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OLL 'Wi It m^m >. ^-^I^OC «^cri^ ^f c c cc. c e<,S ^acc < .. c CCCC r- c cc<c -.. c cc/C ^ 4^ ^^ <^ c-r cccc c ^<c:'x^ '»^. c .--•^acja' •ere- Ctv<i;Ci (Si •?rrc . cc C^ cc C CC f c cc <:':.<c ccCC r c^ c ^ *,■ <iK. ■ C<! ■■ cc cc < * _^c r'' . '^^<«:^ . ^ «5^ c ( . c ccc ■ c^ ^<£X: <^ cc c ::<:cc<rc ^v«:cc«r c ^c:^ «3ixc c c^ iJif ' c^t,c <;< c ^s: ^ «s^ ^ c: ^r ^<cc ,^ : <^cc<« <SCCJ8I .,c_^c< 0< V *^ ^i< c< < cc ' < ■ THE LONDON AND EDINBURGH PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. CONPUCTED BY SIR DAVID BREWSTER, K.H. LL.D. F.R.S. L. & E. &c. RICHARD TAYLOR, F.L.S.G.S. Astr.S.Nat.H.Mosc.&c. AND RICHARD PHILLIPS, F.R.S. L. &E.F.G.S. &c. " Nee aranearum sane textus ideo melior quia ex se fila gignunt, nee noster vilior quia ex Jilienis libamus ut apes." Just. Lips. Monit. Polit. lib. i. cap. 1. VOL. VIL NEW AND UNITED SERIES OF THE PHILOSOPHICAL MAGAZINE, ANNALS OF PHILOSOPHY, AND JOURNAL OF SCIENCE. JULY-DECEMBER, 1835. LONDON: PRINTED BY RICHARD TAYLOR, RED LION COURT, FLEET STREET, Printer to the University of London. SOLD BY LONGMAN, REES, ORME, BROWN, GREEN, AND LONGMAN; CADELL; BALDWIM AND CRADOCK; SHERWOOD, GILBERT, AND PIPER; SIMPKIN AND MARSHALL; WHITTAKER AND CO.; AND S. HIGHLEY, LONDON : BY THOMAS CLARK, AND ADAM AND CHARLES BLACK, EDINBURGH; SMITH AND SON, GLASGOW; HODGES AND M'ARTHUR, DUB- LIN; AND G, G. BENNIS, PARIS. The Conductors of the London and Edinburgh Philoso- phical Magazine and Journal of Science beg to acknowledge the editorial assistance rendered them by their friend Mr. Edward William Brayley, F.L. & G.S., Librarian to the London Institution; which commenced in the year 1823, in vol. Ixi. of the First Series, and has been regularly conti- nued from that time. ^v m December 1st, 183.5. ^cf (o ^ TV ^'7 TABLE OF CONTENTS. NUMBER XXXVIL— JULY, 1835. Page Mr. Weaver on the Gold-workings formerly conducted in the County of Wicklow, Ireland 1 Prof. Encke on Olbers's Method of determining the Orbits of Comets 7 Prof Young on the Summation of slowly converging and di- verging Infinite Series • 25 Dr. Hodgkin on the Importance of Studying and Preserving the Languages spoken by Uncivilized Nations, with the view of elucidating the Physical History of Man 27 Dr. Hussey's Catalogue of Comets 36 Mr. Beke on the Geological Evidence of the Advance of the Land at the Head of the Persian Gulf 40 Mr. Murchison on the Silurian System of Rocks 46 Proceedings of the Geological Society 52 ■;i— Zoological Society 64 Astronomical Society 69 — at the Friday-Evening Meetings of the Royal Institution of Great Britain •.•!• ?;• 70 Cambridge Philosophical Society, Lent Term, 1835 70 British Association for the Advancement of Science : Meeting at Dublin in August next 71 Mr. Brayley on the frequent Deficiency of the Ungueal Pha- lanx in the Hallux of the Orang Outang 72 On Oil of Cinnamon, &c ." 74* Atomic Weight of Aluminum 75 Mr. Charlesworth on Veins of Crystallized Carbonate of Lime in Fossil Wood 76 Phaenomenon resembling the Mirage, seen in the Regent's Park 77 Mr. A. Dunn's Experiments to ascertain the Existence of Lead in the Atmosphere of a White-lead Manufactory .... 77 Decomposition of Cyanuret of Mercury by Iron .....' 78 Sulj)huret of Zinc and Iron 79 Scientific Books : — The Second Edition of the First and Second Reports of the British Association for the Advancement of Science 79 Transactions of the Linnaean Society, Vol. XVII. Part ii. . , 79 Meteorological Observations made by Mr. Thompson at the Garden of the Horticultural Society at Chiswick, neai London , and by Mr. Veall at Boston 80 a 2 IV CONTF.NTS. NUMBER XXXVIII.— AUGUST. Page Mr. Charlesworth's Observations on the Crag-formation and its Organic Remains ; with a View to establish a Divi- sion of the Tertiary Strata overlying the London Clay in Suffolk 81 Dr. Hodgkin on the Importance of Studying and Preserving the Languages spoken by Uncivilized Nations, with the view of elucidating the Physical History of Man (concluded) , . 94? Mr. F. Watkins on Magneto-electric Induction 107 Mr. Talbot on the Nature of Light 113 Suggestions respecting the ensuing Meeting of the British As- sociation for the Advancement of Science 118 Prof. Encke on Olbers's Method of determining the Orbits of Comets [continued) 123 New Books : — Royle's Illustrations of the Botany and other branches of the Natural History of the Himalayan Moun- tains, and of the Flora of Cashmere 132 Proceedings of the Royal Society 136 . Geological Society 142 Zoological Society 1 52 Composition of Iodide of Iron 156 Parallel, by Newton, upon the Corpuscular Theory, to Mr. Talbot's Explanation of certain Phaenomena of Incandes- cence, upon the Undulatory Theory 157 Scientific Association of Germany : Meeting at Bonn from September 17th to 27th 157 M. Liebig on Phosphuret of Azote 158 Crystalline *Forra of Kupferbliithe 159 Temperature of Vapours from boiling Solutions 159 Meteorological Observations 160 NUMBER XXXIX.