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Se Wer ane co & ed hn AS - ow igi ; Ye eRe aN \ M wy yi WW SEC fe8 We 43) ww SX . wis a RN ee ane oy “yy WW Guts yen seiner “cin oem OS i ene S, a] Op POF ON ies \ MW Wiad “wut WS Saunt pe ae PN iyo we Ss ey ‘ add SU AN Ww aS a) wy Sy YN JV i aay aeiy wg’ wW WAS WY Www 9 Gow) Vigihe yew TE > Ne Se Be Red WW SNS iri AMEN SS ST d Wi; t Ce Sere N SSS a AARNE Bree ye y y wore We ci He Ww | My ¢ Deed ie awe Ves Se ee ben SW AY WY iW rare v IN a) Ww ve wi We VO vn Neal N30" WV oe ne Sere) Wy = 3 ive ni Nees wees ns a) > AAR hd or Sas yw eee ma Sas pa ' Wl : at ait va ‘yh 14 if iy ye!’ NW WN Wi 4) WSR AY je iW ai: SNe ; = ju Hw. Sa < W ak 25 Teh , Ne wee ae AM | ae Sey rial NS psa a a Na i) ma ary Ne < ae wy Vee (CO £é ° N Ny \ SN ye (Ee wil Wetec ia “ &&e Ss, iN yin: Vivian NS : & a eang Sr DR et Eg" ju WINS es eee oe can Y. ANY wry Sy “oh Wt a Sea ent ne er \y Wis, YY ia y, Sew Vad , 7 yA ce y TRANSACTIONS i oF nie CAMBRIDGE PHILOSOPHICAL SOCIETY. ESTABLISHED Novemser 15, 1819. eae -. VOLUME THE SECOND. —_»—— CAMBRIDGE: PRINTED AT THE UNIVERSITY PRESS ; AND SOLD BY sy. & J J DEIGHTON, AND T. STEVENSON, CAMBRIDGE; AND T. CADELL, STRAND, LONDON. M.DCCC.XXVII. THANSACTEONS eared \ CAMBRIDGE PHILOSOPHICAL SOCIETY. Vou. II. Parr LI. Ne. I: XII. XIII. CONTENTS OF Vou. II. Parr I. —>_ PAGE On the Distribution of the Colouring Matter, and on certain peculiarities in the Structure and Optical Properties of the Brazihan Topaz: Osh), Dike, RUMEN Ma Ase ach, Se en Ge as Nee OER 1 On the Rotatory Motion of Bodies: by W. WueEweEit, Esq............ 11 On the Phenomena connected with some Trap Dykes in Yorkshire and Durham: by Professor SEDGWIOCK ... 0.02... eeeeeeeeee eee ces ee. 21 A new Demonstration of the Parallelogram of Forces: by J. Kine, Esq. 45 On the Developement of Electro-Magnetism by. Heat: by Professor ELECTS dot OSE aire ae ge fe PC li apna 47 Extract from a Memoir on a Peculiar Connexion which exists between the Magnetism evolved by a Single Galvanic Combination, and the relative Magnitude of’ the Opposing Surfaces of that Combination: PE Nap SURE HORN SERED, aly. Shas cee oc oy... 77 On an Apparatus for Grinding Telescopic Mirrors and Object Lenses : Bee RWW, Cacti, ott on tales. ee 85 On the use of Silvered Glass Jor the Mirrors of Reflecting Telescopes: by G. B. Airy, Bethe scat sa tate aa: Mee Ur tice eye thee kee 105 An Account of some Experiments made in order to determine the Vi elocity with which Sound is transmitted in the Atmosphere : by Dr. GREGORY 1 19 On the Association o Trap Rocks with the Mountain Limestone For- mation in High Teesdale: by Professor MEDGW Knees 139 On the Angle made by two Planes, or two straight Lines, referred to three oblique Co-ordinates: by W. WHEWEIL, Esqie. soc. oc- osc... 197 On the F: 7gure assumed by a Fluid Homogeneous Mass, whose Particles are acted on by their mutual Attraction, and by small extraneous Forces : by G. B. Airy, | SET cB Ger a ae i SE a 203 On the Determination of the General Term fa New Class of Infinite Series: by C. Banmaa, Eeq......0s0:. 0s. qbee"0 SoLnbccerDoee 218 Qo x ADVERTISEMENT. Tuer Society as a body is not to be considered responsible for any facts and opinions advanced in the several Papers, which must rest entirely on the credit of their respective Authors. NeXT. XV. XXII. XXIII. CONTENTS Vou. I. Parr Il. On the Principles and Construction of the Achromatic Eye-Pieces of Telescopes, and on the Achromatism of Microscopes: by MGcaP ID PANLEOL PREIS alate Pare Pat A chetecvaye's minic = tdeWe'e 2 2) wie ere) erat An account of a Whale of the Spermaceti tribe, cast on shore on the Yorkshire Coast, on the 28th of April, 1825: by James ALDER- SONS pbs piss Avaleslaieialeia(elnlsi sin ahelaielsfaiw’= elim nisl sl auxiecicie . On a peculiar Defect in the Eye, and a mode of correcting it: by Ge Re OE heeo.cun nan OA Ge Cece cere e de cccevsecses A general Demonstration of the Principle of Virtual Velocities: Gyitie Revel. POWER). che cale(s isis « avelatelalee (sais shel stalevele On the Forms of the Teeth of Wheels: by G. B. Arry, Esq..... Observations on the Ornithology of Cambridgeshire: by the Rev. MUN NSS ciaiehioss ein Ete nerieysinyecoreie anece ee ee ed On the Influence of Signs in Mathematical Reasoning: by CHARLES BaBpace, Esq.....-...-- S56. PE Rae Re Pee: On Laplace’s Investigation of the Attraction of Spheroids differing little from a Sphere: by G. B. Arry, Esq.........-000-+ On the Classification of Crystalline Combinations and the Canons by which their Laws of Derivation may be investigated: by the Rev. W. WHEWELL......4. Reasons for the Selection of a Notation to designate the Planes of Crystals: by the Rev. W. WHEWELL.....00-.05 005 aveiciene PAGE 227 253 325 380 391 427 Contents, Vou. If. Parr II. Exrracts from Communications made to the Society : PAGE I. Onan Artificial Formation of Plumbago: by J. Auperson, Esq. with Remarks, by Professor CUMMING++ ee ese ees cee eeeeeereenees 44 II. Experiments on Percussion: by B. BEVAN, Esq..seeseeeeessereeees 444 III. Computed and Observed Variations of Magnetic Intensity: by R. W. Roru- MAN, Esqe. ssc cs cccerceccesescveneseceres cane ast. 00 td x44 Donations to the Library.......+. SBRA GI ae Bist aie Geicieisae sa ieuine sik, a AAO Miatseiiine a iolaccetis braula nieieletatatele) gisielivlereivie # ereierstkie ) <400 I. On the Distribution of the Colouring Matter, and on certain peculiarities in the Structure and Optical Properties of the Brazilan Topaz. By DAVID BREWSTER, L.L.D. F.R.S. L. & E. AND HONORARY MEMBER OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. [Read May 6, 1822.]} In the early experiments which I made upon Topaz,* I em- ployed some very fine plates of the Blue Topaz of Aberdeenshire with the natural polish of their cleavage planes. ‘The colourless Topazes of New Holland, which I subsequently examined, dif- fered very slightly from those in their optical properties; and in comparing with them the Topazes of Brazil, I proceeded no farther than to investigate the absorbent power of the latter, and the changes which were superinduced by heat on the matter with which they were coloured.+ The Brazilian Topazes which I used in these experiments were all crystallized, and did not exhibit any of the phenomena which I have since discovered during the examination of a large quantity of these crystals, which accident put into my possession. RAE Oh ie eee ES 1 Eee Bae ii) ee) 2 * Phil. Trans. Lond. 1814. p. 202. + Id. Id. 1819. pp. 19, 125, &e. Vol. If. Part L. A 2 Dr. BrewsTER on the Optical Properties I. On the Distribution of the Colouring Matter in Topaz. In order to obtain results that might be considered as general, I ground and polished the flat summits of a great number of Brazilian Topazes, and having exposed them to polarized light, I observed the following structures. 1. The internal part of the crystal was almost always of a different colour from the external part, as will be afterwards more particularly described, but the pink colouring matter was confined to two small rhomboidal prisms placed at the acute angles of the prism of Topaz as shewn in Fig. 1. When the shorter diagonal of the rhomb is in the plane of primitive polarization, these small rhomboidal prisms, which are of a pale pink tint by common light, assume a deep and brilliant pink colour, and they become of a faint blue approaching to a Lilac colour when the longer diagonal is m the same plane. In some specimens this complementary tint, in place of beng Lilac, is yellow of different characters; but in every specimen which I have examined, the pink, whenever it exists, is the colour which appears when the shorter diagonal is in the plane of primitive polarization. 2. In some crystals the red colouring matter is confined to a triangular prism as in Fig. 2. sometimes occupying the apex of the acute angles of the outer rhomb, and sometimes the same angles of the inner rhomb. In the specimen represented in Fig. 2, about one half of the pink portions becomes lilac before the other half, and before the longer diagonal comes into the plane of primitive polarization; thus indicating a species of hemitropism, or a want of parallelism in the principal sections or neutral axes of the different portions of the thick prism. This effect is shewn in Fig. 9, where no fewer than four different colours appear at the same time. of the Brazilian Topaz. 3 3. In some specimens the colouring matter is deposited on the faces of a rhomb parallel to the natural faces of the prism, as shewn in Fig. 3; and this space sometimes extends outwards to the natural faces as in Fig. 4, and sometimes inwards to the centre, as in Fig. 5. 4. The pink colouring matter frequently occupies only a part of the external rhomboidal space, as in Fig.6; and at other times it is arranged as in Fig.7, both the obtuse and the acute angles being free from the pink colour. 5. In some crystals which are very perfect, the pink colouring matter is uniformly distributed as in Fig. 8. 6. Among the mass of Topazes, amounting to some thousands, which I inspected, there were only éwo which possessed a decidedly green colouring matter. These crystals are represented in Figs. 10. and 12. In Fig.10. the outer portion is of an orange yellow colour, and the inner prism is a mixture of green and pale pink. In Fig. 12. the outer portion is a mixture of green and pale pink, and the inner portion a light orange yellow. When the longer diagonals of both these prisms are in the plane of primitive polarization, the green tint is a maximum as shewn in the figures. For the sake of comparison, I have represented in Fig. 11. the distribution of the colouring matter im some of the finest and most valuable of the Scotch Topazes. II. On the Tesselated Structure of the Brazilian Topaz, and the singular Superposition of its external Lamine. In the year 1819 Mr. Herschel informed me that he had observed in some Brazilian Topazes a hemitrope or tesselated structure, “It is a structure he remarked which never occurs “in fine crystals; but when properly examined, inferior spe- *“‘cimens often present the phenomenon represented in the figure “‘ (Fig. 13.), the principal sections of the different parts making A2 4 Dr. Brewster on the Optical Properties “angles of 20°+. In these the central portion is often of a “high yellow colour, and manifests the phenomenon of absorp- “tion of polarized pink and yellow rays alternately, while the “‘ external border is nearly colourless in all positions, and pre- * sents no such phenomenon.” The tesselated structure is so common in the Brazilian Topaz, that I am more disposed to regard it as an _ essential character of that mineral, than as an accidental formation. Out of the great numbers which I have examined, there is not one in an hundred which is free from this hemitropism; and in some very fine crystals, in the examination of which this structure had escaped my notice, I have since detected it by more careful methods of observation. In these cases, however, the crystal is rather to be considered as a compound than as a _ hemitrope crystal, for the separated portions have their principal sections much nearer to coincidence, than in less perfect specimens. The hemitropism of the Brazilian Topaz is of a very sin- gular kind. The tesselae are not turned round one half or any determinate portion of a circle, as this term implies, but the principal sections of different laminz form different angles with one another, and hence we may distinguish these specimens by the more appropriate name of polytrope crystals. This curious formation, which has not hitherto been observed in doubly refracting structures, will be understood from Fig. 14, where ABED, CBEF are the two external tessela at one of the obtuse angles of the rhomboid. If we suppose that these tessela are divided into four lamine, 1, 2, 3, 4, and that MN is the principal section, or one of the neutral axes of the central portion of the crystal contiguous to DEF, then the laminz 1, 1, have their principal section in the direction aa’ forming a very small angle with MMW; the lamine 2,2 have their principal section in the line 64’, and so on to the superficial lamine 4, 4, of the Brazilian Topaz. 5 which have their principal section in the direction dd’ inclined from 10° + to 22° + to MN, the inclination differing in different crystals. The lines aa’, bb’, cc’, dd’ are also the principal sections of the corresponding laminz on the side NC. In like manner the principal sections aa’, 86’, v7, 60 of the lamine in BCFE are the principal sections of the corresponding laminz on the other side AN. As the laminze however are infinite in number, the principal sections have every possible direction between MN and dd’. III. On the Optical Structure and Properties of Brazilian Topaz. Having found that the Brazilian Topaz exercised a super- ficial action upon light different from the colourless Topaz ot New Holland, I was induced to compare the relative intensities of their polarizing axes. In the blue Topaz of Aberdeenshire, and the colourless Topaz of New Holland, the inclination of the resultant axes of double refraction is about 65°, and the system of coloured rings round each axis deviates from the tints of Newton’s scale, the red ends of the rings being outwards*. In the Topazes of Brazil, the inclination of the resultant axes varies in different specimens; I have found it in some crystals 50° 5’, and even so low as 43°; and, what is very remakable, the one resultant axis is often more inclined than the other to the natural surfaces of the Jaminz, an effect which no doubt arises from the peculiarities of crystallization already described. In one specimen where the axes formed an angle of 50° 5’, the one axis was inclined only 22° 37’ to the axis of the prism, while * In the Phil. Trans. 1814, p. 204. and Plate VII. Fig. 1. I have described the order of the tints, from which it will be seen, that the rings are red at one end and blue at the other, and that the colours do not originate at the centre of the rings. 6 Dr. Brewster on the Optical Properties the other had an inclination of 27° 28’. The tints of the Brazilian Topaz deviate more than those of the other crystals from the colours of Newton’s scale, and are produced by a polarizing force of inferior intensity. The phenomena of the coloured rings in Scotch Topaz which I have described in the Phil. Trans. for 1814, are neither seen with the same distinctness, nor to the same extent, in the Topaz of Brazil. This circumstance arises from the difference in the inclination of their resultant axes. In the Scotch Topaz the angle formed by their axes is such, that when light incident along the one axis is reflected in the direction of the other, the angle of reflection is almost the same as that of maximum polarization for Topaz, and hence the rings appear with peculiar brilliancy, and exhibit themselves under the new modifications which I have distinguished by the names of the third and fourth Set* in the Paper already quoted. In the Brazilian Topaz, however, the inclination of the resultant axes deviates considerably from twice the polarizing angle, and consequently the preceding phenomena are very indistinctly displayed. The Brazilian Topazes are in general phosphorescent when placed upon a heated iron, although I have found several, especially among the finer crystals, that do not possess this property in the slightest degree. The tesselated crystals display their phosphorescence in a very singular manner. Sometimes it is of a rich orange red colour, and in many cases the external border is phosphorescent, while the interior nucleus discharges no light at all. This phosphorescent light is, in general, most brilliant in the outer lamine, though I have seen in some crystals the greatest intensity of light at the boundary of the * The fourth set of rings which has a peculiar character, is a combination of the first and second set, or of the direct and complementary systems. of the Brazilian Topaz. 7 nucleus and the external tessele. In one specimen, a faint phosphorescent glow appeared and vanished at intervals in the nucleus, while the light shone in the outer border with a bright and permanent lustre. IV. On Substances found in the Brazilian Topaz. In a very great number of Brazilian Topazes there is a white pulverulent substance, which must have been formed at the same time with the mineral, as the most powerful Microscopes cannot detect any aperture or crevice through which it could have been admitted. Upon breaking up the specimens where it occurred, and examining the surfaces of the laminew between which it had lain, I found that they had been acted upon by the substances, as they exhibited that superficial disintegration which is produced by the action of a solvent, and which is identically the same with what is found on many of the summit planes of the crystal, as shewn in Fig. 15. This circumstance proves in the clearest manner the contemporaneous formation of the white powder. M. Berzelius, to whom I had transmitted some specimens of this substance had the goodness to analyse it for me, and found it to be a sort of Marle consisting of silex, alumina, lime, and water. ‘This substance, he remarks, if it were crystallized, would belong to Cronstedt’s family of the Zeolites.” From the frequency with which this matter occurs in the Brazilian Topaz, and the imperfect character of the crystallization which accom- panies it, I cannot help thinking, that it is nothing more than the uncrystallized ingredients, and that lime is one of the con- stituents of the mineral*. a NUON IME A A ee * When the crystal is placed upon a hot iron, the white portions are more phospho- rescent than the other parts. 8 Dr. Brewster on the Optical Properties In many specimens of the Brazilian Topazes, I have observed another substance of a very singular kind. It is of a brilliant red colour, and in general perfectly transparent. Some- times it appears in thin plates between the lamine, and some- times in long stripes parallel with the axis of the prism. By holding the neutral axis of the crystal in the plane of primitive polarization, and examining these red portions with polarizing Microscopes, I have found parts of them crystallized so as to produce four sectors of light round a black cross. When the crystal is broken, the surfaces of these red films have a high metallic lustre like Realgar or Cinnabar; but I have not been able to collect enough of the substance to determine what it is. V. On the probable Difference in the Chemical composition of the Brazilian and other Topazes. When I had ascertained the very marked difference between the optical properties of the Brazilian Topazes, and those of Saxony, New Holland and Scotland, I could not for a moment doubt that a difference would be found in their chemical com- position. I accordingly sent specimens, that I had examined, to M. Berzelius, with the request that he would favour me with an analysis of them. This distinguished chemist, however, had previously analysed other specimens of the same minerals, and he informed me, “that his analyses gave exactly the same re- “sults for the New Holland and Brazilian Topazes, with the “exception of a small quantity of hydrate of iron, with which “the latter was coloured, and which, when decomposed by heat, “gave the fine pink colour to burned Topaz in consequence of “‘ the oxide of iron being set at liberty.” High as M. Berzelius’s authority undoubtedly is, I cannot avoid placing the most unbounded confidence in the general principle, that every difference in optical structure is accompanied of the Brazilian Topaz. 9 with a difference in chemical composition; and I am therefore disposed to ascribe the present apparent exception to the imper- fection of chemical analysis. This opinion, indeed, may be placed upon a firmer basis by a reference to some experiments of that excellent Chemist, the late Rev. William Gregor, which have recently come to my knowledge. In a letter to the Rev. John Rogers, dated Nov. 21, 1811, he states that some pieces of Topaz from St. Michael’s Mount consisted of silica and alumina in the proportion of 3.5 to 3.1, and a small proportion of lime. ‘‘The proportion of ** silica and alumina,” he adds, “‘ does not agree with that in the *“Saxon Topaz; but still I see a great difference between the ** Saxon and Brazilian Topaz, as to the relative proportion of “those ingredients. In neither of them do I see any lime re- ““ cognized; but I must say, notwithstanding the analysis of * Klaproth, that I extracted a portion of Lime from the Bra- “‘ zilian Topaz, by means of a simple acid.” In another letter to the same gentleman, dated April 24, 1812, he says, that “subsequent experiments confirmed him in. “the opinion, that Brazilian Topaz contains a small proportion “of Potash; and Dr. Paris, from whose biographical Memoir of Mr. Gregor these particulars are taken, states that the ex- periments by which he established this fact were the last he performed ; and that he himself was enabled to bear testimony to the presence of crystallized alum, which he saw Mr. Gregor produce by the action of sulphuric acid upon the pulverized ‘mineral. In order to avoid every possible source of error, the mineral was reduced to powder in a steel mortar, and the sul- phurie acid employed was carefully tested, and found to be perfectly free from any impurity. Edinburgh, March 4, 1822. Vol, Il. Part I. B aut b> scidson ing. 5 0 el Bec ; - jor duisoqang ELS ° wail Dato a etre: separ ge ertigel ks 7 Fe caltanyng svinkss. oto ey 308 eth Conan See ha aR 4 citi ; _ 7” ; 7 i t * ® 7 Pe | i Lransactions of the Cambrulge Fh. Soc: Vol. 2. Fb.7. 6 Scharf Lithog. Printed by Rowngy &Fozfier. II. On the Rotatory Motion of Bodies. By WILLIAM WHEWELL, M.A. E.RS FELLOW OF TRINITY COLLEGE. [Read May 6, 1822.] Tue mechanical Problem of the rotation of a body of any form under given circumstances, is one of some difficulty. It is remarkable, not only for the errors into which Mathematicians of great eminence have been led in treating it, but for being almost the only instance where there has been a permanent difference of opinion among writers, with respect to the results of our elementary mechanical laws in a particular case. In fact, it seems to present the most striking impeachment of the cer- tainty of mathematical investigations which can be found, since the opposite conclusions are not obtained by an abstruse and complicated process, but arise immediately from different methods of applying the same fundamental principles. It may, therefore, be of service to solve the question in a manner which reduces it to a class of problems about which no doubt was ever enter- tained, and such is the object of the present paper. It is easily seen that the motion of a body in any manner whatever about a centre, leads to considerations somewhat com- B2 12 Mr. WHEWELL on the plicated. At any point of time the body may be conceived te be moving about some axis or other; but the position of this instantaneous axis, as it is called, both in the body and in fixed space, may be perpetually varying, as well as the angular velocity about it; and the forces exerted by each particle will vary with these changes. The first solution of the problem, taken thus generally, is due to Dalembert, who published it in 1749 in his Researches on Precession and Nutation. Euler, in the Berlin Memoirs shortly after, put the solution in a more symmetrical and simple form, acknowledging at the same time Dalembert’s prior claim to it. The equations however to which the con- ditions of rotation are reduced seem to have first appeared in the form in which they are now generally presented, in the Berlin Memoirs for 1758. The same subject was pursued in other Memoirs, and more extensively in Euler's ‘‘ Theorta Motus Corporum Solidorum et Rigidorum,” which appeared in 1767. Lagrange, in the Memoirs of the Berlin Society for 1773, con- sidered the subject on principles more general than his prede- cessors. The results, however, of all these different authors, as well as of Frisi and others who afterwards treated the question, agreed, though obtained by a variety of methods. In the Philo- sophical Transactions for 1777, Mr. Landen gave “A new Theory of the Rotatory Motion of Bodies, affected by forces disturbing such motion.” In this Memoir, he expresses himself dissatisfied with the explanations which he had seen on the subject; but it does not appear that he was at that time acquainted with the researches of Dalembert and Euler, and he has not examined particularly the cases in which his conclu- sions are at variance with theirs. He afterwards read the solutions of his precursors in this path of enquiry, and con- vinced himself that they were false; and in the Transactions of the Royal Society for 1785 he stated this, and gave his own Rotatory Motion of Bodies. 13 method applied to the case of bodies of any figure. A leading difference in the results was this: that while according to Euler, Dalembert, &c. in an irregular body not acted on by any extra- neous forces, the angular velocity about the instantaneous axis is variable, as well as the position of the axis itself; Mr. Landen found, that the angular velocity is constant, and that the instanta- neous axis in changing its place in the body, can assume only a series of positions, for all of which the rotatory inertia is the same; so that the trace of the instantaneous pole upon a con- centric spherical surface would generally be an oval, and its projection an ellipse or hyperbola. In the second part of his “© Mathematical Memoirs,’ published in 1789, he resumed the subject, and having occasion to establish propositions analogous to those in Euler’s “ Theorta,” the most regular treatise on the subject which had then appeared, he obtains theorems at variance with those of that author, and expresses rather strongly his astonishment, that Mathematicians so celebrated as his opponents, could fall into mistakes so gross. His death, which took place shortly afterwards, closed the controversy ; but it is said that his opinions remained unaltered. Mr. Wild- bore, in the Philosophical Transactions for 1790, re-considered the subject in a new point of view, and declared against Landen ; and since that time it does not appear that any person has re- vived his ideas, and the foreign elementary treatises all proceed according to the method which Mr. Landen declared erroneous. Perhaps, therefore, it may seem that the question is already sufliciently settled, and that there is no farther necessity to prove the truth of the conclusions of Euler, Dalembert, Lagrange, &c. As, however, the subject is both important and curious, and as Mr. Landen’s mathematical talents are deservedly highly esti- mated, I may be excused for making an attempt to place in a clearer point of view the falsity of his results. The Memoir ef 14 Mr. WHEWELL on the Mr. Wildbore will not, I think, be found so simple and easily intelligible as might be wished; and the following method appears to me to possess these advantages in a great degree. It would be taking up too much time to trace Mr. Landen’s mistakes from their first principles; the general foundation of them may be stated to be the assumption which he makes, that if one force will produce the same effect as another in affecting the motion round the avis, it may be substituted for that other ; not observing that forces equivalent in that respect will not necessarily produce the same effect in other respects. The error of principle with which he charges other writers is “the resolving a force productive of rotatory motion into three forces, and considering each of these forces as acting separately on the body impelled.” (Math. Mem. xiv, p. 79). Now there can be no doubt that with respect to a single point in motion, we may resolve the forces which act upon it in the direction of three co-ordinates, and consider the motion of the body in the direction of each co-ordinate as affected only by the force in that direction; and we shall find that this reasoning will lead us to conclu- sions on the subject of rotation. The case which TI shall take is where the system is acted upon by no extraneous forces whatever. Let three lines at right angles to each other pass through a point O, and at the extre- mities of the lines, suppose. three material points m, m’, m’. Let these points be joimed by lines mm’, mm", m'm'. If now the six lines Om, Om', Om", mm', mm", m'm', be supposed to be rigid rods without weight, the system will be perfectly unalterable in form, and the whole mass will be collected at the points m, m', m’. Let this system be supposed to move any how about the point O, being left to itself, and not acted on by any force ; then the only forces which act on the poimts m, m', m’, are the tensions of the six rods just mentioned; and by resolving these, Rotatory Motion of Bodies. 15 we may obtain the motion of the system, by applying to m, m, m’, the formulz for the motion of a point. Let the three lines Om, Om’, Om’, be equal; then mm’ will make angles of 45° with Om and Om’; and the tension of mm’, which acts as an equal moving force on m and m’, being re- solved, it will be seen that the foree on m parallel to Om’ is equal to the force on m’ parallel to Om. In the same manner, we may obtain two other equations, by considering the tension of mm" and of m'm’. And these three equations, when put in terms of the quantities which are generally used in expressing rotatory motion, give us the three equations of motion. Let m, m', m", be referred to Ox, Oy, Oz, three fixed co- ordinates. The co-ordinates of m are xz, y, 2 2 U . , , ™m L,Y, &; ” m DU. Bs and it is easily seen, that these quantities are also = cos. mOx, cos. mOy, cos. mOz, cos. m’Ox, cos, m’Oy, cos. m'Oz, cos. m’Ox, cos. m’Oy, cos. m'Oz; (because Om = Om! = Om'= 1). We shall then have these equations (Poisson, Tr. de. Mee. No. 361*.); 1. O=aa + yy +52 vty +2 =1 O= an" +y7" +22"? (a). Also, 2? +y° +27 =1} (6). O=a'x" + yy" +22" wy? 4 2 =] * And in the same manner we might obtain these equations, vy + avy + ay! = 0) ate? +2 = 1 od “bh a? + a of (@’). y +y” +y"= r (0). yetyd +y'2"=0 Pat eta 1 16 Mr. WHEWELL on the The only forces which act upon the points, are the ¢ensions, or reactions of the six rods Om, Om’, Om’, mm’, mm", m'm’. And if these be resolved at each point m, m’, m’, into three rectangular components X, Y, 2, X’, Y’, 3’, X”", Y”, 3", we shall have eX, Oa Vy eee de em? ade m ” e m es, xX! Cy Ss gee gee ac): dt m? dt’ m? dt ad’ a!’ ty = dy" te y"’ diz! gi dt md?) wun 2 at m’ Now let the force which acts on m, and which is composed of X, Y, &, be resolved parallel to Om, Om’, Om". And since x is the cosine of the angle which Om makes with X, we shall have Xx for the part of the force arising from X, acting on m in the direction Om. Similarly, Yy and <2, are the parts arising from Y and &. Hence, the whole force on m, in direction Om is Xx + Yy + &z, vues. OM IS Ad -+-.Yo" 4 32’, Sreeae Om" is Xx" + Vy" + 82". Similarly, the force on m’ in direction Om is X'x + Y'y + &z, Riker waicere:s Om! is X’e + Vy + &2', isiaysncts « Om! aS Mae oe May” ees And the force on m“ in direction Om is X"x + YV'y + 8"z, A Crete 2... Om!’ is X"a! + Y"%y' + "2, ec pederee Om is Xs" + Wig as, Rotatory Motion of Bodies. 17 But the force on m parallel to Om’ arises entirely from the LI ‘ tension of mm’, and is = a Similarly, the force . tension of mm’ on m! parallel to Om is = ———7,—— . Hence, we have Xe + Vy + 32 =Xa + Vy + Ss. Similarly, X'g!' 4+- Y'y’ + 8/3" — X"z! + Y"y' + B"'z!, and Xx + Y"y + B"z = Xx" + Yy" = Sz" Or, putting for X, Y, 3, X’, &c. their values from (c), m (a'd'x + y/d'y + edz) =m (ad'a' + yd’y + 2d’) m! (ala! + yy! + 2'dx!) = ml! (dea! + Ly! meal @, m! (ada +yd?y"+ zd°2") =m (ade + yay + 2d’z) which are three equations of motion. Differentiating the first of equations (a), we have ada’ + ydy' + zdz' + adz +y'dy + dz =0. Hence, if we make* vdxe + y'dy + dz =rdt, we have xda' + ydy' + zdzv =—rdt Sim’. if 2’da'+y"dy'+2"dz'=pdt........ vd’ +y'dy"+2'dz"= =a (e) and if eda” +ydy" +2dz"=qdt ........ a"dx +y"dy +2"dz = —qdt Now, take the three equations, rdx + ydy + zdz =0 a’dx + ydy + zdz = rdt } GS). ade +y"dy + 2"dz = — qdt Multiply by 2, 2’, x’,.and add; we then have, observing equa- tions (a’) and (0’), * This notation is the same as that used by Lagrange, Poisson, &c. with very little alteration. Vol. (1. Part I. (e is Mr. WHEWELL on the dx = (a'r — aq) dt. Similarly, dy = (y'r_ — y''9) 4t, and dz = (sr — 2"q) dt. Hence, dx = (adr - a"dq + rda' — qdz") dt, d'y = (y'dr — y'dq + rdy' — qdy") dt, @z = (dr — 2"dq + rdz — qdz") dt; And multiplying these equations by 2’, y', #, and adding, observing the simplifications which result from (a), (5), and from the equations vdx +y'dy + dz =0 adx +ydy +2dz =0 (8); ada" + y"dy" fe "de!" = re) which arise from differentiating (0); we shall find o, abe +y dy +x%@2= {[dr—q(a/da" +y/dy" + 2'dz") dt} = (dr+pqdt) dt. Similarly, we should find ald?e +y"d?y +2"d*2 =(dp+qrdt) dt, ede" +ydy’+2d°3" =(dq+prdt) dt. Also, ada +ydy +2d'x' =(-dr+pqdt) dt, ada" +7 Cy" +2d'3" =(—dp+qrdt) dt, x'dx +y'd’y +2"d*x =(-dq+prdt) dt. Hence, equations (d) become m (dr+pqdt) =m (-—dr+pqdt), m (dp + qrdt) = m'(—dp + qrdt); m (dq +prdt) = m'(-dq + prat), (m + m’) dp + (m — m’) qrdt=0 “. (m + m’)dr + (m — m) pqdt = 0 (i). (m' +m) dq +(m" — m) prdt =0 Rotatory Motion of Bodies. 19 Let m+tm = G; m’ +m = B, m +m! = A*; -m—m = B— A, m—m'=C—B, m'—m=A-C; (k), which are the equations of Eulert. And hence it appears, that in this cdése the equations coincide with those which are always given for the motion of a solid body by foreign Mathematicians, and differ from the results which Mr. Landen obtained by his method. “. Cdr +(B- A)pqdt Adp + (C — B) qrdt Bdq+(4-C)prdt ll oo 0 ll The equations above obtained agree with the general equa- tions for the motion of any solid body of which A, B, C, are the moments of inertia with respect to the three principal axes. Hence it appears, that whatever the body be which revolves about a given centre, we may always take a system consisting of three material points, such, that their motion shall be exactly - similar to that of the body. Mr. Landen had observed, that it is always possible to substitute for a solid body a system of eight material points placed at the angles of a parallelepiped, whose centre is the centre of motion; but I am not aware of its having been noticed, that these may be replaced by three points. If we wished, in our system of points, to have their centre of gravity coincident with the centre of motion, we may conceive each of the lines mO, m0, m'O" to be produced, and an equal distance and an equal point to be taken on the oppo- Se ea eee ee * It is manifest that m4+-m’ or C is the moment of the system round the axis Om’; and similarly, B and A are the moments round Om and Om. + Euler, Theoria Motus Corp. Rig. Prop. 90. Poisson, Dynamique, Art. 383. c2 20 Mr. WuHEWELL on the Rotatory Motion of Bodies. site side. The three new points may of course be made to move in exactly the same manner as m, m', m’, and these three pairs of points may represent any solid body whatever. Having thus obtained the equations of motion, the deductions from them follow in a manner which has been sufticiently explained by many authors who have treated on the subject, and on which, therefore, it will not be necessary here to dilate. W. WHEWELL. TRINITY COLLEGE, May 1, 1822. III. On the Phenomena connected with some Trap Dykes in Yorkshire and Durham. By tHe Rev. ADAM SEDGWICK, M.A. F.R.S. M.G.S FELLOW OF TRINITY COLLEGE, AND WOODWARDIAN PROFESSOR IN THE UNIVERSITY OF CAMBRIDGE, [Read May 20, 1822.] Tue various phenomena presented by trap rocks have long jyoduction. engaged the attention of Geologists. Different ages have been assigned to them, founded on their union with older or newer strata, and distinctive characters have been pointed out by which it has been attempted to separate the several formations from each other. As observations have become more widely extended, many of the conclusions founded on such characters have proved to be fallacious; and it is now generally admitted, that the mineralogical composition of any system of trap rocks gives us little informa- tion respecting its antiquity or probable associations. When strata rest conformably upon each other, in such a way as to indicate a continued succession of depositions, we can immediately determine, at least, their relative antiquity, and may often adopt some natural or artificial arrangement which will greatly facilitate their description. But formations, which appear as dykes and overlying masses, afford no such facilities for correct classification ; and the only general conclusion which we can arrive at respecting Trap dykes in the coal-fields, 22 Professor Sepewick on Trap Dykes them is, that they are newer than the beds into which they have intruded. It is on this account that different observers have formed completely different views respecting the classification of certain formations of trap; each, im ambiguous cases, having adopted that opinion which happened to fall in with his favorite theory —In determining the origin of any one of these formations, it seems essential to inquire, (1) In what manner it is associated with other rocks. (2) What minerals enter into its composition. (3) What effects are produced by its presence. Satisfactory an- swers to these questions have been obtamed from so many quarters, that the discussions in which they have originated will perhaps soon terminate. It is my intention in this communication to bring together some facts, connected with the subject, which fell under my observation during last summer. Dykes and overlying masses of trap are of such ordinary oc- currence in many of our coal-fields, that they have sometimes been regarded as true members of the great coal formation. Should it, however, appear, that they have not originated in the same causes which formed those innumerable layers of sandstone, shale, ironstone, &c. which enter into the composition of the coal strata; but that they have been subsequently driven m among these beds by the irregular action of powerful disturbing forces ; we shall then be compelled to regard them, not as the subordinate members, but as the intrusive associates of the great coal forma- tion. In confirmation of this opinion it may be stated’; (1) That in many extensive coal-fields there are no traces of any beds or dykes of trap. (2) That in other places, such beds or dykes pass beyond the bounds of the coal-fields, and traverse indifferently all the newer strata which cross their line of direction. The facts presented by the north coast of Ireland afford several illustrations of the truth of this assertion. Mr. Winch, in the fourth volume of the Geological’ Trans- actions, has given many interesting details respecting the in Yorkshire and Durham. 23 dykes*. which intersect the great. coal basin of Northumber- land and Durham. They are. in some instances filled with clay and. roGnded pebbles or shattered fragments of sandstone, mixed with other materials derived from the neighbouring rocks, and their whole appearance plainly indicates the violent nature of the forces by which the solid strata have been cleft asunder... In other instances, the fissures are filled with a variety of basalt, which rises like a great partition wall through all the beds of the formation. .(Geol. Trans. vol. IV. p.21—30.) It is the opinion of Mr. Winch that these basaltic dykes never pass up into the magnesian limestone which reposes immediately on the coal strata. Thus, for example, the cliff of Tynemouth castle is intersected by a basaltic dyke which does not penetrate the capping of magnesian limestone. Every one who is acquainted with the details of English Geology. must have remarked, that our newer strata, down to the magnesian limestone inclusive, are generally unconform- able to, all the older rocks. Thus in numberless instances, more especially in the West of England, we find some of the newer strata fillmg up the imequalities, or resting on the inclined edges, of the coal measures. In all such cases, the fractures and contortions of the lower formation must have taken place prior to the deposition of the supermcumbent hori- zontal beds. Now if it appear, that masses of trap are not only the common associates of such fractures and dislocations, but sometimes the very instruments by which they have been pro- duced ; it follows, almost of necessity, that the dykes we have been describing will not generally be found among the horizontal beds which repose upon the disturbed strata. Such a‘rule as this * In the North of England the term dyke is not confined to the description of those fissures which have been filled with trap, but is extended to all the great faults and disloca- tions which intersect the strata in a nearly vertical direction. A want of attention to this extended use of the word has given rise to occasional mis-statements and false inferences. Cockfield Fell and Cleveland dykes. Dykes near Egglestone in Upper Tees- dale. 24 Professor Sepewick on Trap Dykes may, however, admit of many exceptions. For no reason can be given @ priort, why the same forces, which produced the great fissures in our coal formations, should not again come into action in successive epochs in the natural history of the earth. Accord- ingly, it is found that basaltic dykes are not confined to any particular set of strata, but may occasionally appear among the newest secondary rocks. The facts exhibited by the north coast of Ireland have been already alluded to. The great dyke which starting from Cockfield Fell, in the county of Durham, crosses the plain of Cleveland, and terminates in the eastern moors of York- shire, leads us to a similar conclusion. This dyke, which preserves such an extraordinary continuity, forms a striking feature in all the geological maps of the district. Some good general descriptions have already been given of it*. My principal object in this paper will be, to place before the Society, in a connected point of view, those facts which appear to bear on the question of its origin. I shall afterwards notice some phenomena which are exhibited in High Teesdale, and seem to throw light on the same question. A mass of trap occupies the lower part of the left bank of the river Tees exactly opposite to the entrance of the Lune. It may be traced without difficulty for three or four hundred feet, close to the edge of the water; and it at length disappears under Egglestone bank; where it rests upon, or abuts against a bed of slate clay. The prolongation of the trap to the other side of the Tees is rendered highly probable by the appearance of a bed of similar character in the left bank of the Lune immediately under Lonton Chapel. But the accumulation of diluvium pre- vents this connexion from being established by direct evidence. The imperfect denudation on the left bank of the Tees did not * See the Geological Survey of the Yorkshire Coast by Young and Bird, p. 171. + Geological Transactions, vol. IV. p- 76. in Yorkshire and Durham. 25 allow me to ascertain the exact relation which the trap on that side of the water has to the contiguous strata. Above Egglestone bank another mass of trap, to all appearance immediately con- nected with that which has been described, crosses the road about a mile to the north-west of the village. It there assumes the unequivocal characters of a dyke, ranges (as nearly as I could discover from very imperfect data) E. by N. and a few hun- dred yards above the road crosses the western branch of the rivulet which runs past Egglestone. A quarter of a mile farther up the same branch of the rivulet, a second dyke crosses its bed and seems to range about S. E. by 8. From what has been stated it appears probable that these two dykes unite, or intersect each other. Their concourse will probably be found on the moor above the new smelting-house. The former, where it is seen above Egglestone, is about forty feet wide, and cuts through a bed of coarse grit, provincially called firestone. The latter is about sixty feet wide, and is associated with gritstone and a band of indurated shale which has been much quarried for whetstones. It would certainly be very interesting to trace these dykes as far as possible through the eastern moors, as there can be little doubt of their connexion with some of those masses of trap which traverse the great coal-field. My own observations were much too limited to complete this task. I however found on Woolly Hills, in the Woodland Fells, several quarries opened in a dyke which, from its position as well as in its structure, seemed to form a connecting link between the trap of High Teesdale and some of the dykes which traverse the country near Cockfield Fell*. * It is stated by Mr. Winch (Geological Transactions, vol, IV. p- 76.) That « at Egglestone, three miles below Middleton, a very strong vein of basalt may be seen crossing the Tees in a diagonal direction.” | suspect that he here alludes to the mass of basalt above. mentioned, which appears on the left bank of the Tees opposite to the entrance of the Lune, Vol. UL. Part I. D as Cockfield Fell dyke. 26 Professor Sevewick on Trap Dykes Proceeding some miles farther to the S.E. we come to the north-western termination of Cockfield Fell dyke, which is seen in a quarry by the side of the brook which runs past Gaundlass Mill. In that single locality it assumes a compound form, bemg made up of three distinct and nearly vertical masses of trap alter- nating with a variety of indurated slate-clay. The following is a transverse horizontal section of the whole dyke. (1) On the south-west side, common coal shale, which, as it approaches. the dyke, becomes much indurated and has a vertical cleavage. In this state it is provincially termed pencil. (2) Trap one yard. (3) Pencil about four or five yards, but of variable thickness and much shattered. (4) Trap two yards. (5) Pencil half a yard. (6) Trap about seven yards. (7) Coal shale resembling No. (1). These entangled masses of coal shale are probably not pro- longed far beyond the quarry, as they are seen in no other section. The dyke afterwards ranges through the coal works which are opened in Cockfield Fell about half a mile to the north of the village; and its farther progress in a direction about E.S.E. is marked in Blackburn quarry and Crag-wood. Near the former place it is intersected by a cross course, and heaved several yards out of the line of its direction. To the S.E. of Crag-wood, it would perhaps be impossible to trace it at the surface; but the vein of trap which runs along the high ridge of coal strata between Bolam and Houghton-le-side, agrees so well in character and direction with the masses above-mentioned, that it has generally been assumed as the prolongation of them. as I in vain endeavoured to discover the traces of a dyke farther down the river. If this con- jecture be right, it will be necessary to remove the dyke (which in the map accompanying Mr. Winch’s Memoir is made to cross the Tees below Egglestone) to a place considerably to the N. W. of its present position. When so represented, it will be seen, by an inspection of the map, that the basalt in Teesdale and tbe neighbourhood of Cockfield Fell are much more nearly in a straight line than they have been represented. in Yorkshire and Durham. 27 In the quarries which they are now excavating near Bolam, Botam quarry. the vertical dyke is unusually contracted in its dimensions; but on reaching the surface, it undergoes a great lateral extension, especially on the south-west side, so that the works are con- ducted in a perpendicular face of columnar trap more than two hundred feet wide. The changes produced by this overlying columnar mass are highly instructive, and will be described in their proper place*. The old excavations, in the direction of Houghton-le-side, shew that the trap is there confined to a fissure nearly forty feet wide, which, with a slight undulation in its direction, bears to a point about S.E. by E. There is another locality, the mention of which must not be Sandstone on omitted, though I think it probable that it is not in the line aan the great dyke. In this opinion I may, however, have been misled by the maps of the district, in which many of the places are laid down entirely out of their true bearings. At Wackerfield- lane-end, half a mile W.N.W. of Hilton, a mass of trap appears to range east and west, and may therefore join the leading dyke which intersects the country still farther to the east. The exca- vations in that place would not deserve any particular attention, were it not for the important fact, that at their western termina- tion horizontal beds of sandstone are seen to rest immediately upon the upper surface of the dyke. I have been informed that masses of trap occur on the north-east side of the quarries of Bolam ; but I had no opportunity of examining them with a view of ascertaining their probable connexion with the principal dyke. From all these facts we may infer—(1) That from Gaundlass Mill to Houghton-le-side, a distance of about ten miles, the dyke of trap is uninterrupted—(2) That it may be connected with other dykes, which appear still farther to the north-west nearly in the * See Plate II. Fig. 4. D2 Dyke in Lower Teesdale. 28 Professor Sepewick on Trap Dykes same line of direction, and through them with the dykes in Upper Teesdale —(3) That it probably gives out some Jateral branches connecting it with other masses of trap in the same district. It may farther be observed, that all this portion of the dyke, how- ever modified by local circumstances, dips towards a point on the north-eastern side of its general line of direction, so as to make with the horizon an angle perhaps in no instance less than eighty degrees. The high ridge of coal strata, extending from Bolam to Houghton-le-side, forms a kind of abutment which encroaches considerably on the line of the magnesian limestone. The present collocation of the two formations might lead to a conjecture that a great fault, ranging along the line of demarca- tion, had thrown the magnesian limestone down below its natural level. But the supposition is not necessary ; for the appearance of the limestone below the level of the ridge may be only an indication of its unconformable position. In the low region of the magnesian limestone we lose all traces of the basalt from Houghton-le-side to Coatham Stob. From the last mentioned place it may be traced through the quarries of Preston across the Tees; and very large ex- cayations have been made in a corresponding quarry at Barwick on the right bank of the river. The mineralogical character of this dyke, its direction, and its dip, agree so well with the one which ranges through Cockfield Fell ; that no one has, I believe, denied the probability of their being continuous*. The great distance between Houghton-le-side and Coatham Stob in which no trap has been discovered; and still more the fact, that the basaltic veins in the great coal-field do not generally pass up * Should any one maintain that the dykes of Cockfield Fell and the plain of Cleveland have a distinct origin; he may, perhaps, draw an argument in favour of his own opinion, from the great thickness of the vein of trap in the quarries of Preston, Barwick, Langbargh, &c. In this one respect there is undoubtedly a considerable difference between them. in Yorkshire and Durham. 29 into the magnesian limestone; have led some to imagine, that the prolongation of the dyke of Cockfield Fell is for several miles concealed beneath the beds of that formation. These basaltic veins, which do not penetrate the magnesian limestone, prove one of two things. Either that they took their present form before the deposition of the limestone; or that they were injected from below, but not with sufficient energy to break through the superincumbent limestone. Neither of these suppositions can apply to a great dyke intersecting an enormous mass of secondary strata which are newer than the magnesian limestone, and probably rest upon it. If therefore we admit the identity of the Cockfield Fell and Cleveland dykes; we must suppose that in the whole interval, between Houghton-le-side and Coatham Stob, it is con- cealed by a thick covering of diluyium: an opinion which no one will have much difficulty in admitting who has observed the enormous accumulation of transported materials in all the neigh- bouring district *. At Preston the trap emerges from beneath nearly fifty feet Range of the of diluvian brick earth; and would probably have remained the Eaten” concealed, had it not been laid bare in the bank of the river." On both sides of the Tees it is more than seventy feet wide, and ranges through horizontal strata of sandstone in a direction about S.E. by E. These horizontal strata must be referred to the new red sandstone formation, though they exhibit but faint traces of the usual ferruginous tinge. From Barwick the dyke passes through the quarries of Stainton, Nunthorp, and Langbargh, to the foot of the Cleveland hills; making in its progress a con- siderable flexure to the north. At Stainton the north face of the dyke is interrupted by a fissure about five feet wide, which is filled with light coloured argillaceous materials, with a transverse SS ee ee eee * See Plate II. Fig. 1. 30 Professor Sepawick on Trap Dykes slaty texture. These substances bear no resemblance either to the sound or decomposing specimens of the dyke itself*. On the east side of Nunthorp it gradually rises above the level of the neighbouring country, and might be mistaken for a gigantic artificial mound, had not the quarries exposed its interior struc- ture. A well defined ridge, about four hundred feet above the level of the neighbouring plains, marks its passage over the south flank of Rosebury Fopping. Still farther to the east it is traced by a gap in the outline of the moors: for the upper beds of sand- stone appear to have been shattered and carried off, and the dyke only rises to the highest level of the great bed of alum- shale. After passing through this gap and descending into Lownsdale, we found the trap forming a mass of bare rock which rose twenty or thirty feet above the vegetable soil. From thence it may be followed without difficulty many miles down the valley of the Esk, im a line bearing about E.S.E. Afterwards, by the turn of the valley at Egton Bridge, it is once more brought through the high moorlands; and its course is marked in that desolate region by a low ridge resembling an ancient Roman road. A quarry which is opened at Silhoue, near the seventh milestone on the road from Whitby to Pickering, proves the whole thickness of the dyke to be about forty feet, and its incli- nation and direction nearly the same as in the other localities. Beyond this place it continues to thin off, but it may be traced, though not without some difficulty, as far as a small rivulet about two miles to the east of the road. The exact point of its termi- nation has perhaps not been ascertained; but there does not seem to be any good reason for supposing that it is continued to the German ocean; as no vestige of it has been seen in any part of the cliff where it might be expected to appear. “ See Plate II. Fig. 3. in Yorkshire and Durham. 3] No other dyke has, I believe, been yet described, which Extent and intersects so many secondary formations, and preserves such an extraordinary uniformity of direction and inclination. The whole length, reckoning from the quarry at Gaundlass Mill, is more than fifty miles: and if any one should object to this, as includ- ing a considerable space in which the continuity is not apparent ; there will still remain from Coatham Stob a distance of about thirty-five miles, through which it is almost certain that the trap ranges without any break or interruption. Perhaps it might with more justice be objected, that the first computation falls below the truth ; in consequence of the probable extension of the dyke to the N. W. through the Woodland Fells and Egglestone Burn te the banks of the Tees. Should this supposition be ad- mitted, we shall have an uninterrupted dyke extending from High Teesdale to the confines of the ,eastern coast; a distance of more than sixty miles, The angle at which it cuts the strata is of course variable, and in many places cannot possibly be ascertained. At Barwick, near the Tees, its inclination to the horizontal beds of sand- stone is more than eighty degrees; and the angle at which it intersects the beds of shale and sandstone in the eastern moors is still greater; occasioned, perhaps, by the south-eastern dip, which generally prevails among the strata in that region*. Secondary fermations, when interrupted in the manner above described, seldom preserve the same level on the oppo- site sides of their line of separation. Thus at Cocktield Fell, the coal-beds on the north side of the dyke are eighteen feet below the corresponding beds on the south side. In the excava- tions at Preston and Barwick there is no indication of any great change having been produced in the relative level of the beds of sandstone: nor can any conclusive evidence be obtained on this * See the Survey of the Yorkshire coast by Young and Bird. Structure of the dyke. 32 Professor Sepawick on Trap Dykes subject from the obscure sections exhibited by the quarries in the eastern moorlands. Perhaps, as a general rule, the greatest dislocations are produced by those fissures into which trap is not intruded: such at least appears to be the case in the great coal- field of Northumberland and Durham. The injected masses of trap may be supposed to have acted as a kind of support, and to have partially hindered the broken ends of the strata from sliding past each other. Notwithstanding the great length of the Cleveland dyke, and the different nature of the rocks with which it is associated, it undergoes very little modification in its general structure. Its prevailing character is that of a fine granular trap rock of a dark bluish colour. This colour is imdeed, with some unimportant exceptions, so constant in all the sound specimens, that the dyke is provincially termed blue-stone by the men who are employed in working the quarries. It breaks into irregular, sharp, angular fragments; and on a recently exposed surface there generally may be seen a number of minute brilliant facets: but the con- stituent parts are never sufficiently distinguished from each other to give it the appearance of a green-stone. The essential ingre- dients of the rock are, if I mistake not, pyroxéne and felspar, in which respect it agrees with the greater number of trap dykes which have been carefully examined, as well as with a great many varieties of recent lava. The principal modifications, of course, arise from the variable proportions of these essential in- gredients. Among the prevailing and nearly compact portions of the dyke, there are some larger crystals of felspar and car- bonate of lime; very rarely, however, in such abundance or order of arrangement, as to give any decided appearance of por- phyritic structure. Good specimens of amygdaloid are not common; where they do occur the nodules are chiefly composed of carbonate of lime. In one or two instances we found chal- cedony filling the hollows of an imperfect amygdaloid. Tron in Yorkshire and Durham. 33 pyrites may be mentioned among the minerals frequently associated with the dyke. It is found disseminated through the substance of some decomposing varieties in considerable abundance; and small spangles of it may occasionally be seen in the sound spe- cimens, especially among the larger crystals of felspar before mentioned. All the dark sonorous specimens act strongly on the magnet; but some of the light-coloured varieties, which contain a great excess of decomposing felspar, do not sensibly affect it. The dyke is generally separated by a number of natural partings into large blocks, which are amorphous, prismatic, or globular. Near the centre they are sometimes of such entire irregularity as to defy all description. Not unfrequently, however, in the midst of this confusion we may observe traces of a prismatic form; and where this arrangement is most complete the prisms are always transverse to the dyke. Good examples of this form may be seen in the quarry of Preston, and in other localities above described*. The sides of the dyke are generally occupied by clusters of minute horizontal prisms, which are often seen in great perfection even where the central mass is amorphous. In the great quarry of Bolam, where the trap has extended laterally over the horizontal beds of sandstone and coal shale, the capping of basaltic rock is divided into rude columns which are perpen- dicular to the strata on which they rest; and, therefore, nearly at right angles to the prismatic blocks which lie across the leading dyke. This arrangement is exactly similar to that which takes place among some masses of ancient lava near Mount Vesuvius tf. * See-Plate II. Fig. 1. + Altered beds of coal in contact with trap sometimes exhibit a similar arrangement. Thus at Coley Hill (Geological Transactions, vol. 1V. p. 23.) a small bed of coal abuts against a dyke of basalt, and near this contact, the coal is deprived of its bitumen, and arranged in Vol. I. Part I. E beautiful Effects of de- composition, 34 Professor Sepewick on Trap Dykes Traces of the globular structure are often visible, especially where the trap passes into an earthy state: for many of the larger blocks, whether prismatic or amorphous, decompose in concentric crusts, which easily fall off and expose the hard spherical nuclet. These balls are particularly abundant in the old quarry of Coatham Stob, and are associated with some blocks of a light grey colour, which have an earthy fracture. Both these varieties are interesting. Some of the balls contain a considerable quan- tity of olivine, which is, if I mistake not, a very rare mineral in all the other localities. The light-coloured blocks have a super- abundance of decomposing felspar, and are partially porphyritic. Carbonate of lime exists in the form of distinct crystals, and is also disseminated through the mass; and in some instances small spherical concretions of compact felspar are found in a congeries of very minute crystals, giving to such specimens the appearance of an amygdaloidal structure. In other cases the concretions effer- vesce when first plunged into acids, are opaque from the admix- ture of impurities, and do not possess the characters of a simple mineral. In this dyke, as in almost every similar formation, the effects produced by decomposition are exceedingly varied. The compo- nent parts, from the centre to the surface, are in some quarries hard and sonorous. In others, the sides are invested with a fer- ruginous earthy matter which only penetrates to the depth of a few inches, and gradually passes into a sonorous granular rock. Not unfrequently, a decomposing crust of more considerable thickness covers the surface even of the blocks which are derived from the center of the dyke. A number of white spots, probably beautiful small horizontal prisms. Under the overlying mass in the quarry of Bolam, the carbonaceous shale is rudely prismatic; and in one or two places where this structure is best exhibited, the prisms are nearly vertical. in Yorkshire and Durham. 35 resulting from decomposing felspar, are often disseminated through these earthy masses, and enable us to separate them from other argillaceous materials with which they are sometimes in contact. It would be a laborious, and not a very profitable task, to attempt a minute account of phenomena like these, which vary with every different locality. It now remains to describe some of the effects produced by etets pro- the intrusion of the dyke. These effects will of course vary igor ne” with the substances which are acted on. In some of the quarries °“* «vhich have been already described, the trap passes through horizontal beds of slate-clay, and the changes produced by its presence are in all these cases strikingly similar. At Nunthorp and Langbargh* these beds of slate-clay belong to the great allum-shale formation (Lias), and are easily identified by the belemnites, pectinites and other characteristic fossils which are imbedded in them. On approaching the dyke they become much indurated, and are divided by a great many vertical fissures, which, when combined with the ordinary cleavage, separate the strata into rhomboidal fragments. In all such cases the rifts and fissures are coated over with oxide of iron. In other instances, the true horizontal cleavage entirely disappears; and the indu- rated masses might then be easily mistaken for beds which had been tilted out of their original position. The alteration pro- duced in the coal-shale at Gaundlass Mill is exactly analogous to what has been described, though not so strikingly exhibited. In the quarry at Barwick, on the right bank of the Tees, the vein of trap is well denuded, and the south side of the sec- tion exposes a great many horizontal beds of sandstone, which are separated into prismatic blocks by a number of natural transverse fissures. Close to the dyke this structure disappears ; * See Plate II. Fig. 2. E 2 36 Professor Sepawick on Trap Dykes the sandstone is much more compact, and breaks into amor- phous fragments. It must however be allowed that in some other localities the sandstone did not, under similar circumstances, appear to have undergone any modification. Perhaps, as a general rule, none of the changes above de- scribed are well exhibited, where the portion of the dyke, in contact with the horizontal beds, assumes the appearance of a wacké. Should this observation be sufticiently verified, it would seem to indicate, that the earthy texture of the dyke is, in some cases, rather due to its original mode of aggregation, than to any subsequent decomposition. I may, however, assert unequi- vocally, that I never saw any beds which are easily susceptible of modification (such as coal or carbonaceous shale) in immediate contact with the trap, without haying undergone a remarkable change. The overlying trap at Bolam bears no resemblance to a sub- stance which has been tranquilly deposited on the inferior strata ; for it is separated from them by a broken indented superficies which has exposed many distinct beds to its immediate action. Some of the massy columns rest on a bed of shale partially converted into a substance resembling Lydian stone, which rings under the hammer, or flies in all directions into a number of sharp splinters. Others are supported by a bed of impure coal or carbonaceous shale, in the upper part of which are found shapeless masses in various states of induration, mixed irregu- larly with angular pieces ef trap, and an earthy substance like soot or pounded charcoal. Where the carbonaceous ingredients are most abundant, the parts of the bed in immediate contact assume the appearance of coke derived from the artificial dis- tillation of impure coal, and not unfrequently separate into a number of minute prisms*. An impure carbonaceous powder is * See the Note to p. 33. in Yorkshire and Durham. 