates at this place of contact ; under this supposition therefore _ r(w—u) (PNG IER, whence we find ru! “= Paring i From this equation it results that the electroscopic force in the body M will constantly be smaller than it was at the touched place before contact ; and also that both will approximate the more to each other, the greater r is in comparison to R. If we regard R as a constant magnitude, the relation of the electroscopic forces u and w to each other depends solely upon the mag- nitude of the space which the electricity occupies in the circuit ; we can therefore bring the electroscopic force of the body M nearer to its greatest value solely by increasing the capacity of the circuit, either by a general increase of its dimensions, or by attaching anywhere to it foreign masses. Upon the nature of these masses, when they are merely conductors of electricity, and do not give rise to new tensions, none of this effect, in my opinion, depends, but solely upon their mag. nitudes. If the attached masses occupy an infinitely great space, which case occurs when the circuit has anywhere a com- plete deduction, then the electroscopic force in the body M will constantly be equal to that which the place of the circuit touch- ed by it possesses. To connect these effects with the action of the condenser, we have merely to bear in mind, that a condenser, whose magnitude is R, and whose number of charges is m, must be con- sidered equal to a common conductor of the magnitude m R, yet with the difference that its electroscopic force is m times that of the common conductor. If, therefore, we designate by wu the electroscopic force of the condenser, which is brought into connexion with a place of the circuit whose force is w', we obtain mr ~ r+mR’ whence it follows that the condenser will indicate m times the force of the touched place when r is very great in comparison with m R; but that it will have a weakening action so soon as r is VOL. Il. PART VIII. 21 Pap 470 OHM ON THE GALVANIC CIRCUIT. equal to, or smaller than R. Masses attached anywhere to the circuit will accordingly make the indications of the con- denser approximate to its maximum in proportion as they are greater, and a circuit touched at any place will constantly pro- duce in the condenser the maximum of increase. The preceding determinations suppose that one plate of the condenser remains constantly touched deductively. We will now take into consideration the case where the two plates of an insulated condenser are connected with various points of a galvanic circuit. In the first place, it is evident that the two plates of the condenser will assume the same difference of free electricity which the various places of the circuit with which they are in contact require unconditionally, from the peculiar | nature of galvanic actions. Consequently, if d represents the © difference of the electroscopic force at the two places of the cir- cuit, and w the free electricity of one plate of the condenser, then u + d is the free electricity of the other plate, and every- thing will depend on finding, from the known free electricities — existing in the plates of the condenser, those actually present in them. If, for this purpose, we call A the actual intensity of electricity in the plate, whose free electricity is w+ d, then A —u—d represents the portion retained in the same plate ; mm the same manner B — uw designates the portion of electricity re- tained in the plate, whose free electricity is vu, when B represents” the actual intensity of the electricity in this plate. If now we represent by 2 the relation between the electricity retained by one plate, and the actual electricity of the other plate, the fol- lowing two equations arise: \ A—u—d+nB=0, B-u+nA=0, from which the values A! and B result, as follows: d+u(1—n) ' er l—n? ’ _u(l=n)—nd 3) 1—n? ; : But from the theory of the condenser, it is well known tha cpm! Mee : l—-n= feat if m is the number of charges of the condenser; : ; 1 3 ; if, therefore, we substitute ee for 1 — n? in the expressions fo OUM ON THE GALVANIC CIRCUIT. 471 A and B, and at the same time 1 — x for n, which is per- mitted when m, as is usually the case, denotes a very large number, we obtain A=md+iu, B=—md+iut+id. Or when m is a very large number, and m not much greater than d, we may, without committing any perceptible error, place A=md, B=—wmd, in which is expressed the known law, that when two different places of a voltaic pile are brought into connexion with the two plates of an insulated condenser, each plate takes the same charge as if the other plate, and the corresponding place of the pile, had been touched deductively. At the same time our con- siderations show that this law ceases to be true when uw can no longer be regarded as evanescent towards md. This case would occur if, for instance, two places, near the insulated upper pole of a voltaic pile, constructed of a great number of elements, came in contact with the plates of the condenser, while the inferior pole of this pile remained in deductive connexion with the earth. The determinations hitherto given respecting the mode in which the galvanic circuit imparts its electricity to foreign _ bodies, and which appear to me to leave nothing more to be _ wished for in the explanation of this subject, might, however, : give rise to researches of a very different kind, aaa of no slight interest. For it is placed beyond all doubt, both from theore- tical considerations, as well as from experiments, that electricity in motion penetrates into the interior of bodies, and its quantity accordingly depends on the space occupied by the bodies; while, on the other hand, it is no less ascertained that static elec- tricity accumulates at the surface of bodies, and its quantity therefore is dependent on the extent of surface. But it would hence result, that in the closed galvanic circuit, r in the above formulz would express the volume of the circuit; in the open circuit, on the contrary, the magnitude of its surface, on which point, in my opinion, experiments might decide without great difficulty. 22. We have hitherto kept in view a circuit on which the surrounding atmosphere exercised no influence, and which has 212 472 OHM ON THE GALVANIC CIRCUIT. already arrived at its permanent state, and we have treated it at a length which it merits from the abundance and importance of the phenomena connected with it. However, not to let even here the other circuits pass entirely unnoticed, we will briefly indicate the method to be pursued for the most simple case, and thus point out the path to be followed, although only at a distance. If it is intended to take into consideration the influence of : an

\—> sa > ef! — eB! Prt BP which equation, for 6 = 0, i. e. when it is not intended to take into consideration the influence of the ates passes into a —x 70 1 >, anid @) tor? gy A eae 3 (2 ee ‘). It is easily aces that the ats of the second member to the right in the equations which have been found for the deter- mination of wu, becomes smaller and smaller as the time increases, * See Journal de l’ Ecole Polytechnique, cap. xix. p. 53. OHM ON THE GALVANIC CIRCUIT. 477 and that it at last entirely vanishes ; the permanent state of the circuit has then occurred. This moment can, as is evident from the form of the expression, be retarded by a diminished power of conduction, and in a far greater degree by an increased length of the circuit. This expression found for w, however, holds perfectly only so long as the circuit, which we have supposed, is not induced by any external disturbance to change its natural state. If the circuit is at any time compelled by any external cause, for instance, by deductive contact at any place, to approximate to an altered permanent state, the above method has to undergo some changes, which I intend to develope on another occasion. I will, moreover, observe, that it is in this last class of galvanic circuits, in which the peculiar phenomena of dry piles, and, in general, of circuits of unusually great length, have to be sought for; to which class likewise belong the circuits of very great length employed in the experiments of Basse, Erman, and Aldini, if the influence of their greai length be not annulled by an in- creased goodness of conduction, or by an increased section. C. Phenomena of the Electric Current. 24. According to what was advanced in paragraph 12, the magnitude of the electric current, in a prismatic body, will in general be expressed for each of its places by the equation Bc cage d x’ where S denotes the magnitude of the current, and ~ the elec- troscopic force at that place of the circuit whose abscissa is 2, while represents the section of the prismatic body, and x its power of conduction at the same place. To connect this equa- tion with the general equation found in § 18 for any circuit, composed of any number of parts, we write it thus: du dy dy dw S=xo and substitute for a the value = resulting from that general equation, and for - the value = easily deducible from the same paragraph, both which values are valid for each place, 478 OHM ON THE GALVANIC CIRCUIT. situated between two points of excitation, we then very simply obtain A iz where L denotes the entire reduced length of the circuit, and A the sum of all its tensions. By means of this equation we ob- tain the magnitude of the electric current of a galvanic circuit, — composed of any number of prismatic parts, which has acquired _ its permanent state, which is not affected by the surrounding atmosphere, and the single sections of which possess in all their points one and the same electroscopic force; in this category . are comprised the most frequently occurring cases, on which account we shall dissect this result in the most careful manner. Since A represents the sum of all the tensions in the circuit, and L the sum of the reduced lengths of all the individual parts, there results, in the first place, from the equation found, the following general properties relative to the electric current of the galvanic circuit. I. The electric current is decidedly of equal magnitude at all places of a galvanic circuit, and is independent of the value of the constant c, which, as we have seen, fixes the intensity of the electroscopic force at a determined place. In the open circuit the current ceases entirely, for in this case the reduced length L acquires an infinitely great value. II. The magnitude of the current, in a galvanic circuit, re- mains unchanged when the sum of all its tensions and its entire reduced length are varied, either not at all, or in the same proportion; but it increases, the reduced length re- maining the same, in proportion as the sum of the tensions increases, and the sum of the tensions remaining the same, in proportion as the reduced length of the circuit dimi- nishes. From this general law we will, moreover, particu- larly deduce the following. F 1. A difference in the arrangement and distribution of the individual points of excitation, by a transposition of the parts of which the circuit consists, has no influence on the magnitude of the current when the sum of all the tensions remains the same. Thus, for instance, the current would remain unaltered in a circuit formed in the order copper, silver, lead, zinc, and a fluid, even when the silver and lead Ss = OHM ON THE GALVANIC CIRCUIT. 479 change places with each other; because, according to the laws of tension observed with respect to metals, this trans- position would, it is true, alter the individual tensions, but not their sum. 2. The intensity of a galvanic current continues the same, al- though a part of the circuit be removed, and another pris- matic conductor be substituted in its place, only both must have the same reduced length, and the sum of the tensions in both cases remain the same; and vice versd, when the current of a circuit is not altered by the substitution of one of its parts for a foreign prismatic conductor, and we can be convinced that the sum of the tensions has remained the same, then the reduced lengths of the two exchanged con- ductors are equal. 3. If we imagine a galvanic circuit always constructed of a like number of parts, of the same substance, and arranged in the same order, in order that the individual tensions may be regarded as unchangeable, the current of this circuit in- creases, the length of its parts remaining unaltered, in the same proportion in which the sections of all its parts in- crease in a similar manner, and the sections remaining un- altered, in the same proportion in which the length of all its parts uniformly decrease. When the reduced length of a part of the circuit far exceeds that of the other parts, the __ magnitude of the current will principally depend on the dimensions of this part ; and the law here enounced will assume a much more simple form, if, in the comparison, attention be solely directed to this one part. The conclusion arrived at in II. 2. presents a convenient means for the determination of the conductibility of various bodies. If, for instance, we imagine two prismatic bodies, whose lengths are / and /’, their sections respectively w and a’, and whose powers of conduction are x and x’, and both bodies possess the property of not altering the current of a galvanic circuit when they alternatively form a portion of it, and both leave the individual tensions of the circuit unchanged, then l I! Tap ee consequently 480 OHM ON THE GALVANIC CIRCUIT. the powers of conduction, therefore, of both bodies are directly proportionate to theirlengths, and inversely proportionate to their sections. If it is intended to employ this relation in the deter- mination of the powers of conduction of various bodies, and we choose for the experiments prismatic bodies of the same section, which indeed is requisite for the sake of great accuracy, their lengths will enable us to determine accurately their conductibi- lities. 25. In the preceding paragraph we have deduced the magni- tude of the current from the general equation given in § 18, A u=yy-Ote and have found that it is expressed by a the coefficient of y. i ye ae ar To ascertain the value rf it is in general requisite to possess an accurate knowledge of all the single parts of the circuit, and their reciprocal tensions; but our general equation indi- cates a means of deducing this value likewise from the nature of any single part of the circuit in the state of action, which we will not disregard, as it will be of great service to us here- after. If, namely, we conceive in the above equation y to be increased by any magnitude Ay, and designate by A O the corresponding change of O, and by Au that of u, there results from that equation Au= >A fy AO, A Auw+AO0_ L Ay i we find, therefore, the magnitude of the electric current by adding to the difference of the electroscopic forces at any two places of the circuit the sum of all the tensions situated between these two places, and dividing this sum by the reduced length of the part of the circuit which lies between these same places. If there should be no tension within this portion of the circuit, then AO = 0, and we obtain A_ Aw i ey. 26. The voltaic pile, which is a combination of several similar and we thence find OUM ON THE GALVANIC CIRCUIT. 481 simple circuits, merits peculiar attention in this place, from the numerous and varied experimental results obtained by its means. If A represent the sum of the tensions of a closed galvanic circuit, and L its reduced length, the magnitude of its current is, as we have found, A L Now, if we imagine m such circuits perfectly similar to the former, but open, and constantly bring the end of each one in direct connexion with the commencement of the next following one, in such a manner that between each two circuits no new tension occurs, and all the previous tensions remain afterwards as before, then the magnitude of the current of this voltaic com- bination, closed in itself, is evidently nA nL consequently equal to that in the simple circuit. This equality of the circuit, however, no longer exists when a new conductor, which we will call the interposed conductor, is inserted in both. If, namely, we designate the reduced length of this interposed conductor by A, then, when no new tension is produced by it, the magnitude of the current in the simple circuit will be A L+W and in the voltaic combination, consisting of n, such elements a ao ctoass “ nL+A vip A therefore in the latter circuit it is constantly greater than in the former, and, in fact, a gradual transition takes place from equality of action, which is evinced when A disappears, to where the voltaic combination exceeds 7 times the action of the simple circuit, which case occurs when A is incomparably greater than nu. If by A we represent the relative length of the body upon which the circuit is to act by the force of its current, then from the observations just brought forward it results that it is most advantageous to employ a powerful simple circuit when A is very small in comparison to L; and, on the contrary, the voltaic pile, when A is very great in comparison with L. 482 OHM ON THE GALVANIC CIRCUIT. But how must, in each separate case, a given galvanic appa- ratus be arranged so as to produce the greatest effect? Let us suppose, in solving this problem, that we possess a certain mag- nitude of surface ; for instance, of copper and zinc, from which we can form, according to pleasure, a single large pair of plates, or any number of smaller pairs, but in the same proportion, and, moreover, that the liquid between the two metals is constant- ly the same, and of the same length, which latter supposition means nothing more than that the two metals between which the liquid is confined retain, under all circumstances, the same distance from each other. Let A be the reduced length of the body upon which the electric current is to act, L the reduced length of the apparatus when formed into a simple circuit, and A its tension; then, when it is altered into a voltaic combination of x elements, its present tension will be # A, and the reduced length of each of its present elements x L, accordingly the reduced length of all the 2 elements z* L, consequently the magnitude of the action of the voltaic combination of # elements is vA Fo METRY, % when 2VA.L ip Se = We hence see that the apparatus in form of a This expression acquires its greatest value simple circuit is most advantageous, so long as A is not greater than L; on the contrary, the voltaic combination is most use- ful when A is greater than L, and indeed it is best constructed of two elements when A is four times greater than L, of three ele-— ments when A is nine times greater than L, and so forth. 27. The circumstance that the current always remains the same at all places, affords us the means of multiplying its ex- ternal action, as in the case when the current influences the magnetic needle. We will, for perspicuity, suppose that, im order to test the action of the current on the magnetic needle, each time a part of the circuit be formed into a circle of a given radius, and so placed in the magnetic meridian that its centre coincides with the point of rotation of the needle. Several such distinct coils, formed of the circuit in exactly the same manner, will, taken singly, produce, on account of the equality of the current in each, equally powerful effects on the magnetic . OHM ON THE GALVANIC CIRCUIT. 483. needle; if we imagine them, therefore, so arranged near one another, that though they are separated by a non-conducting layer, they are yet situated so close together that the posi- tion of each one toward the magnetic needle may be regarded as the same, they would produce a greater effect on the magnetic needle in proportion as their number increased. Such an ar- rangement is termed a multiplier. Now, let A be the sum of the tensions of any circuit, and L its reduced length ; let also A be the reduced length of one of the interposed conductors formed into a multiplier of » convolu- tions; then, if we represent the reduced length of one such convolution by a, A = A, the action of the multiplier on the magnet needle will be proportional to the value nA CL aix But the action of a similar coil of the circuit, without the multi- plier, is, according to the same standard, A LL and we will suppose the portion of the circuit, whence the coil is taken, to be of the same nature as in the multiplier; accord- ingly the difference between the former and the present effect is nu—(L+na) A i a e which is positive or negative according as x L is greater or less than L + na. Consequently the action on the magnetic needle will be augmented or diminished by the multiplier formed of n coils, according as the n times reduced length of the circuit, without interposed conductor, is greater or less than the entire reduced length of the circuit with the interposed conductor. If 7 A is incomparably greater than L, the action of the mul- tiplier on the needle will be ; A =e To this value, which indicates the extreme limit of the action by means of the multiplier, whether it be strengthening or weakening, belong several remarkable properties, which we will briefly notice. It is constantly supposed that the multiplier is formed of so many coils that the magnitude of its action may, 484 OHM ON THE GALVANIC CIRCUIT. without committing any sensible error, be considered equal to the limit value. Since the action of a coil of the circuit is ay while the ac- L tion of the multiplier, in connexion with the same circuit, is Neen : é ; = it is evident that the two actions are in the same ratio to each other as the reduced length a and L; if, therefore, we are acquainted with the two actions, and with one of the two re- duced lengths, the other may be found, and in the same manner one of the two actions may be deduced from the other, and the two reduced lengths. Since the limit of the action of the multiplier is 2, it in- creases when A is invariable in the same proportion as the sum of the tensions A in the circuit increases; we may, therefore, by comparing the extreme actions of the same multiplier in various circuits, arrive at the determination of their relative tensions. At the same time we perceive that the extreme action of the mul- tiplier increases, when several simple circuits are formed into a voltaic combination, and, indeed, in direct proportion to the number of the elements. In this manner it is always in our power, in cases where the multiplier in connexion with the | simple circuit produces a weakening effect, to cause it to in- | dicate any increase of force whatever. | If we call the actual length of a coil of the multiplier J, its conductibility x, and its section w, then A= = and conse- | ! quently the extreme action of the multiplier xW. TY? whence it results that in the same circuit the extreme actions of two multipliers of coils of equal diameter, are in the rativ to | each other of the products of their conductibility and their sec- | tion. These extreme actions are, therefore, in two multipliers, | which differ only in being formed of two distinct metals, in pro- portion to the conductibility of these metals; and when the | multipliers consist of similar convolutions, and of one metal, their extreme actions are proportional to their sections. ! But all these determinations are based upon the supposition | that the action of a portion of the circuit on the magnetic | OHM ON THE GALVANIC CIRCUIT. 485 needle, under otherwise similar circumstances, is proportional to the magnitude of the current. But long since direct ex- periments have established the correctness of this supposi- tion. 28. We will now proceed to the consideration of a multiple conduction existing at the same time. If, for instance, we imagine an open circuit, whose separated extremities are con- nected by several conductors, arranged by the side of each other, it may be asked, according to what law is the current distributed in the adjacent conductors? In answering this question, we might proceed directly from the considerations con- tained in § 11 to 13; but we shall more simply attain the re- quired object from the peculiarity of galvanic circuits ascertained in § 25, in which case we will, for the sake of simplicity, sup- pose that none of the former tensions is destroyed by the open- ing of the circuit, nor a new tension produced by the conductor which is introduced. For if a, 2’, ’, &c. represent the reduced lengths of the con- ductors brought into connexion with the extremities of the open circuit, and « the difference of the electroscopic forces at the ex- tremities of the circuit, after the conductors have been intro- duced, the same difference will also occur at the ends of the single adjacent conductors, since, according to the supposition we have made, no new tension is introduced by the conductor. Since now, according to § 13, the magnitude of the current in the circuit must be equal to the sum of all the currents in the adjacent conductors, we may imagine the circuit to be divided into as many parts as there are adjacent conductors; then, ac- cording to § 25, the magnitude of the current in each adjacent conductor, and in the corresponding part of the circuit, will re- spectively be 2 a a a ru, YI!’ nP whence, in the first place, it results that the magnitude of the current in each adjacent conductor is in inverse ratio to its re- duced length. If we now imagine a single conductor of such nature, that, being substituted for all the adjacent conductors in the circuit, it does not at all alter its current; then, in the first place, «, according to § 25, must retain the same value, and, if we designate by A the reduced length of this conductor, must moreover be VOL. Il. PART VIII. 2K &c., 486 OHM ON THE GALVANIC CIRCUIT. 1 1 1 ] A We deg a From the preceding explanations we may conclude, that when A denotes the sum of all the tensions, and L the entire reduced length of the circuit without adjacent conductors, the magnitude of the current, while the adjacent conductors are in connexion with the circuit, will be expressed in the circuit itself by A . L + A’ in the joint conductor, whose reduced length is a, by puso, eee A” in the joint conductor, whose reduced length is 4’, by pete ole, L+A ‘2? in the joint conductor, whose reduced length is a", by Beate L+A ‘A and so on, where for A its value obtained from the equation 1 1 | RET PTE has to be placed. 29. That in the above the galvanic current is found to be of equal magnitude at all places of the circuit, arises from the value of as deduced from the equation being constant. This circumstance no longer happens if we start from the equations given in § 22 and 23. In all these du. ae Shae sous cases Ts is dependent on 2, which indicates that the magnitude of the current is different at different places of the circuit. We may hence draw the conclusion, that the electric current is only of equal intensity at all places of the circuit, when the circuit has already assumed a permanent state, and the atmosphere has no sensible action upon it. This property likewise appears)) best adapted to enable us to find out, by experiment, whether OHM ON THE GALVANIC CIRCUIT. 487 the atmosphere exercises a perceptible influence on a galvanic . circuit, or not, we will therefore enter into this case at greater length. Since, according to § 12, the magnitude of the electric cur- rent is given by the equation du Ss =xX*w. ax’ we have only in each separate case to obtain the value of _ from the equation found for the determination of the electro- scopic force, and to place it in the one above. Thus, for a cir- cuit which has assumed its permanent state, but upon which the surrounding atmosphere exercises no sensible influence, according to § 22, ; Th l—e—Bt © %. Bt 4 e—Bi? where a represents the tension at the place of excitation, and 6 the sum of the electroscopic forces immediately adjacent on both sides of the place of excitation. We hence obtain eft 4 efx et — e—ha — nop (ga ae eae This expression gives the magnitude of the current at each place of the circuit ; but the law, according to which the alvera- tion of the current at various places of the circuit is effected, may be rendered more easily intelligible in the following man- ner. If, for instance, we differentiate the equation du N) = EG eat we obtain the equation dS. du. hen = X*w ada 3 and by multiplying both together, ds8 d*u Be du x2 wo d - 2 If we now substitute for nas its value 6” u, as obtained from P ad?u the equation 0 = da — Bu, we have 488 OHM ON THE GALVANIC CIRCUIT. and we hence obtain by integration S? = c2 + x? a? 8’ ¥, where ¢ represents a constant remaining to be determined. If we designate by w the smallest absolute value which w occupies in the circumference of the circuit, and by S! the corresponding value of S, and determine, in accordance with this, the constant c, we obtain S? =* S/?2 = x2 w B? (u? ae we It may easily be deduced from this equation, that the current of a circuit, which is influenced by the atmosphere, is weakest where the electroscopic force, without regard to the sign, is smallest, and that it is of the same magnitude at places with equal but opposite electroscopic forces. APPENDIX. ON THE CHEMICAL POWER OF THE GALVANIC CIRCUIT. On the Source and character of the Chemical Changes in a Gal- vanic Circuit, cnd on the Nature of the Fluctuations of its Force dependent thereon. 30. In the present Memoir we have constantly supposed that those bodies, which are under the influence of the electric cur- rent, remain unchangeable; we will now, however, take into consideration the action of the current on the bodies subjected to it, and the alterations in their chemical constitution thence resulting in any possible manner, as also the changes of the current itself produced by reaction. If what we here give does by no means exhaust the subject, nevertheless our first attempt shows that we are advancing in this path towards im- portant conclusions respecting the relation of electricity towards bodies. To proceed on sure ground, let us return to what has been enounced in § 1 to 7, and connect our present considerations with those expressions and developments. We will suppose, therefore, two particles, and designate by s their mutual di- stance, by w and w! their electroscopic forces, which we admit to be of equal intensity in all points of the same particle ; then, as may easily be deduced from what has been previously stated, OHM ON THE GALVANIC CIRCUIT. 