Le te Noe mee gee ee Seaeeny & pe eee Benes See Awe pF fin FOR, or < 2b pte e ars ney! ; mei | ro png Poon Clank wo mo SOR SA sy Ae ee NS Pe ; * ey ee ai re. ong en ead. ie Ee, le el Lene 7 eerbd A me ie i ts dae des ny AE hs CL rh ti + oe = ee REPORT OF THE SIXTY-NINTH MEETING BRITISH ASSOCIATION ADVANCEMENT OF SCIENCE HELD AT DOVER IN SEPTEMBER 1899. * pen! LONDON: JOHN MURRAY, ALBEMARLE STREET. 1900. Office of the Association : Burlington House, London, W. “oe . ase ¥ : PRINTED BY SPOTTISWOODE AND CO., NEW-STREET SQUARE LONDON CONTENTS. eget Page OszseEots and Rules of the Association ..2..seeeseeeee veeees psanuress POR aT TREN 9 28 < ‘Places and Times of Meeting, with Presidents, Vice-Presidents, and Local Secretaries from COMMENCAMENL ..vscersssesssersescereeecesrereessseceeeeeseces xl Trustees and General Officers, from 1831 ..........seeeeeees SRLS PE hii Presidents and Secretaries of the Sections of the Association from 1832... lili List of Evening Discourses ............ssecsessecserecsssscancssdscocssanersecsecaees Ixxi Lectures to the Operative Classes ....... Sa Aornonnicl isbn Senyiqodenk, Reduce radar ariopee Ixxv Officers of Sectional Committees present at the Dover Meeting ........... » Ixxvi Treasurer’s Account ..... AngpaSoRbicoencad cadoLabodUL AdcodousocHetUnEdbpe idosdddenbdeT ad Ixxviii Table showing the Attendance and Receipts at the Annual Meetings ...... Ixxx Officers and Council, 1899-1900 ..... ESE Srecb ohcean gocnec sedanesbieaas ears Yelk ees! Report of the Council to the General Committee ........scseeeseeesetsnneeeeees . Ixxxiii Committees appointed by the General Committee at the Dover Meet- Ang in September 1899 ............:ssssecceseseseeeeesscsssssseseeseenaseesese snes a) XCIV Gemiianications ordered to be printed 4 GHLETAD nogsha Wor ashi aftnasiier tins Sects ciii Resolutions referred to the Council for consideration, and action if esirable .......:sssseecesssssteesseesteseeesesesseeenereteeeasecneeenesenneseneeeeeens ciii Change of Hours of Meetings, &C. .....sseeeeeeseeeeeeeeees cbt tte cili Synopsis of Grants of Money .......sessees sees obpechacoc chon a Someta atin eet caste civ Places of Meeting in 1900 and 1901........2...ssccosccesseseerccnsesencsevscensscvans evi General Statement of Sums which have been paid on account of Grants for Bae eae AICO SOS igh sero athoadclek hes eat erastawas Sey susgeigy oss cial cha ins he os evil General MeetingS .......sccceseecsecsesseees Resco na etodoaec SchgoaB warn cceaye case peGX Sl G apie by the President, Sir Micnann Foster, K.0.B., Sec.R.S.........0066 3 a a AZ ° lv REPORT—1899, REPORTS ON THE STATE OF SCIENCE. [An asterisk * indicates that the title only is given. The mark t indicates the same, but a reference is given to the journal or newspaper where it is published in extenso. | Page Corresponding Societies Committee.—Report of the Committee, consisting of Professor R, Mexpota (Chairman), Mr. T. V. Hozmes (Secretary), Mr. Franois Garton, Mr. G. J. Syxons, Dr. J. G. Garson, Sir Joun Evans, Mr. J. Horxinson, Professor T. G. Bonnny, Mr. W. Wui1taxer, Sir Cutu- BERT Ppex, Mr. Horace T. Brown, Rev. J. O. Bevan, Professor W. W. Warrant Rev. P:R, Re Sreppine s2.2ii.2.00.cc.ccstcadsceteuseaest ee stammen’s 27 Radiation from a Source of Light in a Magnetic Field,—Preliminary Report of the Committee, consisting of Professor Grorcr Francis FirzGEratp (Chairman), THomas Preston (Secretary), Professor A. ScHusTER, Pro- fessor O. J. Lodge, Professor 8. P. THomprson, Dr. Geratp Moztoy, and HPS AVY Go), GDENIHY.scwscsevessinsecectiesorteassaesesecevess sos save Orcas eaeuee an CmemmeD . 63 Determining Magnetic Force at Sea.—Report of the Committee, consisting of Professor A, W, Rtcrpr (Chairman), Dr. C. H. Lees (Secretary), Lord Ketyin, Professor A. Scouster, Captain E. W. Creax, Professor W, Srrovup, Mr. C. V. Boys, and Mr. W. Watson, appointed to investigate the Method of determining Magnetic Force at Sea .........sssesecseeseeeeeeves . 64 Meteorological Observatory, Montreal.—Report of the Committee, consist- ing of Professor H. L. Catnenpar (Chairman), Professor C. McLzop (Secretary), Professor F. Apams, and Mr. R. F. Srupart, appointed for the purpose of establishing a Meteorological Observatory on Mount Royal, Montreal ORMACA ccc scvecacdedessatcudereeccoves cocuncdeenoneeereanbeat s san seeeenee wpcteee MOD Tables of the G (7, v)-Integrals.—Report of the Committee, consisting of Rev. Rozsert Harwey (Chairman), Professor A. R. Forsyru (Secretary), Dr. J. W. L, GuaiIsHER, Professor A. Lope®, and Professor Kart Prarson. (Drawn up by Professor KARL PHARSON,) .......cceeeeeeees sopsneostisn@neencselsanke eam 65 APpPENDIX,—Table of F (7, v) and H (7, v) Functions. By Miss Aticp DSCs coco. tre ccpvecceceueune sate ee eee REET Oo eee 71 Report on the Progress of the Solution of the Problem of Three Bodies, By EL NV TRACER Cre scesconecarcaversceeccsteascocecsskeassscacr¢sstenenen ape On Solar Radiation.—Report of the Committee, consisting of Dr. G. Jony- stoNE Stoney (Chairman), Professor H. McLurop (Secretary), Sir G. G. Sroxes, Professor A. Scuuster, Sir H. E. Roscozn, Captain W. pp W. Asney, Dr. C. Cures, Professor G. F. FitzGpratp, Professor H. L. CaLLENDAR, Mr. G. J. Symons, Mr. W. E. Witson, and Professor A. A. RaMBAU‘, appointed to consider the best methods of recording the Direct HntenbityOMnsOlat MAGIAUION +... ccetss-tec.ccecctnovcereccescecasveseescscsnaes treReaee 159 Electrolysis and Electro-chemistry—Report of the Committee, consisting of Mr, W. N. SHaw (Chairman), Mr. E. H. Grirrirus, Rey. T. C. Firz- patrick, Mr. 8, Skinner, and Mr. W. C. D. Wuernam (Secretary), appointed to Report on the Present State of our Knowledge in Electrolysis and Electro-chemistry .,......csece0ee rastaedevees Revecesns isaadedcdateavores donee OD CONTENTS. Vv Page Tables of Certain Mathematical Functions—Report of the Committee, con- sisting of Lorp Ketvin (Chairman), Lieut.-Colonel ALLAN CUNNINGHAM (Secretary), Dr. J. W. L. GuaisHer, Professor A. G. GREENHILL, Professor W. M. Hicks, Major P. A. MacManon, and Professor A. Loner, appointed for calculating Tables of Certain Mathematical Functions, and, if necessary, for taking steps to carry out the calculations, and to publish the results in HIMBA CCOSSUDIC LOLM iid, soda waciiscvdsaas ce Ped-ostigdare dest audea dst susedavehaed secs vessbies Seismological Investigations.—Fourth Report of the Committee, consisting of Professor J. W. Jupp (Chairman), Mr. Joun Mine (Secretary), Lord Ketvi, Professor T. G. Bonney, Sir F. J. BRamwett, Mr. C. V. Boys, Professor G. H. Darwin, Mr. Horace Darwin, Major L. Darwin, Professor J. A. Ewine, Professor C. G. Knorr, Professor R. Mretpona, Mr. R. D. OtpHam, Professor J. Purry, Professor J. H. Poynrrye, Mr. Crement Rep, Mr. G. J. Symons and Professor H. H. Turner. (Drawn up by the Secretary, Mr. JOHN MILNE.) ........:ccccsecsceeeseeseweeees ~ I. On Seismological Stations already established. By J. Mrtnz ... II. Notes respecting Observing Stations and Registers obtained from THE) SAMO. sy DY Dc MUCLNE yi sanes a cectiocotiaassniiayctesnapteeleseswaiie wats III. Discussion of the Preceding Registers. By J. MItne ............... IV. Varieties of Earthquakes and their respective Durations. By PANETUN Tc seccscnesestosner™ sca pooh inHsticpacppadeucuereucrieNadasoro sere NV. duarinquake benoes,, By. MILNE 9. csqcsleaccvvaspavesessnacecn «canis VI. Earthquake Precursors. By J. MILNE ...........cececcesceeceeee sewers VII. On Certain Disturbances in the Records of Magnetometers and the Occurrence of Earthquakes. By J, MIrnz........ fens acasoreoetae Sas Rebbe Worm oke ROpOUES, at.) cacyasscatadery Monde naehy ac eeatetath ads dmematon eene cat Photographic Meteorology.—Report of the Oommittee, consisting of Mr. G. J. Symons (Chairman), Mr. A. W. Crayprn (Secretary), Professor R. Mrt- pota, Mr. Joun Horxrinson, and Mr. H.,N. Dickson, appointed to apply Photography to the Elucidation of Meteorological Phenomena, (Drawn TDA CS SECIOLALY )) cc seasons s qosteactgndeneanan4adoeseeritet cctavetaaseeaaants in ves Experiments for Improving the Construction of Practical Standards for use in Electrical Measurements.—Report of the Committee, consisting of Lord Rayteien (Chairman), Mr. R. T. Guazesroox (Secretary), Lord Kutyin, Professors W. E, Ayrton, J. Prrry, W. G. Apams, Oriver J. Lopes, and. G. Carry Foster, Dr. A. Murruean, Sir W. H. Prec, Professors J. D. Everetr and A. Scuusrer, Dr. J, A. Fremine, Professors G. F. Frrz- GERALD and J. J. THomson, Mr. W. N. SHaw, Dr. J. T. Borromtiery, Rev. T. C. Firzparricx, Professor J. VirtAmu Jones, Dr. G. JoHNSTONE Stoney, Professor S. P. Tompson, Mr. J. Renniz, Mr. E. H. Grirrirus, Professor A. W. RickpR, and Professor H. L. CALLENDAR............... Aer AppenDix I—On the Mutual Induction of Coaxial Helices. By PL OrAwHVAMUEMUGH: ye evade! S12 TUESDAY, SEPTEMBER 19. ‘1. *The Anthropogeography of Certain Places in British New Guinea and Sarawak.ey by Aq CU, AAdagsonnbdsappe-.obenagenaare coated agentes. 873 MONDAY, SEPTEMBER 18. 1. Final Report on the Ethnographical Survey of the United Kingdom ...... 874, 2. On Recent Ethnographical Work in Scotland. By J. Gray, B.Sc.......... 874 3. Report on the Mental and Physical Condition of Children in Elementary Sea Ne OR ao aer nd Po eseaadioet de « ein otaecin Cb « duSp.as dsnnnryt Alona nde Sicenac. eae 875 4, On Recent Anthropometrical Work in Egypt. By D. Maclvrr, B.A. ... 875 So, Oo a CONTENTS. XYXill Page . “Notes ona Collection of 1,000 Egyptian Crania, By Professor A. Mac- RSE ei enn arenes tc EY. CoS ect ws darcell oot ES Seed t eww, on dudsbinad > Sapmuee eames 892 2, The Physiological Effects of Extracts of the Pituitary Body. By Pro- fessor H. A. SCHAFER and SWALE VINCENT .........cccececseesceesensecceseees 894 3. 7On the Theory of Hearing. By A. A. GRAY ......cccceecseceereeeeect seen 894 SATURDAY, SEPTEMBER 16. 1. On the Resonance of Nerve and Muscle. By F.C. Buscw ............... w. 894 2. The Propagation of Impulses in the Rabbit’s Heart. By H. KRronEcKER AHO Ae ESUNCHY socal iecdednoscraesnnsAcsecsrosdeecavereelewec deri eset ese hare eeeaeeee 895 8. Concerning Fibrillation and Pulsation of the Dog’s Heart. By F. C. NSUSOH. hece Jucssac.cddacoccecsrapecstosnmatee eden Cees sevceeesdeanevpmestegete aomeemmaee 896 4, On the Effects of Successive Stimulation of the Visceromotor and Vaso- motor Nerves of the Intestine. By J. L. Buncu, D.Sc., M.D. ............ 897 5. On Stimulation and Excitability of the Anemic Brain. By Writiiam J. CUES! oink. scons evseans svcatenebs so cacuisaesaccseissrcieccisa' Nottingham, 1893. RULES OF THE ASSOCIATION. XXX1iL Committee of the Section, and entered on the minutes accord- ingly. &. Papers which have been reported on unfavourably by the Organ- ising Committees shall not be brought before the Sectional Committees. ! At the first meeting, one of the Secretaries will read the Minutes of last year’s proceedings, as recorded in the Minute-Book, and the Synopsis of Recommendations adopted at the last Meeting of the Association and printed in the last volume of the Report. He will next proceed to read the Report of the Organising Committee.? The list of Communi- cations to be read on Thursday shall be then arranged, and the general distribution of business throughout the week shall be provisionally ap- pointed. At the close of the Committee Meeting the Secretaries shall forward to the Printer a List of the Papers appointed to be read. The Printer is charged with publishing the same before 8 a.m. on Thursday in the Journal. On the second day of the Annual Meeting, and the following days, the Secretaries are to correct, on a copy of the Journal, the list of papers which have been read on that day, to add to it a list of those appointed to be read on the next day, and to send this copy of the Journal as early in the day as possible to the Printer, who is charged with printing the same before 8 4.M. next morning in the Journal. It is necessary that one of the Secretaries of each Section (generally the Recorder) should call at the Printing Office and revise the proof each evening. Minutes of the proceedings of every Committee are to be entered daily in the Minute-Book, which should be confirmed at the next meeting of the Committee. Lists of the Reports and Memoirs read in the Sections are to be entered in the Minute-Book daily, which, with all Memoirs and Copies or Abstracts of Memoirs furnished by Authors, are to be forwarded, at the close.of the Sectional Meetings, to the Assistant General Secretary. The Vice-Presidents and Secretaries of Sections become ea officio temporary Members of the General Committee (vide p. xxxi), and will receive, on application to the Treasurer in the Reception Room, Tickets entitling them to attend its Meetings. The Committees will take into consideration any suggestions which may be offered by their Members for the advancement of Science. They are specially requested to review the recommendations adopted at preceding Meetings, as published in the volumes of the Association, and the com- munications made to the Sections at this Meeting, for the purposes of Selecting definite points of research to which individual or combined exertion may be usefully directed, and branches of knowledge -on the state and progress of which Reports are wanted ; to name individuals or Committees for the execution of such Reports or researches ; and to state whether, and to what degree, these objects may be usefully advanced by the appropriation of the funds of the Association, by application to Government, Philosophical Institutions, or Local Authorities. In case of appointment of Committees for special objects of Science, it is expedient that all Members of the Committee should be named, and 1} These rules were adopted by the General Committee, Plymouth, 1877. ? This and the following sentence were added by the General Committee, Edis- burgh, 1871, 1899. b XXXIV REPORT—1899. one of them appointed to act as Chairman, who shall have notified per- sonally or in writing his willingness to accept the office, the Ohairman to have the responsibility of receiving and disbursing the grant (if any has been made) and securing the presentation of the Report in due time; and, further, it is expedient that one of the members should be appointed to act as Secretary, for ensuring attention to business. That it is desirable that the number of Members appointed to serve on @ Committee should be as small as is consistent with its efficient working. That a tabular list of the Committees appointed on the recommendation of each Section should be sent each year to the Recorders of the several Sec- tions, to enable them to fill in the statement whether the several Committees appointed on the recommendation of their respective Sections had presented their reports. That on the proposal to recommend the appointment of a Convmittee for a special object of science having been adopted by the Sectional Committee, the number of Members of such Committee be then fixed, but that the Members to serve on such Comnvittee be nominated and selected by the Sectional Com- mittee at a subsequent meeting.' Committees have power to add to their number persons whose assist- ance they may require. The recommendations adopted by the Committees of Sections are to be registered in the Forms furnished to their Secretaries,and one Copy of each is to be forwarded, without delay, to the Assistant General Secretary for presentation to the Committee of Recommendations. Unless this be done, the Recommendations cannot receive the sanction of the Association. N.B.—Recommendations which may originate in any one of the Sections must first be sanctioned by the Committee of that Section before they can be referred to the Committee of Recommendations or confirmed by the General Committee. Notices regarding Grants of Money.? 1. No Committee shall raise money in the name or under the auspices of the British Association without special permission from the General Committee to do so; and no money so raised shall be expended except in accordance with the Rules of the Association. 2. In grants of money to Committees the Association does not contem- plate the payment of personal expenses to the Members. 3. Committees to which grants of money are entrusted by the Association for the prosecution of particular Researches in Science are ap- pointed for one year only. If the work of a Committee cannot be completed in the year, and if the Sectional Committee desire the work to be continued, application for the reappointment of the Committee for another year must be made at the next meeting of the Association. 4. Hach Committee is required to present a Report, whether final or in- terim, at the next meeting of the Association after their appoint- ment or reappointment. Interim Reports must be submitted in writing, though not necessarily for publication. 3 Revised by the General Committee, Bath, 1888. 2 Revised by the General Committee at Ipswich, 1896. RULES OF THE ASSOCIATION. XXXV 5. In each Committee the Chairman is the only person entitled to call on the Treasurer, Professor G. Carey Foster, F.R.S., for such portion of the sums granted as may from time to time be required. 6. Grants of money sanctioned at a meeting of the Association expire on June 30 following. The Treasurer is not authorised after that date to allow any claims on account of such grants. 7. The Chairman of a Committee must, before the meeting of the Asso- ciation next following after the appointment or reappointment of the Committee, forward to the Treasurer a statement of the sums which have been received and expended, with vouchers. The Chairman must also return the balance of the grant, if any, which has been received and not spent ; or, if further expenditure is con- templated, he must apply for leave to retain the balance. 8. When application is made for a Committee to be reappointed, and to retain the balance of a former grant which is in the hands of the Chairman, and also to receive a further grant, the amount of such farther grant is to be estimated as being additional to, and not inclusive of, the balance proposed to be retained. 9. The Committees of the Sections shall ascertain whether a Report has been made by every Committee appointed at the previous Meeting to whom a sum of money has been granted, and shall report to the Committee of Recommendations in every case where no such report has been received. : 10. Members and Committees who may be entrusted with sums of money for collecting specimens of Natural History are requested to re- serve the specimens so obtained to be dealt with by authority of the Association. 11. Committees are requested to furnish a list of any apparatus which may have been purchased out of a grant made by the Association, and to state whether the apparatus will be useful for continuing the research in question, or for other scientific purposes. 12, All Instruments, Papers, Drawings, and other property of the Asso- ciation are to be deposited at the Office of the Association when not employed in scientific inquiries for the Association. Business of the Sections. The Meeting Room of each Section is opened for conversation shortly before the meeting commences. The Section Rooms and approaches thereto can be used for no notices, exhibitions, or other purposes than those of the Association. At the time appointed the Chair will be taken,’ and the reading of communications, in the order previously made public, commenced. Sections may, by the desire of the Committees, divide themselves into Departments, as often as the number and nature of the communications delivered in may render such divisions desirable. 1 The Organising Committee of a Section is empowered to arrange the hours of meeting of the Section and of the Sectional Committee, except for Saturday. b2 XXXVi REPORT—1899. A Report presented to the Association, and read to the Section which originally called for it, may be read in another Section, at the request of the Officers of that Section, with the consent of the Author. Duties of the Doorkeepers. 1. To remain constantly at the Doors of the Rooms to which they are appointed during the whole time for which they are engaged. 2. To require of every person desirous of entering the Rooms the ex- hibition of a Member’s, Associate’s, or Lady’s Ticket, or Reporter’s Ticket, signed by the Treasurer, or a Special Ticket signed by the Assistant General Secretary. 3. Persons unprovided with any of these Tickets can only be admitted to any particular Room by order of the Secretary in that Room. No person is exempt from these Rules, except those Officers of the Association whose names are printed in the Programme, p. 1. Duties of the Messengers. To remain constantly at the Rooms to which they are appointed dur- ing the whole time for which they are engaged, except when employed on messages by one of the Officers directing these Rooms. Committee of Recommendations. The General Committee shall appoint at each Meeting a Committee, which shall receive and consider the Recommendations of the Sectional Committees, and report to the General Committee the measures whith they would advise to be adopted for the advancement of Science. The ex officio members of the Committee of Recommendations are the President and Vice-Presidents of the Meeting, the Genera] and Assistant- General Secretaries, the General Treasurer, the Trustees, and the Presidents of the Association in former years. All Recommendations of Grants of Money, Requests for Special Re- searches, and Reports on Scientific Subjects shall be submitted to the Committee of Recommendations, and not taken into consideration by the General Committee unless previously recommended by the Committee of Recommendations. All proposals for establishing new Sections, or altering the titles of Sections, or for any other change in the constitutional forms and funda- mental rules of the Association, shall be referred to the Committee of Recommendations for a report.! If the President of a Section is unable to attend a meeting of the Committee of Recommendations, the Sectional Committee shall be authorised to appoint a Vice-President, or, failing a Vice-President, some other member of the Committee, to attend in his place, due notice of the appointment being sent to the Assistant General Secretary.? 1 Passed by the General Committee at Birmingham, 1865. 2 Passed by the General Committee at Leeds, 1890. RULES OF THE ASSOCIATION. XXXVil Corresponding Societies.' 1. Any Society is eligible to be placed on the List of Corresponding Societies of the Association which undertakes local scientific investiga- tions, and publishes notices of the results. 2. Application may be made by any Society to be placed on the List of Corresponding Societies. Applications must be addressed to the Assistant General Secretary on or before the 1st of June preceding the Annual Meeting at which it is intended they should be considered, and must be accompanied by specimens of the publications of the results of the local scientific investigations recently undertaken by the Society. 3. A Corresponding Societies Committee shall be annually nomi- nated by the Council and appointed by the General Committee for the purpose of considering these applications, as well as for that of keeping themselves generally informed of the annual work of the Corresponding Societies, and of superintending the preparation of a list of the papers published by them. This Committee shall make an annuai report to the General Committee, and shall suggest such additions or changes in the List of Corresponding Societies as they may think desirable. 4, Every Corresponding Societyshall return each year, on or before the 1st of June, to the Assistant General Secretary of the Association, a schedule, properly filled up, which will be issued by him, and which will contain a request for such particulars with regard to the Society as may be required for the information of the Corresponding Societies Committee. 5. There shall be inserted in the Annual Report of the Association a list, in an abbreviated form, of the papers published by the Corre- sponding Societies during the past twelve months which contain the results of the local scientific work conducted by them; those papers only being included which refer to subjects coming under the cognisance of one or other of the various Sections of the Association. 6. A Corresponding Society shall have the right to nominate any one of its members, who is also a Member of the Association, as its dele- gate to the Annual Meeting of the Association, who shall be for the time a Member of the General Committee. Conference of Delegates of Corresponding Societies. 