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Rate cpus. ite) we entoe ee aes Deite * . he Wi be * i” “Hedin ger 7 se aie ‘ath dite a siticaty oe oe the aie Pieaes WOR AEE BE! ns Melb ey eel i 54m at iG RG ed way fie toda Wed We Mg Mey geen he] ay oe wie wa CRP La thee aii i a ‘ah et we we et pictur Hanis PR wae ys "RAGA re il a un / pa HS at ( a Wht CHANT Ag Aa Hue gone catwadene 4 ins “€ sed wea Pe ue Te ae th RY mo sah ’ » Piva pi Se Ha eG hy yt ml Wiig 6 A MEH Mh walt onk de Muara sont LAR re) te fo" pen ee car Hk ese rere ibe ’ ” Sef yee beh Ki 1, tall the re ee | We the ne Ny ee Ld ites med eo dd HAE He meh NE Heh ed Lm toed ed ee mee, 44d OL Ge ad ae et He de key hf ‘ay 4 ithe x ote Gt tiara sen ringer iit mr 4 ‘ ination f Wstfoe AL Moe 9 at ee LS AL yy Pas bs vee. bs bea tte jen yi hb Rett Us fit wt | A aa Hey ott eye fs 4 mite he M HE tani ae on eS wide hinah Carey va pe i ei «9 are 4 as) a te Ah Ae a ay oe ee eral aie « ohh hea ae RH fog +: I ARR Wa ea phe eb tn ty Bash . va if fey Al oy Sie ea i tae eet p D Navigon We Sela cucions eae Hw vi ot ate rh 4 eH wa bi ay sist . Ht osx oe ped pera Watts. aM Wa a et ua ree , i a Ht bani fa viet mH \ tne e Be ne Penny nf EE EA ee ih GoW “ ’ Westie Ree dtde WS He dy tee dae W eS h bok OAS, sha Aid darted 9 Oe 4 Tal b nat JE ANY W wea iwi fag ait ex a “ Pe i nay me 84 Wn 4 Waa ya it A Hei ere WE « MiYedaN teem 4 t * pK wha Oa ANS RIN nin MU Th a 1 Ve 4 1 iene Hh IY ee ‘ety tbe thei (ts gue usesieih a Wis «orm VY baa ngent Wedded BY + Wath: Wort Mb a rt ie fay Weg \e “iy A eel bi ; Oe rN, i Ree vat: i ay W ett ¥ aH « fale a Pit HON shale Ii Vp « Heeb tet qe ft Yetie Mi A mane Rede Soba Leia (he RNY A Ce Hs 5 | ity dita 4 and Ve way ma 4 % 4 “a ehil i { pa 2 & \ & ‘ye i i VOL. XIII. LONDON: PRINTED BY TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET, MDCCCLXIY. CONTENTS. VOL. XIII. ———=—>—_— Page A General Catalogue of Nebule and Clusters of Stars for the Year 1860:0, with Precessions for 1880-0. By Sir J. F. W. Herschel, Bart., F.R.S. .. 2 Note on Kinone. By A. W. Hofmann, LL.D., F.R.S......... Researches on the Colouring-matters derived from Coal-tar.—I. On Aniline- yellow. By A. W. Hofmann, LL.D., F.R.S.......... Researches on the Colouring-matters derived from Coal-tar.—II. On Aniline-blue. By A. W. Hofmann, UL.D., FLR.S. .......ccceeeeeeee OD Account of Magnetic Observations made between the years 1858 and 1861 inclusive, in British Columbia, Washington Territory, and Vancouver Wlaad.s By Captain KR. W. Haig, Rea oii osc ele ese eees pay ts) On Plane Water-Lines. By W. J. Macquorn Rankine, C.E., LL.D., HiRes. & H., Assoc, Inst: NA. 6. fie eae Ce ncoarticocecl capo ncic) os 2 5 On the degree of uncertainty which Local Attraction, if not allowed for, occasions in the Map of a Country, and in the Mean Figure of the Earth as determined by Geodesy: a Method of obtaining the Mean Figure free from ambiguity, from a comparison of the Anglo-Gallic, Russian, and Indian Arcs: and Speculations on the Constitution of the Earth’s Crust. By the Venerable J. H. Pratt, Archdeacon of Calcutta.......... Soeaeo [dle On the Meteorological Results shown by the Self-registering Instruments at Greenwich during the extraordinary Storm of October 30, 1863. By James Glaisher, F.R.S., FR.A.S. 2.0... Mi ofeias.cuat Beaks) ev ajiiioteye: etepsl eo; avele » Anniversary Meeting :— PEC ROE Oty AWGN OTS erate ran) fol eyelslle) aisle) ale ole che se ahaicieae out OTe PAL 21 List of Fellows deceased, &e. .......enceeveee Md: ivsyhhe cues aie, apayet ike 21 ———_————— elected since last Anniversary ..........0.0. iS ea wae Pea Oibeds) OF THeyETESIGeMb) vigialasiicsiale « vlleisis)s/N af 6 o/elaie « s aysleepateaseusrers haze Presentation of the Medals .......... eueverels) ate selaiolel'sl «cresh ener anhenaes . 3 Election of Council and Officers ..... SCOR LLN C LAOH GORDIE GeO nitro 39 imamelal Stabemembse. 11h sos aleintarsiele’s lelQal lacs aladeta Oldie blengwie 40 & 41 Changes and present state of the number of Fellows........+++se00. 42 On the Spectra of some of the Chemical Elements, By William Huggins A AAD pd ovata) 3 oes see. Hehe «6 ey nlvah Stan em : On the Acids derivable from the Cyanides of the Oxy-radicals of the Di- and Tri-atomic Alcohols, By Maxwel) Simpson, A.B., M.B., FL.R.S. ...... 44 a2 iv Page George Biddell Airy, F.R.S., Astronomer Royal ...........+.sceees 48 On the Sudden Squalls of 80th October and 21st November 1863. By Balfour Stewart, M.A., F.R.S., Superintendent of the Kew Observatory. (GEIS Teh Ea in ese RREI ue Sain nin tit henley eain tye nha o's ve 60 date 51 On the Equations of Rotation of a Solid Body about a Fixed Point. By Walliam Spottiswoodes MAL, PARIS ta. os Pea os sce ane sisicasere come Experiments, made at Watford, on the Vibrations occasioned by Railway rains passing through a Tunnel. By Sir James South, LL.D., F.R.S., one of the Visitors of the Royal Observatory of Greenwich .........00+ 65 Extract of a Letter to General Sabine from Dr. Otto Torell, dated from Copenhagen, Dec. 12, 1863 ....... gtcsbweeeescoees cae sae en 83 Results of hourly Observations of the Magnetic Declination made by Sir Francis Leopold M‘Clintock, R.N., and the Officers of the Yacht ‘ Fox,’ at Port Kennedy, in the Arctic Sea, in the Winter of 1858-59; and a Comparison of these Results with those obtained by Captain Maguire, R.N., and the Officers of H.MLS. ‘ Plover,’ in 1852, 1855, and 1854, at Point Barrow. By Major-General Sabine, R.A., President ......... . 84 Examination of Rubia munjista, the Hast-Indian Madder, or Munjeet of Commerce. By John Stenhouse, UL.D., FP.R.S. ... ....2 ss eee 86 On the Magnetic Variations observed at Greenwich. By Professor Wolf, ORAM Maing nee ts cect asec s ees etsvne eas + doer 87 A Description of the Pneumogastric and Great Sympathetic Nerves in an Acephalous Foetus. By Robert James Lee, B.A. Cantab. ............ 90 On the Conditions, Extent, and Realization of a Perfect Musical Scale on Instruments with Fixed Tones. By Alexander J. Ellis, B.A., F.C.P.S... 93 On the Osteology of the genus Glyptodon. By Thomas Henry Huxley, BIST adh hep sive ciel oaibiv sit Unee owitecies b ae bb srs ee 108 On the Great Storm of December 3, 1863, as recorded by the Self-registering Instruments at the Liverpool Observatory. By John Hartnup, F.R.A.S., Director of the Observatory. vo... sce css ce cem ote gs oe p ot oe 109 On the Criterion of Resolubility in Integral Numbers of the Indeterminate Equation f=ax?+a'z?+a"2'?4+2b2'x"+2b'rx"+2b"2'x =0. By H. J. Stephen Smith, M.A., F.R.S., Savilian Professor of Geometry it the Wmiversity Of Oxford “ae os cet oe es toe os 20 110 Results of a Comparison of certain Traces produced simultaneously by the Self-recording Magnetographs at Kew and at Lisbon; especially of those which record the Magnetic Disturbance of July 15, 1863. By Senhor Capello, of the Lisbon Observatory, and Balfour Stewart, M.A., F.R.S. CEs EE eee se Fig See ee SN aare aoe piece pl sean oe lil Experiments to. determine the effects of impact, vibratory action, and along- continued change of Load on Wrought-iron Girders. By William Fair- bain, L/D, FOR Sond tt) wa tae ae Ie ig Se DS Se - 1 On the Calculus of Symbols.—Fourth Memoir. With Applications to the Theory of Non-Linear Differential Equations. By W. H. L. Russell, A.B. 126 On Melecular Mechanics, By the Rey. Joseph Bayma, of Stonyhurst College, Lancashire .,....... BAe te ORR Soe EGR oA Skies Se 126 v Page On some further Evidence bearing on the Excavation of the Valley of the Somme by River-action, as exhibited in a Section at Drucat near Abbe- Milles “by Joseph Prestwich, WBS. oes csess cece te eesneerees oe» 135 A Contribution to the Minute Anatomy of the Retina of Amphibia and Reptiles. By J. W. Hulke, F.R.C.S., Assistant-Surgeon to the Middlesex and the Royal London Ophthalmic Hospitals ............cseeeeeees 188 Notes of Researches on the Acids of the Lactic Series.—No. I. Action of Zinc upon a mixture of the Iodide and Oxalate of Methyl. By FE. Frankland, F.R.S., Professor of Chemistry, Royal Institution, and PUPA, Ccctnc een eis Wieved Soa UAW TRN LA. SRE ine eat ne SUNG s 140 On the Joint Systems of Ireland and Cornwall, and their Mechanical Origin. By the Rey. Samuel Haughton, M.D., F.R.S., Fellow of Trinity College, LUTET I 5 Gey Sika Gh OO ee eae) DO ee Bee Oe een re 142 On the supposed Identity of Biliverdin with Chlorophyll, with remarks on the Constitution of Chlorophyll. By G. G. Stokes, M.A., Sec. R.S..... 144 Continuation of an Examination of Rubia munista, the Kast-Indian Madder, or Munjeet of Commerce. By John Stenhouse, LL.D., F.R.S. ........ 145 On the Spectra of Ignited Gases and Vapours, with especial regard to the different Spectra of the same elementary gaseous substance. By Dr. Julius Plucker, of Bonn, For. Mem. R.S., and Dr. J. W. Hittorf, of Bene Note alka ae us lcig bina’ She nyora-ateie hel See ctBEs Gam winte’n Abie owas 153 On the Influence of Physical and Chemical Agents upon Blood; with special reference to the mutual action of the Blood and the Respiratory Gases. By George Harley, M.D., Professor of Medical Jurisprudence in Univer- See Me Te ONO ses ea ert oo 4 70's Mad shag» pin ¥Po0 & Sie pp Ahan eh 157 Researches on Radiant Heat.—Fifth Memoir. Contributions to Molecular Eerie Sane a yh shiny A yt, TER Ss aha) obs3-pieie = yi 6 bi oy s, eiores oho che, dial shalle » ores din vhs 160 Remarks on Sun Spots. By Balfour Stewart, M.A., F.R.S., Superintendent SCM ENV OUSELVALOLY — savrn ties oh exe Ses 6 Gui wep oe ne te bam oly cies ae 168 Description of an Improved Mercurial Barometer. By James Hicks...... 169 On Mauve or Aniline-purple. By W.H. Perkin, F.C.S. ......eeeeeees 170 On the Functions of the Cerebellum. By William Howship Dickinson, PMS Ose aft in bs: sinibls. 0» bite hire SicaaiGreh Sirilel whieh olsfasaldsipiateinlers abalews otters 177 An Inquiry into Newton’s Rule for the Discovery of Imaginary Roots. By Meee VESteDn FEU 5 cai avis heWcvalloke Ay Goslsva alam alate oe Weasels ap biol sls 179 Description of a Train of Eleven Sulphide-of-Carbon Prisms arranged for Spectrum Analysis. By J.P. Gassiot, FLR.S. oo... eee eee ences 183 The Croonian Lecture.—On the Normal Motions of the Human Eye in rela- tion to Binocular Vision. By Professor Hermann Helmholtz, For. Mem. Lhd: Cap cc adil ies IGE Ora eNO DIC Ce On ara Aiea re SIRE ara aaa ara 186 On the Orders and Genera of Quadratic Forms containing more than three Indeterminates. By H.T. Stephen Smith, M.A., F.R.S. wo... ee ee eee 199 On some Phenomena exhibited by Gun-cotton and Gunpowder under special _ conditions of Exposure to Heat. By F, A. Abel, F.R.S............ 1» 204 On Magnesium. . By Dr. T. L.:Phipson,-FiC.S. ci 0e caves das eae a LE -. vi é Page On the Magnetic Elements and their Secular Variations at Berlin, as observed by Acsiirman:. 005. 6 eee Be PO ERS Saath en es ee Sil 2s On the Action of Chlorine upon Methyl. By C. Schorlemmer, Assistant in the Laboratory of Owens College, Manchester .......cceeeseececcees 225 On the Calculus of Symbols (Fifth Memoir), with Applications to Linear Partial Differential Equations, and the Calculus of Functions. By W. H. Tepassell, “ASB 5 5 e Mee a Shere Ook a ue 0-0 Plate ole © alee een ee 227 Second Part of the Supplement to the two Papers on Mortality published in the Philosophical Transactions in 1820 and 1825. By Benjamin Gompertz, FIRS. 0% oe. aoe « oiiee 0 ois, seis viene nla ss = ee 228 Investigations of the Specific Heat of Solid and Liquid Bodies. By Hermann Kopp, PED... 65. «30 cieweine nos toe 6 0 ain cleh as © 2 © 0\= eles 229 On some Foraminifera from the North Atlantic and Arctic Oceans, includ- ing Davis Strait and Baffin Bay. By W. Kitchen Parker, F.Z.S., and Professor T. Rupert Jones, F.G.S. .........00- os oe oe 0 eae eer 239 Note on the Variations of Density produced by Heat in Mineral Substances. By Ord) die Phipsen, FCS i os eek tee ee eee ay oo ee 240 On the Spectra of some of the Fixed Stars. By W. Huggins, F.R.A.S., and William A. Miller, M.D., LL.D., Treasurer & V.P.R.S. .....c0ccenee 242 A Second Memoir on Skew Surfaces, otherwise Scrolls. By A. Cayley Meo aslo nic,e © Gis a Als cre Ore owls, ose vis @ ave #18 62-0, 07ose ome ele 244 On the Differential Equations which determine the form of the Roots of Algebraic Equations. By George Boole, F.R.S., Professor of Mathe- mataes'in Queen's College, Cork’: o.oo. i is aes oe 245 A Comparison of the most notable Disturbances of the Magnetic Declination in 1858 and 1859 at Kew and Nertschinsk, preceded by a brief Retrospec- tive View of the Progress cf the Investigation into the Laws and Causes of the Magnetic Disturbances. By Major-General Edward Sabine, R.A.., President of the Royal Society... ..0. 0... 60 ae ee cee + cists eee 247 On the degree of uncertainty which Local Attraction, if not allowed for, occasions in the Map of a Country, and in the Mean Figure of the Earth as determined by Geodesy: a Method of obtaining the Mean Figure free from ambiguity by a comparison of the Anglo-Gallic, Russian, and Indian Ares: and Speculations on the Constitution of the Earth’s Crust. By the Venerable J. H. Pratt, Archdeacon of Calcutta .......ccseeeeees 253 Annual Meeting for the Election of Fellows. ......6.00000++s00 oe enie . 276 Description of the Cavern of Bruniquel, and its Organic Contents —Part I. Human Remains. By Professor Richard Owen, F.R.S. ...........065 277 On Soules Binary Quadratic Forms. By H.J. Stephen Smith, M.A., ee Inquiries into the National Dietary. By Dr. E. Smith, F.R.S. .......... 298 On some Varieties in Human Myology. By John Wood, F.R.C.S. ...... 299 Researches on Isomeric Alkaloids. By C. Greville Williams, F.R.S. .... 3803 On the Synchronous Distribution of Temperature over the Earth’s Surface. iy, Henry, G. Hennessy, 1,320; )0,5) ie oleae ole 0.5 at, sole Miele .teiseaee 312 vu Page Experimental Researches on Spontaneous Generation. By Gilbert W. Child, MORON, Leg RE SELES Ut 55 sks ce hea 680 bow dee Gales Court vo. 31d On a Colloid Acid, a Normal Constituent of Human Urine. By William ROR rie aa DI ESE ila. coos «ic ciel vs ood vt sc oce eres ds, « wal bila gee Rae ls Hels 314 Further observations on the Amyloid Substance met with in the Animal Peony. by Lopert M Donnell, M.D. co.cc es sces kobe beseenteenees ol7 Description of a New Mercurial Gasometer and Air-pump. By T. R. Semmemsemcreteee Oh Deeg MGT TIE Bi Sa00 Vino abe a bln Saeiald aca gied vs. Seivleis wlaudle Ole 321 On the Distal Communication of the Blood-vessels with the Lymphaties ; and on a Diaplasmatic System of Vessels. By Thomas Albert Carter, nas ia. 2g aoe, orale & ob, cha aretsinscokels refs: 3 boseee sob eee penne Aérial Tides. By Pliny Earle Chase, A.M.,S.P.A.S...... slip ait Ra 329 On the Microscopical Structure of Meteorites. By H.C. Sorby, F.R.S. .. 333 On the Functions of the Cerebellum. By W. H. Dickinson, M.D......... 334 On the Properties of Silicie Acid and other analogous Colloidal Substances. rpms Graham, PRS. cece ee tence es ee ned ines Hee boee sects 335 Researches on the Colouring-matters derived from Coal-tar.—III. Diphenyl- eens. W. Homann, LL.D ., FoR: 5 i0 occu aise 00 sess ges ote 341 A Table of the Mean Declination of the Magnet in each Decade from January 1858 to December 1863, derived from the Observations made at the Magnetic Observatory at Lisbon ; showing the Annual Variation, or Semiannual Inequality to which that element is subject. Drawn up by the Superintendent of the Lisbon Observatory, Senhor da Silveira...... 347 On Organic Substances artificially formed from Albumen. By Alfred H. Sn LES 2 Se lg ee OR ee ee ere pene eh ee ee mr ie 350 _ On the Reduction and Oxidation of the Colouring-matter of the Blood. By Pence Are CGy Eee ae as lessen vc ele vasivs cavuscee tens vanes 855 Further Inquiries concerning the Laws and Operation of Electrical Force. erent SCiVe EIRTTIS, EO EU.O. (7. seis s'v'e 'e ee e'e e's c'e Valve ee ce wee e eee ne 364 On a New Class of Compounds in which Nitrogen is substituted for ameter cba Peter Gress. coe sst feeiis Cees ob eels Wee tite Gare bale 375 New Observations upon the Minute Anatomy of the Papille of the Frog’s #onene.. By, Lionel S. Beale, M.B., F.R.S,, FLR.C,P. oe. cere eens 384 Indications of the Paths taken by the Nerve-currents as they traverse the caudate Nerve-cells of the Spinal Cord and Encephalon. By Lionel S. Peeeemcte oh be. H.R. (Plate IM.) . occas ots s caine ba Nees 386 On the Physical Constitution and Relations of Musical Chords. By Puen Se PNiG Bb. BC ES ccc wats ee yt awe wees habe ee eens 6 392 On the Temperament of Musical Instruments with Fixed Tones. By Ree meee PUNT, Heb ciey EC yes e sieiscre ay sie cys peels vincca eae ae ait 404 On the Calculus of Symbols.—Fourth Memoir. With Applications to the Theory of Non-Linear Differential Equations. By W.H. L. Russell, a Lorian IE as SU An Ae I Oe ee OR SRR ce te ntie o b On the Calculus of Symbols.—Fifth Memoir. With Jey to Linear Partial Differential Equations, and the Calculus of Functions. By W. mlb, Housel, (A Bite tances eas ern elena Ra itaae Vill Page Comparison of Mr. De la Rue’s and Padre Secchi’s Eclipse Photographs. iby Warren De la. Rue, PBB... cc ccec cn ceverse sce ss es see 442 On Drops. By Frederick Guthrie, Professor of Chemistry and Physics at the Royal College, Maurrtaus 1707.2... a ee alo e's cle ae ee 0 2s sit On Drops.—Part II. By Frederick Guthrie, Professor of Chemistry and Physics at the Royal College, Mauritius. (Plates IV. & V.).......... 457 On the Chemical Constitution of Reichenbach’s Creosote-—Preliminary Notice. By Hugo Muller, Ph.D. a. os. og anne. sale: o = 484 Remarks on the Colouring-matters derived from Coal-tar.—No. IV. Phe- nyltolylamine: ‘By ‘A. ‘W. Hofmann, LL.D., F.R.S.. . 5... .252eeeeeeee 485 On the Spectra of some of the Nebule. By W. Huggins, F.R.A.S. ;—a Supplement to the Paper “On the Spectra of some of the Fixed Stars,” by W. Huggins and W. A. Miller, M.D., Treas. and V.P.R.S........... 492 On the Composition of Sea Water in different Parts of the Ocean. By Dr. — George Forchhammer, Professor in the University of Copenhagen. .493 & 494 Anniversary Meeting :— Aeport of Auditors... oc. was ees ow oe fe se sss «ee 494 aiaat, of Fellows decéased, Qe... v.00... os ec ae ‘Se ons 2a 495 ——_—_—_— elected since last Anniversary ..cs.cceececcseceeee 495 First Report of the Scientific Relief Committee ............. ee eee A495 Address of the President) 0.5 o.048 vdien ld ows Shek sew eee ae 497 Presentation of the Medals. .2..2.4. 6sas 0d des cd ow aeete ee Cee 505 Election: of Council sand: Officers) 4ic4:50).:05) s00 bs Cole 3 oa 517 Pynancial Statement oi... ec os wees as 5 os oe ee ee oe eee 518 & 519 Changes and present state of the number of Fellows .............. 520 Researches on certain Ethylphosphates. By Arthur Herbert Church, M.A. Oxon., Professor of Chemistry, Royal Agricultural College, Cirencester. . 520 A Dynamical Theory of the Electromagnetic Field. By Professor J. Clerk Wilaxwell, ECES. . cc 0 ac ces esc ce eee a ean cle sig Oh slsine 5th ee 531 On the production of Diabetes artificially in Animals by the external use of @old:), By Henry Bence Jones, M.D., F.RS.. ccies)d 0,34 0 ote eee 537 On the Action of Chloride of Iodine upon Organic Bodies. By Maxwell Simpson, M.B., FBS. 2... eee os Pons ad See bn elblWls § ool oteie Olea eee aa 540 On Fermat’s Theorem of the Polygonal Numbers, with Supplement. By the Right Hon. Sir Frederick Pollock, FURS. 0......4. 0.222252 tere oe 542 On the Structure and Affinities of Hozoon Canadense. In a Letter to the President. By W. 5. Carpenter, M_D., RS... : sae eee 545 On the Functions of the Foetal Liver and Intestines. By Robert James Lee, B.A. Cantab., Fellow of the Cambridge Philosophical Society ..., 549 Completion of the Preliminary Survey of Spitzbergen, undertaken by the Swedish Government with the view of ascertaining the practicability of the Measurement of an Arc of the Meridian. In a Letter addressed to Major-General Sabine by Captain C. Skogman, of the Royal Swedish Navy: dated Stockholm, Nov. 21, 1864. (Plate VI.)........... tr On the Sextactic Points of a Plane Curve. By A. Cayley, F.R.S......... 553 1x Page On a Method of Meteorological Registration of the Chemical Action of ‘ Total Daylight. By Henry E. Roscoe, B.A., FLR.S. 20... cece eee eee 555 Obituary Notices of Deceased Fellows :— A IATLNTESE. CIOTRUIGT AEE A nee tes Oe ag SEM i iteamiae Ce OSM MAY COOPER. 5). secre onal mica qiaiege els. alate u16 sapere vince + oeeGiarw i Ji@e Te LEN ae ar ire ne Para No RAE Ge sMels nionate ct memes oa iii emma rR nN aLeTe VESEY oie: caval e asta elatcle aia! ees bake dhal'a 01's Wisral asta a ak eg iil Bape MMU ENE UNV cscs 5. a te cine acs) pe ai eats ena ah seis « wal oka “ancieieard « Muaparer omngale Vv Perpre lim caOe eer is ea hank Has Beas ye eld alee oie kluc, es learn nein V eee EO eS urate whe nie Oy gles b's wise ae ware eae age vi Rea xamiral John Washington . 2.6.5.6 08 see gee eee wees ecemes vil Reese MaMcmete DESPLOEZ ie) cus sds. ais cise Cm eles ea sins oe a em iie a e swe viii Peebrearcum bit S@Merhely voc. hs ca cee esa nh «ccc ees deletes wy et we sas 1x Saeed Christian Bumilrer fe. cosa. cent oO Mem sinew ds nfo Xvi SSCS Bee von oid aioe & wai ayecaes wjaei a's e's ao we gestae ane 561 ERRATA. Page 153, for 8S. W. Hittorf read J. W. Hittorf. Pages 201 & 202. The words “If a quadratic form........ Phil. Trans. vol. exliii. p. 481)” should have been printed as a foot-note in explanation of the term “index of inertia.” Page 330, for Flangergues read Flaugergues. Errata in Opituary (Vou. XIT.). Page xxxvi, line 8 from bottom, for Poisson read Brisson. ? In this Volume the following pages are to be cancelled: xxxvili, line 4 from top, for son read husband. NOTICE TO THE BINDER. Pages 83, 227 bo (eae , 6406 a ee 100-00 These analyses were performed on specimens prepared at different times. This acid is soluble in water, alcohol, and ether. It has a pure acid taste. It melts at about 135° Cent., and at a higher temperature suffers decompo- sition. The free acid gives an abundant white precipitate with acetate of lead, soluble in strong acetic acid. It is not precipitated by lime-water. The neutralized acid yields a bulky white precipitate with corrosive subli- mate, and a pale brown with perchloride of iron. Copper salts give a bluish-white precipitate. Chloride of barium is not affected. The forma- tion of this acid may be explained by the following equation :— C, H, 0, Cy, +2 a 0,)4+4H0=0,,5° | O,,+2NH,. I have also analyzed the silver-salt of this acid. As it suffers decompo- sition at the temperature of boiling water, I was obliged to effect its desiccation by placing it 2m vacuo over sulphuric acid. Itis slightly soluble in water. ‘The numbersit yielded on analysis agree very well with the formula H Cro ag, fOr aT a ; Theory. Experiment. ee fF Il. CT Ua G67 16°61 See 47,5 Bs 1:39 aes | OR ay, — = Ag,” 252255461 AueDs 60-67 100-00 1868. | Oxy-radicals of Di- and Tri-atomic Alcohols. AZ The ether of this acid is readily prepared by passing hydrochloric acid gas through its solution in absolute alcohol. On evaporating the alcohol an oily residue was obtained, which was washed with a solution of car- bonate of soda and distilled. ‘The greater portion passed over between 295° and 300° Cent. The analysis of this portion gave numbers which indicate the formula C,, (CH ) 0,,:— 4 5/2 Theory. Experiment. Le eS I II. Crt... 02:94 54°61 54°32 ec Ze 8°09 6°91 Ore Oo 22 —. — 100-00 This ether suffers partial decomposition during distillation; hence the discrepancy between the theoretical and experimental numbers in the first analysis. ‘The specimen which served for the second was not distilled at all, but simply purified by solution in ether. It is a colourless neutral oil with a very acrid taste. It is somewhat soluble in water. Heated with solid potash it yields alcohol, and the acid is regenerated. I regret to say I have not succeeded in obtaining the cyanide (C, H, O, Cy,), which gene- rates this acid, in a state of purity. The compositions of the ether and silver-salt of this acid prove it to be bibasic. It is highly probable that the basicity of an acid produced in this way depends on the atomicity of the radical in the cyanide which generates it. If this be so, the cyanides of the mono-, di- and tri-atomic radicals of the glycols and glycerines should then yield by decomposition with potash respectively mono-, bi- and tri-basicacids. If it would he pos- sible to prepare the acid C, H, O, from the cyanide C, H, 0, Cy, it would be interesting to examine its bearing on this point. Would it prove mono- basic or bibasic ? This acid bears the same relation to pyrotartaric that malic bears to succinic acid. :— Succinic acid ..,. C, H, O, Pyrotartaric acid .... C,,H,0O, Malice acid'’.!. 5.2. C, H, O,, New acide a wevsteureny quelle, @ 8 10 It has the composition of the homologue of malic acid. Whether it is actually the homologue of that acid or not I cannot yet say. I propose to call it oxy-pyrotartaric acid. Formulated according to the carbonic acid type it is thus written :— 2HO, C, H, 0, 1é oo O,. We may now, I think, safely answer in the affirmative the questions put at the commencement of this Paper. The cyanides of the oxy-radicals of the di- and tri-atomic alcohols can be formed, and the action of the potash on these cyanides is analogous to its action on the ordinary cyanides. 48 Mr. Airy—Analysis of 177 Magnetic Storms. [Dee.17, The foregoing research was finished many months ago, but I delayed publishing it in the hope of being able to announce at the same time the formation of lactic acid by a similar process. I find, however, from the ‘Annalen der Chemie und Pharmacie’ of last month that I have been anticipated by Wislicenus, who has succeeded in forming lactic acid in the manner I have just described, _ December 17, 1863. Major-General SABINE, President, in the Chair. The following communications were read :— I. “First Analysis of 177 Magnetic Storms, registered by the Magnetic Instruments in the Royal Observatory, Greenwich, from 1841 to 1857.” By Gzorer Bippeit Airy, Astronomer Royal. Received November 28, 1863. (Abstract.) The author first refers to his paper in the Philosophical Transactions, 1863, “On the Diurnal Inequalities of Terrestrial Magnetism as deduced from Observations made at the Royal Observatory, Greenwich, from 1841 to 1857.’ These results were obtained by excluding the observations of certain days of great magnetic disturbance ; it is the object of the present paper to investigate the results which can be deduced from these omitted days. The author states his reasons for departing from methods of reduction which have been extensively used, insisting particularly on the necessity of treating every magnetic storm as a coherent whole. And he thinks that our attention ought to be given, in the first instance, to the devising of methods by which the complicated registers of each storm, separately con- sidered, can be rendered manageable; and in the next place, to the discus- sion of the laws of disturbance which they may aid to reveal to us, and to the ascertaining of their effects on the general means in which they ought to be included. The author then describes the numerical process (of very simple cha- racter) by which, when the photographic ordinates have been converted into numbers, any storm can be separated into two parts, one consisting of waves of long period, and the other consisting of irregularities of much “more rapid recurrence. He uses the term “ Fluctuation ”’ in a technical sense, to denote the area of a wave-curve between the limits at which the wave-ordinate vanishes. 'The Waves, Fluctuations, and Irregularities, as inferred from separate treatment of each storm, constitute the materials from which the further results of the paper are derived. Table I. exhibits the Algebraic Sum of Fluctuations for each storm, with the Algebraic Mean of Disturbances, and Tables II. and III. exhtbit the 1863.] = Mr. Airy—Analysis of 177 Magnetic Storms. 49 Aggregate or Mean for each year, and the Aggregate for the seventeen years. The Aggregate for the Northerly Force is negative in every year. That for the Westerly Force is on the whole negative ; the combination of the two indicates that the mean force is directed about 10° to the east of south. That for the Nadir Force appears negative, but its existence is not certain. Some peculiarities of the numbers of waves with different signs are then pointed out. For Westerly Force and also for Nadir Force, the numbers of + waves and —waves are not very unequal; but for Northerly Force there are 177-++ waves and 277—waves. In Nadir Force it is almost an even chance whether a storm begins with a + wave or with a —wave; and the same with regard to its ending; in Westerly Force the chances at beginning and ending are somewhat in favour of a +-wave; butin Northerly Force two storms out of three begin with a —wave, and ten storms out of eleven end with a —wave. | The beginnings and ends of the storms are also arranged by numeration of the combination of waves of different character in the different elements (as, for instance, Westerly Force + with Northerly Force —, Northerly force + with Nadir Force +, &c.); but no certain result appears to follow, except what might be expected from the special preponderances mentioned above, leaving the relative numbers of the combinations a matter of chance in other respects. Tables IV., V., VI. exhibit the Absolute Aggregates of Fluctuations and Absolute Means of Disturbances without regard to sign. In interpreting these it is remarked that the large — mean force in the northerly direction necessarily increases the Aggregate and diminishes the Number of Waves. With probable fair allowance for this, it appears that the Numbers of Waves are sensibly equal, that the Sums of Fluctuations are sensibly equal, and that the Means of Disturbances are sensibly equal for Westerly Force and for Northerly Force. But the Number of Waves for Nadir Force is less than half that for the other forces ; while the Sum of Fluctuations is almost three times as great as that for the others, and the Mean of Disturbances almost three times as great. Attempts are made to compare the epochs of the waves in the a directions, but no certain result is obtained. : Tables VII., VIII., IX. exhibit for each storm, and for each year, and © for the whole period the Number of Irregularities, the Absolute Sum of Irregularities, and the Mean Irregularity. It appears that the value of Mean Irregularity is almost exactly the same in the three directions, that the number of irregularities is almost exactly the same in Westerly Force and in Northerly Force, but that the number in Nadir Force is almost exactly half of the others. It is certain that the times of Irregularities in the Westerly and Northerly directions do not coincide. There appears some reason to think that Nadir Irregularities frequently occur between Westerly Irregularities. In Table X. the Aggregates of Fluctuations and Irregularities are arranged 50. Mr. Airy—Analysis of 177 Magnetic Storms. [Dec. 17, by months, but no certain conclusions follow. In Table XI. the Wave- disturbances and the Irregularities are arranged by hours ; for the Wave- disturbances results are obtained which may be compared with those of previous investigators; in Table XII. it is shown that these may be repre- sented by a general tendency of wave-disturbances, different at different hours, which general tendency is itself subject to considerable variations. For the Irregularities it is found that the coefficient is largest in the hours at which storms are most frequent. It does not appear that any sensible correction is required to the Diurnal Inequalities of the former paper on account of these disturbed days. The author then treats of the physical inference from these numerical conclusions. And in the first place he states his strong opinion that it is impossible to explain the disturbances by the supposition of definite galvanic currents or definite magnets suddenly produced in any locality whatever. The absolute want of simultaneity (especially in the Irregu- larities), and the great difference of numbers between the Waves and Irregu- larities for the Nadir Force (in which the Irregularities are just as strongly marked as in the Westerly and Northerly, and the Wave-disturbances are much more strongly marked), and those for the other Forces, appear fatal to this. It is then suggested that the relations of the forces found from the investigations above, bear a very close resemblance to what might be ex- pected if we conceived a fluid (to which for facility of language the name ‘Magnetic Ether” is given) in proximity to the earth, to be subject to” occasional currents produced by some action or cessation of action of the sun, which currents are liable to interruptions or perversions of the same kind as those in air and water. He shows that in air and in water the general type of irregular disturbance is travelling circular forms, sometimes with radial currents, but more frequently with tangential currents, some- times with increase of vertical pressure in the centre, but more frequently with decrease of vertical pressure; and in considering the phenomena which such travelling forms would present to a being over whom they travelled, he thinks that the magnetic phenomena would be in great measure imitated. The author then remarks that observations at five or six observa- tories, spread over a space less than the continent of Europe, would pro- bably suffice to decide on these points. He would prefer self-registering apparatus, provided that its zeros be duly checked by eye-observations, and that the adjustments of light give sufficient strength to the traces to make them visible in the most violent motions of the magnet. For primary reduction he suggests the use of the method adopted in this paper, with such small modifications as experience may suggest. Broo Roy-Soc Vol. XII PUL | a J ee ee ae Nov? 21%1863. 1 ™ _ | | emt. ee | | | | 5G Hl i a 25: | | | 29 | | | Ee La Of Positive (i) Ay ‘tt i] y' Nees INS) | | | i ! | | Ke YE ' {' AA / v WW, 4 y) fa chy > {-- | Positive if} 1 i] | i j | { St 6h. ZA aes TAI OTFACES Oy e OITHHAES TALS | | Negative i 1863.] Mr. B. Stewart—Squalls of 30th Oct. and 21st. Nov. 51 II. “On the Sudden Squalls of 30th October and 21st November 1863.” By Barour Stewart, M.A., F.R.S., Superintendent of the Kew Observatory. Received December 10, 1863. The 30th of October was windy throughout, and in the afternoon there was a very violent squall. The barograph at the Kew Observatory, as will be seen from Plate I. which accompanies this communication, records a very rapid fall in the pressure of the atmosphere, which appears to have reached its lowest point about 3° 9" p.m., G.M.T. At this moment, from some cause, possibly a very violent gust of wind, the gas-lights in the room which contained the barograph went out, and were again lit in a quarter of an hour. During this interval the barometer had risen considerably ; and indeed the baro- graph curve, although unfortunately incomplete, presents the appearance of an extremely rapid rise. It may therefore perhaps be supposed that there was a very sudden increase of pressure accompanied with a violent gust of wind at the moment when the gas went out, which would be about 3" 9™ p.m., as above stated. In a paper communicated to the Royal Society on November 23, Mr. Glaisher has remarked that at Greenwich the time of maximum de- pression of the barometer was 3° 30™ p.m., while at the Radcliffe Observa- tory, Oxford, it was 2” 30" p.m. This would indicate a progress of the storm from west to east, in accordance with which Kew should be some- what before Greenwich as regards the time of maximum depression. This anticipation is therefore confirmed by the record of the Kew barograph which has been given above. The indications of the Kew self-recording electrometer during this squall show that about 2" 39™ p.m. the electricity of the air, which before that time had been very slightly negative, became rapidly positive, then quickly crossed to negative, became positive again, and once more crossed to nega- tive about 3" 3™ p.m., recrossing again from strong negative about 3" 51™ p.M., after which it settled down into somewhat strong positive. It is well, however, to state (what may also be seen from Plate I.) that the variations of this instrument between 3" 3™ p.m. and 3" 51™ p.m. were so rapid as not to be well impressed upon the paper. At Kew there is often occasion to move the dome, so that we cannot well have an instrument which records continuously the direction of the wind ; but we have a Robinson’s anemometer, which records the space tra- versed by the wind, and thus enables us to find its velocity from hour to hour, though not perhaps from moment to moment. A reference to Plate I. will show an increase in the average velocity of the wind during this squall. A somewhat similar squall took place in the afternoon of Saturday, November 21st, about 4 o’clock. In this case the Kew barograph presents a rapid (and, in the curve, 52 . Mr. Spottiswoode—LEquations of Rotation Re [1863, ragged) fall of the atmospheric pressure, which reached its minimum about 4» 45" p.m. There was then a very abrupt and nearly perpendicular rise of about five hundredths of an inch of pressure, or rather less, after which the rise still went on, but only more gradually. Through the kindness of the Rev. R. Main, of the Radcliffe Observa- tory, I have been favoured with a copy of the trace afforded by the Oxford barograph during this squall, in which there appears a very sudden rise of nearly the same extent as that at Kew, but which took place about four o’clock, and therefore, as on the previous occasion, somewhat sooner than at Kew. This change of pressure at Oxford was accompanied by a very rapid fall of temperature of about 8° Fahr. The minimum atmospheric pressure at Kew was 29°52 inches, while at Oxford it was 29°28 inches. It will be seen from the Plate that at Kew the electricity of the air fell rapidly from positive to negative about 4" 30™ p.m., and afterwards fluctu- ated a good deal, remaining, however, generally negative until 5" 22™ p.M., when it rose rapidly to positive. We see also from the Plate that there was an increase in the average velocity of the wind at Kew during the continuance of this squall. To conclude, it would appear that in these two squalls there was in both cases an exceedingly rapid rise of the barometer from its minimum both at Oxford and at Kew, this taking place somewhat sooner at the former place than at the latter; and that in both cases the air at Kew remained negatively electrified during the continuance of the squall, while the average velocity of the wind was also somewhat increased. . The Society then adjourned over the Christmas recess to Thursday January 7, 1864. “On the Equations of Rotation of a Solid Body about a Fixed | Point.” By Witi1am Srortiswoops, M.A., F.R.S., &c. Received March 21, 1863,* In treating the equations of rotation of a solid body about a fixed point, it is usual to employ the principal axes of the body as the moving system of coordinates. Cases, however, occur in which it is advisable to employ other systems; and the object of the present paper is to develope the funda- mental formulee of transformation and integration for any system. Adopt- ing the usual notation in all respects, excepting a change of sign in the quantities F, G, H, which will facilitate transformations hereafter to be made, let A=im(y? +2"), B= Xm(z2?+2’), C= 2m(2*+y’), —F=myz, —G= Inez, —H=Zmzy ; _ .* Read April 16, 1863: see abstract, vol. xii, p. 523, 1863. ] of a Solid Body about a Fixed Point. - 53 and if p, g, r represent the components of the angular velocity resolved about the axes fixed in the body, then, as is well known, the equations of motion take the form d , 4 a2 nS 46S 7 =F") + (B—O)gr + Hrp—Gpg, | | +H2 ne tp =a —p*)—Hgr+(C—A)rp+Fpq, 7 (1) — 2 +Eo Se esses me To obtain the two general integrals of this system : multiplying the equa- tions (1) by p, g, 7, respectively adding and integrating, we have for the first integral Ap? +Bq’?+Cr?42(For+Grp+Hpqg)=h, . . . «° (2) where 4 is an arbitrary constant. Again, multiplying (1) by Ap +Hg +Gr, Hp+ Bq +F7, Gp+Fq +Cr, respectively adding and integrating, we have for the second integral (Ap+Hq+Gr)’?+ (Hp+Bq+Fr)’+ (Gp+Fq+Cr)?=h*, . (8) where #**is another arbitrary constant. This equation may, however, be transformed into a more convenient form as follows: writing, as usual, . 4=BC—F, 8=CA—-G, @=AB—H2, V=|AHG $=GH—AF, &=HF—BG, %#=FG—CH, HB Fi. (4) A+B+C=S, GFC and bearing in mind the inverse system, viz VA=8C-F, VB=CA-G’, VC=AB—H? VF=GH-AsS, VG=HI-BG, VH= ark CH. - ( 2+8+€=8, we may transform (3) into the following form :— (AS—%B—€)p’?+2(FS + H)qr . +(BS—€C—A)¢°+2(GS + &)rp rete 00) +(CS—@ —%)r?+ 2(HS+ B)pg=/’, which in virtue of (2) becomes — (A-S)p? + (B—-S) P+ (C— SH) +2(For + Grp + Bpg)="—SA. (7) This form of the integral is very closely allied with the inverse or ae form of the first integral (2), and is the one used below. In order to find the third integral, we must find two of the variables in terms of the third by means of (2) and (7), and substitute in the corre- 54 Mr. Spottiswoode—Equations of Rotation [1868. sponding equation of motion. The most elegant method of effecting this is to transform (2) and (7) simultaneously into their canonical forms. If ep y a Bn a, (ee Ya be the coefficients of transformation, and if [] be the determinant formed by them, the terms involving the products of the variables will be destroyed by the conditions (A wit oe asp 6,8, Cy ¥:7,J=0; > (A ee Moe hey ae a2, )=0, (AS Eh eee, eee (A—S... F.... 068, B.C ny.) =, (A—- 3... Ly nL um) =0, (A-S... F... Lae, 0,16 6,6,)=0, from the last two of which we have Byy.—Boy : Pay —By.+ By, +B,y > =Aa+Ha,+Ga,=(A—9)a+ Ba, + fra, i el Hee , : Ha+Ba, +Fo,: Ho+ (GB-—S)a,+ Sa, :Ga+Fa, +Ca,: Gat ffa,+(@—S)a,; whence, 0 being a quantity to be determined, A—-S-—Ao, B dO, (or — G6 = %H —H6, 6@-S-Bo, £F —F@ i 010) 6 —G), £F —Fo, €—S —Ca Proceeding to develope this expression, we have the term independent of @ =V'*—(BC+CA+ 9B)S+ &—2* —(? +62 +B)S =V’—SSV. The coefficient of —é =A{VA—(B+0)5+ 5'}+H(VH+88)+G(VG64+ GS) = pee =V(A?+1?+6)+VS +V (H#?+B?+C’)+VS +V(@+F?+C?)+VS =V {A?+ B?+C?+3(BC+CA+ AB)—F’—G’— H?} =V (84+). 1863. | of a Solid Body about a Fixed Point. 55 The coefficient of — 6° =V\.- Hence (dividing throughout by. V7) (10) becomes 6° + 2.86?+ (S?+$)0+SS— V =0; or, what is the same thing, (0+S)’—S(6+S8)?+4(0+8)-V=0; . . (11) or, as it may also be written, Sees), EF e = H, B= @-ES), 5 G, F, C—(64+S) It will be seen by reference to (9) that the values of 6 determined by this equation are equal to the ratios of the coefficients of the squares of the new variables respectively in the equivalents of (2) and (7). The coefficients of transformation are nine in number ; if therefore to the six equations of condition (8) we add three more, the system will be determinate. Let three new conditions be (A..F... 3a aa, )°=1, (A...F...16 6,6.) =1, e e e e e e (12) (A.B Dy N1Y%2) =1 3 then the variable terms of (2) will take the form of the sum of three squares, and the roots of (11) will be the coefficients of the transformed expression for (7). Or, if 0, 0,, 0, be the roots of (11), (2) and (7) take the forms poi tar +7r=h, 0p°+6,9¢,°+6,772=h—SA. } Rhee Cy In order to determine the values of the coefficients of transformation &, a,, a, we have from (9), (A—S—Abd)a+ H—HO)a+ (G—Ge)a,=0, (4 — H0)a+ (8—-S—BO)a+ (f—FO)a,=0, ai et al By (G&—G0)a+ (F—FO)e+(C€-—S—Coe)a,=0; from the last two of which a: BC -(B+G)S+ S°—(BC+C€B—B+CSH)+ BCE? el — 25F 9 —F° =a:VA+4S+(B+C44+ BS +CC +2F F)0+ Ae =a2:VA+GS+(2V —Hh-G&—-AA+84)04+ 90 =a: VA+AS+(V +84) + A? =a:V(A+60)4+4(S+80+ 6") ; or, writing for brevity $+589+¢=T, 56 Mr. Spottiswoode—Hquations of Rotation [1863. the expression becomes a:V(A+0)+T@ =a: SF &—(FG+GF)0+ FGO ~ CHF + HS + (CH +HS)9—CHe =a: VH+9S+S8H0+ He =a,: VH+TH =a,: VG+TG, whence the system aot, 2a, ah | =V(A+0)+TA = VH +TH = VG +TG :VH +7: V(B+0)4TB : VF 4TH er - C5) : VG +T& : VF +THF : V(C+0)+TE, with similar expressions for 6, 8,; 8,3 > ¥1> Y2. Obtained by writing 0,, T, ; 6,, T, respectively for 0, T. Returning to the equations of motion (1), and transforming by the formule eer Pi+P “ry "yp Q=4,),+2,9,+y,") . -& oe (16) r=0,D, +29, +77 we have (Aa+Ha,+Ge,)p', =[—F(a,—a,*) + (B—C)a,e,+ Ha,a—Gaa,] p,? | NE + (AB-+H6,+G8,)q',+[—F(6?—B,) + (B—C)6,6,+HB,8—G6B,]9,2 + (Ay+Ay,+Gy,)7, +L Fi’ —y,.") + (B-C)y,7.+ 4 yy—Gyy ln? +L -2FC.n—Biy.) + (B—©) 6.7.48.) +H(6,y+By.) -—G(By,+B.ylar, aii [ at 2F (y,2, ro Y2%,) a6 (B Be C) (77%, ch Ya) — AA (y,44+ y%,)— G(ya, + ,4) |p, +[—2F(«,6,—2,8,) + (B—C) (4,8, +2,8,) +H(4,6+48,) —G(a6,+2,C)1p.q -=[o,(Ha+Ba,+Fe,) —a,(Ga+Fa,+Ca,)]p,? (17) +[6,(H6 + BB, +F6,)—6,(G6+ FB, + CB,)]¢,? | +[y{Hy+By +¥Fy.)—y,(Gy+Fy,+Cy jin? - +8, Hy+By,+Fy.)—B\(Gy+Fy+Cy,) + y,(HB+ BG,+FB,)—y,(GB+F6+C£6,)]q7, +[y,(Ha+Ba,+Fa,)—y,(Ga+Fa,+Ca,) +a,(Hy+By,+Fy,)—2,(Gy+Fy,+Cy,)]7,p, + (#,(H8+ BB, +FB,)—a,(G6+FS,+C£,) +6,(Ha+Ba,+Fa,)—B;(Ga+Fa,+Ca,)]p,9,, 1863.] of a Solid Body about a Fixed Point. - 57 with similar expressions for the two other equations. Multiplying the system so formed by y, y,, y, respectively and adding, the coefficients of © p',, 7’, will vanish, and that of 7!, will =1 in virtue of (12); and as regards the right-hand side of the equation, the coefficient of p,* Aa+tHa,+Ga,, a, y Ha+Bea,+Fa,, «4, y, GatFa,+Ca,, a. Yo» which, omitting common factors, (S+O)A+A+(S+0)0, VA+TA+V9, VA+T,A+V0, | — — (S+6)H+®, VH+TB, VH+T.® (S+0)G+G, VG+T& VG+T,G eae, VH+T, ® | +V9| VH+T(S+0)H+B VG+T& VG+T,G VG+T,G(S+0)G+& (S+0)H+% VH+TH (S+0)G+G VG+TG ={(S+6)0V(T,—T) + Va(V —T,(S+6))+Vo,(T(S+0)—V )} (H&—FE) =V(0,—0){T(S+)e—V}(HG:—WG). T(S-+0)—V =(S+0)(°+80+ 8)—V=(S+0){(S+6—(S+0) + 8}—-V =(S+6)?—S(S+6)?+S(S+0)—V =(). Hence, finally, the coefficient of p,? vanishes. So likewise the coefficient of q,” AB+HB,+G8, 6 y | =0. H6+B,+F p, Bay: GB+F 6,+CB, B, xy AytHy,+Gy, y¥ y .|=0. Hyt+By,+Fy n vn GytFy+Cy, wa. Ys Similarly the coefficients of g, r,, and 7, p, will be found to vanish; and lastly, the coefficient of p, ¢, =a {A (B,y.—P.71) + H(6,y—6y.) + G(Py,—B,y)} + a, {H(8,y,—6,7,) +B (B.y—By.) + F (By,—By)} + 4,16 (Byy,— Poy.) + F (Boy— By.) +C(By,— By) } —B {A(1%,—720,) + H(y,4—ya,) + G(ya,—7,.2)} —B,,{H(y, ot Yo) +B (y.e— ya.) a0 F(ya,— ya) * ee 6.6 (ym, a ¥20,) apts (y.e— YO) +C (ya,— Me)ts +V6, —— — And that of 7,’, 58 Mr. Spottiswoode—LEquations of Rotation 1863. which, by reference to (9), may be transformed into rm) {(Ae+ Ha, +Ga,)?+ (Ha + Ba, + Fa,)?+ (Ga+Fa,+Cza,)’ —(AB+H6,+G6,)+ (e+ BG, + Fe,)’+(G6+ FB, +Cp,)"} = {(Ae?+Ba2+Ca,? + 2Fa,, + 2Ge,0 + 2Haa,)S —(AP’?+BB?2+Cp,?+ 2FB,6,+2G68,6+ 2H66, )S +(A—&)(@— f°) + (B—S)(a—B,7) + (€—8)(a,’—B,”) + 2FF(4,% —B,B,) + 2Gi(a,2 — 8,0) + 27)(ae, —BB,)} ; in which the coefficient of S vanishes in virtue of (12); so that the coeffi- cient of p,, 4g, =D {(A—2, B—S, €—3, Ff, G& HY, a, «,) —(a—8, B—5S, €—S, F, & HVA, By 6.) ; but, by (12), (A-S, B-S, C-S, F, & HYa2,2,)’=8, (A—, B—S, C5, F, G, HYPS,6,)=6,. Hence the coefficient in question = [1 (@—8,), 920% ie eee ee and the equations of motion become Pr =O 8—-8,) ar, g¢=f10,.—)r,9,\ .°. . ee =O 6 —9)r9,. To find the value of [ in terms of A, B, C, F, G, H, we have from (12) Aa+He,+Ge,= [1 -\(6,y,—6.7,)s AB+H6,+G66,= O-My.e,—7.%,)s Ay+ By, +-Gy,=T]~(¢,6,—2,0,), Ga+Ba,+Fa,=(-(6,y —6 y,), HB+BS,+F6,= 0 -y.e —y 4,), Hy+ By, +Fy,= 74.8 —@ 6), Ha+ Fa,+Ca,= 1-6 y,—8,y), GP+FR,+C68,= Oy —- nN), GytFy,+Cy,= O7(¢2,—4,8). And forming the determinant of each side of this system, there results Vee, or V = [ape s ~ e ° e e e e ® (20) — 863.) of a Solid Body about a Fixed Point. | 59 whence the equations of motion (19) become p'=V 391 — 92) Ms Se Or, ee ee a. | OT) r!, =V-2(0—6,) p, g,- In order to compare these results with the ordinary known form, we must make = 0; C—i; H=0, p,=A"p, g,= Bq, = C27; which values reduce (13) to the following : (A*p)?+ (B*g?+ (Cr)? =A, | —(B+C)Ap?—(C+ A)By?—(A+B)Cr=—ShA; which last is equivalent to (A—S)(A*p) + (B—S(B¥y)?-+ (C—S) (C*r)*=h—8h, or A(A%p)?*+ B(B¥q)?+C(C*r =? Also, on the same supposition, V=ABC, 9=—(B+C), 6,=—(C+A), 0,=—(A+B), which, when substituted in the above, give Ap"'=(ABC) 3(B—C)B*C2qr, Bg’=..., Ch'=..., : Ap'=(B—C)gr, By'=(C—A)rp, Cr'=(A—B)pgq, as usual. | It remains only to determine the absolute values of the coefficients of transformation, the ratios of which are given in (15). For this purpose let V(A+0,)+T,4=4, VE+T F=F,, VB+6)+TB=B, VE+TG=G- . . . (22) V(C+0,)+T€=€, VH+T,H=¥.. Then, from (15), a 3 Ro &, ; (A... ... 14,9, ,)? (A... 148,38, F,) (A. LF Coy we 200 Ee cee (AH. Uae GY (A an %F,)? A... CG FC,)” Gi, , LAMP TREES Go LOREY VOL, XIII. i i 60 Mr. Spottiswoode— Equations of Rotation 1863. From these relations it follows that ie 8 C,—s,=0, Cr ),—-A,F,=0, C,A,— G,."=0, oF, B,G,=0, | : A,33,—1,= 0, dD to C,4= 0, which relations may be also verified as follows:— _ &A.—-A,S = (VE+T, G)(VE+TH)—-(VA+T,A+ V6,)(VE+T SF) = VF+VT,(GH +HG —Agf—FA)+T, VF—V0,(VF+T,S) ViVF_-T,(SHF +SF)4T2 F—Vo.F—Vg+St FG; . (23) Since H&+Bs+FC—0, and. (0+S)T—V=0, or 6T=V —ST. Hence &,9,—-A. SF = V FIT —T,.2— Vo} = VF{T6,(S+0,)—V6,} ==): From these relations it follows that the first denominator, viz. (A, B, C, F, G, H, {4,B,G,)? =AG?+BB2+CG&2+7(FB,G,+6G,a,+H4,B,) =4 {AG +B, +CC,+2(FF,+ GG, + HB,} =9,V{A2?+B?+C?4+2(F’+ G?+ H’)+3T,+806,} =4 V{S’—25+3T,+86,} =4,V 30,74+480,+5+4+ 8} =4,V{(S+6,)(S+30,)+ 5}. Hence, writing (S+6,)(S+30,) +S=©,, we have, finally, eS Big eee TE OT RE, AE, From this we may obtain the following system : 1 ee ee TE a Oe AR: 36 0 = =gr |) a A, €, 1863. ] of a Solid Body about a Fixed Point. 61 with similar expressions for 6, 6,, B,; ¥. y» y». obtained by writing the suffixes 1 and 2 respectively for 0. By means of these we may write the equations connecting the variables as follow :— 1 1 1 P i= ©. p+ T + qv } ; 3 es) 46 SF, = aC, pir 2: Sm Go, ¥ e ° ° ° (25) Lastly, to complete the transformations, the values of p,, g,, 7, should be determined in terms of p, g, 7. Now 49, +9.8,4+ G6 F,=(VA+T,.A+ V0,)(VH+T,B) +(VH+T,)(VB+T,B+T9#,) Ces +(VG+T,&) (VF+T,B) =V{(A+B)H+FG}+T.T{(4+%8)i9+ se} + V?H(0,+6,) +V#(0,T,+6,T,) =V?(SH+9)+T,T, (2+ VH)+ VHC, + 9,) + V3H(0,T, + 4,7) =V{V(S+6,+6,) +T,T,}H+(V6,T,+V0,T,+$T,T,+V’) =T,T,{[—(S+6,)(S+6,)(S+ 6,)+ VJH+ [6,(S+0,)+0,(S+6,) +8+(S+6,)(S+6,)]H}, plnCce V=T,(S+0,)=T,(S+0,)=T,(S+6,). Moreover by (11) we have (S+0,)(S+0,)(S+6,)=V, and consequently the coefficient of H vanishes. And it may be noticed, as a useful formula for verification, that, from the relations last above written, we may at once deduce the following : 2 T= V?. Again, the coefficient of # may be thus written: (S+6,4+0,)(S+0,)+(S+0,+0,)(S+0,)+S+(S+0,)(S+6,) - — (S8+86,)(8+86,)— (S+6,)(S+9,) =— (S+0,)(S+0,)—(S+0,)(S+0,)—(S+6,(S+6,) +8 == (), FE 2 62. Mr. Spottiswoode—EKquations of Rotation [1863. in virtue of (11). Hence the whole expression vanishes, or Q6,4+8,8,+GF,=0; . ... . (26) 4,&,+34,F,+ G&€,=0. Moreover, in virtue of (23), we have and similarly Q27+H74+ G=4,2,. Hence multiplying (25) first by @,, 34,, &&, respectively and adding, secondly by ,, B, F, thirdly by & F» C,, we shall obtain the inverse system FP Ap+ Bat Gor } go o=MptBat5r, & 2 2 ee + (7) n= =G,p+5.q+€.7. ; Returning to the integrals (13), we derive (0,—6)9,7 +(0,—0)7>=h’—(S+8 DA, (6,—6,)7,° +(0 —0,)p’?=2—(S+8,)A, (8 —8,)p," + (8, —9,)9, =’ —(S+ 8, )h. Let rere (ames ETON cosy then k?—(S+6,)A .- a=a/ yoee rae and ‘ A peu a ee 1 0,—0 iP —(S+OyA ig (BSE gy. 6,—6 B—-(S+0,)h ie 0, —0, k’—(S+0 )h be Substituting in the equations of motion (21) (e.g. the first of them) and dividing throughout by sin x./#°—(S+0,)h, we have 1863.] of a Solid Body about a Fixed Point. 63 dx 4 1 of or = (S+O)h P = 0, dt Se 0,—0 0,—0, #—(S+6)h : or dy : a si _ 0-6, P—S+0)h . , 7 Oh ae, 0,0 6,6. F—(S+0)A oo & or EE Oe ee ; =: uk 0,—0 #—(S+0,)h . BO ea B—(S+o)hdt ; 0,6, Fa (S40 hm X then x=am(4 /2—* eee y/o BOSTON) and p= a / EAST cos am (\/ 29 vE=CT OH), =a / HSH Oh = sinam (4/2 nh / POST Hh S+Hit-+s) pan EET a ee Oe pre Po STOne+s)- These, then, are the integrals of the equations of motion when no exter- nal forces are acting. The next step is to determine the variations of the arbitrary constants, due to the action of disturbing forces, when, as in the case of nature, those forces are small. With a view to this, it will be con venient to change the arbitrary constants into the following, /#—(S+0)h=m /h—(S+O)h=n, (6—0,)h=m?—n’, (0—0,)k* =(S+0)m’—(S+0,)n? ; also, for brevity, let whence 0,—0, 7 ea Ona Gh ee a bd am(Int +f) =x, ar ee Then the equations of motion become cos am(Int +f), = Fe — ea sinam(Int+/), n — Wire Aam(Int+f). 64 Mr. Spottiswoode—EKgquations of Rotation. [1868. - Now it is known by the theory of elliptic functions that d cos ame : —_—— =— sinamxA ama, dx dsinamx —z—— = cosamxAama, dx dA am an a —k,? smamzcosamz. Whence P,, iQ; R, being the moments of the disturbing forces about the present axes, . : dn d ce axa pmsinndx(ieae + Z) f . dm dn df \ “Jaa a Lsxige tone xan (iS +7) r ie fF " — ipa ae R, = a sin y 008 x( eS +7) From ne we derive dm ae GES a 6—6,P, cos x+/0,—6, Q, sin x, n, of Sets ae eel mix (iit dt = —/0—6,P, sin x +/ 0,—0, Q, cos x; dn sin x cos Ax a =a/0,— TOR, + ha —/0—0,P, siny +,/0,—0,Q, cosy}; or dn R, 0,—0 msiny cosy ——_ , —— Tp=v 9.—- ae 06 cn (xy UY 94.2, sing +4/ 6,8, Q,c08y} R, , 9,—6 1sinx cos wh ra ae Hy te / 0—6,P siny +4/6,—8, —6,Q, cosx} fi/o—0, 0,P, cosy +a/0,—0,Q, sinytdt. And lastly, d, dn a ted = lt ; /0—60 6? sin y +4/0,—0 6,.Q, cosy Re =a ewe —6,Q, sinx}dt 1863.] Sir J. South—Vibrations caused by Railway Trains. 65 “‘Hixperiments, made at Watford, on the Vibrations occasioned by Railway Trains passing through a Tunnel.” By Sir Jamzs Soutu, LL.D., F.R.S., &c., one of the Visitors of the Royal Ob- servatory of Greenwich. Received June 17, 1863*. In the year 1846 an attempt was made to obtain the consent of the Lords of the Admiralty to run a railway through Greenwich Park, distant only 860 feet from the Royal Observatory, which would in the opinion of many competent judges have been most injurious to that Establishment. Such consent their Lordships refused ; but as I was assured on high au- thority that this attempt was to be repeated, and that too with the fullest confidence of success on the part of its projectors and supporters, I deter- mined to make experiments which might bear more decisively on the question of railway tremors, as affecting that Observatory, than those pre- viously made by myself and others. For this purpose it seemed indispensable that the station selected for making them should geologically resemble that of Greenwich, and that the astronomical means employed to detect the existence and determine the intensity of the tremors should be, optically, at least equal to the te- lescope of the Greenwich Mural Circle. | As much importance was attributed by the advocates of this railway to the supposed power of a tunnel to render the vibrations imperceptible, it was also desirable that 7¢ should be one of the conditions of these trials. Having but little more than a popular knowledge of geology, I relied on my old and valued friend the late Mr. Warburton, who had recently been President of the Geological Society, to guide me in the choice of a station ; and it was on his authority that I fixed on the Watford Tunnel and its im- mediate vicinity. There, under a light gravelly soil of 18 or 20 inches deep, lies a bed of gravel of considerable but variable thickness, sometimes compact, at other times loose, and immediately under it chalk with occasional flints, The tunnel, of which the bearing is 41° 19’ to N.W. of the meridian, and by my measurement is 1812 yards long, passes principally through chalk ; its arch is about 24 feet in diameter, the crown of it being about 21:5 feet above the rails. The thickness of the brickwork is about 18 inches; the mean thickness of the chalk above the crown of the arch about 50 feet, whilst that of the gravel, though subject to great irre- gularity, may perhaps be regarded as 14 feet. If so, we have outside the tunnel above the horizontal plane of the rails 87 feet of chalk, flint, gravel and soil, constituting an assemblage of which the power of transmitting tremors must be comparatively feeble. There are five shafts in the tunnel, four of which are circular, 8°5 feet diameter, and one quadrangular, about 26 feet by 34. * Read June 18, 1863: see Abstract, vol. xii. p. 630. 66 ‘Sir J. South—Vibrations caused by Railway [1863. The tunnel runs under the park of the Earl of Essex ; and though I had not the honour of a personal acquaintance with the Noble Earl, nor any introduction to him, yet on learning my objects he transmitted to me by return of post, from Carlsbad, a carte-blanche to erect my observatory wherever I pleased, though it were in the very heart of his choicest game preserves. To him therefore is mainly due whatever benefit may accrue to science or to the Royal Observatory from the experiments recorded in this communication. The point I selected was 302 yards distant from the centre of the line ; and the perpendicular from it on the axis of the tunnel meets that at a pomt 567 yards from the southern or London end of the tunnel, 1245 yards from the Tring or north end, and 594°5 from the fourth shaft. This is the centre of the Observatory which I erected there: it is of wood, as small as is consistent with the necessary accommodation, both for portability and that it might be less agitated by the wind. It is quadrangular, 12 feet by 10, and its length is in the meridian; the eaves are 8°5 feet, and the ridge of the roof 10 feet above the floor, this last being 4 inches above the ground, which is nearly level with that over the tunnel. The roof is covered with tarpaulins very well secured, so as not to be torn bya gale of wind. In the south and west sides are four windows, which can be opened or shut at pleasure, to light the Observatory by day, or to see powder or other signals at night. In the roof is no opening ; but in its northern side there is one which can be shut as required: it is little larger than what is absolutely necessary to allow the reflected rays from the Pole-star to pass uninterruptedly to the observer’s eye through its whole revolution. At its centre, parallel with its sides and resting on the undisturbed gravel 4 feet below the surface, is a mass of brickwork laid in excellent Roman cement, 8 by 3:5 feet at bottom, 7 by 3°5 at top, its length running east and west. On this stand two piers of similar brickwork, 18 inches by 14, and 46 inches higher than the floor: they are capped by two Portland stones of similar horizontal section 8 inches thick. In the interior faces of these stones are firmly fixed the Y-plates, which carry the Ys on which the in- strument’s pivots rest. Eighteen inches north of the brick massive, but in the same plane with its base, is the centre of the base of another pier, brought up alsoin Roman cement, 24 inches from N. toS., 18 from E. to W.; and it rises 12 inches above the floor. The upper surface is perfectly horizontal, and serves to support a vessel which contains mercury. Both this pier and the massives are insulated from the floor, and touch the ground only at their bases. The mercury-vessel was 18 inches by 42, with its length in the meridian. The transit-instrument of the Campden Hill Observatory is far too pre- cious to be exposed to the risks of such an expedition ; I therefore had one constructed which might be considered an excellent substitute. The object- glass (which under favourable circumstances will bear a power of 1000) is 87 inches focus and 4°75 aperture. The transverse axis is 31 inches ; and 1863. | Trains passing through a Tunnel, 67 the Y has sufficient azimuthal motion to enable me to follow the Pole-star in its whole course, so that at any hour (if clear) I could have the reflected image of the star in the mercurial vessel ready to testify against the tre- mors caused by a train. Supported by timber passing into the ground, but unconnected with the floor and convenient to a writing-desk which occupies the S8.E. angle of the building, stands a journeyman clock. It is set by my excellent gold pocket- chronometer, Molyneux No. 963, and rarely deviates from that more than one- or two-tenths of a second in three or four hours. The clock of the Watford Station was compared with the chronometer, going and generally returning, for the purpose of identifying particular trains. These details will, I hope, suffice to prove that every precaution was taken to obtain accurate results, and that those which I did obtain may be fairly considered as identical with what would have been found in a first-class observatory under the same circumstances of locality and traffic. I was at my post to commence observations on December 22nd, 1846 ; but that and the three following nights were starless. The 26th was fine, but, owing to the irregularity of the trains, and the want of well-organized signals, I could only satisfy myself that all was in good working order, and that the trains caused great disturbance. For thirteen following nights I was at my post, but in vain; all was dark, with the thermometer from 22° to 31°. On January 11th, 1847, it cleared, and I observed seven trains with decisive results, being able to announce their presence before it was known to my assistants, who were on the watch outside the observatory. The Pole-star’s image as reflected from the mercurial surface, when no train was near, appeared As a very small, perfectly steady disk, thus . . . + » « @ (l.) which as the train approached broke up into a quin- aan tuple, thus— . oS Supe a. psa 4 ee 2 if “i Geer (2s) hes As the disturbance increased, the form be-) __ came linear at right angles to the length } :*see@@@@e@eeee- (3,) of the mercury-vessel, thus— @Se@eo0e- W i i @ hen the train was considerably advanced \ LeepeoeeGOeocee-: (4) in the tunnel, a cross formed, thus— . 22006809 ©608006800000090 @eeagee0ececeeeo (5.) observatory, three parallel lines of disks ap- SCSSCOSGORSOOOD, And when near the perpendicular from the [CE ECORI LISS Saimme ihe 5000 ae aaanaae 68. Sir J. South—Vibrations caused by Railway [1863. still parallel to No.3. As the tremors became more distant, these trans- formations of the image take place in a reverse order, until the star resumes its original disk-like form. These results were strongly conspicuous even in a fully illuminated field, and equally so whether the magnifying power was 60, 200, or 750. The phenomena are very striking, from the contrast between the smaller images, which are blue, while the larger ones are reddish, and from the sudden way in which they break out. The nights of the 13th and 14th were fine, and so thoroughly confirmed my previous observations that I felt it my duty to lose no time in informing the late Lord Auckland, then First Lord of the Admiralty, of the preceding details and of my conclusions from them, that a tunnel did not prevent great tremors from being propagated from it when a train was traversing it, certainly to the distance of 643 yards, and probably much further. The impression which these facts made on his Lordship he expressed in the following letter. Copy of a Letter from the Earl of Auckland to Sir James South. “Admiralty, January 26th, 1847. *‘Srr,—lI have to return you many thanks for the very interesting report which you sent to me of your experiments upon the distance to which the vibration caused by steam-carriages within a tunnel extend ; and I cannot but admire the enterprise and ability with which these expe- riments were conducted. They would be quite conclusive if the question of carrying a tunnel through Greenwich Park were again to be agitated. ‘I am, very faithfully yours, “To Sir James South, &c. &e.” ‘* AUCKLAND.” The reserve with which I spoke of that further distance arose from the circumstance that I was not in possession of the exact measurements of the tunnel and the position of its shafts. I had twice applied for them in vain to the railway authorities, and was obliged at last to execute the measures myself*. This consumed some time, and the observations were not com- pletely resumed till February 24, 1847. The process was this. About 600 yards before the entrance of the tunnel a rocket was fired as a signal for attention. At the instant that the engine passed the south end of the tunnel, one of Lord Essex’s game-keepers fired one barrel of his gun, and the other about a second after, which was necessary to distinguish this from the shots of poachers, who were often at * This delay was not occasioned by any want of courtesy on the part of the Directors or other officers ; from whom, especially from Mr. Creed, their Secretary, I received the heartiest cooperation. He not only directed all the officers along the line to aid as far as possible my investigations, but pressed on me free passes for myself and my assistants. I was also indebted to Captain Bruyeres for the character of the trains, and to Mr. Stubbs, the Superintendent of the Watford Station, for the zeal with which he followed out the Seeretary’s instructions at much —* inconvenience. 1863.] Trains passing through.a Tunnel. 69 work around me. Similar shots were fired when the engine was at the centre of the 4th shaft (which could be seen from above). Whe times of these sig- nals were taken by an assistant. During this time I was at the telescope, and noticed the second when any peculiar phase of disturbance appeared. The computation of the distance of the engine from the eye at a given time is very simple. From the known distance of the south end of the tunnel and the 4th shaft from the eye, we know the times taken by the sound of the gun to reach the observatory. The temperature was during the whole series so near 32° that the velocity of sound for that temperature, 363-13 yards, may be used without sensible error. The effect of wind must also have been insensible. Hence the signal from the south entrance was 1%*77 too late, that from the shaft 15-84. Correcting the times and dividing by their difference the distance of the shaft from the entrance, 1162 yards, we have the velocity of the train (which, however, I have given in miles per hour, as affording a more familiar measure of the disturbing power). Then the difference of the time of phase and corrected time of entrance gives the place of the engine on the line, and the perpendicular is given. In the following record of the observations, the first column contains the number, the second the times, the third the facts observed, and the fourth gives the distance, then follow occasional remarks. In the disturbances, I specially recorded as most definite the cross (4), and the arrangement of bars of parallel stars (5). The slighter disturbances which precede or follow the former were seldom entered, though quite sensible. 1847, February 24.—I. No.| Time. Observations. Yards. Remarks. hm s 1 | 7 18 43 |Cross very distinct...... 845 |Velocity 11-00, miles an hour; weight 7 19 21 |Shaft gun. of train 77°5 tons ; twelve carriages. 7 22 57 |South gun. 2|7 23 8 jLost sight of cross...... | 74 | ia ete 34s Gf CLOSS). oocccieies sues ceseeke | 699 |Velocity 16°6 miles; train 69°5 tons, 34 8 | Shaft gun 231 feet long; ten carriages. Ther- 36 31 | South gun. mometer 24°. 4 36 48 | Lost sight of cross...... 780 III. 5 17 44 40 |Cross; star very faint...) 680 |Velocity 13°8 miles. Star invisible to 44 44 |Shaft gun. the naked eye. Train 58s tons; 47 38 |South gun. engine 14°5 tons; length 185 feet. 6 | 7 47 42 jLost cross ..... SOCCER 678 70 Sir J. South— Vibrations caused by Railway [1863. 1847, February 24.—IV. No. | Time. Observations. Yards. Remarks. hm i s 7 59 6 |Shaft gun. Velocity 11°4 miles; train 89°5 tons; 8 2 30 |Star became visible. engine 18 ditto; length 308 feet. 8 2 34 |South gun; star bright. Wind E. Therm. 24°. 7 | 8 3 10 |Cross disappeared ...... 834°5 1847, February 27.—I. 7 28 o |Shaft gun. Velocity 15°4 miles; train 54 tons; 29 7 |Cross first seen, but star engine 14°5 tons; length of train very faint. 172 feet. 30 34 |South gun. 8 | 7 30 44 |Cross lost; star very TE Os ree ee ae es 722 iT; 9 | 7 44 43 |Cross seen ...... seeeeeee- 736 {Velocity 25° 6 miles ; train 49°5 tons ; 7 44 51 |Shaft gun. engine 14°5 tons; length of train be) 45 8 |Cross very strong ...... 470 | 150 feet. II 45 27 |Line very strong ...... 326 46 24 |South gun. 12 A6n6 |CEOSS 1OSt Ge2.0<228eee ees 915 Lif. io 7250.25 | Cross SCCN. . serosa ees 706 |Velocity 17°6 miles ; train 270°5 tons ; 56 31 {South gun. two engines 29°5 tons; length of 14 57 45 |Cross very strong ...... 314 | train 663 feet; 37 carriages. 58 46 |Shaft gun. 15 BOSS JICEOSS NOSE. “sca. ctaamsenen 736 IV. 16 | 8 3 36 {Cross very strong ...... 736 |Velocity 31°7 miles; train 112 tons; 3 44 |Shaft gun. engine 21 tons; length of train 17 A. HG \Cross veby HWE 6.0202 377 394 feet; carriages17. Wind N.E. 18 4 14 {Triple line, upper and Thermometer 26°. lower stars blue ...... 319 4 59 |South gun. 19 5 25. CLOSS 1OSt i sscwtonewase+! 1086 V. 20 | 8 10 56 |Cross SeeN ............».- 727 |Velocity 18°7 miles; train 51°5 tons; Ir 8 |South gun. engine 12°5 tons; length of train 2k 11 56 (Triple line strong ...... 322 | 187 feet. A train of empty catile- 13 15 |Shaft gun. waggons. Cross lost from cloud...| lost Trains passing through a Tunnel. 71 1847, March 11.—I. No.| Time Observations. Yards. hm s 22 | 7 18 44 |Cross very distinct ...... 802 19 6 |South gun. 28 2G Shaft gun, cloud. EE. | 825 3 |Shaft gun, cloud. 26 15 {South gun, cloud cleared Remarks. Velocity 17°73; train 147°5 tons; eu- gine 12°5 tons; length of train 355 feet. Velocity 33°0 miles; train 122 tons; engine 21 tons ; length of train 416 23 26 32 jCross lost by cloud...... g21 | feet. Cross so strong, but for the cloud it might have been seen 15 or even more seconds longer. 1847, March 12.—I. f 24. | 6 56 22 ‘Cross very distinct...... 822 |Velocity 28°33 miles; train 68 tons; 56 38 South gun. engine 15 tons; length of train 231 25 56 52 Cross very strong ...... 461 | feet; many carriages but mostly 26 57 16 Star tossed about 3 or 4 empty, many wheels and axles; of its diameters ...... 302°4 | agitation excessive. Seemed to 58 2 |Shaft gun. keep time with the jolts of the 27 58 8 (Cross PGS acres ae 766 | train. IE: See yee ES Cross plain ......... 4 tht yoo +7) White oe, oak kt 4 5 6 7. Table II. shows the tones associated with the manuals in each stop ; capital letters indicate white manuals, small letters black, and small * Singers and performers on bowed instruments and trombones can produce any scale whatever. Other instruments are more limited in range and would require special treatment, similar to the “crooks” of the horns and the various clarinets. 1864. ] ~ on Instruments with Fixed Tones.’ 101 capitals red*. By this arrangement the fingering of every key would be the same. The performer would disregard the signature except as naming the pedal, and play as if the signature were natural. Table V. would inform him whether the accidentals belonged to the key, its dominant, or any other key ; and if they indicated another key, he would change the pedal. It would be convenient to mark where a new pedal had to be used ; but no change would be required in the established notation §. Mr. Poole’s organ, which suggested the above arrangement, has 11 stops, from 5) to 52, and only 12 manuals, which appear to be associated with the following tones on each stop: Black .. {2 ({2#) (44) 15¢ ra, White.. 1 2 3 4 5 6 7 The two manuals whose notes are put in parentheses are inadequately de- scribed. Mr. Poole’s scale does not include the synonymous minor chords, which he plays by commatic substitution. Another method of realizing such a scale is by additional manuals and additional boards of manuals. Thus three boards of manuals, each with 23 manuals, containing the tones in Table V. cols. III. to VIII., lines 4 * On examining Table II. it will be found that 10 different tones lie on each pair of manuals, so that there are only 70 different tones. The two missing tones are, necessarily, ttf # (the acute fourth of the key of + C4), and {{0) (the grave seventh of the key of {C>); and to this extent the scheme is defective. It would probably be more convenient to the instrument-maker to use all the70 tones in this arrangement than to take the inferior number 45 due to schismatic substitution. A full-sized harmonium at present employs from 48 to 60 vibrators to the octave, so that the mechanical difficulties to be overcome in introducing 70 are comparatively slight, By omitting the two very unusual keys of {CD and +Cz, the 8 tones denoted by ttdD, tF9, ttf, a7) and t Dz, gx, t Bt, ttb in Table IL. would be saved, and the number of yibrators required would be reduced to 62, nearly the same as that actually in use. As each new key introduces 4 addi- tional tones, and the key of C has 14 tones, the number of vibrators required for any extent of scale is readily calculated. Thus for the 11 keys from 5 flats to 5 sharps, or {D?, A), ED, BD, F, C, G, D, A, E, B, which is Mr. Poole’s range, and is sufficiently extensive for almost all purposes, only 4x 104+ 14=54 vibrators to the octave would be required, distributed over 11 stops (exclusive of the tem- pered notes) ; and such a number of vibrators and stops is in common use. § If in Table V. we reject the marks +, {, consider 16 4=27 C, 64 F=81 C, 128 B= 243 C, yoda! paotees leaving the value of the other letters un- 2048’ 2187’ changed, the Table will represent the Pythagorean relations expressed by the usual notation (which is quite unsuited to the equally tempered scale). The chords thus formed were too dissonant for the Greek or Arabic ear to endure, although Drobisch and Naumann (Joc. cit. ad finem) desire this system to be ac- knowledged as “the sole, really sufficient acoustical foundation for the theory of music ” (als einzige, wahrhaft gentigende akustische Grundlage der theoretisch-mu- kalischen Lehre), 12 102 Mr. A. J. Ellis on a@ Perfect Musical Scale [Jan. 21, . to &, 7 to 11, and 10 to 14 respectively would be nearly complete. The manuals might be similar to those on General T. Perronet Thompson’s Enharmonic Organ, which has 3 boards, with 20, 23 and 22 manuals re- spectively, and contains the chords in Table V. cols. III., lines 6 to 11; IV. 6 to 12; V., VI., VIL., 5 to 12; VIII. and IX., 6 to 12 (four chords belonging to col. IX., lines 6 to 9, are not in the Table, but can be readily supplied, as well as the additional lines 0, —1, named below). ; Euler’s “genus cujus exponens est 2”. 3’. 5°,” as developed in his Tentamen, p. 161, must be considered as adapted for an instrument with two boards of ordinary manuals, such as some harmoniums are now con- structed. His “soni primarii’’ would occupy the lower, and his “soni secundarii”’ the upper board. If to these we add their schismatic equiva- lents, inclosed in brackets, and distinguish white and black manuals by capital and small letters as in Table II., Euler’s scheme will appear as | follows, where the notation interprets his arithmetical expressions of pitch y (‘soni’), and not his notes (“signa sonora”’), which are too vague. | | Euier’s Dousie ScHEME. Upper Board. Schism. Equival... [{C, td), tD, &, FD, tF 9), 1G, @, Bb,w, Clg “Soni Secundari”. BH, ct, Cx, dt, tH, Ei tft, &x, ge tae ee 2 Lower Board. . “Soni Primarii” . -C, ite D, tat, B& F ftG, tot 4 of Bae Schism. Equival... [{D9), ttdb, EDD, te), +F9, tG0>, tg), ADD, ta, LBD, to, tab } Although it is evident from his notation that Euler regarded schismatic equivalents as identities, he has not especially alluded to them. The above scheme would contain Table V. col. V., lines 0 to 14, and the major third +F4 +A# in 15 (with the schismatic error of ¢B) ({DF for By tD TL), cok, VI. J'to 15; “Vil-9 to 24- VITI. .10 to 24. 1X. 18 to 24 —lto5; IV.0to6. It would be therefore nearly complete in major scales, but would have only fd, a, e, 6, £3, c#, g# minor, and their compa- ratively useless schismatic equivalents. It would have no single complete key, and would therefore require many commatic substitutions in modula- tion, and the use of the Pythagorean major third in the major chords of the comparatively common minor scales of tf, te, tg. If only the “ soni primarii”’ of the lower board are used the substitutions become very harsh, as for example 44 D F, D F¥ A for B) {D F, D F4 +A. | Euler’s “soni primarii”’? may be compared with Rameau’s scale *, which was as follows, C, ted, 1D, +, L, F, iff, G, tot, A, +0), B, * Traité de Harmonie, 1721. The values of the tones are determined from his arithmetical expression of the intervals, 1864. | on Instruments with Fixed Tones. 103 and. therefore only contained the following perfect harmonies, and ¢wo perfect scales, 4 major and a minor :— ay jam} EAC td fa {D{Fit A +E) G BD), etbg CEG ace A{CHE ao GC B egd E{GLe. Prof. Helmholtz has tuned an harmonium with two boards of manuals, somewhat in Euler’s manner, as follows :— Hetmuyoutz’s DousLe SCHEME. Upper Board. Schism. equiv. [{C, d),{D, &, FD, F, g),tG, &,A, 0, CD] Tones tuned.. BY, tet, Cx, dt, tH, tER tft, Fx, of, GX, tat, TB. Lower Board. Tones tuned.. —C, cH, D, jdt, H, FE, f£G, tot ta, af, B Schism. equiv. [{D)), tdp, EDD, te), tF), +£F, tg?, Abb, tad, oDd, +6), £00]. This scheme has nearly the same extent and the same defects as Euler’s. The concertina, invented by Prof. Wheatstone, F.R.S., has 14 manuals to the octave, which were originally tuned thus, as an extension of Euler’s 12-tone scheme. | C, ted, D, tag, £, te, £,f% G, tot, A, ta), B, +09. It possessed the perfect major and minor scales of C and Z. The harsh- ness of the chords +B) DF, D FA, for Bp tD F, D F4+A has, however, led to the abandonment of this scheme, and to the introduction of a tem- pered scale. I have taken advantage of the 14 manuals to contrive 4 dif- ferent methods of tuning, so that 4 concertinas would play in all the common major and minor scales. Two of these I have in use, and find them effective and very useful for experimental purposes. The following gives the arrangement of the manuals in each, together with the scales possessed by each instrument, major in capitals, and minor in small letters. Where commatic substitution makes the dominant chord too flat in major scales, parentheses () are used ; where it makes the subdominant chord too sharp, brackets [ |] are used. Minor scales in brackets have only the subdominant tone too sharp. The major chord GBD and the tone C being common to all four instruments, determine their relative pitch. The method of tuning these and all justly intoned or ¢eleon* instruments is very simple. C being tuned to any standard pitch, the fifths above and below it are tuned perfect. To any convenient tone thus formed, as C itself, form the major thirds above, * A convenient name, formed from réAcov dudornpa, a perfect interval. 104 My. A. J. Ellis on a Perfect Musical Scale [Jan. 21, as E, tG#, t By, &c., and below as +4), +F?, &c., and then the fifths above and below these tones. The names of the tones thus tuned are apparent from Table V. This tuning is much simpler than any system of tem- perament, and can be successfully conducted by ear only, taking care to avoid all beats in the middle octave ¢ to ¢’. ScHEME FoR Four CONCERTINAS. 1. +49 Concertina. Manuals..C 0), D db, Ete, F tf, Gg, A tad, B tb, Scales....Dp, +b, tH), (tB)), [FL C; (MLA « 2. tBp Concertina. Manuals..C te, D ct, tE td, tF ft, G tg, tA to, B tb, Scales....t 2), +B), tF (tC), [@], D; [el, g, a. 3. C Concertina. Manuals..C tet, {Dd, E td, F ft. G tot, A ta, BD, Scales....F, C, G, (D), [A], E; [td], a e. 4. D Concertina. Manuals..C ct, D td, E te, Et ft, G ot, tA af, B Td, Seales....G, D, +A, [TE], [B), Fe; [el], 6, fF. In Table III. the first column shows the number of degrees of any tone, two tones whose degrees differ by one-sixteenth being schismatic equivalents. The second column contains the notes of the tones. The third column contains the logarithm of the ratio of their pitch to that of c, whence the ratio itself, the absolute pitch, and the length of the monochord are readily found. In the fourth column # marks Euler’s primary, and £” his secon- dary tones; H, H* the tones on Helmholtz’s lower and upper board; 7, the 40 tones of General T. Perronet Thompson’s Enharmonic Organ ; P, the 50 tones of Mr. Poole’s Euharmonic Organ; ¢, the 72 tones of Table V., cols. III. to VIII. ; s, the 24 tones out of these 72 which may be played as their adjoining schismatic substitutes without injuring the har- mony; se, the 3 tones which, if played as their schismatic equivalents, would produce a slight but sensible error; ¢, not followed by either s or se, the 45 tones which form the minimum number of a justly intoned or teleon scale; et, the 12 tones of the equally tempered scale. The seven tones of the major scale of C are printed in capitals in the second column. 1864. } _on Instruments with Fixed Tones. 105 Taste |.—Principal Musical Intervals. Name. Example. Ratio. or Log. |Deg. LS 001. Gan eer Gis ec 131 eet ‘00000 | 0 AP eM. 3 i. o's 8 fee tee © 32805 : 32768/38.5 :2% |-00049| 0’ Ti» Diaschisma .....4.... ¢ : Bt 2048 : 2025 21; 3? . 5700491 | 1% jj (Gita Bae, Cee TG SG 81 :80 BP 22°. 8 00540) Is TF Pythagorean Comma § TtBE: ec 531441 : 52488 312;219 ~—|-00589| 1 Pe Piccigee ts. .cces--| . Ds tet 128 : 125 97:53 —_|01080| 2 tt Minor Semitone ...... tof: c 25 :24 62°: 23. SOL (soe (Ged Bias re TB 256 : 243 oP Foe 02263 | 4 f Sharp, or Greater Limma| c#: 135 :128 (383.5 :2? 02312 | 4' T (pt Equal Semitone*,,.. oe c 4/2 21 #24 + 34 , 5702509 | 4,7, Greater Semitone . P Guests 16 :15 24:3 .5 |'02803 | 5 Greatest Semitone ...... Teo, ¢ 2187 : 2048 37:24 = |-02852| 5 Greatest Imma” <.,.:.6.|. Td % eg 2h 3 28 i. 33 ; 6? ‘03343 | 6% MMmOE TONG oc ec ness hss a 0 10 :9 2eO oe 04576 | 8' Create LONG y iy si0is 8405 ss d -i5,¢ 9:8 3? ; 23 ‘(05115 | 9 Extended Tone .......... oe tet Sif BAS BT) ‘05799 |102 Contracted 3rd .......... gues dl 716 7 :2 ,8 |:06695 112 Pythagorean Minor 8rd .. i530 2 +27 DP 5 38 07379 [13° MEMOE AUG pews ri vee vse: es 6 6:5 2 ae 55 ‘07918 14 ESFOG MG says ies oe as qe 5:4 5 323 ‘09691 |17' Pythagorean Major 3rd ..| te : c¢ 81 :64 34 ; 26 ‘10231 |18 Fourth, or Perfect 4th.... Pee 4:3 aoe "12494 |29 False 4th Merete Ce ke eittete’s eas 6 27 +20 3? +2? . & |:18033 |23° Contracted 5th ...5..55.. A 7:5 7 3:5 14613 |258 Diminished 5th ........ ae! 64 : 45 98 +32 5 ("15297 |a7" Balse Of asasiisisices. Rae AO 26, (2920. + 38 ‘17070 |30' Bowal St. recs ris cenee ie 3)-e A/a: 1 #214 + 87 -. 5 |'17560 [31 Fifth, or Perfect 5th i & gs £6 os ?2 eg ‘17609 |31 Pythagorean Minor 6th ,. e :{fE 128 : 81 27 +34 19872 |35 Miao, Gthy «9 ree een ak oe c: EK 8:5 oF. 2.0 ‘20461 |36* LETC ih hae eee dq be 5:3 SES ‘22185 |39' Pythagorean Major 6th ..| fa :¢ 54 +32 2.33 : 25 29794 |40 Diminished 7th.......... 1,2 5G2 128 : 76 27:3 = , 5723215 |41" Hxtended 6th: 0.05.6 .6 e's d :gF eg Dagar t ‘23408 |413 Pemect TW i. vaso... af: G 7:4 7:2 |-94804 |494" DMR ACHE gs ice 6 osc ss 1 eal 16:9 24 : 32 ‘24988 |4.4” Acute Minor 7th ........ +b? : ¢ 9:5 37:5 — |-25527 |45° TAROT TOW Nah est wet ore ea ee or) bie 15 :8 BS a5, 228 ‘27300 |48' JEN Sarin r aes (: Geta G: ait an L 30103 |53 § Hence the symbol {J for Pythagoras, with the + (comma) prefixed. * Approximately. =) QR =| rac > eet Mr. A. J. Ellis on a Perfect Musical Scale 106 el eee | peer | een: | ee | | | NS | | | afd) tat fa qt q | hx fx | at a | at An a qi} dat | at | dq x] dats at a gat | ca qqx | dat Coa oe qt | cat tv | @ (AL? ode fa 4 |xs D4 V+ | #3 o4 ve | ot | BD vt | 2h | Bo Vix} 3+ | #9 V+ | fer | 9 ot ee fee co ee Weel eth) Vv «| ¢ Bi Vv | Ge ra) i uf a iW 0 Vv ott Cote vi stt | got v | ae r9) 1) 0t 6 XJ at XJ Ata xj T+ ty at me nt gL | Ga I+. | Fal ee a ABE NE Ht A a ui J at J At J at gor | at | gat J | ni 8 | L mL} Hopd wkd | for aap juut a be a H+ me Ihesaaet Cee Gxt | a a | Cc a at | Gx at | ou at | cox | ca cqe | a ae | 9 q [dojs qove Jo e30u -Loy oy} syteur (x) Ystloyse poxyord v : spentreut yor stoz}0] [[euIs “pas speqrdeo qpeuus ‘opeya quesordor speqideg | ap ‘a+ pt |ta pt a pt |ta tpt | a apt dx tpt | a je) ay at rae at cP at pit | dat ptt | cdt« ptt | cat Gptt | cat 90) ‘at t g xo ik (kee IB TS: Oo Oi eee seed ners © to ees? Means G a Ore |) se ediens | all 9 hy ae CURTIS (C all Oe aly oe aaseciiens. 7 tort fe) OORT G Bie er | Tot Ox |e remae Ny fot 9 OR LEE oe SN) 0 of ‘@) eeeee eevee SIV SG q@ ot eee SSR RNG q ot Be os ENB SD © ot eocrveve S}eT G SIG ASO RMS oe Ue ae kes oft | Godel tt" SsyBH Z fo 9 "tes ngrodute 7, S T ‘doyg Jo oureyy “MUNIUOULIBTT 10 UvSIQ woajaz 10 Spouoquy Asng v uo sdojg—']] TIAVY, 1864. | on Instruments with Fixed Tones. 107 | Taste III. General List of Musical Tones. Deg. | Note. | Log. Remarks. Deg. | Note. | Log. Remarks, 0 C |-00000|E, H, T, P, t, et.) 19 |tte |-10770 {t. 0' | +b# |-00049 It, s. 20" | ttf |-11365 |t. Tr | te |-00540/T, t. 20'" | tet |-11464 |T. 2" | ttt |-01233 |T. 204 | cf |-11810/P. 27 |ctdd |-01579 |P. Fe Tee | Ost IM. Pt. 3' |tta) |-01724 It, s. 21" | ett |-12003 |E2, H, P, t, se. a ie tet |.01773 |B, T, P, t. 22 F |-12494 |K, T, P, t. 4 td) |-02263 |P, t, s. 22' | +tett |-12543 |H2, t, s. 4! 02312 |E, H, T, P, t. es See t pn toe | ie 19543 let = tet. 45. 02509 |et. REALE EES me lee (O28 o3 «| te |-130383 (7, t. 5 dp |-02803 T, t. 232 |zted |-14073|P. 5 tet |-02852 |H2, P, t, s. 95" | tft |-142967 |T, P, t. 7" |+td |-04036 {t. 26 te) |-14757 IP, t, s. 7Z | 2d |-04431)P. 26! f+ |-14806 |E, H, T, P, t. 8' | td |-04576/T, P, t. eT 8” | ex 04625 |B, He, t,s, | 202 | IIB [15051 let. ‘H gs: | ja (05017 let, oT ep) |:15297 |T, t. ae 28 Se SH +ft |-15346 |E?, H2, P, t, s. 9° eDD |-05066 |t, s. 28° | +tft |-15886 |t. 9 D |-05115|E, H,T,P,t. | 29" |tte |-16530|t. 10° | td |-05655 It. 29" | tf |-16579 |T. 11" | ttdt |-06349 |T. 294 | go |-16925 |P. 11z | ze? |-06695|P. S00 ie det betZ070 1, Pete 12’ | te? |-06839|t, s. SOMO ese 7 | 7109 RPS He PSG, 9. 12” | tdt |-06888/E, H, T, P,t. | 3 aD |-17560 It, s. 13 e) |-07379|T, P, t. ee 13' | dy (07498 (Er, H2, P,t,s, | °0t | le [17560 et=adp. iat | jaz [07526 let. 31 G |-17609 |E, H, T, P, t. BA 32 tg 18149 It. 14 | ted |-07918)T, t. 33'” | tte# |-18843 |T. 14 | td# |:07967 |t, s. 332 | gad |-19189|P. 16" | te {|-09151)T, P, t. 34' | tad |-19833 |t, s. 164 |cte |-09547 |P. 34" | tot |-19382|E, H, T, P, t/ 1% ff) |-09642|t, s. 35 a) |-19873 |T, P, t. int BE | 096g Ee T, Pets 7 35° ot |-19922|E2, H2, P, t, se. 172 | |je |-10034 let. let 120068 let. r f? |-10181|t, s. 36° | tad |-20412/T, t. 18 | te |-10231|K2, H2, T, P, t. | 36 tot |:20461 It, s. 108 Prof. Huxley—Osteology of the genus Glyptodon. [Jan. 28, Tas eE III. (continued). Deg. | Note. | Log. Remarks. Deg. Note. | Log. Remarks. 45° | thd |-25527/T, t 452 | cto) |-26567 |P. 47! 1. th. | 267611, Pe 48 te) |-272511t, s. 48! B - |-27300|E, H, T, P, t. 38" | ta |-21645 /T, P, t. 882 |cta |-22040 PP, 39' | A |-29185\E, T, P, t. 89" | gx |-29984 A, t, s. 393 lla |'22577 let. 40° bob |-22675 It, s. 40 ta |-29794 |E 4Y | tta |-28264 It. 42" |\ttbd |-23908 It. 42" | tat [23958 |T: P P 48.7. | |Ib . |:27594 |et. AQ’ cp |:27791 It. 49 th |-27840|E 50’ |+tb {28380 It. bl) | tha -2G024ar T P 494 | chp |-24304 51" | tht |-29078 43' | thd |-24448 P, t, s. 514 | ge |-29419 |P. 43" | aft \O44O7 IE IPs 4° 152! | te |) 20568 eae 44. bD |-24988 /T, P, t. 52” bi |-29612 |E2, H2, P, t, s. 44' | +tatt |-25037 |E?, H2, P, t, s. STEE 1 | llatt |-25086 jet. January 28, 1864. Major-General SABINE, President, in the Chair. The following communications were read :— b) I. “On the Osteology of the genus Glyptodon.” By THomas Henry Huxtey, F.R.S. Received December 30, 1863. In 1862 the author communicated to the Royal Society an account of the more remarkable features of the skeleton of a specimen of the extinct genus Glyptodon which had been recently added to the Museum of the Royal College of Surgeons ; and he then promised to give a full description of the skeleton, illustrated by appropriate figures, in a memoir to be pre- sented in due time to the Royal Society. The present communication con- sists of Part I., and Sections 1 and 2 of Part II., of the promised memoir. Part I. contains the history of the discovery and determination of the remains of the Hoplophoridz, or animals allied to, or identical with Glypto- don clavipes. Part IT. is destined to comprehend the description of the skeleton of Glyptodon clavipes (Qwen)—Hyplophorus Selloi? (Lund); and the Sections 1 and 2 now given contain descriptions of the skull and the vertebral column. The preliminary notice already published in the Proceedings (Dec. 18, 1862, vol. xii. p. 316) will serve as an abstract. | To face p. 108. Masi Minor. Masor. Minor. y,| VIII. EX, X, ee es eA Miata hie lila alee My L| cpp Ep ita te 4 5 | bb Bo 1 ie : 3 e o 3 4 tA yl tb) tid i ee G G ( 7 : 8 nig et |tCz ime {G 1d {ex ia 10 ie tbg | 11 D Ttdt ff x| 12 tes< ls bt | Et tGx B 14 f x| Bg {Dx He ae ‘ ae 15 eX iH XG ADS a Cx iid ke okX) axl em we | Cs) es. GK) axe | OX » eraiealZ IX Xx Magsor. E>) Gp E +E) Gb ae oo tA Cc TE tp +E) + +p | ch tebb gD t +c Il. Ii. By HA tH 1B ThE HB 4DE tt Tasie IV.—Schismatic Equivalents. [To face p. 108, qe =tBt Whence and 4|d = te x q ie =b btct = db fe = dx tt Bie bh cx = td qf = tet 4 tab = pe = ep fle = tf x q b= d btae = ted ‘ja = gx 4] te? = td bey =tf qb = tax | i =e se =i q 2 =te bifz = 9D be = tab | 4 tgp = fe b fx = tg ibd = ey q abb= ¢ bet = be =t# 4] fa) = tgp ten = tad tee | a eae | pee ey s= a = 7a ph = heen bib = @ b= te) lb b: = ite btbz = c Taste V.—Related Systems. Minor. JOR Minor. Magsor. Minor. Mason. Minor. IV. VI. VII. VIII. IX, X. fp abb td |tkp aoe Ob |tta> $8 tab ltt) 44 tab [tte ttab ite | | > tl : ebb t +d i fob tab oh tS BN} {tO ny) HE TD tie foo bbb t ee {D? | te? ig) {bd| tH 1G 1B) tte 19 te tap £ {D AD| tb) tab te | 4B) $ID tPF [tte tb tt Sie et tF tA tc [tta te + @ 5 ig BD e & e tO tH ig ua te te bp f iD F ig bb id | iG ¢B {D |te te tb e i? e@ ji A © f a | iD tre th td tft a Se ee , tb d |G B D b | E ¢G# B | tet “ec tet |tCHt {GH |tatt tet te eee |S fen ‘ B ibe F i > ae togtpe De tek deh ine ta te te tA OftE he e a At Of | t ft af |TDETFX At |tht tdi tt te tg tbh {tH Gr +B Cr ET G aft ct et | AT{Cx Epiffx af tex > ia iff HB DETEH| gh tb at BE Dg! et bt | ERIGx Be ltex ef 4 Deaton |p me rca) ia) SE rk gad] i fx|BEEDx Fx{tex be tax tot tte t +On +H +G: dag +At Cx tER fx tat cx Fx Ax Ox|tdx fx ax ton ttb ti +Gt +BE tD#| t +bt | TEx Gx tBr) cx tet Bx Cx EX Gx|]ax ex ex Iv. Vv. Vil. VIM. DX. x 1864.) Mr. Hartnup—Record of ihe Storm of Dec. 3, 1868. 109 Il. “On the Great Storm of December 3, 1863, as recorded by the Self-registering Instruments at the Liverpool Observatory.” By Joun Hartnup, F.R.A.S., Director of the Observatory. Com- municated by General Saprnz, P.R.S. Received January 21, 1864. [This Paper is accompanied by a diagram, which is deposited, for reference, in the Archives of the Royal Society, and of which the author gives the following explanation. | The accompanying diagram exhibits the strength and direction of the wind, the height of the barometer, and the rain-fall for three days pre- ceding, two days following, and during the great storm of December 3, 1863, as recorded by the self-registering instruments at the Liverpool Ob- servatory. The barometer-tracing is a facsimile of the original record pro- duced by King’s self-registering barometer ; the force and direction of the wind and the rain-fall have been taken from the sheets of Osler’s anemo- meter and rain-gauge ; the time-scale for the anemometer has been slightly increased to adapt it to that of the barometer, and the scale of wind-pres- sure for each five pounds has been made uniform, instead of leaving the spaces greater or less according to the strength of the springs as in the original record. The tracings of the recording-pencils for the direction of the wind and the rain-fall are faithfully represented, but it is scarcely possible to copy the delicate shadings and every gust recorded on the original sheets by the pencil which registers the force of the wind ; all the heavy pressures are, however, correctly represented, and may be taken from the diagram as accurately as from the original sheets. The figures at the bottom of the diagram show the readings of the dry- and wet-bulb thermometers and the maximum and minimum thermometers as recorded at the Observatory during the six days; the wet- and dry-bulb thermo- meters were read each day at 8 and 9a.m. and at 1, 3 and 9 p.m.; the registering dry thermometers were read and readjusted each day at | P.M. The time marked on the diagram for all the instruments is Greenwich mean time. For four days previous to the 30th of November the barometer had been high and steady, the readings ranging from 30°13 in. to 30°33 in., the latter at noon on the 29th being the highest ; from this time to midnight the fall was slow and pretty uniform; from midnight November 29 to midnight December 5 the changes of barometric pressure, the strength and direction of the wind, and the rain-fall are shown on the diagram. The fall _ of the barometer on the day of the great storm was rapid from midnight to 6 A.M.; heavy rain and hail fell from 3" 30™ to 7" 20™; and from 5" 50™ to 6° 45™ it was nearly calm, during which time the wind shifted from E. through S. to W. Between 6°45™ and 8" 15™ the pressure of the wind increased from 0 to 16 lbs. on the square foot, and at about twenty-five 110 Prof. H. J. 8. Smith—Criterion of Resolubilty, &c. [Jan. 28, minutes past eight it increased from 16 to 43 lbs. in the short space of two or three minutes ; the barometer, being at its minimum, suddenly rose about three-hundredths of an inch, and during the heaviest part of the storm it con- tinued to rise at the rate of about one-tenth of an inch an hour. The oscil- lations in the mercurial column, as will be seen by the diagram, were large and frequent during the storm, one of the most remarkable being imme- diately after 10" a.m. and nearly coincident with two of the heaviest gusts of wind ; the depression in this case amounted to between four and five hundredths of an inch, the rise following the fall so quickly that the clock moved the recording-cylinder only through just sufficient space to cause a double line to be traced by the pencil. III. “On the Criterion of Resolubility in Integral Numbers of the Indeterminate Equation f= aan? + ala! + allal2 1 Vhg! gl! +2blag" + Db" e!ev—0.” By H. J. StepHen Smiru, M.A., F.R.S., Savilian Professor of Geometry in the University of Oxford. Received January 20,1864. It is sufficient to consider the case in which / is an indefinite form of a determinant different from zero. We may also suppose that f is primitive, i. e. that the six numbers a, a’, a”, 6, 6’, 6” do not admit of any common divisor. We represent by © the greatest common divisor of the minors of the matrix of f, by AQ? the determinant of f, and by QF the contravariant of f, 7. e. the form (62°—ala")a2+ 2.605 QA? will then be the determinant of F, and Af its contravariant. By Q, A, and QA we denote the quotients obtained by dividing Q, A, and OA by the greatest squares contained in them respectively; w is any uneven prime dividing Q, but not A; 5 isany uneven prime dividing A, but not O; and @ is any uneven prime dividing both © and A, and consequently not dividing QA. We may then enunciate the theorem— «The equation f=0 will or will not be resoluble in integral numbers dif- ferent from zero according as the equations included in the formulee “ @-6. 0-0 CB-60) The symbols (5); (=) and (=) are the quadratic symbols of Legendre; the symbols (5) 5 4)” (Z) , ¢ ) are generic characters of f (see the Memoir of Eisenstein, “‘ Neue Theoreme der hoheren Arithinetik,” in his ‘Mathematische Abhandlungen,’ p. 185, or in Crelle’s Journal, vol. xxxv. p. 125). The theorem includes those of Legendre and Gites on the resolubility 1864.] Magnetograph-traces at Kew and Lisbon. 111 of equations of the form az®*+a'z’*+a"z'2=0 (Legendre, Théorie des Nombres, vol. i. p. 47; Gauss, Disq. Arith. arts. 294, 295, & 298). Itis equally applicable whether the coefficients and indeterminates of f are real integers, or complex integers of the type p+ qi. It will be observed that if f, f', f" ...are forms contained in the same genus, the equations /=0, f/=0, f’=0, &e. are either all resoluble or all irresoluble. IV. “Results of a Comparison of certain traces produced simulta- neously by the Self-recording Magnetographs at Kew and at Lisbon ; especially of those which record the Magnetic Disturb- ance of July 15, 1863.” By Senhor Carztto, of the Lisbon Observatory, and Batrour Stewart, M.A., F.R.S. Received January 14, 1864. The National Portuguese Observatory established at Lisbon in connexion with the Polytechnic School, and under the direction of Senhor da Silveira, has not been slow to recognize the advantage to magnetical science to be derived from the acquisition of self-recording magnetographs. Accordingly that institution being well supported by the Portuguese Government, de- spatched Senhor Capello, their principal observer(one of the writers of this communication), with instructions to procure in Great Britain a set of self- recording magnetographs after the pattern of those in use at the Kew Ob- servatory of the British Association. These instruments were made by Adie of London, and when completed were sent to Kew for inspection and verification, and Senhor Capello resided there for some time in order to become acquainted with the photographic processes. The instruments were then taken to Lisbon, where they arrived about the beginning of last year, and they were forthwith mounted at the Observatory, and were in regular operation by the beginning of July last. It had been agreed by the writers of this paper that the simultaneous magnetic records of the two observatories at Kew and Lisbon should occa- sionally be compared together, and the opportunity for such a comparison soon presented itself in an interesting disturbance which commenced on the 15th of July last. The curves were accordingly compared together, and the results are embodied in the present communication. We shall in the first place compare the Kew curves by themselves, se- condly the Lisbon curves in the same manner, and lastly the curves of the two Observatories together. Comparison of Kew Curves. The disturbance, as shown by the Kew curves, commenced on July 15th, at 9" 13™5 G.M.T., at which moment the horizontal-force curve recorded an abrupt augmentation of force. The vertical component of the earth’s magnetic force was simultaneously augmented, but to a smaller extent ; while only a very small movement was visible in the declination curve. 112 Senhor Capello and Mr. B. Stewart—Comparison of (Jan. 28, The disturbance, which began inthis manner, continued until July 25th, if not longer; but during the period of its action there was not for any of the elements a very great departure from the normal value; probably in this respect the declination was more affected than either of the other com- ponents. While frequently there is an amount of similarity between the different elements as regards disturbances of long period, yet there is often also a want of likeness. If, however, we take the small but rapid changes of force, or peaks and hollows, as has been done by one of the writers of this paper in a previous communication to the Royal Society (Phil. Trans. 1862, page 621), we shall find that a disturbance of this nature which in- creases or diminishes the westerly declination at the same time increases or diminishes both elements of force. This will be seen more distinctly from the following Table, in which + ‘denotes an increase and — a diminution of westerly declination, horizontal, and vertical force respectively, and the pro- portions are those of the apparent movements of the elements on the pho- tographic paper. Taste I. : . | Vertical-force Date a a hs Declination. a ae change =unity bs each instance. a 1363. July 17 2 46°5 —1I'o —I'9 —I'o 17 2. 495 —I'l —2°0 —I'o 17 3 21°5 —tI1‘o —2”°0 —t1‘'o 17 7 Sa"G not similar. —2°O —I'o 17 16 13°0 +3°5 +2°2 +1°0 18 24,.22°5 +3°0 +2°0 + 1'0* 19 © 35°5 not similar. +13 +10 19 2 %3°5 +11 +1'9 +1'o Ig 2 38°70 Ard a7 +10 19 S225 + 1°0 +1°9 +t1'o I9 {| 17 51°0 +2°8 +2°0 +1of 19 | 18 o'o +3°6 +2°0 +1'of Ig | 20 29°5 ar ai aan +1'o 20 3 21°0 +10 +1°6 +10 20 | 18 52°5 +40 Ez +1°0 21 © 22°0 —16 —2°3 —I‘0 2 | 21 to —14 —2'1 —1r°o 2. 5 21 5 38°0 +1°4 +2°'0 +10 22 | I9 20°5 —4'0 —2°0 —10 22 | 39.3275 —3°4 —2°'0 —I'o 22] 2% 40°O +3°5 +2°0 | +10 23 | 18 34°5 +3°5 +2°2 +1'0 23.| 19 26°5 +3°4 +2°0 +1ot 24 3 355 +1'2 +2°0 +10 24.| 16 44°5 +32 +2'0 +rot * Doubtful. + Vertical force too small to be accurately measured, but horizontal- force change reckoned =2"0, 1864. | Magnetograph-traces at Kew and Lisbon. = = ‘118 From this Table it will be seen that the signs are always alike for the ‘different elements, and also that the small and rapid movements of the horizontal force are double of those of the vertical force—a result in con- formity with that already obtained by one of the writers in a previous com- munication. On the other hand, the declination peaks and hollows do not bear an invariable proportion to those of the horizontal and vertical force, but present the appearance of a daily range, being great in the early morn- ing hours, and small in those of the afternoon. Indeed this is evident by a mere glance at the curves, which, it so happens, present unusual facilities for a comparison of this nature. Comparison of Lisbon Curves. 1. Declination- and vertical-force curves.—The peaks of the waves, or the elevations in the curve of declination, are always shown in hollows or depressions in the vertical-force curve, and vice versd. We have never seen an instance to the contrary either in the curves under comparison or during the whole time of the operation of these instruments. This curious relation is exhibited in a Plate appended to this communication, from which it will be seen that we have not only a reversal, but also a very nearly constant ratio between the ordinates of the two curves. At Lisbon therefore an increase of westerly declination corresponds to a diminution of vertical force, and vice versd; also an almost constant proportion obtains between the corresponding changes of these two elements. 2. Bifilar and Declination Curves.—July 15. A great disturbance, which at 8" 37™ Lisbon mean time, or 9" 13"°5 Greenwich mean time, abruptly and suddenly augmented the horizontal force. The curve of the declination continues nevertheless nearly undisturbed for about 30 minutes after this, and only at 9" 41™5 G.M.T. it com- mences to descend very slowly. July 16.