37 sometimes found in the crevices between the basaltic columns, several feet above the beds on which they rest. In addition to the substances above described, I found be- neath the trap some thin white porcellanous fragments, which appeared to be derived from an indurated bed of fire-clay—a well known associate of the great coal formation. All these phenomena so exactly resemble the effects pro- duced by fire, that I am unable to describe them without using language which may be thought hypothetical by those who deny the igneous origin of trap dykes. In Cockfield Fell the coal-works have been conducted on both sides of the dyke, and the extraordinary changes produced by its influence have been recorded by practical men who had no theory to support, and who founded their opinions upon actual observation. The works are not now carried on in the immediate neighbourhood of the dyke; but I procured so many specimens of the substances which had been taken from the altered coal- beds, that I have no doubt of the general accuracy of the accounts which have been published. Close to the dyke, the main coal is converted into a substance resembling soo¢, and at some distance it passes into a more solid substance, which the miners call cinder. At a still greater distance it retains a part of its bitumen, and about thirty yards from the trap it does not differ from the ordinary pit-coal of the district. It is stated, (Hutchinson’s History of Durham) “ that immedi- “ately above the cinder there is a great deal of sulphur in “angular forms of a bright yellow colour. The cinder burns *‘ clear, without smoke, and affords very little sulphurous effluvia.” Were there no other examples of corresponding phenomena feneous origin it would perhaps be unsafe to draw any direct conclusions from” "° “** the facts which have been stated. But in different parts of the British Isles, similar effects appear, in instances almost without number, to have been produced by the operation of similar General sum- mary. 38 Professor Sepawick on Trap Dykes causes: so that the igneous origin of a large class of trap dykes seems to be established by evidence which is almost irresistible. It is urged to no purpose, that Lydian stone and glance-coal occur in places which have never been influenced by volcanic action. The assertion may be true, but is of no value in deter- mining the question; unless it can be shewn, that substances, similar to those derived from the sides of the dykes, are found in other parts of the same district which are removed from their influence. This however is not the case, for the enormous ex- cavations which have been carried on in the great coal-basin of Northumberland and Durham have, with one ambiguous excep- tion*, made us acquainted with no similar substances excepting those which appear to have been produced by similar agents. It may be proper briefly to enumerate some of the facts which are established by a detailed examination of the great dyke, and which will, perhaps, be considered to place its origin out of all doubt. (1) It is more recent than the formations which it traverses. For it occupies the interval between beds which were evidently once continuous; but have been subsequently broken up and severed by some great convulsion. (2) It was consolidated prior to the last great catastrophe which formed the beds of superficial gravel, and excavated the secondary vallies. In proof of this we need only state, that it partakes of all the inequalities of the districts through which it passes, rising with the hills and falling with the vallies, so that *See the Geological Transactions, vol. IV. p. 27. The case is obviously ambiguous, because the effect of a large mass of trap on a bed of coal may be propagated to a considerable distance. The very change described by Mr. Winch may therefore have been effected by a mass of trap which is not exposed in the workings. We must carefully distinguish between the phenomena here described, and the effects of those dislocations which so commonly inter- sect the coal strata. In these latter instances the coal beds are often deteriorated on both sides of the line of fault by the mere mechanical effects of the rupture. in Yorkshire and Durhan. 39 im many of the lower regions it is buried in diluvium. On this subject there is, I believe, no difference of opinion. (3) There is every reason to believe that it has been filled from below. For there exists no trace of any upper bed from which its materials could have been supplied; and in one place, horizontal beds of sandstone rest on the top of a mass of trap which is probably connected with the dyke. We may further state, that many dykes of similar origin wedge out before they reach the surface *. (4) The dyke has once been in a fluid state. For it. is moulded to all the flexures of the chasm which it {fills up. The same assertion is also proved by its crystalline texture. (6) The materials of which it is composed are the same with those which abound in a great many varieties of recent lava, On this subject there is perhaps no difference of opinion. For the Wernerians at one time asserted, that recent lava was derived from the igneous fusion of trap rocks of aqueous origin. (6) The effects produced by the dyke are such as might be expected from the intrusion of a great mass of ignited matter. This assertion is fully established by the facts which have been already stated. If, therefore, similar effects have originated in similar causes, we must conclude, that this dyke, as well as all the other simi- lar masses in the great Durham coal-field, are the undoubted monuments of ancient volcanic action. It is a matter of fact, which is independent of all theory, that an enormous mass of strata has been rent asunder ; and it is probable that the rent has been prolonged to the extent of fifty or sixty miles. If we exclude volcanic agency, what power im nature is there capable of producing such an effect? By sup- iy pee et ee ee ee a * See Professor Henslow’s paper on the Isle of Anglesea; Dr. Mac Culloch on the Hebrides, &c, &c. Conclusion. 40 Professor Sepaewick on Trap Dykes posing such phenomena the effects of volcanic action, we bring into operation no causes but those which are known to exist, and are adequate to effects even more extensive than those which have been described. Combining this observation with the facts described with minute detail in the preceding parts of this paper, we obtain a chain of evidence, in favour of the igneous origin of a certain class of trap dykes, not one link of which appears to be defective. It is not to be denied, that the associations of trap rocks may in other cases present great difficulties to the igneous theorist. But these difficulties are not the present subject of consideration. I have confined myself, as far as possible, to a statement of facts, and I have only attempted to record such conclusions as a review of those facts appeared fully to justify. Trin. Coll. March 12, 1893. P.S. Before this paper was sent to the press, I received two letters from my friend Mr. Wharton, of Oswald House, near Durham, communicating some very interesting facts connected with the appearance of a basaltic dyke; which ranges from the escarpment of the magnesian limestone (at Quarrington Hill, a few miles to the east of Durham) through the great coal-field, in a direction about W.S.W. It is found along this line at Crowtrees, Tarsdale, Hett, Tudhoe, Whitworth, and Constantine farm. From the last mentioned place, it passes along the same line of bearing, through the collieries of Bitchburn and Hargill Hill, to a spot near the confluence of Bedburn Beck and the river Wear, where it is well exposed on the surface of the ground; and it is known to pass up the Bedburn Beck valley towards Egglestone Moor. If prolonged a few miles in the same direction, in Yorkshire and Durham. 41 it must meet the line of the Cockfield Fell dyke within a short distance of Egglestone; and may, perhaps, be a prolongation of one of the masses of trap described in a former part of this paper. This dyke is laid down in none of our Geological maps. Indeed its existence was probably unknown before Mr. Wharton ascertained its continuity, by examining the thickness, the dip, and the bearing, of several masses of trap, which appeared in separate quarries, but in the same general line of direction. That its further extension towards Egglestone Moor, and its probable connexion with the trap of High Teesdale, should be correctly determined, is certainly an object of considerable interest. The following facts appear of most importance in illustrating the natural history of this dyke. (1) The trap, in colour, fracture, and external form, is similar to that of Cockfield Fell. It often parts into irregular prismatic blocks with well defined angles, and four or five plane sides covered with an ochreous crust. (2) The width of the dyke appears to increase in its progress westward. Thus, at Crowtrees quarry it is six feet and a half wide—at Tarsdale quarry nine feet and a half—at Bitchburn bank fifteen feet—and still farther west it is seventeen feet wide. (3) It dips to the north at an angle which brings it up in a direction which is nearly perpendicular to the coal strata; which, on the north side of the dyke, are found about twenty- four feet above the level of the corresponding beds on the south side. (4) In the collieries situate in its line of direction (viz. Crowtrees, Bitchburn, and Hargill Hill) the seams of coal near the dyke are charred, or converted into a hard mass of cinders ; in consequence of which, the works have in some cases been partially abandoned. Vol. Wl. Part I. F 42 Professor Sepawick on Trap Dykes (5). The dyke appears to decrease in width as it rises towards the surface. Thus, in Crowtrees colliery, the width of the dyke, where it is cut through at the depth of fifteen fathoms, is nearly twice as great as at the surface. (6) It does not appear at Quarrington Hill to cut through a bed of sand and pebbles, which lies between the highest beds of the coal-formation and the magnesian limestone. The importance of these facts in confirming the theoretical views given in the preceding paper, is too obvious to need any explanation. Mr. Winch asserts (Geological Transactions, vol. IV. p. 25.) ‘* that he has never been able to trace any of these basaltic veins ‘into the magnesian limestone, and is almost certain that, with ‘other members of the coal-formation, they are covered by it.” The dyke just described affords some additional evidence in support of this opinion. Moreover, it appears, in its general relations, to agree so exactly with the Cocktield Fell dyke; that I now cannot help suspecting, that this latter also belongs to the class of ‘* basaltic veins” which do not pass up into the magnesian limestone, though I inclined to a different opinion when the pre- ceding paper was written. Respecting the prolongation of the Cockfield Fell dyke through the region of the magnesian limestone, there are con- flicting probabilities which lead to directly opposite conclusions. The near agreement in the direction and dip of the Cockfield Fell and Cleveland dykes, has generally been supposed to afford sufficient evidence for their continuity. If this opinion be adopted, we must, I think, be compelled to admit the existence of a dyke through all the imtermediate district*.—On the contrary, there is no direct evidence for the existence of any trap associated with *% See the observations at p. 29 of this paper, in Yorkshire and Durham. 43 the magnesian limestone; and the relations of all the analo- gous formations in the coal district seem to prove, that the Cockfield Fell dyke cannot pass out of the limits of the coal- formation. If we adopt this latter opinion, we must admit, that the dykes of Cockfield Fell and Cleveland (notwithstanding the agreement in their line of direction) belong to two distinct epochs. After all, the question is only one of local interest; and as far as regards the leading object of this paper, of no importance what- soever. Through the kindness of T. R. Underwood, Esq. of Paris, I have become acquainted with the results of an examination of specimens from several English trap dykes by Professor Cordier. I will subjoin his description of such specimens as were derived from localities alluded to in the preceding paper. No. 1. From Preston quarry in the Cleveland dyke. Mimosite, fine grained, imperfectly porpheroidal from the salient crystals of Pyroxéne. It is a Basalé of the ancient mineralogists. The specimen contains a great abundance of dark-greenish grey Felspar, mixed with a very small quantity of Pyroxéne and titaniferous Iron. Some points of Pyrites are to be seen. The paste also envelops laminar crystals of Felspar, having a con- siderable lustre, which give the paste a scaly appearance which distinguishes it from Basalt. No.2. From Coaly Hill dyke near Newcastle. Mimosite, small grammed, passing into Xerasite. Many of the cavities con- tain green-earth. It is imperfectly porpheroidal. The crystals of Felspar very brilliant. No. 3. From Walbottle Dean dyke. This has a more decided character of a Dolerite, very fine grained, the Felspar whiter than in the others. F2 44 Professor Sepawick on Trap Dykes, &c. As these distinctive terms are not generally adopted by English mineralogists; it may be proper to state that Mimosite and Dolerite are granular rocks. Xerasite and Basalt are com- posed of the same elements, but microscopic, and having the appearance of a paste. Plate! 2. De OARS SS SSS Tite EA tiene calor : 3 yi { 4 4 — E — | ASelguwick del Printed by Chullman dad: IV. A new Demonstration of the Parallelogram of Forces. By JOSHUA K'ING, Esq. M.A. FELLOW AND TUTOR OF QUEEN’S COLLEGE, [Read April 14, 1823.] Ler two equal forces, each of which is represented by p, act upon a material point inclined to each other at an angle 20, and let r be their resultant; which will evidently bisect the angle 20; for there is no reason why it should be inclined to one force at a greater angle than to the other. Now for every value of 6, r will vanish when p vanishes, but only upon that supposition. Again, for every value of p, r will 30 5a 2n+1 vanish when 0 = +“, or + Q> OF +> or &es es) bat 2 ae upon no other supposition: Hence the factors p, ? 2 92 Qa -2*) &c. will enter into the expression for r, but no other, except quadratic factors having impossible roots: we may there- fore suppose r=k.p.(1-=%). 0-29) A - oe &e. =k.p. cos. 0. 46 Mr. Kine on the Parallelogram of Forces. Now (1). & cannot involve p, for the angle of inclination remain- (2). (3). (4). ing the same, the resultant must necessarily be pro- portional to the component. k cannot involve 6: for if it does, it must evidently be such a function of @ as has no possible root: it must moreover involve no negative powers of 0, for then would r+ become infinite when @ vanished: nor yet any odd powers of 6, for then would r be altered by changing the sign of @: it must therefore involve the even powers of @ only; and as all the roots are impossible, the terms must all have the same sign, and consequently such a value of @ may easily be assumed as will make &.p. cos. @ greater than 2p, which is impossible. cos. @ cannot be raised to any power as cos. 6\”, for then the equation could not be made identical both when @=0 and when 6 = = the factors composing cos. @ cannot be raised to dif- ferent powers, for the equation could not then be made identical when @ equals any term of the series To determine k, let 6=0, in which case r ought to be equal to 2p; Eos? pus leperame. .¢ sit == 2: Hence r = 2p.cos. 0. The resultant of two unequal rectangular forces, and con- sequently of any two forces whatever inclined to each other at any angle, may easily be deduced from the expression for the resultant of two equal forces by processes already known, which require neither elucidation nor abbreviation. V. On the Developement of Electro- Magnetism by Heat. By tHe Rey. J. CUMMING, M.A. F.RS. M.GS. PROFESSOR OF CHEMISTRY IN THE UNIVERSITY OF CAMBRIDGE. {Read April 28, 1823.] Tue property which the Tourmaline and a few of the erys- tallized gems possess, of exhibiting the opposite electricities by _the action of heat alone, has hitherto been considered as peculiar to those bodies ; but a recent experiment by Dr. Seebeck of Berlin, has proved that this power, so far from being confined to non- conductors, as from analogy might have been suspected, is pos- sessed by one, at least, amongst a class of substances, which are, comparatively, perfect conductors both of heat and electricity. The experiment is described in these words: ‘‘'Take a bar ‘‘of antimony about eight inches long and half an inch thick, “connect its extremities by twisting a piece of brass wire round ‘them, so as to form a loop, each end of the bar having several “coils of the wire. If one of the extremities be heated a short “time by a spirit lamp, electro-magnetic phenomena may be ex- “hibited in every part of ES On repeating this experiment, it appeared to me highly pro- bable that other metals might possess the same property, and in prosecuting this enquiry, I have been led to some results, 48 Professor CummtnG on the Developement which, I hope, will be both as new, and as interesting, to this Society, as they have been to myself. My first object was to ascertain whether in this, as in some late electro-magnetic experiments, the effect depended on the wire being coiled round the bar; but by repeated trials, both on an- timony and other metals, it was found to be indifferent, in what manner the wire and bar were connected, provided that, in all cases, the metallic contacts were complete: I have therefore, in general, made the connection, either by soldering, rivetting, or casting the bar upon the wire. The result was, that not only all the metals, including fluid mercury, but likewise plumbago and charcoal, and some, at least, of the metallic sulphurets, possess the property of exhibiting electro-magnetism by heat, differing, however, both in quantity and quality. If, for instance, a bar of bismuth, having copper wires at each end, be heated at one extremity; on placing the wires in the mercurial caps of the galvanoscope, the heated end produces a deviation on the compass needle, in the same direction as a wire from the silver disk, in the common galvanic circuit: with antimony it is the reverse. (Fig. 1 and 2). These metals may, therefore, so far be considered, as positive and negative with respect to each other. On examining the other metals with copper wires, I found that they might be distinguished into two classes, the heated end of the one, and the cooler end of the other exhibiting the silver or positive electricity. (Table I). When wires of other metals are used, there are modifications, which appear to me some of the most singular circumstances in these experiments. If the bar be of copper, the deviation becomes negative or posi- tive, accordingly as the wires are platina or silver; or if the extremities be considered as positive and negative with the one, they are negative and positive with the other. The same effect was produced with a bar of zine, and zinc or copper wires; and of Electro-Magnetism by Heat. 49 with silver, platina, and palladium, as the wires were silver or platina. In these instances it seems remarkable, not only, that the bar appeared to change its electrical states with different wires, but that electricity or rather electro-magnetism should be exhibited when the bar and wire were of the same metal. In the first case, it might be supposed that the electricity was excited by the contact of dissimilar* metals; as, in the galvanic circuit, copper is positive to zinc, but negative to silver. But this hypo- thesis is inapplicable to cases of the second description. If the effect depended on the contact of dissimilar metals, it would be greatest between those which are opposed in the galvanic circuit, and would cease when the bar and wire were of the same sub- stance. On making the trial with a bar of zinc and silver wires, the deviation was not greater than that by the same bar and wires of zinc. Again, platina and silver are both positive with reference to copper, yet the deviations were opposite; and silver and copper bars acted strongly with silver and copper wires respectively. As in all these instances, to prevent ambiguity, the wires were not soldered, but rivetted to the bars, I cannot but conclude, that the hypothesis of electricity being excited by the contact of dissi- milar metals, is, whatever plausibility it may possess in other circumstances, inapplicable to the case of its developement by heat. The following experiment is, I think, decisive. — Two wires, each composed, the one half of platina, the other of silver, soldered together in the middle, were rivetted into a bar of brass. When the silver ends were connected with the brass, the deviation was positive; on reversing each wire, and therefore con- necting the platina and brass, it was negative; still retaining the platina contact, but shortening the platina wire to about half an inch, the deviation again became positive. In every case the brass was in contact with a metal highly positive with respect * Dissimilar as to their galvanic relations. Vol. Uf. Part I. G a 50 Professor CUMMING on the Developement to itself, yet the deviations in the two last were in opposite direc- tions, though the contact of brass and platina was the same in both. If these experiments be referred to the hypothesis which ac- counts for electrical excitation by the oxidation of the metals, they seem equally adverse to it. Not to repeat the instances of its production where the heated bar and the wires were of the same metal, and in consequence, similarly, if at all, oxidated ; it can scarcely be imagined, that an elevation of temperature of not more than two or three degrees, should cause a difference of oxidation; and it is to be remarked, that the effect is produced, whether the temperature be elevated or depressed. On placing one end of a bar of bismuth in a freezing mixture, or even by allowing a few drops of ether to evaporate from its surface, there was produced a considerable deviation on the needle of the gal- vanoscope; that extremity which remained at the temperature of the room acting as the heated end of the bar in the other instances. Having ascertained that in all the perfect conductors of elec- tricity, electro-magnetism may be excited by the unequal distri- bution of heat, my next endeayour was to determine the direction in which this peculiar influence is exerted, and the mode of its propagation. If a bar of antimony, AB (Fig. 3.), having its ends connected by copper wire Aab B, be heated at one extremity and presented to the compass, the deviation, in every part both of the bar and the wire, is of the same nature, and therefore the current of electricity (if there be such a current) is throughout in the same direction. The effect is similar, whatever metal be employed; but, as will be seen by reference to Table I, the direction of the current in some is opposite to that in others. If two bars of the same or of similar* metals, equal in power, be * Similar, as to their developement of electro-magnetism by heat. of Electro-Magnetism by Heat. 51 connected at their heated extremities; or, which is the same thing, if a single bar be heated at the middle, no effect is produced, the equal and opposite currents counteracting each other: if under the same circumstances, the metals are dissimilar, the effect is that arising from the joint action of their conspiring currents. In some respects this arrangement is analogous to that of the galvanic circuit; heat in one case acting the part of an acid in the other; but there is one material difference between them. In the first, the metallic circuit is complete, and the current is, as has been already observed, in the same direction throughout every part of it. In the second, the circuit is interrupted, and the current through the acid is opposite to that through the wire.* (Fig. 4 and 5). If the bar of antimony AB above mentioned be broken un- equally into two parts ab, cd, (Fig. 6.), and these be connected by a copper wire; on heating one part and cooling the other re- gularly throughout, no effect is produced, however short the interval may be between them.t If the parts ab, cd be (Fig. 6.) again * This may be readily shewn by connecting S and Z in Fig. 5. by a fine wire. If two wires, the one platina, the other bismuth, be connected with the galvanoscope, on immersing their other extremities (not in contact) for a short time in nitric acid, electro- magnetic effects are produced by their galvanic action; on making the contact the effect still continues, and in the same direction, but arises from heat; if platina and iron be used the second action is contrary to the first. + As electricity is excited by the contact and separation of two polished disks of zine and copper, Mr. Herschel suggested to me, that it might be desirable to try what effect would be produced, by heating one of the disks previously to its application to the other. If their thickness be inconsiderable in comparison with their surface, the electro-magnetism elicited is searcely, if at all perceptible; if they be of considerable thickness, and both bismuth or both antimony, it is greatly increased; and again is materially diminished if one be antimony and the other bismuth. he reason is obvious: in the first case each disk receives almost instantaneously an uniform temperature throughout; the second case becomes that of two bars of the same metal, having their extremities at unequal temperatures in contact, and their electric currents in the same direction; in the third case they are opposed. All these instances are decidedly unfavourable to the supposition of electro-magnetism being evolved by the contact of dissimilar metals, G2 52 Professor CUMMING on the Developement soldered together, with a thin plate of copper interposed, they no longer act as one, but as two distinct bars. When heat is applied at the extremity a, the deviation is, as usual, negative; at e¢ the same; but if at 6, the deviation is positive, the extremity a be- coming the cooler end, and the part cd merely conducting the electricity: but as the bar cools, ab, and ¢ the extremity of the other part, gradually assume the same temperature, and conse- quently the bar acts negatively as at first. It appears then, that when the bar was entire, the heat was not merely conducted from one extremity to the other, but, by some means modified in its progress, and that, for the production of this species of electricity, there is required the juxta-position of two particles of the same metal at different temperatures. If therefore, a cylindrical bar, unequally heated, be supposed to be divided into an indefinite number of circular laminz, each will act, as a layer of hot particles upon the lamina on one side, and of cold upon that on the other, and the total effect of the bar will arise from the aggregate action of these lamine. By soldering wires to a long rod of bismuth (Fig. 7.), the parts of which were alternately hot and cold, it was found that the action of the whole exceeded that of any two portions taken se- parately, and as the only condition appears to be, that there should exist a difference of temperature between two adjoining particles, it may be inferred, that if it were possible to increase these divisions sine limite, each bar would act as an assemblage of an indefinite number of small plates; as the common magnet may be conceived to be composed (if the expression may be allowed) of an indefinite number of atomic magnets.* * The connection of the wires may be either 1 and 2, 2 and 3, 3 and 4, or 1 and 4, or 1 and 3 may be placed in one cup of the galvanoscope, and 2 and 4 in the other, in which case the effect is the greatest. The deviation caused by connecting 1 and 4 is not affected by connecting at the same time 2 and 3; this seems unfavourable to the sup- position of circulating currents. of Electro-Magnetism by Heat. 53 In the beginning of this paper it was mentioned that all the metals possess the power of exhibiting electro-magnetism by heat, but in different degrees. Even when these are the greatest, they are much inferior to what can be readily produced in the common galvanic apparatus; it was consequently necessary, in detecting and forming a comparison of their relative powers, to use the delicate instrument described in the First Volume of our Transac- tions*; by which some, though not in all cases, an accurate measure of them could be obtained. For the more minute effects a compass was employed in the galvanoscope, having its terrestrial magnetism neutralized, which gave a deviation of from 10° to 20° with two disks of zinc and copper of 4> inches diameter, excited by spring water, and which was readily sensible to the galvanic action of zinc and copper wires, excited by nitric acid, whose diameters were less than 5, and whose excited surfaces were con- sequently between ;; and 3, of an inch. With this instrument, two rods of zinc and copper of 3 inch diameter and apart, excited by equal parts of muriatic acid and water, gave a deviation of 40°; a bar of bismuth 4+ inches long by + broad, and ? thick, gave 70° of deviation at the melting point of the bismuth, and 10° when the difference of the temperatures of its extremities was 12° of Fahrenheit. The slip of palladium before mentioned, which was 23 inches long by % broad, and weighed 35 grains, gave a deviation of 70° positive with silver, and 10° negative with platina wires, when heated red hot by a spirit lamp: a slip of platina * The spiral wire of the instrument was originally of copper, but silver wire of the same diameter is more efficacious. If the spiral parallelogram be vertical, and the wire of silver of = inch diameter, it acts very powerfully, and presents little or no impediment to a view of the needle; for this reason I prefer the vertical to the horizontal form of the spiral. The needle is neutralized by placing a powerful magnet North and South on a line with its centre; and another, which is much weaker, East and West at some distance above it: by means of the first, the needle is placed nearly at right angles to the meri- dian, and the adjustment is completed by the second. 54 Professor CumMING on the Developement of the same dimensions, and under the same circumstances, gave deviations of 65° positive, and 4° negative respectively. These are selected as being some of the most remarkable results, and as serving, at the same time, to form a comparison between the effects of electro-magnetism as excited by heat, and by the usual process. The results of similar experiments on -some metals and metallic alloys are given in Table IT. The metals which were found to be most powerful im their action, and at the same time most readily prepared for experi- ment, were bismuth and antimony. These were sufliciently ener- getic to have their effects estimated by a compass needle 4 inches in length, the terrestrial magnetism of which was not neutralized, and therefore the deviations caused by different increments of tem- perature may be employed as data for the corresponding electro- magnetic action. With this compass, rods of zine and copper, each = inch diameter, and ; apart, excited by equal parts of mu- riatic acid and water, gave a deviation of 27°. A circular rod of bismuth 43 inches long and + inch diameter, gave a deviation of 21° at the melting point of bismuth, 12° at the temperature of 180°, and 5° at 100°, the cooler extremity being in water at 60°. A si- milar bar of antimony: gave 19° with the utmost heat of a spirit lamp, and a bar of platina which weighed 565 grains, and was 7 inches long by 4 broad, gave 19° at a red heat. Hence it appears, that, at the same temperature, bismuth is more powerful than an- timony, and antimony more powerful than platina: the other metals, with the exception perhaps of iron, were inferior to platina. Were it possible to acquire the same accumulation of power in this species of electro-magnetism as is obtained by the use of large plates in the galvanic, or by numbers in the voltaic appa- ratus, it would be an interesting object of research to compare the electricities thus differently excited. For this purpose bars were cast, differing both in length, breadth, and thickness, but of Electro-Magnetism by Heat. 55 an increase in either of these dimensions was not attended with an equivalent increase of power. A cylindrical rod of bismuth, 9 inches long by 3 diameter was rather more powerfal than another of 25 inches by 2; but, between this, and a thin plate 43 inches long by 1 broad and } thick, the difference was scarcely perceptible. By increasing the surfaces in contact, or rather, as it afterwards appeared, by adding to the conducting surface of the connecting wire, there was a slight addition to the effect produced. The cylinder ef bismuth of 25 inches by 2, having a plate of copper of the same diameter soldered upon it, with four connecting wires, (Table III.) was equally, if not more powerful, than the longer cylinder, and on the whole, this seems to be the best form ; yet the gain of power, by increasing either the diameters of the bars or of the surfaces in contact, is not such as to promise any advantages by the use of large metallic bars, analogous to those obtained by the employment of large plates in the galvanic apparatus. When two metallic rods in connexion, were heated at the same time, there was some accumulation of power; which appeared to be greater when they were in sequence than when the wires from both the heated ends were placed in one cup of the galvanoscope, and the wires from the cooler ends in the other. (Fig. 8 and9). A bar AB which gave a deviation of 16°, when placed in sequence, as Fig. 8, with another CD, whose deviation was 21°, gave a deviation of 25°, but when connected, as in Fig. 9, gave only 23°. As in this experiment there was some, though not a consi- derable increase of power, a battery was formed of eight plates, four of antimony, and four of bismuth, placed alternately, and connected in sequence, (Fig. 10.). This (Table IV.) at the mean temperatures of 175° and 100°, gave 17° and 73° of deviation ; a single plate of bismuth, at the same temperatures, gave the deviations 73° and 23°, and one of antimony gave 67° and 2°; the effect therefore of the eight plates is but little more than double 56 Professor CumMinG on the Developement that of the single plate of bismuth*. A battery of 6 plates of bismuth, (Fig. 11.) gave at 140° a deviation of only 83°; but two plates alone of the same battery, gave a deviation of between 9° and 10°. This diminution of power arose, as appears by Table V. from some of the plates being inferior to the others; as is the case in the voltaic battery, when the plates are of different di- mensions or differently excited, the weaker having a tendency to reduce the others to their own standard. The greatest deviation I have as yet produced was by a double bar 7 inches long, com- posed of two bars, the one antimony, the other bismuth, soldered together at the middle. This, at the melting point of bismuth gave a deviation of 36° with the same compass which shewed a deviation of 28° with the zinc and copper rods excited by dilute muriatic acid. As these rods were capable, though slightly, of magnetizing a needle inclosed in a spiral wire, and of exciting the limbs of a frog, I endeavoured, but without success, to effect the same by the double bar of antimony and bismuth. Whether this failure were owing to a want of sufficient action, or to some peculiarity in the electro-magnetism excited by heat must be de- termined by the application of a more powerful apparatus; but it does not seem improbable that it may be owing to some pecu- liarity in this mode of excitationt. There is something perhaps analogous in common electricity; a large battery may be dis- * The bars of bismuth and antimony, having their extremities connected by copper wires, were fixed in a wooden trough, in such a manner that one half of their length projected below it. The trough was then filled with hot water or sand, and placed upon another vessel containing cold water. The mean of four thermometers in the upper part was assumed as the temperature of the heated extremities, and those experiments alone retained, in which the thermometers did not differ materially from each other. As the melting point (475°) of bismuth, the more fusible metal, is at nearly the temperature (500°) of boiling linseed oil, it is evident that a considerable increase of power may be obtained, if the trough be filled with ice, and placed.upon a vessel of oil heated nearly to ebullition, + Possibly from its low intensity, of Electro-Magnetism by Heat. 57 charged through a spiral wire, without magnetising a needle in- closed within it, provided the discharge be made gradually, and without a shock. The tourmaline, which has been mentioned as analogous to the metals, in receiving polarity by heat, I found, even when strongly excited, to have no magnetic action, either when two silver wires were coiled round its extremities and con- nected with the galvanoscope, or when the same wire was con- tinued throughout; but it has not, I believe, been noticed, though it might have been expected, that in the last case the wire prevents the tourmaline from acquiring the opposite polarities. There is a singularity in the electric properties of the tourmaline, which was pointed out to me by Professor Henslow in the Abbé Hauy’s Traité de Mineralogie: if it be exposed to a low temperature, its extremities assume the opposite electricities; on increasing the temperature, the electric polarities diminish, at length altogether cease, and are afterwards resumed, but in the opposite states; that end which was positive becoming negative, and vice versa. Very unexpectedly, I discovered a similar phenomenon in the metallic electro-magnetism. As bismuth and antimony are oppositely af- fected by heat, and nearly to the same extent, it seemed probable that in certain proportions they might neutralize each other, and a compound bar would be produced, in which the electro-magnetic effects would cease. This is certainly the case, but these bars possess, what may be called a moveable zero. I have four bars, with different preportions of antimony and bismuth: the first slowly exhibits with the large compass 4° deviation, as antimony, then, on continuing the heat, returns to zero, passes through it, and deviates 5° as bismuth; the second deviates 3° as antimony, returns to zero, deviates 4° as bismuth, and again returns to zero as the bar begins to melt; the third deviates through 7°, and just returns to zero at its melting point; the fourth deviates 7° in the same manner, but returns only to 4°. These results were ob- Vol. Il. Part I. H 58 Professor CUMMING on the Developement tained with copper wires; if wires of silver or platina be used, the effects are similar, but the positions of the zero are changed. (Table VI.)* The proportions of these bars are between four and six of bismuth to one of antimony ; but the metals require so many repeated castings before they are thoroughly incorporated, and their union is so much affected by fusion, that it is nearly impossible, without destroying and analyzing them, to determine accurately their composition. If it be supposed that the formule, which express the relation between the increase of temperature and the magnetic effects, be different in the two metals, it is obvious that though in equilibrio at one temperature, they will not be so at any other. This would account for the phenomena of the first, third and fourth bars, supposing them to be rather mixtures than alloys; and perhaps the anomaly of the second bar may be resolved by supposing the metals in this to be partly mixed and partly forming an alloy ; in which case there would be three substances, each following a different law. With the same metal, when the differences of temperature at its extremities are not considerable, they correspond very nearly with the deviations: yet proceeding in an increasing ratio, which augments as it approaches the melting point; but not so rapidly with bismuth as with antimony. The deviations of two similar rods of antimony and bismuth, from 65° to 240° of heat, were 10° and 14° respectively, (Table VII.) Taking the deviations at intervals of 34° in the one, and 4° in the other, the corres- ponding differences of temperature will be 42, 47, 81, and 47, 60, 68. If the melting point of bismuth be assumed 475°, and that of antimony 800°, the deviations corresponding to the last 235° of * The order of deviations with copper wires, as shewn by the small compass, was (Mable Vib): get; parser tienes a erttars,- aisle ciara 0.36 neg. 0.25 pos. O. With platina wires they were ................. 0.35 neg. 30 neg. 65 neg. of Electro-Magnetism by Heat. 59 bismuth will be only 7°, and 9° for the last 560 of antimony ; or, in other words, the deviation in bismuth by 175°, reckoning from 65° to 240°, is double that by 235°, from 240° to 475°; and, in anti- mony, the deviation by the first 175° exceeds that by the last 560°. - The experiments which have hitherto been detailed, rest upon the supposition that the agency of heat in exciting the magnetic electricity is effective on the bars alone, and that the action of the wires is solely to conduct the electricity thus excited. This is not the fact. When the extremity of a bar is heated in connexion with a wire, the wire is itself in the same state with the bar, having its extremities at different temperatures: the total effect is therefore the sum or difference of that of the bar and wire, accordingly as their relations to heat and electricity are different or the same. If a bar of bismuth be connected with the galvanoscope by wires of antimony, it is obvious, from the experiment of the double bars, that its effect is increased by the conspiring action of the antimony ; had the wires been of platina, the effects would, for the same reason, have been diminished. This would be the case, not merely from platina being either a better or worse conductor of heat than bismuth or antimony, but because its electrical properties, as de- veloped by heat, are similar, though inferior to those of bismuth ; and therefore when heated in contact with bismuth, its action is in the opposite direction. As the metals differ materially, not only in the nature, but in the strength of their action, it may happen, when the energy of the bar is weak in comparison with that of the wire, that their joint action will be nearly the same as that of the wire alone. This was the case in the experiment with the bar of brass, heated with silver and platina wires*. For, though when brass and silver, or brass and platina wires of the same size are * This experiment would be yet more decisive with wires of gold and platina. H 2 60 Professor CuMMING on the Developement used, the brass is negative with’ respect to both, yet, the electric energies of silver and brass differ so little, that, by mcreasing the size of the brass, it may be made to overcome the action of the silver, and give a positive deviation, as in the first case of the experiment. The action of platina is so much more energetic than that of brass, as not to be overcome by the increased dimensions of the bar, and therefore the deviation is negative. In the last case, where the platina was shortened to half an inch, it became heated throughout, and communicated heat to the silver wire in contact with it; consequently the effect of the platina disappeared, and this case became similar to the first. To have made the series (Table I.) given in the first part of this paper, an accurate re- presentation of the electro-magnetic relations of the metals, the bars should have been connected with the galvanoscope, by a substance (if there be such an one) that should merely conduct, without modifying the electricity*. The electric energy of copper is so much inferior to that of many of the metals, that, when the dimensions of the bars considerably exceed those of the copper wires, the series thus obtained may be considered as tolerably accurate ; but, with others, as lead and tin for instance, it is ma- nifestly imperfect. The most complete scale seems to be that, which may be formed by using bars and wires of the same di- mensions, i.e. by heating equal wires of the different metals in contact, taken two and two together. The series given in Tables VIII. and IX. was formed in this manner, so far as it was prac- ticable. The ends of the wires to be examined, were placed, one in each cap of the galvanoscope; the other ends were then connected and dipped into a capsule filled with boiling mercury ; * If two small rods of antimony and bismuth, properly adjusted, were soldered together longitudinally, they might, perhaps, at a fixed temperature, be considered as a neutral conductor, of Electro-Magnetism by Heat. 61 by this means all the metals which can be procured in the form of wires may be compared in a few minutes. The others were connected with similar pieces of those metals which had been already examined. The series, Table IX. is so constructed, that every substance may be considered as positive to all below, and negative to all above, and therefore any metal in the upper part of the series will form a circuit with all below it, similar to, though less powerful than that of bismuth and antimony. Platina and iron form an arrangement of this description: equal wires of these metals of +, inch diameter gave, with the large compass, a deviation of 16° by the heat of a single spirit lamp*. As these metals admit of being welded together, and of sustaining an intense heat, I at first imagined, that an instrument might have been constructed on this principle, as a pyrometer; but, the slow increase of devi- ation at elevated temperatures, appears to present an insuperable objection. If either series, whether that formed by metallic bars with copper wires, or that made by the comparison of equal wires of different metals, be referred to the common galvanic series, to those of the conductors of electricity or heat, or to the order of specific gravities, it will be found that there is no correspondence between them: the electro-magnetic relations of the metals, as developed by heat, can be determined by experiment alonet. As this property cannot be previously inferred from any known dis- tinctions of the metals considered collectively, so it does not appear to be dependent upon any peculiarity, such as crystallization, in * Tf silver and iron wires be heated in connection, the deviation attains a maximum, diminishes on increasing the heat, and again attains the former maximum in cooling. + The series is consistent with itself, that is, if on experiment, a metal be found to be positive to one, and negative to another substance in the series, it may be inferred, without further trial, that it is positive to all below the first, and negative to all above the second, 62 Professor CuMMING on the Developement each taken separately. If it were, it would be affected by altering the internal structure. As bismuth is a metal easily cast, and which has a strong tendency to crystallization, I cast two bars of the same dimensions; one in a charcoal mould, which was cooled as slowly as possible; the other in iron placed in cold water, and therefore cooled almost instantaneously: but I could discover no difference either in the nature or quantities of the deviations they produced, when heated to the same temperature. Again, a glass tube eight inches long was filled with mercury, and copper wires being passed through corks at each end it was placed, the one half in hot sand at 170°, and the other in water at 60°*. The deviation was 9° positive, which at 150°, and at 115°, be- came 6° and 3°. It appears then, that, in the same metal, the magnetic effects are not varied, whether the crystallization be more or less perfect; and that they may be exhibited in a fluid metal, where, of course, there can be no crystallization. When, however, two metals are combined, the change of structure thus produced, is attended with a change of electro-magnetic proper- ties. The alloy of bismuth and tin in Table I. is negative with copper wires, though each metal separately is positive, and one of them to a high degree; in other mstances, as zine with lead, the deviation of the predominating metal seems to be increased rather than diminished, by its union with another, whose deviation, separately considered, is opposite. Leaving these theoretical considerations, I wish to call the attention of those who may be disposed for a further investigation of this subject, to the facilities which this mode of exciting electro- magnetism affords, for examining some points, which are as yet undecided. * In this and other instances it appeared to be immaterial whether one half of the bar were heated, or merely the extremity. of Electro-Magnetism by Heat. 63 In determining the effects of the common galvanic apparatus, an almost insuperable difficulty occurs from the loss of power, arising from the gradual saturation of the acid, and oxidation of the plates. In the electro-magnetism excited by heat, provided that the extremities of the bars are kept at an uniform temperature, (which there is no difficulty in doing) the power remains unaltered. There is therefore, by this process, no impediment in determining the relative conducting powers of different metals, or the effects of different dimensions of the same metal. Some experiments, I had previously made, led me to imagine that the conducting powers of metals were materially affected, not only by the diameters of the wires, but by their lengths; the law of which it would be desirable to ascertain. For this purpose, the circuit from a bar of antimony kept at a steady temperature, was made through a connecting copper wire 32 feet in length, the devi- ation was 7°; 16 feet of the same wire gave 10°; 8 and 4 feet gave 154° and 20°. Allowing for the unaveidable inaccuracies of an ex- periment, in which the divisions of the compass were estimated by the eye, it seems that the conducting power of the wire diminishes in a much lower ratio than its length increases. A slight change in the deviations making them 6°, 11°, 16°, 21°, would form an arithmetical series corresponding to the diminutions of length tn geometrical progression. The diameter of the wire used in this experiment was + inch; when 8 feet of wire of t was used, the deviation, which had been 151°, was reduced to 64°, and with platina wire of ;4 it became certainly not more than 3°. When the minute quantity of electri- city developed in this experiment is considered, it might have been expected that the platina, and much more the copper wire of 4 inch, would have conveyed it without loss; yet I found that, even the larger wire of +, was not sufficient, but that the deviation was still augmented by employing wire of 4. This seemed the limit, for 64 Professor CuMMING on the Developement the effect was not increased by using two such wires at the same time; but a silver wire of half the diameter was, in consequence of its superior conducting power, at least equally efficacious, (Table X.) These experiments are confessedly given as mere appreximations, which, hereafter, I hope to rectify, by the aid of a more powerful and accurate apparatus. The new subjects and modes of experiment arising from this hitherto untried department of science, and the light it promises to throw upon all enquiries connected with heat and electricity, have, I fear, led me to encroach too much upon the indulgence of those who may not have the same inducement as myself to excite their attention: I shall therefore conclude, with one quzre, which suggested itself to me whilst writing this Paper, and which, whether true or false, seems to have at least as much plausibility, as some theories that haye been recently advanced upon this subject. Magnetism, and that to a considerable extent, it appears is ex- cited by the unequal distribution of heat amongst metallic, and possibly amongst other bodies. Is it improbable that the diurnal variation of the needle, which follows the course of the sun, and therefore seems to depend upon heat, may result from the metals and other substances which compose the surface of the earth, being unequally heated, and consequently suffermg a change in their magnetic influence ? of Electro-Magnetism by Heat. TABLE I. A List of Subtances heated at one extremity, in contact with Copper Wires ; the Wires being small in comparison with the Substance examined, ex- cepting in the cases marked *. Positive. Bismuth. Mercury. Nickel. Platina. Palladium. Cobalt. Silver. Tin. Lead. Copper. Brass. 1 Nickel + 1 Iron*. 1 Tin+4 Antimony. Solder (common). Pewter. Galena §. Negative. Antimony. Iridium and Osmium*. Rhodium. Gold. Zinc. Tron. Arsenic. 1 Bismuth + 1 Zinct. 1 Bismuth + 1 Tin. 1 Zinc +1 Tin. 1 Zinc + 1 Lead. 4 Zinc + 1 Antimony. 1 Nickel + 1 Palladium*. 1 Nickel + 2 Platina*. Printer’s Type. Fusible Metal. 1 Ditto+ 1 Arsenic. Zinc + Tin + Copper f. Sulphuret of Antimony §. Plumbago. Charcoal. * None of these specimens weighed more than J a grain. + Not an alloy but a mixture, yet was negative whether the heated part appeared to be zine or bismuth. A magnetic compound, capable of polarity, composed of copper two atoms, zinc and tin each one atom.—From Mr. G. Spillsbury. § The sulphurets of lead and antimony alone were examined, probably other sulphurets, phosphurets, &c. would give similar results. Vol. Il. Part I. I Metal, Platina Rod, 18 } inches long, 4 diam. Bar 7 inches r 4, weight 565 grains, Foil 30 grains- ae Palladium 35 grains, iSitver Bar 7 >) er Bar 7 inches by i} wt, 300 grs Silver wire ;',. Ditto .3,. Mercury in glass tube, 6 inches by 4. Professor CuMMING on the Developement TABLE II. Deviations observed with the small Compass neutralized. Brass bar*, 17 inches by 5 and + J Wire. Dev. | Heat. : < Spirit Silver. 70 pos, tinea Brass. 60 pos.| Ditto. Silver. 70 pos. Ditto. Copper. 70 pos.| Ditto. Silver. 70 pos.| Ditto. Platina. 4 neg. | Ditto. Silver 70 pos ; Sue ; ‘| Clamp. Platina. 20 neg.| Ditto. : E Spirit Platina. 50 neg. ; lamp. Silver. 35 pos.| Ditto. Copper #5. |10 neg.| Ditto. Ditto 35. 10 pos.} Ditto. 16 pos.| 170° Copper. 12 pos.| 150 6 pos.| 115 — : Spirit Silver, 60 pos. ha Silver and Pla- tina, Silver in 4/40 pos.| Ditto. contact, Ditto a 20 neg,| Ditto, in contact. Ditto Platina of Z inch in con-¢} 35 pos.| Ditto. tact. Platina, 20 neg.| Ditto. Copper. 35 pos.| Ditto. Zinc. 65 pos.| Ditto. Brass. 45 pos.| Ditto. Metal t. Wire. | Dev, Heat. Copper. Copper.| 16 pos.| Spirit lamp, Platina. 18 neg. Ditto. Silver. | 40 pos. Ditto. Zinc. | 30 pos. Ditto. Zinc. Zinc. | 4pos. | Spirit lamp. Iron. | 6 pos. Ditto. Silver. | 4 pos. Ditto. Copper.|45 neg. Ditto. Platina./50 neg. Ditto. Tin. Copper.| 10 pos.| Solder melted. Zinc. | 27 pos. Ditto. Lead. Copper. 25 pos.| Solder melted. Iron. Copper.|45 neg.| Spirit lamp. Tron. | p——| 2 Ditto. Brass. |40 neg.| Solder melted. Nickel impure, |Copper-.| 35 pos.| Solder melted. Pewter. Copper.| 25 pos. melted. Plumber's solder. | Ditto, | 24 pos. Ditto, Printer’s type. | Ditto, (32 neg.) Spirit lamp. Fusible metal. Ditto, | 5 neg. melted, Ditto Arsenic $, | Ditto, |20 neg, Ditto. 1do.41Type metal.| Ditto, |60 neg. Ditto. 1 Bismutli+-1 Tin. | Ditto, |60 neg.) Solder melted. Antimony+Tin . | Ditto, | 8 pos. Ditto. 1Zinc+1Tin, | Ditto. |22 neg. Ditto, 1 Zinc+1 Lead. | Ditto. |40 neg. Ditto, Zinc+ Antimony ¢.| Ditto. |50 neg. Ditto, Copper-+ Zine+ : : Tin, magnetic. Ditto, |10 pos. Ditto. * The deviation was not affected by heating either the middle of the bar, or of the silver wire by another spirit lamp at the same time. + The metals in this column were square bars 6 inches long by { Proportions uncertain, 3 inch, of Electro-Magnetism by Heat. 67 TABLE III. With small Compass. 180 160 130 120 105 Temperatures. Deviations. A and B, cylinders of antimony and bismuth, each 9 inches long by Z, 133 F plate of bismuth 43 inches by 1 inch and z: B2@, cylinder of bismuth 24 inches by 2. B 2*, the same with four connecting wires. The cold extremities in water at 60°. TABLE IV. Battery of Antimony and Bismuth with the large Compass. Ditto Tempera- a 1 plate, 1 plate tures. See eae Giplates, | 4 plates. | 2 plates. Bismuth. Antimony. 175 170 160 150 140 135 130 | | ' “I © ie SE — aooddod- oo = ooo PIONIN Pls [OPA Nin BIH NIH 125 120 115 110 100 95 90 Aan BIRRIORIS oral Ale | Nin PR BO ROP] RH F/R AIR aa an ao | RJOP[ORIn oonrnroood- | | Deviations. The cold extremities of the plates in water at 57°. 12 68 Professor CuMMiNG on the Developement TABLE V. Battery of Bismuth, with the large Compass. 6 plates. | plate 6. | plate 5. | plate 4.|plate 3.|plate 2.|plate 1. f 45. )243.| 244-4546. | 243455. Deviations, The cold extremities in water at 60°. TABLE VI. Alloy of Bismuth and Antimony, with the small Compass. Copper ‘wer, |Copper and = Silver. La ire. iulver, Melting. 0° a 25 pos. 280 1 pos. 260 0) 250 5 neg 235 7 neg. SSS 200 27 neg. 25 neg. 190 32 neg. - —— 160 36 neg. 57 neg. 140 32 neg. 36 neg. 105 24 neg. 60 (0) Temperatures. Deviations. The colder extremity in water at 60°. of Electro-Magnetism by Heat. 69 TABLE VII. Deviations of Cylindrical rods of Antimony and Bismuth, each 44 inches long by + inch diameter, taken separately and jointly, with the lar ge Ooinnaass ee) 240 220 210 200 190 180 170 160 150 140 130 120 110 100 90 es = o _ it) wn] co |. el | ee fo fo eel af wf alo vl. alo |, tl~ FKFHOWORANAAYNNDDOO we ce fo ake mrPohagqwagleo! = / af ee eee Temperatures. Deviations. The greatest deviation of Bismuth beginning to melt......21°. Ditto of Antimony with two Spirit lena Shane didig Saar minis alley The bar of Platina, ; 19. Deisilver, , (FableTI.) red-heat ..eeseess ces { F The colder extremities in water at 65°. 70 Professor CuMMING on the Developement Bismuth... Bismuth. TABLE VIII. Relative Series of Electro-magnetics by Heat. | Palladium. | f+ | = Mercury... Nickel. ... | Gold. | + | Plumbago. Antimony. Platina.... Palladium. . Cobalt..... | Silver P+ ft fff fa | + | coe: ifol+}+ i+ [+]+ [+ +/+] + t]o]+ +|+|+ Rhodium... [+ {+ [+] Brass....« ol+fif+]+ | | | Arsenic... . ifel+/+i+l+{+}+f+i+f+4l4 [ett | +|+|4+|+ Antimony.. of Electro-Magnetism by Heat. Series of _ Electro-magnetics, by Heat (a). TABLE IX. Comparative Series. Series of Conductors Voltaic Series (d), of Electricity (c). of Heat (d). Bismuth Mercury. Nickel. \ Platina. Palladium. Cobalt. Mea Silver. Tin. Lead. Rhodium. Brass. Copper. Gold. Zinc. Charcoal. \ Plumbago. Iron, Arsenic, Antimony. 2 From Table VIII. > Sir H. Davy, Elements of Chemistry: (*) being inserted from experiment, Charcoal. Platina. Gold. Silver, Antimony *. Copper. Lead. Tin. Tron. Bismuth *. Zinc. Silver. Copper. Lead. Gold. ae Zinc. Tin. Platina. Palladium. Tron. * From Table X. and Sir H. Davy, Phil. Trans. 1821, ® Thomson’s Chemistry. Vol, I, Silver. Gold. Tin. Copper. Platina, Iron. Lead, 71 72 Professor CumMinG on the Developement TABLE X. Conducting powers of Copper Wires of different diameters. Double bar of Antimony and Bismuth with the large Compass ; the colder ex- tremity in Water at 65°. Diameters. Length of wires 16 4 inches. seo iy Seti coe) 8 15.6 | 15.3 8.7 Length of wires 1+ inches. Deviations. When 33 inches of the wires of 4; and -5 were used, the observed deviations were 6°.4 and 3°; with 16+ inches of the same wires, they were 6.4, and 4.27; and with 14 inches were 6.4, and 5.98; therefore, the difference of their conducting powers, which is inconsiderable with short wires, increases very rapidly with their length. Hence it is obvious that wires which do not exceed =!,inch in diameter, are improper in the construction of instruments for detecting minute Galvanic action ; since their loss of conducting power may be such, as to counterbalance any advantage arising from the multiplying action of the spiral. Six wires of +1; 5 each 164 inches lon Four of ..++s> gave the same deviations as one of 4 >'5 ‘ = Three of .. + a> =ty each 1 + inches long. of Electro-Magnetism by Heat. 73 TABLE XI. Conducting powers of different Metals, with the same Bar and Compass. Tempe- rature. Platina.| Iron. 220 : Sei ceed Wes 195 i Diameters of the wires zy Inch. 170 | 10. : 23 ; 160 ; py, : 6 | Length 164 inches. 140 = Diameters =; inch. Length 1 + inches. 11.8 11.45 | 11.3 Deviations. Hence the order of conductors is Silver, Copper, Gold, Zinc, Brass, Platina, Iron. TABLE XII. Deviations by equal differences of Heat, at different Temperatures. Temp. of extremities. Temp. of extremities. Hot. Cold. Cold. Hot. 190 110 180 107 170 104 153 100 140 99 The deviations produced by given differences of heat are therefore increased by raising the temperature of the whole bar. Vol. If. Part I. K 74 Professor CumMinG on the Developement APPENDIX. Sixce this paper was read to the Society, it occurred to me, that, as the juxta-position of two particles of the same metal at different tem- peratures, was the sole condition requisite for eliciting electro-magnetism, it might be exhibited by the minutest metallic specimens. Portions of bismuth and antimony, each weighing one grain, were therefore placed on a silver disk connected with the galvanoscope; on touching the upper surfaces of each separately, with one end of a heated silver wire, the other extremity of which was placed in the other cup of the galvanoscope, the needle deviated through 90°, positive and negative respectively. By this method I was enabled to examine the compound ore of iridium and osmium, of which the largest specimen did not exceed * of a grain, and to verify, in a few minutes, results, for which the laborious process of casting bars of the different metals had been previously requisite *. As the effect of the electro-magnetism developed by heat, is perfectly analogous to that caused by galvanic excitation, in its tendency to produce the rotation of a magnetic bar, it is evident, that, if the magnet be fixed and the apparatus at liberty to revolve, it will, in like manner, exhibit the converse experiment. The instruments formed of Platina and Silver wires, which are represented by Fig. 12 and 13, were constructed for this * If a heated metallic wire be applied to a plate of the same metal, there is in all cases a deviation; the nature of which seems to depend upon some peculiar property in the metal itself. Copper, Zinc, Bismuth, for instance, are positive; Platina, Silver, Iron, Brass, Antimony, are negative. of Electro-Magnetism by Heat. 75 purpose. Platina and iron are obviously unsuitable; and taking into con- sideration the fact of small wires being far more energetic than larger in proportion to their bulk, I believe no other arrangement of metals would be found so efficacious. The wire parallelogram AbcB, Fig. 12, deviates from right to left, or vice versa, accordingly as the north or south poles of the magnet are presented to it; its weight, together with that of the magnet, is rather less than seven grains. The apparatus, Fig. 13*, when heated by a Spirit lamp, exhibits a perpetual rotation, and is analogous to the instrument invented by Ampire for producing the same effect by galvanic excitation. * A small wire is soldered at d in the opposite direction to B, D,C to preserve the equilibrium ; notice of this was accidentally omitted in the plate. Kk 2 ADDITIONS ann CORRECTIONS To the preceding Paper. Page 14. after line 8. add, If one part of antimony and six of bismuth, both in powder, be mixed together and inclosed in a glass tube, they exhibit the same phenomena as the alloy. P. 17. lines 5, 6. for positive, negative; read negative, positive. Addition to Note * same page. If gold, silver, copper, brass or zinc wires be heated in connection with iron, the deviation which is at first positive, becomes negative at a red heat. If the experiment be made by dipping wires not previously connected, in boiling mer- cury, the deviation at the instant of contact, depends, in some cases, upon the order in which they are immersed. This effect is pro- duced by copper with gold, silver, zinc, brass and plumbago, but not with platina, tin or iron; by zinc with gold, silver, brass, iron and plumbago, but not with platina or tin; and by brass with gold, silver, and tin, but not with platina or iron. Note + same page, for below, above ; read above, below. Table VIII. dele the vertical and horizontal columns marked silver, and in the columns for gold read silver. In the horizontal column for rhodium, znseré tin and _ brass negative, and copper positive. Table IX. add galena above bismuth; after manganese dele silver ; insert brass between lead and rhodium; for brass read gold, and for gold read silver. P. 31. line 5. for deviates read revolves. line 8. after exhibits read another arrangement for producing. Ba____4ip wld SMet C Malimandels Lithography Sa dah Leelee ee : " I n ‘ % t i LULA. Platina La. Stlver Support Magnet x! Seepport a agnet Lamp .. Platina Silver od Fe Mos > ANNONA AAA < < - = mor ee LS CO 4H S LS A S OU w A be, — oe > see erage ene eat a ust de shee ie = salem —1; bt al — _ mon aie v7 +67 | SS eae SS ae aS ee v? +b? ee FS the equation to the curve to be traced on the pitch tool. Ex. III. Let AV be an ellipse; ne, 2b" Pas 5? Saree maa cre we n=—s4- In circle nm =— 1; b? les iA 2 2 m—n v v— Bb ‘orp ae are v—b z= a jah Therefore, if the point under consideration is the extremity of the major axis, v is greater than 6, and the construction is possible; but if the point be at the extremity of the minor axis, the construction is impossible. If the original figure be a paraboloid with latus rectum (A), and it be required to form it into a paraboloid with latus rectum (a), it may be shewn nearly in the same manner that the curve < . Yl traced upon the pitch tool must be a common parabola with latus rectum ( a —*) . Telescopic Mirrors and Object Lenses. 