7. The Conference of Delegates of Corresponding Societies is em- powered to send recommendations to the Committee of Recommen- dations for their consideration, and for report to the General Committee. 8. The Delegates of the various Corresponding Societies shall con- stitute a Conference, of which the Chairman, Vice-Chairmen, and Secre- taries shall be annually nominated by the Council, and appointed by the General Committee, and of which the members of the Corresponding Societies Committee shall be ex officio members. 9. The Conference of Delegates shall be summoned by the Secretaries to hold one or more meetings during each Annual Meeting of the Associa- tion, and shall be empowered to invite any Member or Associate to take part in the meetings. 10. The Secretaries of each Section shall be instructed to transmit to 1 Passed by the General Committee, 1884. XXXVI1il REPORT—1899. the Secretaries of the Conference of Delegates copies of any recommen- dations forwarded by the Presidents of Sections to the Committee of Recommendations bearing upon matters in which the co-operation of Corresponding Societies is desired ; and the Secretaries of the Conference of Delegates shall invite the authors of these recommendations to attend the meetings of the Conference and give verbal explanations of their objects and of the precise way in which they would desire to have them carried into effect. 11. It will bethe duty of the Delegates to make themselves familiar with the purport of the several recommendations brought before the Confer- ence, in order that they and others who take part in the meetings may be able to bring those recommendations clearly and favourably before their respective Societies. The Conference may also discuss propositions bear- ing on the promotion of more systematic observation and plans of opera- tion, and of greater uniformity in the mode of pubiishing results. Local Committees. Local Committees shall be formed by the Officers of the Association to assist in making arrangements for the Meetings. Local Committees shall have the power of adding to their numbers those Members of the Association whose assistance they may desire. Officers. A President, two or more Vice-Presidents, one or more Secretaries, and a Treasurer shall be annually appointed by the General Committee. Council. In the intervals of the Meetings, the affairs of the Association shall be managed by a Council appointed by the General Committee. The Council may also assemble for the despatch of business during the week of the Meeting. (1) The Council shall consist of ! . The Trustees. . The past Presidents. . The President and Vice-Presidents for the time being. . The President and Vice-Presidents elect. . The past and preseut General Treasurers, General and Assistant General Secretaries. . The Local Treasurer and Secretaries for the ensuing Meeting. . Ordinary Members. “I for) er HO De (2) The Ordinary Members shall be elected annually from the General Committee. (3 There shall be not more than twenty-five Ordinary Members, of ' Passed by the General Committee at Belfast, 1874. RULES OF THE ASSOCIATION. 6:6:46.¢ whom not more than twenty shall have served on the Council, as Ordinary Members, in the previous year. (4) In order to carry out the foregoing rule, the following Ordinary ’ Members of the outgoing Council shall at each annual election be ineligible for nomination :—1st, those who have served on the Council for the greatest number of consecutive years; and, 2nd, those who, being resident in or near London, have attended the fewest number of Meetings during the year —observing (as nearly as possible) the proportion of three by seniority to two by least attendance. (5) The Council shall submit to the General Committee in their Annual Report the names of the Members of the General Committee whom they recommend for election as Members of Council. (6) The Election shall take place at the same time as that of the Officers of the Association. Papers and Communications. The Author of any paper or communication shall be at liberty to reserve his right of property therein. Accounts. The Accounts of the Association shall be audited annually, ky Auditors appointed by the General Committee. “beat ‘29180 90110 ‘Sap “bsq ‘aosspoyH ydosor "CW “bsg ‘uoystyerg Wopsog ‘S'U'a “bsq Aeysreg es1005 eRe ee bb ae eee ee deer etdeeectonns ouulreyoury redyourrd “Ao £194 Om, se eessseecure rT “bsg ‘lt109 uyor "ad ‘uosurqoy “YW WL ASW OU seeeeeorssranougie Jo [Avg OUL “uojdurey AON JO smubavyy oy, "eSST ‘9% JSNSNY “WVHONINUIG ‘oR S'U'd “VN ‘LUQOOUVH NONUHA “AL ‘ATH OGL "SSSI ‘0% JUIN ‘ANAT-NO-FIISVOMUN ‘SO “bsq “CONGHET STAN ae Oe Sloe eases eeieisinlsuevine ne “oR ‘ "Sua ‘qano0r8 UOTIGA "AA “AVY ODL ‘on “S'D'd “SU ‘GNVTUXENOHLYON FO TNA ous ‘ony Sood “Ds ‘mosuspy uyor Seen meee ee erate eeee eee eeeees ‘Vesa’ “SU ‘Ueyang jo doysig o"uL *joodraATT WOryNy -198uy [eAoy ‘sorg “bs ‘rexT8 A\ "N Udesor *Dsy ‘vLLIND OOB[TB AL “ULLAL "CW ‘TIerL Tossejorg pager ens eas Mabe starr re ee St eee Seam eeu “M “AO "1881 "TL roqmeydog “TooduHArT Hee e weer rere eres ssee ‘sya ‘ ‘sa ‘ ‘qeg ‘uoztesq Aor op durud Us Tteteeeseceesecss® * UOpUuoTy JO APISTOAIUA OU} JO IOT[e ee Soa ctu uN mal | oeeee Pee eee eee eee eee ee “rsd mf uv wou aoe oa | ae ‘uozed uyor "Sp u“S"1'd‘ bat ito onde) “spd “Sud ‘NOLONITHOG JO TUVa UL ‘bsq ‘uepuoaoH “TA (‘Sa “Ce “bsg ‘preyond ‘Of 'S*)'T Sa ‘arwoqAaop “gM bet ; “988I ‘2e ysnsuy “IorsIug "OR SW “CN ‘Auoqnu rossajorg [seer st estes er sseesesses sees “Sug ‘uoyduvyqs0N Jo smbavjyy out) *** “SW “TO'd “ANMOGSNVT JO SINOUVNN eUL @ See eee) on ei area ‘on “ST TTOANOTLAL “AL a "OERT ‘or gsndny ‘xrTanq CO jo qwkoy “uorsy ‘uoyMUBA “AL “A HS SVU “SU ‘UmozueuxG uMOOSTA GQ'IT ‘GXOTT LSOAOUd ‘ATH OU a “agri ‘00g “WosUIqoNT TYOL aIg { seeeaeeeeeeanseceerecssaneeceseseneeneuss sper tOsUtqony "MT "AOU { escecesereeee Jet ar Cee sapere oe ieee Es tom OO WP LSU ‘SaqlOg TOssojorg [tert sete ees esse ste seers te eeeeee sero Goria SaIsMolg PIAUC TIS onaig aD MNVRMNG, TLYDOMGDYR oe aie a 3) i= ‘Sua Teg ee seme ee reer esas eeesecesersessseesed ‘sad OT O'd “Eset ‘aos WHOL | F "2e8I ‘92 eune ‘apatuaNvp “ota “WT ‘saopsueH qossojorg “aay ) om ‘qeioy tomoucaysy “Sad “bsg ‘Airy ap) ‘sy’ A “S'Wd'A “VW ‘HOIMDGES WYCYV ‘AGU eTL *oap ‘ ‘Sw “VN Fee eee Peet ees mM eres ae pe sc acre gpien PAE Ml mee eine : {38st ‘el oung § auoaxo ‘oR SW “CW ‘Auoqueg tossojorg [stteeststseeeeeereeeeereree coop Or ap TSU ‘Waqsmoug praed ag) oR “SHE “SU “C'd ‘ONVIMONG “MA “ATE OL : WA “WIT ‘SATA TOSSOJOTT | sy eeeeeveveveversennes genng Gauri Oy BACON “1881 ‘22 toquieydeg ‘uu0X Poe es “org “bsg “unt ery Saath SO “S'a'd “VW ‘yanoo yA WOME A “AM “AO { oR SDA SUA “T'O'C ‘NVITIMZLIA THY ous ‘saluvilayoas 1v9071 "SLN3CISaad=3S IA “SLN3GIS3ed “UIWAOUIWULOD 871 Wolf ‘sDI.LHIILIIG QDIOT PUD SspWapisatJ-0004 ‘squaprsalg yun ‘uounwossy ysrpug oy; fo burooyy fo sowry pup soon oy; buanoys qn], xl xli PAST PRESIDENTS, VICE-PRESIDENTS, AND LOCAL SECRETARIES. 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Sip She “POST ‘PI tequioydeg ‘HIV veeees ss UBUlqglod ploy ‘UO FUSTY OGL (SHA “TOA “WW “aed “IIGAT SHTUVHO UIS Pee eee eee eee eee eee eee ee UOSTON. yer ‘U0 USN ou oO Pee eee eee ee ee eee eee eee es yyeq jo smbivyq 049 SIQON Jsoy on, s{sisi9ia}e\4i¢\0\81a ¢» sieinisieisivinie.0/sinie\o\s(6i0\eie\s\9\ainio/s\n,0/01 vials sie(ainau un eam Garr aog -IOMOY JO JUBMONIVT-pIO'T ‘ATOLIO PUL FIOV Jo [IVY ory “MOF GUSH OUT, Settee ee eeeeeeeeseeseeeeseees scunrer GCrTy “bsp UITeqIIBT USI, Fenn ee ee ener eeenene ills BC Ge emipAsHp BramaT, sate bang oe MENTED “AL | gurmy Jo oqngNsuUT wroqIION oY Jo uOpIsorT “bsq “pooM SLIOYOIN a yu 8 ee tN Hemmer ter ee eter eneeee a[ASvoMoN Jo 1OLey_ “bsy [log UBIYMO'T OVEST SH S[GON “VW fienteccnvcceercsnsccene apex, [vog aq Jo uvuraegD “bsg ‘ope, YSN ed "SOU “SW “TOE “TT ‘TAT soplvyO ag Ce ‘VIN “avg ‘aesjoaory, *O 109% Ig “S981 ‘90 48NSNY “ANAT-NO-HILSVOMAN OTOP ee seen “SU “ony “rp ‘ONOULSNUV "M UIs sin dialelejeielnie via aicieisiie esiciee Srceny soa “TOE “W'It ‘Sex0Ig “D *4) IOSsOJOIg sete eeeeeeee [efor IMOUOSY “SW “'T'O'a Wi “Dsiy ‘Ary ‘aD iano pe Sia Spee Et 5 "SW “VIN ‘SIIB “r AeY ouL sie eleYals{elei sie w)sia/Sleis]aisinis'scrvrecy “mod wit ‘yoIMSpag ossajorg *Aoy ouL ‘ *Z98I ‘T 1040900 ‘apaIuaNvoO oOo eee eee eee ee ee ee ee ree ry esplaquiep jo Aq18 *IOATUQ) oY4 Ul AYdosopy [e}USMLIOdx GY pus [einjyeNy Jo Aossejorg ULiuosyour “'S' yd “VW ‘SITIIA ‘UY ‘AAU OUL "V'W ‘s19LI8] "WW “N “A0Yf OWL, “VW ‘dUlvATT ‘ *) 10ssajorg ‘S'la “SW CV ‘a0ysurqug "0 °O Aossajorg | ospraqurey ‘osaT[0N Aqruray, Jo 1oyswy “SUT “A 'C “ToT AA “AN “AOR OUT, pIerelbha sta eta isva lela acelin sian ATT jo uvoq ames ‘aA poor) AAIV AL “aay AOA OUT, seeesees OSpriqmiuy Jo AzISIOAIUN OYJ JO AO[[eOUVYD-ooIA 944 *ADY OT, PROP ee eee ee ee ee ee eeeeee “T'O'USUL' ISHS Bt § “Dsq *qaI0M4IT_A Ydoso r) eeee ee ee ee eens “TO sUL TW “WW A Ae ae i ‘aosurnspoy, cg 1OSSdJOIg seaeeeteecesesensestseesceeeersenetteetesrsseteeeneree® JQASQqD “UBT ‘008 "TU 7 9IT ‘Satd “Sd “ATT “bsg ‘omor qyooserg somer ee . . “bs *JouI0, BUI: sou Ms cua acasigteoisl Bhaceade theEMe ae fe degen tl are ar eatnORtE "1981 ‘fF toqmeydag “ua ISHHONVIL Weteaeseeneeeseeeeuseesres souarey Carper poomsay unuvtuag aig) 8 ‘SW “HO “ATT “beg ‘NUIVaUIVa WVITIIA teseeree gun grrr “dN “qaug ‘uoyoedg AorN sedieyy ep did Ig te eeeeeneeeeeeseercugr Maurer “Cid ‘aqsoyouvyy JO doysig ploy oy, ene eeeeereeeeereesssenesees coun Srp “ayy ‘Aapueqg ploy aT, Cove seen rene ererenaccssscescorasasessecunmnr Ty ‘@LOUUSOT A Jo [Avg O1, "saluvlayoas 1V9071 *SLN3SCISSYd-J9IA ‘SLNBGISaud *V'q ‘a00soy "| "WH Jossojorg "VW “bsa ‘emosusy | *bsq ‘PION PA.GIy "SD “V'a “bsg ‘ongsqred "Cd “A xlv “V's ‘ouny *y “Iq *Aoy “Wy ‘sursary “H Aruey *AayT *bsq ‘uostiey pleursayy *qaqystueg “AA ‘Ad UOMITS "WY “AO OT, *bsq ‘Su1Mog “pO uror ‘SV Wa “bso ‘ena “Ss Aruery *[[PMOH Spurpy, UouEH “Ady *"V'W ‘uoydwo1g ydasor *A0qT ‘aid Arjeq preuog “Aq *bsq ‘uostapuy yoLy4Ed ‘bsg ‘Svo0py oye] uysny uyor *bsq “unl *uosiapueH *f “VW ‘UUTIVO.IU “Tf AO OUT "ST “SVU “bsg ‘emo ‘fp prepa *U0s}1OQOY “AC. 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"bsg ‘uosqoMaH “AD Bw cakec écoteme S “OT “so wr “Sad “aT *“TO'd “@O'M ‘NOLIVD syINNOd IS NIVLdVvO ee rs espliqmep jo AqIsaoATU og} Jo pumas WS “SW “ATT “WeysarsyeM PLOT wOH ISTY OL eee eee ee yloyng yo AyunN0p oy} Jo quUBMAM>YT-pIOT “V"TY [OIA Jo smMbaeyY O47 “WORT ysoW OTT, li REPORT—1899. TRUSTEES AND GENERAL OFFICERS, 1831—1900. TRUSTEES. 1832-70 Gok R. I. MuRcuIson (Bart.), BS. 1832-62 aes TAYLOR, Esq., F.R.S. 1832-39 C. BABBAGE, Esq., F.R.S. 1839-44 F. BAILy, Esq., FRS. 1844-58 Rev. G. PEACOCK, F.R.S. 1858-82 General E. SABINE, F.R.8. 1862-81 Sir P. EGERTON, Bart., F.R.S. GENERAL 1831 JONATHAN GRAY, Esq. 1832-62 JOHN TAYLOR, Esq., F.R.S. 1862-74 W. SPOTTISWOODE, Esq., F.R.S. 1872 Sir J. LuBgock, Bart., ¥.R.S. 1881-83 W. SPOTTISWOODE, Hsq., Pres. RS. 1883 Lord RAYLEIGH, F.R.8. 1883-98 Sir Lyon (now Lord) PLAYFAIR, F.RB.S. 1898 Prof. A. W. RUcKER, F.R.S. TREASURERS. 1874-91 Prof. A.W. WILLIAMSON, F.R.S. 1891-98 Prof. A. W. RUCKER, F-.R.S. 1898 Prof. G. C. FostER, F.R.S. GENERAL SECRETARIES. 1832-35 Rev. W. VERNON HARCOURT, E.RBS. 1835-36 Rev. W. VERNON HARCOURT, F.R.8., and F. Baty, Esq., E.R.S. 1836-37 Rev. W. VERNON HARCOURT, ¥.R.S., and R. I. MuRcuHISoN, Esq., F.R.S. R. I. Murcuison, Esq., F.R.S., and Rey. G. Peacock, F.R.S8. 1839-45 Sir R. I. Murcuison, F.R.S., and Major KE. SABINE, F.R. s. 1845-50 Lieut.-Colonel E. SABINE, F.R.S. 1850-52 General E. SABINE, F.R.S., and J.¥F. ROYLE, Esq., F.R.S. 1852-53 J. F. Roy yy, Esq., F.R.S. 1853-59 General E. SABINE, F.R.S. 1859-61 Prof. R. WALKER, F.R.8. 1861-62 W. HopKIns, Esq., F.R.S. 1862-63 W. Hopkins, Esq., F.R.S., and Prof. J. PHILLIPS, F.R.S. 1863-65 W. Hopkins, Esq., F.R.S., and F. GALTON, Esa., F.R.S, 1865-66 F. GALTON, Esq., F.R.S. 1837-39 | 1897 1866-68 F. GALTON, Esq., F.R.S., and Dr. T, A. Hirst, F.R.S. 1868-71 Dr. T. A. Hirst, F.R.S., and Dr. T. THOMSON, F.R.S. 1871-72 Dr.T. THOMSON,F.R.S.,and Capt. DouGuas GALTON, F.R.S. 1872-76 Capt. DoUGLAS GALTON, F.B.S., and Dr. MICHAEL FOSTER, FE.RBS. 1876-81 Capt. DoUGLAS GALTON, F.R.S., and Dr. P. L. SCLATER, F. BS. 1881-82 Capt. DoUGLAS GALTON, F.RS., and Prof. F. M. BALFOUR, F.RB.8. 1882-83 Capt. DouUGLAS GALTON, F.R.S. 1883-95 Sir DouGLAS GALTON, F.R.S., and A. G. VERNON HARCOURT, Esq., F.B.S. | 1895-97 A. G. Vans HaRcovuRkt, Esq., F.R.S., and Prof. E. A. ScHArer, F.R.S. Prof. E. A. ScHAFER, F.R.S., ana Sir W. C. RoBERTS-AUSTEN, K.C.B., F.R.S. ASSISTANT GENERAL SECRETARIES. 1831 JOHN PHILLIPS, Esq., Secretary. 1832 Prof. J. D. Fores, Acting Secretary. 1832-62 Prof. JOHN PHILLIPS, F.R.S. 1862-78 G. GRIFFITH, Esq., M.A. J878-80 J. E. H. Gorpon, Esq., B.A., Assistant Secretary. G. GRIFFITH, Esq., M.A., Acting Secretary. 1881 1881-85 Prof. T. G. Bonney, F.B.8., Secretary. 1885-90 A. T. ATCHISON, Esq., M.A., Secretary. 1890 G. GRIFFITH, Esq., M.A. Acting Secretary. 1890 G, GRIFFITH, Esq., M.A. lili Presidents and Secretaries of the Sections of the Association. Date and Place 1832. 1833. 1834. 1835. 1836. 1837. 1838. Presidents | Secretaries MATHEMATICAL AND PHYSICAL SCIENCES. COMMITTEE OF SCIENCES, IL—MATHEMATICS AND GENERAL PHYSICS. Oxford Cambridge Edinburgh Liverpool... Newcastle 1839. Birmingham 1840. 1841. 1842. 1843. 1844, 1845. 1846. 1847. 1848. Glasgow .. Plymouth Manchester Cambridge Southamp- ton. Oxford Swansea ... 1849. Birmingham 1850. . Glasgow . Edinburgh . Ipswich ... . Belfast...... eee eeeee . Liverpool... . Cheltenham tenes Davies Gilbert, D.C.L., F.R.S. Sir D. Brewster, F.R.S. ...... Rev. W. Whewell, F.R.S. tev. H. Coddington. Prof. Forbes. Prof. Forbes, Prof. Lloyd. SECTION A.—MATHEMATICS AND PHYSICS. Rey. Dr. Robinson Rev. William Whewell, F.R.8. Sir D. Brewster, F.R.S. ...... Sir J. F. W. Herschel, Bart., F.R.S. Rey. Prof. Whewell, F.R.S.... JProfs Morbes,, WRES we. ve. > His Grace the DUKE oF DEVONSHIRE, K. ford. LL.D., F.R.S. : Sir ALEXANDER BINNIE, M.Inst.0.E., F.G.S. The Most Hon. the Marquis dr Ripon, K.G., | Professor RiickmrR, M.A., D.Sce., Sec.R.S. G.0.8.1., D.0.L., F-R.8. Dr. T. E. THorpe, Sce.D., F.R.S., Pres.0.8. The Right Rey. the Lorp BisHor or Ripon, D.D. Dr. N. Bopineron, M.A. The Right Hon: Lord MASHAM. Professor L. O. MIALL, F.R.S. GENERAL SECRETARIES. Professor BH. A. Scudrer, LL.D., F.R.S., the University, Edinburgh. Professor Sir W. C. RopERTS-AUSTEN, K.C.B., D.O.L., F.R.S., Royal Mint, London, E. ASSISTANT GENERAL SECRETARY. G. GrirriTH, Esq., M.A., Harrow, Middlesex. GENERAL TREASURER, Professor G. CAREY Foster, B.A., F.R.S:, Burlington House, London, W. LOCAL SECRETARIES FOR THE MEETING AT BRADFORD. RAMSDEN Baccuus, Esq. | J. E. FAWCETT, Esq. | FREDERICK STEVENS, Esq. LOCAL TREASURER FOR THE MEETING AT BRADFORD. W.C. Lupron, Esq., MAyor OF BRADFORD. ORDINARY MEMBERS OF THE COUNCIL, ARMSTRONG, Professor H. E., F.R.S. Marr, J. E., Esq., I'.R.S. Bonak, J., Esq., LL.D. Poutron, Professor H. B., F.R.S. OnrxEAk, Captain H, W., R.N., F.R.S. PREECE, Sir W. H., K.O.B., F.R.S. Darwi, F., Esq., F.R.S. Prick, L. L., Hsq., M.A. Darwin, Major L., Sec.R.G.S. SHAw, W.N., Esq., F.R.S. FREMANTLE, Hon. Sir C. W., K.0.B, TEALL, J. J. H., Esq., F.R.S. GASKELL, Dr. W. H., F.R.S. T HISELTON-DykR, Sir W. T., K.C.M.G., F.R.S. HALLIBURTON, Professor W. D., F.R.S. THOMSON, Professor J. M., F.R.S, Harcourt, Professor L. F. VERNON, M.A. TILDEN, Professor W. A., F.R.S, HERDMAN, Professor W. A., F.R.S. Tyxor, Professor HE. B., F.R.S. Kettin, J. Scorr, Esq., LL.D. WHITE, Sir W. H., K.O.B., F.B.S. LonGeE, Professor O. J., F.R.S. WOoLrE-BARRY, Sir JOHN, K.C.B., F.RS, MacManon, Major P. A., F.R.S. EX-OFFICIO MEMBERS OF THE COUNCIL. The Trustees, the President and President Elect, the Presidents of former years, the Vice-Presidents and Vice-Presidents Elect, the General and Assistant General Secretaries for the present and former years, the Secretary, the General Treasurers for the present and former years, and the Local Treasurer and Secretaries for the ensuing Meeting. TRUSTEES (PERMANENT). The Right Hon. Sir Joun Lupsock, Bart., M.P., D.C.L., LL.D., F.B.S., F.L.8. The Right Hon. Lord RAYLEIGH, M.A., D.C.L., LL.D., F.R.S., F.R.A.S. Professor A, W. RiickEn, M.A., D.Sc., Sec. B.S. PRESIDENTS OF FORMER YEARS. The Duke of Argyll, K.G., K.T. Lord Rayleigh, D.C.L., F.R.S. | Sir J. S. Burdon Sanderson, Bart., Lord Armstrong, C.B., LL.D. Sir H. E. Roscoe, D.C.L., F.R.S, | F.R.S. ; : Sir J. D. Hooker, K.C.S.1., F.R.S. | Sir F. J. Bramwell, Bart., F.R.S. | The Marquis of Salisbury, K.G., Sir G. G. Stokes, Bart., F.R.S. Sir I. A. Abel, Bart., K.0.B., F.R.S. Lord Kelvin, G.C.V.O., F'.R.S. F.R.S. Lord Lister, D.O.L., Pres.R.S. Prof. A. W. Williamson, F.R.S. Sir Wm. Huggins, K.O.B., F.R.S. | Sir John Evans, K.O.B., F.R.S. Sir John Lubbock, Bart., F.R.S. | Sir Archibald Geikie, LL.D.,F.R.S.| Sir William Crookes, F.R.S. GENERAL OFFICERS OF FORMER YEARS. F. Galton, Esq., F.R.5. G. Griffith, Esq., M.A. Prof. A. W. Williamson, F.R.S. Prof. Sir Michael Foster, K.C.B., | P. L. Sclater, Esq., Ph.D., F.R.S. | A. Vernon Harcourt, Esq, I'.R.S. Prof. T. G. Bonney, D.8e., F.R.S. | Prof. A. W. RUCKER, Sec.R,8. Sec.R,5S. ; AUDITORS. Dr, D. H. Scott, F.R.S. | Sir H. Trueman Wood, M.A. | Dr. Horace Brown, F.R.S. REPORT OF THE COUNCIL. Ixxxill REPORT OF THE COUNCIL. Report of the Council for the Year 1898-99, presented to the General Committee at Dover on Wednesday, September 13, 1899. Tne Meeting this year will be memorable from the fact that for the first time in the history of the Association the time and place of meeting have been fixed in conjunction with and in response to an invitation of the French Association for the Advancement of Science, with the object of affording an opportunity for the members of the sister Associations to exchange visits and to participate in scientific discussions in their several Sections. To carry out this intention, arrangements have been made for our Association to receive a visit from the members of the French Asso- ciation on Saturday, September 16, and the Association Frangaise has, on its part, invited the members of the British Association to pay a return visit on the following Thursday, and has expressed a desire that some of our members should join in an excursion to places of interest which has been planned for the following days. The Council have to deplore the loss by death of Sir Douglas Galton, who, for twenty-four years, occupied the responsible office of General Secretary, a post which he resigned only on becoming President of the Association in 1895, at the Ipswich Meeting. The Council desire to place on record their sense of the invaluable services rendered by Sir Douglas Galton to the Association. The Council regret to announce that a vacancy has been caused in the list of Vice-Presidents for this meeting in consequence of the lamented death of Lord Herschell. An invitation was received from the Vice-Chancellor of the University of Cambridge to nominate a delegate to represent the Association at the Jubilee of Sir George Gabriel Stokes, which was celebrated on June 1 and June 2. The President, Sir William Crookes, was appointed to represent the Association, and to present the following Address :— To Str Groree GasrRiet Sroxes, Bart., D.CL., LL.D., ERS, The Council of the British Association for the Advancement of Science desire to offer you their cordial congratulations on the completion of fifty years of your tenure of the Lucasian Professorship in the University of Cambridge. You have been a Member of the Association for more than half a century, and have served it in many capacities during that period. You were appointed Secretary of the Section of Mathematical and Physical Science in 1845, and continued in this laborious office until 1851. In the two following years you were a Vice-President of the Section, and became President in 1854 and again in 1862. Many times Vice- President, you were President of the Association in 1869, at the meeting in Exeter, and have been a permanent Member of the Council for the last thirty years. Your services to the Association, and to the cause for which it exists, are far from being fully told by a mere enumeration of the offices you have held. In 1852 you gave an Evening Lecture to the Members at the Belfast Meeting on a branch of Optics which has been chiefly elucidated by your own researches ; and from 1845, the first year of your membership, till the meeting last year at Bristol, the Reports of the Association have been enriched year by year by your contributions. Your celebrated reports on ‘Researches in Hydrodynamics,’ published in 1846, and on ‘Double Refraction,’ in 1862, are constantly referred to as classical writings by the cultivators of those branches of Physics, and have conferred abiding lustre on the publications of the Association. Of your other conspicuous services to the cause of Science it is almost needless to e2 lxxxiv REPORT—1899, speak, but your association with the Royal Society as Secretary for thirty-one Yeats, and subsequently as President, has given you a place which is without a parallel among those who, dutihg the last half-century, have fostered the progress of Science. That you may long continue among our leaders in the advance of knowledge is the earnest desire of the Association. The following letter has been received from the Secretaries of the Royal Society :— The Royal Society, Burlington House, London, W., November 29, 1898. DEAR SrR,—We are directed by the President and Council of the Royal Society to inform you that a Committee, consisting of Fellows of the Royal Society acting in conjunction with representatives of the Royal Geograpliical Society, was formed some time since for considering the steps that should be taken for organising an expedition to the Antarctic regions. As you are probably aware, an appeal to H.M. Government to organise such an expedition has met with no encouragement, and the Royal Geographical Society has consequently taken steps for raising a fund for the purposes of such an expedition by private subscription. To this fund the Royal Society hopes to be able to con- tribute through the medium of the Government Grant for Scientific Research, and at a recent meeting of the Antarctic Committee the following resolutions were passed, which we are directed to bring to your notice, and to request you, so far as they concern the British Association, to lay them before the Council of that body at the earliest opportunity : ‘(1) That the Treasurer of the Royal Society be requested to apply to the Government Grant Committee for a grant of 1,0007. (payable in instalments), in aid of an Antarctic Expedition. ‘(2) That an application be also made to the Council of the British Asso- Giation for a graht of 1,000/. for the same purpose.’ We remain, very faithfully yours, M. Foster, ARTHUR W, RUCKER, Secretaries, R.S, The General Secretary of the British Association. After due consideration the Council have resolved to recommend thé General Committee to contribute the sum of 1,000/. to the National Antarctic Expedition, and that the grant be given out of the accumulated funds of the Association, and not out of the sum allocated to annual grants. The following resolutions, referred to the Council by the General Committee for consideration and action if desirable, have been considered and acted upon as follows :— (1) That having regard to the letter of December 15, 1897, from Sir E. Maunde Thompson, the Council be requested to take further action with regard to a Bureau of Ethnology, by renewing the correspondence with the Trustees of the British Museum. _ The following statement, in response to a letter from the President, has been received from Sir E. Maunde Thompson :— British Museum, December 1, 1898. DEAR Si1rR,—In reply to your letter of the 23rd ultimo, with reference to the establishment of an Ethnographical Bureau in connection with the British Museum, I beg to say that unforeseen delays in carrying out certain rearrangements affecting space within our walls have hitherto prevented the Trustees from taking up the matter. Now, however, a room has been found which may serve as an office for making a start with the scheme. REPORT OF THE COUNCIL. Ixxxv But while the Trustees have accepted in principle the proposal, I beg to observe that the desired end would scarcely be obtained without the influential co-operation of the British Association, upon which I assume the Trustees may rely. May I then suggest that, as a preliminary step in maturing the scheme, one or more members of the British Association should be appointed to confer with the officers of the British Museum as to the most advisable course to follow? A Committee, consisting of the President, the President-Elect, the General Officers, Mr. Francis Galton, and Professor Tylor, was accordingly appointed for the purpose of conferring with the officers of the British Museum, as proposed by Sir E. Maunde Thompson. The President has also been in correspondence with the Marquess of Salisbury regarding this matter, and the Council have the pleasure to announce that satisfactory arrangements have been made for the establishment of such a Bureau, and that Lord Salisbury has directed that reports prepared by officers in the various Protectorates under the administration of the Foreign Office be forwarded to the British Museum. (2) That the Council be requested to consider the desirability of representing to the Colonial Government that the early establishment of a Magnetic Observatory at the Cape of Good Hope would be of the highest utility to the Science of Terrestrial Magnetism, especially in view of the Antarctic Expeditions which are about to leave Europe, and that the Observatory should be established at such a distance from electric railways and tramways as to avoid all possibility of disturbance from them. The question having been considered, the Council requested the President to make the necessary representation to the Colonial Govern- ment, and the following letter was accordingly sent to Sir Alfred Milner, the High Commissioner and Governor of Cape Colony, for presentation to the Government :— British Association for the Advancement of Science, Burlington House, W., March 1899. S1r,—I have the honour to inform you that at the Annual Meeting of the British Association for the Advancement of Science, held last September at Bristol, an International Conference met for the purpose of discussing questions connected with Terrestrial Magnetism. One of the resolutions, which was adopted by the Conference in the following terms, was referred to the Council of the Association for further consideration :— ‘That the Council be requested to consider the desirability of representing to the Colonial Government that the early establishment of a Magnetic Observatory at the Cape of Good Hope would be of the highest utility to the Science of Terrestrial Magnetism, especially in view of the Antarctic Expedi- tions which are about to leave Europe, and that the Observatory should be established at such a distance from electric railways and tramways as to avoid all possibility of disturbance from them.’ I have been requested by the Council to inform you that they have considered this resolution, and have decided to transmit it to you for your favourable con- sideration. If you should require any further information in regard to this proposal, I shall be glad to furnish it. I am, your obedient servant, WILLIAM CROOKES, President. The Council have received the following minute of the Government of Cape Colony through the High Commissioner :— ! The correspondence is given in the Appendix, p. Ixxxix, Ixxxyvi REPORT—1899. Prime Minister’s Office, Cape Town, May 13, 1899. Ministers have the honour to acknowledge the receipt of His Excellency the Governor and High Commissioner's Minute, No. 71, of the 19th ultimo, forwarding for their consideration a copy of a letter from Sir William Crookes, President of the British Association for the Advancement of Science, urging the establishment of a Magnetic Observatory at the Cape. In reply thereto, Ministers have the honour to state that they have much sym- pathy with the suggestion to establish a Magnetic Observatory, and do not overlook the scientific and practical aspects of the project, but do not regard as practicable the immediate provision by this Colony of funds for the carrying out of the scheme. (Signed) W. P. SCHREINER. (3) That the Council be requested to consider the advisability of urging Her Majesty’s Government to place at the disposal of the Seismological Committee of the British Association a suitable building for the housing of apparatus for continuous seismological observations. A Committee, consisting of the President, the President-Elect, the General Officers, Professor Riicker, Professor Ewing, and Professor Judd, was appointed to report on this resolution. The Committee, having received and considered a memorandum, drawn up by Professor Milne, on the position and requirements of the Seismological Investigation Committee of the Association, reported that in their opinion it is desirable that a Central Station should be established, and recommended the Council to request the Government to place a suit- able building at the disposal of the Seismological Committee which could be used as a station for carrying on observations, and would serve as a centre for the stations (now twenty-three in number) in various parts of the world which, at the request of the Committee, have been supplied with seismographic apparatus of the pattern they have recommended. The Council decided to reappoint the Committee for the purpose of reporting further on the best situation for the proposed Central Seismological Station, and on the cost of its maintenance. (4) That the Council be requested to urge strongly on the Indian Government the desirability, in the interests both of administration and of science, to promote an inquiry, under the direction of skilled anthro- pologists, into the physical and mental characteristics of the various races throughout the Empire, including their institutions, customs, and tradi- tions, and a carefully organised photographic survey. A Committee, consisting of the President, the President-Elect, the General Officers, Sir John Evans, Professor Tylor, Mr. F. Galton, Mr. C. H. Read, and Mr. J. L. Myres, which was appointed to consider this question, reported that in their opinion the resolution in its present form is of too comprehensive and costly a character to justify the Council in submitting it to the Indian Government. A more definite and less ambitious scheme would in their opinion be more likely to be entertained by the Indian Government. (5) That the Council be recommended to issue the collected Reports on the North-Western Tribes of Canada in a single volume at a moderate price, reprinting so many of the Reports as may be necessary. The Council, having been informed that a sufficient number of separate copies of the Fifth and the following Reports of the Committee on the North-Western Tribes of Canada were in stock for supplying those REPORT OF THE COUNCIL. lxxxvii Libraries, Public Institutions, and persons who required copies for com- pleting sets, resolved that the Reports be not reprinted. (6) That the Council be requested to bring under the notice of the Admiralty the importance of securing systematic observations upon the Erosion of the Sea-coast of the United Kingdom, and that the co-opera- tion of the Coastguard might be profitably secured for this purpose. A Committee, consisting of the President, the President-Elect, the General Officers, Sir Archibald Geikie, Mr. Whitaker, Captain Creak, Mr. A. T. Walmisley, and Professor L. Vernon Harcourt, having been appointed to report on the above resolution, recommended that the Council inquire whether the Admiralty would be willing to arrange that observa- tions of a simple character on changes in the sea-coast be recorded and reported by the Coastguards. The Committee pointed out that if the Admiralty consented to carry out this proposal it would be necessary to appoint a committee for the purpose of drawing up a scheme of instruc- tions for the observers, making arrangements for starting the work, and subsequently examining from time to time such localities as may seem to require special attention. This recommendation having been adopted by the Council, the President was requested to approach the Admiralty upon the subject, and in response to his letter the following reply has been received from the Admiralty :— Admiralty, March 25, 1899. S1r,—In reply to your letter of the 15th instant, inquiring if instructions can be given to the Coastguard to watch and report any changes taking place round the shores of the British Islands, Iam commanded by my Lords Commissioners of the Admiralty to inform you that they see no objection to this proposal, as the required observations can be made by the men in the ordinary course of their duty. On the receipt, therefore, of the instructions referred to in your letter, their Lordships, if they concur in them, will cause them to be issued accordingly. Forms on which it is desired that the reports shall be made should also be drawn up for communication to the Coastguard. I am, sir, your obedient servant, EVAN MACGREGOR. The President of the British Association for the Advancement of Science. The following have been appointed a Committee to carry out the necessary arrangements as to the despatch of forms, the receiving and tabulating reports, and such inspection of coast erosion or upheaval as may from time to time appear desirable, viz. :—Sir Archibald Geikie, Captain Creak, Professor L. Vernon Harcourt, Mr. W. Whitaker, Mr, A. T. Walmisley, and the General Officers. (7) That the Council be requested to take into consideration whether any alterations in the hours of meeting of the Sectional Committees and of the General Committee on the first day of the Annual Meeting of the Association are desirable, and to report to the General Committee at the Dover Meeting. A Committee, consisting of the President, the President-Elect, the General Officers, Sir Douglas Galton, Mr. Francis Galton, Mr. A. G. Vernon Harcourt, Professor Bonney, Professor Riicker, Professor Oliver Lodge, Sir W. T. Thiselton-Dyer, Professor Herdman, Professor Hudson Beare, and Dr. H. Forster Morley, was appointed to consider this resolution, and as a result of their inquiries the Council has resolved to recommend to the lxxxviii REPORT—1899. General Committee that the meeting of the General Committee be held in future at 4 p.m. on the first day of the Annual Meeting, instead of at 1 p.m. as has heretofore been customary, and that the Organising Commit- tees of the Sections should meet at 2 p.m. on that day instead of at 11 a.m., and should, until the Sectional Officers are definitely appointed by the General Committee, exercise the functions of Sectional Committees, with power to add to their number. The Report of the Corresponding Societies Committee for the past year, together with the list of the Corresponding Societies and the titles of the more important papers, and especially those referring to Local Scientific Investigations, published by those societies during the year ending June 1, 1899, has been received. The Corresponding Societies Committee, consisting of Mr. Francis Galton, Professor R. Meldola (Chairman), Dr. J. G. Garson, Sir J. Evans: Mr. J. Hopkinson, Mr. W. Whitaker, Mr. G. J. Symons, Professor T. G. Bonney, Mr. T. V. Holmes, Sir Cuthbert Peek, Mr. Horace T. Brown, Rev. J. O. Bevan, Professor W. W. Watts, and Rev. T. R. R. Stebbing, is hereby nominated for reappointment by the General Committee. The Council nominate the Rev. T. R. R. Stebbing, F.R.S., Chairman, and Mr. T. V. Holmes, Secretary, to the Conference of Delegates of Corresponding Societies to be held during the Meeting at Dover. The Council have received Reports from the General Treasurer during the past year, and his accounts from July 1, 1898, to June 30, 1899, which have been audited, are presented to the General Committee. In accordance with the regulations the retiring Members of the Council will be :— Boys, C. Vernon, Esq., F.R.S. Thompson, Professor S. P., F.R.S, Meldola, Professor R., F.B.S. Unwin, Professor W. C., F.R.S. Reynolds, Professor J. Emerson, M.D., F.RB.S. The Council recommend the re-election of the other ordinary Members of the Council, with the addition of the gentlemen whose names are dis- tinguished by an asterisk in the following list :— *Armstrong, Professor H. E., F.R.S. Marr, J. E., Esq., F.R.S. *Bonar, J., Esq., LL.D. Poulton, Professor E. B., F.R.S. Creak, Captain E. W., R.N., F.R.S. Preece, Sir W. H., K.C.B., F.R.S. Darwin, F., Esq., F.R.S. Price, L. L., Esq., M.A. Darwin, Major L., Sec. R.G.S. Shaw, W. N., Esq., F.R.S. Fremantle, The Hon. Sir C. W., K.C.B. Teall, J. J. H., Esq., F.R.S. Gaskell, Dr. W. H., F.R.S. Thiselton-Dyer, Sir W. T., K.C.M.G., Halliburton, Professor W. D., F.R.S. F.R.S. Harcourt, Professor L. F. Vernon, M.A. Thomson, Professor J. M., F.R.S. Herdman, Professor W. A., F.R.S. Tilden, Professor W. A., F.R.S. Keltie, J. Scott, Esq., LL.D. Tylor, Professor EH. B., F.R.S. *Lodge, Professor Oliver, F.R.S. White, Sir W. H., K.C.B., F.R.S. MacMahon, Major P. A., F.R.S. *Wolfe-Barry, Sir John, K,C.B., F.R.S, REPORT OF THE COUNCIL. lxxxix APPENDIX TO THE REPORT OF THE COUNCIL Bureau of Ethnology for Greater Britain Foreign Office: May 24, 1899. S1r,—I am directed by the Marquess of Salisbury to transmit to you the annexed correspondence which has passed between this Department and the British Associa- tion for the Advancement of Science, respecting the establishment of a Bureau of Ethnology for Greater Britain in connection with the British Museum, and the desire of the Association to obtain from Her Majesty’s Officers in the various Protectorates under the administration of the Foreign Office information of an ethnological character with respect to the numerous uncivilised races with whom they come in contact. Lord Salisbury is of opinion that Her Majesty’s Officers should be encouraged to furnish information desired by the Bureau, so far as their duties will allow of their doing so, and I am to request you to inform Officers under your administration accordingly. All reports which may be drawn up in answer to questions forwarded by the Bureau should be forwarded under flying seal through the Foreign Office. I am, sir, your most obedient, humble servant, (Signed) MARTIN GOSSELIN, H.M.’s Commissioners in the Uganda and Hast and Central Africa Protectorates, H.M,’s Consul-General in the Somali Coast Pro- tectorate. Foreign Office: May 24, 1899. S1z,—I am directed by the Marquess of Salisbury to transmit to you for your information, and for such effect as you may be able to give to the instructions ccn- tained in it, a copy of a despatch which has been addressed to Her Majesty’s Com- missioners in the Uganda, and East and Central Africa Protectorates, and Her Majesty’s Consul-General in the Somali Coast Protectorate, on the subject of pro- curing information for the Bureau of Ethnology which is about to be established in connection with the British Museum. I am, sir, your most obedient, humble servant, (Signed) MARTIN GOSSELIN. H,M.’s Acting Agent at Zanzibar. H.M.’s Consul at Brunei. INcLOSURE 1, Letter from the President of the British Association to the Marquess of Salisbury :— Burlington House: March 30, 1599. My Lorp,—I have the honour to inform you that a proposal to establish a Bureau of Ethnology for Greater Britain has been discussed at several recent meetings of the British Association for the Advancement of Science, and that the Council of the Association were subsequently requested to consider the possibility of establishing such a Bureau. The Council appointed a Committee to consider the proposal, and having adopted the report of the Committee, requested the Trustees of the British Museum to allow the proposed Bureau to be established in connection with that Institution. The Trustees have expressed their willingness to undertake the working of the Bureau, oud the necessary space for its establishment has now been provided at Blooms- ury. In forwarding to your Lordship copies of the report of the Committee appointed by the Council, I would desire to call special attention to the following paragraph xe REPORT—1899:; viz.: ‘The collecting of the necessary information for the Bureau could be done with but little expense and with avery small staff only, if the scheme were recognised and forwarded by the Government. If instructions were issued, for instance, by the Colonial Office, the Foreign Office, the Admiralty, and the Intelligence Branch of the War Office, to the officers acting under each of these departments, not only that they were at liberty to conduct these inquiries, but that credit would be given to them officially for good work in this direction, there is little doubt that many observers qualified by their previous training would at once put themselves and their leisure at the disposal of the Bureau.’ If the proposed Bureau is to work successfully, it is necessary to have the ap- proval and co-operation of the several Departments of the Government concerned with the primitive races to be dealt with. The Council have reason to believe that a large proportionjof the officers now employed in dealing with these savage people would gladly undertake scientific work of the character required by the Report, if only they could be assured that such work would not be regarded unfavourably by the authorities at home. There is reason to believe that such an impression exists, but itis probably the result of some misunderstanding ; and,in order to make the matter quite clear, I would venture to ask from your Lordship an expression of opinion favourable to the terms of the paragraph above quoted. The Report itself gives in concise form a statement of the benefit likely to accrue from the establishment of such a Bureau, as to the general principle of which I feel sure the British Association may count upon your Lordship’s entire sympathy. I am, my Lord, your obedient servant, (Signed) WILLIAM CROOKES, President. INCLOSURE 2. Report of the Committee appointed by the Council to consider the Sollowing Resolution :— ‘That it is of urgent importance to press upon the Government the necessity of establishing a Bureau of Ethnology for Greater Britain, which, by collecting information with regard to the native races within, and on the borders of, the Empire, will prove of immense value to science and to the Government itself.’ A central establishment in England, to which would come information with regard to the habits, beliefs, and methods of government of the primitive peoples now existing would be of great service to science, and of no inconsiderable utility to the Government. 1. The efforts of the various societies which have during the last twenty years devoted themselves to collecting and publishing ethnological information have necessarily produced somewhat unequal, and therefore unsatisfactory, results. Such societies had, of course, to depend upon the reports of explorers, who usually travelled for another purpose than that in which the societies were interested ; and such reports were naturally unsystematic, the observers being mostly untrained in the science. Again, whole regions would be unrepresented in the transactions of the societies, perhaps from the absence of the usual attractions of travellers, e.g. big game or mineral riches. This has been to some extent corrected, at least as to the systematic nature of the reports, by the publication of ‘ Anthro- pological Notes and Queries’ by the Anthropological Institute, with the help of the British Association. If it be admitted that the study of the human race is an important branch of science, no further argument is needed to commend the gathering of facts with regard to the conditions under which aboriginal races now live, and, if this work is worth doing, it should be done without REPORT OF THE COUNCIL. xc delay. With the exception, perhaps, of the negro it would seem that none of the lower races are capable of living side by side with whites. The usual result of such contact is demoralisation, physical decline, and steady diminution of numbers ; in the case of the Tasmanians, entire dis- appearance. Such will probably soon be the fate of the Maories, the Andamanese, the North American Indians, and the, blacks of Australia. While these exist it is possible to preserve their traditions and folk-lore, and to record their habits of life, their arts, and the like, and such direct evidence is necessarily more valuable than accounts filtered through the recollection of the most intelligent white man. It is scarcely necessary to enlarge upon this point, as no one will seriously question the value to science of such information. But it does seem necessary to urge that no time be lost. 2. As to the benefit to the Government of these inquiries, the history of our relations with native tribes in India and the Colonies is rich in examples. No one who has read of the ways of the African can doubt that a thorough study of his character, his beliefs and superstitions, is a necessity for those who have to deal with him. And what is true of the natives of Africa is also true, in a greater or less degree, of all uncivilised races. Their ideas of common things and common acts are so radically different from those of civilised man that it is impossible for him to understand them without a special training. Even in dealing with the highly civilised natives of India it is most necessary that an inquirer should be familiar with their religion, and with the racial prejudices which the natives of India possess in common with other civilised nations. A training in knowledge of native habits is now gone through by our officers, traders, and missionaries on the spot ; and by experience—some- times dearly bought—they, after many failures, learn how te deal with the natives. By the establishment of such a Bureau as is here advocated much might be done to train our officers before they go out, as is now done by the Dutch Government, who have a handbook and a regular course of instruction as to the life, laws, religion, &c., of the inhabitants of the Dutch Indies. The experience thus gained would then mature rapidly, and they would become valuable servants to the State more quickly. The collecting of the necessary information for the Bureau could be done with but little expense und with a very small staff only, if the scheme were recognised and forwarded by the Government. If instruc- tions were issued, for instance, by the Colonial Office, the Foreign Cflfice, the Admiralty, and the Intelligence Branch of the War Office, to the officers acting under each of these departments, not only that they were at liberty to conduct these inquiries, but that credit would be given to them officially for good work in this direction, there is little doubt that many observers qualified by their previous training would at once put themselves and their leisure at the disposal of the Bureau. The Bureau itself, the central office, would be of necessity in London —in no other place could it properly serve its purpose—and preferably, for the sake of economy and official control, it should be under the administration of some existing Government office. But the various interests involved make it somewhat difficult to recommend where it should be placed. The Colonial Office would obviously present some advantages. The British Museum has been suggested, with good reason, Xcll REPORT—1899. and there appears to be no insuperable difficulty if the trustees are willing to undertake the responsibility of controlling such a department. The staff would not be numerous. A director accustomed to deal with ethnological matter would necessarily direct the conduct of the inquiries, and until the material assumed large proportions two or three clerks would probably suffice. If the value of the results were considered to justify it, the increase of the area of operations over the world would probably call for additional assistance after the Bureau had been at work for a few years. The Bureau of Ethnology in the United States aims chiefly at pub- lishing its reports, but its area is limited to America. The scope of the present proposal is so much wider that the Committee think it better not to deal with the question of publication at present. INCLOSURE 3, Letter from the Foreign Office to the British Association — Foreign Office, April 7, 1899. S1r,—I am directed by the Marquess of Salisbury to acknowledge the receipt of your letter of the 30th ult, on the subject of the establishment of a Bureau of Ethnology for Greater Britain; and I am to request that you will inform his lord- ship whether it is correctly understood that what the British Association for the Advancement of Science desires, so far as this Department is concerned, is that Her Majesty’s officers in the various Protectorates administered under the Foreign Office should report on occasion to the best of their ability on the ethnology of the various native races in those Protectorates, If this be the correct interpretation of the wishes of the British Association, Lord Salisbury would be obliged if some more precise definition can be furnished as to the points to which attention should be directed, with a view to framing instruc- tions for the guidance of the officers concerned. I am, sir, your most obedient, humble servant, (Signed) MARTIN GOSSELIN, The President of the British Association for the Advancement of Science. INCLOSURE 4, Letter from the British Association to the Foreign Office :— Burlington House, London, May 3, 1899, S1z,— I have to acknowledge the receipt of the letter from Sir Martin Gosselin of April 7, with regard to the proposed establishment of a Bureau of Ethnology for Greater Britain in connection with the British Museum. The purpose of the British Association in applying to the Foreign Office has been correctly understood so far that it is desired to obtain from the agents and officers of the Foreign Office information of an ethnological character with respect to the numerous uncivilised races with whom they come into daily contact. But it is not contemplated to give the Foreign Office any trouble in conducting these inquiries, The officers of the Bureau will prepare the questions and forward them to the various officers, who, it is hoped, may be willing to furnish the answers. All the material thus gathered will be systematically arranged in the British Museum, so as to be available both for scientific research and for the purposes of the Government. The Council of the British Association felt, however, that before entering into communication with those officers it would be wise to ask for Lord Salisbury’s approval of the scheme, in order that the gentlemen who were disposed to undertake such work as is contemplated by the Bureau might be assured that the work would be favourably regarded by their Department. REPORT OF THE COUNCIL. xtill In the event, therefore, of the scheme meeting with the approval of Lord Salisbury I would venture to ask his lordship to be good enough to express this approval in such terms that the letter can be used in opening the correspondence with the agents of the Foreign Office. I am, sir, your obedient servant, (Signed) WILLIAM CROOKES, President. The Under-Secretary of State for Foreign Affairs. Inctosure 5. Letter from Foreign Office to the British Association, May 24, 1899 :— Foreign Office, May 24, 1899. S1r,—With reference to your letter of the 3rd instant, I am directed by the Mar- quess of Salisbury to transmit to you for your information, copies of despatches which have been addressed to Her Majesty’s Commissioners in the Uganda and East and Central Africa Protectorates, Her Majesty’s Consul-General in the Somali Coast Protectorate, Her Majesty’s Acting Agent at Zanzibar, and Her Majesty’s Consul at Brunei, on the subject of procuring information for the Bureau of Ethnology which is about to be established in connection with the British Museum. I am, sir, your most obedient, humble servant, (Signed) MARTIN GOSSELIN. The President of the British Association for the Advancement of Science, Burlington House, W. xciv REPORT—1899. CoMMITTEES APPOINTED BY THE GENERAL CoMMITTEE AT THE DovER MEETING IN SEPTEMBER 1899. 