—At about 13° 6" G.M.T., avery regularly shaped prominence of some duration occurs in the declination, but is quite invisible in the hori- zontal force. July 17.—We see in the bifilar curve half-a-dozen small peaks repro- duced in the declination in the same direction, but to a smaller extent. July 18.—One or two accordant peaks. A large prominence of some duration in the declination at about 17° 56" G.M.T. is reproduced as a slight depression in the horizontal force. July 19.—A reproduction in the declination of several small peaks of the horizontal force ; nevertheless there are others also small which one does - not see there, or only reproduced to a small extent. Not much accordance between the great and long-continued elevations and depressions. July 20.—An accordance between the small peaks. July 21.—The same. July 22.—The curve is well marked with small peaks. Coincidence of several small peaks, but a want of agreement between the more remarkable 114 Senhor Capello and Mr. B. Stewart—Comparison of [Jan. 28, peaks. The peaks of the horizontal force more developed than those of the declination. July 23.—The same appearance of the horizontal-force curves. One remarks on 22nd and 23rd that the small peaks of the declination and horizontal-force are more numerous and more developed in the morning hours. July 24.—Agreement between the small peaks. A strong disturbance about 103" G.M.T., no agreement between the waves. A well-marked pro- minence of declination (154) does not alter at all the horizontal-force curve. We derive the following conclusions from the comparison which we have made between the Lisbon curves :— 1. The waves and the peaks and hollows of declination are always repro- duced at the same instant in the vertical force, but in an opposite direction ; that is to say, that when the north pole of the declination-needle goes to the east, the same pole of the vertical-force magnet is invariably plunged below the horizon, and vice versd. During five months of operation of these instruments there has not been an example of the contrary. 2. The more prominent disturbances of the horizontal force do not in general agree with those of the declination or vertical force either in dura- tion or time. It is certain that when one of the two elements (bifilar or declination) is disturbed, the other is also; and sometimes one appears to see even for several periods of one of the curves, an imitation of the general march of the other; but when this is examined a little more minutely, and rigorous measures are attempted, one easily perceives that the phases do not arrive at the same time, but sometimes later and sometimes earlier, without any fixed rule. | In the same curve one generally sees contradictions of this kind. Nevertheless it is certain that the agreement in direction and time is more complete when the elevations or depressions are of shorter duration. 3. The small peaks and hollows are generally simultaneous for the three curves. The direction of these is the same for the horizontal force and declination, while that for the vertical force is opposite. The ratio in size of the peaks and hollows is generally variable between the horizontal force and the declination, while it is always constant be- tween the latter and the vertical force. Our next deduction requires a preliminary remark. It has been shown by General Sabine, that if the disturbances of declination at various places be each divided into two categories, easterly and westerly, these obey dif- ferent laws of daily variation, this difference not being the same for all stations. This would seem to indicate that for every station there are at least two simultaneous disturbing forces acting independently, and superposed upon one another. This interesting conclusion, derived by General Sabine, appears to be ENDER Game pati ir ad HGP RINS FH eS . SAE) ea PRIOR HEP [To face page 115. | TaBLE [I.—Comparison of the time of the | hm |hm |hm |h m|h KEW. jisi3.- 9g 10 9 15 9.34 — |e Aga aa July 15 ...|4 Lisbon...... Baa7 8 42 Ota 9 43/11 | Differences | 0 33 ora3 Ge 2) 6 340 'or OW cccenee 5: O 23 G32 I 24 I 40) 2 Sa oa (5) as) # Ve usbon ac aga 23 15H. oO 46 7 2m Differences | 0 39 °o 38 ° 38 © 38] o RC Weer I 30 x 32 T 37 1 5012 17 ...|4 Lisbon...... O57 I oO £83 1 22 Differences | 0 33 O 32 © 34 © 34| 04 KW osesaes. I 23 1 48 es 5 52| 64 7S t2.| 4 uisbon..-... ° 47 I iI I 29 5 151464 Differences | o 36 O 37 o 36 G37 1-0 3 (18) | Mow 2i2.2. Xo O 14 2ate) 3 21 44 FO ox (18) (18) ! Lisbon...... 22, On| 23 ak I 338 2, Aan \ Differences | o 34 O 33 O 34 Oo 341509 Kewitie- ss O 12 2 42 3°19°5 | 4 Ts aS es (19) | ins Wdisboma: 25 23 40 219 2 45°5 | Boaz eee Differences | 0 32 on ge 8) 5 O 33) 84 WEWecc cass I 32°5 | “2-00°S | 5 36°5.) F oolmo rs 21 ...|4 Lisbon...... O 57 Te24cs ss CO 6 22/10 14 Differences | 0 35°5 | © 36 Oo 36°5 | © 381) ola TOW iis oe as I 24 4 02 5 6 8 26.9295 22 eels SOO. fas. © 51 3 29 AS 7 53112 2 Differences | 0 33 O193 6.133 OQ 33) soa Kiewieissc:. 232, 3 11 3°35 7 Agee 23 +..| Lisbon...... TS 7, 2237, epee 6 29) beg = | Differences | 0 35 O 34 © 34 © 35sno 3¢ WCW -2 es 3 29°5 | 4 21 4 50 5 2sls s¢ ANS Lisbon...... 2 56 3 49 4 19 453) B32 Differences | 0 33°5 | © 32 O38 © 32) © 34 1864. ] Magnetograph-traces at Kew and Lisbon. 115 verified by the behaviour of the Lisbon curves. From the relation, always invariable, between the waves of declination and vertical force, as well as from the almost total absence of agreement between these two curves and the horizontal force, one has a right to conclude — 1. That there is approximately only one independent force which acts at Lisbon, if we consider the vertical plane bearing (magnetic) east and west. Now the ratio of the disturbing forces for the vertical foree and declination is, in units of force, between 26 : 48 and 26: 36. This would give the inclination of the resultant between 29° and 36°. 2. The absence of agreement in time, and the variability in direction, between the waves of the horizontal force and those of the declination and vertical force, appear to lead to the conclusion that there is another dis- turbing force besides that already mentioned, which acts in the direction of the magnetic meridian and almost horizontally. Comparison of the Kew and Lisbon Curves (14-24 July). 1. Horizontal force (north and south disturbing force).—The curves of the horizontal force at Kew and at Lisbon exhibit a very great simili- tude *, as will be seen at once from the Plate appended to this communi- cation. Almost all the waves and peaks and hollows are reproduced at both places. At the same time one does not see the same resemblance during the great disturbance of 15th July. In the commencement, and for the first four hours, there is a resemblance for all the waves, but from that time until 193° G.M.T. one remarks little agreement between the different elevations and depressions. But from 193" until the end of the dis- turbance the likeness reappears. There are, however, one or two cases of small resemblance in the other curves, but these are of short duration. In order to demonstrate the similarity between the two curves, reference is made to Table II., in which the principal points are compared together with respect to time; that employed being the mean time for both stations. From this Table it will be found that the average difference between the local times of corresponding points is 343, while that due to difference of longitude is 35"°3. We attribute this apparent want of simultaneity to various causes :— (1) Loss of time in the commencement of movements of the registering cylinder. (2) Difficulty in estimating precisely the commencement of certain curves. (3) It was only in the morth of August that the exact Lishon time of the astronomical observatory was obtained by a telegraphic connexion. (4) To these must be added the uncertainty in estimating the exact turning-point of an elevation or depression of a blunt or rounded form. * We speak of the Kew curves reversed so as to have their base-lines above, the dis- position of the registering arrangement at Kew being the opposite of that at Lisbon. This reversal has been made in the Plate which accompanies this paper. VOL. XIII. KR [To face page 115. Tasxe I1.—Comparison of the time of the principal corresponding points of the Horizontal-Force Curves at Kew and at Lisbon. July 15... 7) taas Sites 19 20... 21... 22 «.. PI eel TM Differences OS8F ORY OWF OPN OD h 9 9 ° I (eo) te} wo OHR OHM OFF OBWW OFM OUN On m 34 I 33 24 46 38 37 K) 34 h fe) 9 ° w - ON OFM OF 013K ODN OW m a7) 43 34 40 h II II ° rs OND OFPN OW m 43 9 34 m 43 10 WW of OXvP Ow w aS h 19 0) ° 4 ONnR OW m 42 fo) 32 09 31 38 51 16 35 h m 20 40 20 9 © 31 4 42 h m OwF 0” oc 20 28 h m 18 43 18 6 @ sh7/ 5 43 5 99 © 33 21 58 21 24 © 34 21 52 21 19 @ 3h33 18 41 13 8 2) 233} h m h m 6 59 6 25 © 34 hm h m 7 22 6 48 @ Mt hm h m| II oo IO 25 O35) II 12 10 38 © 34 hm Ir 51 Ir 18 ©) 543 116 Senhor Capello and Mr. B. Stewart—Comparison of [Jan. 28, The following Table exhibits approximately the proportion between the disturbance-waves of the horizontal force at Lisbon and at Kew. TABLE ITI. Proportion between the disturbance-waves of the horizontal force reduced at both places Date. to English units (Lisbon wave = unity). July 15. Variable between 1 : 1°3 and 1: 1°9 16. . 1 21°83 and 1-43 Wee ae 1o-el 26 18. 1: 1°9 and 1: 2°5 19. 156 18. as 175 19, ae T2125 20. a 1 4°7 21. a | As te PAs aS Lo14 Ze. “ Le bs DA. . 12 146 — Dn Mean [To face page 116. Curves at Kew and Lisbon. TasiE [V.—Comparison of the times of the principal corresponding points of the Declination Curves at Kew and Lisbon. [To face page 116. Ke Wersasenvern July 15 ...|4 Lisbon......... Differences .. 1);/60c09n000700 20 Lisbon......... Differences ( Kew Ae a0bO0RETou 17 «14 Lisbon......... Differences Kew .......e cee 18 .../4 Lisbon... Differences Kew............ 19 .../4 2 Lisbon......... KE) ce} 2m 9211) TE} TA mooe cor Differences ... Kew....secccens PP on Lisbon......... Differences ... Kew........005 23 ...|4 Lisbon......... Differences Wewasecs <} Orw ornw 00 wo = RP OWNW OF cy OWP OWW CH h 17 17 fe} 3 3 ° on OnD ODN OF m 47 038 39 51 16 35 45 h 18 17 ° 12 C) oman OMmnm ON h 18 18 ° 14 m 53 16 37 II 37 34 20 h m 19 49 19 12 O83 7) 16 22 15 45 ©) EW/ 3 59 h 25 21 ° 18 m 51 17 34 26 52 34 40 h 20 ~ ow m h m 21 16 20 42 hm h m 21 4 20 31 h m 18 37 18 6 © 31 h m 19 51 19 20 © 31 1864. | Magnetograph-traces at Kew and Lisbon. 117 ‘It would thus appear that the declination at Kew, judging from the waves, _ is subject to greater disturbing forces than at Lisbon in the proportion of 16:1. This ratio is not, however, quite so great as that for the hori- zontal force. 3. Vertical disturbing force.—The curves of vertical force are nearly quite dissimilar. Sometimes the general march of the curves appears to coincide durmg some time; but in these cases we do not find an appre- ciable general agreement for the majority of the various points of the wave, On the other hand, the small peaks and hollows of the Kew curves are generally reproduced in those of Lisbon, but in the opposite direction, that is to say, a sudden augmentation of the vertical force at Kew corresponds to a sudden diminution of the same at Lisbon, and vice versd. In Table VI. we have a comparison of the principal points of the vertical- force curves with respect to time. Tasie VI.—Comparison of the time of the principal corresponding points of the Curves of Vertical Force at Kew and Lisbon. Kews .:...: hm hm hm |hm hm hm July 15 .../4 Lisbon No similarity. Differences Weiyeeie ak. 2 21 ON 07 | Bea 50 VASHON... I 47 20 45 Differences | 0 34 On 32 1G pees 2 AGTO iiss 3220 Ti Suh 16) 115 Pes Lasbon:..... Padi: id ahs 2 19°5 2 49 gi OE 15 39 Differences} 0) 33°5 | © 32°5 Oo 31 O 34 O 32°5 GOW. «shi: 7%O0 9*33 Dye ples 15 ...|4 Easbon...... 6 29 g 00 20 50 Differences | o 31 Ona O32 RCW i ace: Ox IO Zale, 2 Or hy a aor 17, AGUS Ey uso: S 1G Sis | hon eee 23 38 tr 36°5 2) OB, 2 46 17 18 yin HLs Differences} 0 32 © 35°5 0.3475, 0, 335 © 3m°5.| @ 33°5 TEOW. csies os 3 19°5 18 51 22 03 20M Lisbon. -.:,. 2 46°5 18 18°5 eat Differences | 0 33 Or 325 O 32 HEE WE cci08 i 3 2 10 5 36°5 6 43 | 2 te nen AS bOM: ss. I 36 5 O15 6 09 Differences | o 34 O 35 O 34 ONES. ae s2n 5 06 8 24 1256.) TOc1H}, org iam, |e 48s Be | 5 ISOM can: Ang Wo 5y 2 922 18 46 Tied, BS) 2; Bee) Differences | 0 33 @' 33 O 34 oO 33 O 34 © 34°5 Hews 3.2: 18 33 18 37 ps icy. all DH i. Lisbon...... 18 oI 18 05 222, Differences | 0 32 O 32 O 32 Kew......... 2° 29°5 4 10 5 59 16 43 24 ...|4 Lisbon...... 2G 8) 37 Ge ZOrs | TOSS Differences | 6) 4975 fo) 33 « (ionig’s O 34°5 | ee ne * Only the points marked with this sign are in the same direction, all the athena are in the opposite direction ; that is to say, an augmentation of force at Kew corresponds to a diminution of the same at Lisbon, and vice versa. 118 Senhor Capello and Mr. B. Stewart—Comparison of [Jan. 28, From this Table it will be seen that the average difference between the local times of corresponding points is 33™°1, while for the horizontal force this was 34"°3, and for the declination 34™°0, the mean of the three being 33™°8. The measurements from which these numbers were obtained were made at Lisbon independently for each element: another set of measure- ments, made at Kew, but of a less comprehensive description, gave a mean difference in local time of 33"°7, which is as nearly as possible identical with the Lisbon determination. We have already observed that we attri- bute the difference between 33"°8 and 35-3, the true longitude-difference of local times, to instrumental errors, and not to want of simultaneity in the corresponding points. In Table VII. we have a comparison in magnitude and sign of the peaks _and hollows at the two stations. From this Table it will be seen that the magnitude of these is generally greater at Kew than at Lisbon. The curious fact of the reversal in direc- tion of the vertical-force peaks between Kew and Lisbon has been already noticed. We shall now in a few words recapitulate the results which we have obtained. 1. In comparing the Kew curves together for this disturbance, the peaks and hollows of the horizontal force always bear a definite proportion to those of the vertical force, the proportion being the same as that ob- served in previous disturbances. On the other hand, the declination peaks and hollows do not bear an invariable proportion to those of the other two elements, but present the appearance of a daily range, being great in the early morning hours, and small in those of the afternoon. The peaks and hollows are in the same direction for all the elements. 2. In comparing the Lisbon curves together, the elevations of the decli- nation-curve always appear as hollows in the vertical-force curve, and vice versd, and there is always a very nearly constant ratio between the ordi- nates of the two curves. The horizontal-force curve, on the other hand, presents no striking likeness to the other two. We conclude from this that there are at least two independent disturbing forces which jomtly in- fluence the needle at Lisbon, but that the declination and vertical-force elements are chiefly influenced by one force. The peaks and hollows are generally simultaneous for the three curves. The direction of these is the same for the horizontal force and declina- tion, while that for the vertical force is opposite. The ratio in magnitude of the peaks and hollows is generally variable between the horizontal force and the declination, while it is always constant between the latter and the vertical force. : 3. When the Kew and Lisbon curves are compared together, there is a very striking likeness between the horizontal-force curves, one perhaps somewhat less striking between the declination-curves, and very little like- ness between the vertical-force curves. It is perhaps worthy of note that 119 Magnetograph-traces at Kew and Lisbon. 1864.] 0°066 o7041 K (17 and 18) For the other days ursebeacrnde ceeteeedie—=O1OHO Foo cadausaacheaucneeeeeini=sOro2l0 aT O°O41 K izontal force ... Declination ......... K=0'024 ... Vertical force ...... K=0'024 ... 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Oo o00I00.0 | $zo.0-+ eels : 0Z100.0 | o£0.0-+ | goroo.0 |$¥o.0+ gr 09000.0 | $10.0-++ | gg100.0 |oL0.0+ obo00.0 | @10.0— | 0z100.0 joSo.0o— 7 Zr Ane oF000.0 | 010.0— | 96000.0 | ov0.0 — f oF000.0 | O10.0— | 96000.0 | ov0,0— } “qrum : “yun A ysysug | PML | qsysug | PM “U0gsl'y “MOM "TOT}CUIpII ‘TOGSTT PUL MOY JV SMOT[OF] pur syvog oy} Jo udig pus opnylucep— ]TA FIAV], 120 Magnetograph-traces at Kew and Lisbon. [Jan. 28, the Lisbon horizontal-force curve, in which we may suppose two indepen- dent forces to be represeated, is probably on the whole the most like the corresponding Kew curve. Corresponding points occur at the samie abso- lute time for both stations. The disturbance-waves for the horizontal force and declination are greater at Kew than at Lisbon. The Kew peaks and hollows are simultaneously produced at Lisbon in all the elements, but to a smaller extent than at Kew; also the direction is reversed in the case of the vertical force, so that a sudden small increase of vertical force at Kew corresponds to a diminution of the same at Lisbon. The writers of this paper are well aware that before the various points alluded to in their communication can be considered as established, a more extensive comparison of curves must be made. But as the subject is new and of great interest, they have ventured thus early to make a preliminary communication to the Royal Society. They will afterwards do all in their power to confirm their statements, which in the meantime they submit to this Society as still requiring that proof which only a more prolonged in- vestigation can afford. Note regarding the Plates. Increasing ordinates denote increasing westerly declination, and also in- creasing horizontal and vertical force. The following are the scale coefficients applicable to the different diagrams :— Horizontal force, Kew. One inch represents 0:041 English unit. Ditto Lisbon. > 4; is 0:035 3 for July 15 Ditto do. a fe 0:066 ae for July 17. Ditto do. a * 0:041 ,. for the other curves. Declination do. os * 0:040 ¥, Vertical force do. es a 0°026 * ARORA IY callie rn cc arta ik, coh WN ot iy ect cumin LAL Sia Proc feoy-500. Vol XHL PULT. Dectanatim, Lusbon ie CMT. s October 8. | = | = | ~ \ S =~ ! 2 Vertical Force, Lisbon | : Ss ee s S SHON Socrates | SS eclnatiy, Lisbon, SGI, 22% 87" GMT t I 1 1 ' ¥ ' ! 1 1 ! 1 i) | ee eee Vertucal, force, Lishor: | Dechination, Lisbon: Hor. Force, Hew. Vertical Force, Lisbon: /. Baswre, lth; . "it a ey Ah. ae Oetcbein G0: PR Li Ee @ aati 22 - Lisbon. bs is = 3 ra A x Vertical Force, Lisbon: } | D SCAT, SMG ML. Sil Zh. P50 CML. LEGMT. LW Duby UY 5 LHD bby Uh PY? hee Lt oL FL Oct?8. Th UOQSLT "09.L0,J LOH ' THD nulB UM : yp LHe Dubted U4ay es way “ODIO toy Fis 0a eres MOQXNT, BILD LO LWIutlyo 14 if u“ : ie ae MO OIL] LOL Zoe se Tioulgg were §— MOET P07 WO 1709 aly? 06 2 oe oa | UD gS a0 MOT re L 9 £ as Mal L : F, m9 t99 eA LHI uu 4 < 2 S LHD Fiubly AL ‘ MOY 2240 sop] THO uhh AY MOY “0M “LOFT 1864.] 121 February 4, 1864. Major-General SABINE, President, in the Chair. The following communication was read :— “Tixperiments to determine the effects of impact, vibratory action, and a long-continued change of Load on Wrought-iron Girders.” By WixiiaM Farreairn, LU.D., F.R.S. Received January 2C, 1864, (Abstract.) The author observes that the experiments which were undertaken, nearly twenty years ago, to determine the strength and form of the Tubular Bridges which now span the Conway and Menai Straits, led to the adoption of cer- tain forms of girder, such as the tubular, the plate, and the lattice girder, and other forms founded on the principle developed in the construction of these bridges. It was at first designed that the ultimate strength of these structures should be six times the heaviest load that could ever be laid upon them, after deducting half the weight of the tubes. This was considered a fair margin of strength ; but subsequent considerations, such as generally attend a new principle of construction with an untried material, showed the expediency of increasing it; and instead of the ultimate strength being six times, it was in some instances increased to eight times the weight of the greatest load. The proved stability of these bridges gave increased confidence to the engineer and the public, and for several years the resistance of six times the heaviest load was considered an amply sufficient provision of strength. But a general demand soon arose for wrought-iron bridges, and many were made without due regard to first principles, or to the law of proportion ne- cessary to be observed in the sectional areas of the top and bottom flanges, so clearly and satisfactorily shown in the early experiments. The result of this was the construction of weak bridges, many of them so ill-proportioned in the distribution of the material as to be almost at the point of rupture with little more than double the permanent load. The evil was enhanced by the erroneous system of contractors tendering by weight, which led to the introduction of bad iron, and in many cases equally bad workmanship. The deficiencies and break-downs which in this way followed the first successful application of wrought iron to the building of bridges led to doubts and fears as to their security. Ultimately it was decided by the Board of Trade that in wrought-iron bridges the strain with the heaviest load should not exceed 5 tons per square inch; but on what principle this standard was established does not appear. The requirement of 5 tons per square inch did not appear sufficiently definite to secure in all cases the best form of construction. It is well VOL. XIII. L 122 Mr. Fairbairn—Ezperiments on the Effects [Feb. 4, known that the powers of resistance to strain in wrought iron are widely different, according as we apply a force of tension or compression ; it is even possible so to disproportion the top and bottom areas of a wrought-iron girder calculated to support six times the rolling load, as to cause it to yield with little more than half the ultimate strain or 10 tons on the square inch. For example, in wrought-iron girders with solid tops it requires the sectional area in the top to be nearly double that of the bottom to equalize the two forces of tension and compression ; and unless these proportions are strictly adhered to in the construction, the 5-ton strain per square inch is a fallacy which may lead to dangerous errors. Again, it was ascertained from direct experiment that double the quantity of material in the top of a wrought- iron girder was not the most effective form for resisting compression. On the contrary, it was found that little more than half the sectional area of the top, when converted into rectangular cells, was equivalent in its powers of resist- ance to double the area when formed of a solid top plate. This discovery was of great value in the construction of tubes and girders of wide span, as the weight of the structure itself (which increases as the cubes, and the strength only as the squares) forms an important part of the load to which it is subjected. On this question it is evident that the requirements of a strain not exceeding 5 tons per square inch cannot be applied in both cases, and the rule is therefore ambiguous as regards its application to different forms of structure. In that rule, moreover, there is nothing said about the dead weight of the bridge ; and we are not informed whether the breaking- weight is to be so many times the applied weight plus the multiple of the load, or, in other words, whether it includes or is exclusive of the weight of the bridge itself. These data are wanting in the railway instructions ; and until some fixed rinciple of construction is determined upon, accompanied by a standard measure of strength, it is in vain to look for any satisfactory results in the erection of road and railway bridges composed entirely of wrought iron. The author was led to inquire into this subject with more than ordinary care, not only on account of the imperfect state of our knowledge, but from the want of definite instructions. In the following experimental researches he has endeavoured to ascertain the extent to which a bridge or girder of wrought iron may be strained without injury to its ultimate powers of resist- ance, or the exact amount of load to which a bridge may be subjected with- out endangering its safety—in other words, to determine the fractional strain of its estimated powers of resistance. To arrive at correct results and to imitate as nearly as possible the stram to which bridges are subjected by the passage of heavy trains, the apparatus specially prepared for the experiments was designed to lower the load quickly upon the beam in the first instance, and next to produce a considerable amount of vibration, as the large lever with its load and shackle was left suspended upon it, and the apparatus was sufficiently elastic for that purpose. 1864.] | of Impact, &c. on Wrought-Iron Girders. — 123° The girder subjected to vibration in these experiments was a wrought-iron _ plate beam of 20 feet clear span, and of the following dimensions :— Area of top... 6.5.6 ssceeecs eee ee 4°30 Square inches. Fives Gl DOCCON 6 BUEN as 0 Ses seey 240 i Area of vertical web .......... ve os 290 ye Total sectional area ........ cece ce 8°60 Z, Depth ssw... ss Pe Nee bbe Ea les 16 inches. DION, a a's ww u's we c- f CWhe DOTS Breaking-weight (cileulated).. : wwcece’ 12 tons: The beam having been loaded oil 6 6643 Ibs. equivalent to one-fourth of the ultimate breaking-weight, the experiments commenced as follows :— Experiment I. Experiment on a wrought-iron beam with a changing load equivalent to one-fourth of the breaking-weight. Number of | Deflection Date. changes of | produced by Remarks. Load. Load. 1860. March 21 ...... ° O17 Strap loose on the 24th March. eepEIe f 2.24.20. 202,890 O17 Strap broken on the 2oth April. May Tees... ceeees 449,280 o°16 | May 14 ......008 596,790 0°16 The beam haying undergone about half a million changes of load by working continuously for two months night and day, at the rate of about eight changes per minute, without producing any visible alteration, the load was increased from one-fourth to two-sevenths of the statical breaking- weight, and the experiments were proceeded with till the number of changes of load reached a million. Experiment IT. Experiment on the same beam with a load equivalent to pelos ett of the breaking weight, or nearly 33 tons. 3 Number of | Deflection, | Date changes of | in inches. Remarks. Load. 1860. May 34) ..023..0 fe) 0'22 In- this experiment the number of JHE GQ)’. 25 uk. 236,460 o'21 although the beam had already un- dergone 596,790 changes, as shown in the preceding Table. JUNE 26 ..cccc0e- 403,210 0°23 ‘The beam had now suffered one mil- ; | lion changes of load. Ls es eae 85,820 0°22 changes of load is counted from o, EZ 124 My. Fairbairn—Ezperiments on the Effects [Feb. 4, After the beam had thus sustained one million changes of load without apparent alteration, the load was increased to 10,486 lbs., or 2ths of the breaking-weight, and the machinery again put in motion.. With this addi- tional weight the deflections were increased, with a permanent set of *05 inch, from *23 to *35 inch, and after sustaining 5175 changes the beam broke by tension at a short distance from the middle. It is satisfactory here to observe that during the whole of the 1,005,175 changes none of the rivets were loosened or broken. The beam broken in the preceding experiment was repaired by replacing the broken angle-irons on each side, and putting a patch over the broken plate equal in area to the plate itself. A weight of 3 tons was placed on the beam thus repaired, equivalent to one-fourth of the breaking-weight, and the experiments were continued as before. Experiment IIT. Number of 6 Permanent Date. changes ee set, in Remarks. filioad.9| °° ~~ al inmehes. 1860, ; August 9 ss... TSO) Visaeaesene He ovremeaay .. |Dhe load during these changes was equivalent to 10,500 lbs., or 4'6875 tons at the centre. With this weight the beam took a large but unmea- sured set. August 12 ...... 12,950 During these changes the load August 13 ...... 25,900 0°22 % in the beam was 8025 lbs., or 3°58 tons. August 13 sss 25,900 o'18 fo) Load reduced to 2°96 tons, or December 1 ...| 768,100 o'18 O’OI _ th the breaking-weight. 1861. Mareh2. cscs cee 1,602,000 0°18 O'OI IVI GA. a cians ont 2,110,000 O°17 O'OI September 4 ...| 2,727,754 O°17 oor October 16...... 3,150,000 O°17 O‘oI At this point, the beam having sustained upwards of 3,000,000 changes of load without any increase of the permanent set, it was assumed that it might have continued to bear alternate changes to any extent with the same tenacity of resistance as exhibited in the foregoing Table. It was then determined to increase the load from one-fourth to one-third of the breaking-weight ; and accordingly 4 tons were laid on, which’ increased the deflection to *20. 1864. ] of Impact, &c. on Wrought-Iron Girders. 125 Experiment IV. Number of : Permanent Date. changes ne set, Remarks. of Load. in inches. : 1861. October 18...... fo) 0°20 November 18...| 126,000 0°20 ° December 18...) 237,000 0°20 1862. MEMEO OF yese-| 313,000 | Psiiececsl | cei weeeee | Broke by tension across the bottom web. Collecting the foregoing series of experiments, we obtain the following summary of results. Summary of Results. | = Weight Strain cS on mid- Sapien per sq. pa Dede a Date. dle of the) 4) if inch |P& *4: tion, in Remarks, iE beam, in | “22°85 lon bot-| 2 | inches. fe} p24.) 0b, Load: on top. 7, tons. tom. 1| From March 21 to May 596,790) 4°62 | 2°58 | *17 14, 1860... 2| From May 14 to June 403,210] 5°46 | 3°05 | ‘23 26, 1860... 3| From July Broke by tension a short 25 to July GI 7a of Fee 4:08 35 distance from the cen- 23, 1860... tre of the beam. Beam repaired. 4| Aug. 9, 1860) 4°68 158| 7°31 | 4°08 .... | The apparatus was acci- RP Aue: 11 & 12) > 3°58 25.JAZ) 5G | giz | 522 dentally set in motion. 6 13, 1860 to 3,124,100) 4°62 | 2°58 18 Oct. 16, 1861 From Oct. 18, 1861 toJan.| + 4°00 413,006) O25") | 374G |)! "20 9, 1862 From Aug. 2°96 Broke by tension as be- fore, close to the plate riveted over the pre- vious fracture. From these experiments it is evident that wrought-iron girders of ordi- nary construction are not safe when submitted to violent disturbances equivalent to one-third the weight that would break them. They, however, exhibit wonderful tenacity when subjected to the same treatment with one- fourth the load ; and assuming therefore that an iron girder bridge will bear with this load 12,000,000 changes without injury, it is clear that it would require 328 years at the rate of 100 changes per day before its security was affected. It would, however, be dangerous to risk a load of one-third 126 The Rev. J. Bayma on Molecular Mechanics. [Feb. 11, the breaking-weight upon bridges of this description, as, according to the last experiment, the beam broke with 313,000 changes ; or a period of eight years, at the same rate as before, would be sufficient to break it. It is more than prohable that the beam had been injured by the previous 3,000,000 changes to which it had been subjected ; and assuming this to be true, it would follow that the beam was undergoing a gradual deterioration which must some time, however remote, have terminated in fracture, February 11, 1864. Major-General SABINE, President, in the Chair. The following communications were read :— I. “On the Calculus of Symbols.—Fourth Memoir. With Applica- tions to the Theory of Non-linear Differential Equations.” By W.H.L. Russexrz, A.B. Communicated by Professor Cayipy. Received July 31, 1863. (Abstract. ) In the preceding memoirs on the Calculus of Symbols, systems have been constructed for the multiplication and division of non-commutative symbols subject to certain laws of combination ; and these systems suffice for linear differential equations. But when we enter upon the consideration of non- linear equations, we see at once that these methods do not apply. It becomes necessary to invent some fresh mode of calculation, and a new no- tation, in order to bring non-linear functions into a condition which admits of treatment ky symbolical algebra. This is the object of the following memoir. Professor Boole has given, in his ‘Treatise on Differential Equa- tions,’ a method due to M. Sarrus, by which we ascertain whether a given non-linear function is a complete differential. This method, as will be seen by anyone who will refer to Professor Boole’s treatise, is equivalent to find- ing the conditions that a non-linear function may be externally divisible by the symbol of differentiation. In the following paper I have given a nota- tion by which I obtain the actual expressions for those conditions, and for the symbolical remainders arising in the course of the division, and have extended my investigations to ascertaining the results of the symbolical division of non-linear functions by linear functions of the symbol of differ- entiation. Il. “On Molecular Mechanics.” By the Rev. JosepH Bayma, of Stonyhurst College, Lancashire. Communicated by Dr. Suarpey, Sec. R.S. Received January 5, 1864. The following pages contain a short account of some speculations on molecular mechanics. They will show. how far my plan of molecular 1864.] The Rev. J. Bayma on Molecular Mechanies. 127 mechanics has been as yet developed, and how much more is to be done before it reaches its proper perfection. Of course I can do no more than point out the principles on which, according to my views, this new science ought to be grounded. The proofs would require a volume,—and the more SO, aS existing wide-spread philosophical prejudices will make it my duty to join together both demonstration and refutation. But there will be time hereafter, if necessary, for a complete exposition and vindication of the principles on which I rely; at present it will be enough for me to state them. The aim of ‘molecular mechanics” is the solution of a problem which includes all branches of physics, and which may be enunciated, in general terms, as follows :— ‘From the knowledge we gain of certain properties of natural substances by observation and experiment, to determine the intrinsic constitution of these substances, and the laws according to which they ought to act and be acted upon in any hypothesis whatever.” In order to clear the way for the solution of this problem, three things are to be done. First. From the known properties of bodies must be deduced the essen- tial principlés and intrinsic constitution of matter. Secondly. General formulas must be established for the motions of any kind of molecular system, which we conceive may exist 7m rerum natura. _ Thirdly. We must determine as far as possible the kinds of molecular systems which are suited to the different primitive bodies; and be pre- pared to make other applications suitable for the explanation of pheno- mena. Of these three things, the first, which is the very foundation of mole- cular mechanics, can, I think, be done at once. The second also, though it requires a larger treatment, will not present any great difficulty. The third, however, in this first attempt, can be but very imperfectly accom- plished ; for sciences also have their infancy, nor am I so bold as to ex- pect to be able to do what requires the labour of many: I shall only say so much as may suffice to establish for this science a definite existence and a proper form. In order to give an idea of my plan, I will now say a few words on each of these three points. I. PrincreLes oF MotecutaAR MECHANICS. First, then, (to say nothing of the name of “‘ molecular mechanics,” which will be justified later,) in all bodies we find these three things, extension, inertia, and active powers, to one or other of which every property of bodies may be referred. In order therefore to arrive ata clear idea of the con- stitution of natural substances, these three must be diligently investigated, Extension.—I have come to the following conclusions on this head, which, 128 The Rev. J. Bayma on Molecular Mechanics. ([Feb.11, I think, can be established by evident arguments drawn from various con- siderations. 1. All bodies consist of simple and unextended elements, the sum of which constitute the absolute mass of the given body. The extension itself, or volume, of the body is nothing but the extension of the space in- cluded within the bounding surfaces of the body ; and the extension of space is nothing but its capability of being passed through (percurrzbilitas) in any direction by means of motion extending from any one point to any other. 2. There is no such thing possible as matter materially and mathema- tically continuous—that is to say, such that its parts touch each other with true and perfect contact. There must be admitted indeed a continuity of forces ready to act; but this continuity is only virtual, not actual nor formal. 3. Simple elements cannot be at once attractive at greater, and repulsive at less distances. To this extent at least Boscovich’s theory must be corrected. Ifan element is attractive at any distance, it will be so at all distances; and if it be repulsive at any distance, it will be repulsive at all distances. This is proved from the very nature of matter, and perfectly corresponds with the action of molecules and with universal attraction. 4. Simple elements must not be confounded with the atoms of the chemist, nor with the molecules of which bodies are composed. Molecules are, according to their name, small extended masses, 2. e. they imply volume ; elements are indivisible points without extension. Again, molecules of what- ever kind, even those of primitive bodies, are so many systems resulting from elements acting on each other; consequently elements differ from molecules as parts differ from the whole; so that much may be said about separate elements, which cannot be said of separate molecules or chemical atoms, and vice versa. Element, molecule, body have the same relation to each other in the physical order, that cndividual, family, state bear to each other in the social order; for a body results from molecules, and molecules from elements holding together mechanically, in a similar way to that in which a state results from families, and families from individuals bound together by social ties. So much regarding extension; for I do not now intend to proceed to the demonstration of these statements, but simply to he down what it is I am prepared to prove. Inertia.—There would scarcely be any need of saying anything on this head, were there not some, even learned men, who entertain false ideas about it, and from not rightly understanding what is said of inertia by physical philosophers, throw out ill-founded doubts, which do more harm than good to science. I say, then, 1. Inertia implies two things: (a) that each element of matter is per- fectly indifferent to receiving motion in any direction and of any intensity from some external agent ; (4) that no element of matter can move itself by any action of its own. | 1864.] The Rev. J. Bayma on Molecular Mechanics. 129 2. It follows as a sort of corollary from this, that to be inert does not _ signify to be without active power; and that the very same element, which on account of its inertia cannot act upon itself, may, notwithstanding this inertia, have an active power, by which it may act upon any other element whatever. 3. Inertia is an essential property of matter, and is not greater in one element than in another, but is always the same in all elements, whether they are attractive or repulsive, whether their active power is great or small. 4, That which is called by natural philosophers the vis inertia is not a special mechanical force added on to the active forces of elements, but is the readiness of a body to react by means of its elementary forces, against any action tending to change the actual condition of that body. These four propositions will remove many false notions, which give rise to confusion of ideas and impede the solution of many important questions. Active power.—The questions relating to the active power of matter are of the greatest importance, since on them depends nearly the whole science of nature. On this point I am convinced, and think I can prove, that 1. No other forces exist in the elements of matter except locomotive or mechanical forces; for these alone are required, and these alone are sufficient, to account for all natural phenomena. So that we need have no anxiety about the vires vcculte of the ancients, nor need we make search after any other kind of primitive forces, besides such as are mecha- nical or locomotive. Hence chemical, electric, magnetic, calorific and other such actions will be all reduced to mechanical actions, complex indeed, but all following certain definite laws, and capable of being expressed by mathe- matical formule as in general mechanics. Hence in treating of molecular mechanics we do not make any gratuitous assumption or probable hypo- thesis, but are engaged ona branch of science founded on demonstrable truths, free from all hypothesis or arbitrary assumption. 2. There are not only attractive, but also repulsive elements; and this is the reason why molecules of bodies (as bemg made up of both sorts) may at certain distances attract, and at others repel each other. 3. Simple elements, in the whole sphere of their active power, and con- sequently also at molecular distances, act (whether by attracting or repel- ling) according to the inverse ratio of the squares of the distances. This proposition may seem to contradict certain known laws, as far as regards molecular distances ; but the contradiction is only apparent, and this appear- ance will vanish when we consider that the action of elements (of which we are now speaking) is not the same as the action of molecules. From the fact that cohesion, e. g., does not follow the inverse ratio of the square of the distance, it will certainly result that molecules do not act according to this law, and this is what physical science teaches: but it does not fol- low that elements do not act according to the law. This truth is, as all must see, of the utmost importance, since it is the foundation of molecular 130 The Rev. J. Bayma on Molecular Mechanics. [Feb. 11, mechanics, of which it would be impossible to treat at all, unless the law of elementary action at infinitesimal distances were known. ‘This truth universalizes Newton’s law of celestial attraction by extending it to all elementary action, whether attractive or repulsive, and makes it applicable not only to telescopic, but also to microscopic distances. It is clear there- fore that I am bound to prove this law most irrefragably, lest I construct my molecular mechanics on an insecure foundation. 4, The sphere of the activity of matter is indefinite, in this sense, that no finite distance can be assigned at which the action of matter will be null. It by no means, however, follows from this that the foeee of matter has an infinite intensity. 5. The natural activity of each element of matter is exerted tmmediately on every other existing element at any distance, either by attracting or repelling, according to the agent’s nature. Thus, e. g., the action which the earth exerts on each falling drop of rain is exerted immediately by each element of the earth on each element of the water (notwithstanding the dis- tance between them); itis not exerted through the material medium of the air, or of ether, or any other substance. The same must be said of the action of the sun on the planets. This proposition, however, it is evident, holds only for the simple action of the elements, 2. e., attractive or repulsive. For it is clear that complex actions causing vibratory motions, such as light or sound, are only transmitted through some vibrating medium. This conclusion is also of immense importance, because it solves a question much discussed by the ancients about the nature of action exerted on a distant body, and removes all scruples of philosophers on this head. 6. Bodies do not and cannot act by mathematical contact, however much our prejudices incline us to think the contrary; but every material action is always exerted on something at a distance from the agent. 7. There is another prejudice which I wish to remove, 2. e. that one motion is the efficient cause of another motion. It is easily shown that this mode of speaking, though sometimes employed by scientific men, is incorrect, and ought to be abandoned, because it tends to the destruction of all natural science. Motion never causes motion, but is only a condition affecting the agent in its manner of acting. For all motion is caused by some agent giving velocity and direction; but the agent gives velocity and direction by means of its own active power, which it exerts differently according as it is found in different local conditions. Now these local conditions of the agent may be differently modified by the movement of the agent itself. The impact of bodies, the change of motion to heat, the communication of velocity from one body to another (always a difficult question), and other points of a like nature can only be satistactonily explained by this principle. These are the principal points that have to be discussed, defined, and demonstrated in order that molecular mechanics may be established on solid principles. 1864. ] 7 The Rev. J. Bayma on Molecular Mechanics. 131 II. MaruematicaL EvoLuTION OF THESE PRINCIPLEs. After establishing principles, we must proceed to investigate the formulas of motion and of equilibrium, first between the elements themselves, then between the several systems of elements. The difficulties to be overcome in establishing the principles were chiefly philosophical: the difficulties which occur in the present part are mathematical, and can only be over- come by labour and patience. As long as we confine ourselves to two elements, the mathematical for- mula expressing their motion is easily found. Thus, if there are two attrac- tive elements of equal intensity, and if v be the action of one for a unit of distance in a unit of time, 2a the distance between them at the beginning of motion, x the space passed through by one in the time ¢, the equation of motion will be =a / 78 ( vaca) +0. tang = hee) And since it is clear, from other considerations, that these two elements must vibrate together indefinitely in vibrations of equal times and constant extent, the time of one oscillation will easily be found from the above formula. _ If the two attractive elements have unequal forces, or if one be attractive and the other repulsive, or both repulsive, the equation of motion may easily be obtained. But when we have to do with a more complex system of elements, after obtaining the differential equations corresponding to the nature of the system, it is scarcely possible to obtain their integration, as will appear from the examples which I shall give below. Consequently, if we wish to deduce anything from such equations, we must proceed indirectly, and a long labour must be undertaken, sometimes with but slender results. This material difficulty will be diminished, or perhaps disappear, either by some new method of integration (which I can scarcely dare to hope for, though it is a great desideratum) or by certain tables exhibiting series of numerical values belonging to different systems. But there occurs another difficulty in these systems. For since the agglomerations of simple elements can be arranged in an infinite variety, and it would be neither reasonable nor possible to treat of all such agglo- merations, we must limit the number of them according to the scope we have in view, i. e. according to the use they may be of in explaining natural phenomena. Even this isavery difficult matter. How I have endeavoured to overcome this difficulty I will briefly explain. First. I considered that the molecules of primitive bodies, such as oxygen, hydrogen, nitrogen, &c., cannot reasonably be supposed to be trregular— a conclusion which, though I cannot rigorously demonstrate, yet I can render probable by good reasons. Consequently, while treating of primi- tive systems I may confine myself to the examen of forms that are regular. 182 The Rev. J. Bayma on Molecular Mechanics. [Feb. 11, Secondly. I divided these regular systems into different classes according to their geometrical figure. Of these I have investigated the tetrahedric, octahedric, hexahedric, octohexahedric, pentagonal-dodecahedric, and icosa- hedric. I then divided these classes into different species, viz. pure centrata, centro-nucleate, centro-binucleate, centro-trinucleate, &c., also into acen- trate (without centre), truncate, &c. To enumerate the whole would take too long; indeed I only mention these to show how in such a multiplicity of systems I endeavoured to introduce the order necessary for me to be able to speak distinctly about them. | Lastly, besides classes and species, it was requisite also to consider cer- tain distinct varieties under the same species. And in this way I seemed to myself to have embraced all the regular systems of elements possibly conceivable. Thirdly. The several parts of which any system of elements can consist are reduced by me to a centre, nuclei toany number, and an external enve- lope. And thus I obtained not only a method of nomenclature for the dif- ferent systems (a most important point), but also a method of exhibiting each system under brief and intelligible symbols. Thus, e. g., the tetra- hedric system pure centratum (2. e. without any nucleus), in which the centre isan attractive element, and the four elements of the envelope repul- sive, will be represented thus, m=A+4R, in which expression m signifies the absolute mass of the system (in this case m= 5), A represents the attractive centre, and 4R the four repulsive ele- ments of the envelope. The letters A and R are not quantities, but only indices denoting the nature of the action. In a similar way, the following expression m=R+6A+8R! denotes a system whose centre R is repulsive, whose single nucleus 6A ig octahedric and attractive, and whose envelope 8R’ is hexahedric and repulsive: m, which, as before, indicates the absolute mass of the system, here =15. This will suffice to show how the different species and varieties of the afore-mentioned systems may be named and expressed. Then I had to find mechanical formulas for the motion or equilibrium of the several systems; for it is only from such formulas that we can deter- mine what systems are generally possible in the molecules of bodies. Speak- ing generally, no system pure centratum, of whatever figure it be, can be admitted in the molecules of natural bodies, whether gaseous, liquid, or solid. Let v represent the action of the centre, and w that of one of the elements of the envelope for a unit of distance ; and let 7 be the radius of the system, i. e. the distance o ‘any one of the elements of the envelope from the centre ; 1864. ] The Rey. J. Bayma on Molecular Mechanics. 133 _ the general formula of motion for any system pure centratum (expressed as above by m=A-+ 7R) will be ar de where M signifies a constant, ay the actions which tend to increase 7 are taken as positive. If the system is tetrahedric, M=0:91856 = (v— Mw), pi octahedric, M=1:66430 a hexahedric, M=2°46759 re octohexahedric, M=4:11170 3 icosahedric, M=4:19000 pentagonal dodecahedric, M=7°82419. Now none of these varieties satisfies the conditions either of solid, liquid, or gaseous bodies; because they either will not resist compression, or they form masses which are repulsive at all great distances; or if they could constitute gaseous bodies, they do not allow the law of compression to be verified, which we know to hold for all gases. Passing on to the systems centro-nucleata, the formulas will differ according to the several figures of the nuclei andenvelope. Taking, e. g., the system m=R+6A+8R’, which is hexahedric with an octahedrie nucleus, and taking v, v', w to represent respectively the actions of the centre, one element of the nucleus, and one element of the envelope; taking also 7 and 0 for the radii of the nucleus and envelope, the equations of motion for such a system will be ay v—M'’ PU en Be OND 7 i ae +4w dt Ee aJ (oer BEY 4 2er “) ti (0° ‘ee Zen Oy creme G ee =a) dp __vt+Mw _ 3, prvi p—rVi dt? 2 p a/ (ee ay We (FER a. He where M=2:46759, and M'/=1°66430. The conditions of equilibrium will be obtained by making the two first members equal to zero. What systems of this class (centro-nucleata) can satisfy the conditions of solid, liquid, or gaseous bodies, is exceedingly difficult to determine, for reasons which I have above touched on, viz. that the formulz of these systems are not integrable, and we have consequently to proceed indirectly with great expenditure of time and trouble. It seems to me, however, as far as I can judge, that some of these systems may be found im rerum natura. Passing to another class of systems (centro-binucleata), we shall have three equations to express its laws of motion. Taking, e. g., the system m=A+4R+4A'+ 42, - 134 The Rey. J. Bayma on Molecular Mechanics. (Feb. 11, which is tetrahedric with two tetrahedric nuclei; taking 2, v', ’, w for the respective actions of the elements acting from the centre, first and second nuclei, and envelope ; taking 7, 7", p for the radii of the two nuclei and the envelope, the equations of motion will be as follows: dy! Ras Mo'—v — lt i 37! —r"! dé 72 (t+r"') ("+9 Dplyll ) V 3 ' Rene ee on ea —_—_— oO Teen BY pall ine re d*r = Me Me 2 4g 1 37!!—r! a aE (er? /( eae 2rrity he ae iid 355 oe 4 3r'—p (e+r a (eee ee, 3 dp _Mw—v, y ee dt’ p oa a) e+e ey 24 Fi “) 3p—7"! =o SE wary | (e+ ey (etary in which equations M=0°91856. The discussion of these equations and similar ones will afford a useful occupation to mathematicians and natural philosophers. Whatever conclu- sions may be drawn from them cannot fail to throw great light on the question of the nature of bodies. It is evident that we might go further and pass on to ¢rinucleate, qua- drinucleate, &c. systems; but the number of equations will increase in proportion, together with the difficulty of dealing with them. It is not enough to consider the laws of motion and equilibrium in each system separately, but it is also necessary to know what action one system exercises on another, whether like or unlike, placed at a given distance. For since many of the properties of bodies depend on the relation which the different molecules bear to one another, e. g., liquidity, elasticity, hard- ness, &c., it isnot enough to know what is the state of a system of elements (i. e. a molecule) in itself, but we must investigate also how several such systems (or molecules) affect each other. Now in this ulterior investigation it is clear that the difficulty increases exceedingly, since the equations become exceedingly complex. Here also then may natural philosophers pf! 1864.] On the Excavation of the Valley of the Somme. 135 find matter for industry and patience. I have done a little in this subject, ‘but not enough to deserve any special mention. In order, however, to diminish the difficulties, the investigation may be provisionally restricted te the mutual actions of the envelopes, neglecting for the time that of the nuclei, which may be considered as a disturbing cause, for which some correction may afterwards have to be made. So much then for the mathematical and theoretic development of mole- cular mechanics. There remains the third part, which, though the most laborious of all, will yet give the greatest pleasure to scientific men; since it is less dry, and opens a way for attaining the end aimed at in the natural sciences. Of this third part I will add a few words. Ill. AppLicaTION OF THE PRINCIPLES OF MOLECULAR MECHANICS. [Under this head the author points out the various properties of bodies which would have to be explained, and of which he conceives an explana- tion might be afforded could the mathematical calculations be effected which are required for the elaboration of his theory, and enunciates the following conclusions as deduced from his explanation of the impact of bodies. | 1. If a body does not contain any repulsive elements, it cannot cause any retardation in the movement of any impinging body. 2. Again, if the medium through which a body moves contain no repul- sive elements, no retardation of its motion can take place. 3. Ifa medium does contain repulsive elements, retardation must neces- sarily take place. 4. Consequently, as the planets in their movements through the ether do not suffer any loss of velocity, it must be concluded that the eether does not contain any repulsive elements at all, and that its elasticity must be explained without any recourse to repulsive forces. This last inference is somewhat wonderful, and decidedly curious: but- after much consideration it appeared to me so natural, and so well harmo- nizing with other truths and scientific theories, that I ceased to hesitate about its adoption and gave it a most decided assent; whether wisely or not, I leave others to judge. Ill. “On some further Evidence bearing on the Excavation of the Valley of the Somme by River-action, as exhibited in a Section at Drucat near Abbeville.’ By JosrrpnH Prestwicn, F.R.S. Received January 29, 1864. On the occasion of a late visit to Abbeville, I noticed a fact which appears of sufficient interest, as bearing upon and confirming one of the points treated of in my last paper, to induce me to submit a short notice of it to the Royal Society. It occurs in a tributary valley to that of the Somme, but necessarily forms part of the general phenomena affecting the whole basin. 136 The small stream (the Escardon) which joins the Somme at Abbeville flows through a nar- row chalk valley extending afew miles north ot Abbeville. Three miles up this valley is the village of Drucat; and on the hill above the village, and about 100 feet above the stream, is a small outlier of high-level gravel which I have before described, and which is remark- able for the number and size of its sand- and gravel-pipes penetrating the underlying chalk. One of these which I measured was. 22 feet across at the top and 18 feet at a depth of 30 feet, and I estimated its depth at not less than 100 feet from the surface. It was filled in the usual way with sand and gravel in ver- tical cylindrical layers. M. Boucher de Perthes has two flint implements which are reported to have come from the pit; but I never myself found any there, or any mammalian remains. The sand and gravel is clean and light-coloured, and very similar in character to some of the beds at Menchecourt, and in so far has the appearance of a fluviatile gravel, and, like it, is overlain by a variable bed of loess. This bed was supposed to form an isolated outlier; but on my last visit I found another bed, though of coarser materials, on a hill of the same height on the opposite side of the valley, above I’ Heure. The valley at the foot of the hill on which the -Drucat gravel is worked is about a quarter of a mile wide. A lane leads direct down the slope of the hill from a point near the gravel to the valley ; anda roadside cutting exposes a section of calcareous tufa or travertin several feet thick, and containing in places numerous land shells, of recent species, and traces of plants. Half a mile beyond, the bed is of sufficient importance to be worked for building-purposes. This bed is overlain by the valley loess, and is in places intercalated with it; it commences a few feet below the level of the gravel at about 70 feet above the valley, and continues to near the foot of the hill. Mr. Prestwich on the Excavation /HIGHLEVEL GRAVEL CA [Feb. 11, “FIRST OUTB URST™ OF THE TUFA” SPRING E OW ARID ‘A ce ate en ET Mabe LEVEL OF THE VALLEY AT’ THE Bs Ox OleRE. SSS SS —- . 4—4, Line of present water-level. Y. Gravel-pipe. Now it is well proved that in all purely chalk districts the line of water- level proceeds from the level of the streams and rivers traversing the dis- 1864.] | of the Valley of the Somme. 137 trict, in a slightly inclined and continuous plane rising on either side under the adjacent hills with a slope varying from 10 to 40 feet in the mile, the latter being an extreme case. If we take a mean of 20 feet, as the gravel- pit is not above one-third of a mile from the valley, the rise in the water underneath would not probably exceed 10 feet above the level of the stream. The chalk formation is so generally fissured and permeable that I know of no instance of a line of water-level or of springs occurring above the ge- neral line dependent upon the level of the adjacent rivers. It is also well known that strong springs are common at the foot of the hills along many of our chalk valleys, as, for instance, that at Amwell, those at Carshalton, and many along the valley of the Thames. These springs are more or less calcareous, often highly so. It is evident that the travertin at Drucat has been formed by a deposit from a spring of considerable volume ; and it further appears that it flowed while the loess was in the course of formation. For the tufa could only have been formed at or near the level of the spring; so that its continued deposit down the slope of the hill shows the spring to have been gradually lowered as the valley became deeper, and while subject to the continued -inundations which deposited the loess. The line of present water-level in the chalk here is about 90 feet below the summit of the hill, as proved by a well in an adjacent farmhouse, and at the gravel-pit they have gone down 60 feet without reaching water. But the level of the upper part of the tufa shows the line of water-level or of springs to have been at one time 70 feet above the valley, which could only have happened when the bottom of the valley was on a level 60 to 70 feet higher than itnowis. The gradual deepening of the valley is indicated by the gradual lowering of the spring until it reached to within from 20 to 30 feet of the present valley-level, when it became extinct. Further, we have in the adjacent bed of high-level gravel evidence of the origin of this important spring; for the sands and gravel- beds are not only very thick, but they are also perfectly free from calca- reous matter and very permeable, and they show in their numerous gravel- pipes how great must have been the volume and solvent power of the rain- water which at one time percolated through them. The water, after pass- ing through the gravel and acting upon the underlying chalk to form these large vertical cavities, would, upon reaching the original line of water-level, have flowed off horizontally and escaped in a strong spring at the base of the adjacent slope. It there parted with its excess of the carbonate of lime, and so formed the calcareous tufa. This case furnishes therefore new and good evidence on two points:—first, on the connexion of the sand- and gravel-pipes with the percolation of fresh water through calcareous rocks ; and secondly, on the condition of the former land surface and of the springs, only possible on the hypothesis of former higher levels of the bottom of the valley and of its gradual excavation. VOL. XIII. M 138 Mr. Hulke on the Minute Anatomy of the Retina. [¥eb. 18, February 18, 1864: Major-General SABINE, President, in the Chair. The following communications were read :— I. “A Contribution to the Minute Anatomy of the Retina of Amphibia and Reptiles.” By J. W. Hurxs, Esq., F.R.C.S., Assistant- Surgeon to the Middlesex and the Royal London Ophthalmic Hospitals. Communicated by W. Bowman, Esq. Received February 4, 1864. (Abstract.) The animals of which the retina was examined were the frog, the black and yellow salamander, the edible turtle, the water- and the land-tortoise, the Spanish Gecko, the blindworm, and the common snake. The method adopted was to examine the retina (where possible) immediately after decapitation of the animal, alone and with chemical agents; and to make sections of the retina hardened in alcohol or in an aqueous solution of chromic acid, staining them with iodine or carmine, and adding glycerime, pure and diluted, to make them transparent. The following is a summary of the results of the examination. 1. The rods and cones consist of two segments, the union of which is marked by a bright transverse line. 2. Kach segment consists of a membranous sheath and contents. £'3. The outer segment, or shaft, is a long narrow rectangle (by inference, a prism or cylinder). It refracts more highly than the inner segment. Its contents are structureless, and of an albuminous nature. It is that part which is commonly known as “ the rod.’ It is smaller in the cones than in the rods, and in the cones narrows slightly outwards. 4, The outer ends of the shafts rest upon the inner surface of the choroid, and their sides are separated by pigmented processes, prolonged from the inner surface of the choroid between them to the line that marks the union of the shaft with the inner segment. The effect of this is that the shafts are completely insulated, and rays entering one shaft are pre- vented passing out of it into neighbouring shafts. 5. The inner segment of the rods and cones, or body (the appendage of some microscopists), has a generally flask-shaped form, longer and more tapering in the rods, shorter and stouter in the cones. It is much paler and less conspicuous than the shaft. It fits in an aperture in the membrana limitans externa. Its inner end always encloses, or is connected by an intermediate band with an outer granule which lies in or below the level of the membrana limitans externa. Its outer end, in cones only, contains a spherical bead nearly colourless in the frog and blindworm, brilliantly coloured in the turtle and water- and land-tortoises, and absent from the common snake and Spanish Gecko. In addition to this bead, where present, and the outer gra- 1864.] Mr. Hulke on the Minute Anatomy of the Retina. 139 nule, the body contains an albuminous substance which in chromic acid pre- parations retires as an opaque granular mass towards the outer end of the body. The inner end of the body is prolonged inwards, in the form of a pale, delicate fibre, which was sometimes followed through the layer of inner granules into the granular layer. It does not appear to be struc- turally connected with the inner granules. It is essentially distinct from Miller’s radial fibres, and bears a considerable resemblance to the axis- eylinder of nerve. That it ever proceeds from the outer granule associated with the rod- or cone-body is doubtful, from the consideration (a) that where the body is large, and the granule lies within at some distance from its contour, the fibre is seen to leave the inner end of the body distinct. from the granule, and ((@) that the fibre appears to proceed from the outer granule only where the body is small, as in the frog, and where the granule does not lie within the body but is joined to this by a band. Ritter’s axial fibres are artificial products. 6. The ‘outer granules” are large, circular, nucleated cells. Each cell is so intimately associated with a rod- or cone-body that it forms-an integral part of it. 7. The intergranular layer is a web of connective fibre. It contains nuclei. 8. The inner granules are roundish, in chromic acid preparations poly- gonal cells. They differ from the outer granules by their higher refraction, by the absence of a nucleus, and by receiving a deeper stain from carmine. They lie in areole of connective tissue derived from Miiller’s radial fibres, and from the intergranular and granular layer. They are more numerous than the outer granules, and consequently than the rods and-cones. 9. The granular layer is a very close fibrous web derived in part from Miiller’s radial fibres, and from other fibres proceeding from the connective frame of the layer of inner granules. It transmits (a) the radial fibres, (() fibres proceeding radially outwards from the ganglion-cells and bundles of optic nerve-fibres, and (y) fibres passing inwards from the rod- and cone- bodies. 10. The ganglion-cells communicate by axis-eylinder-like fibres with the bundles of optic nerve-fibres, and send similar fibres outwards, which have been traced some distance in the granular layer. 11. In the frog and Spanish Gecko the author has a few times traced fibres proceeding from the bundles of optic nerve-fibres for some distance in a radial direction in the granular layer. 12. Miller’s radial fibres arise by expanded roots at the outer surface of the membrana limitans interna, pass radially through the layers, contributing in their course to the granular layer, to the areolar frame of the layer of inner granules, and end in the intergranular layer and at the inner surface of the membrana limitans externa. They are a connective and not a nervous tissue, and do not communicate between the basilary element and ganglion-cells. M 2 140 Messrs. Frankland and Duppa on the [Feb. 18, 13. The orderly arrangement of the several layers and their elementary parts is maintained by a frame of connective tissue which consists of— 1, an unbroken homogeneous membrane bounding the inner surface of the retina, the membrana limitans interna; 2, a fenestrated membrane which holds the rods and cone-bodies, the membrana limitans externa, first correctly described by Schultze; 3, an intermediate system of tie-fibres— Miiller’s radial fibres—connected with which in the layer of inner granules are certain oblong and fusiform bodies of uncertain, nature; 4, the inter- granular layer; 5, an areolated tissue, open in the layers of outer and inner granules, and very closely woven in the granular layer. 14. No blood-vessels occur in the reptilian retina. II. “ Notes of Researches on the Acids of the Lactic Series.—No. I. Ac- tion of Zinc upon a mixture of the Iodide and Oxalate of Methy].” By E. Franxxanp, F.R.S., Professor of Chemistry, Royal Insti- _ tution, and B. F. Dupra, Esq. Received February 10, 1864. . In a former communication by one of us*, a process was described by which leucic acid was obtained synthetically by the substitution of one atom of oxygen in oxalic acid by two atoms of ethyl. The relations of these acids to each other will be seen from the following formulee +t :— C, H . C,H, Cee : Cee 0 OH 2 OH OH OH Oxalic acid. Tienes acid. This substitution of ethyl for oxygen was effected by acting upon oxalic ether with zincethyl. On distillmg the product with water, leucie ether came over, which on treatment with an alkali yielded a salt of leucic acid. We have since found that this process may be much simplified by gene- rating the zincethyl durmg the reaction, which is effected by heating a mixture of amalgamated zinc, iodide of ethyl, and oxalic ether in equivalent proportions to the necessary temperature. The operation may be considered complete when the mixture has soli- dified to a resinous-looking mass. This, treated with water as in the for- mer reaction and distilled, produces quantities of leucic ether considerably greater than can be obtained from the same materials by the first mode of operating. Thus the necessity for the production of zincethy] is entirely obviated, the whole operation proceeds at the ordinary atmospheric pressure, and a larger product is obtained. We find that this process is also applicable to the homologous reactions with the oxalates and iodides of methyl and amyl. By it we have obtained * Proceedings of the Royal Society, vol. xii. p. 396. t The atomic weights used in this paper are the following :—C=12, O=16 and Zn=65. 1864. ] Acids of the Lactic Series. 141 numerous other acids belonging to the lactic series, which we have already more or less perfectly studied, and the history of which we propose to lay before the Royal Society as our researches proceed, reserving for a later © communication our views regarding the constitution of this series of acids, and the theoretical conclusions arrived at in the course of the inquiry. In the present communication we will describe the application of this re- action to a mixture of iodide of methyl and oxalate of methyl. = Two equivalents of iodide of methyl were mixed with one of oxalate of methyl, and placed in contact with an excess of amalgamated and granu- lated zinc in a flask, to which an inverted Liebig’s condenser, provided with a mercurial safety tube, wasattached. The flask was immersed during about twenty-four hours in water maintained at a temperature gradually rising from 70° C. to 100° C. as the reaction progressed towards com- pletion. At the end of that time the mixture had solidified to a yellowish gummy mass, which, on distillation with water, yielded methylic alechol possessing an etherial odour, but from which we could extract uo ether. The residual magma in the flask, consisting of iodide of zinc, oxalate of zine, and the zinc-salt of a new acid, was separated from the metallic zine by washing with water. It was then treated with an excess of hydrate of baryta and boiled for a considerable time; carbonic acid was afterwards passed through the liquid until, on again boiling, the excess of baryta was completely removed. To the filtered solution recently precipitated oxide of silver was added until all iodine was removed. The solution separated from the iodide of silver was again submitted to a current of carbonie acid, boiled, and filtered. The resulting liquid, on being evaporated in the water- bath, yielded a salt crystallizing in brilliant needles possessing the peculiar odour of fresh butter. This salt is very soluble in water and in alcohol, but nearly insoluble in ether, and perfectly neutral to test-papers. On being submitted to analysis, it gave numbers closely corresponding with the formula 3 CH, Ce 4. @ |OH | OBa The acid of this salt, for which we provisionally propose the name di- methoxalic acid, is obtained by adding dilute sulphuric acid to the concen- trated solution of the baryta-salt and agitating with ether. On allowing the ether to evaporate spontaneously, prismatic crystals of considerable size make their appearance. These yielded, on combustion with oxide of copper, results nearly identical with those required by the formula 142 The Rev. 8S. Haughton on the Joint Systems [Feb. 25, Dimethoxalic acid is a white solid, readily crystallizing in beautiful prisms resembling oxalic acid. It fuses at 75°°7 C., volatilizes slowly even at com- mon temperatures, and readily sublimes at 50° C., being deposited upon a cool surface in magnificent prisms. It boils at about 212° C., and distils unchanged. Dimethoxalic acid reacts strongly acid, and unites with bases, forming a numerous class of salts, several of which are crystalline. In addition to the baryta-salt above mentioned, we have examined the silver- salt, which is best formed by adding oxide of silver to the free acid, heating to boiling, and filtering, when the salt is deposited in star-like masses of nacreous scales as the solution cools. On analysis, this salt gave numbers closely corresponding with those calculated from the formula LO Ag Attempts to produce an ether by digesting the free acid with absolute alcohol at a temperature gradually raised to 160° C. proved abortive, traces only of the ether being apparently formed. Thus the final result of the action of zinc upon a mixture of iodide and oxalate of methyl is perfectly homologous with that obtained by the action of zincethyl upon oxalic ether. In the methylic reaction, however, no compound corresponding to leucic ether was obtained. ‘This cannot create surprise when it is remembered that dimethoxalic ether approaches closely in composition to lactic ether, which is well known to be instantly decom- posed by water. We have sought in vain to obviate this decomposition of dimethoxalic ether by adding absolute alcohol in place of water to the pro- duct of the reaction. February 25, 1864. Major-General SABINE, President, in the Chair. I. “Qn the Joint Systems of Ireland and Cornwall, and their Me- chanical Origin.” By the Rev.Samurt Haventon, M.D.,F.R.S., Fellow of Trinity College, Dublin. Received February 8, 1864. (Abstract.) This paper is a continuation of a former paper ‘On the Joints of the Old Red Sandstone of the Co. Waterford,” published in the ‘ Philosophical Transactions’ for 1858, and contains the results of the author’s observations for some years, in Donegal, the Mourne and Newry Mountains, Cornwall, and Fermanagh, with deductions from theory. The author establishes the existence in Waterford of a Primary Conjugate System of Joints, and of two Secondary Conjugate Systems, lying at each side of the Primary at angles of 27°5' and 37°11’. - 1864. | of Ireland and Cornwall. 143 _ In Donegal there exists a Primary Conjugate System, and a Secondary System, making with the Primary an angle of 32°24’. Inthe Moume and Newry Mountains there is a Primary Conjugate System, and two Secondary Systems at each side of the Primary, making angles of 31°46’ and 30°56’. In Cornwall there is a Primary and also a Secondary Conjugate System, making an angle of 27° 28’. And in Fermanagh there are Primary and Secondary Systems, forming an angle of 31°1’. Having given, in detail, the observations on which the preceding results are founded, the author says :—“‘ Collecting together into one Table the results of the preceding observations, we find the following Table of Primary and Secondary Joints (True Bearings) :— Name. Waterford.| Donegal. | Mourne. | Cornwall. | Fermanagh. : N. of E. | N. of E. | N. of E. | N. of E. | N. of E. Primary System (A) eeccessececaece 32° 26’ | 26° 16’ | 39° 40’ | 32° 34! 21° 30’ : : W. of N.| W. of N.| W. of N.| W. of N. Ww. PN. Primary Conjugate (C) Caditameatels ta 31° 37’ | 29° 35/ een 32° 55! ate : , N. of E.| N. of E. | N. of E. N. of E First Secondary (A’) Puuveuseudtine oc { 58° 11’ | 58° 49’ Soe 40’ Aaa 54° Q! ni A080) (alt beens boo le ee foe 2. EL eee Conjugate to First Secondary (C’) | ee — bey pee os were : Second Secondary (A”) ae | —« = ae rt —=— Conjugate to Second Secondary E. of N. | W. of N.| W. of N. Pe eettrcticececscdnccesassccccens 4° 30' 7° 38! 6° 30! The only remarkable agreement as to direction of jomts disclosed by the preceding Table is that between Waterford and Cornwall. If we compare together the Primary and Secondary Joints in each locality, we find the following Table of Angles between Primary and Secondary Joints :— | Waterford. Donegal. Mourne. Cornwall. | Fermanagh. Between Primary (A, C) and First Secondary GAN fal hits cannanicrendan +27° 5’ | +32° 24’ | +31° 46’ —— +31° 1' Between Primary (A, C) ‘and Second Secon- Gary CA)’ ©") .4,...0% —37° ll’ — —30° 56’ | —27° 28’ —- ee es SS ee This Table discloses a very interesting and unexpected result; viz. that in Waterford, Donegal, Mourne, and Fermanagh, the angle between the Primary and first Secondary Joint-Systems ranges between the narrow limits of 27°5’ and 32° 24’, and that in Waterford, Mourne, and Cornwall, the angle between the Primary and second Secondary Joint-Systems ranges from 27° 28’ to 37° 11’. 144 Supposed Identity of Biliverdin with Chlorophyll, &c. [Feb. 25, The paper concludes with a brief deduction of the observed laws of Conjugate and Secondary Joints from known mechanical principles. II. “On the supposed Identity of Biliverdin with Chlorophyll, with remarks on the Constitution of Chlorophyll.” By G. G. Sroxzs, M.A., Sec.R.S. Received February 25, 1864. I have lately been enabled to examine a specimen, prepared by Professor Harley, of the green substance obtained from the bile, which has been named biliverdin, and which was supposed by Berzelius to be identical with chlorophyll. The latter substance yields with alcohol, ether, chloro- form, &c., solutions which are characterized by a peculiar and highly di-. stinctive system of bands of absorption, and by a strong fluorescence of a blood-red colour. In solutions of biliverdin these characters are wholly wanting. 'There is, indeed, a vague minimum of transparency in the red ; but it is totally unlike the intensely sharp absorption-band of chlorophyll, nor are the other bands of chlorophyll seen in biliverdin. In fact, no one who is in the habit of using a prism could suppose for a moment that the two were identical ; for an observation which can be made in a few seconds, which requires no apparatus beyond a small prism, to be used with the naked eye, and which as a matter of course would be made by any chemist work- ing at the subject, had the use of the prism made its way into the chemical world, is sufficient to show that chlorophyll and biliverdin are quite distinct. I may take this opportunity of mentioning that I have been for a good while engaged at intervals with an optico-chemical examination of chloro- phyll. I find the chlorophyll of land-plants to be a mixture of four sub- stances, two green and two yellow, all possessing highly distinctive optical properties. The green substances yield solutions exhibiting a strong red fluorescence; the yellow substances donot. ‘The four substances are soluble in the same solvents, and three of them are extremely easily decomposed by acids or even acid salts, such as binoxalate of potash ; but by proper treat- ment each may be obtained in a state of very approximate isolation, so far at least as coloured substances are concerned. The phyllocyanine of Fremy* is mainly the product of decomposition by acids of one of the green bodies, and is naturally a substance of a nearly neutral tint, showing however ex- tremely sharp bands of absorption in its neutral solutions, but dissolves in certain acids and acid solutions with a green or blue colour. Fremy’s phylloxanthine differs according to the mode of preparation. When pre- pared by removing the green bodies by hydrate of alumina and a little water, it is mainly one of the yellow bodies ; but when prepared by hydro- chloric acid and ether, it is mainly a mixture of the same yellow body (partly, it may be, decomposed) with the product of decomposition by acids of the second green body. As the mode of preparation of phylloxantheine * Comptes Rendus, tom. ]. p. 405. 1864. | Dr. Stenhouse on Rubia munjista. 145 is rather hinted at than described, I can only conjecture what the sub- stance is; but I suppose it to be a mixture of the second yellow substance with the products of decomposition of the other three bodies. Green sea- weeds (Chlorospermee) agree with land-plants, except as to the relative proportion of the substances present ; but in olive-coloured sea-weeds (Me- lanospermee) the second green substance is replaced by a third green sub- stance, and the first yellow substance by a third yellow substance, to the presence of which the dull colour of those plants is due. The red colouring- matter of the red sea-weeds (Rhodospermee), which the plants contain in addition to chlorophyll, is altogether ‘different in its nature from chloro- phyll, as is already known, and would appear to be an albuminous substance. I hope, before long, to present to the Royal Society the details of these researches. “Continuation of an Examination of Rubia munjista, the East- Indian Madder, or Munjeet of Commerce.” By Joun Sren- House, LL.D., F.R.S. Received December 21, 1863 *. In the former, preliminary notice of the examination of the Rudia mun- jista t, the mode of extracting munjistine from munjeet, and a number of its properties, have been already described. I now proceed to detail some results which have been subsequently obtained. When munjistine is extracted from munjeet by boiling solutions of sul- phate of alumina, as the whole of the colouring matter is not extracted by a single treatment with the sulphate of alumina, the operation must be repeated five or six times instead of two or three as was formerly stated. During the boiling of the munjeet with sulphate of alumina, a large quantity of furfurol is given off. I may mention, in passing, that the most abundant and economical source of furfurol is found in the preparation of garancine by boiling madder with sulphuric acid. If the wooden boilers in which garancine is usually manufactured were fitted with condensers, furfurol might be obtained in any quantity without expense. In addition to the properties of munjistine already described, I may mention that acetate of copper produces in solutions of munjistine a brown precipitate but very slightly soluble in acetic acid. When bromine-water is added to a strong aqueous solution of munjistine, a pale-coloured flocculent precipitate is immediately produced ; this when collected on a filter, washed and dissolved in hot alcohol, furnishes minute tufts of crystals, evidently a substitution-product.. Unfortunately these crystals are contaminated by a resinous matter, from which I have been unable to free them, and therefore to determine their composition. When munjistine is strongly heated on platinum-foil, it readily inflames and leaves no residue; when it is carefully heated in a tube, it fuses, and crystallizes again on cooling. If heated very slowly in a Mohr’s apparatus, * Read January 14. See Abstract, page 86. tT Proceedings, vol. xii. p. 633. 146 Dr. Stenhouse on Rubia munjista. [1864 munjistine sublimes in golden scales and broad flat needles of great beauty ; these have all the physical characters and the same composition as the original substance. If the sublimation be continued for a long time at the lowest possible temperature consistent with its volatilization, the wees of it is obtained with scarcely any loss. The following are the results of the ultimate analysis of different samples of munjistine :— I. -314 grm. of munjistine yielded *732 grm. carbonic acid and +106 grm. of water. II. :228 grm. of munjistine yielded ‘535 grm. carbonic acid and -0765 grm. water. . III. +332 grm. of munjistine yielded +7795 grm. of carbonic acid and "1125 grm. of water. IV. -313 grm. of munjistine yielded °734 grm. of carbonic acid and "1095 grm. of water. Theory. f. II. III. TY. C,,—96 64: 00 63°60 64:00 64°04 63°97 H,= 6 4-00 3°77 375 3°76 3°89 O, =48 32°00 32°63 32°27 32°20 32°14 The carbon in No. I. is rather lower than that of the other three; this is owing to the specimen not being quite free from alumina; moreover it was burnt with oxide of copper, the others with chromate of lead. No. III. is the sublimed munjistine. All the analyses were made on specimens pre- pared at different times. ; Lead Compound. When aqueous or alcoholic solutions of munjistine and acetate of lead are mixed, a flocculent precipitate of a deep orange-colour falls, which changes to scarlet on the addition of a slight excess on acetate. The best method of preparing it is to dissolve munjistine in hot spirit and add to the filtered solution a quantity of acetate of lead insufficient to precipitate the whole of the munjistine, then to wash thoroughly with spirit, in which the lead compound is but slightly soluble, and dry first in vacuo, and then in the water-bath. I. *836 grm. lead compound gave 407 grm. oxide of lead. II. -625 grm. lead compound gave ‘302 grm. oxide of lead. III. -428 grm. lead compound gave *2075 grm. oxide of lead. IV. -523 grm. lead compound gave *253 grm. oxide of lead. V. °2705 grm. lead compound gave °3445 grm. of carbonic acid and ‘0445 grm. water. VI. °5350 grm. lead compound gave 6830 grm. carbonic acid and -0920 grm. water. Theory. 1 Il. Ill. Iv. V. VI. C,, =486 Sa De Sark eta. «(0 vee. Bete ey 4 OG fee eee Hy = 25 T° O20 eae Sees ai EWE 3H 1°83" tase O,, =200 14°55 6. a) ae .6PbO 669°6 48°70 48: 70 48°32. 48: 50 48° 38 1864.) Dr. Stenhouse on Rubia munjista. 147 _ All the specimens were prepared at different times, except IV. and V., which are analyses of the same specimen. The lead compound therefore seems to approach nearly to the somewhat anomalous formula 5(C,, H, O,) +6PbO, being a basic lead-salt ; it is, however, perfectly analogous to the lead compound of purpurine, 5(C,, H,;O;) +6PbO, described by Wolff and Strecker*. _ From these analyses of the lead compound and also from the ultimate analyses of munjistine itself, it is pretty evident that its true formula is C,, H, O,: Neither sublimed munjistine nor that obtained by crystallization from aleohol, when dried at the ordinary temperature in vacuo, loses weight at 110°C. It is not improbable, however, that the gelatinous uncrystallizable precipitate, which separates on the cooling of boiling saturated aqueous solutions of munjistine, is a hydrate. From some experiments made on a considerable scale, I find that ordinary madder does not contain any munjistine. In order to ascertain this fact, a considerable quantity of garancine from Naples Roots, and likewise some which had been subjected to the action of high-pressure steam according to Pincoff and Schunck’s process, were treated with boiling bisulphide of car- bon, and the product obtained on evaporating the bisulphide repeatedly extracted with large quantities of boiling water ; the solution, when acidu- lated with sulphuric acid, gave an orange-red precipitate from which I was unable to obtain any munjistine. Professor Stokes succeeded, however, in detecting the presence of alizarine, purpurine, and rubiacine in it f. The production of phthalic acid from alizarine, purpurine, and munjistine, together with a comparison of their subjoined formule, indicates the very close relationship between these three substances, the only true colouring principles of the different species of madder with which we are acquainted. Mligarine ol Ble eet C,, H, O,, Purpurme 2 avs Shee ee. C,, H, O,, Munjistineé ......:..... C,,H;O,. Two other very convenient sources of phthalic acid are—first, the dark red resinous matter, combined with alumina, which is left undissolved by the bisulphide of carbon in the preparation of munjistine; secondly, the large quantity of green-coloured resinous matter which remains behind after extracting the alizarine from Professor Kopp’s so-called ‘‘ green aliza- rine” by means of bisulphide of carbon. I have repeated Marignac’s and Schunck’s experiments of distilling a mixture of phthalic acid and lime; and, like both of these chemists, I observed a quantity of very aromatic benzol to be produced, which, by the action of strong nitric acid, readily yielded nitrobenzol, and from this, by the action of reducing agents, aniline. The only impurity in the benzol from phthalic acid appears to be a minute * Annalen der Chemie, Ixxv. p. 24. + He has since informed me that he has succeeded in demonstrating the absence of munjistine. 148 Dr. Stenhouse on Rubia munjista. [1864. quantity of an oil, having an aromatic odour, resembling that produced from cinnamic acid by the action of hypochlorite of lime. Tinctorial power of Munjistine and Munjeet. Prof. Runge stated, in 1835, that munjeet contains twice as much avail- able colouring matter as the best Avignon madder. This result was so unex- pected, that the Prussian Society for the Encouragement of Manufactures, to whom Professor Runge’s memoir was originally addressed, referred the matter to three eminent German dyers, Messrs. Dannenberger, Bohm, and Nobiling. These gentlemen reported, as the result of numerous and carefully conducted experiments, that so far from munjeet being richer in colouring- matter than ordinary madder, it contained considerably less. This conclu- sion has been confirmed by the experience of my friend Mr. John Thom, of Birkacre, near Chorley, one of the most skilful of the Lancashire printers. From a numerous series of experiments I have just completed, I find that the garancine from munject has about half the tinctorial power of the garancine made from the best madder, viz. Naples Roots. These, however, yield only about 30 to 33 per cent. of garancine, while munjeet, according to my friend Mr. Higgin, of Manchester, yields from 52 to 55 per cent. Taking the present prices therefore of madder at 36 shillings per ewt., and munjeet at 30 shillings, it will be found that there will be scarcely any pecuniary advantage in using munjeet for ordmary madder-dyeing. The colours from munjeet are certainly brighter, but not so durable as those from madder, owing to the substitution of purpurine for alizarine. There is, however, great reason to believe that some of the Turkey-red dyers are employing garancine from munjeet to a considerable extent. When this is the case they evidently sacrifice fastness to brilliancy of colour. By treating such a garancine with boiling water, and precipitating by an acid in the way already described, its sophistication with munjeet may very readily be detected. The actual amount of colouring matter in munjeet and the best madder is very nearly the same; but the inferiority of munjeet as a dye- stuff results from its containing only the comparatively feeble colouring matters, purpurine and munjistine, only a small portion of the latter being useful, whilst the presence of munjistine in large quantity appears to be positively injurious. So much is this the case, that when the greater part of the munjistine is removed from munjeet-garancine by boiling water, it yields much richer shades with alumina mordants than before. PURPUREINE. Action of Ammonia on Purpurine. When purpurine is dissolved in dilute ammonia and exposed to the air in a vessel with a wide mouth in a warm place for about a month, ammonia and water being added from time to time as they evaporate, the purpurine almost entirely disappears, whilst a new colouring-matter is formed which dyes unmordanted silk and wool of a fine rose-colour, but is incapable of 1864. ] Dr. Stenhouse on Rubia munjista. 149 dyeing vegetable fabrics mordanted with alumina. If, however, strong ammonia be employed to dissolve the purpurine, considerable heat is pro- duced—a rise of temperature of as much as 20°C. taking place if the bulb of a thermometer be immersed in finely divided purpurine and strong ammonia poured on it. The purpurine employed in these experiments was prepared by Kopp’s process, and I am indebted for it to my friend Professor Crace Calvert. The solution of the new substance, purpureine, is filtered to separate dust, &e., as well as a black substance insoluble in dilute ammonia; it is then added to a considerable quantity of dilute sulphuric acid, boiled fora short time, and allowed to cool. When cold, the impure purpureine is collected on a filter, well washed, and dissolved in a small quantity of hot alcohol. The spirituous solution is again filtered into a quantity of very dilute boil- ing sulphuric acid, about 1 part acid to from 50 to 100 of water; when cold, the precipitate is collected and again well washed.: A crystallization out of boiling very dilute acid now renders it quite pure. This somewhat long and tedious process is necessary to free it from an uncrystallizable black substance, a part of which is separated when the crude purpureine is dis- solved in alcohol, and a part is left behind at the last crystallization. This compound being in its mode of formation and physical properties very analogous to orceine, I have called it purpureine. When crystallized by the spontaneous evaporation of its alcoholic solution, or from boiling dilute sulphuric acid under peculiar conditions of aggregation, it presents a fine iridescent green colour by reflected light ; whilst under the microscope it appears as fine long needles of a very deep crimson colour. As obtained by the process above described, it has, however, but little of the iridescent appearance, being of a brownish-red colour with a faint tinge of green, It is almost insoluble in cold dilute acids, and is in great part precipitated from its aqueous solution by common salt, thus greatly resembling orceine. It is almost insoluble in bisulphide of carbon, very slightly so both in ether and in cold water, much more so in hot, and very soluble in spirit both hot and cold and in water rendered slightly alkaline. Itis'readily soluble in cold con- centrated sulphuric acid, and is precipitated unaltered by water; on heat- ing, however, it is destroyed. Its aqueous solution gives a deep-red precipitate with chloride of zinc; with chloride of mercury a purple gelatinous precipitate ; and with nitrate of silver a precipitate of a very dark brown colour slightly soluble in am- monia, I have been favoured with the following optical examination by Professor Stokes :— * Its solutions show bands of absorption just like purpurine in character, but in some cases considerably different in position. The etherial and acidulated (acetic acid) alcoholic solutions show this strongly. The tint is so different in purpurine and its derivative, that the intimate connexion revealed by the prism would be lost by the eye. A drawing of the spectrum for pur- purine would serve for its derivative (purpureine), if the bands were simply pushed a good deal nearer the red end.” 150 Dr. Stenhouse on Rubia munjista. [1864. I. -3435 grm. pupureine gave ‘8230 grm. carbonic acid and +1240 erm. of water. II. -340 grm. purpureine gave 813 grm. carbonic acid and *123 erm. of water. III. +336 grm. purpureine gave ‘01552 grm. nitrogen. IV. ‘535 grm. purpureine gave ‘02453 grm. nitrogen. Theory. I, If. Ill. IV. Cy, = 396 65°13 65°36 64422 , H,, = 24 3°95 4:01 4:02 ohne Nour 28 4:60 pia tex 4°62 4°58 O,, = 160 26°32 608 100-00 The formula therefore appears to be C,, H,, N, O,,? Nitropurpureine. When purpureine is dissolved in a small quantity of moderately strong nitric acid, spec, grav. about 1°35, and heated to 100° C., it gives off red fumes, and on being allowed to cool, a substance sevarates in magnificent scarlet prisms somewhat like chromate of silver, only of a brighter colour ; it is quite insoluble in water, ether, and bisulphide of carbon, and very slightly soluble in spirit, but soluble in hot moderately strong nitric acid, from which it separates on standing fora considerable time. If boiled with strong nitric acid, it is slowly decomposed. When heated, it deflagrates : from this circumstance, and considering its mode of formation, it is evidently a nitro-substitution compound ; I have therefore called it nitropurpureine. Owing to the small quantity which I have hitherto been able to procure, I have not yet determined the composition of this beautiful body, which is finer in appearance than any of the derivatives from madder I have as yet met with. Action of Ammonia on Alizarine. The alizarine which was employed for the subjoined experiments was obtained by extracting Professor KE. Kopp’s so-called green alizarine* with bisulphide of carbon. It yields only about 15 per cent. of orange-red ali- zarine. This was crystallized three times out of spirit, from which it usually separates as a deep-orange-coloured crystalline powder. Unfortunately this alizarine still contains a quantity of purpurine, from which it is impossible to purify it either by crystallization or sublimation. Accordingly, when treated with ammonia by the method already described for purpurine, while it yields a substance analogous to purpureine, the product is impure, being contaminated with purpureine, This mixture has been examined by | my friend Professor Stokes, who finds that it contains purpureine, derived from the purpurine present as an impurity in the alizarine employed, and another substance very like alizarine.in its optical properties, probably a new substance (alieareine), bearing the same relation to alizarine that * I amalso indebted to Professor Calvert for the “ green alizarine.” hain cae 1864.) Dr. Stenhouse on Rubia munjista. 151 purpureine does to purpurine*. The following is an extract from a letter I received from Professor Stokes :— ‘It would be very unlikely @ priorz that such a simple process as that of Kopp should effect a perfect separation of two such similar bodies as alizarine and purpurine ; and as I find his purpurine is free from alizarine, it would be almost certain @ priori that his ‘ green alizarine’ would contain purpurine, and the two would be dissolved by bisulphide of carbon, and might very well afterwards be associated by being deposited in intermingled crystals, if not actually crystallizing together.” : Action of Ammonia on Munjistine. This reaction with munjistine was only tried on a very small scale, but the results were by no means satisfactory. The munjistine was completely destroyed, the greater part being changed into a brown’ humus-like sub- stance, insoluble in ammonia,—the remainder forming a colouring-sub- stance, analogous to purpureine, but not crystalline. It dyed unmordanted silk a brownish-orange colour. : The combined action of ammonia and oxygen, therefore, on the three colouring-substances alizarine, purpurine, and munjistine, is to change them from adjective to substantive dye-stuffs. I think it not improbable that if this archilizing process were applied to various other colouring mat- ters, they would be found capable of undergoing similar transformations. Action of Bromine on Alizarine. A boiling saturated solution of alizarine in alcohol is mixed with about six or eight parts of distilled water, and to this when cold about one or one and a half parts of bromine water are added, when a bright yellow amor- phous precipitate is produced. After standing twelve or sixteen hours, the solution is filtered ; and if the clear filtrate be now carefully heated so as to expel the spirit, a substance of a deep orange-colour is deposited, consisting of very fine needles, which are contaminated with a small quantity of resin ifa great excess of bromine has been employed. These needles are soluble in spirit and ether, insoluble in water, and soluble in bisulphide of carbon, from which they crystallize by spontaneous evaporation, in dark-brown nodules. With soda they give the same purple colour as alizarine. They ; dye cloth mordanted with alumina a dmgy brownish red, very different from the colour produced by ordinary crystallized alizarme. The follow- ing optical examination is from a letter of Professor Stokes :— “* Bromine Derivative of Alizarine.” **T can hardly distinguish this substance from alizarine. The solutions * Since this paper was communicated to the Royal Society, I find by a notice in Kopp and Wills’s ‘ Jahresbericht’ for 1862, p. 496, that a similar experiment upon alizarine had been made by Schiitzenberger and A. Paraf. The vroduct of one preparation which they obtained, and to which they have given the name of alizarinamid, yielded a formula C4) Hy; NOj., and another preparation gave the formula Cg) H33 Nz O.4, both being, when dry, nearly black amorphous substances. It appears, therefore, from the results of MM. Schiitzenberger and Paraf’s experiments, that these gentlemen were not more successful in obtaining a pure product from the action of ammonia on alizarine than I have been. 152 Dr. Stenhouse on Rubia munjista. [1864, in alcohol containing potassa show three bands of absorption just alike in appearance. By measurement it seemed probable that the bromine sub- stance gave the bands a d¢t/e nearer to the red end; but the difference, if real, was very minute. The fluorescent light of the ethereal solution was, I think, a trifle yellower in the bromine substance, that of alizarine being more orange.” The following are the results of the ultimate analysis of the brominated alzarine dried at 100° C.:— I, +375 grm. of substance gave ‘207 grm. bromide of silver. II. -703 grm. of substance gave *389 grm. bromide of silver. III. -401 grm. of substance gave *221 grm. bromide of silver. IV. *543 grm. of substance gave *300 grm. bromide of silver. V. *3575 grm. of substance gave ‘695 grm. of carbonic acid and ‘0760 grm. of water. VI. 454 grm. of substance gave ‘8790 grm. of carbonic acid and *0965 grm. of water. Theory. I. at. III. IV. V. VI. ©, 860° 5294-00000, 20 oe eee H,= 16 2°35 .... oh eee Br, = 160° 23°53. 