—SEPTEMBER. Mr. Lubbock on the Double Achromatic Object Glass 161 Mr. C. B. Rose's Sketch of the Geology of West Norfolk 171 Mr. Dalmahoy on the greater calorific Effect of the Sun's di- rect Rays in high than in low Latitudes 182 On Taylor's Theorem 188 Mr. Carter on the Ancient and Modern Formation of Deltas 4 in the Persian Gulf by the Euphrates and Tigris, in answer to Mr. Beke 192 Mr. G. B. Jerrard on certain Transformations connected with the finite Solution of Equations of the Fifth Degree 202 CONTENTS. V Page Prof. Encke on Olbers's Method of determining the Orbits of Comets (continued) 203 Proceedings of the Royal Society 207 . ■ Geological Society 212 . Zoological Society 222 Mr. Sturgeon's Explanatory Facts 231 N. S. Heineken on a Method of ascertaining, approximately, the Radii of Curvature of unequally convex Lenses 234 Arsenic in English Sulphuric Acid 235 Appearance of Halley's Comet 236 Congres Scientifique at Douay 237 Dublin Meeting of the British Association for the Advance- ment of Science 237 Succinic Acid and its Combinations 238 New Locahty of Plenakite 239 Meteorological Observations 240 NUMBER XL.— OCTOBER. Mr. C. Rumker on a new Method of reducing Lunar Observa- tions for the Determination of the Longitude 241 Sir David Brewster's Observations relative to the Structure and Origin of the Diamond 245 Mr. Carter on the Ancient and Modern Formation of Deltas in the Persian Gulf by the Euphrates and Tigris, in answer to Mr. Beke 250 Dr. H. Hudson's Proposed Method for inferring the Dew- point from the Indications of the Wet-bulb Hygrometer . . 256 Dr. Apjohn's Experimental Investigation of a Formula for ^ inferring the Dew-point from the Indications of the Wet- bulb Hygrometer '2.Q6 Mr. C. B. Rose's Sketch of the Geology of West Norfolk {continued) 274 Prof. Encke on Olbers's Method of determining the Orbits of Comets {concluded) 280 OflBcial Report of the Proceedings of the British Association for the Advancement of Science, at the Dublin Meeting, August 1835 ; Mathematics and Physics 289 Proceedings of the Geological Society 316 Zoological Society 328 On the Existence of Arsenic in Phosphorus 331 Mr. J. F. W. Johnston's Composition of Oxychloride of Anti- mony ^ 332 Collection of Minerals and North American Fossils 334 Letter from Mr. Watkins 335 Analysis of Wolfram 335 Meteorological Observations 335 VI CONTENTS. NUMBER XLI.— NOVEMBER. Page Mr. Faraday's Reply to Dr. John Davy's " Remarks on certain Statements of Mr. Faraday contained in his 'Researches in Electricity.' " ^37 Note by M. Ampere on Heat and Light considered as the Re- sults of Vibratory Motion i ^42 Prof. Forbes's Note relative to the Polarization of Heat 34-9 Mr. S. Woodward's Remarks upon the Crag Formation of Norfolk and Suffolk 353 Sir G. S. Mackenzie on certain Points in Meteorology and Magnetism 355 Dr. H. Johnson on Divergence as the Cause of Motion in Plants 357 A. R. on certain coloured Bands observed in examining New- ton's Rings 363 Mr. Nixon on the Verification of Captain Lloyd's Levelling Instrument by the Greenwich Mural Circle 364? Mr. W. S. B. Woolhouse on the Enharmonic Organ ; in Reply to an Article in the Westminster Review 366 Mr. John Taylor on a New Rotative Steam- Engine 369 Mr. C. B. Rose's Sketch of the Geology of West Norfolk (continued) 370 Dr. Boase's Inquiry into the Nature of the Structure of Rocks 376 Reviews and notices respecting New Books: — Spiller's Trans- lation of the Solution of Numerical Equations by Sturm . . 384« Official Report of the Proceedings of the British Association for the Advancement of Science, at the Dublin Meeting, August 1835 385 Proceedings of the Royal Society, Prof. Faraday's Experi- mental Researches in Electricity., Tenth Series 411 Geological Society 412 ___ Zoological Society 417 ■- Entomological Society 420 M. Poggendorff's Vindication of Prof. Faraday's right of priority to the discovery of definite electrolytic action .... 421 Kupfer-Antimonglanz 422 Remarks on a peculiar state of polarity induced in soft Iron by Voltaic Magnetism, by E. M. Clarke, Magnetician .... 422 Letter of Professor Rosenberger to the Editor of the Astro- nomische Nachrichten 423 Observations on the Assay of Silver in the humid way, by M. Gay-Lussac 425 Analysis of Pyroxylic Spirit. — Methylene 427 Composition of Pyromucic Acid 429 Analysis of Gadolinite 430 Rhodizite, a new Mineral 431 Mr. Saull's Geological Collection 431 Meteorological Observations 432 CONTENTS. Vll NUMBER XLII.— DECEMBER. Page Sir G. S. Mackenzie on the Theory of the Parallel Roads of Glen Roy 433 Sir D. Brewster on certain Peculiarities in the Double Refrac- tion and Absorption of Light exhibited in the Oxalate of Chromium and Potash 436 Mr. R. W. Fox's Remarks on the Laws of Magnetic Attrac- tion 439 Dr. J.Inglis's Extracts from a Prize Essay on Iodine 441 Dr. Boase's Inquiry into the Nature of the Structure of Rocks 445 Prof. Young's Concise Method of determining the Function X.J in the Application of Sturm's Theorem 454 Mr. Lubbock on Bernoulli's Theory of the Tides 457 Mr. R. Fitch on the Coralline Crag of Ramsholt and Orford. . 