99 The chief difficulty is to find a suitable motion for grinding. To grind one paraboloid upon another by a rectilineal motion, or by any motion largely compounded with a rectilineal motion, is evidently impossible. And to grind only by a rotatory motion about the axes, would entirely destroy the surface, by producing rings. Still it may be possible to combine these motions, sufl- ciently for practical purposes, by making the crank, and therefore the rectilineal motion, extremely short, and the circular motion more rapid. The rotatory motion is increased by removing the cogged wheel (s), and putting in its place a small pully with a string passing round the circumference of the fly wheel. If the cogged wheel had twenty cogs, and the pully now substituted be one-fifth of the diameter of the fly wheel, the rotatory motion will be increased a hundred times; or the leaden tool will revolve ten times in a minute, and the mirror twenty times; hence their re- lative motion is thirty revolutions in a minute. The quantity of rectilineal motion should be the least possible, that is sufficient to prevent the formation of annular streaks on the mirror. As far as respects the grinding of conoids, this subject is only theoretical ; so that appropriate improvements and directions must be sought from a few experiments. NOTE On the Attempts to grind Lenses and Mirrors by Machinery, and to give them a Parabolic Form. wrererrcccrsceerrer en en errr We find very early notices of attempts to apply machinery to grind- ing the object-glasses of telescopes. In the first number of the Philosophical Transactions (1665), mention is made of glasses produced this way by Campani in Italy. Campani at Bologna, and Devini at Rome, were about that time disputing on the relative merits of the lenses which they produced, and the preference was generally given to the former. A machine employed by him is said to be still preserved in the apartments of the Institute at Bologna, and was found there by M. Fougeroux; it is not mentioned what peculiarities it had, but it was something of the nature of the common lathe, and probably, like almost all that have been since invented, was used only to assist, and not to supersede the work of the hand. Hooke in his Micrographia, published 1665, describes a machine for performing the whole work. In this, the glass and the tool turn each about its axis, these axes being inclined at a small angle to each other, and consequently, the motions of each particle of the glass and the tool being oblique with respect to each other. This method, however, appears liable to the objection which we shall find to apply to almost all those which have been proposed without experimental proof of their sufficiency. Namely, that each particle of the glass is ground by a surface whose motion, relatively to it, is, at similar points of the revolution, always the same, so that the inequalities of friction would act in the same manner in each successive revolution; and these cycles of action would probably cause the friction to affect the parts differently, according to their distance from the axis. It was also objected to by 101 Auzout, himself a constructor of telescopes in France, on the ground that it was impossible to give that steadiness and firmness to its structure which were requisite for grinding glasses of any long focus; an effect which Hooke promised as a consequence of its adoption. In 1668 we find a description of machine by Mancini, which consists, however, only of a long arm, one end of which is fastened to the ceiling, while the glass is fixed to the other end worked ona plane. In 1676 Borelli was celebrated as a constructor of telescope-glasses, but we are not aware of the method which he employed. To 1719 belongs a work of Leutman’s, entitled, “Remarks on Glass-Polishing, in which are described improved machines for bringing glasses to greater perfection by the help of three motions.” This Work I have not seen, and cannot therefore be certain what is meant by the three motions here mentioned in the title. In 1741 Mr. Jenkins published in the Philosophical Transactions his method of grinding glasses spherical. It consists in making the glasses, fixed on the surface of a sphere, of which they are segments, revolve round one axis, while a hemispherical cap, which is fitted to the globe, and which grinds them, revolves on an axis at right angles to the former. This, though in appearance much different from Hooke’s machine, is nearly the same in principle, differing only in having the axis inclined at a different angle, and seems to be liable to the same objections. Huyghens’s machine is described in Smith’s Optics. It only serves to produce pressure, and to reduce the action of moving the glass backwards and forwards on the tool, to the action of turning a winch; the care of avoiding inequalities, by changing the relative position of the glass and tool, is left to the operator. Nollet’s machine, among the Machines Approuvées par Academie Royale des Sciences for 1733, (tom. VI. p- 127), is merely to give a rotatory motion to the tool by the feet, while the glass is ground upon it by the hand. Another machine for grinding glasses is described in the Machines Approwvées for 1736. It is by M. de Parcieux, and seems more likely than 102 the preceding to answer in practice. It gives a circular motion to the tool, while the part carrying the glass has an eccentric motion, or towrnotement, and also another motion in consequence of its being loose in the circle which carries it. The description is accompanied by a report of Cassini, de Mairan, and de la Chevaleraye, who say that they have examined the glasses produced by this machine, and that several of them seem to be good. I find also in a catalogue of works on this subject, the title of a ‘ Description of a new invented and very convenient machine for grinding optical glasses,” this is in the Berlin Journal for the Diffusion of Infor- mation, tom. 1V. p. 92. With respect to parabolic, elliptic, and hyperbolic figures, we find also very early pretences to the art of grinding them. In 1668, Smethwick in England, and De Sons at Paris, claimed the possession of the means of doing this. In the Philosophical Transactions for Nov. 1669, is a method, given by Sir Christopher Wren, for grinding hyperbolic glasses ; which is theoretically a most elegant application of a theorem of solid geometry, discovered by that great mathematician, though practically it would probably not succeed, being exposed to the same objection which we have already mentioned as applicable to Hooke’s. In 1726, in the Miscellanea Berolinensia, (tom. Ill.) we find Hertel’s method of grinding parabolas, ellipses, and hyperbolas, which consists only in making the glass or mirror revolve upon its axis, while it is cut by means of a tool directed by a gage to the proper form. This it is obvious could never produce a figure free from annular inequalities. In 1777, a valuable paper of Mr. Mudge, on grinding specula, appears in the Philosophical Transactions, in which however no machinery, pro- perly speaking, is used. He there gives a method of producing a parabolic, 103 or an approximately parabolic form, by a particular method of giving the strokes after the polishing begins to take place. This appears to answer the purpose in practice, and corresponds with the anticipation of Newton, who suggested that finally we should obtain a method, not mathema- tical, but mechanical, of giving the parabolic form to mirrors. From what has been said, it will be seen, that the method proposed by Mr. Cecil, differs materially from all hitherto proposed. Addition to Nore, p. 96. Tue different conclusions in this note for the grinding surface necessary to obtain the different conic sections, are reducible to the simple condition of the absolute value of the ordinate in the curve (sec y*) traced upon the pitch tool: or what is the same thing, to the time of grinding, the absolute value of the ordinate being given. It will be most convenient to assume the ordinate so as to make the grinding surface continuous at the circumference of the pitch tool, (see Fig. 5.) The figure of the mirror, which is at first truly spherical, will migrate successively into an elliptical spheroid, a paraboloid, a hyper- boloid, according to the time of grinding. ‘The times of producing these effects will be respectively proportional to v* — 6°, v*, v> + b*. WAC: ERRATA. Page 96, Line 9. For Fig. 6. read Fig. 5. —— — Note. Line 5 from the bottom, for ax + ma‘? read aa’ + nx’. —— 98, Note. Lines 7, 8, 13, 14 from the bottom, » v*-+ 5? ah ihe for = rea va W. Lowry, scudp VIII. On the use of Silvered Glass for the Mirrors of Reflecting Telescopes. ByvG: B. AIRY, B.A: OF TRINITY COLLEGE. FELLOW OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. [Read Nov. 25, 1822.] Tue idea which probably first occurred to the inventors of reflecting telescopes was that of constructing their reflectors of silvered glass; but it appears to have been immediately rejected. The difficulty of grinding and polishing metallic specula was then so great, and the formation of glass reflectors so easy, that some serious disadvantages must have appeared to be connected with the use of the latter, or Newton would not have bestowed so much labour on the construction of metallic mirrors. The princi- pal objections appear to have been these: if the surfaces were ground to equal radii there would be a confusion of the images produced by reflection at the two surfaces of the glass: if ground to different radii, refraction would be introduced, and consequently dispersion, which it was intended by the construction of the reflecting telescope to avoid. Besides these, the loss of light by reflection from glass is perhaps greater than that by reflection from polished metal. For these reasons, it would seem, the use of glass reflectors has been entirely neglected, and opticians have endeavoured to improve the reflecting telescope only by improving the composition of the speculum-metal. Vol. If. Part 1. O 106 Mr. Arry on the use of Silvered Glass It appears surprising that no attempt has been made to remove the inconveniences attached to the use of glass reflectors. It occurred to me some time since that in Gregory’s or Cassegrain’s telescope, supposing the mirrors constructed of silvered lenses the radii of whose surfaces were different, the chromatic aberration of one mirror might be corrected by that of the other. For example, suppose the great mirror to be a meniscus silvered on its convex side: since the convergence of the rays is caused partly by the refraction of the meniscus through which they pass twice, and partly by the reflection at the silvered surface, the violet rays will converge sooner than the red, but the aberration will not be so great as if the convergence were occasioned entirely by refraction. Now if the small mirror be so constructed that its chromatic aberration may be equal to that of the great mirror but in the opposite direction, that is, so that the focal length for violet rays may be greater than that for red rays, as much as the focal length of the great mirror for red rays is greater than that for violet rays, the rays of all colours will after the second re- flection converge to the same distance from the small mirror, and will therefore form an image free from chromatic aberration. This may be effected by using for the small mirror a concayo- convex lens, silvered on its convex side. For since the refraction tends to make the rays diverge, and the reflection to make them converge, the convergence produced, being the excess of the latter above the former, is greater for red than for violet rays, and con- sequently the focal length is greater for violet than for red rays. In this manner by a proper adjustment of the surfaces, the rays of all colours m each pencil may always be made to converge and form an image at the same distance from the small mirror. It is yet desirable to correct if possible the spherical aberra- tion by the use of spherical surfaces only. The method by which the chromatic aberration is corrected naturally suggested a mode for the Mirrors of Reflecting Telescopes. 107 of correcting the spherical aberration also: namely, by making the spherical aberration of one mirror equal and opposite to that of the other. From the investigation it appears that the determi- nation of the radii of the surfaces proper for this purpose depends on the solution of a cubic equation, and therefore the correction of the spherical aberration is always possible. This will enable us to employ so large an aperture that the objection founded on the loss of light will be entirely removed. In the construction of the telescope it is necessary to attend to another consideration, the nature of which may be thus explained. The apparent distance of an object seen through a telescope from the center of the field of view, depends on the angle made by the axis of the pencil of rays, when it enters the eye, with the axis of the telescope. Now when the axis of the pencil which is reflected from the great mirror is incident on the small mirror, the refraction of the concavo-convex lens, which forms the small mirror in the Gregorian construction, will disperse the axes of the differently coloured pencils: the axis of the red pencil will meet the first eye-glass at a greater distance from the axis of the telescope than that of the violet pencil. The refraction of the first eye-glass would make these axes intersect at a distance greater than its focal length. If then no other eye-glass were interposed, the axes of red and violet pencils would enter the eye in different directions, and the image formed by the red rays would appear more distant from the center of the field of view, than the image formed by the violet rays: that is, objects near the edge of the field of view would be coloured. This will be removed by placing a concave lens near the eye, which will cause the axes of pencils of all colours to enter the eye parallel to each other. By similar reasoning a combination of eye-glasses may always be found, which will produce an image perfectly free from colour. 02 108 Mr. Arry on the use of Silvered Glass These investigations were prepared for the notice of the Philosophical Society, when I discovered that the same idea had been published in the Philosophical Transactions for 1740, by Mr. Caleb Smith. He has given without demonstration some for- mulz for the correction of colour, but the correction of spherical aberration he leaves, as leading into too intricate investigations ; the achromatism of the eye-piece does not seem to have occurred to him. He speaks of having made one experiment which gave him the greatest hopes of success. As the construction has not been remarked by any of the friends to whom I have mentioned it, and as the subject is one of considerable interest, I am induced to think that a statement of the principles, and an investigation of the formule for the construction of telescopes on this plan will be not unacceptable to the Philosophical Society. (1). In the following articles we shall always consider the radii of convex surfaces as positive. We shall denote by x the ratio of the sine of incidence to the sine of refraction out of air into glass for mean rays, and by 6” the variation of this ratio arising from unequal refrangibility. For the alteration in focal lengths, &c. arising from chromatic aberration we shall use 6, and for those produced by spherical aberration we shall use d. Our approximation will be extended to the first power of én, and the second power of the apertures; and all our investigations will be made on the supposition that the telescope is Gregorian. (2). A mirror is formed of a double convex lens silvered on one surface: rays are incident from a given point in the axis; to find the focus of reflected rays. (3). 1%. For first refraction. Let C be the center of the sur- face 4D; B the given point; BD an incident ray, DE the refracted the angle ACD=0; AD=a. Then sn BDF =n.sin EDC; or = sin 0 = pes sin 0; or BC.ED=n. BD. EC. payee edinley Re. eal, SISTA) Cyrene eh Gers Ml Cates (neglecting in the expression for cos @ the powers above the second) flea MN Net «3 SHI RDP S TiN ChE 1 z a Stee bea A= Ht pep a7 Dt CTP) 5 3's since a = AC x 0 = Af a 1 a Similarly, ED = ae) Hence the equation becomes Crip Getta) =A a) Gy taD-5)- or (r+D) (1-2.7=2.£) =n (r—2) CEES: r+D n—1 D = > and «= rT——. n n n Let a=0; r+ D=n(r—-x2); r—x 2 Substituting these approximate values in the coefficient of =: (r+D) €é - =. (r+D) (n=1 .7—D).£) =n (7-2) ( +D (D+r)£), a? (r+D) (n—1 r—D)+(r+D).D.n*) 2 me J r+D n hence r—2 = {1 = 110 Mr. Arry on the use of Silvered Glass r+D_ a@ (r+D)’.(n—1 r+n’—1D) n 2 n° = Date as and 2 =2—4y—— 4 2 2 (rp Dy. (rtm FID). : 1 : (4). 2”. For the reflection ; 2 be the distance of the focus after reflection from the reflecting surface, measured in the same direction ; r the radius of that surface; then in the formula just found making = —1, putting —R for r (since the rays are incident on the concave surface) and —2 for D, we find 2 2 y=—-2R-2 + >.2(R+2)*.(—R)= ~2R-2->.2R.(R+2). (5). 3”. For refraction at emergence; let : be the distance of the focus of emergent rays, measured in the opposite direction ; 1 in the same formula put B for n,-—r for r, y for D; 1 ol = — =e (y-r).(-r+2 + 1 -y) rod, oe n n° ——_—_— a: — ——— =n-1.r—ny—>-.n.n—1 (y-r)y.(n+1.y—nr). (6). From (3) R+2 approximately = “—t r+R— = n—1 D n r+ — n therefore by (4), 7 Sy Ey ia {2 (r+ DY. (+m F1 D)F2R. (ee (7). From (6), y-r approximately = —2R - =" _ 2n?—1 n+1 + D: n n 7 n+1.y-nr=— 2.n+1.R- for the Mirrors of Refiecting Telescopes. 111 n—1 n> hence by (5), s=2.n—1 r+anR-D+ >} (r+D)?.(r+n+1.D) (n—1.r+nR—D)?* ——— 20-1 Dy»? SS ae Se EN ies Pe r-—) x (2.0F1.R+ 71, -"**p)h n n (8). The thickness is neglected, as its intreduction would not sensibly alter either term of the expression for 2. (9). Let @ be the distance of the focus of emergent rays from the surface; then n=8; therefore da = — a’dz, and da= — a%éz, nearly. Let © and F denote the values of 2 and a, when the incident rays are parallel; then dF = — F?dZ, and 5F=-— F°8 8. (10). That the chromatic aberration in-a telescope may be destroyed, it is necessary that the images formed by rays of different colours after reflection from the small mirror be equally distant from that mirror. If now we suppose rays of all colours diverging from the place at which the second image is formed, to be incident on the small mirror, the rays of each colour after reflection will converge to the point in which an image is formed by rays of the same colour reflected from the great mirror. Let a be the distance of the first image from the small mirror, F the focal length of the great mirror, 4 the distance between the mirrors, 7 the distance of the second image from the small P 1 5s : : mirror, and = the radii of the unsilvered and silvered surfaces of the small mirror; since b=a+F we have 5a+édF=0. But by (7), - = 2.n—1.p+2np'—D, neglecting at present spherical aberra- tion; therefore da = — 2a° (p+p') dn, since by supposition D is the same for all colours: similarly })F=—2F? (r+R)én. Substituting 112 Mr. Airy on the use of Silvered Glass these values in the equation da+ *=——. where h and / are independent of n: n ea pa ee eer ae ee tes eee ee , ce. ie ee eae Daily ic np +2n—1p; - 4 —2f(p'+p) on. Substituting and dividing by 5x we get at length g 2Qpb — — = b,— —_ = b a (4--2.pF+2 bp.n—1.p +7) a+ (-5 494-2 bp.n—1 Pp et (1-7-2601 +p) 4 — : o pb Sh SS SS — = eA hae n—1.p—+p—2b.n—1.p'+p-g When & has this value, objects near the edge of the field of view will not be coloured. Jor the Mirrors of Reflecting Telescopes. 115 C 1 , Lee Fe , a sarees (16). Since p= Ine +2.H—-1.p> 2p +2.n—1.p+¢, we may change this expression into the following: “be apb.p' =a G- 2b )g which is rather more convenient for practical application. (17). To construct a telescope on these principles it will be most convenient to assume the values of F anda; Db will generally be rather greater than the distance of the two mirrors; and / will be found by the equation D + hes? fi To determine p, »’, r, R, we have then the four equations ~— 1 2np+2.n—-1.p=- 7 p ra -+(7) 2nR+2.n—1.r peels a’ (p+p')+F° (r+ R)=0.. (10) -1 § —— ,(n—1. ‘— D)? at {7 (0+ Dy’ .(ptntiD)+2p Cm betnp ~D) +n.n—1. (2p ee aren a (2.041 Pert il ste p-"** v)} 4+ FF om 1 yap (al .r+nR) n +n.n—i (2R+ r) (2 Pres et seg #1} =0....(11). From the three first equations ld etd p= ont he 116 Mr. Arry on the use of Silvered Glass a neta Se oS ree te a Wee a =a We Wr i a we get for the determination of p the Substituting these values in the fourth, observing that D + - 4 following equation : = n+3.F*—3 a’) at 2 3.a* D*+1.a@ F* ——— ee eee Te FDP Hoe: 2a FB G- D) a a*\2 ; nee (2741.F-2) ee 3 @ (ag) x a ! 8 P ans + F? mn Gey re F2 4} [Ga ri). C8) soy er? (18). This equation att a cubic has at least one possible root; and p’, r, R, are then determined by the expressions Baa Grr). Thus it is always possible to correct the spherical and chromatic aberrations, except F* =a’; a case which can never occur in practice. for the Mirrors of Reflecting Telescopes. 117 (19). The coefficients of the equation of (16), may then be found, and by assuming values of g and gq, the corresponding values of k may be found. (20). The place of the second image may now be found ; and if Ses fe 3 the value assumed for Dp Was not sufficiently accurate, the process may be repeated, and a more correct value found for p, &c. The equation of the eye-piece will probably not require alteration. (21). It is scarcely necessary to observe that the small mirror must not be changed as in telescopes with metal reflectors; the power must be altered only by changing the eye-piece. (22). The investigations have all been made on the sup- position that the telescope was Gregorian. For one of Cassegrain’s construction it is merely necessary to take a and f negative. The advantages which telescopes on this construction might be expected to possess over those in use may be estimated from the following statement. The principal difficulty in the construc- tion of the achromatic lens arises from the irregularities in flint- glass, which render it almost impossible to make an object-glass of large diameter. As only one kind of glass is necessary for the construction here proposed, the artist evidently has the power of selecting that which will be most easily found free from irregu- larities. From the different laws of dispersion in different kinds of glass the exact correction of colour by the achromatic lens is impossible; but as in the proposed construction the same kind of glass is used in every part of the telescope, it is not liable to the same objection. Considerable difficulty is found in adapting to each other two lenses of crown and flint; for this telescope the radii, &c. might be calculated with the certainty of removing all aberration. For the object-metal in reflecting telescopes of 118 Mr. Airy on the use of Silvered Glass, §c. large aperture, the parabolic form is absolutely necessary, and this can be given only by the best workmen. The advantage of this construction in which none but spherical surfaces are used is sufticiently obvious. The difference in the quantity of light reflected, if there is any, appears so small that it offers a very slight objection; especially as we can increase the aperture with- out fear of indistinctness for aberration. I have constructed two Cassegrain’s telescopes on this plan, whose object-mirrors are 4 inches in aperture, and 20 inches in focal length. From some cause with which [ am unacquainted the image of a star or planet is surrounded with radiations which make the telescope quite useless for practical purposes, and render it extremely difficult to pronounce any thing on the success of the principle. I have not however been able to observe the slightest appearance of colour: of the spherical aberration I can- not, in consequence of the radiation, speak so decidedly, but I am certain that if there is any it is very small. But the success of a principle of this kind is not to be determined from one or two experiments; several trials should be made, and every endeavour used to overcome the difficulties which always occur in instru- ments made on a new construction; and even if it should appear at present to fail, some improvement in the theoretical principles or in the practical application may make it useful hereafter. It is my intention to make new trials; and to attempt the cor- rection of the defect at present existing; and the results of my experiments will be communicated to the Society should this paper appear worthy of their attention. G. B. AIRY. Trinity CoLuece, Nov. 25, 1822. IX. An Account of some Experiments made in order to determine the Velocity with which Sound is transmitted in the Atmosphere. By OLINTHUS GREGORY, LL.D. ASSOCIATE ACAD. DIJON, HONORARY MEMBER OF THE LITERARY AND PHILOSOPHICAL SOCIETY OF NEW YORK, OF THE NEW YORK HISTORICAL SOCIETY, &c. SECRE- TARY OF THE ASTRONOMICAL SOCIETY OF LONDON, AND PROFESSOR OF MATHEMATICS IN THE ROYAL MILITARY ACADEMY AT WOOLWICH, [Read Dec. 8, 1823.] Tue theoretical investigations of different philosophers, im order to ascertain the velocity with which Sound is transmitted through the atmosphere, however ingenious and elegant some of them may be, seem to rest too much upon gratuitous assumptions, to allow any cautious enquirer after physical truth, to receive them unhesitatingly, except so far as they may be confirmed by accurate experiment. Unfortunately, too, the results of experi- ment present irregularities both formidable and perplexing; since many of them cannot well be imputed to any want of skill, or caution, in the conductors of the enquiry. 120 Dr. Grecory on the Feet per Second, Thus, Mr. Roberts assigns a velocity of......-..-. Pee 1300 Mir: Bowle mere es alas is ne oe he ee cg 1200 Mr. Walker, and Duhamel .............-....-. 1338 Mersenne, in his Treatise de Sonorum Natura, (@ausiset Piechibusse- eee eee eee ee 1474 The Florence Academy .........-.-+-++-e +00: 1148 Cassini de Thury (Mem. Paris Acad. ann. 1738) 1107 i.) F073 eR aes io DenGuuepyaoice 6 © Herricks meareuomme HU 27 Derham. <8 6 hee oe Ie ae A HWY Te UV) Sateen ene fr welts ip ou olgtekaiaeee oie 1109 | ETE 2) Ae RRR ORS Say ice Onan oder De eet eee 1130 Arago, &c. from experiments in June 1822, give 337.2 metres, at the temperature of + 10° Cente TAde ne eon crea eas eee 1106.32* The theoretical formula most generally adopted, especially by Continental philosophers, is this :— Velocity in horizontal direction = 333.44 met. ,/1+.00375¢; the metre being = 3.2809 English feet, and ¢ denoting the mdication of the temperature upon the centigrade thermometer. T am inclined, however, to think that this can only be regarded as an approximative formula; and that we are not yet in a state to receive otherwise, than as an approximation, any theorem which simply includes the variations of temperature. The air is subject to various classes of changes, indicated by the barometer, ther- mometer, hygrometer, and anemometer respectively, as well as others probably, for the ascertaining of which we have not yet any appropriate instrument. If we could select these, one by one, ad libitum, and carry experiments first through a moderate * This is the last result of which I had heard, previously to the commencement of my own experiments. OO a Te Velocity of Sound. 121 range upon the barometric scale, all the other probable elements of modification remaining constant; then, through a sufficiently extensive range upon the thermometric scale, the others remaining invariable, and so on; the question would soon be set at rest: but this is impossible. It becomes desirable, therefore, to augment the number of recorded facts, as they result from accurate ex- periments, in order that, at some future (and it is hoped, no very remote) time, a cautious investigator may so select, compare, and classify them, as to deduce a more comprehensive and accurate theorem than is yet known. With a view to contribute, though in a small degree, to this purpose, I now present an account of a few experiments made by myself in the course of the present year. My objects were, to ascertain the velocity with which the sound passed over the surface of the earth, over the surface of water ; under different temperatures ; m a quiescent state of the atmosphere, and in windy weather; by day and by night; the velocities of direct and reflected sound ; and the velocities of sounds of different intensities and produced by different means. As yet the expe- riments have not been carried to their projected extent ; but while I record the results thus far obtained, I look forward with hope, that, in another year or two, I shall be able to complete them satisfactorily. The instrument with which I measured the intervals of time, was one invented and made by Mr. Hardy, by means of which, with a little previous practice, I could measure an interval accu- rately to a tenth of a second, and approximatively to a twentieth ofa second. The velecity of the wind was ascertained by means of an anemometer ; and the barometer and thermometer were of the best construction. I employed no hygrometer, (much as I wished it); for as yet, I am not acquainted with any in whose results I should be inclined Vol. \X. Part 1. Q 122 Dr. Grecory on the to confide. With regard to the distances between the stations at which the sound was emitted and heard, they were in some cases taken from the Ordnance Map of Kent, and verified by new ope- rations; in others they were determined by actual and careful measurement: in others by trigonometrical operations with accurate instruments. The whole were conducted with care; and it would be useless to enter into the detail of them. Fripay, January 3, 1823. A musquet was fired from the battery near the Royal Artillery Barracks, and the interval of time between the flash and sound was observed at two different distances on the mortar-range, di- rection nearly north and south. January 3, half past 2, P.M. barom. 29.7 inches, Fahr. therm. 45°, rather moist atmosphere, but no rain; very gentle wind blowing in direction nearly perpendicular to that of the range. Distance from musquet to my station 3600 feet. Sta rounds fired: in one the interval of time employed by the sound in passing over the 3600 feet was doubtful: in the other five the intervals were 3”.25, 3”.3, 3'2.5,, 3.2, 3”.26, the mean of these is BOE 3600 3.252 Same day, three o’clock p.m. barom. 29.64 inches, Fahr. therm. 45°. atmosphere, wind and weather as before. Distance from musquet to station 3600 feet. Five rounds fired, intervals, 3”.2, 3”.2, 3.3, 3”.3, 3’.25; their mean 3”.25. 3600 : Spr As feet, velocity of sound; therm. 45°. =1107 feet, velocity of sound; therm. 45°. Same day, half past 3, p.m. barom. 29.64 inches, Fahr. therm. 45’. atmosphere, wind, and weather, as before. Distance from musquet to station 2100 feet. Eight rounds fired EE Velocity of Sound. 123 Interval between flash and report in one case doubtful: the others were 1.88, 1”.88, 1”.9, 1.9, 1.9, 1.9, 1.91; the mean 1’.896, 2100 1.896 Thursday, January 9, three quarters past 7, Pp. M. dark, but clear, star-light, frosty night. Barom. 29.82 inches, Fahr. therm. 27°. Dry ; no wind. Musquets fired from the battery, as before, distance 3600 feet. Six rounds fired, one doubtful. The other intervals between observing the flash and hearing the report, were 3”.25, 3’.28, 3.3, 3”.3, 3”32; mean 3”.29, 3600 3.29 The sound of the same charge, fired from the same musquet, = 1108 feet, velocity of sound; therm. 45°. = 1094.2 feet, velocity of sound; therm. 27’. was heard much more intensely, on this clear frosty night than in the day-time of January 3, at the same distance 3600 feet. Same day, January 9. Being anxious to extend the experi- ments to greater distances, I had previously applied to General Ramsey, of the Royal Artillery, the Commandant of the Garrison here, for the use of cannons as well as musquets, these, with his accustomed courtesy and kindness, he immediately ordered to be at my disposal, whenever I should need them in the course of my experiments. On the morning of this day, therefore, I chose a station for the gun, on the side of Shooter’s Hill, between Severn-Droog Castle and the 8 mile-stone on the Dover road. I selected three other stations from which the gun could be seen with a good Theodolite telescope; one of these was at the entrance of the lane turning from the Dover road to Charlton, between ‘‘the Sun in the Sands,” and the 7 mile-stone; the second in the Kidbrook Lane which turns off from the Dover road between the 6 mile-stone and ‘the San in the Sands;” and the third on Blackheath, nearly in a Q2 124 Dr. Grecgory on the continuation of the western wall of Greenwich Park towards the windmills. These three stations are probably 200 feet above the high water-mark in the Thames at Woolwich; and the statien at which the gun was placed is still more elevated. The distances, as accurately measured, were, from the Shooter’s Hill station to that in Charlton Lane 6550 feet; from Shooter's Hill to that in Kidbrook Lane 8820 feet ; from the Shooter’s Hill station to that on Blackheath, 13440 feet. The gun employed was a six pounder, the charge of powder eight ounces. The serjeant-major who remained at the gun, was directed to order the men to commence firing at a certain minute by his watch, (which was previously made to agree with mine) and then to fire regularly a certain number of rounds at intervals of two minutes ;: this was the practice throughout the experiments, the gun was always pointed towards me, at a very small elevation, except. it be otherwise expressed. January 9th, noon. Barom. 29.92 inches; Fahr. therm. 33° ; wea- ther dry, wind scarcely perceptible, a clear cloudless frosty day. Six rounds fired. Interval of passage of sound from Shooter's Hill to Charlton Lane, 5”.9, 6’.0, 5’.9, 6”.0, 6.0, 6.0, their mean 5”.92, distance 6550, 6550 5 Fae 1098 feet, velocity of sound; therm. 33°. Same day, January 9, half past 12, barom. 29.86 inches; Fahr. therm. 33°; weather dry, wind scarcely perceptible. Six rounds fired. Result in reference to one, very doubtful. Intervals of pas- sage of sound from Shooter’s Hill to Kidbrook Lane, were 7”.95, 8”.0. 8”.0, 8.0, 8”.05; their mean 8’. Distance 8820 feet, 8820 ; —g = 1102; feet, velocity of sound; therm. 33°. Same day, January 9, quarter past 1, Pp. M. barom. 29.82 inches ; Fahr. therm. 33°; weather dry, wind scarcely perceptible. Five Velocity of Sound. 126 rounds fired, intervals of the passage of sound between the stations at Shooter’s Hill and Blackheath, 12”.2, 12”.25, 123, 12’.24, 12”26; mean 12”.25, distance 13440 feet, 13440 122 + (1098 + 11024 + 1097) = 1099: feet, mean velocity from the sixteen rounds ; therm. 33°. Monday, February 17, noon. Barometer 29.98 inches; Fahr. therm. 35°. Air humid; but neither rain nor sleet; very gentle wind, N. E. by E. Employed bells on the mortar-range on Woolwich Common, lying nearly north and south. A bell rung at the north station, was heard by a soldier at the south station, who immediately rang another bell, having his arm elevated for the purpose. I stood by the soldier who rang the first bell, and measured the interval of time between the sound of the first bell, and the sound of the second bell, when trans- mitted from the other station. By several preceding experiments, I estimated the time which =1097 feet, velocity of sound; therm. 33°. elapsed between the moment, when the man with the second bell heard the sound from the other, and struck the clapper against his own bell, finding it to be one-fifth of a second, this, therefore, I deducted from the intervals which marked the passage of sound, before I recorded them, as below. Distance between the two bells 1350 feet; whole distance tra- versed by the sound 2700 feet. Intervals elapsed (corrected as above) in five experiments ; 2”.5, 2’.48, 2”.44, 2.46, 2”.42 ; mean 2".46, 2700 2.40 Same day, quarter past 12, barom. therm. wind and weather as before. Distance between the two bells 1650 feet; whole distance 3300 feet. Intervals elapsed in four experiments; 3”.0, 3”.0, 3.0, 3”.0, = 1098 feet, velocity of sound; therm. 35°. 126 Dr. Grecory on the 3300 Y —- =1100 feet, velocity of sound; therm. 35°. Same day, half past 12, barom. therm. wind and weather as before. Distance between the two bells 1800 feet ; whole distance 3600 feet. Intervals elapsed in five trials, 3’.25, 3’.24, 3”.26, 3.25, 3”25; mean 3”.25, 3600 35 = 1108 feet, velocity of sound; therm. 35°. +(1098 + 1100+ 1108) = 1102 feet, mean velocity from this day’s experiments; therm. 35°. Friday, May 23. This morning there was a tolerably brisk wind blowing from the S.W. by W. nearly in the direction of my Charlton’ and Kidbrook stations from Shooter’s Hill. Of this I gladly availed myself, as the morning was in other respects favourable, in order to ascertain what would be the effect of such a wind upon the velocity. Cloudy, air humid, but no rain. I measured the velocity of the wind frequently with an ane- mometer, and found it vary between 22 and 26 feet, the mean 24 feet. The gun a six pounder, charge 8 oz. of powder. 11, a.m. gun at Shooter’s Hill, sound heard at Charlton Lane, distance 6550 feet. barom. 29.66 inches. Fahr. therm. 58°, air humid. Sta rounds fired ; the intervals were 6’.1, 6’.05, 6”.0, 6”.05, 6”.0, 6’.04; their mean 6".037. = 1085 feet, velocity of sound, when opposed by the wind. Same day, quarter past 1, P.M. barom. 29.67 inches, Fahr. therm. 60°; air dryer. Gun at Charlton Lane. Sound heard at Shooter's Hill. Distance 6550 feet. Five rounds fired : the intervals were, 5".65 doubtful, 5”.8, 5.78, 5”.76, 5".78, omitting the first, the mean interval of the other four is 5”.78. stitel tee pee Velocity of Sound. 127 6550 5.78 z (1085+ 11335) =11094 feet inferred velocity of sound indepen- dently of the wind; therm. 59°. And (11333 — 1085) =24+ inferred velocity of the wind at the times of the experiment, supposing it to be nearly the same at both = 11335 feet, velocity of sound, when aided by the wind. times. This agrees quite as nearly as could be expected with the mean velocity of the wind determined by the anemometer. Same day, May 23, half past 11 a.m. barom. 29.67 inches. Fahr. therm. 58°: air humid; wind as before. Before the gun was removed from Shooter’s Hill, sia rounds more were fired. The intervals in which the sound reached Kid- brook Lane, were 8".1, 8.125, 8”.13, 8’.15, 8.1, and one very doubtful. The mean of these is 8’.121. Distance 8820 feet. 8820 8.121 Here the sound was but just audible, the wind diminishing its =1086 feet, velocity of sound, opposed by the wind. intensity exceedingly. | Same day, therefore, the gun was removed to Kidbrook Lane, while I went back to Shooter’s Hill. Half past 12, barom. 29.67 inches; Fahr. therm. 60°; air dryer ; wind as before. Stax rounds were fired. The intervals between the flash and the report were 7’.8, 7.7, 7”.8, 7.78, 7’.78, and one very doubtful ; mean 7”.77. 7 = 1136 feet, velocity of sound, when aided by the wind. = (1086+1136)=1113 feet, inferred velocity of the sound inde- pendent of the wind ; therm. 59°. 4 (1136—1086)=25 feet inferred velocity of the wind, nearly as before. The same day, May 23, in the afternoon, the wind subsided, so as not to exceed 6 or 8 feet per second, while the temperature of the air remained nearly the same. I anxiously availed myself of 128 Dr. GreGory on the this opportunity to ascertain the velocity of the sound, when scarcely affected by the wind. Mortars and howitzers were firmg from the battery, the former at an angle of 45°, the latter at low angles for Ricochet practice. At 3; P.M. when the barom. was at 29.68 inches, Fahr. therm. at 60°, the sun shining, I took a station 3100 feet from the battery, and in a direction nearly perpendicular to that of the wind, then gently blowing. I observed the intervals between the flash and the report, for six rounds, of which the first three were with howitzers, the next three with mortars; these were successively 2”.77, 2".76, 2”.79, 2”.79, 2”.8, 2”.8; their mean 2”.786. ae =1112 feet, velocity of sound ; therm. 60°. In these latter experiments the sound was very distinct and sharp: the result, though drawn from a short distance serves to confirm the preceding results on the same day. Thursday, August 7. On this day, which was cloudy, but with intervals of sunshine, I employed the same 6 pounder as before: sometimes with charges of 8 oz. of powder, at others, when the distance required it, with 12 oz. The wind was quite brisk, varying in velocity from 30 to 35 feet, as determined by an anemometer. At eleven o’clock a.m. barom. 29.80 inches; Fahr. therm. 66° ; air dry, cloudy, but sun shining; wind nearly opposing the motion of the sound, and having a velocity of 30 feet. Sta rounds were fired from Shooter's Hill. The intervals occupied in the passage of sound from thence to Kidbrook Lane, distance 8820 feet, were 8”.1, 87.15, 8".16, $”.13, 8”.13, 5”.12; their mean, 8.13. 8820 S13 7 1085 feet, velocity of sound, when opposed by the wind. Same day, August 7, quarter past 1, p.m. barom. therm. wind and weather as before. The gun being placed in Kidbrook Lane, I went to the station Velocity of Sound. 129 on Shooter’s Hill. Six rounds were fired, and the intervals occupied in the transmission of the sound were 7.7, 7".75, 7'.68, 7".67, 7”.72, 7.68 ; their mean 7”.7. 8820 7-7 about the same velocity as the former. 4(1085+ 11454) = 11154 feet, velocity of sound; therm. 66°, 4(11454 — 1085) =304 feet, velocity of the wind. Same day, August 7, half past 11, a.m. barom. 29.80 inches; Fahr. therm. 64°, the wind blowing in the same direction as before, with (an estimated) velocity of 30 feet; air dry, cloudy, no sun. The same 6 pounder gun was fired from the Shooter’s Hill station with a charge of 12 oz. of powder, and I took a station on Black- heath 20 feet farther than on January 9, its distance being 13460 feet from the gun. Six rounds were fired, one of the intervals was very doubtful ; the others were 12”.4, 12’.38, 12”.42, 12.38, 12".4, 12”.4; their mean 12” .396. 12.396 Being fearful of bringing the gun to Blackheath, in the vicinity of so many carriages as were incessantly passing, I could not here avail myself of the benefit of comparing the above intervals with those in which the direction of the transmission should be reversed. I venture, therefore, to add the velocity of the wind to that of the sound, as obtained by the experiment, and thus obtain 1085.8+30=1116 feet nearly, for the velocity of sound, the therm. standing at 64°. Monday, August 18. On this day, the same 6 pounder gun was placed upon the wharf by the side of the Thames in the Royal Arsenal, and I took a station at the opposite extremity of the Gallion’s Reach, not far from the mouth of Barking Creek ; the Vol. (1. Part 1. R =11454 feet, velocity of sound, when aided by a wind of = 1085.8 feet, velocity of sound when opposed by the wind. 130 Dr. GreGory on the distance from the gun was 9874 feet, the time of high water there, on that day, was about 11 o’clock, a.m. At half past 11, a.m. barom. 29.84 inches; therm. 66°; air dry, sky rather cloudy; very gentle wind nearly perpendicular to the line of transmission of the sound: Sita rounds were fired with the muzzle of the gun towards me: the intervals were 8’.8, 8”.84, 8’.86, 8.86, 8’.83, 8.85; their mean, 8”.84. At three quarters past 11, A.M. barom. &c. as before, six more rounds were fired, the gun muzzle being directed from us (up the river) in a horizontal angle of about 140 degrees: the intervals were 8.86, 8.84, 8”.82, 8”.82, 8’.85, 8”.86; their mean 8”.841, 9874 aT 17 feet, vel. of sound ; therm. 66° over a surface of water. Although there was no perceptible difference in the mean in- tervals occupied by the transmission of sound, in the two different directions of the gun, yet there was a considerable modification of the intensity; the sound being much weaker when the gun muzzle was directed westerly, up the river, than when it was pointed down Gallion’s Reach, towards the place where I stood. In the former case, too, besides the first report, which was marked and distinct, though comparatively feeble, there was a series of audible re-percussions, at intervals of about a tenth of a second, and gradually dying away: these, I conjecture, were reflected sounds from the faces of stone houses and other buildings standing on, or near the side of the river, at Woolwich. Same day, August 18, one o’clock, p.m. barom. 29.82 inches, Fahr. therm. 66°, fair, but cloudy; scarcely any wind: I took a station on the Essex bank of the Thames perpendicularly opposite the large storehouse on Roff’s Wharf at Woolwich, in order to ascertain the interval occupied by both the direct and the reflected transmission of the sound from a musquet fired by my side, and Velocity of Sound. 131 returned in an echo from the front of the said storehouse. The distance from my station to the front of the storehouse, determined carefully by a trigonometrical operation, was 1523 feet. Of eight rounds fired from the musquet, I failed twice in the appreciation of the interval between the sound and the returning echo, from a very wrong estimate of its probable duration ; and that from an erroneous impression as to the time observed by Dr. Derham in a similar experiment.* Of the remaining sia rounds, the musquet pointed across the river, the intervals were 2.7, 2".75, 2.74, 2”.72, 2".75, 2".74; their mean 2”.73. Next, three rounds were fired, the musquet being pointed directly from the river; the intervals were 2".7, 2”.73, 2".76; mean ‘as before. Lastly, four rounds were fired along the bank, at an elevation of about 45°; the intervals were 2.75, 2’.7, 2".73, 2.74; mean as before. Distance occupied by the direct and the reflected sounds 3046 feet. direct, half reflected; therm. 66°. The near agreement of this with the former result on the same day, serves to confirm the opinion that direct and reflected sounds move with the same velocity. Thursday, August 21, three o’clock P.M. barom. 29.86 inches, Fahr. therm. 64°; clear sunshine; wind scarcely perceptible, westerly. * He made it 3 seconds, by means of a half-second pendulum. My erroneous recollection of his experiment led me to anticipate an interval of between 4 and 5 seconds. I could not account for the supposed discrepance, until after my return home, when on examining Derham’s paper, and computing the real breadth of the river from my trigonometrical operation, I found the correspondence of the two experiments to be quite as great as could be expected, considering the different natures of the chronometers employed, and the yarying breadth of the river. R 2 132 Dr. GreGcory on the Mortars were firing from the battery, and I took a station 3900 feet south of it. I observed the intervals between the flash and the report in six successive rounds: they were 3".5, 3.5, 3”.48, 3.52, 3.5, 3.5, respectively ; the mean being 3".5. 3900 _ 7800 3.5 These are all the experiments in reference to the velocity of sound, as transmitted through the atmosphere, which I have yet been able to make. Their chief results may be brought into one view as below. =11142 feet, velocity of sound, therm. 64°. Feet Velocity of sound, Fahr. therm. 27° ....... 1094.2 - ditto BS ate cere 10992 —— ditto SOs ists 1102 ditto AB he ste c's 11072 —————_— ditto 50 aeeeecmemeneas 1109+ ditto GO. ce: 1112 : 0 11142 —. ditto it ae eee 11116 : 1116 — ditto Ca Soe Hie Of these results, some have been obtained in the day-time, others in the night; some when the sound has been transmitted over the surface of the earth, others when it has been transmitted over the surface of water; some are the result of direct sound, others of both direct and reflected sound; some from the report of can- nons, others of musquets, others from the sound of bells. Were these the only experiments on the subject that had ever been made, I should not regard them sufficiently extensive to justify me in deducing from them even an approximative rule. But as they have been made with great care, I may at least venture to present a rule, which, while it includes with only slight discrepancies Velocity of Sound. 133 all the preceding results, is simple enough to be easily recollected by practical men; and may, perhaps, be employed in our own climate. It is this:— At the temperature of freezing, 33°, the velocity of sound is 1100 feet per second. For lower temperatures deduct For higher temperatures add From the 1100 to the 1100 Fahr. therm. ; the result will show the velocity of sound, very nearly, ‘at all such temperatures. Thus, at the temperature of 50°, the velocity of sound is, half a foot. t for every degree of difference from 33° on 1100 x $(50 — 33) =11083 feet. At temperature 60°, it is 1100+3(60—33) =11133 feet; agreeing with the experimental result quite within the limits of a practical rule. ; The theorem 333.44 met. ,/1+00375 t, before cited, gives nearly 1094 feet for the velocity at the freezing point ; and 1114 feet for the temperature 10° centigrade, or 50° Fahrenheit: thus occasioning a greater augmentation to the velocity in the higher temperatures, than my experiments seem to indicate. The above practical rule, so far as it may be entitled te con- fidence, may be useful, Ist, to the military man in determining the distance of an enemy’s camp, of a fortress, a battery, &c. 2d, to the sailor, in determining the distance of another ship, &c. 3d, to the land surveyor in ascertaining the length of base lines, &c. in conducting the survey of a lordship or county; 4th, to the philo- sophic observer, in appreciating the distances of thunder-clouds during a storm. Yet, in either of these applications, the rule must be regarded as approximative only ; because, few practical men 134 Dr. Grecory on the can be expected to possess a time-measurer for less intervals than tenths of seconds (if indeed, so small) : and an error of a tenth of a second, will occasion a mistake of from 37 to 40 yards in the estimate of the distance. Beyond this, however, the error need scarcely ever extend; because a mean of 5 or 6 careful experiments will usually give the interval to a degree of correctness far within the limits just specified. Indeed, an error of from 30 to 40 yards in a distance of three or four miles, will, on most occasions, where such approxi- mative estimates are required, be of but small consequence. When the distance exceeds four miles, this method of approximating to it can only be employed under favorable circumstances of a very quiescent atmosphere, &c.: on which account, I felt scarcely any desire to extend my own experiments to stations more remote from each other, than those which I selected on Shooter’s Hill and Blackheath. Combining the results of experiments here recorded with those which have been formerly deduced by Derham and others, we aie I think, conclude unhesitatingly : ist, That sound moves uniformly; at least, in a horizontal direction, or one that does not deviate greatly from horizontality. 2d, That the difference in intensity of a sound makes no ap- preciable difference in its velocity.* 3d, Nor, consequently, does a difference in the instrument from which the sound is emitted. 4th, That wind greatly affects sound in point of intensity ; and that it affects it, also, in point of velocity. 5th, That when the direction of the wind concurs with that of the sound, the swm of their separate velocities gives the apparent * The consecution of the notes in a tune, notwithstanding the difference in their intensity, being uninterrupted when heard at a distance, furnishes an elegant and decisive confirmation of this proposition. Velocity of Sound. 135 velocity of sound; when the direction of the wind opposes that of the sound, the difference of the separate velocities mtist be taken. 6th, That in the case of echoes, the velocity of the reflected sound, is the same as that of the direct sound. 7th, That, therefore, distances may frequently be measured by means of echoes. 8th, That an augmentation of temperature occasions an aug- mentation of the velocity of sound ; and vice versa. (See Newton, Principia, Lib. 2. Prop. 50. Parkinson’s Me- chanics, Vol. IT. p. 148.) The enquiries with regard to the transmission of sound in the atmosphere,* which notwithstanding the curious investigations of Newton, Laplace, Poisson, and others, require the farther aid of experiment for satisfactory determination, are, I thik, the follow- ing: viz. Ist, Whether hygrometric changes in the atmosphere have much or little influence on the velocity of sound? 2d, Whether barometric changes in the atmosphere have much or little mfluence? 3d, Whether, as Muschenbroek conjectured, sound have not different degrees of velocity, at the same temperature, in different regions of the earth? And whether high barometric pressures would not be found (even independently of temperature) to produce greater velocities ? ; 4th, Whether, therefore, sound would not pass more slowly be- tween the summits of two mountains, than between their bases? 5th, Whether sound, independently of the changes in the air’s elasticity, move quicker or slower near the earth’s surface, than at some distance from it ? * Tsay nothing in this Paper, of the transmission of sound through the gases, along metallic conductors, &c. These furnish a most interesting department of separate enquiry. 136 Dr. GreGory on the (See Savart’s interesting papers on the communication of .so- norous vibration. ) 6th, Whether sound would not employ a longer interval in passing over a given space, as a mile, vertically upwards, than in a horizontal direction? and, if so, would the formula which should express the relation of the intervals include more than thermometric and barometric coefficients ? . 7th, Whether, or not, the principle of the parallelogram of forces may be employed in estimating the effect of wind upon sound, when their respective velocities do not aid, or oppose each other in the same line, or nearly so? 8th, Whether those eudiometric qualities, generally, (whether hitherto detected or not) which affect the elasticity of the air, will not proportionally affect the velocity of sound? and if so, how are the modifications to be appreciated ? To the experimental solution of some of these enquiries I hope to devote myself at no very remote period: but others of them, it is evident, can only be satisfactorily answered, if ever, by means of a cautious classification of skilful experiments made by various philosophers in different parts of the globe. OLINTHUS GREGORY. Royal Military Academy, Woolwich, October 25, 1823. Postscript. Since the above paper was drawn up, a friend has favoured me with the perusal of Mr. Goldingham’s account of his experiments in reference to the velocity of sound, made at Madras. From this very interesting Dissertation I shall venture to transcribe the following Table of the mean motion of sound for each month of the year, at Madras. Velocity of Sound. 137 Mean height of Velocity in a Second. Barometer. |Thermometer.| Hygrometer. meet lt neeeetel © Dey: Feet. January ..... 30.124 79.05 6.2 1101 February .... 30.126 78.84 a 1117 30.072 82.30 1134 30.031 85.79 1145 29.892 88.11 1151 29.907 87.10 1157 29.914 86.65 1164 29.931 85.02 1163 September . .. 29.963 84.49 1152 October 30.058 84.33 1128 November .. . 30.125 81.35 1101 December ... 30.087 79.37 1099 These results serve, as far as they go, to confirm the suspicions which I have long entertained, that the velocity of sound is some- what different in different climates ; and that hygrometric changes have more influence than has usually been imputed to them by theorists. The velocity varies from 1099 to 1164 feet, while the barometric range does not exceed a quarter of an inch, and the thermometer varies only from about 78° to 88°. But the indications of hygrometric change are considerable, passing from 1 to nearly 28 degrees. Unfortunately, however, we are not able to make such satisfactory deductions from these curious experiments as they might have furnished, had Mr. Goldingham described the con- struction of his hygrometer, and the fixed points, or the extent of its scale. Royal Military Academy, Nov. 8, 1823. Vol. II. Part I. Ss X. On the Association of Trap Rocks with the Mountain Limestone Formation in High Teesdale, &c. By tue Rey. A. SEDGWICK, M.A. F.R.S. & M.G.S. WOODWARDIAN PROFESSOR, AND FELLOW OF TRINITY COLLEGE, {Read May 12, 1823; March 1, and March 15, 1824.] Section [. On the great Calcareous Chain of the North of England. In a paper which was read to the Society last year*, [ Trap of High described some of the principal phenomena exhibited by the ia great dykes of Cockfield Fell and Cleveland. I also brought forward some facts which made it probable, that either the dykes above-mentioned, or other masses similarly associated with the coal formation, were prolonged into High Teesdale, and connected with the beds of trap which form so extraor- dinary a feature in that region. Having concluded, on evidence which seemed quite irre- sistible, that this intrusive class of rocks was of igneous origin, it became the more necessary to examine the trap of High Tees- dale, provincially termed the great Whin Sill +: especially since * The paper alluded to is printed in this Volume, p. 21, &e. + The word Sill, is, in some of the mining districts of the north of England, synonymous with stratum. By the Whin Sill is, therefore, understood a large tabular mass of trap imbedded in, and nearly parallel to, the other strata. $2 Calcareous Chain of the North. 140 Professor Sepawick on the it has often been triumphantly appealed to, as affording a proof that such formations, may, at least in some instances, be of aqueous origin. With this view, I ascended High Teesdale in the autumn of 1821, and, although interrupted by torrents of rain which rendered the banks of the river in many places inaccessible, succeeded in examining some localities which promised to throw light on the true relations of the trap. Last summer I again passed through that ®alley, and had an opportunity of verifying the observations of the preceding year. My time was, however, too limited to enable me to complete the task which I had un- dertaken, What is now offered to the Society, must, therefore, be considered as an imperfect sketch drawn from observations, directed, almost entirely, to the elucidation of the true relations of the trap to the contiguous strata. Under such circumstances it will be important carefully to separate such appearances as are doubtful or hypothetical from those which are plainly exhibited; and to draw our conclusions, respecting the origin of the basaltic rocks, from those facts only which are established on the clearest evidence*. Those who are acquainted with the geological features of England, must have remarked the chain of calcareous hills which runs nearly north and south through a part of Yorkshire, Durham, and Northumberland. The rocks forming all the higher parts of this chain are composed of limestone, sandstone, and shale (slate-clay) repeated in numerous alternations. From their continuity and prevailing characters, they are undoubtedly mem- bers of one formation—the mountain or metalliferous limestone. Yet, in districts considerably remote from each other, it is only * Since the Paper was first read to the Society, the author has made a third visit to High Teesdale, and as all his previous observations have been carefully revised, he has considerable confidence in the general accuracy of the details which are now offered to the public. Geology of High Teesdale. 141 possible to institute a general comparison of the subordinate beds; as all the minuter features are modified or changed by the action of local causes. From one end of the mountain ridge to the other, the beds on its eastern flank, dip to a variable point between north-east and south-east, and gradually pass under some deposits con- ‘nected with a part of the great coal formation. Between the mountain limestone and coal formation, there is not, however, any precise line of demarcation; as in many instances, they are decidedly interlaced with each other; the beds of coal and carbonaceous shale appearing among the upper beds of the lower series. It is, in part, to this cause that we must attribute certain discrepancies in our best geological maps. Thus in the map to which I referred in a former Paper (supra, p. 25.) Mr. Winch extends the region of the limestone to Winston on the Tees, while Mr. Greenough places the demarcation near Eglestone, more than ten miles farther up the river. The line marked out by Mr. Greenough agrees better with the general features of the district. But on the other hand, it may be contended, that thin beds of limestone alternate with the sandstone and shale in the immediate vicinity of Winston Bridge *. The eastern flank of the calcareous chain is intersected by many transverse vallies, in which the waters of the mountain- torrents unite and fall down into the lower carboniferous region ; and from which they either find an immediate passage into the sea, or descend into the great plain of the new red sandstone. Vallies which traverse beds of the same formation, nearly in the * The discrepancy is not, however, so great as might appear at first sight, for Mr. Greenough places some of the lower beds of the Coal-measures, and some of the upper beds of the Lead-measures, in a separate class under the name of the Millstone Grit. This classification may be good for a general map of England; but no one would, I think, have adopted it, who had taken his type of the great carboniferous series from this part of the north of England. Transverse Vallies. Stratification and Structure. 142 Professor Sepawick on the same direction, must necessarily agree in some of their charac- teristic features. Amidst many beauties and many objects of interest which are common to them all, each possesses some pecu- liarities of structure which would well deserve a separate illus- tration. They all agree in exhibiting the most obvious traces of powerful denuding forces. The vallies are all nearly perpendicular to the general bearing of the strata, and often exhibit, on their opposite sides, a succession of beds which tally with each other even in their minutest subdivisions. In not one of them is there an indication of dislocation, contortion, or subsidence, sufficient to account for the present inequalities of the surface. On the con- trary, the salient and re-entering angles which determine the present directions of the descending waters, and still more the various ramifications of the rivulets near the higher parts of the chain, plainly indicate the action of diluvian torrents. Any one who considers such a conclusion as doubtful or hypothetical, has only to examine the enormous beds of transported materials which are accumulated on the sides of the transverse vallies, and near the gorges where they first enter into the plains. He will there find the broken fragments of the rocks which once filled up the inequalities of the higher region, not only agree- ing, in every respect, with the strata from which they have been separated, but heaped up in those very places which first offered a lodgement for them, after being propelled by the de- scending currents. There are other phenomena, originating in the physical structure of the country, which are common to almost all the vallies before alluded to. For some distance above the places where the rivers first enter on the lower region, the waters descend down planes which are less inclined than the neighbouring strata. Hence, in ascending any of these trans- yerse vallies, we pass over the outgoings of a succession of Geology of High Teesdale. 143 strata; and where the denudations are considerable, sometimes reach the lower part of the series of beds, which forms the foundation of the mountains. But on ascending towards the last ramifications of the rivers, the inclination of the vallies always begins to exceed that of the strata, so that we again cross some of the beds, which, by their natural rise, had ascended from the region we had left behind. For example:—In following the course of the Tees towards its source, we first traverse the plain of the new red sandstone, and then cross the line of the mag- nesian limestone, and enter on the carboniferous series. Con- tinuing to ascend, we pass over the lower beds of the coal formation, and at length reach the beds associated with the metalliferous limestone. The successive members of the new series occupy the banks of the river for several miles. But on reaching the upper part of High Teesdale, we may ascend, by one of the ramifications of the river, towards the top of the chain, and find a part of the same series of strata which we had before passed over, presented to us in an inverted order. The structure we have described, is evidently favourable to a minute examination of the geological relations of each district, and the existence of a great many rich metalliferous veins in the same region, has directed the attention of practical men to the phenomena of stratification. Hence, many excellent sections, both of the calcareous, and carboniferous series, which form the eastern flank of the mountain chain, have been already given to the public*. Sections of this kind, made in situations which are considerably remote from each other, give us every infor- mation respecting the analogies of structure and composition ; but, as was before observed, seldom enable us to identify the subordinate members of the prevailing formations. * See Forster's Section of the Lead-measures. Winch’s Paper on the Geology of Northumberland and Durham, Geological Transactions, vol. IV. &c. High Tees- dale. Denudation. 144 Professor S—EpGwick on the Secrion II. On the Structure of High Teesdale. Tue descriptions given in the preceding Section, are suffi- ciently general to be applied to the whole system of the vallies which traverse the eastern flank of the great calcareous chain of the north of England. I shall now direct my attention, almost exclusively, to the phenomena which are exhibited in High Teesdale. As a matter of convenience, this part of the valley will be supposed to commence near the village of Eglestone. For above that place the dale begins to assume an austere aspect, which differs greatly from the softer and more picturesque fea- tures of the lower banks of the river. There also commences a series of phenomena to which I wish principally to direct the attention of the Society. That High Teesdale has been formed by denudation, is proved unequivocally, by the whole contour of the valley, by the ramifications of the tributary streams, and by the accumu- lations of diluvial matter, in almost every place, where it was possible for it to find a lodgement*. This assertion is, however, by no means intended to exclude the supposition of pre-existing inequalities and fractures, which may have enabled the diluvian torrents to pass in one direction rather than another. But facts of this kind are, for the most part, too much removed from direct observation to be deserying of much attention. * A person, practised in observations of this kind, would, from the mere external form of the country, generally succeed in pointing out the places where diluvial-gravel has been much accumulated. Transported materials, containing large blocks of trap, limestone and sandstone, are in great abundance near the junction of the Lune and the Tees. On the right bank of the Tees, between Lonton and Eglestone, the gravel is in some places not less than a hundred feet thick; but it gradually becomes comminuted, and in some places passes into a coarse sand, Geology of High Teesdale. 145 Among the circumstances of most importance in the struc- ture of High Teesdale, appear to be the following: (1.) The want of correspondence in the strata, on the two sides of all that part of the valley, which extends five or six miles above Eglestone. (2.) The manner in which masses of trap are, in this part of the valley, associated with the other strata; more especially the appearance of a great bed of trap (the great Whin-Sill of the mining district) first on the south side of the valley, and afterwards in the banks, and in the bed of the river. (3.) The appearance of a great transverse faulé, which inter- sects the whole valley about a mile above the High Force (in a direction which is about N.N.W. and E.S.E.), and throws the whole system of strata on the south-west side of its range, twenty or thirty fathoms above their natural level: thus ex- hibiting in the highest part of Teesdale, a repetition of the phe- nomena which appear at the junction of the trap with the other strata. J I. To these facts I now proceed to call the attention of the General se-- Society, in the order just pointed out. In the hope of making oe these complex phenomena better understood, I shall subjoin a detailed section of the strata in High Teesdale; and shall add a section of the upper beds of the same series, obtained from the lead-works of Old Langdon, in the High Moors, which extend to the north-west of Middleton. Vol. II. Part I. T 146 Professor SEDGwick on the (1.) A general Section of the principal Strata in High Tees- dale, commencing with the highest Bed of the Series*. No. Fath. Ft. No. ° Fath. Ft. L. Millstone Grit... «3... 6 0 | 28. Plateae st siccereiaee ce Laas 254 a pHSotas oo" te Sf Oe P2OrEAAME dr ectedais ayes mae steed ee ee O 3. Pieh State Sil. .e. ees. SerO SOeMbiatereiiid..c cs see state LO 4 Plate Rie. ER. Menge 2 0 || 31. Limestone............... 1 °® 5. Low Slate Sill .. 4 © || 32. Plate and Coal........ 1,0 6; iPlates3.¢.5 -0 RE SMM ease High Brigstone Hazle.. 