23°49 23°54 23°45 23°51 eee OPE Ae WES ee ee.) Rae see 680 100-00 From this somewhat anomalous formula, C,, Hy, Br, 0,,=C,, H, O,, 2(C,, H, BrO,), I was for some time inclined to think that it might be a mixture of bromi- nated. alizarine with free alizarine ; but as all the six samples analyzed were prepared at different times, itis highly improbable that such uniform analy- tical results could be obtained if they were from a mere admixture of sub- stances. The existence of a brominated compound is also confirmed by its dyeing properties, which differ so remarkably from those of alizarine. Action of Bromine on Purpurine. When pure purpurine is dissolved in spirit mixed with a considerable quantity of water, and an aqueous solution of bromine added, as in the case of alizarine, a yellow amorphous precipitate is produced. The solution separated from this by filtration, when heated to expel the spirit, gives no precipitate whilst hot; but on cooling, a very small quantity of a brown resinous powder is deposited. From this it is evident that the presence of a small quantity of purpurine in alizarine will not interfere with the pro- duction of pure brominated alizarine, if the precaution be taken to collect it from the solution whilst it is still hot. I think it right to state that the experiments and analyses detailed in the preceding paper have been performed by my assistant, Mr. Charles Edward Groves. I cannot conclude this paper without again acknowledg- ing the essential services I have received from Professor Stokes, who kindly submitted the different products obtained by me to optical examination. 1864.) 153 March 3, 1864. Major-General SABINE, President, in the Chair. In accordance with the Statutes, the names of the Candidates for election into the Society were read, as follows :— Alexander Armstrong, M.D. leeming Jenkin, Esq. William Baird, M.D. William Jenner, M.D. Sir Henry Barkly, K.C.B. Kdmund C. Johnson, M.D. Henry Foster Baxter, Esq. Prof. Leone Levi. Sir Charles Tilston Bright. Waller Augustus Lewis, M.B. William Brinton, M.D. Sir Charles Locock, Bart., M.D. John Charles Bucknill, M.D. Kdward Joseph Lowe, Esq. Lieut.-Col. John Cameron, R.E. The Hon. Thomas M‘Combie. T. Spencer Cobbold, M.D. Sir Joseph F. Olliffe, M.D. The Hon. James Cockle, M.A. — George Wareing Ormerod, M.A. Henry Dircks, Esq. Thomas Lambe Phipson, Esq. Alexander John Ellis, B.A. John Russell Reynolds, M.D. John Evans, Esq. William Henry Leighton Russell, William Henry Flower, Esq. B.A. | Sir Charles Fox. William Sanders, Esq. | George Gore, Esq. Col. William James Smythe, R.A. George Robert Gray, Esq. Lieut.-Col. Alexander Strange. Thomas Grubb, Esq. Thomas Tate, Esq. Henri Gueneau de Mussy, M.D. Charles Tomlinson, Esq. Wilham Augustus Guy, M.B. George Charles Wallich, M.D. George Harley, M.D. Robert Warington, Esq. Sir John Charles Dalrymple Hay, | Charles Wye Williams, Esq. Bart. Nicholas Wood, Esq. Benjamin Hobson, M.B. Henry Worms, Esq. William Charles Hood, M.D. The following communication was read :— “On the Spectra of Ignited Gases and Vapours, with especial regard to the different Spectra of the same elementary gaseous sub- stance.” By Dr. Junius Pricxer, of Bonn, For. Memb. RB.8., and Dr. S. W. Hirrorr, of Munster. Received February 23, 1864. (Abstract.) In order to obtain the spectra of the elementary bodies, we may employ either flame or the electric current. The former is the more easily managed, but its temperature is for the most part too low to volatilize the body to be VOL. XIII. N 154 Drs. Pliicker and Hittorf on the Spectra of _[Mar. 3, examined, or, if it be volatilized or already in the state of gas, to exhibit its characteristic lines. In most cases it is only the electric current that is fitted to produce these limes ; and the current furnished by a powerful in- duction coil was what the authors generally employed. In the application of the current, different cases may arise. The body to be examined may be either in the state of gas, or capable of being vola- tilized at a moderate temperature, such as glass will bear without softening, or its volatilization may require a temperature still higher. In the first two cases the body is enclosed in a blown-glass vessel con- sisting of two bulbs, with platinum wires for electrodes, connected by a capillary tube. In the case of a gas, the vessel is exhausted by means of Geissler’s exhauster, and filled with the gas at a suitable tension. In the case of a solid easily volatilized, a portion is introduced into the vessel, which is then exhausted as highly as possible, and the substance is heated by a lamp at the time of the observation. In the third case the electric current is employed at the same time for volatilizing the body and render- ing its vapour luminous. If the body be a conductor, the electrodes are formed of it ; but the spectrum observed exhibits not only the lines due to the body to be examined, but also those which depend on the interposed gas. This inconvenience is partly remedied by using hydrogen for the interposed gas, as its spectrum under these circumstances approaches to a continuous one. If the body to be examined be a non-conductor, the metallic elec- trodes are covered with it. In this case the spectrum observed contains the lines due to the metal of which the electrodes are formed, and to the inter- posed gas, as well as those due to the substance to be examined. Among the substances examined, the authors commence with nitrogen, which first revealed to them the existence of two spectra belonging to the same substance. The phenomena presented by nitrogen are described in detail, which permits a shorter description to suffice for the other bodies examined, On sending through a capillary tube containing nitrogen, at a pressure of from 40 to 80 millimetres, the direct discharge of a powerful Ruhmkorff’s coil, a spectrum is obtained consisting, both in its more and in its less re- frangible part, of a series of bright shaded bands: the middle part of the spectrum is usually less marked. In each of the two parts referred to, the bands are formed on the same type; but the type in the less refrangible part of the spectrum is quite different from that in the more refrangible. In the latter case the bands have a channeled appearance, an effect which is produced by a shading, the intensity of which decreases from the more to the less refracted part of each band. In a sufficiently pure and magnified spectrum, a small bright line is observed between the neighbouring channels, and the shading is resolved into dark lines, which are nearly equidistant, while their darkness decreases towards the least refracted limit of each band. With a similar power the bands in the less refrangible part of the spectrum are also seen to be traversed by fine dark lines, the arrange- 1864.] Ignited Gases and Vapours. 155 ment of which, however, while similar for the different bands, is quite different from that observed in the channeled spaces belonging to the more refrangible region. If, instead of sending the direcé discharge of the induction coil through the capillary tube containing nitrogen, a Leyden jar be interposed in the secondary circuit in the usual way, the spectrum obtained is totally differ- ent. Instead of shaded bands, we have now a spectrum consisting of bril- hant lines having no apparent relation whatsoever to the bands before observed. If the nitrogen employed contains a slight admixture of oxygen, the bright lines due to oxygen are seen as well as those due to nitrogen, whereas in the former spectrum a slight admixture of oxygen produced no apparent effect. The different appearance of the bands in the more and in the less refracted portion of the spectrum first mentioned suggested to the authors that it was really composed of twospectra, which possibly might admit of being sepa- rated. This the authors succeeded in effecting by using a somewhat wider tube. Sent through this tube, the direct discharge gave a golden-coloured light, which was resolved by the prism into the shaded bands belonging to the less refrangible part of the spectrum, whereas with a small jar inter- posed the light was blue, and was resolved by the prism into the channeled spaces belonging to the more refrangible part. By increasing the density of the gas and at the same time the power of the current, or else, in case the gas be less dense, by interposing in the secondary circuit at the same time a Leyden jar and a stratum of air, the authors obtained lines of dazzling brilliancy which were no longer well defined, but had become of appreciable breadth, while at the same time other lines, previously too faint to be seen, made their appearance. The number of these lines, however, is not unlimited. By the expansion of some of the lines, especially the brighter ones, the spectrum tended to become continuous. Those spectra which are composed of rather broad bands, which show different appearances according as they are differently shaded by fine dark lines, the authors generally call spectra of the first order, while those spectra which show brilliant coloured lines on a more or less dark ground they call spectra of the second order. Incandescent nitrogen accordingly exhibits two spectra of the first, and one of the second order. 'The temperature produced by the passage of an electric current increases with the quantity of electricity which passes, and for a given quantity with the suddenness of the passage. When the tem- perature produced by the discharge is comparatively low, incandescent nitrogen emits a golden-coloured light, which is resolved by the prism into shaded bands occupying chiefly the less refrangible part of the spectrum. At a higher temperature the light is blue, and is resolved by the prism into channeled bands filling the more refrangible part of the spectrum. Ata still higher temperature the spectrum consists mainly of bright lines, N 2 156 On the Spectra of Ignited Gases and Vapours. [Mar. 8, which at the highest attainable temperature begin to expand, so that the spectrum tends to become continuous. The authors think it probable that the three different spectra of the emitted light depend upon three allotropic states which nitrogen assumes at different temperatures. By similar methods the authors obtained two different spectra of sul- phur, one of the first and one of the second order. The spectrum of the first order exhibited channeled spaces, like one of the two spectra of that order of nitrogen; but the direction in which the depth of shading in- creased was the reverse of what was observed with nitrogen, the darker side of each channeled space being in the case of sulphur directed towards the red end of the spectrum. Selenium, like sulphur, shows two spectra, one of the first and one of the second order. Incandescent carbon, even in a state of the finest division, gives a continuous spectrum. Among the gases which by their decomposition, whether in flame or in the electric current, give the spectrum of carbon, the authors describe particularly the spectra of cyanogen and olefiant gas when burnt with oxygen or with air, and of carbonic oxide, carbonic acid, marsh-gas, olefiant gas, and methyl rendered incandescent by the electric discharge; they likewise describe the spectrum of the electric discharge between electrodes of carbon in an atmosphere of hydrogen. The spectrum of carbon examined under these various conditions showed great varieties, but all the different types observed were represented, more or less com- pletely, in the spectrum of cyanogen fed with oxygen. The authors think it possible that certain bands, not due to nitrogen, seen in the flame of cyanogen, and not in any other compound of carbon, may have been due to the undecomposed gas. The spectrum of hydrogen, as obtained by a small Ruhmkorff’s coil, exhibited chiefly three bright lines. With the large coil employed by the authors, the lines slightly and unequally expanded. On interposing the Leyden jar, and using gas of a somewhat higher pressure, the spectrum was transformed into a continuous one, with a red line at one extremity, while at a still higher pressure this red line expanded into a band. The authors also observed a new hydrogen spectrum, corresponding to a lower temperature, but having no resemblance at all to the spectra of the first order of nitrogen, sulphur, &c. Oxygen gave only a spectrum of the second ointer: the different lines of which, however, expanded under certain circumstances into narrow bands, but very differently in different parts of the spectrum. Phosphorus, when treated like sulphur, gave only a spectrum of the second order. Chlorine, bromine, and iodine, when examined by the electric discharge, gave only spectra of the second order, in which no two of the numerous spectral lines belonging to the three substances were coincident. The 1864.] Influence of Physical and Chemical Agents upon Blood. 157 authors were desirous of examining whether iodine would give a spectrum of the first order the reverse of the absorption-spectrum at ordinary tem- peratures. The vapour of iodine in an oxyhydrogen jet gave, indeed, a spectrum of the first order, but it did not agree with what theory might have led us to expect. In the electric discharge, arsenic and mercury gave only spectra of the second order. The metals of the alkalies sodium, potassium, lithium, thallium show, even at the lower temperature of Bunsen’s lamp, spectra of the second order. Barium, strontium, calcium in the flame of Bunsen’s lamp show bands like spectra of the first order, and in each case a well-defined line-like spectra of the second order. On introducing chloride of barium into an oxyhydrogen jet, the shading of the bands was resolved into fine dark lines, proving that the band-spectrum of barium is in every respect a spectrum of the first order. Spectra of the first order were observed in the case of only a few of the heavy metals, among which may be particularly mentioned lead, which, when its chloride, bromide, iodide, or oxide was introduced into an oxy- hydrogen jet, gave a spectrum with bands which had a channeled appear- ance in consequence of a shading by fine dark lines. Chloride, bromide, and iodide of copper gave in a Bunsen’s Jamp, or the oxyhydrogen jet, spectra with bands, and besides a few bright lines. The bands in the three cases were not quite the same, but differed from one another by additional bands. Manganese showed a curious spectrum of the first order. When an induction discharge passed between electrodes of copper or of manganese, pure spectra of these metals, of the second order, were obtained. March 10, 1864. Major-General SABINE, President, in the Chair. The following communication was read :— “On the Influence of Physical and Chemical Agents upon Blood ; with special reference to the mutual action of the Blood and the Respiratory Gases.’ By Grorcr Hartzy, M.D., Professor of Medical Jurisprudence in University College, London. Com- municated by Dr. SHarpey, Sec. R.S. Received March 8, 1864. (Abstract.) This communication is divided into two parts. The first is devoted to the investigation of the influence of certain physical agencies, viz. simple diffusion, motion, and temperature, and of the conditions of time and the age of the blood itself. The second part includes the consideration of 158 Dr. Harley on the Influence of Physical and [Mar. 10, the influence of chemical agents, especially such as are usually regarded as powerful poisons. The paper commences with a description of the apparatus employed, and the method followed in conducting the inquiry; and the details of the several experiments are then given. The following is a brief statement of the results. Part I, 1. The experiments on diffusion showed that venous blood not only yields a much greater amount of carbonic acid than arterial blood, but also absorbs and combines with a larger proportion of oxygen. 2. Motion of the blood was found to increase the chemical changes arising from the mutual action of the blood and the respiratory gases. 3. The results of the experiment on the influence of time led to the con- clusion that the blood and air reciprocally act on each other in the same way out of the body as they do within it, and that their action is not in- stantaneous, but gradual. 4. It was ascertained that a certain degree of heat was absolutely essential to the chemical transformations and decompositions upon which the inter- change of the respiratory gases depends. The higher the temperature up to that of 38° C. (the animal heat), the more rapid and more effectual were the respiratory changes; whereas a temperature of 0° C. was found totally to arrest them. 5. The influence of age on the blood was found to be very marked, espe- cially on its relation to oxygen. The older and the more putrid the blood becomes, the greater is the amount of oxygen that disappears from the air ; and although at the same time the exhalation of carbonic acid progressively increases with the age of the blood, yet its proportion is exceedingly small when compared with the large amount of oxygen absorbed. 6. The average amount of urea in fresh sheep’s blood was ascertained to be 0°559 per cent., and its disappearance from the blood during the putre- factive process was very gradual, there being as much as 0°387 per cent. in blood after it was 304 hours old. Part II. The chemical agents employed were animal and vegetable products and mineral substances. 1. The effect of snake-poison was found to be an acceleration of the transformations and decompositions occurring in blood, upon which the absorption of oxygen and the exhalation of carbonic acid depend. 2. The presence of an abnormal amount of uric acid in blood was also found to hasten the chemical changes upon which the absorption of oxygen and exhalation of carbonic acid depend. 3. Animal sugar, contrary to what had been anticipated, retarded the respiratory changes produced in atmospheric air by blood. 1864. ] Chemical Agents upon Blood. } 159 4, The influence of hydrocyanic acid was studied both upon ox-blood and human blood, and found to be the same in each case, namely, to arrest respiratory changes. 5. Nicotine was also found to diminish the power of the blood either to take up oxygen or give off carbonic acid gas and thereby become fitted for the purposes of nutrition. 6. The effect of woorara poison, both on the blood in the body and out of it, was ascertained to be in some respects similar to that of snake-poison, namely, to increase the chemical decompositions and transformations upon which the exhalation of carbonic acid depends; but differed in retarding, instead of hastening, the oxidation of the constituents of the blood. 7. Antiar poison and aconite were found to act alike, inasmuch as both of them hastened oxidation and retarded the changes upon which the exhalation of carbonic acid depends ; in both respects offering a striking contrast to woorara poison, which, as has just been said, diminishes oxida- tion and increases the exhalation of carbonic acid. 8. The effect of strychnine on the blood, both in and out of the body, was studied, and found to be in both cases identical, namely, like some of the other substances previously mentioned, to arrest respiratory changes. Moreover, in one experiment in which the air expired from the lungs of an animal dying from the effects of the poison was examined, it was ascer- tained that the arrest in the interchange of the gases took place before the animal was dead. 9. Brucine acts in a similar manner as strychnine, but in a much less marked degree. 10. Quinine also possesses the power of retarding oxidation of the blood, as well as the elimination of carbonic acid gas. 11. Morphine has a more powerful effect in diminishing the exhalation of carbonic acid gas, as well as the chemical changes upon which the absorption of oxygen by blood depends. Under this head the effects of aneesthetics upon blood are next detailed ; and in the first place, the visible effects of chloroform upon blood are thus described :—If 5 or more ‘per cent. of chloroform be added to blood, and the mixture be agitated with air, it rapidly assumes a brilliant scarlet hue, which is much brighter than the normal arterial tint, and is, besides, much more permanent. When the mixture is left in repose, it gradually solidi- fies into a red-paint-like mass, which when examined under the micro- scope is frequently found to contain numerous prismatic crystals of an organic nature. If the blood of an animal poisoned from the inhalation of chloroform be employed in this experiment, the paint-like mass will be found to be composed in greater part of the crystals just spoken of; the crystals in this case being both larger and finer than when healthy blood is employed. Chloroform only partially destroys the blood-corpuscles. Its chemical action is to diminish the power of the constituents of the blood to unite with oxygen and give off carbonic acid. 160 ‘Prof. Tyndall—Contributions to Molecular Physics. [Mar.17, The action of sulphuric ether upon blood differs in many respects from that of chloroform. In the first place, ether has a powerful effect in de- stroying the blood-corpuscles, dissolving the cell-walls’ and setting the contents free. In the second place, ether prevents the blood from assum- ing an arterial tint when agitated with air. The higher the percentage of the agent, the more marked the effect. In the third place, ether neither diminishes the absorption of oxygen nor the exhalation of carbonic acid by blood; and lastly, it has a much more powerful effect in causing the con- stituents of the blood to crystallize. For example, if an equal part of ether be added to the blood of a dog poisoned by the inhalation of chloroform, as the ether evaporates groups of large needle-shaped crystals are formed. Under the microscope the crystals are found to be of a red colour and prismatic shape. Alcohol acts upon blood somewhat like chloroform ; it arrests the che- mical changes, but in a less marked degree. Amylene was found to act like ether upon blood, in so far as it did not diminish the absorption of oxygen or retard the elimination of carbonic acid. It differed, however, from ether in not destroying the blood-cor- puseles. In the last place, the action of mineral substances is stated, viz. : 1. Corrosive sublimate was found to increase the chemical ees which develope carbonic. acid, and to have scarcely any effect on those ae pending upon oxidation ; its influence, if any, is rather to diminish them than otherwise. 2. Arsenic seems to retard both the oxidation of the constituents of the blood and the exhalation of carbonic acid. 3. Tartrate of antimony increases the exhalation of carbonic acid gas, while it at the same time diminishes the absorption of oxygen. A. Sulphate of zinc and sulphate. of copper both act like tartrate of antimony, but not nearly so powerfully. Lastly, phosphoric acid was found to have the effect of increasing the chemical transformations and decompositions upon which the exhalation of eaybonic acid depends. March 17, 1864. Major-General SABINE, President, in the Chair. The following communications were aa — I. “ Researches on Radiant Heat.—Fifth Memoir. Contributions to Molecular Physics.” By J. Tynpatz, F.R.S., &c. Received March 17, 1864. (Abstract.) Considered broadly, two substances, or two forms of substance, occupy universe—the ordinary and tangible matter of that universe, and the 1864.] Prof. Tyndall—Contributions to Molecular Physics. 161 intangible and mysterious ether in which that matter is immersed. The natural philosophy of the future must mainly consist in the examination of the relations of these two substances. The hope of being able to come closer to the origin of the ethereal waves, to get some experimental hold of the molecules whence issue the undulations of light and heat, has stimulated the author in the labours which have occupied him for the last five years, and it is this hope, rather than the desire to multiply the facts already known regarding the action of radiant heat, which prompted his present investigation. He had already shown the enormous differences which exist between gaseous bodies, as regards both their power of absorbing and emitting radiant heat. When a gas is condensed to a liquid, or a liquid congealed to a solid, the molecules coalesce, and grapple with each other, by forces which were insensible as long as the gaseous state was maintained. _ But though the molecules are thus drawn together, the ether still surrounds them: hence, if the acts of radiation and absorption depend on the indi- vidual molecules, they will assert their power even after their state of aggregation has been changed. If, on the contrary, their mutual entangle- ment by the force of cohesion be of paramount influence in the interception and emission of radiant heat, then we may expect that liquids will exhibit a deportment towards radiant heat altogether different from that of the vapour from which they are derived. ‘The first part of the present inquiry is devoted to an exhaustive examina- tion of this question. The author employed twelve different liquids, and operated upon five different layers of each, which varied in thickness from 0°02 of an inch to 0°27 of aninch. The liquids were enclosed, not in glass vessels, which would have materially modified the heat, but between plates of transparent rock-salt, which but slightly affected the radiation. His source of heat throughout these comparative experiments consisted of a spiral of platinum wire, raised to incandescence by an electric current of unvarying strength. The quantities of radiant heat absorbed and trans- mitted by each of the liquids at the respective thicknesses were first deter- mined; the vapours of these liquids were subsequently examined, the quantities of vapour employed being proportional to the quantities of liquid traversed by the radiant heat. The result of the comparison was that, for heat of the same quality, the order of absorption of liquids and that of their vapours are identical. There was no exception to this law ; so that, to determine the position of a vapour as an absorber or radiator, it is only necessary to determine the position of its liquid. This result proves that the state of aggregation, as far, at all events, as the liquid stage is concerned, is of altogether subordinate moment—a con- clusion which will probably prove to be of cardinal moment in molecular physics. On one important and contested point it has a special bearing. If the position of a liquid as an absorber and radiator determine that of its 162 = Prof. Tyndall—Contributions to Molecular Physics. [Mar. 17, vapour, the position of water fixes that of aqueous vapour. Water had been compared with other liquids in a multitude of experiments, and it was found that as a radiant and as an absorbent it transcends them all. Thus, for example, a layer of bisulphide of carbon, 0-02 of an inch in thickness, absorbs 6 per cent., and allows 94 per cent. of the radiation from the red- hot platinum spiral to pass through it ; benzol absorbs 43, and transmits 57 per cent. of the same radiation ; alcohol absorbs 67, and transmits 33 per cent., and it stands at the head of all liquids except one in point of power as an absorber. The exceptionis water. A layer of this substance, of the thickness above given, absorbs 81 per cent., and permits only 19 per cent. of the radiation to pass through it. Had no single experiment ever been made upon the vapour of water, we might infer with certainty from the deportment of the liquid, that weight for weight this vapour transcends all others in its power of absorbing and emitting radiant heat. The relation of absorption and radiation to the chemical constitution of the radiant and absorbent substances was next briefly considered. For the first six substances in the list of those examined, the radiant and absorbent powers augment as the number of atoms in the compound molecule augments. Thus, bisulphide of carbon has 3 atoms, chloroform 5, iodide of ethyl 8, benzol 12, and amylene 15 atoms in their respective molecules; and the order of their powers as radiants and absorbents is that here indicated—bisulphide of carbon being the feeblest, and amylene the strongest of the six. Alcohol, however, excels benzol as an absorber, though it has but 9 atoms in its molecule; but, on the other hand, its molecule is rendered more complex than that of benzol by the introduc- tion of a new element. Benzol contains carbon and hydrogen, while alco- hol contains carbon, hydrogen, and oxygen. Thus, not only does the idea of multitude come into play in absorption and radiation, that of com- plexity must also be taken into account. The author directed the parti- cular attention of chemists to the molecule of water; the deportment of this substance towards radiant heat being perfectly anomalous, if the che- mical formula at present ascribed to it be correct. Sir William Herschel made the important discovery that beyond the limits of the red end of the solar spectrum, rays of high heating power exist which are incompetent to excite vision. The author has examined the deportment of those rays towards certain bodies which are perfectly opaque to light. Dissolving iodine in the bisulphide of carbon, he ob- tained a solution which entirely intercepted the light of the most brilliant flames, while to the extra-red rays of the spectrum the same iodine was found to be perfectly diathermic. The transparent bisulphide, which is highly pervious to the heat here employed, exercised the same absorption as the opaque solution. A hollow prism filled with the opaque liquid was placed in the path of the beam from an electric lamp; the light-spectrum was completely intercepted, but the heat-spectrum was received upon a 1864.] Prof. Tyndall—Contributions to Molecular Physics. 168 sereen, and could be there examined. Falling upon a thermo-electric pile, its presence was shown by the prompt deflection of even a coarse galvanometer. What, then, is the physical meaning of opacity and transparency, as regards light and radiant heat? The luminous rays of the spectrum differ from the non-luminous ones simply in period. The sensation of light is excited by waves of ether shorter and more quickly recurrent than those which fall beyond the extreme red. But why should iodine stop the former, and allow the latter to pass? The answer to this question, no doubt, is, that the intercepted waves are those whose periods of recurrence coincide with the periods of oscillation possible to the atoms of the dis- solved iodine. The elastic forces which separate these atoms are such as to compel them to vibrate in definite periods, and when these periods syn- chronize with those of the ethereal waves the latter are absorbed. Briefly defined, their transparency in liquids, as well as in gases, is synonymous with discord, while opacity is synonymous with accord between the periods of the waves of ether and those of the molecules of the body on which they impinge. All ordinary transparent and colourless substances owe their transparency to the discord which exists between the oscillating periods of their molecules and those of the waves of the whole visible spec- trum. The general discord of the vibrating periods of the molecules of com- pound bodies with the light-giving waves of the spectrum may be inferred from the prevalence of the property of transparency in compounds, solid, liquid, and gaseous, while their greater harmony with the extra-red periods is to be inferred from their opacity to the extra-red rays. Waiter illustrates this transparency and opacity in the most striking manner. It is highly transparent to the luminous rays, which demon- strates the incompetency of its molecules to oscillate in the periods which excite vision. It is as highly opaque to the extra-red undulations, which _proves the synchronism of its periods with those of the longer waves. If, then, to the radiation from any source water shows itself to be emi- nently or perfectly opaque, it is a proof that the molecules whence the radiation emanates must oscillate in what may be called extra-red periods. Let us apply this test to the radiation from a flame of hydrogen. This flame consists mainly of incandescent aqueous vapour, the temperature of which, as calculated by Bunsen, is 3259° C., so that if transmission aug- ments with temperature, we may expect the radiation from this flame to be copiously transmitted by the water. While, however, a layer of the bisulphide of carbon 0°07 of an inch in thickness transmits 72 per cent. of the incident radiation, and every other liquid examined transmits more or less of the heat, a layer of water of the above thickness is entirely opaque to the radiation from the flame. Thus we establish accord be- tween the periods of the molecules of cold water and those of aqueous vapour at a temperature of 3259° C. But the periods’of water have already been proved to be extra-red; hence those of the hydrogen flame 164 Prof. Tyndall—Contributions to Molecular Physics. [Mar. 17, must be extra-red also. The absorption by dry air of the heat emitted by a platinum spiral raised to incandescence by electricity was found to be insensible, while that by the ordinary wndried air was 6 per cent. Sub- stituting for the platinum spiral a hydrogen flame, the absorption by dry air still remained insensible, while that of the undried air rose fo 20 per cent. of the entire radiation. The temperature of the hydrogen flame was as stated, 3259°C., that of the aqueous vapour of the air was 20° C. Suppose, then, the temperature of our aqueous vapour to rise from 20°C. to 3259° C., we must conclude that the augmentation of temperature is applied fo an increase of amplitude, and not to the introduction of periods of quicker recurrence into the radiation. 3 The part played by aqueous vapour in the economy of Nature is far more wonderful than hitherto supposed. ‘To nourish the vegetation of the earth, the actinic and luminous rays of the sun must penetrate our atmosphere, and to such rays aqueous vapour is eminently transparent. The violet and the extra-violet rays pass through it with freedom. ‘To protect vegetation from destructive chills, the terrestrial rays must be checked in their transit towards stellar space, and this is accomplished by the aqueous vapour diffused through the air. This substance is the great moderator of the earth’s temperature, bringing its extremes into proximity, and obviating contrasts between day and night which would render life insupportable. But we can advance beyond this general statement now that we know the radiation from aqueous vapour is inter- cepted, in a special degree, by water, and reciprocally, the radiation from water by aqueous vapour ; for it follows from this that the very act of noc- turnal refrigeration which produces the condensation of aqueous vapour upon the surface of the earth—giving, as it were, a varnish of liquid water to that surface—imparts to terrestrial radiation that particular character which disqualifies it from passing through the earth’s atmosphere and losing itself in space. And here we come to a question in molecular physics which at the present moment occupies the attention of able and distinguished men. By allowing the violet and extra-violet rays of the spectrum to fall upon sul- phate of quinine and other substances, Professor Stokes has changed the periods of those rays. Attempts have been made to produce a similar result at the other end of the spectrum—to convert the extra-red periods into periods competent to excite vision—but hitherto without success. Such a change of period the author believed occurs when a platinum wire is heated to whiteness by a hydrogen flame. In this common experiment there is an actual breaking-up of long periods into short ones—a true rendering of invisual periods visual. The change of refrangibility here effected differs from that of Professor Stokes, first, by its being in the opposite direction, that is from lower to higher; and secondly, in the circumstance that the platinum is heated by the collision of the molecules of aqueous vapour, and before their heat has assumed the radiant form. 1864.] Prof. Tyndall—Contributions to Molecular Physics. 165 But it cannot be doubted that the same effect would be produced by radiant heat of the same periods, provided the motion of the ether could be rendered sufficiently intense. The effect, in principle, is the same whether we consider the platinum wire to be struck by a particle of aqueous vapour oscillating at a certain rate, or by a particle of ether oscillating at the same rate. By plunging a platinum wire into a hydrogen flame we cause it to glow, and thus introduce shorter periods into the radiation. These, as already stated, are in discord with water; hence we should infer that the trans- mission through water will be more copious when the wire is in the flame than when itis absent. Experiment proves this conclusion to be true. Water, from being opaque, opens a passage to 6 per cent. of the radiations from the flame and spiral. A thin plate of colourless glass, moreover, transmitted 58 per cent. of the radiation from the hydrogen flame; but when the flame and spiral were employed 78 per cent. of the heat was transmitted. For an alcohol flame Knoblauch and Melloni found glass to be less transparent than for the same flame with platinum spiral immersed in it; but Melloni afterwards showed that this result was not general, that black glass and black mica were decidedly more diathermic to the radiation from the pure flame. The reason of this is now obvious. Black mica and black glass owe their blackness to the carbon diffused through them. The carbon, as proved by Melloni, is in some measure transparent to the extra-red rays, and the author had in fact succeeded in transmitting between 40 and 50 per cent. of the radiation from a hydrogen flame through a layer of carbon sufficient to intercept the light of the most bril- liant flames. The products of combustion of the alcohol flame are carbonic acid and aqueous vapour, the heat of which is almost wholly extra-red. For this radiation the carbon is in a considerable degree transparent, while for the radiation from the platinum spiral it is in a great measure opaque. By the introduction of the platinum wire, therefore, the transparency of the pure glass and the opacity of its carbon were simultaneously aug- mented ; but the augmentation of opacity exceeded that of transparency, ~ and a difference in favour of opacity remained. No more striking or instructive illustration of the influence of coinci- dence could be adduced than that furnished by the radiation from a car- bonic oxide flame. Here the product of combustion is carbonic acid; and on the radiation from this flame even the ordinary carbonic acid of the atmosphere exerts a powerful effect. A quantity of the gas, only one- thirtieth of an atmosphere in density, contained in a polished brass tube four feet long, intercepted 50 per cent. of the radiation from the carbonic oxide flame. For the heat emitted by solid sources, olefiant gas is an in- comparably more powerful absorber than carbonic acid; in fact, for such heat the latter substance, with one exception, is the most feeble absorber to be found among the compound gases. For the radiation from the hydro- gen flame, moreover, olefiant gas possesses twice the absorbent power of 166 Prof. Tyndall—Contributions to Molecular Physics, [Mar.17, carbonic acid; but for the radiation from the carbonic oxide flame at a common tension of one inch of mercury, while carbonic acid absorbs 50 per cent., olefiant gas absorbs only 24. Thus we establish the coincidence of period between carbonic acid at a temperature over 3000°C., the periods of oscillation of both the incandescent and the cold gas belonging to the extra-red portion of the spectrum. It will be seen from the foregoing remarks and experiments how impossible it is to examine the effect of temperature on the transmission of heat, if different sources of heat be employed. Throughout such an examination the same oscillating atoms ought to be retained. The heating of a pla- tinum spiral by an electric current enables us to do this while varying the temperature between the widest possible limits. Their comparative opacity to the extra-red rays shows the general accord of the oscillating periods of our series of vapours with those of the extra-red undulations ; hence, by gradually heating a platinum wire from darkness up to whiteness, we gradually augment the discord between it and our vapours, and must there- fore augment the transparency of the latter. Experiment entirely confirms this conclusion. Formic ether, for example, absorbs 45 per cent. of the radiation from a platinum spiral heated to barely visible redness; 32 per cent. of the radiation from the same spiral at a red heat ; 26 per cent. of the radiation from a white-hot spiral, and only 21 per cent. when the spiral is brought near its point of fusion. Remarkable cases of inversion as to transparency occurred in these experiments. For barely visible redness formic ether is more opaque than sulphuric ; fora bright red heat both are equally transparent, while for a white heat, and still more for a nearly fusing temperature, sulphuric ether is more opaque than formic. This result gives us a clear view of the relationship of the two substances to the luminiferous ether. As we introduce waves of shorter period, the sulphuric augments most rapidly in opacity ; that is to say, its accord with the shorter waves is greater than that of the formic. Hence we may infer that the molecules of formic ether oscillate as a whole more slowly than those of sulphuric ether. When the source of heat was a Leslie’s cube filled with boiling water and coated with lampblack, the opacity of formic ether in comparison with sulphuric was very decided; with this source also the position of chloro- form, as regards iodide of methyl, was inverted. For a white-hot spiral, the absorption of chloroform vapour being 10 per cent., that of iodide of methyl] is 16 ; with the blackened cube as source, the absorption by chloro- form is 22 per cent., while that by the iodide of methyl is only 19. This inversion is not the result of temperature merely ; for whena platinum wire heated to the temperature of boiling water was employed as a source, the iodide was the most powerful absorbent. Numberless experiments, indeed, prove that from heated lampblack an emission takes place which synechro- nizes in an especial manner with chloroform. This may be thus illustrated. For the Leslie’s cube coated with lampblack, the absorption by chloroform 1864.] Prof. Tyndall—Contributions to Molecular Physics. 167 is more than three times that by bisulphide of carbon ; for the radiation from the most luminous portion of a gas flame the absorption by chloroform is also considerably in excess of that by bisulphide of carbon; while for the flame of a Bunsen’s burner, from which the incandescent carbon particles are removed by the free admixture of air, the absorption by bisulphide of carbon is nearly twice that by chloroform ; the removal of the incandescent carbon particles more than doubled in this instance the relative transparency of the chloroform. Testing, moreover, the radiation from various parts of the same flame, it was found that for the blue base of the flame the bisulphide was the most opaque, while for all other portions of the flame the chloroform was most opaque. For the radiation from a very small gas flame, consisting of a blue base and a small white top, the bisulphide was also most opaque, and its opacity very decidedly exceeded that of the chloroform when the flame of bisulphide of carbon was employed as a source. Comparing the radiation from a Leslie’s cube coated with isinglass with that from a similar cube coated with lampblack, at a common temperature of 100° C., it was found that out of eleven vapours all but one absorbed the radiation from the isinglass most powerfully ; the single exception was chloroform. It may be remarked that whenever, through a change of source, the position of a vapour as an absorber of radiant heat was altered, the position of the liquid from which the vapour was derived was changed in the same manner. It is still a point of difference between eminent investigators as to whether radiant heat up to a temperature of 100° C. is monochromatic or not. Some affirm this, others deny it. A long series of experiments enables the author to state that probably no two substances at a temperature of 100°C. emit heat of the same quality. The heat emitted by isinglass, for example, is different from that emitted by lampblack, and the heat emitted by cloth or paper differs from both. It is also a subject of discussion whether rock- salt is equally diathermic to all kinds of calorific rays,—the differences affirmed to exist by one investigator being ascribed by others to differences of incidence from the various sources employed. MM. De la Provostaye and Desains maintain the former view, Melloni and M. Knoblauch main- tain the latter. The question was examined by the author without changing anything but the temperature of the source. Its size, distance, and sur- roundings remained the same, and the experiments proved that rock-salt shared in some degree the defect of all other substances ; it is not perfectly diathermic, and it is more opaque to the radiation from a barely visible spiral than to that from a white-hot one. The author devotes a section of his memoir to the relation of radiation to conduction. Defining radiation, internal as well as external, as the com- munication of motion from the vibrating molecules to the ether, he arrives by theoretic reasoning at the conclusion that the best radiators ought to prove the worst conductors. A broad consideration of the subject shows at once the general harmony of the conclusion with observed facts. Organic substances are all excellent radiators ; they are also extremely bad 168 Mr. Balfour Stewart on Sun Spots. [Mar. 17, conductors. The moment we pass from the metals to their compounds, we pass from a series of good conductors to bad ones, and from bad radiators to good ones. Water, among liquids, is probably the worst conductor ; it is the best radiator. Silver, among solids, is the best conductor; it is the worst radiator. In the excellent researches of MM. De la Provostaye and Desains the author finds a striking illustration of what he regards as a natural law—that those molecules which transfer the greatest amount of motion to the ether, or, in other words, radiate most powerfully, are the least competent to communicate motion to each other, or, in other words, to conduct with facility. II. “ Remarks on Sun Spots.” By Batrour Stewart, M.A., F.R.S., Superintendent of the Kew Observatory. Received March 8, 1864: In the volume on Sun Spots which Carrington has recently published, we are furnished with a curve denoting the relative frequency of these phe- nomena from 1760 to the present time. This curve exhibits a maximum corresponding to 1788°6. Again, in Dalton’s ‘ Meteorology’ we have a list of auroree observed at Kendal and Keswick from May 1786 to May 1793. The observations at Kendal were made by Dalton himself, and those at Keswick by Crosthwaite. This list gives— For the year 1787 ....27 aurore, For the year 1790 .... 36 aurore; ISS 2 Oo Ff 17912 aie WSO AG = 179224 2a 3 showing a maximum about the middle, or near the end of 1788. This corresponds very nearly with 1788°6, which we have seen is one of Car- rington’s dates of maximum sun spots. The following observation is unconnected with the aurora borealis. In examining the sun pictures taken with the Kew Heliograph under the superintendence of Mr. De la Rue, it appears to be a nearly universal law that the faculee belonging to a spot appear to the left of that spot, the motion due to the sun’s rotation being across the picture from left to right. These pictures comprise a few taken in 1858, more in 1859, afew in 1861, and many more in 1862 and 1863, and they have been carefully examined by Mr. Beckley, of Kew Observatory, and myself. The following Table expresses the result obtained :— No. of cases of No. of cases of No. of cases of No. of cases of fa- Year. _—facula to left facula to right facula equally on culz mostly be- of spot. of spot. both sides of spot. tween two spots. ‘bere chee ARNE SPU SR te OR sees Re A O rach se eee ee Oj. fe) ae es AC My ge aoe ss Sim etre th set Os ate eee 3 Oat tes e Oe ee eee Ihe auh hea Oe ee ee 0 1862.2. Of es eee EP a a T RE 3 1863 Sits AG. Fs be OW Ose ee A gepress ae cnn ts 2 RS Gd eran oe Ba. UG f iO) ee 2 Siete Ryedale tele 1 1864..] “Mr. Hicks on an Improved Barometer. 169 III. “Description of an Improved Mercurial Barometer.” By James Hicks, Esq. Communicated by J. P. Gasstor, F.R.S. Received March 16, 1864. Having shown this instrument to Mr. Gassiot, he wished me to write a short description of it, which he thought would be of interest to the Royal Society. Some time since I constructed an open-scale barometer, with a column of mercury placed in a glass tube hermetically sealed at the top, and per- fectly open at the bottom. The lower half of the tube is of larger bore than that of the upper. If a column of mercury, of exactly the length which the atmosphere is able at the time to support, were placed in a tube of glass hermetically sealed at the top, of equal bore from end to end, the mercury would be held in suspension ; but immediately the pressure of the atmosphere increased, the mercury would rise towards the top of the tube, and remain there till, on the pressure decreasing, it would fall towards the bottom, and that portion which the atmosphere was unable to support would drop out. But if the lower half of the tube be made a little Jarger in the bore than the upper, when the column falls, the upper portion passes out of the smaller part of the tube into the larger, and owing to the greater capacity of the latter, the lower end of the column of mercury does not sink to the same extent as the upper end, and the column thus becomes shorter. The fall will continue until the column is reduced to that length which the atmosphere is capable of supporting, and the scale attached thus registers what is ordinarily termed the height of the barometer. From the above description it will be evident that, by merely varying the proportion in the size of the two parts of the tube, a scale of any length can be obtained. For example, if the tubes are very nearly the same size in bore, the column has to pass through. a great distance before the necessary compensation takes place, and we obtain a very long scale, say 10 inches, for every 1-inch rise and fall in the ordinary barometer. But if the lower tube is made much larger than the upper, the mercury passing into it quickly compensates, and we obtain a small scale, say from 2 to 3 inches, for every inch. To ascertain how many inches this would rise and fall for an ordinary inch of the barometer, I attach it, in connexion with a standard barometer, to an air-pump receiver, and by reducing the pressure in the air-pump I cause the standard barometer to fall, say 1 inch, when the other will fall, say 5 inches; and so I ascertain the scale for every inch, from 31 to 27 inches. It was on this principle that I constructed the open-scale barometer, which has since been extensively used. But having been asked to apply a vernier to one of these barometers graduated in this way, I found this im- practicable, as each varied in length in proportion as the bore of the tube varied, so that every inch was of a different length. VOL. XIII. oO 170 Mr. Perkin on Mauve or Aniline-Purple. [1864. I have now remedied this defect, and made what I believe is an absolute standard barometer, by graduating the scale from the centre, and reading it off with two verniers to the ;5,th of an inch. The scale is divided from the centre, up and down, into inches, and subdivided into 20ths. To ascertain the height of the barometer graduated in this way, take a reading of the upper surface of the column of mercury with the vernier, then of the lower surface in the same way, and the two readings added to- gether will give the exact length of the column of mercury supported in the air, which is the height of the barometer at the time. There is another advantage in this manner of graduating over the former, that if a little of the mercury drops out it will give no error, as the column will immediately rise out of the larger tube into the smaller, and become the same length as before; but by the former scale the barometer would stand too high, until readjusted, which could only be effected by putting the same quantity of mercury in again. : I have introduced Gay-Lussac’s pipette into the centre of the tube, to prevent the possibility of any air passing up into the top. The Society then adjourned over the Easter Recess to Thursday, April 7th. “On Mauve or Aniline-Purple.’” By W. H. Perxin, F.C.S. Com- municated by Dr. Srennousz. Received August 19, 1863*. The discovery of this colouring matter in 1856, and its introduction as a commercial article, has originated that remarkable series of compounds known as coal-tar colours, which have now become so numerous, and in consequence of their adaptibility to the arts aud manufactures are of such great and increasing importance. ‘The chemistry of mauve may appear to have been rather neglected, its composition not having been established, although it has formed the subject of several papers by continental chemists. Its chemical nature also has not been generally known; and to this fact many of the discrepancies in the results of the different experimentalists who have worked on this subject are to be attributed. The first analysis I made of this colouring matter was in 1856, soon after I had become its fortunate discoverer. The product I examined was purified as thoroughly as my knowledge of its properties then enabled me, and the resultst obtained agree very closely with those required for the formula I now propose. Since that time I have often commenced the study of this body in a scientific point of view, but other duties have prevented me * For abstract see vol. xii. p. 713. + The substance I examined was doubtless the sulphate, of which I made two com- bustions :— No. I. gave 71°55 per cent. of carbon and 6:09 per cent. of hydrogen. No. II. gave 71°60 5 % Beh) o Theory requires 71:5 4 y 5°5 ” 1864.] Mr. Perkin on Mauve or Aniline-Purple. 171 from completing these investigations; but, although unacquainted with its correct formula, its chemical characters have necessarily been well known to me for a considerable time. When first introduced, commercial mauve appeared as an almost perfectly amorphous body ; but now, owing to the great improvements which have been made in its purification, it is sent into the market perfectly pure and crystallized. On adding a solution of hydrate of potassium to a boiling solution of commercial crystallized mauve, it immediately changes in colour from purple to a blue violet, and after a few moments begins to deposit a crystalline body. After standing a few hours, this crystalline product is collected on a filter, washed with alcohol once or twice, and then thoroughly with water. When dry, it appears as a nearly black glistening substance, not unlike pulverized specular iron ore. This substance, for which I propose the name Mauveine, is a powerful base. It dissolves in alcohol, forming a blue violet solution, which imme- diately assumes a purple colour on the addition of acids. It is insoluble, or nearly so, in ether and benzole. It is a very stable body, and decomposes ammoniacal compounds readily. When heated strongly it decomposes, yielding a basic oil, which does not appear to be aniline. The following analyses were made of specimens dried at 150° C. :— I. -301 grm. of substance gave ‘8818 of carbonic acid and ‘162 of water. II. *2815 grm. of substance gave °8260 of carbonic acid and °145 of water. Direct Nitrogen determination. III. °3435 grm. of substance gave41°0c.c. N at23°C. and 766 mms. Bar. __41°0 cub. centims. (766-0 millims. —20°9) ot 824°1 millims. 37°7 X 0012562 grm.='04735 grm. of N. These numbers correspond to the following percentages :— y! =37°7 cub. centims. I. II. III. CMON sesso cee -19'9 80:0 — Elydrogene. ona... (5°98 5°72 INTET OREM) sca s ee a 13°75 The formula, C,,* H,, N,, requires the following values :— Theory. Mean of experiment. BSI TS Ce sien ire 24 80°19 79°95 Bg i's Shnwtoe tere 24 5°94 5°85 UN GH ede Persia es 56 13°87 13°75 404 100° Hydrochlorate of Mauveine.-—This salt is prepared by the direct combi- nation of mauveine and hydrochloric acid. From its boiling alcoholic solu- tion it is deposited in small prisms, sometimes arranged in tufts, possessing * C=12, o 2 172 Mr. Perkin on Mauve or Aniline-Purple. [1864.. a brilliant green metallic lustre. It is moderately soluble in aleohol, but nearly insoluble in ether. It is also, comparatively speaking, moderately soluble in water. Different preparations dried at 100° C. gave the following numbers :-— I. °306 grm. of substance gave *8255 of carbonic acid and *162 of water. II. -308 grm. of substance gave *8275 of carbonic acid and *163 of water. III. °310 grm. of substance gave ‘8345 of carbonic acid. IV. °3165 grm. of substance gave 851 of carb. acid and °16525 of water. V. :2447 grm. of substance gave °6603 of carb. acid and *1356 of water. VI. -627 grm. of substance gave *205 of chloride of silver. VII. -560 grm. of substance gave °195 of chloride of silver. VIII. -69 grm. of substance gave ‘2266 of chloride of silver. Direct Nitrogen determination. IX. °3497grm. of substance gave 40 c.c. N at 20°C. and 777°2 mms. Bar. 40 c.c. 72—17°4 aos Ve are e.c. at 0° C. and 760 millims, Bar. 37°2 cub. centims. x 0012562 grm.=-04673 grm. N. These numbers correspond to the following percentages :-— i Le III. IY: (Carbon -% osc... 73°5 73°27 73°4 73°3 Hydrogen.....-...- 5°88 5:88 ae 58 Nitrogen, 225.2 °-% 4 ~ ass SAE, (Chiorme. ©. 26 yee = eee sa Bee V. VI. Vi. . VOL ae arbor. oie es sae 73-50 eee Hydrogen .........- 16 ——— Nitroven o.).4 425... ;- Se ee es Ghilomue: 2s ts 3s —— 8:08 8:06 8:1 Bes. These numbers agree with the formula C,, H,,N,H Cl, as may be seen by the following Table :-— Theory. Mean of experiment. eras 7 Fatt 324-7355 73°41 EL so Sia cee 25° 5°67 0°93 Nc Baie tees ons 56° 12°73 13°30 AO Ro. eee i, 2 35°95 8°05 8°07 440°5 100-00 I have endeavoured to obtain a second hydrochlorate containing more acid, but up to the present time have not succeeded. Platinum-salt.—Mauveime forms a perfectly definite and beautifully crystallme compound with bichloride of platinum. It is obtained by mixing an alcoholic solution of the above hydrochlorate with an excess of an 1864. | Mr. Perkin on Mauve or Aniline- Purple. 173 alcoholic solution of bichloride of platinum ; from this mixture the new salt separates as a highly crystalline powder. I have generally preferred to use cold solutions in its preparation; but if moderately hot solutions be em- ployed, the salt will separate as crystals of considerable dimensions. This platinum-salt possesses the green lustre of the hydrochlorate, but, on being dried, assumes a more golden colour. It is very sparingly soluble in alcohol. The following numbers were obtained from various preparations dried at 100°.C. :— I. -44125 grm. of substance gave ‘072 of platinum. IT. -4845 grm. of substance gave °079 fe III. -511 grm. of substance gave -083 IV. -510 grm. of substance gave °083 V. °6345 grm. of substance gave *1035 VI. -618 grm. of substance gave °101 ” VII. °31275 grm. of substance gave 60525 of carbonic acid and ‘118 of water. VIII. -30675 grm. of substance gave °595 of carb. acid and*110 of water. IX. °3795 grm. of substance gave 27 of chloride of silver. These results correspond to the percentages in the following Table :— 33 I II II. IV V. VI. Carbon .... —— aa —-- wae oe —— Hydrogen .. —— oe a —_ ss ——— —— Chlorine .... —— we a ——— — ——— Platinum .. °16°31 16°3 16°24 16°27 16°3 16°3 VII. VIII. IX. Carhons sa 664 52°77 52°86 a Piydrogen. .. 4 4°19 3°98 Chlorine ...... — —— 17°6 Platinumys3 . 2... sess The formula, C,, H,, N, H Pt Cl,, requires the following values :— Theory. Mean of experiment. C. 324 53°09 52°81 [2 ene 4-09 4°19 N, 56° 9-2 = Pt 98°7 16°16 16°28 Cl, 106°5 17°46 17°6 610°2 100:°00 Gold-salt.—This compound is prepared in a similar manner to the _ platinum-salt, only substituting chloride of gold for chloride of platinum. It separates as a crystalline precipitate, which, when moist, presents a much less brilliant aspect than the platinum derivative; it is also more soluble than that salt, and when crystallized appears to lose a small quan- tity of gold. The following results were obtained from a specimen dried at 100° C, :— 174 Mr. Perkin on Mauve or Aniline-Purple. [1864. I. ‘47175 grm. of substance gave °1245 of gold. II. *35525 grm. of substance gave °094 of gold. III. :309 grm. of substance gave °495 of carbonic acid and *101 of water. Percentage composition :— iF II. IIl. Carbon —— ——— 43°68 Hydrogen — — 3°6 Gold 2673 26°46 The formula, C,, H,, N,, H AuCl,, requires the following percentages :— Theory. Mean of experiment. FE Ce ee 43°53 43°68 ji Bees 25 3°34 3°6 Ngan. S056 7°44 —— AM eta OT 26°61 26°38 1 ae 19-08 es 744 100:00 - Hydrobromate of Mauveine.—This salt is prepared in a similar manner to the hydrochlorate, which it very much resembles, except that it is less soluble in alcohol. Analysis of preparations dried at 100° C. gave the following numbers :— I. *3935 grm. of substance gave 1515 of bromide of silver. II. :450 grm. of substance gave *173 of bromide of silver. III. -3265 grm. of substance gave -79675 of carb. acid and ‘158 of water. IV. °35125 grm. of substance gave °86075 of carbonic acid and °1675 of water. Percentage composition :— I. II. III. IV. Carbon ——. —— 66°55 66°8 Hydrogen —— 5°37 5°29 Bromine’ 16°38 16°37 cae These numbers agree with the formula C,, H,, N, H Br, as shown by the comparisons in the following Table :— | Theory. Experiment. My Tae aS cs . 324 66°8 66°67 |b RR ern) 745) 5°15: 5°33 Neciiwin ob 11°56 ee Broidg-«7 0 16°49 16°37 485 100-00 Hydriodate of Mauveine.—In preparing this salt from the base, it is necessary to use hydriodic acid which is colourless, otherwise the free iodine will slowly act upon this salt. It crystallizes in prisms haying a 1864.] Mr. Perkin on Mauve or Aniline-Purple. 175 green metallic reflexion. It is more insoluble than the hydrobromate. The products used in the subjoined analysis were recrystallized three times, and dried at 100° C. I. :5115 grm. of substance gave *22575 of iodide of silver. II. -248 grm. of substance gave ‘549 of carb. acid and °10975 of water. III. -2985 grm. of substance gave 663 of carb. acid and °1265 of water. IIV. °2765 grm. of substance gave °615 of carb. acid and °1145 of water. Percentage composition :— I. II. III. IV. Carbon —— 60°46 60°57 60°65 Hydrogen 4°9 4°7 4°7 Todine 23°8 The formula, C,, H,, N, HI, requires the following values :— Theory. Experiment. eG REN Be » O24> 60°89 60% 56 LE eT 25° 4°69 4°7 IN ee Mae 508 10°54 —. I Sn aaa 23°88 23°8 532°] 100-00 Acetate of Mauveine.—This salt is best obtained by dissolving the base in boiling alcohol and acetic acid. On cooling, it will crystallize out; it should then be recrystallized once or twice. This acetate is a beautiful salt, possessing the green metallic lustre common to most of the salts of mauveine. ‘Two combustions of specimens dried at 100°C. gave the fol- lowing numbers :— | I. °28325 grm. of substance gave °778 of carb. acid and °153 of water. If. *29275 grm. of substance gave *806 of carb. acid and ‘1645 of water. Percentage composition :— I. Il. Carbon 74-9 75°0 Hydrogen 6-0 6:2 These numbers lead to the formula C. 15 bs N, 0,=C,, H,, N, C, H, O,, as shown by the following Table :— Theory. Experiment. ee aa US Celene Ns, 79° 74°95 ee 2G 6° 6:1 ING warmer oO 12°06 —— O, . 32 6°94 — 464 100-00 Carbonate of Mauveine.—The tendency of solutions of mauveine to combine with carbonic acid is rather remarkable. If a quantity of its solu- 176 Mr. Perkin on Mauve or Aniline-Purple. [1864. tion be thrown up into a tube containing carbonic acid over mercury, the carbonic acid will quickly be absorbed, the solution in the mean time pass- ing from its normal violet colour to purple. To prepare this carbonate, it is necessary to pass carbonic acid gas through boiling alcohol containing a quantity of mauveine in suspension. It is then filtered quickly, and car- bonic acid passed through the filtrate until nearly cold. On standing, this liquid will deposit the carbonate as prisms, having a green metallic reflexion. A solution of this salt, on being boiled, loses part of its carbonic acid and assumes the violet colour of the base. When dry this carbonate rapidly changes, and if heated to 100° C. loses nearly all its carbonic acid and changes in colour to a dull olive; therefore, as it cannot be dried’ with- out undergoing a certain amount of change, its composition is difficult to determine. However, I endeavoured to estimate the carbonic acid in this salt by taking a quantity of it freshly prepared and in the moist state, and heating it in an oil-bath until carbonic acid ceased to be evolved. The residual base was then weighed, and also the carbonic acid, which had been collected in a potash bulb, having been previously freed from water by means of sulphuric acid. The foilowing results were obtained :— I. 1°88 residual base obtained ; *190 carbonic acid evolved. II. 1:375 residual base; °1385 carbonic acid evolved. +190 of CO, is equal to -268 of H,CO,; this, added to the residual base, will give the amount of substance experimented with, viz. 2°148. The amount of CO, obtained from this quantity, therefore, is 8°8 per cent. Calculating the second experiment in a similar manner, the amount of carbonate operated upon would be 1°5702 grm.; the percentage of CO, obtained is therefore equal to 8°8. A carbonate having the formula (C,, H,, N,), H, CO, would contain 5:1 per cent. of CO,, and an acid carbonate having the formula O,,H,,N,, H,CO, would contain 9:4 per cent. of CO,. Considering that this salt when prepared begins to crystallize before it is cold, probably the first portions that deposit are a monocarbonate, while the larger quantity which separates afterwards is an acid carbonate. Hence the deficiency in the amount of CO, obtained in the above experiments. I hope to give my attention to this remarkable salt at a future period. In the analysis of the salts of mauveine great care has to be taken in drying them thoroughly, as most of them are highly hygroscopic. I am now engaged in the study of the replaceable hydrogen in mauveine, which I hope will throw some light upon its constitution. From its for- mula I believe it to be a tetramine, although up to the present time I have not obtained any definite salts with more than 1 equiv. of acid. When mauveine is heated with aniline it produces a blue colouring matter, which will doubtless prove to be a phenyle derivative of that base. A salt of mauveine when heated alone also produces a violet or blue compound. These substances 1am now examining, and hope in a short time to have the honour of communicating them to the Society. Oe 3QiA ci fir i a! : 4, i BU, LOU g, sik 4 1864. | ‘ Pe OR April 7, 1864. Major-General SABINE, President, in the Chair. The Rev. Dr. Salmon was admitted into the Society. The following communications were read :— I. “On the Functions of the Cerebellum.” By Wit1t1am How- sHie Dickinson, M.D. Cantab., Curator of the Pathological Museum, St. George’s Hospital, Assistant Physician to the Hos- pital for Sick Children. Communicated by Dr. Bencz Jones. Received March 8, 1864. (Abstract.) The paper is divided into two Parts; the first gives the results of expe- riments on animals; the second, of observations upon the human being. Part I. Assuming that the great divisions of the brain preserve each the same function through the vertebrate kingdom, it is maintained that experiments which can be performed only on such of the lower animals as are very tenacious of life, will afford deductions of universal application. The method of proceeding with regard to each species was to remove, first, the whole encephalon, with the exception of the medulla oblongata ; then in a similar animal only the cerebrum was taken away. The only difference between the two cases was in the fact that one animal had a cerebellum, and the other had not. A comparison was believed to show, in the powers which one had more than the other, the function of the organ the possession of which constituted the only difference. Finally it was ascertained in each species what is the effect of taking away the cerebellum alone. The use of the organ was thus estimated in two ways—by the effect of its addition to the medulla, and of its subtraction from the rest of the nervous system. The species so treated are arranged in an ascending scale, according to the comparative weight of the cerebellum. The field-snake, frog, sala- mander, toad, land-tortoise, eel, water-tortoise, pike, perch, tench, dace, carp, gold-fish, rudd, loach, and gudgeon were subjected to these opera- tions ; besides which, many experiments of a less systematic character were made upon birds and mammalia. The results are these :— In Reptiles, with the exception of the snake, the cord, together with the medulla oblongata, is sufficient to give the power of voluntary or sponta- neous motion-—limited, but usually enough to allow of feeble locomotion. With the addition of the cerebellum, all actions dependent on the will appear to be naturally performed. VOL. XIII. P 178 Dr. Dickinson on the Functions of the Cerebellum. [April 7, The removal of the cerebellum shows that the cerebrum by itself is unable to give more than a limited amount of voluntary motion, and that of a kind deficient in balance and adjustment. It is therefore inferred that the cord, together with the medulla oblon- gata, is a great source of spontaneous motor power, in which function both the cerebrum and the cerebellum take part, the cerebellum to the greater extent ; it also appears that a certain harmony in the use of the muscles depends on the possession of the latter organ. Regarding Fishes, the cord and medulla oblongata seem unequal to the performance of voluntary motion. When the cerebellum is added, the powers become so far extended that movements are made in obedience to external stimuli. Generally speak- ing, a determined position is maintained and locomotion accomplished, without the use, however, of the pectoral fins. If the cerebellum only be taken away, there is a loss of the proper adjustment between the right and left sides; so that oscillation or rotation takes place. All the limbs are used, but apparently with a deficiency of sustained activity. It is therefore concluded that with Fishes, as with Reptiles, the power of intentional movement is shared by both cerebrum and cerebellum; the former in this case has the larger influence. Such movements as depend on the cerebrum are destitute of lateral ba- lance, are sudden in being affected by any external cause, and are emotional in their character. Such as depend on the cerebellum are mutually ad- justed, of a continuous kind, and less directly under the influence of con- sciousness. The same facts were supported by experiments on the higher orders of animals: in these it seemed that the cord and medulla are insufficient to excite voluntary movements. The muscles, as with fishes and reptiles, acknowledge a double rule, from the cerebrum and from the cerebellum. The anterior limbs are most subservient to the cerebrum ; the posterior to the cerebellum. The limbs on one side are in connexion chiefly with the lobe of the opposite side. The absence of the cerebellum destroys the power of lateral balance. From the negative results of the experiments, it is inferred that the cere- bellum has nothing to do with common sensation, with the sexual pro- pensity, with the action of the involuntary muscles, with the maintenance of animal heat, or with secretion. The only function which the experiments assigned to the cereuellenl is such as concerns the voluntary muscles, which receive therefrom a regu- lated supply of motor influence. Each lateral half of the cerebellum affects both sides, but the one opposite to itself most. The cerebellum has a property distinct from its true voluntary power, which harmonizes the action of the voluntary muscles, and has been de- scribed as ‘‘ coordination.” 1864.) Prof. Sylvester on Newton’s Rule, &c. 179 The voluntary muscles are under a double influence—from the cerebrum and from the cerebellum. The anterior limbs are chiefly under the influ- ence of the cerebrum; the posterior, of the cerebellum. Cerebellar move- ments are apt to be habitual, while cerebral are impulsive. The cerebellum acts when the cerebrum is removed, though when both organs exist it is under its control. Part IT. From an analysis.of one case of congenital absence of the cerebellum, one of disease of the whole organ, and 46 of disease of a portion of it, the fol- lowing deductions are stated :— The only faculty which constantly suffers in consequence of changes in the cerebellum, is the power of voluntary movement. When the organ is absent or defective congenitally, we have want of action in the muscles of the lower extremities. When the entire structure is changed by disease, we have loss of volun- tary power, either general throughout the trunk, or limited to the lower limbs—which results are about equally frequent. From the manner in which the paralysis was distributed in cases of disease of a part of the organ, it is inferred that each lobe is in connexion as a source of voluntary movement with all the four limbs, but in the greatest degree with the limbs of the opposite side, and with the lower more than with the upper extremities. The occasional occurrence of logs of visual power, and alterations of the sexual propensity, is referred to the conveyance of irritation to the corpora quadrigemina in one case, and the spinal cord in the other. From both sources of knowledge it is concluded that the cerebellum has distinct offices. It is a source of voluntary motor power to the muscles supplied by the spinal nerves. It influences the lower more than the upper limbs, and pro- duces habitual rather than impulsive movements. Each lobe affects both sides of the body, but most that opposite to itself. Secondly, the cerebellum has a power which has been described as that of “coordination,” which is similarly distributed. Finally, it is suggested that the outer portion of the organ may be the source of its voluntary motor power, while its inner layer is the means of regulating its distribution. II. “An Inquiry into Newton’s Rule for the Discovery of Ima- gimary Roots.” By J. J. Sytvuster, F.R.S., Correspondent of the Institute of France. Received April 6, 1864. (Abstract.) In the ‘ Arithmetica Universalis,’ in the chapter ‘“ De Resolutione Equa- tionum,”” Newton has laid down a rule, admirable for its simplicity and generality, for the discovery of imaginary roots in algebraical equations, and P2 180 Prof. Sylvester on Newton’s Rule for [April 7, for assigning an inferior limit to their number. He has given no clue towards the ascertainment of the grounds upon which this rule is based, and has stated it in such terms as to leave it quite an open question whether or not he had obtained a demonstration of it. Maclaurin, Campbell, and others have made attempts at supplying a demonstration, but their efforts, so far as regards the more important part of the rule, that namely by which the limit to the number of imaginary roots is fixed, have completely failed in their object. Thus hitherto any opinion as to the truth of the rule rests on the purely empirical ground of its being found to lead to correct results in particular arithmetical instances. Persuaded of the insufficiency of such a mode of verification, the author has applied himself to obtain- ing a rigorous demonstration of the rule for equations of specified degrees. For the second degree no demonstration is necessary. For cubic equa- tions a proof is found without difficulty. For biquadratic equations the author proceeds as follows. He supposes the equation to be expressed homogeneously in 2, y, and then, instituting a series of infinitesimal linear transformations obtained by writing x+y for 2, or y+Az for y, where h is an infinitesimal quantity, shows that the truth of Newton’s rule for this case depends on its being capable of being shown that the discriminant of the function (1, +e, e, te, 1 (a, y)* is necessarily positive for all values of e greater than unity, which is easily proved. He then proceeds to consider the case of equations of the 5th degree, and, following a similar process, arrives at the conclusion that the truth of the rule depends on its being gapoble of being shown that the discriminant, say (D) of the fanenen (a: ee, 7°, 0, 1K, y)’s which for facility of reference may be termed “the (e, 7) function,” is necessarily positive when e*— en’ and n*—ye* are both positive. This discriminant is of the 12th degree in e,y. But on writing a=en, y=e’ +7’, it becomes a rational integral function of the 6th degree in x, and of the second degree in y, and such that, on making D=0, the equation represents a sextic curve, of which z, y are the abscissa and ordinate, which will consist of a single close. It is then easily demonstrated that all values of e, 7 which cause the variable point x, y to lie inside this curve, will cause D to be negative (in which case the function e, n has only two imaginary factors), and that such values as cause the variable point to lie outside the curve, will make D positive, in which case the e, 7 function has four imaginary factors. When the conditions concerning ¢, » above stated are verified, it is proved that the variable point must be exterior to the curve, and thus the theorem is demonstrated for equations of the 5th degree. The question here naturally arises as to the significance of the sign of D when such a position is assigned to the variable point as gives rise to ¢magi- nary values of e, 7, which in such case will eS conjugate quantities of the form \+ipn, \—tp respectively. The curve D will be divided by another sextic curve into two portions, for one of which the couple e, 7 corresponding to any point in its interior is 1864. | the Discovery of Imaginary Roots. 181 real, and for the other conjugate. This brings to view the necessity of there being in general a theory for equations with conjugate coefficients, which for greater brevity may be termed conjugate equations, analogous to that for real equations in respect of the distinction between real and imagi- nary roots in the latter. A conjugate equation is one in which the coeffi- cients, reckoning from the two ends of the equation, go in pairs of the form pxiq, with the obvious condition that when there is a middle coefficient this must be real. Such an equation may be supposed to be so prepared that, when thrown into the form P+7Q, P and Q shall have no common algebraical factor; and when this is effected, it may easily be shown that the conjugate equation can neither have real roots nor roots paired together of the form \+-7p, A—7z respectively. How, then, it may be asked, is the analogy previously referred to possible? On investigation it will be found that the roots divide themselves into two categories, each of exactly the same order of generality,—viz. solitary roots of the form e, and assoct- ated roots which go in pairs, the two roots of each pair being of the form pe, 1 ga respectively ; so that, following the ordinary mode of geometrical representation of imaginary quantities, the roots of a conjugate equation may be denoted by points lying on the circumference of a circle to radius unity (corresponding to solitary roots), and points (corresponding to the associated roots) lying in couples on different radii of the circle at reci- procal distances from the centre, each couple in fact constituting, accord- ing to Prof. W. Thomson’s definition, electrical images of each other in respect to the circle. If the circle be taken with radius infinity stead of unity (so as to become a straight line), then we have the geometrical eidolon of the roots of an ordinary equation, the’ solitary roots lying on a straight line, and the associated or paired (imaginary) roots on each side of, and at equal distances from the line. In the inquiry before us, whether the variable point belong to the real or conjugate part of the plane of the D curve, it is shown to remain true that the number of associated roots will be two, if it lie inside the curve, and four if it lie outside. The author then suggests a probable extension of Newton’s rule to conjugate equations of any degree. In conclusion, he deals with a question in close connexion with, and arising out of the investigation of this rule, relating to equations of the form 2+(ax+6)"=0, to which, for convenience, he gives the provisional name of ‘ superlinear equations ”’ (denoting the function equated to zero as a superlinear form), and esta- blishes a rule for limiting the number of real roots which they can con- tain, which is, that if such equation be thrown under the form A, (@te)™+A, (@te,)"+ ..-. HA, (@+e,)"=0, and ¢,,¢,,. . - ¢, be an ascending or descending order of magnitudes, the equation cannot have more real roots than there were variations of sign in the sequence A,,A,5-*+ + + Aw (—)"A,- 182 Prof. Sylvester—Discovery of Imaginary Roots. [April 7, This theorem was published by the author, but without proof, in the ‘Comptes Rendus’ for the month of March in this year. But the method of demonstration now supplied is deserving of particular attention in itself; for it brings to light a new order of purely tactical con- siderations, and establishes a previously unsuspected kind of, so to say, algebraical polarity. The proof essentially depends upon the character of every superlinear form being associated with, and capable of definition by means of a pencil of rays, which may be called the type pencil, subject to a species of circulation of a different nature according as the degree of the forn: is even or odd, which he describes by the terms “‘per-rotatory”’ in the one case, and “ trans-rotatory ”’ in the other ; so that the types themselves may be conveniently distinguished by the names “‘per-rotatory”’ and “‘trans- rotatory.’ By per-rotatory circulation is to be understood that species in which, commencing with any element of the type, passage is made from it to the next, from that to the one following, from the last but one to the last, from the last to the first, and so on, until the final passage is to the element commenced with from the one immediately preceding. By trans-rotatory circulation, on the other hand, is understood that species in which, com- mencing with any element and proceeding in the same manner as before to the end element, passage is made from that, not to the end element itself, but to its polar opposite, from that to the polar opposite of the next, and so on, until the final passage is made to the polar opposite of the ele- ment commenced with, from the polar opposite of its immediate ante- cedent. The number of changes of sign in effecting such passages, whether in a per-rotatory or a trans-rotatory type, is independent of the place of the element with which the circulation is made to commence, and may be termed the variation-index of the type, which is always an even number for per-rotatory, and an odd number for trans-rotatory types. A theorem is given whereby a relation is established between the variation-index of a per-rotatory or trans-rotatory and that of a certain trans-rotatory or per-rota- tory type capable of bemg derived from them respectively; and this purely tactical theorem, combined with the algebraical one, that the form f(a, y) cannot have fewer imaginary factors than any linear combination of = o leads by successive steps of induction to the theorem in question, but under a more general form, which serves to show intuitively that the limit to the number of real roots of a superlinear equation which the theorem furnishes must be independent of any homographic transfor- mation operated upon the form. ‘The author believes that, whilst it is highly desirable that a simple and general method should be discovered for the proof of Nzwiton’s rule as applicable to equations of any degree, and that the strenuous efforts of the cultivators of the New Algebra should be directed to the attainment of this object, his labours in establishing a proof applicable as far as equations of the 5th degree inclusive will not 1864. ] Mr. Gassiot—Prisms for Spectrum Analysis. 183 have been unproductive of good, as well on account of the confirmation they afford of the truth of the rule, towards the establishment of which on scientific grounds they constitute the first serious step yet made, as also, and still more, by reason of the accessions to the existing field of algebraical speculation to which they have incidentally led. Ill. “ Description of a train of Eleven Sulphide-of-Carbon Prisms arranged for Spectrum Analysis.” By J. P. Gasstor, F.R.S. Received March 17, 1864. The principles which should regulate the construction of a battery of prisms have been alluded to in the description of the large spectroscope now at Kew Observatory, which has a train of nine dense glass prisms with refracting angles of 45°*. While for purposes of exactitude, such as mapping out the solar spec- trum, flint glass stands unrivalled; yet when the greatest amount of dis- persion is the desideratum, prisms filled with bisulphide of carbon present obvious advantages, on account of the enormous dispersive power of that liquid—the difference of its indices of refraction for extreme rays being, according to Sir David Brewster, as 0°077 against 0-026 for flint glass. In the fluid prisms of the ordmary construction, the sides are cemented on with a mixture of glue and honey. This cement, on hardening, warps the sides, and confusion of the spectral lines is the consequent result. To obviate this source of error, it has been proposed to attach an additional pair of parallel sides to such prisms, a thin film of castor-oil beg interposed between the surfaces. The outer plates are then secured by means of Sealing-wax, or some cement, at the corners. In the battery of prisms now about to be described, Mr. Browning has dispensed with this attachment at the corners, which is likely to prove prejudicial, and has secured the second sides in their proper position by extremely ight metal frames which clasp the plates only on their edges. Thus arranged, the frames exert no pressure on the surfaces of the plates, and are quite out of the field of view, and they can be handled with- out any fear of derangement. On account of the lower refractive power of bisulphide of carbon, as com- pared with flint glass, a refractive angle of 50° was given to the fluid prisms. Eight such prisms would cause a ray of light to travel more than a circle, and would be the greatest number that could be employed had the ordi- nary arrangement been adopted. In place, however, of giving to the fluid prisms two pairs of parallel sides, Mr. Browning, taking advantage of the difference between the re- fractive and dispersive properties of crown glass and bisulphide of carbon, has substituted a prism of crown glass having a refracting angle of 6° for one of the outer plates of each prism—the base of this crown-glass prism being brought to correspond with the apex of the fluid prism, thus :— * Proceedings, vol. xil. p. 536. 184 Mr. Gassiot on a Train of Eleven [April 7, Crown-glass prism. By this means the angle of minimum deviation of the prisms is so much decreased, that eleven of them thus constructed can be used in a circle instead of eight. An increase of dispersive power, due to refracting angles of 150° of the bisulphide of carbon, is thus gained, minus only the small amount of dispersion counteracted owing to the dispersive power of the crown-glass prisms being employed in the contrary direction. From the well-known low dispersive power of this medium, however, this loss is inconsiderable, amounting to scarcely more than a fifteenth of the power gamed. Owing to the minimum angle of deviation being lowered, the further advantage is also secured of a larger field of view being presented to the telescope by the first and last prism of the train. Each prism, in addition to the light metal frame referred to, has a separate stand, furnished with screws for adjusting the prisms, and securing them at the angle of mini- mum deviation for any particular ray. The prism stands within a stirrup fur- nished with a welled head. By this arrangement the prisms can be removed and replaced without touching their sides —a matter of some importance, as all fluid prisms show different results with every change of temperature. For the sake of simplicity, the metal framing of the prisms, and the various adjusting-screws, have been omitted in the last sketch. The very unfavourable state of the weather prevented any observations * Direction of ray as it would pass through two pair of parallel sides. + Direction of ray as altered by interposing the crown-glass prism. 1864.] Sulphide-of-Carbon Prisms for Spectrum Analysis. 185 being made on the solar spectrum with these prisms until Saturday the 12th inst. The results then obtained may probably not be considered devoid of interest. They are as follows :— The prisms were arranged so as to enable that portion of the spectrum to be observed in which the well-defined D line of Fraunhofer is situated. This line, long since resolved as double, presented an angular separation of 3! 6’, measured from the centre of one to that of the other principal line, this measurement being made by Mr. Balfour Stewart by means of the micrometer attached to the telescope; the value of the divisions of the micrometer he had previously determined relatively to the divided circle of the spectroscope. A centre line (clearly defined and figured in Kirchhoff and Bunsen’s map) was distinctly visible, and nearly equidistant from the centre towards the violet ; five clearly defined lines were perceptible, as also two faint lines on each side of the principal lines, between the centre line of Kirchhoff towards the red. Several faint lines were also perceptible. The lines as represented in the diagram were drawn by Mr. Whipple, one of the assistants in the Observatory, as they were observed by him about 3.45 p.m Some of these may possibly be due to the earth’s atmo- sphere, but the five most refrangible lines were observed at an earlier period of the day by Mr. Stewart, Mr. Browning, and myself. The great angular separation of the double D line to 3! 6" is a proof of the power of this arrangement of the sulphide-of-carbon prisms, and offers the means of mapping out the entire solar spectrum on a scale not hitherto attained. Received April 6, 1864. Note.—Since the preceding observations were recorded, an inspection has been made of the region of the spectrum towards the refrangible side of double D; and, from the comparisons made with a map of lines ob- tained by means of the battery of glass prisms with that given by those of the sulphide-of-carbon prisms, many new lines are produced in addition to those observable by the former, while the battery of glass prisms itself gives a number of additional lines to those that are depicted in Kirchhoff’s map. 186 [April 14, April 14, 1864. Major-General SABINE, President, in the Chair. The Croonran Lecture was delivered by Prof. Hermann Hetm- uoitz, For. Memb. R.8., ‘‘On the Normal Motions of the Human Hye in relation to Binocular Vision.” The Motions of the Human Hye are of considerable interest, as well for the physiology of voluntary muscular motion in general, as for the physiology of vision. ‘Therefore I may be allowed to bring before this Society the results of some investigations relating to them, which I have made myself; and I may venture perhaps to hope that they are such as to interest not only physiologists and medical men, but every scientific man who desires to understand the mechanism of the perceptions of our senses. The eyeball may be considered as a sphere, which can be turned round its centre as a fixed pomt. Although this description is not absolutely accurate, it is sufficiently so for our present purpose. The eyeball, indeed, is not fixed during its motion by the solid walls of an articular excavation, like the bone of the thigh; but, although it is surrounded at its posterior surface only by soft cellular tissue and fat, it cannot be moved in a per- ceptible degree forward and backward, because the volume of the cellular tissue, included between the eyeball and the osseous walls of the orbit, cannot be diminished or augmented by forces so feeble as the muscles of the eye are able to exert. In the interior of the orbit, around the eyeball six muscles are situated, which can be employed to turn the eye round its centre. Four of them, the so-called rect: muscles, are fastened at the hindmost point of the orbit, and go forward to fix themselves to the front part of the eyeball, passing over its widest circumference—or its equator, as we may call it, if we con- sider the foremost and the hindmost points of the eyeball as its poles. These four recti muscles are from their position severally named superior, inferior, internal, and external. Besides these, there are two oblique muscles, the ends of which come from the anterior margin of the orbit on the side next the nose, and, passing outwards, are attached at that side of the eyeball which is towards the temple—one of them, the superior oblique muscle, being stretched over the upper side of the eyeball, the other, or inferior, going along its under side. These six muscles can be combined as three pairs of antagonists. The internal and external rec¢i turn the eye round a perpendicular axis, so that its visual line is directed either to the right side or to the left. The supe- rior and inferior vecfi turn it round a horizontal axis, directed from the upper end of the nose to the temple; so that the superior rectus elevates the visual line, the inferior depresses it. Lastly, the oblique muscles turn the eye round an axis which is directed from its centre to the occiput, so 1864. | On the Normal Motions of the Human Eye. 187 that the superior oblique muscle lowers the visual line, and the inferior raises it; but these last two muscles not only raise and lower the visual line ; they produce also a rotation of the eye round the visual line itself, of which we shall have to speak more afterwards. ) A. solid body, the centre of which is fixed, and which can be turned round three different axes of rotation, can be brought into every possible position consistent with the immobility of its centre. Look, for instance, at the motions of our arm, which are provided for at the shoulder-joint by the gliding of the very accurately spherical upper extremity of the humerus in the corresponding excavation of the scapula. When we stretch out the arm horizontally, we can turn it, first, round a perpendicular axis, moving it forwards and backwards; we can turn it, secondly, round a horizontal axis, raising it and lowering it; and lastly, after having brought it by such motions into any direction we like, we can turn it round its own longitu- dinal axis, which goes from the shoulder to the hand; so that even when the place of the hand in space is fixed, there are still certain different posi- tions in which the arm can be turned. Now let us see how far the motions of the eye can be compared to those of our arm. We can raise and lower the visual line, we can turn it to the left and to the right, we can bring it into every possible direction, through- out a certain range —as far, at least, as the connexions of the eyeball permit. So far the motions of the eye are as free as those of the arm. But when we have chosen any determinate direction of the eye, can we turn the eye round the visual line as an axis, as we can turn the arm round its longi- tudinal axis? This is a question the answer to which is connected with a curious peculiarity of our voluntary motions. In a purely mechanical sense, we must answer this question in the affirmative. Yes, there exist muscles by the action of which those rotations round the visual line can be per- formed. But when we ask, ‘“‘Can we do it by an act of our will?”’ we must answer, “‘ No.” We can voluntarily turn the visual line into every possible direction, but we cannot voluntarily use the muscles of our eye in such a way as to turn it round the visual line. Whenever the direction of the visual line is fixed, the position of our eye, as far as it depends upon our will, is completely fixed and cannot be altered. This law was first satisfactorily proved by Professor Donders, of Utrecht, who, in a very ingenious way, controlled the position of the eye by those ocular spectra which remain in the field of vision after the eye had been fixed steadily during some time upon any brightly coloured object. I have used for this purpose a diagram like fig. 1: the ground is grey paper, and in the middle, along the line a 4, is placed a narrow strip of red paper on a broader strip of green paper*. The centre of the red strip is marked by two black points. When you look for about a minute steadily and without moving your eye at the centre of the diagram, the image of the * (Green is represented in the figure by white ; red by the central dark stripe. 188 Prof. Helmholtz—Normal Motions of the [April 14, coloured strips is projected on the nervous membrane of your eye; those parts of this membrane on which the light falls are irritated, and in con- sequence of this irritation, their irritability is exhausted, they are fatigued A mm i. i ma and they become less sensitive to that kind of light by which they were excited before. When you cease, therefore, to look at the coloured strips, and turn your eye either to the grey ground of the diagram, or to any other part of the field of vision which is of a uniform feeble degree of illumination, you will see a spectrum of the coloured strips, exhibiting the same apparent magnitude but with colours reversed, a narrow green strip being in the middle of a broader redone. The cause of this appearance is, that those parts of your retina which were excited formerly by green light are less affected by the green rays contained in white or whitish light than by rays of the complementary colour, and white light, therefore, appears to them reddish; to those parts of the nervous membrane, on the other hand, which had been fatigued by red light, white light afterwards appears to be greenish. The nervous membrane of the eye in these cases behaves nearly like the sensitive stratum in a photographic apparatus, which is altered by light during the exposure in such a way that it is impressed differently afterwards by various agents ; and the impression of light on the retina may be, perhaps, of the same essential nature as the impression made upon a photographic plate. But the impression made on the living eye does not last so long as that on sensitive compounds of silver ; it vanishes very soon if the light be not too strong. Light of great intensity, like that of the sun when directly looked at, can develope very dark ocular spectra, which last a quarter of an hour, or even longer, and disturb the 1864. ] Human Eye in relation to Binocular Vision. 