463 Mr. Charlesworth's Reply to Mr. Woodward's Remarks on the Coralline Crag; with Observations on certain Errors which may affect the Determination of the age of Tertiary Deposits 464 Dr. Apjohn's Experimental Investigation of a Formula for inferring the Dew-point from the Indications of the Wet- bulb Hygrometer {concluded) 470 M. Melloni on the Transmission of Calorific Rays through Diathermal Bodies 475 Mr. G. B. Jerrard on the Transformation of Equations {conti- nued) , , , 478 On the Cause of the " Coloured Bands" observed by A. R. . . 474 Official Report of the Proceedings of the British Association for the Advancement of Science, at the Dublin Meeting, August 1835; — Geology and Geography; Zoology and Botany ; Medical Science 480—515 Proceedings of the Geological Society 515 — r Linnsean Society 519 —i Zoological Society 529 SUPPLEMENTARY NUMBER. Proceedings of the Zoological Society {continued.) 521 M.Henry Rose on the Evolution of Light during Crystallization 534 Hydrate of Oil of Turpentine 537 Action of Oxacids on Pyroxylic Spirit 538 Bcrzclius, Preparation of pure Tellurium 539 Prof. Bischoff', Analysis of the Plenakite 510 • • • VIII CONTENTS. Page Mineral Waters 541 Dr. Boue's new Work 541 Meteorological Observations 542 Index 543 PLATES. I. A Map illustrative of Mr. Rose's Paper on the Geology of West Norfolk. II. A Portrait of Auguste Pyrame DeCandolle, Professor of Natural History in the University of Geneva. ERRATUM. lu Mr. Stubgeon*s Paper, p. 231, /orraagnetic electrometer read mag- netic electromotor. THE LONDON AND EDINBURGH PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [THIRD SERIES.] JULY 1835. I. Oil the Gold-workings formerly conducted in the County of WickloWy Ireland, By 1'homas Weaver, Esq.^ F.R.S. F.G.S. M.R.LA. SrcSrc,"^. CINCE my return to Europe, after an absence of nearly *^ three years, among the scientific publications that have attracted my attention, the Journal of the Geological Society of Dublin claimed particular notice. I rejoiced to perceive a growing spirit of geological inquiry in Ireland, and more espe- cially as that spirit is understood to have been mainly excited and promoted by the Trigonometrical Survey of the island now in progress under the able conduct of Colonel Colby, R.E. It was, however, with surprise and regret that I read the following passage in the Address delivered at the First An- nual Meeting of that Society by their President. Speaking of the natural resources of Ireland, the learned author says : " Even native gold was found not many years ago in a neigh- bouring county in large wedges. It was discovered in a limited portion of a stream of about one hundred yards in extent; and though the stream was repeatedly examined up to the mountain from which it descended, not a trace of the metal could be perceived in any other part; yet, notwith- standing this circumstance, the mountain itself was presumed to be the chief depository: it was accordingly perforated quite through, with great labour and at an enormous expense ; and the search conducted in this wa^ having proved unsuccessful, • Communicated by the Author. T7iird Series. Vol. 7. No. 37. July 1835. B Mr. Weaver on the Gold-worJcvms ©' was finally abandoned. The limited portion of the stream in which any gold could be discovered, we might suppose, should have directed to the adjacent banks, and the hills which rise from the spot on either side. These might have been ex- plored at much less expense; and judging by the uncom- monly great size of some of the lumps which had been found, it seems not unreasonable to expect^ notwithstanding the past disappointment, that much will hereafter be brought to light." A judgment thus gravely pronounced from a presidential chair demands attention. From what source the deliverer of that sentence drew his information does not appear ; yet cer- tainly a more strange misapprehension of facts has seldom been embodied in words. In justice to the Government under whose auspices I had the honour to act, as well as to the memories of the individuals concerned in directing the opera- tions (of whom I am the only survivor), it becomes incumbent on me to disabuse the public by recalling their attention to the real circumstances of the case ; and this I might briefly do by a simple reference to the succinct and faithful history I have given of the proceedings in my memoir on the East of Ireland, published in the 5th volume of the Geological Trans- actions, First Series. But as those Transactions are, compara- tively speaking, in the hands of few persons, it appears expe- dient to enter into some details, through the medium of a scientific Journal of extensive circulation; for which purpose I subjoin the following extracts from my memoir on the East of Ireland, merely transposing a few passages for the sake of greater distinctness*. " §§ 105. 