6 0 7, NEONELONE Beene arene FO lt Ap OE ALE cette ght 6 ors Hoeeae er eae, BS. Platesoo ss osesse ceeees 6D O43 5. MalmMestOne:..c cristae ater stes f .3 OPM irEestone «.). 6 ose seit ee 827 O.ll Soamliiatereeyerey vec ae OES LOGE re. See MaRS On (37a How. itriseioue Hazle. kG HO 1.1, sPattigon’s, Hazle: cents 2 0:3) G8e Platemcice. otield. +e ie 46m 12, aR lates. veo - persphetisare sei) yp OnmuOull es Oo Lamestane sen 4+ v°+ 2Quv cos. uv + 2vt cos. vt + 2tu cos. tu...... (1). The angles uv, &c. being measured at the solid angles to which the diagonal is drawn. Let there be lines passing through the origin, parallel to the given lines; the equations to these lines will be respectively, 2 Th = a Ge SSS y = be >» an fle . Let D be the distance of any two points from each other, which * The investigation which follows, is by J. W. Lubbock, Esq. of Trinity College. two Planes, &c. referred to three oblique Co-ordinates. 201 are situated, one on each of these lines; 5, \ the distances of these points from the origin. Then D* = & + 8?-—2080 cos. y..... - (2). But by equation (1), we have D’ = (x#—a)? + (y-y'f + (s—2') + + 2(x—2") (y—y') cos. ry +2(y—y') (—2') cos. y2+2(2—2’) (w—2’) Cos. 22. Also o= a + y +2°+ Qry cos. vy + 2yz cos. YS + 2H COS. ZL, Oo = a? ty” + 2° + Q2'y' cos. ry + 27/2! cos. ¥% + 222’ cos. 2x; putting these values in (2) and reducing, ve +yy +22'+ (ay +2'y) cos.xry+ (yx +y'z) cos. yz + (za +2'z) cos. zx = cos. 97. Vfa°+y?42°+ 2xy cos. vy + 2y2 cos. yX + 22x08. zx}. Vfa4+&c. } And putting for 2, y, «’, y’, their values in z, 2’, and dividing by zz’, we shall find aad +bb'+1+(ab'+a’b) cos. ry +(b+b') cos. yz+ (a+a’) cos. xz Ce a V (a+b? +1+2ab cos. ry + 2b cos. yz + 2a cos. xz). V(a* + &e.) ’ the second factor of the denominator differing from the first only in having a’, b’, instead of a, 6: and thus we have the required angle. When the co-ordinates are rectangular, cos. xy, cos. yz, cos. xz, are each 0, and aad +bb+1 V(ev + b+ 1). V(a* +b” +1)’ which is the known formula. From the above expression, we may deduce the fundamental formula of Spherical Trigonometry. Let the two given lines be in the planes xz, yz, respectively, and both perpendicular to the axis of z. In this case, their angle will measure the inclination of the planes xz, yz, and their equations will be cos. 4 = Vol. II. Part I. Cc 202 Mr. WHEWELL on the Angle made by two Planes, se. But manifestly, in this case, the equations .are 4 & a = ———.,, and y= - : cos. LZ cos. Yz for the two lines respectively. Hence 1 1 a= ———__, b=0, d= 0, = — ——_., COS. VS ‘ cos. Ys and putting these values in cos. y, cos, ry 1-1-1 + ———4*_ Cos. @Z COS. Y% V (acest) V (oe t1-2) | COS. LY — COS. L% COS. YS ore sin. vz sin. y% ’ cos. 7 = Now if a sphere be described about the origin, the co-ordinate planes will cut it, making a triangle, of which the sides are measured by the ‘angles xy, «2, yz, and the angle opposite to xy is ». Hence, the above is the expression for the cosine of the angle of a spherical triangle in terms of the sides. ‘ 4 el XII. On the Figure assumed by a Flud Homogeneous Mass, whose Particles are acted on by their mutual Attraction, and by small extraneous Forces. By G. B. AIRY, B. A. OF TRINITY COLLEGE, AND FELLOW OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, {Read March 15, 1824.] Tue principal difficulty in the solution of this problem, consists in the investigation of the attraction of any spheroid (differing little from a sphere) upon a point in its surface. This has been found by Laplace, in a manner so general, and by an analysis so powerful, that any new investigations might seem entirely unnecessary. But the abstruse nature of that analysis, it must be acknowledged, is such as to make a more simple in- vestigation desirable: and the obscurities which have led La- place himself into error, serve to shew the value of a process which involves nothing more difficult than the common appli- cations of the differential calculus. I venture to indulge in a hope that the solution which I have the honor to lay before this Society, imperfect as it may be, will tend to make this subject more accessible to those who have hitherto been deterred from pur- suing it by the mass of analysis in the Mecanique Celeste. With the exception of the proposition which reduces the discovery of the attraction in any direction to the investigation of the value of a single integral, the process pursued in_ this Paper is entirely different from that of Laplace. The theory cc2 204 Mr. Airy on the Figure of a Fluid Mass given in detail, is restricted to the case in which the disturbing forces are symmetrical about an axis; but the extension of the same principle, which enables us to apply a similar process to any forces whatever, is shortly indicated at the termination. To shew the method of applying the theory to any given case, I have considered the effect which the ring of Saturn produces on his figure (supposing him homogeneous) and arrive at the important conclusion, that the attraction of the ring, on the hypothesis of gravitation, will not explain the peculiarity which has been ob- served in the figure of Saturn. It is not easy to assign any other cause for the singularity of form of that remarkable planet ; and if the observations remain undisputed, the physical expla- nation of this phenomenon may exercise the ingenuity of future philosophers. (1.) Let a, 6, c, be the rectangular co-ordinates of any point of the mass parallel to the axes of x, y, z,: let X, ¥, Z, be the forces acting upon that point in the same directions. Then if a function U can be found such that dU Se duieitedda wat hades Te the equation to the surface of equal pressure passing threugh this point, will be U=C. The condition necessary to the existence of equilibrium is, that a function U can be found which will satisfy these equations. In the case which we propose to con- sider, the forces arise entirely from attractions and centrifugal force; and will therefore, by a well-known theorem, satisfy these equations. The equilibrium being possible, we have only to find the equation of the surface bounding the mass; which we shall attempt by the assistance of the following theorem of Laplace. (2.) Let V be the sum of the products of every attracting particle into the reciprocal of its distance from the attracted acted on by small Forces. 205 point: x, y, 2, the co-ordinates of any point: let the forces be considered positive when they tend to make a, 5, c, increase: call U’, X’, ¥’, Z, those parts of U, X, ¥, Z which arise from these attractions. Pheri poe ae : ; dx dy dz VY \(w—a)* + (y—b)? + (z—c)*}’ F av v-—a **dadxdydz~ {(«—a) + (y—b)? + (z-c)*}3 de X’ dill Bae wobe lay is = da dydv °"dxdydzda’~ dxdydz’ °‘da~~° dV “BF : : , = Z,, (the limits of integration being supposed Similarly a au. independent of a, b, c,) and the value of U’ is therefore V. If then we take the sum of the products of each attracting particle into the reciprocal of its distance from the point whose co-ordinates are a, b, c, and if we add that part of U which originates from the extraneous forces, and make the sum =C, we shall have the equation to the surface. (3.) Our object then, at present, is to find the value of this sum for a point in the surface; and we might suppose the disturbing forces to be any whatever. But in nearly all the cases to which we can apply this investigation, the forces act symmetrically round an axis. Suppose, then, that the forces are symmetrical about an axis: the body will then be a solid of revolution; and, for the sake of simplicity, we will make b=o0. We proceed to investigate that part of Y which arises from the attraction of the particles of the body. (4.) First, for a small pyramid whose vertex is the attracted point. Let p be its whole length, p any variable length, 4 the area of a section perpendicular to its axis at distance 1 from the vertex: the area at distance p is Ap*®; the sum of the products of the masses into the reciprocals of their distances, for the slice included between the lengths p and p+ dp is ultimately Apép; 206 Mr. Arry on the Figure of a Fluid Mass . hence the sum for the pyramid is J,Ap* = 40 ; which for the whole pyramid is aii (5.) To find the length of p and the value of 4, suppose c to be the ordinate parallel to the axis of the solid: suppose the solid divided into wedges by planes passing through the line c; let two of these planes make, with the plane of xz, the angles ¢ and ¢+6¢: suppose the included wedge divided into pyramids by lines on these planes, drawn from the attracted point; let two of them make with c the angles @ and 9+60. Then 4 =sin. 0.80.5¢. Also x = a—p sin. 0 cos. ?; y =p sin. 0. sin. P: = =c—p cos. 0; x, y, 2, being the co-ordinates of the point at which p meets the surface again. Substituting these values in the equation to the surface, the value of p will be had in terms of @ and ¢. (6.) The disturbing forces being small, we will neglect the squares and higher powers of the disturbing forces and quan- tities dependent on them. And as the body, without this dis- turbance, would be a sphere, we will assume for its equation r+y+2=r'+y(z), where x(z) is a function of z involving a small multiplier. Substituting for x, y, z, the values found above, a’ +c°—2p (c cos. 0+ a Cos. p. sin. 0) + p* =7* + x (C—p COS. 4). But @ and c are co-ordinates of a point in the surface: hence (ie tte =r +x (c). Subtracting, p’—2p, (c cos. +a cos. ¢ sin, 6) = x (c—p Cos. 8) — x (€) ; “. p= 2(ccos. 0+ a cos. p sin. 0) + pase DLL Delia ON An approximate value of p is 2(c cos.@¢+acos.¢ sin. 6): let this be v; substituting it in the small term, x(e-—v cos. 0)-x (c) p=vut And & =5 +x (c-v cos. #)—x (0). «By f,Ap is meant what is usually written {4 pdp, the quantity whose differential coefficient, taken with respect to p, is Ap. acted on by small Forces. 207 (7.) The sum of the products of each particle into the reciprocal of its distance, for one of the pyramids into which ‘ Ase A, : the mass is divided, has been found to be ae or ultimately 2 &. sin, 0.50.39. For one of the wedges, then, this. will be soft sin. 6. And it is plain, that one of the lines, which deter- mine the limits of integration for the wedge, is in the direction of the ordinate c; and the other is the tangent of the curve, formed by the intersection of one of its planes with the surface, at the attracted point; the direction of which is determined by making p = 0. 2 be taken between the lmits 6=0, 6=90. This must then be integrated with respect to ¢~ through a whole circumference; and thus the value of that part of V which arises from the Let © be the value of 6 which makes p=0: then af - sin. @ must attraction of the particles, is if St e sin. 6, taken between the limits above-mentioned. (8.) To determine © we observe that p=2 (c cos. 6+4 cos. ¢ sin. 6) — x’ (c).p cos. 0+ XC) cos.’ 0— &e. Jee Ls + P a = 2(ccos. 6+ a@cos. ¢ sin. 0)— x’ (c) cos. 0 + x19) cos.’ 6—&e. where x‘(c), x’ (c), &c. are the differential coefficients of x‘ (c) taken with respect to c. In this expression make p=0, 9=0; then 0=2 (c cos. 9 +4 cos. ¢ sin. 8),— x’ (c) . cos. 8; sea (6) ;.sin. 9 acos.d c—ty’ (c) —acos.? SST COS. O= aa Via cos. pl’ + ¢— 3x’ (c)I'} V {a cos. g\'+e~7Xx (D3 which gives tan. © = — 208 Mr. Airy on the Figure of a Fluid Mass 2 2 (9.) The first part of a sin. @ is 5 sin. @, or 2sin. 6 {c’ cos.? @ + 2ac cos. p.sin.6.cos. 6 + a cos.’ d.sin.’ 6}. The general integral with respect to @ is 2c 4ac : 2a° : Fagn cos.° 6 + =z COs: - sin.’ 6— “3 COS.” p cos. 6 (sin.? 6+ 2). Upon giving to @ the value ©, it will be found that each term involves an odd power of cos. ¢, divided by an odd power of V{acos. p\? + e—F x’ (c)\*}. Now the sign with which this radical is taken, can never alter, for the radical never becomes = 0; if then we put 7+ for ¢, we shall have the same expression with a different sign. Hence, on performing the next integration, these terms will disappear by the opposition of signs, and we may, therefore, reject them at once. On giving to @ the value 0, the expression becomes - the value of the integral is therefore 20 4a : ce + a cos. d. (10.) Integrating this with respect to ¢ through a circum- ference, we have, for the first part of the required expression, +e). (11.) The other part to be integrated is sin. 6.{x(c—v cos. 0)—x (c)}. If we expand y(c—vcos.@) by Taylor’s Series, the m term a” .x(c) 7 dc™ m m ; m = will be Ma el Pikes \™ sin. 6 2)" d™, x (c) Ta... Mm’ dem (€COS. 8 +4 Cos. > sin. 8)”.cos. 0)". sin. 6. acted on by small Forces. 209 The p+ 1" term of (c cos. 0+ a cos. ¢ sin. 0)”. cos. 0\".sin 4 is m.m—1. ..m—p m—p+1 1.2...p c”-P .a? .cos. p\? .sm. gnc cos. 0\°"-?. Now when p is odd —— : 1 —— Josin. 0? +'. cos. 6\?"—? = sin. 6+”. oe COS. O\ae m+ 1 2m—p—1 aM ee eee es 2m—p—3 ——$— ———————————_ PT 2 Sa ee ey + &e. Tome L.2ma1 Some pt+4.pt+2 This vanishes when §6=0; when 6= 9, every term involves an even power of cos. ¢, which is multiplied by cos. ¢\’, and therefore vanishes on being integrated through a circumference at the next operation. No term, therefore, results from the odd values of p. (12.) When p is even, f, sin. 6\?*'. cos. 6"? is sin. 0? ** x 1 z 2m—p—1 ee prea eee eee _O\m—P-3 &e. = + 1 COS. 6 SoS Ser Sy i aae cos ar 2m—-—p—1.2m—p—3....5.3 2m -+-1.2m—1...... p+to.p+s _ 2m—p—1.2m—-p—3......5.3 2m+1.2m—1.....p + 5.pt+3 | eee . COs. 0 fen sin. 6\? + Pet pai sin. 0\?-*? + &e. ‘Def Oe OLE! \ Pp ciara 3.1 When 0@= 9, every term involves an odd power of cos. ¢, and vanishes on integration with respect to ¢. When 6=0, the expres- sion becomes _2m—p—1.2m-p—3....3.1.p.p—2...4.2 2m+1.2m—1..........3.1 2m—p—1.2mM—p—3....3.1.p.p—2....4.2 cy) Ee ee ; Vol. Il. Part I. Dob or the integral is 210 Mr. Arry on the Figure of a Fluid Mass (13.) Multiplying this by cos. 9)’, and integrating with respect to @ through a circumference, since the definite integral of cos. |’ ST Pee oe Sel Ee Se Bee wwii’ 27, we have p-p—2....4.2 - 5 5 2m—p—1.2m—p—3....3.1 cos. d\?. Saleh" LE) SSS SE Solfo 08. St ae oN 7 2m+1.2m—1....p+3.p+1 and the definite integral of the p +1)\" term of m—1,...m—-p+t 1.2....p (c cos. 0+4@ Cos. @ sin. 0)”. cos. 6)”. sm. 0 1s 27 x 2m — p—1 .2m—p— 3. 301 2 Nie wn oe pP+3.p+i1 Hence, we have the following rules for finding the value of Sofy Sim. 9 (x (¢-v cos. #) — x (c) ). 1. Expand c+ a)”, and select those terms (0 PE ro) in which the index of a is even or 0. 2. Multiply the term involving a’ by 2m—p—1.2m—p—3....3.1 2m+1.2m—1......p+3.p+1 3. Collect the terms for different values of p with the same Tale dx (0) value of m, and multiply their sum by ae ee x ct Bear. x 4. Collect the series found by giving to m the values 1, 2, 3, &c. call the sum ¥(c): then 27.W (c) is the integral required. (14.) The expression for ¥ (c) may be put under the follow- ing form, which will probably be found more convenient, ; ~ — 2c, x’(c) 20? x" (c) 2c\° tO) = ali d.>: 6) me heme OS: (Bi sieessl:. 1 5.7 acted on by small Forces. 211 lee ha 1 te Srxeyor oe + 54) () 579 1 “agit + eet +Peg ix (¢) «97113 + &e. , (15.) The whole integral of = sin. 9 is therefore Ba \ (c+ a’) + (ch. This forms part of the equation to the surface on the suppo- sition that the attraction of the matter in volume 1 (collected into a point) at distance 1, is represented by 1; if it be represented =e. by &, the part of the equation is 27.k {- (?+a@)+wy ()}. (16.) Suppose the part of U arising from the disturbing forces to be 27.c/(c), «(c) being a given function of c. Then the equation to the generating curve will be Qa F k(e+a)+hy(c)+e(}=C, or 5k (2 +a") thal (0) +c (c)=C: where the two last terms of the first side are small. (17.) This must coincide with the equation @+c?=r*+ (c). And, therefore, if by means of this equation we eliminate a from the former, the resulting equation must be identically true. The value of a to be substituted in the large terms is r*—c* + (c): in the small terms 7’—c*. Thus we get the equation = kx (0) + ky (e) + e()=C—g kr. Assuming then for x(c) a form with indeterminate coefficients, such as the given form of ¢(c) appears to require, and deter- mining ¥(c), and eliminating a by putting a=r°—c’, every term of the equation multiplying a power of ¢ must be made =0, and thus a number of equations will be obtained sufficient to deter- mine all the constants, except that independent of c. DD2 212 Mr. Arry on the Figure of a Fluid Mass (18.) It will be observed that W (c) is now expressed by this form X10 204 x10 Bel" _ x'(0) 30 lee i ee 1.2° 5 a ae Pe ute pee AO Be ay SLC Lie ay i ©)-g sm to bapa sory. 9 wekeee Ts (r° —c?) ash mead (c) ie; ad oe = 9 1 7 Oe + &e.} 2 (7 16) Ga 1 ss Wk an ()-7 941.13 7 od + &e. (19.) Ifthe disturbing forces were not symmetrical about an axis, it would be necessary to assume for the equation to the surface a + y? + 2 =r? + y(a,y). The values of x, y, z, would be a—p sin. 0.cos. ¢, b- p sin. 0.sin. ¢, ¢—p cos. 6; a, b, c, being the co-ordinates of the attracted poimt. Substitutmmg these in the assumed equation, a value would be found for p, as in the case we have considered: and the integral dh S' to .sin. 8 would be taken C) in the same manner. And we should arrive at a similar equation 2 : shy (a, b) +kw(a, b) + € (a, 6) = o kr?; and assuming a proper form for x(a, b), forming ¥ (4, 4), and eliminating ¢ by means of the equation c*=7*—a?—b*, the coefficients would be determined by the comparison of similar terms. As the applications of this theory are few, we shall content ourselves with thus pointing out the course to be pursued in any given case. (20.) Suppose «(c) to be of the form 4+ Bc’+ De'+Ec+Fe'. Then x (c) will evidently have the form P+Qc?+Re*+Sc&4+ Te'. From this, by the expression in (18), we find y (c) (neglecting the constant term) acted on by small Forces. 213 26 1280 A 432 } 4 SSE i Tagalalgin. 2.5.7.6 ane 3 ate 1120 4 60 2_ 8 4 SS “4 + 9.11.13 ae 5 Rte 112 pye_ 12 bo 8 re. 135 0517 13 17 Hence zk (c) + k¥ (©) Li 1280 ’ 432 cf rae: f hi asasce tt VicRCR TELE ght 55 he 1120 A 60 . EVO +h) sip tparas sag Rh 112 — 210 z +h {tg Ir saz ster ee (21.) The equation 5kx (c) + kw (c) +e (c) = (oa ; kr® gives the following, 1280 : 432 *3 Nome ener SN 5 errant CU ap AC e a alee eas =; at Pgs Sena 1120 GOrts glk ut dit it erate RE ae bag ainllly 112 : Ee) aia — RE 0 13.15.17 es papi tF=0. From these 7 = 2:12 ntl AWS yp — 12 Fr’ . 3.13 £E 25° bo) MMGna Ae 27.16 Fr ‘Se Ex? 99 -D 11) 1Sn bua eel 2 96 Pl yaa Bele Dr? 15 B 5.18 VA PMEEIL Deep OR ke R= 214 Mr. Atry on the Figure of a Fluid Mass (22.) Since C is indeterminate, it is manifest that there is nothing to determine the value of the constant term in x (c) and it is, therefore, quite arbitrary: It may conveniently be taken so as to make x(c) vanish when c=r. This gives x (c)= — Q (r?—c°)— R (r4#—c*) — S (7° —c®) — T (78 — 8) = — (7? —€’) x {Q+ Rr 48r+ Tr +R+ Sr? 4+ Tri.c+ 8 + Tr?.ct + Tc} AVA ce 310509 Br? ox oupa 6 ert 99. Dy 5 bs iB pre ex} 11.83.14.25. 000 267.1010 Gk Was rk 4s, 11013 .14,95 8+) los 1443 Fre 39 E 51 F 14.25° k *10°% 14k“ (23.) As an instance of the application of this formula, let it be required to find the figure of Saturn, as affected by his rotation about his axis, and by the attraction of his ring. Let T be the time of his diurnal revolution: the resolved parts of the cen- trifugal force on a point whose co-ordinates in the plane of his 27\° equator are a, b, will be Te @ and ag b. The part of U which 236589 Fr 519) Er’ \'9 DY (2 2°k arises from this, will hee 7 = ae + 6) = asd ar —c*), and, therefore, the part of «(c) will be = (r?—c), or — are neglecting the con- stant term. (24.) Suppose Saturn’s ring to be a mathematical line, into which is collected a quantity of matter = -th of the matter in 23 the body of Saturn = 2a: let its radius= R. The part of U or of V which arises from this is ark f° 1 bar he 1 3n YOV{R+a*+2aRcos.0+c%} — 3n/40Y{R+7r°4+2aR cos. 0} acted on by small Forces. 215 Expanding this in a series of the form 4+ B cos. 6 + C cos. 20+ &e. and integrating every term through a circumference, we get ae 1 13 84a Re 12-3°.5. 7 | loa. he 3n JSR +r 22.4 (Ro + rt 2°.4°.6.8 (R® +4 ryt 1°.3°.577.-9. 11 me o4agke 4 1?.3°,5°.7°.9.11.13.15 | 256. a DAO .o LO a2 “(Re4r’)? 2° 47.6%.8°.10.12.14.16 (R? ery if we stop ata’. Let e, f, g, h, be the coefiicients of a’, a*, a°, a®; then putting r°-c’ for a’, dividing by 27, and neglecting the term in- dependent of c, we find that this contributes to «(c) the following ark terms = {—(e+2f+3g¢+4h) c+ (f+3g4+6h) c—(g+4h).c+h.c'}. (25.) Our expression will not be very erroneous, if for R we put the mean radius of the ring. Suppose then Boo. The last expression then becomes 2) 09233 .c° + 04849. °,—.01768 .°. + .00295 ie = —.09 oo 9. a 7 “pat: 9 a and adding the term for centrifugal force, _ 09233 _ Do. 04849 | i td ponanee! F- 00295 1 n Tp nr? nr’ nro * 6 Substituting these values in the expression of (22), 7 4476 .1487 c* 0568 c* .0107 a) A OFA =<) {5 - ‘ET? n n Tr nr. n -PS (26.) To exterminate & from the first term of this expression, let ¢ be the periodic time of Saturn’s 7th satellite, s its distance; the motion of this satellite is nearly the same as if all the matter of Saturn and his ring were yg into aries center: hence rk (it: er hencez7s = oie s2f 5 1+- ~); ra 3 2 SEP a3 PCH) = A= 8 oe: (42) = (142) x 419. 216 Mr. Airy on the Figure of a Fluid Mass, &c. Making this substitution in the expression for x(c) we find 862.149 fet 087 ct Ol 2 2 ne ae 2 = @=r—c+y(c)=(7 -e)f1.415 cas an ane 57 an approximate equation to the generating curve of Saturn sup- posing him homogeneous. (27.) This gives an ellipticity = .185, independently of that produced by the attraction of the ring. This is so large a quantity, that the neglect of its square and higher powers must produce sensible errors; those terms, however, which arise from the attrac- tion of the ring, and which it is our more immediate object to investigate, will be little affected by their rejection. And though our supposition of Saturn’s homogeneity is highly improbable, yet if his density be variable, the aberration. from the elliptic figure produced by the attraction of his ring, will be the same in kind (though differing in quantity) as that which would exist, were his density uniform. (28.) An inspection of the equation at the end of (26) will shew, that the theoretical figure of Saturn is flattened between the poles and the equator. It is remarkable, that this deviation from the elliptic form, is exactly the opposite to that given by the obser- vations of Dr. Herschel. This accurate observer, in the Philo- sophical Transactions for 1805 and 1806, has given a great number of his observations, which shew, that Saturn is protuberant between the poles and the equator, and that his longest diameter makes an angle of 43° with the plane of his equator. Here then is a com- plete discordance between theory and observation ; nor is it easy, with our present knowledge of the planet, to suggest any thing by which they can be reconciled. G. B. AIRY. Trinity CoLueceE, March 15, 1824. XIII. On the Determination of the General Term of a New Class of Infinite Series. By CHARLES BABBAGE, Esq. M.A. FELLOW OF THE ROYAL SOCIETIES OF LONDON AND EDINBURGH, AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. [Read May 3, 1824.] Tue subject of investigation on which I have entered in the following Paper, had its origin in a circumstance which is, I believe, as yet singular in the history of mathematical science, although there exists considerable probability, that it will not long remain an isolated example of analytical enquiries, suggested and rendered necessary by the progress of machinery adapted to numerical computation. Some time has elapsed since I was examining a small machine I had constructed, by which a Table, having its second difference constant, might be computed by mechanical means. In considering the various changes which might be made in the arrangement of its parts, I observed an alteration, by which the calculated series would always have its second difference equal to the unit’s figure of the last computed term of the series; other forms of the machine would make the first or the third, or generally any given difference equal to the unit’s figure of the term last computed ; and a further alteration would make the same difference equal to double, or generally to (a) times the digit in the unit’s place: or if it were preferred, Vol I. Part I. Ke 218 Mr. Baspace on the General Term the digit fixed upon might be that occurring in the ten’s place, or generally in the »" place. I did not, at that time, possess the means of making these alterations which I had contemplated, but I immediately proceeded to write down one of the series which would have been calculated by the machine thus altered; and commencing with one of the most simple, I formed the series. Series. Diff. el D> ornvnnaor If uw, represent any term of this series, then the equation which determines w, 1s Au, = unit’s figure of «., an equation of differences of a nature not hitherto considered, nor am I aware that any method has been pointed out for the determining u. in functions of = from such laws. I shall now lay before the Society, the ‘steps which I took for ascertaining the general terms of such series, and of integrating the equations to which they lead. I shall not, however, commence with the general investigation ofthe subject, but shall simply point out the paths through which I was led to their solution, conceiving this course to be much more conducive to the progress of analysis, although not so much in unison with the taste which at present prevails in that science. If we examine the series, and its first differences, it will be of a New Class of Infinite Series. 219 perceived, that the terms of the latter recur after intervals of four, and that all the changes in the first differences, are com- prised in the numbers 2, 4, 8, 6, which recur continually, and the series may be written thus: Series. Diff. 2 2 4 4 8 8 16 6 5 22.= 90 -+- 2 2 24 = 20+4 4 4 28 = 20+ 8 8 36 = 26 + 16 6 42 = 40+ 2 2 10 44 =40+4 4 4 48 = 40+ 8 8 56 = 40 + 16 6 62 = 60+ 2 2 64 = 60 + 4 4 15 68 = 60 + 8 8 76 = 60 + 16 6 82 = 80 + 2 2 If then z be of the form 4v + 7, the value of u, will be 20v + one of | four numbers 2, 4, 8, 16, according to the value of 7, and if i always represents one of the numbers 1, 2, 3, 4, the value of u. will be thus expressed, u, = 20v-+ 2’, As a second example, let us consider the series whose first term is 2, its first difference 0, and its second difference always equal the unit’s figure of the next term; its equation will be A‘u, = unit’s figure of w., EE2 220 Mr. BaspaGE on the General Term and the few first terms are 28 48 4 76 10 110 16 144 182 This series may be put under the form Series. 1 Diff. ts) 2 (0) 2 2 4 6 10 6 1 oe Table of (é). 5 28 20 3 48 ae iin — 2012) eae 76 34 ; 2 110 34 4 144 38 sf 19 10 182 ao == 40 tb ’ 16 222 42 = 40+ 2 a 28 264 46 = 40+ 6 6 Ae 310 46 = 40+ 6 7 76 356 ba = 40a a2 2 i 15 408 60 = 40 + 20 9 ae 468 68 = 40 + 28 536 74 = 40 4+ 34 610 74 = 40 + 34 684 78 = 40 + 38 20 762 80 = 80+ 0 842 gs2= 80+ 2 924 s6 = 80+ 6 of a New Class of Infinite Series. 221 Series. 1 Diff. 1010 86h= 1 80) 41.6 1096 92 = 80+ 12 25 1188 100 = 80+ 20 ,» 1288 108 = 80 + 28 1396 114 = 80 + 34 1510 114 = 80 + 34 1624 118 = 80 + 38 30 1742 120 = 120 +. 0 1862 122 = 120+ 2 In this series it may be observed, that u. when z is less than 10, is equal to the sum of the first differences of all the preceding terms, and if « be greater than 10, it will be composed of four terms, viz., first the sum of the ten first terms of the first difference, multiplied by the number of tens contained in z; secondly, of the sum of the series 40 + 80 + 120 + to as many terms as there are tens in z, this must be multiplied by 10, as each term is ten times added; and thirdly, of the number 40 multiplied by the same number of the tens, and also multiplied by the digit in the unit’s place of 2; and fourthly, of the sum of so many terms of the series as is equal to the unit’s figure of z; this being expressed by (@) signifymg the number opposite a in the previous Table. These four parts, if 2 = 10h +a, are thus expressed, 1* 1808, b.b-1 ae 3" 40ba, Pie 1 (7) These added together produce u,=20b(10b + 2a—1) + (a). This value of u., if diminished by 2, is equal to the sum of z—1 term of the series which constitute the first difference. oe AO 229 Mr. BaspaGe on the General Term This inductive process for discovering the 7» terms of such series, might be applied to others of the same kind, but it does not admit of an application sufficiently general or direct, te render it desirable that it should be pursued further. If we consider any series in which the first difference is equal to the digit occurring in the unit’s place of the corresponding term, as for example, the series 6 6 12 2 14 4 18 8 26 6 32 2 a slight examination will satisfy us, that the value of the digit occurring in the unit’s figure of w., depends entirely on the value of u., at the commencement of the series, and also that when- ever the same digit again occurs, there will, at that point, com- mence a repetition of the same figures which have preceded ; consequently, the first difference at those two points will be equal. In the first example which I have adduced of a series of this kind, it will be found, that this re-appearance of the terminal figure, happens at the 5th, at the 9th, at the 13th terms, &c. or that . This gives for the equation of the series, Au, = Au.+,4, or by integrating u,=U,44+ 5, but when z=1, w, = u, therefore b = 0, and U.,1—U, = O, whose integral is u,=a(— V—-1)*4+b(- V—1)**!40(— V—1)2+*4d(— VY —-1)**3+53. of a New Class of Infinite Series. 223 The four constants being determined, by comparing this value of vu. with the first four terms of the series, we shall find a=0, b=-5,c=$-V-1,d=5+V-1, and the value of vu, becomes u, = 5(z—1) + §-— V—1)(V-1° + § + V—-1)(- V-1);, which expresses any term of the series 9, 4, 85 16, 22, 24, 28, 36; 42, 44, 48. It is necessary, for the success of this method, that we should have continued the given series until we arrive at some term whose unit’s figure is the same as that of some term which has preceded it: now if we consider that this figure depends solely on that of the one which occupied the same place in the pre- ceding term, it will appear that the same digit must re-appear in the course of ten terms at the utmost, since there are only ten digits, and that it may re-occur sooner. The same reasoning is applicable to the case of series whose first difference is equal to any multiple of the digits found in the unit’s place of the corres- ponding term, or to those contained in the equation Au. = ax (unit’s figure of w.), as also to those in which this is encreased by a given quantity, as Au, =a (unit’s figure of uv.) + 0. If the second difference is equal to some multiple of the figure occurring in the unit’s place of the next term, as in the series - Qe DAD NONALO, already given, since the unit’s figure must always depend on the same figure in the first term of the series, and its first difference 2 ty) 10 224 Mr. BaspeaGe on the General Term whenever those two figures are the same, a similar period must re-appear: now as there are only two figures concerned, they can only admit of 100 permutations, consequently, this is the greatest limit of the periods in such species of series.—In the one in ques- tion the period is comprized in ten terms. This reasoning may be extended to other forms of series in which higher differences are given in terms of the digits occurring in the unit’s, ten’s, or other places of wu. or u.,; or elsewhere, but I am aware that it does not in its present form present that degree of generality which ought to be expected on such a subject: probably the attempt to solve directly that class of equations to which these and similar enquiries lead, may be attended with more valuable results. As the term ‘“ wnit’s figure of” occurs frequently, it will be convenient to designate it by an abbreviation; that which I shall propese is the combination of the two initials, and I shall write the above equation of differences thus INTE CMON NTRS on parca gl) This may be reduced to a more usual form by the following method. If S, represent the sum of the a" powers of unity, divided by ten; then OS,+1S,, +28, 424+38,43+48, 44+ 5 S,45+68,46+7 8, 47+85,+t9S,45; will represent the figure which occurs in the unit’s place of any number «: substituting uv. mstead of x, we have ~Au.=0S,_+ DS gi ee a t= oo eer aeete amet Oe an equation in which w. enters as an exponent. From the previous knowledge of the form of the general terms of the series we are considering, it would appear that the general solution of the equations (a) and (4) is u=9s+ cS, + 4 S,., + CoS.42'+ 10-00. C5945: of a New Class of Infinite Series. 225 where the constants must be determined from the conditions. In the further pursuit of any enquiries in this direction, much assistance may be derived by consulting a Paper of Mr. Herschel’s in the Philosophical Transactions for 1818, ‘On Circulating Functions.” Amongst the conditions for determining the general terms of series by some relation amongst particular figures, there occurs a curious class, in which, if we consider only whole numbers, the series becomes impossible after a certain number of terms. Let the equation determining uw. be Au. = B (UFu,_, + UF u,,1). Then the following series conform to this law, Series. Diff. Series. Diff. Series. Diff. 1 3 4 6 1 9 4 5 10 4 10 1 9 14 4 11 1 18 12 3 15 If the law is restricted to whole numbers, none of these series admit of any prolongation; nor have I, with that restriction, been able to discover any series of the kind possessing more than five terms. C. BABBAGE. Devonshire Street, Portland Place, March 29, 1824. Fr TRANSACTIONS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. Vou. II. Parr II. ty - by i > : # ‘ at XIV. On the Principles and Construction of the Achro- matic Eye-Pieces of Telescopes, and on the Achromatism of Microscopes. By GEORGE BIDDELL AIRY, B.A. FELLOW OF TRINITY COLLEGE, OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, AND CORRESPONDING MEMBER OF THE NORTHERN INSTITUTION. [Read May 17, 1824.] Iw the theory of Telescopes no part is more interesting, and in the practical construction scarcely any more important, than the Achromatic Eye-piece. The effects of a badly formed eye- piece are even more disagreeable to the eye than those of a defective object-glass: whether we consider the indistinctness near the extremity of the field of view, the distortion of the image, or the fringes of colours which surround an object when not observed in the center of the field. Important and interesting as the subject appears, we might expect, in works of the highest pretensions, to find it treated in the comprehensive manner it deserves: and simple as are its principles, we might imagine that they would be introduced into our elementary treatises on Optics. But it is remarkable that while all writers have given at great length the theory of the achromatic object-glass, there are not more than one or two books in the English language, in which the achromatic eye-piece is alluded to; and though not a telescope has been made except on this construction for many years, the artist is still obliged Vol. Il. Part II. Ge 228 Mr. Arry on Achromatic Eye-pieces of Telescopes, to work almost without rules, in most instances merely copying the constructions which more able opticians have found to succeed. I flatter myself that an attempt to investigate, on the simplest principles and in the most comprehensive manner, an optical theory of so great importance, avoiding at the same time all unnecessary refinements and useless generalities, will be re- ceived with encouragement by the Society, whom I have now the honor of addressing. The first attempt at improvement on the single eye-glass of the first refracting telescopes was made by Huygens. Instead of a single eye-glass he used two convex lenses, whose focal lengths were in the proportion of 3 to 1, and which were placed at a distance equal to twice the shorter; the lens of greater focal length being that nearer to the object-glass; and the image being formed between the two eye-glasses. His intention was to diminish the spherical aberration of the extreme pencils by dividing the re- fraction into two parts: for, as he found the spherical aberration to vary nearly as the cube of the refracting angle, it was easy to see that the aberration would be reduced to 4th of its former quantity, if the whole refraction, instead of being effected by one refracting angle, were effected by two, each equal ‘to half the former. It will easily be seen that in his construction any pencil is refracted equally by both lenses, and the improvement which he aimed at was therefore completely attamed. But there were other advantages of which he was ignorant, and which were not discovered until the construction had been many years employed. It is smgular, that in his attempt to diminish spherical aberration, he should have hit upon a construction which completely removes also chromatic aberration. This was discovered when the unequal refrangibility of light was established, and the mode of obviating the inconveniences thence arising was invented: and the con- struction of Huygens, commonly known by the term of the and the Achromatism of Microscopes. 229 Huygenian eye-piece, is now in general use for astronomical and reflecting telescopes. While the eye-piece of Huygens was used for astronomical purposes, the eye-piece of three lenses, each contiguous pair being placed at a distance equal to the sum of their focal lengths, as described in our elementary works, was still used for erecting the image. The first notice that we find of any alteration in the erecting eye-piece, is contained in a letter from Dollond to Short, printed in the Phil. Trans. for 1753. In this he stated that he had made achromatic eye-pieces with 4 and 5 lenses for some time, and that, finding some of them to have reached the Continent, he considered it expedient to assert his claim to the invention before it was likely to be disputed. It appears that his object at first was merely to diminish the spherical aberration, on the same principle on which it had been attempted by Huygens: but that, finding that the chromatic aberration might also be corrected by the same construction, he had combined 4 and 5 lenses, with particular attention to the correction of colour. It would seem from this letter, that he had then no theory which enabled him, generally, to make his eye-pieces achromatic: and it is not known whether he afterwards made use of any formula. It is remarkable that in this letter he speaks of an achromatic object-glass, as a thing totally to be despaired of. The attention of philosophers was at this time very generally turned to the consideration of chromatic aberration. Six years before that letter was written, Euler, from a hint given by Newton, had conceived the idea of making a lens truly achromatic. Two years after its date, Dollond found that it was possible to make an achromatic lens. In the next year, 1756, Clairaut gave, in the Memoirs of the French Academy, his first investigations re- lative to achromatic object-glasses: which he continued for several years. They were simplified by D’Alembert, in some papers first GG2 230 Mr. Airy on Achromatic Eye-pieces of Telescopes, published in the Memoirs of the Academy, and afterwards col- lected in his Opuscules Mathematiques, published 1764. And Euler, in the Berlin Memoirs for 1757, gave a great many theorems upon combinations of lenses: which, in a paper in the Turin Me- moirs, Vol. III, published in 1766, are applied to the achromatic eye-piece. These papers, with considerable alterations, were afterwards embodied in his large work on Dioptrics, which was published in 1769 and the two following years. It does not appear that any more original investigations were made, till the Abbé Boscovich in the Memoirs of the Academy of Bologna gave some new theorems which were collected in his Opuscula, published in 1785: those relating to the achromatic eye-piece are principally contained in the second Volume. The subject is treated with great mathematical skill, and the work bears strong evidence of considerable practical acquaintance with optical instruments: the rules given by him are in general well adapted to the use of the working optician : and are, in fact, the foundation of all that have since that time been published. ; During forty years which have elapsed since the investiga- tions of Boscovich were given to the world, no addition I believe has been made to his theory. The only English treatise on this subject that I have seen, is one published by Professor Robison in his Mechanical Philosophy, and in the Encyclo- pedia Britannica, Art. Telescope. Though this writer has not closely copied from Boscovich, yet he seems to have been guided entirely by his theory: his calculations appear to have been made in the same way: however I have seen no treatise, which on the whole seems better adapted to give clear ideas on the construction of telescopes, and useful rules for the assistance of workmen. From this treatise, and from the work of Boscovich, have been extracted the theorems given by Dr. Brewster in the appendix and the Achromatism of Microscopes. 231 to his edition of Ferguson’s Lectures. I ought not to omit that Robison mentions a translation in English of a work by Scherfer, which I have not been so fortunate as to meet with. When it is known that much attention has been bestowed on this theory by Clairaut, Euler, D’Alembert, and Boscovich, it may perhaps appear presumptuous to attempt any improve- ment on what they have done. I will therefore briefly describe the manner in which each of these mathematicians has con- sidered it, and thus, besides pointing out the defects of their operations, I shall have the advantage of more completely elucidating the subject. To explain the principle of the achromatic eye-piece, I will take its simplest case. Suppose a telescope composed of an object-glass and a single eye-glass, placed in the manner usually described under the head of Astronomical Telescope in most of our elementary works. The axis of a pencil of rays coming from the extreme visible point of an object, will pass through the center of the object-glass, and impinge on the eye-glass near its circumference, whence it will be refracted to the eye. But the dispersion attendant on this refraction will separate the ray into its differently coloured primitive rays. The violet pencil will be more refracted than the red, and will enter the eye, making with the axis of the telescope a greater angle than is made by the red rays. The place of the image therefore, as seen by the violet rays, is more distant from the center of the field of view than its place as seen by the red rays; and the object therefore appears coloured. But if a second eye-glass be placed at some distance from the first, the violet rays, after re- fraction at the first eye-glass, will be incident on the second at a point nearer to the center than that at which the red rays are incident on it; and falling therefore on a smaller refracting angle, they will, by the proper adjustment of the lenses, be so 232 Mr. Arry on Achromatic Eye-pieces of Telescopes, much less refracted, as to emerge parallel to the red rays; and the object is now seen without any tinge of colour.. In a way nearly similar it may be shewn, that any number of eye-glasses may be combined so as to form an achromatic eye-piece. The investigations of Clairaut and D’Alembert do not relate to the achromatic eye-piece properly so called. These writers have shewn, that in certain cases a fault of the object-glass may be corrected by the eye-piece, and that all the rays of any one colour may be made to emerge parallel to each other. But the intention of the achromatic eye-piece is, not to make the rays of each colour emerge parallel to each other, (a small deviation from which is quite insensible to the eye), but to make the axes of the differently coloured pencils emerge parallel. Euler in the Turin Memoirs has considered at great length the properties of eye-pieces. But he has paid most attention to the correction of spherical aberration, and his formula for the correction of chromatic aberration is not demonstrated. The greater part too of his work is occupied with the consideration of eye-pieces of five and six lenses; which now are seldom or perhaps never employed. In fact, little information can be gathered from this paper, that is either interesting to the theo- rist, or useful to the workman. In his large work on Dioptrics he has treated the subject far more completely. His method (to which that which I have employed is in some degree similar) is this; he investigates the position and magnitude of every image: from the position and magnitude of the last image, and the position of the eye, he finds the visual angle; he then differentiates this, supposing the focal lengths of the lenses to vary in consequence of the variation of the refractive index for differently coloured rays, and he makes this variation=0. He then makes the position of the eye that which gives the greatest field of view, and eli- and the Achromatism of Microscopes. 233 minating it, he has an equation between the focal lengths of the lenses and their distance from each other. Though the prin- ciple of this process is general and not inelegant, yet the method of introducing and afterwards eliminating the distance of the eye from the last eye-glass, is not so good as might have been expected from Euler. In pursuing the theory, he has loaded it with generalizations and details, to such a degree as to make it almost useless. The thickness of the lenses, in some parts, makes his formule extremely complicated, though that might be safely neglected; in other parts, he has considered the effect of lenses of different sorts of glass, though no one would resort to that construction, when it is possible toe avoid it. And I believe that I do not exaggerate when I say, that, after a general acquaintance with the principles of chromatic dispersion, it would be easier for any one to form a theory for himself, than to select the parts that are useful from the book of Euler. The work of Boscovich is much better calculated to explain the principles of the achromatic eye-piece, than any of those I have before mentioned. And in his investigation he has pur- sued the natural course of tracing the axis of pencils of differ- ently coloured rays through the eye-piece, and making them emerge parallel. But his method is greatly deficient in facility and power. He finds the quantity of mean refraction where a pencil is incident on the first eye-glass, from which he gets the dispersion of the violet and red rays: he then finds the distance at which they are incident on the next lens, and the refractions there; and; continuing this process, makes the last emerging rays parallel. In some places he has expressed the dispersion by an algebraic symbol; in others, by its numerical value; and in some he has calculated the progress of a ray by a laborious trigonometrical process. And this is the last theory pretending to any degree of generality that has yet appeared. 234 Mr. Atry on Achromatic Eye-pieces of Telescopes, Professor Robison, in the books already mentioned, has given for the simpler forms of the eye-piece, a geometrical construction. That an optician should be able to apply an algebraical formula is not improbable; but that he should make a geometrical construction, and deduce a theorem, is almost impossible. In the more complicated cases, he has assumed a particular condition, namely, that the axes of the pencils shall meet on the field-glass. This puts a stop at once to all im- provements: and even for the completion of this solution a geometrical construction is required, not less difficult than the former. The extracts which have been made from this work by other writers, it is not necessary to notice. When engaged in some investigations respecting a peculiar construction of telescopes, the principles of which were laid before the Society about a year and half ago, I found it necessary to obtain a general formula for making it achromatic. At that time I was not acquainted with any of the works that I have described: and the difficulties of the case compelled me to. use a method, which appears to me to be free from most of the objections that can be laid to them. In principle it consists in finding an expression for the visual angle by tracing the axis of a pencil of rays through the eye-piece, and, by a kind of differential process, making its variation, depending on the alteration of the index of refraction, = 0. I have here applied it to eye-pieces of two, three, and four lenses, and have pointed out some of the uses and peculiarities of each construction. The latter part, I hope, may not be without its value: it is upon the achromatism of Microscopes. This, I believe, has not been treated of by any author: and as it is not less im- portant, as far as it extends, than the achromatism of Telescopes, and as its theory is singular, and (I think) not inelegant, I have introduced it in this paper. and the Achromatism of Microscopes. 235 The preliminary investigation for the variation of the power of a lens suggested a mode of treating of the achromatic object-glass, which is, I think, as simple as any that has yet appeared. And a Jaw which I have assumed for the dispersive power of one medium in terms of that of another, reduces at once to mathematical investigation every thing relating to the irrationality of dispersion. Some speculations on achromatic object-glasses I have, therefore, given in the beginning. I have been induced to press this subject on the notice of the Society from a conviction of its importance, and a know- ledge of the little attention usually paid to it. In the date of its invention, the achromatic eye-piece is prior to the object- glass: in its application it is even more general; in its theory it is, I think, more interesting: and in the principles of that theory it is not more difficult. Yet while the object-glass continues to receive the attention of scientific men, the theory of the eye- piece in the space of 40 years has received no improvement: in the elementary works on Optics, the object-glass is described at length, but the eye-piece is entirely omitted. If it should appear that in this paper I have rendered the theory more generally accessible to the inquiring mathematician, or more easily adapted to the purposes of the practical optician, I trust that it will have the approbation of this Society. (1) In the following articles, we shall use the letter x to express the index of refraction for rays of some one colour, which we shall call mean rays, passing out of glass or any other refracting medium into air; and shal] denote the index of re- fraction for rays of any other colour by x+én. The letter 3 pre- fixed to any quantity dependent on z will be employed to denote the alteration produced in that quantity by changing n to n +n. (2) As we shall make great use of the expression for the variation of the power of a lens, we will investigate it here. Let Vol. If. Pare IL. Hu 236 Mr. Airy on Achromatic Eye-pieces of Telescopes, F be the focal length, ; the power, of a lens: the expression for aa , G being a quantity glue: : g 1 iF is required. It is known that P= independent of 7; If SF be required, since 1 F tM ae a plirepug anes Pe = ( 1) + &e.}; FTF ea ’ x on én \2 5 dae a Spa n—1 J: or rejecting the square, &c. of — oF= If the product F should occur, F and f being the focal lengths of : : : Ie i two lenses of the same sort of glass, since 7 is — into 70 “y =) , and ; into ke (a + =. the variation of me as far 2 én 5 Re: n—1- as the first power eee Similarly the variation of ep is eae &e. (3) On the achromatic object-glass. The principle of this is so well known, that we shall not step to describe it. Suppose then a convex lens of crown-glass, and a concave lens of flint- glass, to be placed in contact; let and 7m’ be the indices of refraction for mean rays, for crown and flint-glass respectively ; F and f the focal lengths of the lenses: the power of the com- , nae age . pound lens for mean rays is j — Pa for any other rays it is and the Achromatism of Microscopes. 2937 This may be made the same for rays of all colours, or all may n —1 : é é be made to converge to the same point, if Wl be always = 6 = for then by making iY tg jase dah or f=cF, the power of the compound lens for rays of all colours is 7 a aha fi is found in practice that can be very nearly, but not exactly, n—1 én ke SIT for rays of all colours, ¢ being nearly =1, 5. expressed by c. Object-glasses therefore may be made on this plan very nearly, but not quite, achromatic. (4) Instead of placing the lenses in contact, suppose them to be separated; let their distance = a. The rays which fall on the second lens are converging to a point whose distance is F- a: hence the power after refraction at the second lens is 1 1 F 1 F-a f a Cale F for mean rays. For rays of any other colour, it is F F'n-1 ai n-1 or, by expanding and neglecting eS) &ec., the power is is. F én 1 F-a yo Woe tn—1 f wat HH 2 938 Mr. Airy on Achromatic Eye-pieces of Telescopes, gaa. , the compound Jens, as before, will be achro- matic, upon making F Bot _ ¢e(F- a) Gas Ngee oe (5) The following remark is worthy of notice. In an achro- matic object-glass we have found, when the lenses are in contact, f=cF; when the lenses are separated by the space a, f= aos a) if, eis the lenses are in contact, the focal length of the concave lens be too short, or its refraction too great, or (in the language of opticians) the colour be over-corrected, the object-glass may be made achromatic by separating the lenses. This important theorem is well known to every scientific workman. If the focal length of the concave Jens be too great, no alteration will make the object-glass achromatic. (6) Upon comparing the dispersions of different media, it a quantity evidently less than the former. Hence én : appears that ——— may be represented with an accuracy great on \? Se i+ c= Suppose an object-glass to be formed of three lenses of different substances: let n, n’, n”, be their refractive indices for mean rays, and let é e apnoe 2.n0 bao av Alty ‘ n' +2(2* : n n a. n ) : —— =C ._— > eat a a n-1 n—1 enough for all practical purposes, by the form c. at then if f, f’, 5 be the focal lengths for mean rays, the power of the object-glass for rays of any other colour will be us ra) +7 Ore a) tpt =) eens. ptp) mart Gtp) Ga a and the Achromatism of Microscopes. 239 This will be the same for all colours, if a ea e pig iigt pr which will determine /’ and” from /. (7) This is the principle of Dr. Blair’s object-glass. A_ re- fracting fluid (a solution of some metal in muriatic acid) is inclosed between two lenses of crown and flint-glass. It is found im practice that object-glasses thus constructed do not last, owing to the chemical changes in the fluid, and its corrosive action on the glass. (S) The separation of the lenses promises, in theory at least, to produce beneficial effects. Suppose a convex crown-glass lens whose focal length is F, and a concave flint-glass lens of focal length f# to be separated to the distance a: then (see 4) the power of the combination is 1+ on n—1 1 én’ Fe ie 1+ 7 ) ras én 7¢ n'—1 n—1 oe r. F = (—% “)' si de» pr n' - F—a_ (F—a)?'n + oy ae : i ea Putting f aa th ion c an +e ee. this combi utting for wai e express 7 ease s - nation is achromatic, if i oa LAT erg (F—a)’ f (F- a) : From these equations a= ——-F. The power of e€ cae fe Tee (c+e) the compound lens 1 Pay age ie e 240 Mr. Atry on Achromatic Eye-pieces of Telescopes, This must be positive; ... e were 7 ats, the aperture of the flint lenses would be less than that of the crown by the same part: an advantage by no means inconsiderable, as flint-glasses of 4 inches diameter can be very frequently obtained, while those of 5 inches cannot be procured once in several years. (11) On the achromatic eye-piece. The defect in tele- scopes which occasioned the invention of the achromatic eye- piece, and the manner in which it removes that defect, have been explained in the Introduction to this Paper. We shall merely state in this place, that in all the following instances we shall pursue the same course; we shall find the distance from the last lens at which the axis of a pencil of rays meets the axis of the telescope: we shall then find the distance from the center of the lens at which the ray is incident upon it; and having found the tangent of the visual angle, by dividing the latter by the former, we shall make its variation, depending on the variation of ”,=0. (12) On the eye-piece with two eye-glasses. Let D be the distance of the first eye-glass from the object-glass= 4B, Plate XI. Fig. 1: p its focal length: a the distance of the second eye-glass from the first= BC: qits focal length. Suppose the axis of a pencil of rays coming from the center of the object-glass, to fall upon the first eye-glass at EZ: it will, after refraction, tend to cross the axis of the telescope at the distance : or Dp = BG ee p D beyond that lens, and therefore at the distance Dp D+a.p—Da is Sa D2s or ns = CG and the Achromatism of Microscopes. 2438 beyond the second eye-glass. After refraction therefore at the second eye-glass, it will meet the axis of the telescope at the distance 1 i tL : q (D+a.p—Da) ‘ _ D-p 1 pg + Dq+ D¥a.p—Da D+a.p—Da q And if k be the tangent of the angle BAEZ, then BE=Dk; CF=BE = i pi. Pt4 | dicen . tan c= i. —Pg+ Dq+D+a.p—Da ¥ PY =k{-1 g Seat Somes P q Pq Taking the variation of this to the first power of — and De making it = 0, we have D Dt+a 2Day sn por, pig Moai =0, whence a=—-*4 Or if D be very great, a = Pet This is the rule of opticians. If p= 3q then a = 2q: this is the Huygenian construction before mentioned. (13). If a =q, we must have ena. cee ae 29 — Fy = PAG OF a=g = Da = BG. In this case the lens CF is placed at G: and since its focal length equals the distance between the two lenses, the image, to be distinctly seen, must be formed upon the lens BE. This is inadmissible; because the particles of dust on the glass, since the smallest substance will now intercept an entire pencil of Vol. If. Part If. Ir 244 Mr. Airy on Achromatic Eye-pieces of Telescopes, rays, will prevent the object from being distinctly seen: and because it is impossible to apply a micrometer. Add _ to this, that the eye must be in contact with the lens CF, or it will not receive all the pencils from different points of the object. (14). If a be greater than g, the image is formed between the two lenses; and, since there is always some distortion at refraction by a single lens, the micrometer cannot safely be applied. Ifa be less than q, we shall have, (since g=2a—-p) a~F>p, or a> pb > BG. In this case the pencils cross before they can be received by the eye: this construction therefore cannot be employed im any case. The achromatic eye-piece of two glasses can never there- fore be used with a micrometer. (15). In telescopes with a micrometer, the eye-piece generally consists of two lenses, and the image is formed between EB and the object-glass. As no part but the center of the field of view can be distinctly seen, there is an apparatus which enables the observer to slide the eye-piece across the end of the telescope, and thus move the center of the field to the object. This kind of eye-piece is called by artists the positive eye-piece in contradistinction to that in which the image is formed between the eye-glasses, which is called the negative eye-piece. With the positive eye-piece the lens EB should always be placed as near the first image, and the lens CF as near the intersection of the pencils with the axis, as convenience will allow. , (16). On the eye-piece with three eye-glasses. Let p, g, 7, be the focal lengths of the lenses, beginning with that nearest to the object-glass; let a ‘and b be their distances: and for simplicity, suppose the object-glass so distant, that the axes of and the Achromatism of Microscopes. 245 the pencils incident on the first eye-glass, may be supposed parallel to the axis of the telescope. Then they will cross, or tend to cross, the axis at the distance p from the first lens, or a-p from the second. They will again cross it at the distance oa or — from the second lens, or a—p Sle p — (64=P) _ ab—bp—a+b.q+pq from the third. a—p—4q Ge a cla | They will finally cross it at the distance 1 r (ab—bp—a+b.q+pq) or — LSS ab—bp-a+b.q—ar+pqtpr+gqr 1 rT ab—bp—a+b.q+pq from the third eye-glass. And if m be the distance from the axis at which a pencil is incident on the first eye-glass, mx o—? is the same on the second eye-glass; and ‘ ps ee, Boe eins rg Pod Gop, hebns aM gtd Pp a—p-q q (4—p) PY that on the third eye-glass. Hence the tangent of the visual angle Taking the chromatic variation of this, as far as the first power 3 : 5 : of —., and making it vanish, we have 3ab a 2b 2(a+b) 2a Mt 1 =A pqr qr pr Pq ‘ T or 3ab—2bp—2.a+6.q—2ar+pq+pr+qr=0, 112 246 Mr. Airy on Achromatic Eye-pieces of Telescopes, the general equation for an eye-piece with three eye-glasses- If a,p, 9,7, be given, then pe 2aq+2ar—pq-pr—qr “a 3a—2p—2q ; (17). Suppose, for example, a = p+q. Then 2¢+pqtpr+qr 7 j= See eee Ge apt This formula is given by Boscevich. (18). The eye-piece with three eye-glasses is little used, for this reason. The refraction, which causes the final intersection of the pencils with the axis, must be effected entirely by the last eye-glass; and except its aperture be small, the spherical aberration distorts the image very much: so that it is impossible to have a large field of view. In the eye-piece imstanced above, the course of the axis of a pencil’ is represented in Fig. 2. and it is evident that the inclination of the pencil to the axis of the telescope at Y, is occasioned entirely by the single refraction at X. On this account, the eye-piece with four eye- glasses is now universally employed. (19). On the eye-piece with four eye-glasses. ‘Taking up the investigation of (16), and supposing the focal length of the last eye-glass=s, and its distance from the third=c, the pencils cross the axis at a point whose distance from the last eye-glass r (ab—bp—a+bh.qt+pq) = ¢ = — ee ab—bp—a+b.q-—ar+pqt+pr+qr _ abe—bep—at+b.cq—b+c.artepqtb+c.pr+atb+e.qr—pgr (=8) ab—bp—a+b.qg-ar+pq+pr+qr They finally cross at the distance $ (abe—bep—a+b.cq—b-+-c.ar+-cpq+6+¢.pr+a+b+ c.qr — pqr) +a+6+c.qr+bps+a+b.qs + ars— pqr — pgs— prs— rs abc— bep—a-+b.cq— b-+-c.ar—abs + cpq+-b-Fe.pr Also the distance from the center of the lens at which the axis of a pencil is incident on it is - ab—bp-—at+b.qt+pq t a) nae Pq c—t pqr x {abc—bep-a+b.cq-b+c.ar+cpg+bh+c.pr+at+b+c.qr—pqr}. Hence the tangent of the visual angle, found by dividing this by », is equal to abe bev a--bse-a.b+e abn? te rig dade prs pqs pr rs qs” ps at+tb+ec +— 4+ + — Onl > Bi. oe Bait jaguoin) Making its variation=0 as before, and reducing, we have the general equation of the eye-piece with four glasses, 4abc—3bcp—3.a+b.cq—3a.b+c.r—3abs+2cpg+2.b+e.pr +2.a+b+c.qr+2bps+2.a+6.9qs+2ars—pqr—pqs—prs—qrs=0 If a, b, p, q, 7, 8, be given, _ (6.a+2qg—2a- p.2b—q) .(r+s)+rs (2a—p+q) ~ p(3b—2.q+r)+q(3.a+6—2r)-a(4b—3r) ~ (20). In a common perspective-glass it was found, that a=) 5,9) aHen, (bi— 2.2), Cc = 1-8 i— tee g) = 1,8, = 1,8, s— 1,28 The following numbers have been given for a good eye-piece: P=, gi — 21 re esi — a2 ae 23,00 — Ad cl AO, =v). 248 Mr. Airy on Achromatic Eye-pieces of Telescopes, In one of Dollond’s p = 14,25 lines, g = 19, r= 22,75, s = 14, a= 22,48, b = 46,17, c = 21545. In one of Ramsden’s small telescopes, p= 0,77 inch, gq = 1,025, r= 1,01, s=0,79, a = 1,18, b=1,83, ec = 1,1. It appears from the formula, that the values of c in the several cases should have been 2,31, 37,88, 19,37, 1,12. The course of a ray in the last of these is represented in Fig. 3.; the advantages which it possesses over the eye-piece with three Jenses are sufficiently obvious. (21). If it be required to find where the axes of the differently coloured pencils intersect each other: let z be the distance from the center of any lens, at which the axis of the pencil of mean rays is incident upon it = LM, Fig.4: ¢ the tangent of the angle made by that axis with the axis of the telescope = tan. MNL, and let x be the distance from the lens at which the differently co- loured axes cross: then it is plain from the figure that «dt=6:, or v= e. Thus, to find where the rays intersect after refraction at the second lens, we have, by (12), putting m for Dk, and sup- posing D very great, s=m(i-*): t=m er ma on 1’ 1 24a on ; ag —. : ‘ Ry = a ee Pena} Big nie PI at 28-2 = and the Achromatism of Microscopes. 249 To find the intersection after refraction at the third lens, we have, by (16), b s=m Se ee ee ee ee tt): PY 9 PP par qr pr pg | gab bats 22 it tas | P hence x 3ab_ ab a(a+8) aa, 1 11 POT wy UE PE Pq Big 2? r(bp+a+6.q—2ab) -—3ab+2bp+2.a+b.q+2ar—pq—pr—qr_ (22). Robison has proposed as a general rule in constructing eye-pieces of four glasses, to unite all the differently coloured pencils on the third lens or field-glass. For this purpose, it is merely necessary to make b= arert (25.) These investigations, it is evident, can be extended to five or a greater number of lenses, ‘and the formule, though troublesome, will be as simple as the nature of the subject will allow. And any other problems, relating to the intersection of the rays &c., can be solved on the same principles. On this subject therefore we shall not stop any longer. (24.) On the Achromatism of Microscopes. In the con- struction of the common microscope there is no part similar to the achromatic object-glass of the telescope, for the purpose of making the rays of any one colour from a point of the object enter the eye parallel to each other. The aperture of the object-glass is so small, that the colour arising from this chromatic aberration is not perceptible. The only endeavour is to make the axes of the differently coloured pencils enter the eye parallel to each other: which is effected by properly placing the diaphragm that determines the quantity of light received from the object. Fig. 5. represents the path of the axis of a pencil of rays in the common 250 Mr. Atry on Achromatic Eye-pieces of Telescopes, microscope: at C is the diaphragm which admits a cone of rays smaller than the object-glass. Since the position of this diaphragm determines the part of the object-glass, through which the rays coming from any point 4 of the object must pass, it is evident that the refraction, and consequently the dispersion of the dif- ferently coloured rays will depend upon its position. (25). We shall begin the investigation by finding the effect produced by variation of refrangibility upon the directions of the axes of the pencils from 4, passing through C. Let v be the distance of the object from the object-glass: suppose Bd pro- duced, Fig. 6, to meet the axis of the microscope at a point whose distance from the object-glass = y. Let m be the tangent of the angle made by the pencil, after refraction at the object- glass, with the axis of the microscope. Then it is easily seen that (if w be the distance of the point of the object from that axis,) : - — =1: and since a similar expression is true for rays of all colours, v(5 +3. Pena =i Eliminating w, 3 r) 3 » (5 +82) Sg Aoi =<. or v.8o=(1-2) 2. y ™m Now i St Silene EE Y P Y Pp pxr-i _v én 0 v\ 8m ogy gat ge) ™m And v is always very nearly = p; _ vo dn _v dm dma mz én Set aah, ait) Ot Sate (26). Now the axes of the pencils after refraction at q will and the Achromatism of Microscopes. 