189 perception of external objects very much, as is well known. One must be very careful to avoid the use of too strong a light in these experiments, because the nervous apparatus of the eye is easily injured by it; and the “brightness of these coloured strips when illuminated by common day- light is quite sufficient for our present purpose. Now you will perceive easily that these ocular spectra are extremely well adapted to ascertain the position of the eye-ball, because they have a fixed connexion with certain parts of the retinaitself. Ifthe eyeball could turn about its visual line as an axis, the ocular spectrum would apparently un- dergo the same degree of rotation ; and hence, when we move about the eye, and at last return to the same direction of the visual line, we can recognize easily and accurately whether the eye has returned into the same position as before, or whether the degree of its rotation round the visual line has been altered. Professor Donders has proved, by using this very delicate test, that the human eye, in tts normal state, returns always into the same position when the visual line is brought into the same direction. The position and direction of the eye are to be determined in this case in refer- ence to the head of the observer; and I beg you to understand always, when I say that the eye or its visual line is moved upwards or downwards, that it is moved either in the direction of the forehead or in that of the cheek ; and when I say it is moved to the left or to the right, you are to understand the left or right side of the head. Therefore, when the head itself is not in its common vertical position, the vertical line here under- stood is not accordant with the line of the plummet. Before the researches of Donders, some observers believed they had found a difference in the relative positions of the eye, when the head was brought into different situations. They had used either small brown spots of the iris, or red vessels in the white of the eye, to ascertain the real position of the eyeball; but their apparent results have been shown to be erroneous by the much more trustworthy method of Donders. In the first place, therefore, we may state that the position of the eye- ball depends exclusively upon the direction of the visual line in reference to the position of the head of the observer. But now we must ask, what is the law regulating the position of the eye for every direction of its visual line? In order to define this law, we must first notice that there exists a certain direction of the visual line, which, in relation to the motions of the eye, is distinguished from all other directions of the eye; and we may call it the central or primary direction of the visual line. This direction is parallel to the median vertical plane of the head ; and it is horizontal when the head of the observer, who is standing, is kept in a convenient erect position to look at distant points of the horizon. How this primary direc- tion of the visual line may be determined practically with greater accuracy we shall see afterwards. All other directions of the visual line we may call secondary directions. A plane which passes through the visual line of the eye, I call a meri- 190 Prof. Helmholtz—Normal Motions of the [April 14, dian plane of the eye. Sucha plane cuts through the retina in a certain line; and when the eye has been moved, we consider as the same meridian plane that plane which passes through the new direction of the visual line and the same points of the retina as before. . . After having given these definitions, we may express the law of the motions of the eye in the following way :— Whenever the eye ts brought into a secondary position, that meridian plane of the eye which goes through the primary direction of the visual line has the same position as it has in the primary direction of the eye. It follows from this law that the secondary position of the eye may be found also by turning the eye from its primary position round a fixed axis which is normal as well to the primary as to the secondary of the visual line. [The geometrical relations of these different positions were explained by the lecturer by means of a moveable globe placed on an axis like the common terrestrial globes. | It would take too long to explain the different ways in which dif- ferent’ observers have tried to determine the law of the motions of the eyeball. They have employed complicated apparatus for determining the angles by which the direction and the rotation of the eye were to be measured. But usually two difficulties arise from the use of such instru- ments containing graduated circles, in the centre of which the eye must be kept steady. In the first place, it is very difficult to fix the head of the observer so firmly that he cannot alter its position during a contmuous series of observations, and that he reassumes exactly the same position of the head when he returns to his measurements after a pause,—conditions which must necessarily be fulfilled if the observations are to agree with each other. Secondly, I have found that the eye must not be kept too long a time in a direction which is near to the limits of the field of vision ; else its muscles are fatigued, and the positions of the eyeball corresponding to different directions of the visual line are somewhat altered. But if we have to measure angles on graduated circles, it is difficult to avoid keeping the eye too long in directions deviating far from the primary direction. I think that it depended upon these causes, that the observations carried out by Meissner, Fick, and Wundt agreed very ill with each other and with the law which I have explained above, and which was first stated by Professor Listing of Gottmgen, but without any experimental proof. Happily it is possible, as I found out, to prove the validity of this law by a very simple method, which is not subject to thosesources of error I have named, and which I may be allowed to explain briefly. In order to steady the attitude of the head in reference to the direction of the visual line, I have taken a little wooden board, one end of which is hollowed into a curve fitting the arch of the human teeth; the margin of this hollow is covered with sealing-wax, into which, after it had been ‘softened by heat and had been cooled again sufficiently, I inserted both 1864..| Human Eye in relation to Binocular Vision. We). series of my teeth, so that I kept it firmly between my jaws. The impres- sions of the teeth remain indented in the sealing-wax ; and when I put my teeth afterwards into these impressions, I am sure that the little board is brought exactly into the same position, relatively to my head and my eyes, as it was before. On the other end of that little board, which is kept horizontally between the teeth, a vertical piece of wood is fastened, on which I fix horizontally a little strip of card pointed at each end, so that these two points are situated about five inches before my eyes, one before the right eye, the other before the left. The length of the strip of card must be equal to the distance between the centres of the eyes, which is 68 millimetres for my own eyes. Looking now with the right eye in the direction of the right point of that strip, and with the left eye in the direction of the left point, I am sure to bring the eyes always into the same position relatively to my head, so long as the position of the strip of card on the wooden piece remains unaltered. As a field of vision I use either a wall covered with a grey paper, in the pattern of which horizontal and vertical lines can be easily perceived, or a drawing-board covered with grey drawing-paper, on which a system of horizontal and vertical lines is drawn, as in fig. 1, and coloured stripes are fastened along the line ad. Now the observer at first must endeavour to find out that position of his eyes which we call the primary position. In order to do this, the ob- server takes the wooden piece between his teeth, and brings his head into such a position that his right eye looks to the centre of the coloured stripes, in a direction perpendicular to the plane of the drawing. Then he brings his head into such an attitude that the right end of the card-strip appears in the same direction as the centre of the coloured stripe. After having steadily looked for some time to the middle of the coloured stripe, he turns away his gaze to the end of either the vertical or horizontal lines, ad, ed, which are drawn through the centre of the coloured stripe. There he will see an ocular spectrum of the coloured stripe, and will observe if it coin- cides with the horizontal lines of the drawing. If not, he must alter the position of the strip of card on the wooden bar to which it is fastened, till he finds that the ocular spectrum of the coloured stripe remains horizontal when any point either of the line ad or cd is looked at. When he has thus found the primary direction of his visual line for the right eye, he does the same for the left. The ocular spectra soon vanish, but they are easily renewed by looking again to the centre of the stripes. Care must be taken that the observer looks always in a direction perpendicular to the plane of the drawing whenever he looks to the centre of the coloured stripe, and that he does not move his head. If he should have moved it, he would find it out im- mediately when he looks back to the strip, because the point of the card- strip would no longer cover the centre of the coloured stripe. So you see that the primary direction of the visual line is completely 192 Prof. Helmholtz—Normal Motions of the | April 14, fixed, and that the eye, which wants only to glance for an instant ata peripheral point of the drawing, and then goes back again to the centre, is not fatigued. This method of finding the primary position of the eye proves ms the same time that vertical and horizontal lines keep their vertical or hori- zontal position in the field of vision when the eye is moved from its pri- mary direction vertically or horizontally ; and you see, therefore, that these movements agree with the law which I have enunciated. That is to say, during vertical movements of the eye the vertical meridian plane keeps its vertical position, and during horizontal movements the horizontal meridian. Now you need only bring either your own head into an inclined position, or the diagram with the lines, and repeat the experiment, putting your ‘head at first into such a position that the centre of the diagram corre- sponds with the primary direction of the visual line, and moving after- wards the eye along the lines a 6 or cd, in either a parallel or perpendicular direction to the coloured line of the diagram, and you will find the ocular spectrum of the coloured line coinciding with those black lines which are parallel with ad. In this way, therefore, you can easily prove the law of Listing for every possible direction of the visual line. I found the results of these experiments in complete agreement with the law of Listing for my own eyes, and for those of several other persons with normal power of vision. ‘The eyes of very short-sighted persons, on the contrary, often show irregularities, which may be caused by the elon- gation of the posterior part of those eyes. These motions of our eyes are a peculiar instance of motions which, being quite voluntary, and produced by the action of our will, are never- theless limited as regards their extent and their combinations. We find similar limitations of motion of the eyes in other cases also. We cannot turn one eye up, the other down; we cannot move both eyes at the same time to the outer angle; we are obliged to combine always a certain de- gree of accommodation of the eyes to distance, with a certain angle of convergence of their axes. In these latter cases it can be proved that the faculty of producing these motions is given to our will, although our will is commonly not capable of using this faculty. We have come by experience to move our eyes with great dexterity and readiness, so that we see any visible object at the same time single and as accurately as possible; this is the only end which we have learnt to reach by muscular exertion; but we have not learnt to bring our eyes into any given position. In order to move them to the right, we must look to an object situated on our right side, or imagine such an object and search for it with our eyes. We can move them both inwards, but only when we strive to look at the back of our nose, or at an imaginary object situated near that place. But commonly there is no object which could be seen single by turning one eye upwards, the other downwards, or both of them out- 1864..] Human Eye in relation to Binocular Vision. 193 wards, and we are therefore unable to bring our eyes into such positions. But it is a well known fact, that when we look at stereoscopic pictures, and increase the distance of the pictures by degrees, our eyes follow the motion of the pictures, and that we are able to combine them into an apparently single object, although our eyes are obliged to turn into diverging direc- tions. Professor Donders, as well as myself, has found that when we look to a distant object, and put before one of our eyes a prism of glass the refracting angle of which is between 3 and 6 degrees, and turn the prism at first into such a position before the eye that its angle looks to the nose and the visual lines converge, we are able to turn the prism slowly, so that its angle looks upwards or downwards, keeping all this time the object apparently single at which we look. But when we take away the prism, so that the eyes must return to their normal position before they can see the object single, we see the object double for a short time—one image higher than the other. The images approach after some seconds of time and unite at last into one. By these experiments it is proved that we can move both eyes outward, or one up and the other down, when we use them under such conditions that such a position is required in order that we may see the objects single at which we are looking. I have sometimes remarked that I saw double images of single objects, when I was sleepy and tried to keep myself awake. Of these images one was sometimes higher than the other, and sometimes they were crossed, one of them being rotated round the visual line. In this state of the brain, therefore, where our will begins to lose its power, and our muscles are left to more involuntary and mechanical impulses, an abnormal rota- tion of the eye round the visual line is possible. I infer also from this observation, that the rotation of the eye round the visual axis cannot be effected by our will, because we have not learnt by which exertion of our will we are to effect it, and that the inability does not depend on any anatomical structure either of our nerves or of our muscles which limits the combination of motion. We should expect, on the contrary, that, if such an anatomical mechanism existed, it should come out more distinctly when the will has lost its power. We may ask, therefore, if this peculiar manner of moving the eyes, which is determined by the law of Listing, is produced by practical exer- cise on account of its affording any advantages to visual perceptions. And I believe that certain advantages are indeed connected with it. We cannot rotate our eyes in the head, but we can rotate the head with the eyes. When we perform such a motion, looking steadily to the same poimt, we remark that the visible objects turn apparently a little round the fixed point, and we lose by such a motion of our eye the perception of the steadiness of the objects at which we look. Every position of the visual line is connected with a determined and constant degree of rotation, accord- VOL. XIII. Q 194 Prof. Helmholtz—Normal Motions of the [April 14, ing to the law of Donders; and in altering this rotation we should judge the position of external objects wrongly. The same will take place when we change the direction of the visual line. Suppose the amplitude of such motions to be infinitely small; then we may consider this part of the field of vision, and the corresponding part of the retina on which it is projected, as plane surfaces. If during any motion of the eye the optic image is displaced so that in its new position it remains parallel to its former position on the retina, we shall have no apparent mo- tions of the objects. When, on the contrary, the optic image of the visible objects is dislocated so that it is not parallel to its former position on the retina, we must expect to perceive an apparent rotation of the objects. As long as the motions of the eye describe infinitely small angles, the eye can be moved in such a way that the optic image remains always parallel to its first position. For this end the eye must be turned round axes of rotation which are perpendicular to the visual line; and we see indeed that this is done, according to the law of Listing, when the eye is moving near its primary position. But it is not possible to fulfil this condition com- pletely when the eye is moved through a wider area which comprises a larger part of the spherical field of view. For if we were to turn the eye always round an axis perpendicular to the visual line, it would come into very different positions after having been turned through different ways to the same final direction. | The fault, therefore, which we should strive to avoid in the motions of our eye, cannot be completely avoided, but it can be made as small as pos- sible for the whole field of vision. The problem, to find such a law for the motions of the eye that the sum of all the rotations round the visual line for all possible infinitely small motions of the eye throughout the whole field of vision becomes a minimum, is a problem to be solved by the calculus of variations. I have found that the solution for a circular field of vision, which corresponds nearly to the forms of the actual field of vision, gives indeed the law of Listing. I conclude from these researches, that the actual mode of moving the eye is that mode by which the perception of the steadiness of the objects through the whole field of vision can be kept up the best; and I suppose, therefore, that this mode of motion is produced by experience and exercise, because it is the best suited for accurate perception of the position of ex- ternal objects. But in this mode of moving, rotations round the visual line are not com- pletely avoided when the eye is moved in a circular direction round the primary position of the visual line ; and it is easy to recognize that in such a case we are subject to optical illusions. Turn your eyes to a horizontal line situated in the highest part of the field of vision, and let them follow this line from one end to the other. 1864. | Human Eye in relation to Binocular Vision. 195 The line will appear like a curved line, the convexity of which looks down- ward. When you look to its right extremity, it seems to rise from the left to the right ; when you look to the left extremity of the line, the left end seems to rise. In the same way, all straight lines which go through the peripheral parts of the field of vision appear to be curved, and to change their position a little, if you look to their upper or their lower ends. This explanation relates only to Monocular vision; we have to inquire also how it influences Binocular vision. | Each eye has its field of vision, on which the visible objects appear dis- tributed like the objects of a picture, and the two fields with their images seem to be superimposed. Those points of both fields of view which ap- pear to be superimposed are called corresponding (or identical) points. If we look at real objects, the accurate perception of the superimposition of two different optical images is hindered by the perception of stereoscopic form and depth; and we unite indeed, as Mr. Wheatstone has shown, two retinal images completely into the perception of one single body, without being able to perceive the duplicity of the images, even if there are very sen- sible differences of their form and dimensions. To avoid this, and to find those points of both fields of view which correspond with each other, it is necessary to use figures which cannot easily be united into one stereoscopic projection. In fig. 2 you see such figures, the right of which is drawn with white lines on a black ground, the left with black lines on a white ground. The horizontal lines of both figures are parts of the same straight lines ; the vertical lines are not perfectly vertical. The upper end of those of the right figure is inclined to the right, that of the left figure to the left, by about 11 degree. Now I beg you to look alternately with the right and with the left eye at these figures. You will find that the angles of the right figure appear to the right eye equal to right angles, and those of the left figure so appear to the left eye; but the angles of the left figure appear to the right eye to deviate much from a right angle, as also do those of the right figure to the left eye. When you draw on paper a horizontal line, and another line crossing it exactly at right angles, the right superior angle will appear to your right eye too great, to your left eye too small; the other angles show corresponding deviations. To have an apparently right angle, you must make the vertical line incline by an angle of about 11 degree for it to appear really vertical ; and we must distinguish, therefore, the really vertical lines and the appa- rently vertical lines in our field of view. There are several other illusions of the same kind, which I omit because they alter the images of both eyes in the same manner and have no ‘influs ence upon binocular vision ; for example, vertical lines appear always of greater length than horizontal lines having really the same length. Q 2 196 Prof. Helmholtz—Normal Motions of the [April 14, Now combine the two sides of fig. 2 into a stereoscopic combination, either by squinting, or with the help of a stereoscope, and you will see that the white lines of the one coincide exactly with the black lines of the other, as soon as the centres of both the figures coincide, although the vertical lines of the two figures are not parallel to each other. man Therefore not the really vertical meridians of both fields of view corre- spond, as has been supposed hitherto, but the apparently vertical meridians. On the contrary, the horizontal meridians really correspond, at least for normal eyes which are not fatigued. After having kept the eyes a long 1864.] Human Eye in relation to Binocular Vision. 197 time looking down at a near object, as in reading or writing, I found some- times that the horizontal lines of fig. 2 crossed each other; but they be- came parallel again when I had looked for some time at distant objects. In order to define the position of the corresponding points in both fields of vision, let us suppose the observer looking to the centres of the two sides of fig. 2, and uniting both pictures stereoscopically. Then planes may be laid through the horizontal and vertical lines of each picture and the centre of the corresponding eye. The planes laid through the dif- ferent horizontal lines will include angles between them, which we may call angles of altitude ; and we may consider as their zero the plane going through the fixed point and the horizontal meridian. The planes going through the vertical lines include other angles, which may bé called angles of longitude, their zero coinciding also with the fixed point and with the apparently vertical meridian. Then the stereoscopic combination of those diagrams shows that those points correspond which have the same angles of altitude and the same angles of longitude; and we can use this result of the experiment as a definition of corresponding points. We are accustomed to call Horopter the aggregate of all those points of the space which are projected on corresponding points of the retine. After having settled how to define the position of corresponding points, the question, what is the form and situation of the Horopter, is only a geome- trical question. With reference to the results I had obtained in regard to the positions of the eye belonging to different directions of the visual lines, I have calculated the form of the Horopter, and found that gene- rally the Horopter is a line of double curvature produced by the inter- section of two hyperboloids, and that in some exceptional cases this line of double curvature can be changed into a combination of two plane curves. That is to say, when the point of convergence is situated in the middle plane of the head, the Horopter is composed of a straight line drawn through the point of convergence, and of a conic section going through the centre of both eyes and intersecting the straight line. When the point of convergence is situated in the plane which contains the primary directions of both the visual lines, the Horopter is a circle going through that point and through the centres of both eyes and a straight line intersecting the circle. When the point of convergence is situated as well in the middle plane of the head as in the plane of the primary directions of the visual lines, the Horopter is composed of the circle I have just described, and a straight line going through that point. There is only one case in which the Horopter is really a plane, as it was supposed to be in every instance by Aguilonius, the inventor of that name,— namely, when the point of convergence is situated in the middle plane of the head and at an infinite distance. Then the Horopter is a plane parallel to the visual lines, and situated beneath them, at a certain distance which depends upon the angle between the really and apparently vertical 198 On the Normal Motions of the Human Eye, [April 14, ‘meridians, and which is nearly as great as the distance of the feet of the observer from his eyes when he is standing. Therefore, when we look straight forward to a point of the horizon, the Horopter is a horizontal plane going through our feet—it is the ground upon which we are standing. _ Formerly physiologists believed that the Horopter was an infinitely distant plane when we looked to an infinitely distant point. The differ- ence of our present conclusion is consequent upon the difference between the position of the really and apparently vertical meridians, which they did not know. When we look, not to an infinitely distant horizon, but to any point of the ground upon which we stand which is equally distant from both our eyes, the Horepter is not a plane; but the straight line which is one of its parts coincides completely with the horizontal plane upon which we are standing. The form and situation of the Horopter is of great practical importance for the accuracy of our visual perceptions, as I have found. Take a straight wire—a knitting-needle for instance—and bend it a little in its middle, so that its two halves form an angle of about four degrees. Hold this wire with outstretched arm in a nearly perpendicular position before you, so that both its halves are situated in the middle plane of your head, and the wire appears to both your eyes nearly as a straight line. In this position of the wire you can distinguish whether the angle of the wire is turned towards your face or away from it, by binocular vision only, as in stereoscopic diagrams; and you will find that there is one direction of the wire in which it coincides with the straight line of the Horopter, where the inflexion of the wire is more evident than in other positions. You can test if the wire really coincides with the Horopter, when you look ata point a little more or a little less distant than the wire. Then the wire appears in double images, which are parallel when it is situated in the Horopter line, and are not when the point is not so situated. Stick three long straight pins into two little wooden boards which can slide one along the other; two pins may be fastened in one of the boards, the third pin in the second. . Bring the boards into such a position that the pins are all perpendicular and parallel to each other, and situated nearly in the same plane. Hold therm before your eyes and look at them, and strive to recognize if they are really in the same plane, or if their series is bent towards you or from you. You will find that you distinguish this by binocular vision with the greatest degree of certainty and accuracy (and indeed with an astonishing degree of accuracy) when the line of the three pins coincides with the direction of the circle which is a part of the Horopter. From these observations it follows that the forms and the distances of those objects which are situated in, or very nearly in, the Horopter, are perceived with a greater degree of accuracy than the same forms and distances would be when not situated in the Horopter. If we apply this 1864] Prof. H. T. 8S. Smith on Quadratic Forms. 199 result to those cases in which the ground whereon we stand is the plane of the Horopter, it follows that, looking straight forward to the horizon we can distinguish the inequalities and the distances of different parts of the ground better than other objects of the same kind and distance. This is actually true. We can observe it very conspicuously when we look to a plain and open country with very distant hills, at first in the natural position, and afterwards with the head inclined or inverted, looking under the arm or between our legs, as painters sometimes do in order to distinguish the colours of the landscape better. Comparing the aspect of the distant parts of the ground, you will find that we perceive very well that they are level and stretched out into a great distance in the natural position of your head, but that they seem to ascend to the horizon and to be much shorter and narrower when we look at them with the head inverted: we get the same appearance also when our head remains in its natural position, and we look to the distant objects through two rectangular prisms, the hypothenuses of which are fastened on a horizontal piece of wood, and which show inverted images of the objects. But when we invert our head, and invert at the same time also the landscape by the prisms, we have again the natural view and the accurate perception of distances as in the natural position of our head, because then the apparent situation of the ground is again the plane of the Horopter of our eyes. The alteration of colour in the distant parts of a landscape when viewed with inverted head, or in an inverted optical image, can be explained, I think, by the defective perception of distance. The alterations of the colour of really distant objects produced by the opacity of the air, are well known to us, and appear as a natural sign of distance; but if the same alterations are found on objects apparently less distant, the alteration of colour appears unusual, and is more easily perceived. It is evident that this very accurate perception of the form and the distances of the ground, even when viewed indirectly, is a great advantage, because by means of this arrangement of our eyes we are able to look at distant objects, without turning our eyes to the ground, when we walk. April 21, 1864. Major-General’ SABINE, President, in the Chair. The following communications were read :— I. “On the Orders and Genera of Quadratic Forms containing more than three Indeterminates.” By H.T.SrepHen Smiru, M.A., F.R.S., Savilian Professor of Geometry in the University of Oxford. Received March 22, 1864. Let us represent by 7, a homogeneous form or quantic of any order containing indeterminates ; by (a), a square matrix of order x; by 200 Prof, H. T. S. Smith on Quadratic Forms [April 21, n (a¢ ), its ith derived matrix, 7. e. the matrix of order _ ee the con- le |jz—2 stituents of which are the minor determinants of order z of the matrix (a); and lastly, by /;, a form of any order containing I indeterminates, the coefficients of which depend on the coefficients of f. When f, is transformed by (a), let /; be transformed by (2) ; if, after division or multiplication by a power of the modulus of transformation, the meta- morphic of f; depends on the metamorphic of 7, in the same way in which f; depends on f,, f; is said to be a concomitant of the 7th species of f,. Thus: a concomitant of the Ist species is a covariant; a con- comitant of the (n—1)th species is a contravariant ; if m=2 there are only covariants ; if x=3 there are only covariants and contravariants ; but if 2>3, there will exist in general concomitants of the intermediate species. There is an obvious difference between covariants and contravariants on the one hand, and the intermediate concomitants on the other. The number of indeterminates in a covariant or contravariant is the same as in its primitive ; in an intermediate concomitant, the number of indeterminates is always greater than in its primitive. Again, to every metamorphic of a covariant or contravariant, there corresponds a metamorphic of its primi- tive ; whereas, in the case of a concomitant of the intermediate order 7, a metamorphic of the primitive will correspond, not to every metamorphic of the concomitant, but only to such metamorphics as result from trans- formations the matrices of which are the zth derived matrices of matrices of order x. It is also obvious that, besides the 2»—1 species of concomitance here defined, there are, when is >3, an infinite number of other species of concomitance of the same general nature. For from any derived matrix we may form another derived matrix, and so on continually; and to every such process of derivation a distinct species of concomitance will correspond. The notion of intermediate concomitance appears likely to be of use in many researches; in what follows, it is employed to obtain a definition of the ordinal and generic characters of quadratic forms contaiming more than 3 indeterminates. (The case of quadratic forms containing 3 indeterminates has been considered by Eisenstein in his memoir, ‘‘ Neue Theoreme des hoheren Arithmetik,” Crelle, vol. xxxv. pp. 12} and 125.) Let pH=ng=n f= B= SLA & ‘e pelg=l Pg? 4 represent a quadratic form of m indeterminates; let (A) be the sym- metrical matrix of this form, and (A) the zth derived matrix of (A) ; (A®) will also be a symmetrical matrix, and the quadratic form ge — ¢ fi Se BAO .% Kin tao cele will be a concomitant of the ith species of 7. It is immaterial what 1864.] containing more than Three Indeterminates. 201 principle of arrangement is adopted in writing the quadratic matrix (A), and the transforming matrix (2); provided only that the arrangement be the same in the two matrices, and that in each matrix it be the same in height and in depth. For example, if f.=a, 27+ a, 23+ a, 02 + a,x, + 26, v,2,+ 26, 2, 234+ 26, 2,27,+26,7,27,+26,x, 2,+26, x, x, be a quadratic form containing four indeterminates, the form f, = (;—4, a,) xt + (65-4, a;) xX 5 (65 —4, a,) XS + (62—a, a,) X32 + (b3—a, a,) X3 + (62—a, a,) XB +2(4, b,—4a, b,) Xx, X, a 2(6, Go ay ey) X, X, —2(6, 6,—4, 6.) X, X, ae 2(0, 6; ite 6) X, X; ad 6,—6, b,) X, X, a 2(4, 6,—4, 7) X, X, +2(6, Oa, b,) X, X, me 2(d, O52 O% b,) X, X, — 2(6,6,—a, 6,) X, X, — 2(6, 6,—6, 6;) X, X, as 2(0, 6;—4%, 6,) X, X, a 2(6, b,—4a, 2) X, X, +2(6,6,—a, 6,) X,X, — 2(6,6,—a, 6,) X, X, +2(6; 6,—4,5,) X, X, is the concomitant of the second species of /. The n—1 forms defined by the formula (A), of which the first is the form /, itself, and the last the contravariant of f,, we shall term the funda- mental concomitants of f,; in contradistinction to those other quadratic concomitants (infinite in number) of which the matrices are the symme- trical matrices that may be derived, by a multiplicate derivation, from (AM)... . Passing to the arithmetical theory of quadratic forms—z. e. supposing that the constituents of (A“) are integral numbers, we shall designate by V,, V»--+ Vn the greatest common divisors (taken posi- tively) of the minors of different orders of the matrix (A“)), so that, in particular, v, is the greatest common divisor of its constituents, and VY» is the absolute value of its determinant, here supposed to be different from zero. By the primary divisor of a quadratic form we shall understand the greatest common divisor of the coefficients of the squares and double rect- angles in the quadratic form ; by the secondary divisor we shall understand the greatest common divisor of the coefficients of the squares and of the rectangles ; so that the primary divisor is equal to, or is half of, the secondary divisor, according as the quadratic form (to use the phraseology of Gauss) is derived from a form properly or improperly primitive. It will be seen that V,, VY.» -- +» Va—1 are the primary divisors of the forms uiese= Als pai Tespectively. We now consider the totality of arithmetical quadratic forms, contain- ing 7 indeterminates, and having a given index of inertia, and a given de- terminant. If a quadratic form be reduced to a sum of squares by any linear trans- formation, the number of positive and of negative squares is the same, 202 Prof. H.T.S. Smith on Quadratic Forms _— [April 21, whatever be the real transformation by which the reduction is effected. For the index of inertia we may take the number of the positive squares ; it is equal to the number of continuations of sign in a series of ascending principal minors of the matrix of the quadratic form, the series com- mencing with unity, 7. e. with a minor of order 0, and each minor being so taken as to contain that which precedes it in the series (see Professor Sylvester “On Formule connected with Sturm’s Theorem,” Phil. Trans. vol. exliii. p. 481). The distribution of these forms into Orders depends on the following principle :— «Two forms belong to the same order when the primary and secondary divisors of their corresponding concomitants are identical.” - Since, as has been just pointed out, there are, beside the fundamental concomitants, an infinite number of other concomitants, it is important to know whether, in order to obtain the distribution into orders, it is, or is not, necessary to consider those other concomitants. With regard to the primary divisors, it can be shown that it is unnecessary to consider any concomitants other than the fundamental ones; 7. e. it can be shown that the equality of the primary divisors of the corresponding fundamental concomitants of two quadratic forms, implies the equality of the primary divisors of all their corresponding concomitants. And it is probable (but it seems difficult to prove) that the same thing is true for the secondary divisors also. Confining our attention (in the next place) to the forms corte in any given order, we proceed to indicate the principle from which the sub- division of that order into genera is deducible. If F, be any quadratic form containing 7 indeterminates, and F, be its piconet of the second species, we have the identical equation E— 7 d¥, Fi (@ys Uap 6 0 Br )X EY y Yo» - -y)-4|2 y" a | Hh ile ieee 4 k=1 | oe a Y» Yor 22 e Y, J in which the symbol F, On oe .) indicates that the deter- Yi» Ya vita Yr Uy Voy 0 0 0y Up minants ( ) are to be taken for the indeterminates of 19 99 © 8 99 ra F,, the order in which they are taken being the same as the order in which the determinants of any two horizontal rows of the matrix of F, are taken in forming the matrix of F,. Let i= — J, for every value of z from 1 to n—1; it will be found that, if we form the concomi- tant of the second species of 6, its primary divisor is the quotient Viti. Vi Vivre wed vol. cli, p. 317) is always an integral number. Let 6; be any uneven , which, as has been shown elsewhere (see Phil. Trans. 1864. | containing more than Three Indeterminates, 203 prime dividing = = we infer from the identity (B) that the i s—1 numbers prime to 6;, which can be represented by 0,, are either all qua- dratic residues of 3s’, or all non-quadratic residues of 6;. In the former case we attribute to f, the particular character (= +1; in the latter the particular character (5)=-1. If v,=1, i.e. if the form f, itself do not admit of any primary divisor beside unity (which igs the only important case), the product (Pe + Bent) x (Want Yams) Nee Vn-1 Vn-2 Vn-2 Va-3 “.; whence, inasmuch as every prime that divides V, also n—1 is equal to divides —“” , it appears that a primitive quadratic form will always have n= one particular character, at least with respect to every uneven prime dividing its determinant, and will have more than one if the uneven Vit+1 3 Was Wie Sei The subdivision of an order into genera can now be effected by assign- ing to the same genus all those forms whose particular characters co- incide. But it remains to consider whether the above enumeration of par- ticular characters is complete. It is evident that we might apply the theorem (B) to other concomitants besides those included in the funda- mental system ; and it might appear as if in this manner we could obtain other particular characters besides those which we have given. But it can be shown that such other particular characters are implicitly contained in ours; 7.e. it can be shown that two quadratic forms, which coincide in respect of the particular characters deducible from their fundamental con- comitants, will also coincide in respect of the particular characters dedu- cible from any other concomitant. Again, it will be found that if the determinant be uneven, there are no particular characters with respect to 4 or 8. For this case, therefore, our enumeraticn is complete. But when the determinant is even, besides the particular characters arising from its uneven prime divisors, there may also be particular characters with regard to 4 or 8. There ig no difficulty in enumerating these particular characters; nevertheless we suppress the enumeration here, not only because it would require a detailed distinction of cases, but also because there appears to be some difficulty in showing that the characters with regard to 4 or 8, which may arise from the excluded concomitants, are virtually included in those which arise from the concomitants of the fun- damental set. prime divide more than one of the quotients ~~ 204 Mr. Abel on the Combustion of [April 21, IL. “On some Phenomena exhibited by Gun-cotton and Gunpowder under special conditions of Exposure to Heat.” By F. A. Asst, F.R.S. Received March 29, 1864. The experiments upon which I have been engaged for some time past, in connexion with the manufacture and properties of gun-cotton, have brought under my notice some interesting points in the behaviour of both gun- cotton and gunpowder, when exposed to high temperatures, under parti- cular conditions. I believe that these phenomena have not been previously observed, at any rate to their full extent, and I therefore venture to lay before the Royal Society a brief account of them. Being anxious to possess some rapid method of testing the uniformity of products obtained by carrying out General von Lenk’s system of manu- facture of gun-cotton, I instituted experiments for the purpose of ascer- taining whether, by igniting equal weights of gun-cotton of the same com- position, by voltaic agency, within a partially exhausted vessel connected with a barometric tube, I could rely upon obtaining a uniform depression of the mercurial column, in different experiments made in atmospheres of uniform rarefaction, and whether slight differences in the composition of the gun-cotton would be indicated, with sufficient accuracy, by a corre- sponding difference in the volume of gas disengaged, or in the depression of the mercury. I found that, provided the mechanical condition of the gun-cotton, and its position with reference to the source of -heat, were in all instances the same, the indications furnished by these experiments were sufficiently accurate for practical purposes. Each experiment was made with fifteen grains of gun-cotton, which were wrapped compactly round the platinum wire; the apparatus was exhausted until the column of mercury was raised to a height varying from 29 inches to 29°5 inches. The flash which accompanied the deflagration of the gun-cotton was apparently similar to that observed upon its ignition in open air; but it was noticed that an interval of time always occurred between the first application of heat (or incandescence of the wire) and the flashing of the gun-cotton, and that during this interval there was a very perceptible fall of the column of mercury. On several occasions, when the gun-cotton, in the form of “roving,” or loosely twisted strand, was only laid over the wire, so that it hung down on either side, the red-hot wire simply cut it into two pieces, which fell to the bottom of the exhausted vessel, without continuing to burn. As these results appeared to indicate that the effects of heat upon gun-cotton, in a highly rarefied atmosphere, differed importantly from those observed under ordinary circumstances, or in a very imperfect va- cuum, a series of experiments, under variously modified conditions, was instituted, of which the following are the most important. It was found in numerous experiments, made with proportions of gun- cotton varying from one to two grains, in the form of a loose twist laid double, that in highly rarefied atmospheres (the pressure being varied 1864. ] Gun-cotton and Gunpowder. 205 from | to 8 in inches of mercury) the gun-cotton, when ignited by means of the platinum wire, burned very slowly, presenting by daylight an appear- ance as if it smouldered, with little or no flame attending the combustion. I was at first led by these results to conjecture that this peculiar kind of combustion of the gun-cotton was determined solely by its ignition in atmospheres rarefied beyond a certain limit; and I was induced, in con- sequence, to institute a number of experiments with the view of ascertain- ing what was the most highly rarefied atmosphere in which gun-cotton would burn as in the open air—with a flash, accompanied by a body of bright flame. In order to ensure uniformity in the degree of heat applied to the cotton in these experiments, the platinum wire employed was suffi- ciently thin to be instantaneously melted on the passage of the voltaic current. About fifty different experiments were made with equal quanti- ties of gun-cotton (0°2 grain), placed always in the same position, so that the platinum wire rested upon the material. A tolerabiy definite limit of the degree of rarefaction was arrived at, within which the gun-cotton was exploded instantaneously, as in the open air. When the pressure of air in the apparatus was reduced to 8-2 in inches of mercury, the gun-cotton still exploded with a flash, but not quite instantaneously ; on reducing the pressure to 8 inches, the cotton underwent the slow kind of combustion in the majority of cases; on a few occasions it exploded with a flash of flame. The same occurred in a succession of experiments, until the pressure was reduced gradually to 7*7 inches, when instances of the rapid explosion of gun-cotton were no longer obtained. Although these results were moderately definite when the conditions of the experiments were as nearly as possible uniform, it was found that they could be altered by slight modifications of any one particular condition (such as the quantity of gun-cotton, its mechanical condition, its position with reference to the source of heat, the quantity of heat applied, and the duration of its application). In illustration of this, the following results may be quoted. If the gun-cotton was wrapped round, instead of being simply placed across the wire, its instantaneous combustion was effected in atmospheres considerably more rarefied than with the above experiments. In employing a small piece of gun-cotton (0°3 of an inch long and weighing 0°3 to 0°4 of a grain) loosely twisted, laid across the wire, or upon a support immediately beneath the latter so that the wire rested upon it, the slow combustion established in it by the heated wire, under greatly diminished atmospheric pressure (amounting to 0°6 inch in this and the following experiments), proceeded uniformly towards each end of the piece of twist, until the whole was transformed into gas. But if a piece of the same twist, of considerably greater length (say 4 inches long and weighing about 2 grains), was exposed to heat in an atmosphere of the same rarefaction, the gun-cotton being laid over the wire and hanging, down on either side, it was cut through by the passage of the current, and 206 Mr. Abel on the Combustion of [April 22, the two pieces, falling to the bottom of the vessel, ceased to burn almost immediately. Of a piece of gun-cotton weighing 2°17 grains, there re- mained unchanged 1°80 gr.; the quantity burned amounted therefore to 0-37 gr., and corresponded closely to the quantity which was completely burned in the preceding experiments. (The depression of the mercurial column in this experiment, by the gases generated from the gun-cotton, amounted to 0-2 inch.) A piece of the twist, 13-inch long, was placed across the wire, and supported by a plate of plaster of Paris, fixed immediately beneath it. The current was established to an extent just sufficient to heat the wire to the point of ignition of the gun-cotton, and then interrupted. The twist burned slowly in both directions until about a quarter of an inch was con- sumed on either side of the wire, when the combustion ceased. The same result was obtained in repetitions of the experiment, the wire being at once raised to a red heat, and thus maintained until the gun-cotton ceased to burn. But upon increasing the battery-power, doubling the thickness of the wire, and maintaining the heat, while a similar piece of twist was burning in both directions, the slow combustion continued until the entire quantity was transformed into gas. The same result was obtained by re- peating this experiment with similar and larger quantities of gun-cotton, placed in the same position as before with reference to the wire. In the next experiment, the mass of cotton exposed at one time to heat was increased by doubling a piece of the twist (4 inches long) and laying it thus doubled across the wire, as before. The current was allowed to pass until the wire was heated just sufficiently to ignite the gun-cotton, and then interrupted. In this case the slow combustion proceeded through- out the entire mass of the cotton. The permanent depression of mercury in this experiment was 0°6 inch. It was particularly noticed on this occasion, that, as the decomposition of the gun-cotton crept slowly along the mass, the burning portions or extremities of twist were surrounded by a beautiful green light, more like a phosphorescence than a flame, and in form some- thing similar to the brush of an electric discharge. Eight inches of the twist were laid fourfold over the wire, which was heated just sufficiently to ignite the cotton. The decomposition proceeded, as before, gradually throughout the mass of the gun-cotton, but became somewhat more rapid towards the end, when the green glow, observed at first, was superseded by a pale yellowish lambent flame, very different in appearance from the flame which accompanies the combustion of gun-cot- ton under ordinary conditions. The permanent depression of the column of mercury in this experiment was 1°2 inch. The various modifications in the nature and extent of combustion which gun-cotton may be made to undergo, as demonstrated by the above experi- ments, when exposed to heat in highly rarefied atmospheres under variously modified conditions, are evidently due to the same causes which affect the rate of combustion of fuses under different atmospheric pressures, and which 1864. ] Gun-cotion and Gunpowder. ; 207 have already been pointed out by Frankland in his interesting paper on the - influence of atmospheric pressure upon some of the phenomena of combus- tion. The heat furnished by an incandescent or melting platinum wire is greatly in excess of that required to induce perfect combustion in gun- cotton which is actually in contact with, or in close ‘proximity to it; and the heat resulting from this combustion, which is contained in the products of the change, will suffice to cause the transformation of the explosion to pro- ceed from particle to particle. But if the pressure of the atmosphere in which the gun-cotton is submitted to the action of heat be reduced, the gases resulting from the combustion of the particles nearest to the source of heat will have a tendency, proportionate to the degree of rarefaction of the air, to pass away into space, and thus to convey away from proximity to the cotton, more or less rapidly and completely, the heat necessary to carry on the combustion established in the first particles. Thus, when the heated wire is enveloped in a considerable body of gun-cotton, the ignition of the entire mass is apparently not instantaneous, if attempted in a highly rarefied atmosphere, because the products of the combustion first established in the centre of the mass of gun-cotton escape rapidly into space, conveying away from the point of combustion the heat essential for its full maintenance ; the gun-cotton therefore undergoes at first an imperfect form of combus- tion, or a kind of metamorphosis different from the normal result of the action ef heat upon this material. But the effects of the gradual generation of heated gases from the interior of the mass of cotton are, to impart some of their heat to the material through which they have to escape, as well as gradually to increase the pressure of the atmosphere in the vessel, and thus to diminish the rapidity of their escape; hence a condition of things is in time arrived at when the remainder of the gun-cotton undergoes the ordi- nary metamorphosis, a result which is accelerated by maintaining the original source of heat. If, however, the gun-cotton ke employed in a compact form (in the form of twist or thread), and placed only in contact with the source of heat at one point, the heat will be so effectually conveyed away by the escaping gases, that the material will undergo even what may be termed the secondary combustion or metamorphosis for a limited pe- riod only ; so that, if a sufficient length of gun-cotton be employed, it will after a short time cease to burn, even imperfectly, because the heat essen- tial for the maintenance of any chemical activity is soon completely abs- tracted by the escaping gases. These results may obviously be modified in various ways, as shown in the experiments described : thus, by increasing and maintaining the source of heat independent of the burning cotton, the slow combustion may be maintained through a much greater length of the material until the pressure of the atmosphere is increased, by the products disengaged, to an extent sufficient to admit of a more rapid and perfect metamorphosis being established in the remainder of the material ; or the same result may be attained, independently of the continued application of external heat, by employing a thicker mass of cotton, or by using the 208 Mr. Abel on the Combustion of [April 21, material in a less compact form. In these cases the maintenance of the chemical change is favoured either by radiation of heat to the cotton, and provision of additional heat, from an external source, to the gases as they escape and expand, or by establishing the change in a greater mass of the material, and thus reducing the rapidity with which the heat will be con- veyed away by the escaping gases, or, finally, by allowing the gases, as they escape, to pass to some extent between the fibres of the cotton, and thus favouring the transmission of heat to individual particles of the material. In the description of the two experiments last referred to above, I have stated that some peculiar phenomena were observed to attend the imperfect kind of combustion induced in the gun-cotton in rarefied atmospheres. In order to examine these phenomena more closely, I instituted a series of experiments, in a darkened room, with equal quantities of gun-cotton (4 inch of twist=0°3 gr.) placed always in the same position, across the platinum wire, the only varying element in the experiment being the pressure of the atmosphere in the vessel, which was gradually increased. The following were the results observed :— Experiment 1.—Pressure=0°62 inch. The wire was heated just suffi- ciently to ignite the material; the current was then interrupted. The gun-cotton burned very slowly in both directions, emitting only the small green phosphorescent flame, or brush, already described. Exp. I1.—Pressure=1 inch. In addition to the green glow whieh sur- rounded the burning ends, a very faint yellowish flame was observed hover- ing over the gun-cotton. Exp. WiI.—Pressure=1*5inch. The cotton burned a little faster, and the faint yellowish flame was of a more decided character ; indeed two sepa- rate flames were observed, each following up the green light as the cotton burned in the two directions. Exp. 1V.—Pressure=2 inches. The results were the same as in the preceding experiment, excepting that the yellowish flames became more marked. Exp. V.—Pressure=2°5 inches. The same phenomena, the cotton burning considerably faster. Exp. V\1.—Pressure=3 inches. ‘The same phenomena, the yellow flames increasing in size. Exp. Vil.—Pressure=4 inches. The rapidity of combustion of the cotton increased again considerably ; the other phenomena observed were as before. Exp. VIII.—Pressure=6 inches. The pale yellow fiame had increased in size considerably, no longer forming a tongue, as in the preceding experi- ments, but completely enveloping the burning ends of the gun-cotton. The green glow, though much reduced, was still observed immediately round the burning surfaces. | Exp. 1X.—Pressure=8 inches. The green glow was only just percep- tible in this instance, and the cotton burned very rapidly, almost with the 1864.] Gun-cotton and Gunpowder. 209 ordinary flash ; but the flame was still of a pale yellow. In the preceding experiments clouds of white vapour were observed after the decomposition of the gun-cotton; in this and the following experiments this white vapour was produced in much smaller proportion. | Expts. X. to XV. inclusive.— Pressure=10, 12, 14, 18, 20, 24 inches. The phenomena observed in these experiments did not differ in any im- portant degree from those of Experiment IX. Exp. XVI.—The same pressure (24 inches) was employed as in the last experiment, but the piece of gun-cotton-twist was laid double across the wire. In this instance the gun-cotton burned with a bright yellow flash, as In open air. Hap. XVI1.—Pressure=26 inches. The gun-cotton was laid singly over the wire, as in all experiments but the last. It burned with a flash o bright light, as in open air. It appears from these experiments that gun-cotton, when ignited in small quantities in rarefied atmospheres, may exhibit, during its combustion, three distinct luminous phenomena. In the most highly rarefied atmospheres, the only indication of combustion is a beautiful green glow or phosphorescence which surrounds the extremity of the gun-cotton as it is slowly transformed into gases or vapours. When the pressure of the atmosphere is increased to one inch (with the proportion of gun-cotton indicated), a faint yellow flame appears at a short distance from the point of decomposition ; and as the pressure is increased this pale yellow flame increases in size, and eventually appears quite to obliterate the green light. Lastly, when the pressure of the atmosphere and consequently proportion of the oxygen in the con- fined space is considerable, the cotton burns with the ordinary bright yellow flame. There can be no doubt that this final result is due to the almost in- stantaneous secondary combustion, in the air supplied, of the inflammable gases evolved by the explosion of the gun-cotton. It was thought that the pale yellow flame described might also be due to a combustion (in the air still contained in the vessel) of portions of the gases resulting from the de- composition of the gun-cotton ; but a series of experiments, in which nitrogen, instead of air, constituted the rarefied atmosphere, showed that this could not be the case. The results obtained in these experiments cor- responded closely to those above described, as far as relates to the produc- tion of the green glow and of the pale yellow flame. With rarefied atmo- _ spheres of nitrogen ranging down to one inch of pressure, the green flame was alone obtained; and the pale yellow flame, accompanying the green, became very marked at a pressure of 3 inches, as in the experiments with air. It would seem probable from these results, that the mixture of gaseous products obtained by the peculiar charge which heat effects in gun-cotton in highly rarefied atmospheres, contains not only combustible bodies, such as carbonic oxide, but also a small proportion of oxidizing gas (possibly protoxide of nitrogen, or even oxygen), and that when the pressure of the VOL, XIII. e R 210 Mr. Abel on the Combustion of [April 21, atmosphere is sufficiently great this mixture, which has. self-combustible properties, retains sufficient heat as it escapes, to burn, more or less com- pletely, according to the degree of rarefaction of the atmosphere. A series of experiments instituted with gun-cotton in highly rarefied atmospheres of oxygen, showed that the additional proportion of this gas thus introduced into the apparatus, beyond that which would have been contained in it with the employment of air of the same rarefaction, affected in a very important manner the behaviour of the explosion under the in- fluence of heat. If eight or ten grains of gun-cotton are placed round the platinum wire, and the pressure of the atmosphere of oxygen in the vessel be reduced to four or three (in inches of mercury), the cotton explodes in- stantaneously, with an intensely bright flash, when the wire is heated. In a series of experiments made under gradually diminished pressures, oxygen being used instead of air, it was found that the gun-cotton exploded instan- taneously, with a bright flash, until the pressure was reduced to 1:2 inch ; from this pressure to that of 0°8 inch it still burned with a flash, but not instantaneously ; and at pressures below 0°8 inch it no longer burned with a bright flash, but exhibited the comparatively slow combustion, accom- panied by the pale yellow flame, which has been spoken of as observed when gun-cotton was ignited in air rarefied to pressures ranging from ] inch to 24 inches. The interesting phenomena exhibited by gun-cotton in highly rarefied atmospheres, induced me to make some experiments of a corresponding nature with gunpowder. The same apparatus was used as in the preceding experiments, but a small glass cup was fixed immediately beneath the platinum wire, so that, by bending the latter in the centre, it was made to dip into the cup, and could be covered by grains of gunpowder. Two grains’ weight of small grain gunpowder were heaped over the wire, and the pressure of air in the apparatus was reduced to 0°65 inch. The wire being heated to redness, three or four grains, in immediate proximity to it, fused in a short time and appeared to boil, evolving yellowish vapours, no doubt of sulphur. After the heat had been continued for eight or ten seconds, those particular graims deflagrated, and the remainder of the powder was scattered by the slight explosion, without being ignited. No appreciable depression of the mercurial column occurred during the evolu- tion of the yellowish vapours ; the permanent depression, after the defla- gration, was only 0°15 inch. The experiment was repeated with small-grain gunpowder, amounting to four grains, and the same phenomena were observed, with this difference, that a second slight deflagration followed shortly after the first, probably in consequence of a grain or two of the powder falling back into the cup. A single piece of gunpowder, weighing 14 grains, so shaped as to remain in good contact with the wire, was placed over the latter, being supported by the cup. The pressure of air in the apparatus was, as before, equal to 0°65 inch of mercury. There was no perceptible effect for a short time 1864. | Gun-cotton and Gunpowder. 211 after the wire was first heated to redness; vapours of sulphur were then given off, and slight scintillations were occasionally observed ; after a time the wire became deeply buried in the superincumbent mass of gunpowder, which fused, and appeared to boil, where it was in actual contact with the source of heat. After the lapse of three minutes from the commencement of the experiment, the powder deflagrated. ‘The permanent depression of the mercury column amounted to 1°35 inch. The experiment was repeated with a similar piece of powder, weighing 16 grains ; the same phenomena were observed; and five minutes elapsed between the first heating of the wire and the deflagration of the powder. The experiments were continued with fine-grained gunpowder, and under pressures gradually increased, in successive experiments, from ‘07 to 3 in inches of mercury. The same weight of gunpowder (4 grains) was used in all the experiments. In those made under a pressure of 1 inch, the results observed were similar to those obtained in the first experiments ; single grains of gunpowder were successively deflagrated, burning very slowly, and scattering but never igniting contiguous grains of powder. Eventually, after the lapse of from ten to twenty seconds, 3 or 4 grains were defla- grated at once, the remainder of the powder being thereby projected from the cup. At a pressure of 1°5 inch, the same phenomena were observed, but the successive deflagrations of fused grains of powder followed more quickly upon each other, and the final ignition of several together occurred in about ten seconds after the wire was first heated. At a pressure of 2 inches, at first only one or two of the fused grains were ignited, singly ; and several were deflagrated together after the lapse of five seconds. A larger quantity of the powder was burned, but a portion was projected from the cup as in preceding experiments. At a pressure of 3 inches, no grains were ignited singly; the combustion of the powder was effected after an interval of about four seconds, and the greater portion was burned ; the combustion, though it had gradually become more similar to that of gunpowder in open air, was still very slow. Experiments made with gunpowder in highly rarefied atmospheres of nitrogen furnished results quite similar to those described; nor was any important difference in the character of the phenomena observed when oxygen was substituted for air, except that the scintillations and deflagra- tions of the powder-grains were in some instances somewhat more bril- liant. The above experiments show that, when gunpowder is in contact with an incandescent wire in a highly rarefied atmosphere, the heat is, in the first instance, abstracted to so great an extent by the volatilization of the sul- phur, that the particles of powder cannot be raised to the temperature necessary for their ignition, until at any rate the greater part of that ele- ment has been expelled from the mixture, in consequence of which the por- tions first acted upon by heat will have become less explosive in their cha- racter, and require, therefore, a higher temperature for their ignition than R 2 212 Mr. Abel on the Combustion of [April 21, in their original condition. The effect of the continued application of heat to the powder thus changed is, to fuse the saltpetre and to establish che- mical action between it and the charcoal, which, however, only gradually and occasionally becomes so energetic as to be accompanied by deflagration, because the gas disengaged by the oxidation of the charcoal continues to convey away much of the heat applied, in escaping into the rarefied space. For the same reason, the grains of unaltered powder which are in actual contact with the deflagrating particles are not ignited by the heat resulting from the combustion, but are simply scattered by the rush of escaping gases, at any rate until the pressure in the vessel has been so far increased by their generation as to diminish the rapidity and extent of their expan- sion at the moment of their escape. The disengagement, first of sulphur- vapour and then of gaseous products of chemical change, unattended by phenomena of combustion, when gunpowder is maintained in contact with ared-hot wire in very highly rarefied atmospheres, are results quite in har- mony with the observations made by Mitchell, Frankland, and Dufour, with regard to the retarding influence of diminished atmospheric pressure upon the combustion of fuses. The phenomena described are most strikingly exhibited by operating upon single masses of gunpowder, of some size, in the manner directed above, when the application of the red-hot wire may be con- tinued from three to five minutes (the gases disengaged during that period depressing the column of mercury from 0°5 to 0°7 inch) before the mass is ignited. There is no doubt that the products of decomposition of the gun- powder, obtained under these circumstances, differ greatly from those which result from its explosion in confined spaces or in the open air under ordi- nary atmospheric conditions.. In all the experiments conducted in the most highly rarefied atmospheres (at pressures of 0°5 to 1°5 in inches of mercury), the contents of the vessel, after the final deflagration of the powder, always possessed a very peculiar odour, similar to that of horse- radish, due to the production of some sulphur-compound ; nitrous acid was also very generally observed among the products. It is readily con- celvable that the chemical action established between the constituents of gunpowder, under the circumstances described, must be of a very imperfect or partial character, the conditions under which it is established being un- favourable to its energetic development. In describing the phenomena which accompany. the ignition of gun- cotton in atmospheres of different rarefaction, I have pointed out that, at pressures varying from one to twenty-four in inches of mercury, a pale yellow flame was observed, which increased in size with the pressure of the atmosphere; and that a flame of precisely the same character was pro- duced in rarefied atmospheres of nitrogen. The experiments instituted in nitrogen show that the explosion of loose tufts of gun-cotton in atmo- ‘eho of that gas, even at normal pressures, was arava attended with a pale yellow flash of flame, quite different from the bright fiash produced by igniting gun-cotton in air, Thesame result was observed in atmospheres 1864. | Gun-cotton and Gunpowder. 213 of carbonic acid, carbonic oxide, hydrogen, and coal-gas. In operating with pieces of gun-cotton-twist or thread of some length instead of employing the material in loose tufts, the results obtained in the two last-named gases were very different from those observed in atmospheres of nitrogen, carbonic acid, and carbonic oxide. When ignited by means of a platinum wire (across which it is placed) in vessels filled with either of those two gases, and completely closed or open at one end, the piece of twist burned slowly and regularly, the combustion proceeding much more deliberately than if the same piece of gun-cotton had been ignited in the usual manner in air, and being accompanied by only a very small jet or tongue of pale yellow flame, which was thrown out in a line with the burning surface when the gun-cotton was ignited. The same result was obtained in currents of those gases when passed through a long, wide glass tube, along which the gun- cotton twist was laid, one end being allowed to project some distance into the air. The projecting extremity being ignited, as soon as the piece of twist had burnt up to the opening of the tube through which the gas was passing, the character of the combustion of the gun-cotton was changed from the ordinary to the slow form above described. On repeating this form of experiment in currents of hydrogen and of coal-gas, the ignited gun-cotton burned in the slow manner only a very short distance inside the tube, the combustion ceasing altogether when not more than from half an inch to one inch of the twist had burnt in the tube. The same result was observed when the current of gas was interrupted at the moment that the gun-cotton was inflamed. It was at first thought that this extinction of the combustion of gun-cotton by hydrogen and coal-gas might be caused by the very rapid abstraction of heat from the burning surface of gun- cotton in consequence of the diffusive powers of those gases ; but when the experiments were made in perfectly closed vessels, the piece of gun- cotton-twist being ignited by means of a platinum wire, the combustion also ceased almost instantaneously. These effects, therefore, can only be ascribed to the high cooling-powers, by convection, of the gases in question. It was found, by a succession of experiments, that when nitrogen was mixed with only one-fifth of its volume of hydrogen the combustion of gun-cotton- twist in the mixture was very slow and uncertain (being arrested after a short time in some instances), and that a mixture of one volume of hydrogen with three of nitrogen prevented its combustion, like coal-gas. The slow kind of combustion of gun-cotton, in the form of twist, which is determined by its ignition in currents or atmospheres of nitrogen, car- bonic acid, &c. may also be obtained in a powerful current of atmospheric . air, the thread of cotton being placed in a somewhat narrow glass tube. If, however, the air is at rest, or only passing slowly, the result is uncer- tain. In employing very narrow tubes into which the gun-cotton fits pretty closely, the combustion passes over into the slow form when it reaches the opening of the tube, and occasionally it will then continue throughout the length of the tube. In that case, while the gun-cotton 214 Mr. Abel on the Combustion of [April 21, burns slowly along the tube, with a very small sharp tongue of pale flame, a jet of flame is obtained at the mouth of the tube, by the burning of the gas evolved by the decomposition of the gun-cotton. Sometimes, and especially when wider tubes are employed, the slow combustion will pro- ceed only for a short distance, and then, in consequence of the ignition of a mixture of the combustible gases and air within the tube, the gun-cotton will explode with great violence, the tube being completely pulverized, and portions of unburnt cotton scattered by the explosion. If still wider tubes are employed, the cotton will flash into flame almost instantaneously throughout the tube directly the flame reaches the opening : in these cases the explosion is not violent ; sometimes the tube escapes fracture, and at others is broken in a few places, or torn open longitudinally, a slit bemg produced in the tube directly over the gun-cotton. By using narrow tubes and gradually shortening the tube through which the gun-cotton was passed, pieces of the twist being allowed to project at both ends, it was found, upon inflaming the material which projected on one side, that the slow form of combustion, induced in it as soon as it burned into the tube, was maintained by that portion which burned in the open air on the other side, when the combustion had proceeded through the tube. Eventually, by the employment of a screen of wood or card-board containing a perfo- ration of the same diameter as that of the gun-cotton-twist, through which the latter was partially drawn, the alteration of the combustion of the material from the ordinary to the slow kind was found to be invariably effected. On the one side of the screen, the gun-cotton burned with the ordinary flame and rapidity, until the combustion extended to the perfo- ration, when the flame was cut off and the material on the opposite side of the screen burned only slowly, emitting the small-pointed tongue of pale yellow flame. These results indicate that if, even for the briefest space of time, the gases resulting from the first action of heat on gun-cotton upon its ignition in open air are impeded from completely enveloping the burning extremity of the gun-cotton-twist, their ignition is prevented; and as it is the com- paratively high temperature produced by their combustion which effects the rapid and more complete combustion of the gun-cotton, the momentary extinction of the gases, and the continuous abstraction of heat by them as they escape from the point of combustion, render it impossible for the gun-cotton to continue to burn otherwise than in the slow and imperfect manner, undergoing a transformation similar in character to destructive distillation. These facts appear to be fully established by the following additional experimental results :-— 1. If, instead of employing in the above experiments a moderately com- pact gun-cotton-twist, one of more open structure is used, it becomes diffi- cult or even impossible to effect the described change in the nature of the combustion, by the means described, because the gases do not simply burn 1864..] Gun-cotion and Gunpowder. 215 at, or escape from, the extremity of the twisted cotton, but pass readily be- tween the separated fibres of the material, rendering it difficult or impos- sible to divert them all into one direction; and hence they at the same time transmit the combustion from particle to particle, and maintain the heat necessary for their own combustion. 2. If a piece of the compactly twisted gun-cotton, laid upon the bres be inflamed in the ordinary manner, and a jet of air be thrown against the flame, in a line with the piece of cotton, but in a direction opposite to that in which the flame is travelling, the combustion may readily be changed to the slow form, because the flame is prevented from enveloping the burning cotton, and thus becomes extinguished, as in the above experiment. 3. Conversely, if a gentle current of air be so directed against the gun- cotton, when undergoing the slow combustion, that it throws back upon the burning cotton the gases which are escaping, it will very speedily burst into the ordinary kind of combustion. Or, if a piece of the gun-cotton- twist, placed along a board, be made to burn in the imperfect manner, and the end of the board be then gradually raised, as soon as the material is brought into a nearly vertical position, the burning extremity being the lowest, it will burst into flame. By applying to the extremity of a piece of the compact twist a heated body (the temperature of which may range from 135° C. even up to a red heat), provided the source of heat be not very large in proportion to the surface presented by the extremity of the gun-cotton, the latter may be ignited with certainty in such a manner that the slow form of combustion at once ensues, the heat applied being insufficient to inflame the gases produced by the decomposition of the gun-cotton. By allowing the gun- cotton thus ignited to burn in a moderately wide tube, closed at one end, the inflammable gases produced may be burned at the mouth of the tube, while the gun-cotton is burning in the interior; or they may be ignited and the gun-cotton consequently inflamed, by approaching a flame, or a body heated to full redness, to the latter, in the direction in which they are escaping. It need hardly be stated that these results are regulated by the degree of compactness of the gun-cotton, the size of the twist, and the dimensions of the heated body. Thus a small platinum wire heated to full redness, or the extremity of a piece of smouldering string, will induce the slow combustion in a thin and moderately compact twist; but. a larger body, such as a thick rod of iron, heated only to dull redness, will effect the ignition both of the gun-cotton and of the gases evolved by the combustion of the first particles, so that the material will be inflamed in the ordinary manner. Similarly the red-hot platinum wire, or a stout rod heated to redness barely visible in the dark, if they are maintained in close proxi- mity to the slowly burning surface of gun-cotton, will eventually cause the gases evolved to burst into flame. The more compact the twist of the gun-cotton, the more superficial is the slow form of combustion induced in 216 Mr, Abel on the Combustion of Gun-cotton. [April 21, it, and a condition of things is readily attainable, under which the gun- cotton-twist will simply smoulder in open air, leaving a carbonaceous residue; and the heat resulting from this most imperfect combustion will be abstracted by the gases evolved more rapidly than it is generated, so that in a brief space of time the gun-cotton will cease to burn at all in open air *. The remarkable facility with which the nature of combustion of gun- cotton in air or other gases may be modified, constitutes a most charac- teristic peculiarity of this substance as an explosive, which is not shared by gunpowder or explosive bodies of that class, and which renders it easily conceivable that this material is susceptible of application to the produc- tion of a comparatively great variety of mechanical effects, the nature of which is determined by slight modifications in its physical condition, or by what might at first sight appear very trifling variations of the conditions attending its employment. There is little doubt that the products of decomposition of aunts vary almost as greatly as the phenomena which attend its exposure to heat under the circumstances described in this paper. A few incidental obser- vations indicative of this variation were made in the course of the experi- ments. Thus, in the instances of the most imperfect metamorphosis of gun-cotton, the products included a considerable proportion of a white vapour, slowly dissolved by water, as also small quantities of nitrous acid and a very large proportion of nitric oxide. The latter gas is invariably formed on the combustion of gun-cotton im air or other gases; but the quantity produced appears always to be much greater in instances of the imperfect or slow combustion of the material. The odour of the gases pro- duced in combustions of that class is powerfully cyanic, and there is no diffi- culty in detecting cyanogen among the products. I trust before long to institute a comparative analytical examination of the products resulting from the combustion of gun-cotton under various conditions ; meanwhile I have already satisfied myself, by some qualitative experiments, of the very great difference existing between the results of the combustion of gun- cotton in open air, in partially confined spaces, and under conditions pre- cisely similar to those which attend its employment for projectile or de- structive purposes. I have, for example, confirmed the correctness of the statement made by Karolyi in his analytical account of the products of de- composition of gun-cotton, that no nitric oxide or higher oxide of nitrogen is eliminated upon the explosion of gun-cotton under considerable pressure, as in shells. Coupling this fact with the invariable production of nitric oxide when gun-cotton is exploded in open air or partially confined spaces, there appears to be very strong reason for the belief that, just as the reduc- * By enclosing in suitable cases solid cords, made up of two or more strands, and more or less compactly twisted, I have succeeded readily in applying gun-cotton to the production of fuses and slow-matches, the time of burning of which may be accurately regulated. 1864.] Dr. Phipson on Magnesium. 217 tion of pressure determines a proportionately imperfect and complicated transformation of the gun-cotton upon its exposure to heat, the results of which are more or less essentially of an intermediate character, so, con- versely, the greater the pressure, beyond the normal limits, under which gun-cotton is exploded—that is to say, the greater the pressure exerted by it, or the resistance presented at the first instant of its ignition, the more simple are the products of decomposition, and the greater are the physical effects attending its explosion, because of the greater energy with which the chemical change is effected. III. “On Magnesium.” By Dr. T. L. Pureson, F.C.S. Communi- cated by Prof. G. G. Stoxzs, Sec. R.S. Received March 9, 1864. (Extract.) Iodine and Sulphur.—tI find that iodine can be distilled off magnesium without attacking the metal in the least. In the same manner I distilled several portions of sulphur off magnesium without the metal being at all attacked. Decomposition of Silicie Acid.—Heated for some time in a porcelain crucible with excess of anhydrous silica, the metal burns vividly if the air has access; and a certain quantity of amorphous silicium is immediately formed. Magnesium is therefore capable of reducing silicic acid at a high temperature. The reason why potassium and sodium cannot effect this is simply because these metals are highly volatile and fly off before the crucible has attained the proper temperature. Magnesium being much less volatile than the alkaline metals, takes oxygen from silica before volatilizing. If the silicic acid be in excess, a silicate of magnesia is formed at the same time; if the metal is in excess, much siliciuret of magnesium is produced. The presence of the latter is immediately detected by throwing a little of the product into water acidulated with sulphuric acid, when the charac- teristic phosphoric odour of siliciuretted hydrogen is at once perceived. Decomposition of Boracie Acid.—With boracic acid the phenomena are rather different ; the acid melts and covers the metal, so that it does not inflame even when the crucible is left uncovered. A certain quantity of boron is soon liberated, and the product forms a greenish-black mass, which oxidizes and becomes white in contact with water, and disengages no odoriferous gas in acidulated water. Decomposition of Carbonie Acid.—I thought it would be interesting to try a similar experiment with carbonic acid. Accordingly dry carbonate of soda was heated with a little magnesium in a glass tube over a common spirit-lamp ; and before the temperature had arrived at a red heat I observed that carbon was liberated abundantly, and magnesia formed. Action of Alkalies.—A solution of caustic alkali or ammonia has little or no action upon magnesium in the cold. 218 Prof. Erman—Magnetic Elements at Berlin. [April 28, Precipitation of Metallic Solutions.—Magnesium precipitates nearly all the metals from their neutral solutions. When these are taken in the form of protosalts, even manganese, iron, and zine are precipitated as black powders. Aluminium and uranium (and perhaps chrome) are only pre- cipitated as oxides. Alloys of Magnesium.—I have examined only a few alloys of magnesium. Unlike zinc, magnesium will not unite with mercury at the ordinary tem- perature of the air. With tin 85 parts, and magnesium 15 parts, I formed a very curious alloy of a beautiful lavender-colour, very hard and brittle, easily pulverized, and decomposing water with considerable rapidity at ordinary temperatures. If the air has access during the formation of this alloy, the mixture takes fire; and if the crucible be then suddenly with- drawn from the lamp, the flame disappears, but a vivid phosphorescence ensues, and the unfused mass remains highly luminous for a considerable time. A white powdery mass, containing stannic acid and magnesia, is the result. [With platinum, according to Mr. Sonstadt, magnesium forms a fusible alloy ; so that platinum crucibles can be easily perforated by heating mag- nesium in them. | Sodium and potassium unite with magnesium, and form very malleable alloys, which decompose water at the ordinary temperature. It is probable that an alloy of copper and magnesium, which I have not yet obtained, would differ from brass, not only in lightness, but by de- composing water at the ordinary temperature with more or less rapidity. Uses.—Magnesium will be found a useful metal whenever tenacity and lightness are required and tarnish is of no consequence. The light fur- nished by combustion of the wire has already been utilized in photography at night. In the laboratory it will be found useful to effect decomposi- tions which sodium and potassium cannot effect on account of their greater volatility. April 28, 1864. Dr. W. A. MILLER, Treas. & V.P., in the Chair. The following communications. were read :— I. “On the Magnetic Elements and their Secular Variations at Berlin,” as observed by A. Ekman. Communicated by General SABINE, P.R.S. Received March 1, 1864. All observations and results to be mentioned here relate to Latitude 52° 31' 55” North. Longitude 13° 23' 20" E. from Greenwich. 1. Horizontal Intensity. Denoting by (1800+¢) the date of observation in tropical years of the 1864..] Prof. Erman—Magnetic Elements at Berlin. 219 Gregorian epoch, T the absolute value of horizontal intensity with milli- metre, milligram, and the second of mean time as unities, w the same in unities of the Gaussian constants; the two values of T for 1805°5 and 1828°31 have been deduced from observed w, by T=0°00349216. w. r and 7! denote the observed time of oscillation of two magnets which, since 1853°523 were carefully guarded from the influence of other magnets ; and therefore, marking by C, a, 6, C’, a’, 6’ unknown constants, e the basis of hyperbolic logarithms, and taking ¢,=¢—53°523, each value of r and 7 had to fulfil the equations (2 See Le NR Sa ‘ (1+ae—84) . T° (l+q‘e—6'4) , T In the following list of observed values, the first is due to Humboldt; the twenty-eight following were obtained by Erman :— Date of Horizontal intensity, Se iimes of apeillanon 08) is ae observation. T. Magnet I. Magnet IT. Ut vr’. 1800-+-2¢ | Observed. | Calculated.| Observed. | Calculated.| Observed. | Calculated. 1805:5 16452 G42 Px ay eieeeas Veet te ades ves eae 1828-31 1°7559 ST eer eSies ren ence ce arrcs We ME RCAM ECE a le le. Wyre ec he 1846-13 1-751 LIS 77 Cr EES One bang SSG) inl PMC CR CREME hs MENS on 1849°59 1-784 TELS TEA PNM TEL GD [sth cae cee eee OP eeewaceaen [ol wane uasie LCS 0 8 ai eee hanes a ren a 3°1090 S VOB pees ee > Let 1854-59 1-7900 1:7904 3°1072 31141 8:0082 8:0056 1856°57 1:7900 1:7913 31168 3°1134 8:0954 81036. 1857°54 1-7879 17916 31105 31131 81193 8:1104 1858°53 1:8035 17917 3°1158 31130 8°1364 81126 185960 1°7933 1-79°.2 3°1229 3°1129 81223 81182 MERU srateeccus | becéine ss 3°1043 3°1129 8:0870 8°1135 1861:52 1°7972 Oe ly aie Nee St ocean buna Hache 81258 81138 1862-52 1:7900 a OLE ire de. srek. Guat. tirade 81100 81142 . 1863°80 1°7929 Lol) 31148 31135 8:0975 81151 The calculated values result from the following most probable expressions for T, and for those values of r and 7’ which agree best with the contem- poraneous T. I. T=1°61892+0:0057689 ¢—0:000048119 2’. a eA 17°3633 41+0°07892 ae ty. T with ¢,=f—53°523. TE 117°956 | {1+0°09733 . e— 1168274 T The expression I. appears liable to the probable errors, in first term, +0:00126 in coefficient of ¢, —+0°000065 | of a magnetic unity ; in coefficient of 2”, -+0°00000074 220 Prof. Erman—Magnetic Elements at Berlin. [April 28, and when brought under the form (A) T=1-79183—0-000048119{¢—59-930}?, identical with I., it shows that the horizontal intensity reached in 1859-930 the maximum of 1°79183. _ It ought to be observed, that equal probability has been attributed to the error +l “in 20; that is to say, equal errors to an intensity determined by each of the three methods,—this supposition being at once the most simple and the most conformable to my experience, by nearly contemporaneous repetitions of each class of observation. All my determinations of absolute intensity have been obtained either by one of two, or by.two magnetometers ; the first of which is a Gaussian of large size, by Meyerstein, the second my declination- and transit-instrument by Pistor, completed by the usual graduated holders for pe te magnets, and perfectly adapted to observations in the open air. 2. Inclination. The values of inclination here employed are taken for 1806'0, 1832°5, and 1836°87, from the observations of Humboldt, Rudberg, and Encke ; for the ten other dates since 1825-0, they have been obtained by my own applications of the methods exposed in my ‘ Reise um die Erde,’ Physikal. Beob., tome il. pp. 8-42, to two different instruments—viz. till 1850 to a large and highly perfect one by Gambey, and since that time to a smaller dip-circle by Robinson. The methods of observation leave no room for any constant error in the resulting inclination, as long as no di- rective magnetic force is exerted upon the needle by the instrument itself. In order to free my results from any influence from this improbable (but not impossible) source, I compared, in 1860, three full determinations by the last-mentioned apparatus, with an equal number which I obtained under identical circumstances with a most perfect copy of Weber’s in- ductive inclinometer. The result was an agreement of the two kinds of determinations within the limits of accidental error of the first—that is to say, far below one minute in the inclination; I venture, therefore, to say hat the following numbers must give the absolute value of the element in question with no less certainty than the rate of its secular variation : 1864. ] Prof. Erman—Magnetic Elements at Berlin. 221 Date of Inclination, observation. j 1800+-¢. | Observed. | Calculated. oO i jo} i 1806-0 69 53 69 52-99 1825:00 | 68 49:19 | 68 4462 1828-29 | 68 3455 | 68 34:17 1832-50 | 68 18:08 | 68 21:40 1836:87 | 68 743 | 68 8-84 1838°75 | 68 2:04 | 68 3°66 184620 | 67 43:25 | 67 44-46 1849°65 | 67 35-48 67 36:29 1853-78 67 29°81 67 27-09 185656 | 67 20°50 67 24:26 1857:55 | 67 20°30 67 19-25 1860:60 | 67 15°75 67 13°31 186255 | 67 763 67} 9:69 The system of the above calculated values, which best agrees with the observed ones, results from the expression (B) t= 70° 17!'-42—4"1854¢+0'-018931 2 ; it leaves in each single equation a probable error of + 1'*42; and accordingly in the expression itself the probable errors appear to be in the absolute term +2'°17; in the coefficient of £ +0'1211; in the coefficient of 2? +0'°001591. This expression can be brought under the form (B*) i= 66° 26'09 + (¢?—110°543)’. 0'-018931, which would prove that at the place in question the inclination will come, in 1910°543, to a minimum of 66° 26°09. The aforesaid errors of terms give +2°27 years for the uncertainty of the epoch of this minimum, and +3'-9 for the uncertainty of its value; but as the expression (B) results from observations between 1806 and 1863, its consequences ought not to be extended as far as 1910. 3. Declination. Four results of observations of this element, made by the late astrono- mers Kirch in 1731, Bode in 1784 and 1805, and Tralles in 1819, have been added to my own, which extend from 1825 to 1864. These latter were obtained with the declination- and transit-instrument employed in my voyage, which intermediately was frequently compared and found in per- fect agreement with a large Gaussian magnetometer, whenever the indi- cations of the latter were duly freed from the torsion of the suspending wires and from the want of parallelism between the normal of the employed spe- culum and the magnetic axis of the bar. My observations were all made in the open air, with the exception of the two in 1849 and 1850, which, having been executed in a room, were corrected for the influence of local attractions. As the determination of this latter seemed exposed to a somewhat larger 222 Prof, Erman—Magnetic Elements at Berlin. [April 28, error than the other declinations, in combining the two reduced values with those obtained in the open air, I have given to the two first only a fourth of the weight of the others. A similar allowance for larger pro- bable errors should perhaps have been made in employing the four state- ments of former observers ; but, for want of particulars about the operations they are founded upon, it was more safe to neglect the difference between their weight and that of the others, than to fix it by an arbitrary as- sumption. If, for the moment of observation, there were marked by 1800-+4, as before, the tropical years elapsed since the Gregorian epoch, m the positive excess of ¢ over the next integer, 2 the horary angle of mean Sun, each observed west declination d' had to be brought under the form d'=D+f(t)+9(m,2), D denoting a constant, and f and ¢ two functions, the first of which was to be determined here. In order to form d'—¢(m, x)=d out of each d’, I put o(m, v)=a+a.cosr+y.cos 24+ . cos 32, +6 .sing+6.sin 27+. sin 32, taking the values of a, a, 8, ..-... , £ by interpolation according to m, from the following Table, derived from observations in the Russian observatories at St. Petersburg, Catherinbourg, and Barnaoul in the year 1837 and 1838, and well agreeing with my own determinations of ¢(m, x) for the years 1828 to 1830, and at eight places between latitude 50° and 62° North. m a a. B y: ry € é 0042 | 445. | 54 | se Ibe | 27 | 90 15 Sok ee o193°| = "62 | er | 46 | Soe) Tied 1 Pee ae 0-204 | — 43 | + 54 | 4135 | +411 | +173 | +22 | +62 0-288 |. —108 |} 46.64, [i++ 256] \. 4.14 44.199. it 2 eb eee 0373 | — 88 | 1104 | 4964 | +65 | +182 | +42 | Fae 0-455 | + 14 | +107 | +290 | 471 | +184 | +64 | +445 0-538 | +77 |.4082..) £975 1 a6: | 175 | eee 0623 | + 60 | + 8 |'4221 | +79 | 1187 | 166 >) sie 0-707 | +72 | +91 | 4139 | +51 | +156 | +57 | +34 0-790. | + 68 b+ 78 |ek-69. | pe Bb h1BF 29 «| oot Oras oS ates ME get © yo a 76 “Po ee oe 0050) +) Bll BO) org 1/2074 -ae eg) aaae [eeeae 1042 [od We de 4 eee dbo | OR. GO. siaee a When I supposed in this way that the parameters a,a,...., { of the function ¢(m, x), or ¢ as I will call it for abbreviation, are the same for all moments alike situated in different years, I was well aware that this assumption is but approximative, and that all sufficiently extended and direct investigations of ¢, as chiefly those of General Sabine, have shown a periodicity of about 9:5 years in the total values of this function, But as the laws of such dependence between T and each of the seyen para- Prof. Erman—Magnetic Elements at Berlin, 223 1864. ] meters of @ have not yet been perfectly exposed, I preferred in the pre- sent to treat the latter as mere functions of manda. In the following Table of employed mean declinations for the moments ¢, to each of them is subjoined the value of ¢ by whose subtraction it has resulted from the “momentary value furnished by observation. This arrangement will allow us to appreciate (and, if wanted, to correct for) the influence exerted by any periodical variation of @ upon the final result of my observations. It may, too, be convenient to observe that for some of the following west declinations (D), as well as for the before-mentioned intensities (T) and in- clinations (1), the observations were made in latitude p— Ag, and longitude 1—A/ (where p and / mark the corresponding and above alleged values for my ordinary place), and that then the directly obtained results, viz. d—Ad, T—AT, or :— Ai, have been reduced by Ad=—0-0940 . Ap—0°6103. AZ; AT=—0°7480 . 10-3. Ap+0°2152. 10-3. Al; Ai=+0°7405 . Ap—0°1861 . AZ; the minute of arc being the unity for Ap, Ad, Az, and AJ. These equations, which result from the Gaussian constants with the given p and J, are sufficiently approximated when, as with us, Ap and A/ do not exceed a few minutes. So then were obtained: Mean declination, Momentary Date of declination. d. observation. Mean 1800+. declination, By . observation. Calculated. ie) a i 1731-60 0 12 18:05 | 19 19-85 178400 0 17 59-65 17 46:09 1805-40 0 18 1:35 18 7:86 1819-00 0 17 36°50 17 48:06 1825-79 —1-80 17 24-46 17 28:37 1828-33 —4:08 17 21-35 17 19°34 1834-05 — 1-24 17 2-69 16 55-65 *1849-62 +2°74 15 21-55 15 24:39 *1850°63 +2-96 15 20-48 15 20-47 1853-81 +432 14 55°17 14 58:26 1854:36 — 2-86 15) E05 14 54:19 1856-58 JS) ES: 14 38:13 14 37:40 1857-49 —5°95 14 33°88 14 30:29 1858-54 —5°61 14 21-15 14 21:96 1859-58 —4:87 14 14:24 14 13°59 1861-50 +447 18 53°70 13 57°63 1862°55 +0:12 13 49-83 13 48°72 1863°79 +448 13 36°85 13 37:99 A fourth of the weight of each of the other observed values being given to each of the two marked *, the whole is best represented by d=18° 8':46+ 0'26820 ¢<—0'070665 47, . . . (IV). 224 Prof. Erman—Magnetic Elements at Berlin. A 28, which furnishes the above calculated numbers; and by their corapurigen with the observed ones, the probable errors are— in the absolute term of d +1':94; in the coefficient of in d +0':2932 ; in the coefficient of ¢? in d+0':030669. If, now, instead of employing the variations ¢ (m, x), or ¢ according to observations in the years 1837 and 1838, we assume (1) that the periodi- cal dependence between this function and the date ¢ consists in always changing each parameter proportionally to its mean or primitive value, and then (2) that, as General Sabine has proved, the whole function has nearly reached a maximum in all moments marked by ¢=48-n. 9:5, n being an integer, and (3) that, according to the same philosopher, the least and the largest amount of corresponding variations are approximately as 1: 1:°4, then, ® marking the function of ¢, m, x which in each case must be substituted for ¢, and C a function of M and X, we shall have d=c 12 20+0°20 . sin En C= —45-625)]] | i o=c | 1:20—0-20 . sin (8: 125) | = l3a8tl Ge: To each of the preceding values of d must therefore be added p= be% {0 1163—0:1472. sin E (¢—45°625)) \ By executing this operation, I found that the reduced observations are best represented by (C*) d=18° 8-43 -+-0°26831 . —0°070652. 2’, and that, though scarcely differing from (IV.), this expression is preferable, because the probable error of each of its terms is by nearly =); of its former value smaller than the corresponding one in (IV.) As the expression (C*) is identical with (C) d=18° 8'68—0'-070652{¢—1-8991, we see that, according to my observations, the west declination at the place in question arrived in 1801-899 at a maximum of 18° 8'°68. Putting off for a further article some more general observations on the secular changes of terrestrial magnetism, I briefly resume, as results of my nearly forty years’ observations, that for latitude =52° 31' 55” North, longitude = 13° 23’ 20” E. from Greenwich, 1864. | Action of Chlorine upon Methyl. 225 there have been—between 1805 and 1864, Horizontal intensity =T=1°79183 —0°000048119 {¢—59-930}? ; between 1806 and 1863, Inclination =i=66° 26'-09 + 0':018931 {¢—110°543)? ; and between 1731 and 1864, West declination =d=18° 8'-68—0'-070652 {¢—1-899}?; all results being meant to be just for the date 1800+¢ in years of the Gregorian epoch. N.B. It seems not unworthy of remark, that no evidence of the existence of a third term in the expression for any one of the three phenomena results from the above-mentioned observations; and this, though partly due to the inevitable imperfections of the observations, makes it highly pro- bable that a man’s lifetime, and even a century is but a very small part of the secular period of terrestrial magnetism. If. “ On the Action of Chlorine upon Methyl.” By C. Scuor- LEMMER, Assistant in the Laboratory of Owens College, Man- chester. Communicated by Professor Roscoz, F.R.S. Received April 5, 1864. In a paper published in the Journal of the Chemical Society, New Ser. vol. 1. p. 425, I pointed out the great interest which attached to the study of the lower terms of hydrocarbons, known by the name of the “ alcohol radicals,” inasmuch as the question of the chemical constitution of these bodies requires to be more definitely settled. Having been aided in these researches by a grant from the Council, I beg to lay before the Royal Society the results of an investigation on the action of chlorine upon methyl, which are as unexpected as they are de- cisive. Equal volumes of chlorine and of methyl were exposed in strong well- corked bottles, holding from two to three litres, to diffused daylight in the open air at a temperature of about 5°C. The methyl was prepared according to Kolbe’s method, by electrolysis of a concentrated solution of acetate of potassium, and carefully purified by washing with a solution of caustic potash and concentrated sulphuric acid. The colour of the chlo- rine disappeared rather quickly ; colourless oily drops condensed on the the sides of the bottles, and collected after some time on the bottom as a mobile liquid, the greater part of which volatilized again when the bottles were brought into a warm room. Hence it appears that by the action of one volume of chlorine upon one of methyl, substitution-products are formed, consisting chiefly of a volatile liquid, the boiling-point of which ies between 5° and 15° C. In order to collect these products, the bottles were heated till all the liquid had volatilized, and then opened, with the mouth downwards, under a hot concentrated solution of common salt, to which some caustic soda was added in order to quicken the absorption of VOL. XIII. s 226 Action of Chlorine upon Methyl. [April 28, the hydrochloric acid, of which half the volume of gas contained in the bottles consisted. The bottles were then taken out of the liquid, placed in an upright position, and the mouth provided with a doubly perforated cork, into one opening of which a siphon fitted. Through this siphon a hot concentrated solution of common salt slowly ran in, whilst the gas thus displaced escaped by a bent tube and was condensed in a small tube receiver, surrounded by a mixture of ice and salt. The liquid thus ob- tained was ieft for some hours in contact with a piece of solid caustic potash, in order to remove moisture and the last traces of hydrochloric acid. Subjected to distillation, the liquid began to boil at 11° C., and the boiling-point rose slowly to 30° C., at which temperature two-thirds of the liquid had come over. On continuing the fractional distillation for some time longer, the distillate yielded a few grammes of a colourless mobile liquid, boiling between 11° and 13°C., which, as the following analysis and vapour-density determinations prove, is chloride of ethyl, C,H,Cl. The boiling-pvint of the liquid agrees with that of this com- pound, and it possesses the strong peculiar smell and the property of burning with a white, luminous, green-bordered flame, characteristic of the chloride of ethyl. I. Analysis :— (1) 0°4245 grm. of the substance gave 0°5670 grm. of carbonic acid and 0°3025 of water. (2) 0°1810 grm. of the substance gave 0°3855 grm. of chloride of silver and 0°0165 grm. of metallic silver. Calculated for the formula C, H, Cl. Found. 