489 the repulsive force between these two elements is proportional to the time d¢, to the product ww’, and, moreover, to a function dependent on the position, size, and form of the two particles, which we will represent by F’; we accordingly ob- tain for the repulsive force between two particles the expres- sion F’ uw dt. If we here proceed in the same manner as in § 6, and signify by the moment of action x' between two places, the product of g', which expresses the force produced under perfectly deter- mined circumstances between both, and its mean distance s', so that ae cesitg!s,) 2" and determine g' by putting vu =w' = 1 in the expression F’ uw dt, and extending the action to the unit of time, we have x = b's, whence it follows that ! jae a vr Let us now imagine, as we did in $ 11, the prismatic circuit - to be divided into equally large, infinitely thin discs, and call M’, M, M, those immediately following one another, which belong to the abscisse w + dx, x7, vw —dzx; then, according to what has just been shown, the pressure which the disc M! exerts on the disc M is F’ uu dt; and if we admit that the position, size, and form of the particles remain in all discs the same, the counter pressure, which the disc M, exerts on the disc M, is IY uu, di: the difference between these two expressions, viz. F’ u (Ww —u) dt, gives accordingly the magnitude of the force, with which the disc M tends to move along the axis of the circuit. This force acts contrary to the direction of the abscisse when its value is positive, and in the direction of the abscissz when it is ne- gative. If we substitute for w!—w its value proceeding from the deve- lopments given in § 11 for w' and w, the expression just found changes into the following: 490 OHM ON THE GALVANIC CIRCUIT. du 2 F’ U aig: dxd if: and if we take, instead of the function F’ dependent on the na- I ture of each single body, its value oT? this expression, since s! is evidently here d 2, changes into du 2 x! U has dt 3 or if we reduce the moment of action x’, referring to the magni- tude of the section w, to the unit of surface, and at the same time extend the action to the unit of time, into du 2x wu ae where the present x’ represents the magnitude of the moment of action reduced to the unit of surface. If we write this latter expression thus : in which x denotes the absolute power of conduction of the circuit ; and if we substitute for x w dm, by which, according to dx the equation (4) in § 12, the magnitude of the electric current ! is expressed, the sign S chosen for it, and 7 instead of ~ 5 16.18 changed into 2ius. We hence perceive that the force, with which the individual discs in the circuit tend to move, is proportional, both to their innate electroscopic force, and to the magnitude of the current ; and that this force alters its direction at that place of the cir- cuit. where the electricity passes from the one into the opposite state. And here occurs the circumstance which must not be overlooked, that this expression still holds, even when the elec- troscopic force u of the element M is changed in the moment of action, by any causes whatsoever, into any other abnormal U, while the electroscopic forces of the adjacent particles con- tinue the same; only that in this case the value U must be substituted for wu in the expression 2iuS. It must also be ob- served, that the expression 2iuS which we have found refers to the whole extent of the section w, which belongs to that part of OHM ON THE GALVANIC CIRCUIT. 491 the circuit which we have especially in view; if we wish to reduce this motive force of the circuit to the unit of surface, we must divide that expression by the magnitude of the sec- tion w. With respect to the causal relation between the law of electric attractions and repulsions, and that of the diffusion of electricity, or respecting the mutual dependence of the functions x and x' on each other, we will, for the present, not enter into any further inquiries, as shortly an occasion will present itself for this purpose. We will here content ourselves with the observation, that the above mode of explanation has arisen from the endea- your to render the similarity of the mode of treatment in the doctrines of electricity and heat very obvious. 31. Without pursuing any further these conditions to an ex- ternal change of place of the parts of a galvanic circuit, let us now turn to those changes in the qualitative state of the circuit which are produced by the electric current, i. e. in the internal relation of the parts to each other, and which derive their ex- planation from the electro-chemical theory of bodies. Accord- ing to this theory, compound bodies must be considered as a union of constituents which possess dissimilar electric states ; or, in other words, dissimilar electroscopic force. But this electro- scopic force, quiescent in the constituents of the bodies, differs from that to which our attention has hitherto been directed, in- asmuch as it is linked to the nature of the elements, and can- not pass from one to the other, without the entire mode of ex- istence of the parts of the body being destroyed. If we con- fine ourselves, therefore, in the following considerations, to the case where changes, it is true, occur in the quantitative re- lation of the constituents, and where consequently chemical changes of the body, composed of these constituents, also occur, but where the constituents themselves undergo no alteration destroying their nature, we are able to show the validity of all the laws above developed of electric bodies with reference to their reciprocal attraction and repulsion, only the transition of the electricity from one particle to the other entirely disappears in the consideration of chemically different constituents. A di- stinction here exists with reference to electricity exactly similar to that which we are accustomed to define relative to heat, by calling it sometimes latent, sometimes free heat. For the sake of brevity, we will in like manner term that electroscopic force 492 OHM ON THE GALVANIC CIRCUIT. which belongs to the existence of the particles, which therefore they cannot part with without at the same time ceasing to exist, the electricity bound to the bodies, or latent electricity, and free electricity, that which is not requisite for the existence of the bodies in their individuality, and which therefore can pass from one element to the other, without the individual parts being on that account compelled to exchange their specific mode of existence for another. 32. From these suppositions advanced in electro-chemistry, in connexion with what was stated in § 30, respecting the mode in which galvanic circuits exert a ditferent mechanical force on discs of different electrical nature, it immediately follows that when a disc belonging to the circuit is composed of constitu- ents of dissimilar electric value, the neighbouring discs will exert on these two constituents a dissimilar attractive or repul- sive action, which will excite in them a tendency to separate, which, when it is able to overcome their coherence, must pro- duce an actual separation of constituents. This power of the galvanic circuit, with which it tends to decompose the particles into their constituents, we will call its decomposing force, and now proceed to determine more minutely the magnitude of this force. Employing for this purpose all the signs introduced in § 30, we will, moreover, imagine each disc to be composed of two constituents, A and B, and designate by m and m the latent electroscopic forces of the constituents A and B, supposing the disc M to be occupied solely by one of the two, entirely ex- cluding the other, in the same manner as u represents the free electroscopic force present in the same disc, and equally dif- fused over both constituents. If we now admit, in order to simplify the calculation, that the two constituents A and B, before and after their union, constantly occupy the same space, and designate the latent electroscopic force, corresponding to each chemical equivalent, contained in the disc M, and pro- ceeding from the constituent A, by mz, then n (1—z) expresses the latent electroscopic force present in the same disc M, but originating from the constituent B: for the intensity of the force diffused over a body decreases in the same proportion as the space which the body occupies becomes greater, because by the increased distance of the particles from each other the sum of their actions, restricted to a definite extent, is diminished in OHM ON THE GALVANIC CIRCUIT. 493 the same proportion. But when two constituents combine, by both reciprocally penetrating one another, each extends beyond the entire space of the compound, on which account the inten- sity of the force proper to each constituent decreases by com- bination, in the same proportion as the space of the compound is greater than the space which each constituent occupied before the combination. Consequently if z denote the relation of the space which the constituent A, in the disc M, occupied pre- vious to combination to that space which the compound in the disc M occupies; and also, since we admit that both consti- tuents, before and after the combination, occupy the same ex- tent of space, 1—z will denote the same relation relatively to the constituent B; then, since m and n designate the electro- scopic forces of the constituents A and B previous to combina- tion, mz and n (1 — z) will represent the latent electroscopic forces of the constituents A and B, which correspond to each chemical equivalent of the disc M; and, at the same time, it follows from the above, that the variable values z and 1—z can- not exceed the limits 0 and 1. In order to ascertain the portion of the free electricity w per- taining to each constituent, we will assume that it is distributed over the single-constituents in proportion to their masses. If, therefore, we represent respectively by « and £ the masses of the constituents A and B, on the supposition that one alone, to the exclusion of the other, occupies the entire disc, then « z and 6 (1 —z) will represent the masses of the constituents A and B united in the disc M ; consequently the portions aUzZ TB Wien az+PB(l—z) — a : ql 25 of the free electricity wu appertain to the constituents A and B; instead of which, for the sake of conciseness, we will write aUz,and@U (1—2). If we now take into consideration what was stated in § 30, respecting the motive force of the galvanic circuit, it is imme- diately evident that the tendency of the constituent A to move along the circuit, is expressed by 2i(m+aU) ZS, or that of the constituent B by 2i(n+6U) (1—2z)S. 494 OHM ON THE GALVANIC CIRCUIT. In both cases a positive value of the expression shows that the pressure takes place in an opposite direction to that of the abscisse; a negative value, on the contrary, indicates that the pressure is exerted in the direction of the abscisse. To deduce from these individual tendencies of the constituents the force with which both endeavour to separate from each other, we must remember that this force is given by the twofold differ- ence between the quantities of motion which each constituent would of itself assume, were it associated to the other by no coherence, and those quantities of motion which each con- stituent must assume were it strongly combined to the other. We thus readily find for the decomposing force of the circuit the following expression : ; mB—ne@ a aes atk id Geka 5 from which we learn that the decomposing force of the circuit is proportional to the electric current, and also to a coeffi- cient dependent on the chemical nature of each place of the circuit., If this expression has a positive value, it indicates that the separation of the constituent A takes place in a contrary di- rection to that of the abscissa, that of the constituent B in the direction of the abscisse ; but if this expression has a negative value, it denotes a separation in the reverse direction. It is besides evident, at first sight, that the decomposing force of the circuit is constantly determined by the absolute value of the expression, If « = B, the decomposing force of the circuit changes into 4i.z2(1—z) (m—n).S. Ifmz+n(1—z) = 0, i.e. if the latent electroscopic forces, existing in the united constituents, are equal and opposed; or, what is the same, if the body, situated in the disc M, is per- fectly neutral, in which case m and n have constantly opposite | values, we obtain, for the decomposing force of the circuit, the following expression : mn 47. ; m—n The form of the general expression found for the decompos- | ing force of the circuit shows that this force disappears ; first, | when S = 0, i.e. when no electric current exists; secondly, | OHM ON THE GALVANIC CIRCUIT. 495 when z=0, or z = 1, #.e. when the body to be decomposed is not compound ; thirdly, when m 8 —nz=0, i. e. when the den- sities of the constituents are proportional to the latent electro- scopic forces which they possess, which circumstance can never occur with constituents of opposite electric nature. All the expressions here given for the decomposing force of the circuit refer to the entire section belonging to the respective place ; if we wish to reduce the value of the decomposing force to the unity of surface, the expression must be divided by the magnitude of the section, to which attention has been already called in § 30, in a similar example. 33. If this decomposing force of the circuit is able to over- come the coherence of the particles in the disc, a coherence pro- duced by their electric opposition, this necessarily occasions a change in the chemical equivalent of the particles. But such a change in the physical constitution of the circuit must, at the same time, react on the electric current itself, and give rise to alterations in it, with which a more accurate acquaintance is desirable, and which we will therefore spare no trouble to acquire. For this purpose we will imagine a portion of the galvanic circuit to be a homogeneous fluid body, in which such a decom- position actually takes place; then, at all points of this portion, the elements of one kind will tend to move with greater force towards one side of the circuit than those of the other kind; and since we suppose that, by the active forces, the coherence is overcome, it follows, if we pay due attention to the nature of fluid bodies, that the one constituent must pass to one side, those of the other constituent, on the contrary, towards the other side of the portion, which necessarily produces on one side a preponderance of the constituent of one kind, and on the other side a preponderance of the other kind of constitu- ent. But as soon as a constituent is predominant on one side of any disc, it will oppose by its preponderance the movement of the like constituent in the disc towards the same side, in con- sequence of the repulsive force existing between both; the de- composing force, therefore, has now not merely to overcome the coherence between the two constituents in the disc, but also the reacting force in the neighbouring discs. Two cases may now occur; the decomposing force of the electric cur- rent either constantly overcomes all the forces opposed to it, 496 OHM ON THE GALVANIC CIRCUIT. and then evidently the action terminates by a total separation of the constituents, the entire mass of the one passing to the one end of the portion, and the entire mass of the other consti- tuent being impelled towards the other end of this portion; or such a relation takes place between the forces in action, that the forces opposing the separation ultimately maintain the decom- posing force in equilibrium; from this moment no further decomposition will occur, and the portion will be, in a remark- able state, a peculiar distribution of the two constituents oc- curring, into the nature of which we will now inquire. If we call Z the decomposing force of the current in any disc of the portion in the act of decomposition, Y the magnitude of the reaction by which the neighbouring discs oppose the de- composition by the electric current, and X the force of the coherence of the two constituents in the same disc, then evi- dently the state of a permanent distribution within the supposed portion, will be determined by the equation x VS Z's and it is already known, from the preceding paragraph, that mB—ne S: az+B(l—z)° ” or if we substitute x & for S, Z=4i2 (1—2) du. mB—na BZ=4xuo7.tz (1 — z) ree hee,: Before we proceed further, we will add to what has been above said the following remarks. At the limits of the por- tion in question, we imagine the circuit so constituted, that insuperable difficulties there oppose themselves to any further motion ; for it is obvious that otherwise the two extreme strata of both constituents, which it is evident could never of them- selves arrive at equilibrium, would quit the portion in which we have hitherto supposed them, and either pass on to the adjacent parts of the circuit, or from any other causes separate entirely from the circuit. We will not here follow the last- mentioned modification of the phenomenon any further, al- though it frequently occurs in nature, as sufficiently shown by the decomposition of water, the oxidation of the metals on the one side, and a chemical change of a contrary kind occurring on the metals at the other side of the portion hitherto less ob- OHM ON THE GALVANIC CIRCUIT. 497 served, but placed entirely beyond doubt by Pofi’s remark- able experiments on the reaction of metals. Besides, we will direct our attention to a difference which exists between the distributioh of electricity above examined, and the molecular movement now under consideration. If, for instance, the same forces, which previously effected the conduction of the electri- city, and there, as it were, incorporeally without impediment strove against each other, here enter into conflict with masses, by which their free activity is restricted, a restriction which, whether we regard the electricity de se ipso as something mate- rial or not, must render their present velocities, beyond com- parison, smaller than the former ones ; therefore we cannot in the least expect that the permanent state, which we at present examine, will instantaneously occur like that above noticed, arising from the electric distribution ; we have rather to expect that the permanent state resulting from the chemical equiva- lent of both constituents, will make its appearance only after a perceptible, although longer or shorter time. After these remarks, we will now proceed to the determina- tion of the separate values X and Y. 34. To obtain the value X, we have merely to bear in mind that the intensity of coherence is determined by the force with which the two adjacent constituents attract or repel each other by virtue of their electric antagonism, and consequently, as was shown in § 30, proportional to the product of the latent electro- scopic forces mz and n (1 — 2) possessed by the constituents of the disc M, and is, moreover, dependent on a function to be deduced from the size, form, and distance, which we will desig- nate by 4 4. Accordingly, when we restrict the coherence to the magnitude of the section w, X=—49mnz (1—2) wo. We have placed the sign — before the expression ascertained for the strength of the coherence, since a reciprocal attraction of the constituents only occurs when m and n have opposite signs; when m and have the same signs, the constituents exert a repulsive action on each other, which no longer pre- vents, but promotes the decomposing force. After this re- mark it will at first sight be evident that a positive or negative value must be ascribed to the function $, according as the ex- pression taken for the decomposing force z is positive or nega- 498 OHM ON THE GALVANIC CIRCUIT. tive; the sign of the function $, therefore, changes when the direction of the decomposition is transposed from the one con- stituent to the other. The nature of the function ¢ is as little known to us as the size and form of the elements on which it is dependent; nevertheless, we may, in our inquiries, regard its absolute value as constant, since the size and form of the cor- poreal particles, acting on each other, must be conceived to be unchangeable so long as the two constituents remain the same, and the supposition that the two constituents constantly main- tain for every chemical equivalent the same volume, renders attention to the mutual distance of the chemically different particles unnecessary, as regard has already been paid, when de- termining the electroscopic forces in the disc M, to the relative distances of the elements of each constituent. 35. To determine the magnitude of the reaction Y, which in the disc M opposes the latent electricity of the neighbouring discs to the decomposing force, we have nothing further to do than to substitute in the expression for Z instead of wu, the sum of all the latent electroscopic forces in the disc M. Since now the sum of these latent forces is m z+ (1—2z), we obtain for the determination of the force Y, which is called into existence by the change in the chemical equivalent of the constituents, and which opposes the decomposition, after due determination of its sign, the following equation: Y=4n0 52. i(n—m).2 (1-2). POO. If now we substitute for % Y and Z the values found in the equation X+Y=Z, we obtain, after omitting the common factor 4z(1—z), and az+ 6 (1—2) multiplying the equation by , as the condition of i(mB—na) the permanent state ia the chemical equivalent of the two con- stituents, the equation __ du omn ati (mB —n a) ae x. [zz+B (1—2)] @ which, when we put OHM ON THE GALVANIC CIRCUIT. 499 SMM ing (XO mn i(mB—na) Lae x! (m B—n a)’ passes into du dz oY + bo [2248 (1—2)]—x0(n—m) 5%. (5) This equation undergoes no change, as indeed is required by the nature of the subject, when m, a, z, and n, 8, 1 — z are re- spectively interchanged, and, at the same time, the sign of ¢ is changed, as according to the remark made in the preceding paragraph, must take place, since by this transformation the direction of the decomposition is transferred from one consti- tuent to the other. 36. In order to be able to deduce from this equation the mode of the diffusion of the two constituents in the fluid, 7. e. the value of z, we ought to know the power of conduction x, and the electroscopic force u at each point of the portion in the act of decomposition, the values, however, of which, are themselves dependent on that diffusion. Experience, as yet, leaves us in uncertainty respecting the change of conductibility, which occurs when two fluids are mixed in various proportions with one another, and likewise with respect to the law of tensions, which is followed by different mixtures of the same consti- tuents in various proportion; for, if we do not err, no ex- periments have been instituted relatively to the latter law, and the law of the change produced in the conducting power of a fluid, by the mixture of another, is not yet decidedly esta- blished by the experiments of Gay Lussac and Davy. For this reason we have been inclined to supply this want of experience by hypothesis. We have, it is true, constantly endeavoured to conceive the nature of the action in question, in its connexion with those with whose properties we are better acquainted ; but, nevertheless, we wish the determinations given to be re- garded as nothing more than fictions, which are only to remain until we become by experiment in possession of the true law. With regard to what relates to the change in the power of conduction of a body, by mixture with another, we have been guided by the following considerations. We suppose two adja- cent parts of a circuit of the same section w, whose lengths are » and w, and whose powers of conduction are a and 4; then, when A is the sum of the tensions in the circuit, and L the re- duced length of the remaining portion of the circuit, the mag- Ox w 500 OHM ON THE GALVANIC CIRCUIT. nitude of its current, which results from the above-found for- mulz, is If now a conductor of the length v + w, aud of the power of con- duction x with the same section, being taken instead of the two former, leaves the current of the circuit unchanged, then must wy we oF e aw be. woe? whence we find ab (v +) LS eer bv+aw But it is perfectly indifferent for the magnitude of the cur- rent, whether the entire length v be situated near the entire length w, or any number of discs be formed of the two, which are arranged in any chosen order, if only the extreme parts re- main of the same kind, as otherwise a change might result in the sum of the tensions, consequently also in the magnitude of the current. If we extend this law, which holds for every mechanical mixture, likewise to a chemical compound, the above value found for x evidently gives the conducting power of the compound, where, however, it has been taken for granted that the two parts of the circuit, even after the mixture, still occupy the same volume, for v and w are here evidently proportional to the spaces occupied by the two bodies mixed with each other. If we now apply this result to our subject, and therefore put, instead of v and w, the values z and 1 —z, which express the relations of space of the two constituents in the disc M, we obtain, when a denotes the conducting power of the one consti- tuent A, and 6 the same forthe constituent B; further, x the power of conduction of the mixture of the two contained in the disc M, the following expression for x, Sait ab ~ a+(b—alz 37. Having thus determined the power of conduction at each place of the extent in the act of decomposition, there only re- mains to be ascertained the nature of the function uw at each such place; and since all tensions and reduced lengths in the OHM ON THE GALVANIC CIRCUIT. 501 part of the circuit, in which no chemical change occurs, are unalterable and given, it is, in accordance with the general equation given in § 18, which likewise holds for our present case, only requisite for the perfect knowledge of the function u, that we are able to determine the tensions and reduced lengths for each: place within the extent in which the chemical change takes place. But evidently the reduced length of the disc M is dx. aay ) or if we substitute for x its value just found, a+ (b—a)z Be abw we accordingly obtain the reduced length of any part of that extent, if we integrate the above expression, and take the limits of the integral corresponding to the commencement and end of the part. If now we bear in mind that the integral Cri Oz Sf abw may also be written thus: whe b—a d bw ia aba Sf 7° al when / represents the length of the part, over which the in- tegral is to be extended, and zwdwz expresses merely the space which the constituent A in the disc M occupies; con- sequently /zwdaz, the sum of all the spaces which the con- stituent A fills in the part whose reduced length has to be found, it is obvious that the reduced length of the entire portion, in the act of decomposition, remains unchangeable during the chemical change, since, as we have supposed, each constituent maintains, under all circumstances, constantly the same volume. The same result may also be directly deduced from what was advanced in the preceding paragraph ; however, this unchangeability only relates to the reduced length of the entire portion; the reduced length of a part of it does not in general depend merely on the actual length of this part, but likewise on the contemporaneous chemical distribution of the constituents in the extent, and must therefore, in each separate ease, be first ascertained in the manner indicated. VOL, 11, PART VIII. 21 ? 502 OHM ON THE GALVANIC CIRCUIT. 38. We have lastly to determine the alteration in the _ten- sion of the circuit, which is produced by the chemical altera- tion of the extent, which has hitherto been considered. For this purpose we assume, till experience shall have taught us better, the position, that the magnitude of the electric tension between two bodies is proportional, first to the difference of their latent electroscopic forces, and secondly to a function, which we will term the coefficient of the tension, dependent on the size, position and form of the particles which act on each other at the place of contact. Not only from this hypothesis may be deduced the law which the tensions of the metals ob- serve inter se,—nothing further being requisite than to assume the same coefficient of tension between all metals placed under similar circumstances,—but it likewise affords an explanation of the phenomenon, in accordance with which the electric tension does not merely depend on the chemical antagonism of the two bodies, but also on their relative density, and can for this reason exhibit themselves differently, even in different tem- peratures. For the same reasons which we have already men- tioned in § 34 on the determination of the coherence which occurs between the two constituents of a mixed body, we shall likewise admit here, in the circumference of the chemically variable extent as constant, the unknown function dependent on the size, form and position of the particles in contact, and designate it by ¢’. Since now the latent electroscopic force in the disc M, to which the abscissa # belongs, is expressed by n+ (m—n) 2, and that in the disc M', to which the abscissa # + da belongs, by n+(m—n) z+ (m—n) dz, the tension originating between the discs M and M’ is —¢' (m—n)dz; consequently the sum of all the tensions produced through-j out a portion exposed to chemical change 4! (m—n) (22!) when 2’ and 2” represent those values of z, which belong to the comméncement and end of the extent in question. But the tension of the circuit undergoes, besides the change} just explained, a second one, from the extremities of the che-j OHM ON THE GALVANIC CIRCUIT. 