251 ag aang tend to cross the axis at a distance or from g, or when 1 1 q a incident on r they are converging to a point whose distance from it = -b= S64, they then tend to cross the axis at a-q a= the distance i r(a+b.q—ab) SSS Se OE === a a—-—q —ab+a+b.q+ar—qr r at+b.q—ab from r, or the distance r(a+b.g—ab) ae moss a+b+c.qr—at+b.cq—b+c.ar+abe —ab+at+b.qt+ar—ab ” —qrt+a+b.qgt+ar—ab (=g) from s. They finally cross the axis at the distance s fatb+c.gr—at+b.cg—btc.art+abc} hai —grstatbt+e.qr+at+h.gst+a.rs—ath.cq-b+c.ar—abs+abe from s. And the distance from the axis, at which a pencil -of rays meets g, is am: that at r is mx St tga’, that at s is . e a+b+e.gr—a+bh.cg—b+c.ar+abe . ie - Hence the tangent of the visual angle is a+b a+b @+b.c ‘a.b+c ab _ abc mi—-1+4t°te at ge ate ae et: $ _? q rs qs qr qrs) Vol. If. Part II. Kk 252 Mr. Atry on Achromatic Eye-pieces of Telescopes, Sc. its variation =m eae + Lies oe D = Qathe 2ab+e_ Be on AIAG ig qs qr. qrs Sn-1 a+b+c a+b a a+bhe ab+e ab _ abc ae ee \ + dm {—1 ~ Ss r q Ts qs qr qrs Making this = 0, and putting the value of dm in (25), a+b+c.qr+at+b.qs+ars—2.a+b.cq—2.b+c.ar—2abs+3abe L=p., ——— ————= qrs—a+b+eqgr—a+b.qs—ars+at+h.cg+b+c-ar+abs—abe- (27). It is usual in practice to place the diaphragm in the situation which we have supposed. If it»should be placed on the other side of the object-glass, m would be invariable, and the investigation would be in every respect similar to those for eye- pieces of telescopes. (28). In a common microscope, in which the diaphragm was at a little distance from p towards gq, it was found that a=5, b=2, c=,3, g=2,8, T=1,9, s=1,6. The equation above gives r= —p x,191; hence the microscope was not properly constructed. To complete the theory of eye-pieces, and to shew what are the most advantageous arrangements of their lenses, there remain to be considered the effects of spherical aberration. This part is equally important with that. we have treated, and far more difficult of investigation: and may perhaps form the subject of a future communication. G. B. AIRY. Trinity COLLEGE, April 26, 1824. XV. An Account of a Whale of the Spermaceti Tribe, cast on shore on the Yorkshire Coast, on the 28th of April, 1825. By JAMES ALDERSON, B.A. FELLOW OF PEMBROKE COLLEGE, CAMBRIDGE, AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. [Read May 16, 1825.] So little is still known with respect to the natural history and anatomy of whales, that any opportunity of contributing a few facts to the information already ascertained, is extremely desirable. It is this which has induced me to communicate what I have seen. The subject of the following remarks was seen on the after- noon of Thursday the 28th of April, drifting up from the southward with the ebb tide, off Tunstall in Holderness, and in the course ef the afternoon was landed low in the tide. At the ensuing flood it was floated higher up on the beach, where it was left during the early part of the ebb. It may seem extraordinary, but it is no less true, that it was not generally known in Hull to be on shore until the Tuesday following. Nothing can be more contrasted than the view of the animal perfect, and its skeleton. The enormous and preposterous mass of matter upon its cheeks and jowl bearing no proportion to that of any other animal what- ever, when compared with the bones of the head KK 2 254 Mr. ALpErson on a Whale of the Spermaceti Tribe. On looking over the several works that have been published relating to these animals, there are evidently so many contra- dictions, that it is very difficult to fix on any specific point on which to rest, for forming those distinctions on which the pleasure derived from knowledge depends. This is remarkably the case with the Physeter Macrocephalus, the external orifice of whose breathing tube has been described, as to its termination, so variously*, that it is very uncertain where to place the animal we have had the opportunity of examining ; its form and connexion too, its final cause, its mode of action, are all hitherto unascertained, and although in the following ac- count every thing has not been done, which more favourable circumstances might have afforded, yet I trust that something will have been done worth recording. External and essential characters of the animal. + Length of the animal from the snout to the division of the tail 584 feet. + Distance of the eye from the snout 20 feet 8 inches. + Circumference of the head from the sand on the one side to the sand on the other, taken midway between the eye and the snout, 31 feet 4 inches, (this of course does not include the lower jaw.) * Essential character of the Genus Physeter of Shaw; Dentes in maxilla inferiore Fistula in Capite s. fronte. P. Macrocephalus. P.dorso impinni, fistula in cervice?. + These measurements were taken on Friday the 29th of April, by the Rev. Christopher Sykes, Rector of Roos; an ardent promoter of science. They were taken by means of a tape. « This expression, according to Fabricius, is not quite correct. Shaw's Zoology, Vol. II. P. 2. Dr. Schwediawer, in Phil. Trans, for the year 1783, has given a much more correct account of the ex- ; ternal Mr. ALDERSON on a Whale of the Spermaceti Tribe. 255 + Circumference of the body at the spring of the tail 8 feet. The orifice of the spiracle, or breathing tube, was at the ex- tremity of the snout, and rather on the left side of the median line, somewhat of an f form, and 2 feet 4 inches in length, externally. There were two pectoral fins, in length 54 feet, in breadth 2 feet 9 inches, at the broadest part, with ee furrows be- tween the phalanges. The glenoid cavity of the scapula, (the chord of the arc) measured 9 inches, and the articulating surface of the humerus measured 10 inches. Dorsal fin. This was only the rudiment of a fin; it was composed of the cutis and adipose cellular membrane, like the rest of the external covering of the body, and projected at its greatest height not more than a foot, commencing gradually, and terminating abruptly in a sort of Ne process posteriorly. Sex. Male. Tail, horizontal; the span from tip to tip measured 14 feet ; there were several transverse parallel bands distinctly seen upon it, where the epidermis was removed. The Eye was placed at nearly the greatest lateral projection, a little inferiorly ; the eye-lids were formed of a duplicature of the outer covering, about an inch in thickness; the upper lid pro- jected more than the under, somewhat like a flap, and the opening was about seven inches in length. There was no external ear, but simply a_ small circular opening about nine inches posteriorly to the posterior canthus of the eye, which just admitted the finger. ternal marks of this animal. “It is observed,” he says, “that this species has but one spout (fistula). This spout is not, as has been generally hitherto asserted, in the neck (cervix) of the fish, but in its front, and on the very edge of the head, bending obliquely on the left side, so that whenever he spouts it is always on that side only.” His description, however, of the “ peculiar bony triangular cavity, or trunk, which contains the spermaceti, and which is lodged near the brain, and occupies nearly the whole of the upper part of the head,” in no way agrees with my observations on the anima! cast on shore at Tunstall, 256 Mr. AtpEerson on a Whale of the Spermaceti. Tribe. The Vent was situated at a distance of 12 feet from the spring of the tail, and the penis at a distance of 19 feet from the same place. This latter organ protruded about 15 feet from the body, and was surrounded by a shaggy process of the cuticle. The urethra admitted the point of the finger. Lower Jaw. This is of the form ofa Y, though all its parts are not in the same plane. The bifurcated parts lying in a plane which cuts that in which the stalk of the letter lies, when pro- duced. From the symphysis to the bifurcation it measured 11 feet. articulation 16 feet. The number of teeth in the jaw 47 (visible). Two more were found on cutting down upon the gum on the right side. In the skeleton, therefore, there will be 49 teeth. I should hence infer the animal to be young, though, as they that were uncut were the most posterior of the teeth, it is possible it had reached its full growth. There is a remarkable difference in the posterior teeth compared with the others; they were much smaller, and rather hooked; particularly the last but one, and last but two on the left side, and the last but two and last but three on the right side. The two teeth at the symphysis were much smaller than those near them; they were front teeth, and. were only three inches asunder. The teeth projected about two inches from the gum, and were, with the exception of the last five or six, blunt, with concentric lines on the worn surfaces. The relative position of the teeth, in the two sides of the jaw, varies in different parts of it. Thus, the last tooth on the right side is without a fellow on the left; the succeeding seven cor- respond with and are opposite to, the last seven onthe left side; the six nearest the symphysis correspond in like manner ; the intermediate teeth are alternate. Mr. ALDERSON on a Whale of the Spermaceti Tribe. 257 The tenth tooth from the symphysis is nearly vertical, and the others, with the exception of a few of the last teeth, all tend towards it; that is, those anterior to it look towards the arti- culation; those posterior, towards the symphysis. The distance between the second pair of teeth nearest the symphysis (measured within the teeth at the gum) is 54 inches. The distance between the teeth at the bifurcation 13 inches. In a vertical plane, the greatest depth of the jaw, near to the articulation, measured 2 feet 2 inches; the depth at the sym- physis 24 inches only. The upper jaw presented no teeth, but cavities lined with the mucous membrane of the mouth, and very firm; into these cavities the teeth of the lower jaw fitted, when the mouth was closed. The upper lip appeared to overhang considerably the under. The epidermis was black, and varied in thickness in different parts, in no place exceeding one-third of an inch. When cut into layers, it still preserved its color. The Cutis appeared intimately connected with what may be termed the adipose cellular membrane of the body ; the cells filled with liquid fat and oil readily expressed. The thickness of this layer was different too in different places, reaching to 15 inches on the ridge, whilst at the sides it was not more than 9 or 10. On the head this covering appeared of a more fibrous nature, and had lost its cellular structure as wel] as its oil; it seemed towards the snout solely to afford a surface for the insertion of the numerous tendons which were found in the head: it was here much thinner, and was almost too dense to be cut through by the spade*. The head formed a very_considerable part of the animal, as far as regards its bulk. * A tool in use at the fisheries. 258 Mr. ALpERSON on a Whale of the Spermaceti Tribe. Its commencement, at what may be termed the snout, was very abrupt, increasing in magnitude posteriorly, as far as the junction of the atlas with the os occipitis. _ No measurement, that I could hear of, was taken in this part, and it is the more to be regretted, as it was by far the most pro- minent part of the animal. There was a considerable contraction behind the head, cor- responding to the cervical vertibre. A small* portion of the spear or tooth of the sword-fish, about 5 inches in length, was found enveloped in the adipose cellular membrane, near the ridge of the back, anteriorly to the rudimentary dorsal fin; there appeared too, near this same place, a wound, a fistulous-like opening in the cutis; supposed by the labourers to have been made by a harpoon. Examination of the internal parts. The interior of the head+ contained, on the right side, a cavity or sac, or several sacs, holding spermaceti; the left was occupied by the breathing tube, and nearly all the surrounding and in- termediate parts by long tendons. - The most posterior part of this mass, filling up the large basin, formed by the bones of the cranium, as depicted in Plate XI, Fig. 2 contained a large cavity, lined posteriorly with a membrane, in color yellowish white, and in structure cellular; the convex sur- faces of the cells being towards the cavity, and about an inch in diameter. On being cut into, all the cells appeared to commu- nicate with each other. The structure of the lining membrane of the anterior wall was very different; it consisted of transverse * This is in the possesion of Mr. Hickney of Ridgemont, one of the Society of Friends, aud a great agriculturist. + Still exterior to the cranium, Mr. ALpEerson on a Whale of the Spermaceti Tribe. 259 folds, exactly similar to the lining membrane of the fourth stomach in sheep, in which the secretion of the gastric juice takes place. I traced the communication from this cavity forwards, to- wards the snout, first passing on the left side of the head, under the breathing tube; then crossing over to the right side of the head, and joining with the sac or sacs in which the spermaceti was found; here the lining membrane was altered in appearance; it was more that of a continued mucous surface. The breathing tube commenced at the posterior nares, on the left side of the head, and proceeded on this side to the snout, where it opened externally. Its length was 20 feet, 3 inches; its internal circumference, when cut open, measured 3 feet, 1 inch, near to the external orifice; it, however, became much smaller at its entrance into the foramen allotted to it, leading to the posterior nares. Near to the external orifice, the tube made a turn outwards, so as to propel fluid perpendicularly to the way of the animal, unless acted on by some of the tendons before mentioned. It was single throughout its whole length, and was lined with a continuation of the epidermis, much thinner, however, than in the snout, and becoming still thinner, as well as losing its black color, as it preceeded to the posterior origin of the tube, where it is connected with the trachea. As the whole of this mass was torn from within the basin formed by the bones of the cranium, by the assistance of horses, this communication of the breathing tube and the trachea, could not be made out satisfactorily. I am not convinced that there was not a communication between the posterior nares, and the peculiar cavity before described ; in which cavity, it is probable, the secretion of the spermaceti takes place. I am not, however, disposed to make any further con- yectures on this point: dubia pro falsis adhibenda. Vol. Il. Part I. Lu 260 Mr. Atprerson on a Whale of the Spermaceti Tribe. Along the sides of the sacs containing the spermaceti, as well as those of the breathing tube, ran innumerable thick and strong tendons*, terminating in the snout; evidently an apparatus for moving the snout, and the lips of the external orifice of the breathing tube. The snout overhung considerably the lower jaw. I am not aware that this was measured; indeed from the very first, the lower jaw was dislocated, and projected on the left side of the animal, from under the upper lip, at nearly a right angle to the body. The Eye.—This organ was not examined until the 8th of May: the parts offered for dissection consisted simply of the ball of the eye, together with the sheath which contains the straight muscles of the eye, enveloping the optic nerve, and were in no state to give any satisfactory result. The connexions with the adjoining parts had all been removed, through the ignorance of the workmen. The cornea was very much sunk and flaccid, and the conjunctiva had a dull bluish cast. On clearing away the insertions of the cut muscles, the ball of the eyet was measured. Its transverse diameter measured 2.35 inches. Diameter in the direction of the axis of the eye measured 2.25 inches. This latter measurement taken on the supposition that the cornea was originally flat. The color of the iris, as well as could be judged from its decomposed state, was bluish-brown; very dark; the pupil was transverse, as in ruminating animals. * Some of these were drawn out of the head by the by-standers, upwards of nine feet in length. + These measurements were all taken by means of a pair of graduated callipers. Mr. ALpERSON on a Whale of the Spermaceti Tribe. 261 The Chrystalline lens was nearly spherical, and rather more convex anteriorly than posteriorly. The transverse diameter measured .45 inch, Diameter in direction of the axis .375 The sclerotic coat was cut into about its middie, and there measured .7 inch in thickness, becoming thinner as it proceeded forwards to join the cornea, where it turned inwards. Where the optic nerve enters the sclerotic coat, this latter only measured .5 inch in thickness, differmg in this point from the same part in the balena, which, according to Cuvier, is an inch and a half in thickness. The density of this coat is very great, so much so, as previously to its being cut into, to give the idea of its being formed of cartilage, or corn of bony matter. The internal cavity, containing the humours and the iris, was very small, and not of the same shape as the exterior of the eye-ball, on account of the varying thickness of the scle- rotic coat; the depth of the interior cavity, from the anterior part of the.sclerotic coat, measured only .8 of an inch. We learn from Hunter, that the pigmentum nigrum covers the back part of the iris, and the corpus ciliare, but that it does not extend farther back. I should have felt great diffi- culty in stating its limits, as the sclerotica was certainly tinged with it posteriorly to this part. The tapetum presented a very beautiful appearance ; its color was a green, formed by an admixture of blue and yellow, with a slight predominance of blue; it was speckled with lighter colored spots throughout. Posteriorly to the transverse axis of the eye, was inserted into the sclerotica a thick mass of muscle, enveloping the optic nerve. This mass was itself enclosed in a dense fibrous sheath, LL2 262 Mr. ALpERSON on a Whale of the Spermaceti Tribe. (+ inch in thickness), and was of great length, owing to the extraordinary breadth of the head at this part. The diameter of the mass, including the sheath, was about two inches at its insertion into the sclerotica, but gradually decreased as it ap- proached the bony canal in the frontal* bone, through which the nerve passes to the brain. The sole use of these muscles must be to draw in the eye. The optic nerve itself appeared to have a vascular tunic. The Heart.—This viscus was furnished with a pericardium, and in structure was exactly similar to that of man. It was very flaccid, and the parietes, when examined on the 9th of May, lay in contact. Its weight was 171\bs. The descending cava measured 9 inches in diametert: the ascending cava was not with the heart, for it was not examined in situf. Diameter of the pulmonary artery 122 inches; thickness of the coat +inch. Breadth of one of the semi-lunar valves of the, pulmonary artery, 5 inches; its length 17 inches. There was no corpus sesamoideum apparent. 1 The diameter of the aorta was 12 inches. Thickness of the coat of the artery 7 inch. * However misplaced the orbit, it is still the frontal bone which dips down to form it: the sutures are well marked in Fig. 1. Plate XIV. + As these diameters were obtained through the medium of the half circumferences, it is probable, from the more yielding nature of the coats of veins, the measurements of the veins will rather exceed the truth, and those of the arteries rather fall short of it. } The heart was examined at the house of Mr. Sawyer, Surgeon, at Hedon, to whom I have to acknowledge myself indebted for his kind and skilful assistance in its examination. ik Mr. ALDERSON on a Whale of the Spermaceti Tribe. 263 In the sinus, behind the valves, the thickness was _ not greater than that of the pulmonary artery. Length of the heart, from the apex to the valves of the aorta, 3 feet 10 inches. The columne carne were very large, and one of the corde tendinez in the tricuspid valve, measured 7 inches in length. Near the middle of the left ventricle, the wall of the ven- tricle measured about 3 inches. The diameter of the coronary artery was 12 inches. On the left ventricle being laid open, its capacity was guessed, by some farming gentlemen present, to contain from 8 to 10 gallons. The heart was destitute of fat. Larynx and CEsophagus.—These parts, of great interest in this tribe, were supposed to have been sent perfect with the heart, but, on examination, it was ascertained that the ceso- phagus was wholly missing; and, in consequence, the parts could not be satisfactorily examined. There remained only a part of the trachea, with the cricoid and thyroid cartilages as in man, nearly, together with what has been termed the pyramid, the connexion of the trachea with the breathing tube. The urinary and genital organs were not examined; they were removed, with other parts of the animal, during the time I was occupied in the examination of the head. The Stomach.—This organ was cut open by the labourers, not with a view to the examination of its structure, but in search of ambergris, of which none was found. Near the termination of the csophagus was found about a bucket-full of the beaks of one of the cuttle fishes. See Plate XIV. Fig. 4. The lungs and liver were but cursorily examined; indeed, the viscera were so quickly removed, with a view to clearing 264 Mr. ALpERSON on a Whale of the Spermaceti Tribe. the bones of the animal, that it was impossible to examine every organ. Mr. Hunter, in his paper on this tribe, has made some curious remarks on the mode in which these animals suck, “which,” he says, “would appear to be very inconvenient for respiration, as either the mother or the young one will be prevented from breathing at the time, their nostrils being in opposite directions; therefore the nose of one must be under water, and the time of sucking can only be between each respiration.” Now, as the external orifice of the breathing tube is on the left side of the median line, and the mamme near the anus, supposing both the mother and the sucker on their right sides respectively, but reversed in position, I see nothing to prevent the sucker continuing his occupation, without any interruption whatever to his respiration, or to that of the mother. It only remains for me to state, that the skeleton will be articulated and preserved at Burton Constable, the seat of Sir Thomas Constable, Bart., to whom the animal belongs, as Lord Paramount of Holderness. The bones are now macerating in pits, where they will haye to remain a considerable time. In the Autumn, probably, the process of articulating will be commenced, and from the known zeal of the steward* of the estate, there is every pro- bability of its being completed in a skilful manner. * Mr. Iveson of Hedon. Kineston upon Hutt, May 14, 1825. Fig. 1. Fig. 2. Fig. 1. EXPLANATION OF THE PLATES. PLATE XII. A VENTRAL view of the Whale, shewing the jaw in its dislocated position; the animal lying on its right side. A dorsal view of the Whale, shewing the external orifice of the spiracle or breathing tube; also the rudimentary dorsal fin. » PuLaTeE XIII. The lower jaw, seen from the articulation, after having been removed from the body. The cranium without the lower jaw, seen from the anterior part of the head, to shew the basin formed by the bones, for the reception of the enormous mass composing the head and throat. The width of this basin measured 5 feet, 11 inches. The width across the orbits 8 feet, 8 inches. The length* from the summit of the cranium to the anterior extremity, 20 feet. PuatTe XIV. A side view of the cranium without the lower jaw. Shewing the articulating surface of the os occipitis, which is received into the concave surface of the atlas. Horizontal diameter of this articulating surface, including, between the condyles, the foramen magnum, 2 feet, 5 inches. Horizontal diameter of the foramen magnum, 8 inches. Vertical diameter of ditto,...............6: 9 inches. This gradually became smaller as it approached the cavity for the brain, when it did not exceed 4 inches in diameter. Transverse diameter of orbit, (horizontal)... 8 inches. * The hypothenuse of a triangle formed by these two points, and the bottom of the basin, io. 4*. EXPLANATION OF THE PLATES. The eye, in the state in which it was delivered to us for dissection. The eye, after a transverse section had been made, shewing the magnitude of the internal cavity of the eye, the thickness of the sclerotic coat, the choroid having been separated from the sclerotic, to which it closely adhered, |and turned aside, containing within it the retina. A representation of a beak of one of the cuttle fishes; it was found in the stomach of the animal. * These are very nearly of the natural size, XVI. On a peculiar Defect in the Eye, and a mode of correcting at. By GEORGE BIDDELL AIRY, B.A._ FELLOW OF TRINITY COLLEGE, OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, AND CORRESPONDING MEMBER OF THE NORTHERN INSTITUTE. [Read Feb. 21, 1825.] "Tue communication which I have now the honour to make to this Society, relates to a peculiar defect of the eye, and the mode of correcting it. On a subject so important, [ trust I shall be excused if I enter into details; as the mal-formation which I am about to describe, though hitherto unnoticed, is probably not uncommon. Two or three years since, I discovered that in reading I did not usually employ my left eye, and that in looking carefully at any near object, it was totally useless: in fact, the image formed in that eye was not perceived except my attention was particularly directed to it. Supposing this to be entirely owing to habit, and that it might be corrected by using the left eye as much as possible, I endeavoured to read with the right eye closed or shaded, but found that I could not distinguish a letter, at least in small print, at whatever distance from my eye the characters were placed. No further remark suggested itself at that time, but a considerable time afterwards I observed, that the image formed by a bright point (as a distant lamp or a star) Vol. If. Part If. Mm 268 Mr. Airy on a peculiar Defect in the Eye. in my left eye, was not circular, as it is in the eye which has no other defect than that of being near-sighted, but elliptical, the major axis making an angle of about 35° with the vertical, and its higher extremity being inclined to the right. Upon putting on concave spectacles, by the assistance of which I saw distant objects distinctly with my right eye, I found that to my left eye a distant lucid point had the appearance of a well defined line, corresponding exactly in direction, and nearly in length to the major axis of the ellipse above-mentioned. I found also that if I drew upon paper two black lines crossing each other at right angles, and placed the paper in a proper position, and at a certain distance from the eye, one line was seen perfectly distinct, while the other was barely visible: upen bringing the paper nearer to the eye, the line which was distinct now disappeared, and the other was seen very well defined. All these appearances indicated that the refraction of the eye was greater in the plane - nearly vertical, than in that at right angles to it, and that conse- quently it would not be possible to see distinctly by the assistance of lenses with spherical surfaces. I found, mdeed, that by turning a concave lens obliquely, or by looking directly through a part near the edge, I could see objects without confusion ; but in both cases, the distortion produced in their figure was such, that I could not hope to make any use of the left eye without some more effectual assistance. My object now was to form a lens which should refract more powerfully the rays in one certain plane, than those in the plane at right angles to it; and the first idea was to employ one whose surfaces should be cylindrical and concave, the axes of the cylinders crossing each other at right angles, and their radu being different. To shew that this construction would effect my purpose, it is only necessary to imagine the lens divided into two lenses by a plane perpendicular to its axis; then it is easily seen Mr. Arry on a peculiar Defect in the Eye. 269 that the refraction of one will not be perceptibly altered by that of the other, and that the whole refraction will be the combi- nation of the two separate refractions. The rays in one plane will be made to diverge entirely by the refraction of one lens, and those in the other plane by that of the other lens. If then r and r be the radii of the surfaces, and 7 the refractive index, and parallel rays be incident, the rays in one plane after re- 5 . . 5 a f a 7 fraction will diverge from a point whose distance is = ae and 1 5 3 . . T those in another plane from a point whose distance is = 1 This construction then was sufficient; but for the facility of grinding, and for the diminution of the curvatures, it appeared preferable to make one surface cylindrical, the other spherical ; both concave. Let r be the radius of the cylindrical surface, R that of the spherical: then the refraction in the plane passing through the axis of the cylindrical surface, being entirely effected by the spherical surface, parallel rays in this plane after refraction will diverge from the distance pbs while the refraction in the plane perpendicular to the axis being caused by both surfaces, parallel rays, in this plane, will on their emergence, diverge from the distance ———1___. == (+8) J cae To discover the necessary data, I made a very fine hole with the point of a needle in a blackened card, which I caused to slide on a graduated scale; then strongly illuminating a sheet of paper, and holding the card between it and the eye, I had a lucid point upon which I could make observations with great ease and exactness. Then resting the end of the scale upon the cheek-bone, and sliding the card on the scale, I found that the point at the distance of 6 inches, appeared a very well defined MM 2 270 Mr. Airy on a peculiar Defect in the Eye. line inclined to the vertical about 35°, and subtending an angle of 2° (by estimation): at the distance of 35inches it appeared a very well defined line at right angles to the former, and of the same apparent length. It was necessary therefore to make a lens, which, when parallel rays were incident, should cause those in one plane to diverge from the distance 35 inches, and those in another plane from the distance 6 inches. Making the expressions above equal to these numbers, and supposing ” = 1,53, we find R=3,18, r=4,45. ‘To prevent if possible the eve from becoming more short-sighted, I fixed upon the values R= 34, r= 4}. After some ineffectual applications to different workmen, I at last procured a lens to these dimensions from an artist named Fuller, of Ipswich. It satisfies my wishes in every respect. I can now read the smallest print at a considerable distance with the left eye, as well as with the right. I have found that vision is most distinct when the cylindrical surface is turned from the eye: and as when the lens is distant from the eye, it alters the apparent figure of objects by refracting differently the rays im different planes, I judged it proper to have the frame of my spectacles made so as to bring the glass pretty close to the eye. With these precautions I find that the eye which I once feared would become quite useless, can be used in almost every respect as well as the other. The publication of this case, I imagine, may be not without utility. I believe it has generally been found, that where the direction of the axis of the eye is distorted, the sight of the eye is defective, but not lost: and the distortion is by many ascribed to the disuse of the eye, which is occasioned by this defect. If it should be found that the defect is at all similar to that which I have described, it can be perfectly corrected. The examination of the defect in the manner which I have detailed is very easy ; and it is merely necessary to write down fully the appearance of Mr. Arry on a peculiar Defect in the Eye. 271 the brilliant point at different distances, in order to enable the theoretical optician to mvent a glass which shall make the vision of the eye distinct. If the defects arise from insensibility of the nerve, or opacity of the humours, they are beyond his power: but any fault in the refracting surfaces it is possible to correct. Since I procured this lens, I have been informed that a foreign artist has made spectacle-glasses with cylindrical surfaces of different radii for general use. What his object can be I am quite unable to imagine; certainly no one whose eyes are not defective can see with them distinctly. With my right eye which (by the method of examination above described) I find to have no other defect than short-sightedness, I am unable to read any thing in the lens made for my left eye. After many inquiries I have not been able to discover that this construction has been used to correct any defect in the eye, or even that a defect similar to that which I have described, has ever been noticed: In laying before this Society the notices of a case which appears at once novel and important, I trust that I shall not be thought to have trespassed unprofitably upon their time. G. B. AIRY. Trinity CoLiece, Feb. 5, 1825. , Spathies rien. a XVII. dA general Demonstration of the Principle of Virtual Velocities. By THe Rev. J. POWER, A.M. FELLOW OF CLARE HALL, CAMBRIDGE, AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. {Read March 21, 1825.] Conceive a machine, of any description whatever, to be kept at rest by forces P, P’, P’....., and let m, m’, m’..... denote the points of their application. Call ayaa’... any indefinitely small spaces, subject to the conditions. of the system, which these poimts are at liberty to run over in the same instant; and £#, pf, B’..... the cor- responding spaces estimated in direction of the forces. Then ashi ak cel will be the cosines of the angles at which P, " Pee are inclined to the spaces a, a’, a’..... ; so that if we resolve these forces into others, a part tangential, and the remainder normal to the above spaces, the former will be ex- pressed by for which we may substitute the following: namely, 274 Mr. Power on the Principle at the point m, ak PB+ PR ah Pp a! t . a at m’, PB+P'8 + P'S’ — PB+ PB a a ” at m’, and so on. But the equilibrium will remain undisturbed, if for the two P P : ; . forces se and a2. (of which the former acts upon m in di- rection of a, and the latter upon m’ in the direction opposed to a’,) we substitute two strings stretched with these forces, and acting in the same directions: again, the strings so stretched, may be conducted over fixed pullies, in such a manner, that their other extremities may impel perpendicularly the arms of a straight lever, divided by its fulcrum in the ratio a: a’. The fulcrum will then re-act with a force equal to the sum of the two tensions, and the whole will remain at rest. Should PB be negative, the strings stretched with the positive forces, must proceed from the two points in directions opposite to what we have just supposed, and be attached to the lever as before. In the same manner we may substitate for the forces the re-action of a second lever, divided in the ratio @ : a’. If we make similar substitutions throughout the system, at the same time substituting for the normal part of the resolved forces the re-action of so many fixed surfaces intersecting them at right angles; there will at length remain a single tangential force, whose numerator is the sum of all the terms P6+P'B'+ P’p"+..., making equilibrium with the machine, which results of Virtual Velocities. 275 from uniting with the original system the series of levers and surfaces we have just introduced. Now it is easy to perceive, that the new connexion we have established, leaves the points at liberty to describe, backwards or forwards, the same indefinitely small spaces, which were allowed them by the original machine, provided they were previously at liberty to describe spaces proportional to a, a’,a’... in the opposite directions: but this will evidently be the case; for a, a, a’.... beimg indefinitely small, the equations to which they are subject, must be homogeneous with respect to these variables, or may be considered so; they will consequently re- main equally satisfied when these quantities simultaneously change their signs. From hence it follows, that, in the combined machine, the slightest tangential force applied to one alone of the points, on either side of it, will necessarily disturb the equilibrium. The equilibrium is, therefore, impossible, unless PPPs PP ps. i=0, which is the symbolic enunciation of the principle in question. The converse of this proposition is equally true; namely, that if the equation PB+PB+P’p’+. Ae .=0, be satisfied for every indefinitely small variation in the position of the system, there will be equilibrium. For if an initial motion were possible, the equilibrium might evidently be restored, by applying new forces in the opposite direction to those points only, whose variations, taken along the initial spaces, (but not necessarily identic with them,) may be regarded arbitrary and independent in other respects, without interfering with those points, whose variations are altogether determined, and their motions constrained, by those Vol. Il. Part Il. Nw 276 Mr. Power on Virtual Velocities. of the former. Let a, a, a’..... be the variations of the first class, and a, a, .... those of the second. Then 8, B, B’.... B, B,...-. being the corresponding spaces estimated in direction of the forces, and F, F’, F’,.... the forces requisite to maintain the equilibrium, we shall have, in consequence of what has been already proved, PB+PB+P"p'+....+P,B,4PB, 4+... V=o aig nip This equation is reduced by the hypothesis to Fa+F'd'+ Fa" +....=0. But a, a, «’.... being arbitrary and independent, we must have, separately, F=0, F’=0, F’=0, &c., and, consequently, the forces py pihpy ap alive | will be in equilibrium by themselves. J. POWER. Crare Hatt, March 19, 1825. XVIII. On the Forms of the Teeth of Wheels. By GEORGE BIDDELL AIRY, B.A. FELLOW OF TRINITY COLLEGE, AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, AND CORRESPONDING MEMBER OF THE NORTHERN INSTITUTE. [Read May 2, 1825.] Tue investigation of the forms proper for the teeth of wheels is a useful and interesting inquiry. The mechanical prin- ciples are very simple, and the geometrical propositions on which it is immediately made to depend, admit of being put in an elegant form. But all the theories which have yet been given, are, I believe, very imperfect. Euler in the New Petersburgh Commentaries for 1760 has treated the subject with great gene- rality; but the analytical method which he has used is very unfavourable for the discovery of the most obvious. properties of the curves. In all the other theories that I have seen, no -forms are mentioned but the involute of a circle, and the epi- cycloid and hypocycloid. In this paper I propose to consider generally the figures which must be given to the teeth of wheels to insure uniformity of action. The curves above alluded to, though probably the most convenient of all, I shall shew are particular cases of a very general construction: and the demon- stration which has usually been given for them, I shall apply to every other case. . That the mechanical effect which one wheel produces upon another, may in all positions be the same, it is necessary that NN 2 278 Mr. Airy on the Forms of the Teeth of Wheels. the line perpendicular to the surfaces of the teeth, at the point of contact, intersect the line joining the centers at a fixed point, which divides that line into two parts, the ratio of which is the mechanical power. When this holds, the proportion of the angular velocities will be constant. For let 4 and B (Plate XV. Fig. 1.) be the centers of the wheels, C the point through which the line of action passes: D the point of contact: upon moving the wheels with the teeth still in contact through a very small angle, D in one tooth will be carried to F, and in the other to G, FG being ultimately parallel to the tangent at D, or perpendicular to CD, and DF, DG, perpendicular to AD, BD respectively. Then, BC AC. BD AD’ ee : FD GD 5 therefore the angular velocities, which are as AD * BD’ will be FD : GD:: sn G : sn F :: sm BDC: sin ADC :: as BC : AC, a constant ratio. If then with centers 4 and B circles be described passing through C, and these circles revolve so as to make the velocities of their circumferences equal, the teeth of the wheels, if properly formed, will be in contact, and the normals to both will pass through C. These circles we shall call the principal circles of the wheels. If the normals from every point of the tooth should be equally inclined to the tangents of the circle at the points where they meet the circle, they evidently would if produced be tangents to a circle, whose radius : radius of circle described :: cosine of inclination of normal with tangent of circle described : 1. In this case both teeth would be involutes of circles. If the inclinations are not equal, we must make use of the following theorem. It is always possible to find a curve which by revolving upon. a given curve, shall by some describing point, in the manner of a trochoid, generate a second given curve: provided that the normals from all points of the second curve meet the first. Mr. Atry on the Forms of the Teeth of Wheels. 279 To prove this let 4B, (Fig. 2.) be the first curve, 4C the second ; from the points C and E, which are very near, draw the normals CD, EF; if a describing point P be taken, and PQ, PR, be made respectively equal to CD, EF, and QR equal to DF, and this process be continued, a curve will be formed, which by re- velving upon Bd, will, by the describing point P, generate the curve AC. For if Q coincide with D, then R will afterwards coincide with F, and so on for all succeeding points, since QR = DF. Also DC= QP, &c. And the angles made by these with the tangents are equal. For the cosines of these angles, drawing DG, : Su bez ; QS, perpendicular to EF, PR, are = and ee in which the numerators are the differences of equal lines, and the denominators are equal. Hence P will describe 4C. And the formation of the curve RQ is always possible, because RQ is greater than RS; for FD is necessarily greater than FG. As an example of this, suppose it were required to find the curve, which revolving on one straight line 4B, (Fig. 3.) would generate another straight line 4C. Since the angles made by the line PQ with the tangent, must be constant, it follows, that the curve weuld be the loga- rithmic spiral, P being its pole. The entire theory of the teeth of wheels, may now be included in this proposition. If the tooth HD, (Fig. 4.) be generated by the revolution of any curve on the outside of the circle HC, and if DK be generated by the revolution of the same curve in the same direction, in the inside of the circle KC, then the normal at the point of contact of the teeth, will pass through C. For let the generating curve be brought to the position LC, so as to touch the cirele HC at C; DC will be the normal of HD at D; and that the teeth may be in contact, the same generating curve in the other circle must touch AC at C; in which case it will coincide with this; D therefore will be in the surfaces of both 280 Mr. Arry on the Forms of the Teeth of Wheels. of the teeth, and CD the normal of both at that point; therefore they will touch at D, and the line of action CD, will pass through the fixed point C. If now we give equal velocities to the cir- cumferences CH, CK, the same will be found at all times to be true. These forms then are proper for the teeth of wheels. Suppose then this problem proposed. Given the form of the teeth of one wheel, to find the form of those of another, that they may work together correctly. The followimg is the obvious solution. Divide the line joining the centers of the circles at C, into two parts, whose proportion is the mechanical power. De- scribe the circles CH, CK. Find the curve which by revolving upon CH, will generate the given tooth HD. Make the same curve revolve in CK, and with the same describing point let it generate KD; KD is the form required. The usual construction of the involute of a circle, would seem to require that the circles 4H, and BK, should be separated. If however DH be the involute formed in the usual way from the circle MN, (Fig. 5.) the normal CM will be inclined at a E : P J constant angle to C4, (since its sine = Lea and the construc- tion given before shews that the involute HD may be generated by the revolution of a logarithmic spiral upon CH, the describing point being the pole of the spiral, and the angle between its radius and tangent, the same as the angle made by MC, with the tangent of the circle at C. In the same way the revolution of this spiral in the second circle will generate another involute ; and hence if the teeth of one wheel be involutes, those of the other wheel must also be involutes. The generating circles of the involutes must have radii proportional to 4C, BC. It will be seen immediately, that we may if we please sup- pose successive parts of the curve described by different generating curves; or we may make one curve revolve on the outside of Mr. Airy on the Forms of the Teeth of Wheels. 281 the circle CH, and another on the inside, making the same curves revolve on the inside and outside of CA respectively, and thus an infinite variety of curves may be found. The construction last mentioned gives forms approximating most nearly to the usual forms of teeth. We may even give different forms to dif- ferent teeth ; but this probably would not be desirable. It may be desirable to know when the nature of the teeth will admit of an alteration in the distance of the centers of the wheels. Suppose then DL and FP, (Fig. 6.) to be the principal circles when the wheels are in the first position; KS and HR, the principal circles when the distance of the centers is increased. Suppose in the first position C was in contact with Z, and M with O; suppose in the second position, G and Q are in contact with E and O; draw normals to all these points as in the figure. Since the wheels in the first position work correctly, by sup- position, the angles at D and N will equal those at Fand P. And if they work correctly in the second position, HG will = KE, &c. HR will = KS, and the angles at H and R will equal those at K and S. By attending to this condition, when the tooth ZO is given, we can always form a tooth CQ, which will work with it in two positions of the wheels. Since the angles at H and R equal those at K and S, the angles at L and T will equal those at F and P; and therefore will equal those at D and N. It is evident that this condition will always be satisfied, if CQ be the involute, and therefore if the teeth be involutes, the distance of the centers may be altered to any degree, allowing the teeth to act on each other. In all, however, that has yet been stated, we have only con- sidered the mathematical conditions of the contact of two curves. That these forms may be applicable in practice, it is necessary that the curvature of the convexity of one tooth, should be greater than that of the concavity of the other, or else that both should 282 Mr. Arry on the Forms of the Teeth of Wheels. be convex. For this purpose we must investigate the curvature at any point. Take then two points on the circle near each other, and the two points of the generating curve which will touch them; join these with the center of curvature of the generating curve, and with the describing point; let ¢, 0, ¥, (Fig. 7.) be the small angles at the center of curvature, the describing point, and the center of the circle; suppose the lines from the describing point, when in contact with the circle, to be produced respectively, and let the angle at their point of intersection = x. Also let a and B be the angles which those lines make with the radii of the circle. Then we shall have OP =a BY Xe ig ee Nt But calling R the radius of the circle, r the radius of curvature, s the distance of the describing point, z the distance of the point of intersection, are are are . cos a are. cosa = —: —————— SS = ——_——_ 5 ie R?’ . rT s > Xx x cosa _1 1 cos a. COs a ee fee ed ; bie COS a Rt s Lo eail Rt? r+s=s = rad. of curvature of tooth; 1,1 _ cosa Ro rT s Lae COs a COS a Lee” or s 1 ° “. curvature = = .———_—__ = - — é $s aye $s ies ie Ri ir Roo? From an examination of this expression, it appears, that when «a is < 90°, r may be positive or negative, but must be less than the radius of the circle in the same direction; when Mr. Airy on the Forms of the Teeth of Wheels. 283 a is >90°, r may be positive or negative, and must be greater than the radius in the same direction. If then, as is the case in general, a be < 90°, that part of the tooth which is without the circle, must be formed by the revolution of some curve upon the circle, and that which is within it by the revolution of some curve within the circle. This kind of tooth is represented in Fig. 4. But if a may be > 90’, the whole of the teeth may be formed by the revolution of a single curve; an instance of this is represented in (Fig. 8.) where the teeth GH and KL are formed by the motion of MN, carrying the describing point P. In the last case, if the curve be a circle equal to one of the circles, one tooth will be reduced to a point, the other will be an epicycloid or epitrochoid, according as the describing point is in the circumference of the circle, or in any other part. It will easily be seen, that where the acting surface of the driving tooth is above the cirele, the action takes place after passing the line joining the centers; when below the circle, it is before passing that line. Now practical men always think it proper, that the action should take place only after passing the line of centers. It is thought necessary that the direction of the friction should be such as to wipe off the dust, &c. from the teeth. For this purpose then, the curve which has been found for the lower part of the teeth, must be considered as a limit which that tooth must not reach. In the case in which the whole is fermed by the revolution of one curve, the whole action takes place after passing the line of centers. To find what the friction really amounts to, we have merely to observe, that in Fig.1. if D be brought to G in one tooth, and to F in the other, GF is the friction, and if BDC =a, FG : FD :: sn ADB : sin a; therefore frictional motion « ee sin ADB j BCD nearly, (the teeth being so small, that DF may be Vol. Ll. Part U. Oo 284 Mr. Airy on the Forms of the Teeth of Wheels. considered as nearly representing the motion of the circumference.) Also the pressure occasioned by a given force in given cireum- 1 : $3 415% stances * — pap: and the mechanical effect of friction is propor- tional to the pressure by which it is caused multiplied by the velocity of the rubbing surfaces; and therefore > aay nearly. The numerator is proportional to the distance from the line of centers; and therefore will be the same for all teeth, when that distance is the same. But the denominator is largest when the face of the tooth is parallel to the radius of the circle. I imagine then that it is advisable to make the teeth work a little before as well as a little after the line of centers. And I should think that a tooth similar to that formed by the union of the epicycloid and hypo- eycloid, is preferable to any other form whatever. For the line of action is always very nearly perpendicular to the radius ; by which means not only is the friction made much less, but also the strain upon the axes is considerably diminished. If it be thought desirable to prevent back-lashing, this can be done by giving proper forms on the same principles to the faces of the teeth, which are not the working faces. But the chance of very greatly increasing the friction, makes the propriety of this consideration very doubtful. The whole of what has been stated with regard to circles, it is evident will apply equally to straight lines. Thus the teeth of rack-work may be formed of a combination of cycloids, in which case those of the wheel must consist of epicycloids, and hypocycloids; they may be straight, which will make those of the wheel the involutes of a circle, (both being generated by the revolution of a logarithmic spiral ;) they may be mere pins, in which case the teeth of the wheel will be involutes, or curves described in nearly the same manner as involutes. In this case, Mr. Airy on the Forms of the Teeth of Wheels. 285 and in the case of trundles if it be required to take account of the diameter of the pins, this will be done by taking a curve, whose normal distance from the curve found by considering them as points, shall at all parts be equal to the radius of the pin. Or the form of the teeth may be found by the general theorem. For crown wheels, as the contrate wheel of a watch, the teeth without sensible error may have the same form as for rack- work. The theory may be extended to bevelled wheels, without any difficulty. There is one case which ought to be mentioned particularly. It may be desired that the teeth of one wheel have plane surfaces passing through the axis of the wheel. Since a straight line is the hypocycloid, in which the radius of the generating circle is half that of the fixed circle, the teeth of the other wheel must be epicycloids, the radius of the generating circle being half that of the first wheel. The action here takes place entirely after the line of centers, and the direction of the action is nearly per- pendicular to that line. I imagine this to be a good construction for pinions with a small number of teeth driven by a large wheel. If each tooth consist of a line within the principal circle, and an epicycloid without it, the radius of the generating circle of each epicycloid, being half that of the other principal circle, a very good form will be produced. The action takes place before as well as after the line of centers, and is always nearly perpendicular to that line. The figure usually given to the teeth of watch-wheels approaches very nearly to this. I have confined my attention entirely to uniformity of action, and uniformity of motion, as I conceive them to be of far greater consequence than the diminution of friction. The friction can never be made = 0, except the point of contact be always in the line of centers ; a condition which may be satisfied by an infinite number of curves, and amongst others by two logarithmic spirals. 002 286 Mr. Airy on the Forms of the Teeth of Wheels. — But the mechanical action and the motion would be dreadfully irregular. I am informed by engineers, that this question is now little more than one of mere curiosity. In consequence of the very extensive use of iron, where wood was formerly employed, the teeth of wheels are now made so small, that it is of little conse- quence whether they have, or have not, the exact theoretical form. Almost all teeth are now made with plane faces passing through the axis of the wheel, and are expected to wear themselves in a short time into proper forms. This is the case with nearly all the modern iron wheels that I have examined; in the wheels of clock and watch-work, some attention to the figure is however thought necessary. G. B. AIRY. Trinity CoLLecE, = April 30, 1825. XIX. Observations on the Ornithology of Cambridgeshire. By tHe Rev. LEONARD JENYNS, M.A. F.L.S. AND FELLOW OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. [Read Nov. 28, 1825.] In drawing up the following paper, it has been my object to collect a few particulars respecting the Ornithology of Cambridge- shire. A first attempt towards any undertaking of this nature must necessarily be very imperfect. From this circumstance I desire that the present may not be considered as a complete catalogue of the birds which are found in this county, especially as the greater part of the observations from which it has been chiefly compiled, have been confined to the neighbourhoods of Cambridge and Bottisham. With the view, however, of rendering it as extensive as possible, and in some measure of supplying the deficiency arising from this cause, I have added from our English authors what- ever notices I could find in their works relating to species which though formerly met with in this district have not lately occurred to my knowledge. There is the greater interest attached to these, as there is reason to believe that many of them have become extremely rare, if not wholly extinct, in consequence of the strik- ing change which has of late years taken place in the face of this county from drainage and enclosure. I have also occasionally benefitted from the information of my friends. Such are the sources from whence I have drawn my materials. Under many 288 Mr. Jenyns on the Ornithology of Cambridgeshire. of the species I have inserted, from personal observation, a few remarks illustrative of their habits and manners; particularly dis- tinguishing such as are indigenous from such as are only periodical or occasional visitants. Possibly some of these observations are not new, but it is perhaps of advantage to the science to have them confirmed in different parts of the country. I have judged it unnecessary in a paper of this nature to give any synonyms or full descriptions, but have annexed to each species a reference to Temminck, in whose excellent Manuel d’Ornithologie* ample information on these subjects. will be found. In the systematic arrangement, with the exception of one or two instances, I have uniformly followed that author. a ORDER I. Rapaces. Genus I. FALCO. Linn. Se.1. F. peregrinus, Temm. Man. d’Ornith. p.22. PEREGRINE Fatcon.—There is a specimen of this bird in the Museum of the Cambridge Philosophical Society, which was shot near Cambridge in the spring of 1823. I have since heard of others: that have been observed at Coton. Sp.2. F. Subbuteo, Temm. Man. dOrnith. p. 25. Hogpy.—A nest of these birds was once found at Cottenham. Sp.3. F. Hsalon, Temm. Man. d’Ornith. p.27. Meruin.—lInserted on the authority of Graves, who in his British Ornithology has figured a specimen which was killed near Cambridge. * | refer to the second edition in two volumes octavo, published at Paris in 1820. Sp. 4. Sp. 10. Mr. Jenyns on the Ornithology of Cambridgeshire. 289 F. Tinnunculus, Temm. Man. @Ornith. p. 29. Kestrit.—This is by far the most common hawk we have. ‘The nest, which consists of little else than a few sticks loosely put together, is often placed on the tops of the tallest spruce firs. The eggs are four or five in number, of a reddish brown colour, stained with darker spots and blotches. These are hatched the latter end of April. F. Nisus, Temm. Man. d@’Ornith. p.56. Sparrow-Hawk.—The males of this species occur much less fre- quently than the females. F. Milvus, Temm. Man. @Ornith. p. 59. Krre.—Not so abundant as the two preceding species. F. Buteo, Temm. Man. d@Ornith. p. 63. Common Buzzarp. F. Lagopus, Temm. Man. d’Ornith. p. 65. RovuGuH-LEGGED Buzzarp.—A specimen of this bird, shot in the vicinity of Cambridge, is in the collection of Dr. Thackeray, Provost of King’s College. F. rufus, Temm. Man. d@Ornith. p. 69. Moor Buzzarpv.— This species is entirely confined to the fens and low grounds, in which situations however it is very plentiful, building its nest amongst the tall grass and rushes. I have had the newly fledged young brought me from Burwell fen, the second week in May: these have uniformly wanted the yellow patch on the crown of the head, so conspicuous in the adult bird. It is at all times a variable species with respect to plumage, being some- times found with the lower half of the abdomen entirely white, and the other parts of the body here and there spotted with that colour. This and the Common Buzzard bear indiscriminately the provincial name of Puttock. F. cyaneus, Temm. Man. d’Ornith. p.72. Sap remen re f This species seems also to be most = mos Rinc-Tait, Female. ne s also to 290 Sp. 11. Spe. 12. Mr. Jenyns on the Ornithology of Cambridgeshire. partial to marshy districts; at least it always breeds in such situations, placing its nest on the ground. In the young birds the difference of plumage between the two sexes is not discernible. Genus II. STRIX. Linn, * With ears. S. Otus, Temm. Man. d@Ornith. p. 102. LonG-EARED Ow1.—This is a rare species. Some years ago a female was taken out of a hollow tree at Bottisham, and was kept alive for a few days, during which time it layed one egg of a dull white colour. It has this year (1825) been shot at Swaffham Prior. S. Brachyotos, Temm. Man. d’Ornith. p.99 SHORT-EARED Ow. — This is only seen with us during the autumnal and winter months, retiring northward in the spring to breed. Though unknown in many parts of England, it is not uncommon throughout the low grounds of Cambridgeshire, where it makes its first appearance towards the latter end of September. I have been informed that in the fens, in the neighbourhood of Littleport, these birds are sometimes found in astonishing plenty, particularly after those seasons which have been most productive of field mice, which appear to be their favourite food and a great object of attraction. In those districts they are known by the name of Norway Owl, being supposed to come over to us from that country *. Their usual haunts are fields of coleseed and turnips, in which situations they may often be put up one after another to the number of fifty or more; but they are never observed in stubbles or amongst trees during the day, though they resort to these last to roost at night, and at such times seem much attached to plantations of spruce firs. * Montagu in his Ornithological Dictionary appears to have been of the same opinion. Spe. 13. Sp. 14. Sp. 15. Sp. 16. Spe. 17. Mr. Jenyns on the Ornithology of Cambridgeshire. 291 * * With smooth heads. S. flammea, Temm. Mann. @Ornith. p.91. WuitE Ow..—The food of this species is entirely confined to fresh field mice, which are devoured whole. During the breeding season, which continues throughout the summer, I have observed that it will often catch shrew mice, and bring them home to its young, but it is worthy of note that these were uniformly rejected afterwards, (probably on account of their strong musky odour,) and might be found entire at the foot of the nest. In one instance that occurred at Ely, I noticed amongst these rejectamenta a mu- tilated specimen of the rare species, the Watershrew (Sorex fodiens). S. Aluco, Temm. Man. d’Ornith. p.89. Brown Ow..— Unlike the preceding this is a very general feeder, preying upon rats, moles, rabbits, small leverets, &c. and is consequently destructive to game. It builds in old trees, and is a very early breeder, frequently hatching by the end of March. It is the only British species that hoots. ORDER II. Omnrtvort. Genus III. CORVUS. Linn. C. Corax, Temm. Man. d’Ornith. p. 107. Raven.—Not so plentiful as formerly. C. Corone, Temm. Man. d’Ornith. p. 108. Carrion Crow. C. Cornix, Temm. Man. d’Ornith. p. 109. Royston Crow. — Plentiful on our downs from October to April. Vol. If. Part I. Pp 292 Sp. 18. Sp. 23. Sp. 24. Mr. Jenyns on the Ornithology of Cambridgeshire. C. frugilegus, Temm. Man. @Ornith. p. 110. Rook. — Varieties of this bird, more or less spotted with grey and white, not unfrequently occur at Bottisham. There are two specimens of this kind from that neighbourhood, in the Museum of the Cambridge Philosophical Society. C. Monedula, Temm. Man. @’Ornith. p.111. JACKDAW. C. Pica, Temm. Man. d’Ornith. p. 113. MaGrir. C. glandarius, Temm. Man. d’Ornith. p. 114. Jay.—This is a rare bird at Bottisham, and only an occasional visitant of that district; though more plentiful in the woodlands. At Gamlingay I have observed them in abundance. Genus IV. BOMBYCIVORA. Tem. B. garrula, Temm. Man. d’Ornith. p. 124. Bonemian Wax-winc.—I have been informed that some few years back two flights of these birds were at different times observed near Cambridge. Genus V. STURNUS. Linn. S. vulgaris, Temm. Man. d’Ornith.. p. 132. SrarLING.—Towards autumn these birds congregate in immense flocks. ——>—_ ORDER TIl. Insecrivort. Genus VI. LANIUS. Linn. L. Excubitor, Temm. Man. d’Ornith. p.142. CrnErEous SHRIKE.—Has been observed to visit Cambridge- shire in the autumnal and winter months. There is a specimen Sp. 25. Sp. 26. Sp. 27. Spe. 28. Mr. Jenyns on the Ornithology of Cambridgeshire. 298 in the collection of Dr. Thackeray, which was found dead at Melbourn in the year 1824, and was supposed to have been recently killed by a hawk. L. Collurio, Temm. Man. d@ Ornith.- p. 147. ReED-BACKED SHRIKE.—This species has occasionally been shot at Cherry-Hinton. Genus VII. MUSCICAPA. ~ Linn. M. grisola, Temm. Man. d’Ornith. p. 152. SporrEeD FLy-catcHEeR.—This is one of our latest summer visitants, never appearing before the middle, and often not till the end, of May. Its food consists entirely of insects, taken on the wing. The method which it adopts for this purpose is somewhat singular, and as I believe peculiar to itself. Taking its station generally on the top of a post, it watches till an insect passes by, when it suddenly darts forward, hovers for a moment in order to secure its prey, and then returns to the same spot again. This operation it will often repeat for a con- siderable length of time without changing its place. Genus VIII. TURDUS. Jann. T. viscivorus, Temm. Man. d’Ornith. p. 160. Misset Turusu.—Tolerably plentiful in the neighbourhood of Bottisham, where it remains all the year round. Its song is very powerful, and in mild weather is often heard as early as the beginning of January, but wholly ceases by the end of May. T. pilaris, Temm. Man. d’Ornith. p. 163. FIELDFARE.—This is one of our winter visitants. It is seldom seen before November, but it remains with us till very late in the spring. At Anglesea Abbey in particular, I have for some years back noticed individuals as late as the middle of May. PP 2 294 Spe. 29. Sp. 30. Spe. 31. Sp. 32. Sp. 33. Sp. 34. Mr. Jenyns on the Ornithology of Cambridgeshire. T. musicus, Temm. Man. d@’Ornith. p. 164. Sonc-TurusH.—Sings from the end of January to the middle of July. T. iliacus, Temm. Man. d’Ornith. p. 165. ReEDWw1NG.— Migratory like the Fieldfare; but generally preceding that species in its arrival. T. torquatus, Temm. Man. d’Ornith. p. 166. Rrvyc-Ovuze..—I have been informed that a single bird of this species was shot on the borders of the county, near Great Chesterford, but I could not learn at what season of the year. T. Merula, Temm. Man. d’Ornith. p. 168. BLAcKBIRD.—Sings from the beginning of February to the end of July. Genus IX. SYLVIA. Lath, S. Phragmitis, Temm. Man. d’Ornith. p. 189. SEDGE-WARBLER.—This and all the other species of this genus, with the exception of the Redbreast (S. Rubecula,) are birds of passage, appearing with us in the spring and departing either before or at the approach of autumn. The Sedge-warbler is first seen the last week in April. It is very plentiful throughout the fens and low grounds of Cambridgeshire, especially where there are osiers and other covert, in which situations it remains closely concealed, rarely exposing itself to view. The nest is suspended at a small height from the ground between the stems of the Arundo Phragmites. During the breeding season it sings incessantly night and day in a somewhat hurried and confused manner, often imitating the notes of other birds. S. Luscinia, Temm. Man. d’Ornith. p. 195. NIGHTINGALE.