1. Receiving Grants of Money. Subject for Investigation or Purpose Making Experiments for improv- ing the Construction of Practical Standards for use in Hlectrical Measurements. [And 3002, in hand.] Seismological Observations. Members of the Committee Chai¢man.—Lord Rayleigh. Seerctary.—My. R. T. Glazebrook. Lord Kelvin, Professors W. HE. Ayrton, J. Perry, W. G. Adams, Oliver J. Lodge, Sir W. H. Preece, Profes- sors J. D. Everett and A. Schuster, Dr. J. A. Fleming, | Professors G. F. FitzGerald and J.J. Thomson, Mr. W.N. Shaw, Dr. J. YT. Bottomley, Rev. T. C. Fitzpatrick, Professor J. Viriamu Jones, Dr. G. John- stone Stoney, Professor 8. P. Thompson, Mr. J. Rennie, Mr. E. H. Griffiths, Professor A. W. Riicker, Professor H. L. Callen- dar, Sir W. C. Roberts-Austen, and Mr. G. Matthey. Chairman.—Prof. J. W. Judd. Secrctary.—Professor J. Milne. Lord Kelvin, Sir F. J. Bramwell, Professor G. H. Darwin, Mr. Horace Darwin, Major L. Dar- win, Professor J. A. Ewing, Professor C. G. Knott, Professor R. Meldola, Professor J. Perry, Professor J. H. Poynting, Pro- | fessor T. G. Bonney, Mr. C. V. | Boys, Professor H. H. Turner, | Mr. G. J. Symons, Mr, Clement | Reid, Mr. R. D. Oldham, and Mr. W. E. Plummer. Radiation from a Source of Light | Chairman.——Professor G. F. Fitz- in a Magnetic Field. Gerald. Secretary.—Professor T, Preston. Professor A. Schuster, Professor O. J. Lodge, Professor 8. P. Thompson, Dr. Gerald Molloy, and Dr. W. E. Adeney. and G. |} Carey Foster, Dr. A. Muirhead, | | 60 bo or (= on 00; 00 COMMITTEES APPOINTED BY THE GENERAL COMMITTEE. 1. Receiving Grants of Money—continued. XCV Subject for Investigation or Purpose To consider the most suitable Method of Determining the Components of the Magnetic Force on board Ship. To establish a Meteorological Observatory on Mount Royal, Montreal. For calculating Tables of certain Mathematical Functions, and, if necessary, for taking steps to carry out the Calculations, and to publish the results in an accessible form. The relation between the Absorp- tion Spectra and Chemical Con- stitution of Organic Substances. Preparing a new Series of Wave- length Tables of the Spectra of the Elements. The Electrolytic Methods of Quan- titative Analysis. The Study of Isomorphous Sul- phonic Derivatives of Benzene. The Nature of Alloys. Members of the Committee Grants Chairman. — Professor A. Riicker. Secretary.—Dr. C. H. Lees. Lord Kelvin, Professor A. Schuster, Captain Creak, Professor W. Stroud, Mr. C. V. Boys, and Mr. W. Watson. Chairman.—Professor H. L. Cal- lendar. Secretary.—Professor C.H.McLeod. Professor F, Adams, and Mr. R. F. Stupart. Chairman.—Lord Kelvin. Secretary.—Lieut.-Colonel Allan Cunningham. Dr. J. W. L. Glaisher, Professor A. G. Greenhill, Professor W. M. Hicks, Major P. A. MacMahon, and Professor A. Lodge. Chairman and Secretary.—Pro- fessor W. Noel Hartley, Professor F. R. Japp, and Professor J. J. Dobbie. Chairman.—Sir H. E. Roscoe. Secretary.—Dr. Marshall Watts. Sir J. N. Lockyer, Professors J. Dewar, G. D. Liveing, A. Schus- | ter, W. N. Hartley, and Wol- cott Gibbs, and Captain Abney. Chairman.—Professor J. Emerson Reynolds. Secretary.—Dr. C. A. Kohn. Professor Frankland, Professor F. Clowes, Dr. Hugh Marshall, Mr. A.E. Fletcher, and Professor W. Carleton Williams. Chairman.—Professor H. A. Miers. Secretary.—Professor H. E. Arm- strong. Dr. W. P. Wynne. Chairman. and Secretary.—- Mr. F. H. Neville. Mr. C. T. Heycock, and Mr, E. H. Griffiths. W. eee 10 0 00 00 30 90 00 ou 20 00 d, 0 30 C0 : XOVi REPORT—1899. 1. Receiving Grants of Money—continued. Subject for Investigation or Purpose To investigate the Erratic Blocks of the British Isles, and to take measures for their preservation. [67. in hand. ] The Collection, Preservation, and Systematic Registration of Photographs of Geological In- terest. To examine the Conditions under which remains of the Irish Elk are found in the Isle of Man. To further investigate the Fauna and Flora of the Pleistocene Beds in Canada. The Excavation of the Ossiferous Caves at Uphill, near Weston- super-Mare. (87. in hand. ] The movements of Underground Waters of Craven. To explore Irish Caves. [Collections to be placed in the Science and Art Museum, Dub- lin.] Members of the Committee Chairman.—Professor E. Hull. Secrctary.—Prof, P. F. Kendall. Professor T. G. Bonney, Mr. C. E. De Rance, Professor W. J. Sollas, Mr. R. H. Tiddeman, Rev. S. N. Harrison, Mr. J. Horne, Mr. Dugald Bell, Mr. F. M. Burton, Mr. J. Lomas, Mr. A. R. Dwerry- house, Mr. J. W. Stather, and Mr. R. D. Tucker. Chairman.—Professor J. Geilkie. Seoretary.—ProfessorW.W. Watts. Professor T. G. Bonney, Dr. T. An- derson, and Messrs. A. S. Reid, EH. J. Garwood, W. Gray, H. B. Woodward, R. Kidston, J. J. H. Teall, J. G. Goodchild, H. Coates, C. V. Crook, G. Bingley, and R. Welch. Chairman.—Professor W. Boyd Dawkins. Secretary.— My. P. M. C. Kermode. His Honour Deemster Gill, Mr. G. W. Lamplugh, and Canon E. B. Savage. Chairman.—Sir J. W. Dawson. Secretary.—Professor A, P. Cole- man. Professor D. P. Penhallow, Dr. H. Ami, and Mr. G. W. Lamplugh. Chairman.—Professor C. Lloyd Morgan. Secretary.—Mr. H. Bolton. Professor W. Boyd Dawkins, Mr. W.R. Barker, Mr.8. H. Reynolds, and Mr. E. T. Newton. Chairman.—ProfessorW.W. Watts. Secretary.—Captain A. R. Dwerry- house. Professor A. Smithells, Rev. E. Jones, Mr. Walter Morrison, M.P., Mr. G. Bray, Mr. W. L. Carter, Mr. W. Fairley, Pro- fessor P. ¥. Kendall, and Mr. J. EH. Marr. Chairman.—Dr. RB. F. Scharff. Secretary.—Mr. R. Lloyd Praeger. Mr. G. Coffey, Professor Grenville Cole, Dr. Cunningham, Mr. A. McHenry, and Mr. R. J. Ussher. 10 10 10 40 00 00 00 00 00 00 COMMITTEES APPOINTED BY THE GENERAL COMMITTEE. 1. Receiving Grants of Money—continued. xevil Subject for Investigation or Purpose To enable Mr. H. M. Kyle and Professor Herdman, or, failing them, some other competent investigator, to carry on definite pieces of work at the Zoological Station at Naples. To enable Mr. Martin T. Wood- ward to study the Embryology of the Mollusca; Mr. 8. D. Scott to investigate the Excretory Organs of the Tunicata; and Mr. G. Brebner to continue his studies on the Reproduction of Marine Algze, and to enable other com- petent Naturalists to perform definite pieces of work at the Marine Laboratory, Plymouth. Compilation of an Index Generum et Specierum Animalium, To work out the details of the Observations on the Migration of Birds at Lighthouses and Lightships, 1880-87. The Periodic Investigation of the Plankton and Physical Con- ditions of the English Channel. To continue the investigation of the Zoology of the Sandwich Islands, with powerto co-operate with the Committee appointed for the purpose by the Royal Society, and to avail themselves of such assistance in their in- vestigations as may be offered by the Hawaiian Government or the Trustees of the Museum | at Honolulu. The Committee to have power to dispose of speci- mens where advisable. 1899. Members of the Committee Chairman.—Professor W. A. Herdman, Secretary.—Professor G. B. Howes. Professor E. Ray Lankester, Pro- fessor W. F. R. Weldon, Pro- fessor S. J. Hickson, Mr. A. Sedgewick, and Professor W. C. McIntosh. Chairman.—Mr. G. C. Bourne. Secretary. — Professor E, Ray Lankester. Professor Sydney H. Vines, Mr. A. Sedgwick, Professor W. F. R. Weldon, and Mr. W. Garstang. Chairman.—Dr. H. Woodward. Secretary.—Mr. F. A. Bather. Dr. P. L. Sclater, Rev. T. R. R. Stebbing, Mr. R. McLachlan, and Mr. W. E. Hoyle. Chairman.—Professor A. Newton. Secretary.—Rev. E. P. Knubley. Mr. John A. Harvie-Brown, Mr. R. M. Barrington, Mr. A. H. Evans, and Dr. H. O. Forbes, Chairman.—Professor E. Ray Lankester. Secretary.—Mr. Walter Garstang. Professor W. A. Herdman, and Mr. H. N. Dickson. Chair man.—Professor A. Newton. Secretary.—Dr. David Sharp. Dr. W. T. Blanford, Professor §. J. Hickson, Dr. P. L. Sclater, Mr. ¥. Du Cane Godman, and Mr. Edgar A, Smith. Q Led b~] =} > w 40 100 00 00 00 00; xXcviil REPORT—1899. 1. Receiving Grants of Money—continued. Subject for Investigation or Purpose To investigate the structure, for- mation, and growth of the Coral Reefs of the Indian Region, with special observations on the inter-relationship of the reef organisms, the depths at which they grow, the food of corals, effects of currents and character of the ocean bottom, &c. The land flora and fauna will be collected, and it is intended that observations shall be made on the manners, &c., of the natives in the different parts of the Maldive group. The revision of the Physical and Chemical Constants of Sea- water. Future dealings in Raw Produce, State Monopolies in other Countries. [Balance of grant unexpended, 132. 138. 6d.] To consider whether the British Association form of Thread for Small Screws should be modi- fied, and, if so, in what direc- tion. [Balance of grant unexpended, 171. 1s, 2d.] { | To co-operate with the Silchester Excavation Fund Committee in their explorations, Members of the Committee Chairman.—Mr. A. Sedgwick. Secretary.—J. Graham Kerr. Professor J. W. Judd, Mr. J. J. Lister, and Mr. §. F. Harmer. Chairman.—Sir John Murray. Secretary.—Mr. H, N. Dickson. Mr. J. Y. Buchanan, and Dr. H. R. Mill. Chairman.—Mz. L. L. Price. Secretary.—Prof. A. W. Flux. Major P. G. Craigie, Professor W. Cunningham, Professor Edgeworth, Professor Gonner, Mr. R. H. Hooker, and Mr. H. R. Rathbone. Chairman.—Professor H. Sidg- wick. Secretary.—Mr. H. Higgs. Mr. W. M. Acworth, the Rt. Hon. L. H. Courtney, and Professor H. 8. Foxwell. Chatrman.—Sir W. H. Preece. Secretary.—Mr. W. A. Price. Lord Kelvin, Sir F. J. Bramwell, Sir H. Trueman Wood, Maj.- Gen. Webber, Mr. R. E. Cromp- ton, Mr, A. Stroh, Mr. A. Le Neve Foster, Mr. C. J. Hewitt, Mr. G. K, B. Elphinstone, Mr. T. Buckney, Col. Watkin, Mr. E. Rigg, Mr. Conrad W. Cooke, and Mr. Vernon Boys. Chairman.—Mz. A, J. Evans. Secretary.—Mr. John L. Myres. Mr. E. W. Brabrook. 10 100 00 00 00 COMMITTEES APPOINTED BY THE GENERAL COMMITTEE. 1. Receiving Grants of Money—continued. Subject for Investigation or Purpose To organise an Ethnological Sur- vey of Canada. Preparing a new edition of ‘Notes and Queries on Anthropology.’ To conduct Explorations with the object of ascertaining the age of Stone Circles. [Balance in hand. ] The Collection, Preservation, and Systematic Registration of Pho- tographs of Anthropological Interest. To co-operate with the Committee appointed by the International Congress of Hygiene and Demo- graphy in the investigation of the Mental and Physical Condi- tion of Children. To examine the Natural History and Ethnography of the Malay Peninsula. The Physiological Effects of Pep- tone and its Precursors when introduced into the circulation. { Comparative Histology of Supra- renal Capsules. Secretary.—Dr. George Dawson. Mr. EK. W. Brabrook, Professor Members of the Committee Chairman.—Professor D. P. Pen- hallow. A. C. Haddon, Mr. E. 8. Hart- land, Sir J. G. Bourinot, Abbé Cuoq, Mr. B. Sulte, Abbé Tan- quay, Mr. C. Hill-Tout, Mr. David Boyle, Rev. Dr. Scad- ding, Rev. Dr. J. Maclean, Dr. Merée Beauchemin, Rev. Dr. G. Patterson, Mr. C. N. Bell, Professor E, B. Tylor, Hon.G. W. Ross, Professor J. Mavor, Mr. A. F. Hunter, and Dr. W. F. Ganong. Chairman.—Professor HE. B. Tylor. Secretary.—Dr. J. G. Garson. General Pitt-Rivers, Mr. C. H. Read, and Mr, J. L. Myres. Chairman.—Dr. J. G. Garson. Secretary.—Mr. H. Balfour. Gen. Pitt-Rivers, Sir John Evans, Mr. C. H. Read, Professor Mel- dola, Mr. A. J. Evans, Dr. R. Munro, and Professor Boyd- Dawkins. Chairman.—Mr. C. H. Read. Secretary.—Mr. J. L. Myres. Dr. J. G. Garson, Mr. H. Ling Roth, Mr. H. Balfour, Mr. E. 8S. Hart- land, and Professor Flinders Petrie. Chairman.—Mr. HE. W. Brabrook. Secretary.—Dr. Francis Warner. Dr. J.G. Garson, Mr. White Wallis, and Dr. W. H. R. Rivers. Chairman.—Mr. C. H. Read. Secretary.—Mr. W. Crooke. Professor A. Macalister, and Pro- fessor W. Ridgeway. Chaiyman. — Professor E. A. Schiéifer. Secretary. — Professor W. H. Thompson. Professor R. Boyce and Professor C. 8. Sherrington. Chairman.—Professor HH. A. Schafer. Secretary—Mr. Swale Vincent. Mr. Victor Horsley. xcin 3 Grants £ 50 40 10 bo ou 20 20 s. d. 00 00 00 00 00 00 00 REPORT—1899. 1. Receiving Grants of Money—continued. Subject for Investigation or Purpose Comparative Histology of Cerebral Cortex. The Electrical Changes in Mam- malian Nerve. Vascular Supply of Secreting Glands. Experimental Investigation of Assimilation in Plants. (62. 6s. 8d. in hand. ] Fertilisation in Pheophycez. Cortesponding Societies Com- mittee for the preparation of their Report. Members of the Committee Gratits & 8. oe Chairman.—Professor F. Gotch. 5 00 Secretary.—Dr. G. Mann. Professor H. H. Starling. Chairman.—Professor F. Gotch. 20 00 Secretary.—Mr. J. 8. Macdonald. Professor E. H. Starling. Chairman.—Prof. KE. H. Starling. | 10 0 0 Secretary.—Dr. J. L. Bunch. Dr. L, E. Shore. Chairman.—Mr. F. Darwin. — Secretary.—Professor J. R. Green. Professor Marshall Ward. Chairman.—ProfessorJ.B.Farmer.| 20 0 0 Secretary.—ProfessorR.W .Phillips. Professor F. O. Bower, and Pro- fessor Harvey Gibson. Chairman.—Professor R. Meldola.| 20 00 Secretary.—Mr. T. V. Holmes. Mr. Francis Galton, Mr. G, J. Symons, Dr. J. G. Garson, Sir John Evans, Mr. J. Hopkinson, Professor T. G. Bonney, Mr. W. Whitaker, Sir Cuthbert EH. Peek, Mr. Horace T. Brown, Rev. J. O. Bevan, Professor W. W. Watts, and Rev. T. R. R. Stebbing. 2. Not receiving Grants of Money. Subject for Investigation or Purpose Members of the Committee To confer with British and Foreign Societies publishing Mathematical and Physical Papers as to the desir- ability of securing Uniformity in the size of the pages of their Transactions and Proceedings. Co-operating with the Scottish Meteoro- logical Society in making Meteoro- logical Observations on Ben Nevis. To confer with the Astronomer Royal and the Superintendents of other Observatories with reference to the Comparison of Magnetic Standards with a view of carrying out such comparison, Chairman.—Professor 8. P. Thompson. Secretary.—Mr. J. Swinburne. Prof. G. H. Bryan, Mr. C. V. Burton, Mr. R. T. Glazebrook, Professor A, W. Riicker, and Dr. G. Johnstone Stoney. Chairman.—Lord McLaren. Secretary.—Professor Crum Browh, Sir John Murray, Dr. A. Buchan, and Professor R, Copeland. Chairman.—Professor A. W. Ricker. Secretavy.—Professor W. Watson. Professor A. Schuster, and Professor H. H. Turner. : COMMITTEES APPOINTED BY THE GENERAL COMMITTEE, ci 2, Not receiving Grants of Money—continued. Subject for Investigation or Purpose. Comparing and Reducing Magnetic Ob- servations, The Present State of our Knowledge in Electrolysis and Electro-che- -mnistry. The Rate of Increase of Underground Temperature downwards in various Localities of Dry Land and under Water. The Application of Photography to the ' Elucidation of Meteorological Phe- nomena, Considering the best Methods of Re- cording the Direct Intensity of Solar Radiation. That Miss Hardcastle be requested to draw up a Report on the present state of the Theory of Point-Groups. The Continuation of the Bibliography of Spectroscopy. The Teaching of Natural Science in Elementary Schools, Members of the Committee, Chairman. —Professor W. G. Adams. Secretary.—Dr. C. Chree. Lord Kelvin, Professor G. H. Darwin, Professor G. Chrystal, Professor A. Schuster, Captain E. W. Creak, the Astronomer Royal, Mr. William Ellis, and Professor A. W. Riicker. Chairman.—Mr. W. N. Shaw. Secretary.— Mr. W. C. D. Whetham. Rev. T. C. Fitzpatrick, Mr. E. H. Griffiths, and Mr. 8. Skinner. Chairman.—Professor J. D. Everett. Secretary.—Professor J. D. Everett. Lord Kelvin, Mr. G. J. Symons, Sir Archi- bald Geikie, Mr. J. Glaisher, Professor Edward Hull, Dr. C. Le Neve Foster, Professor A. §. Herschel, Professor G. A Lebour, Mr. A. B. Wynne, Mr, W. Galloway, Mr. Joseph Dickinson, Mr. G. F. Deacon, Mr. E. Wethered, Mr. A. Strahan, Professor Michie Smith, and Professor H. L. Callendar., Chairman.—Mr. G. J. Symons. Secretary.—Mr. A. W. Clayden. Professor R. Meldola, Mr. John Hopkin- son, and Mr. H. N. Dickson. Chaivman.—Dr. G. Johnstone Stoney. Secretary.—Professor H. McLeod. Sir G. G. Stokes, Professor A. Schuster, Sir H. E. Roscoe, Captain W. de W. Abney, Dr. C. Chree, Professor G. F. FitzGerald, Professor H. L. Callendar, Mr. G. J. Symons, Mr. W. E. Wilson, and Professor A. A. Rambaut. Chairyman.—Professor H. McLeod. Secretary.—Sir W. C. Roberts-Austen. Mr. H. G. Madan, and Mr, D. H. Nagel. Chairman.—Dr. J. H. Gladstone. Secretary.—Professor H. E, Armstrong. Mr. George Gladstone, Mr. W. R, Dun- stan, Sir J. Lubbock, Sir Philip Magnus, Sir H. E. Roscoe, Dr. Sil- vanus P, Thompson, and Professor A. Smithells, REPORT—1899, 2. Not receiving Grants of Money—continued. Subject for Investigation or Purpose Members of the Committee The Promotion of Agriculture: to re- port on the means by which in various Countries Agriculture is advanced by research, by special Educational Insti- tutions, and by the dissemination of information and advice among Agri- culturists. Isomeric Naphthalene Derivatives. To establish a Uniform System of Re- cording the Results of the Chemical and Bacterial Examination of Water and Sewage. To consider the best Methods for the Registration of all Type Specimens of Fossils in the British Isles, and to report on the same. The Collection, Preservation, and Sys- tematic Registration of Canadian Photographs of Geological Interest. To report upon the Present State of our Knowledge of the Structure of Crystals. To study Life-zones in the British Car- boniferous Rocks. To promote the Systematic Collection of Photographic and other Records of Pedigree Stock. Climatology of Tropical Africa. Chairman —Sir John Evans. Secretary.—Professor H. E. Armstrong. Sir Michael Foster, Professor Marshall Ward, Sir J. H. Gilbert, Right Hon. J. Bryce, Professor J. W. Robertson, Dr. W. Saunders, Professor Mills, Professor J. Mavor, Professor Poulton, and Mr. §, U. Pickering. Chairman.—Professor W. A. Tilden. Secretary.—Professor H. HE. Armstrong. Chairman.—-Professor W. Ramsay. Secretary.—Dr. 8, Rideal. Professor F. Clowes, Professor P. F. Frankland, Professor R. Boyce, and Mr. W. J. Dibdin. Chairman.—Dr. H. Woodward. Secretary.—Mr. A. Smith Woodward, Rev. G. F. Whidborne, Mr. R. Kidston, Pro- fessor H. G. Seeley, and Mr. H. Woods. Chairman.—Professor A. P. Coleman. Secretary.—Mr. Parks, Professor A. B. Willmott, Professor F. D. Adams, Mr. J. B. Tyrrell, and Professor W. W. Watts. Chairman.—Frofessor N. Story Maske- lyne. Seeretary.—Professor H. A. Miers. Mr. L. Fletcher, Professor W. J. Sollas, Mr. W. Barlow, Mr. G. F. H. Smith, and the Earl of Berkeley. Chairman.—Mz. J. E. Marr. Secretary.—Dr. Wheelton Hind. Mr. F. A. Bather, Mr. G. C. Crick, Mr. A. H. Foord, Mr. H. Fox, Mr. E. J. Garwood, Dr. G. J. Hinde, Professor P. F. Kendall, Mr. J. W. Kirkby, Mr. R. Kidston, Mr. G. W. Lamplugh, Professor G. A. Lebour, Mr. G. H. Morton, Mr. B. N. Peach, Mr. A. Strahan, and Dr. H. Woodward. Chairman.—My. Francis Galton. Seeretary.—Professor W. F. R. Weldon. Chairman.—Mr. EK. G. Ravenstein. Secretary.—Mr. H. N. Dickson. Sir John Kirk, Dr. H, R. Mill, and Mr. G. J. Symons. COMMITTEES APPOINTED BY THE GENERAL COMMITTEE, cili 2. Not receiving Grants of Money—continued. ee ae Subject for Investigation or Purpose The Present State of Anthropological Members of the Committee Chairman.—FProfessor E. B. Tylor. Teaching in the United Kingdom and | Secretary.—Mr. H. Ling Roth. Elsewhere. The Lake Village at Glastonbury. To enquire into the Effectiveness of the System of Identification by Finger- prints now in use throughout India, and on the Probable Limits of its Applicability. Professor A. Macalister, Professor A. C. Haddon, Mr. C. H. Read, Mr. H. Bal- four, Mr. F. W. Rudler, Dr. R. Munro, and Professor Flinders Petrie. Chairvman.—Dr. R. Munro. Secretary.—Mr. A. Bulleid. Professor W. Boyd Dawkins, General Pitt- Rivers, Sir John Evans, Mr. Arthur J. Evans, and Mr. C. H. Read. Chairman.—Mr. Francis Galton. Seeretary.— Mr. L. Gomme. Colonel R. C. Temple, Mr. C. H. Read, Mr. W. Crooke, Professor Karl Pear- son, and Professor W. F. R. Weldon. Chairman.—Professor E. A. Schiifer. Secretary.—Professor A. B, Macallum. Professor E. Ray Lankester, Professor W. D. Halliburton, and Mr. G. C. Bourne. The Micro-chemistry of Cells. Communications ordered to be printed in extenso, ‘The new Alexander III. Bridge in Paris,’ by M. Amédée Alby. ‘The Dover Harbour Works,’ by J. C. Coode and W. Matthews. Resolutions referred to the Council for consideration, and action af desirable. That in view of the opportunities of ethnographical inquiry which will be presented by the Indian Census, the Council of the Association be requested to urge the Government of India to make use of the Census Officers for the purposes enume- tated below, and to place photographers at the service of the Census Officers. That the Council be requested to represent to Her Majesty’s Government the Bee ance of giving more prominence to Botany in the training of Indian Forest cers. That the attention of the Council be called to the wording of the rule regarding specimens collected by Committees appointed by the Association, with a view to its revision. That the complete investigation of the Ichthyology of the West African rivers promises extremely important scientific results, and that the Council of the Associa- tion be requested to take such means as may seem to it advisable to bring the matter to the notice of the Trustees of the British Museum. Change of Howrs of Meetings, &c. That the Organising Committees meet at 2 P.M. instead of at 11 A.M.; and shall, until the Sectional Officers are definitely appointed by the General Committee, exer- cise the functions of Sectional Committees, with power to appoint members of the Sectional Committees. : That the first meeting of the General Committee be held at 4 p.m. instead of at P.M. That the proceedings of the opening meeting begin at 8,30 P.M, instead of at 8 P.M. as heretofore, Be civ REPORT—1899. Synopsis of Gr cite of Money appropriated to Scientific Purposes by the General Committee at the Dover Meeting, September, 1899. Names of the Members entitled to call on the General Treasurer for the respective Grants are prefixed. Mathematics. *Rayleigh, Lord—Electrical Standards (and £300 in hand)... *Judd, Professor J. W.