503 mically changeable portion, which are in connexion with the other chemically unchangeable parts of the circuit, undergoing a gradual change during the decomposition till they arrive at their permanent state, giving rise at those places to an altered tension. If, for instance, we call ¢ the value of z, which belongs to all places of the extent in question, before chemical change has begun in it, and designate the coefficient of the ten- sion occurring at the extremities of this extent, supposing that it is the same at both ends, by ¢”, and moreover express by p and y the latent electroscopic forces of those places of the che- mically unalterable part of the circuit which are situated adja- cent to the chemically changeable extent, the tensions existing at these places can be determined individually. They are, namely, previous to the commencement of chemical change, the fol- lowing: $" [w — (n+ (m—n) $)], and $" [(n + (m—n) 8) —»]5 and after the permanent state in the decomposition has been attained, if we, as above, let 2’ and 2” be those values of z which belong in this state to those places, they are the fol- lowing : 4" [w— (n+ (m—n) 2')], and $" [(m + (m —n) 2") — 9], _ their sum is therefore in one case . $" (u—»), and in the other o! (4 —») + 9" (m—n) (2" —2/) 5 consequently the increase of tension at those places is $! (m—n) (2! —2'). If we add this change of the tension to that above found, we obtain for the entire difference of the tension, produced by the de- _ composition until the commencement of the permanent state, (@" = 4!) (m—n) (2! —2), which, if we substitute @for $' — 9', changes into ® (n—m) (2 —2’). If now we represent by S the magnitude of the current, and by A the sum of the tensions in the circuit, before any chemical change has commenced, by S’ the magnitude of the current, after the permanent state has been attained; lastly, by L the 2u2 _ 504 OHM ON THE GALVANIC CIRCUIT. reduced length of the entire circuit, which, as we have seen, re- mains under all circumstances the same, it results , A—® (n—m) (2"— 2) | ae ss 3 Ss ‘ : A. ; or, if we write for iis equivalent S, ®@ (n--m) (2 —2’) Sie Sin py Tse ie 2 ae so that, therefore, eli) Az) designates the decrease L produced in the magnitude of the current by the chemical al- teration. . 39. After all these intermediate considerations, we now pro- ceed to the final determination of the chemical alteration in the changeable portion, and the change of the current in the whole circuit produced by this chemical alteration, where, how- ever, we have constantly to keep in view only the permanent state of the altered portion. If we substitute in the equa- du dz have just found, is solely dependent on the fixed and unalter- able values of z, and therefore has to be treated in the calculation ab a+(b—a) 2 tion (& ) given in § 35, for xw its value 8’, which, as we as a constant magnitude; further, for x its value given in § 36, this equation changes into or if we place 8+ PwB=, and Pw (« —f) =Q, into abw(n—m) dz ORR het at eae ae from which, by integration, we deduce the following : (6—aB—aO | iy 2+02z abw(n—m) S a+ (b—a)2’ where ¢ represents a constant remaining to be determined. If we designate by x the abscissa of that place of the chemically changed portion for which has still the same value, which, previous to the commencement of the chemical decomposition, belonged to each place of this portion, for which therefore z =§, OHM ON THE GALVANIC CIRCUIT. 505 and determine in accordance with this statement the constant c, our last equation acquires the following form :— Oe ROL, ieee a+(b—a)z at+(b—a)t" where e denotes the base of the natural logarithms. The fol- lowing consideration leads to the determination of the value x. Since, namely, € represents the space which the constituent A occupies in each individual disc of the changeable portion pre- vious to the commencement of the chemical decomposition, if we denote by / the actual length of this portion, /§ expresses the sum of all the spaces which the constituent A occupies on the entire expanse of the changeable portion ; but this sum must constantly remain the same, since, according to our suppo- sition, no part of either of the constituents is removed from this portion, and both maintain, under all circumstances, the same volume, even after chemical decomposition has taken place ; we obtain, therefore, Mr I Zaz, where for z is to be substituted its value resulting from the pre- vious equation, and the abscisse corresponding to the com- mencement and end of the changeable portion are to be taken as limits of the integral. These two last equations, in combination with that found at the end of the previous paragraph, answer all questions that can be brought forward respecting the permanent state of the chemical alteration, and the change in the electric current thus produced, and so form the complete base to a theory of these phznomena, the completing of the structure merely awaiting a new supply of materials from experiment. 40. At the conclusion of these investigations we will bring prominently forward a particular case, which leads to expres- sions that, on account of their simplicity, allow us to see more conveniently the nature of the changes of the current produced by the chemical alteration of the circuit. If, for instance, we admit a = 4, and a=, the differential equation obtained in the preceding paragraph changes into the following : 0= 2 dx—aw (n—m) dz, whence we obtain by integration 506 OHM ON THE GALVANIC CIRCUIT. aw (n—m)’ when x designates the value of x, for which s = %. Since in this case the value of = constantly changes to the same amount on like differences of the absciss, the abscissa x, which belongs to its mean value {, as it was at all places of the changeable portion previous to the commencement of the chemical decom- position, must be referred to the middle of this portion. If, therefore, 2! and z!', as above, represent the values of z, which correspond to the commencement and end of the variable por- tion, and / the actual length of this portion, it follows, from our last equation, that and piety Giang dS, aw (n—m) and from these two equations results (a—m) (2! — 2!) = Bs 2 aw or, if we put, instead of ae by which here nothing further is expressed than the unchangeable reduced length of the chemi- cally variable portion, the letter A, the following : (n— m) (2"— 2!) = . 2is5 If we place this value of (7 — m) (z'' — 2’) in the equation found ah & (n—m) (2! — 2’) g ag — 2mm nA), and at the same time substitute for = its value S’+ wa, we obtain ) s=8— 5 (8+ 4a), an equation, the form of which is extremely well suited to indi- cate in general the nature of the change of the current pro- duced by the chemical alteration, and the expressions of which coincide exceedingly well with the numerous experiments I have made on the fluctuation of the force in the hydro-circuit, and of which only a small part have been published*. * Schweigger’s Jahrbuch, 1825, Part 1 and 1826, Part 2. r we ARTICLE XIV. Selections from a Memoir on the Expansion of Dry Air. By the late Professor F. RuDBERG. [From Poggendorff’s dnnalen, B. 41. 8. 271.] AMONG the constants in physics there is certainly not one which is usually considered to be determined with greater preci- sion than the expansion of dry air, or of dry gases generally, under a constant pressure, between the standard points of the thermo- meter scale. The numerous experiments made by Dalton and Gay Lussac, almost at the same time, about the beginning of the present century, appeared to show, beyond all doubt, that the amount of this expansion from 0° to 100° C., under a con- stant pressure, was 0°375 of the volume of the air at 0°. Their great skill in experimenting, and the magnitude and number of the services they had rendered science, left no room for any doubt as to the accuracy of this result; consequently, for more than thirty years in all computations in which the expansion of gas occurs, it has been assumed to be 0°375. The constant in question is undeniably of the greatest im- portance in Physics, since it forms the basis of all methods of measuring temperature ; it is used in the explanation of most of _ the phenomena caused by heat; and lastly, is requisite in the reduction of many observations in Physics and other sciences; as, for example, in determining the velocity of sound, in the measurement of heights by means of the barometer, and in com- puting astronomical refractions. This being the case, it will no doubt appear surprising, that the value of this constant, which has been employed up to the present time, is erroneous to no small amount, since, as will be shown in this memoir, it appears to be not more than from 0°364 to 0°365, instead of 0°375. The change of volume produced by heat can be determined, either by heating cold air and measuring the increase of its vo- lume, or by cooling warm air and determining the diminution of its volume. I have adopted the latter method, as being by far the most accurate. In most of the experiments, a glass globe, having a neck made of thermometer tube A B C (fig. 1.), and capable of con- 508 RUDBERG ON THE EXPANSION OF DRY AIR. taining from 120 to 150 grammes of mer- E cury, was used for containing the air. After the end of the tube had been fitted into a hole in a cork at one end of a cylinder, D E, containing chloride of calcium, the air was dried, either by heating the globe strongly over a spirit-lamp, and then suffering it to cool, and repeating the process at least fifty or sixty times ; or else by connecting the end E of the cylinder with an air-pump, and exhausting and re-admitting the air fifty or sixty times. I have ke not observed any difference between these two methods of drying air, but have found one as x effectual as the other. The chloride of cal- cium was fused at a red heat, then poured out upon a cold plate of metal, and as soon as it became solid, broken to pieces, and put into F bottles with ground stoppers while red hot. The globe having been dried in this manner, and remaining in connexion with the chloride of calcium tube, a small opening being made in the cork at E for the air to escape through, was suspended by means of a cork G cut in two, in the boiler F, the upper part of which, as described in my memoir on the construc- tion of thermometers (Poggendorfi’s Annalen, B. 40.), consists of two concentric cylinders, so that the globe and the greater part of the tube were surrounded by steam. After the water had been boiling three quarters of an hour, or an hour, the cy- linder D E was removed, and the boiling continued for about ten minutes longer. The height of the mercury in the barome- ter was then observed, and the tube sealed, the water being kept boiling freely in the mean while. After the ball had been weighed with a balance, which turned with one-tenth of a milligramme, it was firmly fixed to the arm Q (fig. 2.) of a steady support, with the tube passing through a hole, in a metal dish H. The arm Q was then so far de- pressed, that the point of the tube was deeply immersed in the mercury of the trough T. Lastly, the point of the tube was broken off, and in order that all the mercury re- quisite might enter, the ball was suffered to remain in this situation several hours, almost always all night, although I had convinced myself that not more than.a quarter of an hour = RUDBERG ON THE EXPANSION OF DRY AIR. 509° at most was requisite, with even the smallest of the tubes which I employed. Snow was now placed on the metal dish H, and the globe surrounded with it on all sides. The water produced by the melting of the snow escaped through the tube L. As soon as the snow began to melt, fresh snow was carefully added, so that the tempe- rature of the globe was kept at 0° for about two hours, and sometimes even longer. When by this means I was certain that all the mercury had ac- tually entered which at 0° could be forced in by the pressure of the atmosphere, I closed the fine opening of the tube with a very soft mixture of wax and turpentine, which was prepared for that purpose in a little spoon of iron. At the same instant the barometer was observed, in order to de- _ termine the existing pressure of the atmosphere; the snow was then carefully removed, and the difference of altitude between the surfaces of the mercury within and without the globe measured. For this purpose the measuring apparatus N M K was pre- pared. Upon the whole, it depends upon the principle employed in measuring the height of the mercury in Fortin’s barometers. A slider M, embracing tightly the vertical bar, is moved up or down by a screw P, and therefore also the cylindrical ring N, and screw K, which are connected with it. The ring N having first been made accurately horizontal, was depressed, surround- ing the globe, till its under edge coincided with the surface of the mercury in the globe, and the screw K S turned till its point § just touched the surface of the mercury in the trough. It is evident that the difference of altitude between the under edge of the ring and the point S was equal to the difference of alti- tude of the two surfaces of the mercury. After the contacts had 510 RUDBERG ON THE EXPANSION OF DRY AIR. been made as accurately as possible, the measuring apparatus was removed, and the globe, the extremity of the tube being closed with wax, as has been already stated, lifted out of the trough. The difference of altitude of N and S was then accu- rately measured, by means of two graduated scales, placed at right angles to each other, and the globe, with the mercury which had been forced into it, weighed after the wax had been removed. When this was accomplished, the tube was bent at the end, so that it could be dipped into a vessel of mercury, the globe filled with it, and all the air expelled, by carefully boiling. When cold it was placed in snow, and completely filled with mercury at 0°. When no more mercury could be introduced (this was known to be the case by the thread of mercury showing itself at the extremity of the tube), a clean empty vessel was placed un- derneath, to receive the mercury that ran out; the globe taken out, and placed in the boiler. The mercury that escaped be- tween the temperatures of 0° and temperature of boiling deter- mined by the height of the barometric column, was weighed, and the weight of this, added to the weight of the mercury re- maining in the globe, consequently gave the weight of mercury contained in the globe at 0°. From these two weights, and the true expansion of mercury, the true expansion of the glass may be calculated. Let uw be the volume of the globe at 0°, and therefore the vo- lume of the mercury contained in it at that temperature; / the height of the barometric column, in centimetres, at the instant the tube was sealed; T the corresponding temperature of the vapour of boiling water; 100 A the true expansion of dry air from 0° to 100°; 100 G the expansion of glass in volume from 0° to 100°. At the instant the end of the tube was closed with wax, let v be the volume of the air contained in it; g the weight of the mercury contained in it; & the height of the mercury in the barometer ; / the height of the surface of the mercury in the globe above the surface of the mercury in the trough; p the weight of the mercury contained in the globe at 0°. The vo- lumes, pressures, and temperatures of the air at the time the tube was sealed, and at the time it was closed with wax, were asu(1+GT), v; h,k—J; T°, 0° respectively, therefore {VAG T), > bet HAS) =A (1+ AT). RUDBERG ON THE EXPANSION OF DRY AIR. bit But Therefore (AA PES ae dahl IE NE i gaged h Let 7’ be the weight of the mercury expelled from the globe when heated from 0° to T’°; 7 the weight of the mercury ex- pelled when heated from 0° to 100°; 100 M the true expansion of mercury from 0° to 100°; 8, 0! the weights of a unit of volume of mercury at 0° and 100° respectively. Then wen 3 100: "Ty,? (1 +100 M) = 4; the volume of the mercury at 100° = uw (1 + 100 M); the volume of the globe at 100 = u (1 + 100 G); therefore the volume of the mercury at 100° expelled = u.100 (M — G); therefore bu . , — rie ptr tei Enid tg its weight r = J'u. 100 (M S) Shoe Mn (M — G) _ p-100(M—G) Sats 100M * Therefore the true expansion of glass from 0° to 100° 100 G = 100M — zl 4+ 100M). ‘The value of the true expansion of mercury is here assumed to be known. This may be done with confidence, inasmuch as it has been determined, quite independently of the expansion of glass, by the masterly experiments of Dulong and Petit. They found 100 M = 0:0180180. Therefore 100 G = 0:018018 — 1:018018 re __ The following table exhibits the values of 100 M — 100 G for the glass employed, which was potash glass, from the manufac- tory at Reymyra. The first fifteen results were obtained from globes used in experiments upon the melting-points of easily fusible metals; the remainder were obtained from the globes used in determining the expansion of air. They show that the same kind of glass, though made at different times, and there- fore in different meltings, possess the same expansibility. 512 RUDBERG ON THE EXPANSION OF DRY AIR. "015732 "015720 "015732 "015713 *015744 7015761 *015706 °015697 "015754 °015730 OLS jak *015751 "015744 *015711 "015741 015744 "015723 °015737 "015753 *015726 "015735 015720 *015762 "015736 The mean of the twenty-four results gives the difference be- tween the true expansion of mercury and the expansion in vo- lume of potash glass, 100 M — 100 G = 0-015733. Hence the true expansion in volume of the potash glass of Reymyra from 0° to 100°, 100 G = 0°002285. In the following table of the results of nine observations, p and p — q are expressed in grammes, and A, &, / in centimetres. p- p-4Yy. h. k. 1. Re 100 A. 166°6891 . 1383:1409 . 76°528 . 74:277 . 3:98 . 100°20 . 0:3643 173°4432 . 1381-7215 . 76°362 . 77:°584 . 3°81 . 100718 . 0°3654 183°4963 . 148°2124 . 75°702 . 75:965 . 4:69. 99°89 . 0:3644 154-2360 . 120°6356 - 77°230 . 75-910 . 3:50 . 100°45 . 0:3650 174-6862 . 134-9876 . 77:985 . 77:748 . 3°81 . 100°73 . 0°3653 187:4650 . 144:9009 . 76-444 . 76°474 . 3°81 . 100°16 . 0°3636 198:8099 . 172°7273 . 76°442 . 76°271 . 11:70 . 100°16 . 0°3651 184:4872 . 146°6123 . 75°811 . 75°342 . 5°25 . 99°93 . 0°3643 191:1037 . 178-9558 . 75°779 . 76°105 . 16°65 . 99:92 . 0:3645 The value of 100 A. in the sixth line is too small, in conse- quence of the loss of a globule of mercury in one part of the experiment. The mean of the preceding values of 100 A. is 0°3646. Two other experiments were made with cylinders of glass. It was found impossible to boil the mercury contained in them, on account of the smallness of the bore of the tubes which formed their necks. The results are therefore considered less accu- rate. Dp p-4y h. k. 1. ii 100 A. 1158°902 . 946:516 . 76°773 . 76°789 . 7°80 . 100°28 . 0:3646 1196:992 . 991:695 . 76°313 . 75°470 . 7:92 . 100°12 . 0°3662 Meant) J7ieiseseonedasseercest seansacsbene 0°3654 Two other observations were made without previously drying the air with chloride of calcium, in which, however, an exami- nation with a microscope showed that there were no visible drops of water in the globe. These experiments were made merely RUDBERG ON THE EXPANSION OF DRY AIR. 54S" for the purpose of seeing how great an error might be introduced by neglecting to dry the air completely. The results are, De rg. h. ke Ll. T. 100A. 166:4746 . 128°0336 . 75°166 . 75:049 . 4:21 . 99°69 . 0°3840 139-2725 . 106°1248 . 75-964 . 75-201 . 4°325 . 99:99 . 0°3902 The experiment was repeated with the ball used in the last of the above observations, the air having first been perfectly dried. It gave the following results :— p p-¢@. h. k. 1. T. 100A. 1392725 . 107:8192 . 76:440 . 76°185 + 3°725 . 100°16 . 0°3652 From the whole of these observations, I can come to no other conclusion, than that the expansion of dry air, and without doubt of all other dry gases, from 0° to 100°, is not 0:375, but only from 0°364 to 0°365 of the volume of the gas at 0°. 514 ARTICLE XV. Second Series of Experiments on the Expansion of Dry Air between 0° and 100°. By the late Professor F. RupBEre. [From Poggendorff’s Annalen, B. 44. 8. 119.] SINCE the publication of my experiments on the expansion of air (Poggendorff’s Annalen, B. 41.8. 271.), I have had an appa- ratus constructed, by the aid of which one such experiment may be made in the short space of an hour and a half, or two hours. The mean of the results which it has given agree perfectly with those I had previously obtained. I here communicate a short description of the apparatus, and the values of the expansion which it has afforded. The construction of the apparatus enables us to determine the pressures of a given mass of dry air at 0° and at 100°, the spaces occupied by the air in the two cases differing only by the expansion of the receiver. The dry air is contained in the cylinder A B, which com- municates through the slen- der tube DE with the wide tube F, which, together with a second tube H I, about 50 centimetres long, and open at both ends, is cemented into the lid of the box G. The box contains a leathern bag for holding mercury, the ca- pacity of which, as in a baro- meter, can be altered by means of the screw K, so thut the mercury may be elevated or depressed in the tubes. A fine line is traced with a dia- mond point on the slender tube D E at C, up to which the mercury is screwed, as RUDBERG ON THE EXPANSION OF DRY AIR. 515. well when the air in the receiver A B is cooled down to 0°, as when it is heated up to the boiling point of water. In order to measure with accuracy the altitude of the mercury in the tube, a brass scale H I, divided into millimetres, is attached to the tubes. The line which marks the commencement of the divisions at C is so long, that it passes behind both tubes, and thus the altitude of the extremity of the column of mercury in the tube HI, above the mark on DE at C, is easily deter- mined. The air in the receiver AB was dried before the tubes were cemented into the box, in the following manner. The lower end of the tube F was drawn out to a capillary point, and connected with a very wide tube filled with chloride of calcium, which communicated with an air-pump. After the air had been fifty times exhausted and re-admitted, the capillary point was sealed and the tube cemented into the box G, which had been pre- viously filled with dried mercury, and lastly, the sealed end broken off under the surface of the mercury. The capillary depression of mercury at C was determined by experiment before the narrow tube DE was joined to the re- ceiver, and found equal to 1°85 centimetres. The tube F was taken of large diameter, in order to receive the air as it expanded on being heated from 0° to 100°, and so ob- viate the necessity of continually screwing up the mercury. The calculation and the method of observing are both equally simple. When the air in the receiver AB is cooled down to 0°, and the mercury is screwed up to C in the tube D EK, let the mercury stand at M inthe tube HI. At the same instant let h be the altitude of the mercury in the barometer. Let the al- titude of the mercury in HI, above the mercury in DE, or C M = &, and let / be the capillary depression of the mercury in DE; then the pressure of the air in the receiver AB will be A+k—J. When afterwards the air is heated up to the boiling point of water, and the mercury is screwed up to C in the tube D EK, let the mercury stand at Pin the tube HI. At the same instant let A! be the height of the mercury in the baro- meter, and the difference of altitude of the mercury C P =X’; then the pressure of the air in the receiver will be h! + k' — 1. Let T be the temperature of steam corresponding to the baro- metric height h', 100 A. the expansion of air from 0° to 100°, and 100 G. the expansion of glass in volume*from 0° to 100°; 516 RUDBERG ON THE EXPANSION OF DRY AIR. then W+k—l 1+AT= er oa, (1+GT). In the above expression the altitudes /!, k', h, k need not be connected for temperature, because the experiment is completed in the short space of an hour and a half, during which the tem- perature of a room will undergo no sensible variation. The only reduction to 0° requisite, is that of the barometric height /!, in order to deduce from it the temperature T. The experiments which I have made up to the present time, with the above-described apparatus, under very different baro- metric pressures (from 752™™92 at + 17°°4 to 783™™72 at + 18°), gave for 100 A. the follewing values :— 0°3640 0°3640 0°3653 0°3648 0°3656 0°3640 0°3641 0°3643 0°3664 0°3648 0°3648 0°3645. The mean value of 100 A. is 0°36457. Since this mean value is the same as that given by my former experiments made in a manner entirely different, I venture to consider it as fully esta- blished,—that the true expansion of dry air between 0° and 100° centigrade, is only from 0°364 to 0°365 of its volume at 0°. ArTicLeE XVI. On Barometrical Measurement of Heights. By ¥. W. BessEu. {From the Astronomische Nachrichten, Nos. 356, 357.] l. THE atmosphere of the earth is known to be composed of the nitrogen, oxygen, and carbonic acid gases, and of aqueous vapour. These constituents are supposed to exercise no che- mical action on each other; and arbitrary quantities of them, mixed together under circumstances of equal temperature and pressure, occupy spaces equivalent to the sum of the spaces that they would severally occupy. Were we to assume that the constituents of the atmosphere are mixed in the same propor- tion at all times and at all altitudes, we might dispense with the knowledge of what that proportion is, in treating of the condi- tions of their equilibrium; but if we desire to preserve the freedom of founding our investigations on other suppositions also, we must not pass by in silence the mode in which the con- stituents are combined. The proportion of the three gases may not always be exactly the same at a given point of the earth’s surface; but the altera- tions which take place are so small, that they are only discover- able by chemical experiments frequently repeated ; we cannot. therefore, regard the proportion as determinable by observation for each particular case, and we must assume a certain proportion. According to Berzelius, the spaces occupied by the three gases, in the order in which they are named above, are to each other as 77°96; 21°15; 0:07 ; or, one volume of dry atmospheric air at the surface of the earth contains v =0°78605 nitrogen gas v, =0°21325 oxygen gas v,, = 000070 carbonic acid gas. The same great chemist has given the densities of these three gases, under the pressure which gives to the mixture the den- sity D, viz.: VOL. 11. PART VIII. 2M 518 BESSEL ON BAROMETRICAL Nitrogen gas = 0:9691 D=¢.D Oxygen gas =1:1026 D=d, D Carbonic acid gas = 175260 D = d, D. These six numbers require to be slightly altered, in order that they may correspond to the relation Pevd + 0,4, +0, dy « eet. UO) Designating by M, m, m, m,, the masses, and by D, 4, 3, 8), the densities of the mixture and of its constituent parts, we have, on the supposition of equal distribution in the space, Di: M 0 sm = 2): m, = 5/2 ys See oe) further, if P, p, p,, p, denote’ the pressures which the mix- ture and its constituent parts, the latter taken separately, exert on the unit of surface of the enclosing space, we have by Ma- riotte’s law, | beat go oo age) Pt Oy et = 210, P:d,D = pyiby and also Pol = 9.0 =p) 2) — pire Oy thus s=vdD; § = 9,4, Ds; 8, = 0, 4, D. Introducing these values of 8, 8, 8, into the above proportion (2.), we obtain m=vdM, m,=v,d,M, m,=v,d,M, and aa M=m-+m,+m,, we have also the relation (1). To satisfy this relation I have slightly altered d and d,, making the first 0°9711, and the second 1°1048. Biot and Arago determined the density of atmospheric air (é. e. the mixture of the three gases) at the surface of the earth, at the temperature of melting ice, and under a pressure of a column of mercury of the same temperature in the 45th parallel of latitude of 336°905 Parisian lines, to be 10466°8 times less than that of mercury. Under the aforesaid circumstances, there- 1 < fore, D = 104668" As the temperature increases, the specific elasticity of the air, or the space which a given quantity of air occupies, increases also, the pressure remaining equal. Gay Lussac arrived at the remarkable result that the specific elastic force of all gases and a Hees MEASUREMENT OF HEIGHTS. 519 vapours alters equally with equal changes of temperature, and that the alteration is proportional to degrees of the mercurial thermometer. If the elasticity at the temperature of melting ice be 1, and its alteration for a change of temperature corre- sponding to one degree of the thermometric scale = f, its value for a given amount of the thermometer is E=1 + kt. For the temperature of boiling water Gay Lussac found E = 1°375. Besides the three gases the atmosphere contains aqueous va- pour, which is present in variable quantities, determinable only by experiment in each particular case. I propose to return hereafter to this part of the subject ; but I will first consider of atmospheric air unmixed with aqueous vapour. 2. Barometric measurements of height rest on a comparison of the observed pressure of the atmosphere at different heights, with the expression denoting the conditions of its equilibrium. Although this expression has been developed in the Mécanique Céleste, and in several subsequent works, I shall not omit its development here; as it will enable me to introduce a small alteration, as well as to connect what I have further to say. Mariotte’s law requires that to produce equilibrium the density (8) of the air should be in the direct ratio of the pressure (p) which it experiences, and which it consequently exerts in return, and in the inverse ratio of its elastic force; or that be constant. The air is here supposed to be constituted alike at all altitudes. If we take for the measure of p the pressure exerted on an unit of surface, by a column of mercury of 336°905 Parisian lines, at the temperature of melting ice, at the _ Surface of the earth in the latitude of 45°,—for the measure of 8 _ the density of mercury at the temperature of melting ice,—and _ for the measure of E the specific elastic force of air at the same temperature,—and if we make 6 = D for p= 1, and E = 1, we have Bi HTD Lites 5) hye ea pole v0 OR) The pressure of the air at an elevation 2 above the surface of the earth, or at a distance a + 2 from its centre, is the sum of the pressures of all the strata above x. A stratum between the 2m 2 520 BESSEL ON BAROMETRICAL elevations x and 2 + dz has for every unit of its surface the mass ¢.d a; therefure it exerts on this unit the pressure (s) ®(—2.) aa, aQ+ea@ in which (g) is gravity at that part of the earth’s surface which is perpendicularly beneath the point to which » and 8 belong, expressed in terms of gravity in the latitude of 45°. But in order that the diminution of pressure, caused by taking away this stratum from those above 2, may be obtained in terms of the measure applied to p, the above expression must be divided by that measure, which then gives ee PG) 10 ( a ) ; Oe ar505 Nag a) or, if we prefer the use of the toise to that of the Paris line, dp= —O~ ( g \Vde.. (4.) 