—This species is seldom heard with us before the 16th of April. After the young broods are hatched, which usually takes place by the end of the first week in June, its song wholly ceases. Sp. 35. Sp. 36. Sp. 37. Mr. Jenyns on the Ornithology of Cambridgeshire. 295 S. atracapilla, Temm. Man. @Ornith. p. 201. Buackxcar.—The note of this bird much resembles, and is only inferior to, that of the Nightingale. It is usually first heard about the middle of April, but in very mild seasons I have noticed it as early as the 29th of March. It continues in full song till August. S. hortensis, Temm. Man. d’Ornith. p. 206. GREATER PrerrycHars.—This species is not very unfrequent in gardens, copses, and high hedges; though more plentiful some years than others. Its note is soft, possessing much variety, and particularly pleasing; but the individual which utters it, from its extreme shyness and its manner of concealing itself in the thickest parts of the wood, is not often seen. I never heard it before the 1st of May nor after the 18th of July. S. cinerea, Temm. Man. d’Ornith. p. 207. WuitTETHROAT.—Towards the end of April, this species resorts to our hedges in great quantities, where it must often have attracted notice by its very peculiar manners. For the most part it sings concealed, but every now and then it may be observed to rise suddenly from its retreat to a considerable height in the air, and without desisting from its song, to shoot about with some rapidity, accompanying its flight all the while with singular jerks and gesticulations of the wing. After continuing these movements for a greater or less interval, it returns slowly to the bush from whence it sprung, and resumes its former station. I cannot forbear mentioning in this place, that I have at different times been much inclined to suspect, that under the name of Whitethroat, there have been two species hitherto confounded together. What has chiefly led me to this opinion, is the circumstance of my having occasionally noticed amongst these birds certain individuals, which not only differed strikingly from the above in habits and manners, but also in note, and which invariably preceded the others in their arrival by a week or a fortnight. This year in particular, I observed some of these last as early as the first week in April. Their haunts were 296 Sp. 38. Mr. Jenyns on the Ornithology of Cambridgeshire. much the same as those of the common sort, being generally in thick hedges and close copses of underwood: in these situations however they were oftener heard than seen, as they always sculked about in the most concealed spots, and never rose into the air with that peculiarity of gesture which I have attempted to de- scribe above. Their song too was very different, being much superior to that of the common sort, more melodious and varied in its notes, though so soft and inward as to be scarcely noticed unless near: moreover, this was never exerted on wing. That these birds are really distinct from the others, I will not at present presume to decide, as I have not hitherto had an oppor- tunity of comparing specimens of each sort together, which would afford the only means of detecting a specifie difference if such exist between them. I find myself however somewhat corroborated in my suspicions, by the following observation of Montagu. In his Ornithological Dictionary, (Art. Whitethroat) he mentions having more than once killed a bird whose plumage differed in some respects from that of the common Whitethroat, and in one instance from off the nest, which contained four eggs almost entirely white, not nearly so much speckled with brown and ash- colour as those of this bird generally are: and whose weight was also greater. He confesses himself to have been much puzzled on this occasion, and coneludes by hinting at the possibility of its being proved hereafter that there are two distinct species. S. Curruca, Temm. Man. d’Ornith. p. 209. Lesser WuiretHroat.—tThis bird, the Lesser Whitethroat of Latham and Montagu, corresponds so exactly in every particular with the Sylvia Curruca of Temminck, that I have accordingly referred it to that species, though it is very doubtful whether it be the Motacilla Curruca of Linneus. In this country it does not appear to have been generally noticed, nor at all known till Latham first. described it in the Supplement to his Synopsis, which circumstance is probably owing to its being of very local occurrence, and almost entirely confined to the eastern parts of the kingdom. In Cambridgeshire, it is far from uncommon, Sp. 39. Sp. 40. Sr. 41. Mr. Jenyns on the Ornithology of Cambridgeshire. 297 making its first appearance in the last week of April. Like the rest of its tribe it is extremely shy and very difficult to get sight of, though when near easily recognized by its note, which consists of a shrill shivering cry repeated at intervals from the thickest parts of the wood. It resides for the most part in copses and gardens, building its nest in some low shrub at the height of about four feet from the ground. This is of a very loose and flimsy structure, and composed of dry bents with the addition of a small quantity of wool placed in patches on its exterior surface; within, it is lined with a scanty supply of white hairs. The eggs are five in number, white, spotted chiefly towards the greater end with small dots of brown, and larger irregular stains of the same colour. Incubation commences about the 20th of May, and the young broods are fledged in June, but the note of the parent birds is continued till the middle or even till the end of July. Montagu has stated very accurately the several points of difference between this species and the preceding, which, if attended to, will always serve to distinguish them from each other. Latham’s figure, in his first supplement, is incorrect in representing the upper parts of the plumage of a deep brown. whereas they are wholly cinereous. S. Rubecula, Temm. Man. d’Ornith. p.215. Repsreast.—This species continues in song the whole year round, excepting in times of severe frost. S. Phenicurus, Temm. Man. @Ornith. p.220. Repstart.—A very abundant species throughout Cambridgeshire where it arrives the middle of April. It is particularly constant in the time of its first appearance, perhaps more so than any other bird, as I do not ever remember to have noticed its arrival before the twelfth or later than the sixteenth of this month. S. Hippolais, Temm. Man. d@’Ornith. p. 222. Lesser PrrrycHaprs.—Of all our summer visitants this is undoubtedly the earliest, often arriving by the middle, or at latest by the end of Mareh., Although I have generally observed it to be 298 Spe. 42. Sp. 43. Mr. Jenyns on the Ornithology of Cambridgeshire. diffused in tolerable plenty over most other parts of the county, yet, in the neighbourhood of Bottisham, it is of very uncertain appearance, as in some seasons not a single individual is seen there, whilst in others they are abundant. It is a restless and an active bird, and is much attached to spruce firs and other tall trees, from the tops of which it issues its incessant but mono- tonous song, consisting only of two loud piercing notes, which it continues throughout the summer and even till late in September. By this and by its early arrival, it may readily be distinguished from the following species, but as far as respects plumage, the two are so extremely similar, that it is difficult to discriminate between dead specimens. Most authors represent this as being of less size and of a paler colour in its under parts, but I am of opinion, that little reliance can be placed on these marks, as from an examina- tion of a great many specimens of each, I have found them very variable. The only constant character that I have observed, resides in the colour of the legs, which in this are dusky, whereas in the following they are pale brown. S. Trochilus, Temm. Man. @Ornith. p. 224. Wittow Wren.—This is a great deal more plentiful than the preceding, and not so much confined to large trees and woods, being a general inhabitant of hedges, underwood, and a variety of other situations. It appears about the same time as the Red- start, and, as is the case with many of this tribe, the males invariably precede the females, by an interval of several days. Its song consists of seven or eight notes which are modulated in a soft and particularly pleasing, though somewhat plaintive, manner. This is continued without intermission during the breeding season, but generally ceases by the beginning of July. Genus X. REGULUS. Cuwv. R. aurocapillus, Selby. Sylvia Regulus, Temm. Man. d@’Ornith. p. 229. GOLDEN-CROWNED WreEN.—These birds from their diminutive size and solitary habits are not often noticed, and may be easily Sp. 44. Sp. 45. Sp. 46. Sp. 47. Mr. Jenyns on the Ornithology of Cambridgeshire. 299 overlooked, but I believe them to be very plentiful wherever there are plantations of spruce firs, to which trees they seem extremely partial, hanging their nests to the under-surface of the lower branches. Though apparently of so delicate a nature, they remain with us all the winter, and appear to suffer less from severe cold than even many of our hard-billed species. It is not at all impro- . bable that at this season they may derive their chief support from the smaller tribes of Tipulide, many of which are to be found on wing and in a state of activity at all times of the year, and even occasionally when the ground is covered with snow. Genus XI. TROGLODYTES. Cuv. T. europeus, Cuv. Sylvia Troglodytes, Temm. Man. d@Ornith. p. 233. Common Wren.—Like the Robin, this bird sings throughout the year, but its note in the winter months is very weak compared to what it is in the spring. Genus XII. SAXICOLA. Bechst. S. GEnanthe, Temm. Man. d’Ornith. p. 237. Wueat-Ear.—I have occasionally observed these birds on the Devil’s Ditch and the open parts about Newmarket heath, but from their not being in any great plenty, I am unable to say at what period of the year they first visit those districts, or when they with- draw. They breed on the first-mentioned place, depositing their nest in an old rabbit-burrow, or some other hole under ground. S. rubetra, Temm. Man. d’Ornith. p. 244. Wuin-cuat.—Like the preceding a bird of passage, appearing in the middle of April, and departing in the autumn. S. rubicola, Temm. Man. d’Ornith. p. 246. Stonr-cHAT.—This is plentiful, and resides with us all the year on fens and other open grounds. Vol. II. Part I. QQ 300 Sp. 48. Spe. 49. Mr. Jenyns on the Ornithology of Cambridgeshire. Genus XIII. ACCENTOR. Bechst. A. alpinus, Temm. Man. d’Ornith. p. 248. ALPINE ACCENTOR.—The discovery of this addition to the Orni- thology of Great Britain is due to Dr. Thackeray, who observed a pair of these birds in the open space immediately under the east window of King’s College chapel, on the twenty-third of November, 1822: one of them, which proved to be a female, was shot, and is at present in his collection. I am not aware that any others have since been mét with in this country, where indeed it can only be looked upon as an accidental visitant. According to Temminck its native haunts are the Swiss Alps and the mountainous parts of Germany and France. A. modularis, Temm. Man. d’Ornith. p. 249. Hepcr-Accentor.—One of the few soft-billed birds that remain with us the whole year, singing at all seasons if the weather be mild. Genus XIV. MOTACILLA. Linn. M. alba, Temm. Man. d’Ornith. p. 255. Prep Wacraiu.—I had often observed that we see greater num- bers of these birds in the autumn than in any other season of the year, but was not aware of the cause till I learnt from Selby’s Illustrations of British Ornithology, that in the north of England this species is a regular migrant, retiring southward in October, and not re-appearing till February or the beginning of March. This circumstance renders it highly probable that at the time above-mentioned the birds of our own neighbourhood are joined by those which arrive from the higher parts of the country. M. Boarula, Temm. Man. @Ornith. p. 257. Grey Wacramt.— This is the least plentiful of the three British species of Wagtail, and is only seen in Cambridgeshire during the autumnal and winter months, appearing first in October or earlier. In the spring it retires northward to breed. About Bottisham I have noticed it most frequently in January. Mr. Jenyns on the Ornithology of Cambridgeshire. 301 Sp. 52. M. flava, Temm. Man. d’Ornith. p. 260. YeLLtow WacraiL. — This species visits us in the spring and departs in the autumn. It does not appear to be uncommon in many parts of the county, though much more so than the Motacilla alba. I have occasionally seen them in considerable plenty upon the arable lands bordering on Bottisham and Swaffham fens, and likewise in the low meadows about Quy Water. Genus XV. ANTHUS. Bechst. Sp.53. A. pratensis, Temm. Man. d’Ornith. p. 269. Trr-preit.— Equally abundant on the low and fenny as well as on the high and heathy parts of the county, in which situations it is to be found all the year. In the autumn it appears to be subject to a considerable change of plumage, from which cireum- stance some authors have erroneously made two species of this bird. Sp.54. A. arboreus, Temm. Man. d’Ornith. p.271. TREE-PIPIT.—This species very strongly resembles the last in plumage, but may always be distinguished by the curvature of the hind claw, and the greater dilatation of the bill towards its base. In its haunts it is widely different, being entirely confined to woods and plantations of tall trees, and never frequenting the open parts of the country; nor does it remain with us through the winter, but makes its first appearance about the third week in April, and departs at the approach of autumn. Its song, which is delivered on wing in its descent, is heard till the middle of July. ——~+>—_ — ORDER IV. Granivort. Genus XVI. ALAUDA. Linn. Sp.55. A. arvensis, Temm. Man. d’Ornith. p. 281. Sky-Lark.—These birds get together in small companies at the approach of winter, but the flocks are not considerable except in very severe weather. QQ2 302 Sp. 56. Mr. Jenyns on the Ornithology of Cambridgeshire. Genus XVII. PARUS. Linn. P. major, Temm. Man. d’Ornith. p. 287. Great TrrmousE.—In hard weather this and the two following species leave their native woods and resort to the immediate vici- nity of dwelling-houses, in order to avail themselves of what they can pick up. At such times I have observed that they will devour flesh with greediness, and may be caught in great numbers by a trap baited with suet. P. coeruleus, Temm. Man. d@ Ornith. p.289. BiLuE TiITMovseE. P. palustris, Temm. Man. d@’Ornith. p. 291. Marsu TrrMovuseE. _ P. ater, Temm. Man. d@’Ornith. p. 288. Cotr Trrmovusr.—Less frequent with us than any of the other species, though probably often overlooked from its strong resem- blance to the preceding. It may however be easily distinguished by its peculiar note independently of other characteristic marks. P. caudatus, Temm. Man. d’Ornith. p. 296. Lonc-raILED TiTMoUsE.—Very common in woods, constructing its singular nest in cedars, small firs, and trees of that kind. The young broods do not disperse when fledged, but follow the parent- birds through the autumn and winter. Genus XVII. EMBERIZA. Linn. E. citrinella, Temm. Man. d@Ornith. p.304. YELLow Buntinc.—In some places this species is known by the name of Writing Lark, from the peculiar markings on the egg, which have somewhat the appearance of written characters. E. Miliaria, Temm. Man. d’Ornith. p. 306. Common Buntine.—These birds being much attached to open cultivated ground and extensive corn lands, are extremely plentiful in Cambridgeshire where they are called Bunting Larks. ‘Towards the approach of winter they collect together in large flocks, and do not separate till the ensuing spring. Spe. 63. Sp. 64. Sp. 68. Mr. Jenyns on: the Ornithology of Cambridgeshire. 303 E. Scheniculus, Temm. Man. d@’Ornith. p.307. Rerp Bunrinc.—Common in fens and low meadows, but con- fined to such situations. As far as I have observed, the nest is always placed on the ground, and never suspended between the stems of aquatic plants, as described by Bewick and some other authors, who have strangely confounded the manners of this bird with those of the Sedge Warbler (Sylvia Phragmitis.) This error has probably arisen from the circumstance of the two species frequenting the same haunts, and being in a general way both called Reed Sparrows. Genus XIX. PYRRHULA, Briss. P. vulgaris, Temm. Man. d’Ornith. p.338. Burrincu.—This is generally reckoned a very common bird; but I have rarely noticed it in the neighbourhood of Bottisham. Perhaps it is attached to more wooded districts. Genus XX. FRINGILLA, Illig. F. Chloris, Temm. Man. d’Ornith. p. 346. GREENFINCH.—Collect together in large flocks in the winter. F. domestica, Temm. Man. d’Ornith. p. 350. HousE Srparrow.—White varieties of this bird have been occa- sionally observed near Bottisham. F. montana, Temm. Man. @Ornith. p. 354. TrEeE Sparrow.—lI have inserted this species on the authority of Selby, who, in his Illustrations of British Ornithology, mentions having received specimens from the neighbourhood of Cambridge. F. celebs, Temm. Man. d@Ornith. p. 357. CuarrincH.—This species continues in full song from the first week in February to the end of June, after which time it is silent till September, when it reassumes its note for a few weeks if the weather be mild. As far as I have observed, both sexes remain with us all the year, and do not appear to separate at the approach of winter, as they are said to do in other parts of England. 304. Mr. Jenyns on the Ornithology of Cambridgeshire. Sp.69. F. cannabina, Temm. Man. dOrnith. p.364. Common Linnet.—These birds begin to assemble in flocks about the middle of October, which increase in numbers as the weather becomes more severe. Sp.70. F. Spinus, Temm. Man. dOrnith. p.371. Siskin.—Only an occasional visitant during the winter months. Large flocks appeared at Bottisham the last week in January of the present year (1825) and many specimens were killed both male and female *. Spe.71. F. Carduelis, Temm. Man. d’Ornith. p. 376. GOLDFINCH. ——_—_—>—— ORDER V. ZycopacrTytt. Genus XXI. CUCULUS, Linn. Sp.72. C. canorus, Temm. Man. d’Ornith. p. 381. Cucxow.—This species visits us in the middle of April, and is heard till the beginning of July, when it again departs; but the young birds appear to remain for a much longer period, as I have occasionally observed them in September. I never found the egg of this bird myself, but have seen one which was taken at Great Swaffham from the nest of a Hedge Accentor, (Accentor Modularis.) Genus XXII. PICUS, Linn. Spe. 73. P. viridis, Temm. Man. d’Ornith. p. 391. GREEN WOODPECKER. * This species was also noticed in Cambridgeshire by Turner. See his work, De Avibus, p. 56. Mr. Jenyns on the Ornithology of Cambridgeshire. 305 Sp.74. P. major, Temm. Man. d’Ornith. p. 395. Great SporreD WooprecKEeR.—Much less common than the preceding, but has been occasionally shot at Bottisham. Sp.75. P. minor, Vemm. Man. d’Ornith. p. 399. Lesser SporreD WoopreckER.—TI have at different times known several instances in which this bird. has been met with in Cambridgeshire, but it must be esteemed a rare species. The last specimen which occurred to my knowledge was shot at Anglesea Abbey in March 1824, and is now in the Museum of the Cambridge Philosophical Society. Genus XXIII. YUNX, Lunn. Se.76. Y. Torquilla, Temm. Man. d’Ornith. p.403. Wrynecx.—A few of these birds visit us regularly every spring, but they are never plentiful. a ORDER VI. AntIsopactTvut. Genus XXIV. SITTA, Linn. Sp.77. S. europea, Temm. Man. d@Ornith. p. 407. Nuruatcu.—Not uncommon in the neighbourhood of Bottisham. During a certain portion of the year, these birds feed chiefly upon nuts which they break with their bill, after having firmly fixed them in the crevices of the bark of trees. For this purpose they appear to resort frequently to the same spots, as I have observed some old trees in particular whose clefts are full of broken shells, whilst in others not one is to be seen. Genus XXV. CERTHIA, Illig. Sp.78. C. familiaris, Temm. Man. @Ornith. p.410. CoMMON CREEPER.—This bird frequently builds its nest under the loose and decayed bark of old trees; it consists of little else 306 Sr. 79. Sp. 80. Mr. Jenyns on the Ornithology of Cambridgeshire. than a few twigs and small sticks piled rudely together with a layer of feathers upon the top of them. The eggs are very nume- rous, often as many as nine or ten. sees VS ORDER VII. Atucyonss. Genus XXVI. ALCEDO, Linn. A. Ispida, Temm. Man. d’Ornith. p. 423. Common Krne’s-FisHER—Tolerably plentiful in the neighbour- hood of streams and clear waters. During its flight, which is very rapid, it utters a shrill piercing note that may be heard to a great distance. ———— ORDER VIII. CuHeELIDoNEs. Genus XXVII. HIRUNDO, Linn. H. rustica, Temm. Man. @Ornith. p. 427. CHIMNEY SwaLLow.—The arrival of this species in the neigh- bourhood of Bottisham usually takes place about the fifteenth of April, as I have found by many years’ observation, but has been occasionally deferred till the twenty-second, which is the latest that I ever noticed. The first broods are fledged early in August, and towards the middle of that month they begin to collect into large flocks, which increase in numbers as the season advances and the time of departure draws near. This, with respect to the majority, takes place in the beginning of October, but stragglers may be seen a week or two longer. I once observed a white variety of this bird at Ely. Mr. Jenyns on the Ornithology of Cambridgeshire. 307 Spe.81. H. urbica, Temm. Man. @Ornith. p. 428. Hovusrt Martin.—This appears about a weck after the swallow, but is seldom in great plenty before the beginning of May. It however remains with us later than that species, and is occasionally seen through the first week in November *, though the greater part withdraw before that time. Previously to migration they congregate upon the roofs of houses and churches. Sp.82. H. riparia, Temm. Man. @Ornith. p. 429. Sanp Martin.—The only places where I have hitherto observed these birds in Cambridgeshire, are the chalky banks by the side of the road near Quy Water, and some gravel-pits in the neighbour- hood of Bourn Bridge. In the former of these situations I have noticed them regularly every year, and have found the time of their arrival to be about the middle of April, but I am unable to say when they leave us, though I am inclined to suspect that this takes place at a much earlier period than with either of the pre- ceding species. Genus XXVIII. CYPSELUS, Illig. Sp.83. ©. murarius, Temm. Man. d@’Ornith. p. 434. Swirt.—This species is by far the latest in its arrival of all the Swallow tribe, as I never remember to have seen it before the seventh of May. Generally speaking it is equally remarkable for its early departure, withdrawing from most places by the begin- ning of August, and often by the end of July. A singular excep- tion, however, to this last mentioned circumstance takes place with respect to these birds in the neighbourhood of Ely, where the bulk of them hardly ever retire till quite the end of August, and a few individuals may often be observed through the first week in September}. From what cause they are induced to pro- * Dr. Thackeray informs me that he has known three or four individuals stay about the south side of Clare Hall till the eighteenth of this month. + A single bird has also been noticed by Dr. Thackeray about King’s College Chapel for many successive days during the early part of September. Vol. If. Part I. Rr 308 Mr. Jenyns on the Ornithology of Cambridgeshire. tract their stay at that place so much beyond its usual limit I. am unable to say, but the fact itself I regularly noticed during a period of several years that I was in the habit. of residing there for the summer months. Possibly they may. in some measure be influenced by the cathedral and other old buildings. adjacent, in the holes and crannies of which these birds. meet with a retreat peculiarly congenial to their habits, as appears by the immense numbers that annually resort thither in the early part of the season *. Genus XXIX. CAPRIMULGUS, Linn.. Sp.84. C. europzeus, Temm. Man. d’Ornith. p. 436. GoaTsucKER.—I have occasionally observed these birds about Ely, and also in the neighbourhood of Bottisham, but at the last mentioned place they have not of late years appeared in such plenty as formerly. Like the rest of this order they are migratory, arriving about the beginning of June and departing in September. In the dusk of the evening they utter a singular chattering noise somewhat resembling that of a spinning-wheel, by which they may easily be distinguished. From an examination of the stomach, their food appears to. consist of the larger night- flying Phalene, particularly those belonging to the Linnean sec- tion Noctua, and the various species of Phryganea. It is also probable that during a short part of the season they derive much of their support from the Midsummer Dor (Melolontha solsti- tialis) as I have seen them hawking about in places where these insects were abundant. * White, in his Natural History of Selborne, observes that swallows are seen later at Oxford than elsewhere, and enquires whether it may not be owing to the vast massy buildings of that place. See his twenty-third Letter to Pennant. Mr. JENyNS on the Ornithology of Cambridgeshire. 309 ORDER IX. Cotums.«. Genus XXX. COLUMBA, Linn. Sp.85. C. Palumbus, Temm. Man. d@Ornith. p. 444. Rinc-Dove.— These birds are exceedingly abundant in Cam- bridgeshire, where they do an incredible deal of mischief by de- vouring pease, beans and other leguminous plants. They are well known by their cooing notes which are heard incessantly from February to October: After that time they begin to collect together into enormous flocks, which disperse themselves over the country during the day-time to feed, but return regularly home in the evening to roost in their native woods and plantations. Some of these flocks do not wholly separate till very late in the spring, though the greater part pair off for the purpose of breeding by the beginning of March. In the Autumn I have observed that they subsist chiefly upon acorns and beech-mast. Sp.86. C. Ginas, Temm. Man. d’Ornith. p. 445. Stockx-Dovr.—White, in his Natural History of Selborne, men- tions the Stock-Dove as being seen there during the winter months only, appearing in large flocks about the end of November, and departing in February. Whatever may be the case with respect to these birds in the southern counties, with us they certainly remain the whole year, as I have noticed them at all seasons and repeatedly found their nests. They are considerably less plentiful than the Ring-Dove, but have much the habits of that species with which they frequently associate in hard weather. Like them they breed very early in the spring. The nest which is flat and shallow, consists merely of a few sticks put loosely together in the hollow of some old tree. The eggs are two in number, white like those of the Ring-Dove, but somewhat smaller and rather more rounded. As far as I have observed, the Stock-Dove never cooes, but utters only a hollow rumbling note during the breeding season, which may RR2 310 Sp. 87. Sp. 88. Sp. 89. Mr, JENyNs on the Ornithology of Cambridgeshire. be heard to a considerable distance. Montagu in his Ornitholo- gical Dictionary has evidently confounded this species with the Rock-Dove, (Columba livia, Temm.) which is supposed to be the origin of our dove-house pigeon, and is found in a wild state upon some of the steep shores and cliffs of Great Britain, but is not a native of Cambridgeshire. The Stock-Dove and Ring-Dove are indiscriminately called Woodpigeons by the country people. C. Turtur, Temm. Man. d@’Ornith. p. 448. TurTLE-Dove.—Some few individuals of this species visit the plantations in the neighbourhood of Bottisham regularly every spring, and are first seen towards the latter end of May, but they are never numerous, and do not stay with us long, departing again soon after the breeding season is over. The young birds, however, appear to remain for a longer period, as I have had them shot in the month of September. I have also noticed this species at Stetchworth and Wood-Ditton. — - ORDER X. Gatun. Genus XXXI. PHASIANUS, Linn. P. colchicus, Temm. Man. d@Ornith. p. 453. Common PuHeEaAsANT.—Instances have now and then occurred at Bottisham in which the hen of this species had partially assumed the plumage of the cock. This singular change has only been observed in individuals which had reached an advanced age. Such are termed by sportsmen mule-birds. Genus XXXII. PERDIX, Lath. P. rubra, Temm. Man. d’Ornith. p. 485. RED-LEGGED ParTrIDGE.—One of these birds was shot near Anglesea Abbey on the twenty-seventh of September 1821, and is at present in my possession. Mr. Jenyns on the Ornithology of Cambridgeshire. 311 Sp.90. P. cinerea, Temm. Man. d’Ornith. p. 488. CoMMON PaRTRIDGE—A covey of these birds were bred in the neighbourhood of Clayhithe, of which a considerable number were perfectly white. Sp.91. P. Coturnix, Temm. Man. d@Ornith. p. 491. QuaiL.—This species is one of our latest summer visitants as I have seldom noticed it before the beginning of June; but it remains with us till the end of October. In the present year (1825) two specimens were killed at Bottisham so late as on the eleventh of November. ORDER XI. Cursores: Genus XXXIIJ. OTIS, Linn. Sp.92. O. Farda, Temm. Man. d’Ornith. p.506. Great Bustarp.—Formerly these birds were plentiful in the open tracts about Newmarket Heath, and till within a few years single individuals. have occasionally been seen in that neighbour- hood, but they are supposed to-be now almost extinct. Ray and Willoughby mention also Royston Heath as a place frequented in their time by this species. Sp.93. O. Tetrax, Temm. Man. d’Ornith. p.507. LirrLe Busrarp.—Bewick has figured a specimen of this bird which was taken alive on the edge of Newmarket Heath. So far as I am aware no other instance has occurred of its having been met with in this county. 312 Sp. 94. Sp. 95. Mr. Jenyns on the Ornithology of Cambridgeshire. ORDER XII. GriLtatones. Genus XXXIV. OEDICNEMUS, Temm. O. crepitans, Temm. Man. d’Ornith. p. 521. StonE-CuRLEwW.—This species does not appear to be plentiful in these districts. I have ‘seen specimens that ‘were killed in the vicinity of Cambridge, and am informed by Dr. Thackeray, that about two years since he had brought him a young bird which was bred very near that place, but I never observed any myself. It is migratory, and only met with during the summer months. Genus XXXV. CHARADRIUS, Linn. C. pluvialis, Temm. Man. d@Ornith. p. 535. GOLDEN PLover.—Common in the fens as well as the high lands, but appear to breed generally in the last mentioned situations. C. Morinellus, Temm. Man. d@’Ornith. p. 537. Dorteret.—This baits with us for a short time in its passage to and from the North where it probably breeds, being seen here in the spring and autumn only. The largest flocks occur about the middle of September. They frequent the same situations as the preceding species. C. Hiaticula, Temm. Man. @Ornith. p. 539. Rincep PLover.—Great quantities of these birds appeared in Bottisham and Swaffham fens in the months of June and July 1824, which was a remarkably wet season. They are by no means regular visitants of those districts. Montagu asserts in the Supplement to his Ornithological Dictionary that this species resides on the sea shore the whole year, but from the above circumstance it is probable that they occasionally retire inland to breed. Sp. 98. Spe. 99. Sp. 100. Sp. 101. * 6 In Mr. Jenyns on the Ornithology of Cambridgeshire. 313 Genus XXXVI. VANELLUS, Briss. V. cristatus, Temm.. Man. d’Ornith. p. 550. Lapwinc.—A very abundant species. In the autumn they collect into large flocks. Genus XXXVII. GRUS, Pallas. G. cinerea, Temm. Man. @Ornith. p. 557. Crane.—In the time of Ray,* these birds. appear to have visited our fens in large flocks regularly during the. winter months, but they have long since deserted them; nor is it likely, from the altered state of the country in consequence of the improved system of drainage which is now carried on, that they will ever return thither. According to Pennant (Brit. Zool. Vol. II. p. 629.) a single specimen was killed near Cambridge about the year 1773. This I believe to: be the latest. instance on record in which the species has been met. with. A Genus XXXVIII. ARDEA, Linn. A. cinerea, Temm. Man. d’Ornith. p.567, Heron.—In hard weather this species resorts to those brooks and running streams which seldom or never freeze, and at such times is met with in great numbers; but in other seasons it is only occasionally noticed in the neighbourhood of Bottisham, there being no place within a considerable distance where these birds are known to breed. A. stellaris, Temm. Man. d’Ornith. p.580. Brrrern.—These birds are met with in Burwell fen and occasionally on. the moors about Cambridge, but they appear to be getting more scaree every year. Formerly they were plentiful. palustibus Lincolniensibus et Cantabrigiensibus magni horum greges hyberno tempore inveniuntur.” Raii Syn. Meth. Aviym, p.95.. Art. The Crane. 314 Sp. 102. Sp. 103. Sp. 104. Sp. 105. Mr. Jenyns on the Ornithology of Cambridgeshire. Genus XXXIX. RECURVIROSTRA, Linn. R. Avocetta, Temm. Man. d’Ornith. p.590. Scoopinc AvosEtT.—I have inserted this species on the authority of Donovan, who, in his History of British Birds, (Pl. 66.) speaks of it as being common in the breeding season in the fens of Cambridgeshire. It is very probable that this was formerly the case, when our marshes were more extensive than they are at present, but I have not been able to learn that it is ever met with now. ’ Genus XL. NUMENIUS, Briss. N. arquata, Temm. Man. d@Ornith. p.603. Common CurRLEW.—Sometimes seen in small flocks about New- market Heath. N. Pheopus, Temm. Man. d’Ornith. p. 604. WuimsreL.—Has been occasionally exposed for sale in Cam- bridge. Genus XLI. TRINGA, Briss. T. variabilis, Temm. Man. @Ornith. p. 612. Dunuin.—These birds now and then visit our fens during the summer months, and it is not improbable that they may breed in those situations. In the beginning of July 1824, they were very abundant. Several which were then killed and came under my observation, I found to answer, in most particulars, pretty correctly to Temminck’s description of this species, as it appears in its summer plumage. The black, however, on the under parts was very variable in different specimens, some of which were only faintly spotted with this colour, whilst in others the whole of the belly and abdomen were thickly blotched over with large irregular patches of the same, but in no case without some mixture of white. From this last circumstance it is likely, that at the time above-mentioned, the season of incubation was just over, as, according to Temminck, so long as that lasts, the belly Sp. 106. Sp. 107. Mr. Jenyns on the Ornithology of Cambridgeshire. 315 is wholly of a deep black, but immediately after a change takes place in the colour of that part, and the white begins to re- appear. Those birds which were the darkest in this respect, were also a trifle larger than the others, and had the bill somewhat longer. These last characters indicate I believe the female sex. T. minuta, Temm. Man. d’Ornith. p. 624. LirtLe Sanppirer.—lI suppose this to be the Little Sandpiper of Pennant, (Brit. Zool. Vol. II. p. 473.) which he described from a specimen shot near Cambridge in the month of September, and which is the only one I ever heard of. At the same time there is some doubt attached to its identity, as it appears from Temminck, that under that name two species have been often confounded together, the Tringa Temminckii and the Tringa Minuta of the Manuel d’Ornithologie, and Pennant’s description is so short and imperfect, that it is not easy to pronounce with certainty which of these two was before him when it was drawn up, though from the circumstance of the legs being mentioned as black, I am inclined to think it was the latter. Add to this, that Iam not aware of the Tringa Temminckii having ever been met with in England, except the Little Sandpiper, described by Montagu in the Appendix to his Ornithological Dictionary, be of that species, which I think not improbable. That however was shot in Devonshire. T. cinerea, Temm. Man. d’Ornith. p.627. Knor.—According to Montagu these birds were formerly found in the Isle of Ely during the autumnal months, and I have heard of specimens being occasionally met with still. Se.108. T. pugnax, Temm. Man. d’Ornith. p. 631. Pol. | Rurr, Male. Reeve, Female. through the summer and departs at the approach of autumn, but is much more plentiful some years than others. It is found in the Isle of Ely, and occasionally in Bottisham and Swaffham fens. The male bird loses the ruff so soon as the season of incubation is over. 1. Part Il. Ss \ rhis species visits us in the spring, remains 316 Sp.109. Sp. 110. Sp.111. Sp.412. Spe. 113. Mr. Jenyns on the Ornithology of Cambridgeshire. Genus XLII. TOTANUS, Bechst. T. fuscus, Temm. Man. d@’Ornith. p. 639. I insert this species with considerable hesitation. According to Temminck it is synonymous with the Scolopax Cantabrigiensis of Gmelin and Latham, and the Cambridge Godwit of Pennant, (Brit. Zool. Vol. II. p. 447.) which was originally described by this last Author from a stuffed specimen shot in the vicinity of Cambridge. Since that time it appears to have been a great question whether the Cambridge Godwit was admissible as a distinct species. Some have supposed it to be the same with the Greenshank, whilst Montagu, in his Supplement to the Orni- thological Dictionary, suspects it to be only the young of the Redshank. Where the truth lies I shall not presume to decide at present. T. Calidris, Temm. Man. d’Ornith. p. 643. REDsHANK.—These birds were formerly plentiful in the fens, particularly during the summer months, but are rarely met with now. Bewick has figured a specimen which had been sent him from Cambridge. : T. ochropus, Temm. Man. d’Ornith. p. 651. GREEN SANDPIPER.—This is a very rare species. The only specimen that ever came under my observation was shot in the Isle of Ely between Downham and the Hundred-foot River, on the 28th of August, 1821. T. Hypoleucos,.Temm. Man. d@ Ornith. p. 657. Common SANppireR.—These birds are occasionally met with on the banks of the river below Cambridge. T. Glottis, Temm. Man. d’Ornith. p. 659. GreENsHANK.—This is another species which, since the drainage of the greater part of our fens, has» become very rare in Cam- bridgeshire.. There is a specimen, however, in the collection of Dr. Thackeray which was killed in the county. Mr. JenyNs on the Ornithology of Cambridgeshire. 317 Genus XLIII. LIMOSA, Briss. Sp. 114. L. Melanura, Temm. Man. @Ornith. p. 664. Sp. Spe. -115. 116. 117. 118. - 119. 120. Common Gopwir.—Formerly plentiful throughout the fens, where according to Willoughby, it was known by the name of Yarwhelp. It is not very often met with now. L. rufa, Temm. Man. d@ Ornith. p. 668. Rep Gopwit.—I have given this species on the authority of Bewick, who has figured and described a specimen that had been sent him from Cambridge. Genus XLIV. SCOLOPAX, Illig. S. Rusticola, Temm. Man. d’Ornith. p. 673. Woopvcockx.—This bird usually makes its first appearance in the neighbourhood of Bottisham about the end of October, but was once killed as early as on the 18th of that month. It remains with us till the middle or occasionally till the end of March. S. major, Temm. Man. d’Ornith. p. 675. Great Snipe.—A mutilated specimen of this bird was brought to the Cambridge market some time since. S. Gallinago, Temm. Man. d’Ornith. p.676. Common SnriPpE.—Many of these birds, if not all of them, re- main with us the whole year, and breed constantly in Burwell and Swaffham fens. S. Gallinula, Temm. Man. d’Ornith. p. 678. Jack Snipe.—This species is less plentiful than the preceding, and is a regular migrant, appearing first about the end of. September. I never heard of an instance of its breeding with us. Genus XLV. RALLUS, Lunn. R. aquaticus, Temm. Man. d’Ornith. p.683. Water Raiw.—Occasionally met with in the neighbourhood of Bottisham. ' 392 318 Mr. Jenyns on the Ornithology of Cambridgeshire. Sp. 121. Sp. 123. Sp. 124. Genus XLVI. GALLINULA, JSath. G. Crex, Temm. Man. d’Ornith. p. 686. Lanp Raii.—This is a migratory species which visits us in the spring and departs in the autumn, but is by no means plentiful. 2. G. Porzana, Temm. Man. d’Ornith. p. 688. SpoTTED GALLINULE.—Montagu supposes this species likewise to be migratory, and not found in England during the winter. but if so, it must visit us very early in the year, as it has been killed near Bottisham in the middle of March. It frequents the same situations with the Water Rail, (Sp. 120.) but oceurs much more rarely. G. Baillonii, Temm. Man. d’Ornith. p. 692. Caught alive at Melbourn in January 1823, and is in the collection of Dr. Thackeray. This is the only instance on record in which this species has been met with in England. G. chloropus, Temm. Man. d’Ornith. p. 693. ComMMON GALLINULE.—This species generally builds on the ground, but I have occasionally found the nest in trees. In one instance it was constructed amongst the ivy encircling a large elm which hung over the water’s edge, at the height of at least ten feet from the ground, ——>—_- ORDER XIII. PInnatTipepDes. Genus XLVII. FULICA, Briss. . F. atra, Temm. Man. d’Ornith. p.706. Common Coot.—These birds were formerly plentiful in the fens between Ely and Littleport. They are probably still to be met with in other parts of the county, as they are of frequent occurrence in the Cambridge market. Mr. Jenyns on the Ornithology of Cambridgeshire. 319 Sp. 126. Sp. 127. Sp. 128. Spe. 129. Sp. 130. Genus XLVILI. PHALAROPUS, Briss. P. platyrhinchus, Temm. Man. d’Ornith. p.712. Grey PxHaLarore.—Three specimens of this rare bird were shot in the fens near Cambridge in the hard winter of 1819-20. Genus XLIX. PODICEPS, Lath. P. auritus, Temm. Man. d’Ornith. p.725. EarED GRreEBE.—In the collection of Dr. Thackeray: from the Cambridge market. P. minor, Temm. Man. d’Ornith. p.727. LittLe Grese.—Common every where in the neighbourhood of streams, ponds and other pieces of water. ORDER XIV. Pa.mirepes, Genus L. STERNA, Linn. S. Hirundo, Temm. Man. d@’Ornith. p.740. Common TEern.—Found in the Isle of Ely during the summer months. S. nigra, Temm. Man. d’Ornith. p.749. Biack Tern.—Immense flocks of these birds appeared in Bottisham and Swaffham fens in the summer of 1824. Many of the specimens which came under my observation differed con- siderably from each other in their plumage, particularly with re- spect to the colours about the head and throat. According to Temminck, these parts, which in the winter are much varied with pure white, become in the breeding season wholly black, or at least of a very dark ash-colour like the rest of the body; but in 320 Sp. 131. Sp. 132. Sp. 133. Mr. Jenyns on the Ornithology of Cambridgeshire. some of these individuals no such alteration had taken place, the forehead, space between the bill and the eyes, throat, and forepart of the neck being as white as at other times of the year, so that this periodical change of plumage cannot be looked upon as constant. Possibly however it may be confined to one sex. On the 8th of July a nest of this species was taken, which was perfectly flat, placed on the ground, about six inches in diameter, and composed of roots and dry grass, which appeared to have been trodden down so as to be rendered quite firm and compact. The eggs were two in number, of an olive-green colour, thickly spotted and blotched with deep brown, especially towards the larger end. These. had been ineubated some days. Montagu observes that this bird is known in some parts of Cambridgeshire by the name of Car-swallow. Genus LI. LARUS, Linn. L. marinus, Temm. Man. @Ornith. p- 76 Great Buack-BacKED GuLuL.—There is an adult bird of this species in the collection of. Dr. Thackeray, which was procured in the Cambridge market. L. argentatus, Temm. Man. daraith. p. 764. Srrvery Guitui—Towards the middle of December 1824, several Gulls in. immature plumage were shot at Overcote near Swavesey in this county, which I believe to have been the Larus argentatus of Temminck, which is synonymous with the Herring Gull of Latham and Montagu: but owing to the strong resemblance between the young of this species, and those of the preceding, and of the L. fuscus Temm. it was impossible to identify them with complete certainty. One of these specimens is preserved in the Museum of the Cambridge Philosophical Society. L. canus, Temm. Man. d’Ornith. p.771. Common Gutu.—Met with occasionally in the fens, but chiefly during the autumnal and winter months. Its provincial name is Coddy-Moddy. Mr. JeEnyns on the Ornithology of Cambridgeshire. 321 Sp. 134. L. ridibundus, Temm. Man. d' Ornith. p.780. BLACK-HEADED GuLL.—In some seasons these birds frequent our fens in great plenty. Specimens shot near Bottisham in the beginning of October. wanted the black head; from whence it appears that the periodical change which takes place in the colour of that part, is completed before that time. Genus LII. LESTRIS, Illig. Sp. 135. L. pomarmus, Temm. Man. d’Ornith. p.793. A specimen of this rare bird (which has been only very lately discovered in this country) is in the collection of Dr. Thackeray, and was shot near Cambridge. Genus LIII. ANAS, Linn. Sp. 136. A. Anser ferus, Temm. Man. d’Ornith. p. 818. Common W1Lp Goosr.—Bewick observes that many of these birds are known to remain in the fens of Cambridgeshire and to breed there. This may have been the case formerly, but I never heard of an instance myself. 5 Sp. 137. A. Segetum, Temm. Man. d’Ornith. p. 820. BEAN GOOsE. Spr. 138. A. albifrons, Temm. Man. d’Ornith:. p. 821. Spe. 139. WHITE-FRONTED GOOSE. A. Bernicla, Temm. Man. d’Ornith. p. 824. Brenr Goose.—This and the three preceding species are indis- criminately called Wild Geese by the country people, and occa- sionally appear in the Cambridge market under that name, more particularly the A.Segetum.. They are all found in our fens during the winter months, in greater or less plenty according to the severity or the mildness of the season. The earliest flocks which I ever . noticed were seen on the twentieth of October; this, however, is not much before the usual time of their first arrival. 322. Mr. Jenyns on the Ornithology of Cambridgeshire. Sp.140. A. Cygnus, Temm. Man. d’Ornith. p.828. Witp Swan.—Seen occasionally in small flocks about our streams and ditches in very severe winters. Sp.141. A. Tadorna, Temm. Man. d’Ornith. p. 833. SHIELDRAKE.—Not uncommon. Spe. 142. A. Boschas, Temm. Man. d’Ornith. p. 835. Common Wi.p Ducx.—First seen about the middle of October. Sp. 143. A. Strepera, Temm. Man. d’Ornith. p. 837. GapDwaLL. Two of these birds were exposed for sale in the Cambridge market on the twenty-fifth of February 1824. Sp. 144. A. acuta, Temm. Man. d’Ornith. p. 838. Prn-rait. Duck. — Shot near Cambridge, and in the collection of Dr. Thackeray. Sp.145. A. Penelope, Temm. Man. d’Ornith. p.840. WIGEON. Sp. 146. A. clypeata, T’emm. Man. d’Ornith. p.842. SHOVELLER.—This species appears to be of not unfrequent occur- rence in the fens. I have seen specimens from the Isle of Ely, and also from the vicinity of Cambridge. In the market at the latter place it may often be met with. Sp. 147. A. Querquedula, Temm. Man. d’Ornith. p. 844. GarGaney.— In the collection of Dr. Thackeray: from the Cam- bridge market. Se. 148. A. Crecca, Temm. Man. d’Ornith. p. 846. TEAL. Sp. 149. A. nigra, Temm. Man. d@’Ornith. p. 856. ScorErR.—Montagu observes in his Ornithological Dictionary that this species never visits our rivers and inland waters; but I haye been informed by Dr. Thackeray, that large flocks of these birds have been occasionally seen in the fens near Cambridge, from whence he has specimens in his collection. Mr. Jenyns on the Ornithology of Cambridgeshire. 323 Sp. 150. A. ferma, Temm. Man. d@Ornith. p. 868. PocHaRD. Sp.151. A. Clangula, Temm. Man. d’Ornith. p.870. GoLDEN Eyr.—Occasionally met with in the fens near Cam- bridge. Willoughby mentions in his Ornithology (p. 28.) having had one sent him from thence by ‘the name of Shelden. Sp.152. A. Fuligula, Vemm. Man. d’Ornith. p. 873. TurreD DucK.—I have seen specimens of this bird which were killed in the neighbourhood of Cambridge. It is not unfrequent in the market. Sp.153. A. Leucopthalmos, Temm. Man. d’Ornith. p.876. Ferrucinous Ducx.—Dr. Thackeray has a specimen of this very rare duck in his collection, which was procured in the Cam- bridge market. Genus LIV. MERGUS, Linn. Sp.154. M. Merganser, Temm. Man. d’Ornith. p. 881. GoOOsANDER, Male. \ Dun Diver, Female. Sp.155. M. Serrator, Temm. Man. d’Ornith. p. 884. RED-BREASTED MERGANSER. — Both this and the preceding species are in the collection of Dr. Thackeray, from the Cambridge market. It is probable that they only visit this part of the country in very severe weather. Sp. 156. M. albellus, Temm. Man. d’Ornith. p. 887. Smew.—This likewise can only be looked upon as an occasional visitant. I find mention in Willoughby’s Ornithology (p. 337.) of a specimen that was sent to the author of that work from Cambridge; and have myself seen another, a male bird, in the collection of Dr. Thackeray, which was bought in the market at this place in April 1825. 7 Vol. If. Part Il. Tr 324 Mr. Jenyns on the Ornithology of Cambridgeshire. Sp. 157. Sp. 158. Genus LV. CARBO, Meyer. C. Cormoranus, Temm. Man. d@Ornith. p.894. Cormorant. —On the seventeenth of August in the present year (1825). one of these birds alighted on the top of King’s College chapel, and was there shot. It is now in the collection of Dr. Thackeray. I have been informed that it is not unusual for this species to follow the course of rivers to a great distance from the sea. Genus LVI. SULA, Briss. S. alba, Temm. Man. @Ornith. p. 905. GanNET.—This is a much more extraordinary instance of a bird’s being noticed so far from its usual haunts. ‘Two specimens were killed. in Cambridgeshire during the autumn of 1824. The first of these was shot near Fulbourn on the eleventh of October, and is now in the Museum of the Cambridge Philosophical Society. About a week afterwards, the second was killed near Southery fen in the Isle of Ely. Montagu observes that in the autumn these birds leave our northern islands where they breed, journeying southward, and may be seen during their winter migration in every part of the British channel, but that generally they keep far out at sea. I cannot indeed find mention in any author of their being found inland. The above therefore appears to be a solitary instance, and must have been occasioned by some very peculiar accident. It will be readily seen in the foregoing catalogue, that one or two species supposed to be of general occurrence are not inserted, as well as others, which it is not improbable may occa- sionally visit this county; but as these have never fallen under my own observation, and I have been unable to learn any thing respecting them, they are necessarily omitted. I trust, however, that what was stated in the Introduction to this Paper, will sufficiently apologize for its imperfection, and shall conclude by requesting from the Members of this Society, any further infor- mation on the subject they may chance to possess. XX. On the Influence of Signs in Mathematical _ Reasoning. By CHARLES BABBAGE, Esq. M. A. Trin. Cott. FELLOW OF THE ROYAL SOCIETIES OF LONDON AND EDINBURGH, MEMBER OF THE ROYAL IRISH ACADEMY, FELLOW OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, MEMBER OF THE ASTRONOMICAL SOCIETY, MEMBER OF THE ROYAL ACADEMY OF DIJON, CORRESPONDING MEMBER OF THE PHILOMATH. SOCIETY, PARIS, AND OF THE ACADEMY OF MARSEILLES, &e. [Read Dec. 16, 1821.] Ir can scarcely excite our surprise that the earlier geometers, engaged in successfully employing the most powerful instru- ment of discovery which human thought has yet contrived, and seduced by the splendour of the view their science had opened to them, should press with earnestness to enlarge its boundaries by new applications, rather than exert their genius in explaining the causes which have combined to advance it to such unrivalled eminence. On the discovery of those branches which have so completely altered the face of the science, the use of the new acquisitions was too inviting to allow time for any very scrupu- lous enquiry into the principles on which they were founded : satisfied with the accuracy of the results at which they arrived, the desire of multiplying them naturally prevented any return on their steps for the purpose of applying themselves to the less promising task of establishing on secure foundations, principles of whose truth they felt confident. These efforts to extend the reach rather than fix the basis of the new calculus, were undoubtedly to be admired at the period to which we refer: an acquaintance with its extensive bearings ought justly to have no inconsiderable influence on the form in which its elements should be delivered; hence the lapse of TT 2 326 Mr. BaspaGe on the Influence of Signs nearly a century has been required to fix permanently the foun- dations on which the calculus of Newton and. of Leibnitz shall rest. Time which has at length developed the various bearings of the differential calculus, has also accumulated a mass of materials: of a very heterogeneous nature, comprehending fragments of un- finished theories, contrivances adapted to peculiar purposes, views perhaps sufficiently general, enveloped in notation sufficiently ob- scure, a multitude of methods leading to one result, and beunded by the same difficulties, and what is worse than all, a profusion. of notations (when we regard the whele science) which threaten, if not duly corrected, to multiply our difficulties instead of pro- moting our progress. As a remedy to the inconveniences which must inevitably result from the continued accumulation of new materials, as well as from the various dress in which the old may be exhibited, nothing appears so likely to succeed as a revision of the language in which all the results of the science are expressed, and the establishment of general principles which shall curtail its exu- berance, and regulate that which has hitherto been considered as arbitrary—the contrivance of a notation to express new relations. Previous however to this, some observations on the nature of that assistance which signs lend to our reasoning faculties, and on the causes which give such certainty to the conclusions of analysis, may render our future enquiries more intelligible. The nature of the quantities with which the mathematical sciences are conversant, is undoubtedly one of the first of those causes: in Geometry it has been well remarked* that its founda- tions rest on definition, and if this do not altogether hold in alge- braical enquiries, at least the meaning of the symbols employed * Elements of the Philosophy of the Human Mind, Vol. II. p. 150.. in Mathematical Reasoning. 327 must be regulated by definition; and here arises one of the great differences which characterise this science, the definitions themselves being exceedingly simple, comprising but few ideas, whilst in other sciences they are usually much more complicated. In Geometry, definition is the beginning of any enquiry; in meta- physical science, it is frequently the result of one: thus that a triangle is a figure formed by three sides, is a convention on which many of Euclid’s propositions rest, and from this, as a point of departure, numerous deductions are made: on the other hand, our idea, and censequently our definition of beauty, is. only the result of considerable thought and enquiry. In the language of analysis, it is very rare that any symbol possesses more than one meaning; in ordinary language, it is as rare to find a word having but one signification: nor is this the only difference; when an algebraical symbol has more than one mean- ing, they are always well defined and distinct, and should there exist several signs for the same operation, the only difference is in their external form, not the slightest in their meaning; whilst in common language, the meanings of words shade away into each other, and it is frequently difficult, even on mature consideration, to assign the precise limits of the signification of words which are nearly synonymous. Now if this be the case when the words themselves are the especial objects of our thoughts, how open must all reasoning be to inadyertencies when the mind is compelled to occupy itself at once on the various meanings of the signs it uses and on the. train of consequences which it endeavours to deduce by them. The multitude of significations which attach to, many of the words that compose our ordinary language, is a disadvantage which is completely removed from that of analysis. In our rea- soning concerning any objects even of a moderately complicated nature, we are obliged to make use. of the words attached to- those’ 328 Mr. Baspace on the Influence of Signs objects, which consequently recall to the mind the variety of par- ticulars of which they consist, some with more, others with less vividness according to our previous habits of thought; from this cause it sometimes happens that the real ground on which our reasoning depends, is with difficulty kept in view by a laborious effort of the attention, and is in many instances very indistinctly perceived. In the use of algebraic signs this mconvenience entirely vanishes; we can always so arrange them, that that quality on which the whole force of our reasoning turns shall be visible to the eye, whilst the numerous others which contribute to form the expression we are considering, although thrown into the back ground, are still by no means excluded. This species of insulation of the property whose consequences we wish to trace, enables the mind to apply that attention, which must otherwise be exerted in keeping it in view, to the more immediate purpose of tracing its connection with other properties that are the objects of our research. As an example of these ideas, I would mention the word government, upon which we may reason in many different directions, either as it secures domestic liberty, or protects from foreign attacks, as it discourages vice or promotes commerce: in these and in numerous other courses, our reasoning may be pur- sued, and the word government will constantly recur without the possibility of avoiding it but by the most tedious circumlocution, or of restricting the view in which it is regarded but by the most unwearied efforts of attention. The word function in analysis possesses a still more extensive signification than that which has been just mentioned : The sign ¥(«, +) x signifies any symmetrical combination of the two quantities « in Mathematical Reasoning. 329 and ~. That combination is in this mode of expressing it left arbitrary and undefined. The same function / may at the same time be a function of x? 4-11 2 xt + a°7+1 Davia a+ 1? a ; or of a thousand other quantities; all which circumstances al- though deducible from the original expression, are not presented to the eye, because in the consequences which it is proposed to deduce, they are entirely immaterial. If two circumstances in the nature of the function are jointly the ground on which any of its properties depend, they may be separated from the rest and made prominent by several methods. Thus the index x subjoined to a function ¥(@, Y)n may be defined to mean that it is homogeneous with respect to x and y and of the dimensions 2 : these two circumstances are the causes on which the truth of the following property depends. xt being substituted for 7 Y(@, et), = a(t, Wn. In all our attempts at mathematical generalization, it is of great importance to discover and distinguish these immediate causes of successful operations; in almost all cases they lead us at once to the highest point of generality, and very frequently contribute in no inconsiderable degree to simplify the processes of the investigation. This advantage so peculiar to algebraic signs, has been remarked by M. Degerando, from whose writings I have derived much satisfaction by observing the support which many of those views that I had taken previous to my acquaintance with them, received from the reflections of that distinguished philoso- 330 Mr. BaspaceE on the Influence of Signs pher. . “ La troisiéme raison,” observes M. Degerando, ‘est dans la propriété qu’ a l’algébre de ne saisir, dans les idées des quantités, que certains rapports généraux, de ne présentir ainsi 4 notre esprit que les considérations qui lui sont vraiment utiles dans les re- cherches auxquelles il se livre. De la il arrive que notre atten- tion se trouve débarassée d’un grand nombre d’idées accessoires, qui etrangéres au but de ses méditations, n’auroient servi qu’ a la distraire *.” The quantity of meaning compressed into small space by algebraic signs, is another circumstance that facilitates the reasonings we are accustomed to carry on by their aid. The assumption of lines and figures to represent quantity and magni- tude, was the method employed by the ancient geometers to pre- sent to the eye some picture by which the course of their rea- sonings might be traced: it was however necessary to fill up this outline by a tedious description, which in some+ instances even of no peculiar difficulty became nearly unintelligible, simply from its extreme length: the invention of algebra almost entirely removed this inconvenience, and presented to the eye a picture per- fect in all] its parts, disclosing at a glance, not merely the conclusion * Des Signes et l'art de Penser, p. 214. tom. II. + The difficulty which many students experience in understanding the propositions relating to ratios as delivered in the fifth book of Euclid, arises entirely from this cause, and the facility of comprehending their algebraic demonstrations forms a striking contrast with the prolixity of the geometrical proofs. A still better illustration of this fact is noticed by Lagrange and Delambre, in their report to the French Institute on the translation of the works of Archimedes by M. Peyrard. It occurs in the ninth proposition of the 2nd book on the equilibrium of planes, on which they observe, “ La demonstration d’Archimede a trois énormes colonnes in-folio, et n’est rien moin que lumineuse.” Eutochius commence sa note “en disant que le theoréme est fort peu clair, et il promet de l'expliquer de son mieux. Il emploie quatre colonnes du méme format et d’un charactére plus serré sans reussir d’avantage; au lieu que quatre lignes d’algebre suffisent a M. Peyrard pour mettre la verité du theoréme dans le plus grand jour,” Ouvrages d’Archimede traduites par M. Peyrard, p. 415. tom. II. in Mathematical Reasoning. 331 in which it terminated, but every stage of its progress. At first it appeared probable that this triumph of signs over words would have limits to its extent: a time it might be feared would arrive, when oppressed by the multitude of its productions, the language of signs would sink under the obscurity produced by its own multiplication: had these expectations been realized, still its utility would have been extensive, and mankind, whilst they felt grateful for the many stages it had advanced them, must have sought some more powerful auxiliary for their ulterior progress. Fortunately however such anticipations have proved unfounded ; in whatever department of analysis the number of symbols has encreased to a troublesome extent, contrivances have soon occurred for diminishing it without any sacrifice of perspicuity : the incon- venience has always been temporary, the advantage permanent. In later times the generalization and contraction introduced by the use of signs, seems even to have outstepped the discoveries which have resulted from them; and reasoning from the past course of science to its future advances, we may fairly presume that our power of condensing symbols will at least keep pace with the demands of the science. Examples of the power of a well contrived notation to con- dense into small space, a meaning which would in ordinary lan- guage require several lines or even pages, can hardly have escaped the notice of most of my readers: in the calculus of functions this condensation is carried to a far greater extent than in any other branch of analysis, and yet instead of creating any obscurity, the expressions are far more readily understood than if they were written at length: the imstance I shall choose as an example is the equation V(x, y) = (a, y)*. * Transactions of Cambridge Philosophical Society, Vol. I. p.68. Vol. If. Part II. Uv 332 Mr. BaBpaceE on the Influence of Signs To any person acquainted with the notation belonging to the calculus, it is instantly intelligible; yet if it were written out at length, the letters « and y would be each repeated 257 times, the letter y would be found 512 times, whilst the expression would also contain 257 commas and 512 pairs of parentheses; thus comprising in the whole 2307 symbols; and it may be added that it would require a much longer time to understand the meaning of the equation written out at length, than it would to find its general solution. The power which we possess by the aid of symbols of com- pressing into small compass the several steps of a chain of reason- ing, whilst it contributes greatly to abridge the time which our enquiries would otherwise occupy, in difficult cases influences the accuracy of our conclusions: for from the distance which is some- times interposed between the beginning and the end of a chain of reasoning, although the separate parts are sufficiently clear, the whole is often obscure. This observation furnishes another ground for the preference of algebraical over geometrical reason- ing, and is one which had not escaped the notice of Lagrange. ““Chaque membre de phrase est claire et trés intelligible a le considerer seul, mais le tout est si long qu’on a souvent oublié le commencement quand on arrive a l’endroit ou Je sens est complet.” The closer the succession between two ideas which the mind compares, provided those ideas are clearly perceived, the more accurate will be the judgement that results; and the rapidity of forming this judgement, which is a matter of great importance, inasmuch as the quantity of knowledge we can acquire in a great measure depends on it, will be proportionably encreased. M. Degerando has clearly stated this advantage in comparing the decimal arithmetic with that of the Romans. “ La rapidité d’une operation intellectuelle est toujours en raison inverse des efforts qu’on demande a l’attention et a la memoire. Cette operation in Mathematical Reasoning. 