—Seismological Observations... "FitzGerald, Professor G. F. ST Redation 2 in a Magnetic Field * Rucker, Brotsssor A. W.—Magnetic Force on ee Ship ... *Callendar, Professor H. L.— Meteorological Observatory, AVMOYNET Ga fea. eer oe cect < ieasniee costes vols hice eainc.c cet REE REE *Kelvin, Lord—Tables of Mathematical Functions ............ Chemistry. *Hartley, Professor W. N.—Relation between pe a . Spectra and Constitution of Organic Bodies *Roscoe, Sir H. E.—Wave-length Tables ....... *Reynolds, Professor J. E. — Electrolytic Quantitative Analysis Miers, Professor H. A.—Isomorphous age Derivatives of Benzene ..... as stee pobh oeccherane Neville, Mr. F. H. —The Nature of Alloys Hiv ovsaesenbesranene freology. *Hull, Professor E.—Erratic Blocks (#6 in hand)............... *Geikie, Professor J.—Photographs of Geological Interest . “Dawkins, Professor W. B.—Remains of Elk in the Isle of Man .... * Dawson, Sir 7 W. "Pleistocene Fauna and Flora i in i Canada *Lloyd-Morgan, Professor C. —Ossiferous Caves at Uphill (OME HATE) Tee. conc ase pavers phuseresnslea'neirsnate Rca yee ae meneame Watts, Professor W. W.—Movements of Underground IW atersmol Ora VEIL 5 is orn bu nen -dnitp sabwaen pinpigueshashw emmeE eee Scharff, Dr.—-Exploration of Irish Caves .........ccccecene eens Zoology. *Herdman, Professor W. A.—Table at the Zoological Station, BU EILGS Ds oon daisats’naWeinslonlnp sn ponidvelosohons ca kidpes aerate sea ee meeeen *Bourne, Mr. G. C.—Table at the Biological Laboratory, EL ATLOMUEED pci eniot pats sesisiamtre nee sae ewlales a Ct Wats a serail *Woodward, Dr. H.—Index Generum et Specierum Ani- ANALIVITA orc eee tastsincaes cies tacetewsiot vise olen sie vali ciet lois eave cee smeatets *Newton, Professor A.—Migration of Birds .................005 *Lankester, Professor E. Ray—Plankton and Physical Condi- tions of the English Channel.. Sere reed *Newton, Professor—Zoology of the Sandwich Islands......... Sedgwick, Mr. A.—Coral Reefs of the Indian Regions ..... é OREPICG HOT AINL Cus caniiaaic sina osistuacateeneaahiee cngasiwuy's * Reappointed, The oo Soreros oo ooo o;ooeo oo j=) oO Co ocooos oo oO (Se) 1s) oo ooo o!looce oo i=) is! SYNOPSIS OF GRANTS OF MONEY, Brought forward .recissscisnssesevavceve ds erdonecvesescnees Geography. Murray, Sir John—Physical and Chemical Constants of Sea RNMRAE Nhs a0 /ae acizianldeesa A cies « ogoua civaiy's Ueea ska Saas ajah vlawa\genieais «ne Economic Science and Statistics. *Price, Mr. L. L.—Future Dealings in Raw Produce : *Sedgwick, Professor H.—State Monopolies in other Countries (218 138. Gd. ie hard)! v.20... siesceeedies sce deeensenerstan ees Mechanical Science. *Preece, Sir W. H.—Small Screw Gauge (£17 1s, 2d. in hand) Anthropology. *Eyvans, Mr. A. J.—Silchester Excavation ...........cccsseeeenes *Penhallow, Professor D. P.—Ethnological Survey of Canada *Tylor, Professor E. B——New Edition of ‘ Anthropological MOLES AMOR OUCTICSS dx .c5 schvad ae cites Fo wrest dacs ogemaaennas daantines *Garson, Dr. J. G.—Age of Stone Circles (balance in hand)... *Read, Mr. C. H.—Photographs of Anthropological Interest *Brabrook, Mr. E. W.—Mental and Physical Condition of PRICE a, oe choko «sinus andicssmasugnidoute sevice utsendpeoranen Ges oi Read, Mr. C. H.—Ethnography of the Malay Peninsula...... Physiology. *Schafer, Professor E. A.—Physiological Effects of Peptone... *Schifer, Professor E. A.—Comparative Histology of Supra- ANOS UES SE RS ip i co a REM er er Merrie ert 4 *Gotch, Professor F.—Comparative Histology of Cerebral aR 2 state eh cheie dca apoah Shy hilo eto ne bins ana se telcaen) osae 210 0 0 Forms of Vessels ......seesseees 180 0 0 Galvanic Experiments on PROCESS ete nantbesencceasen soo a ow ww gia ooo. Crow oN ecm ONO oo =o a oO C1x 1845, 6) 8. ds Publication of the British As- sociation Catalogue of Stars 351 14 6 Meteorological Observations BUMMNVEIMESS ee cdossccesnscudes 30 18 11 Magnetic and Meteorological Co-operation .......seeesseeeee 1616 8 Meteorological Instruments ab Hdinburghi...s:.sescrrrseees Lees Reduction of Anemometrical Observations at Plymouth 25 0 0 Electrical Experiments at Kew Observatory .........00 43 17 8 Maintaining the Establish- ment at Kew Observatory 149 15 0 For Kreil’s Barometrograph 25 0 0 Gases from Iron Furnaces... 50 0 O The Actinograph ...........s006 15 0 0 Microscopic Structure of Shell Syseececeuvscesescetasoseets 20 0 0 Exotic Anoplura ......... 1843 10 0 0 Vitality of Seeds ......... LSS OP ZO 7 Vitality of Seeds ......... 1844 7 0 0 Marine Zoology of Cornwall. 10 0 0 Physiological Action of Medi- CINCS es Sicecrasscaveamtesse renee 20 0 0 Statistics of Sickness and | Mortality in York............ 20 0 0 | Earthquake Shocks ...... 1843 1514 8 £831 9 9 1846. British Association Catalogue OL ShAlSh ness sacanecedese 1844 21115 0 Fossil Fishes of the London Clayeaecccnenccdecataenatecoeeesacs 100 0 0 Computation of the Gaussian Constants for 1829 ......... 50 0 0 Maintaining the Establish- ment at Kew Observatory 146 16 7 Strength of Materials ......... 60 0 0 Researches in Asphyxia ...... 616 2 Examination of Fossil Shells 10 0 0 Vitality of Seeds ......... 1844 2 15 10 Vitality of Seeds ......... 1845 712 3 Marine Zoology of Cornwall 10 0 0 Marine Zoology of Britain... 10 0 0 Exotic Anoplura ......... 1844 25 0 0 Expenses attending Anemo- TMCLGES seacseseicamatsenerastcsrs Ls ide 6 Anemometers’ Repairs......... 2 3 6 Atmospheric Waves ............ 3.3 3 Captive Balloons ......... 1844 819 8 Varieties of the Human Race 1844 7 6 3 Statistics of Sickness and Mortality in York............ 12 0 0 £685 16 0 Cx 1847, £ 8. da. Computation of the Gaussian Constants for 1829.........+«+ 50 0 0 Habits of Marine Animals... 10 0 0 Physiological Action of Medi- CHES) Gessachecncanmpiecs naan »ar% 20 0 0 Marine Zoology of Cornwall 10 0 0 Atmospheric Waves .....+..++++ 6 9 8 Vitality of Seeds ...........005+ 400 Te Maintaining the Hstablish- ment at Kew Observatory 107 8 6 £208 5 4 1848. Maintaining the Hstablish- ment at Kew Observatory 171 15 11 Atmospheric Waves ......-.+.++ 310 9 Vitality of Seeds ............+6- 915 0 Completion of Catalogue of DUES else da dakoni’aserinnes 40, 0 0 On Colouring Matters ......... b 0. 0 On Growth of Plants ......... 1b 0; 0 £275 1 8 1849. Electrical Observations at Kew Observatory ........... 50 0 0 Maintaining the Establish- ment at GittO..........0...+6. 76 2 5 Vitality of Seeds ............... yxy AL On Growth of Plants ......... 5 0 0 Registration of Periodical PHeENOMENA......;..005reedeeeee 10 0 0 Bill on Account of Anemo- metrical Observations ...... 13°" 9h--O £1 £159 19 6 1850, Maintaining the Hstablish- mhent at Kew Observatory 255 18 0 Transit of Harthquake Waves 50 0 O Periodical Phenomena......... 15 0 0 Meteorological Instruments, PAUAONE re aeidelsispieh el hiciaa'e Hels ele es 25 0 0 £345 18 0 1851. Maintaining the Establish- ment at Kew Observatory (ineludes part of grant in TOG) Se eee 309 2 2 Theory of Heat .................- 0 As gl Periodical Phenomena of Ani- mals and Plants............... De) ID Witaliny iol Reeds. ec... ..c.6s 5 6 4 Infinence of Solar Radiation 30 0 O Ethuological Inquiries......... 12 0 0 Researches on Annelida .,.... 10 0 0 3919 7 REPORT—1899. 1852. s. a Maintaining the Establish- ment at Kew Observatory (including balance of grant for L850) s5,-seareatsptews sce - =» 233 17 8 Experiments on the Conduc- tion Of Heat ........sseeseeree 5 2 9 Influence of Solar Radiations 20 0 0 Geological Map of Ireland... 15 0 0 Researches on the British An- NENG) so cces--scccvevcenscsosnerss 10 0 0 Vitality of Seeds ....ccscseeeees 10 6 2 Strength of Boiler Plates...... 10 0 0 £304 6 7 1853. Maintaining the Establish- ment at Kew Observatory 165 0 0 Experiments on the Influence of Solar Radiation ......... 15 0 0 Researches on the British Ammnelidantc-payapagseasietiar's 10. 0: 10 Dredging on the Hast Coast Of Scotland, ...-.cspspascs-sese0 10 0 0 Ethnological Queries ........+ 5 0 0 £205 0 0 1854, Maintaining the Establish- ment at Kew Observatory (including balance of former STaNt),......ccaseoceere . 33015 4 Investigations on Flax......... 11 0 0 Effects of Temperature on Wrought Iron.......0000+se000 10 0 0 Registration of Periodical PHENOMENA. .....0seerssereree 10 0 0 British Annelida .......s.s008 10 0 0 Vitality of Seeds ..........00008 5 2 3 Conduction of Heat ............ 420 £380 19 7 1855, Maintaining the Establish- ment at Kew Observatory 425 0 0 Earthquake Movements ...... 10" "OSD Physical Aspect ofthe Moon 11 8 5 Vitality of Seeds ...,........+2. TO ig: Map of the World............... 15 0 0 Ethnological Queries ..,,..... 5 0 0 Dredging near Belfast......... 4 0 0 £480 16 4 1856. Maintaining the Establish- ment at Kew Observa- tory :— } 1854, ciae £75 0 0 ee CM ee GENERAL STATEMENT. £ 3. a Strickland’s Ornithological SYNONYMS ......cssesereersenee 100 0 0 Dredging and Dredging PUMITUMaRet at scica'ssicebeveeclecacos 9 13° ‘0 Chemical Action of Light ... 20 0 0 Strength of Iron Plates ...... 10 0 0 Registration of Periodical Phenomena....seccscsssesseeees LO} 0.0 Propagation of Salmon......... 10 0 O £734 13 9 1857. Maintaining the LEstablish- ment at Kew Observatory 350 0 0 Karthquake Wave Experi- MACTIES, << Secvsecaciscevdesrssaccers 40 0 0 Dredging near Belfast......... 10 0 0 Dredging on the West Coast Of Scotland .....0.......0000006 10 0 0 Investigations into the Mol- lusca of California ......... 10 0 0 Experiments on Flax ......... 5 0 0 Watural History of Mada- ACAT ya senea ersestassodcacnesse 20 0 0 esearches on British Anne- LOGIE, SRBgs dade de udnceconebacdsorec 25 0 0 Report on Natural Products imported into Liverpool... 10 0 0 Artificial Propagation of Sal- SONI) cob eee Ce DeDrcnee Corieee 10 0 O Temperature of Mines......... 7 8 0 Thermometers for Subterra- nean Observations...........+ bay 4 Life-boats ..... qnenSConp bance 570) 0 £507 15 4 1858, Maintaining the Hstablish- ment at Kew Observatory 500 0 0 Earthquake Wave LExperi- BROMUS cnctecacdesssslboccdesces ay 2banO) 0 Dredging on the West Coast GF Scotland ......1:.ceccsevcsees 10 0 O Dredging near Dublin......... 5 0 O Vitality of Seed ............008 5 5 O Dredging near Belfast......... 1813 2 Report on the British Anne- MUG gen vese sunt seseranedencvescds 25 0 0 ' Experiments on the produc- tion of Heat by Motion in RULUMOS 9550s Badin, Gacveverss lls 20 0 0} Report on the Natural Pro- ducts imported into Scot- MAME veesssssiseses BOddonELRpBOOAE 10 0 O £618 18 2 1859. Maintaining the Establish- ment at Kew Observatory 500 0 0 Dredging near Dublin.,........ 15 0 O CX1 £ 8. d. Osteology of Birds .........066 50 0 0 Trish PN CALA yd acdtcecaness cee 5 0.0 Manure Experiments ......... 20 0 0 British Medusidee ............+0+ 5 0.0 Dredging Committee ......... 5 0 0 Steam-vessels’Performance... 5 O O Marine Fauna of South and West of Ireland..........0.00. 10 0 O Photographic Chemistry ...... 10 0 0 Lanarkshire Fossils ..........++ 20 0 1 Balloon Ascents.......ccecccesees 39 11 O £684 11 1 1860. Maintaining the Hstablish- ment at Kew Observatory 500 0 0 Dredging near Belfast......... lis G 0 Dredging in Dublin Bay...... 16 0 0 Inquiry into the Performance of Steam-vessels .......0..06 124 0 0 Explorations in the Yellow Sandstone of Dura Den ... 20 0 0 Chemico-mechanical Analysis of Rocks and Minerals...... 25 0 0 Researches on the Growth of PIRI! . iisemoeseicwasehdseveerces 10 0 0 Researches on the Solubility OF (Salts inc ccevemecacesteaedlece 30 0 0 Researches on theConstituents Ob Manned! ce .ccerescecensntae 25 0° 0 Balance of Captive Balloon ACCOUNES re wsnses dates Reakveieer 113 6 £766 19 6 1861. Maintaining the Hstablish- ment at Kew Observatory.. 500 0 0 EKarthquake Experiments...... 25 0 0 Dredging North and East Coasts of Scotland ......... 23 0 O Dredging Committee :— 1860...... £50 0 0 LSU CERAM MIGSA tas © Excavations at Dura Den....., 20 0.0 .Solubility of Salts ............ 20 0 0 Steam-vessel Performance ... 150 0 0 Fossils of Lesmahagow ...... 15 0 O Explorations at Uriconium... 20 0 0 Chemical Alloys .......sseeess 20.0 0 Classified Index to the Trans- BCIIONS a asesvsaessecesrrssnaseee 100 0 O Dredging in the Mersey and Deer aeecrsessanueatwereonieabis 5 0 0 Dip Circle rie, awecesesceee see seans 30 0 0 Photoheliographic Observa- (OMS er so pos cdacanbenstattnn. «a3 50 0 O PIBSONMDICL. oa oS oS eS S olsoo CoOoy Cloico. cor oS So oS "So oo, oo o mS 1886. Electrical Standards...........+ 40 0 DolariRadiaiWOnicssusteckeees suk 9 10 Tidal Observations ............ 50 0 Magnetic Observations......... 10 10 Observations on Ben Nevis... 100 0 Physical and Chemical Bear- ings of Electrolysis ......... 20 0 Chemical Nomenclature ...... 5*0 Fossil Plants of British Ter- tiary and Secondary Beds... 20 0 Caves in North Wales ......... 25 0 Volcanic Phenomena of Vesu- VETS ities sss Bote ceebureretsseucnee 30 0 Geological Record............... 100 0 Paleozoic Phyllopoda ......... 15 0 Zoological Literature Record. 100 0 Granton Biological Station... 75 0 Naples Zoological Station...... 50 0 Researches in Food-Fishes and InvertebrataatSt. Andrews 75 0 o eoocooo so co scooonso REPORT—1899. £ 8. d. Migration of Birds ........... 30 0 0 Secretion of Urine............. «« 10) 70710 Exploration of New Guinea... 150 0 0 Regulation of Wages under Sliding’ Scales *<.:..,..2...... 10 0 0 Prehistoric Race in Greek ISIANGS? Scchevescverseeoccscccvecs 20) 00 North-Western Tribes of Ca- ra70 Fe ase eee rees canes 0s Orr £995 0 6 1887. Solar Radiation ......ses.ssseesss 18 10 Hlectrolysisiteces;scsaseseun. carer 30° Ben Nevis Observatory......... Standards of Light (1886 STAND) <.cccerecneseuben ties see 20 Standards of Light (1887 OUAN Ui ackekeerantseeeeseseraesae 10 Harmonic Analysis of Tidal Observations .....2secesc+ons~ne 15 Magnetic Observations......... 26 Electrical Standards ............ 50 Silent Discharge of Electricity 20 Absorption Spectra ............ 40 Nature of Solution ............ 20 Influence of Silicon on Steel 30 Volcanic Phenomena of Vesu- VIUS) nocsecacsvdesee< tees enwadedss 20 Volcanic Phenomena of Japan (1886 grant) ..,....s.ssecceeee 50 Volcanic Phenomena of Japan (USS 7 eranih) ceoccsstseee sweeps 50 Cae Gwyn Cave, N. Wales ... 20 Erratic BlOCKS— .\..scsssessssose 10 Fossil Phyllopoda ............00+ 20 Coal Plants of Halifax........- 25 Microscopic Structure of the Rocks of Anglesey............ 10 Exploration of the Eocene Beds of the Isleof Wight... 20 Underground Waters ......... 5 ‘Manure’ Gravelsof Wexford 10 Provincial Museums Reports 5 Lymphatic System ............ 25 Naples Biological Station 100 Plymouth Biological Station 50 Granton Biological Station... 75 Zoological Record ........+.000+. 100 Bora (of @hingy wii sc..cceeseseces 75 Flora and Fauna of the Cameroons .........+ suaecsveous 75 Migration of Birds ............ 30 Bathy-hypsographical Map of IBTIDISHUISIOS: Gitevsssevcssctste 7 Regulation of Wages ......,.. 10 Prehistoric Race of Greek Tslands.......« Rieditscchacenestve 20 Racial Photographs, Egyptian 20 £1186 18 lo Io So (OSC OoomoooSoeo & eonseo’ Sc eo -Seooesoco oso ec ices GENERAL STATEMENT. 1888. £ 8. da. Ben Nevis Observatory......... 150 0 0 Electrical Standards...........» 2 6 4 Magnetic Observations......... 15 0 0 Standards of Light ............ (eae a Blectrolysis .......ssseeceereeees 30 0 0 Uniform Nomenclature in Mechanics .......cscesseerscere 10 0 0 Silent Discharge of LElec- HIORELUVN aespssacccsssteoacvbtere ee 9 11 10 Properties of Solutions ..... o0 25) 0) 0 Influence of Silicon on Steel 20 0 O Methods of Teaching Chemis- GDM te ht ocess sscascuiatnrss s*e5" 10 0 0 Isomeric Naphthalene Deriva- GAVOMateeasescecesesqsausars Herren 25 0 0 Action of Light on Hydracids 20 0 0 Sea Beach near Bridlington... 20 0 0 Geological Record ........++++++ 50, 0°20 Manure Gravels of Wexford... 10 0 0 Erosion of Sea Coasts ......... 10 0 0 Underground Waters ......... 5.0 0 Paleontographical Society ... 50 0 0 Pliocene Fauna of St. Erth... 50 0 0 Carboniferous Flora of Lan- cashire and West Yorkshire 25 0 0 Volcanic Phenomena of Vesu- MILI ese . or an sind ao'gansice ers anid. 20 0 0 Zoology and Botany of West AGLUIGH | Sccenerspcesesceseeues-baee 100 0 0 Flora of Bahamas ....... “pepe 100 0 O Development of Fishes—St. PARTE CWS) « ncn dde-ansvsviacdessnens 50 0 0 Marine Laboratory, Plymouth 100 0 0 Migration of Birds ............ 30 0.0 Mora, Of Obing <....... ssscerse 76 0 0 Naples Zoological Station ... 100 0 0 Lymphatic System ............ 25 0 0 Biological Station at Granton 50 0 0 Peradeniya Botanical Station 50 0 0 Development of Teleostei 5 10> (0 Depth of Frozen Soil in Polar REGIONS) .oiccccecescescscaeez ers 5 0 0 Precious Metals in Circulation 20 0 O Value of Monetary Standard 10 0 O Effect of Occupations on Phy- sical Development............ 25 0 0 North-Western ‘Tribes of MIAO ss 5 0 0 Excavations at Oldbury Hill 15 0 0 Cretaceous Polyzoa ..,......+.- 10 0 0 Geological Photographs ...... 71411 Lias Beds of Northampton... 25 0 0 Botanical Station at Perade- TVA cocescsactsncececonsecedpsacs 25 0.0 CXX coe 1S: Experiments with a Tow- TIED pusevs-canccsecovennsseosneee 4 3 Naples Zoological Station pe 00) 40 Zoology and Botany of the West India Islands ......... 100 0 Marine Biological Association 380 0 Action of Waves and Currents in Estuaries ..........s.ec00 150 0 Graphic Methods in Mechani- GAISCICNCC see.ccvsseesessesh=cs LIOKO Anthropometric Calculations 5 0 Nomad Tribes of Asia Minor 25 0 Corresponding Societies ...... 20 0 ‘£799 16 1891. Ben Nevis Observatory......... 50 O Electrical Standards............ 100 0O LC CETONY SIS. \naseeascecesssasbencee 5 0 Seismological Phenomena of Mcp Mees ons senensescnsaseseys sh 10 0 Temperatures of Lakes......... 20 0 Photographs of Meteorological PHENOMENA... .00...c0sesese se. 5 0 Discharge of Electricity from GUNS eaprecmmecsliesrmenasccee ne 10 0 Ultra Violet Rays of Solar MPCCLLUMNS nsascdsosasaane ress 50.0 International. Standard for Analysis of Ironand Steel... 10 0 Isomeric Naphthalene Deriva- LIVES wavnecstibaccansessectesessess 25 0 Formation of Haloids ......... 25-10 Action of Light on Dyes ...... 17 10 Geological Record............... 100 0 Volcanic Phenomena of Vesu- WINS letecccaddtactoccssosecsn raise 10 0 Fossil Phyllopoda............... 10 0 Photographs of Geological AMGEN ES bie wensteae ante tnsentes acc’ 9 5 Lias of Northamptonshire ... 25 0 Registration of ‘l'ype-Speci- mens of British Fossils...... 5 5 Investigation of ElboltonCave 25 0 Botanical Station at Pera- OMI acc setae eseascnnerescvases 50 0 Experiments with a Tow-net 40 0 Marine Biological Association 12 10 Disappearance of Native DIATIES peaecseeere=sesensatesce tee 5 0 Action of Waves and Currents Ai SHISHUALICS 152 feo. sale sinesstea'e 125 0 Anthropometric Calculations 10 0 New Edition of ‘ Anthropo- logical Notes and Queries’ 50 0 North - Western Tribes of Canad anmnuaherestnccecwassses 200 0 Corresponding Societies ...... 25 0 #1, £1,029 10 dad. 9 0 0 0 0 0 0 0 0 8 o!1oo fo] oo oS Soo, sO oe o'o 29) OO'1aiS o (=) ro) =) oo ooo REPORT—1899. Co alo Oo) £6 aoe ies 1892. Zea Observations on Ben Nevis... 50 0 Photographs of Meteorological PHeNOMENS,,..cesesevsocessscese 15 0 Pellian Equation Tables ...... 10 0 Discharge of Electricity from Points’ ..sqdsaeeeet senator cceteees 50 0 Seismological Phenomena of JAPAN \ ic seaaesbateneecncecesevess 10 0 Formation of Haloids ......... 12 0 Properties of Solutions ...... 10 0 Action of Light on Dyed Colours ~cc.deetsyereweress eves 10 0 Erratic BIOCKS. <..cccsseccsess ses 15 0 . Photographs of Geological Interest, isesasssagerensodend se 20.*0' 0 Underground Waters ......... 10 0 0 Investigation of Elbolton Cave. seins. stdue damceeveress ne 25 0 90 Excavations at Oldbury Hill 10 0 0 Cretaceous Polyzoa ........+0+ 10 0 0 Naples Zoological Station ... 100 0 0 Marine Biological Association 1710 0 Deep-sea Tow-net ............... 40 0 0 Fauna of Sandwich Islands... 100 0 0 Zoology and Botany of West India Tslandsisccsesserseessn 100 0 0 Climatology and Hydrography of Tropical Africa ......... «. 50 0 0 Anthropometric Laboratory... 5 0 0 Anthropological Notes and Queries ....... GMb eo teeaee 20 00 Prehistoric Remains in Ma- Shonaland! tin scosecctececessces 50 0 0 North-Western ‘Tribes of Canada. camsecves se: bine «on csl 100 0 0 Corresponding Societies ...... 250h40 £864 10 O 1893. Electrical Standards.......,.... 25 0 Observations on Ben Nevis... 150 0 Mathematical Tables ......... 16 0 Intensity of Solar Radiation 2 8 Magnetic Work at the Fal- mouth Observatory ......... 25 0 Isomeric Naphthalene Deri- WADIV.ESir osteehesieeeseaisceiensicot? 20 0 HITTAGIC IBIOCKS wasecerscscos en's 10 0 Fossil Phyllopoda.............++ 5 0 Underground Waters ......... 5 0 Shell-bearing Deposits at Clava, Chapelhall, &c. ...... 20 0 Eurypterids of the Pentland TG Sites eeaten vier snmeess cataape os 10 0 Naples Zoological Station ... 100 0 Marine Biological Association 30 0 Fauna of Sandwich Islands 100 0 Zoology and Botany of West India Fslands..ss.ys0000 e200: 500 oso coco © CcoocKe | Oo aceo & 8. d. Exploration of Irish Sea ...... 30 0 0 Physiological Action of Oxygen in Asphyxia......... 20 0 0 Index of Genera and Species OPPATIUMAIS \ewenesseslecesep eis 20 0 0 Exploration of Karakoram IVIGHNUAINS) .6.025202csecasec seve 50 0 0 Scottish Place-names ......... (Tn) Climatology and MHydro- graphy of Tropical Africa 50 0 0 Economic Training ............ er 0) Anthropometric Laboratory 5 0 0 Exploration in Abyssinia...... 25 0 0 North-Western ‘Tribes of WAVIACA Iss sohavace (ocsasesnenecas 100 0 0 Corresponding Societies ...... 30 0 0 “907 1b) 6 1894, Electrical Standards............ 25 0 0 Photographs of Meteorological PHENOMENA. fcc. .sedeccsccoces 10 0 0 Tables of Mathematical Func- UTOTIE! » eel oo dacrOt BEQuOADECDASBSTe 15 0 0 Intensity of Solar Radiation 5 5 6 Wave-length Tables............ 10 OG Action of Light upon Dyed MEDIGUIS! Cascccessecesttessvce ssa 5 0 0 Hrratic BIoCKS .......s0.ssseeees 15 0 0 Fossil Phyllopoda............... 5 0 0 Shell-bearing Deposits at RUA Van COs cece, tes caseee tases « 20 0 O Eurypterids of the Pentland esiltecenedsttnccasecsscsenss esse 5 0 0 New Sections of Stonesfield SID THE) pootengacticonpevodacadasen 14 0 0 Observations on Earth-tre- LENGTWE Goéranoorpacodceneneadnee 50 0 0 Exploration of Calf- Hole SEI ESGe CSc COUORUOL AE SeEREERER ATE 5 0 0 Naples Zoological Station ... 100 0 0 Marine BiologicalAssociation 5 0 0 Zoology of the Sandwich LISTER [te ha erecesoneace Anpee 100 0 0 Zoology of the Irish Sea ...... 40 0 0 Structure and Function of the Mammalian Heart............ TO OO Exploration in Abyssinia 30 0 0 Heonomic Training ............ 910 0 Anthropometric Laboratory PA DISEICSIL ts cocess tsncesorceck ae. 0) O Ethnographical Survey ...... LOTOESO The Lake Village at Glaston- PUNY irer nent toed secscnscoasa of as 40 0 0 Anthropometrical Measure- ments in Schools ............ 5 0 0 Mental and Physical Condi- tion of Children............... 20 0 0 Corresponding Societies ..,... 25 0 0 £583 15 6 GENERAL STATEMENT. CXXi 1895. £ 8. d. Electrical Standards............ 25 0 0 Photographs of Meteorological PHENOMENA cs ssccesseiesc sence 10 0 0 Haxth) TremoOrsie wctevccedeectse 75 0 0 Abstracts of Physical Papers 100 0 0 Reduction of Magnetic Obser- vations made at Falmouth Observatory. secc.ccscecdsias 50 0 0 Comparison of Magnetic Stan- GATOR vocaswvesractevedehesckewes 25 0° 0 Meteorological Observations OM Ben NGVIStrcvssses~cpecee se 50 0 0 | Wave-length Tables of the Spectra of the Elements... 10 0 O Action of Light upon Dyed COlOUTS NM eecwewssemes see ccbecireee secGH Formation of Haloids from Pure Materials ............... 20 0 0 Isomeric Naphthalene Deri- VAULVES! ssusseesesaceetoteecedrens 30 0 0 Electrolytic Quantitative An- HIMES) ocotigdqoccdoccoutocapbedr oc 30 0 0 Hrrabic BlOCES) cresscscssvesas ses 10) 40> ©: Palzeozoic Phylopoda ......... 5 0 0 Photographs of Geological In- LSE Un apercecoscssoocooesbtsesor 10 0 0 Shell-bearing Deposits at Clava, (SoCs ergs tMedccaue ates 10 0 0 Eurypterids of the Pentland Ven OU Fe remocorrigdcuncancr tadker osc 3.0 0 New Sections of Stonesfield SS] ERE) Wang er bocstaccsecesceonc ce 50 0 0 Exploration of Calf Hole Cave 10 0 O Nature and Probable Age of High-level Flint-drifts...... 