il eee epee Wea ened. ir Sete arent ee Oe Poe ee ee ee ee =e bee N re Mi nts iA Ie Sarak Wate ot ee sul nt Ss ae veal eB Naa: a MEMOIRS OF THE LITERARY AND PHILOSOPHICAL SOCIETY OF MANCHESTER. 5 nd ed i T™ ¢ MEMOIRS ‘OF THE LITERARY AND PHILOSOPHICAL SOCIETY OF MANCHESTER. THIRD SERIES. FIRST VOLUME. LONDON: H. BAILLIERE, 219, REGENT STREET, AND 290, BRoADWAY, NEW YORK. PARIS: J. BAILLIERE, Rue HAvTEFEUILLE. 1862. po PHanchester : Printed bo Charles Sinung and Co. CONTENTS. 5 pat ARTICLE PAGE I.—On the Origin of Colour, and the Theory of Light. By JOHN Smit, Perth Academy...............cs00008 Sees I II.-— On a Method of Testing the Strength of Steam Boilers. Tey do By U@wadey IBID ED) JOG HShs (6 G5 ooonaostepopobdHedboe 97 III. — Experiments on the Total Heat of Steam. By J. P. Jovzz, A TALE] De 1 MAS BAe oS ad dsc nape Hae Sse ORR SCEr Ee aBERa Har 99 IV.— Experiments on the Passage of Air through Pipes, and Apertures in thin Plates. By J. P. Jourz, LL.D., JL RAIS A552 ei naa Men RUG ASH NSREOSe Spd EOC AOHodriG HARE ED OC CCe eRe 102 V.— Supplementary Researches in the Higher Algebra. By James Cockie, M.A., F.R.A.S., F.C.P.S., Barrister- Ab Ma Wea greee ea eeyiceslecasseetnuerrtacaisecinememeseesosscn vadursese 108 VI. — Remarks on the Australian Gold Fields. By W. S. REWVON Site sere eee eee ce ita sc onatanmiee testi ciaatici te eammnnee ce mes 115 VII. — On the Vestiges of Extinct Glaciers in the Highlands of Great Britain and Ireland. By Epwarp Hutt, E.G.S. of the Geological Survey of Great Britain...... 131 VIII.— A Brief Memoir of the late John Kennedy, Esq. By WALTAMY HATRBATEN, HEISQoi ea. teisceeucesssescuncnecedccess 147 IX. —- Suggestions for a New Form of Floating Lightship, and a mode of estimating the distances of Lighthouses. By ATBRD CURVE Rs Gree ceicateisosliseie cassie esse seccurseedelses 158 X.—On the Influence of Atmospheric Changes upon Disease. By Artuur Ransome, M.B., B.A. Cantab. M.R.CS., and GrorGe V. Vernon, F.R.A.S., M.B.MLS. ......... 164 XI. — On the Phenomena of Groups of Solar Spots. By JosnrH DAXHNDELE, H)EeAC Se: sccusccivenscscinesesssesicccactetineess 186 XII.— On the Rotation of Jupiter. By JosEPpH EAAINDELD, HMR VAC Ses itinane sation tieomenaescl sa veucdsannuetovecesecuneweleee 194 XIII. — On the Estimation of Sugar in Diabetic Urine by the Loss of Density after Fermentation. By Wuittiam Roserts, M.D. London, Physician to the Manchester TRON) bid bad rhane} ev - he Gonpocoeca roomed oo dda GubcabbOsOBaNcnACH Cae 200 V1 CONTENTS. ARTICLE XIV.— On the Alleged Practice of Arsenic-Hating in Styria. By Henry Enrierp Roscoz, B.A., Ph.D., Professor of Chemistry at Owens College, Manchester ...... EodotA000 XV.— On the Production and Prevention of Malaria. By Dr. RVANGUS | SMITH FHORI Ss: ccsccesseocesmesateneseeceeeeeeee XVI.— On the Kaloscope. By Mr. W. H. Heys. Communi- cated by Mr. GEORGE MOSLEY ...........ccecececeeneeses XVII. — On Meteorological Observations, and Observations of the Temperature of the Atlantic Ocean, made on runs from Liverpocl to Gibraltar and from Gibraltar to Liverpool in September 1860. By TxHos. Huet, WR GASS Seo. cevie set atenss vopiesteu che sseassieih eaasuanetantonte XVIII.— On Changes of Density which take place in Rolled Copper by Hammering and Annealing. By Mr. CHARLES O'NEILL ........ sie defer sopsfer een sisloie/aeette gatteeetar XIX. — On the A-faced Polyacrons, in reference to the Problem of the Enumeration of Polyhedra. By Mr. A. Cayizy XX.—On a System of Periodic Disturbances of Atmospheric Pressure in Europe and Northern Asia. By JoszpH BAKENDELD, UR GACS) cc sce snecesceanseseccereGeeeneeeeane XXI.— On the Irregular Oscillations of the Barometer at Man- chester. By G. V. VERNON, F.RAS. .........cccece ees XXII.— On the Theory of Groups and many-valued Functions. By the Rev. Tuo. P. Kirkman, M.A., F.R.S. ......... XXIII. — Remarks on the Theory of Rain. By JosrrH BaxzEn- DELU,(H, RAs scccsasinedes sanaee cess sauwoencieer eee eeeeeeten XXIV.— On the Structure of the Luminous Envelope of the Sun. By Jamus Nasmytu, Hsq., C.E. In a Letter to JOSEPH SIDEBOTHAM, Hisq...........scscecsccccensesees APPENDIX. -- Annual Report of the Microscopical Section of the Manchester Literary and Philosophical Society, 1860- 1861 ee ee ee ee ee er Donations Received. Sessions 1859-60 and 1860-61. Council and Members of the Society, April 30th, 1861. PAGE 208 222 234 238 243 248 257 265 274 399 407 LIST OF ILLUSTRATIONS. Diagrams, illustrating Mr. Smith’s Paper on the “Theory of Mightemeetiveplabesimeeeeeeee mesa ectee cece nee to face page Section of the supposed Geological Formation of the Gold Fields OfaViictorias-Australiaiens. sic ctecccucsocecceme noone Marcncecuses: Plan of a new form of a Floating Light-ship .............ececceeeees Charts, showing the Influence of Atmospheric Changes upon IDF EGREO onsbagealdan cs sae asodosdadcnondad een den pagar SSB SBUEERSSabaS Diagrams illustrative of Mr. Cayley’s Paper on the A-faced Poly- ACTONSY OCs ose cs ce sescnectnaseeeee Pec eic clones tsharcteln cave sis atsmrcleates Diagrams illustrating Mr. Baxendell’s Paper on a “System of Periodic Disturbances of Atmospheric Pressure in HuLope end INOrtherny Asiaiies cs socncensssoseecese enacts Monthly Mean daily amount of the Irregular Oscillations of the Barometer at Manchester and Dublin........................ Illustrations of Mr. Nasmyth’s Paper “On the Structure of the Luminous Envelope of the Sun” SO ee re aed 96 130 162 184 256 264. 272 NOE. The Authors of the several Papers contained in this Volume, are themselves accountable for all the statements and reasonings which they have offered. In these par- ticulars the Society must not be considered as in any way responsible. MEMOIRS OF THE LITERARY AND PHILOSOPHICAL SOCIETY OF MANCHESTER. I. — On the Origin of Colour, and the Theory of Light. By Joun Situ, M.A., Perth Academy. Read October 4th, 1859. PART I. Or every science the elementary laws ought to be pro- foundly studied and accurately known, but of none more so than the science of light; and yet perhaps there is no physical science, whose laws, as at present interpreted, give less satisfaction to the intelligent student of natural philosophy. The theory of light as unfolded by Newton is a mathe- matical description of a physical phenomenon, for the arguments on which it is founded are derived chiefly from the geometrical form taken by that phenomenon; and the wave theory is little else than Newton’s theory divested of the emission hypothesis. But to describe a phenomenon geometrically is one thing, to investigate SER. III. VOL. I. B 2 MR. J. SMITH ON THE ORIGIN OF COLOUR physical principles is another. Should a physical pheno- menon assume a geometrical form, a mathematician may argue upon it and draw from it some valuable infer- ences without understanding more than the geometry of the phenomenon. The reasoning from the premises may be strictly correct, the deductions mathematically sound, and still the conclusions far from being physically true. The reasoning being founded on geometrical, net on phy- sical facts, only explains such phenomena as are described. Hence the general dissatisfaction with the theory of the different refrangibilities of the rays of light as unfolded by prismatic refraction. The theory resolves itself into the geometrical description of a physical phenomenon, it affords no explanation of the physical process ; and as far as the analysis of the physical process is concerned, it is chiefly an attempt to prove that because a ray, after being once refracted, cannot be changed by repeating the same process, it is to be inferred that the ray is reduced to its simple elements; thus assuming, not proving, that the prism is an analytical instrument. 2. To the general student a subject is supposed to be made clear when it is described in mathematical language assisted by geometrical diagrams. But there is nothing in which we are more apt to deceive ourselves, in studying science, than in mistaking the knowledge of the language in which a phenomenon is described for a knowledge of the physical laws, on which the phenomenon depends ; and in the study of nature we often deceive ourselves by generalizing without sufficient investigation; for like the student of science the student of nature is apt to mis- apprehend the language in which nature speaks; he not unfrequently mistakes geometrical laws for physical, and explains the one by the other. He begins to generalize when he discovers that physical laws are in accordance with some of the rules of his previously acquired geo- AND THE THEORY OF LIGHT. 3 metrical knowledge — because, for example, an angle is found to be the measure of force in one instance it is concluded to be the measure of force in others. The laws deducible from the phenomena of the solar spectrum are of this description, and they are certainly very remark- able; but in their application we are constantly compelled to introduce emendations and corrections, to depart from true logical inferences, and to adopt suppositions to satisfy ‘the conditions of other physical problems. 3. I make these observations because I find that how- ever easily ordinary minds are satisfied with the theory of prismatic refraction, all scientific investigators have more or less felt the difficulty of the subject. Neither Sir David Brewster nor Sir John Herschel, as far as I understand them, implicitly acquiesces in the doctrine of Newton, and these are as high authorities on the subject as can be obtained. The author of the article “ Undulaticns” in the Penny Cyclopedia concludes in these words: ‘“ Much stress is laid on the accuracy with which the phenomena of diffrac- tion are accounted for on the undulatory hypothesis ; but while there yet remains unexplained by that hypothesis so important a circumstance as the different refrangibilities of light, which are satisfactorily accounted for on the cor- puscular theory, and while our knowledge of the action of material particles on one another, as well as of the pro- pagation of motion through elastic media is so imperfect, philosophers seem to be fully justified in suspending their judgment concerning the relative merits of the two rival systems.” 4. The experiments on light which I am about to explain lead to other conclusions than have hitherto been obtained from the study of the prism. The data are different, and reasoning from the data the conclusions may be expected to be different. It is only by searching for other data A, MR. J. SMITH ON THE ORIGIN OF COLOUR that any improvement or progress can be made in our knowledge of the laws of light; for, judging from the past history of science, human arguments without experi- mental data would lead us farther and farther from a knowledge of divine laws. Besides, who would rashly encounter, by bare argument, the philosophy of the age; or enter on the study of the prism, without secking new facts, observations and experiments in defence of a theory opposed to that by means of which so many illustrious investigators explain the facts unfolded by prismatic analysis? I have therefore ventured on another field, and one, I apprehend, much more easily cultivated by the mass of mankind. In the study of light by means of the phenomena of the prism, although it has engaged the attention of thousands of investigators, we are still met at the very threshold by the question, What is a prism? Until this question is answered our progress must be very uncertain. If experiment answered the question it would be all well, but the refraction of the rays of light throws no gleam of sunshine on the internal construction of the instrument ;— how then in these circumstances can the prism be considered as an analytical instrument? That it is not such every truly scientific investigator has virtually acknowledged, and still he adopts the inferences derived from its study as the foundation of his inquiries into the physical nature of light. Were the prism an accurate analytical instrument it would leave no doubt on the minds of philosophers as to its indications ; few inqui- rers, however, agree in this respect. Newton saw seven distinct colours, some have seen only four and others three; and even those who agree as to the number of colours in white light differ as to what the fundamental colours are. Of those who allow only three, one party maintains that the three are Red, Yellow and Blue, another that they are Red, Green and Blue; a discrepancy which AND THE THEORY OF LIGHT. 4) demonstrates one thing, that there is no distinct analysis by means ‘of the prism. There is thus room left for a new investigation into the cause of colour. 5. In my inquiries into the nature of light I had no fine spun theory to unfold, with its axioms and postulates, by which to demonstrate theorems and solve problems. I was in search of all these. Hence I feel more disposed to describe the mental process by which I was led to un- dertake the experiments which I have made. I prefer describing the observations and experiments which I made whilst examining the natural phenomena of light, and the arguments which I deduce from them, to formally build- ing up ascience. At all events, experiment alone can be accepted as the basis of any new theory, for, in physical researches, no logical or mathematical analysis can take the place of experiment. Has not the great obstacle to the progress of the science of light for these two hundred years past been the building a science mathematically on insufficient experimental data; for if there is one thing more certain than another in regard to the physical nature of light, it is that the data for mathematical calculation are imperfect, and there is nothing which retards the search after truth more than the application of mathema- tical formule to imperfectly ascertained truths. They are then to a great extent assumptions. It may make our researches appear more learned to apply the exact sciences in the illustration of them; but how can the exact sciences be applied to observations not exactly comprehended? I consider, therefore, the exposition of a series of observa- tions a much more scientific and instructive method of arriving at a physical law than any mathematical assump- tion which, though apparently solving many problems, carefully conceals the physical process and unfolds no method of physical research. Mathematical reasoning is not physical reasoning, and is as easily abused as the 6 MR. J. SMITH ON THE ORIGIN OF COLOUR latter, and indeed is so more frequently, when applied to subjects of natural philosophy. 6. To investigate the physical nature of light, to inquire whether the corpuscular or the undulatory theory be true, may by some be considered as a purely speculative or metaphysical discussion resulting in no practical benefit. It matters not, it is said, as far as mathematics are con- cerned, whether light is an emanation of particles from the luminous body, or the undulations produced by the same body in a highly elastic medium, “for the principal phenomena of light are deducible by strict mathematical reasoning from either supposition.” But until mathema- tics shall be proved to be the only true basis of physical reasoning, such an argument for resting satisfied with what knowledge we have of the nature of light is not at all creditable to science. ‘True philosophy in process of time leads to grand universal laws; false philosophy, or suppositions however ingenious, lead to partial, excep- tional and restricted views. Galileo when he was demon- strating by the use of his telescope that the planet Venus had phases similar to those of our moon, and was conse- quently illuminated by the light of another body and not shining by its own, was doubtless considered by the phi- losophers of the age as being engaged in a much more sublime investigation, because a more popular one, than when he was demonstrating the law of falling bodies. For little was it thought that by solving the problem of the pendulum he was solving the most comprehensive problem ever solved; that he was teaching to another ereat investigator the method whereby he was not only to dissipate the mystery of planetary motion but to. weigh the matter of the universe itself. If then, mathematically, it be a matter of indifference what is the true nature of light, it is not so philosophically. What tends to elucidate the true nature of hght, may AND THE THEORY OF LIGHT. 7 aid in tracing the true theory of heat, magnetism and the other imponderables. 7. But there are other questions respecting the physical nature of light than the decision of the truth of the cor- puscular or undulatory theory. It is equally important to know whether it be a homogeneous or a heterogeneous substance. The grand discovery of comparatively modern times is that of the heterogeneous nature of light; and the facts discovered seem to adapt themselves to, and to be explicable by either theory, for like mathematics, in the opinion of some writers, the compound nature of light is consistent with either theory. But surely it is more than a matter of idle curiosity to decide this question. If light could be proved to be a simple substance, the theory would be rendered more conformable to the recognized simplicity of nature’s laws, and would thereby commend itself more to the human intellect, which estimates the divine authority of a law by the universality of its appli- cation and its simplicity. The sublimity of the law of gravitation lies in these two qualities — its universality and its simplicity. 8. I have already remarked that, from the discrepancy that exists among philosophers as to the composition of light, there is room for a new investigation into the cause of colour. In the present day however, such an investiga- tion could not be made by means of any refracting medium. Refracting media are already explained, or believed to be explained, on arguments conventionally acquiesced in as established truths, or at all events, as truths which it would be presumptuous in any one in this era of philosophy to question, either by logical or mathematical reasoning. I propose therefore I. To give the result of my study of the laws of light without the aid of any refracting substance. 8 MR. J. SMITH ON THE ORIGIN OF COLOUR Il. To attempt an explanation of prismatic pheno- mena by means of the laws which have been discovered. And III. To illustrate my views by experiment. g. Before entering on the mechanical experiments by which the subject is to be illustrated, I wish to direct atten- tion to two or three phenomena which were the cause of, first of all, creating dissatisfaction in my mind with the. received theories of light, and which I consider, when pro- perly understood, will serve as the basis of the true theory. I shall, however, have a difficulty in making myself under- stood, as I shall be obliged to use a term the adoption of which will not be easily acquiesced in by scientific men as having any physical value until the operation it points out is actually brought under the cognizance of the organs of vision, and even then there may be a difficulty in defining it. The first of the phenomena referred to is the following. Experiment I. — Wafer Experiment. 10. A not uncommon experiment is to fix two wafers on a pane of glass or a sheet of paper, about four inches apart, and to look at them with oblique or strained vision until the one is seen to overlie the other, and the two appear to be one. In making this experiment a great many years ago, I tried it with variously coloured wafers. A red and dark blue, nearly black, gave in a favourable light a green for the single image. A black and a red gave also a green, sometimes a brown and at other times a purple, the differ- ence depending on the nature of the colours and also on the luminous state of the atmosphere. Having never seen this phase of the phenomenon no- ticed — and I searched for some notice of it in vain after my attention had been particularly directed to it — I AND THE THEORY OF LIGHT. 9 attempted an explanation, to the best of my ability, ac- cording to the received theories of light, but could not satisfy myself. 11. The phase of the experiment to which I wish parti- cularly to direct attention is this: the image of a red wafer being on one eye, the image of a black wafer on the other eye, the resultant of the two, in a favourable light when the one is made to overlie the other, is neither red nor black, but some other colour. Why another colour? Why not red? According to the corpuscular theory black is not an active principle, it is a negation of colour, and should produce no effect on the red. It exacts too much from our faith to believe that two molecules combine some- where beyond the retina to effect a change of colour before reaching the sensorium. And there is no change in the process of refraction of the red to account for the change of colour or to help us to an explanation. 12. On endeavouring to find an explanation, according to the undulatory theory, I found the same difficulty. Black is not an active principle on the undulatory theory any more than on the corpuscular. If the quality of a colour depends on the length or form of a wave, there is nothing apparently in this experiment to alter the form of the wave, nothing to lessen the number of vibrations, or to reduce its intensity. Why then a change of colour, let the colour be what it may? 13. Trying an explanation of the phenomenon by the theory of complementary colours, there were difficulties equally great to surmount. The retina is not fatigued although the muscles of the eye are,* and there is no re- actionary force to account for the colour on the commonly received interpretation of these optical phenomena; so that I failed to find a solution in whatever way I attempted it. * The eyes may be injured very much in making this experiment. SER. III. VOL. I. Cc 10 MR. J. SMITH ON THE ORIGIN OF COLOUR 14. We are apt to think it easy to understand that red and blue become green, or that red and black make a dark dirty green, for we see it often take place; but when we ask ourselves the physical cause of such an effect as that produced by our experiment, it is not so easy to discover one; for according to the commonly received interpretation of the laws of light there is not a sufficient cause for the effect. Both images appear to be seen, but how can the one be seen through the other, or how can they appear transparent, but on the supposition that there are intervals in the movements of the rays from the different objects ? We have thus as it were the first step in an argument for the undulatory or rather the vibratory motion of light. 15. In this experiment we have two sensations which differ from each other on the retina, and yet before they reach the sensorium are compounded into another sensa- tion of a different character. Hence the question natu- rally arises: If the composition of forces or of colours takes place in the organs of vision so simply, does it not presuppose a greater complexity in the laws of light, as commonly taught, than is to be found in any other of the works of God, to affirm that light is a compound hetero- geneous substance not only requiring continual decompo- sition but perpetual recomposition? Is it not introducing into the theory of light ideas as perplexing as the cycles and epicycles which existed in astronomy before the time of Copernicus ? 16. After carefully considering the phenomenon in all its bearings, the only satisfactory explanation I could arrive at was, that the two insulated spots, one on the one eye and the other on the other eye, could only be per- ceived as one object not as two, for the impression at all times received by two eyes of the same spot is single — hence, although we have two eyes, we have virtually or sensationally but one retina — therefore in this case the AND THE THEORY OF LIGHT. ll change of colour was the effect of the pulsations of light from the red wafer alternating with the sensation of black. But there was a difficulty even here, for the impression of red was not changed when the other eye was completely shut. In answer to this I argued, that there could be no sensation where there was no agitation of the retina. The black wafer could not have been seen where it was, unless the rest of the retina had been agitated with light. In an eye which is shut, every thing is in a state of repose, and there is nothing to interfere with the sensation pro- duced on the other eye, for there is no contending sensa- tion. It is the mixing or uniting of these independent sensations which produces the sensation of change of colour. Black then may be considered as a colour in so far as it is a sensation. 17. If the two retinas are intellectually one, and if each individual retina produces its own impression, it is evident that the number of vibrations on the sensorium must be diminished one half if only one eye is used. The two spots are perceptionally or sensationally one spot, but when considered intellectually or philosophically, the per- ception is discovered to be a compound of two codrdinate sensations, and being codrdinate they are sensationally one. 18. But it may be argued that the wave of red light suffered a change before entering the sensorium, for the fibres of the one eye being in motion and the other at rest or quiescent, a diminution of velocity at the junction would be the consequence. This should not affect the length of the impinging wave, which is considered as the cause of colour, if red is an unchangeable element; but even if it did, this supposition would not account for there being no change when the one eye was completely shut, although the same process must still go on. 19. The unavoidable inference from this experiment 12 MR. J. SMITH ON THE ORIGIN OF COLOUR then is that colour is produced by intervals of light and no- light. Had there been no intervals in the motion of the incident red ray, it is scarcely possible to conceive that there would have been a change of colour. But a change of colour being perceived, it is but natural to suppose that this change must be caused by the colour or no-colour of the spot against which an object is seen in the intervals of pulsation of the light by which it is made visible. Doubtless, as the number of pulsations of the red was not diminished, and only half of the sensatory nerves were ex- cited, there may be two suppositions; we may either infer from this experiment that the number of vibrations to produce green is only half of those of red, or that only one half of the force is necessary to produce the one colour which it takes to produce the other. Indeed both sup- positions may be true. But I am rather inclined to think that this experiment proves that each particular colour does not depend so much on the number of vibrations in a given time, the usual solution, as on the necessity of two inde- pendent sensations. For although the colour is changed, the vibrations from the red object are not changed. When LT look at a red object the vibrations continue the same, and the impression is not altered ; but if I take a red object and make it produce vibrations alternately with a black one, I get the sensation of green, which is exactly what takes place in this experiment, and which I wish to im- press, although there is a difficulty in mastering the physi- cal process. There is this difference in the two processes ; in the wafer experiment the vibrations of red are not re- duced, but when I make red vibrate alternately with black, I reduce the number of vibrations of the red; but the same process goes on, for the red and black, alternating with each other, become virtually two retine looking at different objects with exaxlar vision. 20. With this experiment then before me, and these AND THE THEORY OF LIGHT. 13 arguments in support of the explanation of it, it was almost impossible to resist the conviction, that shadow must perform a most important part in the formation of colour, and take its place in every theory of colour. 21. But how is this to be proved? No one can study light at present without at the same time studying the theory of Newton, and Newton has attempted to prove the very reverse. He has completely abolished shadow from his theory of colour. For, after having reflected, as he says, the various colours of the prism on the “confines of shadow,” he concludes “that all colours have themselves indifferently to any confines of shadow, and therefore the differences of these colours from one another do not arise from the different confines of shadow whereby light is variously modified, as has hitherto been the opinion of philosophers.” This is quite decided. But so is our experiment. The above experiment tells us, that a force which produces red when red is seen by itself, no longer produces red but green, brown, or some other colour, as the case may be, when the red is seen alternately with black or shadow.* TI use shadow in Newton’s acceptation of the word, and in that of the ancients; I shall use it in a modified sense by and by. 22. Newton’s arguments are drawn from the experi- ments which he made with the prism. But the phenomena of the prism, being so difficult of explanation and com- plex, should be compared with other phenomena of a more simple kind, and not with changes perpetually rung upon itself. Such a method of experimenting seems to * T name these various colours as the effect is not constant but depends much on the state of the atmosphere. And this is not to be wondered at in this instance, where there is a physical difficulty in making the experi- ment, since Sir John Herschel says, and every one who has experimented can verify what he says, that the image of the sun produced by the prism ‘‘varies enormously.” 14 MR. J. SMITH ON THE ORIGIN OF COLOUR me much like arguing in a circle, always ending where you began, and, without knowing it, starting again on the same track. One of the most elegant writers on physics we have says: “This solar spectrum, as it is called, formed upon the wall, consists, when the light is admitted by a narrow horizontal slit, of four coloured patches corresponding to the slit, and appearing in the order from the bottom of red, green, blue and violet. If the slit be then made a little wider, the patches at their edges overlap each other, and, as a painter would Say, produce by the mixture of their elementary colours various new tints. Then the spectrum consists of the seven colours commonly enumerated and seen in the rainbow, viz., red, orange, yellow, green, blue, indigo and violet. Had red, yellow, blue and violet been the four colours obtained in the first experiment, the occurrence of others, viz., of the orange from the mixture of the red and yellow, of the green from the mixture of the yellow and blue, and of the indigo from the mixture of blue and the violet, would have been anticipated. But the true facts of the case not being such proves that they are not yet under- stood.” He adds: “No good explanation has been given of the singular fact of refraction.”* Why then constantly appeal to the phenomena of the prism, when so little satis- factory is known about it? The most obvious method of procedure is to inquire in such cases of nature in her ordinary moods, how she performs such and such opera- tions; to inquire how colours are usually produced, and if there are any simple processes which can be explained and received as data for future investigation. 23. Adopting this method of investigation, the first inquiry which suggest itself is, What is Blue? * Dr. Arnott’s Hlements of Physics. AND THE THEORY OF LIGHT. 15 How is this colour produced in ordinary cases in nature ? Experiment IT., or Card Experiment. 24. If we take a piece of white Bristol board and paint on it with lamp black any figure whatever, and hold it up between us and the light, the figure will appear purple in place of black. If the board is then inclined, so as to allow the side nearest to us to reflect a little more of the light entering the room, the purple will now be changed into a faint blue. This demonstrates, not as Newton said that there is blue in black, when he found that he could obtain blue of a certain order by reflecting black on white, but that black or a shadow and the reflected light plus the light transmitted through the card produced the impression of a faint blue. For we have here black in one light appearing as purple, and the same black in an addi- tional light as pale blue. This experiment is adduced as another argument that shadow has much to do with colour, and as a step in the process of reasoning by which I arrived at what I consider the mechanical demonstration of the cause of colour. 25. 1 am aware that this and similar cases are consi- dered as anomalous, for which special explanations must be framed; and solely because it has been considered demonstrated that white light is heterogeneous in its nature. For instance, it will be said of the first phase of this experiment that there was a purple in the lamp black, and that the purple only was transmitted, and in'the other case that the blue only was reflected, the other rays being absorbed. This is merely, however, ringing a number of changes on words without any positive or experimental facts from which to argue. Such arguments have one advantage; they are easy, and can surmount any diffi- culty. 16 MR. J. SMITH ON THE ORIGIN OF COLOUR 26. But in attempting to introduce a new term in explanation of the cause of colour, in attempting to speak of shadows and penumbree, there are great difficulties attending the subject, for optical science is not familiarized to the use these terms are put to, and besides the negative — element is as undefined as the positive; for between the faintest shadow and absolute darkness there are an infinite variety of shades, just as there are between the faintest light and the brightest, if indeed there is a limit to either. But if white light be admitted to be composed of an infi- nite number of rays crossing each other at all angles — or of one great wave — then the interception of any amount of these rays or of any portion of the wave on any given spot is a shadow. There must be a body to intercept the light, and that body may be either opaque or transparent, for the most transparent body does not transmit all the light which falls upon it. ‘This shadow is generally some degree of grey, unless it be illuminated by a vivid ray at an angle to the ordinary ray, in which case it is coloured. But it is so customary to argue on this subject from the phenomena of the prism, that I shall be told that in all coloured shadows this light had been previously refracted, and the shadow did nothing more than prepare a place on which to reflect it, and prevent its being extinguished by a greater light. Whereas, on the contrary, there is more reason for saying that the number of vibrations being reduced by the interposition of a substance causing a shadow, the light reflected from this shadow alternating with the shadow itself produces a blue, purple, or red effect. In our ignorance of the value of the ray imping- ing on the shadow, as well as of the value of the shadow (for it is well known that shadows have different values), we have no authority to call them refracted but from a belief in the received theory. 27, Were a ray of light like an infinitely fine wire AND THE THEORY OF LIGHT. 17 reaching from the sun or luminous body to us, and were there rays of every imaginable colour, —or were a ray of light a constant or uninterrupted succession of molecules thrown from the sun or luminous body, which would be the same in effect as the infinitely fine wire,—there would be no need of shadow in a theory of colour, provided there were also a chromatic elective affinity between each terres- trial object and its special ray. But if light is the effect of pulsations repeated at intervals, darkness or shadow must play a most important part in chromatics, as important a part as it does in the natural landscape or in aerial per- spective. 28, I therefore feel more disposed to speak in the lan- guage of the painter than in that of writers on this subject; for, although the experiments which I have to bring forward demonstrate, as I believe, the vibratory nature of light, I think it will be more intelligible to speak of light and shade, which can be made visible, than of the number of vibrations or form of waves, until these have been demonstrated or exhibited to the sight. 29. Besides, it is almost impossible to believe that there should be a system of rays of different velocities in the luminous ether, or that some of the molecules of ether should be disposed to be differently affected from others. It is possible, however, to conceive, nay, it is almost im- possible to resist the conviction, that all natural objects are formed on strict geometrical principles ; that some are capable of reflecting more rays and others fewer; that some are more sensitive to light than others, and that the varying degrees of light from different substances produce an equivalent effect on the retina. Intensity is the effect of repeated pulsations on the same physical point. Of course it is possible that different degrees of intensity may produce different lengths of waves on the retina, or within the medium of refraction, and thus cause colour; SER, III. VOL, I. D 18 MR. J. SMITH ON THE ORIGIN OF COLOUR but no one has seen these waves or has obtained any proof for their existence. Consequently, it seems to me as use- less to talk of the forms of waves of light as of the forms of waves of gravitation, of the forms of the electric, gal- vanic, or magnetic waves, or of the waves of heat, until something more is known about them. By building a system on hypothesis, we not only bewilder ourselves by making ourselves believe that we are wiser than we are, but we put a drag on science. 30. I fear I may have introduced this part of the subject too soon, by discussing abstract points before I have de- scribed the experiments which lead to such discussion. This could scarcely, however, have been avoided, for the idea of length of wave is one which meets us at the very threshold of the science, and will continually present itself under va- rious aspects, called forth by the nature of the subject and the processes it involves, so that it will be very difficult to treat of pulsations without supposing the existence of waves of different lengths. And as my principal object is to sub- stitute a new element in the place of the different refran- gibilities of rays, or of the lengths of waves, neither the one nor the other can ever be forgotten or omitted to be put in contrast with the proposed substitute, or the nega- tive element, which is the subject of my inquiries. 31. So far, then, as my argument has gone, neither the emission nor uudulatory theory of light, as they are com- monly taught, can account for such phenomena as are exhibited in my experiments; it is, therefore, necessary to look for explanations of a much more general character in order to account for such common and every-day appear- ances. 32. To proceed then with our investigation. The fol- lowing well known experiment, which has received an explanation generally acquiesced in, will help us in the examination of this important subject. AND THE THEORY OF LIGHT. 19 Haperiment UH1., or Candle Experiment. Light a candle in daylight, and observe the shadows it casts on the window blind or on a sheet of white paper : they are sometimes blue, at other times purple. The spot on which the shadow is cast, both before and after the candle is lighted, will be, to all appearance, white, — at least the eye can detect no colour; but after the candle is lighted, the spot on which the shadow falls is seen to be coloured. What part does the shadow perform here? Does it merely remove the light of the candle from the spot, and enable us to contrast the light of the sun with the light of the candle? It does more. It shows in this case also that colour is connected with shadow. 33. The received philosophical interpretation of this expe- riment is to the following effect. I quote from Dr. Young’s works: — “If from the light of a candle we take away some of the abundant yellow light, and leave or substitute a por- tion actually white, the effect is nearly the same as if we took away the yellow light from white, and substituted the indigo which would be left; and we observe, accord- ingly, that in comparison with the light of a candle the common daylight appears of a purplish hue.” Let us look for a moment at this explanation, and see what we can make of it. He says, If we take away some of the abundant yellow light from the light of a candle, and leave or substitute a portion actually white; but he does not tell us how to perform this very nice experiment. True, he puts it in ovr power to leave or substitute a portion actually white; but this only makes the experiment more mysterious and more difficult to perform. It would appear as if the remarks of Dr. Young presupposed some power of working with a compound beam of light as a painter works with his colours on his pallet. He adds — suppo- 20 MR. J. SMITH ON THE ORIGIN OF COLOUR sing this to be done — “ that the effect would be the same as if we took away the yellow light from white, and sub- stituted the indigo which would be left.” This I cer- tainly will not attempt to deny. I would, however, ask, Did any man ever perform such an experiment on light ; or could any one perform, or even understand, it? Who ever saw the white light after the yellow was taken from it ? or, who knows what the light of a candle is after the abundant yellow is taken from it? Is it the light of the sun, or what? But, according to Dr. Young’s theory, there is no yellow to take from white. Even allowing, for the sake of argument, that the yellow — which I sup- pose is some proportion of red and green, according to Dr. Young’s hypothesis — can be taken from the white, are we, therefore, to conclude that indigo would be left? We are thus compelled to make two suppositions, unless we can. prove by experiment that such would be the case. It may be fairly asked, Is common daylight made up of only two colours, yellow and indigo? Of course Dr. Young supposes that it is, for his argument here would require it to be so; whereas he says in another part of his work : “We may consider white light as a mixture of red, green and violet — only in the proportion of two parts red, four green and one violet, with respect to the quantity or intensity of sensations produced.” If white light is com- posed of red, green and violet in these or any other pro- portions, how can we take yellow from it and leave violet ? We might be able to compose yellow of certain propor- tions of these colours; but how we could take yellow from white and leave“indigo, on such a supposition, is not easily to be comprehended, unless we make the bold supposition that all the red plus all the green is equal to yellow, which we are nowhere told is the case, and cannot prove. Yellow and indigo must be by this theory compound, and not primary, colours, and consequently could not be taken or AND THE THEORY OF LIGHT, 21 left by any analytical process known.* Again: it is said, “that common daylight appears in comparison with the light of a candle of a purplish hue.” He does not now call common daylight white, but he virtually says that the white light of day is purple compared with the abun- dant yellow light of a candle. What, then, is white? or, what is the light of a candle? 34. There is no contending against such a method of reasoning as this of Dr. Young’s; the premises are con- stantly shifting for want of intelligible elementary prin- ciples. If every new case such as this, in any theory of colour, requires a new hypothesis or some new supposi- tion, to help out an explanation, there is evidently no science, unless conjecture can be dignified with that name. Science requires a general law which shall apply to every case; and in the illustration of that law we must use lan- guage which can be understood by the ordinary intellect, or it is not axiomatic; we must appeal to experiments which can be seen to be axiomatic or purely analytical. Without axioms we have no basis on which to build our reasoning ; and as the axioms of physical science can only be founded on experiment, to appeal to phenomena which cannot be subjected to experiment, as that of taking away the abundant yellow from the light of a candle, and leav- ing or substituting white, as Dr. Young has done here, is virtually to abandon philosophy to support a theory. 35. No part of Dr. Young’s explanation can be admit- ted as correct, for even the comparison is not of daylight with the light of a candle. The screen is illuminated with the light of day plus the light of the candle, and the sha- dow is a comparison with the light on the screen, not with the light of the candle. The pitch or tone of the luminous ether is raised, and the eye or retina estimates * T suspect indigo and violet, and even purple, are often used as synonymes by writers on light. 22 MR. J. SMITH ON THE ORIGIN OF COLOUR the united effect of the two lights as a unit. It is only by casting a shadow that the light on the screen is known to be physically compound, and it is by the shadow being coloured that the intellect discovers, as in the wafer expe- riment, that there must be vibrations or intervals in the motion of both luminous waves or there could be no colour. Besides, this white light must have a certain intensity or the shadow could not be coloured, for as soon as the daylight begins to lessen, the shadow begins to get grey, even although the screen may appear quite white when the candle is extinguished. 36. But this is not all the experiment. Let us light another candle. The united effect on the screen of the daylight and the two candles will still appear white and as originating from one source, but there will be now two shadows of any opaque body; properly speaking, three, for the daylight also produces a shadow; but we are only speaking at present of the shadows caused by the candles. Make the one shadow partly to overlie the other shadow, then the part where the shadows overlie will be of a much deeper tinge of blue than the other parts. This deeper shadow, when estimated by the eye, appears of more than double the intensity of the others. We need not attempt to count beats or pulsations of the ether at present; but supposing the candle to give the greater shadow, let us imagine the effective pulsation of the light of each candle to be double that of daylight. On the screen the number of pulsations would be increased, and consequently also on the retina; and on this supposition there would be five pulsations where before there was only one, and still the eye would estimate the whole five only as one. The pul- sations would be one of daylight plus four of candlelight, two for each candle. On the darkest shadow there would be one pulsation of light, namely, one of daylight; on the shadows not overlying there would be three pulsations of AND THE THEORY OF LIGHT. 23 light, one of daylight plus two of candlelight. Here we have the light of day as compared with that of two candles of a dark purple or blue, and the light of one candle plus the light of day compared with that of two candles of a lighter purple, which appears to be hardly in accordance with the principles of logic. For, according to this method of arguing, if we remove the daylight entirely, the light of one candle should appear to the light of two candles as a grey, of which it would be difficult to convince our senses. 37. This is a most instructive experiment, one on which the whole theory of light might be based, had it not already received the form generally acquiesced in. But we shall find that it cannot be explained by means of either theory of light. According to the emission theory there should be no change in the colour of the white, for there has been no change in refraction; every thing externally is just as it was, the daylight not being changed by the light of the candle: and according to the undula- tory theory there should be no change in the white light of day, for there has been no change in the wave of light; its intensity has not been diminished nor increased; and the undulatory theory makes no allowance for comparison. One wave may be more intense than another, but it does not on that account, — unless on the doctrine of inter- ference, and interference can only exist under special conditions — thereby diminish the intensity of the less. intense; much less can comparison change one wave into another, or, in other words, according to the recognized principles of both theories, decompose the white light into its constituent parts— change the heterogeneous white into the homogeneous blue. If the power of comparison can accomplish so much, why not let it accomplish the whole? why not make it the cause of colour at once? But this is just another instance of departure from or 24. MR. J. SMITH ON THE ORIGIN OF COLOUR shifting of logical premises to meet an anomaly, not in the nature of light but in the received theory. 38. This experiment fortifies my argument very strongly, and I have less and less hesitation in saying, in opposi- tion to the received theories, that shadow appears to be as important a principle in any theory of colour as centri- fugal force in the theory of gravitation, as important as the term minus in the theory of numbers, as important as the term silence in the theory of music. Colour of the Sky. 39. It is necessary, however, to search for other natural phenomena in order to verify this conclusion, and among these none stand more prominently forward than the colour of the atmosphere. The colour of the air and sky is explained by every writer on optics, and the common theory is universally admitted to be correct. But it is easier to acquiesce in the fancy of the poet when he sings, >Tis distance lends enchantment to the view And robes the mountain in its azure hue, than in the science of the philosopher. As Newton’s opinion is still the opinion of the present day, in place of quoting Newton, I shall quote from Cosmos the opinion of Arago, to which Humboldt agrees. ‘“We cannot explain the diffusion of atmospheric light by the refiection of solar rays on the surface of separation of the strata of different density, of which we suppose the atmosphere to be composed. In fact, if we suppose the sun to be situated in the horizon, the surfaces of separa- tion in the direction of the zenith will be horizontal, and consequently the reflections will also be horizontal, and. we shall not be able to see any light in the zenith. On the supposition that such strata exist, no ray would reach us by means of direct reflection. Repeated reflections would AND THE THEORY OF LIGHT. 25 be necessary to produce any effect. In order therefore to explain the phenomenon of diffused light, we must suppose the atmosphere to be composed of molecules (of a spherical form for instance), each of which presents an image of the sun somewhat in the same manner as an ordinary glass ball. Pure air is blue, because, according to Newton, the molecules of air have the necessary thickness to refiect blue rays. Itis therefore natural that the small images of the sun, reflected by the spherical molecules of the atmosphere, should present a bluish tinge; this colour is not, however, pure blue, but white in which blue predo- minates. When the sky is not perfectly pure and the atmosphere is blended with perceptible vapours, the dif- fused light is mixed with a large proportion of white. As the moon is yellow, the blue of the air assumes somewhat of a greenish tinge at night, or in other words, becomes blended with yellow.”* 40. We are first of all told, on the authority of Newton, that the particles of air are of a size to reflect only blue, and then that the blue is not pure blue, but white in which blue largely predominates. Ifit be white in which blue largely predominates, the particles of air in any case must be able to reflect other rays than the blue by this admission. Then, again, we are told that the moon is yellow, and that the light of the moon mixing with the blue of the air gives the appearance of green. In this case the air cannot be pure; there must be vapours in it which reflect yellow, or how could there be a mixture to produce green? Or, if the light of the moon is yellow, from whence comes the blue? 41. It is in vain to accumulate instances of inconclusive reasoning in the theory of light, stumbling blocks in the path of science. According to this method of arguing, if the moon is yellow, the air when pure should be black or blue. * Vol. iil. p. 88. Bohn. SER. II. VOL. I. a 26 MR. J. SMITH ON THE ORIGIN OF COLOUR If there are blue rays in the light by which the atmosphere is illuminated, then it should be blue on this theory ; if there are no blue rays in the yellow of the moon then the air should be black, as it can only reflect blue rays. But the air has a greenish tinge by night on the admission of Arago. My idea is that this compels him to resort to another supposition, namely, that the vapours of the air must necessarily reflect yellow, and as the air itself reflects only the blue, the blending of the two makes green. Why not at once say that the vapours reflect green, for green is a primary colour on Arago’s theory of light. But this is merely an instance, among many which can be adduced, of the inconclusiveness of such reasoning founded on the proportioning of the molecules of air to the molecules of light. 42. I have taken this passage from Cosmos, as it is one with which it appears Humboldt was satisfied, to show how difficult it is to account for the colour of the air on any principles of interpretation yet known, for no one who seriously considers the argument can acquiesce in its validity ; and I attach more importance to it, as I am convinced that if it can be shown that blue or purple can be produced artificially by merely reducing the motion of a white ray, and by the imtroduction of darkness or shadow, then the argument for the formation of the other colours must also yield to the same principles of reasoning. 43. There is a similar explanation of the colour of the | air by the author of the article “ Optics” in the Library of Useful Knowledge, which, as it is expressed in very beauti- ful and clear language, I shall add to the above from Cosmos. ‘ We have already seen that the red rays pene- trate through the atmosphere, while the blue rays, less able to surmount the resistance which they meet, are reflected or absorbed in their passage. It is to this cause that we must ascribe the blue colour of the sky and the AND THE THEORY OF LIGHT. Daf. bright azure which tinges the mountains of the distant landscape.” He adds: “ As we ascend in the atmosphere the deepness of the blue tinge gradually dies away; and to the aeronaut who has soared above the denser strata, or to the traveller who has ascended the Alps or the Andes, the sky appears of a deep black, while the blue rays find a ready passage through the attenuated strata of the atmosphere.” What is to be remarked here is, that the particles or molecules of air in the higher strata of the atmosphere are not of sufficient thickness, to use Arago’s expression, to reflect blue. They reflect no colour at all; they appear black. If we adhere to the argument of Arago and Humboldt, supported by the authority of Newton, we are, in this case, compelled to acknow- ledge that it is the vapours of the atmosphere in the lower strata which refiect blue, and that the air when pure reflects no colour whatever, or that it reflects dif- ferent colours according to its different degrees of density. On the latter supposition it cannot be the size of mole- cules which is the cause of colour, but their compactness or the distance of one molecule from another, or, in other words, the ratio of the molecules which reflect light to the spaces which reflect no light; or otherwise the ratio of light to the shade, which is exactly the principle on which I explained the card experiment and the candle experi- ment; at all events, that the blue is not owing to the size of the molecules of air is evident from the facts supplied by science itself in these examples. 44. There is a respect due to an old and an established doctrine which even criticism ought to venerate, a respect which demands that if we attempt to pull down we shall also endeavour to build up. But the colour of the sky cannot be experimented on in its native element. Our information must be drawn chiefly from accurate observa- tion, assisted of course by such experiments as can be 28 MR. J. SMITH ON THE ORIGIN OF COLOUR made available for the purpose. But the scientific explana- tion, as we have seen, rests completely on Newton’s theory of light ; so that to object to it may appear to be, virtually, objecting to Newton’s theory of large and small molecules, both of light and of matter, of molecules of the thickness requisite for reflecting blue, red, green or any other colour. This, however, I do not mean to attempt, as nothing is really known of the size of molecules of any kind. Lengthened observation has, however, led me to the conclusion that the colour of the sky, and especially “the bright azure which tinges the mountains of the distant landscape,” can be accounted for on a more easy and simple principle. I have often remarked that it is not the most distant hills that are the bluest. In a mountainous district the inter- mediate are of a darker and more beautiful blue than the more distant; and in a pure atmosphere when many clouds are flying, the transparency of the air being great, the nearer hills are of the deepest blue. 45. There are two ways at least in which this may occur. When the nearest clouds are low they not only throw sha- dows on the hills, but they prevent many of the more per- pendicular rays reflected from the hills from reaching the eye. From this circumstance the hills appear in the shade, but the more distant and horizontal rays being reflected from the hills thus placed in the shade, give them the blue colour which is so often described as the blue colour of the air. I have watched until these clouds have passed away, and until I saw the reflected rays grey and brown from rock and heather. In like manner the shadow cast by a single cloud and illuminated by a distant white ray is frequently seen to be blue, and on a hill white with snow the shadow of such a cloud, thus illuminated, becomes very instructive. For we see at the same time on the snow grey shadows and blue shadows; proving that white seen through black does not become blue, but that an insu- AND THE THEORY OF LIGHT. 29 lated white ray, if I may use the expression, reflected from the shadow, produces blue in its intervals of pulsation on the black ground. Painters, as well as other observers, have often remarked that black seen through white is blue. This, however, is not the case, in their acceptation of the term, as the blue and grey shadows on snow prove. At present I am not only arguing for the necessity of intro- ducing into the theory of colour a negative term, namely, shadow or penumbra, but at the same time the equally necessary element, lengthened intervals in the vibratory motion. 46. It is by such observations that we obtain arguments relating to the colour of the sky, as well as arguments for the necessity of introducing a new term into our researches on the cause of colour. From these observations I con- clude that blue is produced by a sensation or feeling of no- light in the intervals of pulsation between one wave of light and another. 47. I know that scientific readers will be getting impa- tient at these arguments, and it will be objected that I am even departing from my own premises; that I am neither — proving that colour is formed by reflecting light on the “confines of shadow,” as Newton affirmed could not be done, nor am I proving that shadow has anything to do with the formation of colour. For to what do all these arguments tend, if they tend to anything, but to demon- strate the length of a wave —that the darkness I speak of is the hollow of the wave, or the interval between the crest, apex or summit of one wave and another, which apex is the light? 48. That colour cannot be produced by reflecting light on the “confines of shadow” in Newton’s acceptation of the term, or in that of the ancients, I admit; but that shadow is necessary to the formation of colour is also true, although in a different acceptation from that of Newton. 30 MR. J. SMITH ON THE ORIGIN OF COLOUR Shadow is abstraction of some of the rays of light in any luminous medium. To reflect light on the “confines of shadow ” is not to abstract any rays, and consequently is not to alter the nature of the ight. To alter the nature of light we must either add to or subtract from its inten- sity, and, taking the incident light as the unit, colour is produced by abstraction and the co-ordinate sensation of no-light. I have no wish to object to those who pre- fer to speak in the usual philosophical language about waves and undulations; only I think that when I speak of alternate vibrations of light with shadow, such lan- guage 1s intelligible to the common understanding, while the other, or the language of waves, is not even clear to the understanding of the philosopher. For instance: we cannot refrain from speaking of degrees of cold, although cold is, philosophically, the negation of heat; and, gene- rally, I cannot think it unphilosophical to use as a positive term the name of any feeling or sensation that is cognisable by the senses as being distinct from another, and to which every nation and ,every language under hea- ven has given a name. The words which express our feelings or sensations are of themselves the most perfect definitions which can be given of these feclings or sensa- tions; no explanation can add to the accuracy of the con- ceptions suggested by the words which recall them. Such language I call intelligible, if not axiomatic ; it commends itself to every one. Philosophy does not change the sen- sations —it explains them; and it is of the sensations I am now speaking. 49. But, waiving the philosophy of the language, if it is possible to take any colour and making it vibrate alter- nately with darkness or shadow, without refraction or with- out decomposition, convert it into some other colour, then this would be proving not only that light and shadow, or alternate sensations of light and no-light, produce a change e AND THE THEORY OF LIGHT. 3jl of colour, but it would be doing more: it would be proving the homogeneous nature of light ; it would be proving that the luminous ether no more requires three or seven, or any other number of primary colours or waves of different values, to produce all the modifications of light, than the air requires seven or eight constituent principles to pro- duce all the modulations of the gamut. The ancients appear to have thought that shadow or darkness was a positive principle, and so far as it is a sensation it is; and Newton seems to have experimented under that idea, or how could he have thought of reflecting light on shadow ? At all events, the ancients considered that they saw some connection between shadow and colour, and Newton set himself to prove that they were in no way connected. It did not occur to Newton, when experimenting with his rays of light (derived from the spectrum) on the “ confines of shadow,” as he describes it, that he wes not in any way altering the constituent elements of his rays of light. He was not reducing their momentum, nor was he making them alternate with shadow. Wherever change of colour is observed in such phenomena as we have been describing, attentive consideration of the whole circumstances will show us that there are two planes necessary to produce the two sensations of light and no-light. In a luminous atmosphere the shadow must be produced by one ray or a portion of a ray, and illuminated by another. In refracting substances, as will be shown by and by, there are two planes at different angles, one reflecting light, the other no-light; for how otherwise could the intervals of motion in the vibrations of the luminous ether be perceived? But although there are, apparently, two different processes in nature, they are virtually one. My experiments are an attempt to produce the same effect by motion, so as to imitate nature in these two processes. 50. What, then, is a shadow? or, what is darkness? 32 MR. J. SMITH ON THE ORIGIN OF COLOUR Is it not im this case the interval between the vibration of one wave and another? It is more. It is the sensation of no-light. The interval between one wave and another cannot be seen or appreciated; for there is no sensation, no negative, until the ray is bent from its original direc- tion; and if it is then seen, in its intervals of pulsation, against a basis of a negative character, or against a plane reflecting little or no light, the ray is changed in its cha- racter. What is called white light may be changed to any colour. But is not this the existing theory in different words? A moment’s reflection will show that it is very different. The present theory admits only positive terms. What I am explaining demands two independent and co- ordinate sensations in the production of colour. As every, even the most transparent, body casts a shadow, so there is a probability that even the seemingly most opaque body may, notwithstanding, transmit a certain por- tion of light. There is, to all appearance, as great a variety of shadow as there is of what is usually called light; and from this point of view shadow may be considered as a positive principle, not from the darkness but from the light which is in it. After a series of experiments we soon find that it is as difficult to discover the value of a shadow as of a ray of light. They have both eluded the most pro- found researches. We are thus forced to the necessity of viewing shadow as a negative principle, but of different values. In fact we must view it in many cases as a phase of light, although, abstractly, it be perfectly inert. It is this difficulty of not knowing what is hght and what is not light that causes such difficulties in our researches into light.* An instance of this is to be found in Newton’s * Dr. Reid sees no propriety at all in the term passive power: “It is a powerless power, and a contradiction in terms.” Sir William Hamilton says: “This was one of the most celebrated dis- tinctions in philosophy. .. . Power, therefore, is a word we may use both in an active and passive signification, and in psychology we may apply it AND THE THEORY OF LIGHT. 33 observations on black. After much supposition and con- jecture, he adds: “And hence may be understood why blacks are usually a little inclined to a bluish colour. For that they are so may be seen by illuminating white paper by light reflected from black substances. For the paper will appear of a bluish white; and the reason is, that black borders on the obscure blue of the first order described in the eighteenth observation, aud therefore reflects more rays of that colovr than any other.” Newton has nowhere, as far as | remember, told us how to perform this experi- ment; and I confess I cannot understand how it can be done without taking into account other light which must be reflected from the paper. 51. Few people, however, who have made any experi- ments on light, can have failed to make an observation the converse of that of Newton, namely, that white reflected from black, in favourable circumstances or conditions, is blue. Any one who has examined, for instance, the re- flection of a white china plate on a black ground, as black marble or black japan, must have remarked this, for it is avery common phenomenon. ‘The refiection of the white object on the black ground is blue; but no one would infer that a decomposition of the white light has taken place because blue is reflected; it would rather be supposed that, provided the light be sufficiently bright, the portion of it which is reflected from the white china, when insulated from the other portion, produces the observed effect on both to the active faculties, and to the passive capacities of mind.” — Again he says: “There is no pure activity, no pure passivity in nature. All things in the universe are reciprocally in a state of continual action and counteraction. ... Activity and passivity are not, therefore, in the mani- festations of mind distinct independent phenomena. ‘This is a great, though a common, error. They are always conjoined. There is no opera- tion of the mind which is purely active, no affection which is purely passive. In every mental modification, action and passion are the two necessary elements or factors of which it is composed.” — Lectures on Metaphysics, 16th and 17th. SER. III. VOL. I. EF 34 MR. J. SMITH ON THE ORIGIN OF COLOUR being again reflected from the black. Nor could any one suppose for a moment that the white contained a blue of some order, and refiected more of that than of any other colour. Of course I am aware that this argument may be turned round in a circle, and I shall be told over again that the black reflects blue, or is an obscure blue of some order! In order to explain phenomena such as this, in accordance with sound logic, we must suppose that the effect is produced by beats or vibrations of light, and that these beats or vibrations of light are connected with dark- ness or shadow — a proposition which, as I shall presently show, may also be proved experimentally. 52. Colour then, I conceive, cannot be changed “ by reflecting it on the confines of shadow,” which would not indeed in the least degree reduce its momentum ; but if we do reduce the momentum of a ray of light without decomposing it, and by reflecting it at regular intervals with black, that is, giving it by motion, as it were, two different planes — a positive and a negative basis — pro- duce two sensations, alternate indeed, but so rapid as to be considered codrdinate; and if we by this means can change one colour into another, then are we warranted in arriving at the conclusion, that the effect of colour is produced by intervals of light and darkness. 53. Although I find it difficult to define the negative term which. I am labouring to introduce, but which does nevertheless denote something having a real existence, still it is not more difficult to understand the necessity of it than to understand the necessity of intervals of silence in the science of acoustics. There is one advantage, how- ever, in our experiments; what we cannot define or make intelligible in the present state of scientific language, we can show to exist and to have an appreciable value. By and by, when once the experiments have become familiar, the effects of intervals of light and shade will be as easily AND THE THEORY OF LIGHT. 35 understood as intervals of sound and silence, and they will even throw light on the science of sound. White Light. 54. I have all along, in compliance with the common, as well as philosophical, language of the day, spoken of white light as if white were the colour of light. But why should hght have any particular colour? Is it because white is the most brilliant colour, that we call light white ? It is by attributing to light a colour that we are led into inextricable difficulties. Light is ether in motion, and is, we might almost say, no more white than black. Every one who has tried to experiment with light has found it an extremely difficult thing to obtain a white ray. Newton says that the light of the sun is yellow, and he had consequently to modify some of his explanations in accordance with that supposition; and Arago, as men- tioned above, has said the same of the light of the moon. From the yellow light of the sun we can obtain red, orange, and the other colours of the spectrum, for we see the effect daily in the spectrum ; but how do we, on the supposition that the prism analyses light, get the white which is reflected from the silvered back of a mirror, or the daz- aling white of snow? - Of course I do not deny that white light really exists, and that it is the brightest light we know; but I do not admit that light is white, or that white light is a compound of various colours, and conse- — quently do not believe that refraction decomposes white light. Every observation which I have made on the nature of light leads me to the conclusion that it is a homogeneous substance and, being a. homogeneous sub- stance, can have no colour but what it acquires from the form or chemical character of the objects which reflect it, and which introduce the negative element of which I am speaking. But the discussion of this subject will be resumed presently. 36 MR. J. SMITH ON THE ORIGIN OF COLOUR 55. So far, then, as our inquiry has gone, we have seen that in every case in which we have attempted to account for one of the primary colours, namely blue, it has resolved itself into light associated with shadow, from which I conclude that light must produce its effects by beats or pulsations on the eye, and that these beats or pulsations must be at appreciable intervals; for if light were an uninterrupted succession of waves at intervals as small as commonly supposed, there would be no use for shadow; shadow could not be seen. It would not pro- duce a new sensation when refracted or reflected. But it will be asked, can the same principles of reasoning be applied to any other colour? Let us see how they apply to Green. 56. It is not easy to devise experiments on green which are as readily performed as those described above on blue. The experiment with the red and black wafer is not only a difficult one to make, but it is one which, even when successful, is not satisfactory to many minds. 57. There is another experiment of the same kind which, although satisfactory to myself, has the same disadvantage of being difficult to make, and it requires moreover strong sunshine. I have observed, when waving a red silk hand- kerchief m the air, m strong sunshine, that the edges appeared of adark green. Although I have made this expe- riment in the presence of young people with good eyes, and have at the same time told them what to observe, still I cannot say that I have ever got any one who really saw what I directed him to look at. But this experiment proves, when it is successful, that green or change of colour is produced by red light and no-light. Beyond the red, no light is reflected to the eye, and the bright red alternating with darkness produces the sensation of green. Acciden- ERRATUM Page 37, line 8 from bottom, for “reflected” read “ transmitted,”’ for according to hypothesis nothing but transmitted light could enter the diving bell. AND THE THEORY OF LIGHT. 37 tally I had another opportunity of observing this pheno- menon on a larger scale, at which two people were present besides myself. In a court at a little distance two persons were beating a red carpet; the sun was shining at the time. When the one struck it there was a large wave of red; when the other struck it there was a large wave of green. I examined the appearance very minutely, and then desired those who were standing by me to observe, and tell me what they saw. One said he saw nothing remarkable, only the red carpet; the other that he saw red and green alternately. These observations being rather difficult to repeat, I merely adduce them as additional arguments in favour of the theory which I am attempting to establish, and perhaps they may be of use to others also. 58. There is one experiment, however, which is uni- versally admitted to have been performed with extreme accuracy by one of the greatest philosophers of his age, and which may be made available for our purpose. I refer to Halley’s celebrated diving-bell experiment. His atten- tion, when under water, was directed to the colour of his hand, and to that of the water below. The upper part of his hand appeared red, the water and the lower part of the hand green. Newton’s explanation of this is, that water transmits the red, and reflects the green rays. It is difficult to understand this explanation, for to me it appears that the light from both the upper and lower part of the hand was reflected light. ‘The upper part of the hand reflected more of the subdued reddish light with which he was surrounded ; the lower less, for it was in the shade. Jn consequence of the depth of the water below, few of the rays entering the water were reflected thence, and the faint pulsation of reddish light alternating with the darkness below, produced green, in a manner similar to that in which we have seen the colour of the sky is 38 MR. J. SMITH ON THE ORIGIN OF COLOUR produced. Even in this experiment, then, one of the most accurate experiments on record, we see that the colour green is associated with shadow, and is caused by intervals of red and darkness, a conclusion which may be arrived at by other experiments without subjecting the light to refraction, as it is in this imstance. 59. There is another observation strongly corroborative of this conclusion; but since the effect, just as in the pre- ceding instance, might be supposed to be due to refrac- tion, I have great reluctance in bringing it forward at this stage of the investigation, but I found it of great assist- ance in my inquiries into the cause of colour. Often on a frosty night I have remarked on shop windows the light within surrounded with a halo, similar to that which is frequently seen around the moon, and of a similar colour. Between the image of the light and the halo there was a dark ring, which was in many cases quite black, but in others a very distinct green of various hues. The question arises, What is the cause of this green colour? Coasider- able difficulties attended the explanation of the phenome- non. For a long time I could discover no explanation, and to resort to hypothesis was contrary to the rule which I had laid down for myself. I resolved to wait patiently, and observe as carefully as possible, until nature told her own tale. On riding rapidly along, these appearances struck me more frequently. On such occasions I con- stantly observed either the green or the black ring. Aiter continued observation I became convinced that the green was only perceived when another light than that causing the halo was reflected from the window. This I saw was constantly the case when there was a light opposite the one creating the halo. Now this fact led to the same inference as that to which I was conducted by my other observations, viz., that light proceeds by pulsations, and that colour is connected with darknéss or shadow, with AND THE THEORY OF LIGHT. 39 intervals of light and no-light. I could not adopt New- ton’s hypothesis im Halley’s experiment in the diving-bell, and say that the glass reflected the green and transmitted every other colour, for that was not the case. It could only be explained on the same principle by which it has been attempted to account for the blue shadow cast by a lighted candle on the window-blind in daylight. In the phenomena just described, refracting substances, water and glass, play a part; and they afforded only arguments of a secondary kind, since Newton had forestalled the explanation by his theory of refraction, reflection, and transmission. 60. It will be remarked that green is always associated with a red or a yellow colour in these observations. I accord- ingly attempted to produce green by the mere motion of red light alternating with black, and I think successfully. The green obtained by the motion of the red handkerchief was of this description, but it was not of so decided a nature as I could wish, as I could not make it visible to every one. I do not, however, consider that light should be at first red in order to produce green; I only mean to assert that a ray of the value of red, when made to vibrate alternately with black or shadow, produces green in a clear atmo- sphere. Another argument bringing us nearer and nearer to the homogeneous nature of light. 61. I willingly grant, as I formerly remarked, that the combination of two such sensations as red and black, or of yellow and black, producing green, is not apt to strike one as at all remarkable, as we often see the effect produced ; but the fact of its being common does not make the investigation of the cause the less necessary. It must be remarked that the mere reduction of the intensity of a ray will not change its colour. 40 MR. J. SMITH ON THE ORIGIN OF COLOUR Inquiry into the cause of Red. 62. Had we begun to inquire first of all into the nature of red, our researches would have made slower progress even than they have made. It was by observing shadow to be always so closely connected with the colours blue and green that it became necessary to assume that colour was produced by intervals of light and no-light, vibrating at greater intervals than science teaches. It is difficult to see shadow in the usual acceptation of the term in con- nection with red. Whether we look at the ever varying and gorgeous phenomena of the atmosphere, or at the phenomena produced by refracting media, there is little indication of shadow to be seen. At the setting of the sun many of the rays of the lower limb of the sun are intercepted by the earth, and by the density of the atmo- sphere; and those rays which do reach us direct are re- duced to parallelism ; there is little crossing in comparison with the light of noon. Of course, since fewer rays reach the eye there will be fewer vibrations, less intensity of light; but this of itself cannot produce colour unless there is also a negative element present, and as this negative element is not discoverable, there is nothing to point out to the intellect that there must be vibrations or pulsa- tions. It might be an uninterrupted flow of light of a different physical character from the ordinary daylight ; in fact it might be « homogeneous light compared to the light of day, and might be produced by refraction, or by the absorption of other rays differently constituted from it. 63. In a fog at noon the sun is often seen of a deep red, with its intensity of light so reduced as to be looked at without inconvenience. I have often remarked on these occasions that it appeared as if one were looking along a red tunnel. The rays were evidently not crossing each s AND THE THEORY OF LIGHT. 