336°905 a4+2 If we eliminate ¢ by combining the two equations, we obtain dp __ _ (g) 864.D ( a ) dx Bian 336°905 ates HK By the integral of this equation the values of p at two different elevations above the surface of the earth, v = h, and 2 = i’, be- come comparable with each other ; or, if we denote them by P and P’, and employ Briggs’s logarithms, of which the modulus IS fy in P(g) 864. Dom a _ daz. = Bi 336°905 i ( lol Be or if we write 336-905 34 Tes eee Seiwa Ta te ee then P 1 ht a 2 dz lg =-7/, (—-) Boe . (5.) The integration, which still remains to be performed, requires that we know the relation between w and E, or the law according to which the observed heights of the thermometer + and 7’, cor- responding to the temperature of the air at the two heights, pass into each other. We do not know this law in every case, and we have, therefore, no ground for assuming the change of tem- perature to be otherwise than proportioned to the change of ele- . ? MEASUREMENT OF HEIGHTS. 521 vation. In order to correspond approximately to this view, and at the same time to give the integral the most simple form pos- sible, Laplace assumes iy gee ee a+e2 to be constant for all corresponding values of ¢ and 2, and deter- mines the constant i, so that it may satisfy the two observed temperatures t and 7’. Hence (l+2)?+iX=(l ieee on (1+ k7')?+7H! ah ah! See Gah thon 2k (r — 1) )(a+a2t*) H’—H ; where I have written X, H and H’ for —— We obtain thereby i= and = fa) dr= — AG +k?) dt; and further, GQ \a a2 Qk (-,) meee ghee whence the integral taken from h/ to i is [re ee ee i ae We have thus, in accordance with Laplace’s assumption of the law of the change of temperature, transformed the formula (5.) into Ee 1 H'—H log = T oS 1+k (6.) t+!" 2 3. I have hitherto considered the air as dry, and have still to take into account the aqueous vapour which it always contains. Tf, in a circumscribed space, the mixture of the dry constituents of the atmosphere exert on the circumscribing surface the pres- ure p, the aqueous vapour the pressure p,, aud if the specific gravities of the two be respectively denoted by D and d, D, and of the moist air which results from their mixture by Di, then according to equation (1.), 522 BESSEL ON BAROMETRICAL RN ANE SrA: oD! and aye = P; amet, pp; thus i = TL + Pp, ca PtP, or if, to avoid introducing a new sign, we denote the whole press- ure (=p + p,) by p Di=D {1-2 (—d)f veers For moist air, therefore, the equation (3.) is changed into 6.E = {p —p,(1—d)} D, and its combination with (4.) gives O=dp + yp tS share Ge To integrate this equation, we must know the dependence which py, has on the other variable magnitudes. If in a parti- cular case we have no observation determining the amount of aqueous vapour contained in the air, we must found our calcu- lation on the supposition either of a mean state of the atmo- sphere, or of one which may appear more suitable to the actual circumstances. I will first examine the case in which we may suppose that at every point of the atmosphere there exists a de- terminate portion of the maximum quantity of vapour which it can receive in accordance with its temperature. If this maxi- mum of vapour exert the pressure (p,), I then assume P= (Pi)> where by a I understand a constant factor not greater than unity, the value of which is to be determined hereafter. The expression for (p,), at the given ¢ of the centesimal scale, deduced by Laplace (Méc. Céi., iv. p. 273.) from the experiments of Dalton, in the unit of pressure chosen in the foregoing article, = 1¢ (t — 100) 0-0154547— (¢ — 100)? 00000625826. For which we may also write (p,) = 0°0067407 .10 t:0°0279712 — t? 0°0000625826 rier (9.) We have thus, conformably to the supposition, p,= «6 10%'— °F, : MEASUREMENT OF HEIGHTS. 523 where 6 = 0:0067407 a = 0°0279712 c = 0°0000625826. If we now multiply the differential equation (8.) bv 1 fax 1oF we can integrate the product, namely, - Oa ree ah © dX ‘Pp pl ? [p,r0 E Laplace’s assumption of the law of the change of temperature between two elevations, at each of which the temperature is given by observation (Art. 2.), is If we substitute this, and also the expression above given of (p,), we have 2k 2k i t — )t—c#? C=p.10 i” ,? Bib a) 106 *—7i) dle a a By this equation we obtain the relation between the pressures of the atmosphere P and P’ at the heights A and A’, namely, Sp Lae - eae mao. fi) — Pl 1g. vi t—cf _ 2%B(1—d)k -#) ari sy ae dt. If we write T for 1 (r + 7’), and T + z for ¢, then the integral still to be sought is changed into 2k 2k Tare i 4(¢—7) — —2cT z—c.2x? mad ate fT ae gh ) dz. = ales’ : If, for brevity, we write A(r—%) 2k = ees} a@— 7 .—2cT Ju—cx? EC hf OS eee ) dz, — ad Sie e/) We have thus, 524 BESSEL ON BAROMETRICAL or 2k Qk f(s — 0 —-7.(r-7 — ae = Pio * aa +7, 10 rh Ga pe whence it follows that Re ow) Pay Tap? PP 4 ee ~ Pv [47 PP + wv] +4’ and if we take the Briggs’s logarithms of both numbers of the equation, and develope fully to uw? inclusive, 2k P u pt ee ps 7 (Ee 10 but according to the relation between the temperatures and the elevations in Art. 2, whereby we obtain PS) ee ei u log PY Li VCP) eae oe The integral occurring in the expression for u is found by deve- loping the Sextet ee into a series SN {14% 2 (a—74-2e)*-2¢n | +8... In order to estimate in some measure the amount of the second member of this series, we may assume that the centesimal ther- mometer falls a degree for every 85 toises of elevation. Then is this member for H' — H = 2.1000 toises, and for T = 0, = n?.0°0093 for small differences of elevation; it is therefore an inconsiderable part of the first member; and even for the greatest accessible elevations it does not amount to a tenth part of it. The supposition as to the distribution of aqueous vapour in the atmosphere, on which the present calculation rests, has far more uncertainty; on which account I think there can be but little interest in adhering strictly to it by means of a compli- cated calculation. I therefore simplify it by assuming _ 2aB(1—d,)k (r—7) ins li According to Berzelius d, = 0°62, and it has been snown above that lot? -¢™ MEASUREMENT OF HEIGHTS. 525” 6 = 0:0067407, 2k(r—7') H'—-H a oe by 2 Hence Hi (ror* oF 0-002s6l ; I 1+kT and we obtain, by the substitution of this expression in equa- tion (10.), H'—H 0°002561 | aT—cT? be = raat 4i-* very Van If we wish to found the calculation of the difference in height of two points, where the pressures and temperatures of the air have been observed, upon the supposition of a mean state be- tween dryness and saturation, we must make 2 =}. But if we have not an immediate determination of the quantity of aqueous vapour on such occasions, we may obtain in particular cases, by taking other circumstances into account, greater exactness than by making « = 3. If, for example, rain falls throughout the whole space between the two elevations, then a = 1. If the two points are far distant from the ocean, and in a coun- try known to be particularly dry*, it will be more suitable to take less than}. In order to give a direct view of the infu- ence of aqueous vapour on barometric measurement, I will de- velope it further. The increase, which is occasioned in a differ- ence of elevation computed on the supposition of dry air, by ‘the introduction of the consideration of the aqueous vapour, according to equation (11.), is aw ~ law where w is written for 0°002561. oT_er? (H’— H) TP Py 10 : If we neglect the square of this quantity, and make (HH P= P1lo—VU+kN), which can only occasion an error of the order w®, ‘ H’—H . 95 9/7 M1LeT™ eae — 102! +E) | 1077? -¢T™ * Such is the case in a great part of northern Asia, as we learn from Adolphe Erman’s Reise, vol. ii. p. 67, where we have not only the fact, but the geographical relations of which it is the consequence. 526 BESSEL ON BAROMETRICAL and thence the influence of the aqueous vapour H’—H =a aS (H'—H) 1077 + #7) | 19aT—eT.. . (12) If we assume the pressure in this formula at the height A = 1, | or the height of the barometer there = 336,°905, and k = 0°00375, we find the quantities to be multiplied into a@ for dif- ferent values of H' — H and T as they are given in the follow- ing table. T=2(r + 2). H’-H. 0°. 10°. | 20°, ve T T Hic 500 1:36 2°55 4-64 1000 2:90 5-41 9°83 1500 4-62 8-61 15-60 2000 6°55 12°18 22:02 2500 8-70 16°15 29°14 3000 11-10 20°15 37:02 From these numbers we may judge of the influence on the result which may be occasioned by an uncertainty in the value of « in any occurring case. 4. Since the invention of Daniell’s Hygrometer and of August’s Psychrometer, we have the means of ascertaining at all times, with ease and sufficient exactness, the quantity of aqueous vapour contained in the atmosphere. The observation of the psychrometer at both elevations, in addition to those of the barometer and thermometer, is readily made, and dispenses — with any arbitrary supposition in regard to the moisture, as_ that of the thermometer does in regard to the temperature of the air. I will, therefore, examine the rules of calculation which are applicable in cases where the psychrometer has been observed. The psychrometer rests on the comparison of the heights r, and + of two thermometers, one with a moistened bulb, and the other with a dry bulb. If the greatest pressure which aqueous vapour at the temperature ¢ can exert be denoted by ¢ #, and the height of the barometer in Parisian lines by 4, the existing pressure of the vapour eg ee eae ~ 7! 336°905 (m—+) where, if the value of r be positive, m = 640; and if negative MEASUREMENT OF HEIGHTS. 527 (in which case the moistened bulb is coated with ice), m = 715. This formula is given by August, the inventor of the psychro- meter, and rests on the comparison of experiments with certain physical considerations*. The expression for ¢¢ for different values of ¢ is deduced by August from observations on the pressure exerted by aqueous vapour at different temperatures, employing a mean result, which Kimptz has derived from the observations of Dalton, Ure, Schmidt, and Artzberger. But these, with the exception of the two first-named series, are so little accordant with each other, that it may be doubtful whether all the four should be combined. I prefer to adhere to the expression already given, which Laplace derived from Dalton’s observations, to which those of Ure approximate. It is my opinion generally that formule which are well known and extensively applied ought ~ not to be altered until the necessity for the alteration becomes decided, which is by no means the case in the present instance. The researches since made by Arago on the same subject were confined to the elastic force of aqueous vapour at very high tem- peratures, and we cannot be sure that a formula of interpolation, which represents those satisfactorily, is applicable in much lower degrees of temperature. By applying Laplace’s expression, we obtain the pressure exerted by the aqueous vapour contained in the atmosphere ac- cording to the formula which has been already given. | (7 ne i) b, Fee tre m— 7; Pp, = 00064707 .10%% —¢°77 — 0:0016562 and if we divide it by | (p,) = 00064707 . 1077 ~°”, or by the pressure which the vapour would exert if the air were saturated with it, we obtain the proportion denoted above by a, thus: atr,—cr? (13.) Beare = 02455 a Se SSRI 10¢7-°¢ m— t, 10¢7-¢7 To facilitate the calculation of « I subjoin a table, the first sec- tion of which is for all values of ¢ from — 20° to + 30°, con- log 10°'- °" = ft, * Poggendorff Ann, der Physik, vol. Ixxxi. p. 69, and vol. xc. (xiv. of the new series), p. 137. 528 BESSEL ON BAROMETRICAL and the second section 0°2457 log : 10-2#+e? — Ye. We obtain thereby log A=fr,—fr log B = log A + Wr, + log (r — 7) + log 4, and «, which is sought, = A — B. ip ft. Yt. Es ft. yt. —20°/ 94155] 9, | 71086 |, |+5°| 01383 | |. | 64493 | | —19 | 94459 | || 70788 5 | 6 | 01656 Bs + | 64227 | as —18 | 94762 7-491 | 7 | 01927 6:3963 302 | 296 | 271 264 —17 | 9:5064 7.0195 8 | 02198 63699 300 295 | 269 262 —16 | 9:5364 69900 9 | 02467 6:3437 299 293 «| 268 261 | —15 | 9:5663 69607 10 | 0-2735 63176 298 292 266 260 —14 | 9:5961 69315 11 | 0-3001 62916 |~_ | 297 290 | 265 258 —13 | 9:6258 68925 12 | 0:3266 6:2658 ~~ | 295 " | 290 264 257 —12 | 9:6553 6:8735 13| 0:3530|~ | 62401 294 288 | 263 256 —11 | 95847 | O 6:8447 ia 14 | 03793 | 3.5 | 62145 oe 4 4 vo —10 | 9:7140 68160 15 | 0-4055 61890 292 285 260 253 —9| 97432 6°7875 16 | 0-4315 | ~ | 61637 290 285 259 252 — 8 | 9-7722 67590 17 | 0:4574 61385 | ~~ 289 283 258 251 — 7 | 9:8011 6:7307 18 | 0:4832 61134 288 282 257 249 — 6 | 9:8299 67025 19 | 0:5089 6-0885 287 280 255 249 — 5 | 9:8586 66745 20 | 0:5344 6-0636 286 279 254 Q47 — 4| 9:8871 66466 21 | 0:5598 60389 284 278 253 246 — 3| 99155 | 0), | 6618s a 22 | 05851 | 9 | 60143 is — 2| 99438 65911 23 | 06102 5:9899 282 276 | | 95) 243 — 1] 9-9720 65635 | 974 | 24| 06353 5:9656 280 | (6:5361 249 242 0 | 0-0000 bs — | 25 | o-6602 59414 279 | § 65842 | 272 247 241 + 1] 00279 ahi 6°5570 a 26 | 0-6849 an 59173 | og 2| 0:0557 6-5299 27 | 0:7096 58934 277 270 245 239 3 | 0:0884 6-5029 28 | 0-7341 58695 275 _. | 268 | 244 "| 937 4 | 0-1109 6-476] | 29 | 0-7585 5:8458 o74 268 243 235 5 | 01383 64493 30 | 0-7828 5:8223 If we determine in this manner the values of « for two eleva tions, they will seldom be found of equal amount; as the la B33) = MEASUREMENT OF HEIGHTS. 529 of the transition of the one to the other is not known, we are compelled to decide arbitrarily, and it seems to me most suitable to take the mean of the two values of « to be applied in the calculation of the formula (11.). 5 It appears to me needful to examine more closely the dif- ferent suppositions by means of which I have obtained the for- mula (11.). The first assumption in all researches relating to the pressure and density of the atmosphere at undetermined heights, is that of its equilibrium. That this is not strictly cor- rect, is not now said for the first time. Its incorrectness is shown both in the oscillations of the barometer around its mean height at each point of the earth, and in the difference of this height at different points strictly at the level of the sea. The know- ledge of this difference was first obtained by an investiga- tion by Adolphe Erman in 1831*, in which he showed, partly from his own observations made in his travels round the earth, and partly from the observations of others in Northern Asia and America, and on board the Russian corvette Krotkoi com- manded by Captain Hagemeister, first, that in the zones of the trade winds, the barometer stands higher at the boundary most distant from the equator than at the boundary which is nearest to it; and secondly, that the mean height of the baro- meter is different in different meridians. The first result rests on observations collected in passing eight times through the gone of the trade winds; and has since been corroborated in Herschel’s astronomically-memorable voyage to the Cape of Good Hope. ‘The second result rests on a comparison of ob- servations made in the Atlantic and Pacific Oceans; the differ- ences amount to several lines, and leave no doubt that the mean height of the barometer at the level of the sea is different at different points of the earth’s surface, and depends on the geo- graphical latitude and longitude of the place. _ The oscillations of the barometer, which may be regarded as accidental, must cause single barometrical determinations of a difference of elevation to deviate from the mean of several de- terminations; but the mean diversities, which depend on the longitude and latitude, if not known, must produce errors, which will not disappear in the mean even of many observations, * Poggendorff, Ann. der Physik, vol. xcix. (xxiii. of the new series), p. 144, 530 BESSEL ON BAROMETRICAL except in the case when the two points, of which the difference of elevation is to be measured, are in the same perpendicular. It follows, from the knowledge we have obtained of these diver-_ sities, that barometrical determinations of the difference of ele- vation of two points, even if resting on observations repeated for years, remain the more doubtful the more distant the points are from each other. If we imagine surfaces surrounding the earth in which the mean pressure of the atmosphere is equal, then all we obtain by the barometer is the determination of dif- ferences of elevation relatively to these surfaces; but whether — the surfaces at which the two points are situated differ more or less from parallelism with the surface of the earth, remains wholly unknown to us, whilst we are ignorant of the function of longitude and latitude which determines their relative posi- tion. This opens a new view in regard to observations on the pressure of the atmosphere; we have to examine for all points of the earth the height of the [atmospheric] surface at which a determinate [mean] pressure is found; but we cannot as yet determine more nearly, the amount of uncertainty arising from the assumption of this height being everywhere the same. Also the uncertainty, arising from the oscillations considered as accidental, cannot be given more nearly; and even if, for the purpose of learning them more correctly, we were to make long- continued observations at points at different elevations, the dif- ferences which might appear could still only be regarded as caused by the combination of these causes with other as un- avoidably erroneous assumptions. The constitution of dry air has been assumed such as it was supposed to be at the time that Biot and Arago obtained for its density the determination given in the first section. If the pro- portion of oxygen were to be altered by x hundredths, its den- sity would be changed by x . 0:001337, and a difference of ele- vation computed under the assumption of D= would 1 10466°8 require to be altered in the ratio of 1: 1+.0°001337. Hum- boldt and Gay Lussac, in nineteen days, between the 17th Nov. and 23rd Dec. 1804, found no sensible alteration in the proportion of oxygen, which seems to justify the assumption of a constant proportion in the two principal constituents of the atmosphere. In the meatime, however, it is known that Dalton has con- MEASUREMENT OF HEIGHTS. 531 sidered it probable that each of the constituents of the air is compressed by its own superior strata alone, and not by the whole superincumbent mass; consequently, that at different heights each constituent possesses the density which it would have if it existed alone. Hence it must result, that the propor- tions of the mixture would vary with the altitude, and the rela- tion of the atmospheric pressure at different heights would differ from the older assumption adopted in Sect. 2. Barometrical measurements of heights have been proposed as a means of de- ciding between the two assumptions. The attention, which the opinions of so eminent a physicist as Dalton deserve, requires that I should follow out his supposition also. The formula (6.) then is no longer correct for the air generally, but only for each of its constituents ; it applies to each of these according as the specific gravity of each is taken instead of that of the atmosphere itself. If we call the pressure exerted by the three constituents of the air, at the elevations h and h, =P; Pj P,, and p', p/, p,|, and their specific gravities D d, Dd, D d,, and if for brevity we denote by U, eto Ei ete Bil Sylysprkyd then, according to formula (6.), gp =p .10—%4 —Ud p= p10 4 —Ud Pil = Py 10 ‘i and as P=p+p,t+ Dy Pi= pl + p/ + pil p=vP; p,=v,P3 py, =v,P; therefore P= Pv. 10-974 v,.10- U4 4-4,.107 0%} or P= P.107Y {9,108 4-9 4.9, 1080-4) 4» 19UA-4) } instead of which, we may write for brevity Me eg. 2 (14.) The quantity , at all accessible elevations, differs little from 1, as is shown by the following table, calculated according to the values given in the 1st Section, viz. S32 BESSEL ON BAROMETRICAL q v = 0°78605 d =0°9711 v, = 0°21325 d, = 1°1048 v,, = 0:00070 d,, = 1°5260 U. yp. Log y. 0-0 1:6000000 0:0000000 G1 10006840 0365 0:2 1-0008334 1148 0:3 1-0007444 3232 0-4 | 1:0013135 5701 0°5 1-0020375 8840 If the relation of P’ to P has been obtained by observation, and if the U proportionate to the difference of elevation be sought, this table shows that, according to Dalton’s views, it will be found somewhat greater than according to the older sup- position, and in a proportion given by the table, the numbers of which progress nearly as the square of the argument. If we are willing to be content with an approximation which scarcely dif- fers from the truth in all cases of probable occurrence, we may develope (14.) further. We have P U = log 5 + log; and if for log y we write the first member of its development, or : | i {o (l—d)*? +, (1 —d)"+ 2, (1— a,)° } = U?.0:003675, and for U, its expression, H'—H wa Biitg H'—H Fa+kT) FP TV +e The alteration which the adoption of Dalton’s view of the con- stitution of the atmosphere produces in the values of H! — H calculated on the older supposition, is therefore , (=H Ui+ kT) and if H' — H = n. 1000 toises n*.OF-391 = a Teer . ° . . . . . (153)a This difference is much too small for us to hope to obtain by barometrical measurements a decision for or against the reason- 2 0°003675. 0003675, MEASUREMENT OF HEIGHTS. 530 ing on which it is founded: it is far exceeded by the constantly existing disturbances of the equilibrium of the atmosphere, as well as by the uncertainty of the law (applied in the 2nd Sect.) of the variation of temperature between two heights at which the thermometer has been observed. Even the geometrical measurement of the difference of elevation could scarcely be made with sufficient certainty to determine a quantity so small as that upon which a decision between the two assumptions would de- end. 4 6. The necessary following out of Dalton’s supposition, in its relation to barometrical measurements of altitude, gives me an occasion of expressing my own view of this much-discussed sub- ject. The supposition rests principally on the comportment of aqueous vapour when mixed with air, and when by itself. A fluid brought into an empty space gives off vapour until the vapour has attained a density dependent on the temperature of the space. Dalton has determined this density, in the case of the vapour of water, for all degrees of temperature between freezing and boiling water; and has shown by indubitable ex- periments, that the vapour attains precisely the same density when the space is occupied by dry air, of any density whatso- ever, as when it is originally a vacuum. Every attempt to in- crease the density, when the temperature remains the same, fails. If the space, when filled with the densest vapour consistent with the temperature, be contracted in the ratio of 1 : 1 — », a part of the vapour, proportioned to the whole as 2: 1, is converted into fluid: precisely the same change takes place if a space filled with the densest vapour consistent with the temperature, and containing air of any density whatsoever, be contracted in the same proportion: in such case the air undergoes no correspond- ing change, but merely an increase of density in the ratio of 1—n:1. This is the pure result of Dalton’s experiments. They show a difference between vapour and air, assigning to yapour a maximum of density dependent on temperature, which does not exist in the case of air. They show further, that the forces at the surface of the fluid, which cause it to rise in va- pour in a vacuum, are not counteracted by the pressure of the air in contact with it. In respect to the latter point, I may re- mark that Poisson derived from phanomena of another class, i, é. the capillary, that the density at the surfaces of fluids is in- VOL. II. PART VIII. 2N 534 BESSEL ON BAROMETRICAL finitely small. Whether all gases have a maximum of density dependent on temperature (as is known to be the case with car- bonic acid gas), so that they only differ from vapours by the amount of the maximum (or specifically), cannot at present be decided, and is not here touched on. So long as vapours have a less density than the greatest they can attain in the respective temperatures, they are not physically different from gases; they follow Mariotte’s law; and Gay Lussac has shown that they possess the same expansibility by temperature which is common to all gases. So long, therefore, as they have not attained the maximum of their density, they comport themselves, whether alone or mixed with gases, pre- cisely as gases do. A pressure does not produce a change of state in them any more than in gases: that change first takes place, equally whether they are mixed or unmixed, whenever an attempt is made to cause their density to exceed its maximum. This can be done by lessening the space in which they are con- tained, in which case the gases, if present, remain unchanged in consequence of their unlimited compressibility. If, further, a space is filled with a gas which exerts a pressure p upon an unit of surface, the introduction of another gas, which if alone would exert the pressure p, on the same unit, produces no other phy- sical consequence than that this unit now sustains the pressure p+ p,; but it would sustain precisely the same pressure, if, in- stead of the second gas, a vapour were introduced exerting when alone the pressure p, Lastly, different kinds of gases mix with each other, as well as with vapour, in any arbitrary propor- tions. There is therefore throughout, no difference between the phy- sical comportment of a mixture of two gases, and of a gas and vapour; consequently the circumstances of the second mixture can teach us nothing which we might not learn from those of the first. The comportment of the mixture of air and aqueous va- pour, which Dalton’s experiments have fully manifested, is not, therefore, more instructive than that of any mixture of two gases; and a theory which could not be constructed upon the latter, cannot find support in the former. It could not, there- fore, have been deduced from the comportment of a mixture o vapour and air that the air does not press the vapour, unles for the presupposition that pressure changes vapour into fluid; for this presupposition, however, there is no justifying fact. MEASUREMENT OF HEIGHTS. 535 According to this view of our knowledge of vapour, no ground is afforded for the hypothesis that vapour is compressed only by vapour, and not by air; and we lose at the same time the analogy for the similar comportment of the mixture of different gases. Dalton has adduced, in further support of his supposi- tion, a circumstance which is independent of the experiments on aqueous vapour, viz. that a specifically heavier gas mixes with a lighter one, even though the latter should be placed uppermost. It is true that Dalton’s hypothesis explains this fact; but it cannot be maintained that the fact is inconceivable apart from the hypothesis. The ascent of fluids in tubes which are wetted by them might, for example, be explained by the assumption that gravity exerted its action but imperfectly within the tubes; but we know the true explanation is different. If I do not mistake, the small amount of the altera- tion which the constitution of the air undergoes in a space in which there are many persons, whose breathing must diminish the oxygen and increase the carbonic acid gas, has been ad- duced in support of Dalton’s views, as the oxygen must by pre- ference replace itself from the outward air, and the carbonic acid gas must pass to the same in preference, if the several con- stituents of the interior air are compressed only by those of the same nature without. The first experiments of the kind were made by Humboldt and Gay Lussae in one of the Parisian thea- _ tres*; and these gave a diminution of the oxygen of 0:007, with an imperfectly determined content of carbonic acid gas. Dalton _ subsequently repeated experiments of a similar kindy in spaces filled with numerous assemblages, and found the oxygen = 0°20325, whereas in free air he found it 0:2090; there was also more carbonic acid gas than in the free air, and one case, in which it was determined, the amount was 0:01. These experi- ments do therefore show actual alterations in the constitution of _ the air; and it only remains to examine whether they are Jess “thm the alterations to be expected according to the older views. e first-mentioned experiments do not appear to have been made for the purpose of testing these; and all are deficient in the exact data requisite for founding a calculation; i. e. the ubic contents of the room, the air of which was examined,— the number of persons, and of the lights, and the strength of * Gilbert’s Ann. der Physik, vol. xx. p. 88. + Phi. Trans. 1837, part LI. p. 363. 2N2 the latter,—the communications with the external air,—and the temperature at different heights. Nor is the case examined sufficiently simple to be a fit subject for strict calculation. But to obtain an approximate view, I have proceeded from the rule adopted in Prussian towns, which prescribes that in buildings, which are to contain assemblages of people, not less than 100 cubic feet shall be allowed for each person. I have further di- minished this space by one-third, and have taken Davy’s expe- riments*, which show that each person diminishes in one mi- nute the nitrogen by 4°9 cubic inches, and the oxygen by 19°5 cubic inches, and increases the carbonic acid gas by 15:4 cubic inches. If we assume that the diminution of 9 cubic inches is compensated by the necessary inpressing of the external air, on account of the continual augmentation of temperature which takes place, we find from these numbers that the proportion of the three gases of the atmosphere given in the first article, viz. 536 BESSEL ON BAROMETRICAL v = 0'78605 v, = 0°21325 - v, = 0:00070 will be altered in the course of an hour to v = 0°78719 v, = 0°20405 v, = 0°00875. If we deduct from the mixed air the carbonic acid gas, the proportion of the two other gases is at first as 0°7866 : 0°2134, and at the end of an hour as 0°7941 : 0°2059. The calculated result is not so dissimilar from the experiment as to afford a conclusion that the supposition on which the calculation is founded is incorrect. It would, indeed, seem as if the com- parison might rather be alleged against Dalton’s view than in favour of it. I believe that if we desire decisive experiments on this point, they would most easily be obtained by observing the ingress, from pressure, of atmospheric air into a closed space not air-tight, and filled either with one of the constituents of the air, or with both mixed in a different proportion from that in which they exist in the atmosphere. In order to sim- plify the experiments, and to obtain most conveniently the bases of their calculation, the space ought not to be the interior of a building, but that of a bell glass. * According to Gilbert’s calculation, Ann. der Physik, vol. xix. p. 312. MEASUREMENT OF HEIGHTS. 