2C 24 37°21 36°43 DEL co) Z:7D 7°92 Gi 3020 95°04 59°63 64°5 100-00 99°98 II. Determination of the vapour-density according to Gay-Lussae’s method :— Weight of substance employed........ 0°0893 grm. Temperature of alr .........---.--- oC; Height of barometer ................ 739 millims. (1) Temperature of vapour.............. 50° C. Volume of vapour Difference of level Vapour-density calculated from these numbers .... (2) Temperature of vapour Volume of vapour Difference of level Vapour-density calculated .... (3) Temperature of vapour Volume of vapour Difference of level eeerees ee e © © & © @ oe © coeeee ec eee 8B ew oe we oO ee eeeevreereereeee ee ose se eeeeeeeeeeeeeee 46°2 cub. centims. 140°5 millims. 2°24a% 70° C. 48:2 cub. centims. 130:0 millims. 2°244. 80°C. 49°45 cub. centims. 1250 millims. 1864. ] Mr. Russell on the Calculus of Symbols. 227 which numbers give the vapour-density 2°235, whilst the theoretical vapour-density of chloride of ethyl is 2°233. The boiling-point of the residue left after the first distillation rose quickly up to 60° C., whilst nearly the whole distilled over between this temperature and 70°C. By afew more fractional distillations of this latter portion, monochlorinated chloride of ethyl, C,H, Cl,, boiling between 62° and 65° C., was isolated. 0°1270 grm. of this compound gave 0°3530 grm. of chloride of silver and 0°0095 grm. of metallic silver, which corresponds to 71°43 per cent. of chlorine, whilst the formula requires 71°71 per cent. Hight litres of methyl yielded about 8 grammes of the mixed chlorides, or only about one-third of the theoretical quantity of chloride of ethyl which should have been obtained. This is easily explained by the vola- tilization of the liquid, and its solution in large quantities of water, as well as by the formation of higher substitution-products,. in consequence of which a considerable quantity of methyl is left uncombined. From these results it appears that the lowest term of the series of alcohol radicals behaves with chlorine exactly in the same manner as I have shown in the paper above referred to is the case with its homologues ethyl-amyl, C, H,, (which gives chloride of heptyl, C,H,,Cl), and amy], C,, H,, (from which chloride of decatyl, C,, H,, Cl, is obtained). If an excess of chlorine is avoided, the principal products consist of the chlo- rides of monatomic radicals containing the same number of atoms of carbon as the original hydrocarbon contained, whilst at the same time chlorine substitution-products of these chlorides are formed in smaller quantities. As there is no reason why those terms of the series which are placed between C,H,, C,H,,, C,, H,, should show a different deportment, it becomes obvious that, beginning with marsh-gas, C H,, the lowest term in the series C,,Hon+42, the most simple of all hydrocarbons, and one which can easily be obtained from its elements, we are now not only in a position to prepare all the members of this series, but likewise to build up by simple synthesis the series of mono-, di-, and polyatomic alcohols, acids, compound ammonias, ethers, &c. &c. of which each of the marsh-gas hy- drocarbons forms the starting-point. III. “On the Calculus of Symbols (Fifth Memoir), with Applica- tions to Linear Partial Differential Equations, and the Calculus of Functions.” By W. H.L. Russexz, Esq., A.B. Communi- cated by Professor Stoxss, Sec. R.S. Received April 7, 1864. In applying the calculus of symbols to partial differential equations, we find an extensive class with coefficients involving the independent variables which may in fact, like differential equations with constant coefficients, be 228 Mr. Russell on the Calculus of Symbols. [April 28, solved by the rules which apply to ordinary algebraical equations ; for there are certain functions of the symbols of partial differentiation which com- bine with certain functions of the independent variables according to the laws of combination of common algebraical quantities. In the first part of this memoir I have investigated the nature of these symbols, and applied them to the solution of partial differential equations. In the second part I have applied the calculus of symbols to the solution of functional equa- tions. For ‘this purpose I- have worked out some cases of symbolical division on a modified type, so that the symbols may embrace a greater range. I have then shown how certain functional equations may be expressed in a symbolical form, and have solved them by methods analo- gous to those already explained. The Society then adjourned to Thursday, May 12th. 1864. | Mr. Russell on the Calculus of Symbols. 227 which numbers give the vapour-density 2°235, whilst the theoretical vapour-density of chloride of ethyl is 2°233. The boiling-point of the residue left after the first distillation rose quickly up to 60° C., whilst nearly the whole distilled over between this temperature and 70°C. By a few more fractional distillations of this latter portion, monochlorinated chloride of ethyl, C,H, Cl,, boiling between 62° and 65° C., was isolated. 0°1270 grm. of this compound gave 0°3530 grm. of chloride of silver and 0:0095 grm. of metallic silver, which corresponds to 71°43 per cent. of chlorine, whilst the formula requires 71°71 per cent. Eight litres of methyl yielded about 8 grammes of the mixed chlorides, or only about one-third of the theoretical quantity of chloride of ethyl which should have been obtained. This is easily explained by the vola- tilization of the liquid, and its solution in large quantities of water, as well as by the formation of higher substitution-products, in consequence of which a considerable quantity of methyl is left uncombined. From these results it appears that the lowest term of the series of alcohol radicals behaves with chlorine exactly in the same manner as I have shown in the paper above referred to is the case with its homologues ethyl-amyl, C, H,, (which gives chloride of heptyl, C,H,,Cl), and amyl, C,, H,, (from which chloride of decatyl, C,, H,, Cl, is obtained). If an excess of chlorine is avoided, the principal products consist of the chlo- rides of monatomic radicals containing the same number of atoms of carbon as the original hydrocarbon contained, whilst at the same time chlorine substitution-products of these chlorides are formed in smaller quantities. As there is no reason why those terms of the series which are placed between C,H,, C,H,,, C,, H,, should show a different deportment, it becomes obvious that, beginning with marsh-gas, C H,, the lowest term in the series C,,H2,+40, the most simple of all hydrocarbons, and one which can easily be obtained from its elements, we are now not only in a position to prepare all the members of this series, but likewise to build up by simple synthesis the series of mono-, di-, and polyatomic alcohols, acids, compound ammonias, ethers, &c. &c. of which each of the marsh-gas hy- drocarbons forms the starting-point. II. “On the Calculus of Symbols (Fifth Memoir), with Applica- tions to Linear Partial Differential Equations, and the Calculus of Functions.” By W. H. L. Russetx, Esq., A.B. Communi- cated by Professor Stoxzs, Sec. R.S. Received April 7, 1864. In applying the calculus of symbols to partial differential equations, we find an extensive class with coefficients involving the independent variables which may in fact, like differential equations with constant coefficients, be VOL, XIII, Y 228 Mr. Gompertz on the Law of Mortality. [May 12, solved by the rules which apply to ordinary algebraical equations ; for there are certain functions of the symbols of partial differentiation which com- bine with certain functions of the independent variables according to the laws of combination of common algebraical quantities. In the first part of this memoir I have investigated the nature of these symbols, and applied them to the solution of partial differential equations. In the second part I have applied the calculus of symbols to the solution of functional equa- tions. For this purpose I have worked out some cases of symbolical division on a modified type, so that the symbols may embrace a greater range. I have then shown how certain functional equations may be expressed in a symbolical form, and have solved them by methods analo- gous to those already explained. . The Society then adjourned to Thursday, May 12th. May 12, 1864. Major-General SABINE, President, in the Chair. In accordance with the Statutes, the names of the Candidates recom- mended by the Council for election into the Society were read, as follows :— Sir Henry Barkly, K.C.B. William Jenner, M.D. William Brinton, M.D. Sir Charles Locock, Bart., M.D. T. Spencer Cobbold, M.D. William Sanders, Esq. Alexander John Ellis, Esq. Col. William James Smythe, R.A. John Evans, Ksq. Lieut.-Col. Alexander Strange. William Henry Flower, Esq. Robert Warington, Esq. Thomas Grubb, Esq. Nicholas Wood, Esq. Sir J. Charles Dalrymple Hay, Bart. The following communications were read :— I, ‘Second Part of the Supplement to the two Papers on Mortality published in the Philosophical Transactions in 1820 and 1825.” By Bensamin Gompertz, F.R.S. Received March 30, 1864, (Abstract.) _ The objects of this paper are various; but the subject appears to the author more especially important in consequence of the state of competition among assurance establishments, which he holds to be injurious to the interest of those valuable establishments, and to those of the assuring population. The author’s purpose in this paper is greatly to extend the modes of cal- culating valuations, and to improve the methods of calculation hitherto used by actuaries, which are in many cases very laborious, and in some almost impracticable. This part commences with observations on the inge- 1864.] Dr. Kopp on the Specific Heat of Solid Bodies. 229 nious plan of Barrett, which is shown to be capable of improvement and extension. Parts of the excellent work of the late Mr. David J ones, ‘ Ta- bles of Life Annuities,’ published under the care and suggestion of the Society for the Diffusion of Useful Knowledge, founded on those ideas of Barrett, are by the author here improved and extended so as to give methods easier for common purposes, and capable of extension to almost all diffi- culties which are likely to occur in the calculation of the value of property. II. “ Investigations of the Specific Heat of Solid and Liquid Bodies.” By Hermann Kopp, Ph.D. Communicated by T. GRAHAM, Hsq., Master of the Mint. Received April 16, 1864. : ( Abstract.) . In the first part the author discusses the earlier investigations on the specific heat of solid bodies, and on the relations of this property to their atomic weight and composition. In this historical report he gives a com- plete analysis of the various opinions published on the subject. In the second part the author describes the method he has used ies determining the specific heat of solid bodies. This method is based on the method of mixtures. The substance investigated is placed in a glass tube, together with some liquid which does not dissolve it, and the tube is heated im a mercury bath, and then rapidly immersed in a calorimeter containing water. Equalization of temperature takes place rapidly, through the inter- vention of the liquid in the tube. The thermal effect (increase of tempe- rature in the water of the calorimeter) is determined. Preliminary expe- riments give the means of allowing for the thermal effect due to the glass and to the liquid in it, and of thereby obtaining the thermal effect produced by the solid substance. The author gives a complete description of the apparatus and of the mode of using it, and also of the means of determin- ing the ancillary magnitudes which require to be taken into account. The entire method is very simple, and it brings the determination of specific heat out of the restricted sphere of the physical cabinet, with its compli- cated apparatus, within reach of the ordinary appliances of the chemical laboratory. It is also applicable to small quantities, and to such sub- stances as cannot bear a high temperature. The author discusses the possible deficiencies as well as the advantages of this method as compared with those of Neumann and of Regnault. In the third part the author gives his determinations of a very great. number of solid bodies. The specific heat of many of them had been determined by Neumann, or by Regnault ; and the almost universal agree-. ment of the numbers found by their methods and by his own proved the comparability of his results with those of other physicists. Where there is a considerable difference, the cause is discussed. By far the greatest T 2 230 Dr. Kopp on the Specific Heat of Solid Bodies. [May 12, number of the author’s experiments are on substances whose specific heat had not been previously determined ; they extend to all the more important classes of inorganic compounds, and to a great number of organic com- pounds. In the fourth part the author gives a synopsis of the materials at pre- sent available and trustworthy for considering the relations between specific heat and atomic weight or composition. That is, he gives for solid bodies of known composition the atomic formula, the atomic weight, the more trustworthy determinations of specific heat, and (corresponding to these) the atomic heats, or products of the specific heats and the atomic weights. The relations between the atomic heat and the atomic weight or the composition are discussed in the fifth part. A discussion whether the specific heat of a body varies materially with its different physical conditions forms an introduction to this part. The influence which change of temperature of solid bodies exerts on the specific heat is considered. This difference is inconsiderable, as is also the differ- ence of specific heats found for the same substance, according as it is ham- mered or annealed, hard or soft. With dimorphous varieties of the same substance, even where the specific gravity is different, the same specific heat is found in most cases. Great difference had been supposed to exist in the specific heat of a substance, according as it was crystalline or amor- phous. The author shows that, for a great number of substances, there is no such difference, and that in other cases the apparent differences depend on inaccurate determinations of the specific heat. He shows that three sources of error more especially may give too great a specific heat for a substance, or for one of its various modifications :— 1. When the substance is heated to a temperature at which it begins to soften, and thus to absorb part of its latent heat of fusion. 2. If the substance is heated to a temperature at which it begins to pass into another modification, and this change, with its accompanying deve- lopment of heat, is continued in the calorimeter. 3. If the substance investigated is porous, and (as was the case in the earlier methods) is directly immersed in the liquid of the calorimeter, in which case the development of heat which accompanies the moistening of porous substances comes into play. The author arrives at the following result :—From what is at present known with certainty, one and the same body may exhibit small differ- ences with certain physical conditions (temperature, or different degrees of density or porosity); but these differences are never so great as to furnish an explanation of cases in which a body markedly deviates from a regu- larity which might perhaps have been expected for it—always assuming that the determination of the specific heat, according to which the body in question forms an exception to the regularity, is trustworthy and free from foreign elements. The author then discusses the applicability of Dalong and Petit’s law. 1864.] Dr. Kopp on the Specific Heat of Solid Bodies. 23) The atomic heats of many elements * are, in accordance with this law, approximately equal; they vary between 6 and 6°8, the average being about 64. The explanations attempted why this law only approximately holds good, he considers inadequate. In any case there are individual elements which do not obey this law. The atomic heat of phosphorus, for instance, as deduced from direct determinations of its specific heat in the solid state, is considerably smaller (about 5:4) ; and still more so are those of silicium (about 4), of boron (about 2°7), and of carbon (1°8 for dia- mond). A regularity, to which attention has been already drawn, is, that the quotient obtained by dividing the atomic heat of a compound by the number of elementary atoms in one molecule, is approximately equal to 6:4; equal, that is, to the atomic heat of an element according to Dulong and Petit’s law. Thus the atomic heat of the chlorides R Cl and RCl has been found to be 12°8 on the average, and of the chlorides R Cl,=18°5. Now ae and a =6'2. The same regularity is met with in metallic bromides, iodides, and arsenides; and, according to the author’s determinations, it is even found in the case of compounds which contain as many as seven, and even of nine elementary atoms. The atomic heat of Zn K, Cl, ig 43°4, and that of Pt K, Cl, is 55-2; now “246-2 and a6. But the author shows at the same time that this regularity is far from being general. For the oxides of the metals the quotient is less than six, and is smaller the greater the number of atoms of oxygen in the oxide. (From the average determinations of the atomic heats, it is for the metallic oxides RO, =-—-=5°6; for the oxides R, 0, and RB, 0, 2” 5 ==5°4; for the oxides RO,= i> oh) The quotient is still smaller for compounds which contain borow as well as oxygen (for instance, it is 16'S 4-2 for the borates, RB 0,5 it is “{"=3°3 for boracie acid, B, ©,), or which contain silicium (for silicic acid, Si®,, it is MBE say. or hydro- v gen (for ice, H,9, it is or = 2:9), or, finally, which contain carbon and hy- aa : r «we 2029 drogen as well as oxygen (for succinic acid, €, H, O,, for instance, it 1s are =2°6). It may be stated in a few words, in what cases this quotient approximates to the atomic heat of most of the elements, and in what * In accordance with recent assumptions for the atomic weights, H=1; Cl=35°5; 6=16; S=32; B=109; C=12; Si=28. R stands for a monequivalent atom, e. g. As=75; Na=23; K=39'1; Ag=100; RB signifies a polyequivalent atom, e. g.€a=40 ; Pb=207; Fe=56; Co=—63°4; Cr=52°2; Pt=184, &c. 232 Dr. Kopp on the Specific Heat of Sold Bodies. [May 12, cases it is less. It is near 6’4 in the case of those compounds which only contain elements whose atomic heats, in accordance with Dulong and Petit’s law, are themselves approximately =6°4. It is less in those com- pounds containing elements which, as exceptions to Dulong and Petit’s law, have a considerably smaller atomic heat than 6-4, and which are found to be exceptions, either directly, by determinations of their specific heat in the solid state, or indirectly, by the method to be subsequently described. After Dulong and Petit had propounded their law, Neumann showed that a similar regularity existed in the case of compounds, that is, that the atomic heats of analogous compounds are approximately equal. Regnault, as is known, has confirmed Dulong and Petit’s, as well as Neumann’s law, to a considerably greater extent, and for a larger number of compounds, than had been previously done. And Regnault’s researches have more especially shown that the elementary atoms, now regarded as monequivalent, are, as regards the atomic heat of their compounds, comparable with the elementary atoms which are to be considered as polyequivalent. Thus, as re- gards atomic heat, arsenious acid, As, O,, and sesquioxide of iron, Fe, Q,, or chloride of silver and subchloride of copper, Cu Cl, may be classed together. Of the applicability of Neumann’s law, as hitherto investigated and found in the case of chemically analogous compounds, the author’s experimental determinations have furnished a number of new examples. But more interest is presented by his results in reference to the applicability of this law to compounds to which it had not hitherto been supposed to apply. In comparing compounds as regards their atomic heat, their chemical character has been taken into account, as represented by the formule hitherto adopted. Sulphates and chromates, for instance, were looked upon as comparable, but they would not have been classed with perchlorates, or with permanganates. According to more recent assumptions for the atomic weights of the elements, the following salts have analogous for- mulze, and the adjoined atomic heats have been determined :— C@hromate ot leade. 5 eee ae Ph€rO, 29-0 Solpuate of emt’). 2 Pbs 0, 25°8 Permanganate of potass ........ KMnO, = 28°3 Perchlorate of potass .......... K ClO, 26°3 The atomic heats of carbonates, R € O,, of silicates, R Si O,, of metaphos- phates, RP O,, of nitrates, RN O,, are also very near. But not even a common chemical behaviour, such as the bodies in this group possess—that is, a common haloid character—is necessary in order that compounds of analogous atomic composition shall show the same atomic heat. No one would think of considering magnetic oxide of iron as analo- gous to chromate of potass; and yet both have the same atomic structure, and determinations of their specific heat have given approximately the same atomic heat for both. Mapneti¢ oxide of ifon *..:..:".'. 2° Be, ©, 37°7 Chromate of potass.............+. K,€rO 36:4 1864. ] Dr. Kopp on the Specific Heat of Solid Bodies. 239 | And it is not less surprising that arseniate of potass, K As Q,, and chlorate of potass have the same atomic heat as sesquioxide of iron, Fe, Q,, or ar- senious acid, As, G,: with very different characters these compounds have approximately equal atomic heat. But comparability of chemical compounds, as regards the atomic heat, is not limited to the cases in which, as far as can be judged, the individual atoms have analogous construction. We do not regard the atom of binoxide of tin or of titanic acid as analogous in construction to the atom of tung- state of lime or of chromate of lead; nor to nitrate of baryta, or metaphos- phate of lime. But if the formule of those binoxides are doubled or tripled, they may be compared with these salts, and their atomic heats are then approximately equal, as is,the case for compounds of analogous chemical character. The atomic heats are for— Binoxidé of tig. vse. a2.. Ze00 = SiO, 27°6 Witamie a€id's sis. ei eas eee =) Ey S, 27°3 Paneciare of lime ieee ay ah os CaW O, 27°9 Clromate of leadia icici de ce Pb Cr OQ, 29°0 Permanganate of potass .......... K Mn O, 28°3 Perchlorate of potass v1.0 sci eee es K ClO, 26°3 Binoxide of ti... 05. 62's 38n90,= 8n, 0, 41°4 Diane 401d vi vice cds es a PLO. =) 210, 41:0 Nitrate of barytas.c. 2... away. Ba N, 0, 38°9 Metaphosphate of lime .......... CaP,O, 39°4 These results seem to give to Neumann’s law a validity far beyond the limits to which it had hitherto been considered to apply. But, on the other hand, the author’s comparisons go to show that neither Neumann’s nor Dulong and Petit’s law is universally valid. Neumann’s law is only approximate, as is well known. For such analo- gous compounds as, from what we know at present, are quite comparable and,in accordance with this law, ought to have equal atomic heats, Regnault found the atomic heats differing from each other by 745 to 4. In a few such cases there are even greater differences in the atomic heats, for which an adequate explanation is still wanting. But there are other differences in the atomic heats of some compounds which might have been expected to have equality of atomic heat in accord- ance with Neumann’s law—differences which occur with regularity, and for which an explanation is possible. Certain elements impress upon all their compounds the common character that their atomic heats are smaller than those of analogous compounds of other elements. This is the case, for instance, with the compounds of boron: the atomic heat of boracic acid is much less than that of the metallic oxides R, O, and R, O,; the atomic heat of the borates R B Q, is much less than that of the oxides R, O,=(2 B®) ; and the atomic heat of borate of lead, Pb B, 9,, is far less than that of mag- netic oxide of iron, Fe, O,. The same is the case with compounds of carbon, if the alkaline carbonates, R, © Q,, are compared with the metallic oxides R, 0,=(3 BQ), or the carbonates B € O, with the metallic oxides R, O, and 234 Dr. Kopp on the Specific Heat of Solid Bodies. [May 12, BR, O,. It is seen that the compounds of those elements which, in the free state, have themselves a smaller atomic heat than most other elements, are characterized by a smaller atomic heat. This leads the author to discuss whether it is to be assumed that the elements enter into compounds with the atomic heats which they have in the free state. This assumption is only admissible provided it can be proved that the atomic heat of a compound depends simply on its empirical formula, and not on the chemical character or rational constitution. Much — of what has previously been said favours this view of the case. It is also supported by the fact, which the author proves, that similar chemical cha- racter in analogous compounds, and even isomorphism, do noé presuppose equality in the atomic heats, if in one compound an atomic group (a com- pound radical) stands in the place of an elementary atom of another: for instance, the atomic heat of cyanogen compounds is considerably greater than those of the corresponding chlorine compounds, and those of ammo- nium materially greater than those of the corresponding potassium com- pounds. A further support for that assumption is found in the fact that, regardless of the chemical character, the atomic heat of complex com- pounds is found to be the sum of the atomic heats of simpler atomic groups, the addition of which gives the formulze of those more complex compounds. A few cases selected from the comparisons of the author may explain this. The atomic heats have been found,— : For the oxides..... eke i cast RO Liga i For binoxide of tin.......... Sn 0, 13°8 j Motalwforson2 cae «ee RRO, 24-9 . For sesquioxide of iron ...... Fe, O, 26°8 Or, For oxides 2B 0.2 .00600.% =o 222 For binoxide of tn 3 Sn 0,..= RB, 0, 41:4 etal fora 3 Pee R, 0, 63°6 For arseniate of lead,......... Pb, As, O, 65:4 Finally, the author shows, as supporting that assumption, that (as was already maintained) water is contained in solid compounds with the atomic heat of ice. The various determinations of the specific heat of ice give the atomic heat of H, © at 8°6 for temperatures distant from 0°, and at 9:1 to 9-8 at temperatures nearer 0°. The atomic heat has been found (to adduce again a few comparisons) be For crystallized chloride of calcium.... €aCl,+6H,O 75°6 For anhydrous chlorides ............ R Cl, 18°5 Difference for .......0..2 6H,O 571 _ 9.5 For crystallized gypsum ............ CaS 0,+ 2H, 0 45°8 For anhydrous sulphates ............ RSO 264 Difference for’... . 82 H;O 19°7_9-9 2 , . 1864.] Dr. Kopp on the Specific Heat of Solid Bodies. 285 The opinion that the elements enter into compounds with the atomic heats they have in the free state has been already expressed ; but the view has also been defended that the atomic heat of an element may differ in a com- pound from what it is in the free state, and may be different in different compounds. The author discusses the latter view, and criticises the reasons _ which may be adduced for it; he comes to the result that it is not proved and is inadmissible. As the result of all these comparisons and observations, the author arrives at the conclusion, Each element, in the solid state and at an adequate dis- tance from its melting-point, has one specific or atomic heat, which may indeed somewhat vary with physical conditions (different temperature, or different density for example), but not so much as to necessitate such variations being taken into account in considering the relation in which the specific or the atomic heat stands to the atomic weight or composition. For each element it is to be assumed that it has essentially the same specific heat or atomic heat in the free state and in compounds. He then passes on to determine what atomic heats are to be assigned to the indivi- dual elements. As data for determining this he takes (1) the atomic heats which follow from determinations of the specific heat of the elements in the free, solid state; (2) the atomic heats obtained for an element if, from the atomic heat of one of its compounds, which contains beside it only elements of known atomic heat, the atomic heats corresponding to the latter elements are subtracted ; (3) the difference found between the atomic heats of analogous compounds of an element of unknown and of an ele- ment of known atomic heat, in which case the difference is taken as being the difference between the atomic heats of these two elements. The au- thor dwells upon the fact that in the indirect deduction of an element by (2) and (3) the result may be uncertain,—first, because the atomic heats of compounds are frequently not known with certainty, as is seen by the circumstance that analogous compounds, for which there is every reason to expect equal atomic heat, are found experimentally to exhibit considerable differences ; but secondly, because in such deductions the entire relative uncertainty, in the atomic heats for a compound and for that to be sub- tracted from its composition, is thrown upon a small number, viz. the residue remaining in the deduction. The details of the considerations by which the author deduces the atomic heat of the individual elements cannot be gone into; the results simply, which are not all attained with equal certainty, may be adduced. The author adopts the atomic heat 1°8 for ©, 2°3 for H, 2°7 for B, 3:7 for Si, 4 for O, 5 for Fl, 5:4 for P and 8S, 6°4 for the other elements for which or for whose compounds the atomic heat is known in somewhat more trustworthy manner, it being left undecided in the case of the latter ele- ments, whether (in accordance with Dulong and Petit’s law) they have the same atomic heats, or whether the differences in the atomic heats cannot at present be shown with certainty. 236. Dr. Kopp on the Specific Heat of Solid Bodies. [May 12, The author gives for all compounds, whose specific heat has been inves- tigated in a trustworthy manner, a comparison of the specific heats found experimentally with those calculated on the above assumption. The atomic heat of a compound is obtained by adding the atomic heats of the elements in it, and the specific heat by dividing this atomic heat by the atomic weight. The calculated specific heat of chloride of potassium, KCL, is 0172 ; of sulphide of lead, Pb 8, op = 00494 oo borate of potass, K B 9,, it is C,H, 0, it is (4x —— 2°3)+6x4 The Table, embracing 200 compounds, shows, on the whole, a sufficient agreement between the calculated and the observed specific heats. The author remarks that a closer agreement between calculation and observation cannot be hoped for than that between the observed atomic heats of those compounds for which, from all we know at present, the same atomic heat is to be expected in conformity with Neumann’s law, to which im such eases, of course, caleulation corresponds. In only a few cases are differences be- tween calculation and observation met with which exceed these limits or exceed the deviation between the results of different observers for the same substance. The author states that be is far from considering the agree- ment between his calculations and the experimental results as a measure of the accuracy of the latter, since the bases of calculation ate too far from being trustworthy. But he hopes that his Table of atomic heats will soon acquire such corrections, and therewith greater trustworthiness, as was the case with the first Table of atomic weights. Here, the data for the Table were at first but little certain, and the differences between the calculated and observed composition of chemical compounds very con- siderable ; but the Table was the means of corrections being introduced by which these differences were diminished. If calculation ofthe specific heat does not supersede the necessity of experimental determination in the solid state, and does not give a trust- worthy measure for the accuracy of such determinations, it gives a rough control for the experimental determinations, and it indicates sources of error in the experiments which without it would not have been noticed. An instance may be adduced. The author found for sesquichloride of carbon, ©, Cl,, which, according to Faraday, melts at 160°, the specific heat between 20° and 50° to be 0°276 in one series of experiments, and 0°265 in another. Hence the number 0°27 might from this be taken to express the specific heat of the compound. But calculation gives (2x 1°8)+(6 x oe 237 experiments with substance once more recrystallized, gave for the specific heat between 21° and 49° 0°278, confirming the previous determinations. 6:442:74(2x4)_ ee ad 0°209; of tartaric acid, =0°300. 77, a very different number. A third series of a 1864] Dr. Kopp on the Specific Heat of Solid Bodies. 237 It might here appear doubtful whether calculation was not refuted by experiment. The discrepancy was removed by the observation that the substance is distinctly more viscous at 50° than it is at lower temperatures, and by the suspicion that it might at 50° (that is, 100° below its melting- point) already absorb some of its latent heat of vitreous fusion. This was found to be the case; two concordant series of experiments gave as the mean of the specific heat the numbers : Between 18° and 37° Between 18° and 43° Between 18° and 50° ...... 0°277 The first two numbers differ so little that it may be supposed the number found for temperatures below 37° is very near the true specific heat of this compound ; it also agrees well with the calculated number. In the sixth part the author enters into considerations on the nature of the chemical elements. He calls to mind the discrepancy which has prevailed, and still prevails, in reference to certain bodies, between their actual indecomposability, and the considerations, based on analogy, according to which they were held to be compound. kEven after Davy had long proclaimed the elementary nature of chlorine, it was maintained that it contained oxygen. In regard | both to that substance and to bromine and iodine, the view that they are peroxides of unknown elements still finds defenders. That iodine, by a direct determination of specific heat, and chlorine, by indirect deduction, are found to have an atomic heat in accordance with Dulong and Petit’s law, puts out of doubt that iodine and chlorine, if compound at all, are not more so than the other elements to which this law is considered to apply. According to Dulong and Petit’s law, compounds of analogous atomic composition have approximately equal atomic heats. In general, com- pounds whose atom consists of a larger number of undecomposable atoms, or is of more complex constitution, have greater atomic heat. Especially in those compounds all of whose elements follow Dulong and Petit’s law, is the magnitude of the atomic heat a measure of the complication, or of the degree of complication. If Dulong and Petit’s law were universally valid, it might be concluded with great certainty that the so-called ele- ments, if they are really compounds of unknown simpler substances, are compounds of the same order. It would be a remarkable result, if the art of chemical decomposition had everywhere reached its limits at such bodies as, if at all compound, have the same degree of composition. Let us imagine the simplest bodies, perhaps as yet unknown to us; the true chemical elements, to form a horizontal layer, and above them to be arranged the more simple and then the more complicated compounds; the general_validity of Dulong and Petit’s law would include the proof that all the elements at present assumed to be such by chemists lay in the same layer, and that, in admitting hydrogen, oxygen, sulphur, chlorine, 208 Dr. Kopp on the Specific Heat of Solid Bodies. [May 12, and the various metals as elements, chemistry has penetrated to the same depth in that range of inquiry, and has found at the same depth the limit to its advance. But with the proof that this law is not universally true, the conclusion to which this result leads loses its authority. If we start from the ele- ments at present assumed in chemistry, we must admit rather that the magnitude of the atomic heat of a body does not depend on the number of elementary atoms contained in a molecule, or on the complication of its composition, but on the atomic heat of the elementary atoms which enter into its composition. It is possible that a decomposable body may have the same atomic heat as an element. Chlorine might certainly be the peroxide of an unknown element which had the atomic heat of hydrogen ; the atomic heat of peroxide of hydrogen, H ©, in the solid state or in solid compounds, must be =2°3+4=6°3, agreeing very nearly with the atomic heats of iodine, chlorine, and the elements which follow Dulong and Petit’s law. In a very great number of compounds the atomic heat gives more or less accurately a measure for the complication of the composition. And this is also the case with those compounds which, from their chemical deportment, are comparable to the undecomposed bodies. If ammonium or cyanogen had not been decomposed, or could not be by the chemical means at present available, the greater atomic heats of the compounds of these bodies, as compared with analogous potassium or chlorine com- pounds, and the greater atomic heats of ammonium and cyanogen ob- tained by indirect determination, as compared with those of potassium and chlorine, would indicate the compound nature of those so-called compound radicals. The conclusion appears legitimate, that, for the so-called ele- ments, the directly or indirectly determined atomic heats are a measure for the complication of their composition. Carbon and hydrogen, for exam- ple, if not themselves actually simple bodies, are yet simpler compounds of unknown elements than silicium or oxygen; and still more complex are the elements which may be considered as following Dulong and Petit’s law. It may appear surprising, and even improbable, that so-called elements, which can replace each other in compounds, as for instance hydrogen and the metals, or which enter into isomorphous compounds as corresponding elements, like silicium and tin, should possess unequal atomic heats and unequal complication of composition. But this really is not more sur- prising than that undecomposable bodies and obviously compound bodies, hydrogen and hyponitric acid, or potassium and ammonium, should, with- out altering the chemical character of the compound, replace one another, or even be present in isomorphous compounds as corresponding con- stituents. The author concludes his memoir with the following words :—‘“ I have here expressed opinions, in reference to the nature of the so-called ele- ments, which appear to depend upon allowable conclusions from well- 1864.] Messrs. Parker and Jones on Foraminifera. 239 demonstrated principles. It is of the nature of the case,{that with these opinions the certain basis of the actual, and of what can be empirically proved, is left. It must also not be forgotten that these conclusions only give some sort of clue as to which of the present undecomposable bodies are of more complicated, and which of simpler composition, and nothing as to what the simpler substances are which are contained in the more complicated. Consideration of the atomic heats may declare something as to the structure of a compound atom, but can give no information as to the qualitative nature of the simpler substances used in the construction of the compound atoms. But even if these conclusions are not free from uncertainty and imperfection, they appear to me worthy of attention in a subject which is still so shroudedin darkness as the nature of the unde- composed bodies.” III. “On some Foraminifera from the North Atlantic and Arctic Oceans, including Davis Strait and Baffin Bay.” By W. Kircaen Parker, F.Z.8., and Professor T. Rupert Jonzs, F.G.S. Communicated by Professor Huxtuy. Received April 26, 1864. (Abstract. ) Having received specimens of sea-bottom, by favour of friends, from Baffin Bay (soundings taken in one of Sir E. Parry’s expeditions), from the Hunde Islands in Davis Strait (dredgings by Dr. P. C. Sutherland), from the coast of Norway (dredgings by Messrs. M‘Andrew and Barrett), and. from the whole width of the North Atlantic (soundings by Commander Dayman), the authors have been enabled to form a tolerably correct esti- mate of the range and respective abundance of several species of Foramini- fera in the Northern seas; and the more perfectly by taking Professor Williamson’s and Mr. H. B. Brady’s researches in British Foraminifera as supplying the means of estimating the Foraminiferal fauna of the shallower sea-zones at the eastern end of the great “‘ Celtic Province,” and the less perfect researches of Professor Bailey on the North American coast, for the opposite, or “ Virginian” end,—thus presenting for the first time the whole of a Foraminiferal fauna as a natural-history group, with its internal and external relationships. The relative abundance or scarcity and the locations of the several species and chief varieties are shown by Tables; and their distribution in other seas (South Atlantic, Pacific, and Indian Oceans, and the Mediterranean and Red Seas) is also tabulated; and in the descriptive part of the memoir notes on their distribution, both in the recent and the fossil state, are care- fully given. In the description of the species and varieties there are observations made on those forms which have been either little understood, hitherto 240 Dr. Phipson on the Variations of Density [May 26, unknown, or mistaken; and the relationship, by structure or by imitation, of the species and varieties is dwelt upon. For the description of the better-known Foraminifera, the memoir refers to the works of Williamson and Carpenter. The authors enumerate 109 specific and varietal forms, most of which receive descriptive comment, and all of which are figured in five plates (two for the North Atlantic and three for the Arctic Foraminifera) with upwards of 340 figures. The relationships of the Lagene are specially treated of. Uvigerina, Globigerina, and especially some of the Rotaline (Planorbulina, Discor- bina, Rotalia, Pulvinulina) and Polystomella (including Nonionina) are among those which are well represented in the fauna under description, and have received much attention in the memoir. The Society then adjourned over the Whitsuntide Recess to Thursday, May 26. May 26, 1864: Major-General SABINE, President, in the Chair. The following communications were read :— I. “ Note on the Variations of Density produced by Heat in Mineral Substances.” By Dr. T. L. Puirson, F.C.S., &c. . Communi- cated by Professor Tynpatu. Received April 16, 1864. That any mineral substance, whether crystallized or not, should diminish in density by the action of heat might be looked upon as a natural con- sequence of dilatation being produced in every case and becoming per- manent. Such diminution of density occurs with idocrase, Labradorite, felspar, quartz, amphibole, pyroxene, peridote, Samarskite, porcelain, and glass. But Gadolinite, zircons, and yellow obsidians augment in density from the same cause. This again may be explained by assuming that under the influence of a powerful heat these substances undergo some per- manent molecular change. But in this NoteI have to show that this mole- cular change is not permanent but intermittent, at least as regards the species I have examined, and probably with all the others. Such researches, while tending to elucidate certain points of chemical geology, may likewise add something to our present knowledge of the modes of action of heat. My experiments were undertaken to prove an interesting fact announced formerly by Magnus, namely, that specimens of idocrase after fusion had diminished considerably in density without undergoing any change of com- position: before fusion their specific gravity ranged from 3°349 to 3°45, and after fusion only 2°93 to 2°945. Having lately received specimens of this and other minerals brought from Vesuvius in January last by my friend Henry Rutter, Esq., I determined upon repeating this experiment of 1864. ] produced by Heat in Mineral Substances. 241 Magnus. I found, first, that what he stated for idocrase and for a speci- men of reddish-brown garnet was also the case with the whole family of garnets as well as the minerals of the idocrase group ; secondly, that it is not necessary to melt the minerals: it is sufficient that they should be heated to redness without fusion, in order to occasion this change of density ; thirdly, that the diminished density thus produced by the action of a red heat is not a permanent state, but that the specimens, in the course of a month or less, resume their original specific gravities. These curious results were first obtained by me with a species of lime garnet, in small yellowish crystals, exceedingly brilliant and resinous, almost granular, fusing with difficulty to a black enamel, accompanied with very little leucite and traces of grossular, and crystallized in the second system. Specimens weighing some grammes had their specific gravity taken with great care, and by the method described by me in the ‘ Chemical News’ for 1862. ‘They were then perfectly dried and exposed for about a quarter of an hour to a bright red heat. When the whole substance of the specimen was observed to have attained this temperature, without trace of fusion, it was allowed to cool, and when it had arrived at the temperature of the atmosphere, its specific gravity was again taken by the same method as be- fore. The diminution of density being noted, the specimens were carefully dried, enveloped in several folds of filtering paper, and put aside in a box along with other minerals. In the course of a month it occurred to me that it would be interesting to take the specific gravity again, in order to ascer‘ain whether it had not returned to its original figure, when, to my surprise, I found that each specimen had effectively increased in density and had attained its former specific gravity. Thus:— Lime garnet (from Vesuvius). Density after being heated Density determined in red-hot for a quarter of an a month after the Original density. hour and allowed to cool. experiments. Meer tea RLS ee DPOT BR. PEW AG Se ale 3°344 eee Oe Ee ek ye BOBO Ole A aloe a 3°350 rere rr he S22 Pe 2977 9%, ONC a) eee iasoaa The same experiments were made with several other minerals belonging to the idocrase and garnet family, and always with similar results. Now I ask, what becomes of the heat that seems to be thus shut up in a mineral substance for the space of a month? The substance of the mineral is di- lated, the distance between its molecules is enlarged, but these molecules slowly approach each other again, and in the course of some weeks resume their original positions. What induces the change? or how does it happen that the original specific gravity is not acquired immediately the substance has cooled ?* ‘Will the same phenomenon show itself with other families of minerals or with the metallic elements ? * Some minerais , like euclase, that become electric by heat, retain that state for a a 242 Messrs. Huggins and Miller on Spectra [May 26, Such are the points which I propose to examine in the next place ; in the mean time the observations I have just alluded to are a proof that bodies can absorb a certain amount of heat not indicated by the thermo- meter (which becomes Jatent), and that this is effected without the body undergoing a change of state; secondly, that they slowly part with this heat again until they have acquired their original densities; thirdly, so many different substances being affected by a change of density when melted or simply heated to redness and allowed to cool, it is probable this property will be found to belong, more or less, to all substances without exception. IT. “On the Spectra of some of the Fixed Stars.” By W. Hue. cins, F.R.A.S., and Witi1am A. Mitier, M.D., LL.D., Trea- surer & V.P.R.S., Professor of Chemistry, King’s College, London. Received April 28, 1864, (Abstract. ) After a few introductory remarks, the authors describe the apparatus which they employ, and their general method of observing the spectra of the fixed stars and planets. ‘The spectroscope contrived for these inqui- ries was attached to the eye end of a refracting telescope of 10 feet focal length, with an 8-inch achromatic object-glass, the whole mounted equa- torially and carried by a clock-movement. In the construction of the spectroscope, a plano-convex cylindrical lens, of 14 inches focal length, was employed to convert the image of the star into a narrow line of light, which was made to fall upon a very fine slit, behind which was placed an achromatic collimating lens. The dispersing portion of the arrangement consisted of two dense flint-glass prisms; and the spectrum was viewed through a small achromatic telescope with a magnifying power of between 5 and 6 diameters. Angular measures of the different parts of the spec- trum were obtained by means of a micrometric screw, by which the posi- tion of the small telescope was regulated. A reflecting prism was placed over one half of the slit of the spectroscope, and by means of a mirror, suitably adjusted, the spectra of comparison were viewed simultaneously with the stellar spectra. This light was usually obtained from the in- duction spark taken between electrodes of different metals. The dispersive power of the apparatus was sufficient to enable the observer to see the line Ni of Kirchhoff between the two solar lines D; and the three constituents of the magnesium group at 0 are divided still more evidently *. Minute considerable time. The increase of density of Gadolinite and the decrease of density of Samarskite by the action of heat are accompanied by a vivid emission of light, as mentioned in my work on ‘ Phosphorescence’ &c., pp. 31 and 32, where H. Rose’s ingenious expe- riment is described. * Each unit of the scale adopted was about equal to -3,;th of the distance between A and H in the solar spectrum. The measures on different occasions of the same line rarely differed by one of these units, and were often identical, 1864. ] some of the Fixed Stars. 243. details of the methods adopted for testing the exact coincidence of the corresponding metallic lines with those of the solar and lunar spectrum, are given, and the authors then proceed to give the results of their obser- vations. Careful examination of the spectrum of the light obtained from various points of the moon’s surface failed to show any lines resembling those due to the earth’s atmosphere. The planets Venus, Mars, Jupiter, and Saturn were also examined for atmospheric lines, but none such could be disco- vered, though the characteristic aspect of the solar spectrum was recognized in each case; and several of the principal lines were measured, and found to be exactly coincident with the solar lines. Between forty and fifty of the fixed stars have been more or less com- pletely examined; and tables of the measures of about 90 lines in Alde- baran, nearly 80 in @ Orionis, and 15 in 6 Pegasi are given, with dia- grams of the lines in the two stars first named. These diagrams include the results of the comparison of the spectra of various terrestrial ele- ments with those of the star. In the spectrum of Aldebaran coincidence with nine of the elementary bodies were observed, viz. sodium, magnesium, hydrogen, calcium, iron, bismuth, tellurium, antimony, and mercury; in seven other cases no coincidence was found to occur. In the spectrum of a Orionis five cases of coincidence were found, viz. sodium, magnesium, calcium, iron, and bismuth, whilst in the case of ten other metals {no coincidence with the lines of this stellar spectrum was found. ( Pegasi furnished a spectrum closely resembling;that of « Orionis in appearance, but much weaker: only a few of the lines admitted of accu- rate measurement, for want of light; but the coincidence of sodium and magnesium was ascertained; that of barium, iron, and manganese was doubtful. Four other elements were found not to be coincident. In par- ticular, it was noticed that the lines C and F, corresponding to hydrogen, which are present in nearly all the stars, are wanting in @ Orionis and ( Pegasi. The investigation of the stars which follow is less complete, and no details of measurement are given, though several points of much interest have been ascertained. Sirius gave a spectrum containing five strong lines, and numerous finer lines. The occurrence of sodium, magnesium, hydrogen, and probably of iron, was shown by coincidence of certain lines in the spectra of these metals with those in the star. In # Lyre the occurrence of sodium, mag- nesium, and hydrogen was also shown by the same means. In Capella sodium was shown, and about twenty of the lines in the star were mea- sured. In Arcturus the authors have measured about thirty lines, and have observed the coincidence of the sodium line with a double line in the star- spectrum. In Pollux they obtained evidence of the presence of sodium, VOL. XIII. U 244 Mr. Cayley on Skew Surfaces. [May 26, magnesium, and probably of iron. The presence of sodium was also indi- cated in Procyon and a Cygnt. In no single instance have the authors ever ohsortid a star-spectrum in which lines were not discernible, if the light were sufficiently intense, and the atmosphere favourable. Rigel, for instance, which some authors state to be free from lines, is filled with a multitude of fine lines. Photographs of the spectra of Sirius and Capella were taken upon collo- dion; but though tolerably sharp, the apparatus employed was not suffi- ciently perfect to afford any indication of lines in the photograph. In the concluding portion of their paper, the authors apply the facts observed to an explanation of the colours of the stars. They consider that the difference of colour is to be sought in the difference of the constitution of the investing stellar atmospheres, which act by absorbing particular portions of the light emitted by the incandescent solid or liquid photo- sphere, the light of which in each case they suppose to be the same in quality originally, as it seems to be independent of the chemical nature of its constituents, so far as observation of the various solid and liquid ele- mentary bodies, when rendered incandescent by terrestrial means, appears to indicate. III. “A Second Memoir on Skew Surfaces, otherwise Serolls.” By A, Cayiey, Esq., F.R.S. Received April 29, 1864. (Abstract. ) _ The principal object of the present memoir is to establish the different kinds of skew surfaces of the fourth order, or Quartic Scrolls ; but, as preli- minary thereto, there are some general researches connected with those in my former memoir “On Skew Surfaces, otherwise Scrolls’? (Phil. Trans. vol, 153. 1863, pp. 453, 483), and I also reproduce the theory (which may be considered as a known one) of cubic scrolls; there are also some con- cluding remarks which relate to the general theory. As regards quartic scrolls, I remark that M. Chasles, in a footnote to his paper, ‘‘ Description des Courbes de tous les ordres situées sur les surfaces réglées du troisiéme et du quatricme ordres,’? Comptes Rendus, t. lili. (1861), see p. 888, states, “les surfaces réglées du quatriéme ordre. . . . admettent guatorze espéces.’”’ This does not agree with my results, since I find only eight spe- cies of quartic scrolls; the developable surface or “‘torse”’ is perhaps in- cluded as a “‘surface réglée;’’ but as there is only one species of quartic torse, the deficiency is not to be thus accounted for. My enumeration ap- pears to me complete, but it is possible that there are subforms which M. Chasles has reckoned as distinct species. 1864,] — Prof. Boole on Differential Equations. 245 IV. “On the Differential Equations which determine the form of the Roots of Algebraic Equations.” By Gzorez Bootg, F.R.S., Professor of Mathematics in Queen’s College, Cork. Received April 27, 1864. (Abstract.) Mr, Harley* has recently shown that any root of the equation y® —ay+(n—1)x=0 satisfies the differential equation sssd =9 aed (p 2n *) (D 3n ) " (D n 1) n n n (n=) D(D—1) .. (D—2+1) =O) ..f EP) in which e?=2z, and D=-,, provided that x be a positive integer greater d de? than 2. This result, demonstrated for particular values of m, and raised by induction into a general theorem, was subsequently established rigorously by Mr. Cayley by means of Lagrange’s theorem. For the case of »=2, the differential equation was found by Mr. Harley to be 1 me ee es Solving these differential equations for the particular cases of n=2 and n=3, Mr. Harley arrived at the actual expressions of the roots of the given algebraic equation for these cases. That all algebraic equations up to the fifth degree can be reduced to the above trinomial form, is well known. _ A solution of (1) by means of definite triple integrals in the case of n=4 has been published by Mr. W. H. L. Russell; and I am informed that a general solution of the equation by means of a definite single integral has been obtained by the same analyst. While the subject seems to be more important with relation to diffe- rential than with reference to algebraic equations, the connexion into which the two subjects are brought must itself be considered as a very interesting fact. As respects the former of these subjects, it may be observed that it is a matter of quite fundamental importance to ascertain for what forms of the function » (D), equations of the type Ute AV eP ey hi) ng! Bas ent B) admit of finite solution. We possess theorems which enable us to deduce from each known integrable form, an infinite number of others. Yet there is every reason to think that the number of really primary forms—of forms the knowledge of which, in combination with such known theorems, would enable us to solve all equations of the above type that are finitely solvable— os * Memoirs of the Literary and Philosophical Society of Manchester. u 2 246 Prof. Boole on Differential Equations. [May 26, is extremely small. It will, indeed, be a most remarkable conclusion, should it ultimately prove that the primary solvable forms in question stand in some absolute connexion with a certain class of algebraic equations. The following paper is a contribution to the general theory under the aspect last mentioned. In endeavouring to solve Mr. Harley’s equation by definite integrals, I was led to perceive its relation to a more general equa- tion, and to make this the subject of investigation. The results will be presented in the following order :— First, I shall show that if u stand for the mth power of any root of the algebraic equation y” — ay" + — 1 —0, then uw, considered as a function of x, will satisfy the differential equation [D]"u-+ “= so Ser ery —(), in which e=27, D= = and the notation [a]®=a (a—1) (a—2).. (a—6+1) is adopted. Secondly, I shall show that for particular values of m, the above equa- tion admits of an immediate first integral, constituting a differential equation of the n—1th order, and that the results obtained by Mr. Harley are par- ticular cases of this depressed equation, their difference of form arising from difference of determination of the arbitrary constant. Thirdly, I shall solve the general differential equation by definite in- tegrals. Fourthly, I shall determine the arbitrary constants of the solution so as to express the mth power of that real root of the proposed algebraic equa- tion which reduces to 1 when 7=0. The differential equation which forms the chief subject of these investi- gations certainly occupies an important place, if not one of exclusive im- portance, in the theory of that large class of differential equations of which the type is expressed in (3). At present, I am not aware of the existence of any differential equations of that particular type which admit of finite solution at all otherwise than by an ultimate reduction to the form in question, or by a resolution into linear equations of the first order. It constitutes, in fact, a generalization of the form a(D—2)’?+n? 2 DD Aycte e given in my memoir “On a General Method i in Analysis’ (Philosophical Transactions for 1844, part 2). w+ 1864.] General Sabine on Magnetic Disturbances. 247 Y. “A Comparison of the most notable Disturbances of the Magnetic Declination in 1858 and 1859 at Kew and Nertschinsk, preceded by a brief Retrospective View of the Progress of the Investiga- tion into the Laws and Causes of the Magnetic Disturbances.” By Major-General Epwarp Sasine, R.A., President of the Royal Society. Received April 28, 1864. (Abstract.) The author commences this paper by taking a retrospective view of the principal facts which have been established regarding the magnetic disturb- ances, considered as a distinct branch of the magnetic phenomena of the globe, from the time when they were first made the objects of systematic investigation by associations formed for that express purpose, at Berlin in 1828 and at Gottingen in 1834, and dwelling more particularly on the results subsequently obtained by the more complete and extended researches instituted in 1840 by the British Government on the joint recommendation of the Royal Society and of the British Association for the Advancement of Science. The Berlin Association, formed under the auspices of Baron Alexander von Humboldt, consisted of observers in very distant parts of the European continent, by whom the precise direction of the declination-magnet was recorded simultaneously at hourly intervals of absolute time, at forty-four successive hours at eight concerted periods of the year, which thence ob- tained the name of “ Magnetic Terms.’ By the comparison of these hourly observations it became known that the declination was subject to very con- siderable fluctuations, happening on days which seemed to be casual and irregular, but were the same at all the stations, consequently over the con- tinent of Europe generally. This conclusion was confirmed by the Got- tingen Association, established at the instance and under the superintendence of MM. Gauss and Weber, by whom the ‘‘Term-observations ’’ were extended to six periods in the year, each of twenty-four hours’ duration, the records being made at intervals of five minutes. The number of the stations at which these observations were made was about twenty, distributed generally over the continent of Europe, but not extending beyond it. They were continued from 1834 to 1841. The observations themselves, as well as the conclusions drawn from them by MM. Gauss and Weber, were published in the well-known periodical entitled ‘ Resultate aus der Beobachtungen des magnetischen Vereins.’ The synchronous character of the disturbances, over the whole area comprehended by the Association, was thoroughly con- firmed: the disturbing action was found to be so considerable as to occa- sion frequently a partial, and sometimes even a total obliteration of the regular diurnal movements, and to be of such general prevalence over the greater part of Europe, not only in the larger, but in most of the smaller oscillations, as to make it in avery high degree improbable that they could 248 General Sabine—Comparison of Magnetic [May 26, have either a local or an atmospherical origin. No connexion or correspon- dence was traceable between the indications of the magnetical and meteoro- logical instruments; nor had the state of the weather any perceptible in- fluence. It happened very frequently that either an extremely quiescent state of the needle or a very regular and uniform progress was preserved during the prevalence of the most violent storm ; and as with wind-storms, so with thunder-storms, as even when close at hand they appeared to exer- cise no perceptible influence on the magnet. At some of the most active of the Géttingen stations the fluctuations of the horizontal force were ob- served contemporaneously with those of the declination-magnet, by means ’ of the bifilar magnetometer devised by M. Gauss: both elements were generally disturbed on the same days and at the same hours. The magnitude of the disturbances appeared to diminish as their action was traced from north to south, giving rise to the conclusion that the focus whence the most powerful disturbances in the northern hemisphere ema- nated might perhaps be successfully sought in parts of the globe to the north or north-west of the area comprehended by the stations. The imter- comparison of the records obtained at the different stations showed more- over that the same element was very differently affected at the same hours at different stations; and that occasionally the same disturbance showed itself in different elements at different stations. The general conclusion was therefore thus drawn by M. Gauss, that ‘‘ we are compelled to admit that on the same day and at the same hour various forces are contempo- raneously in action, which are probably quite independent of one another and have very different sources, and that the effects of these various forces are intermixed in very dissimilar proportions at various places of observa- tion relatively to the position and distance of these latter; or these effects may pass one into the other, one beginning to act before the other has ceased. The disentanglement of the complications which thus occur in the phenomena at every individual station will undoubtedly prove very difficult. Nevertheless we may confidently hope that these difficulties will not always remain insuperable, when the simultaneous observations shall be much more widely extended. It will be a triumph of science should we at some future time succeed in arranging the manifold intricacies of the phenomena, in separating the individual forces of which they are the compound result, and in assigning the source and measure of each.” In the British investigations, which commenced in 1840, the field of re-= search was extended so as to include the most widely separated localities in both hemispheres, selected chiefly with reference to diversity of geo- graphical circumstances, or to magnetic relations of prominent interest. Suitable instruments were provided for the observation of each of the three magnetic elements; the scheme of research comprehended not alone the casual and irregular fluctuations which had occupied the chief attention of the German associations, but also “the actual distribution of the magnetic influence over the globe at the present epoch in its mean or average state, 1864, | Disturbances at Kew and Nertschinsk. 249 together with all that is not permanent in the phenomena, whether it appear in the form of momentary, daily, monthly, semiannual, or annual change, or in progressive changes receiving compensation possibly, either in whole or in part, in cycles of unknown relation and unknown period.” The magnetic disturbances to which the notices in the present paper are limited, form a small but important branch of this extensive inquiry, and are referred to in the instructions prepared by the Royal Society in terms which are recalled by the author on the present occasion, because they are ex- planatory of the principles on which the coordination of the results ob- tained in such distant parts of the world has been conducted, and the conclusions derived from them established. In pages 2 and 3 of the Report embodying the instructions drawn up by the Royal Society, it is stated that ‘‘ the investigation of the laws, extent, and mutual relations of the casual and transitory variations is become essential to the success- ful prosecution of magnetic discovery .... because the theory of those transitory changes is in itself one of the most interesting and important points to which the attention of magnetic observers can be turned, as they are no doubt intimately connected with the general causes of terrestrial magnetism, and will probably lead us to a much more perfect knowledge of those causes than we now possess.” In the opinion thus expressed, the author, who was himself one of the committee by whom the Report was drawn up, fully concurred; and having been appointed by Her Majesty’s Government to superintend the observations made at the British Colonial observatories, and to coordinate and publish their results, he has endea- voured to show in this paper that the methods pursued have been in strict conformity with these instructions, and also that the conclusions derived are in accordance with the anticipations expressed therein. Inferences regarding the ‘‘ general causes of terrestrial magnetism ”’ must be based upon the knowledge we possess of the actual distribution of the magnetic influence on the surface of the globe, since that is the only part which is accessible to us. In regard to this distribution, the Report itself refers continually to two works, then recently published, as contain- ing the embodiment of the totality of the known phenomena, viz. a “Memoir on the Variations of the Magnetic Force in different parts of the Earth’s Surface,” published in 1838 in the Reports of the British Associa- tion, and M. Gauss’s ‘ Allgemeine Theorie des Erdmagnetismus,’ published in 1839. In both these works the facts, as far as they had been ascertained, were conformable in their main features to the theory, first announced by Dr. Halley in his Papers in the Philosophical Transactions for 1683 and 1693, of a double system of magnetic action, the direction and intensity of the magnetic force being, at all points of the earth’s surface, the - resultants of the two systems. In both these works the Poles, or Points of greatest force (in the northern. hemisphere) were traced nearly to the same localities—viz. one in the northern parts of the American continent, and the other in the northern parts of the Europso-Asiatic continent,—their 250 General Sabine—Comparison of Magnetic [May 26, geographical positions, as taken from M. Gauss’s ‘ Allgemeine Theorie,’ being, in America, lat. 55°, long. 263° E., and in Siberia lat. 71°, long. 116° E. Combining then the expectation expressed in the Report of “a probable connexion existing between the casual and transitory magnetic variations and the general phenomena of terrestrial magnetism,’ with M. Gauss’s conclusion from the Géttingen researches, that “the sources of the magnetic disturbances in Europe might possibly be successfully sought in parts of the globe to the north or to the north-west of the European continent,” it seemed reasonable to anticipate that a connexion might be found to exist between the “ points of origin” of the disturb- ances, if these could be more precisely ascertained, and the critical locali- ties of the earth’s magnetism above referred to. To put this question to the test, the first step was to ascertain in a more satisfactory way than had been previously attempted, the laws of the disturbances themselves. The process by which a portion of the observations ex- hibiting the effects of the disturbing action in a very marked degree may be separated from the others, and subjected to a suitable analysis for the determination of their general laws, has been fully described else- where. The immediate effect of its application was to show that, casual and irregular as the disturbances might appear to be in the times of their occurrence, they were, in their mean effects, strictly periodical phenomena, characterized by laws distinct from those of any other periodical phenomena with which we were then acquainted, and traceable directly to the Sun as their primary source, Inasmuch as they were found to be governed every- where by laws depending upon the solar hours. To those who are familiar with the theory by which the passage of light from the sun to the earth is explained, an analogous transmission of magnetic influences from the sun to the earth may appear to present no particular difficulty. It is when the influences reach the earth that the modes of their reception, distribu- tion, and transmission are less clearly seen and understood ; but these are within our own proper terrestrial domain and sphere of research; and ac- cordingly it was to these that the author’s attention was directed. Where- ever the disturbances had been observed and were analyzed, it was found that those of the declination were occasionally deflections to the east and occasionally deflections to the west of the mean position of the magnet, and those of the horizontal and vertical forces occasionally increased and occasionally diminished the respective forces. The disturbances of each element were therefore separated into two categories, according as they be- longed to one or to the other class. Each category was found to present diurnal progressions, of systematic regularity, but quite distinct from one another, and so far in accordance with M. Gauss’s inference of the existence of various forces contemporaneously in action, independent of one another, and having different originating sources. Confining our view, for simplicity, to one alone of the elements, viz. the declination, its two categories (of easterly and of westerly deflection) presented, wherever they were examined, 1864. ] Disturbances at Kew and Nertschinsk. 251 the same distinctive features ; the local hours or maximum and minimum varied at different stations, but the same two dissimilar forms were every- where presented by the curves representing the two diurnal progressions. Having thus traced apparently two sources in which the disturbances might be supposed to originate, the possible connexion of these with the points of maximum attraction in the two systems of the magnetic terrestrial distribution presented itself as the next object of fitting research. It was inferred that if two stations were selected in nearly the same latitude, but — situated one decidedly on the eastern side and the other decidedly on the western side of one of the points referred to, the curve of the easterly de- flection at the one station would perhaps be found to correspond with the curve of westerly deflection at the other station at the same hours of abso- lute time, and vice versd. 'The Kew photograms in the five years 1858 to 1862 supplied the necessary data for one of the two stations, viz. the one to the west of the point of maximum attraction of one of the two mag- netic systems, whilst Pekin, where hourly observations from 1851 to 1855 inclusive are recorded in the ‘ Annales de l’Observatoire Central Physique de Russie,’ might supply a station on its eastern side. As this comparison might be regarded somewhat in the light ofa crucial experiment, the reliance to which the Pekin observations were entitled was examined by the very deli- cate test afforded by rewriting the observations recorded at solar hours in hours of lunar time, and examining the lunar-diurnal variation thence derived. When this is found to come out systematically and well, and similarly in different years, the observations which have furnished it may be safely regarded as trustworthy.. The Pekin observations corresponded satisfactorily to this test, and in the Philosophical Transactions for 1863, Art. XII., the comparison was made of the Kew and Pekin disturbance- deflections, the result showing that ‘the conical form and single maximum which characterize the curve of the easterly deflections at Kew, characterize the curve of the westerly deflections at Pekin at approximately the same hours of absolute time.”’ Fora further trial of this important result, a second comparison of the same kind was made, being that of the curves of the disturbance-deflections at Nertschinsk from 1851 to 1857, also re- corded in the ‘ Annales de l’Observatoire,’ &c., with those from 1858 to 1862 at Kew. Nertschinsk is about 12° north of Pekin, and is nearly in the same longitude as that station, whilst its latitude is almost identical with that of Kew. The Nertschinsk observations were subjected to the same test in respect to accuracy as those of Pekin, and with a similarly satisfactory result. ‘The comparison of the disturbance-deflections showed a still more perfect accord between the curves representing the easterly deflections at Kew and the westerly at Nertschinsk at approximately the same hours of absolute time. The present paper contains a further comparison of the nearly synchro- nous disturbances at Kew and at Nertschinsk on the days of most notable disturbance at both stations in 1858 and 1859, the comparison being 252 General Sabine on Magnetic Disturbances. [May 26, limited to those two years inasmuch as the Kew record did not commence until January 1858, whilst the hourly observations at Nertschinsk for 1860 and subsequent years have not yet reached England. The deflections at Nertschinsk from the normals of the same month and hour, on forty-four days in 1858 and 1859, are given in a Table similar in all respects to the Table, in the Philosophical Transactions for 1863, showing the deflections on the most notable days of disturbance at Kew in the same years. The com- parison of the two Tables is discussed in some detail ; but it is sufficient to state here that the general conclusions are quite in accordance with those arrived at in the previous comparisons. The steps by which the author was led to a discovery of the decennial variation in the magnetic disturbances, aud to its identification in period and epochs with the variation in the magnitude and frequency of the sun-spots resulting from the observations of M. Schwabe since their commencement in 1826, are too well known to need repetition on this occasion. But they furnish the ground on which, in this paper, he has for the first time sug- gested the possibility that a cosmical connexion of a somewhat similar nature may be hereafter recognized as the origin and source of one of the two magnetic systems which cooperate in producing the general phenomena of the variations of the magnetic direction and force in dif- ferent parts of the globe. The author’s suggestion is, that the one of the two systems which is distinguished by its possessing a systematic and continuous movement of geographical translation, thereby giving rise to the phenomena of the secular change, may be referrible to direct solar influence operating in a cycle of yet unknown duration. The phenomena of the secular change in the earth’s magnetism have hitherto received no satisfactory explanation whatsoever ; and they have all the cha- racters befitting what we might suppose to be the effects of a cosmical cause. Some of the objections which might have impeded the reception of such an hypothesis before we had learnt to recognize in the sun itself a source of magnetic energy, and to identify magnetic variations observed on the earth with physical changes which manifest themselves to our sight in the photosphere of the sun, are no longer tenable. It is true that we do not yet possess similar ocular evidence of a solar cycle of the much longer duration which would correspond to the secular change in the distribution of terrestrial magnetism. But careful observations of the variable aspects of the solar disk can only be said to be in their commencement, and it would be premature to assume that no visible phenomena will be discovered in the sun which will render the evidence of connexion as complete in the one case as in the other. Such evidence, however, is not a necessary con- dition of an existing connexion ; the decennial period would have been equally true (though not so readily perceived by us) if the sun-spots had been less conspicuous. 1864.] Archdeacon Pratt on Local Attraction in Geodesy, &c. 258 “Qn the degree of uncertainty which Local Attraction, if not allowed for, occasions in the Map of a Country, and in the Mean Figure of the Harth as determined by Geodesy; a Method of obtaining the Mean Figure free from ambiguity by a comparison of the Anglo-Gallic, Russian, and Indian Ares; and Speculations on the Constitution of the Harth’s Crust.” By the Venerable J. H. Pratt, Archdeacon of Calcutta. Communicated by Prof. G. G. Stokes, Sec. R.S. Received October 5, 1863*. 1. In former communications to the Royal Society I have shown that Local Attraction, owing to the amount it in some places attains, is a more trouble- some element to deal with in geodetical operations than had generally been supposed. The Mountains and the Ocean were shown to combine to make the deviation of the plumb-line as much as 22'"71, 17°23, 21°05, 34'°16 (or quantities not differing materially from them) in the four principal sta- tions of the Great Arc of India between Cape Comorin and the Himma- layas—viz. at Punnce (8° 9! 31"), Damargida (18° 3' 15’), Kalianpur (24° 7' 11"), Kaliana (29° 30’ 48"); and how much these might be in- creased or lessened by the effect of variations of density in the crust below t was difficult to say. Deviations amounting to at least such quantities as 7°61 and 7°87 were shown to exist in the stations of the Indian Are, arising from this last cause (see Phil. Trans. 1861, p. 593 (4) and (5)). M. Otto Struve has lately called attention to similarly important de- flections caused by local attraction in Russia—and especially to a remark- able difference of deflection at two stations near Moscow, only about eighteen miles apart, amounting to as much as 18”, which is attributed to an invi- sible unknown cause in the strata below (see Monthly Notices of the Royal Astronomical Society, April 1862). 2. It is therefore an important inquiry, What degree of uncertainty does Local Attraction, if not allowed for, introduce into the two problems of. geodesy, viz. (1) obtaining correct Maps of a country, and (2) determining the Mean Figure of the Harth. I have pointed out the effect on mapping in India, as far as determining the latitudes is concerned, in a former paper. I propose now to consider the subject generally with reference to any country, and taking into account the longitudes as well as the latitudes. The effect upon the determination of the mean figure of the earth I discuss at greater length. Bya change, I venture to callit a correction, of Bessel’s method of applying the principle of least squares to the problem, I obtain formulze for the semiaxes and ellipticity of the Mean Figure involving ex- pressions for the unknown local deflections of the plumb-line at the standard- or reference-stations of the several arcs made use of in the calculation. These formulze at once show the great degree of uncertainty which an ig- norance of the amount of local attraction must introduce into the determi- * Read November 26, 1863. See Abstract, vol. xiii. p. 18. 254, Archdeacon Pratt on the effect of nation of the mean figure. After this I obtain formule for the mean figures of the Anglo-Gallic, Russian, and Indian Arcs by the same method, each involving the expression for the unknown local deflection of the plumb- line at the reference-station of the arc concerned. I then show that values of these three unknown deflections can be found which will make the three ellipses which represent the three great arcs almost precisely the same. These deflections are not extravagant quantities, but quite the contrary, being small. I infer, then, that the mean of these three ellipses is in fact the Mean Figure of the Earth, and in this way surmount what was the apparently,insuperable difficulty which our ignorance of the amount of local attraction threw in the way of the solution of the problem. The paper concludes with some speculations on the constitution of the earth’s crust flowing from the foregoing calculations. § 1. Effect of Local Attraction on Mapping a Country. 3. In determining differences of latitude and longitude between places by means of the measured lengths which geodesy furnishes, the method of geodesists is to substitute these lengths and the observed middle latitudes in the known trigonometrical formule, using the axes of the MEAN FIGURE of the earth. It might at first sight appear likely that this would lead to incorrect results, as the actual length measured may lie along a curve dif- ferent to that of the mean form. I propose now to show that no sensible error is introduced by following this course, either in latitude or longitude, if the are does not exceed twelve degrees and a half of latitude, or fifteen degrees of longitude in extent. 4. First. An are of Latitude.—Suppose an ellipse drawn in the plane of the meridian through the two stations, a and 6 being its semiaxes ; ¢ the chord joining the stations; s the length of the arc; r and 0, 7’ and 6’ polar coordinates to the extremities of the arc from the centre of the ellipse; J and /' their observed latitudes; the amplitude of the arc; m its middle latitude: then we have the following formule, neglecting the square of the ellipticity (e), s= 5 (at b)A— - (a—b) sin d cos 2m, r=a(l—esin*/), 7'=a(1—esin’/’), tan0=(1—2e) tanJ, tan 6’ =(1—2e) tan 7’. Now c= +r? — 2rr' cos (0—6') =2rr'{1— cos (@—8')} + (r—7')? =2rr'{1—cos (0@—6')t. By expanding the formule for tan 6 and tan 6’, we have 6=l—esin2/, 0’ =l'—e sin 2l', “. O—6'=/—Il'~e (sin 2/— sin 2/') =1—1' — 2e sin (J—1’) cos (J+ 1’) =)\—2e sind cos 2m; .. 1—cos (0—6')=1— cosA—2e sin” A cos 2m a sin’ {1—2c( + cos) cos 2m}. Local Attraction on Geodetic Operations. 299 Also j rr = a?4{1—e(sin?J-+ sin? 1!) } =a’ { 1— = (2— cos 2/— cos 27’) } =a’{1—e(1—cosd cos 2m) ; °, ado’ sin’ | 1—ef1+(2+ cos A) cos 2mt hs ok c E aa =] = | 2: > sin 5 =i + 5 {1+ (2+ cos d) cos m} } xr abe aC € c aa TG | E+ C+ cos) ons 2m | ta, =sin-! © ateye \ 1+(2-+ cos dA) cos 2m } nae La 2 2 Hence by the first formula, o=a( 1 — 5)\— = ae sin A cos 2m =a(2—e) sin-} - +ae{1+(2+ cosd) cos 2m} tan% ~ S ae sinh cos 2m a =(a+6) sin-!—— +(a—b) { 1+ 1 (1—cos ) cos 2m tan > 2a 2 2 Taking the variation of s with respect to a and 4, considering ¢ as constant, and and m also constant, occurring as they do only in small terms, we shall have the difference in length of two arcs joining the stations and be- longing to different ellipses, only having their axes parallel. Hence sb ( WO a-BO cou ds=(da+ 06) sin = Vp -++ (da—66) { + 5( — cos i) cos 2m \ tan = Since the terms are small, we may use the first approximate value for c and 0; oe ds= (Sa-+38) > —2tan ¥ da-+ (a—8) { 1+ 5 (1—c0s)) cos 2m | tan > =(da-+2b) € —tan 5) +(da—20) 5 tan (1 — cos d) cos 2m =(da+06)P+(da—<éb) Q cos 2m, where 1 2 =(P+Q cos 2m)da+(P—Q cos 2m)éb. J will find the values of éa and 06 which will satisfy this equation and make da?+ 66” a minimum. P= 5\—tan A, and Q= 5 tan 5 4(1— cos) 256 Archdeacon Pratt on the effect of se (#—P4 Rem in ) = a minimum ; P—Q cos 2m -. {(P—Q cos 2m)’+ (P+ Q cos 2m)*}da=(P+Q cos 2m)éds ; ase P+Qcos2m os vas P—Q cos 2m os - P?+ Q? cos? 2m 2’ P?+ Q? cos? 2m 2’ 1 os" 2 2 6a? +067= PQ cos Im 2" This is least when m=O and 90°; then _ P+Q os _ P#Q os _ Qés sa= PLO 9” 0b= PQ? 3° da ~ b= Pie Let one of the two ellipses be equal to the mean ellipse of the earth’s figure, a and 6 being the semiaxes, and ¢a and ¢é the excess (or defect, if negative) of the semiaxes of the other ellipse. The first ellipse is not ne- cessarily the mean ellipse itself, but is only equal to it in dimensions, and parallel to it in position; for the actual are may lie above or below the mean ellipse. The result of this is, that the arc of the mean ellipse which corresponds with s of the actual arc will not necessarily have precisely the same middle latitude, although the chord cis of the same length. But as the middle latitude will differ only by a quantity of tue order of the ellip- ticity, this difference will not appear in the result, because we neglect the square of the ellipticity. I will now make the extravagant supposition that the ellipse to which the are actually belongs deviates from the form of the mean ellipse so much that da ~ 66=13 miles, the whole compression of the earth’s figure. On this supposition I will find how large the are may be so as not to produce a difference in length greater than 1". Put da ~ 6=13, ds=1'=0-0193 mile (1° being 69°5 miles), -, (P?+Q?) +Q=0-0193+13=0-0015, me — tan - + 5 tan’ * (1— eos d)*=0: 00075 tan > (1— cos A). A slight inspection of this equation shows that \ must be small. Expand in powers of A; then : +)(3 x =0" 0015, or (5 :) =0-00135 ; *, A=0°22 (in arc) =0°22 x 57°'3 (in degrees) =12°-6 This shows that in an arc of meridian as much as twelve degrees and a half in length, it would require a departure from the mean ellipse equal to the whole actual compression of the pole of the earth in order to produce so slight a difference in the lengthas 1". Hence we may conclude that the difference in length between the mean arc aud the actual are, joing any two places on the same meridian, is an insensible quantity, since an extra- vagant hypothesis regarding the departure of the form from the mean form will not produce a difference in length of more than 1". This being the Local Attraction on Geodetie Operations. 257 ease, the differences of latitudes calculated from the measured ares of meri- dian with the mean axes, as is done in the Survey operations, will come out free from any effects which local attraction can produce, as that attraction can never be capable of causing so great a distortion in the measured arcs as I have supposed for the sake of calculation. The absolute latitude, however, of the station which fixes the arc on the map will be unknown to the extent of the deviation of the plumb-line caused by local attraction at that place. 5. Second. An Arc of Longitude.—Let S be the length of the are, 7 the latitude, L the longitudinal amplitude or the difference of the longitudes of its extremities, ec the chord. Then S=Leosl{a+(a—6) snl}, e=2cosl{a+(a—S) sin* lt sin SL. When a and @ vary, ¢ and J remain constant, but Sand L vary. Hence dS=6cL cos / {a+(a—6) sin? 7+ +L cos 7 {da+ (da—dd) sin? Zt O={a+(a—b) sin? 7} £08 LoL +2{3a-+ (aa—38) sin? 7} sin 5 Li. By eliminating 6L from these, S= (IL-2 tan 3 L) cos {3a-+(da—26) sin? Z} ; Ho ohana rage a (L—2 tan $ L) cos / I will, as before, find the values of da and 980 which satisfy this equation, and make 6a*+ 667 a minimum. sin’ / da°+ {(1+ sin®/)da—n}?= a minimum ; .. {sin*/+ (1+ sin*Z)?}da=n(1+ sin?Z) ; “. da+ (6a—66) sin? /= =n, suppose. caho (i+ sin? dn nen — sin’l.n sin*/+ (1+ sin?/)?’ ~ sint7+ (1+ sin® 2)?’ ee es ani at Oe - sin* 7+ (1+ sin? i? coslfsin'd + (1 + sin*/)?}{L—2tan3L}? This is least when cos*/ {sin‘7+ (1+ sin?/)?} is greatest, or when J=0; then os jan, ~Ob=0, Ja-—sbene i Oe ay tea dat aye Now put da ~ 66=13 miles, dS= arc 1" of a great circle =0°0193 mile; “, L—2tan5L=0-0193 +13=0-0015. This shows that L must be small: expanding, we have L'=0°018, L=0-262 (in arc) =0-262 x 57°3 (in degrees) =15°, We can reason from this, as before, that the differences of longitudes will be accurately found by using the measured ares of longitude and the mean axes, if the arcs are not longer than 15°. Now ares of this length, and of 258 Archdeacon Pratt on the effect of the length determined in paragraph 4 for latitudes, are never used in sur- vey operations: the great arcs are always divided into much smaller por- tions. Hence the maps constructed from geodetic operations will always be relatively correct in themselves; but the precise position of the map on the terrestrial spheroid will be unknown by the amount of the unknown deflection of the plumb-line in latitude and longitude at the place which fixes the map. In India the effect of the Himmalaya Mountams and the Ocean, taken alone, would throw out the map by nearly half a mile. The calculations, however, which I give in the next two sections of this paper, show that the effect of variations in the density of the crust below almost entirely counteracts that of the mountains and ocean at Damargida in lati- tude 18° 3’ 15”, and the displacement of the map is almost insensible if fixed by that station. Iffixed by the observed latitude of any other station, the map will be out of its place by the local deflection of the plumb-line at that station. This, inthe Indian Great Arc, does not exceed one-thirteenth of a mile at any of the stations where the latitude has been observed. It appears also from those calculations, that, except in places evidently situated in most disadvantageous positions, the local attraction is rarely of any con- siderable amount. § 2. Effect of Local Attraction on the Determination of the Mean Figure of the Earth. 6. The mean radius of the earth is nearly 20890000 feet, the ellipticity i 1 ee : is nearly 300° and it is found convenient to put the semiaxes of the earth’s figure under the form a+b 4) : =f ] — Ait ts | . 2 ( sano) 20890000=20890000— 2089 xu feet, a—b 1 u v [eyes ‘ 2 “600 (1- 10000 =F = 20890000= =| 5; +4178000 5 (1) u and v are quantities to be determined, and the squares and product ; of these may be neglected. ee a—b ] v Also, ellipticity = aan (1 ++ ae The arcs which are actually measured in geodesy do not necessarily belong to precisely the same ellipse: in fact those arcs may not precisely belong to any ellipse. Suppose one of these measured ares is laid along the ellipse of which the axes are given above, and that, small corrections v and a! being added to the observed latitudes of its extremities, the are with its corrected latitudes exactly fits this ellipse. Then 2’—# may be expressed in the form m+au+/3v, where m, a, and GB are functions of the measured length, the observed latitudes, and numerical quantities. Let this be done for all the arcs which have been measured and their subdivi- sions. I shall take the eight arcs used in the chapter on the Figure of the Karth in the Volume of the British Ordnance Survey ; viz. the Anglo-Gallic, Local Attraction on Geodetic Operations. 259 Russian, Indian II. (or Great Arc), Indian I., Prussian, Peruvian, Hano- verian, and Danish Arcs. Suppose M+au+tEe+....%, m +e uth ete, are the corrections of the latitudes of the extremities of the subdivisions of the Anglo-Gallic Arc, 2, being the correction for the standard or reference station inthis Arc. Similarly, let mA-autBv+e, m,+e'u+p'wte,,.... M,+0UtB0+2, m',+a'utP' wa... be the corrections for the divisions of the other Ares. Then the values of uw and v which give the most likely form are those which make the sum of the squares of all these corrections a minimum. The sum of the squares will involve w and v, and also eight quantities 2,...&,. The usual course is to regard, not only u and v, but z,...2, as independent variables, and to differentiate the sum of the squares with regard to each of them in succession, and so obtain as many equations as quantities to be determined. 7. This mode of proceeding is, I conceive, erroneous; as I shall now endeavour to show. The corrections z,....«, are not properly indepen- N N2 mm D VOL, XIII. >.< 260 Archdeacon Pratt on the effect of dent variables, but are functions of u and v, and of the deflections produced by local attraction. In the preceding diagram the plane of the paper is the plane of the meridian in which the arc, of which A B is one section, has been geodetically measured. A is the reference-station of the several portions of the whole arc. A Z is the vertical at A in which the plumb- line hangs. The two curves, of which A’B’ and aé are portions, are a variable ellipse and the mean ellipse having the same centre O and their axes in the same lines, the mean ellipse being what the variable ellipse becomes when the values are substituted for wu and v which make the sum of the squares of the errors a minimum: Z’A A'N! and zAaN are nor- mals through A to these two ellipses; AD, A’m', am are perpendicular to OD. Now, if the earth had its mean form, a plumb-line at A would hang in the normal zA to the mean ellipse; but it hangs actually in ZA. Hence ZAz is the deflection (northward in the diagram) which the plumb-line suffers from the local attraction arising from the derangement of the figure and mass of the earth from the mean. This angle is some constant but unknown quantity ¢, ¢ being reckoned positive when the deflection is north- ward, This quantity ¢ is part of the correction ZAZ', or x, added to the observed latitude of A before applying the principle of least squares. "The other part is zAZ', which I will now calculate: it is the angle between the two normals drawn through A to the variable and the mean ellipses. By the property of an ellipse of which the ellipticity is small, ON=2e.Om, and ON'=2e'. Om’, Also as Om, Om', OD differ only by quantities of the order of the ellip- ticities, they may be put equal to each other in small terms, because we neglect the square of the ellipticities. , 2zAZ'= ZNAN’= ZAN’D— ZAND cot AND—cot AN'D Laps (ND—N'D)AD 1+cot AND'cot AND ~"" AD?+ND.ND (ON’—ON)AD 2(e’-—e)OD.AD , 4 : ~ AD?+D02 _ =tan-! AD?+ DO? =tan—l(e —e) sin 2/ —— fans — Aion Wy . 1 : 5 =(e'—e) sin 2/ Sn 1” / being the observed latitude of A. - Suppose that v and V are the values of v for the variable and the mean ellipses. Then by the third of the formule (1), F sin 2/7 pibeas tos Zane = 5000 sn 17 (8 ~ V)=13""75 sin 21(v—V)=n(v— V) suppose..(2). Hence x==t+n(v—V). Local Attraction on Geodetic Operations. 261 Therefore the sum of the squares of errors, which is to be differentiated with respect to u and v to obtain a minimum, is (n,(v—V)+ t, P+ (m,+a,u+B,v+n,(o—V)+ t,)’ + (m', +a’ ut+'vt+n(o—V) +4)? + (n,(v—V) + t.}? aa (m,+4,u-+ fv +n,(vo—V)+ Dy + (m',+a',ut B'v+n,(v—V) +t, )P+ ees + ; A : : : = a minimum. Let U and V be the values of u and v which belong to the mean ellipse. These values, then, must be put for wu and v in the two equations produced by differentiating the above with respect to u andv, We have a,(m,+aU+6,V+t,)+a,(m',+e,0+6',V+2)4+... +a,(m,+a,U+B8,V+t,)+a,(m',+e,0+,',V+t,)+. : * + ; , 2 , : : ; 7 =0; and t, + (B,+2,)(m,+a,U+B8,V+t) + (6+) (m,+e'U0+6V+t)+.. Li + (6,-++2,)(m,+a,U +3,V +64.) +(B', ae +A ¥+i)+-!, + b 2 2 ry rt) e ° e tr e ay Let (m) be a symbol representing the sum of all the m’s appertaining» to the divisions of the same Arc; and let 2(m) represent the sum of all these sums for all the Arcs; and similarly for other quantities besides m. Then the above equations become X(ma) +2(a7) U+ (eB) V+2t(a) =0 and 2(mB) | +228) | + 2G) | V+ 24(8) +3n(m) { +32n(a) { ~ +3n(8){ + Bini i being the number of stations on the representative Arc. The numerical quantities involved in the first two lines of these equa- tions have been already calculated in the article on the Figure of the Earth in the British Ordnance Survey Volume, from which I borrow the results in Table II. on the following page. The quantities involving z are calcu; lated in Table I., and the results inserted in Table II. with the others. x2 Archdeacon Pratt on the effect of 262 1 F8S2-Z1F—| PZSL-E8TL +] 0086-8FGI +} 80ST-28+ | 4894-601 —| 0916-6ES-+] 06L1-E81—| 9618-9671} “°° ahaa: Sees eet? 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I will now apply the formulze just obtained to determine the Mean Figure of the Earth from the (data afforded by the eight arcs. For conve- nience I shall use the well-known symbol (2:6961987) to mean the number of which 2°6961987 is the logarithm ; and so of other numbers. By sub- stitution from the Table, the formule give (2'6961987) + (2°7323262)U —(2:0404588)V + (1:4873505)é, +-(1°7363431 )t,—(0°1996455 )t, + (1°7556462)¢, + (1'8579353)é, + (0:0501090)é, + (1°8610562)#,+(1°7413092)t,=0,. (3'2469417) +(3:0309997)U —(2°5186555)V + (2°6515733)¢é, + (2°2115282)é, + (1°7938874 )é, + (1'0599608)¢, + (1°5898592)é, + (0°6093596)é, + (1°4249273)¢,+ (1°4176941)¢,=0. Multiplying by the coefficients of V crosswise, and subtracting 80 as to eliminate V, we have (5*2148542) + (5°2509817)U + (4:0060060)¢, + (4°2549986)é, = (2°7183010)z¢, — (5°2874005) — (5:0714585) U — (4:6920321)#, —(4° 2519870)t, — (3°834362)t, ++ (2°2743017)¢, + (2°2765908)¢, + (2°5687645)¢,+ (2° ort ae + (2°2599647 )é, — (3°1004196)¢,—(3°6303180) é,— (2°6498184)¢,—(3°4653861)é, — (3°4581529)t,=0. Putting numbers in the place of logarithms, 164004+178230U+ 10139¢,+17989¢,— 523¢, —193821—117885U— 49208¢,—178644,—6829¢, = 29817+ 60345U— 39069¢,+ 1254,—7352¢, 4+ 188¢,+ 2384,+370¢,+ 240¢,4 1824, — 1260¢,—4269¢,—446¢,—2920t,—2872¢, —1072t,—4031t,— 76¢,—2680¢,— 2690¢,=0. Putting logarithms in the place of numbers, _ — (4'4744639) + (4-7806413)U — (4°5918323)¢, + (2:0969100)¢, = fo Secsobs 6 ds01ha8).,~(o ee — (3°4281348)t,— (34297523) t,=0 Transposing and dividing by the coefficient of U, U=(1-6938226) + (1-8111910)¢, —(3°3162687)#,+ (1-0857642)é, + (2:2495535 )t,+ (28247715)t,+ (5 1001723) + (2:6474933)¢, + (2°6491110)¢. Local Attraction on Geodetic Operations. 