41 other much. These two phenomena, the setting sun and the sun seen through a fog, may be adduced as leading to the conclusion, that red is produced by motion in compa- ratively straight lies, without any intersection of rays. I think that this conclusion is supported by Halley’s diving- bell experiment. But the question, What indication is there of a negative element? still remains unanswered. 64. Were I to adduce arguments founded on observa- tions on the eye, I should be met by the objection, that the eye is not a perfectly achromatic instrument — an assertion, however, which no one can believe unless he also believes in the heterogeneous nature of light; since certain phenomena, connected with the eye, cannot be explained on any other supposition by those who adopt that theory. My eye, I know, is not now perfectly achro- matic, but I remember the time when it was so; and I believe every good eye to be achromatic. But letting this pass, every one who has experimented on light knows that he can convert his eye into a prism, so as to produce all the coloured dispersion which is seen in the prism. If we look at a candle over the edge of anything — the finger will do as well as any thing else — and cover some- what more than one half of the pupil of the eye, yet so as to see the candle distinctly, thus intercepting a mul- titude of the rays, but allowing those which do enter the eye to enter in one direction only, the lower part of the candle will appear red, the middle yellow, and beyond there will be a line of blue. Of course this order depends on what part of the pupil is covered. It is evident that the whole of this red or yellow is not produced by refraction, allowing that the refractive power of the eye is the sole cause of the colour. It is impossible to omit observing that the dispersion of light on the penumbra caused by the object has more to do with the colour than refraction. A minute SER. III. VOL, I. i G 42 MR. J. SMITH ON THE ORIGIN OF COLOUR inspection of this phenomenon will show us that there is a penumbra cast by the object on the eye from the sub- stance interposed between the light and the eye, and that it is this penumbra which is coloured. Is not the red of the atmosphere produced in the same manner? Is it not the light of the sun, reflected from the atmosphere at intervals with the penumbra of the earth cast on the same atmosphere? In the one case we can detect the penum- bra; in the other we can only infer its existence. But the inference is perfectly legitimate. 65. I may be told that to produce this parallelism in rays, refraction is necessary. ‘True, refraction is necessary ; but not for the purpose of decomposing a compound hetero- geneous substance into its constituents parts, but rather for the purpose of compounding an active with an inert or passive sensation. 66. But it will be asked, Is not this what the wave theory teaches? If it is, it teaches it in a very different way. The wave theory assumes different lengths of waves; and these different lengths must be owing to the compound nature of light, or to some affections in matter not under- stood, and only hypothetically assumed. 67. These observations on red are satisfactory only so far as the phenomena are seen in connection with shadow; but combining them with others, they may assist in establishing the elements of the science of light; and it has been considered necessary to refer to them, because the proofs derived from experiment would not be under- stood nor acquiesced in, unless accompanied by preliminary remarks illustrative of the method of investigation, and showing that the experiments were not made accidentally or at random, but for the purpose of developing the laws on which the optical phenomena everywhere surrounding us depend. Having thus gone over the arguments on which the theory AND THE THEORY OF LIGHT. 43 of light, which I am endeavouring to establish, rests— at least so far as they relate to the primary colours, blue, green and red —[ shall for the purpose of recapitulation describe two other experiments. which can be made at any time, and which, to my mind, are beautifully illustrative of the whole theory. Observation, A. In a room with a good bright fire, the flame of which is so strong as to produce a distinct shadow on the ceiling of any object such as the gaselier, suppose we light a candle or a jet of gas, then the candle or gas will also form a shadow of the same object, but the two shadows will be coloured, the one differently from the other. The shadow formed by the jet of gas will be brownish-red, for the flame of the fire which illuminates it is red; the shadow cast by the fire will be blue, for the light of the gas which illu- minates it is white or nearly white. Now if we withdraw one of the lights, say the jet of gas which is easily managed, the ceiling will still be white, as far as the eye can judge, but the shadow is grey because it is no longer illuminated with the gas-light ; and the same thing happens to the shadow from the gas, it also becomes grey when the light of the fire gets dull: thus demonstrating, not that the one light compared to the other is as brown to blue, as we were told in the experiment with the can- dle and daylight by Dr. Young, but that each coloured shadow is some fraction of the unit of force agitating the retina; but that fraction of the unit of force agitating the retina cannot be calculated until we know the value of the unit and the value of each shadow. For that each of the shadows has a value, independently of the illuminating power exerted on it by the other’s light, may be seen by examining each of them separately. It will be observed that neither of them is absolute darkness, or a complete negation of force. 44, MR. J. SMITH ON THE ORIGIN OF COLOUR 68. From this and the foregoing observations I conclude, that the value of a colour is known only when both the active and the passive or negative elements are known, and that there is no colour without the negative as well as the passive element. They are codrdinate elements of colour. A similar observation beautifully illustrative of the above may be made almost every day when the sun is shining. I see every morning on my window blinds, when the weather is favourable, coloured shadows such as I have been describing. Let us subject them to examination. Observation, B. Before a sheet of white paper used as a screen, hold a small opaque body, as a pencil, so that its shadow may fall on the paper. There will be seen two shadows; one blue or purple, depending on the state of the atmosphere, or at times on the distance of the object which casts the shadow on the screen —the other, red, ochre or yellow. These are the same colours which were seen in the candle obser- vation or experiment. ‘There is evidently here a partial analysis of the illuminating medium, the separation of it into two distinct waves. But the question arises, Are these two distinct waves red and blue? I say they are not. But this is a problem not so easily solved. We must, therefore, resort to our former observations for assistance, if they can afford us any. The state of the problem is this. We have two shadows; one, blue or purple — the other, red, ochre or yellow. Whence come these colours? At the time these shadows are seen, some of the rays from the lower part of the sun are still intercepted ; there is, consequently, a great penumbra in the direction of the ray, although not easily detected but by such an experiment. The rays within this penumbra are direct; the rays without it are reflected, ~ AND THE THEORY OF LIGHT. AD as is seen by the direction of the shadow. The shadow produced by the rays within the penumbra is illuminated by the light without the penumbra, and it is, consequently, blue or purple. The shadow cast by the light without the penumbra is illuminated by the stronger and more direct light which is within it, and is red or yellow. This is pre- cisely similar to the fire and candle experiment; but the origin of the two rays is not so easily discovered. Terres- trial reflection has often much to do with the colour of the red ray. I have no doubt that these remarks will be met by the counter assertion, that the light of the atmosphere is blue, and that the other, or the red light, is refracted light, red and blue united making white. In the present state of the science this is quite a legiti- mate objection. Let us, however, test the truth of the usual explanation by the following Observation, C. Suppose we ask the question, Do the two rays, red and blue, which are obtained in the above experiment, make white? If they do, if we unite them again, the resultant should be white. Let us then unite them. It is done as follows. In place of one pencil, or opaque body, hold up two. There will now be four shadows, each object producing two. Let us then cause the red shadow of the one to overlie the blue of the other. These two shadows should by uniting produce white; but it will be found that the compound shadow is not white, it is green. But why has white not been produced by the combination of these two shadows in the manner described? Because, although the two lights have been united, they are each compounded with shadow. There are two waves of light and two shadows, and they are each, as it were, insulated 46 MR. J. SMITH ON THE ORIGIN OF COLOUR from or sections of the great photospheric or ethereal wave. We thus see the value of the negative element when once it has been introduced, and the difficulty, the impossibility of again separating it in any colorific ray. I therefore hold to my argument, that light and shadow are coordinate elements of colour. But perhaps I may be told, that if red and blue do not make white, then red, blue and green, the colours which we have now got, make white. Dr. Young says they do. And if by combining them I could prove that I produced purple, I might then be told that red, green, blue and purple make white; and so on ad infinitum. From this experiment, combined with the former, a good resumé of my whole argument may be obtained. For although in the experiment with the two wafers we could not but infer that green was a combination of two sensa- tions, it did not show us that shadow took any part in the formation of colour; it merely enabled us to draw the inference, that the change of colour could not be accounted for but on the supposition that there is a vibratory motion in light. These experiments, I think, give us solid argu- ments to conclude that light and shadow are codrdinate elements of colour. PART II. EXPERIMENTAL. 69. In my researches into the cause of colour I was led —in order to satisfy the conditions of several prob- lems— to introduce in considering the subject a negative term, not in accordance with the commonly received theories. Newton, in his theory of light, recognises only positive terms, for he considered that he had demonstrated AND THE THEORY OF LIGHT. AT the existence of rays of light in the sun for every colour in the solar spectrum or even in nature. He concludes, prop. 2, bk. i. pt. i. of his Optics, in these words: “ From all which it is manifest that if the sun’s light consisted of but one sort of rays, there would be but one colour in the whole world, nor would it be possible to produce any new colour by reflections and refractions, and by consequence that the variety of colour depends upon the composition of light.” To which he appends the following definition : “The homogeneal light and rays which appear red, or rather make objects appear so, I call rubrific or red- making; those which make objects appear yellow, green, blue and violet, I call green-making, yellow-making, blue-: making, violet-making, and so of the rest. And if at any time I speak of light and rays as coloured or indued with colours, | would be understood to speak not philosophically and properly, but grossly, and according to such concep- tions as vulgar people, in seeing all these experiments, would be apt to frame. For the rays, to speak properly, are not coloured. In them there is nothing else than a certain power and disposition to stir up a sensation of this or that colour. For as sound in a bell, or musical string, or other sounding body, is nothing but a trembling motion, and in the air nothing but that motion propagated from the object, and in the sensorium it is a sense of that motion under the form of sound; so colours in the object are nothing but a disposition to reflect this or that sort of rays more copiously than the rest. In the rays they are nothing but their dispositions to propagate this or that motion into the sensorium, and in the sensorium they are the sensations of those motions under the form of colours.” , In his letter to the Royal Society he says: “ Nor are there only rays proper and particular to the more eminent colours, but even to all their intermediate gradations.” In 48 MR. J. SMITH ON THE ORIGIN OF COLOUR another part of the same letter he says: “These things being so, it can be no longer disputed whether there be colours in the dark, nor whether they be qualities of the objects we see; no, nor perhaps whether light be a body. For since colours are the qualities of light, having its rays for their entire and immediate subject, how can we think those rays qualities also unless one quality may be the subject of and sustain another; which in effect is to call it substance. We should not know bodies for substances were it not for their sensible qualities; and the principal of those being now found due to something else, we have as good reason to believe that to be a substance also. “ Besides who ever thought any quality to be a hetero- geneous aggregate, such as light is discovered to be. But to determine more absolutely what light is, after what manner refracted, and by what modes or actions it pro- duces in our minds the phantasms of colours, is not so easy. And I shall not mingle conjectures with certainties.” 70. The experiments which [ am about to describe appear to me to establish a theory the very opposite of Newton’s. They prove that the sun’s light consists of but one sort of rays, and that there are no such rays as red-making, green-making, or violet-making rays. 71. But it may be said that the undulatory theory of light is directly opposed to that of Newton. In point of fact, however, there is little difference between the corpuscular theory, as explained by Newton, and the undulatory, except in the more rational supposition by the advocates of the latter of an ambient ether to convey the sensations of colour, just as the air conveys the sen- sations of sound. For the undulatory theory adopts Newton’s view of the composition of light, or, what is the same thing, supposes that there are waves of dif- ferent lengths either inherent in the nature of light itself or produced by the nature of the refracting AND THE THEORY OF LIGHT. 49 media, for there seems to be some uncertainty on the subject in the scientific mind. One scientific writer says :* “From the foregoing we conclude, then, that the peculiar colour and refrangibility belonging to each kind of homogeneous light are permanent and original affec- tions not generated by the changes which light undergoes in refractions or refiections ; and therefore that these pro- perties are inherent in the rays previous to their separa- tion by experiments.” Another says: “The dispersion of light into the colours of the spectrum, consequent on its refraction through a prism, is explained by supposing that transparent bodies attract different sorts of light unequally ; their refraction is therefore unequal; and thus, on their emergence from the refracting medium, the rays become dispersed... .. According to the undulatory theory the production of coloured light is supposed to be owing to the different velocities with which the particles of ether vibrate; and thus a distinct sensation is excited in the eye analogous to that which is experienced by the ear from the different velocities with which the undulations of the atmosphere are propagated. As high notes and low notes result from the different velocities of these aerial undulations, so the different colours owe their origin to the unequal velocities with which the ether vibrates .... The different refrangi- bility of light, and the prismatic dispersion consequent upon it, admit of being very easily explained; for a wave that vibrates with a higher velocity than another will suffer a different refraction from that suffered by the latter, if the two waves be transmitted through the same medium, as ex. gr. through glass.” + These arguments presuppose that it has been demon- strated that different velocities exist inherently in the * Treatise on Light and Vision, by the Rey. H. Lloyd, p. 198. + Peschel’s Elements of Physics, pp. 90, 91. SER. III. VOL. I. H 50 MR, J. SMITH ON THE ORIGIN OF COLOUR different rays of light, but the writer just quoted seems also to suppose that matter may modify them, for he adds: “ Lastly, as regards the colours generated by reflection of light, it may be seen how it is possible that the velocity of the incident wave may sustain a different modification from different bodies, in consequence of which modifica- tion of velocity different substances will be of different colours.” * 72. One of the most acute and scientific writers of the present day, a star of the first magnitude in the scientific galaxy, when speaking of this subject says: “It will be shown presently that the deviation of light by refraction is a consequence of the difference of its velocities within and without the refracting medium; and when these velocities are given the amount of refraction is also given. Hence it would appear to follow unavoidably that rays of all colours must be in all cases equally refracted; and that therefore there could exist no such phenomenon as dispersion. Dr. Young has attempted to gloss over this difficulty by calling to his aid the vibrations of the ponderable matter itself, as modifying the velocity of the ethereal undulations within it, and that differently according to their frequency ; and thus producing a difference in the velocity of propagation of the different colours; but to us it appears with more ingenuity than success.”+ I may remark that even this supposition of Dr. Young’s is not an original hypothesis, for Newton, in his 17th question at the close of his Optics, says: “If a stone be thrown into stagnating water, the waves excited thereby continue some time to arise in the place where the stone fell into the water, and are propagated from thence in concentric circles upon the surface of the water to great distances. And the vibrations or tremors excited in the air by percussion continue a little * Peschel’s Hlements of Physics, p. 92. + Ency. Metropol. Article, “ Light,” p. 449. AND THE THEORY OF LIGHT. 51 time to move from the place of percussion in concentric spheres to great distances. And in like manner, when a ray of light falls upon the surface of any pellucid body, and is there refracted or reflected, may not waves of vibra- tions or tremors be thereby excited in the refracting or reflecting medium at the point of incidence, and continue to arise there, and to be propagated from thence as long as they continue to arise, and be propagated when they are excited in the bottom of the eye by the pressure or motion of the finger, or by the light which comes from the coal of fire in the experiments above mentioned? And are not these vibrations propagated from the point of incidence to great distances? And do they not overtake the rays of light, and by overtaking them successively do they not put them into fits of easy reflexion and easy transmission described above? For if the rays endeavour to recede from the densest part of the vibration they may be alter- nately accelerated and retarded by the vibrations over- taking them’ A moment’s reflection will show us that both of these suppositions cannot be admitted into the same theory of colour without rendering it extremely complex, not to say incomprehensible. 73. Sir John Herschel sums up a review of the two theories in these words: “The fact is that neither the corpuscular nor the undulatory, nor any other system which has yet been devised will furnish that complete and satisfactory explanation of all the phenomena of light which is desirable. Certain admissions must be made at every step as to modes of mechanical action, where we are in total ignorance of the acting forces; and we are called on where reasoning fails us occasionally for an exercise of faith.” 74. I have adduced these authorities to show that with the admissions and corrections and exercises of faith which 52 MR. J. SMITH ON THE ORIGIN OF COLOUR we are called on to make, the two systems do not in’ reality differ so widely as might be supposed, —so very little, indeed, that I am warranted in thinking that the arguments which will overturn the emission theory will be apt to sap the foundations of the wave theory; for I do not consider the latter theory to consist in the mere hypothesis of a circumambient ether, but rather in the principles of interpretation which it adopts in the explanation of natural phenomena. For that the two systems are intimately con- nected is manifest when we consider that not only has the theory of the composition of light been adopted from Newton, but that it was Newton who first of all ventured on the calculation of the length of wave proper to each element of the compound. Neither theory can indeed altogether dispense with the assumption of the compound nature of light. Newton’s theory presupposes a difference in the size of the molecules, and consequently a difference in the length of the wave of each molecule, so as to account for the difference in the refrangibility of the differently coloured rays; the undulatory theory takes for granted either an inherent difference in the original rays of the light itself, and considers the ether therefore to be a compound substance ; or it requires a difference im the velocity of the ether within and without the refracting media, sup- posing the light to be homogeneous ; which, however, leaves the phenomenon of dispersion unaccounted for, and con- sequently the advocates of the wave system are compelled to fall back on the theory of the compound nature of light as unfolded by Newton. 75. Both theories recoguise only positive terms; neither admits a negative element, or any inert property in matter in order to account for these phenomena. But one of the strongest arguments in favour of a negative term on the wave hypothesis is founded on the nature of the element itself. I do not mean to discuss the many difficult ques- ~ AND THE THEORY OF LIGHT. 53 tions raised by the opposing theories, my intention being to keep one point, and one only, in view; viz. to show by reasoning and experiment the necessity of introducing a negative term in the explanation of the phenomena of colour, and to follow out the conclusions to be derived from the experiments. It may therefore be advisable to state the argument in favour of a negative term derivable from the ether itself before entering on the experimental part of my subject. 76. It has been customary to investigate the theory of light by the analogy of sound, sound being formed in an ambient finid analogous to the supposed luminous ether ; that is, both beimg produced by a similar physical process, sound and light are supposed to follow the same laws. They may obey the same laws; this I am not disputing ; but these laws must be investigated on different prin- ciples. For allowing that they do obey the same physical laws, still at the commencement these laws operate in a different manner. It would, therefore, not be philoso- phical to assume a complete analogy. For in the theory of light, motion or an undulating ether is given us; we have therefore to investigate the various laws by which motion is diminished or modified so as to produce variety of colour. In the theory of sound, silence or rest is given us ; we have therefore to investigate the laws by which motion is produced in a circumambient fluid so as to give rise to variety of sound or tone. The one deals with an analytical process, the other with a synthetical process. One process consists in the resolution of force, the other in the generation of force. Sound is created by a solid bedy communicating its vibrations to the surrounding atmosphere ; colour is produced by a solid body intercept- ing or reducing the vibrations of a vibrating ether. The one is consequently the inverse process of the other, and it should not be assumed that they follow the same laws 54: MR. J. SMITH ON THE ORIGIN OF COLOUR because they beth result from the vibrations of a circum- ambient ether, until these laws are investigated. Assum- ing, however, the analogy between sound and light, and arguing on the assumption, it follows that the theory of light requires a negative principle to carry out the ana- logy. Sound requires an active element to put inert matter in motion; colour requires an inert or opposing element to modify the motion of an active agent. 77. When we know the length of a string, which when struck under certain conditions produces the key note in music, we can estimate the lengths of the other strings which will produce the notes in the entire scale. These have been studied for many ages, and can be very accu- rately calculated, although the calculation is still subject to a rectification, determined by trial or experiment. Could we estimate the vaiue of a shadow, and did we know the value of the illuminating ether, the same thing might be done with hght. 78. Every known substance reflects light, whether it be opaque or transparent. It could not be seen otherwise. That portion of the light which is reflected must cast a shadow, and where can that shadow fall but in the first place within the transparent substance? and if the sub- stance is a refracting substance a portion of the shadow will appear coloured under certain conditions as wiil be shown by and by. As the distance from the refracting substance increases the light decreases, while the shadow or penumbra which has been cast by the reflected light increases ; they are thus codrdinate elements, as will be presently shown, in the formation of colour. 79. But darkness, or the effect of shadow, may be pro- duced by reflection as well as by the interposition of a solid body. It is chiefly by reflection that darkness is pro- duced in nature, at least such darkness as I am speaking of. The geometrician will have no difficulty in conceiving AND THE THEORY OF LIGHT. 55 how this can be done. ‘Transparent or partially trans- parent bodies composed of different plates or laminz not parallel divide the incident light into different sections ; some reflecting light to the spectator — others from him, or, which is the same thing, producing the effect of darkness. 80. The intensity of these shadows may be conceived as being analogous to the lengths and tensions of musical strings ; and I have little doubt they are so: but I have no data on which to found a precise analogy, and all specula- tion is foreign to my plan. Even were I to resort to the analogy of sound, there are other difficulties than those mentioned; and not the least is the difference of the instruments by which we see and hear. I expect, however, that the discoveries which have been made in light will lead to new methods of research in several other sciences ; and consequently | have no greater desire to follow out the analogy with sound than to apply these discoveries to other sciences. As, however, a comparison has frequently been instituted between the two phenomena of light and sound, I wish to show that if the analogy is to hold good, a negative term must also be introduced into the science. of colour. : 81. None of the theories of light deny to it motion. We may, therefore, assume that light is ether in motion, or the agitation of a circumambient ether. 82. Where there is reflection there is light; where there is no reflection there is darkness. The most convenient substances for representing these two different states are white paper and black velvet, or lamp-black used as a pig- ment. It is with these substances that my experiments were made: but a bit of Bristol board having various figures painted black on it, or cut out of it, I found to serve equally well; for, there being no reflection from the spaces cut out, they appear black. 56 MR. J. SMITH ON THE ORIGIN OF COLOUR 83. By means of the experiments to be presently de- scribed, I think that I shall succeed in establishing the following propositions : I. That colour can be produced by the motion of a beam of white light without the aid of refraction. Ii. That to produce colour it is necessary to produce II. IV. a shadow. That a ray of light of the same intensity, re- peated at different intervals, gives rise to dif- ferent colours. That, taking the intensity of the luminous ether at a given time as the unit, each colour consists in some ratio of the luminous to the non- luminous element, however obtained, and is not a constituent principle of the luminous element itself — that it is not the fraction of an active element, but the ratio of an active to a passive element; or conversely, V. That by reducing or increasing the velocity of the VI. machine, or, as we may call it, the ray, and pre- serving the same intervals of light and shade on the diagram, the same effects will be produced as if the ratio between the hght and shade had been altered, or as if the light were reflected from a new substance. That the refrangibility or reflexibility of a ray does not indicate the colour or quality of a ray. VII. That light, or the luminous ether, is conse- quently a simple uncompounded substance. VIII. That colour is a compound sensation, or the effect of two codrdinate sensations — an active and a passive, a positive and a negative — of motion and no-motion. id AND THE THEORY OF LIGHT. 57 84. Having already, without previously developing any formal theory, exhibited the process of reasoning which compelled me to refuse my assent to the commonly re- ceived theory of light, I will proceed to describe the experi- ments, which served to demonstrate the above propositions, just as they were made, and rude and elementary though they sometimes were. The theory will develop itself out of the experiments, and the above propositions will be proved by the sum of the evidence adduced, although pos- sibly not in the order announced. 85. The first thing then to be proved is, that colour can be produced without the aid of refraction. The wafer experiment, the card experiment and the candle experiment were so very interesting and, according to the view which I had taken of them, so very instructive, that I was resolved to make them the basis of my new experiments — that is, to produce, if possible, the same effects mechanically. But seeing that the whole optical science of the day was so much opposed to my idea, I na- turally felt some reluctance in attempting to verify it by experiment. As, however, neither the emission theory, nor the wave theory, nor the theory of complementary colours, could account for the effects produced, I became disposed to yield to my own convictions, and the more so, as I could see no defect in my process of reasoning, after having subjected it over and over again to the most rigid scrutiny I could apply to it. 86. Accepting it as a well established fact, that light is not perceived until it falls upon and is reflected from some material substance, I concluded that if a beam of white light could be made to move rapidly over a space which reflected little or no light, I should be able to change the colour of the light; for, by thus reducing the number of vibrations, I considered that the effect would be analogous to that produced by a refracting substance. At all events, SER. [11. VOL. I. I 58 MR. J. SMITH ON THE ORIGIN OF COLOUR if light and shade should really produce the effects anti- cipated, even though the experiment should not turn out so satisfactory as to enable me to exhibit any colour I could wish, still, should it even change the tint in the slightest degree, the fact would afford some evidence in favour of my argument. 87. In want of something better I first of all made a piece of pasteboard into a disc, and coloured the whole black, except a narrow ray radiating from the centre to the circumference, which portion was left white. (See plate IV. fig. 1.) I then made the dise revolve on a pivot. The experi- ment was not very encouraging, but it was so far success- ful that, just as the motion was about to stop, or when it was beginning to get slow, I perceived, as I thought, a tinge of brown. I directed the attention of several young persons with good eyes to the appearance, but they did not perceive anything. I was not, however, discouraged by this, as I was aware that it requires some experience of a special kind to observe accurately. From repeating the experiment very often I was more and more convinced of the correctness of my own observations; and I also arrived at the conviction that, if colour was to be obtained by motion, it must be by having a slower and more regular motion than I could command with such an instrument. This, of course, was merely an introductory experiment previously to getting an apparatus which would give a greater command over the ray of light with which it was intended to operate. 88. The apparatus which I ordered was of the following construction. (See plate IV. fig. 2.) I call it The Chromascope. It consists, as will be seen, of one wheel driving another wheel, so as to obtain multiple and steady motion. For AND THE THEORY OF LIGHT. 59 every revolution of the larger wheel the smaller makes eight revolutions. On the pivot of the axle of the smaller wheel there is a small chuck or holdfast, by which the disc or figure, or the substance reflecting the light with which it is intended to operate, is fixed to the machine; this was always a small piece of card of various forms. I tried thin plates of silver, but they did not suit the purpose so well as the bits of Bristol board. I expect, however, all these experiments to be improved by means of more per- fect apparatus, as I have hitherto not been able to devote as much attention to them as I could wish, my time being too much and too laboriously occupied in a most arducus profession. Indeed the whole of the experiments were first of all made under a jet of gaslight, after eight o’clock in the evening, and it was sometimes months before I could get them properly verified by sun light. The ex- periments are therefore still in their infancy, and leave a wide field open for experimenters. My object being to establish a principle, 1 shall leave it to others to work it out. 89. After having got the apparatus, the first experiment T made was an attempt to obtain blue, and if my anticipa- tions were well founded it was thought that it was merely necessary to cui a small slip of card in the form of a paral- lelogram, perforate it, place it on the pivot of the small wheel, and then put it in motion over some place near a window reflecting little light from below. But after all was ready I was almost afraid to put it in motion, lest a theory, which had been to me so long the greatest source of pleasure, should, on being subjected to this test, be at last found to be a mere delusion—a phantasy. Even when I had made the experiment and found it completely successful, I could scarcely believe in its reality. I was most anxious to get others to verify what I saw, and one in whom I had the greatest confidence not only for appre- 60 MR. J. SMITH ON THE ORIGIN OF COLOUR ciation of colour, but for delicacy of taste in regard to harmony of tints, could at first see little or no colour. The experiment, however, was successful; blue was ob- tained by the motion of the slip of white card. go. The theory I now considered to be established. The only desideratum was to acquire some knowledge of details, some knowledge of the ratio which the shade, or rather penumbra, ought to have to the light for any given colour. This I knew could only be obtained by experiment, and I never doubted but that some approximation would be made to it. I did not, however, anticipate the difficulty of the calculation, nor yet the splendid results at which I have arrived in the production of colour by the motion of a white ray. The experiment is essentially only a repe- tition of the wafer experiment, the card experiment, and the candle experiment; it consists in the casting of unre- fracted light on a shadow. Of course the shadow in this case is the darkness over which the white ray moves, and is only cast on the eye as in the wafer experiment, but it is virtually not less a shadow than if it were first cast on a piece of paper. This will presently be made more ap- parent. g1. After this experiment, being otherwise much engaged, T rested for some time quite satisfied with the result, but on repeating it afterwards I began to think that I had made some great mistake, that there had been an optical illusion, for I could now only obtain purple in place of blue. This last experiment was made in bright sunshine and with a blue sky. In making the same experiment in a dark dull day, during slight rain, again the same purple appeared. I had neglected to note the state of the atmosphere in the first experiment. When, however, in a few days the sky began to clear, and white fleecy clouds to make their appearance, the same blue was obtained as at first. The paper, at least the small slip I used, on a dull day or with a bright AND THE THEORY OF LIGHT. 61 blue sky evidently did not reflect enough of light to pro- duce blue. Blue was obtained best when there were white clouds in the atmosphere. This is quite contrary to what was to have been ex- pected. Had blue been obtained when the sky was blue there would have been less confidence placed in the expe- riment; but although it would have created suspicion about the blue, it would not have diminished the value of the fact, that a different shade of colour had been obtained by the motion of a white ray of light. The experiment . turning out as it has done leaves a stronger conviction on the mind than ever, that blue is produced by pulsations of light associated with darkness or shadow, and is not, as has been considered, one of the elements of a compound substance,. but is rather itself a compound. At this stage of the investigation, after reflecting much on the subject but without engaging in any more experi- ments, I was so firmly convinced of the truth of the theory that I very hastily concluded that this experiment alone completely established the vibratory nature of light, and proved, without any question, the homogeneity of the ether. There were no rays in the spectrum so difficult to account for as the blue and the purple, and having dis- covered the origin and formation of these rays, there would be no difficulty, I thought, in accounting for the others, nor of explaining the nature of the prism, the rationale of the rainbow, and the cause of what are called complementary colours; and after much labour and thought I was led to the conclusion, from this and other experiments, that they may be explained in an easy and simple manner. Effects of different velocities. g2. Having remarked in the former experiment, that the same slip of paper, moving with equal velocity in different states of the weather, produced different shades of colour, 62 MR. J. SMITH ON THE ORIGIN OF COLOUR or rather two different colours, a blue and a purple, I now began to experiment on the effects of different velocities. I found that when I reduced the rate of motion the blue often became purple, but I also found that the same effect was produced when the white ray was carried into the shade, or properly speaking by reducing the intensity of the white ray with which I was operating. In the shade I could not by a more rapid motion raise the purple up to blue, for no rapidity of motion could raise the value of the incident light or the light reflected from the paper; although by reducing the pulsations of the incident light, or the light - falling on the paper even in the shade, I could reduce its value. 93. This is the same phenomenon as that of the card painted partly black. In a faint light the black appears through the card as purple; in a stronger light as blue. The one experiment illustrates the other, and both demon- strate the pulsatory nature of light. Of course light must proceed by waves, but there is no proof of length of wave, in the common acceptation of the term according to the wave hypothesis, having any effect in the generation of colour. Nor has there been in this experiment any decomposition of light. The same ray has been used, namely, the white ray reflected from the paper. The frequency of the vibrations has only been increased or diminished. Still it must. be understood that the mere increase or diminution of the number of vibrations will not produce colour, or every pe- numbra would be coloured. But every penumbra is no more coloured than every refracted ray is coloured. Every known substance on which light falls, or through which it is transmitted, casts a shadow, but no shadow can be coloured unless a ray at an angle to the ray which forms the shadow is deflected upon that shadow, as was explained in the candle experiment. Hence the use of refraction. It resolves multiple force ; it does not separate quick rays AND THE THEORY OF LIGHT. 63 from slow rays, nor red rays from blue rays, nor constitute fits of easy transmission and easy reflection. But this is not the place to discuss this subject. 94. The experiments hitherto described led to no arith- metical law. I entertained strong hopes, however, that the difficulty would not be great; but when I first began to experiment in the way described I had no idea that the incident light, or pitch of the ether, was so incon- stant, so continually changing as I have found it to be. I soon became convinced that nature in this as in other in- stances never repeats herself, and that every moment of time, could we trace time backwards, could thereby be distinguished from another. I have every reason, how- ever, to think, that by means of motion we shall soon be able to invent some photometer which shall be available for scientific purposes. 95. After having made an immense number of experi- ments it was found necessary, for the sake of reference, to arrange them under two heads, distinguished by the nature of the motion, viz. lst, Horizontal; 2ndly, Perpendicular motion. In horizontal motion the figure, or disc, moves parallel to the machine; in perpendicular motion the figure moves perpendicular to the machine, or parallel to the axis. The phenomena caused by horizontal motion may be divided into three classes : 1. Class. When light is reflected from the figure, or disc, representing coloured reflection. II. Class. When light is transmitted through holes in the disc, representing transparent coloured media. IIT. Class. Coloured shadows, or penumbre, showing how refraction produces colour. I call these Solar Spectra by reflection. 64 MR. J. SMITH ON THE GRIGIN OF COLOUR I.— HORIZONTAL MOTION. Ist Ciass oF EXPERIMENTS. Coloured Reflection. Theory of light and shade by horizontal motion. 96. If we take a piece of Bristol board (see plate IV. Jig. 3) 1 the form of the parallelogram 4 B, and examine it carefully, the light reflected from every spot, as far as the eye can detect, is the same. The space surrounding the card is all darker than the card, and may be consi- dered as shadow. Perforate a hole in the centre O, and make it revolve on the centre, then the disc formed by the revolution will not be of a uniform colour. There will be a ring on the disc of a different shade for every radius that can be drawn from the centre of motion on the card. The outer ring will have for its radius half the diagonal A B, the next will have for its radius half the side A D or E O, aud the third will have for its radius half the end DB,or OF. The inuer circle formed by the radius O F, equal to half the end, will be white, because there is no mixture of shade during revolution, there is no interval in the revolution of the ray; the next, or the circle formed by the radius equal to half the side A D, will be coloured, for here there is a mixture of light and shade, and the outer circle will have a darker shade of colour, for there is less light compared to the dark, as may be seen by in- specting the diagram. By making the parellelogram, therefore, revolve we get a variety of colour; but colour is only produced where there is light and darkness repeated at intervals, a result not at all anticipated by science; a result, however, quite in accordance with our course of reasoning. : 97. If the parallelogram is made to revolve on the centre O, the ratio of the black to the white is apparent to the eye AND THE THEORY OF LIGHT.. 65 and may be calculated for each ring or circle on the disc the part within the card being white, the part without black. Hence the ratio of the part within the card, in any of the above circles, to the part without, is that of the white to the black. 98. The number of experiments which can be made with a card of this form is very great and very instructive. It can be made to revolve on its centre, on one of its angles, on the middle of a side, on the middle of an end, or on any point removed from the middle of the side or end; and for every possible change there will be a new figure, having a ring for every distinct radius which can be drawn on the parallelogram from the centre of motion, as well as a modification or change of colour for every change of figure. Hvery possible figure, when made to revolve by hori- zontal motion, takes the form of a disc, and the number of rings on the disc can be calculated on the principle here laid down; that is, there is a ring for every distinct radius which can be drawn from the centre of motion to some well-defined point, and between two such points there is often a gradation of tint similar to that formed by a penumbra. Examining carefully this experiment, one would at first suppose there could be no great difficulty in determining the numerical value of a ray. The difficulty, however, is greater than it appears at first. The degrees of light are infinite, and the degrees of shade are infinite, and it is, therefore, no easy matter to discover a means of mea- suring their relative proportions. The smallest change, the most minute modification in a diagram, the least alteration in the pitch of the incident light, or in the angle of the diagram to the incident light, an almost inappreciable variation in the motion of the ray created by the machine, each and all tend to produce a problem extremely difficult of solution. SER. III. VOL. I. K 66 MR. J. SMITH ON THE ORIGIN OF COLOUR Circular Figures. In order if possible to simplify this problem, I tried discs with definite ratios of white and black, and by fixing them at the centre I made them first revolve with Concentric Motion. 9g. These discs were made in this manner: I took a piece of Bristol board, and having described on it several concentric circles, I.cut out with a knife, or painted with lamp black, a portion of each of the rings formed by two of these circles, say a half, a third, two thirds, or three fourths, or any other proportions, as in plate IV. figs. 4, 5, 6 and 7. I then fixed the disc on the machine, and made it revolve in bright light. By this means the white and black portions of the discs were made to pass alternately over the same spot on the retina. The result was, that when revolving neither the white nor the black parts were seen, but the whole disc appeared beautifully coloured with concentric rings, and each ring had its own shade of colour; purples, greens and yellows predominating. When this method of experimenting is adopted each re- volution of the machine may be considered as representing the time or duration of a vibration, and the light reflected from each ring, combined with the velocity, as the mo- mentum of the ray. When the revolutions amount to from six to eight in the second of time, colour is generally formed so as to be seen by the majority of eyes. Some per- sons can even perceive distinct colours when the revolutions are twice or three times as numerous. At thirty-two revolu- tions in the second of time, I see no colour, only the geo- metrical figure generated by the motion and shaded grey, like so many penumbre, forming the ground, as it were, on which the colours are produced when the light is suffi- cient. AND THE THEORY OF LIGHT. 67 All that I expected to effect by this method of operating was to produce some tint of green, blue, or purple; and although these colours were produced, and my theory thereby confirmed, still I did not succeed in discovering any definite law, for shades, extremely like each other, are produced by different proportions of black and white on the rings, as well as by different velocities of the machine. Nor is this to be wondered at when we reflect on the infi- nite variety of greens and other colours which is to be seen in nature. ‘This is indeed an object of research, enough of itself to employ the labour of a whole life. 100. Although I am unwilling to enter on the discus- sion of diversity of shade at present, 1 may mention that it requires fewer revolutions per second to produce purple than to produce blue, and fewer to produce blue than green. But it must not be supposed that a colour depends on the number of revolutions of the machine, for every colour is seen on the same disc at the same time. The coloured diagrams annexed (plates I., I., IT.) will serve to give some idea of the colours; but the difficulty of producing a representation of such colours as are seen on thin plates, soap bubbles, mother of pearl, &c., will be easily understood. Excentric Motion. 101. Although by operating with discs having definite ratios of black and white, as in the above examples, shades of various colours were produced, still there was not so great a variety of colour as I desired. Beautiful, there- fore, as these colours were, they did not satisfy me. I imagined that if I could by any possibility make one disc revolve over another I might greatly modify the co- lour. In order to effect this I tried several plans without success. At last it occurred to me that if an excentric motion were given to those figures, or discs, which I 68 MR. J. SMITH ON THE ORIGIN OF COLOUR already had, it would virtually be making one dise revolve over another. I therefore cut out a portion —a quarter, or a half —of a ring from the centre of each disc, so that the centre of motion might be made to slide along the diameter at pleasure. By this process the white of one ring would gradually pass on to the black of another, and vice versd. In this way I added greatly to the variety of colour, and, as will be seen on examination, doubled the number of rings which were originally described on each disc. Thus the portion aa, plate IV. figs. 4, 5, &c., was cut out, and the centre of motion was fixed at pleasure on any part of the line, or even of the curve. An examination of the figure whilst in motion will show what takes place. The effect is the same as if there were two discs in motion, one passing over the other; one having for a radius the larger portion of the original diameter, the radius of the other being the smaller por- tion of the same. Spiral Figures. 102, The experiment just described having proved very successful, my next device was to employ a sort of spiral figure of white on a black ground. I expected, from the nature of the figure, to obtain very graduated tints. The figures, however, which I at first used were so dark that, although a great variety of colour was produced, the disc appeared like a piece of printed calico, without that gra- dation which I was in search of. Nevertheless I obtained a very interesting set of colours. 103. Although these colours were not exactly what I was in search of, still the experiment was valuable, for it showed that the plan I was pursuing would lead me to the end I had in view. I therefore drew on the Bristol board two or three concentric circles, cut the central one and a great portion of the others away, i order to give AND THE THEORY OF LIGHT. 69 the figure a spiral sort of form, and at the same time drew a number of rings, so arranged that when put im motion the light might gradually increase in proportion to the shade from the inner part of the ring to its circumference, or vice versd. Some point in the inner circumference of the ring was made the centre of motion as o, plate V. fig. 3. Although the figure was of a spiral form before being put im motion, like every other figure moving hori- zontally, the disc when in motion seemed composed of concentric rmgs. This experiment gave me nearly all I desiderated in regard to the production of colour. The arithmetical law involved, however, still eluded research. The investigation became difficult, for the smallest change of the centre and the least alteration in the motion created a new obstacle which could not be overcome but by enu- merating the elements, and these were unknown, for we have no terms to designate the variety and the shade of a colour. 104. This form afforded a bright yellow, and I hence concluded that a regular spiral would give less yellow and more green. I therefore cut out a spiral of the form shown in plate V. fig. 4, having its centre of revolution in o. It is so simple that one feels some surprise at the beautiful figure produced by its motion. One of these discs was made for the purpose of pro- ducing yellow, the other to produce green. In a good light they both exhibit very remarkable phenomena. The effect produced by fig. 4 plate V. when in motion is shown on a reduced scale on plate III. The discs themselves were, in general, about six inches in diameter. Half Discs. 105. It will be observed that half discs are often used in the experiments. I had not experimented long before I found that it was requisite to take much light away, or 70 MR. J. SMITH ON THE ORIGIN OF COLOUR rather to produce great intervals between vibrations, in order to obtain good colours. The plan often adopted was to take a whole disc, paint one half of it completely black, and treat the other half in the manner required. By this means I obtained longer intervals between one vibration and another, and saved time in experimenting, that is in making discs. 106. Definite ratios of white and black give distinct co- lours, because the beats or vibrations are distinct. When the discs are made to revolve with a small degree of excentricity the colours are still distinct, but some of the rings of colour become compounded. When, however, the excentricity is great there is great confusion of colour, proving as I think the comparatively long interval be- tween one vibration and another in nature, in order to produce a distinct impression of colour on the sensorium. (See plate IV. figs. 8 and 9.) 107. Sunlight is much more inconstant in its effects than gashght, although the state of the atmosphere has a very decided influence even on. gaslight as any one who experiments as I have done will find. In gaslight, a ring + white $ black produces green or purple. another green or purple. 33 99 39 33 ve yellow. eho Cole CO calbo 33 A narrow black line in one direction of the light gives a 33 bright red, in another direction green, whilst in sunlight the lines produce a mauve or an olive green, sere to the light or the velocity given to the figure. But I am unwilling to indicate the numerical value of any colour at present. The subject requires much more study than I have hitherto been able to give to it, so as to yield satisfactory results. AND THE THEORY OF LIGHT. 71 Change of Colour by reversing the motion of the Figure. 108. One of the most singular and remarkable of the phenomena connected with these experiments is the change of colour which takes place by reversing the motion. Suppose we have a card (see plate IV. fig. 8) with con- centric rings cut out of or painted on it, and make it revolve by excentric motion, we shall find that one series of rings is coloured reddish-brown, yellow, or some al- lied colour; another series green, blue, or purple. ‘These rings, when the motion is reversed, change colour. This perplexed me very much at first, for it really indicated something like the polarization of light. A close and re- peated examination of the phenomenon indeed convinced me that it must help to remove much of the mystery from what is called polarized light. But this is too wide a field to enter on at present. 109. In order, however, to arrive at the cause of this interesting phenomenon I attempted to analyse one of the semi-discs having concentric rings. . I therefore made another semi-disc, divided it into seven concentric rings, leaving a small ring at the centre in order to get the disc fixed as usual. Having drawn the seven concentric rings, I made a new centre, at a distance of half the breadth of one of the rings from the former centre, and drew from it concentric circles so as to meet or touch the other circles, but not to intersect them, as will be seen in plate IV. jig. 9. Each ring was thus divided into two equal parts, by a circle passing as it were diagonally across the ring and forming two triangles, one of which was painted black, so that when in rapid motion the ring composed of these two triangles, a black and a white, might be considered as condensed into a parallelogram divided diagonally, or the white and black parts might be considered as the two faces 12, MR. J. SMITH ON THE ORIGIN OF COLOUR of a prism, one reflecting light, the other no-light to the spectator. On this disc, when one ring was painted black on the left side of the centre, the next was painted black on the right side, and so on alternately. Hence when in rotation, as the white of any ring diminished the black increased, or vice versd as the white increased the black diminished, and hence the light on the dise appeared as it were stratified, and to come from two different directions, or to be in two different planes. _ Now this dise (plate IV. fig. 9) when fixed on the machine at the centre, 0, produced precisely the same effect as fig. 8 when fixed out of the centre, as at e or é, that is at exactly half the breadth of one of the rings from the centre 0. In both the contiguous rings were of different colours — one being of a yellowish or reddish tint, the next of a purple or greenish tint; and the alternate rings had the same colour. On reversing the motion, in both figures the rings which were yellow became purple, and vice versd the purple became yellow. 110. It will be remarked as a part of this very singular phenomenon, that when the light increases in relation to the black or shadow, it appears redder than when it dimi- nishes. In the latter case it appears green or blue, or purple. In order to analyse the phenomena exhibited by this dise still further, I formed two similar semi-disecs, and drew on them seven concentric rings, each of which was divided into two equal parts, as before. But instead of painting each alternate ring on opposite sides of the disc, I painted one disc only on the right of the centre, the other only on the left. The former disc was thus divided into two parts, one representing the right side, the other the left, as in plate V. figs. 1 and 2. I now expected that. the colours which were oniy repeated alternately on the former disc, would be repeated continuously on these; AND THE THEORY OF LIGHT. 1o that is that one disc would produce only one set of co- lours, the other disc the other; and when the motion was changed that each would represent the colours of the other. I was delighted with the appearance as well as the success of this experiment when seen by gaslight. The same effect is produced in good sunlight. 111. It may be mentioned here that all these experi- ments were originally made by gaslight. I may also add that it would be useless to describe the colours of any of these figures, as they vary so much, not only from the nature of the atmospheric light, but according to the spot which is made the centre of the motion. For instance, when fig. 9, plate IV., already described, was made to re- volve excentrically in daylight, the sun shining white through a haze, and the ground being completely covered. with snow, there appeared two series of the most beau- tiful greens and purples imaginable. 112. It is to be remarked that though the amount of light and shade on these two semi-discs (plate V. figs. 1 and 2) is the same, still, when they are put in motion, the colours are different. The ratio of light to shade would thus appear to be no more the cause of difference of colour than the refrangibilities of the rays, or the lengths of waves. Other circumstances must be taken into con- sideration, such as whether the vibration is one belonging to an ascending or a descending series, to an increasing or a decreasing force. But what is this, if it is not the ratio of light and shade? TI shall return to this subject again under the head of perpendicular motion. Now if we consider the diagonal of a ring, as dividing it into two triangles, one being white, the other black ; and if we suppose the white triangle to represent a beam of light and each revolution of the machine to correspond to a vibration of light, the angular point being the com- mencement of the vibration, and the third side or base SER. II. VOL. I. fv 74: MR. J. SMITH ON THE ORIGIN OF COLOUR its termination, or vice versd ; if we make this supposition, we shall have no difficulty, on examining the motion, to account for the change of colour on the change of motion, however strange the phenomenon may at first sight ap- pear. The direction of the ray of light is an important element in all these phenomena. When the light in- creases, or, which is the same thing, when it comes after the shade during the motion of the disc, then the colour is red or allied to red. When the shade follows the light, or which is the same thing, when the light decreases, the colours are allied to blue; and all colour is only a modifi- cation of these two conditions of the ether or a combination of the two. No more has hitherto been made of the sub- ject of colour, analyse it as much as we please. Every sensation is reduced to one or other of these two con- ditions, infinitely diversified by an infinite wisdom. 113. The experiment just described I consider as not only very remarkable and very beautiful, but also as highly instructive. There will, however, arise in the minds of some who have long and profoundly studied the subject of light, and who are much more conversant with its details than I am, but who are wedded to the science of the day, the question—Is not this, after all, only the effect of the polarization of light? I am not about to discuss this point at present, but I would ask those who are unwilling to accept new ideas to explain these phenomena on any old hypothesis; for until this is done, they must for ever remain a stumbling block in the path of science. They are incontestible facts, not illusions, and cannot be ex- plained away by any empirical law or antiquated scientific conventionalism. Remarks. 114. Although by operating with discs, having definite proportions of white and black on them, colours were pro- AND THE THEORY OF LIGHT. 75 duced, they were not so defined as to afford any basis for a unit of measure. The experiments only proved the neces- sity for assuming a negative element in colour. The experiments by excentric motion introduced a wider range of phenomena, and showed how this wider range was obtained: but the colorific power of a molecule of light is to me still as immeasurable as its weight is im- ponderable. The spiral figures showed perhaps still more distinctly than the excentric how one ray modifies another, and how gradation of tint 1s produced. 115. When treating of the colours produced by re- versing the motion, J have shown how an increasing series of rays produces one species of colour and a de- creasing series another. If then we are still left without a mathematical element for expressing the value of a ray of light, we are not in a worse condition mathematically — at least as far as colour is concerned — than we have ever been. For never since the science of light has been systematically taught has the value of a ray been expressed mathematically. All such calculations are mere empiricism, and of no practical utility. 116. If we take a piece of Bristol board, and draw on it a spiral figure in china-ink or lamp-black, or draw on it several concentric black circles, and make the disc revolve with excentric motion, the black circles or spiral in gas- light will appear a brilliant red. On reversing the motion they will appear green or blue, the effect depending much on the light.* (See plate IV. tig. 9-) * To prove that there are, as it were, two discs in motion, and each different in colour from the other when the motion is excentric, let us make a semi-disc of about six inches diameter, and draw on it several concentric rings. Make a few of the lines darker than others to give variety. Let o be the centre, and 2 the centre of motion. The two discs will have for radius x a and wb, and the two series of rings will be dis- 76 MR. J. SMITH ON THE ORIGIN OF COLOUR 117. Shall we then call the unit of measure for light red ? If this, as seems probable, is the case, what then is green or blue? When I come to consider the phenomena of the prism I shall recur to this subject. To speculate, however, ou the theory of light, as has been hitherto done — making one supposition to help another, and one emendation to mend another — is a waste of time and of talent. Could such a measure as we are in search of be obtained, it would be of incalculable value to science. Hypothesis is dangerous here. Besides, if these experi- ments are as conclusive as I consider them to be, they prove that the mathematical method of reasoning, so often recommended by Newton in his Optics, has as yet done nothing for the science of colour. But I shall be told, that it is better to have a mathematical equivalent for the value of a ray, even although erroneous, than a mere physical process without one. Well, be it so to those who think so. HORIZONTAL MOTION. InD CLASS OF EXPERIMENTS. Transmitted Light, representing Coloured Media. 118. The light in these experiments was transmitted through open spaces in the figure or disc. For the sake of distinction I call the experiments which J am about to describe experiments on transmitted light. In the first class of experiments, when the white of the tinctly marked by their colour — one series being blue or green; the other of a brilliant red, tinged or edged with orange and yellow. In sunlight the colours are more subdued. On reversing the motion, the two series instantly change colour, This may be considered as another or further analysis of fig. 9, or 8, on a smaller scale. By making rings of various breadths, such as are here shown, to combine, we get a great variety of colour. We may be able, perhaps, to compound any colour in process of time by careful experiment and analysis. AND THE THEORY OF LIGHT. 7 paper was made to represent the white ray of light, the black might be paimted on the disc, or the spaces cut out as fancy dictated, for the open spaces appeared black from reflecting no light. In this class of experiments, only discs with open spaces could be used, as the white of the sky was here made to represent the white ray, and was there- fore transmitted through open spaces. The whole disc, after the parts were cut out which were to allow the pas- sage of light, was sometimes painted black, but the shadow cast by the paper itself, when held between the light and the eve, I found to be alone sufficient to produce colour. Half dises could seldom be used in these experiments. In my former experiments it was found that to obtain distinct colours it was necessary to destroy a great portion of the incident light; consequently half discs were pre- ferable to whole ones. In this class of experiments, in order to destroy a necessary portion of the incident light, whole discs were employed in preference to half discs. For not only the light and shade on the discs but the method of using them was different. In the former method the parts cut out were dark; in this they were white. “When employing the former method the light was reflected from the disc; in this the disc was held so as to be made to revolve between the eye and the light, and the light was transmitted. I at first experimented with these discs by making them revolve between my eye and a white cloud, but as there were seldom white clouds to be seen, I was forced to look for something to imitate a white cloud, so that I might be able to experiment at any convenient time. Simple as such an invention may appear to an inventive mind, it was some time before it occurred to me that a white sheet of Bristol board might at times be made to supply the want of a white cloud if there were otherwise a sufficient quantity of atmospheric light. In sunshine or 78 MR. J. SMITH ON THE ORIGIN OF COLOUR in gaslight, then, a sheet of white paper could be used in place of a white cloud, and the different effects produced by changing the angle of the incident light, that is by reflecting on the disc either more light or less light as might be thought proper, became most instructive, much more so than with the light from the white cloud, which is less manageable. 119. The method of proceeding was as follows: having placed the disc on the wheel, and made it revolve before a white cloud, or above a sheet of white paper highly illumi- uated, the holes cut in the disc permitted the light from the cloud or the paper to pass through them as the disc revolved. The light from the cloud or paper being inter- cepted at intervals by the opaque parts of the disc, colour was produced on the principles already explained. The change of colour, which results from a change in the quan- tity of light, caused by a change of the inclination of the paper to the light incident upon it as-well as the disc, explains many natural phenomena. I may mention that a sheet of ground glass hung up before a window may be used instead of the cloud or paper: semi-discs may be employed before such a sheet of glass. In many cases also the light reflected from a mirror answered better than the light of the sky. When the ground is covered with snow these experiments succeed very well. HORIZONTAL MOTION. 3RD Crass oF EXPERIMENTS. Solar Spectra by Reflection, or Coloured Shadows. 120. In the experiments which I am now about to de- scribe I made use of the apparatus employed in the second AND THE THEORY OF LIGHT. 79 class of experiments. It is the shadows which these discs east on the white wall of a room, or on a white sheet of paper, which were investigated. On taking one of the discs of the second class of expe- riments, that is discs with open spaces to transmit lght, and placing it so that it should cast its shadow, either in sunshine or in gaslight, on a sheet of white paper, and then making it revolve, the shadows were seen vividly illuminated. The colours in this class of experiments were more distinct than those of the other two classes. They exhibited, however, precisely the same phenomena under a different phase, but they had this advantage, that they could not easily escape the notice of the most careless observer, for they had the appearance of being painted on paper, in the same way as the solar spectrum appears painted; hence I call them solar spectra by reflection. 121. If the experiments are made under a jet of gas, the shadows will have very marked penumbre before being put in motion, and if when in motion the penumbre, which are now coloured, are carefully examined, it will be seen how closely such phenomena are allied to the colours produced by prismatic refraction. In fact they demon- strate that the colours of the prism are coloured penum- bre. They give a complete proof of the identity of the two phenomena. 122. Newton attempts to prove that the penumbre have nothing to do with the colours of the solar spectrum. He is constantly reverting to the proposition which he is straining his faculties to demonstrate, “That colours are to be derived from some other cause than the new modifi- cations of light by refractions and shadows.” He acknow- ledges that there is a “‘ penumbra,” to use his own words, “made in the circuit of the spectrum Y,” that is before refraction, “ and that penumbra remains in the rectilinear sides of the spectrums P T and pt,” that is after refrac- 80 MR. J. SMITH ON THE ORIGIN OF COLOUR tion. He says: “I placed, therefore, at that hole a lens or object glass of a telescope which might cast the image of the sun directly on Y without any penumbra at all, and found that the penumbra of the rectilinear sides of the oblong spectrums P T and p t was also thereby taken away, so that these sides appeared as distinctly defined as did the circumference of the first image Y.” But what does this prove? It only proves that there was not so much penumbra on the sides of the spectrum when the lens was used, but it does not prove the same as to the ends, the direction in which the light is refracted by the prism. There is little dispersion or refraction on the sides even at first; the dispersion is transverse, or towards the thick part of the prism, not longitudinal, or towards the ends of the prism. But how could Newton himself be satisfied with this experiment when a little farther on in his work he elaborately attempts to prove the impossibility of forming an achromatic glass? He says: “Seeing, therefore, the improvement of telescopes of given lengths by refractions is desperate, I contrived heretofore a per- spective by reflection, using instead of an object glass a concave metal.” This admission of itself proves that Newton was not experimentally in a condition to say that the object glass removed all penumbra. It is known that light, in passing the edge of a solid, always causes a peuumbra greater or less according to the distance of the luminous body from the object, or of the object from the place on which the image is received. This penumbra may be either a graduated grey, or coloured according to circumstances. It is these circumstances which we are now investigating. 123. I shall quote another experiment of Newton’s in reference to this point (Optics, p. 102): “ Experiment 3. Such another experiment may be easily tried as follows: Let a broad beam of the sun’s light AND THE THEORY OF LIGHT. 81 coming into a dark chamber through a hole in the win- dow-shutter be refracted by a large prism A BC, whose refracting dngle C is more than sixty degrees, and so soon as it comes out of the prism let it fall upon the white paper D FE glued upon a stiff plane; and this light, when the paper is perpendicular to it as it is represented in D E, will appear perfectly white upon the paper: but when the paper is very much inclined to it in such a manner as to keep always paraliel to the axis of the prism, the white- ness of the whole light upon the paper will, according to the inclination of the paper this way or that way, change either into yellow and red as in the posture de, or into blue and violet as in the posture 6 «. And if the light before it fall upon the paper be twice refracted the same way by two parallel prisms, these colours will become the more conspicuous. Here all the middle parts of the broad beam of white hight which fell upon the paper did, without any confine of shadow to modify it, become coloured all over with one uniform colour, the colour being always the same in the middle of the paper as at the edges, and this colour changed according to the various obliquity of the reflecting paper, without any change in the refractions or shadows, or in the light which fell upon the paper. And therefore these colours are to be derived from some other cause than the new modifications of light by refractions and shadows. “Tf it be asked what then is their cause? I answer that the paper in the posture d e, being more oblique to the more refrangible rays than to the less refrangible ones, is more strongly illuminated by the latter than by the former, and therefore the less refrangible rays are predo- minant in the reflected light. And wherever they are predominant in any light they tinge it with red or yellow, as May in some measure appear by the first proposition of the first book, and will more fully appear hereafter. And SER. III. VOL. I. M 82 MR. J. SMITH ON THE ORIGIN OF COLOUR the contrary happens in the posture of the paper 6 ¢, the more refrangible rays being then predominant which always tinge light with blues and violets.” * Newton appears to have been too easily satisfied with his experiments on penumbre, for this experiment seems to me to prove the very reverse of what he supposes. I consider that the doctrine of more and less refrangible rays completely fails him here, and I find from a note which I pencilled on this part of his Optics many years ago that this was my first impression. I, however, never ascribed this to anything but my inability to comprehend the profundity of Newton’s doctrine of light, But nature herself speaks to me now in a different language to that of Newton. When we incline the paper to the rays, as directed by Newton in this experiment, we do two things ; we remove a portion of the paper farther from the prism, and thus increase the penumbra; and at the same time we cut off nearly one half of the rays which fell directly on the paper, when it was kept perpendicular, and thereby diminish the hight. My experiments lead to the con- clusion that it is necessary not only to have penumbre in order to obtain colour, but that it is also necessary to diminish the momentum of the incident light, similar to what is done in this experiment of Newton’s. But further, if we consider the paper as an eye looking into the prism, a little study will soon convince us that at every change of the angle the object seen from the the paper is totally different, that the lights and shadows are completely altered. We have merely to trace the lines of the image backwards through the prism to convince ourselves of this. 124. Though additional proof may be required by others, Lam satisfied that my experiment proves that the spectrum is caused by coloured penumbree in the way I have shown, and consequently solves the difficulty of its length compared to its breadth. It even solves the difficulty which Newton AND THE THEORY OF LIGHT. 83 is here contending with, viz.: the cause of the spectrum being white when it first emerges from the prism. 125. Let us now revert for a moment to the phenomena of Class II., caused by what I have called transmitted light, and they will be seen to be analogous to those pro- duced by transparent coloured media. The holes in the card may be considered as representing the pores of the substance, and the change of the angle of the paper re- fleeting the light, or of the figure transmitting the light, may be considered as corresponding to the difference of arrangement in the molecules of different solids, which is the cause of their refracting (as it is usually termed) light differently. The experiments of Class III. are similar to those of Class II.; in one case the images are depicted on paper, in the other directly on the eye. But in those of Class III. the penumbree are seen on the screen before motion is given to the machine. In Class IL., although the pe- numbre are there, before motion is produced, they are not perceived by the eye. I consider the phenomena of Class III. to be similar in many respects to prismatic phenomena, and when we compare those of the second and third classes together, we can understand why pe- numbree are not scen in the prism, and at the same time learn how essential they are to the formation of colour. These two classes of experiments clear away much of the mysterious nature of the prism. When we come to the phenomena caused by perpendicular motion, the sub- ject will be made clearer still. Other difficulties will be solved, but the prism would require a paper for itself. I have not thought it necessary to give any diagrams, coloured or otherwise, to illustrate Classes II. and III. of experiments. They are similar to those already given, as far as colour is concerned. 84. MR, J. SMITH ON THE ORIGIN OF COLOUR PERPENDICULAR MOTION. Experiments in Imitation of Transparent or Crystalline Solids. 126. The experiments hitherto described were all made by employing horizontal motion. By changing the mo- tion from horizontal to perpendicular, | have endeavoured to imitate solids, and show other effects seen in the pas- sage of light through transparent media. It will scarcely be necessary to premise that the revolu- tion of a parallelogram by perpendicular motion produces a cylinder, that the revolution of a triangle produces a cone or a double cone, and that the revolution of a semi- circle produces a sphere. In fact this class of experi- ments affords an immense variety of figures which an ingenious and inventive mind may diversify to any extent. 