537 If no special hypothesis be made as to the molecular consti- tution of gases and vapours, it is plain that a particle of gas must press an adjoining particle of similar or dissimilar consti- tution with equal force (i. e. with the same force with which it endeavours to expand). Without a special hypothesis Dalton’s view contradicts the fundamental propositions of aerostatics. But such a view cannot be maintained unconditionally until proof is adduced that no supposition, such as is here referred to, is mathematically possible. On the other hand, the view which I have developed of the comportment of vapours, does not require to be justified by a special hypothesis. We may regard, as the immediate result of experiment, and as the distinguishing mark between vapours and gases, that the density of vapours cannot be increased beyond a certain degree dependent on temperature. But if we desire to enter likewise on the molecular constitution, it is easily conceivable that there may exist a distance between the ultimate particles of vapours, in which their attractive force is equal to the repulsive force arising from the temperature, so that every decrease of distance renders the attractive force pre- dominant, and consequently unites the particles. he If, notwithstanding what is here said, I have followed Dal- ton’s view in Sect. 5, in respect to the dry constituents of the air, I can the less omit to examine the deductions from it in re- gard to the aqueous vapour. This examination must also be pursued, if we desire to learn whether the observed distribution of the aqueous vapour in the atmosphere can be made to tell for or against Dalton’s hypothesis. I will, therefore, assume - with Dalton the aqueous vapour in the atmosphere to be pressed _ only by its own higher strata, or to form an atmosphere by the equilibrium of its own parts alone. The change of the pressure of the atmosphere of vapour, corresponding to the increase dx _ of the elevation z, is according to formula (4.), _ _ (g) 864.8 ( a )' . PPi 886905. \a-a) 07? | or, according to the notation subsequently introduced, | D) dp, — wiD aX. Its density @, until it reaches its maximum, follows Mariotte’s | | 538 BESSEL ON BAROMETRICAL law, or corresponds to equation (3.), which, for the present case, is §.E=p,Dd,; to which must still be added the condition requisite for equili- brium, that the @ resulting from this equation shall at no eleva- tion exceed the maximum of density corresponding to the tem- perature ; or, according to the notation in Sect. IV., that P, 2 >t. If we eliminate 8, we obtain ETS. P; TR we ee Le ee a similar differential has already been integrated in Sect. 2, assuming the variation of temperature between the two heights at which it was observed, to be that supposed by Laplace. With this assumption, it follows that dX 2k Ligeakd 040 Goa and thence the integral, reckoned from the elevation h, where r is the height of the thermometer, and P, the pressure of the aqueous vapour, is P,_ d, 2k le AT ph me te or aT Da oO, SB AD Geo et ee ee If we assume the pressure P, at the elevation h = «$7, where a cannot be greater than 1, the conditions to be fulfilled require that for each value of ¢, ee La ee agdr.1l0O =— ot; or d, 2k —d, 2k agv.10 “ #2 ot.10 4 7; and if for $7, and $¢, we substitute the expression (9.), d, 2k p d, 2k ced =7*) Tog ota 100 \Wisi —a) tee If we suppose ¢ to decrease without limit with increasing eleva- tion, it would attain a negative value, for which, even with the smallest value of «, the conditions would cease to be fulfilled ; MEASUREMENT OF HEIGHTS. 539 but we must not hence conclude Dalton’s assumption as not re- concilable, under all circumstances, with the existence in equi- librium of an atmosphere of aqueous vapour of which the den- sity is always a positive quantity. The decrease of ¢ does not go on indefinitely, but only as far as the value which it pos- sesses at the limit of the atmosphere; the formula (9.), which expresses the condition, is merely an interpolation formula, and has no justification beyond its more or less satisfactory accord- ance with Dalton’s experiments made between ¢ = 0, and ¢ = 10)°. If we take the logarithm of the two quantities, between which the conditions apply, it follows that 9 lo a =(4. ae ate(r +d) (7 — 2); and we also know that a 1, so that log « must not be positive. Hence it follows that the conditions may be fulfilled, or that the atmosphere of aqueous vapour is possible; also that the value of « (<1), which determines its density at the elevation h, remains arbitrary, if 2 = ota —e(r +i) eel ewieee ea which must be the case up to the limit of the atmosphere ; fur- ther, that in the opposite case, if even at the height h, a oes aH. MP Bhi (6) the existence of an atmosphere of aqueous vapour in equilibrium is possible ; but its density, at the elevation h, is limited by the condition that « must be less than d, 2k a aate(rt)) (7-9. 2. (21) _ for the value of ¢ at the limit of the atmosphere. In a parti- cular case of the decrease of temperature, the atmosphere of aqueous vapour may be at all elevations as dense as the tempe- rature permits; this case requires that dp, a dX dot. ie) hie. Ey or, under the supposition of the customary expression for ¢ ¢, that 540 BESSEL ON BAROMETRICAL au dX = (a—2ct) (1+ kt) dt. Hence follows by integration, k a(r —2) +(e) (2— £2) —sek (89 — 8) = & (XH); for which we may also write x — H = 54 (e-2¢ r) (1+ kr) (r—2) — (G-e+2ckr) (—#'— Lek al = (223) If we introduce into this equation the values of I, d, a, c, k already employed, we obtain the law of the decrease of tempera- ture, which, on Dalton’s supposition, is alone reconcilable with an atmosphere everywhere saturated with aqueous vapours. X —H = 424T0 (1—r .0:00447) (1 + 7.0°00375) (7 — 2) + 0™15 (1—7.0°0463) (r —¢)? — 07-003 (7 —Z)*. If, further, according to Sect. 2, we put 2k H'— 1 i t—et “14+kT’ and designate by (¢) the value of ¢ at the extreme limit of the atmosphere, the condition (19.) becomes’ H'-H oT! 1 4 eT) {a—c (r+ ()} r—T od, > {4247-0 —0T-95 (x + (t))}(1 + KT). The actual change of elevation, which produces a decrease of 1° in the height of the thermometer, is much less, or about = 85 toises; this is irreconcilable, under Dalton’s supposition, with the saturation of the atmosphere with aqueous vapour at the surface of the earth. But if the condition (20.) be fulfilled, or if H'—H —r! < (424T-0—1:9 r) (1+4T); then, according to formula (21.), after substituting in it the ex- Qk pression for —, mes or: tratetr+()) be-O), T_T — ————— MEASUREMENT OF HEIGHTS. 54] or u HH (4k) 7 log « < { —— — [4247-0 — 0795 (r + (2)] (14h 7) | Cray whence follows 1 — Fog a. (1+) t—(t)< ! ; " [424-0 —0°95 (x + (¢))] (1 + 4T)— = a ao If, then, we know both the last members of the denominator and a, we can compute by this formula a value of r—(¢), which, con- tinuing Dalton’s supposition, exceeds the difference of tempera~ ture between the elevation / and the limit of the atmosphere. fh If we take, for example, — = 85 toises and T = 0, and sup- posethe atmosphere at the surface of the earth to be half saturated with aqueous vapour, we obtain approximately 7 — (¢) < 135, which is scarcely equivalent to the usual decrease of temperature in 1200 toises, not to speak of the limit of the atmosphere ; if t—(t)= n .13%5, the extreme value of 2 = 47. Dalton’s sup- position is therefore only reconcilable with a very small quan- tity of aqueous vapour in the atmosphere, and not with that which really exists. If we could, therefore, regard as correct the pre-supposition of the equilibrium of the atmosphere on which we have proceeded, the presence of a considerable quan- tity of aqueous vapour in the atmosphere would be a conclusive argument against Dalton’s supposition. But this equilibrium never really exists, and I am indebted to Professor Neumann for the remark, that the density of aqueous vapour ascending from the surface of the earth must be increased by the resistance opposed to it by the air. 8. Icome now to the examination of the supposition, that the temperature between two elevations at which it has been ob- served varies according to the law which has been assumed by Laplace, Sect. 2. The equation between ¢ and X, which enounces this law, as deducible from Sect. 2, is oX—-H se s Prose (l+ko*=(1 + ke) oa + (1 + kr’) . (23.) _ But we have no reason to regard as unreal moderate deviations, 542 BESSEL ON BAROMETRICAL in the transition of the temperature from r to 7’, from the rule ( prescribed by the equation. It remains, for instance, quite — doubtful whether between the two elevations the true tempera- ’ ture may not differ from the value which would follow from the — rule by a quantity amounting to one-tenth part of r—7/. It is not superfluous, therefore, to investigate further the influence of such possible deviations. I will suppose that the true value of me Apaial Linger A ype a. a ey “ 1+kt where ¢ denotes the height of the thermometer at the elevation a, corresponding to equation (23.), and « is a constant coefficient, greater or less according to the amount of the deviation from the 4 : F we. 3 : law. This expression of fF 8 80 chosen, that it agrees with the — previous one at the two limits, and that the deviation of the temperature which it supposes attains its maximum =a (r — 1’) somewhere about ¢ =4(r +7') orv =i (h +h‘). We obtain thence fs ht dv _ — t) (¢-—7’) (, + =) reat 41- tT— Tv a= and the integral taken from / to fh’, en ~= ak(r—a)h HW —H 2 =the sete} , (408 It does not seem probable that in any case which is likely to occur the value of a would be i a within any very ; if it should reach either of narrow limits, as for instance + + 63 these limits, the consequent correction in the resulting difference of elevation, according to formula (23.), would be ad galrol alse = T4kT ° 4000° So, for example, for a difference of elevation of 1000 toises, for which + — 7’ is usually 12°, the correction would be about + 3 toises. We should be the less inclined to assume that @ must necessarily be very small, as it should not be overlooked ae MEASUREMENT OF HEIGHTS. 543 that the temperature of the air observed on a plain or on a height is always affected by the temperature of the surface of the earth. Hence we see, were it from this cause only, how little fitted ba- rometrical measurements of height are to determine questions, the answers to which depend on small differences between theory and experiment. Possibly observations made late at night might agree better together than those made in the day when the surface of the earth is heated by the sun. 5 It is known that Gay-Lussac found the value here denoted by & = 0°00375, by experiments agreeing almost perfectly with each other; and that Dalton found exactly the same result from his experiments. The object of both these great phy- sicists was to determine directly the increase which an unit of yolume of dry air undergoes, when, the pressure remaining equal, the temperature is increased from freezing to boiling water. The accordance, not only of the several experiments in each series, but also of the results of the two series, has caused the determi- nation of k=0°00375 to be generally regarded as one of the most certain that we possess: and there would be no reason for doubt respecting it at this period*, had it not been for recent experi- ments of Rudberg’s, distinguished by the great care with which they were conducted, particularly in drying the air employed, and which give a considerably smaller value for 4, i. e. 0°003648. Any later determination, contradicting an older one which has become in a degree classic by its intrinsic weight and by its general acceptance and use, ought to be accompanied by a strict examination of the older determination ; and it is only when such criticism shows grounds for distrusting the older, that the more recent should be deemed deserving of preference. Rudberg has not entered into such a criticism. As the dif- ference between the two values of k cannot be explained by the accidental errors of the experiments, as is shown by the * I have myself determined, from my own observation, the value to be em- ployed instead of k in computing astronomical refractions, and have found it 0°0036438 ; but this value must be different from that of k, and ought to be less, as shown in the 7th part of my observations, page xi. The research might have been spared had I possessed observations of the quantity of aqueous va- pour in the air at the time of each observed refraction. It remedied the diffi- ty as far as could be done in the absence of a knowledge of the actual acci- dental state of the atmosphere on each occasion. But it is to be regarded as a contribution to the knowledge of astronomical refraction, and not as a determi- nation of the value of k. 544 BESSEL ON BAROMETRICAL agreement of the partial results in the earlier as well as in the later series, it indicates a constant error, and there can be there- fore no propriety in taking the arithmetical mean of the two de- terminations. I see no other course at present than to employ both, and to await a future decision on the differences which may result therefrom. 10. Having gone through the different assumptions involved in formula (11.), I return to this formula, and will now show its application to barometrical measurements of height. The pressures P and P’ at the elevations 4 and A’, are de- ducible from the barometrical observations there made. If we denote one of the heights of the barometer by 4, the temperature of the mercury and of the scale by which the height is measured by ¢, and assume that the scale is of brass as is usual, we obtain the mass of mercury supported by each unit of surface 53242 +¢ 5550 * 53242 + (t) © 5550 + 2” where (é) signifies the normal temperature of the unit of measure of the barometer-scale, and where the unit of volume of mercury at the temperature of melting ice is taken as the unit of the mass. This mass presses in proportion to the force of gravity to which it is exposed; or with the force a 2 Olas. and the pressure which it exerts is the product of both divided by the Sele unit of intial (=336!:905). Thus we obtain 53242 + ¢ 5550, sarees ag a +h 53242 + () 5550 + 2’ and its Briggs’s logarithm in formula 11, with sufficient approxi- mation, 336°905 (53242 + (?) ) — log b— log Kisser node 1 1 Qe ae he ae ind sous | oe If we put for a the geometrical mean of the two semi-diame- ters of the earth (log 6°5140838, Ast. Nach., 333), and for (¢) the normal temperature of the French standard foot = 16°25, then MEASUREMENT OF HEIGHTS. 545 log P = log b — log 337-008 —¢ . 0:000070095 We obtain thus H ~ 3760707 H'’—H 3760707 where log J, and log 4/ are substituted for brevity for log b—t . 0:00007 and log J/—¢’.0°00007. Further, we obtain, without sensible error, P log po log 6, —log 6) + Vv (PP) = vee, If we introduce log 4, and log 4/ in (11.), and put 7 for I’, this equation becomes (y) (H! — ) as poz log b, — log 6) = i +k) ; 111+ kT) Lh 0°863 eae sf ~ (g).3760707 Vv (b'6/) ; If we change (gy) in the denominator of the second member into 1, which has no sensible influence, and if we take for « the half sum of its values at the two points of observation =4(«+ a’), then the quantity within brackets is 339°17 —kT Deez) 108 ™ LL RCo gee 400°17 V (6,6)) © 39917 —kT If we designate thenceforward 400°17 1. + kT) S557 — kT i 472:67.10°' —°? 39917 —kT uy WY the equation gives log 6, — log b/ Vv H —H =—2—__=++_- —_____ (9) ._ @t+2/)W Te Vv (,6/) 3 as : itt ah Te ah ney ti Rl? he a ~ ath ab Tat 4 Bil athe hn? =f? _ logb,—log bd} V 74 aes + a —_ (9) .y : _ @ +a) Ww . (25.) 546 BESSEL ON BAROMETRICAL This is the most convenient form of the equation (11.) for use. — It cannot be abbreviated further without giving up the power of | bringing into the calculation the quantity of aqueous vapour contained in the atmosphere, as shown by the psychrometer at | each station. The tables for facilitating the computation may also be so arranged, as to render quite inconsiderable the labour of taking the aqueous vapour properly into account. 11. I will now explain the auxiliary tables. They are construct- — ed logarithmically, like the small and very convenient tables of Gauss; but are rather more extensive, because they permit the result to be computed on either supposition of k = 0:00375 or = 0°003648, and also because they enable the influence of the aqueous vapour to be taken into account more completely than is done by the formula of Laplace. If we denote log {log 6, — log b/} by B, 1 pete ULES es eee S 1 Pa (aaaCeTTNT ahlinin on v (6, 6/) 1 ces NLVPA I) Oe ane nn, oe (9) A ‘ h!? he then is the logarithm of he a a = B+ logV + log V' + logG. Table I. contains the value of 9397°74 .400°17 (1 + kT) 399:17 —kT calculated in the first column for k = 0°00375, and in the last column for k = 0:003648. Its argument is2'TT=++7, The second column contains log V = log 172-67 10°" ° B99 17—k'T ” a single column is sufficient for log W, as the difference in the two values of k does not influence the last decimal. If we de- duct from the tabular value of log W the half sum of the loga- rithms of 4, and 6/, the remainder is the logarithm of (a + a’) W v (6, 6/) 7 log W = log MEASUREMENT OF HEIGHTS. 547 yee @ =a’ = +; if a different ult be supposed for « and a!, log (# + al) must be added. Hence is obtained the ar- ipsent of Table II., which contains log V’. Table III., with the argument ¢ = the latitude, contains 1 log G = log (—9-0026967 cos 2 e” which formula rests on the value of the increase of the length of the seconds pendulum from the Equator to the Poles, deduced by Mr. Baily from the combination of all the known pendulum observations. Trans. Ast. Soc. vol. vii. page 94. The sum of B and of the numbers taken from the three Tables, 2 is = log 4 —h—-——+ | ; to obtain from hence h! — h, 12 2 We must add = and subtract ~ which are both given by Table IV., which is to be entered with /! and with h. It may be convenient to recapitulate the notation and rules: b, U' are the heights of the barometer, read off on a scale divided into Parisian lines. t, v' are the heights of the centesimal thermometer attached to the barometer. t,t! are the heights of the centesimal thermometer in the free air. _@, a! are the degrees of saturation of the atmosphere with aque- ous vapour. The calculation of the difference of height of the points where these observations have been made, requires 1. log b, = log 6 — ¢ .0:00007; log b/ = log d' — #'..0:00007. 2. B= log {log , — log d/}. 3. log V and log W, which, with the argument r + r/, are to be taken out of Table I. 4. log V', which is given in Table II., with the argument (a+ a)W “WV (6, 8] 5. log G, which is given in Table III., with the argument of the latitude = ¢. 6. The log of the approximate difference of height = B + log V + log V' + log G. log 548 ON BAROMETRICAL MEASUREMENT OF HEIGHTS. 7. The true difference of height is the approximate, + the differ- ence of the two small corrections which are obtained from Table IV., with the arguments of the greater and — lesser height. I take, as an example, one of D’Aubuisson’s measurements of the height of Monte Gregorio, above a point at an elevation h = 128°3 toises (Traité de Géognosie, i. p. 481.). There be- ing no observation of the psychrometer, I take « = a! = 3. fe} fo) i= 329-013, fo LOIS Ds Fie L995 6! = 268°215, 2105, gi =. 9:9 log Gi Polls ft = 1395 log b, = 25 isse half log b! = 2°42848;. 7.#= 73°5; logd/ = oar geetinere log = 0°088075 B = 8-94485 Table B=8'94485 I. +7! = 29°85 (k=0:00375) logV =3-99782 logW=0-0397 Il. Arg. = 7°5679....... logVi= 161 *.. 2:4718 III. (Jee. ig ee ae logG = —2 7°5679 2°94496 Approximate height ....... 8797-54 IV. h'= 10078, h=1283 ..... 40°31 hi — h = 879785 . D’Aubuisson himself computes the height 87977; from Gauss’s tables we should have 879763. Ifk be taken = 0:003648, we have 17-26 less. If we take the air as dry, we obtain 3724 less; ; and if as saturated, 37-28 more. : BrEssEL. BESSEL ON BAROMETRICAL MEASUREMENT OF HEIGHTS. 549 TABLE I. Argument = r + 1’ (Centesimal scale.) 0-00375 log V. 395747 | 9°3501 | 3:95793 20° 3:99014 398971 3°95832 | 9°3646 | 3:95875 21 | 3:99093 | 9:9229 | 399048 3°95916 | 9°3792 | 3-95958 22 | 399171 | 9-9362 | 3-99124 3°96001 | 9:3937 | 3:96040 23 | 399249 | $:9495 | 3-99200 396085 | 9°4083 | 3:96122 24 | 399328 | 9:9628 | 3-99277 399353 3'99428 399406 3-99484 3°96203 3°96285 9:4227 9°4372 396169 3°96253 396337 | 9°4516 | 3-96366 27 | 399561 3°99504 396420 | 9:4660 | 3:96447 28 | 3:99639 399580 396504 | 9°4803 | 3°96529 29 | 399716 399655 3°96587 | 9 4946 | 3-96610 30 | 3:99794 399731 3°96670 | 9°5089 | 3-96690 31 | 3:99871 399806 3-96753 | 9°5232 | 3:96771 32 | 3:99948 399881 396836 | 9°5374 | 3-96851 33 | 4-00025 3°99956 396918 | 9°5516 | 5:96932 34 | 4:00102 400031 3°97012 4:00179 397001 | 9:5657 35 400106 3°970383 | 9-5799 | 3:97092 36 | 4-00255 400180 3°97165 | 9°5940 | 3:97172 37 | 4:00332 4:00255 400408 400484 4°00329 400403 3°97247 3°97329 96080 9:6221 3°97252 3°97332 397411 | 9-6361 | 397411 40 | 4:00560 400477 397493 | 9:6500 | 3:97490 4] | 4-00636 400551 3°97574 | 9-6640 | 397570 42 | 4:00712 400625 3:97655 | 9°6779 | 3:97649 43 | 4-00787 400699 397736 | 96918 | 397725 400863 400772 3°97817 | 9°7956 | 3:97806 45 | 4:00938 400846 3-97898 | 9°7194 | 397885 46 | 401013 400919 397979 | 9°7332 | 397963 47 | 401088 400992 398059 | 9°7470 | 3-98042 48 | 401163 401066 398140 | 9-7607 | 3:98120 49 | 4:01238 4:01139 3°98220 | 9°7744 | 3-98198 50 | 401313 4:01211 398300 | 9-7880 | 3:98276 51 | 401388 401284 3-98380 | 9°8017 | 398354 52 | 401462 401357 398460 | 9°8153 | 3-98431 53 | 4°01536 401429 3:98539 | 98288 | 3:98509 54 | 401611 401502 Sa 401685 3°98619 9°8424 | 398586 55 401574 398698 | 98559 | 3:98663 56 | 401759 401646 3'98777 | 98693 | 398741 57 | 401832 401718 398856 | 9°8828 | 398818 58 | 4-01906 401790 398935 | 9:8962 | 3-98894 59 | 4-01980 | 0-4068 | 4-01862 _———_| 399014 | 9:9096 3:98971 401933 402053 | 0-4189 : VOL, Il. PART VIII. 20 550 BESSEL ON BAROMETRICAL MEASUREMENT OF HEIGHTS: TABLE Il. TABLE III. TABLE IV. Ul W / Argument = Sire Argument = Latitude. eae ais ( Mise ¢ Height | — Arg. log V’.| Arg. log V’.) Arg. |log V’.|| 4. |log G7] @. |log G’. ||h’and h. 186 | 8:03 | 468 110 | 48 |}— 12} 900 | 0:25 764 | 190 | 8-04 | 479 109 | 49 |— 16|) 1000 |031 7°65 | 194 |8-05 | 490 || 10 | 107 | 50 | — 20]] 1100 | 0:37 7°66 | 199 |8:06 | 502 || 11 | 106 | 51 |— 24]] 1200 | 0:44 7°67 | 204 |8-07 | 513 || 12 | 104 | 52 |— 28] 1300 | 0:52 7°68 | 208 |8°08 | 525 || 13 | 103 | 53 |— 311] 1400 | 0-60 11 |7°69 | 213 |8-09 | 538 || 14 | 101 | 54 |— 35)) 1500 | 0-69 14 |770 | 218 |}810 | 550 || 15 | 99 | 55 |— 39)| 1600 |0-78 17 |771 | 223 |}811 | 563 | 16 | 97 | 56 |— 43} 1700 | 0-88 22 17°72 | 229 |\812 | 576 || 17 | 95 | 57 |— 46} 1800 | 0-99 297 |7'°73 | 234 | 813 | 590 || 18 | 92 | 58 | — 50} 1900 | 1-11 34 |7°74 | 239 |8:14 | 604 || 19 | 90] 59 |— 54} 2000 | 1-22 43 |7795 | 245 |8:15 | 618 || 20 | 87 | 60 |— 57] 2100 | 1-35 T 7-55 | 154 17-95 | 389 || 0| 114] 40 20|| 100 | 0-00 7°56 | 158 |7-96 | 398 || 1 | 114] 41 16|| 200 | 0-01 7-57 | 162 |\7-97 | 407 || 2| 114 | 42 12|| 300 | 0-03 7-58 | 165 17-98 | 417 | 3| 114) 43 8) 400 | 0-05 759 | 169 |7:99 | 427 || 4 | 113 | 44 4| 500 | 0-08 7°60 | 173.|8:00 | 437 || 5 | 112 | 45 0|| 600 | 0-11 7.61 | 177 |8-01 | 447 || 6| 112146/— 41 700 |0-15 7.62 | 181 |802 | 457 | 71 111147/— 8] 800 |0-20 6: 8 9 on An WMH SODUIAATRWWH OS Oso PO OSD HH eH eRe © ~~ > cs 55 251 |8:16 | 632 || 21 | 85 | 61 |— 60}) 2200 | 1-48 69 256 |8:17 | 647 || 22 | 82] 62 |— 64] 2300 | 1-62 87 262 |8:18 | 662 || 23 | 79 | 63 |— 67} 2400 | 176 109 269 |8:19 | 678 || 24} 76 | 64 |— 70}) 2500 | 1-91 275 |\8:20 | 694 || 25 | 73 | 65 |— 73)| 2600 | 2:07 28] |8:21 | 710 || 26 | 70} 66 |— 76) 2700 | 2-23 288 |8°22 | 72 2 67 | 67 |— 79|| 2800 |2:40 295 |823 | 744 || 28 | 64 | 68 |— 82] 2900 | 2:58 302 |8°24 | 761 || 29 | 60 7} 69 |— 85]| 3000 | 2-76 309 |8°25 | 779 || 30 | 57 1 70 |— 87|| 3100 | 2-94 316 {8-26 | 798 || 31 | 54] 71 |— 90]! 3200 |3-13 — 323 |8:27 | 816 || 32 | 504 72 |— 92|| 3300 | 3:33) 331 |8:28 | 835 || 33 | 46 | 73 |— 94|| 3400 | 3°54 338 |8:29 | 855 || 34 | 43 1 74 |— 97)| 3500 | 3°75 346 |8°30 | 875 || 35 | 39) 75 |— 99 354 |8°31 | 896 || 36) 35 | 76 |—101 363 |8:32 | 917 || 37 | 31 | 77 | —102 371 | 8:33 | 939 || 38 | 28 | 78 |—104 380 |8:34 | 961 || 39 | 24) 79 |—106 389 |835 | 983 || 40 | 20] 80 | —107 BI III III INI DAS RAPSPASSDSAAMAAAAMA I HOAAHKAnk Re RRR ARR ROW COON OP WDWHSCOBNIAATAWWH — wo — SEHSSECERHDKEKHDMWODHWOOHVIVIN ArPWWOEMSCHODNAUBRWHHSOMND pho) Or AI TS IN TS ARTICLE XVII. On the Anhydrous Sulphate of Ammonia. By Heineicu Rose. [From Poggendorft’s Annalen, vol. xlix. p. 183.] In attempting to precipitate the excess of sulphuric acid from a solution of anhydrous sulphate of ammonia by means of car- bonate of barytes, I succeeded in obtaining crystals of consider- able size from the fluid separated from the sulphate and excess of carbonate of barytes ; these crystals I took for anhydrous sul- phate of ammonia ; having obtained only a small quantity I did not subject them to analysis, but employed for this purpose the indistinctly crystallized mass, which remained with these crystals after evaporation over sulphuric acid. I found in them only 67°47 per cent. of sulphuric acid instead of 70°03, which, ac- cording to theory, the anhydrous sulphate of ammonia should contain*. I have since separated, in the above-described manner, the excess of acid from larger portions of the anhydrous sulphate of ammonia, and have obtained a greater quantity of these crystals. At the same time I investigated more accurately the action of water on this salt, which had been carefully prepared, and was perfectly neutral. After precipitating the excess of acid by car- bonate of barytes, I satisfied myself that the solution had pre- cisely the same properties as the salt obtained by treating an- hydrous ammonia with anhydrous sulphuric acid. I found also, what I had not been before able to decide with certainty, on account of the small quantity of the salt employed, that the solutions do not contain one salt, but two different salts, pos- sessing very singular properties, and remarkable as to their com- position+. I have also submitted the properties and composition of neu- tral anhydrous sulphate of ammonia to a fresh examination, and have ascertained some facts which will complete my former in- vestigations. 1 have called this salt sulphat-ammonf, for reasons formerly explained; the two salts obtained from its aqueous * Poggendorff’s dnnalen, Bd. 47, 8. 474. + Poggendorff’s Annalen, Bd. 32, S. 81. t Ebendaselbst, Bd. 37, S. 475. 202 552 ROSE ON THE ANHYDROUS SULPHATE OF AMMONIA. 7 a solution, I will, at present, denominate parasulphat-ammon, and — the deliquescent salt; the names of sulphat and parasulphat- ammon are, however, to be considered as merely provisional; I shall very readily withdraw them if the ingenious views of Mr. R. Kane*, which regard ammonia as an amide of hydrogen, — should be more generally adopted. It is indeed true, that, by this hypothesis, the phenomena which the compounds of anhy- drous sulphuric acid with ammonia exhibit with reagents, are capable of more satisfactory explanation than by other theories ; but as to the numerous compounds of ammonia with oxyacids and with water, the opinion of Berzelius, that these combina- tions contain the oxide of ammonium, is more simple and pro- bable, because these salts, considered in this light, are analogous in composition to the salts of other bases. I, Neutral Anhydrous Sulphate of Ammonia—Sulphat-ammon. The principal properties of this compound I have described in a former paper, in which I especially mentioned its action on the solutions of barytic salts, oxide of lead, strontia and lime, and chloride of platina. Other reagents, which instantly in- dicate the presence of ammonia in a solution of sulphate of ammonia, do it imperfectly in a solution of sulphat-ammon. In order to determine this point, equal parts of sulphat-ammon and of sulphate of oxide of ammonium were dissolved each in nine times its weight of water, and both solutions were tested with the same reagents; sulphat-ammon is not perfectly soluble in less water than employed in this experiment. A solution of sulphate of alumina soon produced crystals of alum in the solution of the sulphate of oxide of ammonium, but none were immediately produced in the solution of sulphat-am- mon ; after some time a small quantity was formed, but much less than in the sulphate of oxide of ammonium. A concentrated solution of tartaric acid soon produced a crystalline precipitate with the sulphate of oxide of ammonium, and also after a longer time in the sulphat-ammon. A concen- trated solution of racemic acid, which is a much more sensible test of ammonia than tartaric acid, produced similar effects ; but the quantity of precipitate was much greater in the sulphate of oxide of ammonium. * Researches on the Nature and Composition of the Compounds of Am- monia. ‘l'ransactions of the Royal Irish Academy, vol. xix. p. 1. ROSE ON THE ANHYDROUS SULPHATE OF AMMONIA. 