333 qui consiste a fixer les rapports de ses idées pour Jeur appliquer les méme judgements s’executera donc autant plus promptement qwil nous sera facile de nous rappeler et de remarquer ces rapports*.” The almost mechanical nature of many of the operations of Algebra, which certainly contributes greatly to its power, has been strangely misunderstood by some who have even regarded it as a defect. When a difficulty is divided into a number of separate onest, each individual will in all probability be more easily solved than that from which they spring. In many cases several of these secondary ones are well known, and methods of over- coming them have already been contrived: it is not merely useless to re-consider each of these, but it would obviously distract the attention from those which are new: something very similar to this occurs in Geometry; every proposition that has been previously taught is considered as a known truth, and whenever it occurs in the course of an investigation, instead of repeating it, or even for a moment thinking on its demonstration, it is referred to as a known datum. It is this power of separating the difti- culties of a question which gives peculiar force to analytical investigations, and by which the most complicated expressions are reduced to laws and comparative simplicity. One of the most elegant illustrations of this opinion I shall at present briefly allude to, as a more detailed account of it will be given in a subsequent essay. Among the papers left by the late Mr. Spence, is one on a method of solving certain equations of differences: elimination is the means by which he proposed to * Degerando sur les signes, Tom. Il. p. 196. t Of so much importance is this maxim, that it has been adopted by Des Cartes as one of his principles of philosophizing. “ Diviser chacune des difficultés en autant des parcelles qu'il se pourrait et qu'il serait requises pour les resoudre.” Discours de la Methode. uU2 334 Mr. BaBBaGE on the Influence of Signs accomplish his object, but the results soon became so complicated that little expectation could be formed of succeeding by that means. In this difficulty Mr. Spence introduced into the equation to be solved an arbitrary quantity a, which is merely employed as a letter by whose powers the resulting series may be arranged : if the attempt is now made by continually eliminating, a series arises proceeding according to the powers of a, and equations are found for determining their coefficients: finally, the arbitrary quantity a having performed its office is made equal to unity, and the result is the solution of the equation. The success of this plan depends entirely on breaking into a number of separate parts a very complicated expression, each of these portions being separately reducible to known laws. On resolving into their separate parts a vast variety of questions which have occurred, it has been found that the number of individual difficulties is by no means so large as had been originally supposed; many of very different kinds have been found to depend perhaps on the same integral, or on the solution of the same equation. In proportion to the number of questions which are reduced to these new difticulties, they themselves assume importance, and the celebrity which always attaches to those who remove obstacles regarded as insuperable by their predecessors, in- duces many to attempt the solution of these purely abstract ques- tions. Perhaps these ultimate points of reference may not from their nature admit of a comparison with, or reduction to, existing tran- scendents: the labour and ingenuity employed in the attempt are not however thrown away; relations are discovered by which, from a certain number of particular cases numerically given, all others may be readily calculated, approximations are discovered for determining the cases which are required as data, and, finally, they are arranged in tables and accompanied by rules for their employment, by which, as far as results in pure numbers are in Mathematical Reasoning. 335 required, all questions that are made to depend on them may be considered as solved. The power which language gives us of generalizing our reasonings concerning individuals by the aid of general terms, is no where more eminent than in the mathematical sciences, nor is it carried to so great an extent in any other part of human knowledge. In the transition from Arithmetic to Algebra, when letters began to be substituted for numbers, the first step consisted rather in the circumstance of the possibility of operating on a quantity determined but unknown. Thus if it were proposed to discover such a number, that its square added to three should be equal to four times the number itself; we commence by supposing the number to be represented by x: now it is quite certain, as soon as the question is stated, that there can only exist two numbers fulfilling the condition; x therefore must in reality mean either of these two, and the rest of the process is “+3 =42, @—42+4=1, Ge 2 — 2ty Ii, Th uOr Ne To point out more clearly the force of this observation, we adopt the plan which Vieta introduced into Algebra, that of denoting known quantities by letters: instead of the numbers 3 and 4, let us use the letters a and b; then the process is as follows: “+ a= bez, x — br=—a, bb Bb 2 _—_—_— ipl greg a, b 5° e==+ ——— & Bes here it is true that a and b meant 3 and 4, but as no part of the reasoning employed in any manner depended on _ their 336 Mr. BappaGce on the Influence of Signs numerical value, the result must be independent of it, and is consequently true for all possible values. It may perhaps be contended that by the assumption of « for the number to be found, it was meant to represent number in the abstract, and that such was also the meaning of a and b; but there exists this difference, that it is not in our power to alter the value of x, but we may give to those of a and 4 any numerical magnitude we may please*. The utility of the unknown quantities in Algebra, arises from their capability of being operated on without reference to the determined values for which they are placed, the advantage of employing letters for the known quantities, consists in their similarity to general terms in language, and the consequent ex- tension of the reasoning from an individual case to a numerous species. The light in which this question has been regarded, is purely arithmetical, it may however be placed in another point of view, in which without any change in the quantities concerned, it is still more general in its nature; instead of restrictmg the equation v—bxr=-a to number, it may be considered as indicating that x is composed of a and b in such a manner, that when its value is substituted in that equation, all the terms shall mutually destroy each other. This signification, it is true, is not contained in the original question, but arises from the equation into which it is translated : the language of signs is far more general that that of arithinetic, a circumstance which is not perhaps sufficiently attended to in the application of it to questions of pure number. In one re- spect this generality is not so unexpected, for if a number is required satisfying a certain condition, and if it should happen ct ees b * There is in truth one restriction, namely, that a must always be less than rg but this will be removed when the question is viewed in an algebraical light, and does not in the least affect the argument. in Mathematical Reasoning. 337 that more numbers than one fulfil that condition, there is no reason why the answer should produce one of these numbers rather than another, it must therefore contain them all. The reasoning which is carried on in Geometry is of a general nature, and applies to a species, although it is impossible that the picture presented to the eye can be any thing else than that of an individual; hence, it not unfrequently happens that some pecu- liarity in the figure which is actually employed, either leads us into erroneous conclusions, or when the results are correct, they are supposed to be limited by the individual nature of the figure we have employed. If a line is made use of to represent number, since some other line is the standard unit, it is impossible by such means to represent number in the abstract, but if number is denoted by a letter, there is nothing in the sign which at all indicates the magnitude of that which it represents: it is evident therefore that a property which might lead us into error in the first case, is removed from our view in the second. It will per- haps be objected that the standard unit need not be visible to the eye, since the force of the demonstration is in no way affected by its magnitude, this observation is perfectly correct, and if only one line be considered, and no unity of linear measure be stated, that line may represent length in general, and is to all purposes an arbitrary sign: but the moment any other line is introduced into the diagram, although the unit should not be mentioned, the generality of the former sign is diminished, a relation is instantly established, and whatever may be the unit of length, the ratio of these two lines is fixed and determinate. The position of a line is another circumstance in Geometry which must always remain particular, and this brings with it that of the points formed by its intersection, as well as that of the angles formed by it with other lines, and the attention which the mind must exert to perceive that no part of the reasoning it is pursuing 338 Mr. BappaceE on the Influence of Signs rests on any of these individualities, itself requires a considerable effort. The substances of these observations may be expressed in this conclusion. The reasonings employed in Geometry and in Algebra are both of them general, but the signs which we use in the former, are of an* individual nature, whilst those which are employed in the lutter, are as abstract as any of the terms in which the reasoning is expressed. The signs used in Geometry, are frequently merely individuals of the species they represent; whilst those employed in Algebra having a connection purely arbitrary with the species for which they stand, do not force on the attention one individual in pre- ference to any other. An example of the limitation which geometrical considera- tions introduce, we shall select from a very well known author. In determining the relation between the rectangle under the parts of two lines intersecting each other and cutting a circle, Euclid considers separately the two cases of the point of inter- section being situated within and without the circle, and he shows that in the two figures the rectangle under CP and CQ, is in both cases equal to that under CP, and CQ: the case of one of the lines becoming a * Halley’s paper on the determination of the foci of lenses, would furnish a very apposite example of this principle, and probably few of my readers will fail to recollect instances where the same identical words of a proposition, and the same letters apply to two, three, or more different geometrical figures. in Mathematical Reasoning. 339 tangent is also a separate investigation. Now in the algebraic mode of treating these questions, the three cases are comprised in one formula*. The indication of the extraction of roots by means of an appropriate sign, instead of actually performing the operation, is one of the circumstances which add generality to the conclusions of Algebra, and the same principle of indicating operations, in- stead of executing them, when employed with judgement, con- tributes frequently in no small degree to the perspicuity of the result, and sometimes enables us to read in the conclusion every stage which has been passed through in the progress towards it. Any general rules to direct us in the application of this principle will be difficult to form, because they ought in a great measure to depend on the objects we have in view: it may, however, be stated generally that it is improper to adhere to it, when by an opposite course any reduction or contraction can be made in the formula; thus generally speaking it would be better to write (a+ a)? — Aax+ B’, than and on the other hand, wherever in the course of any reasoning the actual execution of operations would add to the length of the formula, it is preferable merely to indicate them. Some of the advantages which arise from the use of letters to denote known quantities, have already been adverted to; but there are others of considerable value which may now be noticed, and which relate in a great measure to the higher departments of analysis. If a player bet a certain sum of money u, he may either win it and become possessed of +z, or he may lose it and possess —u. If we now suppose that he regulates the amount of his * Book III. Prop. 36. and 37. Vol. Il. Part Il. xX x 340 Mr. BappacE on the Infiuence of Signs second stake by the result of the first, and that he makes it wu-7~ if he had won the first, but w+v if he had lost it; on this second bet he will either win or lose u+v, or U—v, : Supposing him to determine his third stake from his second, in the same manner as he fixed his second from his first, it is clear that according to the determination of his previous bets, he may stake on the third event either of the three sums U+2v, U, U— 2. And generally on the nth event, if he proceed according to this law, there are different bets which he may make according to the order in which the previous ones were decided. Now in any question in which such a mode of play entered, it would be ex- ceedingly tedious to consider separately all these cases, and to repeat the same or nearly the same reasoning for each individual case. This may be avoided by rendering the events indeterminate, for we then find his first profit may be denoted by u(—1)°, in which the letter @ represents any whole number whatever ; if it is an even number he wins, and if an odd one he loses; the same artifice applied to his second stake gives for it u—v(-1)% as a is still undetermined, this will represent that stake truly, whichever event has happened on the first. The result of this second stake may be represented by {2 SAGES) | 1a whether it is lost or gained, and this is still kept undecided by means of the letter D. The third stake will be u—v(—1)*- 2 - 1), in Mathematical Reasoning. 341 and the profit or loss by that bet will be DGS OE ak 4 ale a new undetermined letter being again introduced. By these means his xth stake will be represented by th hea Cra aie Ga Mee ee LS mais. 1) and thus all the x cases become reduced into one expression. The influence of this indeterminateness in contributing to the success of the solution of the following problem is very apparent. Supposing a player to regulate his stakes in the manner just described, and that his chances of winning and losing are equal, what will he have profited after p+q events, p of which have been favorable, and q have been unfavorable ? At a first view it might perhaps be considered that the. data are insufficient for the solution, and that the order in which the events take place ought to be given: the result however of the investigation will prove that the profit or loss is entirely indepen- dent of that consideration. It has already been shewn that his profit on each event will be as follows: u(-1)', ui = 1Y—O4 (A Cady w (Hh) pen (=) Hol Se 2S Mal) ay Wieie yi ae 1)°$ ae u(—1)" =» {(—3)" +(<1)+ (ou + (-0). On examining the composition of that collection of terms which multiply v it will appear that the sum of all the above quantities is equal to uf(—1l + (-1f +...(—1)}, or < C. ; . ; in Mathematical Reasoning. 347 usually occur in this first stage; and it cannot be too strongly recommended, that every part even of the most difficult problems should be fully translated into the language of analysis, before any attempts at simplification are made. In the first stage, it is scarcely possible to see clearly in what degree the results will be affected by a proposed omission; whilst in the second, any quantity which it is conjectured will have little influence on the result, although it adds greatly to the difficulty of calculation, may be kept separate, and the operations to which it is submitted, may be indicated rather than performed. In many of the appli- cations of analysis, and particularly in its treatment of mechanical questions, the principles which regulate the first stage of the process are completely known, and little difficulty is experienced in translatmg them into the language of signs, the difficulties when they occur, usually taking their rise in the solutions of the equations thus produced. A_ similar remark is applicable to optical questions, and indeed to by far the greater part of those which occur in the mixed sciences. II. The second stage in the solution of any problem, generally begins with the equations into which it has been translated, and terminates with their solution. The point at which it commences is not always so well defined as that at which it ends, and this is more particularly the case when the question relates to geome- trical figures, where in some instances, the first and second stages are much intermixed. The difficulties which now occur are purely analytical, and are generally such as have been treated of in works devoted to the sub- ject. The solution of one or more algebraic equations is frequently the object to be obtained: differential equations, or equations of finite differences are another class of analytical expressions to which physical problems are often reduced; many of these can Vol. Il. Part If. Me 348 Mr. Baspace on the Influence of Signs only be resolved approximately, but in proportion to the interest thesé questions have excited, the variety and accuracy of the approximations have been multiplied. This is strongly exempli- fied in the problem of the three bodies, as applied to determining the place of the Moon: the great importance of the question has caused the approximations to be pushed to such an extent, that they have arrived at a degree of inelegance and complexity, which would long since have caused them to be rejected from any other question on the exact solution of which less important in- terests depended. But on this second stage in the solution of a question, it is less necessary to add many observations; the operations which are concerned in it, and the modes of effecting them, being more fully treated of in works of instruction than either of the others. III. The last of the stages into which the resolution of a question has been distributed, has been more neglected than any other. It may perhaps appear singular that the answer to a question, which is of course* the great object of research, should have been passed over without suflicient attention. It is not however of any errors in those results which are usually arrived at that I complain; but it is, that: sufficient instruction is not given in elementary works, as to the full meaning of all the different circumstances which are contained in the result that analysis has presented. In those questions which lead to alge- braic equations, it is not unfrequently the case, that some one or perhaps twe roots are taken as the answer, whilst all the remaining ones are completely neglected. Now a question can never be said to be fully answered until every root of the equation to which it has conducted has been discovered, and its signi- fication with reference to the data of that question been ex- plained. It sometimes happens that superfluous roots have been in Mathematical Reasoning. 349 introduced by the algebraic operations that were necessary to arrive at the final equation: these ought to be pointed out, and the step at which they were introduced should be noticed, and also whether they can admit of any translation into the language of the problem considered. Imaginary roots are very frequently introduced; they sometimes imply impossibility or contra- diction amongst the data; their origin ought to be carefully traced, and such a course will frequently make us acquainted with the maxima and minima which belong to the question. It is still more necessary to attend to all the real roots whether positive or negative, and to explain the various circumstances in the solution to which they refer. It is by no means uncommon with algebraical authors, when they have led their readers through a process which terminates in an equation, to select that root which gives the answer they require, without explaining the signification of the other roots that are equally comprised in it; and this incomplete mode of solution, which is censurable from revealing only a part of the truth, has in some instances caused the most interesting circum- stances attending a question to be entirely overlooked. A singular example of this occurs in several authors who have sought analy- tically the side of a heptagon inscribed in a circle, or the radius of a circle which would circumscribe a regular heptagon whose side is given. In neither of these questions can the equation to which we are led, be reduced below the third degree, and the three roots of the cubic are always real: the largest of the positive roots gives the answer to the latter of these questions for the common heptagon of Euclid: but no reason is stated why this root should be considered as the true answer to the question in preference to either of the others. In the Analytical Institutions of M. Agnesi*, where the first problem is solved, no * Vol. 1. p. 168. English Translation. YY2 350 Mr. Bappace on the Influence of Signs notice is taken of the fact that all the three roots are possible, nor am I aware of its being noticed by any author who has treated of this question: had it been observed and enquired into, the existence of three species of heptagons answering: strictly to the definition, and the knowledge of the star-shaped polygons which were discovered by M. Poinsot, could not have remained so long unknown. If x = OP, denote part of the diameter intercepted between the centre, and a perpendicular from the extremity of the first side of the heptagon, then the usual trigonometrical formule give 5 3 ee eas! i rh) x ar +3 > this contains six roots, of which the three positive are (abstract- ing the signs) equal to the three negative: it may be resolved into the two factors 1 1 i 1 1 1 ees eave oS = 3 st —ir—=)=0 (« Q” ptt) (« ar Be Ya > the second of which is only the first with the signs of its reots changed. The three roots are real and are represented by xz = OP, x= OS, and z = OR, the first of these gives 4B for the side of the heptagon, this is the same as that which has long been known; the other two roots give AC and AD, as the sides of the polygon, and by carrying them round the circle, the two star-shaped heptagons, (Figures 2 and 3,) are produced which have no re-entering angles, and in Mathematical Reasoning. 351 which are in fact comprehended in Euclid’s definition of regular polygons. The sum of the interior angles of the first of these heptagons is ten right angles, the sum of the interior angles of the second is six right angles, and that of the third species is two right angles. These new species of polygons were first noticed by M. Poinsot, in a highly interesting memoir on subjects connected with the Geometry of Situation, read before the Institute in 1809, and subsequently printed in the Journal de VEcole Polytecnique, 10° Cah. Another important business which belongs to this stage of the question, is to examine carefully what changes will ensue from supposing any peculiar relations amongst the data; or from any of the constant quantities becoming infinite or evanescent, such circumstances frequently introduce great simplicity, and when they refer to geometrical questions, are sometimes the means of making us acquainted with general properties, by which the construction of the problem is greatly facilitated. A careful and laborious attention to all the possible modifications of a problem which might result from any relation amongst its data, was considered by the ancient Geometers as an indispen- sible part of its investigation, and the manner in which this was accomplished, was generally little else than a repetition of the whole process under the altered circumstances: when the data are numerous, the length of such a system of operations becomes intolerable, and if more rapid methods had not been contrived, Geometry must have become stationary from the accumulation of the details with which it was thus encumbered. Many instances of the extreme length to which a full investigation of comparatively a very simple problem will lead, occur in the treatises De Sectione Rationis, &c. The advantage of Algebraic language is in this respect very striking: all the data of the questions are embodied in the equation in which its solution terminates, and 352 Mr. Bassace on the Influence of Signs without repeating any part of the process by which that was produced, we can examine with ease all those modifications which any differences in the actual magnitude of the data can introduce into the question under consideration: and moreover the equation itself will suggest to us such relations amongst those quantities as will have the effect of lowering the number of its dimensions, or of rendering it the product of two or more factors. It sometimes happens that by a peculiar relation amongst the data of a question, the number of solutions instead of being limited becomes infinite: thus, if the position of a line is deter- mined by two points, when those points coincide, any line passing through the poit in which they coalesce, will satisfy the con- ditions of the question which becomes to a certain extent inde- terminate: this gives rise to a class of propositions in Geometry which are called porisms. When the data on which questions depend are numerous, it is by no means so easy to discover by Geometrical considerations that relation amongst them in which the question becomes indeterminate, as it is by an Algebraical inquiry where the solution is presented in its most condensed form: one consequence of this is, that such cases have frequently escaped the notice of those who have treated the problems to which they belong in a Geometrical manner. One celebrated and important oversight of this kind occurred in a problem which Newton solved in order to determine the orbit of a Comet. Having four lines given in position, it was required to draw a fifth line which should be cut by the other four into segments, having a given ratio to each other. Of this question Wren, Wallis, and Newton had given solutions, but when Zanotti, Boscovich and other Astronomers made use of them, employing the observed places of a Comet, the results were found greatly in Mathematical Reasoning. 353 erroneous. Boscovich inquired into the reason of this singular result, and having first assured himself of the accuracy of the solutions, he discovered that in a particular relation between the given lines the problem became indeterminate, and admitted of an infinite number of solutions, and that the case of a Comet approached extremely near to this, and consequently that any very small error in observations must produce an extremely large one in the result. As an instance of the curious and elegant properties to which such an examination of the relations of the data contained in the final equation sometimes leads, I shall propose the following problem. A circle whose radius is r being given, and also three points in one of its diameters, at what angle must three parallel chords be drawn through these points so that the sum of the squares of two of them shall be equal to a gwen multiple n of the square of the re- maining one ? Let the distance of the three points in the diameter from the centre be v0, Vy and U3, and calling the angle which is sought 6, we have CP = —vcosé+ /r* — v (sin 6)', and CQ = +vcos 0 + /r — v* (sin 6), hence PQ=2,/r—v (sin OF, and similarly for the other two chords P,Q, =2 aire — v,” (sin 8), and P,Q; = 2 r= co (sm 0)*. 304 Mr. Baspace on the Influence of Signs These values of the chords being employed give 4 (r° — v sin 6) + 4 (7? — v sin @) = 40 (7° — v2 sin 6°) At this step the first of the three stages which have been described terminates; the question is now translated into the language of Algebra, and must be treated according to its rules: the following reductions must then be made 7? — v? sin 6° + r* — v, sin @ = nr — nv,” sin 0? (nv,2 — v2 — v*) sin @ = nr’? — 27°, : n— 2 sm@= +7 / aaa’ The second stage is here concluded by the solution of the equation to which the first conducted us, and we have now to explain the meaning of its two roots, and the modifications which may arise from any peculiar relations amongst the data. The two signs signify that the angle @, may be measured either above the diameter or below it, as is apparent on inspect- ing the figure. As the result contains an even radical, we must enquire if in all cases a solution is possible, and if not, what are the conditions of possibility. For this purpose we observe that the numerator and denominator must be both positive or both negative, consequently nm —2>0, and nv, — v; — v° > 0, or n— 2<0, and nv,2 — v7 — v <0, and also sin @ <1, are the two sets of conditions; in all other cases the question is impossible. From this, however, must be excepted the case of the nume- rator and denominator simultaneously vanishing in consequence of the following relations taking place amongst the data, n—2=0, and nv,* — v1 — v = 0, Or On in Mathematical Reasoning. 3 whence Oy + v Nn = 2; anaes 7 ae under which circumstance the value of sin 6, becomes really in- determinate, not depending even on the vajue of 7 the radius of the circle. This indeterminate case suggests the following porism: Any three points in a straight line being given, another point may be found about which as a centre if a circle with any radius be drawn, and if through the three given points, three chords be drawn in any direction, but parallel to each other; then the sum of the squares of two of them shall be always equal to the square of the third. It is to be observed, that the origin of the lines denoted by v, %, v2, may be changed by removing it to the distance a, then the latter of the two conditions which rendered the problem in- definite becomes 2(v, + a)? = (v + a) + (v + a)’, whence ed 20, > or — 8 j 20+ 2%, — 4%, Before I conclude my observations on this subject, which may perhaps be considered as a digression from that which the title prefixed to this Essay would seem to imply, I shall offer one more illustration of the division of a problem into the several stages which I have pointed out. This examination of all the circumstances attending the equation containing the solution, is stilk more necessary when that equation is a differential one: if it be only capable of in- tegration by means of transcendents or by approximating series, it sometimes happens that some relation amongst the data may be assumed, by which in the one case the transcendents shall Vol. If. Part IL. Zz 356 Mr. Bappace on the Influence of Signs disappear, and in the other, that the series shall terminate. Euler has taken advantage of the former of these circumstances, to discover curves whose indefinite quadrature should depend on a given species of transcendent, whilst the areas of particular portions of them are susceptible of an Algebraic expression *. The integration of the equation is not always sufficient for a complete analysis of a question, for in some cases besides: the general integral, there exists another not included in it, which is known by the name of a particular solution; in order to be secure of not overlooking any such, it must be observed that a change in the magnitude of an index may cause the intro- duction of such a solution. When the complete integral as well as all the particular solutions are found, the interpretation of them according to the circumstances of the question is not always an easy task, nor are any general rules yet established to which we can refer for information. In the theory of curves the in- terpretation of particular solutions is sufficiently well known: they represent the curve which touches all those formed by the complete integral when its parameter varies. In mechanical questions a considerable degree of uncertainty prevails relative to these kind of solutions, as in some instances they seem to have no reference to the preblem which gave rise to them, whilst in other cases its solution can only be fully represented by their assistance; some light has been thrown on this subject by M. Poissont+ in a memoir in which he has explained the theory of particular solutions with great perspicuity. Of whatever kind the equation to which our question con- ducts us, may be, it ought to be regarded, merely in an analy- tical point of view; and all its various roots or solutions, should * (Euler Acta Acad. Petrop.) + Journal de Ecole Polyteenique, Cah. 13. p. 60. in Mathematical Reasoning. 357 be sought after; out of these, by means of some peculiarity in the problem, we must select that individual, which more imme- diately satisfies the particular view of it which we have taken, and the other solutions must be explained if possible, by means of the data, from which we commenced the process; or should that be impossible, their entrance must be traced to some gene- ralization in signs, to which the language of the question was incapable of adapting itself. In the demonstration of the com- position of forces, given in the Mecanique Cceleste, which has sometimes been unjustly censured on account of its analytical nature, this does not appear to have been completely attended to. In the enquiry, to which I refer, x one of the forces is assumed equal to z#(@), where z is the resultant force, and ¢(@) some function of the angle between it and the force x, which function it is required to determine; by changing 2 into y and @ into 5-9 the two values of « and y are found to be 2=29(6), y=2¢(7-8), and the equation th +y¥ = 2? is arrived at: this equation in fact amounts to [er + [e(§-9)] =1, which results* from it, by merely substituting for x and y, their values. * This equation is one of that class whose general solution I have ascertained, and it may be exhibited in either of the following forms oo os o() +o (5-8) o@=\/ 5+ (5-28) x (45-9), ZZ2 358 Mr. Baspace on the Influence of Signs From this equation it appears to me, to be the direct course to deduce the form of ¢: its general solution should first be shown, and then from the peculiar circumstances of the problem, that particular one which belongs to it should be pointed out. In the work, to which I refer, the particular form of ¢ has been deduced at once by properties peculiar to the problem, without any reference to the general solution of the equation. A similar objection may be made to other demonstrations of this celebrated theorem: the equation to which the inyestigation conducts, is usually solved in a manner not sufficiently general. This is the case in a work deyoted to the analytical exposition of the elements of Geometry, pp. 53, 54;* the substitutions employed, although satisfying the conditions, not containing all possible — solutions. M. Poisson has given an investigation of this theorem not quite so open to the objections just stated; by the introduction of two variables and. the employment of one sign of function, the solution is necessarily more restricted in its extent. Equations of that class are frequently contradictory, although in the case re- ferred to, a fortunate property leads directly to the solution. See Poisson, Mecanique, p.14. I cannot conclude this slight criti- cism on a detached passage of the Mecanique Ceeleste, without expressing that respect for its illustrious author, which is shared with all those, who are capable of appreciating the important additions he has made to mathematical science, or who have the happiness of being personally acquainted with him. When any question leads to an algebraic equation, it is usual to resolve it generally, and then to point out amongst its roots that particular one which is sought; if the individual root re- where ¢, is perfectly arbitrary, and X is any symmetrical function of @ and s —0. * Precis d'une nouvelle methode pour reduire 4 de simples procedés Analytiques la demonstration des principaux theorémes de Geometrie. Par. I.G.C. Paris, An. vi. in Mathematical Reasoning. 359 quired were discovered by some artifice without the solution of the equation, we could not feel assured that it alone ful- filled all the conditions, and we should, by arriving at an equa- tion and rejecting its use, have the semblance of generality without its reality; nor do I perceive any reasons which should induce us to change our course, when we have to consider equa- tions of a more comprehensive nature. Any enumeration of the ‘causes which contribute to give such extensive power to the employment of algebraic signs, would be justly considered incomplete, if no notice were be- stowed on the symmetry that ought always to prevail, where the calculations in which it is employed are in any degree complicated. In its least restricted signification, symmetry is applied to two things, which although sometimes connected, are yet, in many instances, totally independent: it either refers to a resem- blance between the systems of characters assumed’ to represent the data of a question, or it implies a similarity of situation, between certain of the letters, which are found in an analytical formula. An attention to it in either of these senses, has a direct and very beneficial influence in relieving the memory from a considerable burthen. In the first case, its precepts would direct us to assume similar letters as the representatives of similar things: thus, if we have two series, and propose to find another, which consists of the product of the corresponding terms of the two former; if we assume for the first of two series at+b+c+d+.. the terms of the second ought to be 4+B4+C+D+.. or which is still more convenient ad+b4+ce+d + 360 Mr. BappaGE on the Influence of Signs and the series, whose sum we wish to find is then denoted by aA+bB+cC+dD+.. or by aa'+bb+cck+dd+.. The assumption of A,+A,+A,+.. Ay + 4+ A+ Sc would have been equally proper, and the result equally clear: but had we assumed for the two series at+tb+c+d+.. Ba Pa ’ the resulting series aA,+bA,+c4,+dAy+.. would have been devoid of that symmetry, which forms so pro- minent a character in the former cases. ' The plan of accenting letters, in order to represent quantities which stand in similar relations, adds, when employed with discretion, much to the perspicuity of the formulz in which it is used; but like many other innovations, whose tendency is on the whole decidedly beneficial, an attempt to extend it beyond its proper limits, has been productive of inconveniences as considerable as those which its introduction was proposed to remove. Indices in various positions have been substituted in many cases for the system of accentuation, and the admirers of this scheme, pursuing it with equal ardor, have not been more fortunate in avoiding the confusion, which a multitude of signs, diftermg but by the slightest shades, can scarcely fail of producing. The taste of the geometer is not less strongly tried by the choice of the letters in which he conducts his reasoning, than his skill and ingenuity are by the artifices he invents to surmount the in Mathematical Reasoning. 361 difficulties opposed to him; in the one case, the elegance which a judicious selection produces, carries the reader pleasantly and almost imperceptibly through an abstruse calculation, whilst the latent cause, which gives facility to his progress, is rarely appreciated, because it is spread uniformly over the whole ques- tion: in the other, whose essence often consists in some happy substitution, which is always concentrated in some point, the effect is too remarkable to escape the least attentive enquirer, and its success too striking not to command his admiration. Unlimited variety in the use of signs, is as much to be depre- cated as too great an adherence to one class of them; the one conceals the appearance of relations that really exist, whilst the other, affecting to display them too clearly, fails from its want of distinctness. It is difficult to point out models of imitation, even amongst the most eminent; the same writer, who at one time might be safely trusted as a pattern of correct judgement, has indulged at another in the most unexampled imnovation ; completely setting at defiance many of those principles, whose authority I have endeavoured to establish. The fate, which has attended the greatest of these proposed reformations, though sanc- tioned by the illustrious name of Lagrange, is no slight testimony of the validity of those views, which it is the object of several of these Essays to advocate*. Having delivered a theory of the diffe- rential calculus, which rested on principles entirely new, he intro- duced it to the world, clothed in a language of his own creation. The value of the present was too great to allow of its utility being impeded by the garb with which it was encumbered; and the labor of acquirmg the language was compensated by the truths which it revealed. Time, however, and experience, con- vinced even its immortal author, that the language of signs rests * This relates more particularly to an Essay on the Principles of Notation, which is not yet published. 362 Mr. BappaGE on the Influence of Signs on principles, which cannot be neglected without danger, or vio- lated with impunity ;—the authority of the greatest geometer of the age, failed to make converts to the language he had invented, whilst the justice of the view he had taken, was admitted, and his explanations almost universally adopted. Whilst the language, in which the Theory and the Calculus of functions are conveyed, is pointed out as a warning, not to be neglected by the most successful, that of the Mechanique Analytique of the same author, may perhaps be held up to imitation, with fewer limitations than any other work of equal magnitude. In returning to the notation of Leibnitz, Lagrange has ‘in this work, reduced the whole theory of mechanics to the dominion of pure ‘analysis, and in the choice of his symbols has frequently displayed that happy selection, which so much facilitates the process of reading and comprehend- ing analytical formule. The value of symmetrical symbols is greater in proportion to the complexity of the operations, and the number of quantities, which are concerned; but unless their selection is attended to at the outset of our studies, it is not to be expected that a correct taste can be acquired, I would therefore recommend a degree of attention to this subject, which is not usually bestowed on it by elementary writers: Some instances I shall select from the simplest applications of Algebra to Geometry. The equation of a right line is usually written thus: y=ar+h, which is sufficiently convenient. M. Biot in his Geometrie Ana- lytique * has employed this notation, as also has M. Hachette in his Introduction to the admirable work of Monget: both authors in treating of lines referred to three co-ordinates have denoted it thus: yYy=art+a, © P30: + Application de l’Analyse a la Geometrie, 4™* ed. p. 2. in Mathematical Reasoning. 363 the difference is apparently trivial, but the convenience or incon- - venience of notation frequently depends on differences as trifling, It may be observed, that in the first equation, a denotes the tangent of an angle, and 4 an absolute line; two things which have no relation to each other, and which are therefore justly represented by dissimilar signs. In the second equation, the line and the angle, are both represented by the same letter of different alpha- bets; a circumstance, which will infallibly suggest some idea of a relation that does not exist. When two straight lines enter into the question, other reasons present themselves: they may be represented in any of these four ways; y=ar+hb y=ar+ta y=ar+b y=ar+h y=ar4+t y=br+By y=axr+B y=cur+d and if we seek the ordinate of their point of intersection it will be ab’—a'b aBp—ba aB —ab ad —ch Shit laletigl eo Weep te Be. fm a? ae ee the latter of these expressions is quite devoid of all symmetry in regard to its letters, and the larger the number of lines about which we reason, the more confused will such a mode of expressing them render the result. In the first and third mode, it is sufticient to remember that the letter a, under all its forms, represents the tangent of an angle, and that the letter >, in every form, always represents a particular ordinate: with this principle in our mind, we can see at a glance, however numerous the lines introduced, to what property of them each individual letter refers; whereas in the last method, we must, in order to discover the meaning of any letter, refer back for each individual one, to the original trans- lation into algebraic language. The third plan will suffice, where only a few different lines are concerned, but its application is limited by the smallness of the number of different alphabets we can command. The second method may be defended on the ground, that the tangents are denoted by one class of letters; Vol. Il, Part Il. 3A 364 Mr. BappacE on the Influence of Signs namely, Italics, whilst the Greek letters are reserved for lines ; perhaps it might still be improved by interchanging these signi- fications of the two alphabets. Before I pass on to the consideration of the second species of sym- metry, I shall select from the Arithmetica Universalis, an example, in which the choice of the letters employed seems to have been made without any rule; and shall subjoin to it, the same problem expressed in a language consistent with the views I am illustrating. This course will render more apparent the advantages of attending even to the letters which we select to represent the quantities. “The velocities of two moving bodies 4 and B being given, and also their distance, and the difference of the times of the com- mencement of their motion, to determine the point in which they will meet. Let 4 have such a velocity that it will pass over the space ¢ in the time /; and Jet B have such that it will pass over the space d in the time g, and let the interval between the two bodies be e, and that of the times when they begin to move be h. CaselI. Then if both move in the same direction, and if 4 is farther distant from the point of meeting, call that distance x, from this take away the interval e, and there will remain x-e for the distance of B, from the same point. And since 4 passes over the space c in the time f, the time in which it will pass over the space » will be“. And so also, since B passes over the space d in the time g, the time in which it will pass over the space z-e, will be oe Now since the difference of these is supposed to be x, in order that they may become equal, add / to the smaller time; namely, fe. to the time ae (if B moved first) and it will become in Mathematical Reasoning. 365 fey Sage c d and by reduction y = C08 +edh _ getdh “Ser =are Aue Chie > but if 4 began to move first, add h to the time ane and it will become Be 7 Be +h _J2 F d Cc and by reduction apg a le cdh —— cg —df . Case II. If the bodies move in opposite directions, and if x is, as before, the distance of the body 4 from the poimt of meeting, then e — x will be the distance of B from the same point, and 2 the time in which 4 will pass over the distance x; and 8" will be the time in which B will perform the distance e — x. To the less of these times add the difference h, namely, to if B first began to move, and thus we shall have 5 LS Be “ c d and by reduction _ ege —cdh "eg + dfy If 4 began to move first, add h to the time Sass, and it will become eg-st,, Sf d c’ SAaQ 366 Mr. BappaceE on the Influence of Signs and by reduction gos CSBch cdh eg +df ° This same question with the following data may be solved in nearly the same way, v = velocity per second of 4, s = the space one (A) is distant from the other B, t = the time mm seconds one (B) starts before the other. Since the velocities are both positive, the bodies move in the same direction, and x being the distance of the point where they meet from the place of 4, the number of seconds which 4 will take to pass over it, will be found from the ratio SY Py eS v’ the distance of the other body from the same point will be x —s, and the time it takes to arrive will be <>}, but the other body began to move ¢ seconds before B, therefore ¢ added to the 5 : . rT—s time of its motion must equal Saree x 2s ~+t=—--—-: vD ie) o hence s fe vit t+—=a([--~), ° v v Vv and s ees t ; a ag s + tv He oe o—v vy ov If A move in an opposite direction, its velocity must be accounted negative; hence in that case v' becomes — v’, and in Mathematical Reasoning. 367 s — tv’ v+v- If A begin to move after B, the time ¢ must be made negative, and then these two cases become s — tv’ = i= ? and s — tv’ merorwe’ the former of these referring to the case of the bodies moving in the same direction, and the latter to that of their direction being opposite. There are some restrictions which ought to be noticed, if the velocities of the two bodies are equal, or v = » the first and third cases show that the bodies can never meet. To this there : : a ; ) is however an exception, if v =», and also s =/v’, then = an indefinite expression and « may have any value: the signi- fication of this is that both bodies move in the same direction and with equal velocities, since v' =», and that the hindermost B, which starts t seconds before the other, is situated at such a distance s from it, that it arrives at the point where the other is, exactly as it begins to move; this appears from the equation s=¢v’, it is therefore obvious that the two bodies will be at the same point at every part of their progress and for every value of x. Whenever x is negative they can never arrive at the same point in the di- rection in which they move. If however we conceive that they had been moving at the same rate prior to the point of time at which we consider them, the negative value assigned to « marks a point through which they both passed at the same moment. These two modes of translating the same question into Algebra, and of re-translating the result into ordinary language, 368 Mr. BapsaceE on the Influence of Signs give rise to several observations. The assumption of v to re- present the velocity of one body per second, instead of the plan pursued in the first solution, was productive of two advan- tages: first, it substituted one letter instead of two; and secondly, it is so usual in all mechanical problems to make that letter denote velocity, that it is in such cases associated with it in the mind. The next assumption of v' for the velocity of the other body, possessed both these advantages, and tended to make the result more apparently symmetrical in case it was susceptible of that species of arrangement. The selection of the letters s and t, to represent space and. time, was adopted with a similar view of making the signs recal the thing signified. In pure analysis there is but little room for taking advantage of this species of connection, but in all the mixed questions to which it is applied, it may be extensively employed. The general principle is, that either the initial or some prominent letter should be selected from the word which denotes the thing we wish to represent. The beneficial effect of this arrangement is felt a little in the first stage of the solution; it has no influence on the second; but in the last stage it saves consider- able trouble by obviating the necessity of constantly referring back to enquire what particular letters represent. Tn the first solution, there are in fact two distinct cases, in which the reasoning is repeated from the beginning to the end; and each of these cases has two sub-species: so that there are, in fact, four cases treated of in the Arithmetica Universalis, This defect has been remedied in the second solution, where it has been shown that the four cases are included in one formula, to which it is only necessary to give the proper interpretation, and every circumstance connected with the problem is brought to light: Two causes seem to have contributed to produce this separation into cases: in the first place, the extreme generality of the language of Algebra may in Mathematical Reasoning’. 369 not have been sufliciently noticed: the ancients were accustomed to divide their problems into cases, and the habit of treating these separately, may have produced the same cautious treatment of a question when resolved by methods of a far more general and com- prehensive nature: such indeed would naturally be the case, until the degree of generalization introduced by the new method, was fully ascertained. In some instances, the action of another cause may be traced, and one that is more easily removed than that which arises from a want of confidence in the method employed: it may be referred to the imperfect manner, in which the very first elements of the application of Algebra te Geometry and to Mechanics, have been communicated to us: im order to explain clearly the result, it is quite indispensible that we should be perfectly familiar with the principles which regulate these: without them, we may frequently put the question into an algebraic form, because we view it only in one light; and from the nature of the language made use of, that one, whichever it may be, virtually embodies the whole; but when we require to retranslate our conclusions, as they contain every possible case, we must of course be able to translate each indi- vidual. To display in a more prominent point of view, the reason why there are so few failures in expressing the conditions of a problem in Algebra, and so many in giving the full account of all which the solution informs us, let us imagine a problem pro- posed that admits of ten cases; if we are capable of translating any one of these into an equation, such is the comprehensive nature of Algebraic language, that though they are all contained in that single expression, we may be ignorant of the manner of treating nine cases, and competent to manage one of them. If this happen- ed, we might succeed in resolving the equation, and discovering all its roots, which would answer all the ten cases; yet it is scarcely probable, with such moderate knowledge, we should succeed in explaining in common Janguage, the meaning of many of them. 370 Mr. Baspace on the Influence of Signs The advantage of selecting in our signs, those which have some resemblance to, or which from some circumstance are as- sociated in the mind with the thing signified, has scarcely been stated with sufficient force: the fatigue, from which such an ar- rangement saves the reader, is very advantageous to the more com- plete devotion of his attention to the subject examined ; and the more complicated the subject, the more numerous the symbols and the less their arrangement is susceptible of symmetry, the more indispensible will such a system be found. This rule is by no means confined to the choice of the letters which represent quantity, but is meant to extend, when it is possible, to cases where new arbitrary signs are invented to denote operations. In the formation of some of the most common algebraic signs, this maxim has been attended to ; but although in many individual instances it has been admitted, it is still desirable that it should be recognised as a general principle. The sign of equality was obviously adopted from the circumstance of the same relation existing between its two parts, as that which it indicates between the two quantities which it separates, and the propriety of this selection has confirmed its use, although Girard employed = to denote difference, and Descartes used «< to represent equality. In the two signs representing greater and less than a>b and b + &e. “id Lb Tr Secondly, the equation dV 27h Paes =F +1V must be proved. Thirdly, by the application of this equation, the equation as. £4. oc ey FL Aan’ .y = Tan == ae a + &e.. must be demonstrated. Fourthly, the form of ¥, supposed a rational function of x, J/1i—n. cos, and ./1—x”°.sin », must be found. Fifthly, it must be shewn that y, if a rational function of the same quantities, can be resolved into such a series, and only into one. Lastly, by substituting this in the equation above, the values of U, U, U®, &c. are to be found. The theory, after this poimt, presents no. difficulties which it is necessary to consider here. G. B. AIRY. Trinity CoLLece, April 29, 1826. XXII. On ‘the Classification of Crystalline Combina- tions, and the Canons by which their Laws of Derivation may be investigated. By tHe Rev. W. WHEWELL, M.A. F.R.S. FELLOW AND TUTOR OF TRINITY COLLEGE, AND SECRETARY OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. [Read Nov. 13, 1826.] INTRODUCTION. Iv is possible so to classify crystallme forms, and so to consider them with respect to certam fundamental forms, that our reasonings, with réspect to the laws by which their planes are determined, shall be greatly simplified and facilitated. For this purpose all crystalline forms are supposed to be de- Chespisesaen rived from right pyramids as fundamental forms; and they are classified into four systems, according to the fundamental form. 1. The Rhombohedral, in which forms are derived from a pyramid, having for its base an equilateral triangle. 2. The Square-Pyramidal, in which forms are derived from a pyramid, having for its base a square. 3. The Oblong-Pyramidal, in which forms are derived from a pyramid, having for its base a rhombus. 4. The Octahedral, in which forms are derived from a re- gular octahedron. In all cases all the planes are supposed to be formed which belong to the symmetry of the figure. The Laws of crystalline derivation are two; the first, or Laws of De- law of right derivatives; and the second, or law of scalene de- Poti na riwatives. First. Second. Notation. > 392 Mr. WuHEWELL on the Classification By the law of right derivatives a form is deduced by retaining the base, and changing the axis in any ratio (p : 1). The number p is called the index of right derivation, or the first index. By the law of scalene derivatives a form is deduced as follows. A plane is made to pass through one angle of the base, parallel to the slant side passing through the next angle, and is determined by the manner in which it cuts the diagonal, or perpendicular drawn in the base, through the next angle. In the square and oblong-pyramidal systems the plane would intercept, in the diagonal produced, a line which is in a given ratio (m : 1) to the semi-diagonal. In the rhombohedral system it is in such a position, that it would intercept in the perpendicular produced a line which is in a given ratio (m : 1) to the perpendicular. Thus in the square and oblong-pyramidal system (Fig. 2.) let MC'=m . MC, and let C'V’ be parallel to CV. The figure bounded by all such planes as BC’V’ will be the one derived by the second law from BCI. In the rhombohedral system let CM be the perpendicular of the triangle (Fig. 1.). Let MC’=m.MC, and C'V’ parallel to CV. The figure bounded by all such planes as BC’V’’ is derived by the second law from BCY. In the octahedral system the process is the same as in the square pyramidal, with the understanding, that any one of the lines drawn from the center to the angles may be made the axis. In all these cases the number m is the index of scalene de-. rivation, and the form is said to be derived according to m. By means of these two laws of derivation may be obtaimed any form or face whatever, of which these respective systems are susceptible. Let O, P, Q, R represent the fundamental form in the octa- of Crystalline Combinations. 393 hedral, oblong-pyramidal, square-pyramidal, and rhombohedral systems respectively. Let pO, pP, pQ, pR, represent forms derived by the first law from each of these forms, according to the index p. Let pOm, pPm, pQm, pRm, represent forms derived from pO, pP, pQ, pR, by the second law, according to the index m. In the rhombohedral system, if R be turned round its axis through half a circumference, it is said to be in a éransverse position, and is designated by R’. And the derivatives of R’ are marked pR’, pR’m, &c. In the oblong-pyramidal system, if one diagonal (WC, Fig. 2.) be called the principal diagonal, the other (7B) is called the transverse diagonal. And when the derivation is made on the transverse diagonal, it is marked with P’, as pP’m. ‘ 2m .. ; It is easy to see that mR at when m=0, is an equilateral six-sided pyramid, with the same axis as the fundamental form. 1 1 : In the same manner mO = and mQ >? When m=0 are equilateral four-sided pyramids, in a diagonal position to O and Q: mP — when m=0, is a prism with its axis in the direction of the principal diagonal. These. forms are, for the sake of abbreviation, marked thus, Rr, Or, Qr, Pr Sometimes one half of the number of faces resulting from any law is omitted, according to some principle of alternate selection. In this case the former is said to be hemithedral, and is indicated by writing h before its symbol. Hemihedral forms are sometimes plagthedral, arith the letters rand | are used to indicate the obliquity to the right or the left, as will be shewn. : Theorems of Crystal- lometry. 394 Mr. WuHeEwELL on the Classification It is easily seen that pR is a triangular pyramid, and, when its faces are repeated, a rhombohedron; that pRm is a six-sided pyramid, the hexagonal base of which has alternately equal angles (V'AUBSCT, Fig. 1.). Also pP and pQ are four-sided pyramids: pPmisa four-sided pyramid (V'4'BC’, Fig. 2.), and pQm an eight- sided one, with the angles of its base alternately equal (/’’BSC, &ce. Fig. 2.). The derivations of O are less obvious, but need not here be explained. In the combinations of different forms, the intersections of the planes bounding the figure are called the Edges of Combination. These lines have various parallelisms and relations, which, with other properties of the forms, are enunciated ‘in the following theorems : A. Iftwo forms are derived by the first law according to different indices, their edges of combination are horizontal (the axis being vertical). B. If two forms are derived from the same form by the second law, their edges of combination are parallel to the slant edge of that form. C. In the following series, each form truncates the edges of the following one, making parallel edges of combination ; ane Wa ee OR Ry eRe, Jock Roo Ry SR, PRG ER aR, eR ey, 1Qr,3Q, Qr, Q, 2Qr, 2Q, 4Qr, Px) sPorieP’rs Wee D. If a be the axis of O, P, Q, R, the axis of pO, pP, pQ, pR will be pa. E. If 6 be the principal semi-diagonal of the base of O, P, Q, pOm, pPm, pQm will have for their axes pma, and mb for the intercepted part of the principal semi-diagonal. F. The bases of pOm, pQm, are octagons with alternately equal of Crystalline Combinations. 395 angles. The radii of the circles passing through the original 4 and the derived angles (MB and MS, Fig. 2.), are 6 and ss = 2 x The axis of pRm is ch pa. H. If b be the radius of the circle circumscribing the base of R, the base of pRm is a hexagon with alternately equal angles, and the radii of the circle passing through the original and the derived angles (OB and OS, Fig. 1.), are 6 and a I, The acuter and obtuser edges of pRm make with the axis angles of which the tangents are respectively 2 b 2 b 3m—1 pa “"° 3m-+1 pa’ The demonstrations of the preceding theorems will be given in another place. To complete the subject, it will also be requisite to give methods of transforming the symbols, which are employed in other systems of notation hitherto proposed, (those of Hauy, Weiss, Mohs, &c.) into this system. This will be done by means of certain formula, which we may call formule of trans- formation. At present our object is to shew how, by means of the the- orems above enunciated, with the additional assistance of a few subsidiary theorems derived from them, we may reason from the parallelisms and other properties of crystalline forms, so as to obtain the indices which belong to those forms; and the rules for drawing these inferences may be called Canons or DeRtva- TION. Our object, therefore, will be to divide the combinations of crystalline faces into classes, so that each class may offer some peculiarity in the position of its edges and faces, by which we may recognize the laws from which they result. These peculiarities Vol. WI. Part I. $3 E 396 Mr. Wuewe tt on the Classification of combination will be exemplified in the figures referred to; and we shall also give instances where these combinations, either alone or united with others, have been observed among minerals. For this latter purpose, the references will be made to Mohs’s Crystal- lography, the name of the mineral and the number of the figure being mentioned. Hence, in the following tables, the principal columns are those two which, for the sake of convenience in printing, are put the last. The one containing the symbol of the combination, and the other the resulting properties. The columns which are placed before these, contain the Class and Number of the Combination ot which we speak, the examples of its occurrence with their symbols, .and such other observations as are requisite for the purpose of distinguishing among resembling combinations, or any other circumstances proper to be remarked. As the words Edges of Combinations occur very frequently. the abbreviation E. C. is used to represent them. The preceding Theorems are referred to by means of the letters (4), (B), &c. by which they are here distinguished. The Subsidiary Theorems are marked with the class and number of the combination to which they belong. SUBSIDIARY THEOREMS. Rhombohedral System. Crass II. Comb. 4. Let pR’ and gRm be the forms. Then, in order that pR’ may truncate the acute terminal edges of gRm, we must have the truncating pyramid pR’, equivalent to one which has its axis (OV’, Fig. 1.) equal to that of the truncated re of Crystalline Combinations. 397 pyramid gRm, and which has the edges of its base passing through the points 4, B, C, from which the derivation of gRm proceeds. Or, we may reduce the axis and the linear dimensions of the trun- cating pyramid to one half of these dimensions, and its form will remain the same. Hence, its base being that of the fundamental pyramid (in a transverse position, R’), its axis will be half that of 3m—-1 the derived pyramid gRm. Now the axis of gRm is qa; ‘ pp Bias hatred 3 d 2BM—1 . pa=5——— 9a, and p=——— This, .if,.m— 3, p=2¢; if m=\, p=—2 93 if m=2, p= 29. II. 5. Let pR and qRm be the forms. In order that pR may truncate obtuse edges of gRm, the pyramid pR must be equi- valent to one which has the edges of its base passing through the _ points S, and its axis equal to that of the truncated pyramid OV’ (Fig. 1.). Now, diminish the linear dimensions of pR in the ratio b(3m—a) | of OS : OK, or of : : id (see H). Its base then becomes am+1 2 3m+1 2(3m—1) ae that of the fundamental form R, and its axis becomes 3m-—1 the axis of gRm. But the axis of the pyramid gRm is ga 3m+1 4 3m+1 A m=2,p=iq; if m=3, p= 49; if m=5, p=4g. Il. 6. Let pR and gRm be the forms. In order that the more acute terminal edges of the pyramid gRm, may coincide with the edges of the rhombohedron pR, the axes of the two forms must be equal ; q. Thus, if (see G). Therefore, pa= ga, and p= 3m—1 32 wet ‘ 3m—1 pPa=—>— 94, and p= 3E2 398. Mr. Wuewe Lt on the Classification II. 7. Let pR’ and qgRm be the forms. In order that the obtuse edges of the pyramid gRm may replace the edges of the rhombohedron, we must have the axes coincident, and the base of a rhombohedron equivalent to pR’ must have its angles at those obtuse edges (at S, Fig. 1.). Or, increasing the linear dimen- sions of the rhombohedron in the ratio of OS to OC, or (see #). m 1 3m-1 : 3m+1, its axis becomes © t oF ; into the axis of gRm. The 3m-—1 H 3m+1 _ 3m+i 5 qa. ence, pa= aves qa, p= Moy Waa q: axis of gRm is IV. 5. Let pRm and qR’'n be the forms. If we suppose the angle which the acute terminal edge of pRm makes with the axis to be the same as the angle which the obtuse terminal edge of gk’n makes, and the forms to be in a transverse position, it is manifest, that the faces meeting at the last-mentioned edge will truncate the faces meeting at the former edge, and will make parallel edges of Sac aan Now, the tangents of these angles Sieh 2 e ——— —_ and 3m+1 pa an LS (see I): and these will be equal if (3m + 1)p=(3n—-1) q. Hence, we have these corresponding values of these indices : BS m=3, 5, mi ee m=2, 3, 4 a we g=2) n=2, 3, 4)) q=1) n=5, 7, 5 q=5) n=2))° V.1. Let pRm, gRr be the forms. In the isosceles Sess gRr, the tangent of the angle which the terminal edge makes with the axis, is : OF (Fig. 1.). Therefore, in order that the obtuse terminal edges of pRm may truncate these, we must have (see /) 1 OB By Obs .: OV) Gap OK? we ae of Crystalline Combinations. 