127. Those who understand the principles already ex- plained will easily understand what is to follow. The principles are entirely the same, but in some of the experiments colour, from the nature of the generated figure, is,seen more distinctly than even in the former experiments to be caused by very sensible and easily counted beats. Although the principles involved are extremely simple, it may be advisable before proceeding further to advert for a moment to The Theory of Colour by Perpendicular Motion. 128. Let us take in our hand a white card A B, (see plate V. fig 5,) made of Bristol board, and let us examine what we see when we hold it by one end. [If it is kept between the eye and the floor the whole of the card appears white, and the space around it dark. Divide the end into two equal parts at a, place the middle point exactly on the pivot of the wheel, and make it rotate. No AND THE THEORY OF LIGHT. 85 colour will be seen, for the one half fills up the space occupied by the other during rotation; there is no mix- ture of light and shade. The white does not encroach on the dark space surrounding the card. 129. Now change the centre of revolution of the card from the middle of the end to some point nearer one of the sides, say to 6, at a distance of one fourth of the end from the side 4 C. A portion of the larger division equal to the difference between it and the lesser, in this case equal to one half of the end of the card, will in every revolution encroach upon the surrounding dark space, and will consequently produce colour; or in every revolution the opposite side of the portion 6 f will fill up the space C e, which is equal to it, and the opposite side of Ce will fill up its equal space 0 f, and hence there will be, so to speak, two beats of light on Ce or 6 /f for every beat on a D, or its equal C H when in revolution. 130. But if we paint C e black, only the middle pertion 6 f will have double beats, and if we paint the reverse or other side of bf black, making 6 still the centre of revo- lution, the whole card will have the same number of beats. On this principle a great many diagrams were formed. 131. Continuing to operate with the parallelogram, it may now be made } or 75 of an inch in breadth. If we place the edge A Cor PD B quite perpendicularly on the centre, and in a line with the axis of rotation, we shall obtain a line of greatly concentrated light. I first at- tempted to obtain red in this way, and often did, but have been more successful in getting a delicate pink, approaching to a rich reddish-purple.* * When Bristol board is used it is difficult to make it remain perpen- dicular for any time, as the centrifugal force almost instantaneously gives it an angular motion. Some other material should be used. I have tried silver, but my instrument is not well adapted for pine it firmly, and I have not experimented much with it. For perpendicular experiments, the part @ of fig. 2, plate IV., is taken off and 6 is put in its place. 6 is composed of two plates, with a screw to hold fast the cards used. 86 MR. J. SMITH ON THE ORIGIN OF COLOUR 132. If the edge be removed from the centre, there will be less concentration of light when in motion. The edge will have a larger circle of revolution, and we shall now have greens and other allied colours, such as olives of various shades. This mdicated something like a system of waves, but attentive consideration of the phenomenon would rather lead to the belief that the angle of reflec- tion had some effect, and no doubt it has, m modifying the light, but not in decomposing it, or resolving it into its simple elements. There is not only a different quantity of light reflected from the edge in motion at every angle, but the circle of revolution 1s greatly widened, and the ratio of the dark to the light immensely increased. As far as I can perceive, therefore, the very same law is in operation here as in the case of the plane discs — the law of vibra- tions or pulsations, and vibrations or pulsations which can be counted. It must be inferred, however, that if the motion of light is so exceedingly rapid as it is found to be, the length of wave must also be exceedingly large, otherwise the vibrations would be so frequent as com- pletely to destroy colour. 133. In conducting experiments by means of perpen- dicular motion, it is not necessary that the figure should have the form of a parallelogram; I only selected the parallelogram as the easiest figure for illustrating the theory and the plan of operation. Any figure may be used, or figures may be cut out of the card, or painted on it, as in the former experiments. Different figures may also be painted on different sides of the card, and a mixture of tints as well as very pleasing eflects produced in the manner already described. 134. The number of various figures which may be made in this way is endless. I will content myself with making a selection of such as will enable us to expla some of the more familiar optical phenomena. AND THE THEORY OF LIGHT. 87 Experiments. Take a card and describe on it a semi-circle; it may be either cut out of, or painted on it. (See plate V. jig. 6.) Fix it on the machine, so that the diameter a 6 may be per- fectly in a line with the axis of motion. When in motion the semi-circle will appear a circle. There will be a line, a 4, of bright colour along the diameter, equal in breadth to the thickness cf the card, if the semi-circle is cut out. Should the diameter a 0 be removed in the smallest degree from the centre of motion, the diameter of the generated figure will show various shades of purple and green. On each side of the diameter, the semi-circles of which the generated circle appears to be composed, will be coloured, not unlike a peach, beginning at the diameter with purple, then approaching to yellow, and verging to green towards the circumference. This is another demonstration of the correctness of my views with regard to the change of colour which takes place on reversing the motion of some of the plane figures, for in this experiment the light on each semi-circle visibly proceeds from two different quarters. One, however, is more apt to be struck with the phenomenon when hori- zontal motion is employed. 135. To vary this experiment, in place of a half, cut out a whole circle. In this case there will be a black line along the diameter where there was formerly a bright line, but it will be shaded off on each side towards the circum- ference as before. When the figure thus generated is minutely examined it will be seen to consist of a hollow globe, and the shadows from the edges of the card will distinctly appear on the back of the generated figure. The shadow on the aerial figure formed by the card is very interesting. If a circle painted black be used this ap- pearance will be wanting. 38 MR. J. SMITH ON THE ORIGIN OF COLOUR 136. Another variety of this experiment may be made by cutting out two semi-circles, making the circumference of the one touch the other, and fixing the figure so that one of the diameters may be in a line with the axis. (See plate V. fig. 7, and coloured diagram, plate II.) The figure is fixed on the axis of the machine at a. The appearance is precisely the same as before, as re- gards colour. The form differs in having two semi-circles besides the circle. The semi-circle, which has its diameter fixed on the axis of motion, will produce a circle, the other a semi-circle on each side of the circle. The whole figure when in motion has the appearance of two cylinders, one within the other. The reason is obvious. In this figure there are two small circles cut out at a distance (from the centre of each of the larger circles) equal to the radii of the larger circles. These were made to assist the eye in following the motion of the larger circles, and as the figure generated is rather remarkable they have been retained. Observe the difference in colour of these small circles in the outer cylinder and of those in the inner cylinder. In the outer the colour is bluish, in the inner red, showmg a bright line up the centre caused by the small bit of Bristol board connecting the two semi-circles, as may be seen by examining the gene- rating figure. (See plate V. fig. 7.) In the inner cylin- der the motion of the smail circle is traced receding as far as the eye can follow it. It will be observed that the colour is when receding nearly the same as in the outer cylinder, but partaking more of green. 137. Take another card, and cut out parallelograms, all of the same or of different breadths. (See plate V. fig. 8.) The parallelograms on the right and left of the axis will be coloured alike, as.in the semi-circles. -I have often imagined the colour of these to resemble clouds, and the different parallelograms to represent the stratification of AND THE THEORY OF LIGHT. 89 the atmosphere. Others may be prevented from forming the same idea in consequence of the too exact geometrical forms which the different strata assume in the experiment. Supposing the beam of light from the card to generate a prism, or the section of a lens, one on each side of the axis, we may then imagine the parallelograms on the left side of the axis to be figures seen through the left side of the axis of the prism, and those on the right to be figures seen through the right side of the axis; that is through the two angles on the right and left of the axis of the prism. They obey the same law as regards colour. Ts not this a most important step towards an explanation of prismatic refraction ? 138. These experiments may be infinitely varied. There is, however, another which I cannot avoid mentioning. On a piece of pasteboard draw two circles. (See plate V. fig. 9.) The outer circle may be three and a half inches in diameter, the inner a quarter of an inch or so less, which will give a ring about an eighth of an inch broad. Draw two lines, one on each side of the diameter, so as to inclose a space of about 1, of an inch broad. Cut the circle, a a a, out of the board, and the segments 0c out of the circle, leaving the space d d as a diameter. Fix the figure at o on the diameter, either perpendicularly or at an angle, and examine the different effects. When made to revolve in sunshine the colours at times are most exquisite. The whole figure is as elegant as imagination can conceive. It is not so opaque as mother of pearl, or as soap bubbles, but it partakes of all their evanescent colours. Attention is particularly directed in this experiment to the shadows which are seen on the generated figure, cast by the edges of the figure itself. They are very remarkable, and illustrate, I suspect, many of the phenomena which are to be seen in crystals and SER. III. VOL, I. N 90 MR. J. SMITH ON THE ORIGIN OF COLOUR sections of crystals, when light is allowed to fall upon them in one direction only. 139. There is another variety of this experiment. Make a similar figure, but in place of cutting out any of the segments, cut the arc of one completely through, but the chord or diameter only half through the thickness of the board, so as to enable it to be bent backwards. Bend it so as to be at right angles to the other segment. The effect of this is very beautiful, and the generated figure extremely elegant. It may be mentioned, by the by, that all the gene- ated figures are beautiful, and may be called gems of art. 140. From a great variety of coloured diagrams three have been selected to be lithographed as specimens of the figures as well as of the colours produced by these experi- ments. Plate J. is a representation of the effect produced by fig. 10a and 10b, plate V. In a paper such as this where the subject is only briefly, and for the first time presented to the public, I could have wished that some more simple figure, more like a cut crystal, had been selected. But as it is, let us consider A and B as the obverse and reverse sides of a card, and that they represent, when in motion, a transparent solid body, being cut so that when held with the obverse to the sight the dark part appears as a plane or planes representing no-light, and the white part as a plane reflecting light; and, vice versd, when the reverse is held to the sight the part which was white be- fore appears now to be black or dark, and what was dark before to be white. Centre the card exactly on the point 0, so that when in motion it may be perfectly perpendicular, the obverse repeating itself precisely on the reverse and the reverse on the obverse, and we shall obtain a figure coloured as in plate I. The principle on which the colours are produced is as easily understood from this as from a figure of more crys- talline form. The different colours are obviously produced AND THE THEORY OF LIGHT. 91 by gradations of light and shade, the pulsations of light being repeated at greater intervals than science teaches, and do not depend on rays having different degrees of refrangibility, nor on rays or waves, some of which are repeated in a given tune more frequently than others. 141. These experiments, taken in cwmulo, show us light in two different states, in an increasing and in a decreas- ing state. ‘To the former belong red and its allies, orange and yellow; to the latter blue and its allies. The sensations produced by these two states of the ether may appear astonishing to some because they are novel, and make what were formerly only ideas or concep- tions of the mind appear, as it were, palpable to the senses. For everything which is seen is considered to be palpable or material. But on due reflection they are not more wonderful than other operations of nature. Who would anticipate that heat and cold are but modifications of one law? For we say that the sensation of heat arises from an increasing or absorbing operation; that of cold from a decreasing or radiating operation. We even speak of colours as warm and cold. We call red and its allies warm, blue and its allies cold. Green is warm when it partakes largely of yellow, cold when blue predominates ; the colours arranging themselves, according to our sensa- tions, under the law which we have discovered. But we must lay speculation aside for the present. The metaphysics of every operation ends ultimately in the unconditioned ; the final cause merges in the infinite. 142. At this stage of our inquiry there naturally arises an extremely interesting question, viz: How does nature produce these penumbre, this gradation of light and shade of which we have been speaking? ‘To enter into all the minutiz, and to attempt an explanation of the whole circle of optical phenomena, would be impossible ; every one must apply the principles of the science for himself, 92 MR. J. SMITH ON THE ORIGIN OF COLOUR and I have no doubt that in the hands of skilful investi- gators many things will be made plain that are at present hidden and mysterious. I may, however, repeat what I have oftener than once remarked, that a wave of light when it falls on a substance composed of transparent plates placed at different angles to the incident light, has its force decomposed by the different plates or lamine. The wave or pencil, in falling on the first surface, is partly reflected and partly transmitted, and the ray transmitted by the first plate may be also partly transmitted and partly reflected by the next, and so on any number of times ; but the second plate not being at an angle to re- flect all the light incident on the first, performs the part of shadow to the vibrations of light on the first surface ; and again, at another angle, the first surface may act as a shadow to some other of the plates which form the reflecting substance, or vice versd; causing a change of colour at every change of angle, and giving to the sub- stance a changeable or flitting appearance of colour. In this way there is an analysis of the pencil of incident rays, and thus colour is produced in a manner similar to that of our experiments. Just as our experiments deprive the in- cident light of many of its rays, so do these lamin or plates in a similar manner perform the same operation. They give us our two elements of colour, light and no-light ; the two coordinate principles, the active and the passive. yy) 143. When we combine the conclusions deducible from these experiments with what we know of the geometrical construction of some of the natural objects surrounding us, we learn with what beautiful, with what sublime simplicity nature performs operations which science complicates and involves ; how phenomena which in the hands of a human architect —in the hands of a Newton — are made to ap- pear intricate and abstruse, are in reality produced with an ease and an elegance —if such artistic expressions can be AND THE THEORY OF LIGHT. 93 allowed —-which no created being, no one but a divine architect could contrive or conceive. For example: The phenomena of thin plates which in the hands of Newton have received a philosophical importance, from having led him to invent his celebrated theory of fits of easy transmission, and fits of easy reflection, is by the experiments on the chromascope brought more within the range of investigation. In producing these phenomena nature simply places two or more transparent plates in different planes, or at different angles, to the incident light, the one plane reflecting no-light, or virtually causing a shadow, the other plane reflecting light, and colouring that shadow, by this simple expedient performing the same operation which we attempt to do by our diagrams in motion. Our motion produces on the light not only the effect of vibrations, but the light and shade, when in motion, produce an effect similar to that resulting from the geometrical construction of a natural body, composed of thin plates. In our method of operating, a slow motion gives rise to one colour, and a quick motion to another , but the natural process yields the same results by different means. Nature analyses force; it*does not resolve a com- pound substance into its constituent parts; on the con- trary, it may be rather said to compound a simple sub- stance; or in other words, natural objects are all con- structed so as to reflect or transmit more or less of this impinging force, which we call light, and all the colours in nature are produced by the combination of one simple uncompounded substance with the negative element, which is now demonstrated to be not only an essential but a coordinate element of colour with the positive. 144. The celebrated phenomena of soap bubbles, which have amused the young and the old by the charming play of colours on their surface, receive also an easy solution by the same process of reasoning. 94. MR. J. SMITH ON THE ORIGIN OF COLOUR Newton explained the play of colours on soap bubbles by their ever varying thickness. He protected them, and watched them with care, and the result of his reasoning, and the uses he put it to, are generally known. I do not mean to follow him in his observations, or in his argu- ments, for our principles of reasoning have nothing in common. I wish, however, to say that twenty years ago, I examined most carefully the same phenomena, and although I have not repeated my observations since then, they made such an indelible impression on my mind that nothing can remove it. When examining the play of colours on the bubbles IT remarked, what every one who has attended to the subject must have done, that there was a motion caused by the action of gravity, and also a motion attributable to that of the air within the bubble. The air, within and without, being of different temperatures, there must necessarily be motion, either until the bubble bursts or until an equilibrium is estab- lished; or should there be sufficient tenacity in the infusion, these waves or circles remain stationary for a time. These two motions, or even one of them at a time, are quite sufficient on our principles of reasoning to explain the difficulties attending this hitherto intricate subject; for it becomes only another case of the same kind as that of thin plates. The motion within makes the little waves or facets move in a different direction to those on the outer surface of the bubble. We have, therefore, two transparent planes or surfaces reflecting light, the one from the eye, the other to the eye; the one causing a shadow, and reducing the number of vibrations of the incident light, the other illuminating that shadow. From these two notable phenomena of thin plates and soap bubbles is to be derived the law which reguiates the production of colour in many other natural objects. 145. The doctrine of complementary colours, as taught AND THE THEORY OF LIGHT. 95 by science, does not explain the phenomena of the two waters. The phenomena of the two wafers, however, as now developed by experiment, afford a simple and easy solution of the nature of complementary colours. When any spot on the retina is affected to an unusual degree by looking for some time at a vivid colour, that spot of the retina gets excited beyond any other portion of the retina. When, therefore, the eye is shut, the extra excitement continues, and as the excitement is a vibratory excitement, in the intervals shadow or rest (if I may use the expres- sion) is perceived or felt, and the image becomes differ- ently coloured to what it was; and when the eye is again opened the excitement remains, but the colour changes with the colour of the spot on which the image is seen. There is a diminution of force in the one case and an augmentation of it in the other. My experiments, I humbly conceive, elucidate the phe- nomena we have been considering. In the whole discus- sion of the subject I have not been compelled to resort to any supposition, to require any exercise of faith. I have not even asked the belief in a single axiom. The experiments have supplied axioms and arguments, and the most complete analytical proof which can be given — the production of phenomena which can only be explained by my argument; and I think that they completely set aside the theory which is based on the doctrine of the different degrees of refrangibility of the rays of light, and must modify to a great extent the wave theory as at present taught.* * Where I use the expression ‘‘number of vibrations,” “momentum” might be thought more appropriate, but neither conveys the exact idea wished to be communicated. For “increasing or decreasing force” £ might have used “converging or diverging rays,’ but these terms are used in a sense not identical with the received one. The “motion of a white ray” may be considered as synonymous with the “periodical interruption of a white ray.” Ke) (op) DESCRIPTION OF THE PLATES. PLATE T., coloured, is fig. 10a, and 10d, plate V., as seen when in motiou . PLATE IZ, coloured, is fig. 11, plate V., as seen when in motion. PLATE TIT, coloured, is fig. 4, plate V., as seen when in motion. PLATE IV.—Fig. 1, vide page 58. Fig.2. Form of machine. Small wheel makes eight revolutions for one of large; a, nut for fixing on the discs; 6, nut for fixing figures when the motion is perpendicular. They are moveable, being merely screwed to the spindle. Fig 3. Diagram to explain theory of horizontal motion. Figs. 4, 5, 6, 7, 8, 9, and figs. and2zin plate V., various forms of dises and half discs described. ifig. 8,a dise with several concentric rings, white and black alter- nately. Fig. g, and figs. 1 and 2 in plate V., are an analysis of the preceding jig. when made to move by excentric motion. Fig. 9 is a semicircle similar to fig. 8. Several concentric circles are drawn. Then with another centre, one half the breadth of one of the rings from the former centre, other circles are drawn so as to divide each ring into two, diagonally. Half of each ring is painted black. If the half of the first is painted black on the right, the half of the next is painted black on the leit, and so on alternately. When revolving on first centre it produces an effect similar to fig. 8 when excentric. Hach alternate ring has a different colour. PLATE V.—Tigs. 1 and 2 ave a further analysis of figs. 8 and 9, plate VI. Fig. 1 produces one series of colours; jig. 2, the other series; or by reversing the motion, the one disc produces the same effect as the other. Fig. t when moving from left to right, as the hands of a watch, produces a reddish-brown, shading off to yellow. #%g. 2, when moving in the same direction, produces a kind of gloomy green. In the one the light begins bright and becomes fainter; in the other it begins faint and becomes brighter. Fig. 3. A sort of spiral to produce gradation of colour. From the predominance of the white, when the light is increasing, the ring on the rims is a bright yellow; at times very bright. Fig. 4. A spiral, for a similar purpose as fig. 3, only the shade pre- dominates, and the rim is consequently of a beautiful green. These produce splendid effects, both by perpendicular and horizontal motion. (See plate ITZ.) Fig. 5. Diagram to explain the theory of perpendicular motion. Figs. 6, 7, 8, 9, ave merely varieties of fig. 5 ; the explanation of which must be kept in view when studying the effects produced by these figures. Tiga: 1o@ aud 106, when revolving produce the appearance repre- sented on plate I.; A and B are the obverse and reverse of the card. Fig.11, when put into motion produces the coloured image shown on plate JI, The obverse and reverse of this card are identical, as the black parts are cut out. The horizontal discs are reduced in size; those by which the experi- ments were made were about six inches in diameter. 11.— On a Method of Testing the Strength of Steam Boilers. By J. P. Joutz, LL.D., F.R.S., &e. Read November 29th, 1859. In the course of my experiments on steam, I had to employ pressures which I did not consider absolutely safe unless the boiler was previously tested. The means I adopted, being simple, inexpensive and efficacious, may, I think, be recommended for general adoption. My plan is as follows :—'The boiler is to be first entirely filled with water, care being taken to close all passages leading there- from. A brisk fire must then be made under it, and, after the water has become moderately heated, say to go° Fahr., the safety valve must be loaded to the pressure up to which the boiler is intended to be tried. Bourdon’s cir- cular gauge, or other pressure indicator, is then to be constantly observed ; and if the pressure arising from the expansion of the water goes on increasing continuously, without sudden decrease or stoppage, until the testing pressure is attained, it may be inferred that the boiler has sustained it without having suffered strain. In testing my own boiler, the pressure ran up from zero to sixty-two pounds on the inch in five minutes. Itrose - more rapidly at the commencement than towards the ter- mination of the trial, owing to leakage, which was con- siderable, and of course increased with the pressure. But as there was no sudden alteration or discontinuity in the SHRe Ut. vO. ft. 0 98 MR. J. P. JOULE ON TESTING STEAM BOILERS. rise of pressure, it was evident that no permanent alter- ation of figure or incipient rupture had taken place. In the so-called testing by steam pressure it is impos- sible to be sure that a boiler has not thereby suffered strain; and there is therefore no guarantee that it will not burst if subsequently worked at the same or even a somewhat lower pressure. It is to be hoped that this practice, objectionable on account of its uselessness as well as its danger, will be immediately abandoned. In the ordinary hydraulic test the water is introduced discontinuously, and therefore the pressure increases by successive additions, rendering it difficult to be sure that strain is not taking place. This system also requires the use of a special apparatus. The plan I recommend is free from the objections which belong to the others, and the facility with which it may be employed will probably induce owners to subject their boilers to those periodical tests the necessity for which fatal experience has so abundantly testified. Observations of Pressure every minute. EXPERIMEDT I. EXPEeRIMEnt II. \ Temp* at commencement, 97° F. Temp* at commencement, 126° F. | Pressure in lbs. Pressure in lbs. rie) O° 2°9 2°8 4°4 S19 6:0 3°8 ba) 12°6 9°05 16°I i te 20°38 12°35 26°1 13°85 31°38 15‘! 38° 16°8 44° 20°1 A9°9 24°8 54'8 31 59°4 37°2 63°8 d 44°2 Temperature at conclusion, 139° F. | 99 Ill. — Experiments on the Total Heat of Steam. By J. P. Journ, LL.D., F.B.S., &e. read November 29th, 1859. Tue total heat of steam is understood to mean that which is evolved when the steam is condensed into water of the freezing temperature. It is a mixed quantity, and consists of: ist, the heat due to the change of state from vapour to water, or the true latent heat; 2nd (as I showed long ago)* the heat arising from the work done on the vapour in the act of condensation; and grd, the heat evolved by the water during its descent from the temperature of con- densation to the fixed temperature chosen, viz., 0° centi- erade. The importance of a correct determination of the total heat can hardly be over estimated, and it is fortunate that one of the most eminent physicists of modern times has made it the object of long and elaborate research. It is not my design to attempt to improve upon the experi- ments of M. Regnault, but having had the opportunity of making some determinations in a different manner from that he employed, I think my results may not be thought without interest. In Regnault’s experiments the steam was passed into globes and a worm immersed in the water of a calori- meter. By the use of an artificial atmosphere connected * Transactions of British Association, Birmingham, 1849. 100 MR. J. P. JOULE ON EXPERIMENTS with the worm, the operator was enabled in all cases to eperate under similar circumstances as to the relative pressures of the steam and atmosphere. in my own experiments, a vulcanized india-rubber tube, eight inches long, was attached to the nozzle of a short pipe (furnished with a stop-cock), connected with the top of an upright boiler. To the end of the india-rubber tube a brass nozzle was attached. The stop-cock was left con- stantly open. In making an experiment the brass nozzle through which the steam was blowing was suddenly plunged into a can of water and then, after two or three minutes, suddenly removed again. The weight gained by the can indicated the quantity of water condensed, which, with the observations of temperature before and after the experiment, afforded the means of computing the total heat of the steam. The requisite corrections were readily made and not of large amount. They arose from the heat lost by the steam by conduction in passing from the boiler to the can, the thermal effects of the atmosphere on the can itself, and the evaporation of the water from the can which took place before the weighing was accomplished. The data for these were derived from observations made after each experiment. The following table comprises the results I obtained : ON THE TOTAL HEAT OF STEAM. 140351 575 16° Time | Weight Temperature gurms of water of water. Weisat n can, ner including aT iStean Total pressure Total heat Reg- | | steam can con- inhinghes | of nault’s} ee Reduced eae aie etoa tosh of mercury. steam. zesultes iduced,in| sp. heat | ment. | ment, | grains. 15 minutes, of water. W(t —t) 44 W. t tv. w. Gaia ear ost 2 140351 6°362°143°44.3° 8700] 40°0 ) '641°64 ) 2% | 140351 Po elspe oe 12089 36"4 | pas | i Zen. | UAT Oe ee 5°|12577| 39°95 : 42°73 628-4216 23-441 2 |140351|4°893°l43'592°| 9205| 38:25 [ 3725 eee ee 24 | 140351 57096 '|53°909° T1686] 3671 | 640'16 | 2 | 140351 |5°139 |52°814°\11574| 35°8 J 630°94 ) 2 | 140351 /6°384°|48'228°| 9835] 57°3. | 645°37 | 2 140351654 58°325° ee 57°6 | ee 3 | 140351 |6°399°|55°083°| 11609] 52°2 { 12, |643°66 ( 6) le 5 -gul 2 |140351)6'54° [48:048"| 9775] 546 [5757 [6q4r03 ( 044771947 97 2 140351 5°096"|58°347° 12800] 6073 | 642°24 | 2 | 140351|5°371°/56°523'| 12103] 63:1 J po 2 140351 6°529° 63°043" 13490|105°r ) 651°02 2 | 140351 aces AI | 655-27 | 2 140351 |6°054 |55°205 | 11276/115°3 i oO°21 : ai. 2 140351 |5°342°|53°257°| 11191 |109°2 pe 58 654°18 Goss 45|649°6 : 2 140351 |5°574 154033 |11266/112°6 | 657°383 | 2 a 109°9 J 64°891°| 14141 ee J In the above experiments the steam was condensed at twice the rapidity it was in those of Regnanlt. I had also an advantage in the size of my boiler, which was eight feet high by two feet ten inches in diameter, whereas his was only two feet seven inches by two feet one inch in diameter. Owing, however, to the small number of my experiments, the results at which I have arrived can only be regarded as confirmatory of those of the French physicist. I be- lieve, nevertheless, that the simple method I have adopted may be resorted to with advantage whenever it shall be required to obtain a further increase of accuracy. 102 MR. J. F. JOULE ON EXPERIMENTS ON THE 1V. — Experiments on the Passage of Air through Pipes, and Apertures in thin plates. By J. P. Jourz, LL.D., F.R.S., &e. Read April 3rd, 1860. Sir Isaac Newton, Polenus, Daniel Bernoulli and others have observed that water, when it is made to flow out of a vessel through a hole cut in a thin plate, becomes con- tracted in diameter and increased in velocity at a short distance from the hole, the ratio of the diameter of the stream at its narrowest part to the diameter of the hole being, according to Newton’s experiments, as twenty-one to twenty-five. The phenomenon is occasioned by a cou- course of the particles of water as they enter the orifice, and may, as Venturi has shown, be obviated by employing a short pipe instead of a hole in a thin plate. Air, and other fluids are known to comport themselves in the same manner as water. The subject is one of con- siderable importance, and as I have had an opportunity of trying some experiments on it, I trust they will be found of sufficient interest to warrant my offering them to the notice of the Society. The principal part of my apparatus was a large organ- bellows, which, by means of weights laid on the top could be worked at pressures varying from 1°44 to 5°65 inches of water as indicated by the difference of level of water in a bent glass tube. A circular hole, two and a half PASSAGE OF AIR THROUGH PIPES AND APERTURES. 103 inches in diameter, was cut in the chest, and in this could be placed, or to it affixed, thin plates with holes, or other means for the egress of air. The method of experimenting was to note the time in seconds and tenths occupied by the running down of the bellows. The capacity of the bellows being known to be 29,660 cubical inches, this observation gave the quantity of air issuing per second, plus the unavoidable leakage of the bellows. The amount of the latter was ascertained by observing the time in which the bellows ran down when the hole was made tight, and being subtracted from the gross effect, gave the quantity which actually passed through the orifice. Theoretically the quantity of air emitted in a given time ought to be proportional to the size of the aperture in the thin plate, multiplied by the square root of the pressure, or, in other words, the quantity emitted per square inch of aperture divided by the square root of the pressure ought to be a constant quantity. My observa- tions to confirm this law were made with circular holes in thin tinned iron, measuring 0°535, 1045, aud 1°61 inch in diameter respectively. The pressure and temperature of the air in all the experiments were about 29°8 in. and 4° cent. © = RI STO DTT EI AE OI Cubic inches of Air Cubic inches of Air Diameter of Pressure, in inches discharged per second, dischargeu per second, } aperture. of water. reduced to one square divided by square root inch aperture. of pressure. 1°44 496 413°3 5°6 10333 436°8 1°44 541°4 | 451°2 5°6 1058°4 447°5 1°44 589°5 4g1'2 5'6 1132°7 478°7 The last column of the above table shows the accuracy of the law so far as pressure is concerned, but seems to indicate a slight increase of the quantity issuing per square inch as the aperture becomes larger. 104 MR. J. P. JOULE ON EXPERIMENTS ON THE The law is also verified so far as pressure is concerned by the following tabulated experiments with tubes of various lengths and diameters, but all terminated by a short piece of wide pipe, three inches long by two and a half inches in diameter, which was inserted into the bellows :— - STREET ETT, REED: G@ubiaGnenenoEwaan Cubic inches of Air dis- Length & diameter precenre® discharged per second eee Mer apres of tubes. per ees inch, divided by square root f aperture. of pressure. 44 and 0°875 dt 502'9 me 3°52 9°9°7 434°7 20 and o'98 1°44 671°5 559°6 2552 1049°4 55971 ae landito4 ¢ 1°44 710°6 592°2 9 ry nie) lel 59571 At an early period of the research it was found that a very slight bur or projection on the edge of the hole in a thin plate produced a remarkable change in the quantity of effluent air. I had holes of the respective least dia- meters, 0°535, 0°75, and 1°61 inch, cut out of a thin plate of tinned iron by a brace-bit. A slight bur projected to one-fortieth of an inch beyond the plain surface. The following experiments were then made, using a pressure of air equal to 1°44 inch of water :— PR REE A IT PAST ORAS ANT ONE EY TES SEER ro Cubic inches of Air per second, reduced to one square inch of aperture. Position of the bur with respect Hole of diameter Hole of diameter Hole of diameter to the bellows. 0°535 inch. 0°75 inch. V6l inch. Outwards 597°8 579°2 647°4 Inwards 529°3 524°4 584°7 A hole one inch square, without any bur, gave 567 cubic inches per second. The influence of a tube in increasing the quantity of effluent air has been already adverted to. It was a matter of considerable interest to determine the length of tube which would produce the maximum effect. In my first experiments to determine this point, [ employed a tube PASSAGE OF AIR THROUGH PIPES AND APERTURES. 105 o'98 of an inch diameter, and terminated at one end by a piece of wider tube, three inches long and two and a half inches in diameter— Cubic inches of air per second, reduced to square inch Length of tube of of aperture of narrow tube. ce came ter: Air entering by the short length Air entering by the narrow of wide tube. tube. 40 inches. 642°7 HOS 5 666°7 10 ” 714°2 4 5 759'4 728 2s 787°7 723°3 ar ree 806°5 730°4 ae) 810°9 646°5 Sue ha 803°7 578 flapping sound. aes 749°5 flapping sound. 546 Fn 685°5 547°5 area) 666°6 541°4 In an experiment in which a flange with a hole of one and a quarter inch diameter in its centre was placed on the wide end, the quantity of air entering by the nar- row tube reduced to the length of three-sixteenths of an inch was increased from 546 to 600. The next experimeuts were made with a tube, og2 of an inch outside and o°8 inside diameter, successively reduced in length. The inner sharp edge was removed at one end, and the outer edge at the other end of the tube. It will be seen that the greatest quantity of air flowed when it entered at the end from which the inner edge had been removed. Cubic inches per second, per square inch of aperture. Length of tube of : . 5 : {R43 : Air entering the end from Air entering the end from UMN GRE which the inner sharp edge which the os: sharp edge was removed. was removed. g 44. inches. 513°5 473°2 24. 564 538 T2533 589°6 573°4 ADT ss 660 637°4 Duss 685°2 668 Ty 9 645 726°2 66372 zo 45 699°6 594 Boss 594°8 526 SER. III. VOL. I. Le 106 MR. J. P. JOULE ON EXPERIMENTS ON THE My last experiment was with a hollow cone, the sides of which formed an angle of 60°, and the opening at one end was three inches, and at the other o'625 in diameter. Using a pressure of 1°44, the quantities of effluent air per second per square inch of narrowest aperture were, ac- cordingly as the air entered at the broad or narrow aper- tures, 666°1 and 510°7 respectively. The height of a column of air of the density and tem- perature of that used in the experiments, which would give a pressure of 1°44 inches of water, is 88°93 feet. The formula for very small pressures is v = »/2gh. Thus the theoretical velocity in the absence of disturbing causes would be 75°64 feet per second, which gives 907°7 cubic inches issuing per second through an orifice one inch square. Calling this theoretical efflux unity, the above experiments give— For apertures in thin plates - -— - —- -— ‘6074 For a tube of the same diameter as length — °7676 For a similar tube with a wide entrance tube ‘8933 I have not been able to detect any effect due to vibra- tion of the issuing stream. By placing the end of a tube composed of thin metal, four feet long and one inch dia- meter, at about half an inch distance from an aperture in a thin plate of one inch diameter, musical tones were produced, which by increasing the pressure gradually, — ascended in harmonics through a scale of many octaves. The same musical effects could be produced, using a con- stant pressure of air, by moving the tube nearer the aper- ture through the space of a tenth of an inch. Savart and Masson have adduced facts of this kind to prove that air rushing out of an aperture has a vibratory motion. Al- though I do not admit this conclusion, there can be no doubt that the vibration constituting sound, produced by such methods as above indicated, will be able to travel PASSAGE OF AJR THROUGH PIPES AND APERTURES. 107 back to the air rushing through the orifice if its velocity be not greater than 1090 feet per second. I have failed, however, to discover any sensible infiuence from this cause on the velocity of efflux. It occurred to me also to try whether the air issued with a rotary motion; but such experiments as I have been able to make with vanes have led me to no decisive conclusion, although there can be no doubt that many circumstances might cause such vor- tices, the operation of which would be to diminish the velocity of efflux. 108 MR. JAMES COCKLE : SUPPLEMENTARY V.— Supplementary Researches in the Higher Algebra. By James Cocke, M.A., F.R.A.S., F.C.P.S., Barrister-at-Law. Read November 29th, 1859. § 33. 1 am desirous of submitting to the Society a Supplement to the “ Researches” which I have had the honour of lay- ing before it, and which appeared in the last volume of its Memoirs. And, first, the relations a+d=5, @&+@=15, &+e=10, ab+cd=—(ac+bd)=5, (1) -$(?)=p may conveniently be added to those given in § 2, and it will also be convenient to establish the notation Fy 067. a De Tg (OY Nia ed ae. § 34. We may pass in two ways from v*(wz+ay) to U,, and either cyclically or by forming (epimetric) functions into which each square and product enters once and once only, and which contain no cube or higher power. In the former case the operand is subjected to the processes which Mr. Harley denotes by 3’, and %’,: in the latter, using the notation of the theory of intercbanges,* we are led by 5" to U, and U,(“?). And we may write * On this theory see Mr, Jerrard’s Essay, &c.; his Reflections, &e. (Phil. Mag. June 1845); my Observations, &c. (Ibid. May 1857); &e. RESEARCHES IN THE HIGHER ALGEBRA. 109 S=U, Y=2"=U (=U where 3" corresponds to the cycle vzvyw. Under the cyclical aspect 3’ is singularly efficient as an instrument of calculation, and a transformation, to which I was con- ducted some time since, enables us to demonstrate with great ease and brevity a proposition asserted in § 28. Mr. Harley’s notation (which appears in the same volume of the Memoirs) at once enables us to write Sef) =D” HAZv™2” 4 PS vMy" + Bem” +US"v™w” =U At UO) Consequently T= Soff) +f OLS LOO +LPO Jason pee apt PES BOP?) +H" )$;, and I+J and IJ are invariable under interchanges of m and » and powers of 2 and, inasmuch as the latter is not symmetric, I?+ J*, and, therefore, (%) are functions of 7. § 35: We may, no doubt, obtain relations connecting epi- metrics of various forms, and indeed I have actually obtained them (vid. Phil. Mag. February 1854 and August 1656). But the following results, obtained with the greatest ease, show the practical superiority of the cyclical process : (S/0ow)?=S' vw . S’vw =! (vw? + ov*wz) COT ae died (2) har —tcal (7 17-2 4 (20) Svs! 0?(w+ 2) =5'03(w + z) + 27(u?) ae xy) = >'3(w + 2) + 27(w*) — $7(u) P+ 23'v'wz, or, {7(u)P=>'v(w+z)+ear(w)+23'vwe; . . (21) consequently, combining (20) and (21), we find BHA: (Westra) etn G Ks) =O! Piet hvers Vary int eeayiiied) oie saip (22) These results, which have reference to the trinomial quintic but may readily be generalised, facilitate a trans- formation now to be discussed. 110 MR. JAMES COCKLE: SUPPLE MENTARY § 36. lhet'y =pu--u*, o,—f(2"). and: 0, —7°(0"), then O(u’) = @( put wu’) = (a p+ by) (a2,p + 62) (asp + 83) (ap + by) = Ap*+ Bp*+Cp?+ Dp+H, say, amd. @Al=-0,050,0,— 0 (4), fu =—0\0,0,0,— Un, B= aA, (dqb3 + Oy0s) + Ogg G10, + bya) =pt(u)J —pr(u)I= — op’r(u)S"v3(w + 2) = 2 57 (U)T(W), in virtue of (22) and of the relation p’= 5. Again, C= 4,040.65 + 2030.0, + (a,b, + 3,4) (dab3 + 6245) = pt(u) (d.3— b,6,) + IJ = pt(u) (23 — 6,5,) + O(u’) = O(u’) — 5r(u)far(u>) — Sv*w*?; and D=)dyb,(dqb3 + bya) + byb3( a,b, + 4a) = 6,b,5 + b,b,1= 57(u?) { a7(u*) — Sv*w*t. These results furnish us with the development of @(u’) and the only observations* which it seems necessary to make upon them are that the formule of § 34 give T= -J=3'v5(w+2), and that from Badg= (2+ 0)3'9(0? +25) + (E+ 2) S008 +9?) = — pt(v) + (t+ 4) S0°w* we pass at once to bb, — bb, = — piat(ui) - Svew*. § 37: | I now proceed to other points which appear to me to place in a clearer light the relation in which these researches stand to the theory of quintics. Let a, 5, ¢ and @ be any four symbols, and X a rational function of them. The most general substitution that can be applied to X is one consisting of three successive binary inter- changes, and either side of the equivalence * See also my Observations, &c., Phil. Mag. May 1859. The transformed equation is there exhibited. RESEARCHES IN THE HIGHER ALGEBRA. lll GN a) may be taken to represent such a substitution. Let us make § 38. Next, let 0 be any symmetric function of X. The sub- stitution 2” or its equivalent /*”’*” (im an integer) has, as we see, no effect upon 0. Therefore 6=01=6 =0l'. § 39 From @=6/ and the transformations afforded by the theory of interchanges we find Cee 0) (2) ( 25) (42 (02) (26) (22) CEA) = 0(G5)\(C3) 16 (4) (22) — (C2) (28) 54) = 8) (28) = 0) (8) = 00) (2) GG 6 (C2) (c2) 0 (C9)\(24)) C2) Ca) From 0=@f and the transformations afforded by the theory of interchanges, we find 0 a") ae a in} & SS 3 is) SS = 8 LOPS ILS Sf 7, S a R SS is} g —— From 0=6? adjoined to 0=@/ we obtain A("8) = 8") = 8(")(°8) =8("")(%), and all the interchanges are now reduced to binary ones, two of which are equivalent one to the other. Conse- quently @ may be represented as the root of a sextic 112 MR. JAMES COCKLE: SUPPLEMENTARY 4 D6 + D,d'+ ahs + D,=0, whereof the coefficients are symmetric functions of a, b, ¢ and d. 9 42. The conjugate interchanges now play an important part. To the equation E=R{G, O(")t, where R denotes a rational and symmetric function, apply every possible interchange. The result will be known by examining the effect of the binary changes alone, and we find consequently & cannot receive, by permutation of a, b, ¢ aud d more than the three values RYO, O(%%)}, RIO(), (22), BAO(™), 4°23 which we may call &, &, and & respectively, and which are the roots of a cubic that may be written &+ BE —-yE+d=0, and whereof the coefficients are symmetric in a, 6, c and d. § 43. Write @, in place of @, and let Gye 0 (Ge) Gg— U(r), Gu =U Cal Cle of etl Gane the foregoing discussion shows that if E=O,4+4,, &=0,405, &=A;+ 4, EF = O9, — (0,0, + 6295+ 6505), o¥ y=9,0, + 0,05 + 0,4; — Ds, and y is symmetric in @, b, ¢ and d. then RESEARCHES IN THE HIGHER ALGEBRA. 113 § 44. Now there are two conditions essential to the applica- tion of these results to the theory of quintics : 1°, The symbol y must be the root of an equation with known coefficients. 2°, The symbols D must be known. § 45- The first will be attained if we assume X,=a,4+ aa, + bx, 4+ cv,+ da, for, the interchanges of 2, x3, v7, aud #, being equivalent to corresponding interchanges of a, 0, c and d, the expres- sion y will be symmetric with respect to those four roots and, therefore, a rational function of the remaining root Z,, and the root of a determinate equation of the fifth degree. § 46. It will be remembered that @ is symmetric in X, but that all is otherwise arbitrary. Hence, in order to attain the second condition, we may give to @ any form consistent with that symmetry. One mode of seeking to attain it is by giving a maximum of symmetry to @ with respect to 2, in other words, by endeavouring to construct a SYMMETRIC Propuct. We are thus led to the sextic in 0. § 47. These results justify the form which Mr. Harley gave to the factors of the resolvent product, and by which he has so greatly simplified my discussion (compare Phil. Mag. December 1852 and March 1853 with his Memoir). Viewed in the light of the present Memoir, the Method of Symmetric Products has (beyond invoking the aid of interchanges) no special relation to other methods in the theory of equations. If we suppose the resolvent product SER. III. VOL. I. Q 114 MR. JAMES COCKLE : SUPPLEMENTARY, ETC. to vanish it may be regarded as a modification and simpli- fication of the theory of Lagrange and Vandermonde ; but it is needless to assume this evanescence, and indeed by connecting it with Euler’s process far more striking formulze are obtained. Considered as a process of elimi- nation I shall give it a discussion elsewhere.* * See a paper “On Equations of the Fifth Degree,” which will appear in the forthcoming Diary. 115 VI. — Remarks on the Australian Gold Fields. By W.S. JEvons. Read November 15th, 1859. I. The Geological Characteristics of the principal Gold Districts. To prepare the way for some general conclusions as to the geological source of gold, I shall commence with brief descriptions of those more important gold-producing locali- ties which I lately enjoyed an opportunity of visiting. Benpico. — The celebrated Bendigo consists of a long shallow valley— say ten miles long, and from one to three miles wide. On both sides it is distinctly bounded. by hilly ranges of moderate and rather uniform height, from which spur-ranges often advance towards the middle of the valley, while corresponding gullies* or smaller valleys run up a short distance into the surrounding country. The ranges are entirely composed of a rather soft schis- tose or slaty rock, varying in colour from red to yellow, and decidedly belonging to the Silurian era, but the strata are cloven, here and there, by remarkable walls or dykes of quartz rock, usually called reefs, which always run in a nearly north and south direction, independently of the form of the ground. The quartz is often of a pure milk- white colour, but also, in parts, contains considerable * Gully is an universal term in Australia for any hollow, whether it be a precipitous ravine of great size or a trifling inequality of the ground. 116 MR. W.S. JEVONS: REMARKS quantities of the sulphides of iron, arsenic and copper, which are known by the common name of mundic, and are a pretty sure indication that gold is present in a pro- portion highly profitable to the miner. At the same time, these sulphides prove a great obstacle to the process of amalgamation, causing a serious loss both of mercury and — of gold. The lower parts of the valley are partially filled up with detritus which has evidently been washed down by streams of water from the neighbouring hills. Such detritus con- sists of red and white mottled clays, more or less mixed with sand and gravel, among which quartz pebbles are conspicuous. The deposits of gold dust which gave to Bendigo its reputation lie, or rather used to le, principally in the very lowest strata of the alluvial detritus (technically called by geologists the auriferous drift), but are especially accumu- lated in the deepest channels or hollows worn into the schistose bottom rocks at some former time. Thus, while a large area of ground throughout the valley has proved auriferous, the most valuable claims usually lie in a broad line which does not often coincide with the present Ben- digo Creek, or line of drainage. This line or lead of rich gold deposits has more lately been traced two or three miles below the mouth of the valley, where the present surface of the ground is nearly level. Forest Creex, which lies close to the new and con- siderable town of Castlemain, is another large, rich and well known digging. In general nature it so much re- sembles Bendigo that no separate description is necessary ; the schistose ranges, however, are here more bold, and assume a characteristic shape, resembling a very obtuse pyramid. Iam not aware that quartz reefs have yet been opened here. Creswick’s Creek, formerly of high reputation, and a ON THE AUSTRALIAN GOLD FIELDS. 117 host of smaller gold-bearing localities may be described almost in the same terms —as valleys or gullies among low schistose hills. But not uncommonly the auriferous drift assumes so level an upper surface as to be denomi- nated a flat. Batuaarat. — I come now to the celebrated Ballaarat, the great richness and still more the complicated and remarkable structure of which render it by far the most interesting of the Australian gold fields. We shali be greatly assisted, too, in understanding its geological forma- tion by the excellent geological map produced under the direction of Mr. Selwyn, government geologist of Victoria. From this map it will be seen that Ballaarat consists of a somewhat basin-shaped accumulation of alluvium, or auriferous drift, surrounded by detached and bold hills of schistose or slaty rock, between which run various gullies filled by extensions of the alluvium. All the inequalities of the surface and the continuity of the depressions may - be clearly traced by the contour lines. A very large part of the total area of alluvium is distinctly auriferous, and a large part indeed has been turned up or undermined by the gold diggers in their eager search. But on the map only those spots of ground are marked as gold workings where the deposit was especially rich; now it has been found and is shown that these richest deposits occur in broad continuous lines or bands called leads of gold, which occupy the deepest depressions of the schistose bottom rock. The leads are all branches of one main lead or body ; and while the elevation of the bottom of this main lead is about one thousand two hundred feet above the sea, the branch leads universally show a more or less gradual rise of level. It may be observed, too, that the leads show scarcely any coincidence with the present gullies or water- courses, and do not bear a very distinct relation even to the position of the present schistose hills. 118 MR. W. S. JEVONS: REMARKS A fresh feature is now to be noticed which, although unconnected geclogically with the gold, is of some import- ance to our subject, because it often conceals the golden strata and obstructs the miner in reaching them. I allude to a great sheet of basalt or, in fact, lava, which at a very recent geological period has overflowed the beds of auri- ferous drift often to a thickness of fifty feet. At Ballaarat this overflow has only partially taken place, and the viscid current of lava appears to have been stayed in its progress so as to form a plain bounded by a steep slope. A large and irregular area of country in Victoria, perhaps one hundred miles in length, is covered by this basaltic rock. Melbourne stauds upon the edge of the basalt, and its houses and stores are built of this hard blue vesicular rock. Standing upon the higher parts of that city, the view extends inland over an immense sheet of lava often nearly as level as the sea, and only covered by a thin layer of fine brown soil. Extinct volcanoes also, unmistakeable in form and nature, occur in several parts of this basaltic country. Thus Mount Buninyong, lying seven miles east of Ballaa- rat, is an obtuse cone of basalt, becoming more and more scoriaceous towards the summit, whence I brought several specimens of decided scoria taken from a slight depression which was, undoubtedly, an imperfect crater. From this mountain a very extensive view was obtained over a coun- try generally fiat. A mile or two to the north was seen Mount Warrenheip, the counterpart of Buninyong, and, like it, an ancient volcano; while Mount Blackwood, Mount Clarke, and other eminences lying at various distances and points of the compass, could also be distinguished as of similar origin by their isolated conical or rounded form. Again, twenty or thirty miles to the north of Ballaarat there stand, scattered over the basaltic plain, a numerons group of very smooth-rounded eminences known as the ON THE AUSTRALIAN GOLD FIELDS. 119 Bald Hills, or Hills of Lava. They are of basaltic rock, which, it is my impression, is not more vesicular or scori- aceous than in other parts of the country ; thus these hills differ from volcanoes proper. It is no easy matter to con- ceive the cause of these singular Bald Hills, or the source of the immense body of lava which now forms the surface of a large part of Victoria, and must once have been a liquid mass, glowing in the open air. Among the various ancient rocky strata of the globe, it is well known that overflows and protrusions of volcanic rock are often met with, while I am not aware that an ancient (?.e. primary, secondary, or tertiary) volcano, embedded below more recent strata, has ever been recognised. Is it possible that in Victoria we find the type of the ancient volcanic eruption in which lava was all abundant, while modern voleanoes with lofty cones are distinguished by the dis- charge of much gaseous matter ? But, to return from this digression: for many years — indeed nearly up to the time of the geological survey of Ballaarat — the gold miners had no thought that rich de- posits of gold lay concealed beneath the basaltic overflow ; but greater experience, combined with increased means of penetrating the hard rock and of sinking deep and expen- sive shafts, enabled them to trace out several fine leads of gold, extending quite under the new town of Ballaarat up to the lake or swamp indicated on the plan. Numerous companies of co-operative miners have lately taken claims on these leads and, with the aid of steam engines, pumps and other machinery, are raising and washing great quan- tities of gold. I descended one of these mines (on the Malakoff lead), being lowered by steam power down a well-constructed shaft to a depth of three hundred feet. From the bottom of the shaft a gallery had been driven to reach the lead of gold, which was then followed to the limits of the claim. 120 MR. W. S. JEVONS: REMARKS The lead was here contained in a deep, well-marked channel in the schistose bottom rocks, now filled up by a most peculiar alluvium, consisting of quartz gravel, fine black clay, and an abundance of lignite or blackened fossil wood,—complete trunks of trees, indeed, being sometimes encountered by the miners. The gold dust lay most thickly in the very lowest part of the channel, where it was not difficult to detect the shining grains and pick them out with the fingers; they were partly entangled in the crevices and joints of the bottom rock, but smaller quantities of gold dust lay on the sides of the channel, or were disseminated through the alluvium. We need only see such a mine as this to be con- vinced that these singular leads of gold lie in the water- courses, or river-beds of a former age, where streamlets and rivers ran down from hills now washed away. The history of alluvial gold can here be read as plainly as in a book; great masses, perhaps mountains, of schistose rock have in the course of ages become disintegrated; the quartz reefs which the hills contained were then broken down and filled the river bed with smoothly worn gravel, while the particles or nuggets of gold thus liberated from their quartzose matrix sought, by means of their high specific gravity, the lowest possible position, and there accumulated. Nature has in this way for many ages been performing the self same operations to which the gold miner has recourse in extracting and separating his gold. A large space of time indeed must have elapsed since the lower parts of the alluvium and the large quantity of gold which they contain were deposited, because we find them covered by two or three hundred feet depth of other alluvium since accumulated, not to speak of the basaltic layer, which proves that extensive volcanic eruptions have both commenced and become quiescent in the intervening period. ON THE AUSTRALIAN GOLD FIELDS. 121 Yt will be noticed that a number of quartz reefs, known to be auriferous, are marked upon the Government Map from actual survey, their parallel and meridional direction being strikingly shown. Many other reefs however exist, some perhaps unauriferous, but mostly untested or imper- fectly tested for gold. Western Gop Fizetps or N.S. Watzs.— In the west- ern district of New South Wales, where, it will be remem- bered, Mr. Hargreaves made the first practical discovery of Australian gold, the prevailing strata are of a hard slate rock belonging to the Silurian era. Quartz reefs are found abundantly penetrating the strata. The country, accord- ingly, is of a more rugged and mountainous character than in Victoria, rapid streams or torrents flowing from among the steep stony ranges. The auriferous alluvium generally consists of gravel mixed with a little clay, lying in or close to the present water-course ; but there exist many places where the stream has in the course of time worn for itself a new channel, and left in its former place large accumulations, even hills, of auriferous drift. Making allowance for the harder nature of the prevailing rock, the same geological principles prevail in these diggings as in those above mentioned. Marysoroues. — This small town is situated in the midst of many gold-producing localities of inferior import- ance, but I have only to notice here a small hiil, close to the town, consisting of schistose rock penetrated by a con- siderable quartz reef. A mass of feldspathic porphyry is also to be found close to the reef, aud sometimes pene- trated by leaders or thin veins of quartz in a very curious manner. Ovens Goup Disrricr. — In this district are comprised. many separate localities, such as Jackandandah, Woolshed, Indigo, &c.; but for my present purpose, I need only de- scribe those auriferous flats or gullies lying within a few SER. III. VOL. I. R 122 MR. W. 8S. JEVONS: REMARKS miles of Beechworth, the principal town, which is situated upon the May-day Hills, one thousand seven hundred feet above the sea. The diggings in question lie considerably above the level of the surrounding country, and we can look for the source of the contained gold only to the high- est parts of the hills. Now the May-day range is com- posed of granite, and no stratified rocks were to be seen; so that it is surprising, and almost unaccountable, to find among the auriferous alluvium clay containing an abun- dance of white quartz gravel, evidently the detritus of schistose hills penetrated by auriferous reefs. We are driven to suppose that Silurian strata once covered the granitic range, but have since been entirely disintegrated and removed. Mount TarrencowerR. — Here we meet very different geological features. The gold produce is derived from several very fine and rich quartz reefs which traverse a bold mountain range, principally if not entirely composed of Plutonic rocks. Unimportant alluvial diggings lie at the eastern base of the mount. I descended and examined two of the quartz gold claims or mines, mere rude narrow spaces, where part of the dyke of quartz had been ab- stracted from between the walls of the containing rocks. In these cases, the reef did not crop out at the surface but shafts had been sunk to a depth of fifty or sixty feet in order to strike upon it. The foundations of Mount Tarrengower are undoubtedly of granite, which is to be seen on the western slope reach- ing nearly half way up, but the higher parts, it must be especially remarked, are composed of a hard hornblendic rock more nearly resembling basalt than granite. Nume- rous auriferous quartz reefs traverse this hornblendic part of the mount, one reef cleaving the very summit, while another reef attains the extraordinary thickness of thirty feet. It is true, indeed, that close to those reefs which ON THE AUSTRALIAN GOLD FIELDS. 123 are now chiefly worked, a small quantity of soft whitish rock occurs, containing leaders or thin veins of quartz; but I am not sure whether this is to be considered Silurian schist rock. Even if it prove that there are here the re- mains of Silurian strata, formerly existing in much larger mass but now carried away by pluvial action to supply the alluvial drift of the surrounding country, it is still true that in another part of the mount the close conjunction of quartz reefs, hornblendic rock and granite is clearly estab- lished. ApELonG Creex is in New South Wales, not far from the borders of Victoria, and is remarkable for the most splendid single quartz reef yet opened, I believe, in Aus- tralia. The reef traverses bold steep ranges of a granitoid rock for a distance of a mile, and probably much further, in a line very nearly straight and not diverging more than two or three degrees of azimuth from the true meri- dional direction, as | ascertained by means of a prismatic compass. This great reef is one continuous wall of quartz, crop- ping out at the surface and inclining, as it descends, ten or fifteen degrees from the perpendicular towards the west. The thickness of the quartz varies from almost nothing to six feet or more, “ making” or “dying out,” as the miners say (2.e. becoming thicker or thinner), in a most capricious manner. But the most instructive circumstance to the geologist is that a variety of igneous rocks occur close along- side the wall of quartz, appearing to support it, or at least to form bands parallel to it. The accompanying specimens will best show the nature of these rocks; they comprise felspar in a state of purity, and a series of hornblendic rocks containing successively larger proportions of white quartz. Hornbleude was evidently a predominant mineral in the neighbourhood; but so various were the rocks which the miners had encountered, and so difficult was it 124 MR. W. S. JEVONS: REMARKS to learn particulars from them, that to arrive at the true nature and relative positions would have required a long personal examination, and also more geological experience than I possess. Several other reefs parallel to the main reef, but of much smaller magnitude, were near, and were accom- panied in a similar manner by hornblendic rocks. The richest parts of the reef were thought to be those where the quartz contained large quantities of the sulphides of iron, arsenic and copper. Only very small quantities of gold had ever been ob- tained from the bed of the creek near, although the reef was proving highly remunerative to some thousands of persons engaged in mining and crushing quartz, in order to separate the gold by amalgamation. The yield was perhaps five (5) ounces of gold per ton, as much as ten (10) ounces being sometimes obtained. Brarpwoop. — A very distinct kind of gold field is that near Braidwood, in the southern part of New South Wales. The country is a remarkable plateau of granite, uniformly elevated two thousand feet above the sea level. At Jem- baicumbene gold is found in the wide shallow bed, or in the banks of a creek, surrounded by granite hills of very moderate elevation, or by flat lands from which granite boulders everywhere crop out. } A fine gold dust, quite free from large particles or nug- gets, is found at the creek, lying on the granite bottom rocks, amid a detritus of purely granitic origin, including a fine white sand composed of minute perfect topaz crystals, which remain mixed with the gold in spite of all washing. Black magnetic iron sand is also found here in abundance, and quartz occurs in large crystals. In the bed of the creek, indeed, I noticed some pebbles of extraneous origin; but I have not the least doubt that. the Braidwood gold is derived directly from the surround- ON THE AUSTRALIAN GOLD FIELDS. 125 ing granite country, being originally disseminated through that Plutonic rock. ARALUEN is a long romantic ravine, bounded by un- broken granite ranges. Gold is found in the river bed amid granite boulders and detritus, and is evidently de- rived from the same source as that of Jembaicumbene, which is only six miles from the head of the Araluen valley. T must mention that in the northern part of New South Wales (now within the new colony of Queensland), which I have never visited, gold is found in conjunction with quite a different set of rocks, viz. serpentine and its con- geners. Such was the singular isolated mass of gold which occasioned the wild rush of ten thousand diggers to the tropical Fitzroy river a year since, only to meet there with hardship and. complete disappointment of their golden hopes. At Bingara and other parts of the north, diggings of a somewhat similar character were discovered and described by the well known geologist, the Rev. W. B. Clarke. IT. Speculations as to the Source of Gold. From the order in which I have described the gold fields, the reader will almost perceive the view which I take of the geology of gold. It is doubtless the granite which originally contains the gold in a very finely dis- seminated state, as it is almost proved to exist at Braid- wood. Now quartz is one of the chief constituents of granite ; feldspar and hornblende are the other chief con- stituents. It has been proved,* too, that quartz is the * “ As a general rule, quartz, in a compact or amorphous state, forms a vitreous mass, serving as the base in which felspar and mica have crystal- lized ; for, although these minerals are much more fusible than silex, they have often imprinted their forms on the quartz. This fact, apparently so paradoxical, has given rise to much ingenious speculation. We shovld naturally have anticipated, that, during the cooling of the mass, the flinty 126 MR. W. S. JEVONS: REMARKS portion which longest remains plastic, and that it often becomes segregated or collected together into fissures passing through the granite.* I believe that a similar action on a much larger scale is the cause of all quartz reefs in Australia. Great cracks or fissures are produced in the granite by some cause with which we are unacquainted, passing upwards, also through the overlying stratified rocks. The more liquid quartz filters out of the surrounding granitic mass, carrying with it the gold and the metallic sulphides, and, soon filling the vacant fissure, is forced upwards and penetrates as far as possible the overlying strata. We have thus an explana- tion why hornblende is found in immediate connection with those quartz reefs which lie amid granite; for as the portion would be the first to consolidate; and that the different varieties of felspar, as well as garnets and tourmalines, being more easily liquified by heat, would be the last. Precisely the reverse has taken place in the pass- age of most granite aggregates from a fluid to a solid state, crystals of the most fusible minerals being found enveloped in hard, transparent, glassy quartz, which has often taken very faithful casts of each, so as to preserve even the microscopically minute striations on the surface of prisms of tour- maline. Various explanations of this phenomenon have been proposed by MM. de Beaumont, Fournet, and Durocher. They refer to M. Guadin’s experiments on the fusion of quartz, which shew that silex, as it cools, has the property of remaining in a viscous state, whereas alumina never does. This “gelatinous flint” is supposed to retain a considerable degree of plasticity long after the granitic mixture has acquired a low temperature.” — Lyell’s Manual of Geology, fifth edition, p. 567. * “Veins of pure quartz are often found in granite as in many stratified rocks; but they are not traceable, like veins of granite or trap, to large bodies of rock of similar composition. They appear to have been cracks, into which silicious matter was infiltered. Such segregation, as it is called, can sometimes be shown to have clearly taken place long subsequently to the original consolidation of the containing rock. Thus, for example, I observed in the gneiss of Tronstad Strand, near Drammen in Norway, the annexed section on the beach. It appears that the alternating strata of whitish granitiform gneiss and black hornblende-schist were first cut through by a greenstone dyke, about two and a half feet wide; then the crack, a@ 6, passed through all these rocks, and was filled up with quartz, The opposite walls of the vein are in some parts incrusted with transparent crystals of quartz, the middle of the vein being filled up with common opaque white quartz.” — Ibid. p. 576. ON THE AUSTRALIAN GOLD FIELDS. 127 plastic quartz is forced out, of course the hornblende and feldspar are left behind to form the walls of the reef. Among Silurian rocks we never, I believe, find hornblende near the quartz, and only in the instance of the feldspathic porphyry at Maryborough, do we find any plutonic rock at all present. The gold districts of Victoria, including the southern parts of New South Wales, I regard as a great mass of auriferous granite, upon which lie Silurian strata of a slight thickness. Granite protrudes bodily from the sur- face in very many places, of which the May-day Hills, Mount Alexander, Mount Tarrengower, Adelong and Braidwood are only a few of the best known. In other parts the liquid quartzose constituent of granite has alone been forced upwards, filling a system of fissures which have an almost invariable direction from north to south. I am in no way called on to explain the cause of these parallel fissures, for the same parallelism is well known to be a characteristic of all systems of mineral veins, or fissures filled by mineral ores, in Cornwall and elsewhere. Some of the opinions above expressed were entertained many years ago by the Rey. W. B. Clarke, of Syduey, a geologist who had discovered gold in New South Wales many years previously to Mr. Hargreaves’s practical dis- covery, and who is better acquainted with the geology of Australia than any other person. TIT. On the future Supply of Gold. The question, Will the supply of gold from Australia increase, remain constant, or diminish? is of evident and high importance. I will therefore conclude with a few remarks by way of answer. The rich diggings of Ballaarat, Bendigo and Forest Creek were all discovered in the course of one or two years after attention was first drawn by Mr. Hargreaves 128 MR. W. S. JEVONS: REMARKS to the presence of gold in the Australian rocks. Since then most favourable localities have been well prospected, or tested for alluvial gold by the many thousands of experienced diggers with whom Victoria now abounds. I regard it as certain, therefore, that no alluvial deposits, equally rich and equally accessible, will ever be laid open in future. But it is by no means certain that great and rich leads of gold may not exist in many places still con- cealed by the great thickness of the overlying alluvium, or by a portion of the basaltic formation or sheet of lava. Such leads would be comparable to the new and deep leads beneath the new town of Ballaarat, and would be discovered only by a very lucky chance, or by a skilful and expensive search. From the mode in which schistose hills, bearing fine auriferous reefs, here and there only just appear through the basaltic formation, as at Clunes and Melbourne, we may consider it very possible that complete gold districts may lie hidden, perhaps for ever, at considerable depths beneath other parts of the basaltic plain. Not much, perhaps, is to be hoped from the future discovery of alluvial gold. The chief diggings of Victoria have, during the past eight years, been so vigorously undermined and turned up by small parties of miners that scarcely a square yard of untouched auriferous ground is left—at least such ground as would have formerly been considered profitable to men without experience and means of undertaking large works. But these diggings are by no means exhausted. The miners, learning the advautages of co-operation and the use of machinery, find that masses of alluvium, only slightly auriferous, now afford a handsome and also a steady profit. Ground exhausted in the old style is beimg reworked in a much more complete manner, and even the tailings, or refuse from the cradles of previous diggers, is valuable to those provided with more improved washing apparatus. ON. THE AUSTRALIAN GOWD FIELDS. 1239 As science, capital and co-operation are more and more introduced, auriferous land of less and less richness will successively become profitable, and of such land the quan- tity, it may be stated, is certainly very great. It follows then that the supply of gold, even from the present alluvial diggings, would diminish but slowly, un- less indeed other social circumstances, as may well happen, should interfere. The many thousands of Chinese diggers, with their quiet plodding and, so to say, mean industry, assist to maintain and steady the supply. Yet it must certainly be allowed that the deposits of alluvial gold will begin to fail gradually in a very moderate period of time. The answer is otherwise, I believe, when we take inte account the quartz reefs which are real gold mines. The number of reefs now known to be auriferous is great; of the rest some may be devoid of gold, but the majority probably contain a proportion of the valuable metal which, either now or in some years to come, as machinery im- proves, will repay extraction. As reefs are now worked by small parties of miners, each owning a small claim and sinking an independent shaft, even the richest reefs would. cease to be profitable at a certain depth, from the great comparative expense of draining and working so small a mine. But if a whole reef were in possession of one large monied company, two or three shafts and a single estab- lishment would suffice ; powerful engines and pumps would be employed to drain the mine, and the quartz reef might be followed as deeply as copper veins or coal beds. There is no known reason, beyond a mere fancy in the minds of some geologists, that reefs should become impoverished in sinking; the fancy is now proved to be contrary to fact. Even a few large reefs then would, I believe, yield a considerable and constant supply of gold for many years to come. For how much more may we not hope when greater experience is attained m quartz mining, now 3ER. III. VOL. I. s 130 MR. W. S. JEVONS: REMARKS, ETC. so new an employment ; when improved machinery is brought into use for the rapid, complete and cheap extrac- tion of the gold from the quartz matrix; when capital is attracted in great sums to the pursuit; and when the search for new auriferous reefs, becoming more keen, is rewarded, as I believe it will be, by abundant discoveries ? Our conclusions, expressed as shortly and truthfully as possible, are : 1. That no great and recurring discoveries of alluvial gold are to be expected; so that the yield of alluvial gold must notably yet gradually fall off.* 2. That the supply of gold from its quartz matrix is subject to entirely different laws; that we at present know of no limit to the amount procurable with the aid of capital; and that that amount, whatever it be, will probably remain constant for a long period of time; that, in short, the supply of gold from Australia will prove as inexhaust- ible as the supply of tin and copper from the Cornwall mines, or as the supply of almost any other metal from its most common source. * Since this paper was written, much sensation has been produced by the discovery of the Kiandra gold field in the neighbourhood of the Snowy Mountains. Of course further discoveries may take place in regions pre- viously unprospected and almost unexplored. Thus the northern parts of the Australian Cordillera may contain rich gold deposits. But the opinions expressed above will ultimately apply to all such new regions, as they already apply partially to the principal gold fields now worked for nearly ten years. i3l VII.—On the Vestiges of Extinct Glaciers in the Highlands of Great Britain and Treland. By Epwarp Hutt, F.G:S. of the Geological Survey of Great Britain. Read February 7th, 1860. Iv is with considerable diffidence that I venture to read the following paper before the Society, as it is one in which there is little that is new, and certainly nothing of pressing or immediate interest. But having during the last summer made a visit to the southern slopes of the Westmoreland mountains, with the object of tracing in some detail the impressions which it was generally known the ancient glaciers had left behind, I would fain hope that the communication of the results might not be unin- teresting. 