106. The discovery of native gold in the Ballin- valley stream at Croghan Kinshella mountain was accidental ; and, as it subsequently proved, the principal depot of the gold in that stream extended about twelve hundred yards below the ford. In other quarters the gold obtained was compara- tively of small amount. The largest piece found in any of the other streams was on the Coolbawn side. It weighed 2J ounces. But in all these cases the gold was accompanied by most of the metallic substances that occurred in the Ballin- valley stream, and which will be enumerated hereafter. Under the Act of Parliament for managing the undertaking, the di- rectors established regular stream-works ; and up to the un- happy period of the rebellion in May 1798, when the works • The publication of this article has been delayed in the hope of re- covering the plate of the Geological Map of Croghan Kinshela mountain, which would have thrown a clearer light on the subject ; but as, unfortu- nately, that plate is not forthcoming, I must necessarily refer in this respect to the 3th volume of the Transactions of the Geological Society, First Series. formerly conducted in the County of WicJdo'w, 3 were destroyed, Government had been fully reimbursed its advances, the produce of the undertaking having defrayed its own expenses, and left a surplus in hand. In the year 1801, the operations were resumed, when the directors proposed to Government not to confine its views to the mere collection of the alluvial gold, but to extend the researches, directing them more particularly toward the discovery of auriferous veins. They were justified, as they conceived, in this proposition by the following considerations : " 1. The well-known fact, that in the various quarters of the world, in America, Africa, Asia, and Europe, auriferous veins often occur in the mountains adjacent to the districts that abound in massy alluvial gold*. '' 2. The peculiar circumstances under which the native gold ofCroghanKinshela was found. — It occurred in massy lumps, and in small pieces, down to the minutest grain. One piece weighed twenty-two ounces, another eighteen ounces, a third nine ounces, and a fourth seven ounces. The gold was found, accompanied by other metallic substances, dispersed through a kind of stratum composed of clay, sand, gravel, and frag- ments of rock, and covered by soil, which sometimes attained to a very considerable depth^ from twenty to fifty feet, in the bed and banks of the different streams. By the operations in the stream-works, (which are too well known to professed miners to need detail on the present occasion,) all the metallic particles thus disseminated were collected in a concentrated mass; and in those at Ballinvalley it was constantly found that the gold was attended by magnetic ironstone, sometimes in masses of half a hundred weight, by magnetic ironsand, by cubical and dodecahedral iron pyrites ; and, in small pieces and grains, by specular iron ore, brown and red ironstone, iron ochre, fragments of tinstone crystals, wolfram, gray ore of manganese, pieces of quartz and chlorite, and sometimes fragments of quartz crystals. I observed and collected also some specimens, which show that the gold, magnetic ironstone, and wolfram were each of them frequently intermingled with quartz; and I have also a few specimens which exhibit the gold not only incorporated with iron ochre, but ramifying in slight threads through wolfram. Some of the gold, though very rarely, occurred crystallized in octohedrons, and also in ** * See, for example, the Travels of Ulloa and Humboldt in America ; Molina*s Chili ; Park's Travels in Africa; Marsden's Sumatra; Turner's Thibet; Abhandlunge^i uber Bergbaukunde \ Dr. Clarke's Travels in Europe, Asia, and Africa, vol. iv. ; Robilant on Piedmont, in Journal des Mines, vol. ix.; and D'Aubuisson in vol. xxix." B2 4 Mr. Weaver on the Gold-workhigs the elongated garnet dodecahedron, of which I retam speci- mens. " 3. The ascertained fact that several of the contemporaneous veins of quartz, contained in the mountain, were metalliferous, yielding magnetic ironstone, iron pyrites, copper pyrites, blende and iron ochre, with chlorite and quartz crystals. " Under these considerations, the measures proposed by the directors were : to continue the stream-works, progressively advancing toward the head of the several streams; to examine more closely the solid mass of the mountain, by means of trenches to be cut in various directions down to the firm rock ; to explore more fully the veins already known, and those which might be discovered near the surface by the trenches; and lastly, to try these veins, at a considerable depth from the surface, by means of a gallery or level, to be driven into the mountain, in a direction nearly at right angles with the ge- neral range of the veins, selecting that quarter where they ap- peared to be most numerous. " These measures receiving the sanction of Government, they were carried into effect; and numerous trials were made by driving and sinking on the veins previously known and subsequently discovered. By the Ballin valley trench alone twenty-seven veins of quartz were found, varying from nine inches to four feet wide, in a distance of 700 fathoms ; and in the same manner, by the Ballinagore trench, eighteen quartz veins were discovered in a distance of 600 fathoms. These veins partly range and dip with the slaty rock, but they also ramify and terminate in strings which intersect the latter. They are evident!