265 Now 2089=(3'3199384), -, 2089U =(3:0137610) + (3°1311294)é, —(0°6362071)#, + (2° crepe _ +(1°5694919)¢, + (211447099 )é, + (0°4201107)¢, + (1:9674319)¢,+(1°9690494)¢, = 1032'24-1352:42, —4°3t,+254°5t,+37°1t,+ as open : +92-8t, + 93°1z,. Transposing the term in V in the first UD of this paragraph and dividing by its coefficient, we have V=(0-6557399)+ (0°6918674)U + (1:4468917)é, + (1°6958843)é, —(2'1591867)t, + (3°7151874)t, + (3°8174765)t, + (2°0096502)¢, + (3°8205974)z, + (3°7008504)¢, = (0°6557399) + (1°4468917)¢, + (1°6958843)¢, — (2'1591867)#,+ (3°7151874)¢, + (0°3856900) + (0°5030584)¢,— (2:0081361)4,+ (1°7776316)#, + (2°9414209)¢, + (3°8174765)t,+ (20096502)t,+ (3'8205974)t,+ (3°7008504)¢, + (1°5166389)¢, + (3°7920397)t,+ (1°3393609)t,+-(1°3409784)z,. Now 417800=(5°6209684), » 417800V =(6-2767083) + (5:0678601 )¢, + (5°3168527)t,—(3-780155 1 )t, +(3°3361558)z, + (6:0066584) + (6-1240268)z, —(3-6291045 )z, + (5°3986000)t, + (4:5623893)z, + (3°4384449)¢, + (3°6306186)¢, + (3°4415658)t,-+ (3°3218188)¢, + (5°1376073)t, + (3°4130081 )t, + (4:9602293)t, + (4:9619468)é, =1891073+ 116912¢,+207421t,— 6028¢,+ 21682, +1015450+1330537t,— 4257t,-+ 2503804, + 36508¢, + 2906523 + 14474492, + 203164¢, + 2443522, + 38676¢, + 2744¢,1+4272t,4 2764¢,+ 2098¢, + 137280f, + 2588¢,+ 91249¢,+4 91611¢, + 142024¢,+ 68602, + 940132, + 937092, Substituting the values of 2089 U and 417800 V above deduced in the formule (1) of paragraph 6, we have 24° — 20888968 —1352-4¢, + 4°3¢,—254-5f,—37°1,— 189-5t,— 268, 92°8¢,—93°1t,, a—b- a+6 aoe “ao +417800 V } = a $23795491 + 1446097¢, + 203168¢, + 244098¢, + 386392, + 139884¢,+ 6857t, + 93920¢,+ 93616¢,} = 89659 + 2410°2¢, + 33862, + 406-82, + 64°44, + 233'1t, +1144, 4 156°5¢, + 156°0#, ; 266 Archdeacon Pratt on the effect of +, a= 20928627 + 1057°8¢,+342°9¢, + 152°3t,4+27°3t,+93°6t,4-8'8f,. +63°7t,+62°9t,, 6 == 20849309 —3762°6¢, —334°3t,—661° 3t, —101:5¢,—372°6¢, —14-0¢,—249°3£,—249'1f,. From these we may easily deduce the ellipticity e= _— 7 {1+0°0608¢, + 0-0085¢,+ 0°0103¢,+ 0°00162,+0°00592, + 0:0003¢, + 0°0039¢, + 0:001639¢ a These formule for the semiaxes and ellipticity of the mean figure of the earth show us that the effect of local attraction upon the final numerical results may be very considerable: for example, a deflection of the plumb- line of only 5” at the standard station (St. Agnes) of the Anglo-Gallic are would introduce a correction of about one mile to the length of the semi- : major axis, and more than three miles to the semi-minor axis. If the de-_ flection at the standard station (Damargida) of the Indian Great Are be what the mountains and ocean make it (without allowing any compensating effect from variations in density in the crust below, which no doubt exist, but which are altogether unknown), viz. about 17’24, the semiaxes will be subject to a correction, arising from this cause alone, of half a mile and two miles. This is sufficient to show how great a degree of uncertainty local attraction, if not allowed for, introduces into the dekecdnatios of the mean figure. As long as we have no means of ascertaining the amount of local attraction at the several standard-stations of the arcs employed in the cal- culation, this uncertainty regarding the mean figure, as determined by geodesy, must remain. § 3. Comparison of the Anglo-Gallic, Russian, and Indian Arcs, with a view to deduce the Mean Figure of the Earth. 9. The first three of the eight arcs which have been used in the calcula- tion, viz. the Anglo-Gallic, Russian, and Indian, are of considerable length ; and as the a priori probability appears to be that the earth nowhere departs much from its mean form, it seems not unlikely that by the following de- vice we may overcome the difficulty poimted out in the last paragraph. I will deduce expressions for the semiaxes of the mean figure of each of these three arcs by the method there given. If reasonable values can be assigned to the expressions for the deflection of the plumb-line from the normals to these three ellipses such as will make the axes the same, we shall have a very strong argument in favour of those being the actual de- flections in nature, and of the figure thus deduced, as common to the three arcs, being in fact the mean figure of the earth. 10. In the previous calculation ¢ has represented the angle which the plumb-line makes, in the plane of the meridian, with the normal to the mean ellipse of the earth. I shall now use T as the angle which the plumb- line makes, in the plane of the meridian, with the normal to the mean Local Attraction on Geodetic Operations. 267 ellipse of the particular are under consideration. I shall begin with the Anglo-Gallic are. Proceeding precisely as in paragraph 8, we have (20752575) + (2°1905197)U, — (15506429) V, + (1°4873505)T, =0, (2:9341091) + (2°5805290)U, — (2°1951856)V, + (2°6515733)T,=0, (4:2704431) + (4°3857053)U, + (3°6825361)T, — (4°4847520)—(4°1311719)U, —(4°2022162)T,=0, 18640 + 24306 U,+ 4814T, — 30532—13526 U,— 15930 T, —11892+ 10780 U,—11116 T,=0, or —(4°0752549) + (4:0326188)U, — (4:0459485)T, =0 ; . U,=(0°0426361)+(0°0133297)T,, 2089=(3°3199384), 2089U, = (3°3625745) + (3°3332681)T, = 2304°5 + 2154°1T,. By the first of the equations in V,, we have V,=(0°5246146) + (0°6398768)U, +(1:9367076)T, =(0°5246146)+(1:9367076)T, + (0°6825129)+(0°6532065)T,, 417800=(5°6209684) ; -”.417800V,=(6°1455830) + (6°3034813) + {(5°5576760)+ (6:2741749)}T, = 1398244 + 2011320+ {361140+1880074}T, = 3409564 + 2241214 T, ; a, +6, *, 1 =20887695—2154"1 T, uae 5 = Gq {24297259 + 2239060 T,} = 40495 + 3731°8 T, ; » a,=2092819041577:7 T,, 6,=20847200—5885:9 T,, 1 ee a5e4 +0°0921 T,). 11. I proceed to the second, the Russian arc. (2°5869948) + (2°5257337)U,—(2:0497688)V, + (1°736343])T,=0, (2:9188361)+(2°8042007)U, — (2°2548066) V, + (2°2115282)T,=0, (4°8418014) +(4°7805403)U,+ (3°9911497)T, — (4'9686049)— (4°8539695)U,—(4°2612970)T,=0, 69471 +60331 U,+ 9798 T, —93026—71445 U,—18251 T, —23555—11114U,— 8453T,=0, or 3 — (4°3720831)—(4:0458704) U,—(3'9270109)T,=0 ; “. U,=—(0°3262127)—(1'8811405)T,, 2089=(3°3199384), 2089 U,= —(3°6461511)—(3°2010789)T,= —4427:4—1588°8 T,,. é€ 268 Archdeacon Pratt on the effect of — - By the first of the equations in V,, we have - -V,=(0°5372260)+(0:4759649)U,+(1- 6865743), = (0°5372260)—(0°8021776) + {(1°6865743)—(0-3571054)}T, 417800 =(5-6209684) ; -. 417800 V, =(6:1581944)—(6-4231460) + {(5°3075427)—(5:9780738)}T, = 1439443— 2649391 + {203022—950766}T, = — 1209948—747744 T, ; . are = 20894427 + 1588-8 T,, Mos = 5h {19684479 —746155 T,} =32807—1243°6 Ty, a,=20927234+4+345:2T,, ,=20861620-+4 2832-4 T,, —0-0379 T,). 12. The following is the calculation for the Indian are :—= —(1:1055647) +(1°6681529)U,-+ (1°5646471)V,—(0°1996455)T,=0, (0°5798179)-+(1°3777851)U,+(0°9140680)V, +(1°:7938874)T,=0, ~ —(2:0196327) + (2°5822209 )U,=(1°1137135)T, — (2°1444650)—(2°9424322)U,—(3°3585345)T,=0, = iO5-35920.—) 13 1 —139=876 U,—2283 T, —244—494 U,— 2296 T,=0, or — (2°3873898) — (2°6937269)U,—(3'3609719)T,=0; *, U;=—(1'6936629)—(0°6672450)T,, 2089=(3°3199384), 2089 U,=—(3-0136013) — (3°9871834)T, = —1031-8—97 09-2 T;. By the first of the equations in V,, we have V,= (15409176) — (0°1035058)U, + (2°6349984)T, = (1°5409176) + (1°7971687) + {2°6349984) + Oe 417800= (56209684), 417800 V,=(5° 1618860) + (5-4181371) + {(4°2559668)+(6- si7iouaee = 145173 +4 261901 + {18029 + 2464445}T, _ = 4070744 2482474 T, ; Local Attraction on Geodetic Operations. 269 ” ae TORS ft = oe ~~ {21298106 4+ 2492183 T,} =35497 +4153°6T, ; o al T,, 6,=20855535-+5555'6 T,, ST eri 70). 294°8 13. I have now, if possible, to find values of T,, T,, T, which will make these three ellipses, which measure the Anglo-Gallic, the Russian, and the Indian ares, the same; that is, a,=a,=a,, 6,=6,=0,. These give the four following equations :— 15777T;— 345:2T,+ 956=0,) (3'1980244)T, —(2°5380708)T, + (29804579) =0, 5885°9T,+ 2832°4 mi1440040, bo. (3°7698129)T, + (3:4521546)T, + (4°1589653) =0, 1577-7 T; 138628 T,+ 1661=9,{ | (3:1980244)T, —(4:1418509)T,, + (32203696) =0, 5885°9T,+ 5555°6T,+ 8335=0,/ &(3-7698129)T, +(3°7447310)T,+(3-9209056) =0, The most likely solutions of these four equations connecting the three quantities T,, T,, T, which we are seeking are found by the method of least squares. This leads to the three following equations :— 2(6°3960488) pa »— (7°3398753) \t T, + (61784823) + (6°4183940) +-2(7°5396258)J +(7-2219675)) +4-(7-5145439)/ +.(7-9287782)+-(7-6907185)—0. (5°7360952) } T,—(5°0761416) ) T,-+(5-5185287) + (72219675) +(6-9043092)) +(7-6111199)=0, (7:3398753) | T, —(8°2837018) | T,+(7:3622205) +(7°5145439)) +(7:4894620)/ +(7-6656366)=0, or 2% 2489 = 545 —21871 + 1508 + 2621 4234644 +16671 32700 = 484875 449059 742667, +16126T, +10829T, -++138063 =0 (48707900) (4:2075267) (4:0345884) (5°1400773) ip = 119 = 330 21871 =192177 +23026 +16671 +8022 440843 432700 + 30865 +46306 17216T, +7903T, +41173=0, 54571T, = 161312T; +69332=0 (42359323) (38977920) (4:6146125) (4°7369619) (5-2076667) (48409337) ; T, = — (0°3381403)T, —(0-7168205), T= (1°5292952)T, + (16332670) ; *. 16126T.= —(4-5456670)T, — (49243472), 108297, = (35638836), (3°6678554)- = —35129T, —84013, =3663T +4654 ; *. {74266 _ 35129-+36631T, + 138063 — 8401344654 =0, 42800T,+58704=0, T,=—1'37, To= —2°18T, —5:2] = 42°99 —5°21 = —2'""22, T3=0°338T, +-0°430 = —0°463-+0°430= —0'033. 270 Archdeacon Pratt on the effect of When these are substituted in the semiaxes, they give = 20928190—2161=20926029, a,=20927234—766 = 20926468, a= 20926529 — 457 =20926072, b, = 2084720048064 =20855264, %,—20861620—6288 — 20855332, b,=20855535— 183=20855352. These three results are remarkably near each other; they differ from their average, 20926189 and 20855316, in no case by so much as 300 feet, and in most cases by much less. I think, then, that we may safely infer that this average ellipse is in fact the mean figure of the earth. This being the case, T,, T,, T, are the same as z,, ¢,, ¢,; and therefore the deflections of the plumb-line in the meridian at the standard stations of the Anglo- Gallic, Russian, and Indian ares are 1'°37, 2''-22, 0-033, all in the south- ern direction*. 14. The values, then, which I would assign to the semiaxes and ellipti- _ city of the Mean Figure of the Earth are as follows :— a=20926180, 6==20855316 fect, e=—! _. 295°3 If these are substituted in the formule (1) of paragraph (6), we have U=—0°'3581 and V=0°8819. § 4. Speculations regarding the constitution of the Earth's Crust. 15. If the reasoning in the last section, which has led to so satisfactory a result, be correct, I think we may draw some useful inferences regarding the constitution of the earth’s crust. By substituting the values of U, V, ¢,, ¢,, ¢, im the formule similar to m+aU+6V-+¢ for the fifty-five stations of the eight arcs, which will be found at p. 766 of the Ordnance Survey Volume, every one of the results will be small. These results are the corrections of the latitudes of the stations in referring them to the mean ellipse; that is, they are the deflec- tions of the plumb-line in the meridian at those stations owing to local attraction, or the attraction arising from the departure of the actual figure of the earth from the mean figure. Fifteen of these formulee I here select, adding one new one for Dehra about 56 miles to the north of Kaliana, the northern extremity of the Indian arc. They are as follows :— * The numerical calculations in paragraphs 7 to 13 inclusive have been tested at the Government Trigonometrical Survey Office in Calcutta, es | Local Attraction on Geodetic Operations. 271 From the Anglo-Gallie Are. Detes-|Ctelate oe a tions. |attractions. counted for. (1) Barcelona...... +1:440—3°0644 U+0°0553 V—1°37= |+ 2:22 (2) Dunkirk ...... +0°767+0°4115 U—0-0765 V—1°37 = |—0°84 (3) High Port Cliff. +1:778 +0°2532 U—0:0450 V—1°37 = |+1°28| +3:29 | —2:01 (4) Week Down ...| +1°747+-0°2539 U—0°0452 V—1:37= |+0°25 | +1:98 | —1:°73 (5) Boniface Down) +1°967+0°2559 U—0:0455 V—1°37= |+0°46| +2°42 | —1:96 (6) Dunnose ...... — 0499+ .0°2613 U—0:0466 V—1°37= |—2:00| —0°54 | —1-46 (7) Blackdown ...| +4:279+ 0:2859 U—0-0513 V—1°37= |+2°76 (8) Burleigh Moor.| —1:814+1°6845 U—0°4137 V—1°37= |—4:15 | —4:°55 | +0°40 | (9) Cowhythe ...... —6°915+42°8048 U—0°3340 V—1°37= |—9°31| —5:50?|} —3°81 (10) Ben Hutig ..... +0:095 +3°1173 U—0:9708 V—1:37= |—3°25| —2:01 | —1:24 (11) Saxavord ...... +4°403+3-°9370 U—1°3699 V—1°37 = |+0°41 From the Russian Are. (i) Fornen: .....+.-. +11°826+7:3799 U—2°5821 V—2:22= (13) Fuglenees ...... +10°008+9°1231 U—3°8418 V—2:22= +4°69 +1:13 From the Indian Are. (14) Panne. ......... + 0°625 —3°5622U —3°1853V —0°033=|—0°94 | +22°71 | —23°65 (15) Kaliana......... + 0°403+4°1251U +2°7756V —0°033 =|+ 1°34 | +34°16 | —32°82 (16) Dehra*......... +53°796+4:4215U —0°1010V —0:033=|+52°09 I have inserted the formula of Cowhythe from p. 771 of the Ordnance Survey Volume. I have also added two columns, in one of which are given the deflections of the plumb-line arising from attraction at those of the stations for which it has been calculated. For those of the Anglo-Gallic Are, I refer to the Ordnance Survey Volume, sect. xi. p. 625; and for those of the Indian Arc to my paper in the Philosophical Transactions for 1861, p. 593. I would observe that not only in the two stations of the Indian Arc, but in those I have selected from the Anglo-Gallic Are (all of which are near the sea-shore), allowance is made for deficiency of density and attraction of sea-water. In the stations (3), (4), (5), (6) the effect of the sea for about 9 miles south of the coast is taken and estimated at +-0!!-27 (see Ordnance Survey Volume, p. 631); in station (8) for 36 miles north, and estimated at —0!'"39 (p. 642); in station (9) for 50 miles north, and estimated at —0'°70 (p. 664); in station (10) for 50 miles north, and esti- mated at —0'-64 (p. 662). It is of importance to bear this inmind. For stations (14) and (15) the effect of the sea the whole way to the south pole * This is calculated by the formule at p. 737 of the Ordnance Survey Volume, from the following data obligingly furnished me by Major Walker, Superintendent of the Govern- ment Trigonometrical Survey of India, viz. Astronomical latitude of Dehra 30° 19’ 19”. Distance of parallels of Dehra and Damargida 4463510-7 feet. The latitude of Damargida is 18° 3' 15", 272 Archdeacon Pratt on the effect of is taken, and estimated at +1971 and +6"-18, the effect of the mountain mass on the north being +3'-00 and 27-98. 16. The first thing I observe in the results given in the last paragraph _is the very small amount of the resultant deflections at the two extremi- ties of the Indian Are—Punnee close to Cape Comorin, and Kaliana the nearest station to the Himmalaya Mountains; whereas the effect of the Ocean and the Mountains has been shown to be very large. This shows that the effect of variations of density in the crust must be very great, in order to bring about this near compensation. In fact the density of the crust beneath the mountains must be less than that below the plains, and still less than that below the ocean-bed. If solidification from the fluid state commenced at the surface, the amount of contraction in the solid parts beneath the mountain-region has been less than in the parts beneath the sea. In fact, it is this unequal contraction which appears to have ’ caused the hollows in the external surface which have become the basins into which the waters haye flowed to form the ocean. As the waters . flowed into the hollows thus created, the pressure on the ocean-bed would be increased, and the crust, so long as it was sufficiently thin to be influ- _enced by hydrostatic principles of floatation, would so adjust itself that the pressure on any couche de niveau of the fluid shculd remain the same. _ At the time that the crust first became sufficiently thick to resist fracture under the strain produced by a change in its density—that is, when it first eeased to depend for the elevation or depression of its several parts upon the principles of floatation, the total amount of matter in any vertical prism, drawn down into the fluid below to a given distance from the earth’s centre, had been the same through all the previous changes. After this, any further contraction or any expansion in the solid crust would not alter the amount of matter in the vertical prism, except where there was an ocean; in the case of greater contraction under an ocean than elsewhere, the ocean would become deeper and the amount of matter greater, and in ease of a less contraction or of an expansion of the crust under an ocean, the ocean would become shallower, or the amount of matter in the vertical prism less than before. It is not likely that expansion and contraction in the solid crust would affect the arrangement of matter in any other way. That changes of level do take place, by the rising and sinking of the sur- face, is a well-established fact, which rather favours these theoretical con- siderations. But they receive, I think, great support from the other fact, that the large effect of the ocean at Punnce and of the mountains at Ka- liana almost entirely disappear from the resultant deflections brought out by the calculations. The formule of paragraph 15 show that when we get close to the mountain-mass, as at Dehra, which is at the foot of the mountains where they first rise rapidly above the plains, the resultant deflec- tion is very great; the less density of the crust down below the sea-level drawn under the mountain-mass has here a very trifling influence. This is as it should be, if the depth of this less density is considerable ; Local Attraction on Geodetic Operations. 273 whereas at Kaliana, and stations still further off, the attraction of the mountain-mass above the sea-level, and the deficiency of attraction from the crust below that level, would nearly counterbalance each other. Thus, if the thickness of the crust below the plains is 100 miles, and the amount of matter in the crust under the plains equals that of the crust and moun- tains together in the mountain-region, then the deflections at Kaliana, Kalianpur, and Damargida, instead of being 27°98, 12'°05, 6"°79, arising from the mountains alone, are reduced to 1'°54, —0':06, —0':06 (see Philosophical Transactions for 1858, p. 759), which are all insigni- ficant compared with the large deflections caused by the mountains alone. This theory, that the wide ocean has been collected on parts of the earth’s surface where hollows have been made by the contraction and therefore increased density of the crust below, is well illustrated by the existence of a whole hemisphere of water, of which New Zealand is the pole, in stable equilibrium. ‘Were the crust beneath only of the same density as that beneath the surrounding continents, the water would be drawn off by attraction and not allowed to stand in the undisturbed posi- tion it now occupies. 17. I have, in what goes before, supposed that, in solidifying, the crust contracts and grows denser, as this appears to be most natural, though, after the solid mass is formed, it may either expand or contract, according as an accession or diminution of heat may take place. If, however, in the process of solidifying, the mass becomes lighter, the same conclusion will follow—the mountains being formed by a greater degree of expansion of the crust beneath them, and not by a less contraction, than in the other parts of the crust. It may seem at first difficult to conceive how a crust could be formed at all, if in the act of solidification it becomes heavier than the fluid on which it rests; for the equilibrium of the heavy crust floating on a lighter fluid would be unstable, and the crust would sooner or later be broken through, and would sink down into the fluid, which would overflow it. If, however, this process went on perpetually, the descending crust, which was originally formed bya loss of heat radiated from-the surface into space, would reduce the heat of the fluid into which it sank, and after a time a thicker crust would be formed than before, and the difficulty of its bemg broken through would become greater every time a new one was formed. Perhaps the tremendous dislocation of stratified rocks in huge masses with which a traveller in the mountains, especially in the interior of the Himmalaya region, is familiar, may have been brought about in this way. The catastrophes, too, which geology seems to teach hhaye at certaim epochs destroyed whole species of living creatures, may have been thus caused, at the same time breaking up the strata in which those species had for ages before been deposited as the strata were formed. These phenomena must now long have ceased to occur, at any rate on a ‘very extensive scale, as Mr. Hopkins’s investigations on Precession appear 274 Archdeacon Pratt on the effect of to prove that the crust is very thick, at least 800 or 1000 miles; and this result, I understand, has been recently confirmed by Professor W. Thomson in a paper “On the Rigidity of the Earth.” 18. These theoretical considerations receive, I think, some confirmation from an examination of the calculated deflection of the plumb-line at sta- tions near the sea-shore. It is for this reason that I have collected the thirteen examples from the Anglo-Gallic and Russian Arcs in paragraph 15, all of which are near the coast. The evidence they furnish, however, is not to be compared in weight with that of the Indian Arc, already con- sidered. In some instances the local attraction of the surrounding country and of the ocean for a certain distance has been calculated, as already stated. These results I will take into account, except the allowances for the ocean as noted at the end of paragraph 15, which I deduct in the fol- lowing arrangement of the stations. The Stations at which the Deflection is towards the Land. : ad Ry Deflection. it (1) Barcelona, lat. 41 23, S.E. coast of Spain, «as. wae ae +2:°22 (2) Dunkirk, .;, 51. 2, N.N.W. France, ~... 9h — 0°84 (7) Blackdown, ,, 50 41, S. Ss Dotsetcs.u+ eda ae ee +2°76 (9) Cowhythe, ,, 57 41, N. > ~bantf, —3°81+0:70=—3:1l (10) BenHutig, ,, 58 33, N. sy Sutherland, — 1:24+0°64=—0°60 (12) Tornea, 65,50, Se is Lapland,>. 22% cee +4:69 The Stations at which the Deflection is towards the Sea. Deflection. (3) High Port Cliff, 50 36, S. coast of Isle of Wight, — 2-01 —0-27= —3-28 (4) Week Down, 50 36, _ a —1:73—0°27=—2:00 (5) Boniface Down, 50 36, —1°96—0°27= — 2:23 (6) Dunnose, —-50 37, \ ' —1-46—0-27=—1°73 (8) Burleigh Moor, 54 34, N. coast of Yorkshire, +0°40+0°39=-+0-79 (11) Saxavord, GO 50. ING 4 sass ones «0.2001» eee +0°41 (13) Fuglences, ZOrAO, Nee cvciat REMINISCE tec pee ones eed The theory I have proposed, that contraction of the crust has formed the basins in which the sea has settled, can hardly be expected to apply so completely to such confined sheets of water as the Mediterranean south of Spain, and the Gulf of Bothnia. Here there may be an actual defici- ency of attracting matter in the water, not altogether compensated for by increased density of the crust below. These hollows may have been formed during the breaking up of the crust and subsequent removal of portions by currents, and not chiefly by the contraction of the crust. Thus the deflec- tions at the stations (1) and (12) towards the land may be sufficiently accounted for, even if the land about Barcelona and Tornea does not rise sufficiently high to produce them. The deflection at station (2) is small. It seems probable that even if the North Sea hds been produced according Local Attraction on Geodetic Operations. 275 to the theory of contraction of the crust, the parts near Dunkirk may have been somewhat hollowed out by the scouring of the tide through the Straits of Dover, so as to give the land, low as it is, every advantage in deflecting the plumb-line south. I have no means of knowing the cha- racter of the ground north of station (7) on the coast of Dorset.- There is no difficulty, however, in accounting for the north deflection at that place, and even for a greater deflection, if the attraction of the country north of it is as much as the attraction of the land on Burleigh Moor on the north coast of Yorkshire. To this station I shall revert. With regard to sta- tions (9) and (10), I gather the following information from the Ordnance Survey Volume. “At present there are no sufficient data for calculating exactly the disturbance”’ at Cowhythe (p. 662). It is supposed not to exceed 6" (p. 664); but the calculation is not made for any part of the mountains further south than 50 miles. The south deflection to be ac- counted for, viz. —3''11, may in part be thus explained; or, even if, as before, the North Sea is supposed to have been formed by the contraction of the crust, the confined portion between the north coast of Aberdeen and the Orkney Islands*may have been formed by the removal of the superfi- cial strata by currents so as to produce a deficiency of attracting matter. So with respect to the other station, Ben Hutig, the unaccounted-for de- flection, which is much smaller, viz. —0°60, may be easily explained, as the effect of the land has not been calculated further off than about 3 miles (pp. 660, 661). Thus, on the whole, the deflections at those coast-stations, where itis towards the land, can be pretty well accounted for, without call- ing in aid the deficiency of attraction of water and supposing that the crust below the ocean is not condensed. The seven coast-stations of the second list, where the deflection 1s towards the sea, seem to bear individual testimony to the truth of the theory, that the crust below the ocean must have undergone greater contraction than other parts of the crust. The four stations (3), (4), (5), (6) on the south coast of the Isle of Wight all have deflections southwards; and their mag- nitudes diminish in the order that the distances from the sea increase,—that order being (3) High Port Cliff, (5) Boniface Down, (4) Week Down, (6) Dunnose (see the Contour Map of Isle of Wight in the volume of Plates accompanying the Ordnance Survey Volume). The amounts of the deflec- tion seem almost tc prove too much for the theory. Still they are all in the direction of the ocean, and seem certainly to indicate that there is a redundance of matter, and not a deficiency, in that direction. Blackdown (7) is somewhat further inland than Dunnose is. If, then, the ocean and crust together do really produce the outstanding deflection southward at Dunuose, we shall have to suppose that the north deflection at Blackdown in the first list of coast-stations, arising from the land, is not much less than 2°76+1°73=4°49, which is a little less than the calculated deflection at Burleigh Moor on the coast of Yorkshire, and is therefore not an unlikely amount. The other three coast-stations, (8), (11), (13), all bear out the VOL. XIII. Aa 276 Effect of Loeal Attraction on Geodetic Operations. - theory: though the three deflections are all small, they are towards the sea, the largest of them being at Fuglences, which is very near to the North Cape, and has a large expanse of ocean above it. 19. The least that can be gathered from the deflections of these coast- stations is; that they present no obstacle to the theory so remarkably sug- gested by the facts brought to light in India, viz. that mountain-regions and oceans on a large scale have been produced by the contraction of the materials, as the surface of the earth has passed from a fluid state to a con- dition of solidity—the amount of contraction beneath the mountain-region having been less than that beneath the ordinary surface, and still less than that beneath the ocean-bed, by which process the hollows have been pro- duced into which the ocean has flowed. In fact the testimony of these coast- stations is in some degree directly in favour of the theory, as they seem to indicate, by excess of attraction towards the sea, that the contraction of the crust beneath the ocean has gone on increasing in some instances still fur- ther since the crust became too thick to be influenced by the principles of floatation, and that an additional flow of water into the increasing hollow has increased the amount of attraction upon stations on its shores. Murree, Punjab, August 20, 1863. Local Attraction on Geodetic Operations. 275 to the theory of contraction of the crust, the parts near Dunkirk may have been somewhat hollowed out by the scouring of the tide through the Straits of Dover, so as to give the land, low as it is, every advantage in deflecting the plumb-line south. I have no means of knowing the cha- racter of the ground north of station (7) on the coast of Dorset. There is no difficulty, however, in accounting for the north deflection at that place, and even for a greater deflection, if the attraction of the country north of it is as much as the attraction of the land on Burleigh Moor on the north coast of Yorkshire. To this station I shall revert. With regard to sta- tions (9) and (10), I gather the following information from the Ordnance Survey Volume. “At present there are no sufficient data for calculating exactly the disturbance”? at Cowhythe (p. 662). It is supposed not to exceed 6" (p. 664); but the calculation is not made for any part of the mountains further south than 50 miles. The south deflection to be ac- counted for, viz. —3’'-11, may in part be thus explained; or, even if, as before, the North Sea is supposed to have been formed by the contraction of the crust, the confined portion between the north coast of Aberdeen and the Orkney Islands may have been formed by the removal of the superfi- cial strata by currents so as to produce a deficiency of attracting matter. So with respect to the other station, Ben Hutig, the unaccounted-for de- flection, which is much smaller, viz. —0°60, may be easily explained, as the effect of the land has not been calculated further off than about 3 miles (pp. 660, 661). Thus, on the whole, the deflections at those coast-stations, where itis towards the land, can be pretty well accounted for, without call- ing in aid the deficiency of attraction of water and supposing that the crust below the ocean is not condensed. The seven coast-stations of the second list, where the deflection is towards the sea, seem to bear individual testimony to the truth of the theory, that the crust below the ocean must have undergone greater contraction than other parts of the crust. The four stations (3), (4), (5), (6) on the south coast of the Isle of Wight all have deflections southwards; and their mag- nitudes diminish in the order that the distances from the sea increase,—that order being (3) High Port Cliff, (5) Boniface Down, (4) Week Down, (6) Dunnose (see the Contour Map of Isle of Wight in the volume of Plates accompanying the Ordnance Survey Volume). The amounts of the deflec- tion seem almost to prove too much for the theory. Still they are all in the direction of the ocean, and seem certainly to indicate that there is a redundance of matter, and not a deficiency, in that direction. Blackdown (7) is somewhat further inland than Dunnose is. If, then, the ocean and erust together do really produce the outstanding deflection southward at Dunnose, we shall have to suppose that the north deflection at Blackdown in the first list of coast-stations, arising from the land, is not much less than 2°76+ 1°73=4:'49, which is a little less than the calculated deflection at Burleigh Moor on the coast of Yorkshire, and is therefore not an unlikely amount. The other three coast-stations, (8), (11), (13), all bear out the VOL, XIII. x 276 The Annual Meeting. [June 9, theory: though the three deflections are all small, they are towards the sea, the largest of them being at Fuglences, which is very near to the North Cape, and has a large expanse of ocean above it. 19. The least that can be gathered from the deflections of these coast- stations is, that they present no obstacle to the theory so remarkably sug- gested by the facts brought to light in India, viz. that mountain-regions and oceans on a large scale have been produced by the contraction of the materials, as the surface of the earth has passed from a fluid state to a con- dition of solidity—the amount of contraction beneath the mountain-region having been less than that beneath the ordinary surface, and still less than that beneath the ocean-bed, by which process the hollows have been pro- duced into which the ocean has flowed. In fact the testimony of these coast- stations is in some degree directly in favour of the theory, as they seem to indicate, by excess of attraction towards the sea, that the contraction of the crust beneath the ocean has gone on increasing in some instances still fur- ther since the crust became too thick to be influenced by the principles of floatation, and that an additional flow of water into the increasing hollow has increased the amount of attraction upon stations on its shores. ) Murree, Punjab, August 20, 1863. June 2, 1864. - The Annual Meeting for the Election of Fellows was held this day. Major-General SABINH, President, in the Chair. The Statutes relating to the Election of Fellows having been read, General Boileau and Sir Andrew Scott Waugh were, with the consent of the Society, nominated Scrutators to assist the Secretaries in examining the lists. | The votes of the Fellows present having been collected, the following gentlemen were declared duly elected into the Society :— Sir Henry Barkly, K.C.B. William Jenner, M.D. William Brinton, M.D. Sir Charles Locock, Bart., M.D. T. Spencer Cobbold, M.D. William Sanders, Esq. Alexander John Ellis, Esq. Col. William James Smythe, R. A. John Evans, Esq. Lieut.-Col. Alexander Strange. Wilham Henry Flower, Esq. Robert Warington, Esq. Thomas Grubb, Esq. Nicholas Wood, Esq. Sir John Charles Dalrymple Hay, Bart.. June 9, 1864. i Major-General SABINE, President, in the Chair. — Mr. W. Sanders; Mr: R. Warington; Dr. Jenner; Mr. J. Evans ; 1864.] Prof. Owen on the Cavern of Bruniquel. . oF Lieut.-Col. Strange ; Mr. W. H. Flower ; Dr. Cobbold ; Col. W. J. Smythe ; Sir J. C. Dalrymple Hay, Bart.; and Mr. A. J. Ellis, were admitted into the Society. Pursuant to notice given at the last Meeting, MM. Claude Bernard, Jean Bernard Léon Foucault, and Adolph Wurtz, all of Paris, were balloted for and elected Foreign Members of the Society. The following communication was read :— "i Description of the Cavern of Bruniquel, and its Organic Glonteriti —Part I. Human Remains.” By Professor Richarp Owen, F.R.S., &. Received May 12, 1864. (Abstract.) ‘In sis communication the author gives an account of the Cavern of Bruniquel, Department of the Tarn and Garonne, France, in the state which it presented when visited by him in January 1864, and a description of the human remains discovered therein by the proprietor, the Vicomte de Lastic St. Jal, in 1863, and subsequently by the author in January 1864. The circumstances under which these discoveries were made are minutely detailed, and the contemporaneity of the human remains with those of the extinct and other animals with which they are associated, together with the flint and bone implements, is shown by the evidences of the plastic condition of the calcified mud of the breccia at the time of interment, by the chemical constitution of the human bones, corresponding with that of the other animal remains, and by the similarity of their position and relations in the surrounding breccia. Among the principal remains of the men of the flint-period cesiciinGan are the following :—1st, the hinder portion of the cranium, with several other parts of the same skeleton, which were so situated in their matrix as to indicate that the body had been interred in a crouching posture, and that, after decomposition and dissolution of the soft parts, the skeleton had yielded to the superincumbent weight ; 2nd, an almost entire calvarium, which is described and compared with different types of the human skull, shown to be superior in form and capacity to the Australian type, and more closely to correspond with the Celtic type, though proportionally shorter than the modern Celtic, and the form exhibited by the Celtic cranium from Engis, Switzerland; 3rd, jaws and teeth of individuals of different ages. After noticing other smaller portions of human cranium, the author proceeds to describe minutely the lower jaw and teeth of an adult, and upper and lower jaws of immature individuals, showing the characters of certain deciduous teeth. The proportions of the molars are not those of the Australian, but of other races, and especially those of ancient and modern Europeans. As in most primitive or early races in which masti- cation was little helped by arts of cookery or by various and refined kinds of food, the crowns of the molars, especially of m 1, are worn down beyond x2 278 Prof. Smith on Complex Binary Quadratic Forms. (June 16, the enamel, flat and smooth to the stumps, exposing there a central tract of osteodentine without any sign of decay. The paper is illustrated by a view and plans of the cavern, and by figures of the principal human remains, and of two implements of bone on which the Vicomte de Lastic had discovered, on removal of the breccia, outline figures of the head of a reindeer and the head of a horse in profile. The description of the various remains of the animals killed for food, and of the flint- and bone-implements applied to that and other purposes, will be the subject of a future communication. June 16, 1864. Major-General SABINE, President, in the Chair. Dr. Brinton; Professor Boole; Mr. T. Grubb; Sir Charles Locock, _ Bart.; and Mr. Nicholas Wood, were admitted into the Society. — The followmg communications were read :— I. “On Complex Binary Quadratic Forms.” By H. J. STEPHEN Suir, M.A., F.R.S., Savilian Professor of Geometry m the University of Oxford. Received May 18, 1864. The purpose of this note is to extend to complex quadratic forms some important investigations of Gauss relating to real quadratic forms. We shall consider in order (I.) the definition of the Genera, (II.) the theory of Composition, (III.) the determination of the number of Ambiguous Classes, (IV.) the representation of forms of the principal genus by ternary quadratic forms of determinant 1. For the comparison of the numbers of classes of different orders, we may refer to a paper by M. Lipschitz (Crelle’s Journal, vol. liv. p. 193); and for the principles of the theory of complex numbers and complex quadratic forms, to Lejeune Dirichlet’s Memoir, “‘ Recherches sur les formes quadratiques 4 coefficients et 4 indétermindées complexes” (Crelle, vol. xxiv. p. 291). I. The Definition of the Genera. Let f=(a, 6, ec) be an uneven* primitive form of determinant D, and m= ax+ 2bxy+cy?, m! =ax'?+2b2'y'+cy? twonumbers represented by /- The generic characters of f are deducible from the equation (ax? + 2bxy+ cy") (ax? + 2b2'y'+cy”) = (ave! + Bf ay’ +2'y|+ cyy’P—D(ay'—2'y)’, * A primitive form (a, 4, c) is uneven, semieven, or even, according as the peat common divisor of a, 2b, c is 1, 1+, or (1+2)?; é.¢., in Lejeune Dirichlet’s nomencla- ture, according as (a, b, c) is of the first, second, or third species. In this paper, when we speak of an uneven, semieven, or even form or class, we shall always suppose the ota or class to be primitive. A semieven number is a number divisible by 14-4, but not y (1+4)%, 1864.] Prof. Smith on Complex Binary Quadratic Forms. 279 or, as we shall write it, mv = P?— DQ?. Thus, supposing that p is an uneven prime dividing D, and that m and m’ are prime to p, the numbers prime to p, which are represented by f, are either all quadratic residues of p, or else all non-quadratic residues of p ; in the former case we attribute to f the character [4] = +1, in the latter P the character H =—], Pp Again, to investigate the supplementary characters relating to powers of the even prime 1+7, let m=:-+7' be an uneven number, p and p’ representing real numbers, and for brevity, let (a1 ee =o c= ee + yp!)2— ny ee Bs fe 1)5 =. The values of the units, or characters, a, (3, y depend on the residue of m for the modulus (1+7)’, as is shown in the following Table. TABLE I. m = a= p= y= Se peer ae +1] +1 +1 a Myers oset See's +1 +1 —l BE Bsa iccu big ens atalee ioe BURG Ss as eae +1 —1 —1 +(1—2%) es As Kia). Dea an ne) ee —l ae us + (142%) be lites ails en ol a +(2—1).. | ia) | An inspection of the Table shows that, of the sixteen uneven residues of (1+2)’, eight have the character w=1, and eight the character w=—1, -w representing any one of the seven characters a, 6, y, Py, ey, 4B, apy. It will also be seen that any character of a product of two uneven factors 280 = Prof. Smith on Complea Binary Quadratic Forms. [June 16, is found by multiplying together the corresponding characters of the factors; so that, conversely, according as any character of a product of two uneven factors is +1 or —1, the two factors agree or differ in respect of that character. The next Table assigns the supplementary characters proper to any given determinant; they depend on the residue of the determinant for the ‘modulus (I+7)’. TaBue II. — Characters. = Characters. +(1+42)... B ig | Y +(1—2)... a [3 7 a + (3+7) apy +3 Y a(3—2) By +31 a Le tee ee a, Y + (1—27) y RO ie wc ae' Y +(2+2) ay 2(1+3) ap, y + (1422) y 2(1—i) y + (2—i) ay A, atslels a, 0. tO a, Bs Y | Of the eighteen propositions contained in this Table, it will suffice to enunciate and demonstrate one. “Tf D=+(347), mod (1+7)’, and f is an uneven form of deter- minant D, the uneven numbers represented by jf, all have the character aQBy=-+1, or else all have the character aGBy=—1.” In the equation P?—DQ?’=mm', let us suppose that m and m’ are un- even ; then P is uneven because D is semieven ; also Q?==+1, +27, 4 or 0, mod (1+7)’, according as the index of the highest power of 1 +7 dividing Q is 0, 1, 2, or >2. If Q is uneven, mm'==+37 or +(2+2), mod (1+72)’; if Q is semieven, mm! =+(1+27), mod (1+7)’; if Q is even, mm! ==+1, mod (1+7)*; 7. e. in all three cases mm! has the character ay=1, and mand m! both have the character aBy=-+1, or else both have the character aBy=—1. We add a third Table for the purpose of distinguishing between the possible and impossible genera. In this Table S? is the greatest square. dividing D, P is uneven and primary*, I is the index of the highest power of 1+7 dividing 8S, @ represents an uneven prime dividing P, o an uneven prime dividing S but not P. For brevity, the symbols w and o are written instead of E ] and E |: * By a primary uneven number we understand (with Lejeune Dirichlet) an uneven number p+p’é satisfying the congruences p= 1, mod 4, pp’ =0, mod 2. 1864.] Prof. Smith on Complex Binary Quadratic Forms. 281 . Taste III. (i) D=PS’*, P=1, mod 4. I=0, lla 0, ¥ I=2 a a, Y,& I>2 w a, 5 &, [. (ii) D=PS*, P=1+ 22, mod 4. f=) ae o i —=2 @, ¥ o, a I>2 @,¥ a, a, Gi) D=7zPS’, P=1, mod 4. Veal) Wy & o it) 2 Dm, 0, Y I>2 wD, & 6, y; B (iv) D=7zPS’, P=1+2:, mod 4. jfe=() BD, ay o I=1,2)a,a,¥ o [>2 WD, a, 7 GBs (v) D=(1+7)PS’, P=1, mod 4. I=0 w, [ o a @, Pp 0,7 I>1 @, (3 Os Ys Be (vi) D=(1+72)PS’, P=1-+ 22, mod 4. 1==0 a, By a it @, 2, Y o. I>1 @, 2s Oy Os (vii) D=72(1+72)PS’, P=1, mod 4. I=0 aw, a3 o T=1 @, a3 o,Y I>l |a,a, p G, Y: (viii) D=i(1+72)PS*, P=1+ 2i, mod 4. [=0 @,aGby |e c=) a, a3, y |o I>1 BD, a, ps Y|\G. The characters preceding the vertical line by which the Table is divided are not independent, but are subject to the condition (arising from the laws of quadratic residues) that their product must be a positive unit. To show that this is so, let D=:"(1 +i)? PS’, where @’ and 3’ are each either 0 or 1; also let y’=0, or 1, according as P==1, or ==1+ 27, mod 4; If m is a number prime to (1+72)D and capable of primitive representation * - * If m=a2°+2b2y+cy?, the representation of m by (a, 3, c) is said to be primitive when the values of the indeterminates are relatively prime. 282 Prof. Smith on Complex Binary Quadratic Forms. [June 16 by f, the congruence w=D, mod m, is resoluble ; and its resolubility im- plies the condition [=| =| “|x a ]* E |=. But, by the m 70 laws of quadratic residues, H =. [|= So [= |= "lS » and the condition just written becomes a” cm y" = =1, which is coinci- dent with that indicated in the Table.’ Thus (as in the real theory) one- half of the whole number of assignable generic characters are impossible* we shall presently obtain a different proof of this result, and shall also show that the remaining half correspond to actually existing genera. For the characters of a semieven form f, it is convenient to take the characters of the numbers represented by 7 ; and for the characters of } an even form, the characters of the numbers represented by _ The d following Table will serve to form the complete generic character in each case, For a semieven form. | (GG) PS, PL. nod. 1—0 | ze luc. (i) D=PS’, P=1-+ 27, mod 4. 0 Pa, y |. For an even form. I=0 | w | Oo. II. The Theory of Composition. The theory of composition given in the ‘ Disquisitiones Arithmeticz’ is immediately applicable to complex quadratic forms. There are, how- ever, a few points to which we must direct attention. (1) If m,, m,, m, are the greatest common divisors of a, 2b, ¢; a, (1+7)b, c; a, 6, c, we have G) 7 =m—ms (ii) m,=m,=(1+2)m,, (il) m,=(1+72)m,=(1+7)?m,, according as (a, 6, c) either is, or is derived from, (i) an uneven, (ii) a semi- even, (ill) an even primitive. Hence the order of a form is given when ™, and mm, are Meaty — if Fis compounded of f and 7’, and if M, M, M,, mm, m,, m', m', m’, refer to F, f, f respectively, the order of F is com- pletely penis oe the two theorems, “M, is the product of m, and * The determinant is supposed not to be a square. 1864.] Prof. Smith on Complea Binary Quadratic Forms. 283 ue casei is the least common multiple of 1 and “2.” (Gauss’s 5th M, M, nv, and 6th conclusions, Disq. Arith. art. 235.) It will be found that Gauss’s proof of these theorems can be transferred to the complex theory; only, when f and /’ are both semieven, or derived from semieven primitives, the proof of the sixth conclusion is incomplete, and, while showing that F cannot be derived from an uneven primitive, fails to show whether it is derived from a semieven or from an even primi- tive. But, in the same way in which Gauss has shown that M, is divisible by m, xm’, it can also be shown that M, is divisible by m, x m',*; 7. @., in the case which we are considering, M, is divisible by M,, because m,=m,, m',=m',, and m,m',=M,. Therefore M,=M,, and F is derived from a semieven primitive in accordance with our enunciation of Gauss’s sixth conclusion. (2) In the real theory, when two or more forms are compounded, each form may be taken either directly or inversely ; but, however the forms are taken, the determinant of the resulting form is the same. In the complex theory, not only may each of the forms to be compounded be taken in either of two different ways, but also the determinant of the re- sulting form may receive either of two values, differing, however, only in sign ; and it is important to attend to the ambiguities which thus arise. If a complex rational number 7 be written in the form (1 +7) = where Ais 0, 1, 2, or 3, wis any positive or negative integer, and P, Q are primary uneven complex integers, we may term 7’ the sign of x. Let F, of which the determinant is D, be transformed into the product f, x f,x ... fa, by a substitution [X, Y] linear and homogeneous in respect of h binary sets ; we have, as in the real theory, / equations of the type e& @YiiedX: d¥ine ede Y Et. Bie de, dy, dy, de JD” Fy d, representing the determinant of f,. Let J (= dY dX d¥ 7 dx, dy, dy, dw, Fae ke if A so that x’, =i ift "is the sign of n,, we shall say that f, is taken with the sign 7 ‘We can thus enunciate the theorem, “Forms, compounded of the same forms, taken with the same signs, are equivalent.”” If f, f,«--+fp are given forms which it is required to compound, the signs of d,, d,,... 4, must be all real, or else all unreal; and the sign of D will be real or unreal accordingly. The value of D (irrespective of its sign) is ascertained as in the real theory; but it may receive at our option, in the * Disq. Arith. art. 235. The proof that 2(4d'+A) and 2(4b'—A) are divisible by m,xm',, may be employed ike mutandis) to show that (1 +7) (00'+A) and (1 4) (b4'— A) are divisible by m, Xm’, 284 Prof. Smith on Complex Binary Quadratic Forms. [June 16, one case, either of the two real signs, and in the other case either of the two unreal signs. And whichever sign we give to D, the form f; may be taken with either of the two real signs, if the sign of . is +1, and with either of the two unreal signs, if the sign of = is —1. In the important case in which d,, d,... all have the same sign, we shall always suppose D to have that sign, and f,, f,... to be all taken with the sign +1. Adopting this convention, we see that the class compounded of given classes of the same determinant, or of different determinants having the same sign, is defined without ambiguity. (3) By the general formulee of M. Arndt (Crelle, vol. lvi. p. 69), which on account of their great utility we transcribe here, we can always obtain a form (A, B, C) compounded in any given manner of two forms (a, 6, ¢) and (a’, b’,c’), of which the determinants d and d’ are to one another as two squares. , q po sil aM», meh pe pe Oia, te mod A iu Be bn’ +b'n_ __ bb’ + Dnn | ‘ BD C— 7 In these formule D is the greatest common divisor of dm? and d’m?, m and m’ representing the greatest common divisors of a, 2b, c, and a’, 2b’c’; m and »’ are the square roots of = and pH is the greatest common divisor of an’, a'n, and bn'+b'n. The signs of D, 2, and n’ are given, because the manner of the composition is supposed to be given; to we may attribute any sign we please, because the forms (A, B, C) and (—A, B, —C) are equivalent. (4) If F= (A, B, C) is compounded 2 two primitive forms f and . and if M is the highest power of 1+7 dividing A, B, C (so that M is 1 or 1+2, or (1+7)*), the complete character of the primitive form a F is obtained by the following rule :— “Tf w is any character common tof and Oe vi F will have the cha- racter w= +1, orw=—1, according as f and /” aan or differ i in respect of that character.”’ In comparing the characters of f and 7’, it is to be observed that if w ie w’ are two supplementary characters of 7, and w X w’ a supplementary cha- racter of f’, w x w’ is to be regarded as a character common to f and 7. 1864.] ~ Prof. Smith on Complex Binary Quadratic Forms. 285 (5) Let us represent by (1), (c), and (2)* respectively the principal uneven, semieven, and even classes of determinant D; 7. ¢. the classes con- taining the forms (1,0, —D), ( aly a Ta) aad (2%, je _D-2 ). the aa of the last two classes implying the congruences D==1, mod 2, D==i" , mod 4, respectively. Employing the formule of M. Arndt, we find ( f) x (1)=(f), if (f) is any class of determinant D; (f) x (c) =(1+2)(f), if f is derived from a semieven or even ee i ROD =2i(f), if f is derived from an even primitive; and, in particular, (1) x ()=(), («) x (¢) =(1 +2) (e), (3) x (2) =22(2). Also, if (f) and (f~') are two opposite primitive classes, (f) x ( f)*=(1), or (1 +%)(c), or 2i(3), according as f and f~* are uneven, semieven, or even. Hence the three equations (/,) x (¢)=(/2), (A) X M=A +072), (K) x (@) = 2 fa)» in which (f,) and (f,) are given primitive classes, uneven in the first, semi- even in the second, and even in the third, are respectively satisfied by the uneven, oo. and even classes (¢)=(f,) X ( hye (gy 2) XI; let l4+2 ° (=X G7 (F2) X ae 2 ’, but by no other classes whatever. Again, let D=Am?’ and let the aa (mp, mq, mr), (L1+2]mp, mq, [1+7]mr), (2emp, mq, 2umr) represent classes derived by the multiplier m from uneven, semieven, and even primitives of determinant A; in all three forms we suppose p prime to 2D; in the second and third we suppose q uneven, and A=1, mod 2; in the third we suppose A=??*, mod 4. The formule of M. Arndt will then establish the six equations, — (m, 0, —Am) x (p, mq, mr) =(mp, mq, mr), (0 +i}m, m, —m a ;) X(p, mg, Lim?r)=([1+i]mp, mg, i +i}mn), 72k (2im, i'm, ume a x (p, mq, —4mr) = (2ump, mq, pitied (a +i]m,m, —m +x ([1-+2]p, mq, [1 +7 ]mr) =(1+7)X((1+2]mp, mq,[1+2]mr), (zim vem, —m ie -) x (1 +7]p, mq, 2e[ 1-2 mr) =(1 +2) x (2ump, mq, 2umr), lel 20 2 (2m, vem, — ) X (Qip, mg, 2im?r) = 2i x (2ump, mq, 2mr). * Tt is often convenient to symbolize a class by placing within brackets a symbol representing a form contained in the class; thus (f) may be used to symbolize the class contain ng the form /. 286 Prof. Smith on Complex Binary Quadratic Forms. [June 16, From these equations, which contain a solution (for complex numbers) of the problem solved for real numbers in art. 250 of the ‘ Disquisitiones Arithmeticee,’ we may infer the following theorems (Disq. Arith. art. 25] and 253) :— «The number w of classes of any order © isadivisor of the number » of uneven classes of the same determinant D; and, given any two classes n . 3 ° of order ©, there are always — uneven classes which compounded with one @ of them produce the other.” «Tf D=1, mod 2, and if the classes of Q are derived from semievyen or even primitives, w is a divisor of the number 7x! of semieven classes of determinant D ; and, given any two classes of order Q, there are always ” semieven classes which compounded with one of them produce 1+7 ‘, times the other.” | “Tf D=+1, mod 4, and if the classes of © are derived from even primitives, w is a divisor of the number 7” of even classes of determi- ” nant D; and, given any two classes of order Q, there are always ” even 7 aye classes which compounded with one of them produce 2: times the other.”’ III. Determination of the number of Ambiguous Classes. Any form (A, B, C), in which 2B==0, mod A, is called by Gauss an ambiguous form; but in the investigation which follows we shall for brevity understand by an ambiguous form an uneven form of one of the four types : + (i) (A, 0, C), (i) (1421 B, B, ©), (11) (2B; i, ©), . Gv)" > (27 BB: ©). To determine the number of uneven ambiguous classes of any determi- nant D supposed not to be a square, we shall determine, first, the number of ambiguous forms of determinant D; and secondly the number of ambi- guous forms in each ambiguous class. (1) Let » be the number of different uneven primes Hee D. The number of ambiguous forms of the type (1) is 4 x 2", or 8 x 2", according as D is, or is not, uneven. For we may resolve —D into any two rela- tively prime factors, and may take one of them (with any sign we please) for A, and the other for C. There are no ambiguous forms of the type (ii), unless D==7, mod 2, or =0, mod (1+7)*. For in the equation D=B (B—[1+7] C), if Bis uneven, we have D=7, mod 2, because C must be uneven; if B is semieven or even, we have D==0, mod (1+7)’. If D=:, mod 2, we resolve D into any two relatively prime factors X and Y, and writing B=X, B—(1+72) C=Y, we find bee a at F ‘whieh is in- a 1864.] Prof. Smith on Complex Binary Quadratic Forms. =——- 287 tegral because X and Y are uneven, and uneven because X is not —Y, mod 2. Thus if D =z, mod 2, there are 4 x 2" ambiguous forms of the type (ai). Again, if D=0, mod (1+7)*, we may resolve D in any way we please into two factors having 1+2 for their greatest common divisor ; we find in this way 8x2" ambiguous forms of the type (ii). There are no ambiguous forms of the types (ili) or (iv), unless D=1, mod 2, or == 2, mod 4, or =0, mod(1+7)’. Forif in the equation D=B(B—2C), we suppose B uneven, we find D=1, mod 2; if B is semieven, B°=2/, and 2BC=2(1+7), mod 4, whence D==2, mod 4; lastly, if B is even, D=0, mod (1+). The same reasoning applies to the equation D=B(B—2iC). If D=1, mod 2, we resolve D in every possible way into the product of two factors relatively prime; let D=XxY be such a resolution, then D=7X x —7Y is another ; and it will be seen that according as the last coefficient in the two forms [2% X, <> *|- [ iX, is uneven or not uneven, so the last coefficient in the two forms | 20x = ae | E — 9% iX S| 5) 5) 9 2 9 3 is not, or is, uneven; 7. e. there are 2 x 2" ambiguous forms of each of the types (il) and (iv). If D= 2, mod 4, we resolve D in every possible way into two factors, of which 1+ is the greatest common divisor ; we thus find 4 x 2» uneven forms of each of the types (iil) and (iv). Lastly, if D=0, mod (1+7)’, we resolve D in every possible way into two factors of which 1+2 is the greatest common divisor, and we obtain 8x2 forms of each of the types (iii) and (iy). The result of this enumeration is that if D be uneven, or semieven, or = 2, mod 4, there are 8 x 2" ambiguous forms; if D==2, mod 4, or =0, mod (1+7)*, but not mod (1+2)’, there are 16x 2"; and if D=0, mod (1+7)’, there are 32x 2". On comparing this result with Table IIL., it will be seen that in every case there are four times as many ambiguous forms as there are assignable generic characters. (2) Let f=(a, 6, c) be any form of an ambiguous class ; if (I)= is an improper automorphic of /, X, p, v satisfy the equations Be oe — heen st ak, ag sys 8 ey GN) NL taatity 20% agen es [o-", |. tol. SUM) - X—Y i ¥, —} and, conversely, if X, u, v satisfy the equations (1) and (2), (I)= can is an improper automorphic of f. Let a, y, p, g (of which a Ae y are relatively prime) be a system of integral numbers satisfying the equations pa=n, PUES 1, } ee (3) qa=p+l, gy=y; 288 Prof. Smith on Complex Binary Quadratic Forms. [June 16, and let 2=0, 1—i, 1, or —2, according as 0, 1—?, 1 or —i satisfies the congruences p+6a=0, mod 2, q+6y=0, mod 2, which are simultaneously resoluble, and admit of only one solution, because a and y are relatively prime, while gza—py=2. Then it willbe found hea by the proper transformation | (= a, + (p+) 2 (qtby) | fis transformed into an ambiguous form ¢, which will be of the type (i), (ii), Gii), or (iv), according as §=0, 1—7, 1, or —7. It will also be seen that, subject to the condition that a and y are relatively prime, there are always four, and only four, solutions of the system (3), represented by the formula ta, id , ra inte, There are thus four cee ee | included in the formula (J), two of them transforming f into the same ambiguous form ¢, and the other two transforming f into the same form taken negatively. The four transforma- tions (J), and the two ambiguous forms ¢ and —¢, we shall term respect- ively the transformations and the ambiguous forms appertaining to the im- proper automorphic (I). If we now form the transformations appertaining to every improper automorphic of f, it can be proved (A) that these trans- formations will all be different, and (B) that they will include every proper transformation of f into an ambiguous form. (A) As the four transformations appertaining to the same improper au- tomorphic are evidently different, it will be sufficient to show that if (J) and (J') appertain to the improper automorphics (I) and ce the supposi- tion (J)=(J’) he (1)=(I'). From the equations axa, y=y, ptba=pl+O'a’, gqt+oy=q'+6'y' (which are equivalent to the symbolic equation (J)=(J')), combined with the system (3), and with a similar system containing the accented letters, we find (6—6')e?=r'—2r, (0-8 )ay=p'—p, (0-8) Y=v —r; whence again (§—6’) (aa +2bay+ey*?)=0, by virtue of equation (2). The coefficient of §—6' is not ZERO, for D=0°?—ac is nota square; there- fore §— a'=0; i.e. = Ny p= py v= Y', oS Gee (B) Let oS form ; according as ¢ is of the type (1), (41), (iil), or (iv), let 6=0, 1—2, 1, or —t; let also A=2a3—Oa’, p=ad+Py—flay, y= 2yd—by’; into an ambiguous then \f we =(T) is an improper automorphic of f; for pe —rv=(ad— —ByyY=1, and \a+2ub+re=0, because of the ambiguity of the form into which / is transformed by 1864.] Prof. Smith on Complex Binery Quadratic Forms. 289 ” B . Also ie Bp | appertains to (I); for, writing p and q instead of a, | __|a, 3(p+0a) is ' Y> 3| F | Y> i(q-+6y) 9 Q, YY Ps q (of which a and y are relatively prime) being four numbers which satisfy the system a, (3) 5, :2~.¢. ri i It follows from (B) that, if we calculate the ambiguous forms ¢ and —¢ appertaining to every improper automorphic of f, we shall obtain all the ambiguous forms to which fis equivalent ; it remains to see how many of these. ambiguous forms are different from one another. If (1) 26—6a, and 26—6y, we have appertains to (I), an improper automorphic of /. = ? wt is any given improper automorphic of f, all its similar auto- cant morphics are contained in the four formule (Tye x (I), (T)**1x (1), (Tx aay we | ore =100 0, —1 | x), where & is any positive or negative number, and (T)= (Te 1 x t, —u, 6, —u,c u,a, t, +u, 6}? [é,, u,] representing a fundamental solution of the equation ¢’—Dw’?=1. Similarly, if (J) represent the four transformations, appertaining to (1), by which f passes into ¢ or —¢, all the proper transformations of f into ¢ or —g¢ are included in the formula (T)*x (J). We shall now show that the four transformations included in the formula (T)* x (J) appertain to the improper automorphic (T)**x(I). Writing a= (t,— buy, )a—cupy, Pe=(4—bu,) p—eu,g, ype auzat (t+ bu,)y, = auypt (t+ bu,)q, Nop = (toy — Stlo,) A— Claes rhc: es (toy —FUtg,) P—CUgyY = AU yA + (toy + ba, ps Vo, = AUox et (toy, + Butgy.)¥5 we find immediately L ba ) ky —A Tye x (J) = | MF Pet k) |) (T)2 x (1) = | M2” 2k ( ) ( ) Vhs & (a+ 5 yx) ( ) \ ) V2k,» — P2k Also attending to the equations (2) and (3), and to the relations : ty =O,— Dury Uap = 2b; Uys we obtain, after substitution and reduction, Pp ye =Aoys Py Vor=Hor— 1, We %e=Ho Tl, ge Ve=Vo i.e. (T)*x (J) appertains to (T)?* x (1), if (J) appertains to (I), | It follows from this result that the ambiguous forms appertaining to (1) and to (T) x(I) are the same ; for fis transformed into the same forms by (J) and (T) x (J); and conversely, if the ambiguous forms appertain- 290 Prof. Smith on Complex Binary Quadratic Forms. [June 16, ing to two different automorphics (I) and (1’) are identical, an equation of the form (I')=T™ x (I) will subsist ; for if (J) and (J’) are the transfor- mations appertaining to (I) and (1’), smce by hypothesis (J) and (J’) transform I into the same form, we must have an canny of the form C= Cry" x (): nt (J') appertains to (1’), and (T)* x (J) to (Ee x (1) ; therefore (1') = Ce x (I), by what has been shown above (A). If then we calculate the eight ambiguous forms appertaining to the four improper automorphics (bo, )xO Dx@, (Th, \x xO ‘these eight forms will be the only ambiguous forms equivalent tof. Thus every uneven ambiguous class contains eight ambiguous forms. Combining this result with the preceding we obtain the Theorem, “The number of uneven ambiguous classes is one > halt of the whole number of assignable generic characters.”’ The number of semieven and even ambiguous classes is determined by the two following Theorems :— ; 7 «When D= +1, mod 4, there are as many even as a ambiguous classes.” BS Sie «When D=1, mod 2, there are as many semieven as uneven ambiguous classes, or only half as many, according as there at altogether as many semieven as uneven classes, or only half as many To prove the first of these theorems, let D==<", mod 4, and let Meat 2=(24#, 5), it is evident from the principles of the composition of forms that if (@) is a given semieven ambiguous class, the equation (3) X (¢)=(1+2) x (/) is satisfied by one and only one even ambiguous class (f) ; in addition to this we shall now show that, if (/) is a given even ambiguous class, the same equation is satisfied by one and only one semieven ambiguous class (¢) ; from which two things the truth of the theorem is manifest. First, let the whole number of even classes be equal to the whole number of semi- even classes*; then the equation (2) x (¢) =U. +2) x (/) * That if D==-+1, mod 4, there are either as many semieven as even classes, or else three times as many, is a theorem of M. Lipschitz (Crelle, vol. liv. p. 196), of which it is worth while to give a proof here. The number of even classes is to the number of semi- eyen classes, as unity to the number of semieven classes satisfying the equation (2) x(@)=(+4) x(f), f representing any given even form. ‘To investigate the semieven classes satisfying this equation, apply to f a complete system of transformations for the modulus 1 +2, for example, the transformations . see | 1+2, 0 142, 1 0, 1+ |, DO divly 0, 1 > 1864.] Prof. Smith on Complexe Binary Quadratic Forms. 291 is satisfied by only one semieven class (@) ; and this class is ambiguous, for the equation is satisfied by the opposite of (¢) as well as by (¢) itself ; therefore (¢) and its opposite are the same class, or (@) is an ambiguous class. Secondly, let the number of semieven classes be three times the number of even classes ; then the equation (2) x@=A+9 (Pf) is satisfied by three and only three different classes (¢) ; but it is also satis- fied by the opposites of these classes ; therefore one of them is necessarily an ambiguous class. Let that class be (¢,) ; the other two are defined by the equations (+2) @)=(%) x (ho), A+2) (62) =) X (45); and cannot be ambiguous classes; for by duplication we find (G) x @)=A+9 (@), ) XG)=UA+4 (a) 5 whereas every semieven ambiguous class produces (1+7)o, by its duplica- tion *, The second theorem may he proved as follows. Let S=(U1+¢]p, % [1 +¢]r) be a semieven form of determinant D; and let : D—1 a=((1+) ii, Ta) we suppose that pis uneven. The equation (c,) X (¢)=(/) is satisfied by one uneven class (¢,), or by two (¢,) and (¢,), according as the forms d,=(p, q 2ir), and ¢,=(2ip, g, 7”), if 7 is uneven, or the forms $= (p, q 2ir), and ¢,=(2ip, [1+7] p+q, p+[1—cz]q+7), if r is even, are or are not equivalent}. If any one of the forms f, ¢,, ¢, is ambiguous, the others are so too; the same thing is therefore true for the classes (/), (%,), (¢,). Thus the number of semieven ambiguous classes is equal to or and divide the resulting forms by 1+7; of the quotients, one, or three, will be semieven, according as D==-++1, or -++-5, mod (1+7)°. It will be found that each of these semi- even forms satisfies the equation 2X¢=(1+7)x/; and, conversely, every semieven form ¢ satisfying that equation is equivalent to one of these forms; for, from any trans- formation of (1+-2)f into =X ¢, we may (by attributing to the indeterminates of = the values 1, 0) deduce a transformation of modulus 1-+72 by which / passes into (1+-2) ¢ ; 2.€., 6 is equivalent to one of the forms obtained by the preceding process. It only remains to show that when there are three of these forms, they constitute either one or three classes, but never two. For this purpose it is sufficient to consider the three semi- even forms o,= (1 +2, 1, -75). g,, and o,, obtained by the preceding process from the form >. These forms satisfy the equations o,Xo,=(1+2)o,, o,Xo,=(1+)e,, 6, Xo,=(1+7)o,, o,Xo,=(1+2)o,; from which it follows that any one of the suppo- sitions 6,=0,, 6,=0), 6)=0, Involves the other two. * For the definition of the classes («,), (o,), (¢,) see the preceding note. t The forms ¢, and ¢, are obtained by applying to f a complete set of transformations of modulus 1+, dividing the resulting forms by 1+-7, and retaining only those quotients which are uneven forms, VOL, XIII, Z 292 Prof. Smith on Complex Binary Quadratic Forms. [June 16, is one half of the number of uneven ambiguous classes, according as the classes (¢,) and (¢,) are identical or not; z.e., according as the whole number of semieven classes is equal to or is one-half of the whole number of even classes. The demonstration in the ‘ Disquisitiones Arithmeticee,’ that the number of genera of uneven forms of any determinant cannot exceed the number ~ of uneven ambiguous classes of the same determinant, may be transferred without change to the complex theory. We thus obtain a proof (inde- pendent of the law of quadratic reciprocity and of the theorems which determine the quadratic characters of ¢ and 1+) of the impossibility of one-half of the whole number of assignable generic characters ; and from that impossibility, as we shall now show, the quadratic theorems are them- selves deducible. (1) If p is an uneven prime =1, mod 2, there are two genera of un- even forms of determinant p: of these one is the principal genus, and has the complete characters (4)=1, y=1; the other, containing the form (7,0, +2p), has the particular character y= —1; whence it follows that every uneven form of determinant p, which has the character y=-+1, is a form of the principal genus, and has the character [- | =+1. Again, 1\. ‘ fp=1,mod 4, the form{ 2i, i, )is an uneven form of determinant p; oad 20 this form has the particular character y= —1, because —Pte ==7, mod 2 Z it is therefore not a form of the principal genus; but it has the character (4) =1, because 2¢ is a square; therefore, if p==1, mod 4, every uneven form of determinant p has the character E | =+1. i (2) There is but one genus of forms of determinant é, and its complete character is a=-+1; there is also but one genus of forms of determinant 1+7, and its complete character is B=+1. (3) Let » and g be uneven primes of which the imaginary parts are even ; to prove the law of reciprocity, it will suffice to show that if an =I, then rea =1. The equation fi =1 implies the existence of a con- gruence of the type w°—p==0, mod gq, and consequently of an uneven form of determinant p, and of the type @ Ww, nee ) This form has the fi character y=+1, because g==1, mod 2; it therefore has the character l= ef] (4) To prove the equatioa B == (— 1 sh) Me in which we may sup- 1864. | Prof. Smith on Complex Binary Quadratic Forms. 293 pose that the uneven prime p is primary, it will suffice to show (i) that : 1 a (Np—1) oNiete h a (Np—1) if =| PALA then (1) i aeting et) =1, then a ‘ a 5 : w—7 —{!=1. (4) Let [= | =1; then, if w°—7=0, mod ( » W; ) fe @ p St Srenale is a form of determinant 7; it therefore has the character a=1, Zz. é. (cols pamela B (ai) Let Gay ine then p=1, mod 4, and the form (7, 0,7) is an uneven form of determinant p; it therefore has the character (Z)= +1; whence [= | =+1. Pp P (5) Similarly, if p=p,+¢p, is an uneven and primary prime, to prove (potpi)?—1 the equation E& —(——1 8 we shall show, (1) that if Ee ills (Potp1)?—-1 (po+p1)?—1 1+72 ene ol)—. © ..—=1; (ii) that if (—1). 8. =], then [-s*|=1. (i) Let | =1; then there is a form of determinant 1+7 and of ] d ; the type (p, w,° —"—"); this form has the character B=+1; there- ype | p p (po+p1)?2—1 (po+p1)?—1 fore (—1) & =~ =+1. (ii) Let (—1) ® =-+1; then p is either==1— 27, or==1, mod (1+2)’; if p=(1+2)"44+1—2, ([1+7]’, @ 1—2/7) is an uneven form of determinant p; this form has the character y=-+1, and consequently it also has the character E ] =-+1; there- *\ 3 fore | = [Se] =+1; if p=(1+7)’£+1, one or other of the forms ([1+7]’, 1, —#), and ({1+2]’, 1+[1+2]*, 1—4) is an uneven form of determinant p, having the character [z | =]; therefore in this case vo [14] = [C2] IV. The representation of Binary Forms of the principal Genus by Ternary Forms of Determinant 1. The solution of the general problem, “To find the representations (if any) of a given binary by a given ternary quadratic form,’’ depends, in the case of complex as of real numbers, on the solution of the problem of equiva- lence for ternary forms. Extending the methods of Gauss to the complex theory, we find the necessary and sufficient condition for the primitive* * If a matrix of the type transforms a ternary into a binary quadratic form, the representation of the binary by the ternary form is said to be primitive when the three determinants of the matrix are relatively prime. Z 2 294 Prof. Smith on Complex Binary Quadratic Forms. ! [June 16, representation of a binary form f of determinant D by a ternary form of determinant 1 to be, that f should be a form of the principal genus ; or, if D= +1, mod 4, that f should be a form either of the principal genus, or else of that genus which differs from the principal genus only in having the character y= —1, instead of y=+1. Again, because the reduction of Lagrange is applicable to complex binary forms, the reduction of Gauss* is applicable to complex ternary forms. It is thus found that the number of classes of such forms of a given determinant is finite ; and in particular that every form of determinant 1 is equivalent to one or other of the forms —2’?—y’—z2* and x? +iy’+ iz’, of which the former cannot represent num- bers==2, or==1+7, mod 2; and the latter cannot primitively represent numbers==2, or==2(1+7), mod 4. The method of reduction itself sup- * If F=ax?+a'y?+a" 22+ 2bye+20'nz+2b’xy is a ternary form of determinant A, and Aw?+ A’y?4 A’”22?+2Byz+2B/xz+2B’ ry its contravariant, by applying the reduc- tion of Lagrange to the form ax?+2d’xy+a’y?, we can render N.a<2 VN. A” (Dirichlet in Crelle’s Journal, vol. xxiv. p.348); and by applying the same reduction to the form A’y?+2Byz+A”2?, we can render N.A”7<2/N.aA. The reduction of Gauss consists in the alternate application of these two reductions until we arrive at a form in which we have simultaneously N. a2 AGN. AONE AvN=2 YN.aA; and consequently N.aX<44/N.A, N.A’<4 hy ee If A=l, we have N.aX4, N.A”’=4; whence a and A” can only have the values 0, +1, +7, +(1+2), 4(1—2), +2, +22; and it will be found, on an examination of the different cases that can arise, that the reduction can always be continued until a and A” are either both units, or both zero. In the former case, by applying a further transformation of the type if es rtd 0, L ad 2 0; } Ose the coefficients 5, 0’, b’’ may be made to disappear; and we obtain a form equivalent to F, and of the type ex?+-e’y?+6’’2?, «, e’, e” representing units of which the product is —l. In the latter case the form obtained by applying the reduction of Gauss is of the type ay? al 22 +-2byz+20'we2 ; whence a’}’2=1, so that 0’ is a unit which we shall call ¢; and the form e7y +a"z?-+4-2byz+2exy, by a transformation of the type 1, 0, p’ 0, 1, 0, 0, 1 is changed into one of the four forms 6v?+2crz, e®%y2+-22+2exz, ey2+72? + 2enz, e7y?-+-(1-+7)2?426xe; of which the first two by the transformations 67-12, B90; te 1 0, 0, —e ei, «2, € . e—lz, 0, O 0, —7, —1 0, 7% & are changed into the form —x?—yv?—z’; the last two by the transformations 0, -e O ée—1, e-1, e-1(1—2) —e-l, 0, O ; —6, —6, 6 0, —te?, —1 0, —1, 7 are changed into #?-+-2y2+i22, (See Disq. Arith. art, 272-274.) 1864.] Prof. Smith on Complex Binary Quadratic Forms. ~. 295 plies a transformation of any given form of determinant | into one or other of those two forms. If D=z, or 1+2, mod 2, no binary form of determinant D can be re- presented by —x*—y’—z’, because D cannot be represented by the con- travariant of that form, 7.e. by the form —z2?—y’—z’ itself. Conse- quently, if D==7, or 1+7, mod 2, the binary forms of its principal genus are certainly capable of primitive representation by x?+ iy? +72”. If D==1, mod 2, no form of the principal genus can be primitively represented by 2*+72y’?+7i2°. Let f=(a, 4, c) be such a form, and let us suppose, as we may do, that & is even, so that ac==1, mod 2, and a=c=1, mod 2 (the supposition a==c=z is admissible, because f is of the principal genus) ; if possible, let the prime matrix a, a! ; p! all, Bl | (of which A, B, C are the determinants) transform 2*+7y?+ 72’ into f; we have the equations a=a’+ ia!’ + ia!”, c=6? +73 +73'", D=A’—7B’—:C’, from which, and from the congruences D=a==c=1, mod 2, we infer the incompatible conditions «! + 7a!'=/!' +76"=0, mod1+7, A=1, mod1 +27; i.e. fis incapable of primitive representation by a?+7y’?+ 72°. If, there- fore, D=1, mod 2, the forms of its principal genus are capable of primi- tive representation by —a2*—y’—z*. We may add that when D= rl, mod 4, the forms of that genus which differs from the principal. genus only in having the character y=—1, instead of y=+1, are capable of primitive representation by x*+?y?+72’, but not by —a’—y’—2’. Lastly, let D=0, mod 2. If D=2, or=2(1+2), mod 4, D cannot be primitively represented by «?—7iy?—iz*, the contravariant of x? +7y*+ 72’; i.e. no form of determinant D can be primitively represented by 2? +2y? + 22’; so that forms of the principal genus are certainly capable of primitive repre- sentation by —2°—y’?—2*. But if D=2z, or=0, mod 4, the forms of the principal genus are capable of primitive representation by both the ternary forms —2*—y’—2* and w?+iy?+iz*. For if f=(a, 6, c) be a form of the or =1, mod 2; so that a ternary form cf determinant 1 and of the type Spe? + 2qye+ 2¢q' x2 will be equivalent to —x#’—y’— 2’, or ie x’ +iy?+i2*, according as p’==0, or ==1, mod 2, on the one hand, or p!"==7, or =1++7, on the other hand. Again, if (%, k') is a value of the expression / (a, —0, c), mod D, (in which we now suppose a uneven and 6 semieven or even), des 7} 5 ae is another value of the same expression; and it can be shown* that when * If f+ p’2?+2qy?+2¢’xz is a ternary form of det. 1, derived from the value (4, k’) of the expression 4/ (a, —%, c), mod D, & is the coefficient of yz in the contravariant Observing that a==1, mod 2 form. Hence a=k*?—D(q’?—ap”), or ap” =y?4+= D 296 Prof. Smith on Complex Binary Quadratic Forms. [June 16, D=2:, or 0, mod 4, one of the two forms of determinant 1, and of the ype f+ ple + 2qye+ 2q' 22, which are deducible by the method of Gauss from those two values, satisfies the condition p''==0, or ==1, mod 2, while the other satisfies the condi- tion p=7, or 1+7, mod 2; that is, fis capable of primitive representation by both the forms —a*—y’?—2’ and a°+iy?+i2°. The preceding theory supplies a solution of the problem, “‘ Given a form of the principal genus of forms of determinant D, to investigate a form from the duplication of which it arises.’ Let f=(a, 6, c) be the given form, and let us suppose (as we may do) that a and c are uneven. When D=z, or 1+7, mod 2, let a, f i al, 3 all, Bll be a prime matrix (of which the determinants are A, B, C) transform- ing a°+7y?+iz? into (a, —6, ce); and let @ represent the binary form (C—iB, A, iC—B); then the matrix B+i8", BB, ii) gy a'+ia", a, a, —t(a! —ia!') transforms f into ¢x@*; and is a prime matrix, for its determinants -C—7B, 2A, and 7C—B are not simultaneously divisible by any uneven prime (because A, B, and C are relatively prime), and are not simul- ¢7=0, or 1, mod 2, we see that p”=0, 1, or==¢, 1+2, mod 2, according as "=" =o, A, D \2 oo ) Pada ed ance Bat nL an OLE erer 8 whieh aetaaene gruous to 1+2, mod 2, if D=0, mod. 4, and to 7, mod 2, if D=2i, mod 4, since & is a— k2 D evidently uneven in either case. From this it appears that if PD \2 (é+T) then oe VARET a 2q'az, p Lh or 1, mod 2, and in the other p” =z, or 1+7, mod 2. * This assertion may be verified by means of the identity (G19 Gos) ( Pot” +P LY! +PLY AD LY’ )Y +(GoPs+Pods— Vi PoP Jo) (Po! LP BY +P.L'YAP LY’) X (Jot! +9, 2Y' +9.2'Y +9,2'Y’) +(P) P2—PoPs) (Go®L! FU 2Y +Q2'Y+G0'Y'P = [(Po% PQ) £?+( Pods —P39o TP Yo Po )2Y + (2193 —P3)Y 7] X [Pods Pio) 2” + ( Pods —PsVot Por — P1 Vo) 2Y + (Pos — Ps2)9"] 5 in which we have to replace the quantities Po Pi Po Ps % %1 G2 Vs =0, 1, mod 2, =7, 1+2, mod 2; that is, in one of the two forms f+ p"2?-+-2qyz+ by the elements of the matrix (Z). 1864.] Prof. Smith on Complex Binary Quadratic Forms. 297 taneously divisible by 1-+2, because (Z) is congruous, for the modulus 1 +2, to the first or second of the matrices OHO 1000.1 Ne FOO Ge ee a & 0, ee) and tn i op (4) according as a==2, c=1, or a=1, c=7, mod 2. Consequently ¢ is a form the duplication of which produces f. When D==1, or=0, mod 2, let the prime matrix transform —2*—y’?—z<* into (a, —6, c). As we cannot have simultaneously a=Bf, a'=Pf', a=", mod (1+72), we,may suppose that a and ( are incongruous, mod (1+). If¢=(B+iC, 7A, B—7C), the matrix Gala 22, ip, nna i (Z) anal. wowace. eae yk ee cae transforms f into ¢ xX ¢, and is a prime matrix, being congruous to one or other of the matrices (Z') for the modulus 1+7, in consequence of the two suppositions that a and ¢ are uneven, and that wand 6 are incongruous, mod (1+z2): so that f arises from the duplication of ¢. From the resolubility of this problem we can infer (precisely as Gauss has done in the real theory) that that half of the assignable generic cha- racters which is not impossible corresponds to actually existing genera. We can also deduce a demonstration of the theorem that any form of de- terminant D can be transformed into any other form of the same genus, by a transformation of which the coefficients are rational fractions having denominators prime to 2D. For every form which arises from the dupli- cation of an uneven primitive form—that is, every form of the principal genus—represents square numbers prime to 2D, and is therefore equivalent to a form of the type (x. [ls p =} But (1, 0, —D) is transformed Pees into (x ii N H probable. A mixture of diphenylamine and chloride of benzoyl, when heated, fur- nishes a thick oil, which solidifies on cooling. Washed with water and alkaii, and recrystallized from boiling alcohol, in which it dissolves with difficulty, the new compound is obtained in beautiful white needles. Ana- lysis has confirmed the theoretical anticipation, C, H, C,,H,,NO=C,H, {N. | C,H, 0} This substance has become the starting-point of some experiments which I shall here briefly mention, but to which I intend to return hereafter. On addition of ordinary concentrated nitric acid, the benzoyl-compound liquefies and dissolves. From this solution, water precipitates a light- yellow crystalline compound, OF H., N, O, = C, H, (NO,) N, C, HyQ: oon ht 7 5 which dissolves in alccholic soda with a scarlet colour, splitting on ebulli- tion into benzoic acid_and reddish-yellow needles of nitro-diphenylamine, C,H, ‘ C,, LB N,0,= be ee (NO,) N. If, instead of ordinary nitric acid, a large excess of the strongest fuming nitric acid be employed, the solution deposits, on addition of water, a crys- 1864.] Da Silveira on the Mean Declination of the Magnet. 347 talline compound of a somewhat deeper yellow colour, containing probably C, H, (NO,) C,, H,, N, 0, = C,H, (NO,) $ CHO 7 5 This substance dissolves in alcoholic soda with a most magnificent crimson colour. Addition of water to the boiling liquid furnishes a yellow crystalline deposit, benzoate (?) of sodium remaining in solution. The yellow powder is dinitro-diphenylamine. From boiling alcohol, it crystallizes in reddish needles, exhibiting a bluish metallic lustre. The analysis of the compound has led to the formula C, H, (NO,) C,, H, N,0, = C, (HE (N03 |. The chemical history of these compounds will be the subject of a special communication. XVI. “ A Table of the Mean Declination of the Magnet in each Decade from January 1858 to December 1863, derived from the Observa- tions made at the Magnetic Observatory at Lisbon; showing the Annual Variation, or Semiannual Inequality to which that ele- ment is subject.” Drawn up by the Superintendent of the Lisbon Observatory, Senhor pa SitverRA, and communicated by Major-General Sapine, R.A., President of the Royal Society. Received June 6, 1864. I have much pleasure in communicating to the Fellows of the Royal Society a copy of a Table which I have received from the Superintendent of the Magnetic Observatory at Lisbon, containing the mean values of the Declination in each Decade from the commencement of 1858 to the close of 1863, with corrections applied for the mean secular change, and showing, in a final column, the difference in each decade of the observed from the mean annual value derived from the 216 decades. This Table is a counter- part of Table VII. in Art. XII. of the Philosophical Transactions for 1863, p. 292, differing only in the substitution in the Lisbon Table of de- cades for weeks, and the addition of the year 1863. This general confirmation by the Lisbon Observatory of the annual variation to which the Declination is subject, “ the north end of the magnet pointing more towards the East when the sun is north of the Equator, and more towards the West when the sun is south of the Equator,”’ is very satisfactory. Inthe Lisbon Table the disturbances have not been elimi- nated. io} | : eo ae | : “ G-cO I— | £20 Fe | 22e0— | SFE Z | BFE 6 9-68 SI | 0-21 23 | 0-9 82 { Baz re | B.e2ze | “E *S ¥8S O— | 60 72 | IF O— | 8-09 FZ | 3-0F 6 HES SI | F6222 | 99 82 | B82 SE | 39 Le | “gh jerreeeske~ & ITI | $99 86 | GOS O— | OL HZ | F-02 OL | BOF ST | 06 ZS | BIE 8B | 96 SE | Sar 9¢ Se 3 | j s 0-66 O— | 881 72 | 4-69 0— | G81 ce | 9-Gh¥ OL | LE 9T | 8-66 e% | O-FG 82 | 0.46 FE | GIT LE ‘¢ . €-90 O+ | 1-71 9% | 6:80 I— | 0-€2 93 | 9-98 11 | 96 SI | FS Fo | FOZ 08 | FOG PE | Z-8T 62 4 meee THCY = 1-20 O+ | 6-60 G2 | L-8I I— | 0-82.92 | 8-S¢ ZL | 3&2 ZI | 98h €@ | G29 08 | 0-6 FE | BLE 6E I > 0c O+ | 8zP az | @421—| Lot Zz | o8r pt | 861 ZT | 80T Gz | 9-9¢ Te | 0.9% rs | ZLT 68 3 Ss 0-0F O+ | 8247 GS | FOE T— | ZFS L42 | BSS IL | O42 8I | HLE GS | Och ze | Os GSE | ZTE OF "Z “*°** UDICIN $ 140 OF | GIT GS | 99h I- | GLE 9% | BIE IL | 29 BI | OSF Go | BOT Te | 208 FE | 9.99 6E ‘I ‘Ss 8:70 O+ | 9-21 G2 | 87S I— | FL L242 | 8-01 ZI | 0-42 81 | 80F 98 | 8-FE Te | £-OF Ge | O-IL 8g "e > WIL OF | 2-61 GS | 0-70 2— | 3EZ 12 | F-2E ZI | 822 8l | 36192 | HB Te | £88 GE | G-ZT OF 4 *** ATenIQoyy Q 948 O+ | SP GS | TEI 2—- | G86 22 | 9-6 FL | 0-81 6I | FES 9S | BOL OF Gr 9 | 9-FL IP “4 ~ S 160 I+ | GIT 93 | €22 2— | BEE 82 | O-8F SL | FLT GL | HIF 6Z | 9-61 OG | FZI LE | 80 IP = : = Ley 0+ | GOS G3 | GIE Z— | 0-32 84 | BSS SI | 9-81 06 | 82¢ 92 | 96G Of | £88 9f | GTS TF 43 ‘e+ Avenues % VLE OF | 2SF GZ | L£0F c— | 6-66 8% | O-LZEL | 88002 | OF 92 | 2ze 08 | Gz Ze | 0-80 ZF gt Ss 4 1 4 / 4 4 4s 4 4 / aA 4 Ww / 4 / aA J 4} / S "srvak "sales te te +12 tol® + 12 = oie +o1@ +o1Z g XIS 0q} ‘osueyy) 3 \ : ‘Do jo suvowl *poqoor | TeTHIIS LOF : : E = : i 4 . 2 / ‘opeoa(, “ST IUOTY > ayy wory |-s09 suvayy| payvartog | “SUN S981 Z981 1981 0981 gest |} “saet D sooualayiq, | 4 3 = "SO8T toquiesagy 0} gggy Arenuve wo. ‘10;earosqQ WOgsI'T 94} Je MOLeUITIEC] 48944 94} JO SULOTHT 349 \f the Magnet. x xi ion O rst x2 = —) -- 1864.] . Da Silveira on the Mean Declinat © i GZ GG 1G GS cG G6 cZG GZ GG cZ6 GG GZ io GG &Z ¥G VG VG ¥G &G &G G-18 o+ €-26 Ot Il 2+ 0-40 2+ 8-F¢ I+ 9-Ch I+ -9§ I+ £6 I+ L810 I+ L0F t+ | 8-L0 0-0¢ 6:01 9-9 v-0¢ 9-FF VLE 8-66 T-8¢ 8-GZ 6-6 GGG VE LG G-1G G-GV GFE LGV 0-46 9-01 T-0¢ 8-6 FiO iG co 6G &@ £% 6G 66 £3 63 &@ FG VG VG VG VG VG ¥G VG ize FG GuiVE VG 9-,,8h 19 61:09 6-/BG iV Ouw4 PV a fen aI SS ge ae Gres OT v8 OT 0-29 2% G1 66 F-69 SE 5-90 8& FES Z 0-SE ET 8-18 06 GGG 96 P-G6 GE VIP SE 0-0¢ 8 0-81 &T GL 16 VIP LZ 9-G1 €& 1-91 S& 68S f GSI SI 0-2F 16 0-66 £2 8-6) o£ GLI GE 9-6¢ L 0- B58 When hydrogen was passed through serum, after the lapse of a day or two a tough elastic product was obtained. In experiments tried by passing hydrogen through albumen greatly diluted with water, I found, after the lapse of a few days, a floceulent de- posit very similar in appearance to the deposit of mucus which often takes place when urine is allowed to stand ashort time. This point, how- ever, requires further investigation. I tried also the effect of passing hydrogen through a portion of intestine inserted into an albuminous fluid. I have not as yet been able to form either the dense hard or viscid frothy substance by this method. I repeated the experiment for the formation of fibrin from albumen, by decomposing the water of its composition by elec- tricity. I must admit this is the most difficult, troublesome, and unsa- tisfactory of all the methods I have employed. I find that the great ten- dency of the poles to form different substances on them, and the great rapi- dity with which they grow together, lead, without the greatest care, to the belief that two different substances, differing only in density, are formed at one and the same pole, so intimately blended are they together. Thus I was led to believe at first sight that a dense hard substance was formed at the oxygen end, and not until I had repeated the experiment many times did I discover that the substance belonged to the hydrogen and not to the oxygen pole, and had grown across from one pole to the other. I have obtained on several occasions fibrin and chondrin at the same time by conducting hydrogen and oxygen derived by the decomposition of water by voltaic electricity through separate tubes. The oxygen passed into slightly acid albumen formed fibrin ; the hydrogen passed into alkaline albumen formed either the chondrin or else the frothy and viscid material. The temperature was kept up at 98° F. in these experiments. On one occasion, however, I happened accidentally to reverse the current (that is to say, the hydrogen was passed into the acid, and the oxygen into the alkaline albumen), when no chondrin or fibrin was formed. The following conclusions I have arrived at after the study of the in- fluence which oxygen and hydrogen gases exert upon albumen when sub- mitted to their action separately at a temperature of 98° F., the normal temperature of the living body. Albumen under the action of oxygen forms, after the lapse of a longer or shorter period, fibrin. The fibrin thus artificially produced is of three distinct varieties, viz., Ist, the granular form; 2nd, a form allied to lymph incapable of being unravelled into fibrils ; lastly, the true fibrillated fibrm. The law which appears to regu- late the state into which the albumen is converted, as far as my observa- tion has gone, is one of molecular aggregation, similar to the electric deposit of metals, as the slower the fibrin is formed the more organized is it in substance. I have observed that when fibrin is rapidly formed it is almost always produced in the granular state; this is particularly the case with fibrin 354 On Organic Substances artificially formed from Albumen. June 16, formed from albumen by the decomposition of the water of its composition by voltaic means. Lymph I consider to be imperfectly formed fibrin more highly deve- loped than the preceding or granular form. It is possible for this arti- ficially formed lymph, under favourable circumstances, to assume a more organized appearance. I have no doubt that the fibrous outgrowths on the intestine would have become larger and more developed if the experiment had been carried on for a sufficient length of time. In fact almost all the fibrin formed round a platinum wire inserted into albumen is at first covered by outgrowths of a soft structure. These outgrowths, at the earliest period of their forma- tion, do not under the microscope present any appearance of fibrils. After the lapse of some time they appear to undergo condensation, and then to organize to such an extent that it would be difficult at first sight to deter- mine whether the substance might not be a portion of fibrous tissue. The alkalies, with the exception of ammonia, prevent entirely the forma- tion of fibrin. Ammonia, although it does not retard its formation, dis- solves it after the lapse of ashort time. The acids and absence of alkaline salts favour its formation. The opposite, however, is the case with the hydrogen products, as an alkaline state favours their production. The action of hydrogen on albumen, as far as my investigations have as yet proceeded, forms substances analogous to chondrin and mucm. I believe that the organic substances, chondrin and mucin, products formed ina living organism, are very closely allied to one another, if not varieties of the same substance, differing only in their mode of aggregation and stages of development, and the amount of water in their composition. Of the exact mode in which hydrogen acts on albumen we are at present ignorant. I have noticed that in some experiments sometimes one, some- times the other product was obtained, even when the same influences were apparently acting on experiments conducted at the same time. Considering the important physiological part that fibrin, chondrin, and mucin play in the living body, the production artificially of substances ‘analogous in their behaviour with reagents to those products formed in a living organism will, I trust, be taken as a sufficient excuse for submitting to the Royal Society a paper so obviously deficient in many parts, but which, nevertheless, it would require a vast amount of both time and labour to carry one step further, 1864.] Reductionand Oxidation of the Colouring Matter of the Blood. 355 XVIII. “ On the Reduction and Oxidation of the Colouring Matter of the Blood.” By G. G. Sroxus, M.A., Sec. R.S., Lucasian Pro- fessor of Mathematics in the University of Cainbridge. Received June 16, 1864. 1. Some time ago my attention was called to a paper by Professor Hoppe *, in which he has pointed out the remarkable spectrum produced by the absorption of light by a very dilute solution of blood, and applied the observation to elucidate the chemical nature of the colouring matter. I had no sooner looked at the spectrum, than the extreme sharpness and beauty of the absorption-bands of blood excited a lively interest in my mind, and I proceeded to try the effect of various reagents. The observa- tion is perfectly simple, since nothing more is required than to place the solution to be tried, which may be contained in a test-tube, behind a slit, and view it through a prism applied to the eye. In this way it is easy to verify Hoppe’s statement, that the colouring matter (as may be presumed at least from the retention of its peculiar spectrum) is unaffected by alkaline carbonates and caustic ammonia, but is almost immediately decomposed by acids, and also, but more slowly, by caustic fixed alkalies, the coloured product of decomposition being the hzematin of Lecanu, which is easily identified by its peculiar spectra. But it seemed to me to be a point of special interest to inquire whether we could imitate the change of colour of arterial into that of venous blood, on the supposition that it arises from reduction. 2. In my experiments I generally employed the blood of sheep or oxen obtained from a butcher ; but Hoppe has shown that the blood of animals in general exhibits just the same bands. To obtain the colouring matter in true solution, and at the same time to get rid of a part of the associated matters, I generally allowed the blood to coagulate, cut the clot small, rinsed it well, and extracted it with water. This, however, is not essential, and blood merely diluted with a large quantity of water may be used; but in what follows it is to be understood that the watery extract is used unless the contrary be stated. 3. Since the colouring matter is changed by acids, we must employ re- ducing agents which are compatible with an alkaline solution. If to a solution of protosulphate of iron enough tartaric acid be added to prevent precipitation by alkalies, and a small quantity of the solution, previously rendered alkaline by either ammonia or carbonate of soda, be added to a solution of blood, the colour is almost instantly changed to a much more purple red as seen in small thicknesses, and a much darker red than before as seen in greater thickness. The change of colour, which recalls the dif- ference between arterial and venous blood, is striking enough, but the change in the absorption spectrum is far more decisive. The two highly * Virchow’s Archiv, vol. xxiii. p. 446 (1862). VOL. XIII. oD 356 Prof. Stokes on the Reduction and Oxidation [June 16, characteristic dark bands seen before are now replaced by a single band, somewhat broader and less sharply defined at its edges than either of the former, and occupying nearly the position of the bright band separating the dark bands of the original solution. The fluid is more transparent for the blue, and less so for the green than it was before. If the thickness be increased till the whole of the spectrum more refrangible than the red be on the point of disappearing, the last part to remain is green, a little be- yond the fixed line 6, in the case of the original solution, and dlue, some way beyond F, in the case of the modified fluid. Figs. 1 and 2 in the accom- panying woodcut represent the bands seen in these two solutions respec- tively. G Fig. 2. 4. If the purple solution be exposed to the air in a shallow vessel, it quickly returns to its original condition, showing the two characteristic bands the same as before; and this change takes place immediately, pro- vided a small quantity only of the reducing agent were employed, when the solution is shaken up with air. If an additional quantity of the reagent be now added, the same effect is produced as at first, and the solution may thus be made to go through its changes any number of times. 5. The change produced by the action of the air (that is, of course, by the absorption of oxygen) may be seen in an instructive form on partly filling a test-tube with a solution of blood suitably diluted, mixing with a little of the reducing agent, and leaving the tube at rest for some time in avertical position. The upper or oxidized portion of the solution is readily distinguished by its colour; and if the tube be now placed behind a slit and viewed through a prism, a dark band is seen, having the general form of a tuning-fork, like figs. 1 and 2, regarded now as a single figure, the line of separation being supposed removed. 1864. ] of the Colouring Matter of the Blood. 357 6. Of course it is necessary to assure oneself that the single band in the green is not due to absorption produced merely by the reagent, as is readily done by direct observation of its spectrum, not to mention that in the region of the previous dark bands, or at least the outer portions of it, the solution is actually more transparent than before, which could not be occa- sioned by an additional absorption. Indeed the absorption due to the reagent itself in its different stages of oxidation, unless it be employed in most unnecessary excess, may almost be regarded as evanescent in com- parison with the absorption due to the colouring matter; though if the solution be repeatedly put through its changes, the accumulation of the persalt of iron will presently tell on the colour, making it sensibly yellower than at first for small thicknesses of the solution. 7. That the change which the iron salt produces in the spectrum is due to a simple reduction of the colouring matter, and not to the formation of some compound of the colouring matter with the reagent, is shown by the fact that a variety of reducing agents of very different nature produce just the same effect: If protochloride of tin be substituted for protosulphate of iron in the experiment above described, the same changes take place as with the iron salt. The tin solution has the advantage of being colour- less, and leaving the visible spectrum quite unaffected, both before and after oxidation, and accordingly of not interfering in the slightest degree with the optical examination of the solutions, but permitting them to be seen with exactly their true tmts. The action of this reagent, however, takes some little time at ordinary temperatures, though it is very rapid if previously the solution be gently warmed. Hydrosulphate of ammonia again produces the same change, though a small fraction of the colouring matter is liable to undergo some different modification, as is shown by the occurrence of a slender band in the red, variable in its amount of develop- ment, which did not previously exist. In this case, as with the tin salt, the action is somewhat slow, requiring a few minutes unless it be assisted by gentle heat. Other reagents might be mentioned, but these will suffice. 8. We may infer from the facts above mentioned that the colouring matter of blood, like indigo, 7s capable of existing in two states of oxida- tion, distinguishable by a difference of colour and a fundamental difference in the action on the spectrum. It may be made to pass from the more to the less oxidized state by the action of suitable reducing agents, and re- covers its oxygen by absorption from the air. As the term hematin has been appropriated to a product of decomposi- tion, some other name must be given to the original colouring matter. As it has not been named by Hoppe, I propose to call it cruorine, as suggested to me by Dr. Sharpey ; and in its two states of oxidation it may conveniently be named scarlet cruorine and purple cruorine respectively, though the former is slightly purplish at a certain small thickness, and the latter is of a very red purple colour, becoming red at a moderate increase of thickness. 9. When the watery extract from blood-clots is left aside in a corked 2Dd2 308 Prof. Stokes on the Reduction and Oxidation (June 16, bottle, or even in a tall narrow vessel open at the top, it presently changes in colour from a bright to a dark red, decidedly purple in small thicknesses. This change is perceived even before the solution has begun to stink in the least perceptible degree. The tint agrees with that of the purple cruorine obtained immediately by reducing agents; and if a little of the solution be sucked up from the bottom into a quill-tube drawn to a capillary point, and the tube be then placed behind a slit, so as to admit of analyzing the transmitted light without exposing the fluid to the air, the spectrum will be found to agree with that of purple cruorine. On shaking the solution with air it immediately becomes bright red, and now presents the optical characters of scarlet cruorine. It thus appears that scarlet cruorine is capable of being reduced by certain substances, derived from the blood, present in the solution, which must themselves be oxidized at its expense. 10. When the alkaline tartaric solution of protoxide of tin is added in moderate quantity to a solution of scarlet crucrine, the latter is presently reduced. If the solution is now shaken with air, the cruorine is almost instantly oxidized, as is shown by the colour of the solution and its spec- trum by transmitted light. On standing for a little time, a couple of minutes or so, the cruorine is again reduced, and the solution may be made to go through these changes a great number of times, though not of course in- definitely, as the tin must at last become completely oxidized. It thus appears that purple cruorine absorbs free oxygen with much greater avidity than the tin solution, notwithstanding that the oxidized cruorine is itself reduced by the tin salt. I shall return to this experiment presently. 