5 Besides entering upon this hitherto almost untrodden ground, I shall venture to present a short outline of simi- lar glacial traces in the other highland regions of the British Islands; and here I would remark that there is a wide field for observation still open, and that what Pro- fessor Ramsay has done for North Wales, we want carried out in the mountains of Kerry and over the greater extent of the Scottish Highlands. From my own observations in the Lake district, I feel confident that, in order to arrive at an adequate idea of the extent and nature of the former glacial geography of the British Isles, it will be necessary 132 MR. E, HULL ON THE VESTIGES OF EXTINCT to examine in detail the flanks and main valleys of the mountain-groups, and the rock-surfaces of the districts by which those mountain-groups are surrounded. It will be necessary to record on accurate maps of suffi- cient scale the directions of the striz, and the levels to which they rise on the flanks of the valleys, the sites of the moraines both lateral and terminal, and the upper limits of the Northern Drift. When this has been accomplished for any of our moun- tain-groups, it will then be possible to construct a map, reproducing with approximate accuracy the glacial system of the district, and the relative position of land and sea during the period. I can answer for it, that to one en- dowed with a love of nature, together with those essential accessories, time and money, the inquiry would afford ampie enjoyment, and the scientific value of the results would not be unimportant. It was in a detailed survey of this kind that I was engaged during the summer; and as it is far from being complete, I hope to be able to devote another season to the northern flanks of the great central range which separates Westmoreland from Cum- berland. As far back as the year 1821 M. Venetz first announced his opinion, founded on ample testimony, that the glaciers of the Alps formerly extended far beyond their present limits.* These views were subsequently confirmed by MM. Charpentier and Agassiz, and are now universally received. But it was not until the year 1842 that Dr. Buckland+ published his reasons for believing that the mountains of Caernarvonshire gave birth to glaciers which descenced along seven main valleys; and that to these agents are to be attributed the polished, fluted and striated rock-surfaces which may be traced at intervals along the * Bibliothéque universelle de Genéve, tom. xxi. + Proceedings of the Geological Society of London vol. iit. GLACIERS IN GREAT BRITAIN AND IRELAND. 133 pass of Llanberris and elsewhere. This opinion, at first received with incredulity, was subsequently confirmed by Mr. Darwin * and Professor Ramsay. The grounds upon which Dr. Buckland rested his con- clusions were precisely those upon which M. Venetz infer- red former extension of the Alpine glaciers. The effects of these streams of ice moving along their channels have now been repeatedly observed not only in central Europe, but in the Arctic regions, where they descend into the sea and give origin to icebergs. These effects consist in the polishing and moulding the bottoms and sides of the val- leys into smooth oval bosses, or roches moutonnées — the production of strize, flutings and scratches (which are gene- rally parallel in a given locality); also, perched blocks and moraines. The combination of these phenomena in any region can only be attributed to the agency of glacial ice, as there is no other known power capable of producing them. When to these is added the dispersion of erratic blocks, or boulders of large size, over a district extending many miles from the parent masses to which they may be traced, we cannot hesitate to refer the transportation of these blocks to floating icebergs derived from glaciers in a manner similar to that which is in operation along the coast of Greenland,{ or amongst the fiords of Tierra del Fuego.§ The only districts where, as far as I am aware, a detailed survey of the glacial striz has been accomplished, are those of Snowdon by Professor A. C. Ramsay,|| and the mountains of Scandinavia by M. J. C. Horbye.{ In the elaborate work of the latter, extending over the whole Nor- * Philosophical Magazine, vol. xxi. p. 180. + Peaks, Passes and Glaciers, 4th edition. £ Quarterly Journal of the Geological Society of Loadon, vol. ix. § Dr. Darwin’s Voyage of « Naturalist. || Peaks, Passes and Glaciers, 4th edition. {| Observations sur les Phénoménes d’érosion en Norvéege. (Christiana.) id4 MR. E. HULL ON THE VESTIGES OF EXTINCT wegian Peninsula, but more especially devoted to the southern part of Norway, the directions of the glacial striations are indicated on beautifully executed maps, with great fidelity, and exhibit in the clearest manner the course of the ice, either as glaciers or bergs, at the period when almost the whole unsubmerged region was overspread by one broad winding sheet of snow and ice. From an inspec- tion of these maps, it is evident that there was a general motion of the ice from the central axis of the chain at every point. Thus at the extreme north, the striz point north; along the western coast, they point west; at the southern extremity, and on both sides of the Baltic, the normal direction is 8.S.E. To this south-easterly drift of the ice it is owing that the plains of Germany, Poland and Russia, as far south as lat. 50°, are strewn with blocks of Scandi- navian granite. Reverting to the British Islands, I shall endeavour to present a short sketch of the glacial vestiges which are to be found amongst the mountains of Killarney in Ireland, of Caernarvon in North Wales, of the Lake district in Eng- land, and the Scottish Highlands. If the account of some of these districts is very meagre, it is because there are but few detailed observations for our guidance. Movunratins or Kurry.— Professor Agassiz, in giving a general sketch of the ancient glacial centres of the British Islands, includes amougst them the mountainous district at the southern extremity of Ireland, at the en- trance to which are situated the far-famed Lakes of Kail- larney.* On approaching this region from the east, it is im- possible not to be struck with the vast accumulation of detritus, with large boulders derived from the rocks of which the mountains are composed. This deposit of the age of the northern drift is spread over the low-lying * Proceedings of the Geological Society, vol. il. GLACIERS IN GREAT BRITAIN AND IRELAND. 135 district of Carboniferous Limestone which extends to the lower lake. On the western and southern sides of this lake the mountains rise abruptly and attain at Carn Tual an elevation of 3404 feet, and here the glacial phenomena are as strongly pronounced as in any part of Wales and Scotland. The Black Valley, one of the most wild and striking, which stretches from the head of the lower lake to the base of Macgiilicuddy’s Reeks, exhibits these appearances in their most marked form. The surfaces of the rocks are here worn into smooth oval bosses, lying with their major axes in the direction of the valley, and extending several hundred feet up the sides. These polished roches mouton- nées, however, assume a singular appearance when traced into the upper lake. They rise above the surface in the form of small oval islands, lying parallel to each other, and, though frequently clothed with luxuriant vegetation, are generally smooth and bare. It is almost impossible to give an idea of these ice-moulded bosses, protruding their naked backs above the calm waters of the lake, bearing some resemblance to a number of up-turned hulls of ships, or to a shoal of whales swimming half out of the water. Nearly all the main valleys present similar appearances. The rocks, wherever freshly exposed, are grooved and stri- ated, as I had several opportunities of observing in the course of a short tour. The picturesque valley of Glen- gariff is specially remarkable for the freshness of the ice- groovings and scratches. Sir H. T. De la Beche* draws special attention to them here, and observes that the scoring and rounding of the bottom and sides of the valley, together with the striations, are unsurpassed by any sumilar examples in Ireland. These striz point W.S.W., stretching along the valley till it is submerged in the sea at Bantry Bay. * Geological Observer, p. 312. 136 MR. E. HULL ON THE VESTIGES OF EXTINCT Professor Agassiz considers that there have been other minor centres of glacial agency in Ireland, as amongst the Mourne mountains in Down and the mountains of Wick- low. Mr. Jukes, in the case of these latter, gives an instance of a boulder of granite twenty-seven feet long, which had been transported seven miles, partly across a wide valley.* Such a boulder, however, might have be- come imbedded in a stranded iceberg from some more northerly district, and have been subsequently carried to its present site. At the same time, Agassiz distinctly states the existence of lateral moraines in the mountain- ous districts south of Dublin and near Enniskillen,+ and considers the Wicklow range as a centre of dispersion for erratic blocks. Norta Watzs.— The years 1841-42 appear to have been remarkably prolific in researches into the glacial phenomena of our islands, for we find Professor Agassiz, Dr. Buckland, and Sir C. Lyell announcing consecutively their convictions of the former existence of a state of things in these islands, which have their analogues only in Greenland, South Georgia, or Tierra del Fuego, at the present day. M. Agassiz pointed to the Caernarvonshire mountains as one of the centres of dispersion of glacial and erratic detritus; and Dr. Buckland speedily followed with details tending to prove that the seven valleys of Snowdonia were once occupied by as many glaciers, dis- charging loads of boulders and gravel over the lower grounds or into the sea, and covering the bottoms and sides of those valleys with flutings and furrows. He also shows that on the northern flanks of this district, boulders - and marine drift coming from Anglesea, Cumberland, or * Manual of Geology, p. 551. + Proceedings of the Geological Society of Londo, vol. iii. p. 330. In the original, the district is said to be situated south-east of Dublin, but is evidently intended for south-west of that city. . GLACIERS IN GREAT BRITAIN AND IRELAND. 137 Treland, and containing, as shown by Mr. Trimmer,* marine shells, have been deposited at an elevation of 1992 feet on Moel Tryfane. The observations of Dr. Buckland were followed by those of Mr. Darwin,+ and more recently by those of Pro- fessor A.C. Ramsay.{ This author has shown that many of the tarns, sach as Llyn Llydaw and Llyn Idwal, have been produced partly through the damming up of the waters by moraines, as Agassiz had previously shown to be the case in the Alps, and Lyell in Forfarshire. The same author, in order to account for the fact that several of the mountain tarns, as those near the summits of Cader Idris, Moel Wynne and Snowdon, are in the form of basins hollowed in solid rock, has suggested an explana- tion which may be called “the scooping theory.” These tarns are generally surrounded through half their circum- ference by precipitous walls of rock; and Professor Ramsay supposes, that solid masses of ice, descending from these heights, charged with imbedded fragments of rock, have actually scooped these hollows, which are so numerous in all mountain districts. But there is one interesting fact brought out by Mr. Ramsay, and which, according to my own observation, is repeated amongst the valleys of the Lake district. Taking the moraine of Llyn Idwal as one of several examples, he shows that it is situated at about 1000 feet below the ele- vation attained by the Northern Drift. Now if this moraine had been formed previous to the deposition of this marine deposit (which attains an elevation of 2300 feet), it would most certainly have been entirely obliterated. It is there- fore evident that moraines of this kind belong to a period * Mr. Trimmer, on the Diluvial Deposits of Caernarvonshire, Proceedings of the Geological Society, vol. i. p. 331. + Philosophical Magazine, vol. xxi. p. 180. £ Peaks, Passes and Glaciers, 4th edition; and Quarterly Journal of the Geological Society, vol. viii. SER. 111. VOL. I. 138 MR. E. HULL ON THE VESTIGES OF EXTINCT subsequent to the Northern Drift. Bearing this in mind, and recollecting the clear evidence which the roches mou- tonnées, frequently enclosed by marine drift, afford of having been formed by glaciers before the deposition of the same formation, we have here a sequence of three dis- tinct, though connected, periods: the first, in which the glaciers descended down the main valleys; the second, when the land of Wales had sunk at least 2300 feet, du- ring which the Till or Drift was spread over the flanks of the mountains; and the third, when the land had been © elevated, and glaciers again descended from the heights ploughing out the Drift, and forming moraines for embank- ments to lakes and tarns. The striations of the rock surfaces of Anglesea appear to be altogether disconnected with the glacier system of Caernarvonshire. The strie and grooves generally range W. 30°S.,* and are probably the result of icebergs strand- ing and scoring the bottom as they floated from the mountains of Westmoreland. Tus Laxy District. — The existence of former glaciers amongst the mountains of Westmoreland and Cumberland having been announced by Agassizt and Buckland,t{ these great observers have left but very slight details of the phe- nomena upon which their conclusions were established. The truth is, that the evidence on this subject for all the highland districts of Britain is of so analogous a nature, and so incapable of being misinterpreted, that a few special cases were enough for their general purpose, that of estab- lishing an interesting and novel theory. Both these. authors, however, notice in several localities on the southern and eastern sides of the district, examples of scored and grooved surfaces, and the mammilar bosses * Ramsay, Quarterly Jowrnal, vol. viil. p. 374. + Proceedings of the Geological Society, vol. iii. pt. il. p. 328. t Ibid. pp. 345 et seq. GLACIERS IN GREAT BRITAIN AND IRELAND. 139 which occur at Penrith and Windermere. It appears to me, however, that Dr. Buckland has extended the glacial theory frequently beyond its true limits, and has mistaken, in the valley of the Eden, Walney Island, and elsewhere, remarkable forms of drift gravel and boulders for glaciai moraines ; and I must altogether dissent from the as- tounding supposition that a glacier stretched from the skirts of Shap Fell across the valley of the: Eden,* by means of which the granite blocks were distributed over the high table-land of Stainmoor Forest and the valley of the Tees. 3 After a personal examination of a large portion of the Lake district, last summer, the details of which I have elsewhere recorded,+ I shall endeavour to present a short outline of the glacial phenomena of this district. The watershed of the country crosses from Bow Fell on the west to Shap Fell on the east; and from this, branch off to the north and to the south a number of deep gorges which unite into larger valleys at some distance from the central ridge. The bottoms and flanks of nearly all these valleys present the usual striations, ranging parallel to their directions; and the rocks are frequently worn into roches moutonnées up to certain heights, well-defined along the sides. There are many interesting examples of perched blocks, such as at Stickle Tarn, where a boulder rests on a rounded boss rising slightly above the surface of the lake. In the lower and larger portions of the valleys we seldom find examples of true moraines, these being nearly confined to the higher portions near the central heights, as in the case of the two Langdales, Stockdale, Easedale, and the Stake Pass at the head of Borrowdale. Generally, the * Proceedings of the Geological Society, vol. ili. pt. i. p. 348. Mr. Bryce also notices the striated and polished surfaces of the rocks near Kendal, but refers them to the action of “ waves and currents charged with detritus.” — Rep. British Association, vol. xix. + Edinburgh New Philosophical Journal, vol. xi. p. 31. 140 MR. E. HULL ON THE VESTIGES OF EXTINCT moraines assume the form of a collection of large boulder- strewn cumuli, or mounds, unlike anything I have ever observed beyond the limits of a glacier district.* There is, however, one remarkable exception to which I shall presently refer. . The rocks of a large district surrounding the interior mountains are remarkably ice-monlded, polished and stri- ated, as far as the head of Morecambe Bay to the south, and the vale of the Eden to the north. The Drift, a marine boulder-clay, rises to the height of 1200 feet on the southern slopes of the hills. Of its ele- vation on the northern flanks I cannot speak from per- sonal observation. The clayey gravels, frequently bright red from the decomposition of the felspathic rocks, which occur at higher elevations, appear to consist of moraine matter. Many of the small lonely mountain lakes, or tarns, have been formed partly by moraines thrown across their out- lets. Amongst the most interesting examples of this kind are Hasedale Tarn, Stickle Tarn and Blea Tarn. The for- mation of some others, such as the tarn that lies between Helvellyn and Fairfield, seems to be referable to “the scooping theory previously explained.” I have already stated that several well-marked moraines may be observed at elevations considerably below the upper limit of the Drift. All those occupying this position must, in consequence, be of more recent date than this marine deposit. Of examples of this class, the most re- markable which has yet come under my notice is the large terminal moraine at the lower extremity of Grisedale. This gorge, one of the most desolate and savage in Cum- * See plates 1, 2, Hdinburgh New Philosophical Journal, vol. xi. p. 31. This form of moraine appears to recur frequently in Scotland, according to the accounts of Sir C. Lyell and Dr. Buckland in the Proceedings of the Geological Society, vol. ili. p. 335, and pp. 340 ef seq. GLACIERS IN GREAT BRITAIN AND IRELAND. 14] berland, descends from the heart of Helvellyn towards the head of Ulleswater. The rocks of porphyry which form the bottom and flanks of the valley, up to an elevation of about 500 feet on either side, are remarkably ice- moulded, affording numerous examples of perched blocks and lateral moraines. Striations are not, however, of fre- quent occurrence, owing to the nature of the rocks. On descending towards the mouth of the valley, the terminal moraine arrests the attention, and appears like a congeries of large rounded hummocks, strewn with boulders, rising up the sides of the valley to about 150 feet above the bed of the river. After the melting of the glacier, this mo- raine, in all probability, produced a lake. But the torrent has hewn a channel and levelled the ground over a breadth of about 100 yards. The position of this moraine is not more than 600 feet above the sea level, or 220 feet above Ulleswater ; and it enables us to measure with exactness the dimensions of the glacier which formed it. Taken from the tarn at the head of the valley, this glacier was 3 miles in length, about 500 feet in depth at its centre, and from 200 to 400 yards in width. On the eastern side it was bounded by a continuous and nearly vertical escarp- ment of bedded trap; but the western side was very irre- gular and indented. The phenomena of this region appear to show: first, a period when glaciers protruded far down the main valleys ; secondly, an interval when the land was snbmerged abont 1200 feet or more, during which the boulder clay was spread over the flanks of the hills and valleys; thirdly, a period when the land had been again elevated, and gla- ciers extended some distance down the minor valleys and ploughed out the Drift. It was a glacier of this third period which has left the terminal moraine of Grisedale. Scortish Hicrianps. — The glacial vestiges of the Highlands of Scotland are on a scale more grand than 142 MR. E. HULL ON THE VESTIGES OF EXTINCT those of the Lake district or Wales, in proportion to the greater extent and loftiness of the mountains and their higher latitude. Ben Nevis in lat. 56° 50’ attaining an elevation of 4368 feet falls only a little short of the snow line, and is said to have patches of snow all the year round in the fissures near the summit. The observations which have been recorded regarding the direction of the striz, although they are lamentably — few, go to prove that the Highlands formed a centre of dis- persion, from which the ice-streams and bergs radiated in every direction from the central range. The southern slopes of the Grampian Hills in Angus and Forfar have received a detailed examination at the hands of Sir C. Lyell, who has recorded his observations in the transactions of the Geological Society.* The stria- tions follow the lines of the main valleys 8.S.E., and several fine examples of lateral and terminal moraines are mentioned. Of these the great transverse barrier of Glen- airn seems to be the most remarkable. The valley of the south Esk here contracts from a mile to half a mile in breadth, and is fianked by steep mountains. Seen from below, this barrier resembles an artificial dam 200 feet high, with numerous hillocks on the summit. Its breadth from north to south is half a mile. Sir C. Lyell considers this to be the terminal moraine of the receding glacier, and considers it probable that it once banked up the river so as to form a lake, which has since been drained by the Esk having cut a channel for itself 30 feet deep on the eastern side. The Sidlaw Hills claim particular attention on account of the examples of transported boulders which they afford. Separated by the great valley of Strathmore from the Grampian range they reach an elevation of 1500 feet. * Proceedings of the Geological Society of London, vol. ill. pt, ii. Pp. 357+ GLACIERS IN GREAT BRITAIN AND IRELAND. 143 They are formed of Old Red Sandstone, and on their flanks at elevations of 700 and 800 feet are strewn blocks of gneiss and mica slate, which have been floated across the intervening space over a distance of fifteen miles. One of these blocks on Pitscanby Hill is thirteen feet long and seven across. ‘This is an example on a much smaller scale of the erratic phenomena of the Alps, where enormous blocks have been transported across the great valley of Switzerland from the Mont Blanc range, and stranded along the flanks of the Jura hills. The Pierre a bot, one of these boulders, is one of the noblest monuments in the world of the transporting power of ice. The Highlands of Perthshire have been examined along their southern watershed by Buckland and Agassiz,* who detail numerous examples of glacial traces in the shape of moraines, roches moutonnées, strize and perched blocks. It is to be regretted that the former observer has scarcely recorded a single observation as to the directions of the strie. M. Agassiz, however, mentions that along the valley of the Forth they range from N.W. to 8.E., or in other words, radiate from the great line of the Gram- plans seaward. Towards the southern extremity of the Highlands the same law obtains, and along the valleys of Loch Lomond and Loch Long the striations point south.t With regard to the former lake I have been informed by Professor Ramsay, that not only its sides but the rock-surfaces of its beautiful islands are completely covered with grooves and furrows. In the western Highlands, Agassiz has observed that the direction of the striz aloug the valleys of Loch Fine and Loch Long are generally 8.H., and at Loch Etive and Loch Leven from east to west. * Proceedings of the Geological Society, vol. iil. pt. i. + Ibid. p. 328. 144, MR, E. HULL ON THE VESTIGES OF EXTINCT The wild district of Inverness-shire and Ross-shire remains yet to be described as far as its glacial history is concerned. From what we know of the adjoining regions, however, we may surmise that its long channel-shaped valleys and arms of the sea, stretching from the coasts far into the mountains, must have presented a series of physical conditions very similar to that of Norway, where the gla- ciers appear to have descended into the sea during the glacial period.* . The phenomena of Sutherlandshire appear to have forced such an analogy on the mind of Sir R. Murchison, when lately exploring this region. He says: “The spectator who, ascending the summit of Ben Stack, looks westward, observes between him and the sea of Scourie Bay a countless quantity of small loughs and tarns interspersed among the hollows of this brown-clad barren waste. On descending to examine the lower tract, he finds the surface frequently rounded off and polished like the roches moutonnées of the Alps; and observing numerous striz usually divergent from the central moun- tains, aud following the lines occupied by the principal lakes or maritime fiords, he can have no doubt that, in the glacial period the north-west of Scotland must have been very much in the present state of Greenland, as de- scribed by Rink: i.e. the central mountains occupied by snow and ice, from which vast glaciers are protruded to the laterai fiords or bays.’’} Glacial vestiges are no less marked over the rugged and inhospitable island of Skye. Professor J. Forbes has carried his observations across from the Alps and Scandi- navia to this remote region of Britain, and, in referring especially to the valley of Cornisk, states that “the grooves and striee are as well marked, as continuous, and as strictly * See Professor J. Forbes’ Travels in Norway, and Phénomenes @ érosion en Norvége, par M. Horbye. + Quarterly Journal of the Geological Society, vol. xv. p. 361. GLACIERS IN GREAT BRITAIN AND IRELAND. 145 parallel to what I have elsewhere shown to be the neces- sary course of a tenaceous mass of ice urged by gravity down a valley, as anywhere in the Alps.”* Before taking leave of this district, I may observe that Professor Agassiz is of opinion that the parallel roads of Glenroy, near the foot of Ben Nevis, are attributable to a lateral glacier having been projected across the valley, near Bridge Roy, and another across the valley of Glen Speane. By this means glacier lakes were formed, along whose margin the stratified terraces of gravel were produced _ which are now seen to line the flanks of the valley at a perfectly horizontal level through several leagues. The subsequent melting of the glaciers has entirely obliterated any traces of the agent by means of which the waters were pent up.t Mr. Darwin, however, takes a different view of the subject, considering that the parallel roads are marine terraces, formed during the submergence of the land to a depth of 1250 feet, their present elevation. Having thus made a rapid circuit of the Scottish High- lands, I must now hasten to a close. I think, however, it is only due to so great authorities as Professor Agassiz and Dr. Buckland, that I should state that they appear to have considered “not only that glaciers once existed in the British Islands, but that large sheets of ice (nappes) covered all the surface of the districts surrounding the Highland groups.” ‘This opinion is founded on the wide extent to which unstratified gravels, perched blocks and polished surfaces in situ are distributed over the districts adjacent to the centres of distribution. I believe, however, that it is now generally allowed that floating ice, or rather swimming ice, has played a more important part in produc- * Edinburgh New Philosophical Journal, vol. xi. p. 76. + Proceedings of the Geological Society, vol. iil. p. 332. Lyell’s Hlements of Geology, sth edition, p. 88. SER. III. VOL. I. U 146 MR, E. HULL ON THE VESTIGES OF PXTINCT, ETC. ing these phenomena than was supposed by the founders of the glacial theory. It is indeed an almost unsolved problem, how we are, in all cases, to distinguish the effects of icebergs charged with stones scraping along the sides and bottoms of the channels through which they float, from the effects of subzerial glaciers. If of large size, and impelled by prevalent winds or currents in one gene- ral direction, they would produce polished, grooved and rounded surfaces on the rocks with which they would come in contact, and leave behind blocks and débris strewn so as to resemble the matter of moraines. At the same time, there are several classes of objects which could only have been produced by suberial glaciers, and others which bear the unmistakeable impress of aqueous deposition. I shall now only repeat, that the great object to be accomplished is the production of maps showing the di- rection of the striz, the position of the moraines, and the limits of the drift, amongst the highlands of Britain. 147 VIII.— A Brief Memoir of the late John Kennedy, Esq. By Wititam Farrparrn, Esa. Read April 3rd, 1860. —_——— Tue subject of the present memoir was born at Knock- malling, in the Stewartry of Kirkcudbright, N.B., on the 4th of July 1769. He was the third of five sons, and his father (who died at middle age) was a laird, living upon the paternal property which had descended through a long line of Kennedys for upwards of three hundred years. Of the education of his sons little can be said, as in such remote mountain districts no regular. school was within reach, and they only received occasional instruction from some young graduate of the kirk, who took up his resi- dence amongst the farmers in the winter months. In this way they were taught to read and write, though unfortu- nately what was learnt in the winter was in many cases lost in the summer, when all able to work were employed in the field, or in watching the sheep and cattle on the bare mountain pastures. Mr. Kennedy was, however, fortunate in receiving the instructions of Mr. Alexander Robb, who first directed his attention to the elementary principles of mechanics and mechanical movements. On the death of his father, the whole management of the farm and of the family devolved upon the mother, who appears to have been a woman of strong good sense, who 148 MR. WILLIAM FAIRBAIRN: MEMOIR brought up her children respectably, and laid the founda- tion of those qualities which distinguished them through life. Mr. Kennedy was occasionally subject (like his father) to low spirits. He seems in early life to have been an observant, thoughtful child, and he always retained a most vivid recollection of the home life of his native district. He frequently, in later years, when speaking of the im- provements which have been effected in the management of land and stock, used to tell how as a boy he felt for the poor cattle in the spring, which, from want of food, had become so weak that when once down they could not get up again, but had to be helped to rise from sheer weakness and want of support. At the age of fourteen he and his brothers, excepting the eldest, who took to the farm, began to look about them for some employment by which they could earn a living; as it was then the custom in the great majority of Scottish families for the younger branches to emigrate, and make their way as best they could in a foreign land. At that time the great openings for employment were England, the West Indies, and America. The late Adam and George Murray, the founders of the well-known firm of that name in this city, who came from the same locality, and had settled as apprentices with Mr. Cannan, the machine maker at Chowbent, excited a strong desire in Mr. Kennedy to see new places and new things. He became ambitious to do something for himself, and to look beyond the still glens and blue mountains by which he was surrounded. With these feelings Mr. Kennedy was engaged as an apprentice to Messrs. Cannan and Smith, at Chowbent. Early in February 1784, he started for England, with his mother’s blessing. She told him “ always to conform to the Presbyterian confession of faith, but (she said) to OF THE LATE JOHN KENNEDY, ESQ. 149 remember that the Roman Catholic religion* and all others were alike, if you were sincere and acted up to their precepts, and they were all equally efficacious to salvation.” With this sage advice and a new suit of home- spun, he left his home mounted on a pony, full of the events of his future prospects in life. After two days travelling he reached Carlisle, where he was met by his master’s partner, Mr. Smith, who was employed im start- ing a carding-engine and one of Hargreaves’ jennies, which were placed in different rooms, as there were no cotton mills at Carlisle at that time. From Carlisle he proceeded, in company with Mr. Smith, through Penrith, Kendal, and Preston, and on the sixth day reached his destination. At Preston he had an opportunity for the first time of hearing a lecture of Mr. Banks on Natural Philo- sophy, with which he seems to have been much interested, and which laid the foundation of his future tastes and desires in these pursuits. Immediately on his arrival at Chowbent he commenced work as a machine maker, and in a short time became an expert workman. The machinery made at that time was limited to carding machines, Hargreaves’ jennies and Arkwright’s water frames, including drawing and roving, then very imper- fectly constructed. At the close of his apprenticeship, which lasted seven years, he came to Manchester on the 18th of February 1791. Here he joined in partnership with Benjamin and William Sandford — who were fustian warehousemen — and Mr. James McConnel, under the firm of “ Sandford, McConnel and Kennedy,” and commenced business with them as machine makers and mule spinners. This was shortly after Crompton’s invention of the mule, and the firm was for many years almost the sole makers of * Tt is supposed that Mrs. Kennedy, a descendant of the old chevaliers and Catholic families, had some predilection for that form of worship. 150 MR. WILLIAM FAIRBAIRN: MEMOIR Crompton’s machine.* It will not be necessary here to give an account of the different machines which consti- tuted the mechanism of a cotton factory in those days. I may, however, here observe that Mr. Kennedy rendered great service to the new system of mule spinning by the introduction of a new motion called the double speed, and which gave to the thread any amount of twist that might be required for the number of counts, or the quality of the yarn that had to be spun. ‘This improvement of Mr. Kennedy gave greatly increased facilities to the spinning of fine yarns, and very soon enabled those engaged in that peculiar manufacture to raise their numbers from fifties and forties up to one hundred and fifties and two hun- dreds; aud since that time numbers as high as fifteen hundred and two thousand have been spun. Still greater improvements have been made since that time, but less in the mule than in other machinery, excepting only the imtroduction of the self-acting mule; the increased fine- ness in the quality of the yarn of the present day being more attributable to carding, combing, and preparatory machines than to any of the improvements since made in the construction of the mule on which it is spun. To show in what way Mr. Kennedy effected his improvements of Crompton’s mule, he gives the following * Crompton completed his mule in 1780, and after contending with many difficulties and annoyances he at last gave it to his competitors for a consideration that they never paid, but left the poor inventor a prey to poverty and the ingratitude of those who had benefited by his discoveries. This is another instance of the robbery —it cannot be designated by a milder term—practised upon the benefactors of the human race by the self- interest alloyed with ingratitude of the possessors of princely fortunes derived not from their own talents but from the higher intellects of those they have utterly forgotten and neglected. Such were the fortunes of Crompton and Cort, whose inventions have, along with those of James Watt, revolutionised the manufactures of iron and cotton, and given to their native country a predominance never yet paralleled in the history of nations. — See an excellent work by Gilbert J. French, Esq., entitled Zhe Life and Times of Samuel Crompton. OF THE LATE JOHN KENNEDY, ESQ. 151 account in his “ Memoir of Crompton :’ — “It was not until 1793 that any attempts were made in spinning fine yarns, say from one hundred hanks upwards, by power, when I observed the process very carefully. The rollers, according to the fineness of the thread, would only admit of a certain velocity per minute; for instance, with two hundreds, the rollers could cnly go ninety at the rate of twenty-five or twenty-six revolutions per minute, and the spindle about twelve hundred. But when the rollers ceased to move, then the spindle was accelerated by the spinner to nearly double its former speed. In what manner the acceleration of the speed of the spindle might be effected by machinery, without the aid of the spinner, had occurred to me, by observing in Mr. Watt’s steam engine that one revolution of the beam (if I may use the expression) acting upon the fly-wheel by means of the sun and planet wheels, produced a double velocity. The difficulty, however, of making the necessary appa- ratus at that time induced me to use the more com- plicated method of four wheels of unequal sizes for pro- ducing the same effect. The description is as follows : — Two of the wheels were less and two larger; upon the rim axis one of the small and one of the large; and the two others were fixed in a frame which carried the axis upon which they were placed, and which had a shank or axis growing to it. This was placed in a vertical position, so that when the carriage was put up an arm projecting from this vertical shank was connected by a wire with a catch, which kept the lying shaft that turned the rollers in gear. In the elongating process the smaller wheel was in contact with the larger wheel upon the rim, but when by the disengagement of the catch the rollers became still or stationary, at that moment the larger wheel, by means of a weight, came in contact with the lesser wheel upon the rim or axis, to which it communicated a double velocity. 152 MR. WILLIAM FAIRBAIRN: MEMOIR The shaft, with its large and small wheels working alter- nately had a pulley with a catch upon it, and was driven by the mill-work; and was forced into a corresponding catch upon the small shaft when the mule was to be set in motion by the steam power. The power in this instance was Savery’s, which was used to raise water upon a water-wheel. There was a worm upon the rim axis, with a wheel upon it, the number of whose teeth determined the revolutions of the rim. The second drawing, which had generally been performed by hand, had also to be performed by the machine itself. This had heen done in a few instances before power had been applied. From the simplest of these methods I took the hint, drawing a shaft from the rim by a strap from a small pulley upon the rim axis, and a large one upon the small axis which had the small pinion upon it; so that when the drawing-out wheel and band were disengaged from the front roller, they fell back into the small pinion whose axis was revolving at a very slow speed, and consequently gave a much slower speed to the second stretch or draw, as it is called, the speed of which was more or less, according to the numbers to be spun. “Messrs. A. and G. Murray at that time, like myself and partners, were machine makers, and to a small extent we were both engaged in fine spinning by hand. They fitted up, upon the principle above described, a few pairs of hand mules, which they had previously made for one of their customers in Derbyshire, who had water power. Mr. Drinkwater, of Manchester, was the most extensive fine spinner at the time of which I speak. He was one of the early water spinners, and in possession of the most perfect system of roving making. His large mill in Piccadilly was filled with mules of one hundred and forty- four spindles each, all of which were worked by mens’ hands. Mr. Owen, the philanthropist, was then his manager, and OF THE LATE JOHN KENNEDY, ESQ. 153 they came to see the new machine in 1793. They approved of it, and thought it practical. Mr. Humphrey, of Glasgow, a good mechanic and millwright, succeeded Mr. Owen as manager, who also approved of the scheme, and got in- structions to apply this steam power to the fine work produced by the mules in Piccadilly mill; and to make its advantages available he coupled two of one hundred and forty-four together, so that he saved one-half of the steam gearing, and obtained a reduction in the price of spinning, the spinner having double the number of spin- dles to operate upon. Mr. Humphrey made an improve- ment in the four wheels already described, by keeping them always in gear with a loose clutch between the two wheels on the rim shaft, which was alternately fastening the small driving wheel, and then relieving it and fastening the larger one which accelerated the speed of the rim, and furnished with a loose and fast pulley as already described. This prevailed for some years, when I thought it might be simplified, which was done by adopting three pullies, namely, one on the small wheel, another on the large wheel, and a loose pulley. The driving strap, which was on the loose pulley when the mule was at rest, was removed to the pulley on the smaller wheel when the rollers were to work, and then to the pulley on the larger wheel which accelerated the rim and spindles, until the thread was completed, when the strap being removed to the loose pulley, the whole machine came to rest, and the thread was put up by the spinner in the ordinary way. I was at this time able to construct the sun and planet wheels for the acceleration of the speed of the spindle in the following way : — The sun and planet wheels had only two wheels and one pulley with a clutch that fastened the sun wheel when the accelerated motion was required. But though this and many other modifications were introduced, the four wheels prevailed. Some of these, for convenience, SER. III: VOL. I. 5 154 MR. WILLIAM FAIRBAIRN: MEMOIR I constructed by making them bevils, and placing their axis vertically to get motion from an upright shaft, which produced the same effect as the spur wheels.’’* In addition to his improvements of the mule, Mr. Ken- nedy was also the pioneer in forwarding the interests of the cotton trade, by improvements of the other machines employed in that manufacture. He was one of the first to suggest and carry out improvements in the roving frame, and the differential motion for winding the roving upon the bobbin owes much of its success to his sagacity and skill. For many years Mr. Kennedy carried on a series of experiments connected with this motion, and although it has been greatly modified in form and construction since his time, it still bears the impress of his mind, and remains the same in principle as when he experimented upon it. This beautiful and ingenious machine is also indebted to the late Mr. Ewart for some useful modifications ; but the perfecting of the differential motion is due to Mr. Henry Houldsworth, of this city, assisted by the late Sir Peter Fairbairn, of Leeds. As a proof of Mr. Kennedy’s sound judgment on ques- tions relating to mechanical improvement, he was consulted on the subject as to whether the Liverpool and Manchester Railway should be worked by locomotive or stationary engines. He was also appointed Umpire in the competi- tion trials at Rainhill in 1830, and to his honour be it stated that this country and all others are indebted to him for advancing the railway system, by his appreciation of the different qualities of the engines, and his correct and just decision in this case. As a spinner Mr. Kennedy was most successful in all his undertakings, and realised a large fortune. He was a most accurate observer, was endowed with a retentive memory, a clear perception, a sound judgment; and every * This was done for Mr. Kennedy by the writer of this memoir in 1823. OF THE LATE JOHN KENNEDY, ESQ. 155 discovery in mechanical science received his cordial sup- port. He was a friend and admirer of Watt, and there were few distinguished men in the scientific world with whom he was not acquainted, and on terms of friendly intercourse. Round his table were at all times to be found men who were noted for intellectual acquirements. With the distinguished men of Manchester and the sur- rounding districts he lived on terms of the closest inti- macy; and although he did not attend regularly at the meetings of this society, in consequence of a dislike to the public expression of his opinions, he nevertheless took a deep interest in its proceedings; and during the whole of a useful life remained the friend of Dalton, Henry, and other men eminent for their discoveries and writings in science. Mr. Kennedy was a man of sterling honesty in all his transactions. He began life at a time when the cotton trade was in its infancy, and he lived to see it attain its present colossal dimensions. .As a man of business he was successful, but it is doubtful whether his tastes and talents would have fitted him for the present system of free trade, and whether he would not have been distanced by more energetic and active competitors. That this would have been the case is more than probable, as he was of a nervous temperament, subject to great depression of spirits, which might have paralysed his exertions, and prostrated him in a contest to which he was unequal. As it was he attained honour and success, and he lived at a time when business matters were easy, and when skill and practical science were much in demand. Mr. Kennedy never pursued business for the sake of money, but for the love of improvements in his favourite mechanical pursuits. To these he devoted nearly the whole of his time, and there was scarcely any discovery in the arts that he did not make himself acquainted with. 156 MR. WILLIAM FAIRBAIRN : MEMOIR He was fond of mechanical discussion, and never lost an opportunity of conversing with imgenious workmen, or those who were entitled to respect for their skill. Hence he visited all the workshops, and had no greater pleasure than when having a “crack” with an intelligent, well- informed workman. To the young he was always kind and communicative, but according to the custom of the times he expected that young people should never dispute the wisdom of their superiors, but thankfully receive the information afforded to them. This feeling was almost universal amongst the immediate successors of Watt, and many of them would admit of no mechanical improvement unless it originated in the school at Soho. This weakness was not always agreeable to rising merit, and was foreign to the great men for whom this partiality was shown, and it often had an injurious tendency, in so far as it damped the energies of the more modest and deserving aspirants. Mr. Kennedy nevertheless freely extended his patronage and friendship to those who were entitled to his confi- dence, and when once given it was never withdrawn. He had a high sense of honour in his friendships, and never allowed another to depreciate qualities for which he per- sonally entertained feelings of respect. In private society Mr. Kennedy had the manners and conversation of a gentleman, acquired, not from his educa- tion, but from his subsequent intercourse with the best society. He had great discrimination, and would never associate with any but those of superior attainments, and hence the attraction of his opinions and conversation. He was full of anecdote, old sayings, and sage remarks, and few could tell a story with more zest. The sayings of others and the anecdotes of early life he gave with a dry humour that never failed to produce a pleasing effect. At the time of his death, which took place on the goth October 1855, at the advanced age of eighty-six years, CF THE LATE JOHN KENNEDY, ESQ. 157 he was the oldest member of the society. He was elected in April 1803, and during a long period of years was a regular attendant at the meetings. His writings were few in number, but he contributed to the society’s memoirs four valuable papers, “On the Rise and Progress of the Cotton Trade,” read November 2, 1815; “On the Poor Laws,” read March 5, 1819, a paper much spoken of at the time, ably reviewed in the Edinburgh, and not without results in the improvements which subsequently took place in the amendment of those laws. The next, entitled ‘‘ Observations on the Influence of Machinery on the Working Classes of the Community,” was read February 10, 1826; and a brief ‘“ Memoir of Crompton,” with a description of his mule, read February 20th 1830, was the last of his literary efforts. In these communica- tions he displayed consummate judgment, and a thorough knowledge of the subjects on which he treated; yet it must be confessed that the views he entertained upon the subject of free trade and national intercourse have since that time been greatly modified, and a totally different system of commercial relations between this country and the rest of the world has been adopted. 158 MR. ALFRED FRYER: SUGGESTIONS IX. — Suggestions for a New Form of Floating Lightship, and a mode of estimating the distances of Lighthouses. By AuFrep FRYER. Read January 1oth, 1860. Tue objections to the present form of floating lightships are well expressed by an eminent authority in pharology as follows: “Floating lights,” writes Alan Stevenson, C.E., “are very expensive, and more or less uncertain, from their liability to drift from their moorings; they should there- fore never be employed to indicate a turning point in a navigation in any situation where the conjunction of lights on the shore can be applied at any reasonable expense.” Objectionable as this method of lighting is, it is in many cases the only one that can be adopted; and no fewer than twenty-six floating lights surround the English coast. As the injury to which they are liable arises from the violent assaults of wind and waves during storms, the problem appears to be, “ What is the form of vessel that shall be most secure from the action of storms?” The following is offered as a solution for cases where there is sufficient depth of water: The proposed lightship somewhat resembles a hydrometer, the mass of which (the bulb) used as dwelling and store rooms, is removed from the action of wind and waves, and the stem presents the smallest possible surface for their violence. The structure, FOR A NEW FORM OF LIGHTSHIP, ETC. 159 which is represented in the accompanying drawing is of wrought iron; the tube or stem at its upper end is of a diameter sufficient only to admit the body of a man easily, and at its lower end the diameter is doubled to give in- creased strength. The lantern, with light-room beneath it, and gallery attached, terminates the superior end of the tube, while its inferior extremity is riveted on to the dwelling place of the attendants, which is the frustrum of an inverted cone with convex ends. To the base of the cone and in its plane a wide annular flange is securely stayed, and the vessel is weighted by means of pigs of iron stowed in the hold and a cast iron sphere suspended beneath, so that the dwelling apartment is sunk beneath the surface of the water, and beyond the reach of the action of the waves. The exact form and dimensions will depend in part on the depth of the water, the range of light required, and the exposure of the situation. The following are the chief particulars of such a lightship as is represented in the diagram : With the exception of the brass-work and glass of the lanterns, the vessel is constructed entirely of wrought iron. The plates of the submerged part and those within reach of the waves are very strong, but thinner ones are used for the upper part. The upper lantern is five and a half feet in diameter, the light-room six feet in diameter and six and a half feet high. The whole structure is one hundred and twenty-seven feet in length, two-thirds of which are exposed, and one-third submerged. The tube, two feet and a half at the top, increases to five feet in diameter at the water-line, and is seven feet when connected with the dwelling. The tube, which is divided vertically into two shafts of unequal size, furnishes means for efficient ventilation. An iron ladder is secured in the larger or downcast shaft, while the smaller or upcast shaft encloses the smoke flue. The dwelling apartment consists of a cir- 160 MR. ALFRED FRYER: SUGGESTIONS cular room eighteen feet in diameter and eleven feet high, and below it is the store-room fourteen feet wide and seven feet in height. The structure weighs fifty-five tons, and when sunk to the required depth displaces five thousand one hundred and thirty feet of water or one hundred and forty-eight tons. ‘To counteract the buoyancy eighty-seven tons of ballast are required, thirty-one of which are suspended from the bottom as a sphere of cast iron seven feet in diameter, while fifty tons of pig iron cover the floor of the store-room ; the remaining weight of six tons is in stores. The upper light, eighty feet above the water, can be seen at twelve miles distance; and the lower light (the object of which will be explained afterwards) being thirty feet above the water, can be seen at about eight miles distance. The centre of gravity is thirty-three feet below the surface of the water, and the centre of buoyancy twenty-six feet; thus the vessel is stable, and exerts con- siderable power to regain the perpendicular if removed from that position. The large disc mentioned above is intended to steady the vessel and to reduce oscillation, as no movement of a vibratory character can take place with- out the flange displacing a proportionate quantity of water. The inertia of a mass of one hundred and fifty tons, and the time and power required for the disc to displace large volumes of water, cause the vessel to be slow to obey the proportionately slight impulse to rise and fall with the waves. The surface on which the waves can beat is very small, and is in fact less than the aggregate surface of the legs of a screw-pile lighthouse. The wind also can exert but little force upon so narrow a tube, which appears scarcely thicker than a large ship’s mast, and like it presents a taper form and convex surface. The structure is moored by three anchors, the cables from which are attached to FOR A NEW FORM OF LIGHTHOUSE, ETC. 161 the edge of the disc, and at equal distances. The vessel here represented cannot float in less than seven fathoms water, though it is possible that with a slight modification of form a lightship could be adapted for certain situations that would rest on the bottom, or float, according to the height of the tide. Where the depth of water will allow of it, the structure would be improved by keeping the centres of gravity and buoyancy further apart than here proposed. This would be best accomplished by enlarging the capacity of the upper part of the dwelling, and placing the ballast still further from the surface of the water. The discomfort that would arise from a residence five fathoms below the surface of the water is found when examined, to be at least as little objectionable as that attendant upon any floating dwelling away from land. The inhabitants are safer than in a lightship of the usual construction er in an ordinary vessel, there being slight danger of being capsized, drifted away, or wrecked. They are safe from the fury of the waves, and protected from the violence of the wind. They do not fear that their vessel may be consumed by fire, or injured by strokes of lightning. From the shape and strength of the lightship there 1s little chance of foundering; but, should it spring a leak, the spherical weight can be released from within, when the vessel will rise to the surface, but still remain vertical; the hatches can then be made up, a signal hoisted, and assistance waited for. The dwelling, even during storms, is free from motion, and secure from noise ; it is, moreover, protected from the frosts of winter, and the intense heat of summer. Some daylight may be intro- duced down the tube by means of reflectors, but this is of little importance, as the services of the attendant on duty are required on the gallery or in the light-room. No better plan could be devised for securing ample ventila- tion. The danger to be apprehended from being run down SER, III, VOL. I. ¥ 162 MR. ALFRED FRYER: SUGGESTIONS by vessels is less to this construction than to pile light- houses or ordinary lightships, as less surface is presented for collision; but if actually struck, the convex surface and extra-strong plates would most likely cause the tube to glide aside uninjured. Objection will not be taken to this plan on the ground of expense. Two thousand five hundred pounds would probably cover the cost, including ballast, but excluding lanterns and illuminating apparatus. As the oscillations of this vessel will be slow and small, it is possible that the dioptric method of illumination could be used, especially if the apparatus were judiciously sus- pended, and furnished with a contrivance for retarding the oscillations. Experiment only, however, can determine this. Accidents not unfrequently occur from the mariner mistaking his distance from alight. Estimating it to be five miles distant he may find, when upon the rock or shoal he wished to avoid, that a slight haze had partially obscured the light and that he was within a mile of it. Double lights, or lights exposed from different elevations in the same tower, are now used in some cases, but solely I believe with a view to prevent confusion, by giving the light a distinctive character. I would propose, in all cases where it is important that the distance from a light should be ascertained, that double lights should be em- ployed. Sailors would learn to estimate their position approximately by the apparent separation of the lights, and the distance could be determined with accuracy by the use of the sextant. The lights here represented are fifty feet apart, and as the height of the lower is thirty feet above the surface of the water, they would be visible at eight miles distance, and the combination would then be clearly discerned as a double light. FOR A NEW FORM OF LIGHTHOUSE, ETC. 163 If Mr. Herbert’s proposal to moor a series of floating lights along the middle of the English channel should find acceptance, the plan here proposed would be specially applicable on account of the depth of water. In that case it is suggested that each vessel should become a telegraph station, and by means of a cable each would be in commu- nication with the others, with the shore, and with London. It is already the duty of light-keepers to report wrecks, vessels in distress, and even vessels sighted. These reports could then be made instantly to head quar- ters, and the information would be received when it was of most value. Injury to the illuminating apparatus, to the vessel, serious illness of a light-keeper, or deficiency of stores, could be reported at once, and the requisite assist- ance could be early afforded. In addition to this each lightship would occupy the post of a sentinel, and the time may come when the information they could give might be of importance to the Government. The paper was illustrated by models of the ordinary and proposed lightships; and the relative steadiness of the latter was shown when the water was agitated, or equal force applied to disturb the equilibrium of each. 164 MESSRS, RANSOME AND VERNON ON THE INFLUENCE X.—On the Influence of Atmospheric Changes upon Disease. By Artuur Ransome, M.B., B.A. Cantab. M.R.C.S., and Grorer V. Vernon, F.R.A.S., M.B.M.S. Read April 17th, 1860. Two hundred years ago the remark was made to Syden- ham,* and the statement holds true now, that “no physician hitherto has attentively considered the force and influence of the atmosphere upon human bodies; nor yet has he sufficiently ascertained the part it plays in prolonging human life.” From very early ages men have observed that certain diseases prevail most during certain seasons, and have ascribed to atmospheric changes an important influence upon health, but, until recently, no solid foundation of accurately observed facts had been laid. Most of the old medical writers deal with the subject — some of them very carefully. Hippocrates devoted one of his works to “ Airs, Waters and Places ;” and his writings upon epidemics, and his aphorisms, abound with remarks on the influence of various states of the air upon the human frame. Since his time many others have very fully noticed the coincidence between these phenomena, as * Letter to Dr. Sydenham, from Dr. Thos. Brady. — Sydenham Society’ s Transactions, vol. ii. p. 1. OF ATMOSPHERIC CHANGES UPON DISEASE. 165 Aretzus,* Sydenham,t Boerhaave,{ Vitet, Ramazzini,§ Baglivi,|| and more modern authors. Still, owing perhaps to the imperceptible and apparently mysterious way in which atmospheric changes take place, and more to the necessity for well organized and simultaneous observations in both branches of the inquiry, little further progress towards a true science of medical meteorology has been made until lately. The first attempt upon a comprehen- sive plan to advance this subject was made in January 1844, by adding meteorological tables, furnished by the Astronomer Royal, to the weekly returns of the Registrar General, these returns being for London only. Somewhat later, monthly returns were obtained from stations in dif- ferent parts of England, and appended to the quarterly returns of the Registrar General; and from the year 1849 these stations have gradually increased in number, and at the present time there are about sixty in England and Wales. Similar returns from about forty-five stations are now added to the monthly and quarterly returns of the Registrar General for Scotland. Some years after these returns were commenced, it was thought that more useful information might be obtained by similar comparisons with respect to disease; and in 1853 an attempt was made by some members of the Pro- vincial Medical Association to compare meteorological tables for different places, with the diseases prevalent in those districts, but unfortunately these records were not continued for more than two years and a quarter. In 1857 the General Board of Health in London took up this question ; and from the week ending April 11¢h * rep. Atuaros ’"Avaywyijs. + Observationum Medicarum. Tt Causes of Disease. § De principium valetudine tuenda, cap. ii. || De eris influxibus investigandis ac perdiscendis, ad morbos dignoscendos et curandos., 166 MESSRS. RANSOME AND VERNON ON THE INFLUENCE 1857, to the week ending November 6¢/ 1858, they pub- lished a carefully compiled weekly return of new cases of disease in London, furnished by the voluntary efforts of upwards of two hundred gentlemen connected with the medical profession. The tables are accompanied by meteor- ological observations made at six stations in and out of London, and although not perfectly accurate, yet they are of great value; it is much to be regretted that they were carried on for so short a period. Hitherto careful collation of the two classes of facts recorded in these tables seems to have been wanting; and in the present paper we have endeavoured to supply the deficiency, and to deduce from the comparison some ge- neral conclusions. We must here state that our inquiry originated in some investigations which were made for the Manchester and Salford Sanitary Association by a committee consisting of Messrs. Curtis, Ransome and Vernon. The method we have employed in making the necessary comparisons of the two series of observations has been as follows : ist. We have projected the medical and meteorological returns upon separate charts, so as to form curves, which represent the prevalence of the disease or the state of the atmosphere at any particular time; and then, by com- paring the two charts, and noticing any evident coinci- dences, we have been led to the conclusions specified in the paper, respecting the following diseases: Diarrhea, dysentery, pneumonia, bronchitis and catarrh, pleurisy, continued fever, rheumatic fever, measles, whooping cough, and scarlatina. DIARRHEA. A high mean temperature (above 60°) would seem to have a powerful influence in predisposing to this disease ; OF ATMOSPHERIC CHANGES UPON DISEASE. 167 when continuous, causing a rapid increase in the number of cases. A temperature below 60° appears to be unfavourable to its progress. The above action is generally most evidently shown when the temperature is above or below the average of the season. In the spring of 1857, from April 11th to June 2of6h, there is a gradual, and at first scarcely perceptible, rise in the diarrhcea curve, the number of cases being compara- tively very small. The temperature in April and the early part of May is much below the average (8° on May 2nd), although, on the whole, gradually rising. From June goth to July 11th the rise of the disease curve to 2,000 cases is more rapid —the temperature is above 60°, and on June 27th 7° above the average. From July 18¢h to August 15¢h there is a very great increase in the number of cases (even to 5,600), but from the latter date until September 12¢h the curve sinks at nearly the same rate to 2,200 cases. The number of cases then continues to diminish, but at a rather slower rate, until October 10¢h, when it is 600. The mean temperature during the whole of this period, from July 18th to October 10fh is above 60°, and con- siderably above the average, sometimes as much as 7°. In the weeks ending July 18¢h and 25th, it is stationary at 68° (the highest pomt this year), but it then gradually falls to 63°5° on August 1522; and after a temporary rise in the week ending August 29¢f it continues to fall until October 10¢h, thus throughout bearing a close relation to the disease. From the week ending October 10¢f (the temperature being 1°5° below the average) the number of cases still 168 MESSRS. RANSOME AND VERNON ON THE INFLUENCE remains low (still diminishing as the temperature falls until January 9th), and it does not again rise until May gond, 1858. In the two weeks preceding May 22nd 1858, the tem- perature is below the average as much as 6°, but it now begins rapidly to rise, and from May 29th to June 26th, 1658, it is considerably above the average (on June 5th, nearly 10°).° In accordance with the rise of the temperature curve the number of cases increases, and continues to increase steadily, as in the preceding year, until July 10¢h, when it is 1,200 (on July 1s¢ 1857 it was 1,400). But a remarkable difference between the two years must now be noticed, as it affords a striking illustration of proposition (8). In 1857 the disease runs on after July 11th to an amazing prevalence, but in the present year (1858) there seems to be a sudden arrest, the number of cases remains almost stationary for a fortnight, and then slightly dimi- nishes until August 7#h. When we inquire into the causes of this difference, we find that whereas in 1857, from June 20th to September 26¢h, the mean temperature never sinks below 60°; in 1858, for the first two weeks in July, the mean temperature is below 60°; and on July 1o¢h nearly 6° below the average. It seems as if the germs of the disease were so far destroyed by the unusual cold, that even the moderate warmth that follows could not again rouse them into activity. The mean temperature in 1858 does not remain above the average, as it did in the preceding year. From August 7/2 to August 28th 1858, the diarrhcea curve rises and falls with the mean temperature, but on August 28¢h the thermometer again sinks below 60°; and although it again rises in September to 63°5°, it is accom- panied by no corresponding increase in the number of OF ATMOSPHERIC CHANGES UPON DISEASE. 169 cases, —the diarrhoea agent has again received a check from which it does not recover. DYSENTERY {1) Seems to be influenced by the variations in the mean temperature, but in less degree than: diarrhcea, the effect not being generally traced in the lesser undula- tions of the curve. . (2) Increased atmospheric pressure seems to be unfavour- able to the progress of the disease, high readings of the barometer being nearly always accompanied by diminished prevalence of dysentery. The dysentery curve rises, on the whole, from the week ending April 11¢h tc the week ending September 12¢h. Fostered by the unusual warmth of the season, the disease seems to gather such strength that for a fortnight after the mean temperature begins to decline, it rushes on to still greater prevalence, and reaches its highest point when the mean temperature has fallen from 68° to 60°5.° The diminishing autumnal temperature, however, seems at length to produce an opposing influence, for the disease from this point gradually subsides, with occasional fluctua- tions, until the week ending January 16¢/. There is then a sudden temporary rise in the disease curve, the mean temperature being now above the average, but having been very variable in the preceding three weeks. During February 1858, there is a rapid increase in the number of cases which is associated with a temperature very much below the average (as though great cold as well as great heat were favourable to the disease) ; but it must be noticed at the same*time that the barometric reading during the month was very low. The disease curve now falls until April 17th, and con- tinues low until June 19fh (nearly the same date as that SER. III. VOL. I. Zz 170 MESSRS. RANSOME AND VERNON ON THE INFLUENCE on which the disease took its first decided rise in the pre- ceding year). The mean temperature has now been very high for a fortnight (from 8° to 10° above the average) ; and the number of cases rapidly increases until July 10¢h, when it may be noticed that the mean temperature falls suddenly to 56° (6° below the average), and the further progress of the disease is checked. After a short rise on July 24¢h (the mean temperature having then again risen 5° above the average for the week) the dysentery curve now gradually subsides, with many fluctuations, until October 2nd; and it may be noticed that the most decided rise is in the week ending Septem- ber 25th, following the unusually high temperature of the preceding week (64° or 6°5° above the average).* PNEUMONIA Seems to be very greatly influenced by the mean tempe- rature, the disease curve rising as the temperature falls, and vice versa. The above statement receives its best illustration in the spring, summer, and autumn of the year 1857. In the early: part of the year, while the temperature remains low, the disease is still prevalent, but as the * Hippocrates, Aph. 22, book iii., speaks of dysentery as an autumnal disease: ‘“‘ With regard to the seasons, if winter be of a dry and northerly character, and the spring rainy, and southerly, in summer there will neces- sarily be acute fevers, opthalmies, and dysenteries, especially in women, and in men of humid temperament.” — Aph. 3, xi. Sydenham mentions dysentery, amongst other diseases, “which com- mencing in August run on to winter.” In the report upon the status of disease, drawn from returns made at the time of the census of Ireland for the year 1851, Messrs. Donnelly and Wilde conclude that diarrhcea and dysentéry prevail more in the summer and autumn than at any other season. “They occur in the season of summer ; next in autumn; less in spring ; least of all in winter.” — Aretzeus, On the Causes and Symptoms of Chronic Diseases, book ii. ch. ix, OF ATMOSPHERIC CHANGES UPON DISEASE. 171 warm weather advances it gradually declines, and remains low throughout the unusually warm summer, being least prevalent when the mean temperature is highest in July and August. The number of cases begins to rise in the latter end of August, and reaches its maximum on Novem- ber 28¢h, the mean temperature being then 42°. During this period, the way in which the two curves of mean temperature and pneumonia supplement one another is very remarkable. From April 11th to November 28th (thirty-four weeks) there are only seven exceptions to this rule, and when we examine these we find that most of them may be accounted for without much difficulty. The first of the exceptions occurs in the week ending May goth, when the disease curve rises considerably, the temperature also rising, but it must be remarked that the temperature is still 7° below the average, and that in the preceding week it was 7°5° below the average. In the week ending May 237d another, but very slight, deviation from our rule may be noticed, — the pneumonia curve continues to descend, while there is a slight fall (half a degree) in the temperature. In the week ending June 20¢h there is a temporary rise in the number of cases, together with the mean tempera- ture, but this seems again to be accounted for by the occurrence of a temperature 2° below the average in the preceding week. In the week ending August 29th there is a slight rise in the number of cases, which cannot be accounted for by any fall of the mean temperature (68° or 9° above the average). (The N.E. winds prevailed this week, fol- lowing a long continuance of 8.W. winds, and the degree of humidity rapidly fell.) In the week ending September 26¢h the number of cases diminishes during a falling temperature, but in the pre- ceding week the mean temperature was 5°5° above the 172 MESSRS. RANSOME AND VERNON ON THE INFLUENCE average. Lastly in the week ending November 7th the onward course of the disease does not seem to be checked by the temporary but unusual heat. In the spring of 1858 the very close accordance between the two curves is not observed. Although the mean tem- perature falls lower than it has yet done, and the number of cases of pneumonia is still very great, yet it never again reaches the height that it did in November. It must, however, be observed that the highest point of the curve this season corresponds with the period of the greatest cold, the week ending March 193¢h having a tem- perature of 95° (6° below the average), the preceding week being still colder (92° or 8°5° below the average). The waves of the disease curve apparently lay behind those of the mean temperature. (The humidity is now diminishing, and N.E. winds very prevalent.) The mean temperature now begins to rise, and the disease diminishes in prevalence on the whole until August 21st, many fluctuations intervening, until with the ad- vancing cold of the autumn an increase again takes place. In the lesser modulations of the two curves from April roth to June 26h (eleven weeks) there is again a very close correspondence, there being only one exception in the week ending May 8th; the disease curve then falling, after a short rise, while the mean temperature continues to diminish. From June 26th to July 24th there is apparently an important departure from our rule. In the weeks ending June 26th, July 3rd, while the mean temperature is falling, the number of cases of pneumonia continues to diminish. Tt. seems probable, however, that this may be owing to the unusual heat of the preceding week (60° or 8°5° above the average), and the discrepancy in July seems to be due to the disease curve rolling up bebind that of temperature, the rise in the pneumonia curve followmg the unusual fall OF ATMOSPHERIC CHANGES UPON DISEASE. | 173 in the mean temperature of the preceding week (to 6° below the average). In the week ending August 21s¢ the departure from our rule is very slight; the disease curve continues to fall together with the temperature, apparently in consequence of the continued influence of the heat of the preceding week, which is 3°5° above the average. In the week ending September 18¢h the disease curve rises, and in that ending October end it falls in accordance with the curve of temperature, but in the latter instance the © preceding high temperature seems to display its influence. Out of the seventy-nine weeks which we have now examined, twenty-three (29 per cent) exhibit departures from exact accordance with our rule; but, as we have seen, most of these are still to be accounted for on the supposi- tion that the mean temperature influences the progress of the disease; but it seems probable that other elements, such as N.E. winds, also exercise some effect. BRONCHITIS AND CATARRH. The curve of these diseases, although drawn from ten times the number of cases, is almost identical with that of pneumonia, its highest and lowest points coinciding exactly with those of the pneumonia curve. It will be unnecessary, therefore, to trace it throughout its course, since it is evidently affected by temperature in much the same way as pneumonia. The correspondence of the mean temperature curve with that of bronchitis is even closer than with that of pneu- monia, the exceptions being only 264 per cent. It may be observed that in the year 1857, when the disease curve marks a deviation from the rule of tempera- ture, it may generally be ascribed to a change in the degree of humidity, the disease curve rising as the amount of moisture diminishes, and vice versd. 174 MESSRS. RANSOME AND VERNON ON THE INFLUENCE The chief discordance between the pneumonia and the catarrh-bronchitis curves takes place in the latter end of September and in October, which may possibly be due to the greater influence of the moisture upon bronchitis and catarrh than upon pneumonia; the degree of humidity at this time rises rapidly. In June and July 1858, the catarrh-bronchitis curve seems to answer more rapidly to the influence of the tem- perature than the pneumonia curve does. PLEURISY. This disease is too irregular in its course to yield any information in the present investigation, as the meteoro- logical elements under consideration do not appear to have any apparent connexion with it.* ConTINUED FEVER. It is difficult to trace any connection between the pro- gress of this disease and the meteorological elements under consideration, but on the whole high temperatures seem rather favourable to its production, and extreme cold is probably opposed thereto. From Aprii 112k 1857, the fever curve, frequently fluc- tuating, on the whole ascends until November 7th, when a sudden fall takes place, and it sinks rapidly until February 13th. In the first part of its course, from May oth to August 29th, it accompanies the rise of the mean tempe- rature, but after the latter begins to full the fever curve goes on rising as steadily as before for two entire months, and is not affected by the advancing cold until the week ending November 14¢h, when the thermometer stands at 45°. * Among the seasons of the year, winter more especially engenders the disease, next autumn, spring less frequently, but summer most rarely. — Areteeus, Causes and Symptoms of Acute Diseases, book i. chap. x. OF ATMOSPHERIC CHANGES UPON DISEASE. 175 As though the heat had called into activity some agent which resisted moderate fall of temperature, but which was destroyed by the cold of November, December, and January.* During the whole time of the gradual increase of the disease the mean temperature is throughout above the average. From February 1856 the fever curve does not rise much until the week ending June 5h, when a sudden increase of 110 in the number of cases accompanies a rise of 12° in the mean temperature. From April 11¢h 1858 to October 23rd, there is the same gradual advance in the disease curve as in the corre- sponding period of 1857, with the exception of the month of June 1858. The temperature this month was excessive, and to this in a great measure must be attributed the sudden rise in the number of cases. The month of July, which followed, had a temperature considerably below the average, and this checked the rapid advance of the disease for a time; but it will be seen that leaving the month of June out of the question (as being abnormal) the curve gradually ascends up to October 23rd, when our observa- tions end. Of the lesser modulations of the fever curve 61 per cent take place in accordance with the variations of the mean temperature, the disease rising and falling with the tem- perature; and in many of those weeks which present deviations these seem to be due to the lagging of the disease curve behind that of temperature ; as in the weeks April 18, 25, May 2, June 6, 13, August 15, 22, 29, and September 5 and 25, 1857; also March 27, May i, June 26, September 4, 11, 1858.+ * Drs. Donnelly and Wilde remark that fever, although very prevalent in spring, seldom rises to its intensity until summer and autumn. + It takes birth when spring passes into summer, and it rises towards 176 MESSRS. RANSOME AND VERNON ON THE INFLUENCE RHEUMATIC FEVER. The curve of this disease is not sufficiently extended to admit of accurate comparison with the meteorological curves, and therefore no decided conclusion can be drawn respecting it. Our data, however, would bear out the observation of Sydenham: “This disease may come on at any time; it is commonest, however, during the autumn.’ — Obs. Med. Vi. 5 (1). MEASLES. In its chief undulations, the measles curve seems to rise with the fall of the temperature, and vice versd; and the influence of this element is best marked when it is above or below the forty-three years’ average. These two propositions will be proved by the following observations. In the spring 1857 the largest number of cases occurs in the week ending May end, when the temperature reaches its minimum (42°) this season, and when more- over it is 7°5° below the forty-three years’ average. The disease curve then gradually declines as the temperature rises until August, when there is a sudden temporary increase in the number of cases (of whooping cough also), and a considerable fall from the July temperature, although the latter is still above the average, and remains so through- out the autumn. After this temporary deviation the temperature rises to 68° (7° above the average), and the number of cases diminishes again, but continues to do so for a fortnight after the temperature begins to fall. From September 12¢h the disease curve rises gradually maturity as the year advances; with the decline of the year it declines also. Finally the frosts of winter transform the atmosphere into a state unpro- pitious to its existence. — Sydenham, Medical Observations, ili. 2 (5). OF ATMOSPHERIC CHANGES UPON DISEASE. War while the temperature falls, and it continues to rise until November 7¢h, and then falls until November 21s¢, as though checked for a time by the temporary rise in the temperature of the preceding week (when it is 8° above the average). The disease attains its maximum in the week ending March 19¢h,* the temperature having reached its lowest point in the week preceding, and being more- over 7'5° below the average. The temperature now rises, and the disease diminishes in prevalence during the month of April: the week ending April 10¢h alone has a temperature below the average, and the number of cases again slightly increases for that one week. During the month of May the disease increases in pre- valence, although accompanied by an advancing tempera- ture which, however, is below the average; but after the week ending June 5th, which has a mean temperature no less than 10° above the average, the disease curve gradually declines until July 10¢h. The temperature in the week ending July 10¢h sinks to 56° (6° below the average), and from July 24th to August 7th it is 2°5° below the average ; and in all these instances a slight rise in the number of cases follows. With these exceptions, however, the disease curve rapidly declines until the week ending October 2nd; this week the temperature sinks to 51° (slightly below the average), and the depression is immediately succeeded by a rapid increase in the number of cases. Comparisons of the daily mean barometer readings, during the period April 1857 to October 1858, tend to show that during the time this disease was most prevalent * “They begin as soon as January; they increase gradually; they reach their height about the 24¢ of March; they then gradually decline, so that, with the exception of a few that may attack isolated individuals, they dis- appear by midsummer.” — Sydenham, Med. Obs. i. 3. y+ Dr. Miihry states that measles in the temperate zone experiences no change with the temperature. SER. III. VOL. I. AA 178 MESSRS. RANSOME AND VERNON ON THE INFLUENCE the fluctuations in the atmospheric pressure were far greater than when it was less rife. Measles seem to be much influenced by the same con- ditions as whooping cough, since it is usually most preva- lent during the same seasons; and yet it is evident that this relation is not exact, since in many of the lesser un- dulations of the measles-curve the variations take place in the opposite direction to those of whooping cough. (This is the case in twenty-nine out of the seventy-five weeks noticed, about 38 per cent.) On comparing the curve from April 1857 to March 1858 with the degree of humidity, it seemed that this element had some effect upon the lesser undulations of the disease- curve, since the number of cases rose and fell with the humidity in 72 per cent of the weeks; but on comparing the second period, from April 1858 to October 1858, this hypothesis is not borne out, and the coincidence may be accidental, since in one half the weeks the variations went with the degree of humidity, in the other half they weut in the opposite direction. Moreover, although in October, while the degree of humidity is rapidly rising, the disease prevails very greatly, yet in March 1658, when the num- ber of cases reaches its maximum, the degree of humidity is very low; and in September and October 1858, when the relative amount of moisture in the air is the greatest, the number of cases is at its minimum. : W Hoorinc CouGH Seems to be much influenced by the extremes of heat aud cold, the curve, on the whole, rising with the fall and sinking with the rise of temperature. The disease remains apparently unaffected by the gra- dually increasing warmth of the spring of 1857, but a decided diminution of the number of cases follows as soon as the mean temperature of the week rises to 67°, which takes place in the week ending June 27th 1857. OF ATMOSPHERIC CHANGES UPON DISEASE. 179 From June 27th until October 37d the temperature re- mains high (above the forty-three years’ average nearly every week), and during this time the disease is at its minimum (between forty and fifty cases per week). The number of cases does not again increase until the sudden fall of temperature in October and November, after which the weekly average remains pretty constant until February 13¢h, unaffected by the great fall of temperature in the week ending January oth 1858, but it again rises rapidly after the extreme cold of February and March (which was much below the forty-three years average). It remains very prevalent during the spring of 1858, but the remarkably warm June appears to check its pro- gress, just as it did in the preceding year. It is important also to notice that an increase in the number of cases again takes place in July, the temperature being much below the average; the year before the curve declined much more regularly and continuously. Durig the summer of 1858 the disease remains almost stationary, as in the preceding year; but while it may be observed that the temperature is never so high as then, the number of cases never sinks so low (seldom below sixty). SCARLATINA. A large amount of aqueous vapour in the air appears greatly to facilitate the formation and action of the peculiar scarlatinal poison, especially when this is accompanied by sudden fluctuations in the atmo- spheric pressure as shown by the barometer; a diminished pressure being favourable to the disease. It is rather difficult to separate the influence of tempera- * Drs. Donnelly and Wilde remark that spring affords rather more than the average amount of small pox, measles, scarlatina, and whooping cough. Census of Ireland for 1851. 180 MESSRS. RANSOME AND VERNON ON THE INFLUENCE ture from that of humidity, but a moderately low temperature seems to be favourable to the progress of the disease, whilst the extremes of both heat and cold seem often to exert a disturbing influence one way or the other; a temperature above the average generally diminishing, cold increasing the number of cases. From May oth to August 8th 1857, the degree of humidity remains low (below 07), although fluctuating considerably, and the number of cases is small; but in the lesser fluctuations the two curves rise and fall together in a remarkable manner. In the seventeen weeks from April 11 to August 8 1857, there are only three exceptions to this observation; the first two exceptions occur together in the weeks ending May 23rd and goth, the number of eases increasing while the degree of humidity falls, and it may be noticed that the first decided rise in the tempera- ture occurs in the preceding week ; the mean temperature then rose from 45° to 56°, and during that week and the next it remained nearly 6° above the average — the baro- meter regularly descending for three weeks. The second exception is in the week ending June 27¢h, and at this time again the perturbing influence of heat seems to act, the mean temperature rises 7°, and is 7° above the average. The humidity increases, and the baro- meter goes down, but the number of cases diminishes. On July 18¢A the scarlatina-curve begins to rise, and on the whole continues to do so until October 31s¢, thus accompanying very closely the degree of humidity ; but in the week ending August 29th there is a sudden fall both in the degree of humidity and in the number of cases, the mean temperature being very high (68°) and 7° above the average. From August 29th there is a steady rise in the number of cases until October grd, but the following week a slight OF ATMOSPHERIC CHANGES UPON DISEASE. 181 fall occurs, and it may be noticed that in the two preceding weeks there has also been a slight fall in the degree of humidity, and in the week ending October 10¢/ there is a great diminution of atmospheric pressure (barometer 29°4™s, the lowest this year), and this is at once followed by a further rise in the degree of humidity and the num- ber of cases. From October 10o¢h until November 21s8¢f, both the curves remain high, but in their secondary undulations, instead of being in accord, they supplement one another. In the week ending November 14¢h, the highest degree of humidity accompanies a decline in the disease curve, but is followed in the week after by an increase in the number of cases. The two curves then decline on the whole until December 5¢, when the returns of disease are discontinued for six weeks. In the spring of 1858 the degree of humidity remains tolerably high, without any great prevalence of the disease ; but here again may be noticed for twelve weeks an almost exact accord between the rise and fall of the secondary waves of the two curves. There are two exceptions to this rule: First, in the week ending March 6, the number of cases continues to fall after the sudden depression of the degree of humidity has ceased. In this week the atmo- spheric pressure 1s again very small (29°6™’), and the week following there is again a sudden rise in both the disease and the humidity curve. In the week ending March ooth there is no material change in the degree of humidity, but the mean tempera- ture rises 13°, and is 6° above the average, and the scarla- tina-curve descends again. During April the number of cases diminishes gradually, and on the whole the humidity-curve declines, but fluc- tuates remarkably, the scarlatina-curve marking these fluc- tuations by slighter variations in accordance with them. 182 MESSRS. RANSOME AND VERNON ON THE INFLUENCE An apparent exception to the rule which we have hitherto noted now takes place. From May 8th the disease-curve begins on the whole to rise, while the degree of humidity with great fluctuations seems to descend until the middle of June (as in the preceding year the secondary undulations corresponding with those of the disease). At the same time, however, it must be noticed that the mean temperature in the beginning of May is very low (46° or 6° below the average), and it does not rise materially until the week ending June 5. For a fortnight after this date the temperature rises, and remains very high (66°), nearky 10° above the average, while the number of cases diminishes during the same time. From June 26¢h the humidity and disease curves on the whole rise until October 23; but from July 10 to July 24 the degree of humidity falls as the disease curve rises ; and here again we may perhaps trace the disturbing influ- ence of temperature, the week ending July 10¢/ having a a mean temperature of 56°5° (6° below the average). In the week ending August 7th the disease-curve rises very rapidly (sixty cases), while the degree of humidity remains low; but the preceding week the mean tempera- ture has been 2°5° below the average. During the four following weeks the variations in tem- perature would seem to have the chief influence upon the disease, the rise and fall of the fluctuations of temperature and scarlatina supplementing one another very closely. In the week ending September 4, the barometer is very low, and the following week the degree of humidity rises considerably, while the temperature remains stationary, but there is a rapid rise in the disease curve. The number of cases again falls greatly in the week ending September 18, probably from the action of the unusual heat, the temperature rising to 63°5° (6°5° above OF ATMOSPHERIC CHANGES UPON DISEASE. 183 the average), while the degree of humidity continues to rise. From this time, however, until October 16th, the hu- midity again appears to exert its influence, and the curves are in accordance. The disease curve reaches its highest point for the year (200 cases) in the week ending October 16th, the degree of humidity rising rapidly until October 23rd, but the temperature not descending much, and remaining 2° above the average. It is interesting to observe the manner in which the curve of scarlatina supplements the curves of whooping cough and measles. ‘‘Thus they vex humanity by turns, as the constitution of the year and the sensible tempera- ‘ture of the air most assist the one or the other.” — Sydenham.* In the foregoing examination into the effects of the several meteorological elements upon scarlatina, it will be seen that we have ascribed to humidity the chief influence, but at the same time have carefully noted the effects of variations of temperature aud pressure of the atmosphere ; but it may be that we have not sufficiently indicated the reasons for our opinion. Without very close comparison it would be very diffi- cult to decide whether temperature or humidity had the greatest influence upon this disease. First, if we take the correspondence of the curves during the same times, we shall find that in 64 per cent of the weekly periods the number of cases rose and fell with the fall and rise of the thermometer, and in 63 per cent with the rise and fall of the degree of humidity; in 42 per cent of the periods * Sydenham states that scarlet fever may appear at any season, but oftenest towards the end of summer. — Med. Obs. vi. 2, 1. He also speaks of one epidemic being driven out by another “ut clayvum a clavo.” — Med. Obs. 1x. 1, 7. 184 MESSRS. RANSOME AND VERNON ON THE INFLUENCE these two elements might act together, the temperature falling as the degree of humidity and disease-curve rise, and vice versd. Of the weeks in which the degree of humidity and temperature rise and fall together, the appa- rent effects, as shown in the rise and fall of the disease- curve, are almost exactly balanced, there being fifteen points of agreement with the temperature, sixteen with the humidity-curve. The fact of accordance between the rise and fall of the curves, however, must be of little im- portance in determining the influence of the element upon the disease, compared with observations upon the actual state of the air at the time of prevalence or absence of the disease. A few instances will, we think, show that the tempera- ture, although by no means inactive, exerts less influence than the humidity. While the degree of humidity is at its lowest point in 1857, between June 6 and July 18, the number of cases is also the least, scarcely rising above thirty. During the corresponding period in 1858, between April 1 and July io, although the disease is rather more prevalent than in the year before, yet the number of cases rarely exceeds fifty, and does not increase until the degree of humidity begins on the whole to rise. Both in 1857 and 1858, when the amount of moisture in the air is greatest, the disease-curve is at its highest point. On the other hand, a low degree of temperature accom- panies both the smailest and largest number of cases in both 1857 and 1858, and the same is true of a high tem- perature; e.g. on July 18, 1857, the mean temperature is 68° while the disease is at twenty-five, and in August 1858, with the mean temperature above 60°, the number of cases remains above 100. Notwithstanding this remark, however, many of our observations will prove that tem- perature has au important modifying action. OF ATMOSPHERIC CHANGES UPON DISEASE. 185 Thus far has our investigation carried us. We already trace, though often but faintly, the influence of the few meteorological elements which we have studied. Taken in connection with large numbers of cases of disease, although a clearly defined and exact accordance cannot be found, still we perceive a certain general relation existing between them, and enough may perhaps have been done to prove the value of such an inquiry. The returns for several of the weeks, especially those ending August 22nd and 29h, 1857, and March 6th, 1858, are very im- | perfect; but after careful examination we do not find that these deficiencies affect the conclusions at which we have .arrived. In each instance, when appreciable, the probable amount of error arising from this source had been marked in pencil upon the chart. Before closing this paper we must state that it seems very desirable that many other branches of meteorological research should be included in the inquiry, and compared with disease; among these may be mentioned winds, elec- tricity of the atmosphere, rain, microscopical and chemical analyses of the atmosphere. Under the last-named head regular series of observations, made at various stations with the sepometer of Dr. R. A. Smith, F.R.S., would be of very great importance. SER. III. VOL. I. BB 186 MR. JOSEPH BAXENDELL ON THE XI. — On the Phenomena of Groups of Solar Spots. By Josepa Baxenpe t, F.R.A.S. Read before the Physical and Mathematical Section, October 13th, 1859. Iv is well known that the spots on the sun generally occur in groups, which differ considerably in size and form, and in the duration of their appearance; and which frequently undergo great changes in short intervals of time. Although they have long engaged the attention of astronomers, little progress has yet been made in the investigation of the laws which regulate their formation and their diversified appearances and changes. It has long been known that they occur principally in two bands or zones parallel to the solar equator, and that they are rarely seen beyond the limits of 35° north and south latitude; and Mr. Carrington has shown that the position in latitude of these zones is subject to considerable and rather sudden changes. M. Schwabe’s long series of observations has proved that there is a pretty regular period of increase and decrease in the ‘number of the solar spots. Mr. Dawes and Professor Secchi have observed that large spots have sometimes a movement of rotation; and Dr. C. H. F. Peters and Mr. Carrington have pointed out that the individual spots of small groups have a tendency to diverge or recede from each other. ‘There are, however, certain phenomena of groups of spots which, so far as I am aware, have not PHENOMENA OF GROUPS OF SOLAR SPOTS. 187 been alluded to in any astronomical work or memoir, but which seem to me to merit attention, as indicating the direction in which future inquiries ought to be made; and I have therefore thought that a brief notice of them might not be unacceptable to the members of this section. The first point to which I would draw attention is the fact that in those groups, which consist of spots differing considerably in size, the largest spot is generally in the preceding part of the group. An examination of such published drawings and descriptions of the solar spots by other observers as have fallen under my notice has con- firmed the conclusion drawn from my own observations ; and in a series-of unpublished diagrams of the solar spots made by my friend Mr. Williamson of Cheetham Hill, in the years 1849, 1851, and during the present year, and which he has kindly allowed me to inspect, I find that out of thirty-one different groups only fowr had the largest spot in the following part of the group, the ratio of excep- tions to the rule being therefore about one to seven. My own observations indicate a much smaller ratio, and I must remark that as Mr. Williamson’s principal object was to show the positions of the groups on the solar disc and not to delineate very exactly their details, many of the groups laid down in his diagrams do not exhibit any very decided difference of size in their components, and are therefore not available for the purposes of this branch of the inquiry. The next feature in the constitution of groups of spots to which I have to allude is, that a great number of these groups may be regarded as consisting of two sub-groups, each of which generally contains one or two spots deci- dedly larger than the rest. In their early stages these sub-groups have often no apparent connexion, but after a certain time small spots break out in the interval between them, and form an irregular line or band which completes 188 MR. JOSEPH BAXENDELL ON THE the group. Some of these new spots occasionally increase in size until they equal or even exceed the principal spots of the sub-groups. In such cases, however, the latter generally diminish, and the whole group undergoes consi- derable changes; but amidst all these changes there appears to be a constant tendency of the forces which produce the spots to concentrate in ¢wo places in the group at some considerable distance apart, aud lying in a direction which is evidently more or less dependent upon that of the sun’s rotation; and it often happens that groups which have apparently quite lost their original binary character again resume it before their final decay and extinction. The preceding sub-group of binary groups is almost invariably the first to appear, and generally the last to disappear, the exceptions in the first case being fewer than in the latter. Mr. Williamson’s diagrams have proved very valuable in affording many cases of groups which exhibit a deci- dedly binary character; and from the descriptions of the solar spots given by former observers there can be little doubt that groups of this class have at all times been of common occurrence. Thus I may quote as an instance an observation by Sir William Herschel, which will be found recorded at p. 272 of the Phil. Trans. for 1801, and which is an excellent illustration of one of the most interesting phases of these binary groups. It is as follows: “January 18, 1801. Between two clusters of openings near each other, there are some, as I suspect, incipient openings: they resemble coarse pores of indentations. January 19, 1801. The incipient openings between the — clusters of yesterday’s observations are completely turned into considerable openings. It seems as if an elastic but not lumimous gas had come up through the pores or inci- pient openings, and spread itself on the luminous clouds, forcing them out of the way, and widening its passage.” PHENOMENA OF GROUPS OF SOLAR SPOTS. 189 Since I first satisfied myself as to the reality of the binary character-of groups, I have found that in the case of spots apparently single and isolated a careful examina- tion with a larger aperture and higher power will often reveal a minute spot at a distance on the following side, which afterwards frequently proves to be the nucleus of a following sub-group; and though I formerly regarded many spots as single and isolated, I now believe that, in fact, such spots are of extremely rare occurrence. The two centres of force or activity in binary groups are sometimes very widely separated, instances not uncom- monly occurring in which the distance between them exceeds ninety thousand miles. | With regard to groups which exhibit anomalous appear- ances and curious and complicated changes, I have some- times observed that they arise from the interference of two smaller groups which have broken out near each other, and have gradually extended themselves until they formed one compound group. I need not remark upon the im- portance of a careful observation of the phenomena of these compound groups. I have alluded to the observations of Mr. Dawes and Professor Secchi on the rotation of spots, and I may add that in all the cases of rotation which have come under my own notice the rotating spot has been the principal member of the preceding sub-group; but I must remark that the observations which I have hitherto made are very far from being conclusive as to whether this rotation is real or only apparent. It will not escape remark with reference to the binary character of groups, that there is considerable difficulty in conceiving the mode of operation by which the forces that produce the spots in a group should first develope them- selves, almost simultaneously, at two points so widely distant from each other; and I must admit that in the 190 MR. JOSEPH BAXENDELL ON THE present state of the inquiry I am not prepared to enter into any theoretical explanation of this remarkable pheno- menon. My observations have hitherto been made without reference to any particular theory, and they have led me to believe that much yet remains to be done in accumu- lating the raw materials of observation before we can obtain results sufficiently well established to form the basis of a philosophical theory of solar phenomena; and at present I do not offer the conclusions given in this communication as anything more than the first results of an attempt to pursue, without reference to any theory, a systematic examination of the phenomena of the solar spots. The following extracts from my journal of observations will serve to illustrate the several points to which I have drawn attention. Greek and Roman letters are used to denote the different spots and groups as laid down in diagrams in the journal; and I have often found it conve- nient to indicate the size of spots by reference to a scale of magnitudes, in which the first magnitude represents a spot of the largest size and the tenth one of the smallest. The telescopes which have been used in the observations are a small but excellent achromatic by Mr. Dancer of twenty-two inches focal length and 2°6 inches aperture; a Newtonian reflector of eighty inches focal length and seven inches aperture; and the excellent equatorially- mounted achromatic of Mr. Worthington’s observatory, which has a focal length of seventy inches and an aperture of five inches. Observations. September 9, 1859, 9/.a.m. Of the siz groups now visible on the sun’s disc, five have their largest spots in the preceding part of the group, and the sixth consists of spots differing little in size. September 22, 5h.p.m. The sun has gone through half a revolution since the observation of the 9th instant, PHENOMENA OF GROUPS OF SOLAR SPOTS. 191 and in all the siz groups now visible the largest spot is in the preceding sub-group. September 15, 4h. 15m. p.m. All the groups now visible have’ an elongated form, and in five of them the largest spot is in the preceding, and the next in size is in the following end of the group, giving each the appear- ance of a double group, the sub-groups being connected by irregular lines of very small spots. In the sixth group (HL) the two largest spots are equal or nearly equal in size, but there is a disposition to the same arrangement. September 16, 84h.a.m. Preceding large spot of H now decidedly larger than following; preceding 7, and following 8 magnitude. September 17, gh. am. A new group has broken out ate; following principal spot 7/h mag.; preceding 74m.; one or two 10th mag. between them; the following principal spot is very black, and has no penumbra. The chief spot in following sub-group of H is also very black. 5h. p.m. Three or four small spots n f following the sub- group of H. Preceding large spot of e is now rather larger and blacker than following large spot; four spots in preceding sub-group. September 18, 114.a.m. Following large spot of H now larger than preceding. September 19, 9f.a.m. Many new spots in H, and several in e, but second principal spot of H and both those of « have diminished in size. A new group has entered disc at J — principal spot 74 mag., second principal g mag., and three or four 10¢h mag. spots. A new group has broken out since yesterday at 0; the following spot is the largest, and is very black; two principal spots and many smaller ones. 47h.p.m. The spots in ¢ are now all nearly of the same size. Counted twenty-five spots in group H. Preceding principal spot of # is now equal to the following, and two spots have broken out near the latter. © 192 MR. JOSEPH BAXENDELL ON THE September 21, 334. p.m. The spots in @ have increased in size and number. The spots in J are larger, and the principal of following sub-group is rather larger than that of preceding sub-group. ‘The spots in ¢ and H have changed considerably, and the leaders in both are again decidedly larger than any of the other spots in the groups. September 22, gf.a.m. There seems to have been a movemeut of rotation in preceding large spot of J since yesterday afternoon. September 24, gh.a.m. Preceding large spot of J appears to have gone round. three quarters of a revolution since gh. am. September 22nd. The rotation, if real, has been in the direction E.N.W.S. as seen from the earth, or to a spectator on the surface of the sun E.S.W.N., the spot being in the northern hemisphere. There seems to have been little or no movement in the large spot of following sub-group. September 25, 114.a.m. @ approaching western limb, and large spot of preceding sub-group is now much larger than any other in the group. The following sub-group contains more spots than the preceding. The position of the longer axis of principal spot of J seems to have altered again, and in the same direction. A ofh mag. spot has broken out at &, far removed from any other spot. 33h. p.m. The distance between the centres of the principal spots of the sub-groups of J is 3’ 29.4”, or more than 96,000 miles. September 26, 54. p.m. Preceding large spot of J appears to have rotated through an angle of about 30° since yesterday forenoon, and in the same direction as before. The large spot in following sub-group seems to have moved very little, and if at all it has been in a con- trary direction. The spot & has increased, and I am pretty sure I caught sight of a minute companion at a distance on the following side. PHENGMENA OF GROUPS OF SOLAR SPOTS. 193 September 27, 97h. am. The large spot of J has rotated still further, and a small spot which was in a pro- jecting part of its penumbra 2 p is now quite detached, with a bright line between. The following large spot seems to have changed little, if at all. & is now a fine double group; principal spot 6¢h mag. September 29, 9f.a.m. Preceding spot of J presents no fresh indication of rotation. Four 84 or 9th mag. spots, which I regard as two new groups o and ¢, have broken out in the space within the groups M, N, O, and P; pre- ceding spots in each a little larger than following. October 1, 44.p.m. Many new spots in o and q, and they now form one compound group, in which, however, there is a tendency to the binary arrangement. October 3, 84.a.m. The compound group (o+¢@) has five 8th mag. spots; the rest are gtk and 10fh mag. It has a very irregular form. October 5, 844.a.m. Many of the spots in following part of the compound group (o+¢@) have united into one, which is a 5th or 6th mag. in size. October 6, 43f.p.m. The compound group (0+ ¢) now consists of a 5¢/ mag. spot with a very large penum- bra, preceded by several smaller spots. There is a common, and, as it appears to me, very sig- nificant feature of binary groups which is often alluded to in the journal, and I will therefore add an observation having reference to it. September 16, 1659, 8$h.a.m. The following sub- groups of D, F, G, and H contain more spots and extend over a larger area than the preceding; but the principal spots in the preceding sub-groups are larger than those in the following. SER, III. VOL. f. cc 194 MR. JOSEPH BAXENDELL ON XIT. — On the Rotation of Jupiter. By Josrru BaxenveE Lt, F.R.A.S. Read before the Physical and Mathematical Section, December 8th, 1859. Dvurine the last winter and spring I availed myself of the opportunity presented by the appearance of several small isolated dark spots on the surface of Jupiter to make a series of observations with the ultimate view of obtaining a re-determination of this planet’s period of rotation; and with the more immediate object of testing the conclusions which have been drawn from the observations of Cassini, Maraldi, Sir William Herschel, Schroeter, &c., to the effect that different spots have different periods of rota- tion; that the time of rotation of the same spot will some- times vary; and that in general spots near the equator move more quickly than those in higher latitudes. Some of the observations were made at my own house with a Newtonian reflector of seven inches aperture and eighty inches focal length; but the greater number were made at Mr. Worthington’s observatory with his excellent equatorially-mounted achromatic of five inches aperture and seventy inches focal length, and Newtonian reflector of thirteen inches aperture and nine feet focal length. The magnifying powers employed in making the estimations of the spots ranged, according to atmospherical circum- stances, from 130 to 301, those most frequently used being 180 and 223. ~ THE ROTATION OF JUPITER. 195 The object of each night’s observations being to deter- mine, as accurately as possible, the time when the meridian of a spot passed through the centre of the disc of the planet, careful eye estimations of the position of the spot were taken at intervals during its transit across the disc; each result was then reduced to the central position by the aid of small tables constructed for the purpose; and the mean of all the reduced results —allowing weights in certain cases—-was taken as the true time of the spot’s apparent meridional conjunction with the centre of the disc. The times thus obtained were afterwards reduced to the epoch of the first observation, November 7th, 10h. 4om. G. M. T., by applying the corrections due to the altered geocentric longitude of Jupiter, and to the changes in the value of the light equation. Attempts were made to determine the positions of the spots by micrometrical measurement; but it was found that the results obtained in this way did not exceed in accuracy (except in the case of a very conspicuous spot and under highly favourable atmospherical circumstances) those obtained from eye estimations, or from carefully drawn diagrams; and, moreover, valuable estimations could often be made when, from frequent interruptions by rapidly passing clouds, complete micrometrical measure- ments were quite impracticable. The number of spots observed was seven, and they were denoted in the journal by the letters A, B, C, &c., in the order in which they were first seen; but the observations of five only were sufficiently numerous and exact to be available for the determination of their periods of rotation They were all distributed in a narrow band or zone in the southern hemisphere, siv of them being nearly on the same parallel of latitude, and the seventh only about fifteen degrees more to the south. It will be evident that this peculiarity of distribution was very unfavourable for an 196 MR. JOSEPH BAXENDELL ON investigation of the differences of velocity of rotation said to be due to differences of latitude of the spots. The following tables contain the corrected Greenwich mean times of the observed meridional conjunctions of the spots A, B, C, E and F, with the centre of the planet’s disc : Spor A 1858. 1859 Nov. 7 iG 40°0 Jan. 18 10 33°0 17 9 2°8 1g 6 14°6 Dec, 2 6 34°2 22 13 4A'S 4 8 14'0 23 9 38°6 20 II 24°9 28 8 A7°5 23 8 Get Feb. 2 4 GGG 1859. 7 7 9°5 Jan. 1 II 2325 It 10 Zits 4 6 25°4 Mar. 10 3 49°6 OT HOW AOI Nov. 12 Ir 44°2 Feb. 2 9 15" Jan. 28 10 16°2 } 10 6 Sareanen arene THE ROTATION OF JUPITER. 197 Dividing the observations of each spot into two groups, finding the mean epoch for each group, and dividing the interval between the epochs by the number of included revolutions, we obtain the followmg values of the mean period of rotation : him 8s. iS: From Spot A, 9 55 467086 +0439 B, 9 55 44821 407578 C, 9 55 39°240 + 1°650 H, 9 55 37°812 + 0°636 F, 9 55 39°858 + 1650 Between the 27th of October 1858 and the 14th of January 1859, Sir William Keith Murray Bart., of Ochter- tyre, made a very interesting series of observations of Jupiter with a fine Munich achromatic refractor of 64 inches aperture, and he kindly forwarded to me exact copies of a series of diagrams, showing the positions of some of the spots; and also extracts from his journal. These observations were, I believe, made without any | special reference to the question of the period of rotation, the observer’s principal object being to record and deli- neate any remarkable or interesting phenomena which might present themselves on the disc of the planet; but as great care has evidently been taken in laying down very exactly the positions of the spots in the diagrams, and in noting the corresponding times, I have reduced and dis- cussed the observations of spots A and B, and have obtained the following mean periods: h. m. 8. S. From Spot A, 9 55 45°96 + 1°33 B, 9 55 43°62 + 1°65 The close agreement between these results and those derived from my own observations will serve to show the degree of reliance which may be placed upon observa- tions of this nature. Had Sir William Keith Murray’s observations been continued a month or two longer there is no doubt the agreement would have been still closer; but unfortunately they were interrupted in consequence of 198 MR. JOSEPH BAXENDELL ON the removal of the Munich refractor to make room for a still larger instrument, a fine equatorially-mounted achro- matic of thirteen feet focal length and nine inches aperture, by Cooke, of York. Reverting to the results of my own observations, the periods of spots B, C, E and F, compared with that of A, give the following differences and probable errors : ; Probable Difference. Bieon. Aandi ther cccemanes 1°265 + o'719 Oran 6°846 + 1°477 Hse eeees 8:274.+ 0°760 I hiee een tas Se 6°228 + 1°477 As some of these differences considerably exceed their probable errors, the conclusion which has been drawn from former observations, that different spots have diferent periods of rotation, is thus fully confirmed. The following table shows the observed distance between the spots A and B on eleven different nights : 1858. ™. 1859. ' ™. 1859. Mm. Dec. 20 = 51°8 Van 225 5523 Feb. 7 52°4 23. 54°2 2357 S5n Ir 54°! Zi 5220 1859. Jani! 0 5055 Feb. 2 5073 Mar.10 58:0 18 604 The probable error of a single distance is 2°7 m., and we may therefore fairly conclude that the changes indicated by the numbers in the table were real, and that the dis- tance between the spots was greater on the 18¢h January and 10th of March than on the goth December and 2nd of February. We have already seen that the probable error of the mean period of B, as derived both from my own and Sir Wilham Keith Murray’s observations, is greater than that of A, and it may therefore be reasonably inferred that the changes in the distance between the spots were principally due to irregularities in the motion of B. The few observations which were made of spot F show that its motion also was not uniform. On the 27th of THE ROTATION OF JUPITER. 199 October, according to Sir William Keith Murray’s obser- vations, it followed A 1h. 55°8m.; on the 12th November, a tolerably good observation of my own gave precisely the same difference; but on the 28¢h January the distance had diminished to 1h. 41°4m., and on the end February it was only 1h. 32°7m. Adopting the position of the axis of rotation of Jupiter given in the introduction to Damoiseau’s Tables Ecliptiques des Satellites de Jupiter, the mean result of three nights’ micrometrical measures gives the latitude of spot A=13° 47’ south; and the mean deduced from fourteen of Sir William Keith Murray’s diagrams is 13° 11S. The spots B, C, and F were all very nearly on the parallel of A, while E, the spot which had the shortest period of rotation, was in latitude about 284° S. It appears, therefore, that the conclusion drawn by Cassini, from his own observa- tions, that spots near the equator of the planet generally move quicker than those in higher latitudes, receives no support from these observations. It is hardly necessary to remark that the results now given afford no certain information as to the period of rotation of the planet itself, as distinguished from that of its spots. On the contrary they seem to me to indicate very clearly —especially when taken in connexion with the results obtained by former observers —that in the present state of our knowledge of the phenomena which take place on the surface of the planet or in its atmo- sphere, any conclusion as to its exact period of rotation based upon observations of the times of rotation of its spots must necessarily be very precarious. 200 DR. ROBERTS ON THE ESTIMATION XIII. — On the Estimation of Sugar in Diabetic Urine by the loss of Density after Fermentation. By Wits1iam Rozerts, M.D. Lond., Physician to the Manchester Royal Infirmary. Read October 16th, 1860. WuHeEn saccharine urine is fermented with yeast its specific gravity, previously ranging from 1030 to 1050, falls to 1009 or 1002, or even below 1000. This result is chiefly due to the destruction of the sugar it contained, but partly also to the generation of alcohol, and its presence in the fermented product. As the diminution of density must be proportional to the quantity of sugar broken up by the ferment, the amouut of this diminution evidently supplies a means of calculating how much sugar any urine contains, always pro- vided that the remaining ingredients of the urine continue unchanged, or become changed in some uniform ratio. In order to ascertain the exact relation subsisting be- tween the density lost on fermentation and the sugar destroyed, experiments were made on the fresh urine of several diabetic patients in the Royal Infirmary. The following procedure was adopted : 1. The amount of sugar per 100 parts was first accu- rately determined by the volumetrical method, with Fehling’s test solution. 2. Next the density was taken by the specific gravity bottle. IN SUGAR OF DIABETIC URINE. 201 3. About 402. of the urine were then placed in a 12 02. bottle, with a drachm or two of German yeast, and set aside in a warm place to ferment, taking care to cover the mouth of the bottle with a slip of glass or a loose cork. 4. In from twelve to eighteen hours fermentation was usually over, and at the end of twenty-four hours the froth and scum had subsided or been dissipated sufficiently to permit the density to be again taken. Operating in this way on a urine passed on the 21s¢ of April, I obtained the following results: Sugar per roo parts by the volumetrical method ... 7°69 Density before fermentation at 60° or D............... 1038°60 Density after fermentation at 60° or D’............... 1005°92 IDexaveniin 7 oy nitere JOE SID) ed to aenponecedoesdancseussdaosogse 32°68 The relation, therefore, between the density lost and the per centage of sugar in this instance was as 32°68 to 7°69, or as I to 0.235; so that by multiplying the density lost into the co-efficient 0°235 we have for product the amount of sugar per 100 parts which this urine contained. That is, Sugar per 100 parts or S=(D-—D’) x 0'235. On repeating the experiment a great number of times with different specimens of urine and different specimens of yeast, the number 0°230 was found to be the more exact co-efficient. The degree of exactitude with which the quantity of sugar may be determined by this method is very great; indeed with the precautions to be mentioned presently it seems susceptible of nearly as much accuracy as the volu- metrical method. The following table places in comparison twenty obser- vations made by the two methods on various diabetic urines, with densities ranging from 1031°52 to 1053°48: SER. III. VOL. I. DD 202 DR. ROBERTS ON THE ESTIMATION TABLES, Sugar per 100 parts Sugar per 100 parts No. by the formula by direct volumetrical Difference. S=(D-D’) x 0°23. analysis. I Feca 7°69 o18 *2 1°47 7°69 0°22 3 6°68 6°66 0°02 #4 6°72 6°66 006 5 5°16 5°18 o'02 *6 5°19 5°18 o'or 7 5°65 577 or12 #8 5°65 Bal O12 9 4°47 4°35 Orr2 *10 4°49 4°35 O14 II 7°85 3°06 O'21 12 5°91 6°10 O'1g 13 D127 11°36 0°09 14 Ir'21 11°36 O'1s 15 5°69 5°68 O'ol 16 811 8°06 0°05 uy) 8:09 8°06 0'03 *18 8'00 3°06 0'06 19 8°29 8°20 0°09 20 7°61 774 O13 These results are so close that the discrepancies may be regarded as due to errors of manipulation rather than to any fault in the method. In pursuing the inquiry further it was found that the volumetrical analysis, in spite of every care in its perform- ance, did not possess the delicacy and certainty required in a standard when minute differences were concerned ; insomuch that, when a discrepancy appeared between the indications of the volumetrical and the fermentation me- thods, it was found impossible to decide in which pro- ceeding the error lay. In order therefore still further to test the constancy of the results, artificial diabetic urines were prepared hy diluting a natural diabetic urime with known volumes of water, or of a healthy non-saccharine urine. Assuming the estimate of the sugar in the original urine to be correct, the quantity of sugar in the dilutions could be ascertained * Those marked with an asterisk are duplicate experiments. OF SUGAR IN DIABETIC URINE. 203 with almost absolute accuracy by a simple calculation. In the following table the results obtained by fermenting these dilutions are placed side by side with the calculated quantity of sugar. PA BD LE LL: Suber, per 00 Sugar per 109 a g -|p; No. Doleh Giornale Dion for ene! Difference. S=(D-D’) x 0°23.| first estimate. 1 | A natural diabetic urine ..... 5°91 = = 2 |The same mixed with ;4, of nae ae Son its bulk of water ......... 5°3 33 3 | The same mixed with 2, of A i i its bulk of water en cle 4°73 oie 4 | The same mixed with 3, of 3 : : its bulk of water aes ane tae Neg 5 | The same mixed with 7, of ; : ; its bulk of healthy urine 5°34 De ae 6 | The same mixed with ~, a : : ; its bulk of healthy urine 477 4°73 ae 7 |The same mixed with ,°, of A f p its bulk of healthy urine igae ay ey 8 |The same mixed with ;°, of Bas S oink its bulk of healthy urine 7 59 Numbers so nearly alike as those in these two columns may be considered as practically identical. In the last experiment only, where the quantity of sugar was under one per cent., was there a sensible discrepancy. Satisfied now with the accuracy of the fermentation method, I was desirous of determining with more cer- tainty and exactitude the required co-efficient; which, from the preceding experiments, using the volumetrical analysis as a standard, was fixed at 0°23. I was unable to obtain grape sugar in sufficient purity to make solutions of known strength, and had recourse to cane sugar; using for the purpose the best loaf sugar of the shops. Solutions were made, varying in strength from 2 to 10 per cent. both with distilled water and healthy urine. But as cane sugar becomes converted into grape sugar under 204 DR. ROBERTS ON THE ESTIMATION the influence of yeast before fermentation begins, and the density of the solution is thereby materially increased, corrections had to be made on both these accounts before the experiments could be fairly compared with those on diabetic urine. In making the first correction cane sugar was taken asC,, H,, O,, and grape sugar as Cy. Hy Ox. In making the second correction allowance was made for the increase of density in accordance with the tables pub- lished by the authors of the report on “Original Gra- vities.’’ * Six solutions were made of cane sugar in water; two containing 10 per cent. and the remaining four containing respectively 8, 6, 4 and 2 per cent. of sugar. The mean co-efficient obtained from these six experi- ments was 0°234. Twelve solutions were similarly made in a healthy non- saccharine urine; two containing 10 per cent., three 8 per cent., and two 2 per cent. The remainder contained respectively 6, 4, 1°4, 1 and 0°6 per cent. These yielded a mean co-efficient of 0'228; and the general mean for the eighteen trials was a fraction under 0°23. These experi- ments therefore confirmed the results previously obtained with the volumetrical analysis as a standard. Equally correct results were obtained in operating on weak solutions, containing only I or 0°6 per cent. of sugar, where the density lost was not more than 3 or 4 degrees, as with solutions containing 10 per cent. of sugar, in which the loss of density exceeded 43 degrees. Having examined the question experimentally, and fixed the co-efficient from multiphed trials under varied condi- tions, it was not without interest to examine a little more closely the several items which go to constitute the “ den- sity lost” in fermentation, and to endeavour to arrive * Report on Original Gravities, by Professors Graham, Hofmann and Redwood, in the Quarterly Journal of the Chemical Society, 1853. OF SUGAR IN DIABETIC URINE. 205 synthetically at similar results to those obtained by the more empirical method of direct experiment. If we take a solution of cane sugar in water, made so that 100 grains are contained in each 1000 grain-measures, its density at 60° will be 1038°64. Such a solution consists by weight of 100 grains of sugar, and 938°64 grains of water. After fermentation there remains 53°74 grains of alco- hol, and 987°18 grains of water. Such a mixture of alcohol and water has, according to Gilpin’s tables,* a density of 993. This figure is very near that obtaimed in actual fermen- tation. The density of the fermented product is, however, a little higher, about 14 or 2 degrees. This slight excess is to be attributed to the escape of a portion of alcohol with the carbonic acid during fermentation, and to a littie soluble matter taken up from the yeast. Some effect probably, though I know not of what kind, is likewise produced by the retention in the liquor of more than its own bulk of gaseous carbonic acid in a state of solution. It was uniformly found, with solutions of equal strength, that those made with urine showed rather more “ density lost” than those made with water. This is an indication either that changes take place in some of the non-saccha- rine constituents of the urine during fermentation — but changes so slight and so constant in their nature that they do not interfere materially with tie accuracy of the mode here proposed for estimating sugar—or else that urines usually reputed nou-saccharine do in reality contain a small quantity of sugar. An excessive quantity of yeast was used in these expe- riments in order to hasten the process of fermentation. The quantity of soluble matter taken up from the yeast was * See Henry’s Chemistry, tenth edition, vol. ii. p. 343. 206 DR. ROBERTS ON THE ESTIMATION very small. Two drachms shaken up with 402. of water, and left in a warm place for thirty-six hours, only increased the density of the water by 0°3°. In the above experi- ments a piece of yeast about the size of a large filbert or small walnut was employed to ferment four ounces of urine, a little more or a little less making scarcely a sensible difference in the results. The fermented urine continues somewhat turbid for a day or two after fermentation is completed, and the degree of this turbidity has an appreciable though slight effect on the density. In the experiments detailed in this paper the second density was taken about twenty-four hours after the addition of yeast, and before the urine had completely cleared. It was found in four trials that if the urine was allowed to rest twelve hours longer the density fell by 0°2° or 0°3°. It is desirable, therefore, where a series of expe- riments is made, and scrupulous accuracy and uniformity are required, always to take the second density at about the same period after fermentation has ceased. The method here proposed for determining sugar is not put forward in rivalry with the accurate and elegant volu- metrical process now usually employed by chemists; but it is believed that it will be of great service to the medical practitioner, who is unaccustomed to delicate chemical manipulations. Any one who has had much practice in the volumetrical method knows that great nicety and con- siderable experience are requisite to insure trustworthy results. Fehling’s test solution also is liable to speedy deterioration unless hermetically sealed from the air. This arises from the conversion of a portion of the tartaric acid of the test ito racemic acid, which, equally with sugar, has a reducing power on the oxide of copper, and when present of course vitiates the results. Jor these reasons it may be surely predicted that the volumetrical method can never come into general use at the bed-side. OF SUGAR IN DIABETIC URINE. 207 The fermentation method, on the other hand, is of ex- ceedingly easy performance, and the taking of densities is an operation to which the medical practitioner is daily accustomed. I shall have an opportunity through another channel to bring this method under the notice of my medical brethren, and to enter more fully on the particu- lars which concern its clinical application. I content myself here with a short sketch of the process, as it may be conveniently carried out in private or hospital practice. 1. The specific gravity of the urine is taken at the ordinary temperature of the ward or bed-room. 2. Three or four ounces of the urine are poured into a 12 02. phial, together with a lump of German yeast of the size of a large filbert. i 3. The bottle is hightly corked, or covered with a slip of glass, and set aside in a warm place to ferment. 4. In about eighteen hours, when the fermentation has entirely ceased, the bottle is tightly corked aud removed to the ward or bed-room so that it may cool to the tem- perature at which the specific gravity was taken the day before. 5. The urine in the meantime clears, and in five or six hours it may be decanted into an appropriate vessel and the specific gravity taken again. 6. The amount of “density lost” is thus ascertained, and the following simple and most convenient rule ex- presses the result of the analysis. Hach degree of “ density lost”? indicates one grain of sugar per fluid ounce of urine. So that in the example already quoted in the earlier part of this paper, where the urine lost 32°68° of density after fermentation, the quantity of sugar indicated was 32°68 grains per ounce, or 653°6 grains per imperial pint. 208 PROFESSOR H. E. ROSCOE ON THE ALLEGED XIV. — On the alleged practice of Arsenic-Eating in Styria. By Henry Enrietv Roscos, B.A., Pa.D., Professor of Chemistry at Owens College, Manchester. Read October 30th, 1860. Ir has been frequently stated that among the peasants of Styria the habit prevails of regularly taking into the system large quantities, from two to five grains daily, of arsenious acid. This extraordinary practice is said to be indulged in for the purpose of improving the health, avoiding the danger of infection, and raising the whole tone of the body. Dr. Taylor shows in his excellent treatise on poisons,* that this widely spread notion is traceable to the state- ments on the subject of Dr. Von Tschudi, in a paper published by him in the Wiener Medicinische Wochen- schrift for October 11th; 1851. The late Professor John- ston repeats these statements of Von Tschudi’s in his popular work on the Chemistry of Common Life, and this seems to be the source whence the assertion has gained some credence in this country. Dr. Taylor points out that Von Tschudi does not give satisfactory evidence in proof of this singular and very improbable statement, inasmuch as he did not either submit the so called * Taylor On Poisons, second edition, 1859, p. 91, &e. Dr. Taylor states that from two to three grains of arsenious acid may be regarded as the smallest dose usually producing death. PRACTICE OF ARSENIC-EATING IN STYRIA. 209 arsenic to a chemical analysis, or examine the urine or feeces of those persons who were said to indulge in the practice of arsenic-eating; and Dr. Taylor suggests that the white powder eaten by the Styrian peasants is not arsenious acid, but probably oxide of zinc. Mr. Keste- ven, im some papers published in the Association Medical Journal for 1856, reviews the whole subject, gives Von Tschudi’s communication in extenso, discusses the arti- cles upon the subject by Mr. Boner which appeared in Chambers’s Journal, protests against the hght manner in which Johnston adopts and theorises upon the unproved statements of Von Tschudi, and concludes that the story of the Styrian arsenic eaters is not only unsupported by adequate testimony, but is inconsistent, improbable, and utterly incredible. Since the publication of Mr. Kesteven’s paper, Mr. Heisch has communicated in the Journal of the Pharmaceutical Society for May 1860, statements concern- ing a number of interesting and apparently well-attested cases of arsenic-eating obtained from persons of position in Styria.* Believing that this interesting question, as to whether or not persons are able, without apparent injury, to take into their systems doses of arsenic usually supposed suffi- cient to produce death, could only be settled by more pre- cise local information than is at present made public, I was supplied, through the kindness of my friend Professor Pebal of Lemberg, with a number of reports addressed by seventeen medical men in Styria to the government medi- eal-inspector Dr. Julius Edler Von Vest of Gratz, respect- ing the alleged practice of arsenic-eating in that district. A notice of some of these reports, as well as of other mat- * Another statement concerning the “ Styrian Poison-EHaters” was pub- lished in a Gritz newspaper, the Tages Post, on March 30th and April 8¢h, 1860, by Dr. Schidler, in which several cases of arsenic-eating which came under his notice are detailed. SER, III. VOL. I. EB 210 PROFESSOR H. £. ROSCOE ON THE ALLEGSD ters affecting this question, was laid before the Academy of Sciences in Vienna a short time ago by Dr. Schafer* of Gratz. I shall have the honour of laying before the: Society the results of these, as well as of other commnu- nicatious which I have received upon this subject. It is almost unnecessary to say at the outset, that, upon a question of this kind, nothing but the most positive and direct evidence can be received as conclusive. My aim will simply be to endeavour to bring forward such evidence as may help to decide the question whether arsenious acid is, or is not, regularly eaten by men in Styria in quantities usually supposed sufficient to produce death. I shall not attempt to discuss or explain the superstitious notions with which many of the reports naturally enough abound ; nor shall I detail the various purposes for which the arsenic is said to be taken, or the various modes in which the dose is said to be administered. For the present pur- pose I have not thought it necessary to translate the re- ports in extenso, but I have extracted from them all such portions as appear to me to bear more directly upon the point under discussion; and I would propose to deposit with the Society the accompanying accurate copies of all the reports and other documents from which these ex- tracts are made, in order that persons interested in the inquiry may have opportunity of referring to the original papers at full length. All the letters received from the medical men in Styria agree in acknowledging the general prevalence of a belief that certain persons are in the habit of continually taking arsenic in quantities usually supposed to produce death. Many of the reporting medical men have not had any ex- perience of the practice; others describe certain cases of arsenic-eating which have not come under their personal * Sttzungs Berichte der Academie d. Wissenschaften, 1860. Band xli. p- 573: PRACTICE OF ARSENIC-EATING IN STYRIA. atl notice, but which they have been told of by trustworthy people whose names are given; whilst others, again, re- port upon cases which they themselves have observed. It is noteworthy that no one of the seveuteen medical men denies, or attempts to disprove, the truth of the generally expressed opinion concerning the arsenic-eating. The first question which I shall endeavour to answer will be —Is or is not arsenious acid, or arsenic in any other form, well known and widely distributed amongst the people of Styria? Through the kindness of Professor Gottlieb of Gratz I received a quantity (about six grammes) of a white substance which the following examination showed to be pure opaque amorphous arsenious acid. The substance being subjected to a careful qualitative analysis gave all the well-known characteristic reactions of arsenious acid; the presence of no other body could be discovered. The specific gravity of the substance at 15°C. was found to be 38059. (Opaque arsenious acid has, according to Taylor, the specific gravity 3°798). Quantitative Analysis. — 06495 grm. of the substance was (dissolved in nitric acid, the excess of acid expelled by heating on the water bath, the liquid then neutralised by ammonia and a solution of sulphate of magnesium, added with which so much chloride of ammonium had been mixed that ammonia produced no precipitate of magnesia. The precipitate thus formed was allowed to stand for twelve hours, then collected on a weighed filter, com- pletely washed with cold dilute ammonia, and after drying at 100° C. for twenty-four hours it was weighed, and re-dried until no further diminution in weight occurred. 1:2465 grms. of the arseniate of magnesium and ammonium was obtained, corresponding to 0'6492 grms. of arsenious acid. ‘The substance, therefore, contained g9‘97 per cent. of As Os, or is perfectly pure arsenious acid. 212 PROFESSOR H. E. ROSCOE ON THE ALLEGED This arsenious acid, which was enclosed in a dirty blue paper bag, was brought by a Styrian peasant woman to the district judge of Knittelfeld in Styria; the woman stated that she had taken the parcel from her farm labourer, whom she had caught in the act of secretly eating the arsenic. The following certificate was forwarded to me with the parcel containing the arsenious acid: (Translation.) — ‘This Hidrach* was delivered to me by a peasant woman from Mittenlobing whose name I am unacquainted with. She saw her farm-labourer secretly eat it, and gave it to me in order that the bad habit might be stopped. Hevrenstein, Bezirksvorsteher. Knittelfeld, 26¢2 April, 1860.” This shows that the substance called “ Hidrach” by the Styrian peasants is arsenious acid, and not, as suggested by Dr. Taylor, oxide of zinc. The following extracts from some of the reports of the medical men before spoken of prove that this Hidrach is well known and widely diffused in Styria; they show also the sources whence the sub- stance is obtained. Dr. Oberhammer writes from Noflach: “ Arsenic is a substance well known to and in great request with the people of this district. It is chiefly and pretty generally used to be given to horses, &c.” He also says: “I myself am unacquainted with any case of arsenic-eating. The clergy would be able to give the most information upon the subject. The clergyman Von der Graden showed me two pieces of arsenious acid which he had taken from a parishioner at the confessional.” Dr. Burghardt of Kindberg says: “ In lower and central Styria there are a large number of glass works in which * “ Hidrach” is the corruption of “ Hiitten-rauch,” by which name ar- senious acid is generally known in mining districts in Germany. PRACTICE OF ARSENIC-EATING IN STYRIA. 213 arsenious acid is much used under the name of ‘ Hidrach,’ or ‘Hittenrauch’ I know that arsenic-eating is most common in those districts, because the arsenic can there most easily be obtained.” Dr. Forcher writes from Gratz: “As long as the cus- toms-frontier to Hungary existed, Hidrach was brought over as a much prized article by smugglers and others into the more remote valleys. The labourers in glass works obtained the arsenic from their master’s stores; the arse- nic works in Salzburg serve as a source of this substance for the neighbouring Alpine districts; in Orblau the red sulphide of arsenic, realgar, is obtained from roasting ar- senical copper ores; and the arsenic works in the Reiche- nau-thal connected with the Cobalt works at Schlogelmthl is another source whence arsenic could be obtained in quantity.” Dr. Schafer of Gratz attributes the large number of cases of poisoning which occur in Styria — for during the period of two years which he has acted as government Toxicologist in that district no less than thirteen cases of arsenical poisoning have come under his notice, — to the wide distribution which this poison enjoys in that country. Dr. Holler of Hartberg mentions the following purposes for which arsenic is employed in Styria. (1.) It is eaten by men. (2.) It is used in the preparation of Styrian cheese. (3.) It is used by pregnant women to procure abortion. (4.) It is given to horses, cattle, and pigs. (5.) It is used asa poison for rats and mice. (6.) It is used for soaking wheat. (7.) The charcoal and lime- burners, as well as tile makers and potters, throw arsenious acid upon their fires. Mr. Heisch in the paper published in the Pharmaceutical Journal gives the following interesting extract from a re- port by Professor Schallgruber of Gratz in the Medicinischer 214 PROFESSOR H. BE. ROSCOE ON THE ALLEGED Jahrbuch d. Gistr. Staates, 1822. The professor says : “The reason of the frequency of these sad cases (of poisoning) appears to me to be the familiarity with arsenic which exists in our country, particularly in the higher parts. There is hardly a district in upper Styria where you do not find arsenic at least in one house under the name of ‘Hidrach.’ They use it for the complaints of the domestic animals, to kill vermin, and as a stomachic to excite an appetite. I saw one peasant show another on the point of a knife how much arsenic he took daily, without which, he said, he could not live; the quantity I should estimate at two grains.” The sale of arsenious acid is strictly forbidden by law in Styria, and according to Dr. Holler two applications for permission to buy this substance were refused by the au- thorities. Hence it is of course somewhat difficult to get accurate accounts as to where the arsenic was obtained. The difficulty of procuring exact information respecting the arsenic-eating is also great, owing to the fact that the habit is carried on in secret and is generally disapproved of. IT shall now proceed to examine the question as to whether arsenic is or is not regularly taken by persons in Styria in quantities usually supposed sufficient to produce death. I. Cases coming under the personal observation of medical men. The most narrowly examined and therefore the most interesting and valuable record of a case of arsenic eating, is that described by Dr. Schafer in the paper above referred to. The case fell under the personal examination of Dr. Knappe of Oberzehring. In presence of this medical man. a wood-cutter thirty years of age, who stated that ke had taken arsenic for twelve years, and had always been in good health, did, on the 22nd of February 1860, eat a PRACTICE OF ARSENIC-EATING IN STYRIA. 215 piece of arsenious acid which weighed 43 grains, crushing it between his teeth, and swallowing it. On the 23rd February he in like manner eat a piece of arsenious acid which weighed 54 grains. During this period he consumed his food with his usual appetite, drank much spirituous liquors, and on the 24¢h went away in his ordinary state of health. He informed Dr. Knappe that he was in the habit of taking the above quantity of arsenic three or four times during the week. ‘The urine which this man passed on the 21s¢ instant, on which day he said that he likewise took a dose of arsenic, as well as that passed on the 22nd, was chemically examined by Dr. Schafer, and in both portions the presence of arsenic was incontestibly proved. ‘The details of the methods of analysis em- ployed showed that the examination was made with due care, and that every reliance may be placed in the results obtained. The following cases came under the personal observation of the reporting medical men; and although no chemical examination of the substance was made, and the weight of arsenic eaten was not determined, yet they appear of great interest, as helping to throw light upon the question. Absiract of statements made by the reporting medical men. (Copies of the original reports in the possession of the Society.) Dr. Holler of Hartberg makes the following remarks :— 1. “I once made an expedition, in company with several friends, to the Alps in the Ens Valley. We were accom- panied by two guides, and remained in a hut overnight. In the morning, on going to the spring to wash, I saw one of the guides carefully shake a white powder from a paper into his hand, and lick it off with his tongue. I asked him what the powder was; he answered that it was arsenious acid. On questioning him as to the reason of his taking 216 PROFESSOR H. E. ROSCOE ON THE ALLEGED arsenic, he stated that when he was forty-five years old he had been overseer of a chamois-hunt, and that in this capacity he had so over-exerted himself that he became very ill, and by the advice of an old hunter began to take arsenic. His health then improved; he was able to return to his duties; and he had continued to take regular doses of arsenious acid for ten years up to the date of my com- munication with him. I have every reason to believe his statement, as he appeared to be a respectable, truthful person, and only gave me this information upon my re- peated cross-examination. 2. Arsenic is used in Styria to flavour cheese, for the purpose, it is said, of giving it stomachie properties. I was once eye-witness of the preparation of such cheese. I happened to be in the Ensthal, in a hut where the dairy- woman was making cheese. A pot of curds was standing ready to be put into the wooden cheese moulds, wien the woman took out from a small box as much powdered white arsenic as covered the end of a table knife, and shook it into an earthen vessel, in which about a pint of water was boiling. She stirred the water with a stick, and after it had boiled for ten minutes she poured the liquid in a thin stream into the curds, so that it was equally distri- buted throughout the mass of about eight quarts. She then kneaded the curds about for a long time, and after- wards filled the wooden moulds. Upon my asking her what the powder was, she said it was “ Hiittenrauch” (arsenious acid), and that the Styrian cheese must be thus prepared in order that no accident may occur. She pointed out to me an inn in the valley where I could buy such cheese. There I went, and after much opposition on the part of the maid-servant, who was afraid that I should poison myself, obtained some of the cheese, but eat only such a quantity as the maid told me would be harmless, a piece the size.of two walnuts. I eat some bread and drank PRACTICE OF ARSENIC-EATING IN STYRIA. 217 some beer to the cheese, and experienced nothing but a slight burning in the throat as from food containing much spices, and afterwards a pleasant warmth in the stomach, and good appetite. I ate such cheese several times again without any evil effects manifesting themselves. Once, however, in the neighbourhood of Murnau, I ate some Styrian cheese, which on mastication appeared to contain sand; I left the remainder untouched, and although I had eaten a piece only as large as a nut, I soon became very unwell, vomited several times, and was seized with colic. This cheese no doubt contained arsenious acid in the form of powder.” Mr. Stern, surgeon, of Kundorf, reports as follows: — “About two years ago I was called in to see a strong field- labourer thirty-four years of age, who it was said had been suffering for twenty-four hours from severe colic. On examination I found that it was a case of violent gastro-enteritis, and as the patient could give no reason for his sudden attack, and all symptoms pointed to poison- ing by arsenic, I asked him if he had not eaten “ Hiitten- rauch,” upon which he answered exactly as follows: About seven years since he had suffered from indigestion and general debility, and being always unwell he had followed the advice of an old man, and had begun to eat arsenic. After this he had gradually improved in health, and had since then regularly taken arsenic, and had been perfectly well until yesterday, when he took rather a larger dose than usual, and was seized with violent pains, which he thought were colic. Upon administration of the usual remedies, especially the hydrated peroxide of iron, the patient recovered in a short time.” Dr. Kropsch of Leoben relates that he was once called in to prescribe for a peasant who had been induced to eat arsenic to improve his health. Symptoms of acute poison- ing set in, and the patient acknowledged that he had SER. III. VOL. I. FF 218 PROFESSOR H. E. ROSCOE ON THE ALLEGED eaten arsenic. Dr. Kropsch relates another case which came under his own observation of a felon upon whose person two pieces of arsenious acid, about two grains in weight, were found concealed. The prisoner begged hard that he might be allowed to keep these the last portions he possessed, and at last managed to swallow both pieces. He gave reasons for having contracted the habit of arsenic- eating, in which he had indulged for a length of time, and said that the quantity which he usually took at once was about two grains. II. Cases related to the medical men by trustworthy persons. The cases of arsenic-eating related by trustworthy per- sons to the medical men are of course by far the most numerous. ‘These cases, though not so decisive as those coming under the personal observation of the reporters, have a certain value as expressing the opinions and expe- rience of a large number of persons whose truthfulness there is no reason to doubt. Dr. Holler, in his report, adds a list of persons in his neighbourhood who have given him information concern- ing the practice of arsenic-eating. He says: “I have had many opportunities during the twenty-four years of my medical practice to collect observations upon this subject. To my own observations I have added those of other trust- worthy persons. As we have not here to support a theory but to prove a fact, which at first sight appears incredible, viz., that persons are able to eat arsenic for many years and yet remain strong and healthy, and that animals to whom this substance is given become fat and strong, I name the persons from whom I have the information in order that all doubt as to the truth of the statements may be cleared away.” Here follow the names and addresses of fifteen persons. Dr, Holler then continues: “I and PRACTICE OF ARSENIC-EATING IN STYRIA. 219 the above-mentioned persons guarantee that together we are acquainted with forty arsenic-eaters. They are all men from thirty to seventy years of age, by occupation charcoal-burners, stable-men, field-labourers, innkeepers and traders.” The following names of and particulars concerning ar- senic-eaters are collected by Dr. Forcher of Gratz. He does not say that he has personal knowledge of any of the cases; he gives them as facts, but only once or twice mentions his authority for the case. 1. Johann Wolch vulgo Gfoller in Mitterbach, took arsenious acid daily, beginning with pieces of the size of a pin’s head, and gradually increasing up to the size of a grain of oats. Died of the effects of the poison. An inquiry as to cause of death was held. Particulars concerning this will probably be found in the district offices of Knittelfeld or Juden- burg. 2. Peter Flockmayer of Mittenfeld, ate arsenic, which he also gave to his horses. Died of typhus. 3. Josef Hiebler of the 1s¢ Grenadier Company of the 27th Infantry Regiment, ate arsenious acid of the size of a grain of wheat. 4. Tuschler in Knittelfeld, took arsenic, and raising the dose, was seized with symptoms of poisoning. He recovered. 5. Maria Windisch, a cowherd in the district of. Obdach, was in the habit of eating arsenic, which she continued to her old age. The quantity taken at once was the size of a grain of wheat. She died in Obdach in 1858. 6. Kutscherwirth in Briick, takes arsenious acid, but conceals both the frequency and quantity of the dose. 7. Heinreich Kittner, Captain in the Engineers, sta- 220 PROFESSOR H. E. ROSCOE ON THE ALLEGED tioned in 1859 at Carlsburg, took a piece of arsenious acid as large as a grain of sand every evening. 8. Herr Wernberger, formerly residing in Tuffer, now probably living at Zilli, took arsenious acid for years. He used to scrape the quantity he required from a larger piece. He was of advanced age, and had always enjoyed good health. g. George Miinzer, now dead, worked in the iron mines in See-thal near Obdach; he took every week a piece of “ Hidrach” of the size of a grain of wheat. This information is derived from his fellow miners. 10. Mathias Sittler, always in good health, died in Obdach at the age of seventy-one; believed that he owed his good health to the habit of taking arsenic several times a week on bread. His son- in-law, Ottenig, blacksmith in Obdach, guarantees these facts. 11. Caspar Graf, pow in service at the forge in Obdach Le formerly ate a small portion of arsenious acid every day. During this period he never was unwell. The foregoing are the most important facts regarding the alleged practice of arsenic-eating in Styria which I have been able to collect. A great deal of other interest- ing matter will be found in the full reports presented to the Society. A most important case of administration of no less than 555 grains of arsenious acid to a horse in twenty-three days, without any evil effects being produced, is related by Dr. Schafer. I have, however, confined myself for the present to the discussion of the two points above mentioned, and I believe that the evidence brought forward is quite sufficient to justify the conclusions : PRACTICE OF ARSENIC-EATING IN STYRIA. rapa I. That arsenious acid is well known to and widely distributed amongst the peasants of Styria. II. That arsenious acid is taken regularly into the system, by certain persons in Styria, in quantities usually supposed sufficient to produce immediate death. 222 DR. R. ANGUS SMITH ON THE PRODUCTION XV.— On the Production and Prevention of Malaria. By Dr. R. Aneus Smurra, F.B.S. Read February 19th, 1861. 1. On the Production of Malaria. By malaria I mean simply bad air rising from land or water not contaminated with impure substances from the habitations of man. In this short paper I by no means propose to myself to enter upon the whole question of the production or prevention of malaria. The subject would require not only more time than can be given to it on this occasion, but much more information than I possess. It has been my intention to examine the matter tho- roughly, but I have not had the necessary opportunities, and I give here only some ideas towards an elucidation of the subject, founded on a few simple observations. It has been shown by numberless writers that malaria is a very frequent evil; that large portions of Europe are more or less subject to it, and considerable districts rendered by it quite uninhabitable. But besides those dreaded evils, such as almost instantaneous illness or slow and lingering death, Dr. Macculloch has shown that large portions of very healthy countries, such as England itself, are subject to emanations which diminish their healthiness to an extent which careful observation can detect, although the sufferers themselves are unable to account for the loss of strength, or look on it only as one of the inevitable evils incident to AND PREVENTION OF MALARIA. 223 humanity. Macculloch says: “I trust to prove in this essay, that the causes of malaria exist under numerous circumstances not at all suspected in our country, and in thousands, tens of thousands of places, even at our very doors” (p. 11, ed. 1827). Again: ‘ Malaria produces in itself a far wider mass of human misery than any other cause of disease; as, for the world at large, it is also the cause of far more than half the mortality of mankind” (p- 31). | In the examination of different districts for organic matter in the air, I have invariably found that low grounds well cultivated have shown more than high grounds or hills, and more than the open sea. I cannot, therefore, doubt that the remarks of Dr. Macculloch ave correct as to the very general occurrence of unwholesome influences caused by the state of the ground. I am inclined to go further, and to say that even in places where we cannot find that health is in any way injured, emanations arise. But it is not sufficient that health should be injured, or at least the average age reduced below the standard of the country, in order that a place may be called insalubrious. It may be that the impulse given to life may be dimi- nished in some districts by substances rising from the ground, whilst in other districts an entire or greater freedom from all noxious exhalations may allow the supply of the purest air, with ali its highest quickening effects. The canse of malaria has been with great certainty traced to the soil. I shall not revert in this place to the arguments and observations which have led to this conclu- sion. It is true that some places are exposed to emana- tions of gases from subterranean reservoirs, but gases such as can be prepared by the chemist, combinations of inor- ganic bodies, are not known to be able to produce the agues and peculiar fevers which we connect with the true malarious district. Atmospheric phenomena, the condition 224 DR. R. ANGUS SMITH ON THE PRODUCTION of electricity for example, affect the health, and so does the amount of water, of cloud and of light; but all these causes seem to be different from that insidious poison arising from those organised bodies found neither deep in the earth nor high in the air, but only on the surface of the ground. It is now many years since I first began to study these subjects; and one or two of the facts which I allude to, and which I consider important, were published fourteen years ago; but their significance has not till lately been sufficiently clear. I observed that the water of soils is sometimes acid and sometimes alkaline; that on the sur- face generally of our soils it is acid to a small extent, whilst below, when the inorganic salts overpower it, it becomes alkaline. We may say as a rule that all inorganic soils give off their drainage water in an alkaline state. The tendency of plants is to render the soil acid, and if they render it very acid the soil is said to be sour. Drainage will draw off the excess of vegetable juices, and lime will also remove them, as well as other free alkalies. In hot-houses and where rich manures are used, the soil is not always acid, but frequently very alkaline. I have not examined a great many, but where forcing is resorted to with strong manures the result must be and is an alka- line condition. This condition depends on temperature. Having observed some very alkaline soils, I went a few days afterwards for a specimen of water flowing from the surface, without allowing it to pass deep into the strata which would have filtered out its organic matter, and given it alkaline earths. Meantime the weather had changed, and the alkaline water had disappeared, having given place to one that was acid. This was in a peaty district. If the soil of our fields were not kept acid, but were kept alkaline and warm, there would arise from it a AND PREVENTION OF MALARIA. 