553 A solution of carbazotic acid acted in the same way ; it im- mediately produced a considerable precipitate with the sulphate of oxide of ammonium, and to a less extent, and after a longer time with the sulphat-ammon. The sulphat-ammon is a homogeneous powder; when exa- mined by the microscope it does not exhibit any appearance of crystallization ; like other powders it attracts moisture from the air, but this is got rid of without any change of properties, by drying in a water-bath, and by fresh exposure it gains as much water as before. Although I have already stated an analysis of sulphat-ammon, yet having, by a method which I shall hereafter describe, ob- tained it in larger quantity and of great purity, I have considered it necessary to repeat the examination. The proofs of the purity of this salt are not only that it scarcely reddens litmus paper, but on the contrary renders it blue (after it has been reddened), but only to a slight degree, and this effect it continues to pro- duce only when kept in a bottle containing ammoniacal gas. When litmus paper, which has been dipped in a solution of sulphat-ammon, sulphate of oxide of ammonium, or most other soluble ammoniacal salts, is dried in the air, it is red- dened. One hundred parts of sulphat-ammon were treated with a so- Jution of chloride of barium ; the whole was evaporated to dry- ness, heated to low redness, treated with hydrochloric acid and water, and there were obtained 203°79 parts of sulphate of barytes. This is the only method by which the whole of the sulphuric acid can be eonverted into sulphate of barytes; but this substance, when so procured, passes through filters, and requires frequent filtration. The sulphate of barytes obtained indicates 70-04 of sulphuric acid, which agrees as nearly as pos- sible with the amount of this acid calculated from the formula S + N H®, or 70°03 per cent. The results of several analyses, confirming this composition, will be subsequently stated. Il. Parasulphat-ammon. I have thus denominated a remarkable salt, which crystallizes in large well-formed crystals from the concentrated aqueous Solution of sulphat-ammon; they may likewise be obtained by combining sulphat-ammon with anhydrous sulphuric acid by a method already mentioned. These are the crystals which 554 ROSE ON THE ANHYDROUS SULPHATE OF AMMONIA. have been described by my brother in Poggendorff’s Annalen, B. XLVII. 476. These crystals are obtained by evaporating the solution ; but, like that of sulphate of oxide of ammonium, it is apt to become acid during the operation, and to have its pro- perties thereby difficultly recognized ; it is better to evaporate over sulphuric acid in vacuo. On further evaporating the mother-water another salt is formed, which differs essentially in its properties from the larger crystals ; but it is difficult to se- parate it from them, especially when considerable quantities of the sulphat-ammon have not been operated upon. This salt attracts moisture from the atmosphere, which is not the case with the crystals of the parasulphat-ammon, this when quite dry suffering no alteration by exposure to the air. Of this salt I shall treat in the following section. The parasulphat-ammon is rather more soluble than the sulphat-ammon ; its solution is neutral to litmus paper. When also preserved for a long time, so that nothing can evaporate and crystallize, it remains neutral. When, however, the salt is moistened with water, it acquires in a short time the pro- perty of reddening litmus paper, and the solution possesses qualities and acts differently with reagents from that of the salt not previously moistened. The acid reaction, which the salt acquires by moistening, probably arises from the expulsion of some ammonia by the water; the carbonic acid of the atmosphere appears also to exert some action; for if a solution of parasulphat-ammon is slowly evaporated, cold, over sulphuric acid, in contact with the air, it often acquires an acid reaction, which is not the case i the evaporation be performed in vacuo; when the crystals of this salt are obtained, no attempt must be made to free them from the solutions by washing with water; they must be dried only by blotting-paper. What particularly characterizes the parasulphat-ammon, and distinguishes it from the sulphat-ammon is, that the solution of the dry salt is not rendered turbid by the salts of barytes or of lead, even when they remain long mixed. This property, it is, however, sometimes difficult to observe, partly because the crystals may contain a portion of the solution from which they have separated, and therefore contain the deliquescent salt; and partly from having been exposed to the atmosphere after moistening, and then yielding a solution which reddens litmus ROSE ON THE ANHYDROUS SULPHATE OF AMMONIA. 555 paper; in both these cases the solutions instantly precipitate the salts of barytes and lead. If hydrochloric acid and a solution of chloride of barium be added to one of parasulphat-ammon, it remains also for some time perfectly clear ; in about twelve hours, however, a precipi- tate of sulphate of barytes is formed; but it does not occur without the hydrochloric acid be present. In the property of not precipitating the solutions of barytic salts in the cold, the parasulphat-ammon very much resembles the compound obtained by M. Regnault, by saturating sulphate of chloride of sulphur S Cl? + 2 s (S Cl) with anhydrous am- monia*, and which he considered as a mixture of sal-ammoniac and a sulfamide (S N H’). The solution of this compound oc- casions no precipitation with the salts of barytes, even when they have been long in contact. M. Regnault did not succeed in separating this peteide from sal-ammoniac by crystalliza- tion ; and he adds, moreover, that the compound which he ob- tained very soon attracts moisture from the air, which, as already mentioned, is not the case with the crystals of parasulphat-am- mon or sulphat-ammon. The results of analyses prove, likewise, that the crystals can- not be regarded as an anhydrous sulfamide; 100 parts dissolved in water, were mixed with a solution of the chloride of barium and boiled. After some time a precipitate of sulphate of ba- rytes appeared, but less in quantity and much more slowly than would have occurred, under similar circumstances, with a solu- tion of sulphat-ammon. The whole was evaporated to dry- ness; the residue heated to incipient redness, left 203°64 parts of sulphate of barytes after treatment with hydrochloric acid and water; this is equivalent to 70 of sulphuric acid. The result of this analysis proves that these crystals possess as exactly as possible the same composition as the anhydrous sulphate of ammonia or sulphat-ammon, If the sulphur in an anhydrous sulfamide S N H? was entirely converted into sul- phuric acid, there would be obtained 80-03 per cent. of sul- phuric acid from the sulfamide employed. One hundred parts of crystals of parasulphat-ammon, which had been formerly prepared, gave, when treated in the same * Ann, de Chim, et de Phys. \xix., 170.: 556 ROSE ON THE ANHYDROUS SULPHATE OF AMMONIA. manner, 204°49 parts of sulphate of barytes, equivalent to 70°29 of sulphuric acid. If we were to regard the crystals prepared — by me, on account of their similarity to the combinations formed by M. Regnault, as a sulfamide, it must be considered as hy- drated, S N H? + H. Since, however, the existence of hydrous amides is not suffi- ciently proved, and even appears in some respects to be impro- bable, I have denominated these crystals parasulphat-ammon, or parasulphammon, on account of their similar per centage composition with sulphat-ammon. In the solution of the parasulphat-ammon the ammonia is still more imperfectly separated by reagents than in a solution of the sulphat-ammon. In solutions of equal strength, one part of each salt to nine parts of water, a concentrated solution of tartaric acid does not effect the formation of supertartrate of am- monia, even after several days in the parasulphat-ammon, while a precipitate, though not an abundant one, is produced in the sulphat-ammon. A concentrated solution of racemic acid occa- sions, after some time, a very small quantity of crystalline pre- cipitate in the solution of parasulphat-ammon, and much smaller than in the solution of sulphat-ammon ; solutions of chloride of platina, carbazotic acid and sulphate of alumina, react in the same manner with the solution of sulphat-ammon. As the presence of sulphuric acid is not indicated in the solu- tion of parasulphat-ammon by the salts of barytes and lead, this is also the case, as might be anticipated, with the salts of strontia and of lime. I have long hesitated whether the crystals of parasulphat-_ ammon should be regarded as distinct from the sulphat-ammon, merely on account of their different crystalline forms. It is well known how difficult it is to obtain perfectly anhydrous sul- phuric acid; and, if it contain only a trace of water, a corre- sponding quantity of sulphate of ammonia is formed on satura- tion with dry ammoniacal gas; and the solution of barytes, being an extremely sensible reagent for sulphuric acid, it might easily happen that the solution of sulphat would be slightly pre- cipitated even in the cold by barytes, owing to its being impure, on account of the presence of sulphate of ammonia. It is, indeed, true, that the solution of parasulphat-ammon acts some- wnat differently from that of sulphat-ammon, with solutions of ROSE ON THE ANHYDROUS SULPHATE OF AMMONIA. 557 barytes and lead and other reagents, and more particularly with tartaric and racemic acids; the sulphat-ammon is more spa- ringly soluble than the parasulphat, and does not so readily become acid when moistened ; these, however, are circumstances of too little importance to allow of our regarding with certainty the parasulphat as a distinct substance from sulphat-ammon, and isomerical with it. The following facts led me, however, to adopt this opinion: when a neutral solution of chloride of barium is added to a cold solution of pure sulphat-ammon, and the sulphate of barytes is allowed to precipitate for an hour, the filtered solution, without being heated again, deposits sulphate of barytes, and this occurs again after repeating the filtration; this is not the case with the parasulphat-ammon ; its solution, after the addition of chloride of barium, remains for months perfectly clear in the cold, when no acid has been added; in performing these experiments equal portions of the isomerical salts were dissolved in similar quan- tities of water. I consider these different actions as an essential difference be- tween these substances ; and the following series of experiments is also decidedly in favour of this difference: 100 parts of sul- phat-ammon weighed 91°42 after drying in a water-bath; it was dissolved in cold water, without any acid, and mixed in the cold with a solution of chloride of barium; in an hour after mixing, the sulphate of barytes was separated by the filter and washed, towards the end of the operation, with warm-water ; it weighed 51°71 parts, equivalent to 18°16 of sulphuric acid: Hydrochloric acid was added to the filtered solution, and it was evaporated to dryness; the residue, moderately heated, treated with water and a little hydrochloric acid, gave 145-7 of sulphate of barytes, equivalent to 51°16 of sulphuric acid; the whole quantity of sulphuric acid, therefore, in 100 parts, eli chata to 69°32 parts, approximating very closely to the quantity con- tained in the sulphat-ammon by calculation. In supposing that the 18°16 of sulphuric acid precipitated in the cold, might be derived from an admixture of sulphate of am- monia with the sulphat-ammon, they would be equivalent to 30°01 of the former salts, and the 51°16 of sulphuric acid ob- tained by evaporation indicate 73-08 of sulphat-ammon, giving an excess of 3°06, which the analyses will not admit of. 558 ROSE ON THE ANHYDROUS SULPHATE OF AMMONIA. The results of two additional experiments are still more deci- sive of the difference between sulphat- and parasulphat-ammon ; 100 parts of the same sulphat-ammon, as already employed, corresponding to 91°42 when dried in the water-bath, dissolved, cold and mixed with a solution of chloride of barium, gave 63°84 of sulphate of barytes, which was separated by the filter half an hour after precipitation, and are equivalent to 22°41 of sulphuric acid; in another experiment the sulphate of barytes separated an hour after precipitation, the sulphate obtained in- dicated 23°49 of sulphuric acid; the quantity of sulphate of barytes, obtainable by evaporation, was not determined in either experiment. It is evident, from these experiments, that the quantities of sulphuric acid, precipitated in the cold by chloride of barium, may differ greatly; the three portions employed were weighed at the same time from the same quantity of the preparation ; the greater or less quantity of the sulphate of barytes obtained in the cold, by a solution of the chloride of barium, undoubtedly depends not only upon how soon it is filtered, but upon the quantity of water in which the sulphat-ammon is dissolved, and the concentration of the solution of chloride of barium. Were we to suppose, that in the last-described experiments, the sulphuric acid precipitated in the cold is derived from the sulphate of oxide of ammonium, there would arise greater con- tradictions than would attend the results of the first-mentioned analysis ; for 22°41 parts of sulphuric acid would correspond to 37°03 parts of sulphate of oxide of ammonium. The different analyses of the sulphat-ammon having constantly given 70°03 per cent., or very nearly, of sulphuric acid, there would be ob- tained by further treatment 47°62 per cent. of the same acid, which corresponds to 68 parts of the sulphat-ammon. But in this case the quantities of sulphate of oxide of ammonium and the sulphat-ammon would amount to 105°03 per cent., and there- fore the analyses would indicate an excess of 5:03 per cent. In the last-mentioned examination of the sulphat-ammon 23°49 per cent. of sulphuric acid were obtained in the cold; if these indicated 38°81 parts of sulphate of oxide of ammonium, and if 46°54 parts of sulphuric acid, obtained by evaporating, correspond to 66°46 parts of the sulphat-ammon, the analyses would have given an excess of 5:27 per cent. ROSE ON THE ANHYDROUS SULPHATE OF AMMONIA. 559 Ill. The Deliquescent Salt. This salt, as already mentioned, is contained in the solution from which the parasulphat-ammon has crystallized ; if this be evaporated to dryness over sulphuric acid in vacuo, imperfect crystals, or crystalline crusts only are obtained, which attract moisture from the air, and eventually deliquesce ; it is very diffi- _ cult to obtain this salt perfectly free from parasulphat-ammon ; it is indeed more soluble, but the parasulphat is not very diffi- cultly so, which renders it impossible to separate them when operating on small quantities ; but in larger quantity I effected their separation in the following manner: I allowed the solu- tion, which had been evaporated to dryness in vacuo over sul- phuriec acid partially to deliquesce by exposure to the air; or added a few drops of water to it, left them for some time in contact, and then evaporated the small portion of the salt [dis- solved], again to dryness, as before, and employed it for analysis. If the solution of the salt contains parasulphat-ammon, and if it has been evaporated very slowly over sulphuric acid, but not im vacuo, the crystals obtained from it become, in a moist state, very readily acid; the crystals of the parasulphat-ammon must therefore be picked out as much as possible from the mass evaporated to dryness, then the deliquescent salt must be dissolved in water, and carbonate of barytes added to the solu- tion to saturate the free acid, and lastly, the solution must be again evaporated in vacuo. The crystals of the salt are too indistinct to admit of their form being determined, and they are usually mere crystalline crusts, and any crystals which may be observed with bright faces are parasulphat-ammon. The solution of this salt instantly precipitates solutions of barytes; but, as happens with the solution of sulphat-ammon, not nearly the whole of the sulphuric acid is thrown down in the state of sulphate of barytes. When hydrochloric acid is added to the solution, more sulphate of barytes is precipitated in the ‘cold, than without such addition; a solution of chloride of strontium produces immediate precipitation in the solution of this salt only when very much concentrated ; this distinguishes the solution from that of sulphat-ammon. If equal quantities of both salts are dissolved in similar quantities of water, both the solutions are not precipitated by a dilute solution of a salt 560 ROSE ON THE ANHYDROUS SULPHATE GF AMMONIA. . of strontium ; after some time, however, if the solutions are not too dilute, precipitation begins in that of the deliquescent salt, while that of the sulphat-ammon remains clear. A solution of the acetate of peroxide [protoxide ?] of lead precipitates the solu- tion of the deliquescent salt in the same way that it does the — sulphat-ammon; a solution of chloride of calcium does not render either solution turbid; both solutions are similarly — affected by chloride of platina, sulphate of alumina, tartaric acid, racemic acid, and carbazotic acid. It is difficult to prevent the solution of the salt from reacting as an acid upon litmus paper, but it is inconsiderable if the salt has been carefully prepared. The salt obtained by evaporating the solution in vacuo was dried at 212° until it ceased to lose weight ; 100 parts of the dried salt dissolved in water, mixed with a solution of chloride of barium, and left in the cold for twenty-four hours, gave 20°42 of sulphate of barytes. Hydrochloric acid being added to the filtered solution, it was evaporated to dryness, and the residue was heated nearly to redness, and treated with hydrochloric acid. The quantity of the sulphate of barytes precipitated was 166°18 parts; the quantity of sulphate of barytes, precipitated in the cold, therefore, amounts to scarcely one-eighth of the whole ; both quantities together gave 64°14 per cent. of sulphuric acid in the salt; this corresponds to a compound of anhydrous sulphate of oxide of ammonium, with half an atom of water, which, calculated according to the formula SN He +3 H gave in 100 parts i Sulphuric, acid seis) a! \cpye. tei A ADAMIONIE,. Jy./\s<>..:8%) (bie, sennetl cee ee WBbeB icc hpnorcy soils see Seats Malan) ieee ree 100° On repeating this experiment with a portion of the salt pre- pared on another occasion by dissolving pure sulphat-ammon, I obtained, by exposure to cold from 100 parts, after adding hy- drochloric acid and chloride of barium to the solution, 106°06 parts of sulphate of barytes, and from the residue obtained by evaporation to dryness, and treating it with hydrochloric acid, . 84:62 parts more of sulphate of barytes were obtained. It will be seen from these experiments that much more sulphuric acid ROSE ON THE ANHYDROUS SULPHATE OF AMMONIA. 561 is precipitated from the salt when cold, if mixed with hydro- chloric acid, than when this is not the case. The quantities of sulphate of barytes, added together, indicate 65°54 of sul- phuric acid in the 100 of salt; the slight excess is unquestion- ably derived from the parasulphat-ammon which the salt con- tained, because it had been prepared from but a small quantity of the sulphat-ammon. When I first prepared the crystals of the parasulphat-ammon, having obtained but a small portion of it, I resolved not to employ them for analysis, but to examine the irregularly crystalline masses obtained by evaporation to dryness, which must consist of a mixture of the deliquescent salt, and the parasulphat- ammon*; analyses confirmed this by finding only 67°47 per cent of sulphuric acid in this mixed substance. The hydrous sulphat-ammon is perfectly analogous to a salt which I ob- tained during my investigation of the compounds of carbonic acid and ammonia‘, and which consists of carbonate of ammonia and half an atom of water, requisite to convert the ammonia [am- monium ?] into the oxide of ammonium. The same is also the case with the hydrous sulphat-ammon. With respect to the car- bonic salt, I have advanced the opinion that it might be regarded as carbonat-ammon with the carbonate of oxide of ammonium. The same view may also be adopted with respect to the hy- drous sulphat-ammon, by regarding it as a compound of sul- phat-ammon with the sulphate of oxide of ammonium S AH? +S N Hi}; the salt may perhaps also be formed by saturating the first hydrate of sulphuric acid 2 S + H, contained in Nord- hausen sulphuric acid, with dry ammoniacal gas. The deliquescent salt unquestionably originates from the parasulphat-ammonia when dissolved in water, and remaining for some time in contact with it. Very pure crystals of the parasulphat-ammon, quite free from the deliquescent salt, when dissolved in water, and evaporated over sulphurio acid in vacuo, always yield a considerable quantity of the deliquescent salt, along with the crystals of parasulphat-ammon. As crystals of the parasulphat-ammon become acid, when exposed to moist air for some time, it seemed to me interesting to inquire into the nature of the alteration which they undergo. Some ex- * Poggendorff’s Annalen, Bd. xlvii. S. 474. + Poggendorfl’s Annalen, Ba, xlvi. S. 3738. 562 ROSE ON THE ANHYDROUS SULPHATE OF AMMONIA. ceedingly pure crystals of the salt were reduced to powder and moistened during several hours, by which the salt acquired an acid reaction, and it was then perfectly dried in a water- bath; 100 parts of the dry residue were dissolved in cold water; the solution reddened litmus paper, but not strongly, and it precipitated solution of chloride of barium. By the method frequently mentioned, I obtained 198°19 parts of sul- phate of barytes equivalent to 68°13 of sulphuric acid ; it fol- lows from this result that the parasulphat-ammon, by moisten- ing with water, is partially converted into the deliquescent salt. The acid reaction arises from the presence of free hydrate of sulphuric acid. It results from these investigations, that, although the sulphat- ammon seems to dissolve in water without decomposition ; yet, when the solution crystallizes, the crystals obtained, notwith- standing they are similar in composition to the sulphat-ammon, possess many properties which differ from it. In the solution of the sulphat-ammon the constituents of water are more readily combined with it by the action of certain reagents, and the compound therefore changes more readily. This is the case with the crystallized sulphat-ammon, or the parasulphat-am- mon, which resists more powerfully the action of such reagents. The conditions of the sulphat-ammonand parasulphat-ammon, may be compared with the vitreous and crystalline state of certain bodies, in which they exhibit different properties. The combinations of anhydrous sulphuric acid with am- monia may be regarded, according to Dr. Kane, as perfectly analogous to the hydrate of sulphuric acid. By supposing that ammonia is an amide of hydrogen, and that the amide combines in a similar manner with other bodies, as oxygen and chlorine, the amide of hydrogen becomes a body analogous to the oxide and chloride of hydrogen. But when sulphuric acid is combined with water or other oxibases, it may possess pro- perties very different from those which belong to it when com- bined with the amide of hydrogen. We have, in fact, of late, become acquainted with a great number of cases, in which the sulphuric acid, when combined with certain substances, as for example, with the oxide of ethule, and other bodies of organic origin, loses some of the peculiarities by which we were pre- viously accustomed to characterize it, especially that of giving an insoluble precipitate with barytic salts. But [hypothetical] ROSE ON THE ANHYDROUS SULPHATE OF AMMONIA. 563. as this opinion may be, the explanation which Dr. Kane gives of the compounds of ammonia with hydrous oxacids is equally so; these are regarded by Berzelius as salts of the oxide of ammonium, and on this view, the analogy of these salts, with those formed with other oxibases, is maintained ; as is also the isomorphism of some salts of potash and the oxide of ammo- nium; and these opinions were rapidly and almost universally adopted. But according to Dr. Kane, this numerous class of ammoniacal salts consists of combinations of acids with two bases, the oxide and amide of hydrogen; and the sulphate of oxide of ammonium becomes on this view analogous to several sulphates, which, at a higher temperature, retain one atom of water. But the perfect analogy and isomorphism of these am- moniacal salts with the salts of potash, are thrown into the back ground by Dr. Kane’s theory, instead of being advanced. Prof. Graham*, for similar reasons, adopts the opinions of Berzelius justly, as he acknowledges the importance of the theory of Dr. Kane. I will direct the attention of the reader to an analogy existing between the compounds of sulphuric acid with ammonia, and of the same acid with bicarburetted hydrogen (the elayl or ztherol of Berzelius) which was long since pointed out by Dumasf. The elayl and the ammonium produce, when combined with hy- drogen, one the hypothetical radicle zthyle, the other the no less - hypothetical radicleammonium ; both radicles may be combined with sulphur, chlorine, bromine and iodine: combined with the elements of water, one yields the base, oxide of zthyle, the other the base oxide of ammonium. Both bases may be com- bined with anhydrous oxyacids; both the bicarburetted hy- drogen, as well as the ammonia, may be united by direct com- bination with anhydrous sulphuric acid; this acid may likewise be combined with oxide of ethyle, a compound contained in the sulpho-tartaric acid and in its salts, and also with the oxide of ammonium. The sulphuric acid forms compounds also with elayl (or rather with etherol), as well as with ammonia, which contain so much water, or its elements, that only half the quan- tity of the bicarburetted hydrogen or the ammonia can be con- verted by it into the oxide of zthyle, or the oxide of ammonium ; the former compound is the oil of wine (sulphate of the oxide of * Elements of Chemistry. By T. Graham, p. 117. + Poggendorff’s Jnnalen, Bd. xii. S. 452. 564 ROSE ON THE ANHYDROUS SULPHATE OF AMMONIA. ethyle—etherol), the latter the deliquescent salt, contained in the mother-water, from which the parasulphat-ammon is separated by crystallization. Much value, however, is not to be attached to these com- parisons, for they merely refer to a certain analogy or combina- tion, which may be even called a remote one, since bicarburetted hydrogen and ammonia differ with respect to the number of their elements. This parallel becomes still more improbable, on account of the different properties of the substances compared, they pos- sessing not the least resemblance to each other. 