399 V.2. Let pRm, qRr be the forms. In order that the acute terminal edges of pRm may truncate the edges of gRr, we must have 1 OB 2 OB 7 OF Geen oy ee eee Square Pyramidal System. Crass II. Comb. 2. In order that the faces of pQr, replacing the summit of gqQn, may be rhombs, we must have ¢Qx coincident with an eight-sided pyramid, derived from pQr (see B). Now, if this deduction’ from pQr be according to m, the form will be 3 —1 the same as one derived from pQ, thus, —_ pQ at. Hence, ' : ies l xs this form and qgQz will coincide if om =n, and _ a = q: 2 whence mm —_ (n—1) qQr, Qn. II. 6. In order that the faces of the four-sided pyramid pQ may truncate the scalene terminal edges of the pyramid qQn, they must make the same angle with the axis which those edges do. But the tangent of the angle which the faces of the pyramid pQ make with the axis is manifestly 7 =n, and p=q(n—1), and the combination is b b nbv2 2np SSS SS ae F —S>—_-_—__——— = — 23 /2.pa Se J/2.pa n+i.qa’ Era And the combination is pQ, 2uF Qn. II. 7. In order that the faces of the four-sided pyramid pQr may truncate the principal terminal edges of the figure gQn, they must make the same angle with the axis which those edges make. But the angle which the faces of the first pyramid make, has for its tangent aa and the angle which the edges of gQn make has : b ity Pebec for its tangent Bails ena And the combination is gnQr, qQn. 400 Mr. WuHewe.t on Crystalline Combinations. Il. 8. In order that the faces of the eight-sided pyramid ¢Qx, meeting at the scalene edge, may replace the terminal edge of the pyramid pQr, the angles which these edges make with the axis must be equal; and equating their tangents, we have (see F) bY2 mBbV2, _ n+l pa ~ n+l .qa’ dip cahiae II. 9. In order that the faces of the eight-sided pyramid, meeting at the principal edge, may replace the terminal edge of the pyramid pQ, the angles which these two edges make with the axis must be equal; and, equating their tangents, we have eS Seg et, PIR IL III. 4. In order that in the eight-sided pyramid pQm, the principal terminal edges may be replaced by the faces, in pairs, of the pyramid gQn, meeting at its principal terminal edges, we must havé the angles which the principal terminal edges make with the axis, equal in the two pyramids. Hence, equating the iiigents,) aedeent tin Syn angents, oma qna’” ale Oblong Pyramidal System. Crass II. Comb. 3,4. When the terminal edges of the py- ramid pP are replaced by the faces, in pairs, of gPnz, we must have the proportion of the axes a and & the same in the two forms; .. by £, p = qn. III. 3. When we have the prism qP’r, cutting pP and pPr so as to make the figure APB’Q, (Fig. 3.) it is required that this figure should be a rhomb. For this purpose it is requisite that the diagonals of the figure 4B’ and PQ should bisect each other in V; .. AB’ =2 AN, and=2B’N, and BM’ =2BN'. Hence AM 2s Ss Now the prism which passes through the line MB! | 2 0N UR § é B'G is pP’r. Hence, the prism which passes through 4B’ is Lp P’r. TABLES OF THE COMBINATIONS OF CRYSTALS WITH THE PARALLELISMS RESULTING, AND THE CANONS OF DERIVATION. Tue combinations of faces which can occur in crystals, are, in the following pages, tabulated and classified according to the parallelisms and other relations of edges which they exhibit. And these relations thus afford the means of recognizing the class to which the forms belong, and consequently the laws by which the faces are derived. The fourth column contains the symbols for these combinations, according to the notation which has been explained. The last column, which is on the opposite page, contains the cor- responding properties; and these are to be looked for in a given crystal, in order to determine its class. These combinations are exhibited in the figures in the Plates, where they are numbered according to the order of the classes in the third column : the letters in the figures refer to the second column. In the second column are examples, in particular minerals, of these combinations; with references to the figures by which they are illustrated in Professor Mohs’s Treatise on Mineralogy; and the nomenclature of that work is adopted in designating these minerals. The frst column contains such other remarks as may be useful for distinguishing and discriminating different forms. Besides the theorems (4), (B), &c. the subsidiary theorems are referred to by the letters (S. 7’) 402 Mr. WHEWELL on the Classification RHOMBOHEDRAL SYSTEM. Example and Figure. Class. Combination. Class I. |pR,gR & pR,gR’ Rh’. Kouphone-spar. £R’,R| No. 1. oR R’ Mohs, Vol. II. 120. x P o"-'R’, "R pR, 2pR’ or 3pR’, pR Rh’. Lime-haloide R,4R] No. 2. oR, ot R II. 115. P m generally pR, gR Rh’. Alum-haloide OR, R| No. 3. OR, gR Loomer, This prism is distinguish- | Rh’. Lime-haloide R, oR | No. 4. pR, oR able from Rr in III. 4. TS 4 eee oc by the position of its faces. Class IT. pR,qRm and pR’, gRm Rh’. Lime-haloide R, R3 | No. 1. pR, pRm If. 116. “Pr OR, R3 | No. 2. OR, pRm Oo © The figure of the faces of | Rh’. Lime-haloide ©R,R5| No. 3. the prism shews it to be ®R, cy andnot © Rr, in which the te- tragons are rhombs, see V.4. No. 4 Rh’. Ruby-blende+ R’, i R3 LOR 3 generally te: gk',qRm 4 of Crystalline Combinations. 403 RHOMBOHEDRAL SYSTEM. Relations of Edges, &c. The forms are in a transverse position. The first truncates the edges of the second (C). The Edges of Combination are parallel to one another; to the terminal edges of the ‘acute rhombohedron; to the inclined diagonals of the obtuse one (C). The forms are in a parallel position. E. C. are horizontal (A). A face perpendicular to the axis. E. C. horizontal. The second form is a regular six-sided prism. E. C, of its alternate faces with upper faces of pR, are horizontal (A); and of remaining alternate faces, with lower faces of pR. remaining E. C. are inclined. cos. hor. edge = 2 cos. incl. edge. E. C. parallel to edges of rhombohedron pR (B); and to lateral edges of pyramid p Rm. Faces of rhombohedron are rhombs contiguous to the apex. E. C. horizontal. The face (OR) perpendicular to axis, is a hexagon with alternate angles equal. The faces of the prism ©R are irregular tetragons divisible by a horizontal line into two isosceles triangles of unequal heights. These haye their more acute angles alternately upwards and downwards. The heights of the two isosceles triangles are as m—1:m-+1. The forms are in a transverse position (being derived from R’ and R). Faces of the rhombohedron replace acute terminal edges of the pyramid. (S. 7’.) E. C. parallel to each other, oral aed» he .. to acute terminal edges of pyramid, ..+eee.. to inclined diagonals of rhombohedron. The lateral edges (e) of the pyramid are inclined in a direction opposite to the edges of intersection (f) of the pyramid and rhombohedron. Vol. II, Part IL. 3F 404 Mr. WHEWELL. on the Classification RHOMBOHEDRAL SYSTEM. Example and Figure. Class. Combination. Class IT. pR,gRm No. 5. 7qR,qRe Rh’. Lime-haloide 4R,R5 Mohs, Vol. 11.116. m y generally 3m+1 < qR, gRm No. 6. &qR, qR2 Rh'. Ruby-blende R,{R3 LT At26se Peeve generally 3m—1 gR,qgRm 2 generally 3Sm+1 gk’, Ge 2 Class IIT. pR, qRr If the angles of a regular prism with a pyram’. term™. be truncated by rhombs, the pyramid and truncating py- ramid are pRr, pR, or it may be pRr,pR’, or both, | Rh’. Quartz R, Rr No. 1. pR, pRr IT. 145. $s -2 pR,pRr, oRr Rh’. Corundum R, 2Rr | No. 2. pR, 2pRr 1 sel ell oh Nc If alternate edges of a re- gular pyramid be truncated, truncating rhombohedron and pyramid are pR, 2pRr. of Crystalline Combinations. 405 RHOMBOHEDRAL SYSTEM. Relations of Edges, &c. ‘The forms are in a parallel position. The faces of the rhombohedron replace obtuse terminal edges of pyramid. (S. T.) E. C. parallel to each other, a aca _.... to obtuse terminal edges of pyramid, as CUE to inclined diagonals of rhombohedron. The lateral edges of the pyramid (e) are inclined in the same direction as the edges of intersection (g) of the pyramid and rhombohedron. Phe forms are in a parallel position. The acute terminal edges of the pyramid in pairs replace the edges of the rhombo- hedron. (S. T.) E. C. parallel to each other, eA obaboviuat to edges of rhombohedron, Soe clad Sse to obtuse terminal edges of pyramid. The forms are in a transverse position. The obtuse terminal edges of the pyramid in pairs replace the edges of the rhom- bohedron. (S. T.) E. C. parallel to each other, «eee... to edges of rhombohedron, hoa to obtuse terminal edges of pyramid. The forms are co-ordinate. The faces of the pyramid in pairs replace the edges of the rhombohedron. E. C. parallel to the edges of the rhombohedron, .+s+e++-. to the alternate terminal edges of the pyramid (B). If we have also the prism © Rr, the faces pR retain their rhombic form. The faces of the rhombohedron replace the alternate edges of the pyramid. The edges replaced are alternate at the two apices. E. C. parallel to alternate terminal edges of pyramid, vseeee oe to inclined diagonals of rhombohedron. Meee eee eee 406 Mr. WHEWELL on the Classification RHOMBOHEDRAL SYSTEM. Remarks. Example and Figure. Class. Combination. Class III. pR, qRr OR, Rr} No. 3. OR, qRr or This position of the faces | Rh’. Emerald-mal.2R,®Rr| No. 4. pR, @Rr of the prism, easily distin- | Mohs, Vol. II. 118.7 s guishable from I. 4. shews > that the prism is © Rr, and not «R. The form of the faces of aR, pRr the prism distinguishes it from Rr, as in VI. 2. Rh’. Fluor-haloide ® R, eRr oR, oRr II.149. e M pRm, gRu If edges of combination of | Rh’. Lime-haloide, R3,R5| No. 1. pRm, pRn two pyramids are parallel to lateral edges, the first index is the same for both. TE GS 3 sy If edges of combination are horizontal, the second index is the same for both pyramids. Rh’, Lime-haloide, +R3,R3| No. 2. pRm, gRm TE 190.4, fb # R3, oR No. 3. pRm, ORn Fito Rh’. Fluor-haloide R3, oRS No. 4. pRm, 2 Rm II. 148. > ¢ of Crystalline Combinations. 407 RHOMBOHEDRAL SYSTEM. Relations of Edges, &c. A horizontal face, which is a regular hexagon. E. C. horizontal. A regular six-sided prism terminated by a rhombohedron. Faces of prism replace lateral edges of rhombohedron. E. C, are parallel to each other, and to lateral edges of rhombohedron. Faces of prism are rhombs with two sides vertical. The lateral angles of the pyramid are truncated by the faces of a prism. E. ©. are parallel to each other. Faces of prism are rhombs with a diagonal vertical. An equiangular twelve-sided prism. Forms are co-ordinate six-sided pyramids. Lateral edges of obtuse pyramid replaced by faces of acute pyramid. E. C. are parallel to lateral edges of both pyramids (B). E. C. are horizontal (A). Two pyramids meeting in a plane perpendicular to the axis. Face perpendicular to axis is a hexagon with alternate angles equal. E. C. horizontal. A twelve-sided prism, terminated by a six-sided pyramid. E. C. of pyramid with alternate pairs of faces of prism, are horizontal. The pairs of faces with which this is the case, are alternate at the two apices. The remaining edges of combination are inclined. Vol. 1. Part Il. 3G 408 Mr. WHEWELL on the Classification RHOMBOHEDRAL SYSTEM. Remarks. Example and Figure. Class. Combination. Class _IV.| pRm, gR'n Rh’. Lime-haloide R3,2R’2| No. 5. pR’5, 2pR3 pe pR3, 2pR 2 2pR' 2, 2pR5 generally, if (3m-+ 1) p =(32 -,1), 9. Class V. pRm, qRr pR3, 5pRr pR5, 8pRr generally, if (3 m+1)p=2q Rh', Iron-ore $R5, 2Rr No. 1. g R3,4Rr{ No. 2 pR3,4Rr if n pR5,7Rr | generally, if (3m—1)p=2q ORm, qRr pRm, oRr Rh’. Ruby-blende R3,®R} No. 4. Mohs, Vol. II. 126. x This position of the edges shews the prism not to be oR, as in II. 3. Rh’. Corundum 2 Rr, 4Rr II. 122. + b of Crystalline Combinations. 409 RHOMBOHEDRAL SYSTEM. Relations of Edges, &c. The acute terminal edges of the more acute pyramid are replaced by the obtuse terminal edges of the more obtuse pyramid in pairs. E. C. parallel to each other and to these terminal edges. (S. 7.) The alternate terminal edges of the isosceles pyramid are replaced by the faces, in pairs, of the scalene pyramid, meeting at its obtuse terminal edges. E. C. are parallel to each other and to these terminal edges. (S. T.) The alternate terminal edges of the isosceles pyramid are replaced by the faces, in pairs, of the scalene pyramid, meeting at its acute edges. E. C. are parallel to each other and to those terminal edges. _ (S. T.) See III. ea Face perpendicular to axis. See IV. 3.) E. C. horizontal. The lateral edges of the scalene pyramid are replaced by the faces of the prism. E. C. are parallel to each other and to these lateral edges. A pyramid with its summit replaced by a more obtuse pyramid. E. C. horizontal. 22 een A regular prism terminated by a regular pyramid. E, C. horizontal. 410 Mr. WHEWELL on the Classification RHOMBOHEDRAL SYSTEM. Dirhombohedral Combinations are those which contain corresponding derivatives of R and R’.. > Remarks. Example and Figure. Combinations. Class VII. | Dirhombohedral. This pyramid is different Rh’, Emerald R, R’~ No. 1. pR, pR’ in position from pRr. Mohs II. 150. s No. 2. | pR, pR’, pRr Rh’. Emerald R3, R: No.'3. pRm, pR'm » ThPr50) 9a ee TS Hemirhombohedral Combinations are those in which half the faces of a rhombohedral combination are omitted in an alternate manner. These are also Plagihedral, if the remaining faces are not sym- metrical with respect to right and lett. Simple lass VIII. Cees Tt Plagihedral. When plagihedral faces Rb). quartz. are thus distributed, the for- Fe R,-{R?, RS, RY, Rs} res : i 2 {pRm, pRn} mula is —, or =. : r 1 p s Bil yp Lae IT. 146. When plagihedral faces Rh’. quartz. | No. 2. are es ie the for- Rr, R, ret Ri, Rs, RYR5} r mula is =, OF >. r ] ,|pRm, pRn} pins Tee ey reo. IJ. 147. of Crystalline Combinations. 41] RHOMBOHEDRAL SYSTEM. Relations of Edges, &c. The form is an isosceles six-sided pyramid. If a=edge of rhombohedron, cos. 8 = J (cos. a— 2). See III. 1. The form is an isosceles six-sided pyramid; its edges at the summit replaced by the faces of another six-sided pyramid (pRr), which are rhombs. See IV. The form is a scalene twelve-sided pyramid. E. C. are parallel to the lines joining the apices with the bisections of the lateral edges of each pyramid. This form is called a Dirhombohedron. eS In this class the plagihedral faces are turned the same way at both apices: that is, _ at both to the right, or at both to the left. The plagihedral faces appear on the alternate angles, and alternately at the two apices. Their edges of combination are parallel to each other and to the lateral edges of the pyramid pR (A). See IV. 1. The plagihedral faces appear on the alternate angles, and simultaneously at the two apices. Their edges of combination are parallel to each other: those at one apex to the lateral edges of the pyramid pR, and those at the other apex to the lateral edge of pR’. (SS SS rR A ER ere 7 412 Mr. WHEWELL on the Classification RHOMBOHEDRAL SYSTEM. Remarks. Example and Figure. Class. Combination. Class IX. Double When plagihedral faces Plagihedral. are thus distributed, the for- ; No. 1 mula ist . | ipRm, pRn} No. 2. 7 Be = {pRm, pRn} When plagihedral faces Rh!. Fluor-haloide. No. 3. are thus distributed, the for- Rr, RR, = {RE RLY = {pRm, pRn} mula is ria te - b Mohs II. 149. SQUARE-PYRAMIDAL COMBINATIONS. Remarks. Example and Figure. Class. Combination. Class I. |pQ,gQ& pQ,qQr No. 1. pQ, 2pQr Square Pyr’. Zircon Qr, Q or pQr, pQ Mohs, Vol. II. 99. ¢ P Square Pyr'. Garnet Q, 2Q PQ, 2"pQ II. 96. ¢ 6 Sq. Pyr'. Lead-baryt £Q, 0Q pQ, 0Q II. 92. 6 a Sq. Pyr'. Tin-ore Q, ©Q PQ eQ II. 102, p 1 of Crystalline Combinations. 413 RHOMBOHEDRAL SYSTEM. Relations of Edges, &c. In this class the plagihedral faces are turned opposite ways at the two apices: that is, twisted to the right at one apex, and to the left at the other. The plagihedral faces appear on the alternate angles, and alternately at the two apices. The plagihedral faces appear on the alternate angles, and simultaneously at the two apices. The plagihedral faces appear on all the angles, and at each apex. Their Edges of Combination are parallel to each other and to the lateral edges of the pyramids pR, pR’. SQUARE-PYRAMIDAL COMBINATIONS, Relations of Edges, §c. The forms are in a diagonal position. The preceding form in these combinations truncates the edges of the succeeding. The Edges of Combination are parallel to each other and to the terminal edges of the more acute pyramid. sb eee e tees acces esessesvesese also to the lines bisecting the lateral edges of the more obtuse pyramid. The forms are in a diagonal position. The preceding form replaces the summit of the succeeding. E. C. horizontal (A). E. C. horizontal (A). A face perpendicular to the axis. A regular four-sided prism, terminated by a pyramid in a parallel position. E. C. horizontal (A). 414 Mr. WHEWELL on the Classification SQUARE-PYRAMIDAL COMBINATIONS. Remarks. Class. Combinations. Example and Figure. Class I. Included in 4 by Mohs. Sq. Pyr'. Tin-ore Qr, ©Q PQ, ~Qr Mohs II. 102. s 1 or pQr, ~Q Sq. Pyr. Tin-ore ©Q, ©Qr oQ, Qr T1025 78 g Class IT. PQ gGQn So also or pQr, pQzr Sq. Pyr'. Garnet Q, Q2 Il. 96:. cs Sq. Pyr'. Garnet Q, Q3,Q4| No. 1. PQ, pQn Q2 is marked (P—1)° in II. 96. ¢ s a |MohsII.1. Mohs, being derived from Qr, by 3. Sq. Pyr': Garnet Qr, £Q3| No. 2. P TH. 96.,iulo" tia MohsII. 1. PQr, n—1 a7 n—1qQr, gQn No. 3. 0Q, gQu MohsII.2. Sq. Pyr’. Zircon @Q, Q3 No. 4. ~Q, gQn II. 99. “4 «x |MohsII.3. If we have also qQ, the | Sq. Pyr'. Zircon ©Qr, Q3 No. 5. eQr, gQn sides of the rhombs are II. 99. s 2 |.MohsII.3. parallel to the terminal edges of gQ. of Crystalline Combinations. 415 SQUARE-PYRAMIDAL SYSTEM. Relations of Edges, &c. A regular four-sided prism, terminated by a pyramid in a diagonal position. E. C. parallel to the rhombic section of the pyramid through its diagonal. Faces of pyramid are rhombs, and faces of prism, if they are complete. Faces of prism are rhombs if they do not meet: if they do, faces of pyramid are rhombs. A regular eight-sided prism. The forms are in a parallel position. The faces of the four-sided pyramid replace the apex of the eight-sided one, and are thombs. E. C. parallel to the terminal edges of the four-sided pyramid. (B). The forms are in a diagonal position. ° The faces of the four-sided pyramid replace the apex of the eight-sided one, and are rhombs. E. C. parallel to the terminal edges of the four-sided pyramid pQr (B). {- P ; Qn is the same as a form derived by the law of scalene derivation from pQr, n+1 n—-1 according to i. (S. T.) A face perpendicular to axis. E. C. horizontal, A regular square prism in a parallel position, with an eight-sided pyramid. The faces of the prism appear as rhombs replacing four lateral angles of the pyramid. The form of these rhombs depends on 7. A regular square prism in a diagonal position with an eight-sided pyramid. The faces of the prism appear as rhombs replacing four lateral angles of the pyramid. The form of these rhombs is similar to that of the section of the pyramid qQ through its diagonal, and E, C. parallel to this section. Vol,.M.~Part I, 3H Mr. WHeweE tu on the Classification SQUARE-PYRAMIDAL SYSTEM. Remarks. Example and Figure. Class. Class II. : No. 6. Sq. Pyr'. Titan-ore Mohs, Vol. II. 100. Hobs e $593 7S P+n,(P+n—3)', &c. 2Qr,Q2| No. 7. of Mohs. u z |MohsII.5 P +n, (P+n— 3), &e. 3Qr, Q2]| No.8. of Mohs. ; v» z_ |MobhsII.6. These are included by | Sq. Pyr’. Garnet No. 9. Mohs in the preceding com- II. 96. 2Q, Q2 | MohsII.7. bination, because he derives bce Q 2 from Qr. 4Q, 2Q2 re and 4Q, Q4 rT 2 Class IIT. From this parallelism of |Sq. Pyr. Zircon Q3,Q4,Q5} No.1. the edges of several pyramids IT. 99s" ee thee we learn that they are co- ordinate (i. e. g =p). Sq. Pyr’. Garnet 2Q2,Q2| No. 2. II. 96. e z z& Combination. 3 PQ, pQ2 2pQ, PQS 3 PQ, pQ4 generally 1 = PQ, pQn 2qQr, gQ2 3qQr,qgQ3 4 qgQr,IQ4 generally ngQr, gQn 3qQr, gQ2 4qQr, gQ3 5 qgQr,qQ4 6gQr, gQ5 generally n+1qQr, gQn 29Q,qQ2 3qQ, gQ3 5¢Q, qQ5 generally . ngQ, gQn pPQm, qQn pQm, pQn pQm, qQm of Crystalline Combinations. 417 SQUARE-PYRAMIDAL SYSTEM. Relations of Edges, §c. The forms are in a parallel position. The faces of the four-sided pyramid truncate the scalene edges of the eight-sided pyramid. (S, 7.) The E. C. are parallel to themselves and to the truncated edges. ++eeeeeeeee-- and to the line bisecting the lateral edges of the four-sided py- ramid. : The forms are in a diagonal position. The faces of the four-sided pyramid truncate the principal edges of the eight-sided pyramid. (S. T.) The E. C. are parallel to themselves and to the truncated edges. veoe.++eeeeeee and to_the line bisecting the lateral edges of the four-sided pyramid, The forms are in a diagonal position. The faces of the eight-sided pyramid, adjacent to its scalene edges, appear in pairs in the place of the terminal edges of the four-sided pyramid. (8S, T.) The E. C. are parallel to themselves and to the replaced edges, «ss eeeseeeeeee and to the scalene edges of the eight-sided pyramid, The forms are in a parallel position. The faces of the eight-sided pyramid, adjacent to its principal edges, appear in pairs in the place of the terminal edges of the four-sided pyramid. (S. T.) The E. C. are parallel to themselves and to the replaced edges. seeveeeesseeeee and to the principal edges of the eight-sided pyramid. The forms are co-ordinate pyramids in a parallel position. E. C. parallel to the terminal edges of pQ. The forms are in a parallel position. E. C. horizontal. 3H 2 418 Mr. WHEWELL on the Classification SQUARE-PYRAMIDAL SYSTEM. Example and Figure. Class. Combination. Class III. Sq. Pyr'. Garnet 2Q2,eQ2| No. 3. pQm, ©Qm Mohs, Vol. II.96.e f In Mohs these are (P)* Sq. Pyr', Garnet 2Q2,Q4| No. 4. pQm, gQn and (P+ 1). i. 06. 7. te when pm=qn Thus Q5,3 Qe 2Q3, 4Q} Class IV. | Plagihedral. Class V. | Hemihedral. 3 (2 = pP- 3) Sq. Pyr. Copper pyr. h’Q, h’2Q, FQ, bi Q3 h pQm In Mohs r *3 ee f h’q/ Qn e II. 178. OBLONG-PYRAMIDAL SYSTEM. Remarks. Example and Figure. Class. Combination. Class I. pP, qP Topaz OP, =P, P, »P | No.1. pP, gP so ™m Mohs, Vol. II. 34. of Crystalline Combinations. 419 SQUARE-PYRAMIDAL SYSTEM. Relations of Edges, &c. An eight-sided prism with an eight-sided pyramid. E. C. horizontal. The faces of one eight-sided pyramid replace in pairs the principal terminal edges of the other pyramid. (S. T.) E. C. parallel to the edges replaced, and to each other, Half the number of faces of eight-sided pyramids, taken alternately. The faces may be corresponding ones at the two summits, or alternate ones. i Half the number of faces of four-sided pyramids taken, and alternately at the two summits. (h and h’.) And, of eight-sided pyramids, the faces corresponding to such faces of the four-sided ones. OBLONG-PYRAMIDAL SYSTEM. Relations of Edges, §c. A pyramid with its summit replaced by another pyramid. E. C. horizontal. 420 Mr. WHEWELL on the Classification OBLONG-PYRAMIDAL SYSTEM. Remarks. Example and Figure. Class. Combinations. Class IL. pP, qPm M.ILIV,V.} or pP, gP’m 5 __P2 is (Pr in Mohs, | Ob. Pyr'. Serpentine P, P2 | No. 1. pP, pPm and is ranged in Class IV. | Mohs, Vol. II. 33. p x Ob. P’. Mel.-glance P, P’3 | No. 2. pP, pP’m II. 30. p a N®. 3 and 4 differ’ from No. 3. qnP,gPn 1 and 2 in that the faces| Fig. to No. 1. replacing edges belong to pP in 1 and 2, to qPn ees Jand q’Pn in 3 and 4. No. 4. qnP, qP’n Fig. to No. 2. Class III. | pP, gPr, q'P’r Mohs VI. Ob. P’. Olive-mal. P, Pr} No. 1. pP, pPr 1 ey p o Ob. P'. Iron-ore P, P’r | No. 2. - pP, pP’r IL. 4. © 0p | M. VII. 1: P,Pr,yP’r| No.3. |pP,pPr,ipP'r p-0-~-n M. VII. 2, teas oP, Pr, P’'r| No.4. |@P,pPr, pP’r zo) p |M.VII.3. If the faces of a horizon- tal prism be rhombs, we can determine them. of Crystalline Combinations. 421 OBLONG-PYRAMIDAL SYSTEM. Relations of Edges, &c. The principal terminal edges of the pyramid pPm are replaced by the faces in pairs of pP. E. C. parallel to each other and to the above principal edges. The transverse terminal edges of the pyramid pP’m are replaced by the faces in pairs of pP. E. C. parallel to each other and to the above terminal edges. The principal terminal edges of the pyramid gnP are replaced by the faces in pairs of gPn. E. C., parallel to each other and to the above transverse terminal edges. The transverse terminal edges of the pyramid guP are replaced by the faces in pairs of gP’n. E. C. parallel to those edges. The transverse terminal edges of the pyramid pP are replaced by the faces of the horizontal prism. E. C. parallel to each other and to these terminal edges. The principal terminal edges of the pyramid pP are replaced by the faces of the horizontal prism. E. C. parallel to each other and to these terminal edges. The faces of $pP’r will appear as rhombs replacing the corners at the upper edge of the compound figure pP, pPr. The form appears as a vertical rhombic prism terminated by a pyramid in a diagonal position. All the faces of the figure are rhombs. 422 Mr. WHEWELL on the Classification OBLONG-PYRAMIDAL SYSTEM. Remarks. Class. Combinations. Example and Figure. Class IIT. Pr, P’r OntIg pPr, pP’r Pr, oP'’r o Pr, gP'r x’ y |M.VIII.2.) or oP’r, pPr This is not to be con- founded with square prisms 0Q, ~Q, and 0Q, oQr. oPr, ofr OP, © Pr, a Pr a y Tue Hemihedral forms in this system, are those which contain the faces on one side of the axis only, at each summit. They are designated, as before, by the prefix h. There are also Tetartohedral forms, which exhibit only one out of the four faces given by each law. There are also forms in which the axis of the oblong pyramid is oblique, and these require other modes of investigation. ; The OcraHEpRAt System, including the combinations of figures sym- metrical in all directions (cubes, octahedrons, dodecahedrons, icositessera- hedrons, &c.), requires to be treated in a manner somewhat different from the preceding, and will not be here considered. The application of the preceding classification will be easily seen. In any proposed form, the relations of the faces and edges, being compared with the preceding descriptions, and with the figures here given, or with those referred to in Professor Mohs’s Mineralogy, will shew to what Class and Number the combinations of its faces may be referred. This will give us data for discovering the symbols of its faces, and, by connecting such data, we are to obtain the laws of the derivation. of Crystalline Combinations. 423 OBLONG-PYRAMIDAL SYSTEM. Relations of Edges, &c. The form is a right pyramid with a rectangular base. A horizontal prism with vertical ends. E. C. parallel to terminal edges of the finite prism. A right rectangular prism, the section of which is an oblong. An example of this application in the Rhombohedral System is given in the Edinburgh Journal of Science for January, 1827. The following examples may shew its use in the Square-Pyramidal and Oblong-Pyramidal systems. Fig. 4, (Mohs I. 67, 68.) belongs to the Square-Pyramidal system. The faces a alone would form some pyramid pQ; the faces f, a pyramid p’Q; the faces b alone, some pyramid in a diagonal position p'Qr; the faces c, an eight-sided pyramid gQn; the faces d and e, prisms ©Qm, and »Qm’. Hence, the general designation of the form is PQ, p'Qr, Qn, ~Qm, ~Qm’, p’Q a b c d e if Let a belong to the fundamental pyramid: therefore, pQ =Q. The combination a, b agrees with p, 1, Class I, No. 4. The combination », e, would have horizontal edges; therefore, in the same manner, oQm’ = oQr. Vol. 11. Part IL. 3I 424 Mr. WHEWELL on the Classification The edges of the pyramid 5 are truncated by the faces a, the edges of combination being parallel. Hence, by C, 6 is 2Qr. The edges of combination of @ and ¢ are parallel to the terminal edges of a. Hence, by B, c is derived, by the second law, from a. Therefore, q = P, gQn = Qn. If the faces a were removed, the combination b, c, would agree with c, a, in II. 2. Hence m—1lg=2, orn=2+1=3: and c is Q3. The combination c, f,; supposing the other faces absent, would agree with xz, r, Il. 9. Hence, p” =3. Therefore, the expression for the form is Q, 2Qr, 3Q, Q3, ~Q, «Qr a bit of) nie d e. Fig. 5, (Mohs I. 72.) is a form belonging to the Oblong-Pyramidal system. It is obvious that it may be thus represented pP, p’P, qPn, p’Pr, p”P’r, oPr, Pm, Pm’ be d c a h We g. o Since we have horizontal edges of combination between 5, e, f, we will suppose the pyramid e to be the fundamental form. Therefore, as in I.1, Pm is oP. Since c would truncate the edges of e, by III. 1, ¢ is Pr, and p’'=1. The faces a would make rhombs with c and e, as in III. 3. Hence, a is iPr. The faces a truncate the edges of b, asin III. 2. Hence, d is iP. The pyramid d would have its edges truncated by e and a respectively. Hence its edges make the same angles with the axis as the faces of those prisms do. of Crystalline Combinations. 425 Now if a be the axis, 6 and c the semi-diagonals of the fundamental form, the axis and semi-diagonals of gPm are respectively gna, nb, c. And hence, we have Cc eee qna a ; whence g =i, n= 2. Since g makes horizontal edges with d, and d is 1P2, g is »P2. Hence, the expression for the form is LPP; $82, Pies Pr. ober olf, oP? Db ve? id c a fa ae o a: W. WHEWELL. Truity CoLLeceE, April 29, 1826. : . Pies atx? os, . . iit'a. ikpep @- a ue Wi nono ig a “ot By Se Sw ; ita are, oe deadhvits ae Ha it, a in i y. “Thaw s ee sein jad My ebe oh Py tes eres ia ‘evi be fi Li rea id ss ee Ha th a a Cy ae af ; adie ia es ae Fabs jie, tenet Fatt Le EARLY oo Seat tes > be ‘ WON_VS cee ae ie ae Se ea uate r we A as neh deed fer : aN, ease. gf ahh wie a Rt Wiley et ve Rei ae :u sated at ti Heer ot tall A Bie echt Sgt a sine Os a Ye Metin Af? vy Siete te He incr’ odd! Sere in ‘ibe te te timucutihinantcucaos bata pair cee ‘ chs or Mis in TR SEMPRE aT SR ‘ ey, 1g ey abs wie, aha het a tebe f olin fry vir tite Wate. naendey” fey ee tte SEAL ae Pee a A, Pa PW xt Bin SRR ae een stevnin rhae sad baat Od 40 Bama, pik oo 47 wi aa ms Rola: yu nutes MEP OG aT TS ris ‘Deets ehh a (lane AHEM toer ai A hac iam age Mk thc Bet Wp aig #4 Pequot oh fy yd} bas Shoes 1a": oo eae Fa a ee | sk — : Z = ; s 4 J,* * — ; = # - S v1) 4 ie “eh L ay x is fa z PY 4 XXIII. Reasons for the Selection of a Notation to designate the Planes of Crystals. By THE Rev. W. WHEWELL, M.A. F.R.S. FELLOW AND TUTOR OF TRINITY COLLEGE, AND SECRETARY OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, [Read Feb. 11, 1826.] In a communication read to the Society in the course of last year, I pointed out that crystals may be divided into four classes, according to their degree of symmetry. Being referred to an axis, they may consist of three similar and symmetrical portions about this axis, or of four, or of two opposite pairs of similar portions. They may also be symmetrical, with respect to three axes at right angles to each other. The merit of this classification has been a subject of controversy between Professor Mohs, and Professor Weiss, both of them persons to whom the science of Mineralogy is deeply indebted ; but to whomsoever it is to be ascribed, it is a division so consonant to the nature of the case, and the mode in which we have to reason on crystalline forms, that there can be no doubt of its being, for the future, the proper and scientific view of the subject. Founded upon this view, it may be mentioned also, that Professor Mohs had given modes of deducing all derived crystalline forms, and of designating them by a notation, such as to render easy and general the process of deducing the Jaws according to which each is formed. Whether his mode of de- duction is simpler than any previously proposed, is a question which must be determined by a consideration of the forms which really occur in nature, and the discussion of it must be reserved 428 Mr. WueweE t on the Selection of a Notation for another opportunity. That Professor Mohs’ method does offer remarkable and valuable facilities for the determination of crys- talline laws, is a fact which may be ascertamed by examining the applications which have been made of it. The object of the fol- lowing pages is to modify the notation, so that, besides answering this purpose, it may possess as much of the scientific beauties of simplicity and generality, as is consistent with the nature of the subject to which it is applied. In speaking formerly of the above four classes of crystals, the terms were mentioned by which previous authors have designated them, and others were suggested. It is, however, reasonable that those who first navigate these new shores of science, should have the privilege of giving the names to the objects they discover, and I shall, therefore, adopt the denominations of preceding writers, with some alterations, which analogy seems to demand. The four classes will be termed the Rhombohedral, the Oblong-Pyramidal, the Square-Pyramidal, the Octahedral.. And the fundamental forms which will be used in deducing the derived forms will be, in each instance, a pyramid. In the first class, an equilateral triangular pyramid; in the second, one with a rhombic base; in the third, one with a square base, and in the fourth, the half of a regular octahedron. The first class is called Rhombohedral both by Mohs, Weiss, and Breithaupt; the last, which I have called Octahedral, in order to refer it to a pyramid like the rest, is what is by these writers called Tessular, Hexahedral, &c. In the names of the second and third, Mohs and Breithaupt differ with each other, and neither appears to attend sufficiently to analogy. Mohs calls the square-pyramidal simply pyramidal, and the oblong-pyramidal, prismatic, in which nomenclature the two terms have in no way a relation corresponding to the rela- tion of the forms, and might, with equal propriety be permuted. Breithaupt calls these two classes rhombic and tetragonal, or to designate the Planes of Crystals. 429 square, which is consistent as far as those two alone are con- cerned, but does not exhibit their relation to the others. I think the terms I have adopted ensure both these objects. The selection of a notation to designate the relations of derived forms in crystallometry is by no means unimportant; for, if it be constructed with a proper symmetry, and a due regard to the analogies which prevail in these forms, it may be made the means of facilitating and compressing, im a remarkable degree, our reason- ings with respect to the laws and connexions of crystallized forms. In fact, it has already been adapted and applied to this end, with great imgenuity and success, in the hands of Professor Mohs; and there is no one to whom the merit of making this great step in the science of mineralogy could more naturally fall, than te one possessed of his talents, perseverance, and extraordmary know- ledge of the subject. In proportion as this process —I mean that of reasoning from our symbols—is valuable and important, it be- comes desirable to perfect our notation as much as possible, and to carry to the greatest extent which they admit, the analogy and simplicity of our symbolical system. It is to be wished that perfection m this point should be attained as early as possible, in order that we may not have afterwards to alter a notation to which men’s minds have become familiarized, and to learn afresh the alphabet of the science, which it is an irksome task to have even once to acquire. These considerations will excuse the at- tempt made in the following paper, and will also account for the liberty which I have allowed myself of pointing out some anomalies in Professor Mohs’ System of Notation. Without some strong reasons, such as those above mentioned, any endeavour to alter a notation, in which so much information is embodied, as is contained in his treatise, would be inexcusable. Nothing, in- deed, can be more likely to produce confusion and unnecessary 430 Mr. Wuewett on the Selection of a Notation labour than the levity and restlessness in proposing new symbols, which some writers allow themselves: and no notation at all deserves consideration, which does not by its structure exhibit the analogies of the things represented, so as either to facilitate our reasonings on them, er, at least, our understanding of them. I hope, however, to be able to shew that what I have proposed in the following pages introduces no alterations, except such as are requisite to maintain the symmetry, and, if I may so speak, the homogeneity of the notation: and, having made it my object to leave no anomalies which appeared capable of removal, I trust the evil of change will be more than compensated by the establishment of a system, which there will be no future necessity to remodel. The merit of this improvement will hardly diminish the obligation due to the first introducer of an efficient notation. In order to point out the reasons for the proposed changes, it may be allowed to introduce, and refer to, some general principles which ought to regulate all scientific notation, and which, though they come with no authority beyond their own reasonableness and utility, will nevertheless, I think, be acknowledged as true by all who give the subject a scientific consideration. They will offer themselves as we review the different points of the system. It will be seen that the indices which I have used, are generally the same as those used by Mohs, and that the difference lies prin- cipally in the mode of writing and combining them: hence, my system has the same means of obtaining results which his possesses, with so much additional symmetry as was consistent with the retention of this property. to designate the Planes of Crystals. 431 1. Mohs designates the fundamental rhombohedron by R, and the principal series of rhombohedrons by ...R-3, R-2, R-1, R, R+1, R4+2... In the same manner, in the rhombic and square-pyramidal systems, he designates the fundamental pyramid by P, and the principal series of derived pyramids by POP, Vn POP PRES Cig This notation violates the principle that the signs + and — cannot properly be used to connect signs of form (R or P) with signs of quantity (2, 3, &c.). Such ideas are altogether heterogeneous, and should not be combined as if they were capable of addition or subtraction. It may be said that in this notation -—2, -1, 1, &c. are not quantities added, but indices. If so, they should be so written and the principal series would be then ee sri eae 2k, &e. which would be free from the above objection. This designation of the principal series is somewhat simpler than ours, which is bd 278 By 27h Ry 2) ety 2 7R; ¢Bee.s where the exponent of the root 2, in this case, is the index in the system previously mentioned. And this simplicity would be a reason for preference, if the series here referred to contained the only forms which are to be taken into consideration. But, in using this notation, Mohs is compelled, when forms occur which are not in this series (subordinate series), to adopt a new convention for expressing them; whereas in our method, the principal, and the subordinate series, are cases of the same general notation. And the law of the principal series is sufficiently evident in our symbols. Vol. II. Part IL. 3K 432 Mr. Wuewe tt on the Selection of a Notation 2. This principal series R-—2, RAR SRYS INR Pa and, P=2, .PiALA\P; 25 4,-P $2. in another system, aré, by Mohs, indicated by the same notation, though they are derived in different ways. In the first series the axis is doubled, and the base turned through 180°. In the second, or oblong-pyramidal system, the axis is doubled simply. In the square-pyramidal system, the base is turned through 45°, and the axis increased in the ratio 1: V2. The principle on’ which these series are constructed by Mohs is, that each member of the principal series shall truncate the preceding member. But in the oblong-pyramidal system this analogy is violated by omitting the alternate members. 3. The subordinate series, cr forms whose axes do not agree with terms in the principal series, are represented thus, mR+n, which is used to indicate that the axis is m times that of the form Rin. It is manifest that since the m thus refers to the whole Rin, the symbol ought to be m.R+n, or rather, as has been said, mR,, or m.,R. But, even with this alteration, it would follow that m and n, indicating operations of the same kind, viz. an alteration of the axis, are different in their mode of writing, and in their effect in calculation; m indicating an in- crease of the axis in the ratio 1: m; while » indicates an increase in the ratio 1 : 2". It is a principle to be adopted in notation, that symbols, which are identical in their meaning, should appear to be so by the application of the rules of algebraical operation; thus, 2.2R and 2°.R are the same in our notation. But, in Mohs’ notation, we have expressions 2°R+1 and R+3, which are identical without the identity appearing in the form of the ex- pression. Vice versd, we have, in the same rhombohedral system, to designate the Planes of Crystals. 433 (P+n)', which is not the same form as P +n, to which the symbol is algebraically equal. It is this principle of identity of meaning, corresponding with identity of numerical value, which mainly assists us in employing symbols as instruments of rea- soning. The algebraical reductions which we make, represent. then different ways of considering the same form. 4. Forms derived from R+ 2 in a certain manner (namely, by the law of scalene derivatives,) are, by Mohs, represented by (R+n), (Pin), &e. It may be observed, first, that there is no sufficient reason in this case for altering the fundamental letter R into P. Mohs does this because the Rhombohedron becomes a six-sided Py- ramid, which change is marked by adopting the new initial letter: but it would have been more important to mark that this derivation was from the rhombohedron; and to leave it to be recollected that its form was a pyramid. The expressions would then be (Rin), (R+n), &e. 5. But, in this notation, the exponents 2, 3, are used merely as indices: and there is no propriety in writing them in such a manner as to represent the powers of the symbols R+n. To refer again to the principle in the third observation, this would have been proper, if, by this means, reduction had given us different but identical symbols: if, for instance, (2 R)’ had been the same as 2°(R)*; but this is not at all the case. These in- dices, therefore, ought to be written where they have no alge- braical meaning, and, therefore, cannot give wrong coincidences. If we write both these numbers and those mentioned in Obser- vation 1. as indices before and after the foot of the letter, we shall have for 3K 2 434 Mr. WHEWELL on the Selection of a Notation (R + 2)°, (R—3)*, &e. these .R, _3R,, &c. which agree very nearly with 2R3 2-*R2, &c. the symbols here proposed. 6. A rule is given, Mohs, Sect. 96, to refer the members of the subordinate series to the members of the principal series, whose axis comes nearest to them. This rule introduces com- plicated expressions unnecessarily, because the subordinate form may have a very complicated ratio to the nearest principal form, though a very simple one to some other. Thus, we have a form in the square-pyramidal system, which is represented by —— P-3; 3 its axis being, therefore, Hence, if we refer it to P, the fundamental form of it will be +P. By this mode of designating the forms (particularly in the square-pyramidal system), we lose sight of the simplicity of the law by which they are deduced. 7. In the same manner in which Mohs changes R into P, when the rhombohedron becomes a pyramid, he changes, in the pyramidal system, P into Pr, when the pyramid becomes a prism. Here, however, some alteration is necessary, because the indices, in this case, become infinite, and, therefore, the symbol incon- venient. I have adopted, in some measure, this part of the no- tation, but with these advantages, that the r added indicates, in our method, a particular process, the analogy of which runs through all the systems; and, also, that the letter P, which recurs in the symbol, marks the fundamental form from which the deduction is made. to designate the Planes of Crystals. 435 8. Analogous to this process is the deduction, in the rhombo- hedral system, of the isosceles six-sided pyramid (Rr) from the rhombohedron. This deduction, however, is, by Mohs, desig- nated in a different manner, namely, by the change of R to P. And it may be again observed that this change from R to P is different from the same change made, by Mohs, in the manner mentioned in Observation 4, where both are in brackets, and indicates a process altogether different. 9. We may observe, also, that Mohs has several other signs, which offend against the rule of having the symbols as homo- geneous, and the conventions, from which they derive their meaning, as few and as general as is possible. Thus, the pro- ’ sodical marks - and © are introduced over the P and Pr to in- dicate the derivation being made from the extremity of the long ov short diagonal. And it may be observed also here, that the comparative length of these diagonals has nothing to do with the derivation; but only the consideration of one as principal, and the other as transverse to it. Again, the crotchets [ ] are used to mark that the limiting prism is in the diagonal position, which notation has no analogy with the other parts of the system, (Sect. 102.) Again, the negative sign, Sect. 128, is used to indicate a second half of a derived form. This also is suggested by no analogy, except in the case where the half form is the same as —1.R according to the definitions, and may, therefore, be properly re- presented by —R. I have employed an accent (as in R’) to indicate this change, which mark is also employed in an analogous manner in the next paragraph. 10. In such cases, where we subject a form to some con- ditions, after we have established the way in which it is deduced ; it seems better that the part of the symbol which represents these conditions should be separate or separable from the rest. Thus, 436 Mr. WHEWELL on the Selection of a Notation to indicate the half of a form xnRm, it will be better to write h nRm before it, than to write it , as is done in this part of Mohs’ notation. The notation = = (right or left) in which hemihedral faces follow each other round the figure, both in the upper and under halves of the form, is convenient and significant, and seems not open to any sufficient objection. , &c. by which he designates the direction As it is only in cases of hemihedral’ combinations that these symbols occur, it does not appear necessary to indicate this by the denominator 2. I have, therefore, written simply - nRm, nRm r r— hnRm. Fethgte r and not z If one of the hemihedral portions arise from the fundamental form in a transverse position, this circumstance is indicated by giving an accent to the corresponding letter of the symbols -, &e. Thus, =, nRm indicates that half the faces of Rm, at one apex, are combined with half the faces of nR’m at the other, both portions being turned to the right. 11. In the octahedral or tessular form, Mohs uses a different conventional letter for almost every different species of form. To this, perhaps, there is no great objection, as, from the regu- larity of these forms, the facility of reasoning does not depend much upon the generality of the symbols. It may be observed, however, that, according to our system, all these forms are pro- vided with symbols, according to the analogy of the other systems of crystallization. 12. The designation of twin crystals by Mohs seems also to designate the Planes of Crystals. 437 deficient in simplicity and symmetry; but this will be considered hereafter. Removing then these objections; placing the indices in the simplest manner, namely, before and after the letter; altering the first index so that the same Jaw shall obtain for principal and subordinate series of forms; indicating each of the four systems of crystallization by a distinct letter, and pursuing this in all the derived forms; we have the above system of notation. 13. We shall notice some of the properties which appear to recommend the preceding system of notation, by their sim- plicity and . uniformity. The fundamental forms are four. These are in our method designated by the successive letters O, P, Q, R, which appear in each of the derivatives of these respective systems, and thus indicate to which class each derived form belongs. These may be called fundamental letters. Every derivative is represented by the letter designating the fundamental form, attended by two indices, or numbers, one of which is always written before and one after the letter. In those cases in which we have only one number or index, as mP, 2R, Q3, the other index is 1, and may be so written, if convenient, for the sake of analogy. Thus, 0O is 001, Q is 1Q1, &e. The index which is written before the fundamental letter al- ways. designates the same law of derivation in all the systems; namely, the derivation of one pyramid from another, by changing the axis in the ratio of the index. The index which is written after the fundamental letter also designates a law of derivation, which is the same in all the systems. 438 Mr. WuHEWELL on the Selection of a Notation This law may be thus expressed. A diametef is drawn to the base of any pyramid; (MC in Figs. 1, 2, 3.) This diameter is increased (or diminished) in the. ratio of the index m (producing MC’). Through the angle (B,) and the extremity of this diameter, parallel to the edge of the origmal pyramid, (CV,) is drawn a plane (BC'V’). The figure bounded by all such planes is taken. By symbols thus constructed, we can represent any plane whatever, and it has been seen in a preceding Paper that the mode of intersection of different planes will afford us the means of recognizing their origin and law. For the sake of convenience, two other symbols are introduced. ist. The letter r after the fundamental letter. For —- Ro, we have put Rr, 3@ LS plata 2 sce ORE Nees 5 25 @ for Q or oS jae c a ane outa ane male's ow ae Ve Oreo 0 or = Oe) Re oo oosisaccasoon) ry The letter r is added to the fundamental letter, when m, the second index, becomes infinite, and the axis of the fundamental pyramid remains unaltered. - 2d. An accent added to the fundamental letter in two cases. R’ is used to designate the fundamental pyramid of the rhombo- hedral system, its base being turned through two right angles, or placed in a éransverse position. to designate the Planes of Crystals. 439 P’ is used to designate the fundamental pyramid of the oblong pyramidal system when the derivation is made from the ¢ransverse angle of the base. Derivatives of R’ and P’ are obtained and designated in the same manner as for R and P. In cases of hemihedral combinations, we have also the symbols r h, 7 > oe &ec. as explained in paragraph 10. W. WHEWELL. Trinity CoLLece, Feb. 11, 1827. Vol. Il. Part II. sL aK : ava) tine Seid tbe: cyecennbink: 4) By dy Petr ‘ind pei ail ie i ‘ie, ee ast tox, 4 r ad! at ah a es Wy, peupsiak ti Ae wihal et onli Baise ae'y-: ‘beni wa ie ae Raa": wich: 4 ai bt gidaag bi i i, EXTRACTS FROM COMMUNICATIONS MADE TO THE SOCIETY. I. On an Artificial Formation of Plumbago. EXTRACT of a Lerrer from James ALDERSON, Esq. to the Rev. Professor Cummine. [Read February 21, 1825.] Tue accompanying specimen of Carburet of Iron formerly composed part of a groove in which a patent perpetual log was made to slide. It was originally cast iron, and was nailed to the coppered stern post of the ship Zoroaster, of the port of Hull. On the return of the ship from India, after an absence of about nine months, it was found as it now remains, converted ito Plumbago, having lost very considerably in weight, but having perfectly retained its original form. Pembroke College, Cambridge, Jan. 22, 1825. Remarks py Proressor Cummina. Oxe of the most striking properties of this artificial Plumbago, is its extreme levity as compared with that of the substance from which it is obtained, SL 442 The specific gravity of soft cast iron is about 6.94, of the hard 7.54; the average may therefore be considered as 7.24 containing about 5 per cent. of carbon; but the plumbago resulting from it, when perfectly dry, floated not only on water, but, for an instant, even on alcohol. With the view of freeing it entirely from the air included in its pores, a portion weighing 226.2, grains was boiled in distilled water, by which, not- withstanding it deposited two grains of ochre, it gained 214.8 grains. Its specific gravity was then taken, and was found to be 1.26. It appears therefore that nearly all the metallic iron had been gradually converted into a soluble muriate, which, as it was formed, was removed by the passage of the vessel through the water; leaving only the carbon in the interstices of which it originally existed. The average specific gravity of the Borrowdale graphite is 2.13 and it contains 41 per cent. of metallic iron; the inferior specific gravity of the artificial specimen made it probable that it contained still less, which was confirmed by deflagration, in the usual mode, with nitre. The exact quantity seems however of no importance; as there can be no doubt that the same process which had removed so great a portion of the iron, would, if con- tinued, have left nothing but the carbon. For the agent being evidently Galvanism, exerted in precisely the same manner as in its recent application by Sir H. Davy, would have acted upon the iron, through the intermedium of the carbon, until it were entirely removed. In regard to the useful application of this substance, there can be little doubt that in many cases it may be substituted for the coarser kind of plumbago; but it is so much softer that it cannot be used for the purposes of drawing, unless as a crayon. The colour of its streak on paper.is not so dark as that of plumbago, and instead of presenting a continuous line, the streak appears, when examined by a lens, to be composed of minute detached portions of carbon, intermixed with specks of metallic iron. i In the ist Volume of the Annals of Philosophy, Dr. Henry of Manchester has given an account of a similar substance found in a 443 colliery near Newcastle, the water of which was strongly impregnated with the muriates of lime, magnesia and soda. The specific gravity of his specimen was 2.155, the decomposition had therefore evidently not been so complete as in the present instance. It had been known, so long since as in the time of Scheele, that though the affinity of iron for muriatic acid is less than that of soda, lime, or magnesia; yet when either of these substances, in the state of a muriate, is present in great excess, it is itself decomposed and a muriate of iron is formed. Hence Dr. Henry seems to have attributed the corrosion of the cast iron solely to the action of the muriates of lime and magnesia. But it has been recently proved by Dr. Marcet that muriate of lime does not exist in sea water; and though, by long continued action, the muriate of magnesia alone might possibly be adequate to the decomposition, without having recourse to galvanic agency; yet, as muriate of soda makes a powerful galvanic circuit between dissimilar metals, which undoubtedly existed in contact in the present instance, and probably in the case cited by Dr. Henry (as the corrosion seems to have taken place at the junction of two pipes) there appears to be no necessity for the rather improbable supposition that chemical affinity alone had effected so complete a de- composition in so short a time. How far the original formation of native graphite may be attributed toa similar cause, is so much a- matter of hypothesis, that I shall not waste the valuable time of this Society by attempting to discuss it. 444 II. Experiments on Percussion, made on a spring of Memei Deal, 20 feet long; Weight 16.87lbs. Weight of falling body 3.222lbs. Deflection by steady weight of 6lbs.=1 inch. By B. BEVAN, Esa. Inches Inches Squares. Fall, Deflection. 2 1.33 1.7689 4 1.68 2.8224 8 2.08 4.3264 16 2.60 6.76 32 3.48 12.1104 After adding to the middle of the spring, 4.85lbs. 2 1.37 4 1.60 8 1.90 16 2.26 32 2.95 After adding 10lbs. to the spring instead of the above: 2 1.23 4 1.36 8 1.65 16 2.35 j 32 3.10 a Se he Wil. Table of the Computed and Observed Variations of the Magnetic Intensity at the Earth's Surface. By R. W. ROTHMAN, Esq. M.A. FELLOW OF TRINITY COLLEGE. Tue formula used in the computation is ./4—3 (sin. dip)’. Observed Computed Intensity. Intensity. 1.0000 1.0000 Differences. J. Carlos del Rio Negro Carichana. .. . en tea TION se Mexico . 2 Db. PRS 2290 eel SAS Dae ats thOON Aa ee London Hie nlgen >) 1 7o 20 eae Christiana... . 1.4959 MiB aaa Arendahl.... NAO se .. 14941 SIG ee sk Hare Island. . 1.6939 1.9547 . Davis’s Straits. 1.6900 1.9584 ... Baffin’s Bay. . pine ore LOO8Sh AY eae cy ial: at: 1.9743. 1,6943 1.9751 1.7383 1.9850 . . . 1.9866 . whl 04570 cris) 1.0484 ; 5 +. + +. +. - ot. ah siete aps wah 5 +. GP Nov. 10, 1825. A LIST OF DONATIONS TO THE LIBRARY ann MUSEUM OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. . 1822. May 6. Nov. 11. Nov. 25. 1823. Feb. 17. Mar. 17. April 14. April 28. oe SS I. Donations to the Liprary. Berzexius on the Blowpipe ........- On the Structure of the Apophyllite Translation of Venturoli’s Mechanics On calculating Tables by Machinery.... . Periculum ad Cippos Punicos......... Diatribe de monumentis Punicis ....... Observations on the Nautical Almanack.. . Tracts on Political Economy ........-- Art of Embalming.........-.-- a ee Sur Ja Lumiére des Ondes.......... Monographia Apum Anglie.......... Treatise on Dynamics .....-..-.-.-- Geological Transactions, new series, Vol. I. Winch’s Geographical disttibution of Plants of Northumberland, &........... -Donors. J. G. Children, Esq. Dr. Brewster. Dr. Cresswell. C. Babbage, Esq. Prof. Reuvens. Prof. Hamaker. J. South, Esq. W. Spence, Esq. Dr. F. Thackeray. — Spooner, Esq. Rev. W. Kirby. Rev. W. Whewell. Geological Society. J. Hogg, Esq. 1823, Noy. 10. Nov. 24. Dec. 8. 1824, Mar. 1. Mar. 15. May 3. Nov. 15. Dec. 13. 1825. feb. 21. Vol. WI, Part II. Donations to the Library. Experiments at Hafod Copper Works. . Jerusalem Delivered, Book 4.......... (reatiseron' Opties/s .\ueulae, » Wells.) cas 508 Address to the Hull Philosophical Society .. Essai Historique sur le Probleme des trois ELT BSR eh evry 8 Maia ie, abate 4ac Sur quelques Observations Astronomiques. . Transactions of Royal Society of Edinburgh, Vol. 1X.) Paptia ccneut asttte., tates 2d. ve Sur les Axes de Rotation.......... pevecs Sur L’Electro-dynamique. .....+.......05 Sur la Theorie du jeu......... payegttle = aittwes Sur une Example de Reunion de Coquilles marines et fluviatiles..............s00. Sur le gres coquiller de Bernchamp........ Essai sur le basin de Vienne....... Solution of the higher order of Equations. . 24 No’. of Tilloch’s Philosophical Magazine 6 Early No’. of the Transactions of the Royal Society. ..e05.-.0-see ee Fi Ghictn Classical Journal, No. 55...... eFarekaiaiemtonel On Slavery in the West Indies............ History,.of Persia... .. ..0:00 000000002 ogee Geological Map of country round Bath.. Transactions of saa ne ae of Edinburgh Vol. X. Part 1.. Bis ageytarsiare Astronomical Tables... ion s\enupeet aed mioteeidta ts. ete Treatise on Mechanics..............2+.- Museum Britannicum..........+e.e0.e200- Memoirs of the Astronomical Society, Vol. I. Part, 2. . sele sie On Flustra Arenosa. aio. oesie senccieence seis View of the College of Calcutta........+.. 3M ® 447 Donors. Swansea Society. J. H. Wiffen, Esy. Rev. H. Coddington. Dr. J. Alderson. A. Gautier. Royal Soc. of Edin. M. Ampere. C. Prevost. Rev: J. Buck. Rev. L. Jenyns. T. Thorp, Esq. Prof. Reuvens. Jas. Stephen, Esq. Sir J. Malcolm. Rev. W. Conybeare. Royal Soc. of Edin. Dr. Pearson. Rev. W. Whewell. Rev. L. Jenyns. Astronomical Society. J. Hogg, Esq. Rev. R. Duffield. 448 1825. Mar. 7. Mar 21. Nov. 28. 1826. Feb. 13. April 10. April 24, May. 8. May 9. Noy. 13 ’ Donations to the Library. Transactions of the Geological Society, new series, Vol, I. Part, 2... 8. 2008 22208 WaBlOUSHE rats. 5 elec eve, clere oot aie aetareian =, oittalerete Portraitvof Saussureso%. (see. tee Transactions of the ate Irish rg rc Vol. XIII... chore tes d Martyn’s History of the Academy at Paris.. No. 439 and Vol. XX XIX. of the Philosophical Wransactionss secre ean crt. 4 OGRE, OY. Imperial Almanack for 1826...-.. SA eee Dissertation sur la ‘Theorie de la Vision... Memoire sur les Apparences visibles....... On the Magnetizing Influence of the Sun’s RRA YBirss: «) in x12 eee ee eater one eae Observations on the Longitude of Paris and Greenwich.. Me Memoirs of ais yaMecwaiiitiat ‘Society of London, Vol! 12 Payteers 3... 224.28. Journal de |’Ecole Polytechnique, Cahier 1 Shit Os moo duoobdeoban cose oon Boieisicia Mathematical Tracts:........060...5....5. Elements of Medical Logic............. Principles of Analytical Geometry........ On the Action of Sulphuric Acid and Naph- thaline:\.. 21 eens My ERE: SREP On Geological Specimens from Australia. . . . On Positions of 458 double Stars........ Catalogue of 838 double and triple Stars Transactions of the American Philosophical Society, Vol. LI. new series. © .3<:.«).- Transactions of Royal caidbe of Edinburgh, Vol. X. Part 2. Privileges of the Teeny of CambridBe!” The English Constitution...........0...4 Donors. Geological Society. Dr. Hagarth. Rev. D. Pettiward. Royal Irish Academy. T. Thorp, Esq. R. Sheepshanks, Esq. G. Maurice. Mrs. Somerville. J.F.W. Herchel, Esq. Astronomical Society. Rey. H. Robinson. G. B. Airy, Esq. Sir G. Blane. Rev. H. P. Hamilton. M. Faraday, Esq. Dr. Fitton. J. South, Esq. American Society. Royal Soc. of Edin. G. Dyer, Esq. 1826. Noy. 13. Nov. 27. Dec. 11. 1827. Feb. 26. Donations to the Library. Asiatic Researches, Vol. XV........... iia Sankalitsi<. d/h) Savard tts oe 3. Account of Botanic Garden of St. Vincent. Article on Heat Observations on 380 double and triple Stars } On observed and computed Right Ascensions taptie SUNN na" items ON ae eee eee ee Plana sur les Perturbations des Planets . . Recherches sur quelques Effleuves terrestres Memoires sur |’Electro-dynamique ...... Sur un nouvelle experience Electro-dynamique Dissertation on Three Javanese Images . . . Sor laePheonedw jen... eee 8 Sur des Axes de Rotation .......... Du Calcul Differential and Integral Theorie des Phenomenes Electro-dynamique . Sur |l’Electro-dynamique D’un Appareil Electro-dynamique....... Sur une nouvelle Experience Electro-dynamique Sur l’Electro-dynamique par M. Savary... . Surtlei@alorique, &c.... 6. ee Various Nos. of l’Analyse des Travaux de P Academie Royale Article on Arithmetic oh 0; 16 Muelt ©, 88, feltie) >| 6) (6. \id ) «ee af omic tis fem ame (efal ie © Nake: Various Nos. of Westminster Review Address to the Zoological Club 449 Donors. Asiatic Society. Dr. Shuter. Rev. L. Guilding. Rev. F. Lunn. J.F.W. Herschel, Esq. and J. South, Esq. J. South, Esq. C. Babbage, Esq. G. B. Airy, Esq. M. Ampere. Prof. Reuvens. M. Ampere. S. Pugh. M. Ampere, and J. Underwood, Esq. Rev. G. Peacocke. Astronomical Society. Rev. W. Whewell. J. E. Bicheno, Esq. 450 1822. May. 21. Noy. 11. 1823. Feb. 17. Mar. 17. April 14, April 28, Nov. 10. Nov. 24. 1824. Mar. 1. May 24. Donations bo the Museum. II. To the Museum. Collection-of, Shellsy oi 4. -..jact). scant Concretion from the Intestines of a Horse. . Specimens of Bulla aperta and B. Heliotidea . Varieties of Helix cingenda ......... British Mnsects: «...:. v's cys.i4d dcayaemean shen Fossil Teeth of the Elephant, &c....... Retinasphalt and Prehnite from Staffordshire . Skeleton models of regular solids ...... Portion of an edible bird’s nest........ Batch Shells oe) dys owteosseeeet-aael? ‘ac Nests of Vespa Britannica .......... Six cases containing specimens of stuffed birds. Specimen of Falco peregrinus and variety of Brinpille ceelebs 4c cnend In barinn a2 Specimens of Ardea Stellaris and Falco Buteo Specimens of Dotterel and Water Crake. . . Large fossil Elk’s Worin apeyiecsiispe aadowe dsb Red-lezred: Rartridee.ty.oukial)) cas nsegersl ol% Asterias Caput Medusez and several British Slip’ stayshaue: Siac) Soe as. oo oe. Sa Kite he xilotienaieS: As dasaenried. 08 Fossil bones from the Kirkdale Cavern... Head of a New Zealander .......... Fossil Wood from Harwich........... Guan from the Honduras and Demerara Pheasantie Gee ccs as tees aero bs Short-horned Owl, variety of Chaffinch, &c. . King-duck .. . we) Sie) re 6" Be by OS aeURt ee) oe ee Donors. Rey. H, Coddington. Dr. F. Thackeray. Miss E. Harwood. Rev. J. C. Ebden. J. Dale, Esq. E. Manners, Esq. F. G, Spilsbury, Esq. N. J. Larkins, Esq. Dr. Ingle. W. Lyons, Esq. Rev. L. Jenyns, Esq. Rev. W. Russell. Rey. Prof. Henslow. Rey. L. Jenyns. J. Hogg, Esq. C. T. Higgins, Esq. S. B. Turner, Esq. Dr. Goodall. Rev. Prof. Henslow. Rey. W. Russell. Rey. L. Jenyns. Rev. J. T. Huntley. Rev. F. Wrangham. E. S. Haswell, Esq. Rey. J. C. Ebden. Rey. Prof. Henslow. Rey. L. Jenyns, Capt. Wilson, 1824. Nov. 15. Dec. 13. 1825. Feb. 21. Mar. 21. April 18. May 16. Nov. 28. 1826, Feb, 26. Nov. 13. Dec. 11. 1827 Feb. 26. Donations to the Museum. Extensive collection of stuffed British Birds . Polecat, and) a0 .cc sets iat sp aie. Two cases of British Insects .......... Santa EKCL ted). ince ORE Ae ok kok es ei oe Baveye Druey. Coal nei. sf. . 3 aia fs Wild Cat of Scotland, white Hare, and Pied 1 ET he etnies ay caked "SS “pane ae D earns Pied ‘variety of Pheasant... 5... ... .”, PAIMME EATMB) oe osha Ree eh heck die SCMRICEUGUY. LAT ch ti nade. ie 4 sfiale racers Insects from New South Wales ....... Large collection of North American Minerals BOSSI M INNES: pe po hetet > Sts: oho i cats ofl a Two polished Chalcedonies.......... Dried Plants of Cambridgeshire ....... Fine lines drawn on polished Steel: and very AIMAUMNOMEIPINA coe aes, 5 o ee ne reat F 44 oT A, ta HSE fa a i . } ‘ pes eran A he abs Se Raa we fp 4 yah A cant it. hal sy Muperate ibe aed a Dad 19s at) ret, ‘sh Nes Bry. Tat a doled sine uid it ered | wer aN, a P te thi sadly i vn ae aes coma sly sa Ry iS tty) cota Br A: mid A + Siuabink ‘aro vider) >to ates ay Sie aut Shand i a. ae wa of cbaquotT ey ee a ile any 4 3 ound Mi, Tht, reas ' niger anne a” any. "7 py Dee! ieee - yior ton lane fide Hog i uve ee Paes | so ip ‘ Hagen 120-4 x . fh Ms ¥ phe ‘a ee 7, : ea “ye ey M Slee Cath eV al ererviin t sitet: ie Py ‘is ' f ys oe . . vl be Te nae } wy x ~e ve = whey ~—e ‘ ne ~. ae nS A ry > a i. to het) j ‘ : m My, ‘- ca (ipa ree. o f; a oahu, xv u - 5 Ae rel « rn yd ’ rs, i aah * a ea ee mae rae ie ee. ee Ref sn ines ion be ies aM ee ed oe ee cere es Tur Author of the Paper “On Achromatic Eye-pieces,” &c., has to apologize for an error in his statement of the principle of Dr. Blair’s object-glass, (in Article 7 of that Paper, page 239). The construction finally adopted by Dr. Blair was, to inclose a mixture of muriatic acid with a muriatic salt between two lenses of the same sort of glass. By varying the proportions of the mixture it was found possible to form a ying prop P fluid in which — was exactly expressed by the formula c on Trorvaction f the Crh. Whit Woe Vil Me Late AL, ‘ P Cambridae Phitosyphiead Lrapractions Vol? Plate te. 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