225 greater amount of exhalations. “If the soil is very alka- line and moist, the conversion of the organic matter into ammoniacal compounds is very rapid. If put some soil not very rich in organic matter into this condition by the assistance of a little ammonia, so as to make it alkaline, and the consequence was the rapid occurrence of a very intense putrefactive decomposition, not in any way differ- ing, as far as could be perceived, from that of ordinary putrefaction of animal and vegetable matter. These nau- seous and unwholesome odours are therefore possible from the ordinary soil of our fields; but any occurrence such as this on a large scale would be disastrous, and the ground is protected from it by an almost constant acidity, which sometimes increases so as to be injurious, forming what is called sour land. This very acid state generally occurs in wet land, where it is probable that alkalinity would be most injurious, but the soil may be found alkaline in a well-manured garden, and where the ground is dry, without apparent imjury.” — Author’s Report on Air and Water of Towns, British Association 1851, p. 69. In this experiment I produced in fact artificial malaria. The conditions were exactly such as the best observers have in most cases described, with the addition of the alkaline state, which will probably be found to play a very important part, but which as far as I know has not been attended to. Not that I say that it is impossible to have malaria without ammonia, but ammonia is the wing on which some of the products of decomposition seem to delight to fly. This experiment, which brings the usual condition of warmth and moisture, shows that there may be many modes by which the organic matter in land may pass from the soil besides the transformation into plants, and besides simple oxidation. The decomposition spoken of shows that from soil numerous products may arise. If we allow that putrefaction begins we know no end to the SER. III. VOL. I. GG 226 DR. R. ANGUS SMITH ON THE PRODUCTION number of substances produced. These products will vary according to the nature of the substances in the soil, according to the season when one or other substance predominates, and when one or other substance acts as a ferment. It is not necessary to go further than this experiment in order to show that there may arise from soil under certain conditions substances producing the most deso- lating diseases. I suppose this artificial state to be the extreme, such as probably has never happened in an exten- sive district. All other conditions of insalubrity are pro- bably stages of this. It may be said that it was well known that soil retained organic matter, and that organic matter could putrefy, but this state of actual decomposition had never before, as far as I know, been observed. By the sense of smell we have found reason to believe that volatile products arise; disease has also led to this indication, and minute quantities have even by careful experiment been observed; but in the experiment I speak of, the amount rising was gross and unquestionable. I will scarcely ven- ture to speak of the quality of these products. To me they strongly resembled those of putrid flesh or blood, and in that case their compcsition would probably ally itself to that of protein; indeed I find that after removing the purely gaseous bodies arising from such decomposition there is left a substance whose carbon and nitrogen bear a relation almost identical with that m protein. But the variety of substances found will probably prove to be extremely great. We can imagine some washed down by the rain, and some removed by the plant. In the same way some classes of malarious diseases may be removed by rain, some caused by more and some by less moisture and heat, and others better removed by the agency of plants and the processes of agriculture. The soil is an extremely complex machine; the balance AND PREVENTION OF MALARIA. 227 of forces in it has not been studied. The surface is acid, the lower part is alkaline; it is a filter, with a power of retention and selection. The flow of air and water is from the surface downwards. The surface therefore becomes first oxidised, and is made acid. This acidity is itself a preventative of great putrefaction, and this oxidation is a destruction of those substances which may have resulted from the rotting masses. When land is well drained these processes will go on with greater rapidity; it is as if the lungs were enlarged. But notwithstanding this oxidation and prevention of decomposition we must not forget that it is useful to counteract another agency of great importance, that by which the plant rots. This rotting is accompanied by grubs, insects, animalcules, and vapours and gases; it closely resembles, and no doubt in part is a suppressed putrefaction. The soil has the power of retaining the products to a great extent and until it has destroyed them, when they are not excessive. If the passage of air through a soil is not sufficient the products of decomposition are not destroyed. If the passage of water through a soil be not sufficient the soluble products are not removed into the soil, but left, and as- sisted to putrefy. It is certainly an object of great interest to be abie to know by the examination of a soil if it could or would produce malaria, or if it were really in the act of producing it. By the view taken most soils, indeed all containing putrescent organic matter, will be subject to malaria when the oxidising influence of the air and the balance of vege- table and chemical life are disturbed. The soils most productive of malaria are low grounds, and moist ones with abundant vegetation. At the same time there are others not described as such, and not very thoroughly investi- gated. If we examine the surface water flowing from such grounds in this country we shall find it to contain animal 228 DR. R. ANGUS SMITH ON THE PRODUCTION or animalcular life in immense variety, and we shall also find that if we allow it to stand vegetable life will show itself in great abundance. ‘The hilly districts of our own island do not contain water of this kind. I was extremely sur- prised, on taking my microscope to the north-west of Scot- land, to find that the ditches were so meagre of living crea- tures. A green pool was scarce, and when found it was small, whilst on examination it was vivified by very few living forms. The whole hills in fact were either void of matter capable of putrefaction, or they were so rapidly washed that the putrefying matter was passed at once into the soil. The moisture, too, produces no corruption ; it is water in rapid motion filled with air, and rapidly traversing the soil, oxidising all the small portions of putrid matter that may be forming in the soil, and fixing the products in the earth. The animal life is repressed, 7.¢. the life of such living things as arise from decomposed plants or animals. But why should this be? Let us push the argument further. If grass grows in such places there must be focd for plants, and elements for the production of infusoria. It is true that grass grows, but not in great vigour, and as the rain passes very rapidly through the soil there must be very little matter kept soluble at one time. The plants must be fed by small although by very frequent. meals. Besides, the soil is not deep; the power of retaining soluble matter is therefore small; the stock of food laid up over the whole district, in other words the material from which such animal life must spring, is ex- tremely scanty. Now I am not prepared to say that all unhealthy districts actually contain a large amount of matter either putrefying or forming into animalcular life, but it certainly is the characteristic of what seems to me to be the soil most blamed; there may be some, as already stated, where other causes arise. This state of soil is that of all our lower lands; the ditch water swarms with life; AND PREVENTION OF MALARIA. 229 muddy wet lands present it in every drop of water; whilst in our high hills, purified by barrenness and disinfecting peat, we have hunger and health. I do not say that there is a necessary connection between malaria and ani- malcules, but as there is a connection between them and the state of the air in our own county, it is probable that the inquiry may be extended profitably to malarious districts. Infusoria and even larger animals indicate the presence of matter which may putrefy, but it is even probable that they may prevent its injurious action on the atmosphere. The microscope then may very soon be brought forward as an instrument for ascertaining the sanitary condition of a country. It is extremely probable that the quality of the decompositions will be known by the microscope to a large extent, and the nature of the disease and its extent be indicated. It was not my intention to have brought forward these views until | had examined in detail this part of the subject, but I fear I shall not soon have an opportunity of doing so, although I have more materials ready than I can find room for in this paper, which is chiefly a practical one. I may readily be misunderstood. I do not say that where most animalcales are there is most disease. Water may exist too putrid for them, and some states of decomposition destroy them. It may be said that it was well known that places con- taining less organic matter were less liable to disease, and that the sea coast and barren hills have long on that account been frequented; but to me at least the reasoning generally and the mode of observing is new, and I feel as if new eyes were given us for observing the soil more minutely. To some extent we can at once tell the quality of waters by the use of the microscope, and have even a clue to the gases and vapours rising. If these waters are taken from the surface of land we have a clue at once to _ the condition of the land. 230 DR. R. ANGUS SMITH ON THE PRODUCTION 2, Prevention of Malaria. If we can imitate the production of malaria, can we not also imitate the mode by which it is destroyed or prevented ? When the products of decomposition are formed in a soil they are removed by natural processes. Mere mixture with the soil will remove or render decomposing matter innocent. The soil will act as a porous body. But the soil may be overburdened. It may be shallow, and its machinery may be of small force; or it may be inefficient from the excess of organic matter over the amount of air passed through it. The first act of disinfection is the action of the soil as a porous body. The next seems to be the act of oxidation by which the soil at the surface is rendered acid. By these means decompositions are confined within the soil itself. We may consider the atmosphere to be in a constant struggle with the vegetable matter of the soil. Substances containing nitrogen are constantly tending to give out ammonia, the ground is constantly tending to convert this into nitric acid, whilst it converts other portions into car- bonic and organic acids. By manuring we assist the tendency to become alkaline, taking the side of vegetation, aud we resist the oxidising tendency of the air. A large portion of our manuring is simply the addition of alkalies, a struggle against the acidifying influence of the air. No wonder it requires so much lime for the fertilizing of acid peat lands. From lands of this kind it is probable that no miasma arises. Moisture lying on rich lands becomes filled with animal and vegetable life, a sure sign of rapid decomposition. Allow this moisture to pass through the soil and this animal life disappears. This is an act of purification by drainage. It may not at all times be possible to obtain the requisite amount of drainage, and there are even cases where the land produces malaria, without, according to the AND PREVENTION OF MALARIA. P43 | greatest authorities, showing need of drainage. We can readily imagine one condition which will produce this result, viz. excessive dew, which will evaporate and carry a large amount of organic matter ;* others may depend on the plants and animals present. The oxidation in the soil may also be aided by the artificial opening of its structure, but this may not always be convenient, and the demand for it may be too laborious. Are we not able to imitate some of the other methods taken by nature to prevent infection ? If we add any acid to the putrid soil of which I spoke, that peculiar mode of decomposition ceases ; if we add any other disinfectants or antiseptics we put a stop to decom- position. If we add antiseptics to the water over soil containing a great variety of animalcules, and giving every evidence of decomposition, we find these instantly dying ; the same, of course, results with grubs, larve, &c. Animal life is arrested, and chemical action is staid or im- peded according to the amount used; but a very large amount must be taken in order to show an injurious effect on vegetation. In this experiment we do in reality pre- vent malaria; we arrest the decomposition of substances in the soil. We cannot call this a theory or a speculation; it may be considered simply as a fact. The use of anti- septics will arrest all animal and vegetable decomposition. and where there is neither of these malaria will not arise. The chemical action which goes on in a vessel containing a few cubic inches will not differ if extended over a surface of miles. I look on the results, therefore, as certain, viz: that by the use of disinfectants malaria will be destroyed. The idea of the disinfection of whole districts rose out of a proposal made long ago to disinfect whole cities by beginning at the root of the evil, the sewers. That was * Since writing this an Italian friend informs me that this opinion is prevalent amongst the scientific men of his country. 232 DR. KR. ANGUS SMITH ON THE PRODUCTION intended for this country, the land of great cities. This plan for the disinfection of districts is intended chiefly to apply to other countries. I may, however, be allowed to give one illustration from this country. Mr. McDougall has used sewage over one hundred acres of land; sheep and cattle have fed on it withont any case of disease. The growth of vegetation was great, the moisture was constant ; there was also the constant presence of decomposed matter, but there was no disease. There was, on the contrary, a large production of healthy vegetable and animal life. This experiment suggests many questions, but I do not intend to enter on any except so far as to show it as a proof that putrefactive decomposition may be arrested without any fear of destroying the life of plants, and that rich water meadows may be used for feeding cattle without fear of rot, even if treated with sewage. The disinfectant used was crude carbolic acid with lime water.* The arrest of decomposition was thus caused by a process which at the same time diminished the acidity. I think the amount required will be small enough to render the process pos- sible over large districts. At the same time also I expect that the idea will lead to other and perhaps even to cheaper modes of producing a like result. As it is to be hoped that an opportunity of using it will occur, it is well not to attempt to speak of details. I may add also at the same time that the extinction of insects obnoxious to a country, — gnats, mosquitoes, and those dreadful enemies of cattle the zimb and the tsetse, — will probably follow the destruction of the obnoxious de- compositions in the soil. Nor do I suppose that only one disinfectant will do the work, although the one I * The value of carbolic acid has elsewhere been shewn by Mr. Mc Dougall and myself; the preparation here used was that patented by Mr. Mc Dou- gall in 1859. He will be able soon to make public many other experiments relating to rot and the destruction of insects. AND PREVENTION OF MALARIA. 233 have mentioned is, as far as I know, the most powerful, and can be made in all countries and climates where coal hes or where trees grow. It is desired to make Rome the capital of Italy. From all I hear it is badly fitted for such from its sanitary con- dition. By the method proposed I believe objection on the plea of health would be removed, and it would rapidly be made fit for habitation. I wish in this paper chiefly to show that decomposition, to a most pernicious extent, is possible in soils; that it is not a mere opinion, but a fact readily demonstrated. Next, to show that decomposition may be arrested artificially to the preservation of health without the destruction of vegetation; and that in these facts we have not only a surer basis in our reasonings on the origin of malaria, but an almost certain process for its ultimate and total destruction.* * Since writing the above I have formed artificial malaria in soil by the use of water only, an exactly analogous case to that of marshes. I call it artificial malaria, although, of course, I have not produced fever or ague with it. Similar soil, when disinfected, evidently resists animal and chemi- cal decomposition longer, yielding at once to the wants of vegetation. It is said that cultivation will cure malaria, but malarious ground is dangerous to touch; our engineers are dying rapidly whilst making the railways in India. The soil and the jungle could both readily be treated before turning them up, or as soon as turned up. I may add that the road to and from India might be made more wholesome by the destruction of the corruption in the bilge water of ships; nothing can be easier, and yet men die on ac- count of this corruption, and people have inquests and commissions upon them instead of commanding its cure. SER. III. VOL. I. HE 234: XVI. — On the Kaloscope. By Mr. W. H. Heys. Communicated by Mr. Grorar Mostery. Read February 5th, 1861. Ons of the objections frequently advanced against the use of the microscope is that it injures the eyes. All observers are aware that they are constantly using a light which is at least unpleasant from its brightness, if not positively hurtful; and attempts have long been made so to modify the light that the observer may work on through a long evening without pain or fatigue. It is hoped that the “ kaloscope” will meet the wants of all those who use the microscope frequently, enabling them to pursue their investigations with the long-desired comfort, and induce many persons to commence micro- scopic studies who hitherto have been deterred through fear that injury to the sight would be inevitable. The kaloscope consists of eight discs of coloured glass, the discs having a uniform diameter of 24 inches. They are mounted upon a stand of 12 inches in height, similar to that used for a bull’s eye condenser, but larger, and are arranged in two sets of four glasses each, an upper set and a lower set, each set being attached to an arm which can be moved about in any direction; and as every separate disc has its own independent motion, it will be obvious that any angle of light may be obtained. The upper set of glasses is fitted to a tube, which slides upon the upper portion of the pillar of the instrument so as to allow of MR. W. H. HEYS ON THE KALOSCOPE. 205 their being moved about without disturbing the position of any of the glasses belonging to the lower set which may happen to be in use at the same time. The kaloscope can be used for a very considerable variety of objects; and a great number, which do not polarize, are made to disclose all the beauties of polarized objects. For instance, suppose the anthers of the mallow, with their pollen, to be the object under view; we place the slide upon the stage, arrange it as a transparent object, and also throw condensed light upon the upper surface ; then placing the kaloscope in position, or so as to allow of a red glass belonging to the lower set being interposed between the lamp and the mirror, the ground upon which the object is seen is coloured crimson. By next adjusting a green disc belonging to the upper set (green being the complementary colour of red) so that its tint shall be thrown through the bull’s eye, a beautiful green hue is cast upon the object, illuminating it in a perfectly novel and most valuable manner, the anthers and pollen appear- ing intensely green, while the ground beneath them is crimson. These colours may be changed for others at the pleasure of the observer. It will soon be discovered by any one using the kalo- scope that nearly all objects viewed by it appear so much in relief that we might suppose them to be viewed through a stereoscope. For observing anthers, jointed hairs, oil glands and vegetable sections in general, but especially sections of wood, this instrument will be found, accordingly, of singular use. Some objects, moreover, which before were attractive only to the scientific botanist, are trans- formed into pictures of such beauty that they cannot fail to rank in future with the most popular and esteemed microscopic preparations. ‘The calyx of the moss-rose is a good example of both these results. Under the ordinary modes of illumination it is a mere entanglement of coarse 236 MR. W. H. HEYS ON THE KALOSCOPE. fibres, with dark beads sprinkled here and there. Viewed by the kaloscope it is instantly transformed into a most lovely stereoscopic vegetable branch, while the glands at the extremities of the twigs glitter like diamonds. Sections of wood, spines of echini, and similar objects, may be viewed in two entirely distinct ways. By the plan already described for the mallow they will be found as beautiful as when examined with the polariscope. By the other arrangement, now to be described, the details are brought out in a wonderful mauner, and better than by any other mode of illumination. The slide being placed upon the stage, the most oblique light possible must be thrown up through it from the mirror. The kaloscope is then placed between the lamp and the mirror in such a position that the object is fringed by the coloured light, while the ground remains intensely black. This method of illumina- tion may be used with great advantage in examining the hairs upon the edges of leaves and petals, and those also which often spring from the filaments of stamens. The following plants supply remarkably beautiful objects of the class in question : Tradescantia Virginica (stamens). Lamium maculatum (stamens). Achimenes (petals). Hypericum pulchrum. Lysimachia vulgaris. Stellaria media. ‘Salix (male flowers). Salvia splendens (hairs on corolla). Asperula odorata (spines on fruit). Narthecium Ossifragum (stamens). © It may be added that it is in many cases sufficient to use a glass of the lower set alone. The pollen of the Malvacex, for example, may be shown with a blue ground from the kaloscope, while the surface MR. W. H. HEYS ON THE KALOSCOPE. 237 is illuminated in the ordinary way with the bull’s eye. It then forms a highly attractive and pleasing object. It is thought that the efficacy of this instrument in connexion with such a variety of purposes cannot fail to render it of much value to the scientific observer. It most certainly enables him to detect details of structure which are inconspicuous without its aid. By the exquisite beauty with which it paints many kinds of objects it will also allure the beginner to continue his researches at a time when probably he might abandon the pursuit as tedious. To all who have refrained from using the microscope through fear of hurting their eyes, it can further be recom- mended as calculated to remove all anxiety ; for when the blue glass of the lower set is used, even if no other end be gained, the light is delightfully toned down, and in a way that can only be appreciated by those who have worked at the microscope for many hours at once. 238 MR. T. HEELIS ON THE TEMPERATURE XVII. — On Meteorological Observations, and Observations of the Temperature of the Atlantic Ocean, made on runs from Liverpool to Gibraltar, and from Gibraliar to Liverpool, in September 1860. By Tuos. Hees, F.R.A.S. Read before the Physical and Mathematical Section, January 31st, 1861. THe runs during which these observations were taken were made in large iron screw merchant steamers, the property of Messrs. Meadows and Bibby. The run out was made in the ship “ Crimean” of (I think) 1006 tons register. The weather during the run was fine and moderate, with smooth water almost the whole way, the little swell that occurred being met with in the southerly part of the Bay of Biscay. The winds along the coast of Portugal were light at 8S. and 8.S.W., with smooth water. The passage home, on the contrary, was stormy and boisterous, the ship (“The Rhone” of 964 tons register) having left Gibraltar in a gale of wind, and having met with bad weather, with a heavy westerly and north- westerly swell, caused (I think) by the approach of the hurricane which ravaged the north of Scotland and the Baltic on the second of October and following days, of which more hereafter. The observations of the air and surface water were taken with a thermometer supplied by Mr. Dancer, and the requisite corrections to reduce them to the readings of OF THE ATLANTIC OCEAN. 239 the standard thermometer at Kew have been applied. Unluckily soon after entering the Bay of Biscay on the return voyage, and when the observations (which were undertaken chiefly for the purpose of ascertaining whether the system of currents known to exist in that Bay could be ascertained by the temperature of the water) were be- coming most interesting, the thermometer was broken, and the return series is thus incomplete. On the run out the gradual deepening of the water is the first thing which tells notably on the thermometer. Indeed the delicacy of the action of the instrument at depths of from 80 to 100 fathoms and upwards is sur- prising. On referring to the table of observations it will be found that the thermometer, showing the temperature of the sea water, fell on the 11¢2 September from 65'1 to 64, although the ship was standing to the southward at the time, and no land was in sight. On examining the chart it was found that the depth had decreased from 130 fathoms to 80. Hence the fall. I cannot account for the two observations made at noon and at g p.m. on the 9th September being identical, espe- cially as the temperature of the air was not the same in both cases, and the second observation was taken nearly ninety miles 8.W. by S. from the first, except by sup- posing the ship to have passed twice through a cold current at two points of its course. The influence of the shoaling of the water is clearly shown in the first observation on the 12th September, which was taken inside the Burlings, and in which the temperature of the water had fallen from 65'1, as shown by the last previous observation, to 60°2. The mixing of the waters of the Tagus with those of the ocean is clearly shown by the fallin temperature, the temperature off the mouth being on the run out 62°525. This brings us directly to a difficulty raised by the 24.0 MR. T. HEELIS ON THE TEMPERATURE second series of observations, which were taken by the writer with the same, or if possible, greater care than those taken on the run out. On the run home, in passing the mouth of the Tagus, the ship was further in the offing than in the run out. Hence it might reasonably have been expected, as only twelve days had elapsed since the former observation, that the temperature of the water would have been higher. It is, however, shown to be lower, the temperatures of the air being 61°11 to 65°35, and those of the water being 60125 to 62°525. The differences of temperature in the two sets of observations are very striking, and show how impossible it is to arrive at any accurate determina- tions without long continued observations. A few words as to the mode in which the observations were made may be desirable. On the assumption that the vessel would not exceed a speed of ten knots an hour, which under steam and sail was considered good speed for the “Crimean,” it was decided to take the observations as far as possible every three hours, which would give one to each half degree ; but the observations were not continued during the whole night, as no practical experience in reading thermometers was possessed by the officers of the ship. The thermo- meter was generally hung to the belaying pins, and under the lee of the mainmast, for the purpose of obtaining the temperature of the air. For the water it was sometimes towed over the taffrail, and at other times immersed in a bucket of sea water drawn for the purpose. I found no difference in the readings obtained by these two methods. The barometer readings were taken from an aneroid which was carefully compared before and after the seri¢s with a standard barometer. The following are the observations : OF THE ATLANTIC OCEAN. 24:1 D Tempera- |) Tempera- Oren, | RECT | ESSE bnarometer. corrected. | corrected. 1860, Sept. 8 61°6 | 572 30°270 | 53°7 N. 4°4n’ W. 58°15 | 58°40 30°237 | Off Smalls. SVMs Su 582 30°200 60'2 57°8 atk 58°6 | 59°15 | 30°145 | 49°27’ N. 6°51’ W. 59°85 | 59°90 | 30°100 : 6o'r 00% Gor | 59°15 | 30°037 10 64°1 | 61'9 29°880 64°2 | 62°2 29°890 | 45°29’ N. 8°0’ W. 65°1 | 62°25 | 297940 Longitude, Latitude, and Remarks. 63 63°1 60 62°6 | 6371 30°000 II 64 65 30°075 G7ar NGiser 30°067 | 41°52’ N. 9°26’ W. 67°r | 64 30°065 64°1 | 65°35 | 30°095 62°85 | 65°1 30°100 12 61°1 60.2 500 65°35 | 62°525 | 30°180 | Inside Burlings. 70°07 | 66:25 | 30°190 | Off Tacus. on 38°12’ N. 9°24’ W. 64°7 64°3 30°200 64°35 | 65°6 39-235 13 66:2 68°15 ot 68°: | 69°1 30°200 i 67°1 | 69°1 30-155 | 5°49’ N. 6°9’ W. The table of observations made on the passage home reads upwards from the bottom, to allow of the more easy comparison of the observations with the preceding ones. The following are the observations : Tempera- ture of Tempera- ture of Air Water corrected. | corrected. Date of Longitude, Latitude, Observation. Barometer. and Remarks. 582 61°620 | 29°490 | 44°2’ N. 9°23’ W. 1860, Sept. 27 | 60°125 | 59°35 | 29°500 60°125 | 62°2 29°820 60'125 | 60°130 | 29°890 64-3 - | 6371 29°940 64°5 641 30'040 | 40°45’ N. 9°27’ W. 65°1 63°1 30°100 61°11 60°125 | 30'170 | Off Tagus. 63°1 61°35 | 30.140 62°4 64°1 30°130 60°2 63'1 30°140 | 37°32’ N. 9°4’ W. 60'125 | 62°1 30°50 61.325 | 67°1 30'020 65°1 67°3 30°020 64°1 ace 30°000 Noon 24 | 62°5 66°9 30°020 | 35°56’ N. 6°42’ W. SER. III. VOL. I. Il 242 MR. T. HEELIS ON THE TEMPERATURE, ETC. From the 27th September the barometer gradually rose, attaining on the morning on the 1st October at 9 a.m. a maximum height of 30°420 inches, from which point it fell under the influence of the hurricane which followed. So early as the morning of Sunday the 30th September the hurricane, which ravaged the north of Scotland and the Baltic on the 2nd October and following days, was visible off Scilly as a low bank of grey cloud gradually rising up in the west and north-west. The sun went down behind this, producing an intense scarlet and very wild effects. On the 1st October a red sky was noted at sunrise, and the same eloud grew and grew until by the time the ship reached Holyhead it had covered the whole sky, and was flecked over with scud, or what sailors call “water casks.” On the evening of this day there was also a wild and red sunset. The barometer had begun to give way as early as noon on the 1s¢ October, and yet the ship- ping on the west of Scotland were, to some extent at least, caught by surprise. We had experienced north-east winds all across the Bay, and along part of the coast. These were, I have no doubt, the drafts setting down to fill up part of the com- parative vacuum caused by the progress of the storm. XVIII.— On Changes of Density which take place in Rolled Copper by Hammering and Annealing. By Mr. Cuanrtes O’ NEILL. Read March sth, 1861. A GENTLEMAN engaged in the application of copper to a certain purpose in the arts believed that it was possible to compress or condense the best commercial rolled copper so as to increase its density and improve it for the pur- poses he had in view, which I may state were in connexion with engraving or etching upon the metal. He devised or applied a powerful machine for the purpose of testing this idea, by means of which a great power could be brought to bear upon the copper by a succession of blows acting through a regulated space on the principle of the genou. The results were not satisfactory, and I was applied to in order to ascertain what was the actual increase of density obtained by a moderate amount of hammering. The first experiments I made did not show that there was any in- erease of density, but pointed in the opposite direction, intimating a loss of density. Nobody was inclined to accept this result, but further experiment confirmed it, and then a regular series of experiments was made to test the point, which resulted in a further confirmation of the earlier experiments. It was afterwards found that anneal- ing the hammered pieces of copper increased their density, bringing them almost exactly to the same density they possessed before hammering. I proceed to give details of several of the experiments. 944. MR, CHARLES O'NEILL ON CHANGES OF DENSITY First Series. I had ten pieces of copper cut from different parts of the same sheet, which was represented as being best rolled copper. The thickness was about 3%, inch, and the pieces weighed from 252 to 320 grains, each being rather more than an inch square. I cleaned them with dilute nitric acid and brickdust, so as to get a clean surface, and then took the density of each piece by a balance which would indicate clearly the rooth of a grain, and all the weigh- ings were made to that fineness. I obtained as follows: Moser asc 8884 NowiGrecss 8°881 Dig uae 8°385 7 8°880 Baa. 8°884 Sic 8°372 Ate ih 8°375 9.82255 8°870 Ree gead 8°884 TOMS osans 8874 Mean density ............ $879 The pieces were then separately hammered in the genou, each receiving abont fifty blows; the blows passed through about ;’— inch, and the pieces being at liberty to extend, they did extend perhaps one-half, but without any cracks or fissures. They were perceptibly warmed by the hammering, though the massive cast iron bed did not allow them to get very hot. After the hammering they were carefully cleaned, and their density again ascertained with the following results : NOM 8°850 No. 6 , 8-866 oA An 8856 a Sere 8°85 5 een $351 al 8°854 Area $°856 Qaeinns. 8855 Bivaesece 8°848 TOm seats 8°863 Mean density ............ 8°855 Loss on density ......... ©'024 The pieces were now annealed by putting them in red hot sand, and letting them cool slowly. They were then cleaned carefully and their density taken as follows: INOsteses ee 8°882 INOssG2 2533.3 8°886 7h) atest 8°885 G eerste 8°384 eA HRHNA 8-883 Soran 8-885 7. ae 8-382 Oped. $°385 Sie ctsinaals $°883 Owes 8'885 IN ROLLED COPPER BY HAMMERING, BTC. Mean density of rolled copper ..........ccccesseveeees 8°879 Mean density of hammered copper .............e0008 $°855 Mean density of annealed copper ..........2 sceseeeee 8°334 The densities of the annealed specimens agree very well, the greatest divergence from the mean being ‘002, but in both the rolled and hammered pieces the error is much greater. But the plus and minus errors nearly balance each other in these cases, and do not, I think, depend upon the manipulation but upon actual differences in the copper. If it is so, it is interesting to note that the ham- mering and annealing have reduced the pieces to a nearly uniform density. I got another piece of sheet copper, thicker than the former one, and said to be of a better quality; and in fact a very close examination of the surface showed that it was free from the minute perforations and crevices which could be perceived in the first sample experimented upon: I had ten pieces cut from the sheet, weighing each from 420 to 520 grains. Their densities were as follows: SIRS eice 8°899 INO Gays es 8°897 Di sanbe $900 F) cdqnoo | OSL BuBnAbaE 8-gor Sete $°897 Aninectete sts $°398 Ou seein 8°899 5 noose $°397 TOW sae $°398 Mean density ............ 8°898 Being hammered the same as before, the densities found Geisaaee 8°379 INO ROMA. 8°882 Bienes 8°879 Fhveitents $°879 Burs 8°876 Ske: 8°876 Ae rset: 8°380 Olas 8°377 G hesdoan 8°377 EOu seats 8°375 Mean density ............ $°378 They were then made red hot, and allowed to cool slowly; carefully cleaned from adhering oxide, and the density of five of the pieces found as follows : INOS UG od nals crane 8890 Dilton aaanen smatuiare tat sees 8°906 § cennoadcacadscdnoang0o0u000009 8-899 His aR cee esice sie escistenister 8°892 Olas eseeena nese nen eecue recs 8°893 Mieanidensity...\.cssesscesseses S896: 946 MR. CHARLES O’NEILL ON CHANGES OF DENSITY Mean density of rolled copper ..............essceee ee 8°898 Mean density when-hammered..................s.0008 8°878 Mean density when annealed ................65 sees 8896 The other five were lost or not weighed. I made an experiment to try if this loss of density was progressive, and depended upon the amount of hammering. I took a piece of square bar of copper, 1'95 inches long and o'5 inch square, and hammered it myself upon an anvil, taking its density in all eight times, each successive hammering being more severe than the previous one. I tabulate them as follows : Sp. gr. of bar copper (weight 1097°35 grs.) ...... 8885 Mammeredislightlya ai acisciecdeccessrscecseecccsseeres 3°887 2nd Hammering, length 2°10 inch ............se0e0e 8°834 3rd Hammering, length 2°20 inch ..............0.0. 8°882 4th Hammering, length 2°50 inch..........sceceee ees 8°876 5th Hammering, length 3°00 inch.,..............00c 00 8°871 6th Hammering, length 3°50 inch..............050.0.+ 8°869 7th Hammering, length 4°50 inch..............cc000es 8°867 8th Hammering, length 5°50 inch............00.2ee00s 3°367 Tt lost five grains in weight during the successive ham- merings, and was several times made very hot. The last hammering cracked the edges, and the recorded density may not be correct on account of the difficulty of taking it. The difference in density was oo18. In the two first series of experiments, by omitting the third decimal when under five and adding one to the second when above five, we get results perfectly alike in each determination of the second series upon which I place the most confidence ;* and the effect of hammering and annealing may be stated as follows: . 2nd Series. st Series. Rolled copper has a density of...... 8:90 8°88 Wihenchammercdst..scccc.cs esas. 8°88 8°86 Hammered and annealed............ 8*90 8°88 The decrease of density, then, is about 1 in 450; that is, the copper has increased in bulk that much. The cubical expansion of copper by heat is about ssd¢o for a * But in fact the weighings justify the three places of decimals. IN ROLLED COPFER BY HAMMERING, ETC. 247 degree of Fahrenheit; an increase of heat of 78 degrees would expand the copper about 745. This increase of heat I ascertained to be produced. If there be not a connexion between the heat produced and the decrease of density I can conceive no cause. But to attribute it to this cause would presuppose a permanent retention of the expanded. state, of which I know no instance. The particular brittle- ness of hammered copper seems to show that the molecules are in quite a different state to those in annealed copper. It was to draw attention to this singular loss and recovery of density, that I have submitted these few facts to the Society. 948 MR. A. CAYLEY ON THE A FACED POLYACRONS IN XIX. — On the A faced Polyacrons, in reference to the Problem of the Enumeration of Polyhedra. By Mr. A. CayLey. Read October 2nd, 1860. Tue problem of the enumeration of polyhedra* is one of extreme difficulty, and I am not aware that it has been discussed elsewhere than in Mr. Kirkman’s valuable series of papers on this subject in the Memoirs of the Society and in the Philosophical Transactions. A case of the general problem is that of the enumeration of the poly- hedra with trihedral summits; and Mr. Kirkman in the earliest of his papers, viz. that “On the representation and enumeration of polyhedra,” (Memoirs, vol. xii. pp. 47-70, 1854,) has in fact, by an examination of the par- ticular case, accomplished the enumeration of the octa- hedra with trihedral summits. A subsequent paper “On the enumeration of x-edra having triedral summits and an (v-1)gonal base,” (Phil. Trans. vol. xlvi. pp. 399-411, 1856,) relates, as the title shows, only to a special case of the problem of the polyhedra with trihedral summits, and in this particular case the number of polyhedra is more completely determined; but the later memoirs re- late to the problem in all its generality, and the above- * Tuse with Mr. Kirkman the expression “ enumeration of polyhedra” to designate the general problem, but I consider that the problem is to find the different polyhedra rather than to count them, and I consequently take the word enumeration in the popular rather than the mathematical sense. REFERENCE TO THE ENUMERATION OF POLYHEDRA. 249 mentioned particular problem of the enumeration of the polyhedra with trihedral summits is not, I think, any where resumed. Instead of the polyhedra with trihedral summits, it is really the same thing, but it is rather more convenient to consider the polyacrons with triangular faces, or as these may for shortness be called, the A faced polyacrons ; and it is intended in the present paper to give a method for the derivation of the A faced poly- acrons of a given number of summits from those of the next inferior number of summits, and to exemplify it by finding, in an orderly manner, the A faced polyacrons up to the octacrons: thus, as regards the examples, stopping at the same point as Mr. Kirkman, for although perfectly practicable it would be very tedious to carry them further, and there would be no commensurate advantage in doing so. The epithet A faced will be omitted in the sequel, but it is to be understood throughout that I am speaking of such polyacrons only; and I shall for convenience use the epi- thets tripleural, tetrapleural, &c. to denote summits with three, four, &c. edges through them. The number of edges at a summit is of course equal to the number of faces, but it is the edges rather than the faces which have to be con- sidered. An n-acron has nm Summits, 3n—6 edges, 2n-4 faces, and it is easy to see that there are the following three cases only, viz. : 1. The polyacron has at least one tripleural summit. 2. The polyacron, having no tripleural summit, has at least one tetrapleural summit. 3. The*polyacron, having no tripleural or tetrapleural summit, has at least twelve pentipleural summits. In fact, if the polyacron has c tripleural summits, d tetrapleural summits, e pentipleural summits, and so on, then we have SER. III. VOL. I. KK 250 MR. A. CAYLEY ON THE A FACED POLYACRONS IN n=c+d+e+ft+gth+f&e. 6n—12=3¢+4d+4+5e+ Of+ 794+ 8h+ &e., and therefore 12=3c+2d+e+o0f-g-2h-&e., or 3ce+2d+e=12+94+2h+ Se. ; whence if c=o and d=o, then e=12 at least. It appears, moreover (since 2 cannot be less than e), that any poly- acron with less than 12 summits cannot belong to the third class, and must therefore belong to the first or the second class. An (n+ 1)-acron, by a process which I call the subtraction of a summit, may be reduced to an n-acron; viz., the faces about any summit of the (n+ 1)-acron stand upon a poly- gon (not im general a plane figure) which may be called the basic polygon, and when the summit with the faces and edges belonging to it is removed, the basic polygon, if a triangle, will be a face of the n-acron ; if not a triangle, it can be partitioned into tri- angles which will be faces of the n-acron. The an- nexed figures exhibit the Siwy a process for the cases of Ne wy a tripleural, tetrapleural and pentipleural summit respectively, which are the only cases which need be cousidered ; these may be called the first, se- cond and third process respectively. It is proper to remark that for the same removed summit the first process can be performed in one REFERENCE TO THE ENUMERATION OF POLYHEDRA. 25] way only, the second process in two ways, the third in five ways; these being in fact the numbers of ways of partitioning the basic polygon. We may in like manner, by the converse process of the addition of a summit, convert an -acron into an (7+ 1)- acron; viz., it is only necessary to take on the n-acron a polygon of any number of sides, and make this the basic polygon of the new summit of the (n+1)-acron, and for this purpose to remove the faces within the polygon and substitute for them a set of triangular faces standing on the sides of the polygon and meeting in the new summit: the same figures exhibit the process for the cases of a tri- pleural, tetrapleural and pentipleural summit respectively, which (as for the subtractions) are the only cases which need be considered. It may be noticed that for the same basic polygon the process is in each case a unique one; the process is said to be the first, second, or third process, according as the new summit is tripleural, tetrapleural, or pentipleural. Now, reverting to the before-mentioned division of the polyacrons into three classes, an (7+ 1)-acron of the first class may by the first process of subtraction be reduced to an n-acron, and conversely it can be by the first process of addition derived from an n-acron. An (n+ 1)-acron of the second class, as having a tetrapleural summit, may by the second process of subtraction be reduced to an n-acron, and conversely it can be by the second process of addition derived from an n-acron. And in like manner, an (n+ 1)- acron of the third class, as having a pentipleural summit, may be by the third process of subtraction reduced to an n-acron, and couversely it may be by the third process of addition derived from an -acron. Hence all the (n+ 1)-acrons can be by the first, second and third processes of addition respectively derived from the n-acrons. It is to be observed that all the (n+1)- 252 MR. A. CAYLEY ON THE A FACED POLYACRONS IN acrons of the first class are obtained by the first process ; the second process is only required for finding the (m+ 1)- acrons of the second class; and these being all obtained by means of it, the third process is only required for find- ing the (n+ 1)-acrons of the third class. Hence the second process need only be made use of when the n-acron has no tripleural summit, or when it has only one tripleural sum- mit, or when, having two tripleural summits, they are the opposite summits of two adjacent faces. In the last-men- tioned two cases respectively it is only necessary to con- sider the basic quadrangles which pass through the single tripleural summit and the basic quadrangle which passes through the two tripleural summits; for with any other basic quadrangle the derived (n+ 1)-acron would retain a tripleural summit, and would consequently be of the first class. The condition is more simply expressed as follows, viz.: The second process need only be employed when there is on the n-acron a basic 3 quadrangle the summits of which are at least of the number of edges shown in the annexed figure, and all the other summits are at least 4-pleural. Again, by the third process (as already mentioned) we seek only to obtain the (7+ 1)-acrons of the third class; the process need only be applied to the n-acrons for 6 which there exists a basic pentagon the VW summits of which are at least of the 4 i he fh. 3 number of edges shown in the annexed figure, all the other summits being at least 5-pleural; for it is only in this a 2 case that the derived (n+1)-acron will be of the third class. The condition just referred to obviously implies that the n-acron is of the second or third class. It is to be noticed that in applying the foregoing principles REFERENCE TO THE ENUMERATION OF POLYHEDRA. 253 to the formation of the polyacrons as far as the 11- acrons we are only concerned with the first and second processes. Consider the entire series of n-acrons, say A, B, C, &c., and suppose that the n-acron A gives rise to a certain number, say P, Q, R, S of (n+1)-acrons, the (n+ 1)- acron P is of course derivable from the n-acron A, but it may be derivable from other n-acrons, suppose from the n-acrons B and C. Then in considering the (m+ 1)-acrons derived from B, one of these will of course be found to be the (n+1)-acron P, and it is only the remaining (n+ 1)- acrons derived from B which are or may be (n+ 1)-acrons not already previously obtained as (n+ 1)-acrons derived from A. And if in this manner, as soon as each (n+ 1)- acron is obtained, we apply to it the process of subtraction so as to ascertain the entire series of n-acrons from which it is derivable, and, in forming the (n+ 1)-acrons derived from these, take account of the (n+ 1)-acrons already pre- viously obtained and found to be derivable from these, we should obtain without any repetitions the entire series of the (n+ 1)-acrons. For merely finding the number of the (n+ 1)-acrons, a more simple process might be adopted: say that an n-acron is p-wise generating when it gives rise to p (n+ 1)-acrons, and that it is g-wise generable when it can be derived from ¢ (n+1)-acrons; and assume that a given n-acron is (4. +%+Y3+ &c.)-wise generating, viz. that it gives rise to y,(m+1)-acrons which are I-wise generable, y, (n+1)- acrons which are 2-wise generable, and so.on; these form- ing the sum = (Jit BYatsyst - +) where > refers to the entire series of the v-acrons, it is clear that every m-wise generable (n+1)-acron will in respect of each of the n-acrons from which it is derivable be reckoned as 3, that is, it will be in the entire sum 954 MR. A. CAYLEY ON THE A FACED POLYACRONS IN reckoned as 1, and the sum in question will consequently be the number of the (7+ 1)-acrons. The figures of the polyacrons comprised in the annexed Tables show the application of the method to the genesis of the polyacrons as far as the octacrons, in which the numbers indicate the nature of the different summits, according to the number of edges through cach summit, , 3a tripleural summit, 4 a tetrapleural summit, and soon. It will be noticed that there is only a single case in which this notation is insufficient to distinguish the polyacron, viz. among the octacrons there are two forms each of them with the same symbol 33445566; the inspec- tion of the figures shows at once that these are wholly distinct forms, for in the first of them, viz. that derived from 3344555, each of the tripleural summits stands upon a basic triangle 456, while in the other of them, that from 3444555, each of the tripleural summits stands upon a basic triangle 566. But the symbol is merely generic, and of course in the polyacrons of a greater number of summits it may very well happen that a considerable number of polyacrons are comprised in the same genus. The following remarks on the derivation of the octa- crons from the heptacrons will further illustrate the method : 1. The heptacron 3335556 has three kinds of faces, viz. 355," 356, 555, the first process consequently gives rise to 3 octacrons. As the heptacron has more than two tripleural summits the second process is not applicable. 2. The heptacron 3344466 has three kinds of faces, z.: 366, 346 and 446, and the first process gives * Tt is hardly necessary to remark that it must not be imagined that in general all the faces denoted by a symbol such as 355 (which determines only the nature of the summits on the face) are faces of the same kind, lout this is so in the cases referred to in the text. REFERENCE TO THE ENUMERATION OF POLYHEDRA. 255 therefore 3 octacrons. The heptacron has only two tripleural summits, and they are disposed in the proper manner; the second process gives therefore I octacron. 3. The heptacron 3344556 has five kinds of faces, viz. 345, 346, 356, 456 and 455, and the first process consequently gives 5 octacrons. The heptacron has two tripleural summits, but they are not disposed in such manner as to render the second process appli- cable. 4. The heptacron 3444555 has four kinds of faces, viz. 355, 455, 445 and 444, and the first process gives therefore 4 octacrons. The heptacron has one tri- pleural summit, and the basic quadrangles 3545 which belong to it are of the same kind; the second process gives therefore 1 octacron. 5. The heptacron 4444455 has only one kind of face, viz. 445, and the first process gives therefore 1 octa- cron. There are two kinds of basic quadrangles, viz. 4545 and 4445, and the second process gives therefore 2 octacrons. The number of octacrons would thus be 20, but by passing back from the octacrons to the heptacrons, it is found that there are in fact only 14 octacrons. Thus the octacron 33336666 has only one kind of tripleural summit 666 (the summit is here indicated by the symbol of the basic polygon) and the octacron is thus seen to be deriv- able from a single heptacron only, viz. the heptacron 3335550 from which it was in fact derived. But the octa- cron 33345567 has three kinds of tripleural summits, viz. 567, 557 and 467, and it is consequently derivable from three heptacrons, viz. the heptacrons 3335556, 3344466 and 3344555, and so on. The passage to the heptacrons from an octacron with one or more tripleural summits is of course always by the first process, but for the last two 256 MR. A. CAYLEY ON THE A FACED POLYACRONS, ETC. octacrons, which have no tripleural summits, the passage back to the heptacrons is by the second process: thus for the octacron 44445555 we have but one kind of tetra- pleural summit 4555; but as opposite pairs of summits of the basic quadrangle are of different kinds, viz. 45 and 55, we obtain two heptacrons, viz. 3444555 and 4444455. The octacron 44444466 has but one kind of tetrapleural summit, viz. 4646, and the pairs of opposite summits of the basic quadrangle being of the same kind 46, we obtain from it only the heptacron 4444455. It may be remarked that for the five heptacrons respec- tively the values of the sum y,+4y,+4y3;+--- are 1+44+4, b414+44+4, 445434141, 14+14+14+14+4, 14+4+5, giving for &(y,+4y.+4y3+ +--+) the value 14, as it should do. 207 XX.— On a System of Periodic Disturbances of Atmo- spheric Pressure in Europe and Northern Asia. By Josera Baxenve xt, F.R.A.S. Read November 13th, 1860. In the course of an investigation of the phenomena of the general disturbances of the atmosphere which I made some time ago, I was led to conclude that moderately accurate determinations of the sums of the irregular oscillations of the barometer for given periods, at different places on the surface of the earth, would afford valuable information respecting the nature of these disturbances, and at the same time throw additional light upon the causes by which they are produced. It is obvious that even the most accurate determinations of the statical element of mean pressure can be of only very limited use in an inquiry of this kind; but notwithstanding the im- portance of the subject, meteorologists have generally neglected to ascertain even approximately the values of the dynamical element as represented by the extent and frequency of the oscillations of the mercurial column. In none of the many volumes of observations which issue from the public observatories of this country and the con- tinent have I yet seen any attempt made to deduce the values of this element. In the Greenwich, and also in some of the Oxford volumes, tables of the principal max- imum and minimum readings of the barometer are given, SER. III. VOL. I. LL 258 MR. J. BAXENDELL ON PERIODIC DISTURBANCES which it might be expected would be useful for deter- mining readily the monthly and annual sums of the oscillations; but the systems on which these tables are formed at the two observatories are so widely different that while the sums of the oscillations for the years 1856 and 1857 by the Oxford tables amount to 70°95 inches and 70°62 inches, by the Greenwich tables they are only 58°22 inches and 56°22 inches respectively, the difference in the one year being 12°73 inches, and in the other 14°40 inches. In the Oxford volumes it appears from the head- ing of the tables that all oscillations above one-tenth of an inch are included; but in the Greenwich volumes it is not stated what limit has been adopted, and as the Greenwich barometric observations are not published in detail, it is impossible to ascertain with accuracy the maximum extent of the oscillations which have been excluded; and the results derived from the tables are therefore not available for comparison with those derived from observations made at other stations. In the present state of practical meteorology accurate values of the barometric dynamical element can be ob- tained for those places only where hourly or bi-hourly observations are made. In by far the greater number of eases observations are made only twice, or at most four times a day; and the hours of observation are generally such as to render it difficult to separate readily the regular diurnal oscillations from the irregular and larger fluctua- tions which we are now considering. Under these circum- stances I have thought it desirable to confine my attention, in the first instance at least, to oscillations derived from observations made once a day only. By this plan the regular diurnal oscillations are completely eliminated, the results obtained are more uniform, and better adapted for comparison with each other ; and a greater number of sets of observations become available for the purposes of our OF ATMOSPHERIC PRESSURE IN EUROPE, ETC. 259 inquiry. I have accordingly employed this method in the discussion of a very considerable number of observations made at various places in Europe and Asia, and have obtained results which appear to me to possess considera- ble interest, as indicating very decidedly the existence of a remarkable and hitherto unsuspected law of disturbance of atmospheric pressure, which extends its influence over the greater portion of the European and a considerable portion of the Asiatic continent. The following table contains the mean monthly and annual sums of the oscillations of the barometer at seven stations in Europe and six in Asia, as derived from obser- vations extending over periods varying from six to fifteen years. The European stations are Dublin, Sandwick in Orkney, Greenwich, Milan, Stockholm, St. Petersburgh, and Lougan, 48°35’N. 39°20’E. The Asiatic stations are Tiflis, 40°42’N. 44°50’ E.; Catherinbourg, 56°49’ N. 60°35’ E.; Barnaoul, 53°20’ N. 83°57’ H.; Irkoutzk, 52°17’ N. 103°35'H.; Pekin, 39°54’N. 116°27’E.; and Nert- chinsk, 51°19’ N. 119°36’ EK. It must, however, be ob- served, with reference to the results given for Greenwich, that as the individual observations of each day at Green- wich are not given in the published volumes, these results are derived from the daily means, and not from single daily observations as in all the other cases: MR. J. BAXENDELL ON PERIODIC DISTURBANCES 260 ‘DIssny ap oujuag anbishygq as10jvatasqg 7 ap saypuupy “Ex pue zx ‘11 ‘or “6 ‘g ‘L “9 ‘poyp “ysuajay nysuaay yhuoy fo phisaaafo °S aMZDbo Ty "2Yq “Qupy puv puoy *z “OpLpqueo'Ty opngysuy "aT gap ayousoug “b *[PUnysogy "suounalasgg yoinuaaty *£ ‘hyawogy uyqng qwhoy ayz fo joummop pure huapnop ysry yohoy fo shurpaov01g *1 R0:8v=| ES.Vy | -Ggusel St.19-) SOLS ry.ce |? 16.Ly" 12.99 |o-E8iSo \e.6g.E4. 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LS. bz gz.b Lo.v Tea LoS 1g. £0.9 00,$ tS.9 Zo.Si 30.9 SP. 09.9 pa Se “LS81 “9581 "6—9981 0} 0} pue St81 LST 6—£F81 ‘savaA OL | ‘sieof OT | ‘sxead TT “YOU ‘yorm | ‘urtqnd -us019 -pues *¢ OFA 7 PaopoUPeas eye Iaqmaday ***L9QULOAO AT “*** 1940909, ** aquiaydag “Casnsny pelea ATM (2 Heenan ep Oe iy Av See ler ady: See Ole PL £renaqoyg res KTen Ue pe ‘DISH UAIYJAONT U2 xIS pun agoAnA ut SUOI{DIS UIIS JV ANIMOLDG IYJ fO SUOYYYIISE ays JO SUNS JOnUUP puw Ajyzuopy Unepy ayy fo NqUT OF ATMOSPHERIC PRESSURE IN EUROPE, ETC. 261 The curves laid down from the numbers in this table are exhibited in the plates which accompany this paper, and a slight inspection will show that in all of them a principal minimum occurs in one of the three summer months, June, July, or August, but generally in July, the warmest month of the year; and in many of them there is a second minimum occurring in one of the two winter months, January or February. With respect to the two maxima which occur between these minima, it will readily be ob- served that the interval between their summits gradually increases as we advance from the eastern to the western stations. Thus, at Nertchinsk, the first maximum occurs in the middle of April, and the second about the second week in November, the interval being nearly seven months , at Irkoutzk the first maximum occurs at the end of March or beginning of April, and the second maximum in the middle of November, the interval being seven and a half months; at Barnaoul the first maximum takes place in the middle of March, and the second in the middle of November, the interval being eight months; at Catherin- bourg the first maximum occurs in the second week of March, and the second about the second week of Decem- ber, the interval being nine months; at Tiflis, a station between the Black Sea and the Caspian, and at Lougan in southern European Russia, the interval is also about nine months; at St. Petersburgh the first maximum occurs in the third week of February, and the second in the second week of December, the interval being thus about nine and a half months; at Stockholm the first maximum occurs in the middle of February, and the second in the middle of December, the interval is therefore ten months; at Milan the interval is also about fen months; but at the western stations, Sandwick in Orkney, Greenwich and Dublin, we have only one principal maximum, which occurs about the second week of January, and which appears to be formed 262 = MR. J. BAXENDELL ON PERIODIC DISTURBANCES by the union of the first maximum of one year with the second maximum of the year preceding, the interval be- tween the two maxima being ¢welve months. It is evident, therefore, that these maxima move across the two conti- nents in opposite directions, the course of the first being from west to east, and that of the second from east to west. If these two maxima were produced by imdependent causes it might be expected that their joint action in the month of January would produce a compound maximum of much greater elevation than either of the separate maxima; but as this is not the case it appears to me very probable that both maxima are produced by the same dis- turbing cause, such disturbing cause taking its rise im Eastern Asia in the month of November, and gradually moving westward until it arrives in the British Islands in January; then, reversing its course, it returns with a diminished velocity to the region of its origin, where it arrives in the month of April, and afterwards rapidly dis- appears under the inftuence of an increasing temperature, to appear again later in the year on the return of a low temperature. The middle latitude of the area over which these peri- odical disturbances of atmospheric pressure take place is about 51°, and the difference of longitude of the extreme east and west stations being about 126°, it follows that the length of this area is more than four thousand six hundred geographical miles; and as the difference of latitude be- tween the extreme north and south stations in Eastern Europe and Western Asia is more than 19°, its breadth must considerably exceed eleven hundred miles. With reference to the probable nature and origin of the disturbing cause it may be remarked that the times of its first appearance and final disappearance in Hastern and Central Asia correspond very nearly with the times of the OF ATMOSPHERIC PRESSURE IN EUROPE, ETC. 263 breaking up of the periodical trade winds or monsoons in the China Sea and Indian Ocean. It is probable, there- fore, that the two systems of phenomena are directly con- nected with and dependent upon each other; and as the changes of the monsoons are known to be sometimes very regular it will become an interesting subject of inquiry to ascertain whether corresponding irregularities occur in the disturbances of the atmosphere in the middle and higher latitudes of Europe and Asia. Although probably not bearing directly upon the subject of this paper, | may perhaps be allowed to draw attention to a very decided convewxity of nearly all the curves in the month of October, in two or three cases producing a small maximum. Indications of a similar irregularity in the month of March are also apparent in some of the curves. These features clearly indicate the operation of a secondary disturbing cause acting durimg those months, but more especially during October, over the whole breadth of the two continents, and therefore probably differing alto- . gether in its nature and origin from the disturbing cause which produces the two great movable maxima. It may be well to state that the mean monthly values given in the table have not been corrected for the dif- ferences in the lengths of the months; but corrected values would not materially affect the general results, and it was not considered necessary in a preliminary inquiry to adopt months of equal length, or to calculate the mean daily amount of oscillation for each month. In concluding this brief account of results which seem to me to indicate clearly the existence of a regular law of atmospheric disturbance in regions of the earth which have hitherto been generally regarded as remarkable for the apparent irregularity of their atmospherical phenomena, I cannot omit to mention the fact that the first application of the method of reduction which has led me to these 264 MR. J. BAXENDELL ON PERIODIC DISTURBANCES, ETC. results appears to be due to Dr. Dalton, who, at page 16 of the second edition of his Meteorological Essays, gives a little table entitled “A table of the mean spaces described by the mercury each month, determined by summing up the several small spaces ascended and descended ; also the mean number of changes from ascent to descent, and the contrary, each month, it being reckoned a change when the space described is upwards of ‘03 of an inch. The means are for five years, at Kendal and Keswick.” There is no remark appended to this table, but in another part of the work the author refers to it in support of his theory of the variation of the barometer as showing that the greatest disturbance of atmospheric pressure takes place in the coldest period of the year, and having apparently satisfied himself on this point it would seem that he never afterwards resumed the subject, and it has since remained almost entirely neglected by meteorologists. The two sets of numbers in Dr. Dalton’s table indicate a maximum of disturbance in the first week of January, thus agreeing very closely with the results which I have obtained for Dublin, Sandwick, and Greenwich. XXI.— On the Irregular Oscillations of the Barometer at Manchester. By G. V. Vernon, F.R.A.S. Read April 2nd, 1861. A paper having recently been read before the Society by Mr. Baxendell, upon the values of the mean monthly and annual amounts of the irregular oscillations of the baro- meter, at various localities upon the globe, it was thought that similar deductions from observations made at Man- chester might be acceptable as additional data upon the subject. All the observations given in this paper were made with standard instruments. The barometer readings used were made daily at eight a.m. In Table I. are given the total monthly amounts of the oscillations and their number. The number of oscillations in each month may be slightly in error, but never to the extent of an entire oscillation: this arises from oscillations beginning in one month and ending in another. As the number of oscilla- tions could not be deduced for the entire year, it was not thought necessary to estimate this number more closely. Table II. contains the mean daily oscillations for each month, and has been formed by dividing the figures in Table I. by the number of days in each month. The maximum amount of oscillation appears te take place in January and the minimum in July, to which may SER. III. VOL, I. MM 266 MR. VERNON ON THE IRREGULAR OSCILLATIONS be added a second maximum which appears to take place in October. Some observations, kindly furnished me by Mr. John Curtis, give the following values for the month of August during the last six years: Amount of Oscillations. No. 1855 mat 0°146 Na 16 1856 ae 0°126 er 18 1857 Be 0°095 ns It 1858 os o°102 ap 15 1859 ane O'1IS ait 12 1860 ies o'159 see 16 Means... o'124 Bao 8 A) The mean daily oscillations, and their number, are then as follows: Mean daily Amount. Inch nches. No. January 466 0°226 Pt 14°9 February ae 0°193 a 14°6 March Hee org Be 14°6 April ba 0'166 ae 14°6 May ie 0°135 ue I4‘l June ae O°r31 we 14°2 July He O'12I ets 15°3 August : ne or124. ve 14°7 September ae Ossi ae 13°9 October aS O°193 ABE TiS November ai O°1g2 me 13°2 December ae 0°206 Sie 5c Mean for the year 0168 14°63 Total for the year 61°41 175°5 The August values being from six years’ observations only, I have not tabulated them in Tables I. and II., as additional years might alter the values to some extent. In the diagram, I have added the curve for Dublin from the reductions given by Colonel Sir Henry James in the volume of observations made at the Ordnance Office, Dublin. The features of the two curves are very similar. The Dublin curve, however, does not show so low a mini- mum as the Manchester one; it gives a somewhat higher maximum, and also higher values, in the months of Feb- ruary and October. The two curves being for different od OF THE BAROMETER AT MANCHESTER. 267 periods, of course somewhat interferes with the exact com- parison. Table III. contains the fall of rain for each month com- pared with my own average for twelve years and Dr. Dalton’s average for forty-seven years. Table IV. contains the mean monthly temperatures and their difference from the twelve years’ average. It is much to be regretted that we have no trustworthy observations of the temperature for as long a period as that of the fall of rain, as differences from a longer average would have been of more value for this part of the investigation. Having obtained the values of the oscillations, it was thought desirable to determine what connection there might be between the amount of oscillation and the fall of rain; also the relation, if any, existing between the mean temperature and the amount of oscillation. In Table V. the falls of rain above the average are com- pared with the corresponding amounts of oscillation, and the falls of rain below the average with their corresponding oscillations. ‘This table shows that a fall of rain in excess of the average, is attended by a considerable increase in the amount of the oscillations in every month but October ; and it is very probable that this month would agree with the others, if taken from a longer series of observations. In Table VI. similar data are given for the mean tem- peratures of each month, according as they are above or below the average. — In this case we find that in February, April, May, June, July, October and December, a temperature below the mean for the month increases the amount of the oscilla- tions, whilst in the remaining months of January, March, August, September and November the converse would appear to hold good. In Table VII. the numbers of oscillations above or below the average are compared with the corresponding falls of 968 MR. VERNON ON THE IRREGULAR OSCILLATIONS rain in precisely the same manner as in the two preceding tables. We find from this table, that a number of oscillations above the average is accompanied by a larger fall of rain than that which accompanies a number of oscillations below the average in the months of January, February, March, June, October, November and December. In the remaining months of the year, April, May, July and Sep- tember, a number of oscillations in excess of the average is accompanied by a diminished fall of rain. Leaving out of consideration the month of June as being perhaps abnormal (observations for this month in 1851, 1857, 1859 and 1860 being also wanting), it would appear that we have two different laws, one for the six winter months and another for the six summer months, — one of these laws being the direct converse of the other. The mean for the eleven months gives the following : Number of Oscillations Fall of Rain. above or below the average. Inches. + 1°96 sie 2°509 — 2°02 ese 2°359 This seems to show that on the mean of the entire year, the number of oscillations does not seem to affect the fall of rain, but only appears to do so when the separate months or seasons of the year are taken separately. The apparent or real relation which appears to exist amongst these various data, seems well worth further in- vestigation with the aid of a longer series of observations. OF THE BAROMETER AT MANCHESTER. 269 TABLE I.— Monthly Amounts and Numbers of the Irregular Oscillations of the Barometer at Manchester, from Observa- tions made daily at 8h. a.m. by G.V. Vernon, F-R.AS. January. | Oscilltns February. . |Oscilitns - |Oscilltns D April. Oscilltns 6 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 Means Year 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 Means Year 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 Inches. Lose 6°965 10°96 6122 6°275 4°036 1249 8°495 5325 4°832 g'I4I Inches. 6-322 4°584 7320 5°677 5818 4142 3°871 4°040 3712 5°938 8°62z0 Inches, 57402 4°479 5715 3°991 Sree 57517 8*020 3°282 7°613 6°164. 8-186 6°995 6'993 May. 57518 June, Inches. 4°856 6°537 Incom 3°497 4°986 4°896 4°923 4°512 Ge2212 5°544 5°336 4°626 51929 July. 4°994 August. Oscilltns Inches. 4°727 3°94 3°885 3°764 4°423 4°9 II 4°729 5°478 3°381 No. 14 14 Il 15 It 14 19 12 13 Oscilltns . |Oscilltns} No. Oscilltns| No, Inches. 3°621 4°983 Incom 4/040 2°838 37369 4°823 4°868 Incom'plete} 2°806 | 15 2°551 18 Incom plete Incom|plete} Incom|plete Inches. ghar ean || a0) 5°407 | 17 3089 | 16 4°446 | 15 3°103 | 16 4°063 | 14 3°804 | 13 Incom plete} 4°265 | 16 47119 | 12 2°637 | 18 3179 | 14 Incom|plete 4-175 September. 3°925 October. 3°749 November. December. Oscilltns No. Oscilltns| No, Oscilltns . |Oscilltns) No, | Inches. 5'O1l 37593 4°108 4°939 4°365 37386 3°805 5°290 II II 14 13 10 15 18 14 Inches. 5°856| 16 7°333 | 18 5°913 | 16 Incom|plete) 5°550 | 13 6°234 5°292 4°907 15 17 15 6°116 | 19 5°961 | 19 Means 4°530 5°985 | 7°351 | 16 6°535 | 16 57006 | 12 Incom|plete 4°439 | 14 5°776 | 14 , 4°160 Inches. 6°349 7°708 4°909 6°105 4°398 $°083 4°568 4°882 4°247 5°174 4°388 Inches. U2 ZALES 5061 | 11 4°210| 14 | 8°572 | 22 5°270| 16 87040 | 21 | 6:014 $°128 4°750 5°765 7°407 6°143 5°456 6°390 270 MR. VERNON ON THE IRREGULAR OSCILLATIONS TABLE I1.— Monthly Mean Daily Amount of Oscillation of the Barometer at Manchester. Year January. | February. | March. April. May. | June. Inches. | Inches. Inches. Inches. Inches. | Inches. 1849 iS eeraan| eee O'174 o'162 O'152 | O'121 1850 0'243 0'226 o'144 0'218 0'126 0166 1851 0°225 0163 0184 ae O'125 auc 1852 0°354 0'252 O°129 O'I17 | O'I21 O°135 1853 0197 | 0°203 0°187 o'166 | o'143 0096 ~ 1854 0°202 0°208 0178 o0°163 | o158 O'II2 1855 0°130 o'169 0'259 o'164 | O'153 o'161 1856 0°2 34 0°133 o°106 O150 | O'177 0°163 1357 0°274 O'144 0'246 O°174 | O*109 we 1853 O'Ir7I O'121 o°199 OPTS hh elie oses 0'093 1859 o'156 0212 0'264 0178 | 0'082 Ei 1860 6295) 1) 0:297).| 2O:22004|) O20 5 4) yee Means 0226 0°193 O'IgI 0166 | 0135 O'131 | | Mean No. of | Oscillations , 4-6 6 ‘% : : fonea sh 14°9 14 14 esesb Ur 14'2 month. | ( Year July. August. |September.} October. | November.! December. Inches. Inches. Inches. Inches. Inches. Inches. 1849 sa i 0°167 0'237 O'212 0'236 1850 O'IOI i O'II7 oO'211 0'°257 0163 1851 O'174 res 0°137 or61 ||" ‘07164: |: -o136 1852 o*100 sis o'165 o'189 | 0'204 0'°277 1853 0'143 Bee 0146 oan O°147 o'170 1854 O*100 ee O'IT3 0°237 0269 | 0'259 1855 O°131 ae O°127 Org 0°139 o°194 1856 0°123 ve 0176 O°143 O'152 0'262 1857 Ops Sais. By ©7179) |) JOt1637 || oss, 1858 0°133 es O°174 0186 O'142 0186 1859 0'085 a o°176 0°197 O°172 0°239 1860 O'103 or o'164 O192 | 0°163 o'198 Means or2r |. 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VERNON ON THE IRREGULAR OSCILLATIONS TABLE IV.— Mean Monthly Temperatures at Manchester, and Differences from the Means. March, April. May. Mean Diffne.|Mean|Diffne.|Mean Diffnc.|Mean|Difinc.|Mean|Difinc.|Mean Diffac. ce] ° ° ° January. February. June. cesypuiees Ieee [ecsee, |4200) mg l4 acd “gle 3°32 te 5 5 Gea anlee 1850|34°4) — 3°9/44°1| + 6'3/40°8) ~— 0°5/48°7| + 2°1/50°5) — 1°3/58°6 +11 [1851/41°4| + 3°1/39°4| + 1°6|42°5) +12] ... |... [S0°7;—1'F] ... eee 1852|40°0) + 1°7|40°0| + 2°2/41°1 — 0°2|48-0| + 1°4/52°7| + 0°9/58'6 + 1°1 | 185 3/40°6) + 2°3/32°8) — 5°0/38°2! — 3°1/45°6] — 1°0|51°6| —0°2/58°6) + 1°1 1854)/37°7| — 0°6|39"0| + 1°2/44°0) + 2°7/49°4] + 2°8152°3] + 0°5/57°1| — O74 f 1855/36°7| — 1°6/28°6| — 9°2/38°2) — 3°1|46°1| — 0°5148°5| — 3°3157°1| —0'4 1 1856/37.7|— 0°6/41°1| + 3°3/41°1| — 0°2/47°2] + 0°6|50°2| —0'°6155°7} — 1°8 | 1857/36°5| — 1°8/39°3| + 1°5|41°8) + 0°5|46°4) — 0°2/52°7| + o-g/61°2| + 3°7 | 1858)37°9) — 0°4|35°3] — 2°5]40°6) — 0°7/46"4] — 0'2]49°9| — 2°1]63°3) + 5°8 | 1859/41°1| + 2°8]4r-1| + 3°3/44°8) + 3°5145°2| — 1°4]54°3) + 2°5|60°2) + 2°7; 1860/37°7| — 0°6/34'5| — 3°3/39°8| — 1°5|44°4| — 2°2/54°5| + 2°7/55°2) — 2°3 Mns|38°3 37°3| 41°3| 46°6 58 57°5 Year July. August. September. | October. | November. | December. Mean Diffnc.| Mean Diffne.|Mean/Difme. Mean|piffne. Mean piffne.|Mean Diffne. 1849|59°8| —077] ... |... [56°12] +.0°9/48°1} — 0°4/43°9) + 2°7|/38°6 —0°3 | 1850|60°4| — ovr] ... |... [53°9)— 1°3/45°6| — 2°9/44°9) + 3°7/39°5| + 0°6 1 1851|/58°2] —2°3| ... |... 15326} — 1°6|50°5) + 2°0/35°6| — 5°6|40-7) + 1°8 }1852|67-9/+7°4| ... |... |§4°3{ —0°9/44°6) — 3°9/45°0 + 3°845°2 +6°3) 11853/58°6)—1'o] ...] ... [54:1] - rr] ...) ... [40°7/ +0°5 35 2h oul 1 1854/60°3|—0°'2] ... |... [5'7°2] + 2°0|4:7°0] — 1°5/40°3| — 0°9/39°8| + 0'9} | 1855/61°8) + 1°3) ... |... [5479] -0°3148°5| 0'0/41°5) + 0°3/35°6| - 3°3I 1 1856|58°7| —1°8] ... | ... |54°2| — 1°0/51°8) + 3°3/39°2] — 2°0138°5| —0°4 | 1857|60°6! + o-162°9) ... |57°7| + 2°5/52°O) + 3°5/43°8| + 2°6/45°2) + 6°3) § 1858/57°9|-—2°6/61°7) ... |59°2| +.4°0|48-2| — 0°3/39°2/ — 2°0|40°8) + 1°94 | 1859|64°3| + 3°38] ... |... |55°2| o°0/48°5| 0°0/39°6| — 1°6/34°1| — 4°3] } 1860|57°6/—2°9] ... |... 151771 — 3'5| +0°'1/40°0| — 1°2/34°5|— 4°43 60°5 Rise 41°2| TABLE V.— Excess and Deficit of Rain compared with the Amount of the Oscillations of the Barometer. Excess of | Amountof | Deficit of | Amount of Month. Rain. Oscillations.. Rain. — |Oscillations. + Inches Inches — Inches Inches January...... 1136 7°887 ogi 4 4°727 February ...| 0.579 5°737 1°292 5°239 Marchi inne 1081 6°965 0'929 5°607 Aprile ces 0°539 5°346 0893 4°501 May. ites secs 0°435 5°478 o'919 3°943 DUNE Fasc ueeeee I°194 37885 1°127 3.676 Daly se sa|) L080 4044 | 1'240 3-639 August ...... on We wa 300 September...| 0'924 4828 0°752 4°396 October.2..2. 0'637 5°565 0950 5°970 November ...| 0°415 6°12 1608 5°192 December ...| 1016 7°626 1°604 5551 Means...}| 0°785 Rope TTT Aco OF THE BAROMETER AT MANCHESTER. 273 TABLE VI.— Excess and Deficit of Mean Temperature com- pared with the amount of the Oscillations of the Barometer. Month Excess of | Amount of Deficit of | Amount of { ORIN Temperature.| Oscillations.| Temperature.| Oscillations.4 or Inches — Inches oO oO January......} 2°48 7°220 1°36 6°863 February ...). 2°77 5°413 5°00 5°702 Marche... 1°88 6°487 1°33 5°532 v\joyall en anoeee 1372 4861 1:06 5071 Maya ecccnses 1°26 3°867 1°30 4484 Tuner. 2°28 3°679 0°95 4°170 DULY Secvvene Buns 3°513 1°68 3°884 August ...... ue ve ite 18 September... 1°72 4°731 1°21 4526 October...... 1-45 5°497 1°28 6°414 November...| 2:26 - | 5600 252) Gogh December ... 2°97 6'066 2°81 6714 | Means...) 2°18 57176 1°74 5°334 | TABLE VII—Monthly Oscillations above and below the aver- age number, compared with the corresponding Falls of Rain. Oscillations 5 Oscillations Month. in Excess. Rain. in Deficit. ean oe Melnches igs) acai January...... 1°81 2°379 3-40 February ... 1-65 2°707 0-83 March ...... 2°23 2°204 1-93 PA rill Renee 1°57 1°646 1:80 IWIN? Goododane 2°90 132 1°38 ABUHY): Gonnaauoe 2°47 3°754 1°95 July ......... Oe) 2°7'75 1°70 August ...... ie Mae sae September... Bas 2°622 2°65 October...... 1°44 | 3°464 2°45 November... 1°80 | 3124 1°20 December ... | 2°87 SER. III. VOL. I. NN 274 REV. T. P. KIRKMAN ON THE THEORY OF XXII. — On the Theory of Groups and many-valued Functions. By the Rev. Tuo. P. Kirkman, M.A., F.R.S. Read April 16th, 1861. § 1. First principles: Factors of a substitution: Permutable substitutions. 1. Let G denote the sum of & different arrangements Aj Ag: + A; taken from the 1-2-3--N permutations of the N elements 1,2,3--N—-1-N. Every pair A,, A, of these & permutations gives a sub- stitution, which is written An Helens and which has reference to a subject B, on which we operate. This subject is any permutation of the N ele- ments. The effect of the substitution on B is to exchange in B, for any letter a, that which stands above a in the substitution. We express that C is the result by writing An B=C, A, If the k arrangements are such that every result of operation by any substitution made with any pair on any arrangement is an arrangement of the system, they form a group of permutations. GROUPS AND MANY-VALUED FUNCTIONS. 