565 ArticLe XVIII. On a Transportable Magnetometer. By WiLuELM WEBER. [This article is translated partly from the Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1838, and partly from manuscript commu- nications from M. Weber to Major Sabine.] A SMALL travelling apparatus for the absolute measurement of the force of the earth’s magnetism has been described in the Resultate for 1836*. That apparatus was not a magnetometer, but rather served as an illustration of the mode in which this measurement, which had previously been executed only with a magnetometer, might be made with an ordinary compass needle. The degree of accuracy attainable with such a small appa- ratus, and the occasions on which it ought to be employed, were examined in the memoir referred to. But for the limitation imposed by the want of time, or by other external circum- stances, it would of course be always preferable to use the magnetometer; the small apparatus being only intended to serve as a substitute, on occasions when the use of the more per- fect instrument is impracticable. It is very desirable to reduce ee. a . aids _ the number of such occasions as much as possible, by devising ‘means of removing the difficulties which often oppose them- selves to the use of the magnetometer ; and this will appear the more desirable, the more we consider the great difference in the degree of precision attainable by the two instruments; and the more we reflect on the importance that would be given to a class of observations in which magnetometers have not hitherto been used, (namely, those made during distant and extensive journeys and voyages,) if they could be rendered susceptible of a higher degree of accuracy, certainty, and completeness. If the final aim of such observations were simply that of constructing magnetic maps on which no ulterior investigation was to be based, the degree of exactness to which such maps should be carried might be arbitrarily determined; and pos- sibly such an amount of accuracy as can be obtained without * Translated in the Scientific Memoirs, Part V. VOL, Il. PART VIII. 2P 566 WEBER ON A TRANSPORTABLE MAGNETOMETER. \ the use of magnetometers might be deemed sufficient. But if 5 these maps are not themselves the final object sought,—if they : are to form the basis of a new investigation,—if determinate — rules and laws are to be recognised,—if the maps are to serve ‘ as the means of comparing experiment with the general theory of the earth’s magnetism,—and if the elements of the theory are to be deduced from them,—then the degree of accuracy to be demanded is no longer arbitrary, but is determined by the nature of the subject. A minor degree of accuracy, such as these maps now possess, has, it is true, served for a first attempt at such a comparison; but in order that they may afford an adequate basis for an amended calculation, they must receive a higher degree of exactness. Such is now the great purpose of the magnetic observations to be made in distant expeditions, and it is this which now gives to such expeditions peculiar im- portance and value. But the greater the importance which thus attaches to such voyages and observations, in consequence of the demands of theory, the more essential it becomes to examine what they are capable of affording. Magnetic observations may be made at places widely remote from each other, either at the same time or nearly so, or alter- nately, so as to lessen the errors occasioned by regarding them as simultaneous. At all the stations, or at the more important at any rate, the observations may be continued with regularity for at least one or more weeks, so as to afford mean values freed in some measure from disturbing influences. But it 1 still more desirable to give to such expeditions the advantage the recent improvements, by furnishing them with magnetome= ters. This would probably be best accomplished, by the per- sons who undertake magnetic expeditions making themselves thoroughly acquainted, both theoretically and practically, with the whole subject of magnetometric measurements, as they would then be able to devise for themselves the best travelling arrangements, But as there are not many opportunities of acquiring this knowledge, the following memoir may be inter- esting and useful to persons who cannot study the subj more thoroughly in other ways. I proceed to describe a transportable magnetometer, which, it unites all the advantages proper to magnetometers, with fi lity of management and compendious construction, appe ' e WEBER ON A TRANSPORTABLE MAGNETOMETER. | 567 well adapted for magnetic expeditions and journeys, and is not more inferior to the magnetometers of fixed observatories, than good portable astronomical instruments are to the larger ones used in fixed astronomical observatories. I shall first give some general remarks on this instrument; then a description of its several parts; and lastly, observations of the Declination, and its Variations, made simultaneously with the transportable magnetometer and with that of the Gottingen Observatory, and a measurement of the Intensity made for the purpose of exhibiting its capability in that respect. § I. General Remarks. The transportable magnetometer, figured in half size in Pl. XXV., fig. 1, requires in general but few explanations, as it _is only essentially distinguished from other magnetometers by its small size, and by its more compendious construction. All the observations which are made with the larger magnetometers may also be made with the one under consideration ; so that the absolute declination, the variations of the declination, and the absolute horizontal intensity, can all be measured by it; the variations of the horizontal intensity can also be observed, by suspending the bar employed in the experiments of deflection, as a bifilar magnetometer. The exactness with which these various measurements can be made is much greater than has et been attained in travelling observations; it suffices for all the purposes of magnetic travellers; and it admits of as much “accuracy and certainty, in proportion to its size, as do the largest magnetometers. The results obtained with the large instrument used in the Gottingen Magnetic Observatory may be depended upon almost to the immediate readings, which are to ;1, of a division of the scale, or to 2 seconds of arc. This supposes the scale to be at least five meters from the mirror of the magnetometer, as other- ise the arc value of the divisions of the scale (which are one ‘millimeter long), would be greater. Such a distance would not answer in journeys, as much time would be lost in bringing all the parts of the instrument into their proper positions. For travelling purposes, the distances ought to be limited so as to admit of the whole apparatus being placed on a table, and they should therefore be about four times less. Consequently, in lieu of the 8-inch theodolite, which is required to do full justice 2. Pi2 568 . WEBER ON A TRANSPORTABLE MAGNETOMETER. ‘3 . to the great magnetometers, one of about three or four inches may be used without disadvantage, being at once more conve- nient and more ceconomical, and still allowing the measurements to be depended upon to within from 10 to 20 seconds of are. In considering the subject further, it will be seen, that admitting the necessity in the travelling apparatus of diminishing the ob- servation distance, a diminution in the size of the magnetometer (which would not be admissible under other circumstances), does in no degree detract from the accuracy of the observations. For with a distance four times less, the degree to which the reading can be depended on (and which it is desired to preserve), is not affected, though the proportion of the magnetic force of the magnetometer to external disturbing influences be lessened in the same proportion. It may be assumed, that the magnetic force decreases as the cube of the linear dimensions of the bar, and external disturbing influences as the square, whence it follows that the bar may be made four times less without diminishing the dependence to be placed on the readings (which is to about the one tenth part of one division of the scale). If, with this di- minution, other arrangements are adopted for guarding against external disturbing influences more carefully than has been hitherto found necessary with the larger magnetometers, there will be no material disadvantage in pushing the diminution in size somewhat further, having in such case only to preserve the degree of dependence which may be placed on the readings. In fact, the length of the bar of 600 millimeters has been reduced to 100 millimeters; and observation has shown that the readings may be equally depended upon; with this differ- ence only, that the divisions, as read off, have a four times greater value of arc than in the case of the larger magnetome- ters, so that one division of the scale is equivalent to 80 seconds of arc instead of 20 seconds. Hence it appears, that by suit- able arrangements, all the advantages of the magnetometer may be secured to magnetic expeditions; of course, without that highest degree of precision attainable only in fixed observa- tories, where nothing is wanting in construction and arrange- ment. The instrument to be described affords these advantages in respect to the absolute declination and its variations, and still more in respect to the absolute measurement of the horizonta intensity; for in the Resultate for 1836, p. 88, it has been WEBER ON A TRANSPORTABLE MAGNETOMETER. 569 shown*, that if both bars are six times smaller, the deflecting bar may be brought six times nearer to the magnetometer, with- out its being necessary to take more exactly into account the distribution of free magnetism in the bars. If, then, the length and breadth be diminished, and the thickness be left unaltered, (the large bars are 600™™ long, 36™™ broad, and 9™™ thick ; and the small bars 100™ long, 9™ broad, and 9™™ thick,) it follows that as much may be gained in the small magnetometer, by in- creasing the angular deflection, as is lost by diminishing the di- stance of observation. In fact, the experiments of deflection admit of a precision which leaves nothing to be desired, and which harmonizes perfectly with the degree of accuracy which is known to be of easy attainment in the experiments of vi- bration. Of course the small magnetometer must be construeted in such a manner that all its parts may form a solid whole, so that their relative position may not be liable to be disarranged by packing, unpacking, or putting up. It must be possible both to set the magnet bar at liberty, and to secure it again while in its case, as is done in the common compass, and the torsion of the thread must not be altered in so doing; the access of air must be quite cut off even from the mirror, which may be observed through a thin plate of mica, if a piéce of plane glass ground parallel is not to be obtained. It is very advantageous _ to make the case entirely of copper, and even of strong ‘copper-plates, not only for the sake of the increased solidity given to the whole apparatus, but also because the case will thus act on the inclosed magnet as a damper, and all the measurements may be made with much greater rapidity. The instrument must be so strong and solid, even when used in the open air, that it may carry two arms, which serve for placing the deflecting bar at equal measured distances east and west. These arms being correctly placed, all the preparations for the ; experiments of deflection which would otherwise be necessary, —namely, placing the measuring bars horizontally, and in a di- rection perpendicular to the magnetic meridian, and finding the corresponding points on either side of the magnetometer,— are spared, and the experiments are rendered much easier, and require less time. * Sci. Mem. Pari V. page 86. 570 WEBER ON A TRANSPORTABLE MAGNETOMETER. § Il. Description of the several parts. Fig. 1 represents the vertical section of the magnetometer in the direction of the magnetic meridian. The magnetic bar which forms the needle is bored through- out its length, and the opening which is turned towards the telescope is provided with a lens, in the focus of which at the other end there is a cross of wires. This cross of wires is seen in the telescope, when (as is required for determining the true azimuth in the measurement of the absolute declination) it is adjusted to distant objects, and then directed to the lens. This arrangement was proposed by Airy, to make it possible to dis- pense with the mirror, and to be able to make, with the same telescope, and without displacing the eye-glass, the astronomical, geodesical, and magnetical observations required in measuring the absolute declination. In making this measurement the needle must be reversed; but in the reversal the optical axis must not alter its relative position in respect to the needle ; this is effected in the closed case by means of a key, turned on the outside, and causing the needle inside to perform half a revolution round its axis of length. But this arrangement is inapplicable to observations which require great changes in the position of the needle, as in the experiments of vibration and deflection in the measurement of the absolute intensity. It therefore appeared advantageous to employ also a mirror, placed in the same manner as in the bifilar magnetometer, close to the axis of rotation of the needle, and above the copper case, and available however great the deflections may be. The copper case is seen to have three openings: the first is into a space containing the mirror, and closed towards the theodolite by a plate of glass, through which the light can pass, in the direction shown in the plate from the scale, to the mirror, and thence back to the telescope of the theodolite. The other two openings are nearly at the same height as the magnetic needle and the telescope of the theodolite. The light entering through one of these apertures illuminates the cross of wires which is stretched across the hindmost end of the hollow needle, passes on to the lens at the other end, and thence, parallel to the horizontal direction marked in the figure, to the telescope of the theodolite with which the cross of wires is observed. The- needle, bored throughout its length, is made accurately cylin= WEBER ON A TRANSPORTABLE MAGNETOMETER. 571. drical, and is inclosed in a cylindrical brass box, on the under surface of which are two small projections which fit into two cavities in the copper case when the suspension thread of the needle is let down. The brass box can be fixed in this position by two screws brought through the upper part of the copper case: the box being thus held fast, the needle may first be drawn out through the opening in the back of the case, and a brass cy- linder of the same form as the needle, inclosing a weak magnet, may be placed in its stead, to try the torsion of the thread. Secondly, for the purpose of measuring the error of collimation, the needle may be turned in the box round its longitudinal axis, by means of a key introduced through the aperture in the back of the case. During the observations the apertures in the front and the back of the case are closed with a plate of mica to a against currents of air. _ Fig. 2 represents a somewhat different and more simple con- struction of the same instrument; the needle is not hollow, is not enclosed in a brass case, fal cannot be reversed. This simplification is admissible when the use of the instrument is to be restricted to the experiments which are to be made i the open air, as detailed in the sequel. In this case the mirror is included in the copper case, and its normal forms a right angle with the magnetic axis of the needle. The glazed open- _ ing in the side of the case does not impair its action as a damper, and the opening may be made of any convenient size. Fig. 3 represents the outside box, in which the instrument is _ packed for travelling, and which serves also for suspending the deflecting bar when it is to be used for the experiments of vibration. A mirror is fixed to the end of the bar, so that it may be observed from a distance with a telescope and scale. The box has a small opening which can be closed with a plate of mica admitting the light. The figure shows the bar suspended in the box, and loaded with two cylindrical weights, made of s, and connected by a silk thread passing over a bar parallel Ho the needle, to keep the centres of gravity of the two weights exactly the length of the bar from each other. The weights serve for the deduction of the moment of inertia. _ The unifilar suspension of the bar can be changed for a bifilar, if the variations of the intensity are to be observed. The box must then be placed relatively to the theodolite and to the mag- netometer in the manner represented in the ground plan, fig. 4, namely, so that, according to the rule laid down in the Resultate 572 WEBER ON A TRANSPORTABLE MAGNETOMETER. for 1837, p. 22*, the line connecting the middle of the bar with the middle of the magnetometer needle may form with the mag- netic meridian an angle of 35° 16!. Thus observations of the variations of the declination and of the intensity may be conve- niently combined in this manner by travelling observers. § III. Examples of Observations and Measurements. Measurement of the Absolute Declination. This measurement resolves itself into three parts: 1. The determination of torsion. 2. The azimuthal determination of the magnetic axis. 3. The azimuthal determination of the true meridian. By the azimuth of a direction is here understood the angle formed by two vertical planes, one in the direction in question, and the other in the direction of the optical axis of the telescope of the theodolite, the alidade being placed on the zero point of the circle. 1. Determination of Torsion. This determination consists of the measurement of the force of torsion, and of the angle of torsion. Force of Torsion. There belong to the magnetometer two needles, the magnetic and the torsion needle, which may be suspended to the same thread, and which differ in the proportion of their magnetic moments (M, m). Designating by T the horizontal part of the earth’s magnetic force, the force of torsion is to be compared with the force M T, as well as with the force m T. Comparison with the force M T. In order to reduce the observations to the same time, the declination was observed in the magnetic observatory simulta- neously with both the observations. Observation of thel Observation Reading of the position of the : : Radius in parts Reduced Torsion Circle. Magnetometer in the Magnedt of the Scns Observation. by the Scale. y- ° ’ ° Fe “u 355 6 275-67 18 29 49 2174 275'67 175 6 237-06 18 30 42 237°31 Hence the force of torsion is given in parts of M T 180° 2174 178 * Sci. Mem. Part VI. p. 270. WEBER ON A TRANSPORTABLE MAGNETOMETER. 573° Comparison with the force m T. Observation of the Reading of the position of Radius Torsion Circle. Gai ar aah ge Differences, Mean. By eee of fo} , fheew Bek SBE litcla IY Te ae 269 15 270°77 ? 329 54 10979 ae 269 15 280°91 167-69 2243°5 eu) uh | ie 269 15 282-12 Hence the force of torsion is given in parts of m T tae 5 FB95)) 16 7°69 7 42-563 PWGOLGS:. onc 2248 e RZ B Angle of Torsion. Observation of the position of Radius the Magnetometer in divisions of by Ge Scale. the Scale. Magnetic needle... 29290 2174 Torsion needle .. 328-67 The distances of the observed divisions of the scale from the zero point of torsion being designated by v and y, then ~ is the angle of torsion sought, expressed in divisions of the scale; and for determining x we have the following equations: 292:90 — x = 32867 — y 12°563 7 = y. Hence the angle of torsion in divisions of the scale is found, Ta 3,095 in seconds of arc 309g " " os - 206265" = 293". a eo a From this determination of the force and of the angle of torsion, the correction on account of torsion to be applied in measuring the declination is found 1 178 This correction is so small that it may be wholly neglected; the more so, as, during the time occupied in the measurement, the declination itself altered two divisions of the scale, so that the angle of torsion for the time of this measurement almost wholly disappeared. - 0G == GG. 574 WEBER ON A TRANSPORTABLE MAGNETOMETER. 2. Azimuthal determination of the Magnetic Axis. In order to reduce the observations to the same time, the declination was observed simultaneously in the magnetic ob- servatory. Azimuth Observation Time. : Azimuth e f the in the Reduced 1839, eee : ‘ of the April 11. pp reas pecan Azimuth. | Magnetic Axis. Befor hoe Pere ln Ault 12; ay \ Li- <¢ 131 22 43| 18 26 2613120 0| o » » After 131 41 29°5 \ 11 375 |132 259| 18 29 9/132 259 reversal 3. Azimuthal determination of the true North. Three visible objects were observed, the positions of which, in respect to the Gittingen Observatory, are given by geode- sical measurements. Distance from the Observator ; Be Observed Azimuth of the Designation : Azimuth, true North, of the Objects. | a South. / West. 33 58 50 ete ual mat) 315.17 5 150 6 14 117 15 15 Hohehagen ..| + 6060:00 | + 12447-70 Gartenhaus ..| + 289°28 - 27°54 | | [Oar | Jacobithurm..| — 710°70 | + 500- 49 | As there is no correction to be applied on account of torsion, we obtain immediately from hence the westerly declination, by deducting the azimuth of the magnetic axis from the azimuth of the true north. 150° 6! 14! —131° 41! 29'""5=18° 24! 44/5, This result corresponds to 115 37™5, 11th April 1839. The declination observed at the same time in the magnetic observa- ORS 1g” Zorg", showing a difference of — 4! 24-5, which probably is only in part due to error of observation, and is in part caused by the influence of the copper case surrounding the magnetometer, which may not be wholly free from iron. Repeated measure- ments, and comparisons with the observations in the magnetic — observatory, may serve to deduce such an influence if it exists, so that it may be taken into account in future measurements. A second measurement actually gave a similar result, namely, = A WEBER ON A TRANSPORTABLE MAGNETOMETER. 575 1839, April 13. In the open air. In the magnetic observatory. 104 31! 18° 18! O! 1°. 93f Sel showing a difference of —5' 36". The mean influence of the copper case in this instrument may therefore be taken as =—5/. Observation of the Variations of Declination. On the 15th April 1839, from 54 25™ to 7) 27™5, the varia- tions of declination were observed alternately, with the mag- netometer in the Géttingen Observatory, and with the small magnetometer. In the following table the four first columns show the immediate results of observation with the two appa- ratus. In the final column the observations with the small magnetometer are reduced according to the proportion of the value of the scale divisions. The two series of observations are exhibited graphically in fig. 5, for the purpose of comparison. It may be seen from this example that the observations of the variations of declination can be made with a portable mag- netometer with much accuracy. : Transportable Magnetometer. 1839. Magnetic 1839. April 13. Error: April 13, Reading ee ae -? (w—244'2) h. m. h. m. 5 25 896-00 5 27:5 244-95 897-44 5 30 895°56 5 32:5 244-20 895°00 5 35 894-66 5 375 244-97 897°50 5 40 896°47 5 42°5 245-20 898-25 5 45 899-56 5 47°5 24618 901-44 5 50 899°52 9 92°5 245-78 900714 5 59 898°78 5 575 246-02 900°91 6 90 900°57 6 25 247°35 905:24 6 5 905°95 6 75 248-04 907-48 6 10 905-00 6 125 249°77 913°10 6 15 916°77 6 17:5 251-77 99°60 6 20 920-00 6 22°5 251:77 919-60 6 25 919-66 6 27:5 251°56 918-92 6 30 916°63 6 325 250°70 916712 6 35 912-72 6 37-5 250°96 916:97 6 40 917-66 6 42:5 251°74 919-51 6 45 927°35 6 47°5 25432 927-89 7 0 941-27 7 25 260°79 948-92 Lo 959-33 ee Jas 265°71 964-91 7 10 964-53 7125 261-27 950°48 7 15 936°38 Ph WA 254-34 927:95 Z 20 922-80 7 225 251-75 919°54 i 25 914-42 7 27-5 250°09 91414 576 WEBER ON A TRANSPORTABLE MAGNETOMETER. Absolute Measure of the Intensity. The measurement of the intensity divides itself into four parts. 1. The determination of torsion. 2. Of the moment of inertia of the deflecting bar. 3. The experiments of de- flection. 4. The experiments of vibration. I will confine my- self in this place, for the sake of brevity, to two parts, viz. the determination of the moment of inertia, and the experiments of deflection, which are especially instructive towards a know- ledge of the instrument. The determination of torsion has been already spoken of in the measurement of declination, and the experiments of vibration are so simple and so well known, that it is sufficient to give their results. 1. Determination of the Moment of Inertia. The deflecting bar is suspended to a thread or wire, and is then vibrated: 1) without a weight; 2) with a weight, the mo- ment of inertia of which is known. Vibrations without a weight. Number of Arc of Reduced time Vibrations. Time. Vibration. of Vibration. h m. Ss. ° ' 0 Gen20s oL27. 8 56 uv 26 7 23 45°49 8 40 6698 61 7 v2] 39:92 8 8 6°695 115 7 33 41°64 7 22 6696 151 7 37 42°80 6 56 6695 186 7 Al 37:19 6 32 Vibrations with a weight. 0 2 18 35°57 8 16 12-058 46 2 27 50°45 6 58 12-039 125 2 43 41°76 5 4 12:019 200 2 58 43:31 3 20 Hence the mean time of vibration without a weight is = 6!"696, and with a weight = 12-039. For determining the moment of inertia of the weight we have the following data: 1) the length 7 of the deflecting bar, or the distance apart of the threads which hang from its two ends and support two equal cylindrical weights ; 2) the mass 2 p; 3) the radius r of the two cylinders. 1 = g3mm-4g 2 p = 500008 Ko= 4mm-60 besigas > WEBER ON A TRANSPORTABLE MAGNETOMETER. 577: If the mass of the cylinder were concentrated in its axis, its mo- ment of inertia would be 1 12 p = 109091000. If the cylinders revolved only round their own axis, their mo- ment would be 7? 'p = 529000. Their moment in the above experiments is to be taken as equal to the sum of 112 p + 72 p = 109620000. Whence therefore the moment of inertia of the oscillating bar may be obtained from the equation mK w(K +K’) 7? ae 7/2 b) Vig where K! signifies the known, and K the desired moment of inertia, z' the time of vibration with a weight, and ¢ the time of vibration without a weight, consequently K = 49103000. In these experiments the needle was suspended to a thread in which the force of torsion was so small as to be insensible. The same series of experiments was repeated with the needle sus- pended by a wire in which the force of torsion was much greater ; the result was almost the same as before, namely, K = 49044000. Finally, in order to furnish a check, the deflecting bar was weighed, and its length and radius were exactly measured : Weight p! = 66670™: Length 7 = 93mm-4Q Radius 7 = 5mm 45, whence its moment of inertia may be calculated. Supposing perfect internal homogeneity, K = 7, 1? p! + 17'? p! = 48982000. The accordance of all these experiments sufficiently shows that the moment of inertia of even such small bars may be deter- mined with great precision. 573 WEBER ON A TRANSPORTABLE MAGNETOMETER. 1 2. Experiments of Deflection. 1839, February 13. Double Deflection. Distance in North . In divisions of Millimeters. Pole. Readings. the scale. Are values. — 55675 | E. | 372-95 W. | 13233 | 34062 Lear-os | 5° 30%3 E. | 373-78 — 453-25 | E. | 47591 | ,.-.- W. | 23-36 tagoa | 44789 10° 9°3 E. | 47658 wid + 453°25 | E. 480:04 W. 31-83 E. 480:27 448-21 ) 448-44 f 448:32 | 10° 112 455675 | E. | 375-93 w. | 135-06 | 34084 \ 240-82 | 5° 30-0 E. | 375-82 ae j Hence the simple deflections vp, v, are obtained for the di- stances Ro, R, (without regard to signs) Up = 2° 45! 4/5, for Ry = 556°75 v, = 5° 5! 7/5, for Ry = 453°25. Consequently, if tang. v be developed according to the powers of R, tang. v = 8305800 R~ ® — 4081300000 R~ ° whence (see Intensitas Vis Magnetice, Art. 21, 22), = = 4152900. With the comparatively great distance of the deflecting bar from the needle (equal to from 5 to 6 times the length of the needle), the determination of the coefficient of the second mem- ber of this equation (which is to be divided by the 5th power of the distance) is uncertain, and it is therefore better to dis- M ft R,? tang. vy = 4146600 R,? tang. v, = 4143200, viz. the mean of which may be taken, consequently, = = 4144900, which differs but little from the above value. regard it. We then obtain for =, two values, mais a Oe WEBER ON A TRANSPORTABLE MAGNETOMETER. 579: If to the results obtained we add lastly the time of vibration t, which was found to be t = 6!-0586*, and if we assume K = 49073500, we obtain 2 MT = = = 13195000, consequently T = 17842. We are not enabled to test and compare this result further, as a simultaneous measurement with the large magnetometer could not be executed at that time. When a new mea- surement of the earth’s magnetic force is made in the Gottingen Observatory, the opportunity of comparison thus afforded will not be neglected. The improvements, (represented in figs. 2, 6, 7, 8,) which, since the above was written, I have caused to be made in the transportable magnetometer, are designed to facilitate the use of the instrument in the open air, as in travelling it will be rare to meet with a suitable building free from iron for the execution of absolute measurements. It is not absolutely necessary that the whole of the observations for these purposes should be made in the open air; and on account of the liability to interruption from weather, it is desirable to reduce the number requiring this exposure as much as possible. In the improved construc- _ tion I have given great care and consideration to this part of the subject, and have found it possible to arrange the obser- vations in such manner that the greater part may be made in a room, including those which would be made to the greatest disadvantage in the open air. Fig. 6. represents the tripod stand, on which the measuring apparatus, fig. 7, and the magnetometer, fig. 2, are to be placed and levelled, as shown in fig. 8. The measuring apparatus, * The bar having been magnetized afresh for the experiments of vibration and deflection, had a shorter time of vibration than in the previous experiments on the moment of inertia. 580 WEBER ON A TRANSPORTABLE MAGNETOMETER. fig. 7, required for the deflection experiments, consists of a copper-plate fitting on the tripod, and carrying the supporters of the deflecting-bar; each of these is formed of two conver- ging tubes connected at their extremities, from whence proceeds a third tube provided with a graduation, and on this the deflect- ing-bar is to be placed: this tube forms also the reading tele- scope, and has the reading scale attached to it. Fig. 8. represents the magnetometer placed on the measuring apparatus, which rests itself upon the tripod: the needle is suspended in a copper case, which acts as a damper in checking the vibrations. The mirror close below the needle is directed to the east. The whole of the eastern side of the copper case can be removed, to give access to the screw to which the suspension is fastened, and by which the inclination of the mirror may be corrected. In the middle of this side is an opening closed by a piece of plane glass, making a small angle with the vertical, in order that the reading telescope, which is directed to the mirror behind the glass, may not see a double image of the scale. For the measurement of the absolute intensity the deflection experiments alone require to be made in the open air; the re- mainder may be made in a room if more convenient; for if the magnetism of the needle, which can be ascertained in a room, be known, the intensity of the earth’s magnetism may be calcu- lated from that of the needle, and from the experiments of de- flection made in the open air*. It should be noticed, how- ever, that the determination of the magnetism of the needle in such cases requires a complete measurement of the intensity to be gone through, including both the experiments of deflection and those of vibration, with and without the weights. The magnetism of the needle should also be determined either shortly before, or shortly after, the deflection experiments in the open air, because it is liable to alteration: and the temperature in the room and in the air should be as nearly the same as possible. The experiments of deflection in the open air require only a * The experiments of vibration might be made in the open air instead of those of deflection; but in such case the instrument would afford less cer- tainty and less convenience. plas WEBER ON A TRANSPORTABLE MAGNETOMETER. 