11. When a little acid, suppose acetic or tartaric acid, which does not produce a precipitate, is added to a solution of blocd, the colour is quickly changed from red to brownish red, and in place of the vriginal bands (fig. 1) we have a different system, nearly that of fig. 3. This system is highly characteristic; but in order to bring it out a larger quantity of substance is requisite than in the case of scarlet cruorine. The figure does not exactly correspond to any one thickness, for the bands in the blue are best seen while the band in the red is still rather narrow and ill-defined at its edges, while the narrow inconspicuous band in the yellow hardly comes out till the whole of the blue and violet, and a good part of the green, are absorbed. The difference in the spectra figs. 1 and 3 does not alone prove that the colouring matter is decomposed by the acid (though the fact that the change is not instantaneous favours that supposition), for the one solution is alkaline, though it may be only slightly so, while the other is acid, and the difference of spectra might be due merely to this circumstance. As the direct addition of either ammonia or carbonate of soda to the acid liquid causes a precipitate, it 1s requisite in the first instance to separate the colouring matter from the substance so precipitated. This may be easily effected on a small scale by adding to the watery extract from blood-clots about an equal volume of ether, and then some glacial acetic acid, and gently mixing, but not violently shaking for fear 1864. | of the Colouring Matter of the Blood. 359 of forming an emulsion. When enough acetic acid has been added, the acid ether rises charged with nearly the whole of the colouring matter, while the substance which caused the precipitate remains in the acid watery layer below*. The acid ether solution shows in perfection the characteristic spectrum fig. 3. When most of the acid is washed out the substance falls, remaining in the ether near the common surface. If after removing the wash-water a solution, even a weak one, of ammonia or carbonate of soda be added, the colouring matter readily dissolves in the alkali. The spec- trum of the transmitted light is quite different from that of scarlet eruorine, and by no means so remarkable. It presents a single band of absorption, very obscurely divided into two, the centre of which nearly coincides with the fixed line D, so that the band is decidedly less refrangible than the pair of bands of scarlet crnorine. The relative proportion of the two parts of the band is liable to vary. The presence of alcohol, perhaps even of dissolved ether, seems to favour the first part, and an excess of caustic alkali the second, the fluid at the same time becoming more decidedly dichroitic. The blue end of the spectrum is at the same time absorbed. The band of absorption is by no means so definite at its edges as those of searlet cruorine, and a far larger quantity of the substance is required to develope it. This difference of spectra shows that the colouring matter (hematin) obtained by acids is a product of the decomposition, or metamorphosis of some kind, of the original colouring matter. When hezematin is dissolved in alcohol containing acid, the spectrum nearly agrees with that represented in fig. 3. 12. Heematin is capable of reduction and oxidation like cruorine. If it be dissolved in a solution of ammonia or of carbonate of soda, and a little of the iron salt already mentioned, or else of hydrosu!phate of ammonia, be added, a pair of very intense bands of absorption is immediately developed (fig. 4). These bands are situated at about the same distance apart as those of scarlet cruorine, and are no less sharp and distinctive. They are a little more refrangible, a clear though narrow interval intervening between the first of them and the lime D. They differ much from the bands of cruorine in the relative strength of the first and second band. With cruo- rine the second band appears almost as soon as the first, on increasing the strength or thickness of the solution from zero onwards, and when both bands are well developed, the second band is decidedly broader than the first. With reduced hematin, on the other hand, the first band is already black and intense by the time the second begins to appear; then both bands increase, the first retaining its superiority until the two are on the * Jf I may judge from the results obtained with the precipitate given by acetic acid and a neutral salt, a promising mode of separation of the proximate constituents of blood- crystals would be to dissolve the crystals in glacial acetic acid and add ether, which pre- cipitates a white albuminous substance, leaving the hematin in solution. 360 Prof. Stokes on the Reduction and Oxidation [June 16, point of merging into one by the absorption of the intervening bright band, when the two appear about equal. Like cruorine, reduced hematin is oxidized by shaking up its solution with air. I have not yet obtained hematin in an acid solution in more than one form, that which gives the spectrum fig. 3, and which I have little doubt contains heematin in its oxidized form; for when it is with- drawn from acid ether by an alkali, I have not seen any traces of reduced heematin, even on taking some precautions against the absorption of oxygen. As the alkaline solution of ordinary hematin passes, with increase of thick- ness, through yellow, green, and brown to red, while that of reduced hematin is red throughout, the two kinds may be conveniently distin- guished as brown hematin and red hematin respectively, the former or oxidized substance being the heematin of chemists. 13. Although the spectrum of scarlet cruorine is not affected by the addi- tion to the solution of either ammonia or carbonate of soda, yet if after such addition the solution be either heated or alcohol be added, although there is no precipitation decomposition takes place. The coloured product of de- composition is brown hzematin, as may be inferred from its spectrum. Since, however, the spectrum of an alkaline solution of brown hematin is only moderately distinctive, and is somewhat variable according to the nature of the solvent, it is well to add hydrosulphate of ammonia, which immediately developes the remarkable bands of red hematin. This is the easiest way to obtain them; but the less refrangible edge of the first band as obtained in this way is liable to be not quite clean, in consequence of the presence of a small quantity of cruorine which escaped decomposition. Some very curious reactions are produced in a solution of cruorine by gallic acid combined with other reagents, but these require further study. 14. Hoppe proposed to employ the highly characteristic absorption- bands of scarlet cruorine in forensic inquiries. Since, however, cruorine is very easily decomposed, as by hot water, alcohol, weak acids, &c., the method would often be inapplicable. But as in such cases the coloured product of decomposition is hematin, which is a very stable substance, the absorption-bands of red hematin in alkaline solution, which in sharp- ness, distinctive character and sensibility rival those of scarlet cruorine itself, may be employed instead of the latter. The absorption-bands of brown hematin dissolved in a mixture of ether and acetic acid, or in acetic acid alone, are hardly less characteristic, but are not quite so sensltiye, re- quiring a somewhat larger quantity of the substance. 15. I have purposely abstained from physiological speculations until I should have finished the chemico-optical part of the subject; but as the facts which have been adduced seem calculated to throw considerable light on the function of cruorine in the animal economy, I may perhaps be permitted to make a few remarks on this subject. It has been a disputed point whether the oxygen introduced into the blood in its passage through the lungs is simply dissolved or is chemically 1864. | of the Colouring Matter of the Blood. 361 combined with some constituent of the blood. The latter and more natural view seems for a time to have given place to the former in consequence of the experiments of Magnus. But Liebig and others have since adduced arguments to show that the oxygen absorbed is, mainly at least, chemically combined, be it only in such a loose way, like a portion of the carbonic acid in bicarbonate of soda, that it is capable of being expelled by indifferent gases. It is known, too, that it is the red corpuscles in which the faculty of absorbing oxygen mainly resides. Now it has been shown in this paper that we have in cruorine a substance capable of undergoing reduction and oxidation, more especially oxidation, so that if we may assume the presence of purple cruorine in venous blood, we have all that is necessary to account for the absorption and chemical combination of the inspired oxygen. 16. It is stated by Hoppe that venous as well as arterial blood shows the two bands which are characteristic of what has been called in this paper scarlet cruorine. As the precautions taken to prevent the absorption of oxygen are not mentioned, it seemed desirable to repeat the experiment, which Dr. Harley and Dr. Sharpey have kindly done. A pipette adapted to a syringe was filled with water which had been boiled and cooled without exposure to the air, and the point having been introduced into the jugular vein of a live dog, a little blood was drawn into the bulb. Without the water the blood would have been too dark for spectral analysis. The colour did not much differ from that of scarlet cruorine; certainly it was much nearer the scarlet than the purple substance. The spectrum showed the bands of scarlet cruorine. This, however, does not by any means prove the absence of purple cruo- rine, but only shows that the colouring matter present was chiefly scarlet cruorine. Indeed the relative proportions of the two present in a mixture of them with one another and with colourless substances, can be better judged of by the tint than by the use of the prism. With the prism the extreme sharpness of the bands of scarlet cruorine is apt to mislead, and to induce the observer greatly to exaggerate the relative proportion of that substance. Seeing then that the change of colour from arterial to venous blood as far as it goes is in the direction of the change from scarlet to purple cruo- rine, that scarlet cruorine is capable of reduction even in the cold by sub- stances present in the blood (§ 9), and that the action of reducing agents upon it is greatly assisted by warmth (§ 7), we have every reason to believe that a portion of the cruorine present in venous blood exists in the state of purple cruorine, and is reoxidized in passing through the lungs. 17. That it is only a rather small proportion of the cruorine present in venous blood which exists in the state of purple cruorine under normal conditions of life and health, may be inferred, not only from the colour, but directly from the results of the most recent experiments *. Were it * Funk’s Lehrbuch der Physiologie, 1863, vol. i. § 108. 362 Prof. Stokes on the Reduction and Oxidation [June 16, otherwise, any extensive hemorrhage could hardly fail to be fatal, if, as there is reason to believe, cruorine be the substance on which the function of respiration mainly depends; nor could chlorotic persons exhale as much carbonic acid as healthy subjects, as is found to be the case. But after death there is every reason to think that the process of reduc- tion still goes on, especially in the case of warm-blooded animals, while the body is still warm. Hence the blood found in the veins of an animal some time after death can hardly be taken as a fair specimen as to colour of the venous blood in the living animal. Moreover the blood of an animal which has been subjected to abnormal conditions before death is of course liable to be altered thereby. The terms in which Lehmann has described the colour of the blood of frogs which had been slowly asphyxiated by being made to breathe a mixture of air and carbonic acid seem unmistake- ably to point to purple cruorine*. 18. The effect of various indifferent reagents in changing the colour of defibrinated blood has been much studied, but not always with due regard to optical principles. The brightening of the colour, as seen by reflexion, produced by the first action of neutral salts, and the darkening caused by the addition of a little water, are, I conceive, easily explained; but I have not seen stated what I -feel satisfied is the true explanation. In the former case the corpuscles lose water by exosmose, and become thereby more highly refractive, in consequence of which a more copious reflexion takes place at the common surface of the corpuscles and the surrounding fluid. In the latter case they gain water by endosmose, which makes their refrac- tive power more nearly equal to that of the fluid in which they are con- tained, and the reflexion is consequently diminished. There is nothing in these cases to indicate any change in the mode in which light is absorbed by the colouring matter, although a change of tint to a certain extent, and not merely a change of intensity, may accompany the change of conditions under which the turbid mixture is seen, as I have elsewhere more fully explained +. No doubt the form of the corpuscles is changed by the action of the reagents introduced ; but to attribute the change of colour to this is, I ap- prehend, to mistake a concomitant for a cause, and to attribute, moreover, the change of colour to a cause inadequate:to produce it. 19. Very different is the effect of carbonic acid. In this case the ex- istence of a fundamental change in the mode of absorption cannot be ques- tioned, especially when the fluid is squeezed thin between two glasses and viewed by transmitted light. I took two portions of defibrinated blood ; to one I added a little of the reducing iron solution, and passed carbonic acid into the other, and then compared them. They were as nearly as possible alike. We must not attribute these apparently identical changes to two totally different causes if one will suffice. Now in the case of the iron * Physiological Chemistry, vol. ii. p. 178. + Philosophical Transactions, 1852, p. 527. 1864. ] of the Colouring Matter of the Blood. 363 salt, the change of colour is plainly due to a deoxidation of the cruorine. On the other hand, Magnus removed as much as 10 or 12 per cent. by volume of oxygen from arterialized blood by shaking the blood with car- bonic acid. If, as we have reason to believe, this oxygen was for the most part chemically combined, it follows that carbonic acid acts as if it were a reducing agent. We are led to regard the change of colour not as a direct effect of the presence of carbonic acid, but a consequence of the removal of oxygen. There is this difference between carbonic acid and the real re- ducing agents, that the former no longer acts on a dilute and comparatively pure solution of scarlet cruorine, while the latter act just as before. If even in the case of blood exposed to an atmosphere of carbonic acid we are not to attribute the change of colour to the direct presence of the gas, much less should we attempt to account for the darker colour of venous than arterial blood by the small additional percentage of carbonic acid which the former contains. The ascertained properties of cruorine furnish us with a ready explanation, namely that it is due to a partial re- duction of scarlet: cruorine in supplying the wants of the system. 20. Iam indebted to Dr. Akin for calling my attention to a very in- teresting pamphlet by A. Schmidt on the existence of ozone in the blood*. The author uses throaghout the language of the ozone theory. If by ozone be meant the substance, be it allotropic oxygen or teroxide of hy- drogen, which is formed by electric discharges in air, there is absolutely nothing to prove its existence in blood; for all attempts to obtain an oxi- dizing gas from blood failed. But if by ozone be merely meant oxygen in any such state, of combination or otherwise, as to be capable of producing certain oxidizing effects, such as turning guaiacum blue, the experiments of | Schmidt have completely established its existence, and have connected it, too, with the colouring matter. Now in cruorine we have a substance ad- mitting of easy oxidation and reduction ; and connecting this with Schmidt’s results, we may infer that scarlet cruorine is not merely a greedy absorber and a carrier of oxygen, but also an oxidizing agent, and that it is by its means that the substances which enter the blood from the food, setting aside those which are either assimilated or excreted by the kidneys, are reduced to the ultimate forms of carbonic acid and water, as if they had been burnt in oxygen. 21. In illustration of the functions of cruorine, I would refer, in conclu- sion, to the experiment mentioned in § 10. As the purple cruorine in the solution was oxidized almost instantaneously on being presented with free oxygen by shaking with air, while the tin-salt remained in an unoxidized state, so the purple cruorine of the veins is oxidized during the time, brief though it may be, during which it is exposed to air in the lungs, while the substances derived from the food may have little disposition to combine with free oxygen. s the scarlet cruorine is gradually reduced, oxidizing thereby a portion of the tin-salt, so part of the scarlet cruorine is gradually * Ueber Ozon im Blute. Dorpat, 1862. 364 Sir W. Snow Harris on the Laws and Operation [June 16, reduced in the course of the circulation, oxidizing a portion of the sub- stances derived from the food or of the tissues. The purplish colour now assumed by the solution illustrates the tinge of venous blood, and a fresh shake represents a fresh passage through the lungs. XIX. “ Further Inquiries concerning the Laws and Operation of Elec- trical Force.” By Sir W. Snow Harris, F.R.S., &c. Received June 8, 1864. (Abstract.) 1, The author first endeavours to definitely express what is meant by quantity of electricity, electrical charge, and intensity. By quantity of electricity he understands the actual amount of the un- known agency constituting electrical force, as represented by some arbi- trary quantitative ‘electrical’ measure. By electrical charge he under- stands the quantity which can be sustained upon a given surface under a given electrometer indication. lectrical intensity, on the contrary, is ‘the electrometer indication’ answering to a given quantity upon a given surface. 2. The experiments of Le Monnier in 1746, of Cavendish in 1770, and the papers of Volta in 1779, are quoted as showing that bodies do not take up electricity in proportion to their surfaces. According to Volta, any plane surface extended in length sustains a greater charge,—a result which this distinguished philosopher attributes to the circumstance that the elec- trical particles are further apart upon the elongated surface, and conse- quently further without each other’s influence. 3. The author here endeavours to show that, in extending a surface in length, we expose it to a larger amount of inductive action from surround- ing matter, by which, on the principles of the condenser, the intensity of the accumulation is diminished, and the charge consequently increased ; so that not only are we to take into account the influence of the particles on each other, but likewise their operation upon surrounding matter. 4. No very satisfactory experiments seem to have been instituted showing the relation of quantity to surface. The quantity upon a given surface has been often vaguely estimated without any regard to a constant electro- meter indication or intensity. The author thinks we can scarcely infer from the beautiful experiment of Coulomb, in consequence of this omission, that the capacity of a circular plate of twice the diameter of a given sphere is twice the capacity of the sphere, and endeavours to show, in a future part of the paper (Experiment 16), that the charge of the sphere and plate are to each other not really as 1:2, but as 1:/2, that is, as the square roots of the exposed surfaces; so that we cannot accumulate twice the quantity of electricity upon the plate under the same electrometer indication. 5. On a further investigation of the laws of electrical charge, the quan- tity which any plane rectangular surface can receive under a given intensity 1864.) of Electrical Force. 365 is found to depend not only on the surface, but also on its linear boundary extension. Thus the linear boundary of 100 square inches of surface under a rectangle 37°5 inches long by 2°66 inches wide, is about 80 inches ; whilst the linear boundary of the same 100 square inches of surface under a plate 10 inches square is only 40 inches. Hence the charge of the rec- tangle is much greater than that of the square, although the surfaces are equal, or nearly so. 6. The author finds, by a rigid experimental examination of this question, that electrical charge depends upon surface and linear extension conjointly. He endeavours to show that there exists in every plane surface what may be termed an electrical boundary, having an important relation to the group- ing or disposition of the electrical particles in regard to each other and to surrounding matter. This boundary, in circles or globes, is represented by their circumferences. In plane rectangular surfaces, it is their linear ex- tension or perimeter. If this Joundary be constant, their electrical charge (1) varies with the square root of the surface. If the surface be constant, the charge varies with the square root of the boundary. Ifthe surface and boundary both vary, the charge varies with the square root cf the surface multiplied into the square root of the boundary. Thus, calling C the charge S the surface, B the boundary, and » some arbitrary constant depending on the electrical unit of charge, we have C=p S.B, which will be found, with some exceptions, a general law of electrical charge. It follows from this formula, that if when we double the surface we also double the boun- dary, the charge will be also double. In this case the charge may be said to vary with the surface, since it varies with the square root of the surface, multiplied into the square root of the boundary. If therefore the surface and boundary both increase together, the charge will vary with the square of either quantity. The quantity of electricity therefore which surfaces can sustain under these conditions will be as the surface. If/ and 6 represent respectively the length and breadth of a plane rectangular surface, then the charge of such a surface is expressed by p ¥ 216 (1-6), which is found to agree perfectly with experiment. We have, however, in all these cases to bear in mind the difference between electrical charge and electrical - intensity (1). 7. The electrical intensity of plane rectangular surfaces is found to vary in an inverse ratio of the boundary multiplied into the surface. If the sur- face be constant, the intensity is inversely as the boundary. If the boun- dary be constant, the intensity is inversely as the surface. If both vary alike and together, the intensity is as the square of either quantity ; so that if when the surface be doubled the boundary be also doubled, the intensity will be inversely as the square of the surface. The intensity of a plane rectangular surface being given, we may always deduce therefrom its elec- trical charge under a given greater intensity, since we only require to de- termine the increased quantity requisite to bring the electrometer indica- tion up to the given required intensity. This is readily deduced, the 366 Sir W. Snow Harris on the Laws and Cperation [June 16, intensity being, by a well-established law of electrical force, as the square of the quantity. 8. These laws relating to charge, surface, intensity, &c., apply more especially to continuous surfaces taken as a whole, and not to surfaces divided into separated parts. The author illustrates this by examining the result of an electrical accumulation upon a plane rectangular surface taken as a whole, and the results of the same accumulation upon the same sur- face divided into two equal and similar portions distant from each other, and endeavours to show, that if as we increase the quantity we also increase the surface and boundary, the intensity does not change. If three or more separated equal spheres, for example, be charged with three or more equal quantities, and be each placed in separate connexion with the electrometer, the intensity of the whole is not greater than the intensity of one of the parts. A similar result ensues in charging any united number of equal and similar electrical jars. A battery of five equal and similar jars, for example, charged with a given quantity =1, has the same intensity as a battery of ten equal and similar jars charged with quantity =2; so that the intensity of the ten jars taken together is no greater than the intensity of one of the jars taken singly. In accumulating a double quantity upon a given surface divided into two equal and separate parts, the boundaries of each being the same, the intensity varies inversely as the square of the surface. Hence two separate equal parts can receive, taken together under the same electro- meter indication, twice the quantity which either can receive alone, in which case the charge varies with the surface. Thus if a given quantity be disposed upon two equal and similar jars instead of upon one of the jars only, the intensity upon the two jars will be only one-fourth the intensity of one of them, since the intensity in this case varies with the square of the surface inversely, whilst the quantity upon the two jars under the same electrometer indication will be double the quantity upon one of them only ; in which case the charge varies with the surface, the intensity being con- stant. If therefore as we increase the number of equal and similar jars we also increase the quantity, the intensity remains the same, and the charge will increase with the number of jars. Taking a given surface therefore in equal and divided parts, as for example four equal and similar electrical jars, the intensity is found to vary with the square of the quantity - directly (the number of jars remaining the same), and with the square of the surface inversely (the number of jars being increased or diminished) ; hence the charge will vary as the square of the quantity divided by the square of the surface; and we have, calling C the charge, Q the quantity, 2 and S the surface, ta which formula fully represents the phenomenon of a constant intensity, attendant upon the charging of equal separated sur- faces with quantities increasing as the surfaces ; as in the case of charging an increasing number of equal electrical jars. Cases, however, may possibly arise in which the intensity varies inversely with the surface, and not in- — Se 1864.] of Hlectrical Force. 367 versely with the square of the surface. In such cases, of which the author gives some examples, the above formula does not apply. 9. From these inquiries it is evident, as observed by the early electricians, that conducting bodies do not take up electricity in proportion to their sur- faces, except under certain relations of surface and boundary. If the breadth of a given surface be indefinitely diminished, and the length indefinitely in- creased, the surface remaining constant, then, as observed by Volta, the least quantity which can be accumulated under a given electrometer indi- cation is when the given surface is a circular plate, that is to say, when the boundary is a minimum, and the greatest when extended into a right lime of small width, that is, when the boundary is a maximum. In the union of two similar surfaces by a boundary contact, as for example two circular plates, two spheres, two rectangular plates, &c., we fail to obtain twice the charge of one of them taken separately. In either case we fail to decrease the intensity (the quantity being constant) or to increase the charge (the intensity being constant), it being evident that whatever de- creases the electrometer indication or intensity must increase the charge, that is to say, the quantity which can be accumulated under the given in- tensity. Conversely, whatever increases the electrometer indication de- creases the charge, that is to say, the quantity which can be accumulated under the given intensity. 10. If the grouping or disposition of the electrical particles, in regard to surrounding matter, be such as not to materially influence external induc- tion, then the boundary extension of the surface may be neglected. In all similar figures, for example, such as squares, circles, spheres, &e., the elec- trical boundary is, in relation to surrounding matter, pretty much the same in each, whatever be the extent of their respective surfaces. In calculating the charge, therefore, of such surfaces, the boundary extensions may be neglected, in which case their relative charges are found to be as the square roots of the surfaces only ; thus the charges of circular plates and globes are as their diameters, the charges of square plates are as their sides. In rectangular surfaces also, having the same boundary extensions, the same result ensues, the charges are as the square roots of the surfaces. In cases of hollow cylinders and globes, in which one of the surfaces is shut out from external influences, only one-half the surface may be considered as exposed to external inductive action, and the charge will be as the square root of half the surface, that is to say, as the square root of the exposed surface. If, for example, we suppose a square plate of any given dimen- sions to be rolled up into an open hollow cylinder, the charge of the cylinder will be to the charge of the plate into which we may suppose it to be ex- panded as 1: 72, In like manner, if we take a hollow globe and a circular plate of twice its diameter, the charge of the globe will be to the charge of the plate also as 1: ¥ 2, which is the general relation of the charge of closed to open surfaces of the same extension. The charge of a square plate to the charge of a circular plate of the same diameter was found to be 1: 1°13; 368 Sir W. Snow Harris on the Laws and Operation [June 16, according to Cavendish it is as 1: 1°15, which is not far different. It is not unworthy of remark that the electrical relation of a square to a circular plate of the same diameter, as determined by Cavendish nearly a century since, is in near accordance with the formules C= /S above deduced. 11. The author enumerates the followimg formule as embracing the general laws of quantity, surface, boundary extension, and intensity, prac- tically useful in deducing the laws of statical electrical force. Symbols. Let C= electrical charge ; Q= quantity ; E= intensity, or electrometer indication ; S= surface, B= boundary extension, or perimeter ; A= direct induction; 6= reflected induction; F= force; D= distance. Formule. C «8S, when §S and B vary together. C « Q, E being constant, or equal 1. Ca wh S, B being constant, or equal 1. Ca VA B, S being constant, or equal 1. C «,/S.B, when S and B vary together. Ea = (Q being constant), for all plane rectangular surfaces. 1 B’ E« = B being constant, or equal 1. E « _,S being constant, or equal 1. Ea 2. when S and B vary together. S? | Ca WA E « Q’, S being constant, or equal 1. Q? Ca sr In square plates, C « with side of square. In circular plates, C « with diameter. In globes, C « with diameter. A, or induction « S, all other things remaining the same. The same for 6, or reflected induction. In circular plates, globes, and closed and open surfaces, E x4; oras a S’ A F (=E) « Q’. ForEa =, S being constant. 2 Generally we have F « D™ 1864. | of Electrical Force. 369 12. The author calculates from these laws of charge for circles and globes, a series of circular and globular measures of definite values, taking the circular inch or globular inch as unity, and calling, after Cavendish, a circular plate of an inch in diameter, charged to saturation, a circular inch of electricity ; or otherwise charged to any degree short of saturation, a circular inch of electricity under a given intensity. In like manner he de- signates a globe of an inch in diameter a globular inch of electricity. In the following Table are given the quantities of electricity contained in circular plates and globes, together with their respective intensities for dia- meters varying from °25 to 2 inches; a circular plate of an inch diameter and 4th of an inch thick being taken as unity, and supposed to contain 100 particles or units of charge. Diameters, Circle. Globe. or 7 | Se units of charge. || Particles. | Intensity. || Particles. | Intensity. 0:25 25 0:062 39 0:124 0:50 50 0:250 70 0-500 0-75 75 0-560 105 1:120 1:00 100 1-000 140 2°000 1:25 125 1560 175 3°120 1:40 140 1-960 196 3°920 1:50 150 2250 210 4-500 1:60 160 2°560 224 5120 1:75 175 3060 245 6:120 2:00 200 4-000 280 8:000 13. The experimental. investigations upon which these elementary data depend, constitute a second part of this paper. The author here enters upon a brief review of his hydrostatic electrometer, as recently perfected and improved, it being essential to a clear comprehension of the laws and other physical results arrived at. In this instrument the attractive force between a charged and neutral disk, in connexion with the earth, is hydrostatically counterpoised by a small cylinder of wood accurately weighted, and partially immersed in a vessel of water. The neutral disk and its hydrostatic counterpoise. are freely suspended over the circumference of a light wheel of 2°4 inches in diameter, delicately mounted on friction-wheels, so as to have perfectly free motion, and be susceptible of the slightest force added to either side of the balance. Due contrivances are provided for measuring the distance between the attracting disks. The balance-wheel carries a light index of straw reed, moveable over a graduated quadrantal are, divided into 90° on each side of its centre. The neutral attracting plate of the electrometer is about 14 inch in diameter, and is suspended from the balance-wheel by a gold thread, over a similar disk, fixed on an insulating rod of glass, placed in connexion with any charged surface the subject of experiment. The least force between the two disks is immediately shown by the movement 370 Sir W. Snow Harris on the Laws and Operation [June 16, of the index over the graduated arc in either direction, and is eventually counterpoised by the elevation or depression in the water of the hydrostatic cylinder suspended from the opposite side of the wheel. The divisions on the graduated quadrant correspond to the addition of small weights to either side of the balance, which stand for or represent the amount of force between the attracting plates at given measured distances, with given measured quantities of electricity. This arrangement is susceptible of very great accuracy of measurement. The experiment requires an extremely short time for its development, and no calculation is necessary for dissipation. The author carefully de- scribes the manipulation requisite in the use of this instrument, together with its auxiliary appendages. He considers this electrometer, as an in- strument of electrical research, quite invaluable, and peculiarly adapted to the measurement of electrical force. 14. Having fully described this electrometer, and the nature of its indi- cations, certain auxiliary instruments of quantitative measure, to be em- ployed in connexion with it, are next adverted to. - First, the construction and use of circular and globular transfer measures given in the preceding Table, by which given measured quantities of elec- tricity may be transferred from an electrical jar (charged through a unit- jar from the conductor of an electrical machine) to any given surface in connexion with the electrometer. The electrical jar he terms a guantzty- jar, the construction and employment of which is minutely explained ; as also the construction and employment of the particular kind of unit-jar he employs. 15. Two experiments (1 and 2) are now given in illustration of this method of investigation. Experiment | developes the law of attractive force as regards quantity ; which is found to vary with the square of the number of circular or glo- bular inches of electricity, transferred to a given surface in connexion with the fixed plate of the electrometer, the distance between the attracting sur- faces being constant. Experiment 2 demonstrates the law of force as regards distance between the attracting surfaces, the quantity of electricity being constant ; and by which it is seen that the force is in an inverse ratio of the square of the distance between the attracting plates, the plates being susceptible of per- fect inductive action. From these two experiments, taken in connexion with each other, we derive the following formula, F « as calling F the force, Q the quantity, and D the distance. It is necessary, however, to observe that this formula only applies to electrical attractive force between a charged and neutral body in connexion with the earth, the two surfaces being susceptible of free electrical induction, both direct and reflected. 16. The author now refers to several experiments (3, 4, 5, and 6), show- ing that no sensible error arises from the reflected inductive action of the 1864. ] of Electrical Force. 371 suspended neutral disk of the electrometer, or from the increased surface attendant on the connexion of the surface under experiment with the fixed plate of the electrometer; as also, that it is of no consequence whether the suspended disk be placed immediately over the fixed attracting plate of the electrometer, or over any point of the attracting surface in connexion with it. 17. Having duly considered these preliminary investigations, the author now proceeds to examine experimentally the laws of surface and boundary as regards plane rectangular surfaces, and to verify the formule C=,/ S.B, and B=: in which C=charge, E=intensity, S=surface, and B= boundary. : For this purpose a series of smoothly-polished plates of copper were em- ployed, varying from 10 inches square to 40 inches long by 2°5 to 6 inches wide, and about 3th of an inch thick, exposing from 100 to 200 square inches of surface. The charges (1) of these plates were carefully determined under a given electrometer indication, the attracting plates being at a constant distance. Experiment 7. In this experiment, a copper plate 10 inches square is compared with a rectangular plate 40 inches long by 2°5 inches wide. In these plates the surfaces are each 100 square inches, whilst the boun- daries are 40 and 85 inches, The boundaries may be taken, without sen- sible error, as 1 ; 2, whilst the surfaces are the same. On examining the charges of these plates, charge of the square plate was found to be 7circular inches, under an intensity of 10°. Charge of the rectangular plate 10 circular inches nearly, under the same intensity of 10°, The charges therefore were as 7: 10 nearly, that is, as 1: 1-4 nearly ; being the square roots of the boundaries, that is, as 1: V2. Experiment 8. A rectangular plate 37:5 inches long by 2°7 inches wide, surface 101 square inches, boundary 80°5 inches, compared with a rect- angular plate 34°25 inches long by 6 inches wide, surface 205 square inches, boundary 80:5 inches. Here the boundaries are the same, whilst the surfaces may be taken as 5:2. On determining the charges of these plates, charge of the rectangular plate, surface 101 square inches was found to be 8°5 circular inches under an intensity of 8°. Charge of the plate with double surface =205 square inches, was found to be 12 circular inches under the same intensity of 8°; that is to say, whilst the surfaces are as 1: 2, the charges are as 85: 12 nearly, or as the square roots of the surfaces, that is, as 1: 4/ z, Experiment 9. A rectangular plate 26°25 inches long by 4 inches wide, surface 105 square inches, boundary 60°5, compared with a rectangular plate 40 inches long by 5 inches wide, surface 200 square inches, boundary 90 inches. Here the surfaces are as 1: 2 nearly, whilst their boundaries are as 2:3. VOL, XIII. 25 372 Sir W. Snow Harris on the Laws and Operation [June 16, Charge of the rectangular plate surface =105 square inches, 7 circular inches under an intensity of 10°. Charge of rectangular plate surface 200 square inches, 12 circular inches, under the same intensity of 10°. The charges therefore are as 7:12 nearly, or as 1: 1°7, being as the square roots of the surfaces multiplied into the square roots of the bouneaee# very nearly. Experiment 10. A square plate 10 inches square, surface 100 square inches, boundary 40 inches, compared with a rectangular plate 40 inches long by 5 inches wide, surface 200 square inches, boundary 90 inches. Here the surfaces are double of each other, and the boundaries also double each other, or so nearly as to admit of their being considered double of each other. Charge of square plate 6 circular inches, under an intensity of 10°. Charge of rectangular plate 12 circular inches, under the same intensity of 10°. The charges, therefore, are as the square roots of the surfaces and boundaries conjointly, according to the formula C=‘ S.B, as also verified in the preceding experiment 9. A double surface, therefore, having a double boundary, takes a double charge, but not otherwise. Neglecting all considerations of the boundary, therefore, the surface and boundary varying together, the charge in this case will be as the surface directly. 18. The author having verified experimentally the laws of surface and boundary, as regards plane rectangular surfaces, proceeds to consider the charges of square plates, circular plates, sphéres, and closed and open sur- faces generally. Experiment 11. Plate 10 inches square, surface 100 square inches, boundary 40 inches, compared with a similar plate 14 inches square, sur- face 196 square inches, boundary 56 inches. Here the surfaces are as 1:2 nearly, whilst the boundaries are as 1: V 2 nearly. In this case charge of square plate, surface 100 square inches, was found to be 8 circular inches under an intensity of 10°. Charge of the plate, surface 196 square inches, 11 circular inches, under the same intensity of 10°. Here the charges are as 8:11, whilst the surfaces may be taken as 1: 2, that is to say (neglecting the boundary), the charges are as the a roots of the surfaces, according to the formula C= VS. On examining the intensities of these plates, they were found to be inversely as the surfaces ; thus 8 circular inches upon the plate, surface 100, evinced an intensity of 10°; 8 pace inches upon the plate, surface 196, evinced an intensity of 5° only, or 3 the former, according to the formula ai et E=<.- Experiment 12. A circular plate of 9 inches diameter, surface 63-6 square inches, compared with a circular plate of 18 inches, or double that diameter, surface 254 square inches. Here the surfaces are as 1: 4, whilst the boundaries or circumferences are as | : 2. Charge of 9-inch plate, 6 circular inches, under an intensity of 10°. ee 1864. ] , of Electrical Force. . 373 Charge of 18-inch plate, 12 circular inches, under the same intensity of 10°. Here the charges are as 1 : 2, whilst the surfaces are as 1:4; neglecting the difference of boundary, therefore, the charges, as in the preceding ex- periments, are as the square roots of the surfaces. On examining the intensities of these plates, they were found to be in- versely as the surfaces; thus 6 circular inches upon the 9-inch plate evinced an intensity of 10°, as just stated; 6 circular inches upon the 18-inch plate had only one-fourth the intensity, or 2°5; beimg inversely : ] as the surfaces, according to the formula Be Experiment 13. A circular plate of 9 inches diameter, surface 63°6 square inches, compared with a circular plate of 12°72 inches diameter, surface 127-2 square inches. Here the surfaces are as 1: 2. Charge of 9-inch plate (surface 63°6 square inches), 5 circular inches, under an intensity of 8°. Charge of 12°72-inch plate (surface 127°2 square inches), 7 circular inches, under the same intensity of 8°. The charges here are as 5:7, whilst the surfaces are as 1:2; that is to say (neglecting the boundaries), the charges are as the square roots of the surfaces. On examining the intensities of these plates, they were found to be, as in the preceding experiments, inversely as the surfaces. Experiment 14. Comparison of a sphere of 4°5 inches diameter, viene 63°5 square inches, with a sphere of 9 inches, or double that diameter, surface 254 square inches. Charge of sphere of 4°5 inches diameter (surface 63°5 square inches), 4 circular inches, under an intensity of 9°. Charge of sphere of 9 inches diameter (surface 254 square inches), 8 circular inches, under the same intensity of 9°. Here the charges are as 1: 2, whilst the surfaces are as 1:4. The charges, therefore, are as the square roots of the surfaces, or as 1:44. On examining the interieities of these spheres, they were found to be as the surfaces inversely, or very nearly ; being as 2°°5 and 9° respectively. Experiment 15. Circular plate of 9 inches diameter compared with a sphere of the same diameter. Here the actual surfaces are 63:6 square inches for the plate, and 254 square inches for the sphere, being as 1: 4. We have to observe, however, that one surface of the sphere is closed or shut up, consequently the exposed surfaces, electrically considered, neglect- ing one-half the surface of the sphere as being closed, are as 1:2; and the exposed surface of the plate is exactly one-half the exposed surface of the sphere. Charge of plate 8 circular inches, under an intensity of 12°. Charge of sphere 11 circular inches, under the same intensity of 12°, The charges, therefore, are as 8: 11, or sa 1:1°4; the exposed surfaces being as 1:2, The charges, therefore, are as the square roots of the exposed surfaces. On examining the intensities of the plate and sphere, they were found 2E2 B74 On the Laws and Operation of Electrical Force. [June 16, to be in an inverse ratio of the exposed surfaces, as in the former expe: riments. Haperiment 16. Comparison of a sphere of 7 inches diameter with a circular plate of 14 inches, or double that diameter. In this case the inner and outer surface of the sphere, taken together, are actually the same as the two surfaces of the plate. The inner surface of the sphere being closed, however, as in the last experiment, the surfaces of the sphere and plate, electrically considered, are therefore not equal, and the surface of the plate is twice the surface of the sphere. The surfaces, therefore, open to external induction are as 2: 1. On examining the charges of the plate and sphere, they were found to be as 10: 14, or as 1:1°4; charge of sphere being 10 circular inches, under an intensity of 20°, and charge of plate being 14 circular inches, under the same intensity of 20°. The charge of the sphere, therefore, as compared with the charge of the plate, is as 1: ¥ 2, that is, as the square roots of the exposed surfaces. On examining the intensities of the sphere and plate, they were found to be, as in the preceding experiments, in an inverse ratio of the exposed sur- faces. We cannot, therefore, conclude, as already observed (4), that the capacity of the plate is twice that of the sphere. 19. The following experiments are further adduced in support of the preceding :— Experiment 17. A copper plate 10 inches square, compared with the same plate rolled up into an open hollow cylinder, 10 inches long by 3:2 inches diameter. Here, as in the last experiments, although the surfaces are actually the same, yet, electrically considered, the plate has twice the surface of the cylinder, one surface of the cylinder being shut up. _ On examining the charges of the cylinder and plate, they were found to be, as in the preceding experiments, as 1: V2; that is, as the square roots of the exposed surfaces, and the intensities in an inverse ratio of the sur- faces, which seems to be a general law for closed and open surfaces. Experiment 18. A hollow copper cube, side 5:7 inches, surface 195, compared with a hollow copper sphere of diameter equal side of cube, surface 103 square inches nearly. On examining the charges of the sphere and cube, they were found to be as 9: 10 nearly ; charge of the sphere being 9 circular inches, under an intensity of 10°, and charge of cube being 10 circular inches, under the same intensity of 10°. The charges of a cube, and of a sphere whose dia- meter equals the side of the cube, approach each other, notwithstanding the differences of the surfaces, owing to the six surfaces of the cube not being in a disjointed or separated state. 20. The author observes, in conclusion, that the numerical results of the foregoing experiments, although not in every instance mathematically exact, yet upon the whole were so nearly accordant as to leave no doubt as to the law in operation. It would be in fact, he observes, assuming too much to 1864. ] My. P. Griess on a New Class of Compounds. 375 pretend in such delicate experiments to have arrived at nearer approxima- tions than that of a degree or two of the electrometer, or within quantities less than that of ‘25 of a circular inch. If the manipulation, however, be skilfully conducted, and the electrical insulations perfect, it is astonishing how rigidly exact the numerical results generally come out. XX. “Ona New Class of Compounds in which Nitrogen is substituted for Hydrogen.” By Prrrr Griess, Esq. Communicated by Dr. Hormann. Received June 2, 1864. (Abstract. ) All the bodies in which nitrogen is substituted for hydrogen which I have discovered during the last few years* may be divided into two dis- tinctly different classes. The first class comprises those bodies which are obtained when three atoms of hydrogen in two atoms of an amido-com- pound are replaced by one of nitrogen from nitrous acid. The members of the second group are formed by the action of nitrous acid upon one equi- valent of an amido-compound only. The following equations will best show these various reactions :— a 2(C, H, NO,) + NHO,=C¢,, H,, N,9,+2H,0 en —~— OE = Amidobenzoic Nitrous Diazoamido- acid. acid. benzoic acid. II. 6, H, N, 0, +NHO0,=C, H, N, 0,42H,0. Picramic acid Diazo-dinitro- (amidonitrophenylic phenylic acid. acid). I have hitherto chiefly examined into the constitution of the bodies that form according to the first equation (diazoamidobenzol, diazoamidobenzoic acid), and have only incidentally explored the field of bodies which equa- tion II. opens up. I have had occasion since to study more closely several representatives of the latter class of compounds, which are derived from aniline (amidobenzol) and analogous organic bases; and since the results which I obtained cannot but excite some interest, I may be permitted to submit them briefly to the Royal Society. Nitrate of Diazobenzol, C,H, N,, NHO,. This compound is most readily obtained by passing a rapid current of nitrous acid gas through a solution of nitrate of aniline, saturated in the cold, until aniline ceases to be separated by the addition of solution of caustic potash to the liquid. On diluting the solution then with three times its volume of alcohol, and adding a sufficient quantity of ether, nitrate of diazobenzol separates in long white needles. In order to remove a trace of a colouring substance, the crystals are redissolved in a small quantity of * Ann. der Chem. und Pharm. vol. exiii. p- 201; vol. cxvii. p. 15 vol. cxxi. p. 257. Proceedings of the Royal Society, vol. x. p. 591; vol. xi. p. 263; vol. xii. p. 418. 376 Mr. P. Griess on a New Class of Compounds | [June 16, cold dilute alcohol and precipitated by ether. The following equation expresses the reaction :— ©, H,N, NHO,+ NHO,=€, H,N,, Be 3+2H, 9. we —<— A Ms ; ~~ Nitrate of aniline. Nitrous Nitrate of oe acid. benzol. Nitrate of diazobenzol may also be prepared from diazoamidobenzol, a substance described by me on a former occasion*, by treating an etherial solution of the latter with nitrous acid, C,, H,, N, +NHO,+2NHO,=2(G, H, N,, NHO,)+2H, 9. Cente Oe | __9 Se — —S Diazoamido- Nitrate of diazobenzol. benzol. The new compound dissolves very readily in water, more difficultly in alcohol, and is almost insoluble in ether. On heating, the solutions are decomposed with evolution of gas. The dry substance explodes with the greatest violence when gently heated, and it is necessary to observe great precautions whilst working with it. ‘The chemical analysis could not, for the same reason, be performed by the usual methods. Its composition was, however, readily established by studying the products of decomposi- tion to which boiling with water gives rise, according to the equation ey EN Lee. + H,O=C,H,O + N, + NHO,. “Nite ate of diazo- Phenylic Nitro- Nitric benzol. acid. gen. acid. Sulphate of Diazobenzol, C,H, N, SH, 9,,. This salt forms when a highly concentrated aqueous solution of the former compound is treated with a sufficient quantity of cold sulphuric acid diluted with its own bulk of water. ‘The solution is treated, as before, with three times its volume of alcohol, and ether added, which causes the sulphate of diazobenzol to separate in a layer of a very concentrated aqueous solution at the bottom of the vessel. On placing this latter solution over sulphuric acid, crystallization ensues after a short time. The crystals are freed from the mother-liquor by washing with absolute alcohol. In this manner large white prisms, which rapidly deliquesce in moist air, are ob- tained, and which are decomposed with slight deflagration when heated by themselves. Hydrobromate of Diazobenzol, C,H,N,, HBr. This compound is obtained in small white soft plates when an etherial solution of diazoamidobenzol is mixed with an etherial solution of bromine, (C,, H,, N,+6Br=C, H,N,, HBr+ C, H, Br, N+2(HBr). —-,-—" SS + —— 7 ——_~,- -- Diazoamido- Hydrobromate of Tribromaniline. benzol. diazobenzol. * Ann. der Chem. und Pharm. vol. cxxi. p- 258. 1864. | in which Nitrogen is substituted for Hydrogen. B77 Hydrobromate of diazobenzol is very unstable. The beautifully white erystals change so rapidly that in a few moments they acquire a reddish colour, and in a few days the decomposition is almost complete. They explode on heating almost with the same violence as was experienced with nitrate of diazobenzol. | Dibromide of Hydrobromate of Diazobenzol, ©, H,N,, HBr, Br,. On adding excess of bromine-water to an aqueous solution of any one of the compounds previously described, an orange-coloured oil is obtained which rapily solidifies, after the mother-liquor has been removed, to small orange- coloured plates. The crystals of dibromide are obtained in a perfectly pure state by washing with a little alcohol. This compound is rather difficultly soluble in cold alcohol and ether ; and the solutions are rapidly decomposed, particularly on the application of heat. Platinum-salt of the Hydrochlorate of Diazobenzol, ©, H, N,, HCl, PtCl,. This salt forms beautiful yellow prisms which are almost insoluble in water. The gold-salt, C,H, N,, HCl, AuCl,, can be recrystallized from alcohol, and is obtained in very fine golden-yellow brilliant plates. It has thus been sufficiently shown that diazobenzol deports itself like an organic base, being capable, like aniline, of forming salts with various acids. It possesses, however, also the property of combining with the hydrates of the metals, thus playing the part of a weak acid. Compound of Hydrate of Potassium with Diazobenzol, C,H, N,, KHO. This body is obtained when a concentrated aqueous solution of nitrate of diazobenzo] is treated with excess of concentrated aqueous potassa. By evaporating on the water-bath, the liquid solidifies, when sufficiently eon- centrated, to a magma of yellow crystals consisting of nitre and the com- pound of hydrate of potassa with diazobenzol. The crystalline mass is pressed between porous stones, and thus partly freed from moisture. By dissolving in absolute alcohol and treating with ether, the new compound of hydrate of potassium with diazobenzol is obtained in a pure state, crystallizing in small soft white plates, which rapidly become reddish, especially in the moist condition. It is very readily soluble in water and alcohol; the solutions, however, decompose slowly, and deposit a reddish amorphous body. Heat does not seem to accelerate this decomposition materially. Compound of Hydrate of Silver with Diazobenzol, C,H, N,, AgHO. This substance is obtained in the form of an almost white precipitate when a solution of silver is added to an aqueous solution of the former compound. It is verystable. Similar compounds are obtained with lead- and zinc-salts. ue 378 Mr. P. Griess on a New Class of Compounds [June 16, Diazobenzol, ©, H, N,. This substance is obtained when an aqueous solution of the compound of hydrate of potassium with diazobenzol is neutralized with acetic acid. It separates as a thick yellow oil of very little stability. After a few moments an evolution of gas ensues, and the diazobenzol is rapidly con- verted into a reddish-brown viscid mass. Diazobenzol is soluble in acids, as well as bases, with formation of the saline compounds previously described. By acting in the cold with aniline upon nitrate of diazobenzol*, the following change takes place :— G, H,N, NHO,+2¢, H, N= Ga HAN, +, H, N, NHO,. NOE -TREIAS bs. Vecesy, adceite ghOEy 5Get Tia Nitrate of diazo- Aniline. PRON ea Nitrate of aniline. benzol. benzol. ee I was formerly of opinion that diazoamidobenzol must be viewed as a double compound of diazobenzol and aniline. The above equation seems to confirm this view. New compounds analogous to diazobenzol-amidobenzol are obtained by the action of other organic bases upon nitrate of diazobenzol, viz. diazo- ici ele wh benzol-amidobromobenzol, C be NP? by the action of bromaniline. °\ Br Naphthalidine and nitrate of diazobenzol combine directly and form nitrate of diazobenzol-amidonaphtol, crystallizing in magnificent large green prisms. The action of amido-acids upon nitrate of ee: is analogous to that of the aniline; ez. g7., 6, H,N, C,H,N,, NHO,+2C,H,NO,= 1é ee 4+. HI, NO,, NHO,. —— -+- W- -~-— +e — Ss =a Nitrate of diazo- Amido-acid. Diazobenzol- Nitrate of amido- benzol. amidobenzoic acid. benzoic acid. Diazobenzol-amidobenzoic acid separated quickly as a yellow crystalline mass on mixing the aqueous solutions of both substances. It is purified by recrystallization from ether in the form of small yellow plates. It combines with bases and forms saline bodies. Bichloride of platinum precipitates from an alcoholic solution a yellowish-white crystalline pla- tinum-salt of the composition C,, H,, N, O,, 2HCl, 2PtCl,. Similar double acids to the one just described are obtained by the action of amidodracylic acid, amidoanic acid, &c., upon nitrate of diazobenzol. Fwedeen Compounds of Diazobenzol. These peculiar compounds are formed when aqueous ammonia, as well as certain organic bases, are made to act upon the dibromide of diazobenzol. * It is self-evident that for this and similar experiments sulphate and hydrobromate of diazobenzol may also be employed. 1864.] in which Nitrogen is substituted for Hydrogen. 379 Diazobenzolimide, C,H, H ie) N, is obtained according to the equation C, H, N, HBr,+4NH,=C, H, N,+3NH, Br. — ae SS —>_-—" —__-—~+---—_—“ Dibromide. Ammonia. Diazoben- Bromide of zolimide. ammonium. It forms a yellowish oil, Which must be distilled zz vacuo with the aid of a current of steam. Exposed by itself to a higher temperature, it decom- poses with detonation. It is remarkable for its stupefying ammoniacal- aromatic odour. Ethyldiazobenzolimide, Ce! i: } N, is analogous in its properties, and is formed in a similar manner. Products of Decomposition of Diazobenzol Compounds. The transformations which the molecule of diazobenzol undergoes under the influence of various reagents are numerous. The products often re- present some peculiar classes of entirely new compounds; more frequently, however, they belong to the phenyl- and benzol-group. I will describe a few of them somewhat more fully. It has already been mentioned that, on boiling with water, nitrate of diazobenzol is broken up into nitrogen, phenylic acid, and nitric acid. Hydrobromate of diazobenzol undergoes an analogous decomposition, viz., C,H,N, HBr+H, O=N,+4€, H, O+ HBr. Treated with alcohol, nitrate of diazobenzol is decomposed in the fol- lowing manner :— 2(C,H,N, NHO,) + C,H,O=C,H,+C,H,(NO,),0+€,H,O+N,+H,0. ——— ecak eee ter >—————-+-—- —— “Nitrate of diazo- Alcohol. Benzol. Dinitrophenylic Aldehyde. benzol. acid. On dissolving sulphate of diazobenzol in a small quantity of concentrated sulphuric acid, it gives rise to the formation of a new sulpho-acid which I propose to call disulphophenylenic acid, ©, H, N, SH, 0,4+SH, 0,=C, H,, 8, H, O,+N,,. SS —— ———+- Sulphate of diazo- Disulphophenylic benzol. acid. The excess of sulphuric acid may be removed from the new acid by means of carbonate of barium. The new barium-salt crystallizes in beautiful prisms. Its composition must be expressed by the formula €,H,S,H,Ba,@,. The free acid obtained by the addition of sulphuric acid to the barium-salt is deposited in warty masses of radiating crystals which deliquesce in the air. Disulphophenylenic acid is four-basic, and is capable of forming four series of salts. The silver-salt forms, however, an exception, its composition being expressed by the formula C, H, 8, Ag, .. Disulphophenylenic acid, like phosphoric acid, appears to be capable of existing in different modifications, possessing different powers of basicity. 380 Mr. P. Griess on a New Class of Compounds [June 16, Diazobenzolimide in alcoholic solutions is decomposed by nascent hydro- gen, generated with zinc and sulphuric acid in the following manner :— GC ,H,N,+8H=€, H,N+2NH,. + Diazoben- Aniline. zolimide. On adding to an aqueous solution of nitrate of diazobenzol levigated carbonate of barium, a feeble evolution of gas ensues, which lasts for several days, until the original compound has been completely decomposed. Two new substances are formed, which are very differently soluble in aleo- hol, and can thus readily be separated. ‘The easily soluble compound, which I will call phenyldiazobenzol, crystallizes from alcohol in yellowish warty masses; from water (in which it dissolves very difficultly) in small rhombic prisms. The difficultly soluble one, which I propose to call phe- nyldidiazobenzol, crystallizes in reddish-vellow needles. The following equation expresses the formation of these two bodies :— I 2(6,H,N,, NHO,)+H, O=C,, H,,N,O+N,+2NHO,. ~~ ——4-- Nitrate of diazobenzol. Phenol- diazobenzol. Il. 3(C,H,N,, NHO,)+H, 0=G,, H,,N,O+N,+3NHO,. * MM —+-----—" Phenoldidiazo- benzol. On looking at these formule, it becomes evident at a glance that both compounds contain phenylic acid and diazobenzol ; viz €,H,N,+6,H,0= leet CE te eC), Cree —_-— Diazobenzol. Phenol. a i diazobenzol. 2(, H, NS +, H, of, H,, ay 0 —— Ly, ee eS) Diazobenzol. Phenol. Phenol- diazobenzol. Both compounds are weak acids; the first being capable of forming a well-characterized silver-salt, which is obtained in the form of a blood-red precipitate when an ammoniacal solution of phenoldiazobenzol is treated with nitrate of silver. On heating the platinum-salt of diazobenzol mixed with carbonate of sodium in a retort, chlorobenzol is obtained, the formation of which may be expressed as follows :-— GC, H, N,, HCl, PtCl,=6, H, Cl+N,+PtCl,. Niel ee ee sey rc = Platinum-salt. Chlorobenzol. A similar decomposition ensues when the dibromide is heated with carbo- 1864. | in which Nitrogen is substituted for Hydroyen. 381. nate of sodium, 6, H,N, HBr,=€, 0, Br + N,+ Br,. SEALS aan Dibromide. Bromobenzol. The same change may also be effected by simply boiling an alcoholic solution of the dibromide. The peculiar and often remarkable properties of the diazobenzole-com- pounds have induced me to try whether analogous bodies could not be pre- pared also from bromaniline, nitraniline, dibromaniline, &c. Experiment has fully borne out theory. These analogous diazo-substitutions exhibit, however, so much resemblance to the normal diazobenzol compounds, that I should frequently have to repeat almost literally what has already been said of the latter, were I to describe these compounds in detail. I may be permitted, however, to mention a remarkable and interesting fact which their investigation prominently brought out. There are, as is well known, two isomeric nitranilines—the alphanitraniline of Arppe, and the betanitraniline of Hofmann and Muspratt. This isomerism, I found, extends itself to their respective nitrogeri-substitution compounds, and even to their pro- ducts of decomposition. On heating, ev. gr., the dibromide of alphadiazo- nitrobenzol with alcohol, the following change takes place :-— ©, H, (NO,) N, HBr,=6, H, (NO,) Br+N,+Br,. poyiee | G VST a-dibromide. Bromonitrobenzol. The bromonitrobenzol thus obtained is identical with ‘that prepared by Cooper from benzol derived from coal-tar. It crystallizes in the same form, and fuses, like the latter, at 126° C.; sulphide of ammonium converts it imto bromaniline, which crystallizes in octahedra, and is identical with the bromaniline of Hofmann obtained from bromisatine. Bromonitrobenzol, prepared in a perfectly similar manner by the decomposition of the dibro- mide of betadiazonitrobenzol, possesses, however, widely different properties. The e-bromonitrobenzol just described crystallizes in long needles, whilst the new benzol-derivative, which I will designate by the name of 6-bromonitro- benzol, forms well-developed prisms, the fusing-point of which lies at 56°C. Sulphide of ammonium converts it likewise into bromaniline; but this base differs in its physical properties entirely from the bromaniline obtained by Hofmann. It forms a colourless oil, which combines with acids, and gives rise to a series of beautiful salts, which in their turn differ greatly from the ordinary bromaniline salts in their physical properties. I will distinguish this bromaniline by calling it 3-bromaniline from that obtained by Hofmann, which I will call a-bromaniline. It deserves to be mentioned briefly that there exist likewise two isomeric chloronitrobenzols (alpha and beta) obtained by heating the platinum-salts of the respective diazonitrobenzole with carbonate of sodium, C,H, (NO,) N,, HCl, PtCl,=C, H, (NO,) Cl+ N,-+ PtCl,. io Tar AIS Ce ys a, B Platicame salt, a, B Chloronitrobenzol. 382 Mr. P. Griess on a New Class of Compounds [June 16, Alpha-nitrochlorobenzol furnishes, when reduced by means of sulphide of ammonium, the ordinary (alpha-) chloranilme; beta-chloronitrobenzol yielding a new base of like composition (beta-chloraniline), distinguished from the former by its oily nature. Corresponding diazo-compounds can readily be prepared from the homo- logues of aniline and other analogous bases by submitting them to a treat- ment exactly similar to that which in ease of aniline yielded diazobenzol. Thus I have obtained the diazo-compounds from toluidine, naphthalidiue, and nitranisidine, C,H,(NO,)9. I have abstained from entering more fully into a description of their physical and chemical habitus, as well as the respective products of decomposition to which they give rise, since they offer nothing characteristically new *. All compounds already described have been derivations from monoatomic amido-bases.* I have on a former occasiont had an opportunity of pointing out that the action of nitrous acid upon diatomic bases, such as nitrate of benzidine, is perfectly analogous to that which gives rise to the formation of nitrate of diazobenzol from nitrate of aniline. Whilst, however, in the last-mentioned reaction only one atom of nitrous acid exchanges its nitrogen for three atoms of hydrogen of the original compound, six atoms of hydrogen are exchanged for two atoms of nitrogen when nitrous acid reacts upon nitrate of benzidine. Respecting these compounds I shall only briefly describe a few general properties and a few products of decomposition. Sulphate of Tetrazodiphenyl, 2C,, H, N,, 38H, Q,, crystallizes in white or slightly yellowish-coloured needles, which are very soluble in water, and almost insoluble in strong alcohol and ether. On boiling the alcoholic solution, the following decomposition takes place :— 26,,H,N,, 39H, 0,44H, 0=26,, H,,0,+N,+39H, 9,. as Nagy oe 8 Sulphate of tetrazodi- Diphenylenic acid phenyl. (diphenylene-alcohol). I have already had occasion to describe diphenylenic acid (diphenylene- alcohol) obtained by decomposition, analogous to that of nitrate of tetrazo- diphenyl with water, and I have therefore only to refer to what has been stated on that occasion. The decomposition which tetrazodiphenyl undergoes on boiling with * In a former notice (Proceedings, Jan. 22, 1863) I briefly described the formation of nitrate of naphthol, which by its decomposition with water gave rise to the long-sought- for naphthyl-alcohol, €,, H, 9. + Proceedings, Jan. 22, 1863. 1864. | in which Nitrogen is substituted for Hydrogen. 383 alcohol differs from the previous one, and takes place according to the equation 2C,,H,N,, 38H, 0,+4¢, H, O= 26H +40, H, +38 H,0,+8N. a Eye) (eee Bisse Sulphate of tetrazodiphenyl. Alcohol. Diphenyl. one The dipheny] which results from this reaction is identical with the com- pounds obtained by Fittig from bromobenzol. A comparative examination of the two demonstrates this most unmistakeably. The transformation which sulphate of tetrazodiphenyl undergoes when it is heated with a small quantity of strong sulphuric acid is likewise of great interest. Two new sulpho-acids are formed, which I shall call tetra- and tri-sulphodiphenylenic acid. The following equation explains their formation in the most natural manner :— C,H, N, +48 H, Ss), = Ea Ho §, H eOrnt Ny Tale aaa Tetrazodi- Tetiasnlphediane: phenyl. nylenic acid. G,, H, N,+38H, 0,=¢,, H,, 8, H,0,,+N,. NE aa Trisulphodiphenylic acid. The separation of these two acids is based upon the unequal solubility of their barium-salts. The process is, however, somewhat complicated, and I therefore abstain from describing it. Both acids are capable of com- bining with bases in various proportions. Tetrasulphodiphenylenic acid is octobasic. The lead-salt crystallizes in beautiful needles, and has the com- position €,, H,, S, Pb, O,,. Trisulphodiphenylenic acid appears to be hexa- basic. I have as yet only prepared the former acid in a free state. It crystallizes in white needles, which are readily soluble in water and alcohol. Tetrabromide of the Tetrazodiphenyl, C,, H,N,, 2HBr, Br,, This compound forms crystals of an orange colour with curved faces. On heating with alcohol, it splits up according to the equation ©, H, N, H, Lar =€,, H, Br,+N,+Br,. cee ioe ey) = ae Tetrabromide. Bromodipheny]l. Bromodipheny] crystallizes from alcohol and ether (in which it is rather difficultly soluble) in beautiful prisms which fuse at 164°C. This sub- stance can be distilled without undergoing decomposition. Bromodiphenyl has also been obtained by Fittig (according toa private communication) by the action of bromine upon diphenyl. The platinum-salt of tetrazodiphenyl, C,,H, N,,H, Cl, (PtCl,),, forms small yellow plates, which furnish, when heated with carbonate of sodium, chlorodipheny] closely resembling the analogous bromine-compound. 384, Dy. L. S. Beale on the Minute Anatomy [June 16, : 1 H,N, Tetrazodiphenyl-amidobenzol, € H. N a C H, N 3 This complex body is formed when an aqueous solution of nitrate of tetrazodiphenyl is mixed with aniline. It is deposited in a yellow crystal- line mass, which can be recrystallized from alcohol or ether (in which it is but slightly soluble), and is obtained in lancet-like plates. When heated in a dry state, it is decomposed with slight explosion. Its formation is ex- pressed by the equation C,H, N,, 2NHO,+46, H, N=G,, H, oN,+ 2G, H,N, NHQ,. | : ees Va pee Nitrate of tetrazodi- Aniline. New compound. arn of aniline. phenyl. Pere daphien ne Oo A \n,. This body is obtained in the form of slightly yellowish-tinged lustrous plates, which are very difficultly soluble in cold, readily, however, in hot alcohol and in ether. It combines neither with acids nor with bases. Heated by itself it explodes. The following equation explains its forma- tion :— C,, H, N, H, Br,+8NH, eo , H,N,+6NH, Br. ay Set ay fact pm, Tetrabromide. , Tetrazodi- phenylimide. I have not succeeded in preparing tetrazodiphenyl in a free state, nor have I been able to obtain compounds of tetrazodiphenyl with bases in any- thing like a well-characterized condition. I pass over the abortive experi- ments made by me in this direction. Many of the experiments just described have been carried on in the laboratory of the Royal College of Chemistry, London, others in that of the University of Marburg ; and I take this opportunity of returning my thanks to Prof. Hofmann of London, and Prof. Kolbe of Marburg, for allowing me the use of these institutions. XXI. “New Observations upon the Minute Anatomy of the Papillee of the Frog’s Tongue.” By Lionex 8. Beare, M.B., F.R.S., F.R.C.P., Professor of Physiology and of General and Morbid Anatomy in King’s College, London; Physician to the Hospital, &e. Received June 16, 1864. (Abstract.) After alluding to the observations of Axel Key, whose results accord with his own more closely than those of any other observer, the author refers particularly to the drawings of Hartmann, the latest writer upon the structure of the papillee. According to the author, Dr. Hartmann, owing 1864.] | of the Papille of the Frog’s Tongue. 385 to the defective method of preparation he employed, has failed to observe points which had been seen by others who had written before him, and which may now be most positively demonstrated. Hartmann’s process consisted in soaking the tissue for three days in solution of bichromate of potash, and afterwards adding solution of caustic soda. It can be shown by experiment that many structures which can be most clearly demon- strated by other modes of investigation, are rendered quite invisible by this process. Hartmann’s observations, like those of the author, have been made upon the papillee of the tongue of the little green tree-frog (Hyla arborea). With reference to the termination of the nerves in the fungiform papillee of the tongue of the Hyla, the author describes a plexus of very fine nerve- fibres, with nuclei, which has not been demonstrated before. ‘ Fibres re- sulting from the division of the dark-bordered fibres in the axis of the papilla can be traced directly into this plexus. From its upper part fine fibres, which interlace with one another in the most intricate manner, forming a layer which appears perfectly granular, except under a power of 1000 or higher, may be traced into the hemispheroidal mass of epithe- lium-like cells which surmounts the summit of the papilla. This hemi- spheroidal mass belongs not to epithelial, but to the nervous tissues. It adheres to the papilla after every epithelial cell has been removed; the so-called cells of which the entire mass consists cannot be separated from one another like epithelial cells; fibres exactly resembling nerve-fibres can often be seen between them; and very fine nerve-fibres may be traced into the mass from the bundle of nerves in the papilla. The fine nerve-fibres which are distributed to the simple papille of the tongue, around the capillary vessels, and to the muscular fibres of these fungiform papillz, come off from the very same trunk as that from which the bundle of purely sensitive fibres which terminate in the papille are derived. ‘The fine nucleated nerve-fibres of the capillaries which the author has demonstrated have been traced into undoubted nerve-trunks in many instances, so that it is quite certain that many of the nuclei which have been considered to belong to the connective tissue (connective-tissue cor- puscles) are really the nuclei of fine nerve-fibres not to be demonstrated by the processes of investigation usually followed*. ‘These nerve-fibres in the connective tissue around the capillaries are considered by the author to be the afferent fibres of the nerve-centres of which the efferent branches are those distributed to the muscular coat of the small arteries. The author’s observations upon the tissues of the frog convince him that the nervous tissue is distinct in every part of the body from other special tissues. For example, he holds that nerve-fibres never pass by continuity of tissue into the ‘ nuclei’ (germinal matter) of muscular fibres, or into those * See “On the Structure and Formation of the so-called Apolar, Unipolar, and Bi- polar Nerve-cells of the Frog,” Phil. Trans. 1863, plate 40, fig. 44. 386 Dr. L. S. Beale on the Paths [June 16, of tendon, of the cornea, or of epithelium. He advances arguments to show that the epithelium-like tissue upon the summit of the papilla is not epithelium at all, but belongs to the nervous tissues. Hence it follows that nerves do not influence any tissues by reason of continuity of tissue, but solely by the nerve-currents which pass along them*. The author states that the so-called ‘ nuclei’ (germinal matter) of the fine muscular fibres of the papille are continuous with the contractile material, as may be demonstrated by a magnifying power of 1800 linear; and he holds the opinion that the contractile matter is formed from the nuclei. He adduces observations which lead him to the conclusion that these nuclei alter their position during life, and that, as they move in one or other direction, a narrow line of new muscular tissue (fibrilla) is as it were left behind+. This is added to the muscular tissue already formed, and thus the muscle increases. XXII. “Indications of the Paths taken by the Nerve-currents as they traverse the caudate Nerve-cells of the Spinal Cord and Ence- phalon.” By Lionet 8S. Beatz, M.B., F.R.S., F.R.C.P., Professor of Physiology and of General and Morbid Anatomy in King’s College, London; Physician to the Hospital, &c. Re- ceived May 18, 1864. Although the caudate nerve-vesicles, or cells existing in the spinal cord, medulla oblongata, and in many parts of the brain, have been described by the most distinguished modern anatomists, there yet remains much to be ascertained with reference to their internal structure, connexions, and * The author feels sure that the conclusions of Kihne, who maintains that the axis cylinder of a nerve-fibre is actually continuous with the ‘ protoplasm’ (germinal matter) of the corneal corpuscle, result from errors of observation. The prolongations of the corneal corpuscles, on the contrary, pass over or under the finest nerve-fibres, but are never continuous with them, as may be distinctly proved by examining properly pre- pared specimens under very high magnifying powers (1000 to 5000 linear). The corneal tissue results from changes occurring in one kind of germinal matter—the nerve-fibres distributed to the corneal tissue from changes occurring in another kind of germinal matter. If the connexion is as Kihne has described, a ‘nucleus’ or mass of germinal matter would be producing nervous tissue in one part and corneal tissue in another part; and since it has been shown that the ‘nuclei’ of the corneal tissue are continuous with the corneal tissue itself, the nerve-fibres must be continuous, through the nuclei, with the corneal tissue itself ; and if with corneal tissue, probably with every other tissue of the body. But such a view is opposed to many broad facts, and not supported by minute observation. The nuclei of the nerve-fibres are one thing, the nuclei of the corneal tissues another ; and the tissues resulting from these nuclei, nerve- tissue, and corneal tissue are distinct in chemical composition, microscopical characters, and properties and actions. t “New Observations upon the Movements of the Living or Germinal Matter of the Tissues of Man and the higher Animals,” Archives, No, XIV. p. 150. er aS ee eee ee ae ee = PROCZEDINGS OF THE ROYAL SOCIATY. Scale, 7¢43 of an Bnéglish Inch [+++ s1+s+ss+s 1 X 700. Large candate nerve cell, with smaller cells and nerve fibres, from a thin transverse section of the lower part of the grey matter of the medulla oblongata of a young dog. The specimen had been soaked for some weeks in acetic acid and glycerine. The lines of dark granules resulting from the action of the acid are seen passing through the very substance of the cell in very definite directions. Thus the cell is the point where lines from several distant parts intersect (Diagram. Fig.2). It is probable that each of these lines is but a portion of a cornplete circuit (see Diagram in Fig. 3). A,A,A, large fibres which leave the cell. B, a fibre from another cell, dividing into finer fibres, exhibiting several lines of granules. c,C,C, fibres from a youn¢er candate nerve vesicle. D, fine and flattened dark-bordered fibres. E, three fine nerve fibres running together in a matrix of connective tissue. F, F, F, capillary vessels. i, 3, B. del. 1863. 1864. ] of Nerve-currents in Nerve-cells. 387 mode of development. In this paper I propose to describe some points of interest in connexion with their structure. In the first place, however, I would remark that there are neither ‘cells’ nor ‘ vesicles’ in the ordinary acceptation of these words, for there is no proper investing membrane, neither are there ‘cedl-contents’ as distinguished from the membrane or capsule ; in fact the so-called cell consists of soft solid matter throughout. The nerve-fibres are not prolonged from the nucleus or from the outer part of the cell, but they are continuous with the very material of which the substance of the ‘cell’ itself is composed, and they are, chemically speaking, of the same nature. So that in these caudate cells we have but to recog- nize the so-called ‘nucleus’ (germinal matter) and matter around this (formed material) which passes into the ‘ fibres,’ which diverge in various directions from the cell: see Plate III. (fig. 1). At the outer part of many of these ‘cells,’ usually collected together in one mass, are a number of granules. These are not usually seen in the young cells, and they probably result from changes taking place in the matter of which the substance of the cell is composed. But it is not pro- posed to discuss this question in the present paper. My special object in this communication is to direct attention to a pecu- liar appearance I have observed in these cells, which enables me to draw some very important inferences with reference to the connexions and action of these very elaborate and most important elements of the nervous mechanism. In some very thin sections of the cord and medulla oblongata of a young dog, which had been very slowly acted upon by dilute acetic acid, the ap- pearances represented in Plate III. (fig. 1) were observed. Subsequently, similar appearances, though not so distinct, have been demonstrated in the caudate nerve-vesicles of the grey matter of the brain of the dog and cat, as well as of the human subject. I have no doubt that the arrangement is constant, and examination of my specimens will probably satisfy observers that the appearance is not accidental. ach fibre (a, a, a) passing from the cell exhibits in its substance several lines of granules. The appearance is as if the fibre were composed of several very fine fibres imbedded in a soft transparent matrix, which fibres, by being stretched, had been broken transversely at very short intervals. At the point where each large fibre spreads out to form the body of the cell, these limes diverge from one another and pursue different courses through the very substance of the cell, _ in front of, and behind, in fact around the nucleus. Lines can be traced __ from each fibre across the cell into every other fibre which passes away from it. The actual appearance is represented in Plate III. ; and in the diagram, fig. 2, a plan of a ‘cell,’ showing the course of afew of the most important ___ of these lines which traverse its substance, is given. | I do not conceive that these lines represent fibres structurally distinct from one another, but I consider the appearance is due to some difference in composition of the material forming the substance of the cell in these VOL. XIII. 2F 388 Dr. L. 8. Beale on the Paths [June 16, particular lines ; and it seems to me that the course which the lines take permits of but one explanation of the appearance. Supposing nerve-cur- rents to be passing along the fibres through the substance of the cell, A diagram of such a cell as that represented in Plate IIT. (fig. 1), showing the prin- cipal lines diverging from the fibres at the point where they become continuous with the substance of the cell. These lines may be traced from one fibre across the cell, and may be followed into every other fibre which proceeds from the cell. they would follow the exact lines here represented ; and it must be noticed that these lines are more distinct and more numerous in fully-formed than in young cells. They are, 1 think, limes which result from the —— ; passage of nerve-currents in these definite directions. Now I have already advanced arguments in favour of the existence of complete nervous circuits, based upon new facts resulting from observations upon a, the peripheral arrangement of the nerves in various tissues* ; 4, the course of individual fibres in compound trunks, and the mode of branching and division of nerve-fibrest ; and ce, the structure of ganglion- cellst. I venture to consider these lines across the substance of the cau- date nerve-cells as another remarkable fact in favour of the existence of such circuits ; for while the appearance would receive a full and satisfactory explanation upon such an hypothesis, I doubt if it be possible to suggest another explanation which would seem even plausible. Nor would it, I think, be possible to adduce any arguments which would so completely upset the view that nerve-force passes centrifugally from one * Papers in the Phil. Trans. for 1860 and 1862. Lectures on the Structure of the Tissues, at the College of Physicians, 1860. t “On very fine Nerve-fibres, and on Trunks composed of very fine Fibres alone,” Ar- chives of Medicine, vol. iv. p.19. ‘ On the Branching of Nerve-trunks, and of the sub- division of the individual fibres composing them,” Archives, vol. iv. p. 127. ¢ Lectures at the College of Physicians. Papers in Phil. Trans. for 1862 and 1863. 1864. | of Nerve-currents in Nerve-cells. 389 cell, as from a centre, towards its peripheral destination, as this fact. So far from the fibres radiating from one cell, or from the nucleus as some suppose, in different directions, all the fibres which reach the cell are com- plex, and contain lines which pass uninterruptedly through it into other fibres. Instead of the cell being the point from which nerve-currents radiate in different directions along single fibres, it is the common point where a number of circuits having the most different distribution intersect, cross, or decussate. The so-called cel/ is a part of a circuit, or rather of a great number of different circuits. Diagram to show the possible relation to one another of various circuits traversing a single caudate nerve-cell. @ may be a circuit connecting a peripheral sensitive sur- face with the cell; & may be the path of a motor impulse; ¢ and d other circuits passing to other cells or other peripheral parts. A current passing along the fibre a might induce currents in the three other fibres, 4, c, d, which traverse the same cell. i I conclude that at first the formed material of the cell is quite soft and almost homogeneous, but that as currents traverse it in certain definite lines, difference in texture and composition is produced in these lines, and perhaps after a time they become more or less separated from one another, and insulated by the intervening material. It may perhaps be carrying speculation upon the meaning of minute anatomical facts too far to suggest that a nerve-current traversing one of these numerous paths or channels through the cell may influence all the lines running more or less parallel to it (fig. 3). I have ascertained that fibres emanating from different caudate nerve- cells situated at a distance from one another (fig. 4, a, a) at length meet and run on together as a compound fibre (4, 6, 6), so that Tam compelled to conclude (and the inference is in harmony with facts derived from ob- servations of a different kind) that every single nerve-fibre entering into the formation of the trunk of a spinal nerve, or single fibre passing from a 282 Lie 390 Dr. L. S. Beale on the Paths [June 16, ganglion, really consists of several fibres coming from different and probably very distant parts. In other words, I am led to suppose that a single dark- bordered fibre, or rather its axis-cylinder, is the common channel for the passage of many different nerve-currents having different destinations. It is common to a portion of a great many different circuits. The fibres which result from the subdivision of the large fibre which leayes the cell become exceedingly fine (the ath of an inch in diameter or less), and pursue a very long course before they run parallel with other fibres. As the fibres which have the same destination increase in number, the compound trunk becomes gradually thicker and more distinct. The several individual fibres coalesce and form one trunk, or axis-cylinder, around which the protective white substance of Schwann collects. At the periphery the subdivision of the dark-bordered fibre again occurs, until peripheral fibres as fine as the central component fibres result *. Diagram to show the course of the fibres which leave the caudate nerve-cells. a, a are parts of two nerve-cells, and two entire cells are also represented. Fibres from several different cells unite to form single nerve-tibres, 6, 6,5. In passing towards the periphery these fibres divide and subdivide; tne resulting subdivisions pass to dif- ferent destinations. The fine fibres resulting from the subdivision of one of the-caudate processes of a nerve-cell may help to form a vast number of dark-bordered nerves, but it is most certain that no single process ever forms one entire axis-cylinder. Although it may be premature to devise diagrams of the actual arrange- ment, if I permit myself to attempt this, I shall be able to express the in- ferences to which I have been led up to the present time in a far more intelligible manner than I could by description. But I only offer these schemes as rough suggestions, and feel sure that further observation will * “ General Observations upon the Peripheral Distribution of Nerves,” my ‘Archives,’ iu. p. 234. “* Distribution of Nerves to the Bladder of the Frog,” p. 248. ‘‘ Distribution of Nerves to the Mucous Membrane of the Epiglottis of the Human subject,” p. 249. 1864. ] of Nerve-currents in Nerve-ceils. 391 enable me to modify them and render them more exact. The fibres would in nature be infinitely longer than represented in the diagrams. The cell below ¢ (fig.5) may be one of the caudate nerve-cells in the anterior root of a spinal nerve, that above & one of the cells of the ganglion upon the posterior root, and a@ the periphery. I will not attempt to describe the course of these fibres until many different observations upon which I am now engaged are further advanced, but I have already demonstrated the passage of the fibres from the ganglion-cell into the dark-bordered fibres as represented in the diagram. Fig. 5. Diagram to show possible relation of fibres from caudate nerve-cells, and fibres from cells in ganglia, as, for example, the ganglia on the posterior roots. @ is supposed to be the periphery ; the cell above 6 one of those in the ganglion. The three caudate cells resemble those in the grey matter of the cord, medulla oblongata, and brain. The peculiar appearance I have demonstrated in the large caudate cells, taken in connexion with the fact urged by me in several papers, that no true termination or commencement has yet been demonstrated in the case of any nerve, seems to me to favour the conclusion that the action of a nervous apparatus results from varying intensities of continuous currents which are constantly passing along the nerves during life, rather than from the sudden interruption or completion of nerve-currents. So far from any arrangement having been demonstrated in connexion with any nervous structure which would permit the sudden interruption and completion of a current, anatomical observation demonstrates the structural continuity of all nerve-fibres with nerve-cells, and, indirectly through these cells, with one another. 392 Mr. A. J. Ellis on Musical Chords. [June 16, I venture to conclude that the typical anatomical arrangement of a ner- vous mechanism is not a cord with two ends—a point of origin and a terminal extremity, but a cord without an end—a continuous cireuit. The peculiar structure of the caudate nerve-cells, which I have described, renders it, I think, very improbable that these cells are sources of nervous power, while, on the other hand, the structure, mode of growth, and indeed the whole life-history of the rounded ganglion-cells render it very pro- bable that they perform such an office. These two distinct classes of nerve-cells, in connexion with the nervous system, which are very closely - related, and probably, through nerve-fibres, structurally continuous, seem to perform very different functions,—the one originating currents, while the other is concerned more particularly with the distribution of these, and of secondary currents induced by them, in very many different directions. A current originating in a ganglion-cell would probably give rise to many induced currents as it traversed a caudate nerve-cell. It seems probable that nerve-currents emanating from the rounded ganglion-cells may be constantly traversing the innumerable circuits in every part of the nervous system, and that nervous actions are due to a disturbance, perhaps a varia- tion in the intensity of the currents, which must immediately result from the slightest change occurring in any part of the nerve-fibre, as well as from any physical or chemical alteration taking place in the nerve-centres, or in peripheral nervous organs. XXIII. “On the Physical Constitution and Relations of Musical Chords.” By Arexanper J. Exuis, F.R.S., F.C.P.S.* Received June 8, 1864. When the motion of the particles of air follows the law of oscillation of a simple pendulum, the resulting sound may be called a simple tone. The pitch of a simple tone is taken to be the number of doudle vibrations which the particles of air perform in one second. The greatest elongation of a particle from its position of rest may be termed the extent of the tone. The intensity or loudness is assumed to vary as the square of the extent. The tone heard when a tuning-fork is held before a proper re- sonance-box is simple. The tone of wide covered organ-pipes and of flutes is nearly simple. Professor G. S. Ohm has shown mathematically that all real tones whatever may be considered as the algebraical sum of a number of simple tones of different intensities, having their pitches in the proportion of the numerical series 1, 2, 3, 4, 5, 6, 7, 8, &c. Professor Helmholtz has esta- blished that this mathematical composition corresponds to a fact in nature, that the ear can be taught to hear each one of these simple tones separately, and that the character or quality of the tone depends on the law of the intensity of the constituent simple tones. These constituent simple tones will here be termed indifferently partial * The Tables belonging to this Paper will be found after p. 422. 1864. | Mr. A. J. Ellis on Musical Chords. 393 tones or harmonics, and the result of their combination a compound tone. By the pitch of a compound tone will be meant the pitch of the lowest partial tone or primary. When two simple tones which are not of the same pitch are sounded together, they will alternately reinforce and enfeeble each other’s effect, producing a libration of sound, termed a beat. The number of these beats in one second will necessarily be the difference of the pitches of the two simple tones, which may be termed the beat number. As for some time the two sets of vibrations concur, and for some time they are nearly oppo- site, the compound extent will be for some time nearly the sum, and for some time nearly the difference of the two simple extents, and the zntensity of the beat may be measured by the ratio of the greater intensity to the less. But the beat will not be audible unless the ratio of the greater to the smaller pitch is less than 6:5, according to Professor Helmholtz. This is a convenient limit to fix, but it is probably not quite exact. To try the ex- periment, I have had two sliding pipes, each stopped at the end, and having each a continuous range of an octave, connected to one mouthpiece. The tones are nearly simple; and when the ratio approaches to 6:5, or the interval of a minor third, the beats become faint, finally vanish, and do not reappear. But the exact moment of their disappearance is difficult to fix, and indeed seems to vary, probably with the condition of the ear. The ear appears to be most sensitive to the beats when the ratio is about 16:15. After this the beats again diminish in sharpness ; and when the ratio is very near to unity, the ear is apt to overlook them altogether. The effect is almost that of a broken line of sound, as i the spaces representing the silences. Slow beats are not disagreeable; for example, when they do not exceed 3 or 4 in a second. At 8 or 10 they become harsh; from 15 to 40 they thoroughly destroy the continuity of tone, and are discordant. After 40 they become less annoying. Professor Helmholtz thinks 33 the beat number of maximum disagreeableness. As the beats become very rapid, from 60 to 80 or 100 in a second, they become almost insensible. Pro- fessor Helmholtz considers 132 as the limiting number of beats which can be heard. They are certainly still to be distinguished even at that rate, but become more and more likea scream. Though /*¢ and g* should give 198 beats in a second if ce=264, and the interval is that for which the ear is most sensitive, I can detect no beats when these tones are played on two flageolet-fifes. Hence beats from 10 to 70 may be considered as discord- ant, and as the source of all discord in music. Beyond these limits they produce a certain amount of harshness, but are not properly discordant. When the extent of the tones is not infinitesimal, Professor Helmholtz has proved that on two simple tones being sounded together, many other tones will be generated. The pitch of the principal and only one of these combinational tones necessary to be considered, is the difference of the pitch of its generating tones. It will therefore be termed the differential 394 ‘Mr. A. J. Ellis on Musical Chords. » [Sunee; tone. Its intensity is generally very small, but it becomes distinctly audible in beats. The differential tone is frequently acuter than the lower generator, and hence the ordinary name ‘‘ grave harmonic”’ is inapplicable. As its pitch is the beat number of the combination, Dr. T. Young attri- buted its generation to the beats having become too rapid to be distin- guished. This theory is disproved, first, by the existence of differential tones for intervals which do not beat, and secondly, by the simultaneous presence of distinct beats and differeutial tones, as I have frequently heard on sounding f*, f*4, or even f”, ft together on the concertina, when the beats form a distinct rattle, and the pa tone is a peculiar pene- trating but very deep hum. The object of this paper is to apply these laws, partly physical and partly physiological, to explain the constitution and relations of musical chords. It is a continuation of my former paper on a Perfect Musical Scale*, and the Tables are numbered accordingly. Two simple tones which make a greater interval than 6:5, and there- fore never beat, will be termed disjunct. Simple tones making a smaller interval, and therefore generally beating, will be termed pulsative. The unreduced ratio of the pitch of the lower pulsative tone for which the beat number is 70 to that for which it is only 10, will be termed the range of the beat. The fraction by which the pitch of the lower pulsative tone must be multiplied to produce the beat number, will be termed the beat factor. The ratio of the pitches of the pulsative tones, on which the sharpness of the dissonance depends, will be termed the beat enterval. A compound tone will be represented by the absolute pitch of its primary and the relative pitches of its partial tones, as C (1, 2, 3, 4,....)2 As generally only the relative pitch of two compound tones has to be con- sidered, the pitches will be all reduced accordingly. Thus, if the two primaries are as 2: 3, the two compound tones will be represented by 2, 4, 6, 8, 10,...., and 3, 6,9, 12,15 .... The intensity of the various partial tones differs so much in different cases, that any assumption which can be made respecting them is only approximative. In a well-bowed violin we may assume the extent of the harmonics to vary inversely as the number of their order. Hence, putting the extent and intensity of the primary each equal to 100, we shall have, with sufficient accuracy— danmonies.. Me 262 3 A ne OT Te 8) Oat? Latent...» 100, 50, 33, 25, (20,17, 145° ° 2 ee Intensity... 1003.5 25, 11-86, 45352 It will be assumed that this law holds for all combining compound * Proceedings of the Royal Society, vol. xiii. p.93. The following mis- prints require correction :—P.97, line 7 from bottom, for c? read 6. Table L., p- 105, diminished 5th, example, read f:B; minor 6th, logarithm, read :20412; Pythagorean Major 6th, read 27:16, 3°:2*; Table V., col. VL, last line, read felt tot tk. 1864. ] “Mr. A.J. Ellis on Musical Chords. 395 tones, the intensity of the primary in each case being the same. The results will be sufficient to explain the nature of chords on a quartett of bowed instruments, but may be much modified by varying the relative intensities of the combining tones. On examining a single compound tone, we may separate its partial tones into two groups: the first disjunct, which will never beat with each other ; the second pulsative, which will beat with the neighbouring disjunct tones. Thus. Wisunce.. 1, 2, 3, 4, 5,'6, = 8, -, 10, —, 12, —, —, —, 16, Pulsative. -,-,-, -, -,-, 7, -,9, -—, 11, -—, 13, 14, 15, - Disjunct. . TT are) ese 20, are Seppe. Sh cannys 24, SEED s age Sia 30. Pulsutive. 