275 2. It is suffictent to consider groups of which an ar- rangement is the natural order 123--N of the elements, which is called unity, and may be denoted by (1). Such a group is 1A,A,+-A, 4. Every substitution in this group may be written with unity for denominator ; that is, we have always AG pas: AS eG) A, being one of the & arrangements. And the k substi- tutions of the group are (ey ae CG) 4a. which may be written (1)A)Ay-- Ay. A group of & permutations, of which one is unity, is a model group of the order k. The simplest definition of a group is this—that the product of any two substitutions of the group is a substi- tution of the group; that is, in any model group, A Ae ING Jaf iN ela A, and A, being in the group. It follows that every power of a substitution is in the group. If A’ we say that A, is a substitution of the b order. We have also ee Aaa; that is, every negative power of any substitution is one of the group. In fact, =1, 3. It is important to be able to form readily the pro- duct of two substitutions A,, and A, in a model group. The simple rule is: Pronounce the arrangement A,, with 276 REV. T. P. KIRKMAN ON THE THEORY OF which you operate on A,,, and at the same time write down the consecutive elements of A,, in the same order in which you see 123--N disposed in A,. For example, in the group 12345 31452 43521 54213 25134; we have 31452 +31452=43521 = (31452)” 31452 +43521 = 54213 =(31452)° 31452-54213 =25134=(31452)* 31452 +25134= 12345 = (31452)=(1) BI len aay me ola yd oye oi 4. Every substitution A,, is obtained by operating on unity with cyclical permutations of certain circles of ele- ments, which are called the circular factors of the substi- tution. For example, the substitutions 654213, 312564, 465132, may be written 634521 21,50 1,62 qegiae aust): aaa of which the first has a circular factor of six elements ; the second has two factors each of three elements ; and the third has three factors of two elements. Tt is usual to say that the first has a factor of the sixth order, the second has two(factors of the third order, and that the third has three factors of the second order. We speak also of the order of the substitution 0, which is the number of its different powers, (10@--.--). It is well known that the order of a substitution is the least common multiple of the order of its circular factors. It is useful to be able to see the circular factors without losing sight of the form of the substitution in ‘a model GROUPS AND MANY-VALUED FUNCTIONS. 277 group. We may write the circular factors under the first of their elements which occurs in the substitution thus : 312564, 465132 2 £ 123 5812406bc9a7- 2 c a 9 7 minor cd In the notation of Cauchy the last substitution is repre- sented thus: (1 386 4) (7 0¢) (9a 4) (25), of which the inconvenience is that we lose sight of the form which the substitution wears, viz., 35812406bcgQaz7, in a model group. 5. The product of two substitutions PQ varies in general with the written order of the two factors. But in some cases we have PQ=QP. When this is true we say that Q zs permutable with P. If we change in any way the sequence of the circular factors of the same order, and write the changed sequence over the given one, we have always a substitution permu- table with the given one. Thus let the given one be 230) (O)to) >) al HHO) Ex HN) (el ee made with thirteen elements. We can form Q and Q’, both permutable with P, thus: 804-361 361-804 804-361-¢b-a7 __ 361-804-a7-cb and we have =—=442'9 1) 5.017) 219 Oa 6 0—@ 4281506396c¢7a=Q QP =PQ QP—Pas as it may be proved by the rule given in (3). This is a theorem of Cauchy’s, whose somewhat difficult 278 REV, T. P. KIRKMAN ON THE THEORY OF demonstration may be seen in his “Mémoire sur les arrangements que Pon peut former avec des lettres don- nées,” Exercises d Analyse et de Physique Mathématique, tome troisiéme. And its truth will be more simply evident by what follows on the construction of groups. § 2. Deranged groups: Derived groups. 6. Let G be a model group of k substitutions 1A,A,-+Az_i, and let P be any substitution not found in G. We can write the product GP=P A,P A»P AP ko in a vertical column. None of the & arrangements thus made is in G; for if APA} we have P= AP AGS Ay by (2)5 that is, P is one of the substitutions of G, contrary to hypothesis. The only effect of P on G is to change the arrangement of entire vertical columus of G, as is evident if we compare together the products (1)P; A,P, AGP,“ &c: We see that the m” element of every arrangement in G, by the rule of Art. 3, is placed in the same vertical row. We shall call GP the derangement of G by P. It is known, and easily proved, that the derangement GP of G by P is identical with the derangement of G by A,,P, A,, being any substitution of G. Let @ be an arrangement of the N elements which is GROUPS AND MANY-VALUED FUNCTIONS. 279 neither in G nor in GP. Then no arrangement in GQ can be G; nor can it be in GP; for if A, P=A,Q, we should have An A PQ ANP that is, Q is in GP, contrary to hypothesis. Hence we see that G, GP, and GQ are what Betti has called equal groups, that is, groups of permutations, all whose substitutions are identical, the common model being G. In this way we can partition the entire system of 1-2-3--N permutations of N elements into 1:2.3.°N k equal groups, by adding to G, GP, GQ.- the derangement GR.of G by R, which is not in G+GP +GQ. Hence we see that G has 2h. 1 derangements, and that it has no more, and that these are all equal groups. 7. Let G be a model group of the order k, made with N elements, viz., G=1+A,+A,+-+--+Az-13 and let P be auy substitution not in G. The product PG=P+PA,+PA,4+--+PA,_, is a group of substitutions. In fact it is the derangement by P of the model group PGP1=1+PA,P1+PA,P1+.-:, which is a group, because, according to the definition (2), TRAN Veevts Jee botnet Buia es) ete eve Btn The group PQ is the derivate of G by P. No arrangement of PG is in G; for if A eae we should have ‘ 280 REV. T. P. KIRKMAN ON THE THEORY OF A AG PAS or P would be in G, contrary to hypothesis. If Q be neither in G nor in PG, no arrangement in ~QG can be in G or in PQ;; for if QA,,=PA,, Q= PA, A PA;, contrary to hypothesis. If then we add to G+PQ+QG.- the derivate RG by R, which is not in the preceding groups, we shall partition the IIN permutations of N elements into a groups, which - are G and its ae 1 derived groups. We easily demonstrate the known theorem. The derivate of G by P is the derivate of G by PA, A, being any sub- stitution of G. Cauchy has also proved in the Memoir above quoted that the substitutions AP and PAP are similar substitutions; that is, that they differ neither in the number nor in the orders of their circular factors. We shall say that the model groups G and PGP"! are equivalent groups, when they are not the same group. All that precedes is well known. 8. The derived groups of G PG, QG, RG, -- are derangements by P,Q, R,-- of the equivalent or iden- tical model groups PGP, QGQ7, RGR" -. Let us suppose that one of them is identical with G; or that G= PGE: then GP =PG, -and consequently GP?=PGP=P°G, GROUPS AND MANY-VALUED FUNCTIONS. 281 GP°=P'G, G=PGP. Let P™=1, we have G(i1+P4+P?4+--+P")=(14+P4+P?+--4+P"")G. or Gg=9G=G’, g being the group of the powers of P; and G’ is a model group of the km” order; for ARIA Po =A pep) Ae Ae PoAS A A Pe ALP, which is the test of a group by the definition, (2). We shall call the derangements GP, GQ, GR-- of G, which are also the derivates PG, QG, RG--of G, the derived derangements of G. And we have the known theorem. If PG=GP Ze a@ derived derangement of G, G being of the order k, and P being a substitution of the m” order, there is a group of the km* order composed of G and its derived derangements by the powers of P. It is also known that if PG, RG, QG-- be derived de- rangements of G, the group G with its derangemeuts by the powers and by the products of the powers of PQR-- forms a model group of substitutions. g. Let us suppose that of the equivalent or identical groups above mentioned PGP and QGQU are identical, and different from G. From PGP '=QGQ7 it follows that Q5'PGP{Q=—G: Let Q]P—6} whence P=Q98, 1 =|Pel@a, Cay Petee which gives by what precedes 6Gé=G, SER, III. VOL. I. 8a 282 REV. T. P. KIRKMAN ON THE THEORY OF by which we see that for every group among those under consideration which is identical with PGP, there is one identical with G. Suppose now that EGEai—G,, QGQ'=G’, G’ being a different group from G: we have QGQ7'= QPGP a=". Let us put QP=60; whence comes @= OPPs}, and f= OP Or, and GJ= PO, whence G’—QGQ71=6Ge-. This proves that for every group among those considered, which is identical with G, there is one identical with G’, G’ being any group equivalent to G. We have then this theorem — Turorem A. The group G of the order k made with N elements and its — 1 derived groups are ae derange- ments of G and of its equivalents G,, G., G;.-+-- And there are among these derived groups neither fewer nor more derangements of G than of any equivalent G,, of G. Cor. If the number of groups equivalent to G is a ee including G, G has no derived derangement, and every derangement of G is a group of permutations different from every derivate of G. The same is true of the derived groups and derange- ments of every equivalent to G. The number of the equivalent groups and of their de- 2 rangements is =) which are a system of groups of 2 permutations identical with the system of the ee GROUPS AND MANY-VALUED FUNCTIONS. 283 derangements of the same equivalents, including the equi- valents; for every derivate of G, is a derangement of G, equivalent to G,. If the number of groups equivalent to G is less than that of its derived groups, we know that G has derived derangements. If M be the number of groups equivalent IIN kM ments of G or of any equivalent to G (including G and the equivalent); and each of the equivalent groups forms to G, including G, is the number of derived derange- with its derived derangements a modular group of ae substitutions. Def. A modular group consists of a model G and certain derived derangements of G. A non-modular group is a model group which cannot be written as a model with its derived derangements. § 3. Model groups of the order k, made with N elements, which are the powers of a substitution of the k order. 10. Let N=Aa+B04+Ce+--4J7 be any partition of N, such that AS Bebe Ce J > 0, abc--j being any numbers; and let & be the least common multiple of ABC: -J. We are about to construct a group of the powers of a substitution having a circular factors of the order A (of A elements), 4 of the order B, &c. The elements of the first factor of the order A may be selected among the N elements 123--N in IIN ITA IT(N-A) different ways. 284 REV. T. P. KIRKMAN ON THE THEORY OF The elements of the second factor of the order A cam be chosen among N—-A in IT(N - A) ITA . II(N-2A) different ways. If a=2, we have Mle TIN 1-2 (A)? r(N—2A) different ways of choosing the 2A elements, where we divide by 2, because either of the two circular factors may be considered as the first chosen. In ike manner there are 1 IIN Tia (wA)* II(N—Aa) ways to select the @ circular factors of the order A. In the same way there are 1 IT(N - Aa) IIb (IIB)* (N — Aa— BO) ways to select the 6 factors of the order B from N- Aa elements, and 1 IT(N —- Aa- BB) IIe (IIC)* IT(N — Aa— Bb-Ce) ways to select from the remainder the ¢ circular factors of the order C. Consequently there are 1 TIN ITaI1bIc- - Tj (ILA)“(ITB)°(IIC)°- - (IT3) different ways to choose the cireular factors. We can write each of the circles of A elements in a vertical row headed by its least element in [I(A —1) ways; whence there are (I(A- 1))* (HI(B = 1) (C= 1))°- (I -)¥ = V ways of writing the circles in vertical rows under unity (Art. 4). Wherefore we have thus he IIN ~ Wa Tb. -TTj AcBCe. . ==) J UV GROUPS AND MANY-VALUED FUNCTIONS. 285 skeleton groups, each of which is to be completed by writing under every element of unity repeated cyclical permutations of the factor of which it forms a part, and continuing the horizontal lines so constructed until the next line is unity. 11. It is easy to see that the number of horizontal lines will be 4, the least common multiple of ABC. -J. Every group so formed is composed of k successive powers of a substitution of the order & (Art. 4). But the groups will not be all different. It is well known, and easily proved, that any group of & powers of a substitution Gj=1-P+P24+ P34 ..4 Pe? can be written in this form in R, ways, R, being the number of integers, unity included, which are less than k and prime toit. lor there are R, different substitutions in G similar to P, any one of which, P,, gives G=14+P,4+Pi+Pi+--4+Pi°. Wherefore every group of the UV constructed has been formed R,, times, and we have the exact number of equi- valent groups formed on this partition of N by dividing UV by R,. 12. We have proved the theorems following. Tueorem B. The number of equivalent groups of the order N which are powers of a substitution of the order N IT(N - 1) made with N elements, is , where Ry is the num- N ber of integers, unity included, which are less than N and prime to it. THeorem C. If any partition of N be N=Aa+B04+Cce+--4+J7, A>B, B>C.--J>0, abe-+7 being any numbers, and k being the least common multiple of ABC--J, the number of equivalent groups, which are powers of a substitution having a circular factors of the order A, 6 of the order B, &c., is 286 REV. T. P,. KIRKMAN ON THE THEORY OF we JIN ~ R,Ialb Tc. -j A°B’ Ce. where Ry, 1s the number of integers, unity included, which are less than k and prime to it. 13. Any two G G’ of the W groups are equivalent, that is, we have G'=0G0"". Let MyMyN3° © My, be the m circular factors of G of the order M, and let Mm’, M', M's ++ Mm be the m circular factors of G’ of the same order. We have 0 0',- Oy G7 8'g- Oy C1 Cg: Cee: 6) hy Ay* Ay, 0, b+ +b, C, Ca++ Cpe For example, let N=8=6-1+2-1=Aa+Bé. Two of the W groups are 12345678 12345678 36487215 86714532 42851637 25381476 86573241 64728135 52714683 51362874 76138254 48756231 _@, 0 18265437 O= a,b, 13485726 13975748: G’= OGO1= 13825746 G14275863. § 4. Modular groups formed with N elements of an erder superior to N, which contain the powers of a substitution of the N” order. 14. We have proved, theorem B(12), that there are 2a equivalent groups of the powers of a substitu- N tion of the N™ order made with N elements. GROUPS AND MANY-VALUED FUNCTIONS. 287 The number of derived groups of each of these is II(N-—1); wherefore each group has Ry—1 derived de- rangements, which complete with it a modular group of NRy substitutions (Cor. theorem A). We have then the following theorem — IT(N - 1) Ry equivalent groups each of NRy substitutions, among which are the powers of a substitution of the N* order. The simplest of these groups is 1234-- N-iN TuroremM D. With N elements we can form 2345-: N 1 Ba 50s a ely) 12. SOTO We ee: which may be written ans(it? or G=S(i+c) (mod. N), where 7 is any element of unity and c may have any of the values 012--N-—1. For example, 56781234= ti +4 (mod. 8). Let p>1 be any integer 1, PG=piS(i+e), are the Ry—1 derangements GP=S(i+c) pi; for pi(t+c)=p(tite)=pitpe=pi+ec’ and (i+c)\pi=pite; that is, every substitution in PG is a substitution in GP. The group of RyN substitutions has none of the order N except those of G=S(i+e). For if any of the derived derangements added to G contained Q of the order N, there would be the N —1 derived derangements QG Q?G--Q*"G1 (Art. 8) ; but we know that G has not more than N-2 derived derangements, for Ry—-1 + N-2. We shall presently show how the groups equivalent to the simple modular group G’=S(pito), where p>1 has Ry values and e 1, inferior to N and prime to it, a prime root of the congru- ence 17. We have shown that there are *_4 other xv” —-1==0 (mod. N), where reRy, we can form with N elements ee IN equivalent groups each of Nr substitutions, containing the powers of a substitution of the N“ order. And if there are m prime roots of this congruence of which none is comprised among the powers of another, according to the modulus, we can form eee different groups each of the order Nr, IT(N -1) R N SER. Ili. VOL. I. PRP forming sets each of equivalent modular groups. 290 REV. T. P. KIRKMAN ON THE THEORY OF 18. It may be useful to give examples on the theorems Dand K. Take N=5. We should have by theorem D sto equivalents groups of 20 substitutions. The sim- plest of these is made by adding to G=S(t+ ce) its derivates by 22, 3¢ and 47. This gives the group G’ 17345 2.4135 31425 43215 2.3451 41352 14253 32154 34512 13524 42531 21543 sou25 35241 25314 15432 D234 52413 53142 a4 oe which is found in the Memoir of Cauchy, above quoted, and which has been given by Betti, and first, as I believe, by Galois. I cannot find that any of the equivalent groups has been formed, or that their enumeration has been distinctly affirmed by previous writers. We see that the three added groups are derived de- rangements, by writing them thus: 24035 31425 oo) 35241 42531 54321 41352 53144 15432 52413 142.53 21543 13524 25314 ele: The equivalent groups of the twentieth order are j 12453 G'12534=G) 12534 G'12453=G, 12354 G'12354=G', 12.435 G'12435=G, 12543 G'12543=G’; all of the form Q G’Q71. In order to determine Q, we compare with G any one g of its equivalents, thus: 9=12345 G=12345 24531 23451 #3152 34542 GROUPS AND MANY-VALUED FUNCTIONS. 291 35214 45123 51423 51234 The circular factor 12435 of g is obtained by multiply- ing the circular factor of G by 12.435 _ le enn eo The derived of G by Q is QG= 12435 24351 43512 35124 51243 and the derangement of QG by Q7'=12435 is 7 = 12345 24.531 neo no Ney 35214 SAE wherefore g= 12435 G12435=QGQ. All that we have to do is to write, for each one (c) of the Ry circular factors of G and its derived derangements, 12435(c), and for the first permutation A of each group of five, 12435 A(12435) '=12435 A12435, thus: 12345 23451 41235 34125 2. 2 1 4 4 1 3 2 3 4 2 1 5 5 5 5 The equivalent group 12435 G'12435 is completed thus: 12345 23415 41235 34125=G, PAS31 31542 13524 42513 AOS) MAPA) Bene | Ce Se Son Amat 9 AEA Tae D473 5293454512 53247. The others are 292 REV. T. P. KIRKMAN ON THE THEORY OF 12345 25314 41352 54321=G; 24153 51243 15423 42513 45231 14532 52134 21435 53412 43125 23541 13254 31524 32451 34215 35142 12345 23541 51243 35142=G', 25413 31425 13452 52431 53124 15234 32514 21354 34251 54312 24135 14523 41532 42153 45321 43215 12345 24351 51324 45312=G’, 25134 41235 14532 52431 54213 15423 42153 21543 43521 53142 23415 13254 31452 32514 35241 34125 12345 25143 31542 53241=G, 23514 51324 15234 32154 $5428 1345252413) 24435 BSA SAN D4 tO) E43 O oss At253 \ 42532 AS25430 2: Hach of these consists of a group of powers of a substi- tution of the fifth order and of its derived derangements. 19. Whatever N may be we have always (N-—1)?—1=0 (mod. N). ff then we add to G=S(z+c) the derived Q’G=(N-1)t-G=-iG (mod. N), we have (theorem 1!) a group of 2N substitutions. The first and last of the four groups of five in G’(18) make such a group of ten. All the substitutions of the derived Q’G have the form g=-(#+c)=-i-c (mod. N), and g=- (@+e):— (@+¢)=1+¢-¢=1= (1) : GROUPS AND MANY-VALUED FUNCTIONS. 2938 that is, all these substitutions (Q’G) are square roots of unity. The last derived group of G’, G’, G’, G’, G’s, like the last of G’, is composed of square roots of unity. We have then the theorem following — THeoremM F. With N elements we can form equivalent groups each of 2N substitutions, which are N powers of one substitution and N square roots of unity. Take N=7; we have three roots of the congruence x2*—1=0 (mod. 7), viz., 1,2 and 4. We.have the group (theorem E) S(¢+c) +2i8(¢+c¢) + 4iS(¢+ ce) of the twenty-first order, and this has 119 equivalents. If N=8, we have four roots of x’? -—1=0 (mod. 8), viz., I, 3, 5, 7, of which 3, 5 and 7 are prime roots. We can therefore form three groups (theorem E) each of six- teen substitutions, all containing the same group of eight powers of a substitution. We have only to add, to (G) the eight cyclical permutations of 12345678, the derived groups 31G = 36147258G, 5tG = 52741638G, 71G = 76543218G. The last of these is composed of eight square roots of 3-07 _ Rs of sixteen, each comprising eight powers of a substitution ; unity. There are 3-7:6-5-3-2 different groups and in the group G+ 57G there are all the powers of two different substitutions of the eighth order. § 5. Groups of N(N-1) (N-2) substitutions made with N letters, which comprise substitutions of the N order, where N-1 ts a prime number. 294 REV. T. P. KIRKMAN ON THE THEORY OF 20. Let g=S(pit+tec) (mod. (N-1)) be the group of (N-—1)Ry_, substitutions made with N-1 elements of theorem D and of Art. (15), when N —1 is prime, in which case (N -1)Ry_,)=(N-1)(N-2). Let the final element N be added to every one of the (N-1)(N-2) substitutions of g,. This gives a group made with N elements of the order (N-1)(N-2), g =S(pi+e) (mod. (N-1)) + where 7 is any number o and 1, Mg er) ctr Bibeaeas (Nay ee ‘ ee £8°(1 —1) +7} no. v= as where @ is any primitive root of (N-1), and where the numerators and denominators of the first fractions are residues estimated to modulus (N—1). Every one of the elements 1), or ¥.g=Vng3 (r>1, Sm). In the former case ¥,, will be among the substitutions of Wg; that is, —A+#)(7 —m) +m? ue N m iy _ 18 $mod.(N—-1) , +N, Gan tian. am ON ay SMG tlt noa: (1) 1 N) pite N =| $B* +1 bmoa. (W-1) cane 1) ( a TH Let t= P*(1-—m)+m, mod. (N—-1), and let p(B*(1 —m) +m) +ce= HY +1, which can be satisfied by a value of y, whatever z may be. The effect of the substitution Y,, (in the right member) is to change 2 into BUM 44, and the effect of the same ¥,, (in the left member) is to change 2 into B91 —m) +m. We have then the two equations p(G(1—m) +m) +c= Pp" +1 B+”) 414 =B-1)(1-m) +m; whence comes ¢P(P*(1 —m) +m) + $38 (1—m) +2m—-25=1, which must be true, if ¥,, is among the substitutions of W\g, for every value of z. This is impossible, unless the coefficient of 8* is zero; that is, unless 296 REV. T. P. KIRKMAN ON THE THEORY OF 2p(1 —m)(m—1)=0, mod. (N-1), for a value of m>1,; which is absurd. Wherefore ¥,, is not in Wg, and ¥,,9 is different from V9. Neither is ¥,, among the substitutions of Wg. For if $B) (1 —m) + Mbnoa. wy, N m §B*(1 —m) +: mh inoa. 1) m'N ABET) 7) begets ieee ane IN as ite N =| Tie mm ao r +7} G i +x) P= let i= P*(1-m) +m, and let p(b*(1 —m)+m)+e= P"(1-7r) +r. The effect of the substitution ¥,, is to change 7 into BUM (a—r)tr in the right member, and z into BO) (4 —m) +m in the left member. We deduce from BU (1 7) 477A (1 —m) +m, and from the last written equation, 2p(1—m)(m—7r) =o, mod. (N -1), for the coefficient of 6%. This requires either m=1, or m=r, both contrary to hypothesis. It is thus demonstrated that the N-1 derived groups are all different. It is requisite in the next place to prove that they form with g a group of substitutions. ‘This is established if we can demonstrate the two propositions i Bie epee et for all values of g, r and m; where P,, P,, and P, are substitutions of the form (pitc) N i N’ in which p>o1i1; and if ¥.=¥,, the same thing is: demon- strated by beginning with += 6*+1; andif ¥,=¥, we still obtain three equations (a)()(c), by which we can determine p, c and s. 22. It is necessary, in the next place, to demonstrate that Ce ‘Pe = Pp WV, 5 that is, that (ae aoa) ae pee ( B(1-q)+¢ Ng a +H _(pite N\ (8 @%UG-h +k kN = +H} cae : Bt tate and that p,, ¢, and g are given in terms of p, ¢ and &k. Let c= 6"(1-k) +4. The effect of the substitution P,,%, is to change 7 into se 8) +h, and then to change this into p(B (1—k) +k) +e. P(B"(1—k) +k) +4 =h(1-9) +4, () which can be satisfied by a value of z, whatever be 2. The effect of the substitution ¥,P,, is to change 7 into Be AU) ct a DIBA -A) +h) +e=PCM1-g) tg. (4) The equations (a) (d) give {ar(B"(1-4) +4) + - ght Bp(B (1-4) +4) +Ble- 9) = (1-9), which is to be true for all values of x. This requires that the coefficients of the three powers §” should be zeros, Let so that GROUPS AND MANY-VALUED FUNCTIONS. 299 which three conditions determine p,, ¢, and g in terms of p, cand k. The same process will determine p,, ¢, and q in terms of p and c, if ¥).=W,, and if a beginning is made with 4=P* +1. Wherefore it is demonstrated that Y PL=P, ¥:, and that G=g+¥igt+Pigt+--+Prig is a model group of N(N-1)(N—2) substitutions. 23. Wecan modify the form of the substitutions ¥¥,. - thus. In a Boe eae tN Y Peay ING let 1/32 We have BP erh+i= (i- Dy Buses SG eect 1; wherefore N23 se Nes N y 8 (4 1) + 1 (mod. (N—1)) ala at zat 2 Nigh a a (a-+1) Dy telat get - bee Od ue a (1-7) (i-7) =A" A BV (a-r)4+r=(1-r)(Bi- Br) 14+r an rat = 7)'BY Ab = 7)" ? +P n0a. ww) fe N a ING Ze The group of N(N-1) (N-2) substitutions may be written G={S(pit teat Vit Pit Vat + Pra where ¥, and ¥, (7 > 1) are the substitutions above written. 300 REV. T. P. KIRKMAN ON THE THEORY OF 2 has all values 0, c has all values . Ci1&c.); (A> 2). Let Pm =n bn Cres hin (A elements) Vi Oh. bie Cin. © ahien O70 (B elements) P_-m>= 4m hin Im'* On, J—m = Om Tas er ie Uh. and let De pan gu Jim’ ; be the 2” cyclical permutations of the factors PmP—n* “Ams q—m > (pie). The substitution Ra Pa Pee Pes Par Para" Pi PP, P3-° Gaxi. Gate > pin pie pay. gti girs. PD pe (OS) OO) Oban Opatioy Or is a square root of unity; for, let pi =abcdefg ; we have ph _ agfedcb ph abedefy pt’ _ gfedcba pi — abcdefg pag _ fedcbag pr abcdefy &e. We see that all the factors in these substitutions are of the second order. 25. Further: RG is a derived derangement of G, of which all the substitutions are square roots of unity. For the vertical circles of RG are those of G read in contrary order; and since, if 3802 REV. T. P. KIRKMAN ON THE THEORY OF G=1+P+P?+P*.. pe _ Pi Pr: “Gi Va-2° * : Pi Po? * Fa JWa+2°° we have Rp a we eS Pa Pi, Pa Ps re Ga Gooner which has no factors but of the second. order, as we have just proved. The number of these derivants R, all beginning in the denominator and in the numerator, with the same element 1 of pi=iabe-- 1S At" B? Ce. . J; for every factor P_-2 P_-3* *P—a G—(a+)* °Y-(a+b+41) may have A different or B different exponents, according as it is a p or q, &c. Let M2, A>B, B>C.., and let k be the least common multiple of ABC--J. We can consiruct in this partition of N, TIN R,, & Ia [16- - Tj equivalent groups of 2K substitutions, each of the form G+RG, where G is one of the groups of k powers (of theorem C), and RG is composed of k square roots of unity. § 7: Grouped groups of the first class. 27, Let N=AaBb+--+J7 A>B>C.-.->J be any partition of N. A substitution P is a principal substitution of a group formed on this partition, when it has the form fP=Aa+Bo+--4+J), which means that P has a circular factors of the A”, 0 of the B” order, &c.; and if the form of every other substi- tution Q of the group is fQ=A'V4+B4+C e+, = GROUPS AND MANY-VALUED FUNCTIONS. 305 such that the first of the differences A-A’, a-a’, B-B’, 6-0’, C-C,, c-c’' --, which is not zero, is positive. If they are all zero, Q also is a principal substitution. We are about to enumerate groups having their prin- cipal substitutions defined by this partition of N. A grouped group has elementary groups of which those of the same form are permuted, sometimes as though they were single unchangeable elements, and more frequently so that they suffer certain interior derangements while they are transposed. For example, Es 2143 3412 4321 is a grouped group, in which are seen two elementary 12 groups , | and 3*. 43 Another grouped group is roo 2143 3421 4312. In the first the elementary groups are simply permuted ; im the second they are permuted while one suffers at the same time a certain interior derangement. 28. Let G bea group of theorem C (12); and let Pi Po "Pa be the a circular factors of A” order, Ga41 Va+2 cor Vatb the 6 of the B” order, Pato Tat+o42" “Tatb+e the ¢ of the C™ order, and let hdA+ De =Pn be the 2” cyclical permutation of p,,, &c. J ne zt uy en SHR. ILI. VOL. I. RR 306 REV. T. P. KIRKMAN ON THE THEORY OF Let g be any model group of / substitutions formed with Ot Ope ty elements, such that the first a vertical rows of g written in a column shall contain only the first a elements of unity, the 6 next vertical rows shall contain none but the next 6 elements, &c. Let this group be g=1t+O+.-+0'4.-4(00)4+.. where 1 = 1234 = |] ae © = aByd --On--e-- 6’ = KApY ‘spe (00)= ofyb--0-- where 7 is one of the 6 elements, e one of the c elements, &e. We form with the circular factors of G the substitution following, with the aid of © in g: qa Be PaPy: Ue Get eee Py Pe PS °° Da Vor Gata.** Varo Tarot?” in which the factors of the numerator are those of the denominator in a different order, so that B=n, if B=3 &e., and where the subindices of the numerator are the elements of 0. 29. The effect of the substitution Q in the derived group QG is merely to change the order of entire vertical ranks of G; that is, QG is a derived derangement of G. In fact we see that the A vertical ranks in which the elements of the circular factor p, repeat themselves, in QG, occupy exactly the places of those in which the cir- cular factor p, repeats itself in G; the precise order of those rows being determined by the substitution Da i GROUPS AND MANY-VALUED FUNCTIONS. 307 Let us next construct on 0’, in the same manner, the substitution (ge Cac Pata PrP? + PuGan” We shall have Qa BaF Pots PiPs °° Pa Van -°” for in order that OO’ may be a substitution of g, it is required that aie by «0 : KA oe Pp eo whence we must have Qa Pe bs wg De ane T pi pt .. 9S - Pe Pr Pp whereby we obtain the product QQ’ above written. This proves that the substitutions QQ’-(QQ)-. which are formed on those of g compose with unity a 2 model group, for the reasons by which g is a group. Wherefore, whatever be the system of exponents which we employ in the denominator of Q Q’ &c., the same system being used in them all, and whatever be the auxi- liary group g, we have always a grouped group of &/ sub- stitutions, consisting of G and /—1 derived derangements of G. let us endeavour to enumerate the groups thus con- structed which shall have their principal substitutions of the form Jp=Aa+Bb+Ce+:-4+57 of the principal substitutions of G. This will restrict the number of auxiliary groups g that we can employ. 30. Let KTaO be a circular factor of the order h in © any one of the substitutions of g, and let us suppose it formed with / of the a elements of a+b+te+>:+7. 308 REV. T. P. KIRKMAN ON THE THEORY OF The substitution Q which we form on will have the circular factor Pu Pr et De m Dep. NER Let us suppose De = 0.0 6h P= %,8,.6, ° p= a: O_O, Cn ~P§ = Uy Dg Co where the number of the elements a,6,c, --1s A. The substitution P which is written under unity in the group G may be represented by Jm Comic bm for P puts for any element 7,, in a circular factor the ele- ment j,, which follows in that factor, p,, or 7», &c. The substitution QP is of this form: i QP= by, Cz d, a0), on 7 es It is plain that this circle will be closed when the circle abed-- of the A” order, and the circle xumo--@ of the order h, are both completed; and that the order of this circle of QP will be the least common multiple of A and h. There will be A of these circles in QP beginning with Des Ces cy SC. It follows from this view of the circular factors of QP that the order of this substitution is the least common mul- tiple of the orders of © and P. Let us suppose that P is a principal substitution of the constructed grouped group, which implies that QP shall have no circular factor of order above A made with the Aa elements, nor of order above B made with the Bé elements, &c. It follows that A the order of the circular factor Kuma: > 0 GROUPS AND MANY-VALUED FUNCTIONS. 309 in © is a divisor of A, and that every circular factor in © made with the a or 6 or c-- elements is of an order which divides A or B or C, &c. © is here any substitu- tion of g. And QP, whichever of the principal substitutions of G P may be, is a substitution of the order of P, and a prin- cipal substitution of the grouped group G+QG+QG+..- 31. There is nothing to prevent Q from being also a principal substitution of the group constructed. If it be, we have fQ=Aa+Bo+Cc+--+4J7. But the orders of the circular factors of Q are those of the factors of ©, on which Q is formed. Hence O must have the form fO=Aa,+ Bb,+Ce+--4 5), =a+tob+e+--+ 7, whence it appears that the partition of N on which G is formed is N=A-Aa+B-BO+C-Ce+.--4J3-J7. When Q has the above form (fQ), every substitution of the derived group QG is a principal substitution of the grouped group; and every principal substitution ©, of g will give a derivant Q, such that Q,G has k& principal sub- stitutions. 32. Let Q, be not a principal substitution of the grouped group. The partition of N may or may not be of the form N=A.-Aa &c. above written. If it is of this form, Qa which is not principal is constructed on @a which is not principal in g; and Q,G will have only R, principal sub- stitutions, one for every principal of G. If the partition of N be N=A.-Aa &c., and if the prin- cipal substitution of g has circular: factors of an order below A made with the Aa, letters, or of an order below M made with the Mm, letters, no substitution Q will be 310 REV. T. P. KIRKMAN ON THE THEORY OF principal in the grouped group, and the only principal substitutions in the /—1 derived groups QG, Q@G, QG,-. will be QP Q,P Q,P &c., where P is any principal of G. This gives only (7—-1)R, principal substitutions in the 7—1 derived groups. If the partition of N be not of the form N=A- Aq, &c., no substitution Q can be principal, and there will be only (¢—1)R, principal substitutions in the /—1 derived groups. 33- Let ® be the number of principal substitutions © - of g which have the form, (31), fO=Aa,+Bb,+Ce,+ --+57,; then there will be /-X-—1 non-principal substitutions in g, besides unity. : The number of principal substitutions in the grouped group will be R,+Ak+ (U-X-1)R,=lk + (U-A)R,. When A=0, whatever be the form of the partition of N, there will be only IR, principal substitutions in the grouped group. A grouped group has its normal form, when it begins by the powers of a principal substitution. The number of different groups G of theorem C, formed on the same partition of N, with which the group can commence, is Ak + (1A) RB, Ry, This is the number of repetitions of the same grouped group that we shall make by selecting every one of the W groups of theorem C for our group G, without making any change in the auxiliary group g, or in the system of exponents in the denominators of the /—1 derivants Q GROUPS AND MANY-VALUED FUNCTIONS. dll (29). . Wherefore (S), the sum of our results, must be divided by this number. 34. We have to consider the effect of a variation in this system of exponents. ‘We can always take unity for the first exponent in the denominator of Q, of every circular factor in Q; for M—K-+1 ,,.N—K+1 R—K+1 Pa Pi: Pp Be _ Bu Ps oe: Pr pe pe pn va pe De pea po Ro pee : If the a factors p, p,p;:- make in Q aq, circles of the order Aj, a, of the order A, &c., and if the 6 factors make 6, of the order B, and 4, of the order B, &c., the number of different systems of exponents is At M—-%2— ++ Pb-bimbe— ++ (e—er—ea— ++ JI ha « ; for there are a—ad,—a,—-- of the a factors p,p.-- which are not first in their circles, to which we can give at plea- sure any one of the exponents i Phy ao OUR cae) The first derivant Q is always constructed on ©, a prin- cipal substitution of g, and the denominator in Q is that of all the derivants. This denominator is determined by the form of 9, (28), fO=Aya, + Apt, + Asaz+-- (=a) + B,4,+ B,4,+ Bs b,+ - - (=0) &e. 35. The substitution Q’ which gives the derived group QG is qa PePh "PBI Py Pa’ Pi Pa Po * Pm Pn Pi > Pio’ where abe-- 1 aid lq- +n 1ab--m nl. -t are two of the circular factors of ©’ of the orders 7 and s, on which Q’ is constructed, these factors being formed out of the a elements in 6’. The substitution (i (=/). AM Soo cays Bho dipe-dhene and its principal substitutions have a, factors of the order A,, 5, of the order B,, &c. The group g may be of the form ytOyt+Ay+::, where y is a group of powers of a principal substitution of g. If there be in g any substitutions containing circular factors of orders different from the orders A, A,--B,B,-- seen in the principal substitutions, those orders will all be divisors of A, of B, &c., by our hypothesis (30). Our object is to determine the number of repetitions of grouped groups due to changes in the system of exponents of Q, the same G and g being retained. If the system of derivates QG, QG, Q’G, -- made by one set of exponents, be repeated as Q,G, Q).G, Q,’G, -- made with another set, Q, will be the (1 +7)” arrangement of QG, Q, Py (1 +2)” ap Q’'G, Q”, 3 (1 +7)” 55 Q’G ; and although QG-and Q’G may be repeated, Q’G will net be repeated, unless there be in Q’G a (1+7)™ arrange- ment, in which iB, B>C, &c., k being the least common multiple of ABC--J, and one at least of abe. +7 being >1. Let G be any one of the W groups of powers of a sub- stitution constructible on this partition of N by theorem C (12). Let Pi Pro? Pa Vari War2 ** Taro Vapor * *Va+b+e ** be the circular factors of G of the orders ABC.-.-.- Let pi=p*‘ be the 2” cyclical permutation of p,. Let w equivalent groups g be constructible of / substi- tutions made with at+b+e+--4y7 elements, of which the a elements are consecutive in unity, the 6 elements consecutive &c.; and such groups g that every circular factor in them made with the a elements is a divisor of A, every circular factor made with the 6 ele- ments is a divisor of B, &c.; and also such groups g that every vertical row of the group written in a column shall comprise no elements but of the @ or of the J &c. ele- ments. Let M be the least common multiple of AVA A’) BB -€C A, A, A; BB, C,C, — where JAAS AG: - are the orders of the circular factors of g made with any of the a elements, B,B,-- those of factors made with the 6 elements &c. Let the substitutions following be formed : Pale la. on ie TRE TES 7) OCR Ca neste where aBy--0n--+e-+ is every one in turn of the 7-1 sub- stitutions of the group g, aPy--@ being the a elements, -+ being the 6 elements, ¢-- being the ¢ elements, &c.; and where the terms of the numerator are those of the 3 316 REV. T. P. KIRKMAN ON THE THEORY OF denominator in a different order, so that B=n, if B=3 &c.; and where the exponents of the a factors p, are any- thing we please >o, o, 2, A>B,-B>C, &c., abc.-j being any integers, and k being the least common multiple of ABC..J. The simplest of the W groups of theorem C (12) made on this partition may be represented by G= (ay +6) (a,+0)+ + (4+0)(Ai +0) (B26): (C,+¢)(wi+¢)-+(Y+e), where the variable a, is any one of the A first elements (¢) of unity, ais any one of the next A elements &c., and where 8, is any one (i) of the B elements 1,, 2» 3° Ba, where the subindex 6 affects not the numerical value, and ®, any one (é) of the B following elements, Sc. The constant ¢ has any one of the values OAs Dial. tiki ten GROUPS AND MANY-VALUED FUNCTIONS. 323 We understand by (a@,+¢) (8m+¢) (Y%m+e) the resi- dues, @,-+¢, (mod. A), Bn-+¢, (mod. B), Vn €; (mod. C), &e. eee nuk sae The substitution i+¢ of the group a,,+¢ 1s = times repeated in the column z 11 14+2 i+k-1, and every substitution 7+c of B,,+c¢ is _ times repeated in a parallel vertical column of G. 41. Let M be any one of the numbers AB--J. Let y< M be a primitive root of the congruence Ti toode Mi) 2): We can add to the group (Min si c) y—1 derived derangements Art. (16), which complete with (Hm+e) a regular group. Let us conceive that y is a root either primitive or not of the congruence xz” =1 (mod. X), X being every one in turn of the number AB. -J. Let us add to each of the a+0+-+-+ / elementary groups of G, in the same vertical column, the 7-1 derived groups (On +€)s e being every one of the numbers 1, 2,3--(r—1) and 6,, being every one of a,,8,,7::: We add thus to G= (an +€)(Bm+ ©) (Ym + ©) aie O.G=Y"(Am +6) Y¥(Bm +0) Y(Ym+€)**s B24 REV. T. P. KIRKMAN ON THE THEORY OF which has the same number of horizontal and vertical rows with G; thus adding 0,G O,G.-.. The 7-1 derived groups will complete with G a group of the order Ar, and they are all derived derangements of G. For let (Y"(Gm +61) Y"(Bin + Or) )(Y"(Ym+Er))+* =P (Y(Amt EN Y(Bm+ EVO (Ym+e))-+ = (f 50) be any two of the kr substitutions, c, ¢’ being each > GROUPS AND MANY-VALUED FUNCTIONS. 3825 The derived group Oo; 1G = (Yam +6) (YY (Bin + ©)) ++ (Y*MEm +6 +2m)) > ° where z,, is any one of the numbers . 012:+-H-1, and where 2,2,23::, which are introduced with the vari- ables ¢¢,¢,--, may be the same or different numbers. We thus have the vertical column Y°(Em + 2m) Y? (Em + 2m + 1) Yen emt 2) y" (Em + m+ k—1) when @ is any odd number. The group so formed | G+0,G+0,G.:. is a model group; for let Be (Yim + C1) (YOR Bm + Er) +» (GY (Em + 2m + C1) * ° Be (Yer an +E) YO P(Bin +6) = (YE + 2m +6) + + be any two of the kr substitutions. Their product is Bete) ( (ay, aC aRTD (8B 6p—(4 +e YY Cy We ) a +y 1] Cy 70) “J Cy ITED (@ sae +c’) ( ye +C, ): i Now, by the two equations ye" =1 (mod. EF) yem*' =—1 (mod. E), Z, Vanishes from this product, and we have a substitution of the derived group iets (Gre ©) Goat (Omat ©) Ye aCe) in, viz. that in which c has the value c=c +777", (mod. £). Again: let & be as above, and B= (yan +6) )(Y(Bm+©))> Yen +©))* +s e being any even number. 026 »* REV. T. P. KIRKMAN ON THE THEORY OF The product is 66"— Ce HC.) BG ies ie ty +4 BIDE, + yore, yAtetd(e +c’) (2 Q+D(y + ): ee ay, (2m + Cy) which is a substitution of the derived group Gra, aF ¢)) (OM ised (er a5 c)) fall Ga aer TF 2m €)) aie. viz. that in which C=C py Cy. In like manner the product 6'b— Cs, By, + Cy Ce ae ab 2) +y°C y ey (e, rt Zn +¢ a) x ( +y°C' )- : is a substitution of the derived group (YFP (i, + 6) (YTB + 6)» (YOO Em + 2m + €)) + 5 viz. that im which C=C bUlae aC. 45, We have demonstrated this theorem — TuroreM J. If E be any one of the numbers ABC.- of Art. (41), and y=BE-1 be a primitive root of v"=1 (mod. M), M being any one of ABC-- and at the same time a root ~ primitive or not of xv” =1 (mod. X), X being every one in turn of ABC.-, we can construct on the given partition of N (Art. 41) with this root K—-1, E°w (theorem C, 12) equivalent groups of the order kr, where e is the multiplier of K in the partition of N; these groups being all of the form G+0,G+0,G+.-+0""G, where G is any one of the W groups of theorem C. Take, as an example, N=10:14+8:144:2=26 (k=40). GROUPS AND MANY-VALUED FUNCTIONS, O27 We have the primitive root y=A—1 Of #=1 (mod. 10), and y is a root of the congruences x*=1 (mod. 8) a*=1 (mod. 4). We can form G=(m+e)(Bite)(n+e)(m+e) of the order 40. = 3(mte)) (ate) Bm+et+1)) Blret+e+3)) Q= ((m+e)) (ite) Clnmt+e)) (l%+e4)) O,G = (27 (a, + ¢))(27(8, + ¢))(27(m+6+ 1))(27(%+¢+ 3)) which are, treating abcdefgh as 12345678, and ijklmnpq as 12345678, G=1234567890abcdefghijkimnpg 2345678901 bcdefghajklinpgqm ee eT ae qmn 0,.G= Ae Raat uaT bea on 6925814703 fadgbehcilkjpnmg eee 0,.G= Perce omc ae er shaaelmape 87654321090 bcdefghajklinpgqm OO Rg ee ecu COG oie yen hanes 4185296307fadgbehcilkjpmng ee EON One which is a group of the 160th order. And we can form with this root y=3 sixteen equivalent groups by simply varying 2, and 2. 46. Hach of the groups J of kr substitutions above con- structed is composed of a equivalent groups 995° -g, each 328 REV. T. P. KIRKMAN ON THE THEORY OF kr A tution is repeated a times; of 5 equivalent groups each A of us substitutions which are repeated = times, &c. of substitutions of A elements, of which every substi- If the a equivalent groups were all made with the same A elements 123--A, they could be written =F, I by" In= Ip (=) Po Pipa: + being the circular factors of the A” order deter- mining the @ groups 9) 91 92° The a@ groups are in fact formed with the different sets of A elements LED eS eexowe! 12431: + Ay 12930++ As and we can evidently still write Gn= Io (ae Po Po or Pn i Br Im Po Po Jos if we understand that the derived group Pm —" G Po ue has lost both the elements p and their order; but that the derangement Bre ™ Po has preserved the elements p,, and has exchanged their order for that of po. With this understanding the derivate Pm Po 2 J GROUPS AND MANY-VALUED FUNCTIONS. 329 is identically the derangement On Im Po Y 47. Let I be any model group of the /” order made with Orie Olt C ata cricts elements, such that every vertical row of I’ is composed only of the a or of the J elements, &c. Let aBy:+, OK--, TPrr, *° be any substitution of I, a@y-- being the a elements, Ox--, the 6 elements, mp-- the c elements, &c. Form the substitution Qe Pals Py Sel Oy Gc suctienine ar, "p Pi P2 Ps °° Tati Yat2° *Vato+41 Va4+042° * where 9441 9ai9°* are the circular factors of the 6 equiva- lent groups /,h,--h, made with the Bd elements of N &c. The effect of the factor 2” of Q in the derived group QJ is to change the group Png Pa” In= 5, 0 Po into Pn Dn ant Po a ma) which is the derangement of Pm Pm Ce Pin Pn mp) py Gy) It is then evident that QJ is a derived derangement of J. If we form J+QJ+QJ+--+Q_ 1, employing all the substitutions of I’, we shall have by the reasoning of (29) a model group of kr/ substitutions. In fact the effect of the derivant Q, is to put for any elemen- tary group of J a certain derangement of some other equivalent elementary group of J. SER, III. VOL. I. UU 330 REV. T. P. KIRKMAN ON THE THEORY OF Wherefore we have this Turorem K, Let J be any group of the order kr formed on the partition of N elements N=Aa+Bd+--+Ce, comprising any a equivalent groups made each of A ele- ments, any b equivalent groups each of B elements, the equivalent groups being either with or without repetition of their substitutions of the order kr; and let I’ be any group whatever of the order | made with Gtb+e+e-+7 elements, of which the a first vertical rows contain only the a elements; the b next vertical rows contain only the 6b elements, &c. Every pair of groups JI gives a grouped group of krl substitutions, whose elementary groups are the groups of the order kr which compose J. The equivalent groups of the order kr may be any of the groups formed on the partition of N which have been enumerated in the preceding theorems; that is, any one of them may be determined either by one or by many cir- cular factors. 48. It is difficult to determine how many of the grouped groups of Ari substitutions thus formed may be presented as grouped groups of kv, substitutions formed by the pro- cess of Artt. (37, 38). But there is an enormous number of them which cannot be so presented. If, for example, we form by theorem G (26) a group G+ RG of the sixth order on the partition 93433 and then add to it either two or five derived derangements by the auxiliary groups T1223 Ory — 192 ge 32 221 221213 212 gaol: ed Ws GROUPS AND MANY-VALUED FUNCTIONS. 351 we shall obtain a group of 3-2-3, or of 3-2-6 substi- tutions which cannot be arranged as a group of Artt. (37, 38), that is, as a grouped group of the first class. § 9: Woven groups: Woven grouped groups. 49. It is known that, if N=A+B, any group G made with A elements of the order L can be interlaced with any group G’ of the order L’ made with B other elements, so as to form a woven group of LL’ sub- stitutions made with N elements. For example: the two woven groups 12345 6789 23145 7689 31245 6798 > 12354 7698 23154 31254 can be woven into a group of 24 substitutions. For a second example, on the partition 9=3:3=Aa, we can construct by aid of the auxiliary a2 3 231 312 the grouped group G of kri=3-2-3 substitutions, (48), 123456789 458762193 = 769125438 231589647 584693271 = 697238514 312847956 845971362 = 76314825 132489756 485793162 = 796138425 321856947 = 854962371 =. 967325814 213547689 5548671293 = 679214538, which is G’+ RG’, (26), with two derived derangements. G can be made into a woven group of the order 6’ thus : REY. T. P. KIRKMAN ON THE THEORY OF 123456789 231586749 —- 312846759 123459687 = 231589647 =. 312849657 123457986 =. 231587946 =. 3.128 4.7956 123459786 231589746 = 312849756 123456987 231586947 = - 312846957 123457689 231587649 3.128 4.7659, &e. Let this group of the order 6° be (G); then (G) {1 +458762193 + 7691254385 is a woven grouped group of the order 648. 50. 'Pake the partition N=Aa+Bb4+Ce+--4+J7 ASB, BSC, CSD, &e. Let Mz be the number of groups equivalent to any group G, made with R elements, of Sz substitutions, and such that G, shall not be equivalent to Gz, even though A=B. The model group Gz, is determined by a certain partition R=Ryyr,+ Rr. + Ryr34+ - - and by a certain system of circular factors, and may be any group constructible with R elements on this partition. Let F,, be the entire number of model groups of the order J, constructible with a elements. We can form of the order S, M% equivalent model groups J, with Aa elements, by selecting any one of the M, groups G, to be made with the first A elements 123--A, any other to be made with the next A elements of unity, &c. But this order of the elements is not necessary. We may choose the A elements @ times out of Aa in nt (Aa) ma(mA)* Wherefore there are, (Ta=1:2+3:-@), m(Aa@) ma(mA)* different ways. (M.,)" GROUPS AND MANY-VALUED FUNCTIONS. 300 ways of constructing @ equivalent groups J, with the first Aa elements of unity. By theorem K (49) we can form Bo 7 CAGE E(M 4) wae grouped groups H., each of 8,/, substitutions with the Aa elements, each containing unity. In like manner we can form with the Bd elements F(a) sep grouped groups H, with the next Bd elements, each of S,J, substitutions; &c. But further we need not choose the first Aa elements of unity for the groups J,. There are awN m(Aa)7(N— Aa) different ways to select the Aa elements out of N, and 7 iN ee) 7(Bd)7(N — Aa-— BO) ways to select from N- Aa the Bd elements. Wherefore there are nara m(Aa)m(Bd)m(Ce)-- ways to choose the Aa the Bd the Ce-- elements, and we have =U) UxF, QL)" ae x F,(M,)’ ERP x: _ w?N-F,B,C,- -(M4)7(Mz)?(MC)°- - Ly mambre-+ (mA)*(mB)? (wC)°- - ways of writing under the unity of N elements a grouped group H, of the order SJ, made with Aa elements, a grouped group H, of the order S,/, made with Bd ele- ments, &c. Let SA SzlSel,-- =F; and let the grouped groups 304 REV. T. P. KIRKMAN ON THE THEORY OF H, H, He:- be formed in any one of the Z possible ways. We can weave together these groups so as to form a woven grouped group of & substitutions. But it is evi- dent that before we weave together H,, H, He-- we can weave the a equivalent groups G., which, with their derived derangements, compose the groups J, of 8, substitutions. Then, instead of the S.J, substitutions of — Bya=Jdyt+Q),4+Q)4++++Q, 94, we should have TEL HtI E+ Qs + Qtgt + +Q, ah, a group of the order (8,,)“J,. Then weaving the groups {13 {He} {He} formed in any one of Z possible ways, we obtain a woven grouped group of (S.)"Ie(Sz)"lp(Se)"ler «= substitutions. 51. We have demonstrated the theorem following —. Turorem L. If N M; Sz Ff, J, be the numbers defined in Art. (49), there are constructible on the given partition of N aN -¥F,F,- s FB, (M_,)“ (Mz),°- (M,) mambre ++) (wA)* (7B)? -- (ad) woven grouped groups each of (S.4)“(Sz)"(So)°+ + (Ss)"lalole: +f substitutions. The largest group coustructible on this partition of N is evidently of (A)*(7B)?(aC)°: « (aJ)/rarbre: - 1 substitutions. This group is maximum; it has no derived derangement; and the number of its derived groups is that of its equivalents. For example: on the partition 10=3:2+2:2=Aa+ BA, where we have GROUPS AND MANY-VALUED FUNCTIONS. 335 M;= 1 —\Vio— ie we can form woven grouped groups, each of Bee Dre Dimi Ag substitutions, if S,=3; or each of 67.2?-2-2=576 substitutions, if S,=—6. And on the partition 10=3:14+3'-11+2.-2, Me i ot Vn we can construct, if we take S;=6, 8',=3, IT10 1:1°2+6-6-2? woven grouped groups each of 6-3-27-1-1-2=144 substitutions, all different groups from those above enu- merated. The distinctive characters of grouped groups and woven groups have, of course, been observed before. For the nomenclature, which will be found useful, I am respon- sible, and far from maintaining that it cannot be im- proved. § 10. On the construction of functions of m values. 52. Let G, AG, BG--, be any group of the order L formed with N elements, and its as 1 derived groups. L Let Pi =H UB ay - ar, be the product of the N variables ! VLyl3* Ly raised to any N powers. 336 REV. T. P. KIRKMAN ON THE THEORY OF Let @=P,+P,+ nea: +P, where P,P,-- are made by executing on the subindices of P, the substitutions of G. It is evident that ® will have = values, if the func- tions @,@,-- constructed on the derived groups AG, BG.--, are all different algebraically. When all the exponents aGy- -X are different, the func- tions ® &,,-- will be all different; for there are no two substitutions alike in the series G, AG, BG.-.- And since no derived group DG is identical with any equivalent of G, there will be as many distinct functions ®, of which no one is a value of another, as there are groups equivalent to G. If then ® has fewer than aN values, we must have L either . $=9, or Py= Pp, and the number of different exponents is =P,+P,+--, by the group G, having HN values. L Let ie + i Shy P,+F,= Son &e. be what P+F=S becomes by the substitutions of G. The function {Gi =8,+8,4+8;+--+8; has —— values. L 56. It is of importance that we should demonstrate also the following THrorem Q. If two equivalent groups G and G' give by any system of exponents the same function ®, this function ® has not TIN values. L Let us suppose that the given system of exponents is written over the elements in G and in G’, and that GG —7nG hae where the symbol =, affirms algebraic identity. 340 REV. T. P. KIRKMAN ON THE THEORY OF We must have either RG=,G, or RG == .G, where =|= , denies algebraic identity. If RG =|— ,G, R must exchange an element x, which carries an exponent 8 in G for an element carrying a different exponent. Let us suppose that R has the substitutions XY aq where 2, is an element carrying in P, and in G a different exponent from that of x,. The effect of the operation RG is to change 2, into x, in every vertical row in which x, appears in the function written in a column, and to disturb in the same vertical rows no elements which occupy in R their natural posi- tions. We have the two conditions RG =|= ,G RGR1=,G. Then the derangement of RG by R™ exactly compensates the algebraic disturbance produced by the operation RG, Now this derangement affects entire vertical rows of RG and can affect no row standing under an element of unity not displaced in R~', while the operation RG disturbs ~, in every row in which it appears. It is then impossible that this compensation of algebraic disturbance can take place, and that RG = ,G. Wherefore RG=,G if G’=,RGRe=G: that is, G gives the same algebraic function with its GROUPS AND MANY-VALUED FUNCTIONS. 341 derived RG, and the function ® has fewer than aN L values. Q.E.D. 57. If we take a maximum group G consisting of a nucleus group g and all its derived derangements, and if we select a system of exponents of which some are equal to others, we have only to consider how many of the groups equivalent to G give the same algebraic function g. ! The number of these equivalents is always given, if G be any of the groups constructed by the theorems which have preceded ; and there is no difficulty in determining the groups which will give the same function. Two equivalent groups which give the same ®, and which have e nonrepeated exponents, can always be so arranged that the e vertical rows under those exponents shall be identical in the two groups, and that the tactical difference between the groups shall be limited to vertical rows under repeated exponents. Let the system of N exponents chosen be a a times, 8 b times, y ¢ times repeated, &c., giving N=aa+b6+ecy+:-- We can readily satisfy ourselves, by inspection of the equivalent groups G, AGA, BGB".., or by consideration of the law that governs them, how many there are which give a function @ not given by any other, which function may however be a value of the function given by another. And inspection of the derived derangements of these groups, or consideration of the law which governs their formation, enables us to determine how many of the groups are to be rejected for the reason that the term P does not differ in algebraic value from MP, where MG is a derived derangement of the rejected group G, 342 REV. T, P. KIRKMAN ON THE THEORY OF There remains a certain number 7 of the equivalent groups, to which the system of exponents is applicable when all those to which it is inapplicable have been rejected. It remains to be determined how many of these r groups give distinct functions ©, of which no function is a value of another. Any one G, of these r groups gives the same function with the derangements GO, G02, G9, hse G0, (d) where t=(ra-ab-mc--)—1, which is the number of different arrangements of N letters possible by exchange of letters bearing the same expo- nents. For the function @ given by G,@,, cannot differ algebraically from that given by G,. 58. It will generally be the case that G, and p—1 others of the 7 groups have each m of these ¢ substitutions 6, 02+ °6,. Let us suppose that no group contains more than m of them, which are in G,, 0, 05+ +6, We write — G,=G,0,=G,%=--=G,6,, (e) The number of different derangements in the series (d) is in general reduced by these equations (e); that is, we have reductions of the form G, 6,=Gz, 0, C= Gr 0,, &e. 5 and instead of ¢-m derangements (d@) we find that there are but i-m=t, and that the function ® given by G, is formed also on ?¢ derangements and no more of Gy. Now these ¢’ derangements of G, are each the derived of a group equivalent to G,. We conclude that this func- GROUPS AND MANY-VALUED FUNCTIONS. 343 tion ® is a value of each of the functions constructed on ¢’ others of the r equivalent groups under consideration. We see, further, that the same function @ is constructed on the m+1 forms of the same group G, G,G,9, GzA.°-GaFn 5 and the first term in each of these m+ 41 constructions of @ will have the same algebraic value, since the derange- ments 0,0,--@,, make no algebraic change in Page ay ak The first substitutions written in these m+41 forms of G, are all different. Hence there are in G, m+1 substi- tutions which give the same term P, that is to say, S has every term m+ 1 times repeated, and loses in consequence ee terms. 59. We conclude that the p groups, which contain each the same series of m out of the t=(wa-7b-1e-)—1 substitutions, give P ig Si distinct functions each of em terms, which have each aN values L Ne We have thus disposed of p of the 7 groups, and can dispose in like manner of p’ of the r—p which remain. And we find in general that there are w of the r groups which contain none of the ¢ substitutions 6, O,+ + 0;. Each one G, of these wu groups will give a function , constructible alike on ¢ derangements of G,, which ©, has L terms and = values, and which is a value of each of the functions formed on ¢ others of the wu groups. Thus we determine the number of distinct functions 344 REV. T. P. KIRKMAN ON THE THEORY OF which have a terms and aE values. U+1 L Nothing is gained by permuting the exponents of the term P=x2%z8x. x? If, for example, we construct ® on P= vraigha?. - 29 we merely form @ and its values on the derangements by = of G and of its derived groups, G, AG, BG, &c. In this series of derived groups we find 32g 432g. BEG, &e., Za) 23 3 whose derangements by - are G, AG, BG-- &c.; that is, we have made the functions which are obtained by the first system of exponents, on G and its derivates. O14. Examples of the construction of m-valued functions. 60. The equivalent groups of the eighth order made with four elements are 1234. Oo 1234 2341 3142 4312 3412 4321 2.143 4123 2.413 3421 1432 324 1243 je 2143 3412 3714 4231 Boe 2143 ba12 qe G, G, Gs, which are maximum groups, each consisting of a group of four powers and of its only derived derangement. We have three distinct three-valued functions , by the GROUPS AND MANY-VALUED FUNCTIONS. 345 system of exponents (ays) = 3210; viz., putting 1°2°3 for i= 12272 4 D2 BRAN: Boa Aga? im 1422 TAS 229) a Boe i Dee — 1223 a Be eA ATP o 23471 fe ego i DelwAn NAS 222 in Be a 1252 YAS OeN i DEH Nore BPA? + 1224 A Beat te 2172 1 Aage oe The nine values of these functions of the sixth degree are all different. Let the system of exponents be (aByy). We see that no two of the equivalent groups will give the same function, because no two can be written so as to have their two first vertical rows identical. Hence we know that the three groups will give either different functions, or at least different values of functions. Any one G, of the three groups gives the same function with the derangement G4(1243). The only group which contains 1243 is G,; wherefore the function constructed on this group will be of four terms only. It is (aByy=2100) @,=172+4°34+2714374. The group G, gives the same function with its derange- ment G, (1243) =(1243)Gp, which is a derived group of G,. Therefore , constructed on G, is a value of ®; constructed on G,. It is @,= 1°24-2°343'44-41417444°34 324271. Let the system of exponents be aaS£. No two of the three groups G,G,G; will give exactly the same function, because no two can be written so as to have the same vertical row under a and the same vertical row under @, unity being 12273°4°. Any one of these G, gives the same function algebrai- cally with the derangement G 42134 Gy2143 Gyi243. G, has 2143; Gy has 2143; G; has 2143 and 2134. SER, III. VOL. I. YY 346 REV. T. P. KIRKMAN ON THE THEORY OF Wherefore G, gives a function of two terms only, which is (aaBB=1100) ®=12+4 43; and G, gives a function of four terms which is a value of the function given by Gy, viz., $,=12+234+34+41. 61. The first halves of the groups G, G, G; are groups of the fourth order, and will give six-valued functions. They give by the system of exponents (a6yd=3210) the halves of the functions $,9,9,. The group 1234—=—9 #302 2143 3421 gives no six-valued function by the system of ayy. because P= ata ay x{ =1243P where 12439 is a derived derangement of g. But the groups 1234 and 1234 22 An 3142 3412 | AGe4 At23 |) 2413 give each a value of the six-valued function th toe 17983747 ah 2° 2PAY LY ab 3741727 a 47182737, of which one form is 1724+2734+ 3744471. 62. The six groups each of the 20th order, made with five elements (18) give each a six-valued function by any system aPy6e of five exponents. The system of four exponents aSy6d gives only three functions; for each group Gy, gives the same algebraic GROUPS AND MANY-VALUED FUNCTIONS. 34:7 function with its derangement G,12354, which is the de- rived of another group. The three functions are (a@ydd= 32100) Vy = 19273 + 23471 + 33174 + 493?2 2374 + 47173 + 12422) as BES 3°4°5 + 1°3°5 +.4°275 + 29175 4°5"1 + 3°5°2 + 2°5°3 + 1°5°4 53120 + 5924 + 59971 + 52429 Va= 1293 + 2°5°3 +.4°1°3 + 5°4°3 DP Arn + 5°172 ae 1°5°4. + 4°25 435724 1A Be Geout a ENN 5°3°4. + 4°371 a 19275 ae 1°92 3°1°5 + 3'2°4+ 3°42 + 3°5°t V3 = 19223 + 2334 + 59122 + 32571 2°5°4 + 3°1°4 + 1°3°4 + 5°2°4 5°371 =e 12572) ae 33275 ab Dete2 Brae as 5°4°3 + DENS ae 1°4°5 4°1'5 + 4°2%1 + 4°5°3 + 4°3°2 The eighteen values of these three functions are all dif- ferent. 63. Let the system of exponents be aPyyy. Two of the six groups give values of the six-valued function Vy= 1°23 +2753 2741 +5712 4°52.+ 1°45 5°34+4°31 3°15 + 3°24. The other four groups give values of the six-valued function V5= 1723 +2741 + 37144 4°32 + 2°34 +4713 + 1°42 + 3°21 + 3°45 + 1°35-+4°25 +2715 + 4°51 + 3°52 +2°53+ 1°54 + 5°12 + 5°24 + 5°31 + 5°43 348 REV. T. P. KIRKMAN ON THE THEORY OF The systems of exponents a@yyy and aaB8BB are ex- cluded, because each of the six groups gives the same symmetrical function by these systems. The six groups of 20 (18) comprise each a group of 10. By these we can form twelve-valued functions, which are the halves of the six-valued already constructed. 64. The system aSyyy is not excluded here, but gives by any of the groups a value of the twelve-valued function (aByyy= 2.1000) V=V2+2Z+ 24+ 45+ 51+ 21+ V5+ 54443432; nor is the system aa excluded, for it gives by any of the groups a value of the twelve-valued function v =124234+34+4+ 51. The functions V,V.V3V;v comprise each a twenty-four- valued function and one of its values. There are six twenty-four-valued functions made with the system of exponents (a@yde) ; three with the system (aByod), one with the system (a8@yy), and one with the system (a8yyy), upon the models with which G’,G’,- -G’, Dein, e(Art. 18): 65. The six groups each of 120 substitutions made with six elements are thus formed, theorem F, (23). We first write the group made with five elements G’=S(pi+e) (mod 5) of Art. 18, with the addition of six final, thus ; 123456 241356 314256 432156 234516 413526 142536 321546 345126 = 135246 425316 = 215436 0 (H) 451236 = 352416 = 253146 154326 512346 524136 531426 543210. We next form the derivants (23), (@=2), (20), : 6 = (2°(@=.1)? +1) oa. +7+-= 645231, . 6 Y= (24a aD 2)*(2 a 2). ty 2) vreet 5 +7+ 5 465 132, GROUPS AND MANY-VALUED FUNCTIONS. i 6 Fy=(2"(1— 3)°6-3)°+ 3)moa.s tp +5 = 216543, | 6 V= (2°(1—4)°(6- 4) + Ana s+6 +g = 532614, ee 6 P= (2(1— 5) 5)? + Suns +g +2 = 341265. The five derived groups of (H) (¥\+ %4+ %,4+ %,4 ¥,)H are 645231 452361 523641 236451 364521 465132 651342 513462 134652 346512 216543 165423 654213 542163 421653 532614 326154 261534 615324 153264 341265 412.635 126345 263415 034125 426531 265341 653421 534261 342651 614532 145362 453612 536142 361452 152043 526413 264153 641523 415263 365214 652134 521364 213054 136524 423165 231645 316425 164235 642315 562431 62.4351 243561 435621 356241 541632 416352 163542 635412 354162 625143 251463 514623 146253 462513 256314 563124 631254 312504 125634 132465 324615 246135 401325 013245 2.54621 546321 463251 632541 325461 156432 564312 643152 431562 315642 561243 612453 124563 245613 456123 623514 235164 351024 516234 162354 214365 143025 436215 362145 621435. 349 350 REV. T. P. KIRKMAN ON THE THEORY OF The shortest way to satisfy one’s self that the 120 sub- stitutions form a group, is to write the five derangements BP oa eet eet 2), which will be found to contain exactly the substitutions of the five derived groups above written. The five equivalent groups to the one (J) above formed of 120 substitutions are 1235643123645 =J, 1236453123564 =J, 123465J123465 =J, 1236543123654 =J, 1235463123546 =J;. All these may be readily formed by writing out the derived groups 123564J, 123645J, &c., and then effecting a simple derangement of the three final vertical rows of the derived groups, whereby what follows is easily veri- fied. 66. The 73=6 groups each of 120th order, made with six elements, give six functions by the system of exponents (aByseC), each function of 120 terms. The functions are reduced to three by the system of exponents a@ydee, since each of the six groups G, gives the same function with its derangement (Artt. 57, 58), Gy123465, which is the derived of another group. There are then three distinct functions of the tenth degree, made with the exponents (aSydee= 432100), each function of 120 terms. The system of exponents aSyydd gives the same alge- braic function on (Artt. 57, 58) Gz, G~i23465, Gy124356, and G,124365. None of the six groups has either 123465 or 124356; for none has four letters undisturbed in any substitution. Two of the six groups have 124365; wherefore these give values of the same six-valued function of 60 terms. GROUPS AND MANY-VALUED FUNCTIONS. 351 The function is (aByy65= 321100) Y= 190794 + §°1742 + 4°3°65 + 2°5°31 + 1°2°65 + §°4761 + 67235 + 195724 + 41°62 + 3°5°41 + 3717254 4°5°16 + 6°4°31 + 3°2°54 + 6'1"23 + 655743 + 3°6°24 + 291756 + 5°6°34 + 6°3°51 + 2°9716 4 134753 + 599726 + 234715 + 2°67 14 + 496° 52+ 532713 + 196745 + 3°5°62 + 5°4°32 + 1°36 4 + 2°5°64 + 3°1°64 + 2°4°36 + 2°1°43 + 576721 4+ 9°4712 + 430751 4+ 671754 + 1°4°62 + 41°35 + 591763 + 196723 + 4°3°12 + 4°6°31 + 5°2°46 + 2°3%45 + .5°3°14 + 3°4756-+ 2°6°53 + 396715 + 695712 + 674725 + 672741 + 639724 + 4°5°13 + 42°36 + 1°5°36 4 199752 + 3°2716, The other four groups each give a value of the same function W of 120 terms of the same degree (a@yy56 = 3921100). 67. The system of exponents aa®@yy constructs the same algebraic function on G, and on (Artt. 57, 58) G 7214365 Gy214356 Gy~213465 Gyz124365 G 7123465 Gy124356 Gy213456. Of the six groups there are four which have the substi- tution 214365, and none of the four has any of the other substitutions above written. Wherefore these four groups all give a value of the same six-valued function of 60 terms, which is putting (aaSByy = 221100), Z= 12°34 + 29345 + 3-451 + 0325+ 3°0'45 + 6°5°42 + O'1"53 + O'2"1 4 + 1'4'53 + 05°34 +3125 + 42°31 + 53°42 + 2°34 + 1'2"45 + 476751 + 5°6712 + 1°6723 + 571723 + 471726 + 279716 + 974726 4 271765 + 64°91 + 5°3°61 + 574763 + 1757644 475°12 + 275°14 4+ 672751 tghg't2 +6145 + 1°4°36 + 472°53 + 4°6'25 + 179746 + 279751 + 572764 + 3°6°14 + 372746 352 REV. T. P. KIRKMAN ON THE THEORY OF + 275763 + 679712 + 174725 + 179765 + 571762 + 6423 + 425°23 + 5°31 + 2°5°3I + 2°1"36 + 5717394 + 371742 + 472716 4 1°6742-+ 4°5°O1 + 472%6 5 + 2°6"53 + 3°5°62 + 53°14 + 3°4 65. The other two groups have the substitutions © 124365 213465 214356, and they, consequently, each give a value of the same six- valued function U of 30 terms of the degree (aaGBryy = 221100). This function is U= 172794 4+ 571742 +674°91 + 274715 + 522246 + 5°4°OI + 672795 + 271756 + 629251 + 179764 + 371725 + 376724 + 579226 + 296714 + 526221 + 675743 + 174753 + 1°6745 + 574792 + 974712 + 299216 + 570719 4+ 471762 + 975741 4 571769 + 476°52 + 4°3°65 + 372754 + O'1'23 + 4°2°36. SUPPLEMENT. 68. The preceding investigations and theorems are translated, except Art. (39) since added, from a larger work, which I last year presented to the Imperial Insti- tute of France, in competition for their Grand Prix de Mathématiques for the year 1860, of which the subject was proposed early in 1858: “ Quels peuvent étre les nombres de valeurs des fonctions bien définies qui contiennent un nombre donné de lettres, et comment peut on former les fonctions pour lesquelles il existe un nombre donné de valeurs?” Three Memoirs were presented, but no prize was awarded. Not the briefest summary was vouchsafed of what the competitors had added to science, although it GROUPS AND MANY-VALUED FUNCTIONS. 353 was confessed that all had contributed results both new and important; and the question, though proposed for the first time for the year 1860, was, contrary to the very frequent custom of the Academy, withdrawn from compe- tition. (Vide Comptes Rendus, Mars, 1861). 69. One question of the Academy as above quoted is: “Comment peut on former les fonctions pour lesquelles il existe un nombre donné de valeurs?” The mode of doing this, which, so far as I can gather, has hitherto been considered satisfactory, is thus given in one of the very last contributions to this subject. (Vide p. 10 of Theses presentées & la faculté des Sciences & Paris etc., Mallet-Bachelier, 1860; or vide for a reprint of the same Memoir Le Journal de l’Ecole Polytechnique, 1861, vol. xxul., p. 120). The author is giving an account in his preliminary chapter of the established method of con- structing, on a group of the k” order, with N letters, a function of aa values. “Tout systéme conjugué (=group) peut étre considéré comme le systéme des substitutions inaltérantes d’une certaine fonction quw il est facile de former. On se donne une expression dissymétrique par rapport aux N lettres abcd--; celle-ci, par exemple, aa+8b+ yc -- +; puis on forme une fonction symétrique des valeurs que prend cette expression en y effectuant toutes les sub- stitutions du systéme conjugué; on prendra par exemple le produit (aa+Bb+rye--)(--)(-+). Cette fonction sera invariable par toutes les substitutions du systéme, qui ne font que permuter les facteurs les uns dans les autres; car si P,Q, R-- sont les substitutions du systéme con- jugué et si l’on désigne par F, ce que devient la fonction lorsqu’on y a effectué la substitution a, le produit sera F\FpFe--; aprés la substitution P il sera devenu FF; Foep:-- Mais les substitutions P, P?, Q@P-- sont évidem- SER. III. VOL. I. Zz ODA REV. T. P. KIRKMAN ON THE THEORY OF ment distinctes, et appartiennent toutes au systéme con- jugué: ce sont donc les substitutions 1, P,Q-- écrites dans un autre ordre. D’un autre cdté toute autre substi- tution, altérant essentiellement les facteurs altérera le produit.” Let us try this rule on the simple group abcd 1234 bcda 2341 re which is ee dabe 4123. We may take any asymmetric function of the letters abcd. Take, as directed, the function ad + be. The product which we are instructed to form is FFF QF p= (ad + bc) (ba + ed) (cb + da) (de + ab). It is perfectly true that this is unchanged by any sub- stitution of the group. Let us operate on it with the sub- stitution adcb abcd’ which is not in the group, and which therefore, by the last words quoted, ought to alter the product. We obtain for result, (ab + dc) (da + cb) (ed + ba) (be + ad), which is the same product still. In the same manner we shall find that it is unaltered by any of the three substitu- tions, dcba chad bade abcd’ abcd’ abcd’ none of which is in the group, and which by the rule given ought all to alter the product. This French rule will be found equally misleading, when applied to innumerable other groups, both simple and complex. It would not be quite fair to lay this error to the charge of the author quoted; for he does not pretend to give any GROUPS AND MANY-VALUED FUNCTIONS. 355 of his own results in this first chapter. He has given abundant evidence, in the subsequent parts of his Memoir, that he is competent to deal with this difficult subject, and has here simply repeated an oversight which probably runs through previous French writers, — the oversight of this principle: that in order to prove that a function of N letters has K values, it is required not merely to shew that it is znvariable by the substitutions of a certain group of the order substitutions. I believe that the true theory of the connection between groups and functions to oe constructed on them was first given, and I hope completely given, in the tenth section of the preceding Memoir (Art. 52). 70. I have now to correct an error of my own. ‘This a but also that it is variable for all other error is in the enumeration of equivalent groups in theo- rem D (14) and theorem E (17). My mistake was in the assumption, that no substitution made with N letters of the form pi+c can ever be of the N® order, for p>1. This assumption is confuted by my own remark at the end of Art. 19, page 293, that there are substitutions of the eighth order (N=8) of the form 5%+c, viz. in the derivate 57G. It is necessary here to determine the condition that the substitution pi+c shall be of the N” order, where p1, is prime to N, and ¢ is any of the numbers 012-- (N-1), N being any number whatever. We have, (Art. 15), (let me beg the reader to prefix 15 to the twentieth line of page 287), (pi+c)(po+c)(=)p%i+pe+e(=)(pite) (pitc)(pitpe+c)(=)p'it+p'e+pe+c(=)(pite)§ (pi+ c)(pi+c)"™(=)p™ + p14 p™ e+ eh +pe+e (=)(pit+e)”, 356 REV. T. P. KIRKMAN ON THE THEORY OF where I venture to introduce the new symbol (=) of tac- tical equality, to be employed when we equate a substitu- tion C to the product of its factors, as C(=)AB, or C(=)P”, or AB(=)P”, or CD(=)AB, &c. For it will be difficult to avoid confusion, if we continue to employ the same symbol, =, in the tactical proposition (pi+e)(qit+d)=A, when we mean by A the substitution pgit+tpd+e, and in the algebraic proposition ( pi+e)(qgi+d)=B, or =B, when we mean by B the number pqi? + pdi + qei+ ed. It is above evident, that, whatever p, m, or c may be, : pad Gis (pit e)"(= prt whence, if p, being prime to the composite number N, is also a primitive root of a certain congruence p” =1 (mod. N), (a) for a value of m which is a divisor >1 of N, there is a certain number of powers of pi+e of the form C; (pito(=)i ee *¢(=)ite, (mod. N). —1 That pi+c, may be of the N” order, when c¢, is prime to N, it is required, and it suffices, in addition to the condi- tions (a), that the congruence Pa, ae (mod. N), (that is, e=o) be true for no value of too by hypothesis, and ¢, is =1, or prime to N, ie (mod. N) p-1 is impossible. Therefore pi+c, is of the N™ order. And we thus see that the entire number M of substitu- tions of the N” order in the group G’=S(pi+c) (15) is not less (12) than Ry times the number of terms in the series 1D, Po Ps > of integers 1 of N. It is easy to shew that M is not greater than that number. The derivate pzG is the derived derangement : piG=Gpi=P)+P,+P,4 Py: - 0) G=14+04+0+.. =S(t@+c), and has exactly the same vertical rows; that is, (Art. 6), (ae Oe; are the same substitution. If then @ and P, be of the same order, 6” and P, will be of the same order. There are then as many substitutions P, of an order below the N” as there are powers of 6 below the N”, which number is N—Ry (12). 72. Wherefore it is proved that the number of substitu- tions of the N™ order in the group G’=S(pi+ce) (Art. 15) 1s Ryp, 358 REV. T. P. KIRKMAN ON THE THEORY OF where p is the number of integers 1 p, py ps--, 1 of N. This is true of all the groups both of theorem D (14) and of theorem E (17). The first are all equivalents of the simple group whose NR, substitutions are all of the form pite, (page 287) ; and the number of substitutions of the N” order in this group G’ (page 288), of which no one is a power of another, (viz. 9x p:+, whose powers of the N” order are 0 0" 0 @.. Pp pp - KEK i &c., where abc-+ are all prime to N), is the number p above described. If p>1 in G, it is plain that no equivalent to G’, QG’Q-, can have the powers of O==2 4 = 294-5 Ni, for every substitution of G’=S(pi+c) is given with 0=7+1, whose powers compose the model G (page 287). (Let me here beg the reader to complete the ninth line of page 287 thus: The simplest of these groups is formed on the model G). Tt follows that none of the substitutions, @ y &c. of the N” order in G’, can occur in any equivalent of G’; for G’ can be equally written as the model 1p ¢'d’-- and Ry-1 derived derangements of it; or as the model UX Xs; and Ry—1 derived derangements of it, &c. But every equivalent of G’ will contain, like G’, p sets of powers of distinct substitutions of the N” order, none of GROUPS AND MANY-VALUED FUNCTIONS. 359 which sets will be found in any other equivalent. We obtain, therefore, the number of equivalent groups by dividing by p that given in theorem B (12) of the equiva- lent groups of powers of a substitution of the N” order. 73. The correction required in theorem D, (14), is simply to write — instead of eee in the first line of it, adding the definition, that p is the number of integers, unity included, which are 1 of N. If the paragraph beginning at line nine and ending at line fifteen of page 288 be erased, the fourteenth and fifteenth articles are corrected. The groups of theorem E are all equivalent to the simple group G’=S(pi +o), in which p is every power of any primitive root of the congruence chosen x2” -1=0 (mod. N). The correction required at page 289 is to write in the nineteenth, twenty-fifth, thirtieth and thirty-first lines, IT(N-1 TI(N-1 ee ) for ao ); and to add the definition, that p is the number of the integers 1pp*--p"", which are powers of the primitive root employed of the congruence x" —1=0, when r is a divisor of N, and which are roots of no congruence Dee (mod. N), v-1 in which m1. But I suspect that the only possible form is Nan, where 7 is a prime number. We know that, in all the groups of theorems D and H, no substitution pite is of the N® order, even when p> 1, if ¢ be a factor of N; for it is found in PG=P,+P,+P,;+--, a derived derange- ment of G=S(i+e)=14+04+04 and has the same vertical rows with G, of which every substitution can be written, (6, page 279), G2 PB, 6 PR; wherefore if 6 and P, are of both the N® order, 6% and P, must be both of one order. When pis one of the p num- bers, P}=pi+1 is of the N” order, like 0=i+1; and as 6%, when a is not prime to N, is of an order below the N, P, is not of the N” order. Therefore P, =pita can be raised to the power unity, Pi(=)( pit ay=ypri+” Palsii where 77 o for rj1. I think it will be proved that N aud a must of necessity be powers of the same prime number. 75. It is worth remarking, by way of supplement to the GROUPS AND MANY-VALUED FUNCTIONS. 361 example of Art. (19), that when N=8, 5 is the only primitive root >1 of the congruence “1 L-1 = 0) (mod. 3), and we have 5°=1 (mod. 8); IT7 whence p=2, and there are (theorem D) = equivalent groups of 32, each comprising two sets of powers of sub- stitutions 0 of the eighth order, which have however common powers Gabe when a is a factor of 8. It follows by theorem A, (9), that there are a equi- = 64 made with eight elements. IT valent groups each of 1 When N=g we have only p=4 and p,=7 primitive roots >1 of the congruences 3— ie 2° =1 and you =? (mod. 9). IT8 , Hence p=3, and there are on (theorem D) equivalent groups of 54; whence by theorem A there are i equl- valent modular groups each of ee =g9:6-3, made with nine elements. When N=16-we have only p=5, ~\=9, Po=13; primitive roots >1 of the congruence 13 2 ivand ——*=0 (mod. 16) ; 115 84 groups each of 16-8 substitutions, and consequently by 15 theorem A, ats equivalent modular groups each of 8-4 II = AG 3 made with sixteen elements. 15 whence p=4, and we have (theorem D) equivalent SER. III. VOL. I. 3A 362 REV. T. P. KIRKMAN ON THE THEORY OF I proceed to give a brief sketch of several investigations, all of interest, and most of them of new interest, in this theory, which I hope soon to discuss completely in a second Memoir. 