}' contemporaneous, and most of them are barren, but some were found bearing magnetic ironstone, iron pyrites, and iron ochre, with chlorite. Two of the most power- ful veins occur on the western side of the northern arm of the mountain, one being six feet wide in the broadest part, and the other four feet. They consist principally of massive mag- netic ironstone and quartz, with disseminated copper pyrites, and iron pyrites. A similar vein, four or five feet wide, with the addition of blende, occurs near the extremity of the northern arm, and another, two and a half feet wide, appears in the face of the rock over the Daragh river." In the Geo- logical Map of Croghan Kinshela mountain *, it may be seen that the level driven into the mountain, nearly at right angles * See Plate XI. of the 5th volume of the Geological Transactions. That Map serves not only to elucidate the geological relations of the mountain, but also to explain the nature of the mining operations which were car- ried on there ; the whole being laid down fron! actual survey. formerly conducted in the County of Wicklow, 5 %vitli the general range of the veins, extended 178 fathoms in length. " The mineral substances obtained by these various opera- tions were subjected to experimental processes both by fire and amalgamation ; but in no instance was a particle of gold elicited from them either by the one or the other process. So unsatisfactory a result led to the persuasion, that the gold formed no part of the veins which appear in the mountain ; and hence Government were induced to abandon the enter- prise. " The same conclusion seems to apply to the tinstone, wol- fram, and manganese, in discovering which the mining opera- tions equally failed. What is, then, the primary source of these substances found detached in the beds and banks of the streams ? It is not improbable that they occur more or less scattered and disseminated through the rocks of the mountain, although it must be acknowledged that no discovery leading to this inference attended the researches of Government. It is, however, well known, that such occurrences are not un- common in other tracts* ; and we may, then, conceive the gold and the other metallic substances to have been detached from their native seat, and to have been lodged in their present position, in company with other alluvial materials, at the time of the first formation of soil, upon the retrocession of the ocean. " The total quantity of native gold collected by Government amounted to 944 ounces 4 pennyweights and 15 grains, pro- ducing when sold 3675/. 75. 11^^.; the ingots having yielded from 21 f to 2J J carats of fine gold, the alloy being silver." " § 107. But the discovery of native gold was not con- fined to Croghan Kinshela. Trials were instituted by the di- rectors in another mountain also, namely Croghan Moira, about seven miles distant from the former mountain, and gold was obtained there, though in very small quantity, the largest piece not exceeding two and a half pennyweights in weight. One trial was made on Ballycreen in the stream at the eastern foot of the mountain, and minute particles of gold were found, accompanied by magnetic ironstone, magnetic ironsand, com- pact brown ironstone, cubical iron pyrites, and numerous small garnets. Another trial was made on the western side ** * See, for example, Ulloa's Travels in South America, vol. 1 i. p. 160, English translation; Humboldt's Political Essay on New Spain; Char- pentier's Bcobachtungen uber die Lagerst'dttc der Erze; Von Buch*s Geo- gnostische Bcobachtungen auf Reisen durch JJeutschland und Italien, vol. i. p. 128—130." 6 On the former Gold-workings in the County of JVic/clow. of the mountain, in Ballynacapogue brook, and small particles of gold were obtained, with magnetic ironstone, magnetic ironsand, and fragments of tinstone crystals. A third trial, in Fannanerin stream, on the north-eastern flank of the moun- tain, did not produce any gold." The preceding extracts may serve to show how untenable are the positions advanced in the statement of the distinguished President. They prove, 1. That the discovery of native gold was not confined to a limited portion of a stream of about one hundred yards in extent. 2. That gold was found in other parts, and in other streams proceeding from the same mountain ; and indications to this effect are given in the Map of Croghan Kinshela mountain. 