10 0 0 Table atthe Zoological Station at Naplestissccccesasecsccoseas 100 0 0 Table at the Biological Labo- ratory, Plymouth ............ 15 0 0 Zoology, Botany, and Geology of the Trish Sea............+0« 35 9 4 Zoology and Botany of the West India Islands ......... 50 0 0 Index of Genera and Species Of Animals) . Ps =\ where T is the 1 2 kinetic energy ; let q, be the angle Chik the node (of the plane of the bodies, on the invariable plane) makes with one of the principal axes of inertia of the bodies at their centre of gravity ; and let py=k cos 7, where % is the constant of angular momentum on the invariable plane, and 7 is the angle between the plane of the bodies and the invariable plane. Then the differential equations become dq, °H dp, tH »_4 9 a ip. aaah Bled U=h 2 2) where H is a certain function of the quantities p and g, and H =constant is an integral of the system. When the motion is in one plane, the system reduces to the sixth order, as p, becomes a constant, and q,, now measured from a fixed line in the plane, is determined by a simple quadrature. This reduction is more symmetrical than one given by Perchot and Ebert ‘ in 1899. 1 «Sur une transformation des équations différentielles du probléme des trois corps,’ C. R. lxvi. pp. 710-14. 2 «Jntorno ad alcune trasformazioni delle equazioni differenziali del problema dei tre corpi,’ Atti di Torino, vi. pp. 440-54; ‘Sur le probleme des trois corps,’ C. &, xxviii. pp. 110-13. 3 «Sur la transformation des équations canoniques du probleme des trois corps,’ C. R. cxix. pp. 451-4. 4 ‘Mémoire sur le probleme des trois corps,’ C. #&. Ixxvii. pp. 1071-4, xxviii. pp. 408-10; ‘Mémoire sur le probleme des trois corps,’ Liowville (3), ii. pp. 345-70; ‘Sur Vapplication du probléme des trois corps 4 la détermination des perturbations de Jupiter et de Saturne,’ Journal de ? Heole Polytechnique, xxviii. pp. 245-69. 5 «Satz aus der Storungstheorie,’ Crelle, lxv. pp. 291-2. 5 * Ueber das Problem der drei Korper,’ Credle, |xviii. pp. 390-2. 7 ‘Sur la réduction des équations du probléme des trois corps dans le plan,’ Bulletin Astronomique, xvi. p. 110-16, . PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 125 In 1868 Radau published, first in a series! of notes in the ‘ Comptes Rendus,’ and subsequently in a memoir? in the ‘Annales de 1’Kcole Normale Supérieure,’ his researches on the differential equations of the problem of » bodies. He finds the effect of an orthogonal substitution performed on the variables in the problem, and shows that Jacobi’s substi- tution in the problem of three bodies is a case of this. Two other cases are worthy of mention : firstly, a transformation which is equivalent to referring the second body to the first as origin, the third body to the C.G. of the second and third, the fourth body to the C.G. of the first three, and so on, at the same time modifying the masses ; and, secondly, a transformation which shows the existence of ‘canonical’ points, each of which has, with reference to the motion of (x—1) of the bodies, properties similar to those possessed by the C.G. for the whole system, Considering the case of three bodies, he deduces Bour’s equations, and also a new canonical system of the eighth order. A modification of the transformation of Jacobi and Radau was con- sidered in 1889 by Andrade,* and in 1896-7 Poincaré? gave another transformation which appears to be still better suited for effecting the same reduction. The results obtained by Allégret ° in 1874 are substantially equivalent to some of those in Radau’s papers. Radau’s researches were continued in 1869 in a number of papers," of which that in Liouville’s journal is the most complete ; the author dis- cusses the reduction of the order of a canonical system when one of the coordinates does not enter explicitly in the energy-function, and applies his results to the problem of three bodies, arriving at Scheibner’s system. Hesse’ in 1872 published a fresh discussion of the problem of three bodies, somewhat on the lines of Lagrange’s memoir ; but it was pointed out by Serret * in 1873 that the equations in one of Hesse’s systems were not independent, and consequently his results were invalid. Serret’s paper contains also an exposition, in an improved and symmetrical form, of the essential parts of Lagrange’s memoir. Other reductions of the "Sur un théoréme de mécanique,’ @. #. Ixvi. pp. 1262-5; ‘ Remarques sur le probléme des trois corps,’ idid. lxvii. pp. 171-5; ‘ Sur une transformation orthogonale , applicable aux équations de la dynamique,’ ibid. xvii. pp. 316-9; ‘Sur l’élimination directe du nceud dans le probléme des trois corps,’ ibid. lxvii. pp. 841-3. * ‘Sur une transformation des équations différentielles de la dynamique,’ Annales del’ Ecole Norm. Sup. v. pp. 311-75. ° «Sur une réduction du probleme des » corps, qui conserve 5 ou teed mutuelles,’ C. A, cviii. pp. 226-8; ‘Sur les réductions du probléme des x corps, qui conservent certaines distances mutuelles,’ ibid. cviii. pp. 280-1. * ‘Sur une forme nouvelle des équations du probleme des trois corps,’ ibid. cxxiil. pp. 1031-5; Acta Math. xxi. pp. 83-97. ° ‘Sur une transformation des équations de la mécanique céleste,’ C. R. cxxix. pp. 656-8. ° *Betrachtungen tiber die Flichensitze,’ Ast. Nach. Ixxiii. pp. 337-44 ; ‘Weitere Bemerkungen tiber das Problem der drei Kérper,’ ibid. lxxiv. pp. 145-52; ‘Sur une propriété des syst¢mes qui ont un plan invariable,’ @. R. lxviii. pp. 145-9; ‘Sur une pr etomaiios des coordonnées des trois corps dans laquelle figurent les moments ‘inertie, ibid. cxviii. pp. 1465-9; ‘Ueber gewisse Higenschaften der Differential- gleichungen der Dynamik, Math. Ann. ii. pp. 167-81; ‘Sur une propriété des systémes qui ont un plan invariable,’ Liowville, xiv. pp. 167-230. * «Ueber das Problem der drei Kérper,’ Crelle, Ixxiv. pp. 97-115. * ‘Réflexions sur le mémoire de Lagrange intitulé “Essai sur le probléme des trois corps,”’ C. #, lsxvi. pp. 1557-65 ; and Bull. des Sc. Math. vi. p, 48. distances 126 REPORT—1899. problem of three bodies, which seem scarcely of sufficient importance to be here described in detail, are those of Weiler! in 1869-70, Hill? in 1875, Weiler # in 1879-80, Seydler + in 1884, and Duport® in 1898, The problem of 7 bodies can be reduced from the 6nth order to the (6n-12)th order, just as the problem of three bodies can be reduced from the 18th order to the 6th. This subject has been discussed in the period under review by Allégret ® in 1875 (who fell in errors which were pointed out by Mathieu’ in 1877), by Betti* in 1877, Mathieu in 1877, Ball !° in 1877, Dillner !! in 1877 (who attempted to use quaternions, but made mis- takes which were pointed out by Bruns !* in 1880), and Dillner }3 in 1882-8. Seydler !* in 1885 extended the analysis of Lagrange’s treatment of the problem of three bodies to the case of the problem of four bodies. The system isreduced ultimately to a system of the twelfth order and quad- ratures. The general theory underlying the work of this section has been developed by Lie and Mayer. A special consideration of the problem of three bodies will be found at p. 282 of a paper |’ published by Lie in 1875. In 1887 Bruns !° published a paper which will be analysed later, but which contains a new reduction of the problem of three bodies. Let 9, 72, 73 be the mutual distances of the three bodies ; and let ¢,= Sa,(a,+ty,)/2b\(%,+ty,), where (a, y;, 2) &c. are the coordinates of the bodies when the origin is taken at the centre of gravity, and the 1 «Ueber die Elimination des Knotens in dem Problem der drei Kérper, ete.,’ Ast. Nach. \xxiv. pp. 81-96, Ixxv. pp. 113-28; ‘Notes sur le probleme des trois corps,’ Liowville, xiv. pp. 805-20, 2 «Reduction of the Problem of Three Bodies,’ The Analyst, iii. pp. 179=85. 8 ¢*Ueber die Differentialgleichungen der Bewegung in dem Problem der drei Korper,’ Ast. Nach. xcvi. pp. 161-82; ‘Das Problem der drei Korper in der neuen Stérungstheorie,’ ibid. xcvil. pp. 97-112, 129-44, 161-76, 193-208. 4 «Ueber einige neue Formen der Integrale des Zwei-und-Dreikérper-Problems,’ Sitzungsberichte der Ak. zu Wien, \xxxix. pp. 851-72; ‘O integrovani nékterych rovnic vyskytrujic ich se v problemu tfi téles,’ Sitzwngsberichte d. Ges. der Wiss. in Prag, 1884, pp. 16-29; ‘ Dal&Si ptispévky k integrovani,’ etc., ibid. pp. 106-26. 5 Sur le probléme des trois corps, Bull. Astr. xv. pp. 377-83. 6 «Mémoire sur le probléme des trois corps,’ Liouville (3), i. pp. 277-316. 7 Mathieu, ‘Sur le probléme des trois corps.’ Liowville (3), iii. pp. 216-9; Allé- gret, ‘ Note sur le probléme des trois corps,’ ibid. pp. 422-6; Mathieu, ‘ Réponse a la note de M. Allégret sur le probléme des trois corps,’ Liouville (3), iv. pp. 61-2. 8 ¢ Sopra il moto diun sistema di un numero qualunque di punti che si attraggono o si respingono tra loro, Annali di matematica (2), viii. pp. 301-11. 9 ‘Mémoire sur les équations du mouvement d’un systéme de corps,’ Liowville (3), ii, pp. 5-21. j a Note on a transformation, etc.,’ Monthly Notices, xxxvii. pp. 265-71. 1 «Mémoire sur le probléme des 2 corps,’ Nova Acta R.SS. Upsal. vol. extra ord., 1877, 18 pp. 12 Jahrbuch ber die Lortschritte der Mathematik, 1877, p. 788. 13 ¢Qm integration af differentialeqvationerna i m-kroppiirs problemet,’ Ofversigt af K. Vet.-ah. Forhandlingar, 1882, No. 4, pp. 13-20, No. 8, pp. 9-29; 1886, pp. 173-84, 217-22; 1888, pp. 367-78; ‘ Sur l’intégration des équations différentielles du probléme des V corps, Annali di matematica (2), xi. pp. 56-64. 14 « Ausdehnung der Lagrange’schen Behandlung des Dreikérper-Problems auf das Vierkérper-Problem,’ Abhandlungen der kh. bilm. Gesellschaft der Wissenschaften, (7), i. No. 5, 20 pp. 1s Begriindung einer Invarianten-Theorie der Beriihrungs-Transformationen,’ Math. Ann. viii. pp. 215-303. 16 ‘Ueber die Integrale des VielkGrper-Problems,’ Berichte der hgl. Siichsischen Gesellschoft der Wiss. zu Leipzig, 1887, pp. 1-39, 55-82 ; Acta Math. xi. pp. 25-96. PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 127 invariable plane is the plane of wy, and where a, dy, a3, 0, bo, b3 are con- stants subject only to the conditions Then the problem can be reduced to the Hamiltonian system of the eighth order, dq, __ 6H dp; Aes __¢H py alll okie) the eon di ép? dt &q; Start dln Patri 8s) where ( . SH MtM; 9 PPAQe +98 —T) ei ee +N H= [Prgng! AP = 249645 =f ee {Po(%4 D190) +kb,} D< WMoM1- { 25(ay—Dsq)—?3(a2—bago) — Eas In this, 74 is the constant of angular momentum. Bruns then reduces this to a system of the sixth order by eliminating the time and using the integral H=—A; writing H=H,po+H., where H, and H, do not H,+h involve p , and putting K=— wwe have 1 Bo tec OK. (5541,9) 8) dgo &p, dg ey; which is the required system. It may be noted that a particularly simple case of Bruns’s transforma- tion is afforded by putting a, = —1, ag=1, a,=0, 6; =—1, b,=0, 63=1 ; in this case g is simply the ratio of the two vectors which join the projection of m, to the projections of mz and mz, respectively on the invariable plane. Kiaier’ in 1891, starting from Jacobi’s transformation, likewise reduced the problem to a canonical system of the sixth order. The differential equations of the restricted problem of three bodies were discussed by Tisserand * in 1887 and by Poincaré* in 1890. Both authors reduce the problem to a canonical system of the fourth order ; Tisserand takes variables defined by means of the elements of the instantaneous ellipse described by the particle round one of the bodies, © while Poincaré uses the instantaneous ellipse described by the particle round the centre of gravity of the system. § Il. Certain Particular Solutions of Simple Character. Lagrange‘ in 1772 had shown that the equations of motion of the problem of three bodies can be satisfied by two particular solutions of a very simple character ; in one case the three particles are always at the vertices of a moving equilateral triangle and in the other they are always on a moving straight line. We shall generally call these respectively the motions of Lagrange’s three equidistant particles and three collinear particles, * “Sur la réduction du probléme des trois corps au systéme canonique du sixiéme ordre,’ Astr. Nach. cxxvi. pp. 69-76. * «Sur la commensurabilité des moyens mouvements dans le systéme solaire,’ Bull. Astr. iv. pp. 183-92. * © Sur le probléme des trois corps,’ Acta Math. xiii. pp. 1-270. * «Essai sur le probleme des trois corps,’ Priz de V Académie de Paris, iz. 128 REPORT—1899. The first paper oi the subject in the period under review was published by Routh! in 1875; he showed that the three equidistant particles are stable when the square of the sum of the masses is greater than twenty-seven times the sum of the products of the masses taken two and two together ; a result which, however, had already been stated by Gascheau. The stability was considered from a somewhat more general point of view by Liapunow? in 1889; and Gyldén® in 1884 discussed solutions which differ but little from the three collinear particles. Lagrange’s results have been generalised, and corresponding theorems found for the motion of more than three bodies. An attempt made in this direction by Veltmann‘ in 1875 is open to criticism, but Hoppe® in 1879, and Lehmann-Filhes ° in 1891, discovered solutions in which more than three particles are placed at the corners of a regular polygon or poly- hedron, or on a straight line. Sloudsky” in 1892 claimed to have given some of Hoppe’s results in 1878, in a paper published in Russian. In Hoppe’s paper the masses of the particles are supposed to be equal, which detracts from the value of his results ; in Lehmann-Filhes’s paper the masses are not so restricted. Cases in which the triangle formed by the bodies is isosceles were discussed by Fransen § in 1895, and Gorjatschew ° in 1895-6. § III. Memoirs of 1868-89 on General and Particular Solutions of the Differential Equations, and their Expression by Means of Infinite Series (excluding G'yldén’s Theory). From the time when it was first realised that the motion of the three bodies cannot be represented in a finite form by means of known functions, interest has centred chiefly round that division of the subject to which the present section will be devoted, namely, the derivation, nature, and properties of the infinite series by means of which the problem can be solved. The result of our observations of the heavenly bodies suggests a form into which we may try to put the analytical solution. It is found that the facts can be represented, at any rate for as far back as our records take us, by supposing that the planets move in ellipses round the sun. These ellipses are, however, not fixed, but their elements (the eccentricity, ec.) vary from year to year. Some of these variations, or inequalities, are periodic—that is to say, can be expressed by terms such as a sin (bt + ©), ‘ «On Laplace’s three particles, with a Supplement on the stability of steady motion,’ Proc. Lond. Math. Soc. vi. pp. 86-97. 2 *On the stability of the motion inaspecial case of the problem of three bodies,’ Trans. Math. Soc. of Krakow (8), ii. pp. 1-94. (Russian.) ; 3 «Om ett af Lagrange behandladt fall af tre-kroppars-problemet; Ofversigt af K. Vet.-ak. Forhandlingar, xii. pp. 3-11; ‘Sur un cas particulier du probléme des trois corps, Bull. Astr. i. pp. 361-9. 4 * Bewegung in Kegelschnitten von mehr als zwei Korpern, welche sich nach ‘dem Newton’schen Gesetz anziehen,’ Ast. Nach. \xxxvi. pp. 17-380. 5 «Erweiterung der bekannten Speciallésung des Dreikdrperproblems,’ Archiv der Math. u. Phys. lxiv. pp. 218-23. 6 «Ueber zwei Falle des Vielkérperproblems,’ Astr. Nach. cxxvii. pp. 137-44. 7 ‘Note sur quelques cas particuliers du probléme de plusieurs corps,’ Bulletin de la Soc. Imp. Natur. Moscow, 1892, pp. 437-40. 8 ¢ Ett specialfall af tre-kroppars-problemet,’ Ofversigt af K. Vet.-ak. Forhand. lii, pp. 783-805. 5 Transactions of the Inyp. Soc. of Nat. Moscow, vii. viii. e PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 129 where a; 6, c are constants ; such variations obviously do not in the long tun produce any marked change in the solar system ; while other variations are secular—that is to say, are expressed by terms such as aé¢ + b0? + ...; these variations of course have the effect of continually altering the orbits, leading ultimately to a completely different configuration. The method of the classical planetary theory is to express the solution in this way: differential equations are found for the variations of the elliptic elements, and from them is found an approximate solution, which in the earlier memoirs was of the kind just described. The question naturally arose, What would be found to be the true nature of the secular inequalities if the equations were solved rigorously instead of approximately? The first approximation can be represented by terms like c¢, where c is a constant ; but it is possible that this is only the first term in the expansion of (say) - sin mt, where m is a very small number. If this were the case, the secular terms would really be periodic, though of a very long period. In researches relating to the stability of the solar system, and the expression of the coordinates after long intervals of time, the settlement of this question is of fundamental importance. Although the founders of the planetary theory succeeded to some extent in their approximation in thus replacing secular terms by trigono- metric terms of long period, the most important contribution to the sub- ject previous to the period under review was the method by which Delaunay | discussed the motion of the moon, the essence of which may be described as follows. Let 8, J, P be the three bodies, and let the mass of P be zero ; then the motion of S and J, being elliptic, may be supposed known, and to determine the motion of P we have a system of the sixth order. This can be brought to the form Bi OM Bish ON popes ina} dt cq, dt op, where H is a function of ¢ and of the generalised coordinates p,, ¢,. H may be called the disturbing function, and can be expanded as an infinite series, each term of which consists of 4 function of p,, p., ps; multiplied by the cosine of a linear function of 9¢,, q, g;, & Delaunay then fixes the attention on some particular one of these terms, and shows how to find a transformation from the variables p,, g, to new variables p’,, q’, such that the equations become ap, CHA ity bao bloat rigs wae ee Seater red tee a where H’ is a function of p’,, q’,, ¢ of the same kind as H ; but H’ does not contain any term corresponding to the term in H which is under considera- tion. This transformation has therefore robbed the disturbing function of one of its terms ; by a fresh transformation we can deprive H’ of any other term, and so on. In this way all the important periodic terms are abolished from the disturbing function, and when the residue has become negligeable, the equations are integrated ; and the coordinates are thus expressed in terms of six arbitrary constants, and the time by means of series in which the time occurs only in the arguments of periodic terms. 1 «Théorie du mouvement de la Lune.’ Paris. Vol. i. 1860; vol. ii. 1867. K 130 REPORT —1899, In 1872 Newcomb,! assuming that the coordinates of the planets can be expressed by trigonometric series, as in Delaunay’s theory, proved various properties of the coeflicients, &c., by using the function called by Clausius the virial, which is the mean value of the kinetic energy of the system. This was extended by Siacchi * in 1873, In 1874 Newcomb ? proceeded to justify his assumption regarding the expression for the coordinates as functions of the time. He applies the transformation of Jacobi and Radau to the equations of (+1) bodies, and so obtains a system of the 6nth order. It is assumed that a set of infinite series of the forms Oey ae . . ° p= >K, my (Ay H%QAo+A3Ag + » «+ +43nA3n) can be found, where p; is one of the coordinates and \,=/,+0,¢ (the quan- tities / being 3n arbitrary constants, and the quantities b and K being functions of 3n other arbitrary constants), such that the differential equa- tions are approximately satisfied by these series. Newcomb, then, using the method of variation of arbitrary constants, replaces these series by others of the same form which satisfy the differential equations to a higher degree of approximation. Proceeding in this way, it appears that the problem of three bodies can be formally solved by series of this kind. The year 1877 saw the appearance of a paper * which may be regarded as the beginning of the new era in Dynamical Astronomy. The author, Mr. G. W. Hill, was at the time an assistant on the staff of the American Ephemeris. The first of the novelties in this paper is the abandonment of Kepler’s ellipse. It had hitherto been usual to take, as the first approximation to the orbit of the moon, an elliptic path round the earth ; the orbit, in fact, which the moon would actually describe if the sun did not exist to disturb it ; the actual path of the moon was then found by calculating the per- turbations caused by the sun on this elliptic motion. Hill, however, does away with the elliptic orbit altogether, and takes, as the intermediate orbit or first approximation to the moon’s path, an orbit which includes all the inequalities which depend only on the ratio of the mean motions of the sun and moon, but takes account of no other inequalities. This difference between Hil] and the older theorists may be otherwise stated as follows: the old astronomers first solved the problem of two bodies, and then attempted to solve the problem of the three bodies by suitably varying the solution so obtained ; whereas Hill begins by solving the re- stricted problem of three bodies, and then attempts to solve the problem of three bodies by suitably varying this solution. Suppose, then, that an orbit for the particle is known, which is periodic, z.e. which is such that the two bodies and the particle retake the same relative positions after the lapse of a certain interval of time. Then the coordinates of the particle can be expressed as sums of sines and cosines of multiples of a linear function of the time. We can now consider the small ' “Note sur un théoréme de mécanique céleste,’ @. #. Ixxv. pp. 1750-3. ? «Sur un théoréme de mécanique céleste,’ C. R. Ixxvii. pp. 1288-91. ® «On the General Integrals of Planetary Motion,’ Smithsonian Contribution to Knowledge, 1874, pp. 1-31. **On the part of the Motion of the Lunar Perigee which isa Function of the Seas Motions of the Sun and Moon,’ Cambridge; Mass., Press of John Wilson & on. PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 131 oscillations of the particle about this orbit, when the initial conditions of its motion are not exactly such as to cause it to describe the periodic orbit. These oscillations represent those inequalities in the moon’s motion which depend only on the eccentricity of the lunar orbit and the ratio of the mean motions of the sun and moon; and the period of the oscillations represents the time between two successive perigees of the moon, so that the difference between this period and the period of the orbit gives that part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon—whence the title of the memoir. Let w represent the distance (measured along the normal) of the particle from the periodic orbit, at any time ¢ during the performance of the small oscillations. Then Hill finds that wis given by an equation of the form d?w dt +0w=0 where 9 depends only on the relative position of the two bodies and the particle ; O is therefore a known periodic function of ¢, and can be ex- panded in the form 0=0,+80, cos 2+ 0, cos 444+ ,..., where 0,, 0,, 0, ... are pure constants. (It ought to be stated here that, since all inequalities in the moon’s motion which involve the sun’s parallax are neglected, the distance of the two bodies from each other is supposed to be infinite, and the one of them at infinity is supposed to possess such an (infinite) mass as would correspond to a finite mean motion.) The problem therefore is to solve the differential equation dw det {0 +0, cos 244+ 0, cos 4¢4+ .. .}w=0. Equations of this type had been discussed by the founders of dynamical astronomy, D’Alembert,'! Lagrange,” and Laplace, and have since been discussed by a large number of mathematicians. Tisserand called the equation dw spi {Q,+9, cos 24} w=0, which is a particular case of the above, the Gyldén-Lindstedt equation ; the name does not seem very appropriately chosen, but as it has now ‘become established we shall use it here. The same equation occurs in the Potential Theory as giving rise to the functions appropriate to the Elliptic Cylinder ; it is discussed from this point of view in Heine’s ‘Kugelfunctionen.’ The more general equation above will be called either Hill’s equation or the generalised Gyldén-Lindstedt equation. The theory of these equations is a matter of pure mathematics, and the papers in 1 Opuscules Mathématiques, v. p. 336. ? ‘Solutions de différents problémes de calcul intégral,’ Miscellanea Taurinensia, ili. (@uwvres, i. p. 586.) 3 Guvres, viii. and ix: K 2 182 REPORT—1899. which it has been developed will not be reviewed here; the result important for our purpose is that an integral can be found in the form w= > a, cos{(e+2r)t+a}, %™=-COO where c depends on the coefficients in the equation, and a, and a depend on these coefficients and on two arbitrary constants of integration. For the determination of c Hill devised the following beautiful method :— (oe) Putting e’=Z, we have O= ™\& 9,2?", where the quantities 6, are g ys q n=—CO ee} constants, Hill assumes that w is of the form w= > b,f°*?" and sub- n=—-CO stitutes this value of w in the differential equation. Since the whole coefficient of each power of £ must now be zero, an infinite number of equations are obtained, which involve the 6’s linearly ; on eliminating the é’s a determinant with an infinite number of rows and columns (an idea first introduced by Kotteritzsch in 1870) is obtained, which involves only ¢ and the known quantities 6,. This determinant, equated to zero, furnishes the value of c, and consequently the motion of the lunar perigee. The convergence of the infinite determinant was not considered by Hill ; this gap in the work was filled by Poincaré ' in 1886. Hill’s paper was reprinted,’ with some additions, in 1886. In 1877, Adams,’ referring to Hill’s paper, remarks that he had himself, many years previously, investigated the motion of the moon’s node by a method similar to that used by Hill for the perigee, and had found the same infinite determinant. Tn 1878 Hill4 published in a more complete form the derivation of the periodic solution, which in its character of intermediate orbit had been the foundation of his previous paper. The solution is found by actually substituting, in the differential equations of the restricted problem of three bodies, expansions of the desired form with undetermined coefticients ; these coefficients are then determined as functions of a parameter m, which depends on the ratio which the period of the periodic solution bears to the period of revolution of the two principal masses round each other, 2.e. on the ratio of the mean motions of the sun and moon. By varying m, different periodic solutions are obtained ; the last one of Hill’s solutions (the orbit of maximum lunation) has cusps at the points where the elonga- tion from the sun is a right angle. Hill’s work soon led to further developments. In 1883-4 Poincaré,’ using a theorem due to Kronecker in the general theory of functions, 1 ‘Sur les déterminants d’ordre infini, Bulletin de la Soc. Math. de France, xiv. pp. 77-90. 2 On the part of the motion of the Lunar Perigee, which is a function of the mean motions of the Sun and Moon,’ Acta Math. viii. pp. 1-36. 3 ‘On the motion of the Moon’s Node in the case when the orbits of the Sun and Moon are supposed to have no eccentricities, and when their mutual inclination is supposed to be indefinitely small, Monthly Notices, Roy. Ast. Soc. vol. xxxviii. . 43-9. he. ‘Researches in the Lunar Theory,’ Amer. Journ. Math. i. pp. 5-27, 129-48, 245-61. > «Sur certaines sclutions particuliéres du probléme des trois corps,’ C. 2. xevil. pp. 251-2; Bull, Aste. i, pp. 65-74. PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 1338 proved the existence of an infinite number of periodic solutions in the general problem of three bodies; and in 1887 Bohlin ! applied an idea of Hill’s (viz. using the Jacobian integral to separate off regions of space into which the moon cannot enter) to a more general class of dynamical problems. In the same year (1887) Hill ? discussed the different systems of variables which might be employed in solving a system somewhat more general than the restricted problem of three bodies, namely, that of a particle of zero mass, attracted by two bodies which move in Keplerian ellipses round their common centre of gravity. Poincaré’s* memoirs of 1881-6 on curves defined by differential equations lead to one result of importance in Dynamical Astronomy. In order that the system of » bodies may be stable, two conditions must be fulfilled : firstly, the mutual distances must always remain within certain limits ; and, secondly, if the system has a definite configuration at any instant, it must be possible to find a subsequent instant at which the configuration differs from this as little as we please. It follows from the investigations of this series of memoirs that, if the first of these conditions is satisfied, the second is also. In 1883 Lindstedt * resumed the consideration of the problem which had been treated by Newcomb nine years before, namely, the expression of the coordinates in the problem of three bodies as trigonometric series, whose arguments are linear functions of the time. A fuller account ° of the work was published in 1884. The author starts from the equations of Lagrange’s ‘ Essai sur le probléme des trois corps’ ; the system is reduced to four different equations, each of the second order ; and these are solved by successive approximation. After the rth approximation has been effected, the (r+1)th approximation is obtained by solving four linear non-homogeneous differential equations with constant coefficients. This ean be done by known methods ; but if the solution is carried out in the usual way, termg will arise in which the time ¢ occurs as a factor (these are the ‘secular terms’ of the old planetary theory). Lindstedt therefore modifies the equations in accordance with a method indicated by himself in a previous paper,® and obtains a solution in which ¢ occurs only in the arguments of trigonometric functions. It thus appears that the mutual distances of the three bodies can be expressed as trigonometric series of four arguments, each of which is a linear function of the time. The defects of Lindstedt’s memoir in regard to convergence, &c., will be noticed in connection with other papers. A fresh proof of Lindstedt’s results was given by Tisserand’ in 1884-5, ' ‘Ueber die Bedeutung des Princips der lebendigen Kraft fiir die Frage von der Stabilitit dynamischer Systeme,’ Acta Math. x. pp. 109-30. ? *Coplanar Motion of two Planets, one having a Zero Mass,’ Annals of Math. iii. pp. 65-73. * «Sur les courbes définies par les équations différentielles’; Ziowville (3) vii. pp. 375-422 ; (3) viii. pp. 251-96 ; (4) i. pp. 167-244; (A) ii. pp. 151-217. * «Sur la forme des expressions des distances mutuelles dans le probléme des trois corps,’ C. R. xlvii. pp. 1276-8, 1353-5; ‘Ueber die Bestimmung der gegen- seitigen Entfernungen in dem Probleme der drei Kérper,’ Astr. Nachr. cvii. pp. 197-214. 5 « Sur la détermination des distances mutuelles dans le probléme des trois corps,’ Annales de V Ecole Norm. (3) i. pp. 85-102. ° «Beitrag zur Integration der Differentialgleichungen der Stérungstheorie, Mémoires de ? Acad. de Saint-Pétersbourg, xxxi. No. 4. 7 «Note sur un théoréme de M. A. Lindstedt concernant le probléme des trois corps, C. &. xcviii. pp. 1207-13; ‘ Mémoire sur le probléme des trois corps, Annales de lv Observatoire de Paris, Mémoires, xviii. 134. REPORT—1899, on the lines of Delaunay’s lunar theory ; Tisserand extended Lindstedt’s theorem, and in 1887 Lindstedt! showed how this extension could be derived from his own original paper. An imperfection in Lindstedt’s first paper was removed by Poincaré? in 1886, who, by an ingenious application of Green’s theorem, proved that only one secular term appears in each of Lindstedt’s approximations, and that this can always be removed. In 1889 Poincaré* gave a fresh method of derivation for Lindstedt’s series, by transforming the Gyldén-Lindstedt differential equation into a Hamiltonian system of the fourth order, replacing this by the corre- sponding Hamilton-Jacobi partial differential equation, and solving the latter by a series proceeding in ascending powers of a small parameter, the coefticients being trigonometric series of two arguments. Poincaré observes, however, that in the problem of three bodies this method will not apply if Kepler’s ellipse is taken as the first approximation, and consequently another intermediary orbit must be used. The number of independent arguments required in order to express the coordinates in the problem of 7 bodies, without having the time out- side trigonometric functions, was shown by Harzer* in 1889 to be 3(n—1). The question of the convergence of sums of periodic terms, such as are obtained in Lindstedt’s expansions, had now become a matter of prime importance, Poincaré? in 1882-4 showed, firstly, that if such a series is absolutely convergent for certain values of the time, it is so for all values of the time ; and, secondly, that a function cannot be represented by two different absolutely convergent series of this kind. Further, a function represented by such a series can assume indefinitely great values if the convergence is not uniform. In a further note ° in 1885, he showed that this can happen in two ways: either the function may steadily increase, or its value may oscillate, the amplitude of the oscillations increasing indefinitely. Bruns‘ in 1884 further discussed the series of Dynamical Astronomy : as these are usually obtained by the integration of trigono- metric series, it follows that the coefficients of those terms whose periods are very long will be affected by very small divisors. Bruns shows that this causes the series to fluctuate between convergence and divergence, when the constants, on which the coefficients of the time in the arguments depend, are altered in value by small amounts. Features of special interest present themselves in the planetary theory when the periods of two planets are very nearly commensurate with each other. In this case some of the inequalities of long period rise to im- portance; thus, in the theory of Jupiter and Saturn an inequality with a period of 900 years has a large coefficient; the grandeur of this coefficient is due to the fact that its denominator contains a factor 1 ¢Ueber ein Theorem des Herrn Tisserand aus der St6rungstheorie,’ Acta Math. x. pp. 381-4. 2 «Sur une méthode de M. Lindstedt, Bull. Astr. iii. pp. 57-61. 3 ‘Sur les séries de M. Lindstedt,’ C. R. eviii. pp. 21-4. 4 ‘Ueber die Argumente des Problems der m-K6rper,’ Astr. Nach. cxx. pp. 193-218. 5 ‘Sur les séries trigonométriques,’ C.R. xcv. pp. 766-8, xcvii. pp. 1471-3; ‘Sur la convergence des séries trigonométriques,’ Bull. Astr. i. pp. 319-27. 6 «Sur les séries trigonométriques,’ C. #. ci. p. 131. 7 *Bemerkungen zur Theorie der allgemeinen Stérungen,’ Ast. Nach. cix. pp. 215-22 PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES, 155 5n—2n’ (where 7 and 7/ are the mean motions of Saturn and Jupiter), and this factor is very small, on account of the approximate commensura- bility of m and n’. In certain cases (called librations) the commensura- bility is exact ; thus a linear relation exists between the mean motions of three of Jupiter's satellites. Tisserand ! in 1887 applied Delaunay’s method of integration to dis- cuss the effect of approximate commensurability, showing that commen- surability is not inconsistent with the stability of the system. Bohlin 2 in 1888-9 gave a new method for treating the cases in which terms with small divisors appear likely to endanger the convergence, He considers the equation Os =—3iBy sin (ju), where the coefficients (,; are of the order of the disturbing masses and form an absolutely convergent series, and where the independent variable w is, in the applications to the planetary theory, a multiple of the time, ia If in this equation we write w= — 2, om =—p,, we have ae dw dp F Sarna) ps ; apy agate Bt as —Xif3,, sin (7%¢—jw), which we can write i is pe et) edly DOM ipods Oh dx tp, dx cp, da of’ dx dw’ where H=}p,?—p.—/,; cos (if —jw). The solution of this Hamiltonian canonical system depends in the ordinary way on the solution of the Hamilton-Jacobi partial differential equation sV\2__8V This is now replaced by the equation Ov Non JON in order to mark the fact that g and the f’s are small (in the applications x* is the mass of the disturbing body) ; and Bohlin integrates this equa- tion by expanding V as a power-series in x, V=VotcV, te?7Vo+ Ce It is found that the occurrence of small divisors can be avoided in the series which represent the quantities V,, and hence the original difliculty would appear to have been removed, It is, however, possible that large 1 ‘Sur la commensurabilité des moyens mouvements dans le systéme solaire,’ C. R. civ. pp. 259-65 ; Bull. Astr. iv. pp. 183-92. 2 ‘Ueber eine neue Anniherungsmethode in der Stirungstheorie, Bihang till Kgl. Svenska Vet.-ak. Handlingar, xiv. No. 5; ‘Zur Frage der Convergenz der Reihenentwickelungen in der Stérungstheorie,’ Ast. Nach. cxxi. pp. 17-24. 136 REPORT-—1899, numerators may occur, and so the question of convergence is not definitely settled. The above expansion in powers of « is noticeably similar to that of Poincaré. SIV. Memoirs of 1868-89 on the Absence of Terms of certain Classes from the Infinite Series which Represent the Solution. The distinction between the secular and periodic inequalities of the elliptic elements of a planet’s orbit has already been explained. Laplace in 1773 showed that one of these elements—the mean distance or semi- major axis of the orbit—has no secular inequalities at all, when terms of higher orders than the first powers of the masses and the squares of the eccentricities and inclinations are neglected. Lagrange in 1776 proved that this result still holds when all powers of the eccentricities and inclinations are taken into account ; and in 1808 Poisson showed that it is still true when terms involving squares of the masses are included in the calculations. In the period under review, Tisserand ! in 1875-6 simplified Poisson’s proof by using the transformation of Jacobi and Radau, thus reducing the problem of three bodies to a system of the twelfth order, depending on a single perturbing function. In 1874-5, Mathieu? extended the discussion so as to include terms multiplied by the cubes of the masses. He first, by using Jacobi’s sub- stitution, replaces the sun and three planets by three fictitious planets moving round a fixed centre ; the orbits of these bodies are homothetic with the actual orbits, and consequently the study of the variations of the mean distances in the fictitious orbits will give the required results. The author then, by developing the disturbing function as far as terms of the third order in the masses, shows that the reciprocals of the mean distances have no secular variations of this order. In 1877 Haretu * published an extract from a memoir 4 which appeared in 1883. He uses the transformation by which Tisserand had, in 1875, simplified Poisson’s work, and discusses a memoir published in 1816, in which Poisson believed he had proved the non-existence of secular terms in the mean distances, of the third order in the perturbing masses, when the variations of the elements of the disturbed planet only were taken into account. Haretu shows that Poisson had overlooked a certain class of terms, and proves that secular inequalities arise from these terms, which are not ultimately cancelled ; and hence that the theorem of the invariability of the mean distances holds only to terms of the second order in the disturbing masses. Haretu, however, does not give the explicit analytical expression of the third order terms in the secular inequalities. 1 «Mémoire sur un point important de la théorie des perturbations planétaires,’ Mémoires de lV’ Académie de Toulouse (7) vii. pp. 374-88 ;y Annales de U'Ecole Norm. Sup. (2) vii. pp. 261-74 (merely a reprint); ‘Note sur l’invariabilité des grands axes des orbites des planétes,’ C. R. Ixxxii. pp. 442-5. 2 ‘Mémoire sur les inégalités séculaires des grands axes des orbites des planétes,’ C. R., xxix. pp. 1045-9; Crelie, xxx. pp. 97-127. 8 «Sur Vinvariabilité des grands axes des orbites planétaires,’ C. R. lxxxv. pp. 504-6. 4 «Sur Vinvariabilité des grands axes des orbites planétaires,’ Annales de U’ Obs. de Paris, Mémoires, xviii. (39 pp.). PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 137 In 1889, Eginitis! gave the analytical expression for those secular inequalities of the mean distances which are of the third order in the dis- turbing forces. After showing that they can arise only from a term 1 /(3B" ,\* nai ( “Ot at) j ut where = denotes the aggregate of terms of the first and second order, he finds [ae by substituting the ordinary development of the disturb- ing function, squares it, and shows that secular inequalities arise from the multiplication of terms with the same arguments. He further shows that these secular inequalities are periodic, though their period is very long. The transition from the old planetary theory, with its secular and periodic inequalities, to the new Dynamical Astronomy, in which all terms are periodic, had its effect on theorems such as that now under considera- tion. Tisserand ? in 1888 gave the new formulation of the theorem of the invariability of the mean distances, when the solution of the problem of three bodies is expressed as in Delaunay’s lunar theory. He shows that the theorem does not hold when terms of the order of the fourth power of the ratio of the mean motions are taken into account, and for the general problem of three bodies confirms Haretu’s result that the theorem does not hold for terms of the order of the cube of the disturbing forces. In 1878 Adams? published some curious results relating to one of the expansions in the lunar theory. Let e¢ be the eccentricity of the lunar orbit, and let y be the sine of half the inclination of the moon’s orbit to the ellipse ; these quantities are supposed defined as in Delaunay’s theory : let n be the moon’s mean motion, (1—c)n the mean motion of the lunar perigee, (I—g)n the mean motion of the moon’s node, a the mean distance, and 7 the radius vector. Then the non-periodic part of < can be expanded in the form A+ Be?+Cy?+ Ee! + 2Fe?y?+Gyi+... where A, B, C...are functions of the solar eccentricity and of the ratio of the mean motions of the sun and moon ; similarly the terms in ¢ which involve e? and y? can be denoted by He?+ Ky’, and the similar terms in g by Me?+Ny?. Then Adams’s theorems are that B=0, C=0, EK—FH=0, FN—GM=0. These results are all obtained by one process, which for the case of the first may be thus described : consider two moons, of which the orbit of one has no eccentricity or inclination, and the orbit of the other has no inclina- tion. It is proved that a certain function of the coordinates of the two moons is purely periodic ; and it is shown that this requires the vanishing of the quantity B. 1 ‘Sur la stabilité du systéme solaire,’ C. R. cviii. pp. 1156-9; ‘ Mémoires sur la stabilité du systéme solaire,’ Annales de l’ Obs. de Paris, Mémoires, xix. ? «Sur un point de la théorie de la Lune,’ C. R. evi. pp. 788-93. * Note on a remarkable property of the analytical expression for the constant term in the reciprocal of the moon’s radius vector,’ Monthly Notices, Roy. Astro. Soc, xxxviii. pp. 460-72, 138 REPORT—1899, § V. Gyldén’s Theory of Absolute Orbits. In 1881 Hugo Gyldén, Director of the Observatory at Stockholm, began the publication of a new method for calculating the motions of the heavenly bodies. The method has been made of practical importance by its application, in the hands of Gyldén’s pupils, to the minor planets, and is of theoretical interest from the fact that (as in Delaunay’s, Newcomb’s, and Lindstedt’s memoirs) the time appears only in the arguments of periodic terms. In this report it seems best to give, first of all, a general account of the method, and then briefly to notice the series of memoirs in which Gyldén and his pupils have developed it. Consider, then, a system consisting of the sun and two planets. For convenience one of these will be spoken of as the distwrbed and the other as the disturbing planet. At any instant the motion of the disturbed planet is taking place in a certain (moving) plane, which passes through the sun; this we can call the plane of its instantaneous orbit ; in this plane we take (as an axis from which to measure angles) a line which moves with the plane in such a way that the surface formed by the motion of the line always cuts the plane orthogonally. The angle between this line and the radius vector to the planet can be called the planet’s true longitude, and denoted by v ; the radius vector from the sun to the planet will be denoted by +. The quantities 7, v are given by differential equations of the form af ,dv\ 4-60 aa S M_wy cQ di (" a )aMae Bush ap TN ge (i aaah? where © (which is called the disturbing function) is supposed to be expressed in terms of 7, v, and the quantities which define the moving plane and the position of the disturbing planet, and where M is the sum of the masses of the sun and the disturbed body. . Let the perpendicular distance of the disturbed body from some fixed plane be zr. Then the third differential equation of the disturbed body’s motion may be written in the form 2 > aa + Mzr=a function of the positions of the planets. Now introduce new variables p, n, 8 connected with the old variables by the relations _2(1—n?) pd? — \Ma(1— n?)}} hos seer dt hee ary where @ is a constant called the protometer, as yet undetermined ; as there are only two conditions here to determine the three quantities p, n, S we can impose another condition later. The equations for » and v now transform into oA DB og431_lb an? 1+Sdv — ‘eae 5 serra dv PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE RODIES, 159 and Pep {tg tt csne} Laas cs — {aes aoe tap (as) Py Cae OO where Par a= aay a Also, the equation in x can be written az 4 dz ‘ Oe 4 2 — (148)Q0 +(148)a%, dv? dv where Z, is a certain function of the positions of the planets. Now let us consider the form in which Gyldén wishes to express the solution of these equations. The differential equations will finally be solved by means of sums of periodic terms whose arguments are linear functions of v; these terms may be classified in the following way :— Firstly, there will be terms which vanish altogether when the dis- turbing mass is put equal to zero ; these are called coordinated terms, and correspond to the ‘ periodic inequalities’ of the classical planetary theory. Secondly, there will be terms which, when the disturbing mass is put equal to zero, do not vanish, but coalesce with the terms which represent the Keplerian elliptic motion of the disturbed planet round the sun. These terms involve the disturbing mass in their arguments, but not in their coefficients ; they are called elementary terms, and correspond to the ‘secular inequalities’ of the classical planetary theory. Terms will also occur in the course of Gyldén’s process which involve the disturbing mass in the denominator of their coefficients, and so would become infinite if the disturbing mass were put equal to zero; these are called hyper- elementary, and it is shown that they do not appear in the final result. And, lastly, we have already seen that when the periods of two planets are nearly commensurable, certain terms of long period rise to importance ; these are called the semi-elementary or characteristic terms for the planet under discussion. The quantity p, as already defined, will be composed of both elementary and coordinated terms. Let (p) denote the elementary terms, and let R denote the coordinated terms, so p=(0) +R. Jt will appear that p is of the form ea) (p)=« cos {@! ie c)v —T} + poe cos {1 ee c,)v— Let 5) n=1 where « (called the diastemmatic modulus) and T (called the longitude of the absolute perihelion at the origin of time) are two of the six constants of integration of the problem, «, and T, are functions of the constants of integration, and ¢ and ¢, are small constant quantities of the order of the perturbing forces, 140 REPORT— 1899, We can now define 7 ; let n cos r=K cos '+ 3x, cos {(c—c,) v—T,} n sin w= sin '— 3x, sin {(¢c—c,) v—T,}. Thus 7 contains only elementary terms, and (e)=n cos {(1—c)v—7}. n is called the diastemmatic function, and (l—c)v—7z is called the dia- stemmatic argument. If in the expressions for the coordinates we strike out all the co- ordinated terms, leaving only those which are elementary, these modified expressions for the coordinates will define a new orbit, which will be so near to the true orbit that the difference between them will be only cf the same order of magnitude as the disturbing forces. This new orbit may be called the absolute orbit. The radius vector in the absolute orbit (r) is thus defined by a(1—n?) a =~ *) 1+(e) ° The variable z can be divided into two parts just as p was; thus 2=(z)+Z, where (z) contains all the elementary terms ; (x) is of the form loa) (z)=7 sin {(1+7r)v—O} + >i. sin {(1+7,)v—9,} n=1 where 7 (called the anastemmatic modulus) and © (called the longitude of the absolute node) are two more of the six constants of integration of the problem, and r, ¢,, 7,, ©, are constants depending on them. If this be written in the form (z)=J sin {((1+7)v—0}. J is called the anastemmatic function, and (1+7)v—Q is called the anastemmatic argument. Gyldén (who, however, is not in this particular followed by his pupil Harzer) further introduces a quantity ¢ called the reduced time, which is defined by the equation df at (1—n?)