76. On the didymous factors of the substitution d. With every substitution ¢ made with N=Aa+Bb+Ce+--J7 letters, of the order K, which is the least common multiple of ABC.-J, there is given a certain number of systems of K square roots @,@,@,--d,x, such that the product of any tWO Gd, is a power of d; and every power of ¢ can be written as the product of K different pairs of such didy- mous radical factors of the system. Thus, with the group of powers a, (¢+ 1), (¢+2), (+3: a? CANS 1), =1, , ¢°, ¢: 79 there is given the system of N square roots of unity (254); (1 >8)(- 1) t3) (582) (0 Na) Q a =a a as ay; and we have, (putting 1-7 for @ in a, for the product aa), @,0,(=)t+1, aa,(=)i+2, aya.(=)t+3, &e. Agls(=)it1, Aa,(=)i+2, aa,(=)t+3, &e. AA(=)i+1, A:A(=)i+2, dste(=)t+3, &e. It is easy to form the system of such radicals corre- sponding to any substitution when the partition is N=N.1i1=Aa, and ¢ is of the N” order. Take, for example, the substi- tution 57683241 of the eighth order. We form the skeletons 12345678 12438765=a b= 57683241 8 3 4 6 7 2. 2 7 6 4 3 8 5 GROUPS AND MANY-VALUED FUNCTIONS. 363 in which the first vertical circle is given by 57683241, the second is the same circle reversed, and __ 18472635 _ 15362748 We next complete the vertical circles thus : 12345678 = 12438765 a & 57683241 86754231 a P 34216785 43217658 as h® 68752413 75682314 a p* 21437856 = 21346587 a; @ 75864132 68573142 ag p* 43128567 34125876 a, p’ 86571324 57861423 as. We see that P (= )ayda( =) Ags ( =) aga: + +» (=) 0;05(=) asm P'( =) a1s(=) a20,(=)asa5, &e. ae aes )AxG;(=)aszag, Se. Nie #(= ) Ant (=) d3o(=) A403, Ke. And there are sixteen equations of the form Ax) An( =) Mg, AsUgls( =) My, AM As( = )Az, Ady%(=)ag, &e., or of the form a,a,a,(=)a,, where pgr are consecutive radicals of the system. We shall speak of @,, a, a;-- as of the didymous radi- cals of , that is, of the group of powers of >, and any pair d,,4,, giving a,,d,=—' are didymous factors of ¢’. The whole group both of powers and radicals is given with any two consecutive radicals. Thus, if a,=143256, y= 2.13546, we have the skeletons 123456 143256 413526 213546 5 5 2 4 where 364 REV. T. P. KIRKMAN ON THE THEORY OF 14.32.50 x 213546(=)413526 ; and we complete the group of eight thus : 123456 143256 a @ 413526 213546 a, gf 543216 523416 as f 253146 = 453126 ay. This, as well as the group of sixteen preceding, is one of the groups enumerated in theorem G, (26). The partition of N is here 6=4-1+1-2, and K=4. 77. Whenever the order K of the group 1+6+¢":- is even, the didymous factors a,a,(=)¢” are always two of the same form, but those of ¢’”’*' are two of different forms. Thus a, and a, are of the form 6=2-1+1-4=Aa+Bb, whilst a, and a, are of the form 6=2-2+1-2=Aa+B0. When the order K of the group is odd, the didymous radicals of the system are all of one form. An important point is here to be noticed concerning the radicals of such a group as the one last written, formed on the partition N=Aa+1-2, where the group 1,¢,¢°-- has two letters undisturbed. The derived derangement RG of theorem G, (26), will consist of radicals having the same two letters undisturbed. But if we exchange those two letters, writing, for example, above, 146253 @, for 143256 a, 216543 a’, 213546 ay 526413 a’, 523416 as 456123 a’, 453126 ay, we have still a system of didymous radicals such that the product of any two is a power of ¢. The form @',a’; is 6=2+24+1-2, GROUPS AND MANY-VALUED FUNCTIONS. 365 and that of a’,a’, is O22. And these are the forms which are always seen in the didymous radicals of a substitution ¢ of the sixth order. For example: @’,;=146253 and 216543=a’, 351624 423105 423165 645231 564312 361452 215436 532614 632541 154320, of which the first set contains a’, and the other a’,, are didymous radicals of the powers of ¢,=364521, and of g$,= 245613, both of the sixth order. This modification of the derived derangement RG is not taken into account in the enumeration given by the theorem G, which however gives correctly the number of equivalent groups of the precise form R+RG therein specified. If RG becomes R’G by the modification here noticed, G+R’G is a form of group not specified in that theorem, of which the equivalents can be enumerated. This requires further developement, for which I have not space here at my disposal. 78. Tactical investigation of the groups of NN-1N-2 substitutions formed with N elements, when N-1 is a prime number (theorem F, 23). By a tactical investigation I mean one in which no nu- merical equations or congruences are necessarily used. The discovery of these groups was first published by M. Mathieu; and they were both found and enumerated by myself before I was aware that he had secured a prior claim. His demonstration is effected by an analysis of formidable difficulty (vide Liouville’s Journal, January, 1860) ; and my own (20), though more readable than his, as avoiding imaginaries, is sufficiently abstruse. 366 REV. T. P. KIRKMAN ON THE THEORY OF The following investigation of M. Mathieu’s theorems, when N-1 is the power of any prime number, will be probably found simpler than either. We begin first with N —1, a prime number. By a theorem of Cauchy’s, we can form with the N-1 elements 123--(N-—1) a group G of (N-1)(N—-2) sub- stitutions all of the form G=S(pi+e), where p has any of the N-2 values 123>-N-2, and ¢ any of the N-1 values 0123--N-2, by writing, for any element 7 of unity, pite, (mod. N-1). In this group G there are N —2 substitutions ending in (N-—1), which form a group g of the order N-—2; and G contains also g,, a group of the same order, in which the element 7 is undisturbed. The substitutions of this group Gy are S(pit+r—pr). Thus G can be written as the product of the group g=S(i+ 6) of N-1 powers, 1+04+0+4..4 0", where ae 3 0=2345--(N-1)1, and of N-1 groups g, of the order N-2, in each of which one element is undisturbed. That is, G=NNIr2" Ina: The N-1 didymous radicals of g are ROROK LA < where b= 8, ach 07 ad=F, &e. Oh CxO, od, — 0% bey = 6, &e. C50 = Os Chen 078 NaC And these N-1 radicals are the central powers of the GROUPS AND MANY-VALUED FUNCTIONS. 367 groups g,; that is, if Gu A AGe ANTS: we have dy= Ate), Let the N-1 groups g,9.:- be written, and let each group have written under it its N-—2 didymous radicals, vizZ., AyAgllz++Ay_» under gj, 6,6,5,--by_. under g,, &e. We have a system of N—1 groups /,h,f3-- of the order 2(N-2) in each of which, h,, is a didymous factor of 0, viz. %, Which is permutable with each of the N-2 didy- mous radicals of h,. 79. Every group of m powers = p ¢: : ¢”; when m is even, has one power d*” permutable with each of the m didymous radicals of the group. It is easily seen that if h, be any one of the N-1 groups Ahh: - iets, CH o0 (roy are the remaining N-2 groups. That is, if y be any one of the (N—1)(N—z) didymous radicals in the groups fh: « and 6 be any other, we have always V—=07200m".. OF Ory = 00, for some value of a. The didymous radicals in the group /,, are alternately of the two forms, (Art. 27), N-1=2.5=441.3 =Aa+BB N-2 2 and those in the group h, have all the element 7 undis- turbed. Let us denote by H the entire system of (N —1)(N—-2) N-1=2. +1-1, =Aa+Bé, 368 REV. T. P. KIRKMAN ON THE THEORY OF substitutions of G along with the (N-—1)(N-2) didy- mous radicals of h,h.-- made with N—1 elements, 123-- (N—-1). If we add N as a final element to each of these 2(N — 1) (N-—2) substitutions, G so augmented is still G’ a group of (N-—1)(N—2), and each of the augmented groups /’, is still a group of 2(N —2) substitutions, containing the aug- mented g’,. And it is easily proved that every pair xy of the first N-1 elements is found once, and once only, in the order avy in the c” and d” vertical rows of H written in a column, and once, and once only, in the order yw in the same two vertical rows of H, whatever c and d may be +A*~G, where Ag, &c. are N-—3 derived derangements of 9’, SG Ot - (0=234+-(N—1)i1N) by the substitutions of 9;. This group G’ consists of the cyclical permutations of the N-1 first places, of unity, of A, of A, of A®. &c. The didymous radicals of h’, are 0), @=a,A, d,=a0,A*, a,=—a,A?, &e. If then we write under a, a9, a0, a,0--, the cyclical permutations of the first N—-1 places of a; under dp, 0,05, 0,0°3 (AsO? >)» and under az, a30, a,0°, as0?- «, &c.; we shall form a,G’ the derivate of G’ by a, consisting of (N-1i)(N-2) substitutions all ending in 1. And this is the derivate of G’ by every substitution in a@,G’. In the same way we can complete the derivates 6,G’, c,G’, &e. of G' by OU, by ¢, &c., which have a different final vertical row of one letter. For G' can be written Gao t+ By + Biyo+-:, where BB?B?.- are the substitutions of g’,. 81. It can be proved, by a simple tactical argument founded on the property stated in Art. 79, concerning the position of every duad wy, that if a and 6 be any two of the square roots of H’, 6a,b=c is another. Let 6 be written below a), and under 6 write the sub- stitution a 6 —c; b then c¢ is the third of the series abe.- of the didymous SER. III. VOL. I. 3B 370 REV. T. P. KIRKMAN ON THE THEORY OF radicals of a certain substitution ¢. We have (Art. 76), P= ab=p=be=cd=de, &e. bab=c, chc=d, dcd=e, &e. Hence all the series abc-- can be proved to be present in H’. Let 8 be any primitive root of N-—1. Then one of the sets of N-2 radicals is always obtained by writing as the first line the substitution 18263. -BX-495-3 TgeaBra BA? (mod. (N-1)), made with N-—2 elements, and completing under each element the vertical circle 18% 8B". .6?8, (mod. (N—-1)). We then add the elements N and N—-1: to each of the radicals so obtained. | If under any one (0) of the ~(N~2) thus found, which have no letter undisturbed, we write the didymous radicals f : 0=2345--(N-1)1N, viz. iN—-1 N-2-- 432N (a) N-i1N-2N-3-- 321N (do) &c., we always find more than one of the quotients DM De be Sarre boas} eae ° 5 Go One nes Qo which are substitutions of the order N. Let d= bay =" be of the order N; then we have 0 present in H’ the series abcd: - of N didymous radicals of @; and as no two of dcd-- end in N, and as no two end with the same element, they are found one in cach of the N-1 groups h’,h’,h’;. +; and the N-—1 derivates of G’ above formed are bG’, eG’, dG’, &e. GROUPS AND MANY-VALUED FUNCTIONS. 371 bG’ contains bay=¢, cG’ contains ca=¢"’, dG’ contains da,=¢", &c.; that is, the N—1 derivates are OG’, PG’, PG’. -p* 1G". This, along with what has been already proved, that every substitution of these derivates is of the form OO =O"Cn, completes the tactical demonstration that K=G+4G4+¢G+¢'G+--6*71G is a group of N(N-1)(N—2) substitutions. All these groups K (Art. 23) are non-modular, (9). 82. This group (K) contains N different systems H constructible on N different groups go of the (N— 1)” order, which have each a different letter undisturbed. They all agree in this, that if the N—1 didymous radicals of the group g) be added to those of the groups 9,9,:- each of N-—2, after an exchange, under every one g,, of the fixed letter of g for r, we have in every case the same (N —1)? substitutions of the second order. The connection between these systems H and the groups of the N” order comprised in K lies herein, that these have exactly the same (N— 1)? square roots for their N\” See : (=) powers, and for their didymous radicals. m(N —1) Ry metrically distributed among the w(N —3) ‘groups K; that is, for N=6, the sixty groups L are found ten together in the six groups K. For N=8, the - groups L are found The groups (Li) of the N” order are sym- twenty together in the 120 groups K, We. _ 83. Applications of theorems H, (37), and A, (9). The existence of modular groups has long been known. I know not whether Betti or Cauchy first gave the general ole REV. T. P. KIRKMAN ON THE THEORY OF theorem, that if to a group G you add any number of derwed derangements of G, PG+QG+RG &. = GP+QQ4+QR+.-., such that every power and product of the derivants PQR. - is found in the series PQR: -, the sum M of substitutions so obtained is a group. This principle has remamed since the days of Cauchy little more than a broad and barren generality. It is not an easy step from this proposition to the answer of the following questions : 1. How many derived derangements at the most can be added to the model group G? “11. How many different inferior modular groups can be selected from the maximum modular group? 111. How are we to form these modular groups, both maximum and subordinate ? tv. What is the number in each case of the possible equivalent groups ? vy. How are we to determine, when a non-modular group, which is not simply a group of powers of ¢, is found, the number and order of modular groups of which it is the model ? I am not aware that these questions have been discussed elsewhere than in this Memoir. The first and the last of these questions are, if I mis- take not, sufficiently answered by my theorems, including the corollary to theorem A, (9); and this corollary, along with the theorem H, (37), is an instrument of new and considerable power, as we shall presently see. 84. As an example, which will introduce certain import- ant groups that are known, and a superior group which is probably new, let us take the partition a= 2 AN At Ag, The theorem H, (37), gives thirty for the number of equivalent grouped groups (F) of eight substitutions, all GROUPS AND MANY-VALUED FUNCTIONS. 373 except unity principal and of the second order. We have here Ni VV OS, ee A Nae Bt 105 32nd) 7 Gee By the corollary of theorem A, (9), it follows that there are 130) 7S 8.7.4.3. 30 substitutions in each of the thirty equivalent maximum modular groups V made on this partition. Hence there are thirty distinct transitive functions of thirty values made with eight letters, of the same degree, (57), that is, functions formed on transitive groups. The next and the more difficult question is, How is this maximum group V to be formed on the model F? In the first place, since F is transitive, there are as many substitutions in the sought group V that end in 8 as that end in any other element; and if we collect those terminating in 8, we shall have a group of 7-6-4 substitutions, in which, if we erase the 8, we shall have a group of 7-6-4 made with seven elements. In the same way, if we collect in this group of 7-6-4 the terms ending in 7, and erase the 7, we shall have a group J of twenty-four made with six letters. There are several groups of twenty-four made with six elements. The required group J in this case is the fol- lowing : 123456 435621 652134 342165 126543 564321 2.14356 345612 651243 431265 216534 563412 124305 346521 562143 432156 = 125634 653421 213405 430512 561234 341256 215643 654312, (J) 374 REV. T. P. KIRKMAN ON THE THEORY OF which is half the group of 48 = (72)*3, of Art. 51, made with 6=2-3 elements. This is a modular group. If we multiply it, after the addition of 7 final throughout, by the seven powers of G=7512420, we obtain a non-modular group L of 7-6-4 substitu- tions. The existence of the group just written is easily proved by the test, putting J’ for J with 7 final, L=J’(1+0404+--4+6)=(14+0404--0)J’. We shall return presently to the direct construction of these groups of 7-6-4, which are of considerable interest, by reason of the learned and instructive researches of Galois, who first affirmed their existence, as well as that of the groups of i1-10-6 made with eleven elements (vide Hermite’s Théorie des Equations modulaires, § xiv.), and also by those of MM. Betti of Pisa, Kronecker of Berlin, and Hermite of Paris, who appear all to have constructed these groups. Has anything more been done during the last twenty years in this theory of groups, apart from the recent com- petition, beyond the construction of these groups implicitly discovered by Galois, and Mr. Cayley’s analysis of groups of the eighth order ? It suffices for my purpose here to shew, that a vast number of non-modular groups, aud consequently often of modular groups made by adding to them their derived derangements, is given by the theorems demonstrated in the preceding Memoir. 85. To form the modular group of 8-7-6-4, we add 8 final to every substitution of L, and then multiply L’ thus formed by the group F 12345678 21436587 34127856 GROUPS AND MANY-VALUED FUNCTIONS. 375 43218765 56781234 65872143 78503412 87654321 one of the thirty equivalents formed on the partition (F) 8=2-4=Aa. We thus form 7-6-4-—1 derived derangements of F. For example: the derangements of F by 0=75134268 above written, by a=43562178 the second substitution of J’, and by 0a=31425768 their product in L’, are 75134268 = 43562178 = 31425768 86243157 34651287 = 42316857 57312486 = 21784356 = 13247586 68421375 12873465 = 24138675 31578624. 87126534 75861324 42687513 78215643 = 86752413 13750842 65348712 57683142 24865731 56437821 68574231; and these are also derivates of F. We have then the modular group V=FOJ)’, where J” is the group J of twenty-four angmented through- out by 78 final. © is the group of powers of 0= 75134268, and 6J7= 1) is the group of 7-6-4 substitutions having 8 final. By the group L, omitting 8 final in L’, we can form functions of seven letters having thirty values, and by the group V we make functions of eight letters having thirty values, which are, so far as I know, new. Our group yunar — Ly Ufo 376 REV. T. P. KIRKMAN ON THE THEORY OF consists of a group of 8-7 substitutions FO, multiplied by the group J”. This group of fifty-six is the product of the group F by the powers of €=75134268. The product FO=0F is written above. Every substitution m in it is of the form OX=p, » being a substitution of F, whence Fu=pF is the same derivate of F. Also AOA=Ap=v is another substitution of this derivate. But this is v=AT=DOA, which is therefore, by a known theorem of Cauchy, of the same order with @; that is, every substitution not in F, of the group of the order fifty-six FO, is a substitution of the seventh order. There are therefore 6-8=48 substitutions of the seventh order in this group FO of fifty-six. 86. The number of substitutions of any form N=Aa+Bé+-- having a circular factors of the order A, 6 of the order B, &c., is given by the theorem C, (12), and I believe it to be given nowhere else. - For if W be the number of equiva- lent groups there enumerated, R,W is the number sought of substitutions. Now there are 8-(II5)-6 different substitutions of the seventh order made with eight letters; and every group of fifty-six equivalent to this FO contains forty-eight of them. It follows that there are not fewer than equivalent groups of fifty-six; for every substitution of the seventh order will appear in at least one group of GROUPS AND MANY-VALUED FUNCTIONS. 377 fifty-six. But there is nothing to prevent the number of equivalents from being such that every substitution 0 of the seventh order made with seven letters shall appear more than once among them. Let 7 be the number of such repetition of every 0’; then is 1207 the number of the equivalent groups of fifty-six. Hach of these equivalents contains one of the thirty equivalents of F. Wherefore 12079 30 is the number of different sets of forty-eight substitutions of the seventh order that may be so added to a given group F as to complete a group of fifty-six. The valne of 7 in this case is r=2; and, in fact, if we multiply F, our group of eight, 12345678 21430587 34127856 43218705 &e., by the powers of any one of the eight substitutions fol- lowing, 24865731 4 24861375 Az 24758613 Az 24753108 ry 24685713 Vs 24683175 Ng 24571368 2, 24578631 As; of which the first gives the derangement FO=F), above written, we shall form eight different groups of fifty-six on this group F. There are, consequently, 8-30=240 equivalent groups of fifty-six; whence by the theorem A, (9), there are 240 SER, III. VOL. I. 3C 378 REV. T. P. KIRKMAN ON THE THEORY OF equivalent modular groups of 8 BIG =—8. 7+3 substitutions, made by adding to a group of fifty-six two derived derangements. This group of 8-7-3, like that of 8-7, is subordinate to the maximum modular group V of 8-7-6-4, and forms part of it. V can be written either as the group F with twenty derived derangements, or the group of fifty-six with two. The group of 8-7 is one of the groups included under M. Mathieu’s theorem, that whenever N is a power of a prime number, there is a group of N-(IN—1) substitutions. All these groups of N-(N-1) are modular, and form subordinate portions of the maximum groups given by the theorems H (37) and A (9). The theory of them all is like that of the above groups of 2°. (28-1); and the construction of these groups on any partition N=7', n being a prime number, is easy, as well as the enu- meration of their equivalents, which, being known, the theorem A gives the maximum groups to which they belong. 87. Non-modular groups of N-N-1 N-2 substitutions when N-1=p', p being any prime numéer. These groups were first discovered by M. Mathieu; but nothwithstanding the elegance of his demonstration, by the method of imaginary subindices, there are few readers who will not desire a mode of investigation somewhat less learned. The following is a sketch of a tactical demon- stration of these groups, and is little else than a repetition of that given above for 7=1. The group (L) of N-1 substitutions, which is formed by the theorem H, (37), on the partition GROUPS AND MANY-VALUED FUNCTIONS. 379 N-1=p-p*=Aa, is (L)=1,A,.A3:+ where A,Aq: : are all of the p” order. The substitutions Pies *hn—1 of any of the derived derangements of (L) are such that we have always Ngta= Her Pag = Mes bag = Pos rg ba=Fa> wherefore Ngtadg = beNg = by is a substitution of the de which is similar to 4,; and the derivate is composed of N—-1 similar substitu- tions. The group of M. Mathieu, K=LM,, of (N-1)(N—2) substitutions made with the elements 123--(N-1), which we suppose to be given, is K=L+p,L4+ 404+ 04+ + -u7"L, which can be written also as the product K=L-M,-M,-M;- ~My, where M, is the aes of (N—2) powers of p,. The derivate ae = hy when Nise is integer, that is, when p>2, is always a system of didymous radicals of MAghs °° Let the N-—2 didymous radicals of M, be written below M,. We have now N-—1 model groups M’\M’.M’;- -M’y4 each of the order 2(N—2), and each one having’a different element undisturbed in a vertical row. Let N final be added to every substitution of K, and in M’, so augmented let N be exchanged in all the N-2 didymous radicals with the undisturbed letter 7. The result is M”, which is still a group of N-2 powers with N-2 didymous radicals. 880 REV. T. P. KIRKMAN ON THE THEORY OF The other groups of the order 2(N —2) are yy M”,A7*, Ag M”, AQ", As MAS", &e. ; whence we see that if a be any one of the (N-—1)(N-2) didymous radicals added to K, Antrp =P is another, and we have Ant= B Xm whichever 2,, may be of the substitutions \A,A3- « - Let a,6,¢,:: be the didymous radicals of the group M’,. If we write all the derivates a,=4, + 4,4 + G,rot =: 6, L=6,4+5,4+5,rA.+ +: C, LU=C, FeO, A+ ,Ag+* we have thus formed a,K=a,L+a,¢,L+a,wL+4 :- since Dyn =n ORATION od Oe This derivate a,K will have for its final vertical row the letter » which is undisturbed in M”,. We can thus complete N-1 derivates of K, which along with K make (N-1)(N-2)+(N-1)(N-1)(N-2)=N(N-1)(N-2) substitutions, to every one of which (a) corresponds an- other (8) such that INA moll 0 88. It can be proved by a simple tactical argument similar to that mentioned in Art. 81, that if a and 6 be any two of the square roots of unity of the system of groups of the order 2(N — 2), M”, M’,: +, (which radicals, including ,**—, u**-, &c., when N-2 is even, that is, when p>2, which we suppose to be the case, make up the number GROUPS AND MANY-VALUED FUNCTIONS. 381 (N-1)(N-1)+N-1=(N-1)*), BaB=y is another. Now if we write under p,*%—=a,, which has two letters undisturbed in M”, those didymous radicals of any other group M”,, which have no letter undisturbed, we always find a given number of them £’8"-- such that Baus’ =$ is a substitution ¢ of the N” order. Wherefore (’a,0’=y, yBy=8, Sy8=e, &c., the whole series of didymous radicals of ¢, ¢’--, are in the system of (N-1)? radicals. As no two of these can terminate with the same letter, there is one of them, not a, in each of the N-1 derivates of K that we have formed, which are therefore the derivates BK yK 8K, &e. Now B’K contains B’a,=¢", yK contains ya,=¢~, &e. ; that is, our N -1 derivates are oK+¢@K4+--+"7K. Thus we complete the demonstration, which is to be slightly modified if p=2, that we have formed a group of N-(N-1)(N—-2) substitutions ; and these are the groups given by the important theorem of M. Mathieu, vide Liou- ville’s Journal, January, 1860. As a complete demonstration of these groups, when N-1 is a prime number, has been given in § 5 of the preceding Memoir, and as the elegant demonstration of the more general theorem is easily accessible in the Journal just referred to, it is hoped that the readers to whom the subject is interesting will pardon the brevity with which, under restrictions as to space, I have laid before them the tactical investigation, to which I do not presume to attach. more than a secondary importance. 382 REV. T. P. KIRKMAN ON THE THEORY OF 89. The group of N-(N-1)(N-2), when N-1 is a prime number, is always a maximum. It has no derived derange- ment. The group of N«(N-1)(N-3), when N-1 is a@ power greater than the first of a prime number, is never mazxi- mum. It has always derived derangements, which can be enumerated by the application of the theorems of this Memoir. Thus when N-1=2°=8 3 there are, as has been shewn above, 240 equivalent groups of 8-7 made with eight elements, and consequently 240 groups of fifty-six made with nine elements, in which the ninth element is final and undisturbed. Each of these 240 determines a group of 9-8-7. We have then 240 equivalent groups of 9-8-7; wherefore by my theorem A there are 240 equivalent groups each of TT qi 9'8°7°3 substitutions, of which each one contains one of M. Mathieu’s groups of 9-8-7. If in this group of 9-8-7-+3 we collect the substitutions terminating with 9, we shall, after erasing 9, have the group of 8-7-3 which has been found before, (86), and which is made by adding to the group of 8-7 two derived derangements of it. The number of equivalent groups of nine containing each eight substitutions of the third order is given by the theorem H to be 840. There are, consequently, by the theorem A 840 equivalent modular groups each of a? = 9.8.6 substitutions. M. Mathieu’s group of 3°-(3’-1) is part of this maxi- mum group of 9-8-6, and is given withit. Hach of these GROUPS AND MANY-VALUED FUNCTIONS, 383 groups of seventy-two can be completed in three ways, and in three ways only, into one of M. Mathieu’s groups made with ten elements of (Bir) 3508" = 1) substitutions. Then by the theorem A there are 3-840 equivalent groups made with ten elements each of 710 3-840 and comprising substitutions of the tenth order. M. Mathiew’s group of 10-9-8 is a portion of this maxi- mum group of 10-9+8+2. If in this group we collect the terms ending with 10, and erase 10, we have a group of 9-8-2 made with nine elements. This is a portion of the maximum group of 9-8-6, above found, and contains the group of 9-8 found by M. Mathieu. What is above extracted from the theorems of the pre- ceding Memoir is merely an example of what they will yield for any partition N-1=n", n being any prime. =10-9-8-2, 90. Groups of sN+1 N-N-1 made with N elements. Let N be any prime number. Let 8 be any primitive root of the congruence gN-) — 4 =0 (mod. N). = substitutions all of the There is a group G of N a form Brite, (So), where ¢ has any of the values 012--N-1. This group G=S(0%+¢) can be written as the product of g=S(i+c) of the N®* order, and N groups each of the order -(N—1), Ay, hg, hs: - 2 2 hy, which all contain, besides unity, only substitutions of 384: REV. T. P. KIRKMAN ON THE THEORY OF the -(N — 1) order formed on the partition N=(5N-1)2+1-1=Aa+Bo. 2 To each of these can be added “(N- 1) didymous radi- cals, and this can be done in =(N- 1) different ways, by mu | k One of the groups (h), hy, which has N final undis- theorem G, (26) ; =(N- 1)=+A4%, (25). turbed, is composed of >(N- 1) substitutions B%, fi, 8, [See and any one of its ral : systems of didymous radicals is i, P(Bt), P(B%%) - PAK", gi(=)t*4+ Ae -F°™) (mod. N) if A be determined by the condition gi=c, (mod. N), c being any number (=)3, or that ¢7 is a square root of unity, whence it readily fol- lows that where $(8"1) (8%) (=)A”%, (mod. N), one of the substitutions of hy. The former of these equations affirms the congruence t= (t+ ACG71-28°®)) 7+ AGI + AG - 8) —~A(i* 4 AG 8A) , or that ii a A(i ou jN-9))) == Se oe Ati a A(i— GHN-8)))3N-D | which is 1+A-—AP™) = 14+ A-A(i14+ Ai - 80%), This is true on condition that XN) = (614 Ai — BO-))19-Y = (i)?*, (mod. N). GROUPS AND MANY-VALUED FUNCTIONS. 39) Now whatever element powers >0 of 8”. Then by adding to V=o the Nos equations 2 B°V=o, B*V=o, &e., which will introduce no higher powers of 8", as BX" = 1, we can eliminate linearly all these powers. If the group has any existence, we shall obtain a result containing BC free from mnkh, which will be a congruence Fe=o (mod. N). The integer solutions of this, which are not powers of N-1 8, give each a distinct set of N- didymous radicals, which sets, with their cyclical permutations, will complete the group G=S(@i+c) into as many distinct groups of - (N+1) N-(N—-1) sub- stitutions. We readily find the resulting quadratic for N=7, giving 388 REV. T. P. KIRKMAN ON THE THEORY OF c=3 and c=5. And it is easy to satisfy one’s self that the equation Q is satisfied when N=11, by taking c=7 or c=8 for the determination of A in $7; that is, Q is satis- fied when N=7 by B=-1 and C=-2, or by B=-2 and C=-32, and when N= 11 it is satisfied by B=5 and C=4, or by B= and, C2. 94. But there is no need to attempt the eliminations to which the condition Q invites us. We can much more readily settle the matter by tactical considerations; that is, we can determine whether a given value of ¢ in gi=t i +AGt— FO) =Brt-_ Ce gives one of a system of didymous radicals which will complete the group in question. If the equation Q be satisfied by the system, it is satis- fied when 6” =(8”=(°=1. We must have, putting k=0, and) 69, pili+ h)pii—h)(=)G+2)$6"G-y), whence comes $ili +h) bi(=) (+2) $(BC-y)) 644)(=) 6+2) (G-2). By feeb 'Z, As there are N values of # and only of x, there will be one or more values of f# that will introduce 6’= 8°=1 into the right member. If then the group exists, spi= $ili+h) $i(=)6+2) 66-1) (=) 66-0 F2=™ This affirms that wi, the h’” power of the substitution of the N” order, x, which is determined by the circular factor $i, differs from a certain cyclical permutation $(i-?) of $7, only by the addition of the constant z to every ele- ment; that is, if oi= abcde: -, yisa’b'c'de': «, GROUPS AND MANY-VALUED FUNCTIONS. 389 the circle of differences b-—a, c—b, d-c, e-d.- is exactly the circle of differences ba, c-0', d-c', e-d'-- begun at some other point than 0’—a’. We have only to form the N powers of the substitution dX, by writing under 1 in unity the vertical circle ¢2, of which the first element is always 1, whatever c may be, and then completing the same vertical circle under every element of unity. If there be no power of X whose circle of differences is that of $2, the group sought has no existence. If there be such power or powers, we have found ¢i consistent with equation Q, and as all the unknowns of Q in the right member can be determined in terms of B and C, that is of A in ¢7, we have proof that the group exists, for the system of didymous radicals given with ¢#. g5. It is easy to prove by this method that no such group of <(N+ 1) N-(N-1) exists for N=13 or N=17 or N10; The substitutions ¢i which give the sought groups are fon IN —_7), gi= Oe 1462537, and Gis 501+ 30°= 1432657 5 and for N=11, pi= 3t '—- 20'= 18439072564, and gi= 5 '— 4i°= 10439768524. The four groups thus found of the order * (N+1)N- (N-—1) may be thus expressed as products, denoting by {H},, the group of powers of the substitution H of the m” order. 42345671%,5 1357246239 1426735 t4$ 1462537 }o= FE §2.3456712,$ 1357246295 134.76523,5 14326057 t,= EF" 12345671 5741357246 $55 134705 3 390 REV. T. P. KIRKMAN ON THE THEORY OF 123456789001 $114 3691470258055} 381950724605, x 165189347200},;=E 223456789001 $114 36914.70258a};5$ 2407185936054 x 156298341 700},;=E). The three last factors of F or of F, are a modular group (9) of twenty-four made with six elements (234567). The three last factors of E or of E, are a non-modular group of sixty made with (1234567890). The last two factors of any of the four groups are modular groups also, of eight or of twelve, given by the application of the pre- ceding theorems. All these groups of 8-7-3 and 12-11-5 are maximum and non-modular, whose equivalents can be easily enume- rated, as can those of the inferior groups exhibited. The group F is formed by adding to G=S(2%+c), of the twenty-first order, 7-7-3 substitutions of the form xi=6(2"t— a) +4 2(2"i—a)?+ 8. The group EF” is formed by adding to the same G, 7-7-3 substitutions of the form xix s(2%— a) +3(2"— a) +b, where a or 6 may have any of seven values 0123456. The group E ‘is formed by adding to G’=8(3"-2+ ¢) of the fifty-fifth order, 11-11-5 substitutions of the form Xi=5(3"i- a))* — 4(3"G-— a) +8; and E, is made by adding to the same G’ as many of the form ais iat Hi=3- (3b a))*-2.-(3"-@))* +4, where a and 0 have each any of eleven values. The groups F and E, are those given by M. Hermite at page 63 of his Théorie des Equations modulaires, Paris, 1859. It may not have been observed before, that there is one, and one only, of the equivalents of F or of KE which has for a factor the same cyclical group of 7-3 or of 11-5 substitutions. The functions constructed on F or E will have no value in common with those given by F, or Ky. GROUPS AND MANY-VALUED FUNCTIONS. 391 96. There does not appear to be any general theorem on groups of the order =(N+ 1) N-(N-1) when N is prime, to which the groups above found are to be referred. The true generalization of these theorems will be found in the ¢actical path which has so readily conducted us to the group of 7-6-4 and to the higher group of 8-7-6-4 of which, when augmented by 8 final, it is a factor, viz. by applying the theorem H (37) and the corollary of theorem A (g) to partitions of the form p N=Aa+Bd+-- =A-Aaq,+B-Bd,4+--- Thus we easily prove that there are, taking w=30 for the number of auxiliary groups of eight, 6 ° ° 8—4 Tc ernie equivalent groups of sixteen, containing each fifteen prin- cipal substitutions of the second order; whence by the corollary (9) there are S maximum equivalent modular groups of 16-15-12-8-7-4, made with sixteen elements, which may be treated as we treated the group of 8-7-6-4 in Art. (84). 97. On grouped groups, and the forms of the operative or auxiliary groups. In the preceding Memoir there are (38, 39) examples of the mode of operation on a model group, that is, any group containing unity, by an auxiliary group, whose ele- ments represent each one an elementary group of the model group. We thus produce a vast number of modular groups, by adding to a model a defined series of derived derangements, which I believe cannot be found by any method elsewhere directly indicated. What follows is a brief account of my continuation of the investigation there opened. The exponents of the circular factors described (37, 39) in theorem H are such that, by their variation, only 392 REV. T. P. KIRKMAN ON THE THEORY OF groups are obtained equivalent to that given when all the exponents are unity. Thus, for example, if we operate on the model 123456789 231564897 (H) 312645937 by the auxiliary 1 2°32? 2igia (K) 34 2%, which adds to (H) its derivates by _ 645-897-123 _ 293° ~ 123°645-897 1 23” 1 and by Q, — 897: 123+645 _ 3°41 2° "123-645-897 1 -2°3* we form a group of nine, equivalent to that obtained by operation with the auxiliary 123 231 (hy) 212: But there is a curious extension of this theory, whereby we obtain groups of nine not so equivalent. Let the operative group be 1233 2°31 (ke) ail 22 with ‘the definitions; 2°=2, 9*=3) 47=1. We find that (4,) is a group by the test DHS a tes Weta oY ae lee as SOUS for first, the operation 2731 on the subject 3712? puts 2? for 1, 3 for 2, that is 37 for 2, and i for 2, thats) 17 for au: and the operation 3712” on the subject 2°31 puts 3? for 1, 1 for 2, that is 1” for 2?, and 2? for 3; and secondly, 172738 == 122 is the model, whatever a may be, written in a different GROUPS AND MANY-VALUED FUNCTIONS. 393 order: that is, 122727 == 220504897; 312645978 123456789. The result of operation with (k,) is to add to the model H the two derivates 564789123 645897231 456978312 897123564 978231645 789312456, which complete a group of nine containing three cube roots of 231564897, and three cube roots of 312645987, all six of the ninth order. If we operate with each of MeN 11D" Bi evte D2) i 12 Bir) PU Cunl eA eral ie) dete Watiedichal pdtv ey 37172, 3 172%, 351 23, 38192, 2 152% which are all alike groups, we add to H five more sets of six substitutions of the ninth order; and we thus solve the curious problem in evolution, to find the eighteen cube roots of 231564897 and the eighteen cube roots of 312645978. It will be found impossible to modify the vertical rows of these operative groups, as we can those of Art. 39, so that the same element shall everywhere shew the same exponent. 98. Let the model be 12345078 4 21436587 and the auxiliary Ne 8 Ge ey 3°4°1 2 ”) ABQ 2D ie SER. III. VOL. I. 3E 394 REV. T. P. KIRKMAN ON THE THEORY OF This (X’) is a group, as appears by the tests, 27143°(k’) =(k’)27143”, &e. We have in the case of this model, if 1 =F Qi ct &ec, io 1, Dm). 2k AAR The result of operation by (A’) on K is 12345678 21430587 43127805 34218756 65871234 56782143 78653421 87564312, a group first divined and defined by Mr. Cayley in an elegant little paper in the Philosophical Magazine for 1859. The difficulty of the step from the analytical definition of a group to its actual construction, is shewn by the fact, that Mr. Cayley did not succeed in constructing this group till long after he had published its definition. I hope that one great use of this Memoir will be to facilitate the tactical construction of groups, as well as the enumeration of their equivalents. Hereby the fonctions bien définies (68) of the Paris Prize Question for 1860 will be accurately formed and exhausted. This group of Mr. Cayley’s has six substitutions of the fourth order all square roots of 21436587. If we operate on the same model with mee By as 271 4°3 3°41 2? 4°3°2 1, (4") we obtain the group 12345678 21430587 43128750 GROUPS AND MANY-VALUED FUNCTIONS. 395 34217865 65781243 50872134 87653412 78564321 equivalent to the preceding, which has six more square roots of 21436587. What has just been done isa case of a more general theorem on grouped groups. The two following a eek 2) BE af 438 2h 4%3 37441 2 374 1 24 4 342 1%, 42nd 1, are true groups, if we define that 1, 1a, 2-13, Priuyg, and that no substitution is changed in value by having its four exponents increased each by the same number. For example : 374212, 43921 = 2194291 = 2.19423 = 941434, We may write in either of these groups, for v, any square group uw of powers of a substitution of the (2d- 2)” order made with 2d—2 elements, v being each one of the four elements 1234, and for v” the result of d—1 cyclical permutations of the vertical rows of w. For example: if d=4, we have by the latter of the two auxiliaries the group following : 123456 78g0ab cdefgh wklmn 234561 890ab67 defghe jkimu 345612 go0ab78 efgyhcd klmny 456123 0ab78q fghcde Imniyk 561234 ab789q0 ghedef mniykl 612345 b78q0a hgcdef myklm 0ab789 123456 Imnijk cdefgh ab7890 234561 mnijkl defghe 395 REV. T. P. KIRKMAN ON THE THEORY OF b78g90a 345612 nizklm efghed 7890ab 456123 iklmn fohcde 890ab7 561234 gklmni ghedef goab78 612345 klmnijy hedefy Sohcde ikimn 123456 oab789 ghedef jklmnt 234561 ab7890 hedefg kimniy 345612 678900 cdefgh Imnijk 456123 7890ab defghe mnijki 561234 8q0ab7 efghced nijklm 612345 goab78 Imnyk fohede 7890ab 123456 mnijkl ghedef 89q0ab7 234561 nijklm hedefg goab78 345612 iklmn edefgh oab789 456123 jklmni defghe ab7890 561234 klmniy efghed b7890a 612345. This group contains only one square root of unity, 0=4561230ab78qfghcdelmnijk, with two of its cube roots of the sixth order, six of its square roots of the fourth order, and twelve of its sixth roots of the twelfth order, besides two substitutions of the third order. Or we may consider the above group of twenty-four as constructed by adding to the model group of the sixth order, which begins it, its three derived derangements by Dat Ages 20st 2 te Bae oa aa ae where 1, 2, 3, 4, like p,p,-- in Art. 27, stand for the four circular factors of the model group. And we may, by this method, easily construct an equivalent group of twenty- four on any model group equivalent to the one above employed, whether its circular factors, as above, are, or whether they are not, composed of contiguous elements of unity. GROUPS AND MANY-VALUED FUNCTIONS. 397 I hope soon to have the honour of presenting to the Literary AND PuriosopHicaL Society or MANCHESTER a second Memoir, in which this Theory of Grouped Groups will be discussed in detail. Here I shall merely remark, that it will often be found necessary, in handling the theorem H, (37), to include in the number w of equivalent auxiliary groups certain of those above described, which are not identical, at least as operators, with groups of the ordinary form, in which the same element can, by modification of vertical rows, be made to shew the same exponent, (39), whether unity or some other, wherever it appears. Note on Art. 39, page 321, of the Memoir on the Theory of Groups. _I have neglected to observe, that if the group (g’) of page 319 be transformed, with the understanding that the square groups of the fourth order, written at page 321, are to be substituted for the elements, the transformed group (9’') will beyn23) 23214 3212" 13224 2221+ 213, which difters) from! (@’) by the addition of 3 to the last exponent of each triplet, the definitions being of course in this case 15=1, 2>=2, 3°=3. The two last written triplets of page 321 should be 3°21? and 27193. The reduction of exponents named in the fourth line of page 322 proceeds by the addition of the same number to every exponent in a vertical row of the auxiliary, the definitions beg rtl—y, atl», &e., if ¢ be the number of vertical rows in the square groups to be substituted. AUS Sede 398 REV. T. P. KIRKMAN ON THE THEORY, ETC. CORRIGENDA. Page 279, line 2, for be G, read be in G. 280, 287, 287, 289, 288, 289, 99 39 aE) 9, for AP, read A. complete the line thus: “formed on the model G.” for correction of Art. 14 and Art. 17, see Art. 75. line 22, 29 29 16, 255 23 and sl -for prime, read primitive. for occupy, vead occupy in the same order. for Be =Ce, read Be, =Ce,. Jor tk, read rk. for K, read kh. Sor K(AK + (2-A))Rz, vead k(Ak + (l-A)R; ). for V can be, read It can be. Sor Ma, read ba. Sor 37, read 78. 399 XXIII. — Remarks on the Theory of Rain. By My. Joszeru Baxenve Lt, F.R.A.S. Read March 29th, 1860. Ir has been well established by numerous carefully con- ducted experiments, that the quantity of rain received by a gauge placed on or near the ground is almost invariably greater than that received by a similar gauge placed at a greater elevation in the immediate neighbourhood ; and, in explanation of this remarkable fact, meteorological writers have generally adopted the hypothesis advanced by Pro- fessor Phillips (Report of the British Association for 1833, p- 410), “that the whole difference in the quantity of rain at different heights above the surface of the neighbouring ground is caused by the continual augmentation of each drop of rain, from the commencement to the end of its descent, as it traverses successively the humid strata of air at a temperature so much lower than that of the sur- rounding medium, as to cause the deposition of moisture upon its surface.’ In support of this hypothesis, Professor Phillips remarks “that it takes account of the length of descent, because in passing through more air more mois- ture would be gathered; it agrees with the fact that the augmentation for given lengths of descent is greatest in the most humid seasons of the year; it accounts to us for the greater absolute size of rain-drops in the hottest months and near the ground, as compared with those in the winter and on mountains; finally, it is almost an 400 MR. JOSEPH BAXENDELL inevitable consequence, from what is known of the grada- tion of temperature in the atmosphere, that some effect of this kind must necessarily take place.” Now, although it must be admitted that the tempera- ture of falling rain is generally below that of the air — near the ground, yet if we proceed to determine the temperature of a rain-drop at the commencement of its descent, from its known rate of augmentation in fall- ing, and from its temperature when it arrives at the surface of the ground, we shall obtain a result wholly inconsistent with known facts, and therefore fatal to the hypothesis. The most complete series of observations with which I am acquainted, of the quantities of rain falling at different heights, is that made at York, in the years 1832-5, by Mr. Gray and Professor Phillips at the request of the British Association, the results of which are given and ably dis- cussed by Professor Phillips in the volume of Transactions of the Association for the year 1835. Three gauges were used in these observations: the first was placed on a large grass-plot in the grounds of the Yorkshire Museum; the second on the roof of the Museum, at an elevation of 43 feet 8 inches; and the third on a pole g feet above the level of the battlements of the great tower of the Minster, at an elevation above the ground gauge of 212 feet 103 inches. The total quantities of rain received by these gauges during the three years of observation were as follows : Pst Salleer we wees cee 65°430 inches. Diy Moma eeee ee uenen 52 1OQme ts 26 BaP iG eas ean so aia 28072 3 From these numbers it appears that the ratio of increase of size of a rain-drop is 0°679 for the last 213 feet of its fall, and 0'254 for the last 44 feet. A very able discussion of the whole series of observa- ON THE THEORY OF RAIN. 401 tions, with reference to the temperature of the seasons, led Professor Phillips to the following formula for calculating the difference between the ratios of the quantities of rain received on the ground and, at any height h, the value of the coefficient p depending upon the temperature ?¢’ of the season : Calculated by means of this formula, the mean height of the point at which rain begins to be formed, is 1747 feet; and the height at which the quantity of rain is only one-half of that which falls on the ground, is 356 feet. Assuming the mean temperature of newly fallen rain at York to be 48°; and taking the latent heat of vapour at 1210° at the temperature of 32° Fahrenheit —- the value adopted by Professor Espy in his Meteorological Reports and Essays, —it will be found that a rain-drop cannot acquire the increase of size indicated by the observations, by the condensation of vapour upon its surface, unless its temperature, when at a height above the ground not exceeding that of the top of the tower of York Minster, is below—434° Fahrenheit! From this result it is evident that only a very small portion of the total augmentation of a rain-drop can be due to the condensation of vapour upon its surface, and that by far the greater portion must be owing to the deposition of moisture which has already lost its latent heat, or heat of elasticity, and which is, therefore, not in the sate of a true vapour, although on the other hand, its invisibility in the atmosphere under ordinary circumstances, in the form of cloud or fog, renders it difficult to suppose that it can be in the ordi- nary liquid state. We have just seen that at a height of 356 feet, the quantity of rain is only one-half of that which falls on the ground; and it is evident, therefore, SER III. VOL. I. oF 402 MR. JOSEPH BAXENDELL that a shallow stratum of the lower and comparatively clear atmosphere, supplies as much rain as a densely clouded, and much deeper stratum in the higher regions. As these remarkable results may raise doubts as to the general correctness of the methods of observation, which have been used to determine the quantities of rain at different heights, I may here mention an important fact, for which I am indebted to my friend Mr. Binney, F.R.S. In descending the shafts of deep coal mines, Mr. Binney has observed that the drops of water which drip from the upper part of the shaft increase to an extraordinary size during their descent to the bottom. Evidently the same principle is here in operation as in the case of a rain-drop falling through the atmosphere, and Mr. Binney’s obser- vation affords a valuable confirmation of the general accu- racy of the results of the observations which have been -made to determine the rain-fall at different elevations. That the whole amount of a fall of rain is not derived from the direct condensation of vapour at the time that the fall takes place, is apparent from other considerations than those which depend upon the different quantities of rain at different heights. It is supposed by some meteoro- logists that the mild temperatures of the higher latitudes of western Europe are due to the heat which is liberated by the condensation of vapour during the frequent pre- eipitations of rain which take place on or near the coasts ; but if this view were correct, the mean temperature of rainy days ought to be considerably greater than the mean temperature of the year. A discussion of the Greenwich observations for the years 1852, 3, and 4, and of the Oxford observations for 1855, 6, and 7, with reference to this point, has given the following results : ” ON THE THEORY OF RAIN. 403 Greenwich Observations. Number Mean Temp. Mean Temp. Year of (0) of Rainy Days Rainy Days the Year 1852 152 51°39 50°66 Ks 1853 184 47°62 47°49 1854 145 48°80 48°80 Genera! Means. 160°3 49°27 48°98 Oxford Observations. Number Mean Temp. | Mean Temp. Year fe) ts) oO Rainy Days Rainy Days the Year 1855 "140 49°34 47°10 1856 154 49°09 48°70 1857 146 49°96 50°40 General Means. 146°6 49°63 48°73 It appears, therefore, that the excess of mean tempera- ture of rainy days, over the mean temperature of the year, on an average of three years, is only 0°29 by the Green- wich. observations, and o*go by the Oxford observations ; but as the winds which bring rain come principally from warmer latitudes, the mean temperature of rainy days ought, on that account alone, to be greater than the mean temperature of the year. Dividing the winds into two groups, northerly and southerly, it appears from the Oxford observations that out of 218°5 days of fair weather in the year, the wind was from the northern half of the compass on 131°5 days, and from the southern on the remaining 87; but out of 146°5 rainy days the wind was from the northern half on only 64°5 days, and from the southern on 82. Moreover, the quantity of rain which fell with winds from the southward was nearly four-tenths greater than that which fell with winds from the north- ward. Calculating the mean temperature of rainy days from the mean temperatures of the winds which prevail on those days, the result is 50°05; but we have seen that 4.OA: MR. JOSEPH BAXENDELL the observed mean temperature is only 49°63, or 0°45 less than the computed. It appears, therefore, that a wind accompanied with rain is, in general, sensibly cooler than the same wind attended with fair weather, and that what- ever may be the mode of formation of rain it may be regarded as a cooling process; and this view is borne out by the fact that the mean temperature of the days next after days of rain is sensibly less than that of the days of rain. According to the Greenwich observations the dimi- nution is 0°29, and according to the Oxford observations it is O° 1g. But if the vapour brought by a rainy wind retains its latent heat up to the moment that actual preci- pitation of rain takes place, the sudden disengagement of this heat, although occurring in the higher regions of the atmosphere, ought to have a very sensible effect in raising the mean temperature of rainy days; but as no such effect is produced we may conclude that the greater portion, if not the whole, of the moisture from which the rain is formed, had previously lost all its latent and also a small portion of its sensible heat. The questions now arise — ist, What becomes of the enormous quantity of heat given off by the vapour which is condensed in the atmosphere? and 2nd, As the mois- ture which forms rain is not in the state of a true vapour, is it in the ordinary liquid state, or in some other state not hitherto recognised by meteorologists and chemists ? With regard to the first question, it may be remarked that air nearly saturated with vapour, has probably a greater power of radiating heat than dry air. The upper portion of a wind charged with vapour would therefore undergo a rapid cooling, and as the vapour which loses its latent heat does not immediately affect the transparency of the air, this process would go on unchecked for some time, and would gradually extend to the lower strata; the vapour which had lost its latent heat would also gradually ON THE THEORY OF RAIN. 405 descend and accumulate in the lower atmosphere, until at a certain stage of the process clouds and rain were formed. This view of the subject is supported by the well-known fact, that the rate of decrease of the temperature of the atmosphere with the height, is greater in rainy than in fine weather; and it appears likely to lead to a satisfactory explanation of many important atmospherical phenomena. With respect to the second question, it is difficult to offer any plausible conjecture. There can, however, be little doubt that vapour deprived of its latent heat often exists to a considerable extent in the atmosphere without sensibly affecting its transparency ; and, indeed, it often happens that the atmosphere is unusually transparent immediately before, and even during showers of rain, and when, therefore, it is strongly charged with vapour in this peculiar state. Notwithstanding the cooling by radiation of the upper portion of a warm, moist wind, it is very probable that at a station on the surface of the earth, the temperature would be found to go on slowly increasing, in conse- quence of the continual arrival of fresh warm air, until the moment when rain began to fall; the rise would then receive a check, and if the rain continued, a decided fall of temperature would take place. If, therefore, we take a day of rain, the day before and the day after, the differ- ence of the mean temperatures of the day of rain and the day before, ought to be less than that of the mean tem- peratures of the day of rain and the day after. It will be seen that this conclusion is borne out by the following results of the Greenwich and Oxford observations: Mean Temp. Mean Temp. Mean Temp, of Da of Day of Day before Rain of Rain after Rain 5 = ° ° ° Greenwich observations 49°25 49°27 48°98 Oxford 0» 49°50 49°63 49°44 406 MR. JOSEPH BAXENDELL ON RAIN. Should the supposition, that a considerable portion of the aqueous vapour in the atmosphere may lose its latent heat without becoming visible, as cloud or fog, be held to be inadmissible, it appears to me that we shall then have no alternative but to conclude that the generally received theory of latent heat is inapplicable to meteorological phenomena, — a conclusion at least as questionable as the view which I have ventured to advance. 407 XXIV.— On the Structure of the Luminous Envelope of the Sun. By James Nasmytu, Esq., C.E. In a Letter to JosepH SIDEBOTHAM, Esq. Read March sth, 1861, Tuinkine that it might interest some of your scientific friends to be informed about the remarkable details which I have discovered in the general luminous surface of the sun, as also in the structure of the solar spots, I send you for that purpose a rough but faithful drawing, which I hope may serve to convey a pretty clear idea of the details I refer to. In order to obtain a satisfactory view of these remark- able objects, it is not only requisite to employ a telescope of very considerable power and perfection of defining capability, but also to make the observation at a time when the atmosphere is nearly quite tranquil and free from those vibrations which so frequently interpose most provoking interruptions to the efforts of the observer; without such conditions as I allude to, it is hopeless to catch even a glimpse of these remarkable and delicate details of the solar surface. The drawing I send you represents a spot on the sun, which I had a most favourable opportunity of observing on the 20th July 1860; although it is in some respects a rather remarkable spot, yet it may be taken as a fair 408 MR. JAMES NASMYTH ON THE STRUCTURE average type of those of the larger class in their general aspect. The chief object which I have had in making this drawing is, to exhibit (so far as such a drawing can enable me to do so) those remarkable and peculiar “ willow-leaf” shaped filaments of which I find ¢he entire luminous surface of the sun to be formed. The filaments in question are seen aud appear well defined at the edges of the luminous surface where it overhangs “the penumbra,” as also in the details of the penumbra itself, and most especially are they seen clearly defined in the details of “the bridges” as I term those bright streaks which are so frequently seen stretching across from side to side over the dark part of the spot. I accompany the drawing with a diagram (No. 2) which exhibits in a more definite and clear manner the exact form of those remarkable structural details of the solar surface. So far as I have as yet had an opportunity of estimating their actual magnitude, their average length appears to be about 1000 miles, the width about roo. Diagram No. 2 conveys a pretty clear idea of the man- ner in which these remarkable details are arranged, in forming, as they do, the entire luminous surface of the sun, There appears no definite or symmetrical arrangement in the manner in which they are scattered over the surface of the sun; they appear to he across each other in all possible variety of directions. The thickness of the layer does not appear to be very deep, as I can see down through the interstices which are left here and there between them, and through which the dark or penumbral stratum is ren- dered visible. It is the occurrence of the infinite number of these interstices, and the consequent visibility of a cor- responding portion of the dark or penumbral stratum, that OF THE LUMINOUS ENVELOPE OF THE SUN. 409 gives to the general solar surface that peculiar and well known mottled appearance which has for a long time been familiar to the observers of the sun. You will note that I consider the penumbra to be a true secondary stratum of the sun’s luminous envelope, and that what is termed the penumbra of a spot is simply a portion of it, revealed to us by the removal so far of the external and most luminous envelope. A slight approach to symmetrical arrangement of the details may be observed at the edges of the exterior lumi- nous envelope as it appears surrounding the edge of the spot, and the same may be seen at the edges of the penumbra; the tendency to symmetrical arrangement being a slight approach to a radial formation; the fila- ments tending in their general position, at the parts in question, to the average centre of the spot. As I have before said, nothing like a tendency to sym- metrical arrangement is observed in the filaments that form the entire uminous surface of the sun. Diagram No. 2 conveys in this respect a very faithful represen- tation. I may also here note that, although I have most care- fully watched for it, I have never seen any indication of a vortical or spiral arrangement of the filaments within or about any of the solar spots; this observation appears to set aside all likelihood of any whirlwind-like action being an agent in the formation of the spots, as has been con- jectured was the case. When a solar spot is mending up, as was the case with the one represented in the drawing, these luminous fila- ments, or willow-leaf shaped objects (as IT term them), are seen to pass from the edges and extend across the spots, thus forming what I term “the bridges” or bright streaks across the spots; if these are carefully observed under favourable conditions, the actual form of these remark- SER. III. VOL. I. 3G 4.10 MR. JAMES NASMYTH ON THE STRUCTURE able details of which “the bridges” are composed will be revealed to sight. You will also observe that the details of the penumbral portion of the spot are slightly varied in brightness ; that portion of the penumbra immediately under the bright edges of the external luminous envelope is less bright than the part of the penumbra next the dark centre of the spot. This is not a mere effect of contrast, but an actual variation in brightness. You will also notice that portions of the details of the penumbra are in patches considerably brighter than the rest. This effect appears to me to be due to such portions of the penumbra, or the filaments forming it, being more elevated, and consequently brought up into more close contact with the luciferous atmosphere which I am of opinion surrounds the sun, and excites, by some peculiar action, the willow-leaf shaped filaments into full lumin- osity. This of course is only conjecture at present, but I have some pretty strong grounds for entertaining this view of the subject. I have also indicated in the drawing a portion of the third luminous envelope, which you will observe like a mist underneath some portion of the penumbra. This mysterious object is very difficult to catch a glimpse of, as its comparative brightness is of so very low an order that it is but faintly distinguishable from the darkest portion of the centre of the spot. I do not as yet feel warranted to hazard any conjecture as to the nature and special functions of those remarkable willow-leaf shaped details of the solar surface which I have discovered, and have attempted to describe to you. I hope however to pursue the investigation of this most interesting suvject with all due assiduity this summer, and trust I may be fortunate enough to obtain further insight into their nature. In the meantime, I hope the OF THE LUMINOUS ENVELOPE OF THE SUN. 41] hasty description I have endeavoured to give you may prove in some degree interesting, and excite some of our “observers” to devote a little more of their attention to the glorious centre of our system than, I am sorry to say, has been the case hitherto. Appendty, MICROSCOPICAL SECTION OF THE MANCHESTER LITERARY AND PHILOSOPHICAL SOCIETY. ie . iy He NIAC Bee oe Reese ey ie eee ya i i). me ANNUAL REPORT. 1860-61. Tue third annual report of your Council presents an opportunity for congratulation upon the steady progress of the Section, especially on the more regular attendance at the meetings, and the more interesting nature of its pro- ceedings. The difficulties attending its establishment appear to be overcome, and a career of usefulness is open- ing to it, which may prove important to the progress of microscopical investigation. During the past year two members have been removed by death, Mr. THomrson and Mr. Lone. The former had few opportunities of attending the meetings of the Section, but he took great interest in its proceedings. Mr. Lone was a member of your Council; he was an ardent follower of scientific pursuits, and his loss is deeply felt by all who knew him. One resignation has been accepted. Three new members have been elected. Several gentlemen have become members of the Parent Society in order to be qualified for joiming the Section, and the names of six candidates are now before you for election. Your Secretary has been elected a member of the Council of the Parent Society, which may be regarded as a compliment to the Section, and a proof of the estimation in which it is held. During the Session the Section has held two summer and eight ordinary meetings, at which several papers have been read, much valuable information communicated, and 416 ANNUAL REPORT OF THE many specimens exhibited. A pleasant excursion was made to Croft’s Bank, at the invitation of Mr. Herworrn, whose kind reception, display of objects, and extensive microscopical knowledge, will long be remembered by those who partook of his hospitality. Papers have been read by Mr. J. B. Dancer, F.R.A.S., “On cleaning and pre- paring Diatoms, &c., obtained from soundings.” Mr. W. H. Heys “On the Kaloscope.” And by ycur Secretary upon “ Mr. Datz’s process for the separation of tallow from soundings.” Addresses have been given on important subjects by your Prestpent, by Mr. Binney, Mr. Sipesoruam, and others. Many contributions have been received from gen- tlemen who take an interest in the prosperity of the Section; amongst whom may be named Captain M. F. Maury and Lieutenant Brooke of the United States Navy, Captain ANperson, Mr. W. K. Parker, Dr. Watticu, Professor Acassiz, Dr. Bacon, Mr. Epwarps of New York, Mr. Herworrn, and other distinguished scientific men, whose assistance has been highly valued and duly recorded. The thanks of the Section are due to Mr. Dancer for the unremitting kindness with which he has provided microscopes and objects for use at the meetings; and the Council wish particularly to record their appreciation of his valuable assistance. Your Secretary has originated a method of collecting specimens of the sea-bottom obtained by captains of vessels from the soundings they take in ascertaining their position on approaching land; and many shipmasters have been furnished with envelopes in which to preserve those specimens for this Section. The plan promises to be highly successful ; upwards of eighty specimens have been received from different parts of the world, such as the MICROSCOPICAL SECTION. 417 English Channel, Mediterranean and Red Seas, Coasts of Portugal and Brazil, Deep Atlantic aud Deep North Pacific, Coasts of Japan, &c. &c. Amongst those from the Pacific Ocean are the deepest soundings from which mate- rial has yet been brought up from the sea-bottom, say 3,030 fathoms, or nearly 34} miles; the quantity of material is necessarily small; and so far as yet examined, in this specimen no trace of organic bodies has been found. Arrangements are in progress for the scientific examination and mounting of these soundings, some of which will be laid before you this evening. About 1,200 envelopes have been distributed, mostly amongst captains now out on distant voyages, to the East and West Indies, Coasts of Africa and Australia, as well as to some of the Pacific and Sperm whalers and traders; a few of which may in time be returned with interesting material. It is encouraging to know that other societies are following this example, so that our knowledge of the sea-bottom will soon be vastly increased. Results of unexpected magnitude are likely to follow these humble efforts to obtain specimens from the deep sea. Amongst those captains who were solicited to pre- serve their soundings was Captain James ANDERSON, then of the Cunard steamer “Canada.” In the course of cor- respondence with your Secretary, this enlightened sailor developed a long-thought-of plan for the social advance- ment of his fellow-mariners, to induce them to study natural science in its various branches, and to render their assistance available to scientific institutions throughout the country. Captain Anprrson asked for your assistance to carry out his views. All who heard his letter, were so convinced of the importance of the project that it was unanimously determined, as a first step, the letter should be printed and circulated at the expense of the Section. That has been done to a limited extent, and in conse- SER. III. VOL. I. 3H 418 ANNUAL REPORT OF THE quence a meeting of a few friends was held in the Liver- pool Town Hall on the 30th ultimo. The Mayor, R. 8. Graves, Esq., presided. There were present Colonel Wm. Brown, Dr. Cottinewoop, Captain AnpErRson, Mr. Ratu- BONE, Mr. Mackay, and other eminent shipowners and gentlemen favourable to the scheme. After Captain ANnpERSON had explained his views, your Secretary endea- voured to point out how societies in interior towns could contribute to its success, and participate in its advantages ; how shipmasters would improve themselves by the collec- tion of specimens, and the study of natural sciences in general, but more particularly that of meteorology, to en- able them to shorten voyages, aud to reduce the losses shipowners and underwriters now constantly suffer. All were deeply impressed with the advantages to be derived if a good working plan could be organised. None could at once be formed without some objections; but a committee was appointed to take the subject into consideration, and report thereon. It will be a source of gratification to this Section if, through its instrumentality, the first steps were taken to commence a work the importance of which, if thoroughly carried out, will be considerable. ‘To promote scientific research amongst a numerous class of men and youths whose opportunities of collecting specimens and making scientific observations in all parts of the world are une- qualled, is an object worthy of our attention; and although another generation may be required fully to develop its usefulness, some good may be done even in our day. With such purposes in view, the future prospects of our Section are encouraging; and although in the highly scientific branches of microscopical research we have done but little, it is to be hoped that our professional members may, from their stores of experience and study, contribute more liberally to the general fund. MICROSCOPICAL SECTION. 419 Before the next Session is far advanced the members of the British Association and many distinguished foreigners will be amongst us, and it behoves one and all of our members to make every exertion, that this Section may worthily represent the microscopy of the day, and the city to which we belong. Manchester, 30th May, 1861. 420 ANNUAL REFORT OF THE 9, St. Peter's Square, Manchester, 4th April, 1861. Sir, The Microscopical Section of the Manchester Literary and Philosophical Society have by circular re- quested many Captains of merchant vessels to preserve for microscopical examination, the material brought from the sea-bottom, by the soundings they make in various parts of the world. Amongst others, application was made to Captain ANnpERsoN of the Royal Mail Steam Ship “‘Canada,”’ and he not only entered fully into the spirit of the request, but in course of correspondence with the Secretary, Captain Anderson gave the outline of a project which has occupied his attention for some years past, for cultivating a taste for Natural and Meteorological Science amongst Mariners, and for rendering their assistance available to scientific institutions; he embodied his views in a letter that was read at the meeting of 18th March. So strong was the feeling of the members upon the sub- ject, it was unanimously resolved that Captain Anderson’s letter should be printed and circulated at the expense of the Section, with a view to elicit opinions upon the feasi- bility of the project, and upon the best practical method of carrying it into execution. In the first instance the con- currence of ship owners and merchants will be required ; I shall therefore be glad to be informed if after perusal of the letter, you are disposed to assist the movement. Should sufficient encouragement be obtained it is proposed to call a meeting in Liverpool to organise the plan of fur- ther proceedings. I am, Sir, Yours truly, GEORGE MOSLEY, HON. 8EC. MICROSCOPICAL SECTION. 421 Royal Mail Steam Ship “ Canada,” at Sea. Onan Sin) February 23rd, 1861. * * K I forwarded the packages of sounding envelopes to Commander Maury and to Mr. Osborn of New York; but allow me to express a doubt if much can be hoped from any such method as applying to whalers or others going upon casual voyages, whose commanders have no induce- ment to any such industry. I would not express myself thus had I no other plan to offer; but it will involve filling another sheet, and as it is a hobby with me, let me make an appeal to your Society for a fair consideration. You have on shore Literary and Philosophical Societies, Museums, Free Libraries, Working Men’s Associations, Lectures, all kinds of instruction and diversion — for that too is necessary to relieve the over-taxed brain or irritated temper — besides the influence of social intercourse to smooth rough manners, and make man’s life run pleasantly along. All classes and all ages so generally avail themselves of one or other of these privileges, it is fair to infer that variety of occupation is essential to the healthy tone of an active mind — for any of these sailors have no equivalent, but might have. I am not now writing of such a life as mine, ever varying, or of the few who by strong scientific turn of mind, happily. directed, have succeeded in possessing themselves of a. hobby ever present as a source of recreation and happiness ;. but I am pleading for many fine minds in the mercantile marine of England, now lost to science for want of hobbies. To give them intellectual hobbies would be to raise the whole class in the social scale, and I know of no obstacle to that, but the one fear that possesses all of them, that they might be told to let science alone and mind their own 422 ANNUAL REPORT OF THE business. Sailors, by their education at sea, are naturally submissive to any one having authority, and it remains with ship-owners to give them that intellectual recreation and direction, if they please. The merchants of Manches- ter are themselves ship-owners, or have influence amongst them. Iam in hopes I am writing what will be read by some such, and if there be some philanthropist who will follow it up, he would find a harvest such as no other scheme can offer to science. There is a “ Mercantile Marine Association” in Liver- pool well attended by ship-masters, so that they can be reached as a body at once. I would propose that body issuing diplomas or certificates, such as other Literary Associations use, and giving them to all members who by any industrious application to science or gathering of con- tributions to scientific associations :—or who by keeping a school on board their ships and bringing forward two or more apprentices to pass an examination, as a proof of their assiduity : — or by qualifying themselves in any lan- guage other than their own: — or by a knowledge of Naval Architecture : — or by any kind of industrious application, tend to elevate the mercantile marine of England in the social scale. It is obvious, however, that these diplomas would be of no value unless countenanced by ship-owners. If they would but say to their commanders, ‘ We would give a preference (other circumstances being equal) to men hold- ing such certificates,’ I give it as my strong conviction, that it would be largely responded to, and that they would make better men of their ship-masters. A man with a hobby is always safer both at sea and abroad, than a thoroughly idle man. From the very nature of the duties required of a seaman, by which he may hope to arrive at the position of com- mander, that station is generally reached with no other MICROSCOPICAL SECTION. 423 resources than just what constitute a sailor, and then he finds himself an idle man or comparatively so, after his life of unceasing industry. There are many times when ships are becalmed in the Eastern Seas and Pacific, when soundings might be taken, if any one who had a taste for that pursuit were supplied with the necessary apparatus, partly at the expense of the Association to which he be- longed, and in part by the Society desiring such soundings, we might soon know the nature of the deep sea-bottom. Indeed, for collecting specimens of every thing in the sea and out of it, from the first germ to the finished animal or fabric, whether in Comparative Anatomy, Zoology, Ethno- logy, Botany or Meteorology, no better adapted staff of workers could well be wished for: they carry your fabrics to every corner of the world; and to my thinking there is but one thing necessary —let the owners desire their commanders to pay some attention in their leisure time to some of these pursuits, and approve of their holding the certificate of the Association. It is too much to expect that they will apply themselves, and neither find their pride gratified or their industry ac- knowledged ; but with this or some such scheme there is a great deal of good to be done to the class of men to which I belong. As a matter of social science I hope some of your members may take it up. IT hope you do not think you have found one likely to bore you frequently with such long yarns. I am in earnest on this point, having thought of it for years; and were I not trying to be brief I could go on and shew how I would supply the Museums of Liverpool and Manchester so as to exceed all others in their completeness; but I should like what I have written to be read, and if it is not afterwards thought of, then it is long enough. Yours very truly, JAMES ANDERSON. it ve oN SESSIONS 1859-60 anp 1860-61. LIST OF DONATIONS RECEIVED From 1859 April 19th, to 1861 April 29. From Scientific Societies and Institutions in Great Britain and Lreland, with Colonies. Bato. The Bath and West of England Society of Agriculture, &c.: BERWICK-UPON-[T WEED. The Ber- wickshire Natural- ists’ Club: BirmincHamM. The Institution of Mechanical Engi- neers : Bomspay. The Bombay Branch Royal Asiatic So- ciety: The Bombay Geogra- phical Society: Catcurra. The Asiatic Society of Bengal: SER. III. VOL. I. Journal of the Society, New Series, Vol. VIII. Parts1.and 1. London 1860. 8vo Proceedings of the Club, Vol. IV. 1-4. 8vo Proceedings of the Institution, 1858 Nov. 3; 1859; and 1860, Jan. 25, April 25, Aug. 8 and 9. Sve Journal of the Society, Vols. I.-V., Bombay 1544-57. 8ve Transactions of the Society, Vols. 1.—V.3; VII. Part 1; VIII-XV. Bombay 1844-66. Sve Journal of the Society, New Series, Vol: XUN. 35 2; XID. 1, 2; XXbV. 12 ReXGV Ee XOVTE 1 2 eXG VALET Ne 2 = XVIII. 1,2; XIX-XXV.; XXVI. No.1; XXVII.; XXVIII. Calcutta 1843-59. 8yo 31 CatcuTra. The Asiatic Society of Bengal; The Geological Survey of India : The Dublin Statistical Society: DUBLIN. L The Dublin Univer- sity Zoological and Botanical Associa- tion: The Geological Soct- ety of Dublin: The Royal Dublin Society: The Royal Irish Academy: Epinpureu. The Botanical Soci- ety of Edinburgh: The Royal Physical Society of Edin- burgh: Index to Vols. XIX. and XX. of the Asiatic Researches; and to Vols. I.- XXITI., New Series, of the Journal. Caleutta 1856. 8yvo Evidence of the Hon. A. Eden, taken before the Indigo Commission. Cal- cutta 1860. 8vo Code of Bye-Laws for the Society. Cal- cutta 1851. Svo Memoirs of the Geclogical Survey of India, Vol. I. Parts 1. and 111.; Vol. II. Part 1. Calcutta 1858, 1859. 8vo Annual Report (third year) for 1858-59. 8vo Journal of the Society, Vol. II. Parts x1v.—xvu1.; Vol. ITI. Part xvuqr. ‘Dublin 1859-61. Svo Proceedings of the Association, Vol. I. Paris I.andu. July 1858 and July | 1859. 8vo Journal of the Society, Vol. VIII. Parts m.and 1. Dublin 1859, 1860. S8vo Journal of the Society, Vol. II. Nos. — 9-16.; Vol. ILI. No.17. Dublin 1859, 1860. 8vo Transactions of the Academy. Vol. XXIV. Part 1. Dublin 1860. 4to Transactions of the Society, Vols. I.-V.; and VI. PartsI.and 1. Edinburgh 1844-59. 8vo Proceedings of the Society. Vol. I. Edinburgh 1858. Svo Epinpurex. The Royal Physical Society of Hdin- burgh: The Royal Scottish Society of Arts: The Royal Society of Edinburgh: Fatmoutu. The Royal Cornwall Polytechnic Society: Giascow. The Scottish Ship- builders’ Associa- tion: GreENwicH. The Royal Observa- tory: al) Memoirs of the Wernerian Natural History Society, Vols. VI.; VII.; and VIII. Part 1. Edinburgh 1832, 1838, 18—. 8vo Transactions of the Society, Vol. I. Parts 11.-v.; II-IV.; and V. Parts I-Iv. Edinburgh 1858-60. 8vo Transactions of the Society, Vol. XXII. Parts I. and II. Ato Appendix to the Makerstoun Magneti- eal and Meteorological Observations. (Supp. to Vol. XXII. of the Tran- sactions.) Edinburgh 1860. 4to Proceedings of the Society, Vol. IV. Nos. 49 and 50. 8vo The 3rd—5th, 8th-17th, 22nd, 25th-27th Annual Report of the Society. Fal- mouth v.y. 8vo. Constitution of the Association. Glas- gow 1860. 12mo Proceedings of ditto, pp. 1-130. Svo Astronomical Observations made 1765- 1810, by the Rev. N. Maskelyne B.D. Astr. Roy., Vols. I-IV. London 1776-1811. fol. Catalogue of Stars from Flamsteed’s Observations, by Carolina Herschel. London 1798. fol. Catalogue of 1112 Stars. London 1833. fol. Tables for 23 Principal Fixed Stars. fol. Ditto 40 ditto. 1825. fol. Ditto 40 ditto. fol. Catalogue of Circumpolar Stars, from the Observations of 8. Groombridge F.R.S. London 1838. 4to GREENWICH. The Royal Observa- tory : Leeps. The Geological and Poly- technic Society of the West Riding of Yorkshire : The Leeds Philoso- phical and Lite- rary Society: LiverPooL. The Historic Society of Lancashire and Cheshire: The Literary and Philosophical So- ciety of Liverpool: Lonpon. The British Association for the i vs ‘ ~ + 7 ; . » x ’ iS . . . , o . - ” * ‘ a . . ya ’ . . 4 - . ry » I ¥ y * > \ x X ’ r 1 ul bes) i J , Ve . 2 . 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PL LYPULOLOY BY) JO SUIBUTIISD LNlBLL] yp yo gunn fipop rwupsut spypuopy Uae cull LY HV } | +d sate hand ; Ges 4 Ye TLEUSTRATIONS OF MR. NASMYTH'S PAPER “ON THE STRUCTURE OF THE LUMINOUS ENVELOPR OF THE ° mers ts SO fT 1.