3. That the mountain was not perforated through, nor with great labour, nor at an enormous expense. And the sole object of a level, which was driven 178 fathoms long into the mountain, was to try in depth veins found by trenching at the surface. The moderate expense of such a level passing through a clayslate rock (and that rock dipping toward, and thus fa- vouring the progress of the level,) any competent miner may readily compute. 4. That the adjacent banks and hills rising on either side were attended to ; for in fact the mountain was scarified by trenches on every side, cut down to the live rock ; and many veins were discovered in consequence, upon which workings were conducted, and the produce subjected to repeated ex- periments. These veins were thus explored at the least ex- pense. With this plain statement of facts I forbear comment, merely adding that I am persuaded that few, if any, Government undertakings have ever been conducted in a more oeconomical manner, or at so little charge to the public. The great ob- ject was to determine, according to the approved principles of practical miners, whether auriferous veins existed in the mountain ; and the negative appearing to be proved. Govern- ment were induced to relinquish the enterprise. Prior to the operations of Government, my friends had joined me in a pro- position to embark in the adventure ourselves, yielding to the Crown one fifth part of the produce. Apprehensive that the subject might become too rich, that proposition was rejected. The result of the subsequent researches, however, gave us reason to be content that our offer had not been accepted by Government. [ 7 ] 11. 0?i Olbers's Method of determinifig the Orbits of Comets. By Professor Encke.* "P|R. OLBERS had the kindness to transmit to me last -*-^ year the preceding paper, and to permit me to insert it in this volume. It completes his excellent method of deter- mining the orbits of comets in a point in which greater facility and a quicker approximation to the truth by trials more me- thodically conducted had been desired by many of those who employed this method, and it finishes in some measure the structure, the foundation of which Olbers laid in the year 1797. Dr. Olbers supposes in this addition that his readers are ac- quainted with his former formulae. Although this method was already carried to such a degree of perfection in the first me- moirf , that even the master-hand of the author of the Theo- ria motiis,&cc,, made no essential alterations in it, but only some abbreviations, and although consequently the subject might be considered as exhausted, yet I hope I shall not incur cen- sure if I once more completely explain this method of Dr. Olbers. The development of the necessary formulae will give rise to some remarks, partly regarding the accuracy of the me- thod in general, partly referring to the above important ad- dition lately made to it by Olbers himself. Those remarks are indeed not new, but their being placed together in this publication will, I trust, by no means be deemed improper. Lambert's theorem is a main part of Olbers's method. The manner of solving it given by Olbers admitting of some abbre- viations, I shall begin with explaining this little improvement. Denoting two radii vectores of a parabola, the minimum focal distance of which is q, by r, 7', the corresponding true anoma- lies by V, t/, and the interval of time between them by t, the double area described will be by Kepler's laws, ¥ = Kt\^2q, where K is the constant of the Theoria motus : K = 00172021. On the other hand we have F = rr'^ d v, from u = v, to y = i/. Substituting for r, its value, r = — ^t—x ° cosju^ • Encke's Jahrbuch for 1833, page 264. •f Memoir on the most easy and most convenient Method of Calcu- lating the Orbit of a Comet, by William Olbers. Weimar 1797. 8 Prof. Encke on Olbers's Method of we have, by integration, (1) K/^2 5' = 2(^-(taniiy— tan|u+itanjT/^-Jtan|i;»). Introducing instead of the true anomaly and the smallest distance, the radii vectores and the chord between, we obtain the important advantage of exterminating the elements of the parabola from the equation. In applying the method of co- ordinates it is likewise more convenient to retain distances only in the formula. With this view put (2) 7/ - i; = 2/, and taking Jc for the chord, we have ^2 __ ^2 ^ ^/2 __ 2 r y-' cos 2y5 or = (r + >*')^ — 4 ri*' cosf^ ; consequently 2 cosy* -v/ ;•?•' = + -v^ (r 4- r' + k) {r + r' — k). For the sake of brevity we make r + r^ -\- k = m^ r -\~ r' — k = n^, which is permitted in every case, even with the condition that m and n shall be always positive ; and we have ^2 cosy \/ r f' = + m 71. The upper sign for which, jT ^ 90°, i/ — w Z 180°, may be almost considered as the only one here to be referred to. By the same quantities sinj^may likewise be expressed. For we have siny^= sin 4 (i^ — v)^ = cos i x)^ + cos 4 ^^— 2 cos K'd—v) cos 4 u cos J t/ ; and by introducing r and t^ we obtain sin/2=^+-5^~2y , r + * ' — 2 cosy V^ r ;•' and by substitution from (4), (5) 2 sin/-/^^'' = (m + w) >/ 2 </, where, as above, the upper sign is to be taken ^oxfz. 90°, and the lower one i'ovfy 90°. Separating in (1) the factor contained in the part on the right-hand side we have, Y =:Kt A/Oq^q,q'^ (tan | i/ — tan i tj) (1 + J tan i t/2 + J tan i i/ tan J t; + J tan J v^) ; and placing in this equation instead of 1, its value deduced from this equation, determinitig the Orbits of Comets. 