}, dv M? {1+( i 2 and a quantity c W therefore satisfies the differential equation nie 148 a =o Hy pee ies the integration of this will clearly introduce another arbitrary constant, PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES, 141 which will be denoted by A, and will be called the absolute longitude, or mean longitude for t=0. The six arbitrary constants which have now been defined are the elements which fix the absolute orbit of the disturbed body, namely, A (the absolute longitude), (the longitude of the absolute perihelion), © (the longitude of the absolute node), a (the protometer), « (the diastem- matic modulus), and 7 (the anastemmatic modulus). Having now described the form in which the solution is to be obtained, we can resume the consideration of the differential equations. First, we have to expand in a suitable way the disturbing function 0 and the quantities P and Q. This is effected by means of infinite series, each term of which consists of a product of powers of the various small quantities such as 7, multiplied by a trigonometrical function of the longitude. Next, we have to substitute these expansions in the differential equations for p, s, and W, and integrate these equations. The equations for p and 8 are respectively of the forms dp ae era) 7,3 t(1—B)ps= 3a, cos (ww By), dS as ? = b, cos (A,V—/3,); where the quantities A, are constants, but the quantities a,, b,, (,, contain the unknown variables. These equations are solved by processes of suc- cessive approximation ; only those terms are initially retained which have a considerable effect, and in this way the elementary part (p) is determined. A feature of Gyldén’s treatment of equations such as that given above for p is the use of the horistic function, which is a modification of the term containing the first power of the dependent variable, and is designed to ensure the convergence of the approximations. We may regard the arbitrary constants « and Tas arising from the integration of the equation in p, i and 0 as arising from that in z, a as arising from that in 8, and A as arising from that in W. The principal papers in which Gyldén’s theory has been developed will now be briefly noticed. In 1881 Gyldén published three short papers ! in French and German, and three long memoirs in Swedish.? The deriva- tion of the differential equations of the first Swedish memoir was simplified by Backlund 3 and Callandreau.! Further notes and criticisms on the early part of the theory of intermediate orbits were given in 1882 by Thiele® and Radau.® The 1 «Sur la théorie du mouvement des corps célestes,’ @. R. xcii. pp. 1262-5; ‘ Sur lintégration d’une équation différentielle linéaire du deuxiéme ordre dont dépend l'évection,’ C. R. xciii. pp. 127-31 ; ‘ Ueber die Theorie der Bewegungen der Himmels- k6rper,’ Ast. Nach. c. p. 97. * ‘Undersékningar af theorien for himlakropparnas rorelser,’ Bihang till K. Sv. Vet.-ak. Handlingar, vi. and vii. I wish gratefuily to record my obligations to Mr. W. F. Sedgwick, late Scholar of Trinity College, Cambridge, who has kindly placed at my disposal a manuscript English translation of the Undersékningar, with many corrections of his own. 8 « Ableitung von Professor Gyldén’s Differentialgleichungen fiir die intermediire Bewegung,’ Asi. Nach. ci. pp. 19-22; Professor Gyldén’s ‘Neue Untersuchungen iiber die Theorie der Bewegung der Himmelsk6rper,’ Copernicus, ii. ‘ «Sur la théorie du mouvement des corps célestes,’ C. #. xciii. pp. 779-81. ° ‘Ueber Professor Gyldén’s intermediiire Bahnen,’ Ast. Nach. cii. pp. 65-70. ® «Sur un pojnt de la théorie des perturbations,’ C. RB. xcy. pp. 117-20. 142 REPORT—1899. notion of the absolute orbit and the definitions of elementary and co- ordinated terms are introduced in the second part of the ‘ Undersékningar.’ Gyldén wrote another paper! on this in 1882, and in the same year dis- cussed further? one of the differential equations of his theory of inter- mediate orbits. A long series of papers dealing with the processes for integrating differential equations of the second order by successive approximation, and with the convergence of the developments, was published? by Gyldén in 1882-96. On this see also Harzer.! In 1885 Gyldén® and Shdanow® applied the theory of intermediate orbits, which had been given in the ‘ Undersékningar,’ to the case of the moon’s motion. The problem is made to depend on the solution of the Gyldén-Lindstedt equation, and the results yield an approximation to the motion of the perigee. Andoyer’ also applied Gyldén’s theory to the moon in 1887 ; and Tisserand * in 1888 discussed the Gyldén-Lindstedt equation, and applied his results to Andoyer’s equations. Harzer ° in 1886 applied Gyldén’s ideas to the determination of the motion of those of the minor planets (e.g. Hecuba) whose mean motion is approximately twice as great as that of Jupiter. On account of this approach to commensurability, some of the characteristic terms become very important. Harzer modifies Gyldén’s original procedure in two respects : firstly, in using the true longitude throughout as the independent variable ; and, secondly, in abandoning the use of the ‘reduced time. 1 ‘Ueber die absoluten Elemente der Planeten-Bahnen,’ Ast. Nach. ciii. pp. 49-58. 2 «Sur l’équation différentielle qui donne immédiatement la solution du probléme des trois corps jusqu’aux quantités de deuxiéme ordre inclusivement, C. R. xcv. . 55-8. - 3 *Hine Anna&herungsmethode im Probleme der drei Kérper, Acta Math. i. pp. 77-92; ‘ Untersuchungen tiber die Convergenz der Reihen, welche zur Darstellung der Coordinaten der Planeten angewendet werden,’ ibid. ix. pp. 185-294; ‘ Nouvelles recherches sur les séries employées dans les théories des planétes,’ ibid. xv. pp. 65-189 ; xvii. pp. 1-168; ‘ Ueber die Convergenz einer in der St6rungstheorie vorkom- menden Reihe,’ Ast. Wach. cxix. pp. 321-30; ‘ Bemerkungen tiber die Convergenz der nach der Potenzen der stdrenden Krafte geordneten Anniiherungen im Stérungs- problem,’ ibid. cxxi. pp. 80-94; ‘Sur une équation différentielle du second ordre, non linéaire et a coefficients doublement périodiques,’ C. £. cxxii. pp. 160-5 ; ‘ Remarques ultérieures relativement 4 ma derniére communication 4 M. Hermite,’ ibid. exxii. pp. 585-8; ‘ Om bestiimningen af ojemnheter med mycket lang period i theorien for planeters och satelliters rorelser, Ofversigt af Kongl. Vet.-ak. For. lii. pp. 419-32 ; ‘En transformation .af den differentialekvation, som bestiimmer ojemnheterna med mycket langa periorder i en planets longitud,’ ibid. lii. pp. 503-6 ; ‘ Olika methoder att bestiimma de horistika termerna i den differentialekvation, som férmedlar hiidled- ningen af ojemnheterna ien planets longitud,’ ibid. liii. pp. 421-30. + ¢*QUeber eine Differentialgleichung der Stdrungstheorie,’ Ast. ach. ecxix. . 273-94. aks Sur l’orbite intermédiaire de la Lune,’ (. R. ci. pp. 223-6; ‘Die intermediire Bahn des Mondes,’ Acta Math, vii. pp. 125-72. 6 Recherches sur le mouvement de la Lune autour de la Terre daprés la Théorie de M. Gyldén, Stockholm, 1885. 7 ¢Contribution a la théorie des orbites intermédiaires,’ Annales de la Fac. des Sc. de Toulouse, i. * «Sur une équation différentielle du second ordre qui joue un réle important dans la mécanique céleste,’ did, ii, ® *Untersuchuugen tber einen speciellen Fall des Problems der drei Kérper,’ Mémoires de Saint-Pétersbowrg, xxxiv. No. 12; ‘Quelques remarques sur un cas spécial du probléme des trois corps,’ stronomiska Iakttagelser ooh Undersikningar anstilda pa Stockholms Observatorium, iii. No. 4. y ? PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE bopiEs. 148 Some criticisms on Harzer’s paper (and also on Brendel’s paper of 1889) were made by Charlier’ in 1890 ; and Harzer himself made some correc- tions in 1891.2. Brendel? in 1887 derived Harzer’s equations as a special case of Cyldén’s system. After this several applications of Gyldén’s theories to definite cases were published. Wellmann‘ in 1888 discussed the intermediate orbit of Thetis, and Brendel ° in 1889 found the absolute orbit of planets of the Hestia type, whose mean motion is approximately triple that of Jupiter. Some developments useful in the theory were given in 1889 by Masal," and improvements in the integration of the equations were introduced by Wolf? in 1890. Backlund 8 in 1892 discussed by Gyldén’s methods the motion of the group of small planets whose mean motion is approximately twice that of Jupiter ; in distinction to Gyldén and Harzer he takes the time as the independent variable. The same subject was treated by an improved analysis in a number of papers’ published in 1897-8. Researches in connection with the properties of series such as those occurring in Gyldén’s work were published by Gyldén '? and Backlund !! in 1889, and by Olsson? in 1890 and 1891. 1 Vierteljahrsschrift der Astron. Gesells. xxv. p. 175. 2 «Berichtigung zur Abhandlung “ Untersuchungen tiber einen speciellen Fall des Problems der drei Korper,”’ Ast. Wach. cxxvi. p. 399. 3 «Ueber einige in neuerer Zeit angewandte Formen fiir Differentialgleichungen im Problem der drei Kérper,’ ibid. exvi. pp. 161-6. 4 «Die intermediire Bahn des Planeten @) Thetis nach Herrn Gyldéns Theorie,’ Archiv der Math. u. Physik (2) vi. pp. 353-91. 5*Om Anvindningen af den absoluta Stérungsteorien pa en Grupp af sma planeterna, med numerisk Tillimpning pa Planeten | 46 ) Hestia,’ Astr. Zakttagelser och Under. anst. pa Stockholms Obs. iv. No. 3 ; ‘ Sur les perturbations de la planéte Hestia, d’aprés la théorie de M. Gyldén,’ C. &. cviii. pp. 49-51; Ueber die Anwendung der Gyldén’schen absoluten Stirungstheorie auf die Breitenstorungen einer genissen Klasse kleiner Planeten, Inaugural-Dissertation, Géttingen, 1890. 6 ‘Wormeln und Tafeln zur Berechnung der absoluten Storungen der Planeten,’ Kgl. Svenska Vet-ak. Handlingar, xxiii. No. 7. 7 Sur les termes élémentaires dans Vexpression dw rayon vectewr, Habilita- tionsschrift, Stockholm, 1890. 8 « Ueber die Bewegung einer gewissen Gruppe der kleinen Planeten,’ Wémoires de ? Acad. de Saint-Pétersboury (7) xxxvili. No. 11. 9 *Ueber die Integration der Differentialgleichung des Radius vector einer gewissen Gruppe der kleinen Planeten,’ Budletin de l’ Acad. de Saint-Pétershourg (5) vi. pp. 311-19; ‘Sur la détermination des termes 4 longues périodes dans expression de la longitude des petites planétes du type de Hécube,’ Bull. Astr. xiv. pp. 321-4 ; ‘Deuxiéme méthode pour la détermination des termes a longues périodes dans l’expression de la longitude des petites planétes du type de Hécube,’ ibid. xv. pp. 5-9; ‘Formeln zur Berechnung angeniaherten Bahnen der kleinen Planeten vom Hecuba-Typus, nebst ihrer Anwendung auf den Planeten (18s) Dejopeja,’ Astr. Nach. exly. pp. 241-8; ‘ Ueber die Bewegung der kleinen Planeten des Hecuba-Typus,’ Mémoires de V Acad. de Saint-Pétersbourg (8) i. 10 * Sur les termes 6lémentaires dans les coordonnées d’une planéte,’ C. R. cviii. pp. 79-82, 116-9; ‘Sur la représentation analytique des perturbations des planétes,’ ibid. cix. pp. 395-6. 1 «Ueber die Kleiner Divisoren bei den elementiiren Gliedern in der Theorie der Planetenbewegungen,’ Astr. Nach. exxii. pp, 273-302. 12 ¢Bemerkungen tiber die Integrationsmethoden der Zeitreduction in der 144 REPORT—1899; Gyldén in 1893 published the first volume of a work! which was intended to furnish, in three volumes, a complete exposition of his theory of absolute orbits. His death occurred in 1896 before the second volume was ready, but it is expected that Dr. Backlund, who has charge of Gyldén’s manuscript, will as far as possible carry out the original design. Backlund ? in 1896 described a method, founded on Gyldén’s work, for integrating the differential equation which determines the radius vector in the case of minor planets whose mean motion is nearly twice that of Jupiter. Brendel® in the same year discussed the relation of Gyldén’s series to the gaps in the distribution of the minor planets; in 1897 Brendel‘ announced that he had found an improved process of integration, and in 1898 the same author published the theoretical part > of a memoir in which the motions of the minor planets are discussed by a modified form of Gyldén’s process ; the second part, which is not yet published, is to deal with definite applications to the solar system. § VI. Progress in 1890-8 of the Theories of §§ ILI. and IV. A new impetus was given to Dynamical Astronomy in 1890 by the publication of a memoir ® by Poincaré. The first feature is the introduction of integral invariants. We can regard a system of ordinary differential equations May _ GH ee "hai leider n dt as defining the totion of a point whose coordinates are (x, 22)... 2,) in space of » dimensions. If now we consider a group of such points which occupy » v-dimensional region Z, at the beginning of the motion, they will at any subsequent time ¢ occupy a region Z. A v-ple integral taken over ¢ is called an integral invariant if it has the same value at all times ¢. Thus, in the motion of an incompressible fluid, the volume of the fluid which was contained initially in any given region is an integral invariant. Poincaré gives a number of integral invariants which exist in particular cases, and then proceeds to apply his results to the question of the stability of the motion in the problem of three bodies. There are, he remarks, two senses in which the word ‘stability’ may be taken. It may be taken to mean that variations in the mean distances of the bodies are always restrained within finite limits—Hill and Bohlin have proved that under Gyldén’schen Theorie,’ Archiv f. Math.og Natur, Christiania, xiv. pp. 1-10 (1890) ; ‘Ueber die Convergenz der Annaherungen in der Gyldén’schen Storungstheorie,’ ibid. pp. 232-9 ; ‘Untersuchung tiber eine Gruppe von langperiodisch elementiiren Gliedern in der Zeitreduction,’ Bihang till k. Sv. Vet-ak. Handlingar, xvii. No. 4. 1 Traité analytique des orbites absolues des huit planétes principales, tome 1, ‘ Théorie générale des orbites absolues, Stockholm. : 2 ¢Sur Vintégration de Véquation différentielle du rayon vecteur d’un certain groupe des petites planétes, C. R. cxxii. pp. 1103-7. 3 «Ueber die Liicken im System der Kleiner Planeten und iiber ein Integrations- verfahren im Probleme der drei Korper,’ Ast. Nach. cxl. pp. 145-60. 4 «Ueber stabile und instabile Bewegungen in unserem Planetensystem,’ Jahresbericht der Deutscher Math. Verein, vi. pp. 123. 5 «Theorie der kleinen Planeten,’ erster Theil, Abhandlungen der Kin. Ges. der Wiss. su Gottingen, Neue Folge, i. No. 2. 6 «Sur le probleme des trois corps et les équations de la dynamique,’ Acta Math, sili. pp. 1-220. PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 145 certain conditions the motion in the restricted problem of three bodies is, in this sense, stable,—or stability ' may be taken (as by Poisson) to mean that the system is to pass infinitely often through positions as near as we please to the initial position ; the intervening oscillations may be of any magnitude. The existence of asymptotic solutions (which will be explained later) shows that an infinite number of particular solutions of the restricted problem of three bodies exist, which are not stable in Poisson’s sense of the word. But M. Poincaré now proves that there are also an infinite number which are stable, and, further, that the former are the exception and the latter are the rule, in the same sense as commensurable numbers are the exception and incommensurable numbers are the rule. In other words, the probability that the initial circumstances may be such as to give rise to an unstable solution is zero. The proof of this is made to depend on the following theorem ; when the point (x, 2, .- + 2ny Yiy Yo +++ Yn) Moves so that its coordinates are always finite, and the integral invariant [aa re A eee ie exists, then for every region 7) in space, however small this region may be, there exist trajectories which pass through 7, an infinite number of times ; and, in fact, those points of 7, which do not give rise to such trajectories form a volume which is infinitely small compared with 7». It is thus shown that, when the constant of relative energy in the restricted problem of three bodies lies between certain limits, the motion is stable not only in the sense of Hill and Bohlin, but in the sense of Poisson ; the number of exceptional cases to which this law does not apply being infinitely small in comparison with the number of orthodox cases. The result is, however, not extended to the general problem of three bodies. The author next passes to the theory of periodic solutions. Consider a system of differential equations div, ° aoe G=lI, 2; eee 2); where the X’s are functions of ~,, %,...,, and a parameter p; X,, X,,...X, may also involve ¢, but if so they are supposed to be periodic functions of ¢ with a period 27. A periodic solution of these equations of period T is of the form x=¢(t), (i=1, 2,... 7) where the functions ¢ are such that ¢(¢+T)=¢,((t). (If x, is an angular variable, this may become ¢,(¢+T)=9,(¢) +27, where 7 is an integer.) The meaning of this for our purpose is, of course, that the relative motion of the moving bodies repeats itself at regular intervals T of time. Suppose that, for the value 0 of the parameter ,, these equations admit of a periodic solution, Hi—9,(t)y (G1, 2, . she 90) 1 A discussion of general definitions of stability is given in the second volume of Klein and Sommerfeld’s Theorie des Kreisels. . L 146 REPORT—1899. the period being, for example, 27. The question is now propounded, whether the system admits of periodic solutions differing but little from this, when p has values which, though very small, are different from zero, M. Poincaré finds the answer. By choosing, as initial values of the co- ordinates, certain functions of p, it is in general possible to obtain such periodic solutions. It is further shown that as p varies, periodic solutions disappear im pairs in the same way as the real roots of algebraic equations, This happens when a certain functional determinant is zero. Poincaré next proceeds to define the characteristic exponents of a periodic solution. Suppose that a periodic solution, as above, has been found. Consider a motion differing but little from this, and defined by x=o(t) +6: hoa teay tc W??), where the quantities — are supposed to be so small that their squares and products can be neglected. Then to determine the £’s we have ey. sip OG Wis tan ie dt =XE, te,’ V—" ayeoae n). As these are linear differential equations of a definite type, with periodic coefficients, £; will be a sum of 7 quantities, each of the form > e%'S,,, where the quantities S,, are periodic functions of ¢ with the same period as ¢,(¢), and the ~ constants a, are the roots of a certain algebraical equation. The constants a are called the characteristic exponents of the periodic solution. If they are purely imaginary the é’s will remain small, and the periodic solution is stable ; if not, the solution is unstable. If two of the characteristic exponents are equal, the form of the solution is altered, as the terms of the form te“ now appear. When the original equations have the Hamiltonian canonical form, the characteristic exponents can be arranged in pairs, the exponents in each pair being numerically equal, but of contrary signs. The 7 values of a2 are called the coefficients of stability of the periodic solution considered ; if they are all real, negative, and distinct there is stability. Whén the Hamiltonian function does not involve ¢, one of the m coefficients of stability is zero, so two of the characteristic exponents are zero. The author now shows that the functional determinant already men- tioned vanishes only when one of the characteristic exponents of the original periodic solution is zero ; the theorem already given can thus be put in the more precise form. If a set of equations, which depend on a parameter p, admit of a peri- odic solution when p=0, for which no one of the characteristic exponents is zero, then they also admit of a periodic solution for small values of pu. Poincaré then turns to the periodic solutions of the differential equa- tions of dynamics. For greater definiteness, the system is supposed to have three degrees of freedom ; the equations are taken in the form— dx, oF dy;_ 6F day OR ayy _ OF sg aa ae Sa and F is supposed to be a uniform function of the «’s and w’s independent PROGRESS OF THE SOLUTION OF THE PROBLEM OF THREE BODIES. 147 of ¢, and to be periodic in the y’s of period 27. F is further supposed to depend on an arbitrary parameter p, and to be expansible in the form FEFot+pF+ pF, + F3 +... where F,) does not involve the quantities ¥. When p=0, the equations can be integrated ; x), », «3 are in this case constant, and y=nti+w,, where n,=——? 0a the quantities w, being arbitrary constants of integration. The solution will be periodic if m,, ”,, 3 are commensurable with each other ; the period T will then be the least common multiple of the quantities oF, “4 Qn Qe Dr eee N, Ny Ne In general, it will be possible to choose 2, %, #3 so that 7), no, 5 may have any prescribed values—at least in some domain ; so that there are co? periodic solutions, when p is zero. The author next proceeds to investigate whether periodic solutions of period T exist, when « is not zero. By a process of reasoning similar to that used before, it is shown that, corresponding to any periodic solution which exists when »=0, and whose constants satisfy certain conditions, there exists in general a solution of period T when p has small values dif- ferent from zero. The quantities x, and y;—,t can be expressed as series of ascending powers of p, the coefficients in which are circular functions of Int pT and a method of forming these expansions is given. The results are applied to the restricted problem of three bodies ; a difficulty arises, which in this case is solved by asimple artifice, but which is not so easily disposed of in the general problem of three bodies. Still considering the dynamical system with three degrees of freedom, Poincaré considers a solution U=P(t)+E, Yi=v(t)+n, differing but little from a periodic solution, and writes E=e'S;,, ne 'T,, where §; and T; are periodic functions of ¢. When the periodic solution corresponds to »=0, the six exponents a are all zero ; when p is not zero it is shown that the quantities a, S,, and T; are expansible in ascending powers (not of p, but) of Wp. It is shown that to every set of values of m, and n,. there correspond at least one stable and one unstable periodic solution ; and that there are exactly as many stable solutions as unstable when , is sufficiently small. The next idea to be introduced is that of asymptotic solutions. Re- turning to the general system BX, Gest Fem) let Ate v t L2 148 REPORT—1899. define a known periodic solution. Put w=, +E, The system now becomes a set of m equations to determine the é’s ; if these are solved on the supposition that squares and products of the és can be neglected, the solution is of the form E,= Aye"); + Ace™ poi + eee $A,67ntbyis where the A’s are constants of integration, the a’s are the characteristic exponents, and the ¢’s are periodic functions of ¢. In order to solve the equations when products of the 2’s are not neglected, assume E= bit Mbait «++ EM Pri The equations determining the 7’s can now be written down ; it is proved that they can be solved by assuming each of the quantities 7 to be a series of ascending powers of A,e™’, A,e™,...A,e; the , there are two of these solutions of the second kind corresponding to each value of j ; when p=p they coalesce into a single solution of the first kind, namely, the member for p= , of the set of solutions from which we started ; when