58] ° solid foundation, on which the tripod may be placed and le- velled ; the measuring apparatus, resting on it, carries the de- flecting bar, the telescope, and the scale, each in its due posi- tion relatively to the others; and the whole system can be turned upon the tripod without their displacement. The copper case of the magnetometer fits into the depression a 3, fig. 8, by which its position is fixed relatively to all the other parts. The whole instrument is then turned on the tripod until the middle _ of the scale is seen in the reading telescope, and it is then ready for the deflection experiments. The vernier of the deflecting bar being placed on the zero point of the graduation of the measuring apparatus, the deflec- tion of the needle is observed. The deflecting bar is then re- versed, and the observation repeated. The bar is then removed to the end of the measuring apparatus, and the vernier set to 1000™™ of the graduated scale, when the deflected position of the needle is again observed before and after the reversal of the bar. Let the four observed deflections be called m, m!, n, n!,— the absolute intensity of the magnetism of the needle, previously observed in a room, M,—and the arc-value of a division of the scale, determined also in a room (the torsion being taken into account), «,—then the absolute horizontal intensity of the earth’s magnetism will be 2 M a 008 tan oF _ where v = } arc-tang. } (m — m! +n —7!) a. This simple formula may be employed, because the small dimensions of the needle and bar, relatively to their distance apart, renders the next member (having the fifth power of the distance in the denominator) insensible. Fig. 10 represents the theodolite used in observing the de- clination and its variations; it is provided with a verification telescope, having a small scale at the end: a larger scale is placed above the theodolite, perpendicular to the optical axis of the principal telescope. The observation of the absolute declination may be divided into those parts which must be made in the open air, and those which may be made in a room. Fig. 11. represents in A the VOL, II. PART VIII. 2a §82 WEBER ON A TRANSPORTABLE MAGNETOMETER. : cross-section of the tube of the magnetometer telescope, and ‘ in BC the scale; between A and BC is a transparent space; — the theodolite must be so placed that the observer may look with — the verification telescope through the space D towards the © mirror of the magnetometer needle, and perceive the image of — the scale attached to that telescope; he must first observe the position of the needle by this scale, and thence determine the angle ¢ (fig. 12.), which the optical axis of the verification tele- scope makes with the normal to the mirror of the magneto- meter; he must then bisect objects of known azimuth with the principal telescope of the theodolite, and thence find the angle corresponding on the divided limb to the direction of the principal telescope relatively to the north. These are all the observations required to be made in the open air in determining the declination. The angle x, Fig. 12, corresponding, on the graduated limb, to the parallel position of the optical axes of the two telescopes of the theodolite, can be ascertained in a room; as can also the angle g which the mag- netic meridian makes with the normal of the mirror belonging to the needle. Hence we obtain (x—‘) the angle which the optical axis of the verification te- lescope makes with the true meridian. (x —) — 4, the angle which the mirror-normal of the needle makes with the true meridian. e—{(x—v)—4$} the angle which the magnetic meridian makes with the true meridian. The angle x is found by placing a plane mirror before the verification telescope, and viewing in the telescope the reflected — image of a vertical thread suspended over the middle of the — object glass; a vertical thread is also suspended over the middle of the principal telescope, and the telescope adjusted to its reflected image ; the reading on the circle gives the angle x, | supposing the collimation error of the principal telescope to re- main unaltered when the eye-piece is adjusted to distant ob- jects; otherwise the alteration must be sought by reversing the telescope, and applied as a correction to the reading on the circle. The angle ¢ is determined by directing the principal tele- scope of the theodolite from B (fig. 12.) to C, a second needle suspended in the wooden case, as represented in fig. 3; the WEBER ON A TRANSPORTABLE MAGNETOMETER. 583. verification telescope is directed on the first needle A, in the copper case as in the open air. The needle C is furnished either with a collimator or a mirror, and is capable of reversal. The direction of its magnetic axis is next to be found, 7. e. the angle »., to which the theodolite must be adjusted, in order that the optical axis of its principal telescope may be parallel with the direction of the magnetic axis of the needle ©, whence the angle eg (= — (x — %) + p) is obtained, if the two needles A and C are sufficiently distant apart to exert no sensible influ- ence on each other, so that their magnetic axes may be regarded as parallel. But if this be not the case, it is easy to determine the angle v formed by the magnetic axes of the two needles*, and to add it as a correction to the value, as above, of @; i.e. g=7—(x— 9) + e+» The suspension of the needle in the wooden case is so con- trived, that it may be used either as an unifilar or as a bifilar magnetometer. This contrivance is represented in fig. 9. The variations of the declination and of the horizontal intensity can thus be observed at the same time; the former with the mag- netometer in the copper case, and the latter with the magneto- meter in the wooden case. In preparing for the latter observa- tions, the telescope of the theodolite is to be directed perpendi- cularly to the magnetic meridian, and the magnetometer in the wooden case is to be placed in the same direction. The time of vibration ¢ of the needle, with the unifilar suspension, must be determined, if not already known, which it will generally be, from the experiments of vibration belonging to the measure- ment of the absolute intensity. The unifilar suspension must then be changed for the bifilar without altering the direction of the magnetic axis, and the time of vibration must be observed afresh, the distance apart of the suspension threads being in- __* From the propositions contained in the Resultate for 1837, page 22 et seq., it follows that if A BC = 90°, ACB=a@, and AC =», and if m and m'! denote respectively the magnetism of the needles A and C, m—m! rs T The value of m and m' must be determined by the deflections 3 and 3! of a compass needle placed successively east and west at the distance d, namely mde nl Val sag Re aaa al be y= —sin2e. $ ae 584 WEBER ON A TRANSPORTABLE MAGNETOMETER. : creased until ¢’ is about = 0°6871 ¢. The torsion circle must then be turned until the middle of the scale appears in the field of view of the telescope, and the time of vibration 2” observed. The magnetometer is then in the transversal position proper for observing the variations of intensity, and the value of the scale divisions may easily be calculated from the observed times of vibration ¢, ¢', t’; namely, if « denote the arc-value of a division of the scale in parts of radius, the value of a division of the scale, in parts of the whole horizontal intensity, is Fig. Fig. Fig. t? tz {? mc=,/ (1-25) . Foee: EXPLANATION OF THE FicurREs, PLATE XXV. 1. a, b,c, dis a vertical longitudinal section of the copper case of the mag- netometer, with the needle e,f suspended by a silk thread g,h. The needle is seen to be pierced through its Jength, and provided at the ex- tremity f with a lens; it is inclosed in a copper tube /, J, m, m, and can be turned by means of a key o, p, which is accessible by an opening in the copper case. In doing this the copper tube is held by two screws g, 7, and two projections s, ¢, The mirror w, v is seen above the copper case, near the axis of rotation of the needle. A dotted line indicates how the telescope of the theodolite, fig. 10, is directed, both to the needle and to the lens at its end f, and also to the mirror uw, v. It is also seen how the inclination of the mirror may be regulated by the screw w, that the image of the scale placed above the telescope at a, fig. 10, may appear in the field of view. This fig. is half the size of the instrument — itself. 2. represents a magnetometer, which differs from the one just described in not being adapted for complete measurements of the declination. The collimator is cmitted, and the needle cannot be reversed. The mirror @, 6, c,d is inclosed in the copper case, and is parallel to the plane of the magnetic meridian ; the inclination of the mirror is regu- lated by the screw at e; the copper case forms an unbroken damper round the needle, except at the aperture for the suspension thread; the mirror is observed through a glass plate in one of the sides of the copper case. This figure is also half size. 3. a, b, c, d, e represents the wooden case, in which either of the instru- ments shown in figs. 1. and 2. are packed for travelling. The lid, with the tube a, b, ¢ which is fastened to it, is taken off, the instrument . ae - Fig. Fig. Fig. Fig. Fig. WEBER ON A TRANSPORTABLE MAGNETOMETER. 585 placed inside, and the box closed again. When observations are made, this box serves for suspending a second needle, the time of vibration of which is required for the measurement of the absolute intensity; this second needle f, g is provided at both ends with mirrors, one of which serves for observing the scale. The needle rests on two supports h, &, attached to a small measuring bar m,n, over which passes a thread carrying the weights p, g, which serve to increase the moment of inertia of the oscillating needle. The needle can be turned in the supports h, k, and may be reversed; rendering it available, in absolute measurements of the declination, as an auxiliary needle, when the instrument repre- sented in fig. 2. is used, the needle of which is not reversible. For this purpose, instead of a needle with a mirror, one with a collimator, fig. 13, may be placed in h, k. It consists of a magnetic steel tube a, b, c, d, carrying at the end a, c, an achromatic object-glass ; and at its other extremity a sliding tube of brass e, f, g, h, provided with a glass micrometer in the focus of the object-glass. It will be seen also by fig. 3. that this needle is suspended to two threads, the upper points of attach- ment of which are r ands. The threads are conducted over a roller x to give them equal tension, and are united in one from x to v, forming an unifilar suspension, which may be converted into a bifilar by opening out the apparatus a, @, y, 3, which is done by pressing down the knob w by the screw ¢, and disengaging the threads from the pins 2, y, as repre- sented in fig. 9. Fig. 3. is also half size. 4. A, is the theodolite carrying two telescopes and two scales; one tele- scope and one scale serve for observing the unifilar magnetometer B, and the other telescope and scale for observing the bifilar magnetometer C. The figure gives the angles which the instruments ought to form with each other. 5. is a graphical representation of the variations of the declination ob- served on the 13th of April, 1889, at Gottingen, simultaneously in the magnetic observatory, and with the transportable magnetometer. 6. is the tripod on which the magnetometer, fig. 2, is placed and le- velled. 7. is the apparatus required for the experiments of deflection. a, b,c, d is a copper disc which fits on to the tripod, fig.6; e,f, 9, h, and h, l, m, n, are arms screwed to the copper disc at e, f and /,7; one arm carries the telescope p, g, to which the scale r, s is attached, and upon which the de- flecting bar u,v is to be laid; the other arm carries a tube on which the deflecting bar is also laid, but which could not be conveniently repre- sented in the figure. Between e, f and 4, J the magnetometer (fig. 2.) is placed. 8. is a smaller side-view of the magnetometer represented in fig. 2, in its proper relative position to the measuring apparatus, fig. 7, and rest- ing on the tripod, fig. 6. In this view the needle is seen only by its circular cross-section, and the glass plate is shown, in the side of the case which permits the image of the scale, reflected from the mirror, to be ob. served with the telescope. Fig. 9. is explained in the description of fig. 3. 586 WEBER ON A TRANSPORTABLE MAGNETOMETER. Fig. 10. represents the theodolite with the verification telescope: two scales are seen, one of which, a, is applied in such manner that its middle — corresponds to the prolongation of the vertical axis of rotation of the — theodolite ; the other, b,c, is attached in front of the object-glass of the verification telescope. It is very narrow, in order to intercept the less light. Figs. 11, 12 and 13. are explained in the text. W. WEBER. 587 ARTICLE XIX. An extract from Remarks on the Term-Observations for 1839, of the German Magnetic Association. By W1iLHELM WEBER. (With a Plate.) [From the Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1839. | In concluding this notice, I wish to call attention to the ob- servations made in the high northern latitudes m 1838 and 1839, for which we are indebted to the zeal and perseverance of the French savans, MM. Lottin, Bravais and Martin, and of the Swedish naval officers, Lieutenants Siljestrom and Silje- hook, who joined the French expedition to Spitzbergen and Finmarken: we may derive instruction from these observa- tions in regard to the arrangement of future researches of the same nature in those regions. In looking at the Plate, XXVI., it is obvious at the first glance, that the beautiful accordance in the variations of the magnetic elements, which had been hi- therto invariably observed, from Catania, Rome, Milan, &c., to Upsala in the north, ceases when we proceed still further north ; so that in comparing the curves of Upsala and Alten (in Fin- _marken, lat. 69° 58!) they would scarcely be recognized as be- _ longing to the same term. There is no doubt as to the correct- ness of the observations, as the voyagers undertook the addi- tional trouble of occasionally observing Gambey’s needle simul- taneously with the magnetometer, and the movements of both were found to be in accord. If therefore these observations suffi- ciently assure us of the great difference between the magnetic changes in those more northern districts and in Upsala, the im- portant conclusion follows, that future term-observations in these very high latitudes will only be rendered truly valuable by the establishment of intervening stations, which may show the gra- dual passage of the one system of changes into the other ; or by having a group of several stations around Alten and its vicinity, which will afford a sufficient interest by their mutual comparison independently of others, as it is to be expected that great differ- ences should there manifest themselves at small distances. Such observations would be available for inquiries, for which those 588 WEBER ON THE TERM-OBSERVATIONS AT ALTEN. made elsewhere are little or not at all adapted; in particular we might, by their means, determine most securely whether the forces which cause the variations have their seat above or below the sur- face of the earth. Without this multiplication of stations in its vicinity, observations of the variations at Alten will have a much inferior value, as they differ so greatly from those at the nearest present station, Upsala, of which we may convince ourselves by inspection of the curves of declination and horizontal intensity on the 23rd of February, 1839, shown in Plate XXVI. The three declination curves represent the variations of that element from noon to 10 p.m., Gottingen mean time, at Alten, Upsala, and Gottingen, and are all on the same scale. The two curves of the horizontal intensity are for the same period, and repre- sent the changes at Alten and at Gottingen, which was the next most northern station at which the intensity was observed during that term. We cannot perceive in the two latter curves even that trace of resemblance which is visible in those of the declination. W. WEBER. 589 ARTICLE XX. Results of the Daily Observations of Magnetic Declination during six years at Gottingen. By Dr. B. Gotpscumipr. [From the Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1839. ] IN the volume of the Resultate for 1836, M. Gauss com- municated the results of the observations of the magnetic de- clination, made daily in the magnetic observatory at Gottingen, from the 17th March, 1834, to the 31st of March, 1837, and combined them in various ways for the purpose of determining the march of the declination*. Since that period these obser- vations have been continued uninterruptedly by me according to the same plan, and we have now before us the determina- tions of six years, which I propose to consider in this treatise. To the mean values of the declination for the several months of the three first years, published as above, we have now to add the following :— 1837 to 1838, 1838 to 1839. 1839 to 1840. Month. > _ 3 LE | a io inn | a 8 A.M. 1 P.M. 8 A.M. 1 P.M. 8 A.M. 1 P.M. *) i “l ‘ “ 4 “ 4 “ April......... 21 59:1 | 40 42-2 8 08:9 35 56°7 || 14 43-8 | 28 43:5 May ......... 23 17°3 | 38 35-2 i 439 | 35 46-1 || 15 16-7 | 28 15-0 June ......... 22 46:2 | 38 24:8 7 40°77 | 35 06-2 || 13 54:1 | 27 15-5 Oa 21 33:3 | 36 55-4 8 47-6 | 33 48-2 || 14 27-6 | 28 16-6 August ...... 24 22-2 | 37 51-9 i 43:9 | 34 59-4 || 13 40:9 | 30 07-0 September...|} 25 02-5 | 37 19-1 8 17:1 | 33 17-5 || 13 41:8 | 27 26-5 October...... 25 50:0 | 37 00-2 19 58°7 | 30 48-3 || 14 47-4] 25 53-0 November...|| 25 47-5 | 33 12-7 a 06:6 | 28 14-4 || 16 01:3 | 23 08-9 December ...|| 25 51-4 | 31 14:5 1 34-3 | 26 19-0 |} 16 54:5 | 21 02-6 January...... 25 25°3 | 33 36-2 1 01-6 | 27 35-1 || 15 41:5 | 20 48-6 | February ...|} 23 55-3 | 33 37:8 20 01-0 | 27 29-8 |} 13 53-1 | 22 15-9 | March ...... ¥ : 18 09°6 | 29 52-4 || 11 14:4] 23 42-4 The number of degrees is throughout 18. We will now proceed to combine these numbers in the same manner as was done with the observations of the first three years, beginning with the deduction of the differences between the forenoon and afternoon declinations. These differences, the * Translated in the Scientific Memoirs, vol. ii. Part V. pp- 54 to 65. 590 DR. GOLDSCHMIDT ON THE OBSERVATIONS monthly mean values of which have all the same sign, are exhi- bited in the following table :— 1837 to 1838, 1838 to 1839. 1839 to 1840. é “ “se 4 “se Alpril) scysia.st2se.<. 18 50-1 17 47:8 13 59:7 May. osssd.c-c0steoeds 15 17-9 17 02-2 12 58:3 JUBE\ oscccscncesstes’ 15 38°6 17 25:5 13 21-4 Dl yivccescsvessnsencd 15 22-1 15 00-6 13 49-0 PUD IICi a abeerssarees 13 29-7 16 15:5 16 26-1 September 12 176 15 00-4 13 44:7 October _ ~ _ rrr PPP ROR PP Rr Or Go Hm 9 GO Go bo St bo Sr St ANwAArAnRSUSOwWSS* bo S COorenmnwnott &| SSLSSSERL Slew gl AkASC UNKNone POSSSR PE PPP Pee ee eR OOK 5 © 5 ie) i S bo i ist) rs i—) — So much regularity appears in the numbers in the last column, that we may hope that the mean value of the secular change 4! 15"-4, corresponding to the Ist of April, 1837, may not be far from the truth. We now proceed to consider the mean values of the declina- tion for each twelve months of our six years’ observations, as in the following table :— 8 A.M. Mean, “i 1834 to 1835 ...| 18 37 12° 18 41 19°75 1835 to 1836 ... 33 42-0 38 43-4 4 1836 to 1837 ... 27 20:3 33 47-45 % 1837 to 1338 ... 23 52:5 30 01-25 1838 to 1839 ... 19 26-2 25 31-15 1839 to 1840 ... 14 315 20 03-05 ak 18 26 00:8 18 37 07°8 18 31 343 The mean values of the several years correspond to the middle day of the period comprised ; e. g. October Ist, 1834, &c. The means of the six years give the mean declination for the 1st of ~ April, 1837. Under the supposition that the mean decrease of the declina- tion is proportional to the time, I have calculated by the method of least squares, from the numbers contained in the last column, the following formula for the declination ¢:— 8 = 18° 42! 16-231 —4! 16""756 . ¢, where ¢ denotes the interval elapsed since October Ist, 1834, expressed in years. The values of the mean declination com- 594 DR. GOLDSCHMIDT ON THE OBSERVATIONS puted by this formula, and their deviations from the observed values, are as follows :— Computed Differ- Computed Differ- Declination. ence. Declination. ence. SS ee ee fe) 1834 to 1835... 18 42 16-231|+4 56-481/|1837 to 1838...|18 29 25-963|— 35-287 1835 to 1836... 37 59°475|— 43-925//1838 to 1839... 25 9-207|— 21-943 1836 to 1837... 33 42°719|— 4:731)/1839 to 1840... 20 52°451)+ 49-401 According to this table the mean deviation of a determination of the declination deduced from one year is 48942; the mean error to be feared in the determination of the absolute part of the formula is 34/92; and the mean error to be feared in the determination of the coefficient of ¢ is 11!"53. It is more natural to suppose that the decrease is uniformly accelerating than constant; therefore the declination may be represented by the formula a+ b¢-+ c7°. Giving ¢ the same signification as before, we obtain by the combination of the six data, using the method of least squares, §=18° 41! 31'-442 —3! 09"-514 t—O! 13-453 #; and the values of 8, computed by this formula, as well as the deviations from the observed values, are as follows :— Computed Differ- Computed Differ- Declination. ence, Declination. ence. 1834 to 1835...| 18 41 31°442|4 11°672/|1837 to 1838...| 18 30 01-830|4+ 6-580 1835 to 1836...| 38 08-473|— 34-927||1838 to 1839...) 25 18152|— 12-998 1836 to 1837...| 34 18-604|+ 31-154||1839 to 1840... 20 07°570|+ 4-520 The sum of the squares of the remaining deviations is 2515743 hence the mean deviation of a single year’s determination, so far as it can be derived from six years’ observations, is 2896. The weights of a, 6, ¢ are 1°317,1°376, and 37°34, where the weight of a mean value of the declination, deduced from a whole year, is unity ; with 28'-96 as the mean deviation of such a mean value, we have the mean errors of a, 6 and c, 25-23, 24"-68,. and 4"-74, Our formula gives 18° 52! 38" for the maximum of the declination, and the corresponding ¢ = — 7:043; so that on the 14th of September, 1827, the declination had become retrograde. It need scarcely be remarked, that both these num- bers are uncertain, as the coefficient of ¢?, on which the determi- nation of the time of the maximum principally depends, is un- certain to one-third of its whole value. Unfortunately we have OF MAGNETIC DECLINATION AT GOTTINGEN. 595 no direct determination of the year in which the declination began to decrease in Gottingen. We obtain from the formula the yearly change of the decli- nation — 3’ 22"-967 — 26"-906 ¢, corresponding to the interval 1834 + ¢ to 1835 + ¢, where ¢ denotes the time elapsed since the 1st of October, 1834, expressed in parts of a year. The influence of the season of the year on the mean values of the declination in the several months, has already been noticed in discussing the differences between the forenoon and afternoon declinations ; by comparing the monthly means with the decli- nation deduced from the whole year, we may perceive how great this influence is, and the nature of the effects it produces. This comparison gives the following differences for the three years 1837 to 1840 :— Declination, 8 a.M. Fourth Year. Fifth Year. Sixth Year. Mean. / “ é/ “4 4 “4 / “4 PATEL (co cenacese=s — 2 00-4 —1173 + 0 12:3 — 1} 1-8 MW eecpaess-ass>s — 0 35-2 — 0 42:3 + 0 45:2 — 0 108 BREE ee aciccsscoe —1 06:3 — 1 45°5 — 0 37-4 — 1 09:7 Jus SAgpeeeceoosnees — 2 19-2 — 0 38-6 — 0 03-9 — 1 006 August ......+.. 0 29-7 — 0 42:3 — 0 50-6 — @0 21-1 September ...... +1 10-0 — 1 09-1 — 0 49-7 — 0 16:3 October ......... +1 57:5 +0 32-5 +0 15-9 + 0 55:3 November ...... +1 55-0 +2 40-4 +1 303 +2 01:9 December ...... +1 58-9 +2 08-1 +2 23-0 +2 10-0 January ......... +1 32°8 +1 35-4 +1 10:0 +1 26:1 ob February......... + 0 02:8 + 0 35:8 — 0 38-4 +0 00-1 BVIAITCH oc ccccccecss — 3 06:1 —1 166 —3 17:1 Paice 2 33°3 Declination, 1 p.m. | Fourth Year. Fifth Year. | Sixth Year. Mean. | é 4 “ é/ “4 ca “t DRT ice. .<-20000| +4 32-2 +4 206 +3 08-9 + 4 006 Wis dasc-c000s0 + 2 25-2 +4 10:0 +2 40-4 +3 05-2 STI feo Scuses ses +2 14:8 +3 301 +1 40:9 + 2 28-6 MAUL nc nccnencesese +0 45-4 + 2 121 +2 42-0 +1 53-2 UID UBE \ Fd n = z g =I Os = Ca a> i} <8: i AddaADeEwWSNAE AYE oy Ee i) Teel St Sa Ht GO He SS SI BO OH PANN ISLISE (cess sccr=ss MOCIOUGE fresecsccance November ......... December bo bo ot hye CRM DPNNoOSCOR HEY bo Go bo mR bO 09 GO Oo OT oT SRAAMAD Aww H +11 b+++++4++ COHN WH Oe NR ORE, mwBonwpnrsoneouee GUS St ret Pal et OBS SOF Cn mM OO CN RN Om WOW +++ it4+4+41 001 I+++++ | m= Sra bs 83 Op Qos Be _ The numbers of the first column give the differences between the forenoon declination in the several months, and the mean forenoon declination in the whole year; applied with their sign to the mean declination of the year, they give the mean fore- noon declinations of the several months freed from the secular change, so far as the latter can be derived from six years’ ob- servations. The same remark applies to the second column in respect to the afternoon declinations. If we represent these two columns by periodical functions, we find for the first — 83"-7 cos 6 — 1183 sin ¢ — 45""8 cos 2 4+ 11-2 sin 2 > — 12"-7 cos 3 ¢ —9!"2 sin 3 ¢— 18""5 cos 4 + 13/2 sin 4 > — 11-3 cos 5 —0"3 sin 5 ¢— 4""9 cos 6 4. For the second column we find OF MAGNETIC DECLINATION AT GOTTINGEN. 597 +404 cos + 121""1 sin ¢ + 391 cos 2 ¢ — 522 sin 29 + 78 cos 3 $+0'"2 sin 3 9 + 50 cos4 $ + 7'"2 sin 4 — 10"6 cos 5 ¢ — 26""7 sin 5 g — 2'"4 cos 6 4, where ¢ denotes the number of months elapsed since the middle of April multiplied by 30°. In eleven months we perceive a confirmation of the remark- able result previously deduced from the consideration of the ob- servations of the first three years, namely, that the forenoon and afternoon declinations deviate from their mean values in oppo- ‘site directions. October is the only exception; and, viewing the small amount of the differences in that month, and the degree of uncertainty which still remains, this exception may perhaps disappear when the observations shall have been longer continued. In the four winter months, November to February, the forenoon declination is greater, and the afternoon declina- tion less, than their respective mean values; and both these circumstances contribute, during this portion of the year, to bring the whole difference below its mean value. From March to September the opposite effect takes place. These opposite deviations, moreover, being, on an average, nearly of equal magnitude, nearly counterbalance each other in their means, which are represented in the last column. The mean being also very small in the exceptional month of October, the law -enounced in the Resultate for 1836*, appears to be confirmed, “namely, that the mean result of the declinations observed at 8 a.m. and 1 p.m. does not contain, apart from the irregular anomalies and the secular decrease, any erortant fluctuations oo on season. = Lastly, we have to consider the fluctuations of the declination _ from one day to another. In the, Resultate for 1836+ the fol- lowing definition was given of the term “ fluctuation,” namely, “the difference from the declination of the preceding day at the "same hour;” and, by analogy with what are called mean errors _ of observation, the mean fluctuation, during any given interval of time, is the square root of the mean of the squares of the se- veral fluctuations. It was further remarked, that when several equal intervals, or intervals considered as equal, are united in one, the arithmetical mean of the partial mean fluctuations * Scientific Memoirs, vol. ii. page 62. (Part V.) + Ibid, loc. cit, VOL. II. PART VIII. 2R 598 DR. GOLDSCHMIDT ON THE OBSERVATIONS b must not be taken as the general mean; but we must revert to ‘ the squares, and take the square root of their arithmetical — mean. The results of the last three years, calculated in this ‘ manner, and expressed in seconds, are contained in the follow- ing table. Mean Fluctuation of the Declination during the three years Jrom 1837 to 1840. 8 A.M. 1P.M. Fourth Fifth Sixth Fourth Fifth Sixth Year. Year. Year. Year. Year. Year. AUB e, cosh ees 316 149 162 199 229 152 May tet tescsctet 319 157 266 211 193 176 DUNE) copie east 262 208 205 211 236 159 UU ek ce cee 189 224 214 332 158 183 August ...... 234 119 194 139 209 216 September ...| 232 240 267 215 167 246 October ...... 286 272 267 278 210 205 November ... 145 147 98 257 189 143 December ... 174 84 108 250 129 132 January ...... 302 179 220 208 254 154 February...... 274 133 97 241 217 195 WMiarch -ccc.see 195 271 118 184 145 174 Mean...... 252 192 193 232 198 179 The following table contains the mean values of the fluctua- tions for the several months of the intervals 1834 to 1837, 1837 to 1840, and of the whole interval 1834 to 1840. 8 A.M. 1 P.M. I.to III, | 1V.to VI.| I.to VI. I. to III, | IV. to VI. Aipril, Seutecess 147 223 189 180 196 May, ..s ; : | = os Werec cies stat testitaatt HEE | | Z Ly TH t , ena + t ’ | fs ae ia HH ct : | | \ |) Sai Pee oH | : i “ ‘ it ; EEA HH ! | et N | | | a | | i a EEE ERE Ret i ant a 1 = if : ae : : : 560 ee : : a! Ss : : | if 4 : : 3 I50 . | mao : | . E 40 al va | : } SIO 3 520 a 5 gegaaanas x 300 : | 490 : he ; "Bante lth ; . +f - rion, E ba er sata cok eI a , Se 2P% ti - Fw { = rap on | Sepia ts Tarm of August 77 aN FO S is 2 Ss y y S oy LZ Collinge, Berlin, Lap la, the lMaqgue, Obsirvations ac Upsa 1g, Muntch. » fo at 22. a6 20 To Seeueeesseaueunr=Jenancsiseusu5"aa a HEE EHH HH Hee H tte | PETE rrr A, Milan VWinie ye Marbury We S NS S S S S aN S SU” 77) lorm Wbservatiuns at the [aque Gottingen, Berlin ean ee | _ SS a —_ — Seren tattc Mamet Fol TE PLIX. | Term Y Novi 26” 1836 | Observations al Upsala, breda, Gottingen, Breslau, Fraberg, Laperg, Marburg, Munich, Milan (C 2 co * TA gh sear wh aie fe tae Sees aac eereeeeeeeceeeeeeeaee eter cereevee Entttteee H | a it t Et ia EEE 288)-cecK6o8el 4 sraroraraeeneeti EEE HH tT Hy waa sSBUEEETEERIESEEE : : | eeeeeueeeens HH | mae { a oH | reas a +} ; ik a a FS ee Manors Vol LW. Plate X Fig. 10 Fig7 fgS fig 9 eee rad Chart for the Value of z Saentitic Memowrs. FOUL 230 10250. iY. Ct Pe Pe Avot & i ; Chart, for the Value va = 7 serentafic Memorrs Fol T ae gs FS ER WS SE ed i a ok a ie eh a rt pe a feet t pw es Re Oo 8 A See oe. eee [ 4 a er —— (a t -———-- | tae ee at it SP: | - ‘tf +2 oe ad SA a ie Bes SO ye a eet i ! J 7 : hd a 1 f } } Vet i. } it Disturbances of the hormontal magnetic Intensity (upper tirve), and of the magneltc The muaibers on the lat are the Sexle-divinenss of the extent apparatus, r : ] Decdtinatin / lower Curve) Goltengen 183%. November 13. th. f PML, ie prestter numbers correspond. to yfmiter- snteestitins The nuanibene an de right are the Sante itiarione of the Magnenenster in the in OB rervutety: rnater numbers aarresporad te mare Lasterty Declination | z 7 z z 7 5 foe! 7 ere ia el pat 13 7] 7s 7 PELL) tw aE 7 = ; 7 ww § je00 $00 \ \ sat +\ f \ 60 asen Pun sual Ve Ky Wl V ey a ~ Eaten nes Hea 4 ae 4 ~pa en aa SRReTnnwEm SSSSah@2 ara VARIATIONS oF tHE MAGNETIC DECLINATION VARIATIONS or tHE HORIZONTAL INTENSITY Hi 4 February 2371839 February 23 1839 iam