9 1 + tan i t/ tan J « = ^(^^ = ?£!Z^^, ^ ^ COS 4 ^ COS t w q and likewise for the one factor its value, viz. tan i t/ - tan 4 . = ii^^^ ^smyVr_^ ^ COS ^ V COS 4 w ? we obtain .,3 (6) F = Kjf V 2^=2 rr'sin/ cos/ + fi^^''sin/^ or after substitution from (4) and (5), KtV2q=^±i(m + n)mnV2q + i ^^L^^ V q, therefore 2 K ^ = + (m + w) ?72 w + J (m + w)^ = J (m^ + w^), or (7) {r + 7^ -\-k)^+ (r -^rf — k)^=z6Kt. This equation is solved by trials in Olbers's method. Ap- proximate values of k, r, r' are substituted until the equation is satisfied. As it is evidently quite indifferent what form is given to it, we may just as well deduce the value of one of the variable quantities from it. The powers being expanded in series, the odd terms will destroy one another, (having for the moment only regard to the upper sign,) and we shall have 6K^=3>t(r + r')--i^F(r + r')---^||^P(r-|r')""&c. or _1K^_ _k i_ ( ^ y 1.3.5 / k y (r -{- r')^'' r + r' 4.6 V 4- r'/ 4.6.8.10 V + r'/ k Reversing this series, so that the value of -; -j is de- termined from it, and putting for brevity (8) :, = yj, we shall find (r + 7-')^ r + 7^ ' ^ 24 ' ^ 384 ' 9216 or (9) ^=VFT7r)V +24 ''' + 3-84 ^' + 92r6^ +•••/• * Lambert's Insigniores Orbitcc Cometarum ProprietateSf p. 63. Third Series. Vol. 7. No. 37. Juli/ 1835. C 10 Prof. Encke on Olbers's Method of If we consider the intervals of time as small quantities of the first order, >j will be of the same order. The factor of '/ J, . — n ^^^^» consequently, differ from unity only by quan- tities of the second and higher orders, and may be conveniently represented in a table with the argument >j. The calculation is then very simple. The approximate value of r, r', k being 2 Kt , given, -yj -jr is first calculated, from which ij is imme- diately derived, and this gives by the table immediately the second factor. The comparison of the value of k thus found, with the one assumed, will prove how far that value is correct. Tlie value of this factor, which I will designate by jx, so that (10) ^=l+l,^ + A,. + Jf^,e... may also be expressed by a finite formula, by which the calcu- lation of the table is facilitated, and its use immediately ex- tended to all cases. k We may always put •, = sin y, whence -j=(l + sin7)?^ + (l -siny)^. (r + r)^ Assuming, what we may assume, that y always Z 90°, we may use this formula, (cos J 7 ± sin J y)^ = 1 + sin y, and the above equation will be transformed thus, 6 K ^ (cos i y + sin I yf + (cos f y - sin i yf = —. (r + rr Taking first the upper sign, we obtain f\ \C t ecosiy^sin^y + 2sin^/ = - — -—3, (r + r)^ to which this form may be given, . /sin ly\ ./sinlyy^ 6Kt •'V 'v/2 / \ ^/2 ) 2i(r + 7^)^* Lambert's formula shows at once, as ^ Z (^ + ^)j that in both cases also 6K/Z2^(r + t'^y^. Making, therefore, -; ^ = sin 5, determining the Orbits of Comets, \\^ we get from the last equation • sin ^ y = sin ^Q V2. Of the three roots of the cubic equation, one only fulfills the two conditions, that sin i y ^ sin 45° or Z J V 2, and is positive, viz. the one for which Q has been taken Z 90°. For the lower sign we have this form : /cos Jj/\ __ /cos lyy ^ 6Kf \ V2 J \ V2 J 2^{r -f /^)^' or cos ^ y = sin ^ 5 V2, Here sin ^ d must be Z.\ "^ % and at the same time cos ^ y 7 ^ ^^ 2. These two conditions are only fulfilkd when 5 is between 90° and 135°. It follows from this that when the equation 6 K^ — ; = sm Q, 2^ {r H- r')^ gives a value for h which is /_ 45°, for the same data only one solution is possible, for which xf — v /, 180°. But when ^7 ^5°^ there are two solutions, 5 and 180° -- 5, the latter of which answers to t/ — t; 7 180°. The value of fl may be thus written : and as we have both from the value of sin ^ y as well as from that of cos|^ y sin y = 2a sin ^ 6 a/ cos f fl, and likewise yt = (r + r') sin y, we have the following complete system of formulae : _ 2Kt (11) ^"^^=V8 a = J— I— V^ cos I d ^ sm d ^ , __2K^ ^ - V (r + r') '^^ log2K = 8-53661 U. where By means of these formulae, log jx has been calculated in the C2 12 Prof. Encke o?i OlbeiVs Method of annexed table for all values of >j, from >j = 0 to »j = 3^ V 8, in- creasing by hundredth parts of unity. The logarithms by which these calculations were made being only correct to 7 decimals, the last figure may be wrong one or at most two units, which, however, will have the less influence, as Lam- bert's formula is in this way applied with much greater ac- curacy than in the common way. In the latter, as k is small in comparison of r -f r', a small quantity is obtained as the difference of two greater ones, while there is no such subtrac- tion in the use of the table. The case of r/ — t? 7 180° has not been taken into consideration, as the interpolation between the values of logjx would then be too tedious. Besides, in these rarer cases there is not the same disadvantage in following the common process. Lambert's formula shows most clearly how many observa- tions are required for determining the parabolic orbit. If the distance of a comet from the earth were known for one obser- vation, its place in space, consequently likewise its place with re- gard to the sun, would be given. Every observation which is used introduces, therefore, one unknown quantity ; for two ob- servations two radii vectores and the connecting chord k might be expressed by two unknown distances, but only one equation between r, r' and k would be derived from them. But three observations would give three equations for three unknown quantities, by combining every two of the three radii vectores; this would be consequently sufficient for solving the problem. This application would satisfy one of Kepler's laws, viz. that the times are proportional to the areas described ; but it is quite independent of the second equally important law, that the comet must move in a plane passing through the sun. This latter law gives for the three places of the comet an equation of condition, so that in three complete observations there are four equations with three unknown quantities. Although, there- fore, two observations are not sufficient for determining the parabola, yet three observations are more than enough, and it will be necessary either not perfectly to satisfy one of the data contained in the three complete observations, or only to fulfill one condition resulting from a combination of two data while the other is entirely neglected. Olbers's method is founded on the condition that the three places of the comet shall be in one plane with the sun. It de- termines from it with a close approximation the ratio of two distances from the earth, and consequently also two places of the comet in space, expressed by one unknown quantity. This ratio applied to Lambert's equation between the places and the time would give, if the substitution were actually made, one determining the Orbits of Comets. 13 equation with one unknown quantity, but of a very high de- gree, which would have to be solved by trials. It is therefore preferred to solve the system of those equations by trials. If we denote by x, z/, ar, the coordinates of a point referred to the centre of the sun as the beginning of the coordinates, we have the following three equations for three points in the same plane passing through the centre of the sun: Ax+By+Cz=0 A ^ + By + C s' = 0 A ^'^ + By + C «" = 0. The equation of condition for these nine coordinates, after the ratios-^ and —r- have been eliminated, may be repre- sented in different forms useful for future application. In or- der to find the factors by which each of these equations must be multiplied with a view to eliminate B and C, by which means A likewise disappears, we have to combine these two equa- tions : py -\- 2?'y + p"y = 0 p 2; + 2>' »' + p" a" = 0. We obtain from them p proportional to y 2' •— y^ z" jjf 3/ s" — j/" ^ ptf y^z —yz!, or as these latter values are the double areas of the triangles between the beginning of the coordinates and the points y z\ yf^ ^", &c., respectively in the plane of the y ^, and as these again are proportional to the areas of the triangles in the plane of the orbit, these factors become proportional to [/r"], — [r r"], \r r'], if these symbols stand for the double areas of the three triangles, viz. sun, second and third places of the co- met ; sun, first and third places of the comet ; sun, first and second places of the comets. The negative sign is introduced in order to avoid negative areas, in as much as all areas are sup- posed to originate from the turning of the first radius vector always in the same sense. Hence the equation of condition has three forms : \f r"] J? — [r r"] x' + \t r'] ^" = 0 (12) [r' /'] j/ - [r /•"] y' + [r r'] y'' = 0 [r' r"] ^ — [r r"] ^' + [r r'] ;s" = 0. Let us now designate The three curtate distances of the comet from\ , ,, the earth, by J §9 §9 g 9 1 4? Prof. Encke on 01bers*s Method of The three observed geocentric longitudes of\ , „ the comet, by .......... J ' ' ' The three geocentric latitudes of the comet, by . S, 8', 8", The three longitudes of the sun, by ^j ®'> ^"> The three distances of the sun from the earth, by R, R', R", The three times of the observation, by .... ^, t'y ^", Hence we have X = g cos a — R cos 6 3/ = § sin a — R sin 6 z = § tan S and the analogous equations for the six other coordinates. Substituting these expressions in (12) we get, [/•' r"] {q cos a — R cos 6) — [r r"] (§' cos u' — R' cos 6') + [r r'] (g" cos a"— R"cos S") = 0 [r' r"] (fsin « - R sin 6) - [r r"] (g' sin a' - R' sin 6') + [r r'] (g" sin a" - R" sin 6") = 0 [r'r"]gtanS~[rr"]^'tan5' + [rr'] g" tan S" = 0. 