17, 18,19, —, 21, 22, 23, —, 25, 26, 27, 28, 29, — When any compound tone therefore developes any of the harmonics above the 6th, there may, and probably will, be beats, producing various degrees of harshness or shrillness, jarring or tinkling. These, however, are all natural qualities of tone, that is, they are rooted at once by the natural mode of vibration of the substances employed. But if we were to take a series of s¢mple tones having their pitches in the above ratios, and to vary their intensities at pleasure, we should produce a variety of artificial qualities of tone, some of which might be coincident with natural qualities, but most of which would be new. This method of producing artificial qualities of tone is difficult to apply, but has been used with success by Professor Helmholtz to imitate vowel-sounds, &c. If, however, instead of using so many simple tones, we combine a few compound tones, the pitches of which are such that their primaries might be harmonics of some other compound tone, then the two sets of partial tones will necessarily combine into a single set, which may, or rather must be considered by the ear as the partial tones of some new compound tone, having very different intensities from those possessed by the partial tones of either of the combining compound tones. That is, an artificial quality of tone will have been created by the production of these jovn¢ harmonics. Such an artificial quality of tone constitutes what is called a musical chord. The two or more compound tones from which it is built up are its consti- tuents. ‘The primary joimt harmonic is the real root or fundamental bass of the chord, which often differs materially from the supposititious root assigned by musicians. If the primaries of the constituents are disjunct, and all their partial tones are disjunct, then the joint harmonics will be also disjunct, unless some pulsative differential tones have been introduced. If, however, the constituents have pulsative partial tones, the chord will also have them. Such chords, which are generally without beats, and are only exceptionally accompanied by beats, are termed concords, and they are unisonant or dis- sonant according as the beats are absent or present. Their character therefore consists in having the pitches of their constituents as 1, 3, 5, oras 396 Mr. A. J. Ellis on Musical Chords. [June 16, these numbers multiplied by various powers of 2, that is, as 1, 3, 5, or their octaves. If any of the constituents is pulsative the chord will generally have beats, but may be exceptionally without beats. Such chords are termed discords. ‘Their character consists in having ¢wo or more of the pitches of their constituents as 1, 3, 5, or their octaves, and at least one of them as 7, 9, or some other pulsative tones, or their octaves. What pulsative tones should be selected depends on the sharpness of the dissonance which it is intended to produce, and therefore on the interval of the beat which is created. Thus, since 7: 6=1:'16667 and 8: 7=1°14286 are both near the limit 6: 5= 1-2, the discord arising from 7 would be slight. Some writers have even considered the chord 1, 3, 5, 7 to be concordant. Again, 9: 8=1°'125 is rather rough, but 10: 9=1-11111 is much rougher. Hence, if 9 is introduced, 10 should be avoided, that is, the octave of 5 should be omitted, which generally necessitates the omission of 5 itself, as in the chord 1,3,9. But 11: 10=1°'1 and 12:11=1-09091 are both so sharply dissonant, that if 11 is used neither 10 nor 12 should be employed. Now 10 is the octave of 5, and 12 is both the 3rd harmonic of 4 and the 4th harmonic of 3, and would therefore be produced from 3 and 4. Hence the use of 11 would forbid the use of 3, 4, and 5, that is, of the best disjunct tones. Hence 11 cannot be employed at all. Similarly, 13 : 12=1-08333 and 14: 13=1:07692 are both extremely harsh. The latter is of no con- sequence, because 7 can be easily omitted. But even 15: 13=1°15384 is more dissonant than 7:6. Hence 13 would also beat with the harmonics of 3, 4, and 5. Consequently 13 must be also excluded. All combina- tions in which the differential tones 11 and 13 are developed will also be extremely harsh. As we therefore suppose that 14: 13=1°07692 never occurs, and as 14:12=7:6, the mildest of the dissonances, 14 may be used if 15 is absent, and thus 15: 14=1°'07143 avoided. When 14 and 15 are developed as harmonics of 7 and 5, and not as the primaries of con- stituent tones, their intensity will be so much diminished that the discord will not generally be too harsh. When 15 is used as a constituent, 14 and 16 should be avoided; that is, 7, and 1, 2 and 4, of which 14 and 16 are upper harmonics, should be omitted to avoid 15: 14=1-07143 and 16: 15=1:06667, which may be esteemed the maximum dissonance. By omitting 16 and 18, and thus avoiding 17:16=1°0625 and 18:17= 1:05882 (that is, by. not using 4, 8, or 9 as constituent tones), 17 becomes useful ; for 17 : 15=1°13333 is milder than 9: 81-125, which is by no means too rough for occasional use. The other pulsative harmonics, which are represented by prime numbers, are not sufficiently harmonious for use ; but those produced from 2, 3, 5 (such as 25, 27, 45) may be sometimes useful, provided that the tones with which they form sharp dissonances are omitted. The result of the above imvestigation is that the only pulsative tones suitable for constituents are 7, 9, 15, 17, 25, 27, 45, and their octaves. 1864. ] Mr, A. J. Ellis on Musical Chords. 397 The introduction of any one of these tones in conjunction with 1, 3, 5 and their octaves will therefore form a discord, the harshness of which may be frequently much diminished by the omission of 1 and its octaves for the constituents 7, 15, 17, by the omission of 5 for the constituent 9, and by the omission of 24 for the constituents 25, 27, 45. Using the notation of my former paper, where gz=63: 64, and putting in addition vij=84:85, xj=33: 32, xij =39:40, 17=255:256, and Xv1j=135 : 136, the tones 1 to 18 may be represented by the following notes in terms of C* :— TD 2. 3, 4, 5, 6, rips 8, 9, 10, Pye Gc, e, OG, GOD or vy a” oct 1], We 13, 14, Los 4-16, 17, Meo) Xe, coDorvjat, (6, c', gd) or xviy ct, 18, 20, 24, 29, 27; 45, a’, e, I ee Sh: This notation will show what are the musical names of the constituents of musical chords, and how they may be approximately produced on an organ, harmonium, or pianoforte. - By the ¢ype of a musical chord is meant the numbers which express the relative pitches of its constituents, after such octaves below them have been taken as to leave only uneven numbers, which are then called the elements of the type. By the form of the chord is meant the numbers before such reduction. Thus the type 1, 3, 5 embraces, among others, the forms I, 3, D5 l, 2, 3, D5 2, 3, D5 4, 3, 35 3, 8, LO’; 6, 10, 16 ; 2, Qo; 6, 8, and so on; hence the types of musical chords consist of groups of the elements 1, 3, 5, 7,9, 15, 17, 25, 27, 45. The type of a concord is 1, 3, 5, and of a discord 1, 3, 5, P, or 1, 3, 5, P, P’, where P, P! are any of the numbers 7, 9, 15, 17, 25, 27, 45. Discords may be divided into strong and weak, according as those disjunct tones with which the pulsative tones principally beat are retained or omitted. These discords again may be dis- tinguished into those which have one or two pulsative constituents. The chords may also be grouped according to the number of elements in their type, dyads containing two, triads three, tetrads four, and pentads five. The number of elements in the type by no means limits the numbers of constituents, as any octaves above any of the elements may be added. Hence it is possible to classify all the suitable chords of music according to their type, as in Table VI., where the notes corresponding to each type are added in the typical form only. A simple systematic nomenclature is proposed in an adjoining column, and the names by which the true chords or their substitutes are known to musicians are added for reference. Occa- sionally two forms of substitution are given, as they are of theoretical im- portance, although confounded on some tempered instruments. A mode of symbolizing the chords is subjoined, in which several types are classed under one family. A capital letter shows the root of major chords, either 398 Mr. A. J. Ellis on Musical Chords. [June 16, complete or imperfect, and of strong discords, and a smaller letter gives the root of weak discords, a number pointing out the family. In the minor triad the characteristic number is omitted; thus ¢ is written for 15e, meaning the minor triad g e 6, which is the major tetrad 15 C, or CG EB, with its root C omitted, and is usually called “the minor chord of e,” a nomenclature which conceals its derivation. Although chords of the same type have the same general character, this is so much modified by the particular forms which they can assume, that it is necessary to examine these forms in detail. They may be distinguished as simple and duplicated. In the former the number of constituents is the same as in the type; thus 4, 5, 6; 2, 3, 5 are simple forms of the type 1, 3, 5. In the latter, the number of constituents is increased by the higher octaves of some or all of them; thus 1, 2, 3, 5; 2, 4, 5, 6 are dupli- cated forms of 1, 3, 5 and 2, 5, 6, as they contain the octaves 1, 2 and 2, 4. The mode in which the effect of any or all of these combinations may be calculated is shown in Table VII., which consists of two corresponding parts, each commencing with a column containing the “ No. of J. H.,” or of the joint harmonics resulting from the combination of the harmonics of the constituent compound tones. ‘The next columns are headed by the relative pitch of the constituent tones, and contain their harmonics, never extending beyond the 8th, arranged so that their pitch is opposite to the corresponding number of the joint harmonic. It is thus seen at a glance which harmonics of the constituents are conjunct or tend to reinforce each other, and produce a louder joint harmonic, and also which are disjunct and pulsative. In the second part of the Table the extent of each harmonic of each constituent is given on the assumptions already explained. To find the extent of the joint harmonic, we add the extents of the generating conjunct harmonics, and thence find the intensity by squaring and dividing by 100. The dif- ferential tones must then be found by subtracting the pitches of the pri- maries (or in exceptional cases of higher and louder harmonics). The in- tensity of these differential tones may be called 1 for a single tone, and 4 for two concurrent tones, and this number may be subscribed to the inten- sity of the corresponding joint harmonic, as 0,, 25,. The beat intervals have next to be noted, and the beat factors, which are usually the reciprocal of the relative pitch of the lower pulsative harmonic. Thus for the dyad 3, 4 the beat interval is 2, and the beat factor 4. From this factor, or 1:f, we calculate the range P: p=70f: 10f/=210: 30 mm the present case. This must not be reduced, as‘it shows that the interval is dissonant when the pitch of the lower tone is between 30 and 210. To find the intensity, we add and subtract the extents of the pulsative joint harmonics; in this case 50 and 33 are the extents of the 8th and 9th joint harmonics, and their sum and difference are 83 and 17. Then we take the ratio of their squares, each divided by 100, which gives 69: 3. This result must not be reduced, as it gives not only the relative loudness of the swell and fall, but also the loudness of these in relation to the other 1864. | Mr. A. J. Ellis on Musical Chords. 399 jomt harmonics. It must be remembered that when there are several dis- junct harmonics, their unbroken sound tends to obliterate the action of the beats. There is no sensible silence between the beats unless the tones are - simple and the intensities nearly equal. The intensities of the beats be- tween joint harmonics and differential tones cannot be reduced to figures. It is not large. The history of a beat is therefore given by four fractions, which in this case are the interval 9 : 8, the factor 1: 3, the range 210: 30, and the intensity 69:3. These calculations have been made for concordant dyads in Table VIII., and for concordant or major triads in Table IX. An attempt has been made to arrange the 13 forms of the first, and the 20 forms of the second in order of sonorousness, by considering the distribution of the intensities among the several joint harmonics, the development of pulsative differential © tones, and the nature of the beats, omitting those due to the seventh har- monic of an isolated constituent. It has not been thought necessary to give the history of every beat. The intervals of all the beats are seen at a glance by the list of intensities of the joint harmonics. By Table VIII. we see that the only unisonant dyad is the octave 1, 2*, which will be as unisonant as the constituents themselves. All other dyads are occasionally dissonant. Thus the fifth itself is decidedly dissonant when the pitch of the lower constituent lies between 20 and 140. Ona bass concertina tuned justly, I find the fifth, C*G*, quite intolerable, the fifth, C G, rough, but D +d nearly smooth, and at higher pitches there isno per- ceptible dissonance. The beat interval of the major third is 16:15, and the range of dissonance is much greater. The roughness can be distinctly heard as high ase e; in the lower octaves CH is quite discordant, and C*E* intolerable. This Table VIII., therefore, establishes the fact that con- cordance does zo¢ depend on simplicity of ratio alone; but when the de- nominator of the beat factor is small the range is lower, and therefore the dissonance less felt. Dissonance also arises from the pulsative differential tones 7 and 11, so that if the relative pitches are expressed in terms high enough to differ by 7 and 11, the combination will be dissonant. The ear is also not satisfied with forms in which great intensities of joint harmonics are widely separated by many small intensities. The four last forms in Table VIII., namely, the minor tenth 5, 12, the eleventh 3, 8, and the two thirteenths 3, 10 and 5, 16, should therefore be treated as discords. The Table also suggests how defects may be remedied by introducing new constituents to fill up gaps, or by duplications. Similar observations apply to the triads in Table IX. None of them can be unisonant at all pitches. Some of them, as the last seven, are really discordant. The gaps may be generally filled up by duplication. Thus * That is, within the limits of the Table. Dyads such as 1, 2; 1,3; 1,4; 1, 5; 1, Gare all unisonant; but when the interval is very large, the want of con- nexion between the tones renders them unpleasant. The dyad 1, 8 which developes the differential tone 7 is dissonant. 400 Mr. A. J. Ellis on Musical Chords. [June 16, 1, 3, 5 may be converted into 1, 2, 3, 5, and by thus strengthening the 2, 4, and 8 joint harmonics the finest form of concord is produced. In this way the series of duplications in Table X. was produced. In this Table an example has been added to each form to facilitate trial ; but the great im- perfection of the major third in the ordinary system of tuning pianos and harmoniums materially deteriorates the effect of the chords, which ought to be taken on some justly tuned instrument. The discords may be deduced from Tables VII. and VIII., when pro- perly extended, by supposing 7, 9, 15, 17, 25, 27, 45 to be used in the first, and their effect allowed for in the second. The additional discordant effect of 7 will be necessarily least felt where 7 occurs as a differential tone, but these are not the best forms of either triad or tetrad. In the better forms the dissonances 6, 7 and 7, 8 will always be well developed, and as the latter is sharper, the omission of 8, at least as a constituent tone, is suggested. If 71 is used instead of 7, the omission of 8 becomes more urgent, while 6, 74 will beat less sharply than 6, 7, and therefore almost inaudibly. The real beats of the constituents 6, 74 arise from the har- monics 6.6, 5.74, or 36, 352, which are, however, not so much felt as those of 6.6, 5.7 arising from 6, 7, because 36 : 353=1-0125 is further from 16 : 15=1:°0667 than is 36: 35=1:0286. Hence, when 8 is omitted, the dissonance arising from 7} is less than that arising from 7 itself. When 8 is present, 7 or 7/5 is superior to 74. The use of 17,4 for 17 would hardly create any perceptible alteration of roughness when 18 is absent, and when 18 is present 18: 17;—=1-:0548 is further from 16:15 than is 18: 17=1-0588, and therefore the roughness is not quite so great. Of all discords the least dissonant is the minor triad 3, 5, 15, which is formed from the tetrad 1, 3,5, 15 by omitting the root 1, to avoid the dis- sonance 15,16. When the differential tones derived from the primaries of the constituents are deeper than the primaries, and therefore merely indicate the presence of a pulsative tone, which is only faintly realized by the differential tones derived from the upper harmonics of the primaries, and when the dissonant intervals of the minor tenth and major thirteenth 5, 12 and 3,10 are not present in the constituent tones, this chord may be treated as a concord. But in most positions the minor triad is sensibly dissonant, as shown in Table XI., where an attempt has been made to arrange its 20 forms in order of sonorousness. ‘The pitches of the differ- ential tones are added, and examples. subjoined. The effect of the minor chord is very much injured by the usual tuning of harmoniums, &c. A peculiar character of these and other discords, when the pulsative con- stituent is not the highest, consists in the quality of tone being due to very high joint harmonics, except such as are due to differential tones. The root will be consequently extremely deep when the constituent tones are taken at a moderate absolute pitch. This great depth renders its recogni- tion by the ear difficult. Hence probably the disputes of musicians con- cerning the roots of certain discords, and their error in considering 5 to be 1864. ] Mr. A. J. Ellis on Musical Chords. 401 an octave of the root of the minor triad, so that e, g, 6 or 10, 12, 15 is con- sidered by them as derived from * instead of C*. Chords will evidently be related to each other when one or more of their constituents are identical, and natural qualities of tone will be related which have one or more identical harmonics, or which form parts of related chords. Transitions between related chords and compound tones will be easy and pleasing. Hence, in forming a collection of compound tones for use either as natural qualities of tone (in melody) or as constituents of artificial qualities of tone, that is, chords (in harmony), it is important to select such as will have numerous relations, and will produce the concordant dyads and triads, and the least dissonant discord, the minor triad. Hence, taking the concordant major triad 1, 3, 5 as a basis, we must possess its products by 2, 3, and 5. There must be abundant multiples by 2 in order to take the several forms of the triad and to introduce the duplications. The products by 3 and 5 give 3, 9,15 and 5,15, 25. We have then the tones 1, 3, 5, 9, 15, 25, and their octaves. These give three concordant major triads, 1, 3,5; 3, 9, 15; and 5, 15, 25, each of which has one con- stituent in common with each of the others. We have also the major pentad 1, 3, 5, 9, 15, the nine-tetrad 1, 3, 5, 9, the major tetrad 1, 3, 5, 15, and the minor tetrad 3, 5, 9, 15, whence, by omissions, result the nine- triad 1, 3, 9 and minor triad 3, 5, 15. Each of these triads is related to two of the three major triads. The minor triad is intimately related to all three major triads by having two constituents in common with each of them. The discords involving 7 and 17 would evidently require 1, 3, 5, 7, 17 to be taken as a basis. Neglecting these discords for the present, the above results show that we should obtain a useful series of tones by multiplying 1, 3, 5 successively by 3, and each product by 5, taking octaves above and below all the tones thus introduced. We thus find | MAJOR. Menor. Mid rust NEAT OR Asi | eons lea sane nl ee | i ie Ass Bee Set lolS, 4.9; 45 EK je ae | AK {Cy a 27, 45) 9, 15, 45 B ig ¢.6 | KB. ta FE | d 6 ff) BREID 27, 81, 135/27, 45, 135| 81, 243, 405 tA TECH 128, 256, in order to compare them with the larger numbers. Such mul- tiplications are presumed to have been made in the columns of notes. Hence +4:A=81: 5.16, or F=81 : 80, Be: F—135: 1.128, or #=135 : 128, tCi: C=25: 3.8, or {#=—25 : 24, whence E—S07 or. And in the same way the other ratios in ‘ Proceedings,’ vol. xii. p. 95, are reproduced. 402 Mr. A. J. Ellis on Musical Chords. [June 16, In addition to the chords already noticed, we have now the twenty-seven tetrad, 1, 3, 5, 27, or F CAD, and the twenty-seven triad, 1, 5, 27, or F AD, and all the discords derived from 1, 3, 5, 9, 15, 25, 27, 45. But for those derived from 7 and 17 substitutes must be employed. These are obtained as follows. The chord 9, 27, 45, 1.64 is 9 times 1, 3, 5, 74, so that GD BF approximates to 1, 3, 5, 7 in a manner already tested. Again, 1. 32, 3.32, 5.32, 225 is 32 times 1, 3, 5, 73\,, whence FC A {DE gives the second and closer approximation to 1, 3, 5, 7 already considered. When 7} is used for 7 it will be better to use 1, 3, 5, 74, 88, or one-ninth of 9,27, 45, 1.64, 5.16, that is G B D F A, in place of 1, 3, 5, 74, 9 or one-ninth of 9, 27, 44, 1.64, 81, that is G B D F +A, to avoid the disso- nance 5.74, 4.9, or 352, 36. This will therefore replace the seven-nine pentad 1, 3, 5, 7, 9. . The chord 45, 135, 225, 5.64,3.256 is 45 times 1, 3, 5, 74, 1774, or B Ft {Dz AC, and it forms an excellent substitute for the seven-seventeen pentad 1, 3,5, 7,17. Again, the chord 3.16, 5. 16, 15. 16) 135)or 6 times 3, 5, 15, 164, that is C 4 # F, is a sufficiently close approximation to the rough discord 3, 5, 15, 17. It has already been shown that the alterations in the discords thus pro- duced will be slight, and, under certain circumstances, improvements. The omission of 7, 17 in the base 1, 3, 5 is therefore justified. Their insertion would embarrass the performer and composer by an immense variety of tones very slightly differing from each other, as 64, 63; 135, 136; 255, 256. As it is, the distinction between 81, 80 is the source of much difficulty, - and separates chords such as 81, 243, 405, and 5.16, 15.16, 25. 16, or 80, 240, 400, that is, td +H CZ and 4 # {C¥#, which composers desire to _ consider as identical. It was shown in my former paper (Proceedings, vol. xill. p. 98) that the use of 1, 3, 5as a basis requires 72 different tones, exclusive of octaves. The introduction of 7 in the base would increase this number by 45, and the introduction of 17 by 30, while the mental effect produced would be very slightly different. On the other hand, if instead of 1, 3, 5 as a base, we took 1, 2v, 47, where v, T' are ratios differing slightly from 3: 2 and 5:4, we might avoid the ratio 81: 80, reduce the number of tones to 27, and greatly increase the relations of chords. How to effect this important result with the least dissonant effect will be con- sidered in the following paper on Temperament. The three major triads 1, 3,5; 3,9,15; 9, 27, 45 are so related as to form two major pentads, 1, 3, 5, 9, 15 and 3.1,3.3,3.5,3.9, 3.15. Hence the middle triad forming part of both pentads connects the three triads into a whole, closely related to the middle triad, and therefore to its root. These are called the tonic chord and fonic tone, and the connexion itself is termed tonality. If octaves of these tones be taken, thus, Ligroes 3:2 83 5.8 or FC A, Sus Sy 9 Awe Noe 2 CGE, 9.4, Dies 45 GD B, 1864.] Mr. A. J. Ellis on Musical Chords. 403 and the results be taken in order of pitch, we find, on supplying the second octave 3.16, 2A 27, (OO) 32," 30,40, 45,4 (48 ON Ds CE, ce Gy As BS & In this series any two consecutive tones, except 40, 45 or 4, B, belong to the same major pentad, and these are therefore eminently adapted for suc- cessions of chords. Even 40, 45, or 5.8, 45, belong to two related discords ; for 1, 3, 5, 9, or F, C, A, G, and 1, 3, 5, 27, or F C AD, have each two constituents in common with 9, 27,45, 1.64,or GDBF. The discord 3, 5, 15, 45, or CA EL B, contains both the tones in question. These con- siderations justify the major diatonic scale. ~The last discord contains a minor triad, 3, 5, 15. These minor triads, from their relations to three major triads, are evidently peculiarly adapted to introduce successions of harmonies. ‘Taking then the three minor triads and forming their octaves, thus 3. 64, Dad, 15.8 or c2 ae, ger 10, Eas 8s 454 g eb, 27 .8, 45.2, 1385 @&bf4, we may extend them into a scale, 120% 13552. 144, 160, 180, 192, 216, 240 é; St ip I> a; b, c?, d*, e*, where the chordal relations are even more intimate than before, and by means of the chord 45, 135, 225, 5.64, 3.256, or B Ft {D# ac, already noticed, the major triad, 45, 135, 225, or B F¢ {D, is brought into close connexion with the minor triad, 3, 5, 15, or cae. Practically the use of the minor scale consists of a union of four major triads, 1, 3, 5; 3, 9, 15; 9, 27, 45; 27, 81, 135, forming two major scales, with three other major triads, 5, 15, 25; 15, 45, 75; 45, 135, 225, forming a third major scale, by means of three minor triads, 3, 5,15; 9, 15, 45; 27, 45, 135, the roots of which, 1, 3, 9, are the same as the roots of the first three major triads. There are therefore seven roots to all the chords introduced, namely 1, 3, 9, 27, and 5, 15, 45, or F, C, G, D and A, EH, B, and these seven roots form a major diatonic scale. From these relations spring all the others which distinguish the minor scale together with all its various forms and its uncertain tonality, which is generally assumed to be the relation of the chords to 15 or £, the tonic of the last three major triads, but which evi- dently wavers between this and 3, 9 or C, G, the tonics of the first four major triads, and these three tonics, 3, 9, 15, or CG H, form a major triad. By extending this system of chords up and down, right and left, we arrive at the perfect musical scale in Table V. (Proceedings, vol. xii. p. 108), which is therefore entirely justified on physical and physiological grounds, without any of those metaphysical assumptions or mystical attributes of numbers which haye hitherto disfigured musical science. In that Table the VOL, XIII. 2G 404 Mr. A. J. Ellis on the Temperament [June 16, chords have been arranged in the forms 4, 5, 6 and 10, 12, 15, im accord- ance with the usual practice of musicians. In the present paper the typical 1, 3, 5 and 3, 5, 15 have, for obvious reasons, been made the basis of the arrangement. ; XXIV. “On the Temperament of Musical Instruments with Fixed Tones.” By Atexanper J. Exxis, F.R.S., F.C.P.S.* Received June 8, 1864. In the preceding paper on the Physical Constitution of Musical Chords (Proceedings, vol. xiii. p. 392), of which the present is a continuation, I drew attention to the importance of abolishing the distinction between tones which differ by the comma 81:80, on account of the number of fresh relations between chords that would be thus introduced. The con- trivances necessary for this purpose have long been known under the name of Temperament. I have shown that the musical scale which introduces the comma consists of tones whose pitch is formed from the numbers 1, 3, 5, by multiplying continually by 2, 3, and 5. Hence to abolish the comma it will be necessary to use other numbers in place of these. But this alteration will necessarily change the physical constitution of musical chords, which will now become approximate, instead of exact representa- tives of qualities of tone with a precisely defined root. It is also evident that all the conjunct harmonics will be thus rendered pulsative, and that therefore all the concords will be decidedly dissonant at all available pitches. The result would be intolerable if the beats were rapid. Tem- perament, therefore, only becomes possible because very slow beats are not distressing to the ear. Hence temperament may be defined to consist in slightly altering the perfect ratios of the pitch of the constituents of a chord, for the purpose of increasing the number of relations between chords, and facilitating musical performance and composition by the re- duction of the number of tones required for harmonious combinations. The subject has been frequently treated +, but the laws of beats and * The Tables belonging to this Paper will be found after p. 422. + I have consulted the following works and memoirs. Huyghens, Cosmo- theoreos, lib. i.; Cyclus Harmonicus. Sauwvewr, Mémoires de l’Académie, 1701, 1702, 1707, 1717. Henfling, Miscellanea Berolinensia, 1710, vol. i. pp. 265-294. Snuth, Harmonics, 2nd edit. 1759. Marpurg, Anfangsgruende der theoretischen Musik, 1757. Estéve, Mém. de Math. présentés a l’Acad. par divers Savans, 1755, vol. ii. pp. 113-136. Cavallo, Phil. Trans. vol. lxxviii. Romieu, Mém. de l’Acad., 1758. Lambert, Nouveaux Mém. de l’Acad. de Berlin, 1774, pp. 55-73. Dr. T. Young, Phil. Trans. 1800, p. 143; Lectures, xxxiii. Robison, Mechanics, vol. iv. p. 412. Farey, Philosophical Magazine, 1810, vol. xxxvi. pp. 89 and 374. Delezenne, Recueil des Travaux de la Société des Sciences, &c. de Lille, 1826-27. Woolhouse, Essay on Musical Intervals, 1835. De Morgan, On the Beats of Imperfect Consonances, Cam. Phil. Trans. vol. x. p. 129. Drobisch, Ueber musikalische Tonbestimmung und Temperatur, Abhandlungen 1864. ] of Instruments with Fixed Tones. 405 composition of tones discovered by Prof. Helmholtz have enabled me to present it in an entirely new form, and to determine with some degree of certainty what is the best possible form of temperament. Let the compound tones P and Q, of which P is the sharper, form the concordant interval p:g. Then P: Q=p: gq, or gP=pQ, that is, the gth harmonic of P and the pth harmonic of Q are conjunct. Now let P be changed into P.(1+72), where ¢ is small, and rarely or never exceeds go 0125. Then the gth harmonic of P.(1+¢) will be gP.(1+¢) and will no longer=pQ. The difference between the pitch of these harmonics is gP. (1+4)—pQ=qt.P=pt.Q. Hence the number of beats in a second produced by this change in P will be found by multiplying the lower pitch Q by pé, which is therefore the beat factor, and will be positive or negative according as the pitch of P is increased or diminished, or the interval is sharpened or flattened. The other beats which existed between the joint harmonics of the dyad P, Q may be increased or diminished by this change, but in either case so slightly that they may be left out of consideration in comparison with the beats thus introduced. But the dif- ferential tone which was P—Q becomes Pt+ P—Q, and is therefore a tone which is entirely unrelated to the original chord, and which may be- come prominently dissonant. This is an evil which cannot be avoided by any system of temperament, and is about equally objectionable in all ' systems. It may therefore be also left out of consideration in selecting a temperament. The melody will also suffer from the alteration in the perfect ratios. An interval is best measured by the difference of the tabular logarithms of the pitches of the two tones which form it. ence the interval error e=log [P. (1+¢)+Q]—log [P: Q]=log (1+¢)=n#, if the square and higher powers of ¢ be neglected, and » be the modulus. Hence the beat factor which =pt, will =pe+p, or ape. I call pe the beat meter, and represent it by (3. We may assume that the dissonance created by temperament « 3”. Hence for the same just interval p: g, variously represented in different tempera- ments, the dissonance «e*. That is, the harmony varies inversely as }’, and the melody varies inversely as e*. Hence for the same interval the harmony and melody both vary inversely ase*. The general harmony and melody may be assumed to be best when 3° and Xe” are minima, which will not happen simultaneously. The following contractions for the names of the principal intervals will der k. Sachsischen Gesellschaft der Wissenschaften, vol. iv. Nachtrage zur Theorie der musikalischen Tonverhaltnisse, ibid. vol. v. Ueber die wissen- schaftliche Bestimmung der musikalischen Temperatur, Poggendorn’s Annalen, vol. xc. p. 353. Nawmann, Ueber die verschiedene Bestimmung der Tonver- haltnisse und die Bedeutung des Pythagoreischen oder reinen Quinten-Systems fiir unsere heutige Musik, 1858. Helmholtz, Die Lehre von den Tonempfin- dungen, 1863. I am most indebted to Smith, Drobisch, and Helmholtz. 262 406. Mr. A. J. Ellis on the Temperament [June 16,. be used throughout this paper. See also the last columns in Tables XII. and XIV. Sign. Interval. Example.|| Sign. Interval. Example. Ist. Unison!) ta CE Mind.“ Major second... ¢5) ic a 2nd. | Minor Second....| ef Ilird. | Major Third Gie ord. | Minor Third ....| eg IVth. | Augmented Fourth} fb Ath. | Perfect Fourth Cr Vth. Pertect Pifth |...) 7 eg 5th. | Diminished Fifth..| bf? Vith. | Major Sixth ca 6th. | Minor Sixth e ¢? VIlth. | Major Seventh ch 7th. | Minor Seventh g PP Willie, | Oenave sce. enc? IXth. | Major Ninth cd? || 9th: | Minor Nimth Saye t= = 56 = oe Da et E (a eal @ aa mals P) —=— er —o« bok ee ae —~ pe Whence m=#/, l=dt, m=iot, mm ?=2. 1864. ] of Instruments with Fixed Tones. — 407 Hence all intervals and pitches can be expressed in terms of v. This further appears from arranging the 27 different tones required in tempered scales, in order of Vths, thus app, epP, bb, SO; Oo, 9 d), a), ed, bp, fee. Ga a, ay! ey -0; SH, ch, oF, Up, af, et, b¢, fX, eX, 9X. It will be obvious from Table V. (Proceedings, vol. xiii. facing p. 108), when the signs +{ are omitted, that these 27 tones suffice for all keys from Cp to CH. This also appears from observing that the complete key of C requires 7 naturals, 3 flats and 3 sharps, or 13 tones, and that one flat or sharp is introduced for each additional flat or sharp in the signature of the key. Hence for 7 flats and 7 sharps in the signature 14 additional tones are required, making 27 in all. The rarity of the modulations into dd, 9? or c) minor enables us generally to dispense with the three tones a)b, e)b, 690, and thus to reduce all music to 24 tones. The system of writing music usually adopted is only suitable to such a tempered scale, and there- fore requires the addition of the acute and grave signs (+ {) to adapt it for a representation of the just scale founded on the numbers 1, 2, 3, 5. To calculate the value which must he assigned to v so as to fulfil the conditions supposed to produce the least disagreeable system of tem- perament, it will be most convenient to use logarithms, and to put log v=log3—a#='1760913—w. The above arrangement of the requisite 27 tones in order of Vths, therefore, enables us to calculate the logarithms of the ratios of the pitches of all the tones to the pitch of c in terms of x, by continual additions and subtractions of log »v, rejecting or adding log 2 ='3010300, when necessary, to keep all the tones in the same VIIlve. The result is tabulated in Table XII., column T. From this we imme- diately deduce log m=log d —log ce °0511526— 2x log 7 =log f —log e °0226335+ 5x log # =log ft—log f °0285191— 7x log 6 =log gd —log fy=— 0058851 + 12a. To find the interval errors, the just intervals must be taken for the commonest modulations into the subdominant and dominant keys, as ex- plained in my paper on a Perfect Musical Scale (Proceedings, vol. xiii. p- 97). Asthe method of determining temperament here supposed makes the errors the same for the same intervals in all keys, that is, makes the temperament equal, it is sufficient to determine the interval errors for a single key. Hence the just intervals are calculated in Table XIL., column J, for the key of C, and the interval error is given in column e, in terms of x and k=log $1, the interval of a comma. From these interval errors the beat meters for the six concordant dyads are calculated in column (3. To these are added the values of 3 e? and = £’, also in terms 408 Mr. A. J. Ellis on the Temperament [June 16, of wandk. If for k we put its value ‘0053950, these last expressions become Se?="0009314 —1°1437400x2+4+ 4202? 2° = 00043659 — 5:8158100x+ 19982". Hence Table XII. suffices to give complete information respecting the effect of any system of temperament when wv is known. The following are some of the principal conditions on which it has been proposed to found a system of temperament. I shall first determine the value of w and log v on these conditions, and then compare the results. A. Harmonic Systems oF Equa TEMPERAMENT. I. Systems with two concords perfect. No. 1 (45)*. System of perfect 4ths and Vths. Here x=0, log v=°1760913. This is the old Greek or Pythagorean system of musical tones, more developed in the modern Arabic scale of 17 tones. No nation using it has shown any appreciation of harmony. 7 No. 2 (2). System of perfect [IIrds and 6ths. Here e for III, or s—4v=0, vx=14=:00134875, log v="17474255. Hence log m=log d=‘0484551=3% log 3=3 (log 3+1 42), so that the tempered mean tone is an exact mean between the just major and minor tones. Hence this is known as the System of Mean Tones, or the Mesoftonic System, as it will be here termed. It was the earliest system of temperament, and is claimed by Zarlino and Salinas. See also Nos. 13 and 19. No. 3(23). System of perfect 3rds and VIths. Here e for 3, or —k+32=0, «=3 k=:0017983, loz v=-1742930. Il. Systems in which the harmony of two concords is equal. No. 4 (20). The I1Ird and Vth to the same bass; beat equally and in opposite directions tT. Here @ for L1L+ 6 for V=0, or (54—20 x) —34=0, «=, k=:0011725, log v=1749188. No. 5 (15). The 6th and Vth beat equally, and in the same direction +. Here 6 for 6=£ for V, or —84+4+324=—3z, v= k=:0012381, log v='1748582. * The number preceded by No. points out the order of the system in the present classification. The number in a parenthesis shows the position of the system in the comparative Table XV., which is explained hereafter (p. 418). + That is, one interval is too great, or “beats sharp,” and the other too small, or “ beats flat.” + That is, both “beat sharp ” or both “ beat flat.” 1864. ] | of Instruments with Fixed Tones. 409 No. 6 (21). The IIIrd and 4th beat equally, and in the same direction. Here 6 for III=6 for 4, or 54—20e=40, w=3, k=:0011239, log v="1749674. No. 7 (18). The 6th and 4th beat equally, and in opposite directions. Here 6 for 6+ for 4=0, or (—84432x) +4x7=0, w=2 4=:0011989, log v= 1748924. No. 8 (16). The 3rd and Vth beat equally, and in the same direction. Here 6 for 3=( for V, or —64+18v=—3x, w=? x=-0015414, log v="1745199. See No. 20. No. 9 (13). The VIth and Vth beat equally, and in opposite directions. Here 6 for VI+ for V=0, or (54—152x) —3a=0, v= 3,k='0014986, log v="1745927. This coincides with Dr. Smith’s system of equal harmony, as contained in the Table facing p. 224 of his ‘ Harmonics,’ 2nd ed. No. 10 (9). The 3rd and 4th beat equally, and in opposite directions. Here G for 3+ for 4=0, or (— 64+ 18x) + 4e=0, v=, h=:0014713, log v="1746200, No. 11.(2). The VIth and 4th beat equally, and in the same direction. Here 2 for VI=f for 4, or 54—1l5e=—4a, x=; k=:0014197, log v="1746716. III. Systems in which the harmony of two concords is in a given ratio. No. 12 (24). The beats of the IIvd and Vth are as 5: 3, but im opposite directions. Here G for III: B for V= — 5: 3, or 154—602= 15x, ex=Lh4=:0010790, log v= "1750123. M. Romieu gives this temperament under the title of “ systeéme tempéré de + comma,” Mém. de Acad. 1758. See No. 18. No. 13 (2). The beats of the 3rd and Vth are as 2: 1, and in the same direction. Here 6 for 3: 6 for V=2, or —64+ 18e=— 62, v=zh, as in No. 2. No. 14 (12). The beats of the 3rd and Vth are as 5: 2, and in the same direction. Here G for 3:3 for V=5:2, or —12h+4 36a= — Lda, a= 74,k= "0012694, log v=1748219. See No. 29. IV. Systems of least harmone errors. No. 15 (7). The harmonic errors of all the harmonic intervals conjointly are a minimum. This is determined by putting the sum of the squares of the beat meters, 410 Mr. A. J. Ellis on the Temperament [June 16, or (by Table XII.) 1504°—1078A4x+19982°=a minimum, which gives r= Pe k='0014554, log v=:1746359. Tf we had used the sum of the squares of the beat factors, we should have obtained an equation of 16 dimensions in», which gives logu=*1746387. The difference between the two values of log v is not appreciable to the ear. No. 16 (14). The harmonic errors of the 3rd, IIIrd, and Vth conjoiatly are a minimum. Here (( for 3)?+(G for IIT)?+(G for V)’, or (6k—182x)?+ (54—202)? + 92° =a minimum, which gives 2=298 f=-0015309, log v=:1744404. No. 17 (6). The harmonic errors of the Vth and IIIrd conjointly are a minimum. Here (6 for V)?+(6 for IID’, or 9a°+(54—20x2)=a minimum, x=ti) k='0013190, log v="1747723. AO9 B. Mrxropic Systems or Equa TEMPERAMENT. V. Systems of equal or equal and opposite interval errors. No. 18 (24). The interval errors of the I1Ird and Vth are equal and opposite. Here e for IJI+ e for V=0, or 4—4v=4, x=Hf, as in No. 12. No. 19 (2). The interval errors of the 3rd and Vth are equal. Here e for 3 =e for V, or —k+3v=—a, =H, as in No. 2, No. 20 (16). The interval errors of the [[Ird and 3rd are equal. Here e for []T=e for 3, or k—4a=—k+3x, r=2h, as in No. 8. e e a e e VI. Systems in which the interval errors of two intervals are in a given ratio. No, 21 (17). The errors of the IIIrd and Vth are as 5: 3, but in opposite directions. Here e for II]: e forV =—5: 3, or 34—124=5ea, v=}, k='0015750, log 8= "1745163. This is the theoretical determination of M. Romieu’s anacratic tempera- ment (Mém. de l’Acad. 1758, p. 510), to which, however, he has in practice preferred No. 22. No. 22 (29). The errors of the IIIrd and Vth are as 2: 1, but in opposite directions. Here e¢ for III : e« for V=—2, or 4—4v=2e, x=1k=:0008975, log v='1751938. This is M. Romieu’s anacratic temperament. See No. 21. No. 23 (26). The errors of the I1Ird and Vth are as 1:94: 1, and in opposite directions, 1864:.] of Instruments with Fixed Tones. 41] Here e for IIT :¢ for V=—1°94, or kK—4v=1°94a, z=195 £=-0009683, log v="1751830. This is the temperament calculated by Drobisch (Nachtriige, § 10) from Delezenne’s conclusion (Rec. Soc. Lille, 1826-27, pp. 9 and 10), that the ear can detect an error of ‘2844 in the [1Ird, and -1464 in the Vth, which gives the comparative sensibility as -284 : -146=1°94. No. 24 (20). The errors of the IIIrd and 3rd are as 2 : 5, but in opposite _ directions. Here e for IIT : for e 3=—2: 5, or 5k—20e%=2h—6a, w=3, kh =°0011561, log v="1749352. See No. 27. No. 25 (46). The errors of the 3rd and IIIrd are as 2: 1, but in opposite directions, or the errors of the Vth and 3rd are equal and opposite. Here e for3: efor IIJ=—2, or 2k—6x=h—4z, or else r= —4+32; both give e=4 k=:0026975, log v='1733938. Here the error of the Vth reaches the utmost limit of endurance. VII. Systems of least melodic errors. No. 26 (1). The interval errors of all the melodic intervals conjointly are a minimum. Here the sum of the squares of the 23 interval errors in Table XII., or 32k? —212kae+4202° =a minimum, 2=~7;3, k= 0013616, log v="1747297. No. 27 (20). The melodic errors of the IInd, IlIrd, 4th, Vth, VIth, and Vilth conjointly, are a minimum. Here (e for II)*+(e for IIT)?+(e for 4)*+(e for V)?+(e for VI)?+ (ce for VII)’, or 4x°+ (A—42)’ + 22° + (k—3x)?+(k—5e0)’= a minimum, x= 3h, as in No. 24. This is Drobisch’s ‘‘ most perfect possible” (mdglich reinste) tempera- ment (Poggendorff’s Annalen, vol. xc. p. 353, as corrected in Nachtrige, § 7). It is only the “most perfect possible”’ for the major scale. No. 28 (5). The melodic errors of the 3rd, II[Ird, and 4th conjointly are a minimum. Here (e for 3)’+(e for IIT)’+(e for 4)’, or (—k+43x)?+(k— 4a’) +2°=a minimum, c= 75 k=:0014525, log v="1746388. This is Woolhouse’s Equal Harmony (Essay on Musical Intervals, p- 45). No. 29 (12). The melodic errors of the [IIrd and Vth conjointly are a minimum. Here (e for IIl)’+(e for V)’, or 2° +(k—4z)’= a minimum, r= 44, as in No. 14. This is given by Drobisch (Nachtrige, § 8) as ‘‘ the simplest solution of the problem.” 412. Mr. A. J. Ellis on the Temperament — [June 16, C. CoMBINED SystTEeMs or Equa TEMPERAMENT. No. 30 (4). The combined harmonic and melodic errors are a minimum. By combining the equations of No. 15 and No. 26, we have (539+ 106) & = (1998+ 420)a, or o= 3845, k=-001444, log v="1746439. No. 31 (32). The tones are a mean between those of No. 1 and No. 2. Here e=3 (sum of the two values of w in No. 1 and No. 2)=:0006744, log v="1754169. This is proposed by Drobisch (Nachtriage, § 9). No. 32 (42). The errors occasioned by using the tempered c,d, f, f, g, bbb, ¢), & for the just c, d, e, f, g, a, 6, c are a minimum. Using s for *0004901, and forming the values of these errors by Table XIT., we have42” + (s—8z)?+27+(s—9zx)?+(s—/7z)’=a minimum, r= 24, s='000059084, log v=1760322. This is proposed by Drobisch as a system of temperament adapted to bowed instruments (Mus. Tonbestim. § 57), allowing them to use a system of perfect fifths, and yet play the perfect scale very nearly by substitution. Such a system would be more complicated than the just scale for any in- strument, and would require many more than 27 tones. It is, therefore, un- necessary for the violin, and impossible on instruments with fixed tones. D. Cycuic Systems or EquaL TEMPERAMENT. When it was supposed that the number of just tones required would be infinite, importance was attached to cycles of tones which by a limited " number expressed all possible tones. Hence Huyghens’s celebrated Cyclus Harmonicus, which he proposed to employ for an instrument with 31 strings, struck by levers and acted upon by a moveable finger-board (abacus mobilis), acting like a shifting piano or harmonium. The condi- tion of forming a cycle is not properly harmonic or melodic ; it is rather arithmetic. If log v:log 2 be converted into a continued fraction for any of the preceding values of log v, and y: 2 be any of the convergents, then, putting log 2=2.h, we shall have log v=yh, which is commensurable with log 2, and consequently the logarithms of all the intervals will be multiples of 4, and therefore commensurable with log 2. A cycle of z tones to the octave will thus be formed. If z is less than 27, the number of tones otherwise necessary, the cycle may be useful, otherwise it can only be judged by its merits as an equal temperament. As an historical interest attaches to several of these cycles, I subjoin a new method for deducing them all, without reference to previous calculations of log v. Since log vu=y.h, and log 6=7 log 2—12 log v=(7z—12y) . h, we have only to put 7z—12y= ... —2,—1, 0, 1, 2,...and find all the positive integral solutions of the resulting equations. This gives for 13 27 41 55 69 2— =——?2 ih csi SS SS SP SSS lpia! > g~«29? 46 70’ 94 118 eo 1864. ] of Instruments with Fixed Tones. 413 eal [a oe eu i729: A ge Ge a Sa a ioye LEE 7e—l2y= 0, La Pi 1 = yl4 1 18 25 32 90 46/58) ba 15 29 43 57 f2 19 i= te Ao, gue 96? 50). 7a' 08’ 7z—12y= 3 y 5 12 19 26 33 40 47 54 SS tS ro eee) —=9 9.2133 Ab: 47 69" 61.93. Many of these cycles are quite useless. The following selection is ar- tanged in order of magnitude, from the greatest to the smallest cycle. No. 33 (38). Cycle of 118; A="0025511, log x= 69h="1760259. This is Drobisch’s cycle (Mus. Ton. § 58) representing No. 32. No. 34 (8). Cycle of 93; h=-0032368, log vx=54h='1747872. This may represent No. 2. No. 35 (3). Cycle of 81; A=:0037164, log v=47h='1746708. This may represent No. 11 (2). No. 36 (39). Cycle of 77; h=:0039095, log vu=45h='1759275. This is the same as No. 52. No. 37 (19). Cycle of 74; h=:004068, log v=434="1749200. This is another of Drobisch’s cycles (Nachtrage, § 7) representing No. 27. No. 38 (22). Cycle of 69; h=-004363, log x=404=°1745200. No. 39 (28). Cycle of 67; #= "004493, log v=394="1752270. No. 40 (40). Cycle of 65; h=-0046123, log v=38h=°17598674. No. 41 (27). Cycle of 57; A=-0052812, log x=33h='1742796. No. 42 (30). Cycle of 55; A=:0054733, log v=32h=°1751456. _ This is mentioned by Sauveur (Mém. de l’ Acad. 1707) as the commonly received cycle in his time. Estéve (Joc. cit. p. 135) calls it the Musicians’ Cycle. No. 43 (11). Cycle of 50; A=*0060206, log x=29h=:1745974. This is Henfling’s cycle (loc. ect. P- 281), and is used by Dr. Smith to - represent No. 9. No. 44 (43). Cycle of 53; h=-0056798, log vx=31h="1760800. This is the cycle employed by Nicholas Mercator (as reported by Holder, ‘Treatise on Harmony,’ p. 79) to represent approximately the just scale. He did not propose it as a system of temperament as has been recently done by Drobisch (Musik. Tonbestim. Einleit.). It was the foundation of A414 Mr. A. J. Ellis on the Temperament [June 16, the division into degrees and sixteenths adopted in my previous paper Proceedings, vol. xii. p. 96. No. 45 (37). Cycle of 45; h="006689, log vu=26h='1738940. No. 46 (25). Cycle of 43; A=°0070007, log x=25h=:1750175. This is Sauveur’s cycle, defended in Mém. de l’Acad. for 1701, 1702, 1707, and 1711. No. 47 (10). Cycle of 31; A=*009711, log v= i8h=:1747900. This is Huyghens’s Cyclus Harmonicus, which nearly represents No. 2 (2). It was adopted, apparently without acknowledgment, by Galin (De- lezenne, loc. cit. p. 19). No. 48 (44). Cycle of 26; A=-011578, log vx=15h=-1736700. No. 49 (25). Cycle of 19; h=-0158437, log v=11h='1742807. This is the cycle adopted by Mr. Woolhouse (Essay on Beats, p. 50) as most convenient for organs and pianos. It may therefore go by his name, although it is frequently mentioned by older writers. It is almost exactly the same as No. 3 (23). No. 50 (35). Cycle of 12; 4=°0250858, log u=7h=:1756008. As this is a cycle of twelve equal semitones, it may be termed the Hemi- tonic temperament. It is the one most advocated at the present day, and generally spoken of as ‘‘equal temperament” without any qualification, as if there were no other. It was consequently referred to by that name only in my former paper (Proceedings, vol. xiii. p. 95). For its harmonic character see No. 53. E. DEFECTIVE SYSTEMS oF EQuAL TEMPERAMENT. It has been from the earliest times customary to have only twelve fixed tones to the octave, on the organ, harpsichord, piano, &c., and to play the other fifteen by substitution, as shown below, where the tones tuned, ar- ranged in dominative order, occupy the middle line, and the tones for which they are used as substitutes are placed in the outer lines, and are bracketed. [Abb, Hob, Bob, Fb, Cb, Gp, Dp, Ar] Eb, Bp, F, 0, Gy) D,- Ay BB, FR On ee [Dy, At, FH, BE, Fx, Cx, Gx]. The consequence was, that while the Vths in the middle line were uni- form, the Vths and 4ths produced in passing from one line to the other (as GED for ADE or GED) were strikingly different. Similar errors arose in the other concordant intervals. It is evident that the interval error thus produced must be the usual interval error of the system increased or diminished by the logarithm of the diesis, where log 6=log 9) —log ft= 1864. | of Instruments with Fixed Tones. 415 —°0058851 +12%—=—k—s+127, where s=:0004901*. Such interval errors are termed wolves, from their howling discordance. In Table XIII. will be found an enumeration of all the wolves, with a notation for them, and an expression of their interval errors and beat meters in terms of 4, s, and 2. No. 51 (33). System of least wolf melodic errors. The sum of the squares of the wolf interval errors, or 2h? + 2ks + 6s*—4 (1144 28s) «+ 2662’, isaminimum. Hence 224+ 56s= 2662, or e=:0005495, log v=: 1755418. No. 52 (39). System of least wolf harmonic errors. The sum of the squares of the wolf beat meters, or 25k? + 175s’ + 50ks— (5504+ 3072s) 2+ 136622" , is a minimum. Hence 2754+ 1536s=13662z2, or r=:0001638, log v ="1759275, as in No. 36 (39). No. 53 (35). The wolf interval errors are equal to the usual interval errors, that is, there are no wolves, or there are none but wolves. In this case log 5=0, or, since 6=9) : ft=2": v”, 7 log 2=12 log v. Hence this system is the cycle of 12, No. 50. Wyle . is greater than 1, g? is sharper than f¥, and log v is less than + 15 log 2 eGt, Fares But if dis less than 1, y) is flatter than 7%, and ihe v is less than {5 log 2, or "1756008. The lattes case is, according to Drobisch, fadisfencaule for musical theory and violin practice (Musik. Tonbestim. Einleit.). Since this temperament thus forms the boundary of the two other classes, distin- guished by g? being flatter or sharper than /#, Drobisch terms it the “mean”? temperament (ibid. § 51). It is this property of making g) =/+ which renders this temperament so popular, as the ear is never distressed by the occurrence of intervals different from those expected, and the whole | number of tones is reduced to 12. No. 54 (31). The wolf interval error of the ILIrd is to its usual interval error as 14: 5. This gives —s+8r7: k—4rv=14: 5, or 962=144+5s8, c='0008123, log v="1752790. This is Marsh’s system of temperament ; see Phil. Mag. vol. xxxvi. p. 437, and p. 39 segq. Schol. 8. No. 55 (36). The wolferrors of the IIIrd and Vth conjointly are a minimum. Here (—s+8v)?+(—A—s+llz) is a minimum, whence 114+19s =1852, 2="0003712, log v=:1757201. No. 56 (37). The wolf errors of the Vth and IIIrd are equal and opposite. Here —k—s+lla=s—8za, 192=h+ 2s, e=:0003356, log v= "1757557. No. 57 (34). There is no Vth wolf. Here —k—s+11e=0, r=-0005351, log v=-1755562. * It appears from Proceedings, vol. xiii. p. 95, that s must be nearly the loga- rithm of the schisma or log J. Actual calculation shows that s and log 4 agree to 14 places of decimals, 416 Mr. A. J. Ellis on the Temperament [June 16, No. 58 (41). There is no IIIrd wolf. Here —s+8x=0, =0000613, log v=-1760300. This is almost ex- actly No. 32 (42). No. 59 (42). There is no 3rd wolf. Here s—9x=0, x='0000545, log v="1760368. F. Systems or UnequaL TEMPERAMENT, In a defective equal temperament the same just concordance is represented by two different discordances. As performers limited themselves to twelve tones to the octave, those who found the Hemitonic temperament No. 50 (35) too rough, accepted this variety of representatives of the same con- cordance as the basis of a temperament, hoping to have better IIIrds in the usual chords, without the wolves of the defective temperament. Others conceived that an advantage would be gained by altering the character of the different keys. Thus arose wnequal temperament, properly so called, which must be carefully distinguished from any defective equal temperament with which it is popularly confused. Arrange the twelve unequally tempered chords as follows, where the identical numbers indicate identical chords with different names :— LG kesG, 7. Fe At cet. 7. Go Bp dp. 2.GB d. 8. Ot Ee GH. 8. D? F Ab. 3. D Ft A, 9. Gt BE dy. 9. .c op. 4, A cH e. 10. Dy Fx A. 10. Ep G Bp. 5. E Gt B. ll. At ex et. 11. JB) og gage 6. B dt ft. 12. He Gx By. 12, Fee Let 7, ¢,, Y, be the ratios of the IIIrd, 3rd, and Vth in the nth chord, so that, for example, in the 6th chord dg=T7T,.B, fg=t,.dt, ft=v,. B. Then it is evident from the above scheme that there exist 12 pairs of equations between these 36 ratios, of the form fm ° ‘ Vy and Ato ° Ont4 ° Unto e Un43 (where, when the subscript numbers exceed 12, they must be diminished by 12), and one condition, Vy + Das Uy 6 Uy 0 Oy + Uy 2 Vy + Vy + Vy # Vo = yy» Vig =A" Put log T,,=log $+y,; log ¢,=log $—z,, log v,=log 3—2,, then the above equations become Ln=ULn t+ Yn ’ Yn=h—(Xy+G nti tenpotXn4s)s 0, +0, +0,+0,+4, 44,14, +8, 1% +2,,+%,,+%,,='0058851, which represent 25 equations, where the second set of 12 may be replaced by the following, which are readily deduced from them and the last con- dition :— 1864.] of Instruments with Fixed Tones. 417 WAFYst+Yo=YotYot Y= Yat Yr FY = Yt Yot Yir= '0103000...... (2) Uy= H+ Y1— Yo» Ly =X,+Yy— Yoo Ue=L,+ Y2— Yo Ly y= % FY —Yros Ly =U, Ys— Yas Ly =%y+Ypo— Yip C=", FYs—Y 59 Lyy=L,+Y, —Yro A system of unequal temperament may therefore be determined by arbi- trarily selecting eleven different Vths, or else eight different IIIrds and three Vths. The equations (a) show that if the temperament is not equal (in which case all the y’s are equal, and the interval error of the IIIrd is 4 xX ‘0103000 =0034333, as in the Hemitonic temperament), at least four IiIrds must have their interval errors greater than 3 x ‘0103000, that is, there must be at least four IIIrds in every unequal temperament which are inferior to the very bad IIIrds of the Hemitonic system. Kirnberger, Dr. T. Young, and Lord Stanhope*, in the unequal systems they propose, have each seven IlIrds sharper, and therefore worse than the Hemitonic IlIrds. In one of Prof. de Morgan’s unequal temperaments, six IIIrds are sharper than the Hemitonic; in another four are sharper and four the same; in a third all are the same, but the Vths differ+. Hence nothing is gained over the Hemitonic system in the way of harmony, while the uniformity in the representation of the uniformity of just intonation is entirely lost. In selecting a temperament, therefore, we may dismiss all unequal tem- peraments, as they must be inferior to the Hemitonic in both harmony and melody, and will have no advantage over it in the relations of chords or the number of tones required. Also, if it is considered necessary to play in all keys with only twelve tones, any system of defective equal temperament will be inferior to the Hemitonic, on account of the various and distressing wolf intervals which occur when the music is not confined to the six major scales of Bp, F, CO, G, D, A, and the three minor scales of g, d, a. Hence the two conditions of having only twelve tones (exclusive of octaves) and of playing in all keys, at once exclude all temperaments but the Hemitonic. As, however, organs, harpsichords, and pianofortes with 14, 16, 17, 19, 21, 22, and 24 tones to the octave have been actually constructed and used {, as Mr. Liston used 59§, Mr. Poole used 50, and Gen. T. Perronet Thomp- * Kirnberger, Kunst des reinen Satzes in der Musik. Dr. T. Young, loc. cit. _ Charles Earl Stanhope, Principles of the Science of Tuning, 1806. + De Morgan, loc. cit. p. 129, temperaments Q, R, S. ¢ Mr. Farey (Phil. Mag. vol. xxxix. p. 416) gives the particulars of their scales, builders, and localities. § The following account of Mr. Liston’s organ is deduced from the data of Mr. Farey (Phil. Mag. vol. xxxix. p.418). Scale: ¢ te tet cit d) tet td) tex td cx d td tat & dt te tte .c f) te tfd ttfD tet ct f tet tf tt A P tft to? tfx tg Fx 9 to tok @ gh ta? ttad a ta oD tod tak of wD tat +0) tb bc th +) tot te bY. Chords: Table V. col. ILL, lines 4 to 13; 418 Mr, A. J. Ellis on the Temperament [June 16, son now uses 40 tones to the octave on their justly intoned organs, the condition of having twelve tones and no more, does not seem to be inevit- able. It will therefore be necessary to determine what would be the best system of temperament for the complete equally tempered scale of 27 tones, and how great a sacrifice of musical effect is required by the use of the Hemitonic system. In Table XV. I have calculated for each of the 59 (reducing to 51) systems of equal temperament already named, the interval errors of the Vths, IIIrds, and 3rds, and the sums of the squares of the 23 interval errors and the 6 beat meters of Table XII. I have then arranged the temperaments in order according to each of these five results, and numbered the order. Finally, I have added the five order numbers together and ar- ranged the whole in the order of these sums. The smallest number would therefore clearly give the best temperament, supposing that all the five points of comparison were of equal value. Now the first and second tem- perament on the list, or No. 26 and No. 2, only differ from each other in the fifth, sixth, or seventh place of decimals with respect to these five re- sults, a difference which no human ear, however finely constituted by nature or assisted by art, could be taught to detect. As No. 2, or the Mesotonic system, is determined in the simplest manner, I consider it as the real head of the list. There is, however, little to choose between it and any one of the ten or twelve systems which follow, except in simplicity of construction and comparative ease in realization. The Hemitonic system, however, comes 35th in the list, and the old Pythagorean, recently defended by Drobisch and Naumann (op. cit.), and asserted to be the system actually used by violinists, is the 45th. No one who has heard any harmonies played on the Pythagorean system will dispute the correctness of the position here assigned to it, which fully explains the absence of all feeling for harmony among the nations which use it—the ancient and modern Greeks, the old Chinese, the Gaels, the Arabs, Persians, and Turks. No modern quartett players could be listened to who adopted it. The contest lies, therefore, between the Mesotonic and the Hemitonic systems. The Mesotonic is that known as “ the old organ-tuning,”’ or, since it was generally used as a defective twelve-toned system, as the ‘‘ unequal tem- perament.” Within the limits of the nine scales already named, the superiority of the Mesotonic to the Hemitonic system has long been practically acknow- ledged. But the extremely disagreeable effect of the wolves (more espe- cially to the performer himself) has finally expelled the system from Ger- many altogether, and from England in great measure. On the pianoforte IV.,5 to 14; V.,5to16; VI, 6%015; VIL,7to16; VIIL, 7 to 17; IX.,9 to 13; X.,9 to 13. Tones not forming part of any chord and required chiefly by the system of tuning : +d) tte +f0 +t/p +g) tta? Bp +b) +2. Complete keys: F, C,G, D, +A; E, B, Ft. The keys of £7, BD had their synonymous, and +£, +B their relative minors perfect, 1864. | of Instruments with Fixed Tones. 419 the Hemitonic system is universally adopted in intention. It is, however, so difficult to realize by the ordinary methods of tuning, that “equal tem- perament,” as the Hemitonic system is usually called, has probably never been attained in this country, with any approach to mathematical precision. In Table XIV. I have given a detailed comparison of the Mesotonic and Hemitonic temperaments with each other and with just intonation, for the system of C (Proceedings, vol. xiii. p. 98), from which the great superiority of the Mesotonic over the Hemitonic both in melody and harmony becomes apparent. But this comparison rests upon the preceding calculations, which were founded upon the beats that arise from rendering the conjunct har- monics pulsative. It was therefore assumed that the qualities of tone em- ployed were such as to develope these beats. The result will consequently be materially modified when the requisite harmonics either do not exist or are very faint. Now for the Vth the conjunct harmonics are 2 and 3, Bert Ath v3 5 3 and 4, geo Vith a sdf s 3 and 5, 3 Wed ay x 4 and 5, sj eord is iss 5 and 6, sp Oth 36 53 5 and 8. If then only simple tones are used, as in the wide covered pipes of organs, or such qualities as develope the second harmonic only, such as tuning-forks, to which we may add flutes, which have almost simple tones, no beats will be heard, and any system of temperament may be used in which the ear can tolerate the interval errors. Now Delezenne’s experiments show (Joc. cit.) that a good ear distinguishes in the unison an interval error of 0°28074, » VIlIve bs A 0:31h, 33 Vth 39 99 0°14614, x Ld ird FF if 0°2844, aL 4 i 0°2994, and an indifferent ear perceives an error of 0°561% in the VIIIve, and 0°292k in the Vth. We may say, therefore, generally that the ear just perceives an interval error of 74 in the Vth, and 44 in the other intervals. Now in the Mesotonic system the interval error of the Vth is —14, and therefore just perceptible, but in scarcely any other interval does it exceed tk. Thus it is —14 in the VIIth, 0 in the ITIrd, and +1 in the VIth, and it is therefore in those intervals imperceptible. In the Hemitonic system the error of the Vth is —;/,A, and hence quite imperceptible, but the errors of the VIIth, I1Ird, and VIth are respectively 6-4, Jk, and 7, and therefore perfectly appreciable. It is only in the VIIth that this error is at all agreeable. The sharpness of the IIIrd and VIth is univer- sally disliked. Hence in those qualities of tone which are most favourable to the Hemitonic system, it is much inferior to the Mesotonic. In Table XV. VOL. XIII. 2H 420 Mr. A. J. Ellis on the Temperament [June 16, the Mesotonic stands 2nd in order of melody, inappreciably different from the Ist, and the Hemitonic 39th. If the 3rd harmonic only is developed in the qualities of tone combined, the beats of the Vth are heard, but those of the other intervals are not ‘perceived. The beats of the IIIrd and VIth, which are so faulty on the ‘Hemitonic system, will not be perceived at all unless the 5th harmonic be developed, and will not be much perceived unless it be strongly developed. ‘Now the 5th harmonic is comparatively weak on all organ pipes and on pianofortes, and hence the errors are not so violently offensive on these instruments. If, however, the ‘mixture stops,’ which strengthen the upper harmonics by additional pipes, are employed on the organ, the effect is un- mistakeably bad, unless drowned by din or dimmed by distance. On the pianoforte, however, these intervals, and even the still worse 3rd and 6th, depending on the 6th and 8th harmonics, which are undeveloped on piano- forte strings, are quite endurable. Hence the Hemitonic system, except as regards melody, will not be greatly inferior to the Mesotonic on a pianoforte and on soft stops of organs, but will only become offensive on loud stops. But for harmoniums and concertinas, violins and voices, where harmonics up to the Sth, and even higher, are well developed, the Hemitonic temperament is offensive. The roughness of harmoniums is almost entirely due to this mode of tuning. The beats of the VIth, IIIrd, and 3rd are distinctly heard, and the develop- ment of differential tones is so strong as frequently to form an unintelligibly inharmonious accompaniment*. Concertinas having 14 tones to the octave are indeed generally tuned mesotonically (or intentionally so), thus ¢ cf, d di, e€), Tig ga, aa), bb). They are, however, occasionally tuned hemi- tonically (or intentionally so) to accompany pianofortes, thus ¢ cf, d dq, * The three recognized forms of the common major triad 4, 5,6; 5,6,8; 3,4, 5, or CEG, E Gc, Gce, have the pitches of their tones as 4n,5n, 6n ; 5n, 6n, 8n, and dn, 4n, dn respectively. They produce, therefore, the differential tones n, n, 2n ; Nn, 2n, 3n, and n, n, 2n respectively. If the chords are tempered, the altered unisons 7, x become pulsative, and the other tones disjunct. Now if in Table XIE we put «=log (1+7) and Spe as t?, we shall have very nearly £=#1 -d— At) .C; G=2.(-12).C; e=2C, e=31. Ges At).C; g=3.(1-2).C. The la of pul sative diterencell tones are ae efore H— C=(44—22).C, G—C=(#343H).¢G, and c— G=(3+3t).C, e-—c=(L2-8 S¢)- C. The numbers of beats are a abso- lute Sk of the eee of these pairs of numbers, or of (-,4+“2).C,and (—34+%2).C. The squares of these expressions, and the sum of their squares, will be minima respectively for t=54,, r=00157070, log v=+1745206, which is nearly No. 38 (22); ¢=,4,, c= 0011658, log v= 1749255, which is heey No. 24 (20); and t=,,4,,, e='0013096, log p= 1747817, ae is nearly No. 34 (8). These beats, though perfectly distinct in some octaves, do not appear to be suffi- ciently prominent to serve as a criterion of the relative value of different systems of temperament, or to form the basis of a system, and they have consequently not been introduced into the text. They were noticed and used by H. Scheibler (Der So a a und musikalische Tonmesser, p. 15). 1864.| of Instrumenis with Fixed Tones. | 421 e dt, f ft, g ot, «gt, bag. Hence it is easy to compare the different effects of the two systems as applied to the same quality of tone, for harmonies which are common to both. Having two concertinas so tuned, and a third tuned to just intervals, I have been able to make this comparison, and my own feeling is that the Mesotonic is but slightly, though unmistakeably, inferior to the Just, and greatly superior to the Hemitonic. There are two other points in which the complete Mesotonic system possesses advantages over the Hemitonic. The Mesotonic VIIth is rather flat, but by using the flat VIIIth in its place, when the harmony will allow, the effect of an extremely sharp VIIth is produced, which is sometimes de- sirable in melodies. Thus log Mesotonic VIII? =-28195, which is sharper even than log Pythagorean VII[=°27840. The ordinary and flatter VIIth can be used when necessary for the harmony. Again, by using the German sharp VIth in place of the dominant 7th, that is, by using the chords G? By Dy ce, Dp F Ap b, Ad C Eb ft, ED G BD ck, Bb D F gt, HACd, CHGo, GBD, DAM ACRES x, HGE Bex, B Dt Ft g x, in place of G) B? Dd fo, Do F Ad, Ad CO ED gh, NG Bd, BDFd, FACY, CEGh, GBD, DihAs, ACZ Eg, EGE Bd, B Dz Fx a, when the progression of parts will allow, an almost perfect natural seventh, better than that obtained by using the corresponding just tones, will result, producing beautiful harmony ; for log Mesotonic VIF=°24228, log i=-24304, and log Just VI =-24497. The ordinary sharper 7th can be used when necessary. Neither of these effective substitutions is possible on the Hemitonic system. Considering that singers and violinists naturally intone justly (Delezenne, foc. cit.), and that the interval errors of the Mesotonic system seldom exceed the natural errors of intonation which may be expected from the inability of the ear to appreciate minute distinctions of pitch, it appears desirable to tune harmoniums at least, and perhaps organs, mesotonically. Except as an instrument for practising singers, however (for which purpose it would be superseded by a Mesotonic harmonium), it would be unnecessary to alter the Hemitonic tuning and arrangement of the piano. But it would be best to teach the Mesotonic intonation on the violin in preference to the Hemitonic, as proposed by Spohr*. As, however, it would be useless to tune mesotonically with only 12 tones to the octave, it is necessary to have some practical arrangement for 27, 24, or 21 tones at least. I propose the following plan for 24 tones, and as these are exactly twice as many as on pianos, &c. of the usual construction, I call my arrangement the * “Unter reiner Intonation wird natiirlich die der gleichschwebenden [ Hemi- tonic | Temperatur verstanden, da es fiir moderne Musik keine andere giebt. Der angehende Geiger braucht auch nur diese eine zu kennen ; es ist deshalb in dieser Schule von einer ungleichschwebenden [defective equal, or unequal] Temperatur eben so wenig die Rede, wie von kleinen und grossen halben Ténen [¢ cf & ¢ d? = Gc, that is, = 7), weil durch beides die Lehre von der véllig gleichen Grosse aller 12 halben Tone nur in Verwirrung gebracht wird.”—Violinschule, p. 38. 422 On the Temperament of Instruments with Fixed Tones. [June 16, DurpLex FINGER-BOARD. PLON. FX g9©éX cb dbx eb hb gx abx bp ch ab Oclours et ot ee t ee White Black. Lront€ Llevation. FellowRea Let the black and white manuals remain as at present, and let a yellow manual, of the same form as the black, be introduced between B and C, and E and F. Cut out about the middle third of each black and yellow manual, up to half its width, on the right side only, and introduce a thin red manual rising as high above the black or yellow as these do above the white. Over G, A, and D, each of which lies between two black manuals, introduce three yellow metal manuals (lacquered or aluminium-bronze) shaped like flute keys, and standing at the height of a red manual above the white one, which can therefore, when necessary, be reached below it. ‘The 7 white manuals are the 7 naturals; the 5 black manuals are the 5 usual sharps, ch dz f¥ gt at; the 2 long yellow manuals are the unusual sharps et 62, and the 3 metal yellow manuals are the double sharps f/x gx cx ; and the 7 thin red manuals are the 7 flats, 2), db, 9, f0, 97, a), 69. The shapes of the red and metal manuals were suggested by those of General T. Perronet Thompson’s quarrils and flutals. The 24 levers opening the valves on the organ or harmonium would li side by side, being made half the width of those now in use, and metallic, if required for strength. The organ pipes or harmonium reeds would be arranged in two ranks of 12 for each octave, the first rank containing the 7 naturals and 5 usual sharps, and the back rank containing the 7 flats, 2 unusual and 3 double sharps. The use of this finger-board is accurately pointed out by the ordinary musical notation which distinguishes the sharps from the flats, and is therefore in no respect adapted to the Hemitonic fusion of sharps and flats into mean semitones. ‘(epour ‘Tar) 136 poppe “ “ | peneLay | ogF, F199] GF ‘cL ‘eg "Tg ae apy “ @ | peyeL ee | 09% $499] 9G ‘el ‘eT r 9pour Tox) W9 poppe “ “ | peor JT ‘GT | OLE ‘eT #0 fax qe) LT ‘er ‘e ‘g Com py ta “ “ | pegey rourpy | 9, qP9°H| g1'6 ‘G's TL peysmurog | peter 2 “L | LT GP2T 29D] 212 ‘G'S UW “BUr "TA, TAL OY} FOTO | pear e‘s | 26, eP dp 9) 6° ‘9 "8 *SJUONYTISUOL) OATPESTO OMT, oS ‘(qooproduar) 139 Poppe “BT OY} Jo "GO| pvuty, 1g 16 are) 1 ‘QT ‘penty, snongiodng | prny, gg GG foto 9 Go ‘@ ‘T ‘(qoogroduat) (9 peppetourm =“ “| per, JT 9,11 #,9 [tax 0 9 LI ‘a ‘g ‘Cduar) yz MH em}JO'YO) — PRL LT OLT GP2T 99 LL'G'¢ LOU) LOULTAT MOUMUOD | PLL, OUTTA, Qs lebi qa GT 39 1G. ‘STUOONOD ‘T ’ eure Ny Areurprg ‘oure ny “yskg | ‘Toquikg ‘op durexny | ‘od XT, el ee ee eee (*26¢ ‘d 909) "SployyQ [vorsnyy jo TONvoyIsse[Q—'TA XIAVy, Taste VI.—Classification of Musical Chords. (See p. 397.) Type: Example, Symbol. | Syst. Name. Ordinary Name. I. Concorps. 1,1 C4 C4 Cc 1 Dyad | Unison (Octaye 1, 2). 765) OG C 3Dyad |Twelfth (Fifth 2, 3; Fourth 3, 4). 1,5 Cle Cc 5 Dyad | Majorl/7#h(Ma. 10th2,5; Ma, 31d 4,5; Mi. 6th5,8). 3, 5 Ge c | 8,5 Dyad |Ma. 6th (Mi. 8rd 5, 6). 1, 3,5 C'Ge C | Major Triad |Common Major Chord. Ii. Strone Discorns. 1, One Pulsatiye Constituent. ii C! gb? 7C 7Dyad |Perfect 7th 4,7; Ex- tended tone 7, 8. 3, 7 G gb) 70 3, 7 Dyad | Contracted 3rd 6,7; Ext. 6th 7, 12. 5, 7 e gb? 70 5,7 Dyad | Contracted 5th 5, 7. iL, 8, af CG gb 7C | 3,7 Triad |Imp.Ch.of Dominant7th. MO C1 e gb? 70 | 5,7 Triad . in 1, 5, 7 C* e yvij att 70) 5, 7'Triad | Chord of the Italian 6th. 1, 5,9 Cred 9C | 5,9 Triad | Imp. Ch. of the 9th. 1, 3, 15 OG b 15C | 3,15 Triad || Imp. Ch. of the Mi. 6th, 1,5,15 * Ct eb? 15C | 5,15 Triad inor mode; TL, BY ili 0! G Iga'b 17C | 3,17 Triad pa Ch, of the Minor 1, 6, 17 Cte Iga? 17C | 5, 17 Triad 9th. 1, 3,5, 7 0 Ge gb) 70 7 Tetrad | Ch. of the Dominant 7th. Wy By By 1 C* G e vij att 7'C 7'Tetrad | , 4, German 6th. 1, 3,5, 9 CGed 9C 9Tetrad | ,, 4, 9th. 1, 3,5, 15 Ct Ge hb? 15C | Major Tetrad| ,, 4, Ma. 7th. 1,3,5,17 |C*Ge 1ga 17C | 7 Tetrad |, 4, Mi. 9th imp.) 1, 3, 5, 27 Ot Ge tat 27C | 27 Tetrad | ,, ,, add.Gth,ma.m, 2. Tio Pulsative Constituents. 1,8,5,7,9 |C!Ge cb) @ 79C | 7,9 Pentad | Ch, of the added 9th. 1, 3, 5,7, 17 | C*Ge zbD gdb 170 17,17 Pentad | ,, ,, Mi. 9th, 1, 3, 5,9, 15 |C'Ge db! "15C | Major Pentad| ? ” Ma, 9th. 1, 3, 5, 15, 17) C* Ge b’xvij cf} 15,170 | 15,17 Pentad| ,, ,, augmented 8th. I. Wax Discorps. 1, One Pulsative Constituent. M37 Ge cb) Te 7 Triad | Ch.ofthe See 1, 3,9 LORE 9c 9Triad | 4, 4, 9th (im 3, 5, 15 Geb =G Minor Triad |Common Minor Da 3, 5, 17 Ge 1gay 17c 17 Triad | Ch. ofthe Dim. 7th (imp .). 8, 5, 17 G e xvij cit 17'c 17’ Triad » » Minoradded bth (amperfect), 1, 5, 25 Ce tet 25¢ 25 Triad Superfluous Triad. 1, 5, 27 Cte ta 27e 27 Triad Ch. of the Ma. added 6th (imperfect), 2. Two Pulsative Constituents. 38, 5, 7,9 Ge gb) d? 9c 7, 9 Tetvad | Ch, of the Mi. 7th, ma. m. 8, 5, 7, 17 Ge gb? gdb 17e | 7,17 Tetrad | ,, ,, Diminished 7th, 3, 5, 9, 15 Gedb ‘ec | Minor Tetrad | ,, ,, Mi.7th(mi.m.). 8, 5,15,17 | Geb xvij cf 15,17¢ | 15,17 Tetrad| ,, ,, added 6th (mi. mode), 1,5,15,25 |Ceb te 25¢ 25 Tetrad » 4, augmented 5th, 3,5, 15,45 |Gebf ’45e | 45 Tetrad » _» added 9th (mi, mode). ~ is iS iS ON NRK AAR KS oD oD NOOO HWDOMODNGO = == nid oD 6 SHS v ‘ApISMOYUT ‘QdUvyy] "1OJOB ‘TRALOYUT H Jatoe oy} Jo serpisuozUy *SOTUOTHAIV Py li ae L L & & ve vé 69 69 1z 9G Ov OF OF 0 Sree ores le cotre, |) conte sl) 20 g ¥ £ & % L L T th 8L gL 6 8 8B oz | 9 Ot 6 6 9 $ G 6 G eet e eee G 6 G 9 0G 9 Vv ates IL eeees g g 8 eee 9g eee 9 ° ee 6e (9 eo ae ag ore ik tes les a ae oS NeGs Sal 8G 4G OOT | OOL QOL | OOL SS alee QOOT|‘*°'' | OOT| 9 | | a | nr | ——— EE oa Enel a |e | | RR | QOL | ,OOT | , Bboy “svy) “RIUOT PUNOAWIOD Wo SproYy [woIsNY Jo UoNONIySU0Q— "TTA T1avy, a | RS ee at al ta aed gee GE ar | RET. eas ae oa ear ee SLE | F leer alse a lea oalee 69 | a 7 og | 99 OOr| TI O66 0 | ‘oot dla eat | Index. p: be ‘SOMMOULIVTT JULIO JO AjIstoyUT 0; “ae 100 | 100 10 10 1 2 3 i 5 ) ee eae 1900 | 100 ae Tasie VII.—Construction of Musical Chords from Compound Tones. (See p. 398.) of the Constituent Tones. onics Extent of the Harm Harmonies of the Constituent Tones, CO) S a=) al aint io io: al | aaawtioon@ (See p. 399.) Taste VIII.—Qualities of Concordant Dyads. -zequinyy | AHARDWOAANDROD re a cs AMMDOAWODOWOONDNS 4 ECL || 1c: Ey CC eC ae ar Oe ott ND 09 09 OD HD 19 10 1 di | Bega e|& “eure Ny ‘SOTUOMLIV]T JULOL Oy} JO SorzIsuOzUT § 3 a8 FA | Re feay-ftc| “qyueOPIL, a N : : he . “as . . . . cle le “W700 care - cre S -O -tH -O 10NN Ce Ae 5 ja oom | DiGi: ja cas] : : | easy Es SES iS lis tom Hope acs! Sev} 6 re SH | ao yqueayg | 2 iS Q ion c 5 a: a ne Pans 2 ga 8 ; : : : op Pee es . = e WHS TW) 2:9 1B) 13g: zi aReegece 2 | ox | peur in |S: 2 38 al ae | ae. é |-~ | a an | S 38 | [1:8 | ie e Sus | ie ———— en ne Pe e[oo| msm] i598 ig|8 : ine | io: ia| i ie jenn [onze e[se| -wma[S gg [a ose || ie 2 |e ie tee [> ay len | wean |SSR°* jares | = fois | 100 5 NN | ose [worth] ae8e|= = 3 [males = | |__| gga :| | | Jans | | | | | i Index. 1D © O 10 ro ro COI IS oO ONS PS ION UES hn fs I FK PE Cee ONL EN Index. Taste [X.—Qualities of Concordant Triads. (See p. 399.) log eae l| - x Ex sa yz, | ‘uO T SOTMOULIEPT quIor Jo AjtsuoyUy aS Be Fi 28 as aS spe eaee Eis 6 = o lonw S65 3S) S 16 | SS Lani =) 19 219 | we Soh a Ee | :: s | a Ss = E SS mtowles a loow|o ; =) o|co Sim 30 10 St a [ace |? i: g|g:iis|9s tials ia © = aoe: 7S °@ ].9 2 iene | oO i ; : ey eee | ce) 5S [8 Se || OS E ace ooo t - a ed rao Ct . mlood]|] - Oo : : jid ia EX) 100 “ec n|a2°| jig i]s uns| eg: Eee © |nNaWm 5 EIS Soo 20 aS) Ned EOF 2|5 lee g | Eats E Eetegeae = Sura i z y S eo Oro =10 ok See omns | jae= | ss | Ara} Of [e Theo ne . - in| ol 6 | 295 [E9100 | HOO 1G it c ar 2 ay Inee | “97> ANS E a Bian) siGN ardaalS | 1 a 3 no A = 7 ao 5 ae Seon ROO F Jes) 3 o,4 S| ieee | Ssa E Sait oS N er) aH | ote a4 Sh ilee 3 aa 5 © loda |S (eo FI] SSweHO |] fx © ia ee oat a{4 SS = 1A AHS | 209 : Reiter a = 3 = ap tae mi 5 A a oa, Bho es Sr 19 | hie Sei CN lO @ loon | Sand /SNS :a! :& ics piste a a] 5 a aie Ce gro | oeanso ce) to o);onre 2 P Se Scene SOR Sasa eS |e oes SeTey = S 3 ve ho 1 Id 2 ay |eex | 7S E 16 ia | ia a is Sle teuel22In i | F 5 o SS io : : = © |oox seigizig| ie 2 ig et aa] e 2 al ISS IS J1o tone | io c eee DW} WIOMD! I~ SIS|A SHA] = ai? n-Pnls oth ote! a c on ae, a Zn Teen : 5 : 19 Je : @| 9 : 2 st Joon | “gose RiP Gl] ik In | seHeo- | = |- > 5 3 ri! oS F, 3 = 19 419 5 z : Seo Ee) jew |°°sgs a > Tea lley | ia ©) oCe) reels i E = PS aon 5 r o [oon [S855 [97738 ‘0 ies gent 202 oo | 5 ane 5 5 5 : z = e& | "esas E Be CK |e GShei2 Soave tigs | andes — | q . 5 i Oa a onx j HA rye on [FOS Re OS a 4 SoSSana | seo No.| sue XII.—General Table of Equal Temperament. (See p. 407.) Tf, log of tem- pered pitch. 0000000 0285191 — 7x 0226335 5x 0570382 — 14x ‘0511526 — 2x 0452670-+10x 0796717 — 9x 0737861-+3x 1023052 —4x ‘0964196+-8x 1308245 — 11x -1249887-+x 1534578 —6x 14757224.6x 1819769 — 13x | J, é, log of just pitch. i 0000000 | 0 0053950] —k ‘0177287 | 2k—7x 0280285 | —k-+5x 0457574| k—2x 0511526] —2x 0688813 | 2k—9x 0791812} —k+8x 1249386] x 1503338 "1426675 1480626 B, beat meter. Inter- val. —ooooooor* | ee rr rr | me | | | ee Hig —6k+418x| 3 5k — 20x seal a | a a | A | A | a | | er | eR ce | RTS ee | 1760913 —x 1702057 -+-11x 2046104 —8x 198724844 2331295 —15x | ‘1706961 ‘1760918 1938200 2041199 i | | re ee | ee een | REESE ee | eee 1 ( Pel peme | 3 - peted 4 i note. 5 Saree 7\|t | @ |? dp 9 cx 10 Ses 12 |. 13/2 | 42 ja |? | te Tb eee Sony it as 16 17, eq 18 19 bi £ 20 |. ; fi ry) #x t] ab ‘|g gx 7 ppb ) ap . bp b oy bt ‘2272459 — 3x 2213583-49x 2557630 — 10x ‘24987744 2x ‘2783965 — 5x '2725109-+-7x 0058851 —12x 3010300 | | | 2218486 2272438 ‘2498773 ‘2552725 ‘2676061 ‘27300138 3010300 ee. ee ee eee —1078kx +1998x? Where k=:0053950 and = is arbitrary. ¢ J. Just Intonation i M. Mesotonic Temp H, Hemitonic Temp Wolves of —— 00810 Gee ae 00270 2 00270 — Miner 3a |-90135 3 ihe ©, 00000 6th W onarn 0.4 =e A 00 Bye — De HOS? B) ft =At f 2 Beh = Es cf 00270 ere ay F002 TO TOW cet BD ct = At c B GE = EEG oo405 Ae el FE eb =F ay00876 cf B= Chad ; GEE =GF et ooo00 (00539 Where & 100000 100675 00135 00405 00270 Beat Factor. _ | + -00638 — ‘00392 -- ‘018605 | — 053965 + 00343 — 00197 000000 |-+--059684 +-00490 | — -00589 | + 00686 4.00195 | — 00344 | + 00932 | + 00589 | + -00098 | — 00441} 010547 Taste X.—Duplicated Forms of the Concordant Triad. (See p. 400.) No.| Simple. Duplicated. Simple. Duplicated. Tasie XII.—General Table of Equal Temperament. (See p: 407.) 1/2, 3, 5/2, 8 4, 5/2, 3, 5, 8|2,3,5,61C Ge |C Geel Ge @|O Geg Tem- 2/1, 3, 5/1, 2 3 5/1, 3 4, 5/135, 61C'Ge |O'CGe |O'Gce |O'Ge g Just | pered | log vies. ie Geral eae Bl Inter. 3/3, 4, 5) 3, 4, 5, 6) 3, 4, 56, 8) 3,4,.5,10]G ce|/G ceg |G cee’?!Gce e& note. | ote. pered pitch aan T—J. |beat meter.) yal, 4|2 5, 6|2, 4, 5, 6) 2, 5, 6 8 Ceg/iC ceg |G eg @ [else aitee eael Rale 5 | 3, 5, 8) 3, 5, 6, 8] 3, 5, 8,10 GeceiG ege’|G eee? ce 0000000! 0 T — : 5 te } = || Seendeo 10088980 =k H 6| 4, 5, 6) 4, 5, 6, 8) 4, 5, 610) 45,6,12)c¢ eg |c egelc eg elceg g te ef | 0285191—7x | -0177287| 2k—7x tt 7 | 5, 6, 8/5, 6, 8,10) 5, 6, 8,12 e BS e g cele g cg’ dd | -0226335-+-5x | -0280285] —k+5x 2 8| 1, 3,10] 1, 2, 3,10] 1, 3, 4,10/1,3,6,10]C'Ge|C'CGe2|C'Ge &|C'Gg e& ex | -0570882_ 14x 9/1, 5, 6)1, 2, 5, 6/1, 4, 5, 6 Cteg |C'Ceg |Ciceg 10 | 3, 4,10] 8, 4, 6,10 G ce? |G ec ge? ; z Yiledess|28. 9D : Sues if d | -0511526—2s 10511598 are a 11 | 2, 3,10| 2, 3, 4,10] 2, 8, 8,10 CGe|C Gee|O0 Gee ebb | -0452670+10x ge 12 | 2, 6,12 | 2, 4, 5,12 | 2, 5 812 C e g. Ce eg C e cg? tat dt | -0796717—9x | -0688813| 2k—9x {Ut FO Ps UO A te Ae cdligu bes alg te? | eb | -073786148x | -0791812| —k+3x | —6k+18x| 3 14] 1, 5,12] 1, 2, 5,12) 1, 4, 5,12 | 1,5,8,12] 0% e o2/0'G e 2|C'ce g?| Cre og? 15 | 4, 5,12 | 4, 5, 8,12) 4, 5,10,12 | 4,5,8, 10} ¢ e g*)c¢ e cg?) c ecep?!c e ee 6 é -1023052—4x 0969100] 1—4x 5k —20x oat aaa fp | 0964196-+8x ; 16 | 5, 8,12 | 5, 8,10,12 e c?o?]e ce? p? x 17 | 3,10,16 | 3, 6,10,16 | 3, 10, 12,16 G ec|G g eG eg? ct pint sgees = 18 | 8, 5,16 | 8, 5, 6,16 | 3, 5, 10,16 G ectiG eg ctlG e ect 949¢ : 19 | 5, 616 | 5, 6,10,16 | 5, 6,12, 16 Bee ils coalie a cea ai fey | 124087 | 112 Pela ar a 20 | 5,12) 16 | 5,10, 12, 16 e pect|e e% pct t SRD EET EW ook "lity | -1702057+4-11x an j -2046104—8x | -1988200| 2k—8x Forms of the Minor Triad. ig at 1987248+4x | -2041199] —k+4x | —8k+82x| 6 ox | 2331295 — 15x No.| Form. | Diff. Tones. No. = 9918486| k—3x Bk—15x VI TR ES fa t BO BEEP SOS | 3072488 —8x tVI BS a bob | -2218583+49x 2 | 12, 15, 20 g 8 7 ; 2557630 — 10x 8 | 10, 2, ib 6 12 al at 2498778 7 12, 15 _ | -2498774-42 5| 610,15 G 1s oy | O8774-+25 | 9559795 47 6 | 15, 20, 24 9] > eg? th = ouge fv 7| 6,15,20| 5, X14|G be 16| BIE | Oe eee | 12780018 wat 8| 3,10,15| 5, 7,121/Ge b 19 e) | 2725109-+7x 9 | 12,15,40| 3,25,28]¢ bet 5 | | be | 0058851—12x 10 | 10,15,24 | 5, 9,14] b g 7 e ce? | -3010300 *3010800| 0 vol | 5, 6,15 10]E G 7 ze 28 12 | 8,15, 20 17 |G*b 3 =32k? | =1501° 13 | 15, 20, 48 33] b & 10 —212kx | —1078kx 14 | 15, 24, 40 25} b g? d, c*, tet 10,15, 48 | 20 +420x? | +1998x? 15 | 15,40, 48 83 | b et gt] o tee, sje! 12) 15,90 | “3 SE ee eee ee SS 16] 5,15, 24 E 9 Where £=:0053950 and z is arbitrary. 17 | 615,40 G 6 18] 38,15, 40 G 13 19 | 5,15, 48 E 14 20 ae 15, e 15 J. Just Intonation i M. Mesotonic Temp H, Hemitonic Temp Wolves of Coe ee tN PGE! 4 90135 Gee = Fn C ip Cr EK tg 0st ors BiH, = Di pote Bie Att Bee = EF et ooa70 00270 3rd w s BD Fe= De F00270 Des Ate, BGS = EL G4 og405 th w'00185 ee mee GZf =GP eZ oo000 aero 100589 Where eye 100675 Comparative Tab . (continued). Interval Errors. +:00981 + °00441 — ‘00098 + 00638 — ‘00392 + 00348 — ‘00197 +-00784 | + 00245 | — 00246 | +0040 | — 00049 |—-009304 — 00589 | + 00686 +-00195 | — 00344 | + 00932 + 00392 1+ -015553 — 00147 | Beat Factor. | ‘000000 H ‘010547 Taste XIV. (continued). Notes. Logarithms, Interval Errors, Beat Factor, J M J M H M-J | H—-J M c @ 00000 ‘00000 | -00000] -00000} -00000 Tasie XII te 00540 —-00540 |— -00540 ; tet | | cy 01773 | | 01908 + 00135 |-++ 00786 Wolves of Defective Equal Temperaments. (See p. 415.) ¢ 02312 ‘02509 |— 00404 toota7 db | ab 02803 | 02938 +-°00135 |— 00294 Wolves. Interval erzor, «| Beat meter, 8, ex 03816 a ee eee tid ‘04036 + 00981 b Vth ee td d nee 04846 | ~-05017 |+:00: + ‘00441 Gie=AD g=GH dt........ —k—s+llx | —3k—38s+33: d 05115 —00098 u ye awe eee a ebb ‘05876 I : 4th wolf=4w. ED GE=B) Ab=D# Gt...... lets—Ilx | 4k-4-ds—ddx tat Jat || ax] 06888 | 06753 +-00638 ted |e 07918 | 07783 — ‘00392 }---018605 IUhd wolf= Iw. =| —S GZ s =ADe e { e ‘09691 09691 + 00343} ‘000000 Be? =B dt e te 10231 —'00197 F# B) =F Wile abeosoonune —s-+8x —5s+40x fb 10721 CLF =Cz i ef 11599 6th wolf=6y. tf 11954 + 00599 C Git =C Al f fi 12494} | 12629 + ‘00049 P = . eat ¢ is B att poOoHSTOOLGeD s—8x 83— 64x is 18033 00491 Pet =D ct tf fit 14267 | | 14537 + 00784 fq 14806 15051 |— 00270 |+ 00245 ; 3rd wolf= 3vw. a) |g? 15297 | 15567 — 00246 ED Fe= De Fe Beh = Atiet yoo. ees seen s—9x 63—54x fx 16444 FG? = Ee Gt te 17070 +-00490 g g 17609} | 17474 — 00049 VIth wolf= VIw. tg 18149 — 00589 FH e) =F d abb 18504 CHBDS= CHAT Yee eee —s+9x —5s+45x 3 Git =GH ed tet | et 19382 | -19382 + 00686 ap 19873 + 00195 2 tad { | ap 20412( | 20412 — 00344] -000000 |—:062995 Where &=-0053950, s=:0004901, and a is arbitrary. — pce 8x 21290 fa 1645 +-00932 a a 2185 \ | 22320 + 00392 ta 22724 5 | — 00147 bob ‘23350 at at 24497 | -24998 | ae . 24988 | | 25258 4 Tasiz XIV. (See p. 419.) + 25507 * O0441 Comparative Table of the Mesotonic and Hemitonic Temperaments. ina —— eatin acne CH : 0 00883 8 b (|b 273 ‘27165 |\ * — 00135 |+-'00295 J. Just Intonation in the keys of B), F, C, G, D, or System of C; 33 tones. Fe) 7300 eee M. Mesotoniec Temperament in all keys. No. 2 (2); 27 tones. ———_|—_ arora os cara| H. Hemitonic Temperament in all keys. No. 50 (35); 12 tones, te? by moet 29073 30103 [4+--00540 | +-00540 2 {|e 301034 30103 ‘ 00000 ya Sum of squares........---+ '0004735 |:0008748 i rT eae aments. Error of 3rd. el fi 12 er. Error. 7 | —:0013102 3 | —-0013488 5 | —°0011359 t | —-0011385 3 | —-0010528 2 | —°:0010375 9 | —:0014380 1 | —:0010288 0 | —:0014833 0 | —-0009811 1 | —:0014911 9 | —:0009133 2 | —-0015868 8 | —:0008992 7 | —:0008023 4 | —-0016957 6 | —-0007707 5 | —:0006700 5 | —:-0017983 8 | —:0018811 9 | —-0019267 7 | —:0018775 10 |} —:0020233 ‘3 | —-0016811 1 0 }1 | —:0020580 oat (36 2 | +-0000368 33 | —-0026901 3 | +-0000401 36 | —-0028021 35 =| —-0027025 4 |+-0004421 37 ~=«| — 0029581 38 «=| — 0083718 39 | —-0037465 10 | —-0037897 41 | —:0039235 49 | —-0042814 16 | +-0011969 43, | —-0043882 48 | —-0051988 44. | —-0049036 45 | —-0050812 46 | —:0052111 49 | —:0052177 47 \ —-0052315 50 | —:00538611 26 | —:0018689 51 | —:0053950 34 |+-0026975 Melodic Errors. 0001597 (See p. 418.) Se. 0001527 ‘0001528 (0001541 0001542 0001559 ‘0001562 ‘0001535 0001564 0001541 0001578 ‘00015438 0001601 0001563 ‘0001606 ‘0001648 ‘0001664 0001719 0001639 ‘0001679 ‘0001705 0001983 ‘0001764 ‘0001706 0002329 0001863 ‘0001875 0002375 0002177 0002378 0002563 0002432 0002961 0002794 0003511 0004297 0004897 0004714 0004461 0005939 ‘0009584 0007554 ‘0008164 0008651 0008703 0009185 0006244 ‘0009314 | 0009028 0005648 0008628 | Order. 0001513 ‘0001565 | 0001363 ‘0001361 ‘0001339 0001338 Harmonic Errors.} Comparison. =P. ‘0001709 0001337 0001718 0001542 ‘0001812 ‘0001367 0002029 0001375 0001451 0002525 ‘0001485 0001625 0002625 0002950 0003127 ‘0002936 0003533 0001589 0003587 0004168 0004247 0003858 ‘0006077 0003873 0008306 0007556 ‘0006140 0009600 0013524 0017734 ‘0018255 0019940 0024829 0012335 0026391 ‘0039960 ‘0034669 0037795 0040169 0040292 0040549 "0043004 ‘0019977 0043659 0032183 1864.] Mr. W. H. L. Russell on the Calculus of Symbols. 423 “On the Calculus of Symbols.—Fourth Memoir. With Ap- plications to the Theory of Non-Linear Differential Equa- tions.” By W.H. UL. RussExz, Esq., A.B. Received July 31, 1863%*. In the preceding memoirs on the Calculus of Symbols, systems have been constructed for the multiplication and division of non-commutative symbols subject to certain laws of combination ; and these systems suffice for linear differential equations. But when we enter upon the consideration of non-linear equations, we see at once that these methods do not apply. It becomes necessary to invent some fresh mode of calculation, and a new notation, in order to bring non-linear functions into a condition which admits of treatment by symbolical algebra. This is the object of the fol- lowing memoir. Professor Boole has given, in his ‘ Treatise on Differential Equations,’ a method due to M. Sarrus, by which we ascertain whether a given non-linear function is a complete differential. This method, as will be seen by anyone who will refer to Professor Boole’s treatise, is equivalent to finding the conditidns that a non-linear function may be externally divisible by the symbol of differentiation. In the following paper I have given a notation by which I obtain the actual expressions for these con- ditions, and for the symbolical remainders arising in the course of the division, and have extended my investigations to ascertaining the results of the symbolical division of non-linear honedare by linear functions of the symbol of differentiation. Let F (a, y, Ys Yoo Y3 + + + + Yn) be any non-linear function, in which Yis Yo, Ys, ++» + Y, Aenote respectively the first, second, third, .... th differential of y with respect to (a). Let U, denote fdy,, ¢.e. the integral of a function involving 2, y, 445 Ya ++ with reference to y, alone. Let V, in like manner denote 4 when the differentiation is supposed YY effected with reference to y, alone, so that V, U, F=F. The next definition is the most important, as it is that on which all our subsequent calculations will depend. ‘Ve may suppose F differentiated. (m) times with reference to Y¥,) Yn—1) OF Yn—25 &C., ANA Yn, Yn—1y OF Yn—os &c., as the case may be, afterward equated to zero. We shall denote this entire process by Z™, Z), Z}, &e. The following definition is also of importance: we shall denote the ex- pression d Yr-1 f+, © yt! te aS a a eee as by the is b * Read Feb. 11, 1864; see Abstract, vol. xiii. p. 126. VOL. XIII, 21 & Taste XV.—Comparative Table of Equal Temperaments. (See p. 418.) Error of Vth. Error of Iird. rror of 3rd. Melodic Errors. | Harmonic Eyrors. 0001513 0001565 0001363 0001389 0001338 0001709 0001337 0001718 0001342 0001812 0001367 0002029 0001375 0001451 0002325 =p. 0001361 0001485 0001623 0002625 0002950 0003127 0002936 0003533 0001589 0003587 0004168 0004247 0002375 0003858 Name. Log »v. Envor. Exror. Enror. Order. Least Errors.............-.. 1747297 — ‘0013616 2 | —-0000514 — 0013102 1 M&EsSOTONIC ......-eseeeees 1747426 — 0013488 1 0 —:0013488 2 Equal beats of Vi and4...... 1746716 —'0014197 6 | —:0002838 —*0011359 4 Cycle of 81 ...----..sssseee 1746703 —:0014205 7 | —:0002870 — 0011385 6 Least Errors and Beats ...... 1746439 — 0014474 9 | —:0003946 — 0010528 8 Woolhouse’s Equal Harmony . .| 1746388 —'0014525 | 10 | —-0004150 —0010375 9 Least Beats of V and IIT .| 1747723 —'0013190 3 | +:0001190 — 0014380 3 Least Beats ........++-0000s 1746359 —:0014554 | 11 | —-0004266 —:0010288 } 11 Cycle of 93 ....... ee ee eee 1747872 —:0013039 4 |+-0001794 — ‘0014833 5 Equal and opp. Beats of 3 and 4} 1746200 —:0014713 | 13 | —-0004902 — 0009811 12 Huyghens’s Grate iol scanod ‘1747900 — ‘0013013 5 | +:0001898 —‘0014911 7 Henfling’s Cycle of 50........ 1745974 — 0014939 — ‘0005806. — 0009133 | 14 Drobisch’s Simplest it OTRO 1748219 — 0012694 —‘0003174 —:0015868 | 10 Dr. Smith's Equal Harmony . .} 1745927 — 0014986 — ‘0005994 — 0008992 | 16 Least Beats of 8, Il, V...... 1744404 — ‘0015309 —-0007286 —0008023 | 18 BWqual Beats of 6 and V ...... 1748582 — 0012331 + 0004626 — 0016957 | 13 Equal Errors of Ii and 3 ....}+1745199 — "0015414 —:0007707 —0007707 | 19 Romieu’s Theoretic ........-. 1745163 — ‘0015750 —:0009050 —:0006700 | 16 Equal and opp. Beats of 6 and 4) -1748924 — 0011989 + 0005994 —0017983 | 17 Drobisch’s Cycle of 74.......5 1749200 — 0011713 + :0007108 —‘0018811 20 Drobisch’s least Evrors........ 1749352 —‘0011561 + 0007706 — ‘0019267 21 Bqual and opp. Beats of IIT & Vj 1749188 —'0011725 -+:0007050 —‘0018775 | 26 Hqual Beats of IIT and 4 1749674 — 0011239 + 0008994 — 0020233 Qycleiof 22) 6. cues .| 1745200 — 0015713 — 0008902 —-0016811 Perfect 3rds and VIths 9 f — ‘0017983 — ‘0017982 0 Enrors of III and V eq. and op.}* —‘0010790 +:0010790 — ‘0020580 Sauveur’s Cycle of 43 ........ 1750175 — ‘0010738 +-0011008 —0021736 Woolhouse's Cycle of 19 ,,.... ‘1742807 —‘0018106 — 0018474 +-0000368 Drobisch after Delezenne ...... 1751830 —:0019683 +:0015218 — 0026901 OMOEA, ahoon nao mninccg 1742796 —:0018117 —‘0018518 +:0000401 Oy clevoniG 7a vrniicicanrirsiet 1752270 — ‘0008643 + 0021378 —-0028021 Romiew’s Anacratic ........4+ 1751938 — 0008975 +:0026050 — 0027025 Musicians’ Cycle of 55 ...... 1751456 —‘0019457 --'0023878 +:0004421 Marahisia vein cnieisttaietes wast tsk: 1752790 — ‘0008128 +:0021458 — ‘0029581 Drobisch’s V and III combined] 1754169 — 0006744. +:0026974 — 0033718 Least Wolf Errors .......... 1755418 —°0005495 +:0031970 — ‘0037465 No Vi WiOolBiSscaccreeue tees 1755562 — ‘0005351 + 0032546 —:0037897 FEBMOTONTIC: \torssranteratermatriaess *1756008 — 0004905 + 0034333 — ‘0039235 Least [iL and V Wolves...... 1757201 —:0003712 +-0039102 — ‘0042814 Gyelelof 46) os. vedieicetc esate 1788940 | —:0021973 — 0033942 +:0011969 III and V Wolves eq. and opp. | 1757557 — ‘0003356 + 0040526 — ‘0043882 Drobisch’s Cycle of 118 ...... 1760259 — 0000654 +:0051334 — 0051988 Least Wolf Beats............ 1759275 —:0001638 +:0047398 — ‘0049036 Cycle of65 ....0....0. 6.4.5 1759867 — ‘0001046 + 0049766 —:0050812 INORUISW VOLE rae trcterearee nae 1760300 5 | —:0000613 +:0051498 —-0052111 Drobisch's Violin ...........- “1760322 4 | —:0000591 + 0051586 — 0052177 No 8rd Wolf .............. 1760368 3 | —-0000545 +:005177 —-0052315 N. Mercator and Drobisch ....| ‘1760800 2 | —:0000113 +:0053498 —-0053611 Oycle of 26 .............05- ‘1786700 | 50 | —:0024213 — 0042902 —:0018689 PYTHAGOREAN..... 0... 0000s 1760913 1 0 +:0053950 — 0053950 Error of V and 8 eq. and opp.. .|-1733938 | 51 | —:0026975 —:0053950 +-0026975 0006077 0003873 0008306 ‘0007556 0006140 0002177 0002378 0002563 0002432 0002961 0009600 0013524 0017734 0018255 0019940 “0002794 0003511 0004297 0004397 0004714 0005648 0004461 0005939 0009584 0007554 0008164 0024829 0012335 0026391 0039960 0034669 0037795 0040169 0040292 0040549 0043004 0019977 0043659 0032183 0008628 0008651 0008703 0009185 0006244 0009314 0009028 a 1864.] Mr. W. . LU. Russell on the Calculus of Symbols. 423 “On the Calculus of Symbols.—Fourth Memoir. With Ap- plications to the Theory of Non-Linear Differential Equa- tions.” By W.H.L. Russexz, Esq., A.B. Received July 31, 1863%*. In the preceding memoirs on the Calculus of Symbols, systems have been constructed for the multiplication and division of non-commutative symbols subject to certain laws of combination ; and these systems suffice for linear differential equations. But when we enter upon the consideration of non-linear equations, we see at once that these methods do not apply. It becomes necessary to invent some fresh mode of calculation, and a new notation, in order to bring non-linear functions into a condition which admits of treatment by symbolical algebra. This is the object of the fol- lowing memoir. Professor Boole has given, in his ‘ Treatise on Differential Equations,’ a method due to M. Sarrus, by which we ascertain whether a given non-linear function is a complete differential. This method, as will be seen by anyone who will refer to Professor Boole’s treatise, is equivalent to finding the conditidns that a non-linear function may be externally divisible by the symbol of differentiation. In the following paper I have given a notation by which I obtain the actual expressions for these con- ditions, and for the symbolical remainders arising in the course of the division, and have extended my investigations to ascertaining the results of the symbolical division of non-linear functions by linear functions of the symbol of differentiation. Let F (a, ¥, Ys Yoo Y3 + + ++ Yn) be any non-linear function, in which Yrs Yr, Yo » +» + Yn Aenote respectively the first, second, third, .... mth differential of y with respect to (a). Let U, denote fdy,, 7.e. the integral of a function involving 2, ¥, 445 Ya 00+ with reference to y, alone. Let V, in like manner denote = when the differentiation is supposed Yy effected with reference to y, alone, so that V,U,F=F. The next definition is the most important, as it is that on which all our subsequent calculations will depend. We may suppose F differentiated. (m) times with reference to ¥,5 Yp_1, OY Yn—2> &C., AND Yn, Yn—1» OF Yn—2s &c., as the case may be, afterward equated to zero. We shall denote this entire process by Z), 2}, Z,0%, &c. The following definition is also of importance: we shall denote the ex- pression d d d d d eee dy? a 7 a ee ae by the symbol Y,. * Read Feb. 11, 1864; see Abstract, vol. xiii. p. 126. VOL. XIII. 21 a ee o * eS ED Ear, oc ee a ee AQ4 Mr. W. H. L. Russell on the Calculus of Symbols. Having thus explained the notation I propose to make use of, I proceed to determine the conditions that F may be externally divisible by = or, in other words, that F may be a perfect differential with respect to (7). It will be seen that the above notation will enable us to obtain expressions for the conditions indicated by the process of M. Sarrus. ) It is obvious that if we expand F in terms of y,, in order that the sym- bolical division with reference = may be possible, the terms involving Yn>o Yrs &C. must vanish. | Hence V,? F=0, and consequently | FH=ZF+y,0!,F, where, of course, Z2F, Z',F do not contain y,, Hence we have £U,a2.E) =Y. -1U,12',F =e YnL',F, and therefore F becomes £ (U,30,F) Ste epee a and if R, be the first remainder, R,=Z,F—Y,,_,U,12',F. The condition that this may be divisible by = will be, as before, C—O hence R, becomes Lia Zn —Z)_1Y,1U,12Z',F + Yn (Z'_,ZoF — 2, 1¥n-1 naan) : Now 2 p-1(En POEL, Nn sUya8'yB) = Yn-2U of n a ZF — Y¥_2U non 1 n1U, il). F + Yn (Zins ZnF —Zi, YU, aZ'nF) ; and if R. be the second remainder, we find R.=Z)_:Z),F —Z)_1Y,.U,.1Z),.F = Y¥,-2U pn 2Z'n_1 ZF + Y,-2U,-24naYn aU, ZF : the next condition is V,,_,.R.=0, and therefore R,=Z)_ LZ) ZF — Zyoin 1X nO 1h’, F — Zn_2¥n—2U 2, a ZF +2) oY n—2U 28’ n_1Vn1U ns Z' nF + Yn Z,-2L, ZF —Z', LZ 4 Me Pee AGS *s Diy 2X p20 pod nr ZF + Zn _2¥n—2QU ya 2Z'n-1 nL pr nF). Mr. W. H. L. Russell on the Calculus of Symbols. 425 But : fF VUca(2hy sh COP 2',_ 2h SY Uy ali — Zi, 2X n—2U 2’ ys ZnF + Zin 2V ns nL! ny 1Yn—1 ra Zik) } =Yj-3Un_3(Z'n—2Ln1Z,F — 2’, Ln iY n1U, 1Z',F — Din 2Xy—2U Lin LF + Z'n_2¥n2U py 2L'n1YnUnaZ',F) +Yn-9(Z'n—2Lp_10),F —Z',_ LZ) 1 Y,-1U,_1Z',F — Dn 2X n—2U nl’ ZF + L'n-2¥ n—2U p-2n_1¥ n—1U, 12 ) 3 whence we find R3=Z,_2Ln_1Z,F —Z)_oLn_1Y,1U,1Z',F —Zi-2¥n—2U y_2L'y_1ZnF + Zp_o¥n—2U y_2Z'n 1 Vn 10, Zi nF —Y,,-3U yn 3Z'n2LZn a ZnF + Yn 3U n2Z'n—2Zn—1 Yn1U np 1Z oF + Y,,-3Uy3Z'n2¥n2U 2 Z'n a ZnF a nan aL nan 2 noi na Una ak. Hence we infer the following rule for the formation of R,. Construct the term MUA a opcik _,4U, Li Gee peep) ee oA ne ce wee SN pea ete Dre ee ss Ue eZ In any symbol Z’,, the accent may be changed into a zero, 7. e. we may at pleasure substitute Z?, anywhere for Z’,,; but in such case the previous symbolical factor Y,,_,U,,_; must be omitted. This term is positive or negative according as the symbol Z’ occurs an even or an odd number of times in it; the aggregate of all the terms thus formed constitute the re- mainder R,, and the conditions that F may be externally divisible by a are dz Ve R06; Vio —0; Vion; Wael t= 0, &e. We shall now investigate the conditions that < +P may externally a divide F where P is a function of (7) and (y). As before, V,F=0, and in consequence F=2AF+y,Z',F. Now ‘ a PU, 2,F=Yo1 pond we +PU,2,.F+y,2/2. Hence we shall have R,=2.F—Y,,_,U,,_,Z',F—PU,_,Z',F. We have V7,_,R,=0 in order that this remainder may contain only the first power of y,_,, and R,=Z)_:Z2)F—Z)_, n—1U y1Z',F —Z)_,PU,,_1Z',F y. n~i(Z Ze —Z ith dU pik oe Z',PU,1Z "nk )s 212 426 Mr. W. H. L. Russell on the Calculus of Symbols. since = 2 P) U2 Din aZnF — Zi, natin iA ae —Z', ,PU,4Z',F) = Yo 0U 24, 2a Te YU, 24, 59,50, 52,8 = Y,,2U,, 52’, 4PU,,_,Z',.F +9. (2), ZOE —Z',4Y,1U,:Z',F —Z',_,PU,_,Z',F) + PU Ay OE — PU, 7 2eN, Wn e —PU,,-2Z',-1PU, 1Z',F. Whence we find that = +P divides R, with a remainder, ny R=) {7h Rey, Uae —fi ,PU,G7,.F“Y;_.U, 57, Ze Vp $25, a Mini Un _iZink 4 YU ee) os Pee — PUL F(Z PU. 6A N21 py es + PU Aaa eU, 3a ak: Putting V;_.R,=0, we find in like manner, Rs=Z)_oZ) ZF —Z)_oZn_iV,1U0,1Z',.F DP Fe a PU Lin — Leno \ pio eae +2i2Y, Up Ao V,aU, 17,8 +2, 2%, 2, eee = Zi2PU, f'n Z¥ + APU, 24, 4¥, 50, aa =F ZpeePUpsHi, a. PU, «Zi re Nis Uy hy of) Znk SEN LU 557, eo Leia NG, 4 Uplate ae SY UA ol oagletle Ae te Kg Upeg hl grate Voge quod nv, cath ier OR AR Oo UP ai ee UL aya Ie me Uy A, SY, pU nA PU ak V2 U p2sZ no Pn Un eA Ze mee Ge Oe Ae dl US A ec DINE EE! © —¥ U7, ol Uae Zi Palo Ane — PU,_34', oF, Zn? + PU, sf aoZn a + PU, 24’, oft _-4PU,1Z',F + PU; 7 Ye U2 ee =P eT Sa ee 7). Von ae ke ae Uy eel) oN noe Shi Ua See: UP Sy GaaN 2! Opt oy filam Seal Si PM Zia NO ok Zi yn N yep eed = PU, a oP eA PU aA We see at once that the value of R, in this case can be formed from that calculated in the last example, by writing Pat pleasure for any one or Mr. W. 8. L. Russell on the Calculus of Symbols. 427 more of the symbols Y, and taking the aggregate of the terms so formed. The conditions of division will be, as before, VEE =e Vt 0 Vokes ee Let us now investigate the conditions that F may be externally divisible ad by a We see at once that F, as before, must take the form Z°F+y,Z',F, and also that Z’,F can contain neither y, nor y,_;. Hence we shall have V2F=0, and also V,,_,V,F=0. Now Ee 7) 2 be Ven UE des B® dnt "2 Yn-1¥ n-26 Un-24 n =XY,,_.U,_.Z',.F +y, XZ) nF + ynZ' nF. Hence we shall have R,=2F—XY,,_.U,,_2Z',F —y, 1 XZ',F; when we must introduce the conditions v2, — O07 ani V5 ¥, 2, —0: consequently we shall have RZ! _,7oE— Zo XY, U3 FF + (Z', :Z0F —Z',_ yXY,,-2U nL, F —Z9_ XZ, F )yn_1 Now 5 U,-s(Z'ns2RE ci Zi ya XX n-2U, 2’ .F —Z°_,XZ',F)= XY, 30 n-3Z'n ZF ae XY, 30, 3Z'n_-1 XY n_2U,, FF “Ee a XY,,_3U,,_3Z5 :XZ',F == (XZ, 1Z,F a A ee Ve UAE ere XZ XZ )yn—2 ae (Z',, Zi F— Zin XX n_2U ZF — To XL Ey, Hence Ro=Z)_ ZF —Z)_,XY,,_2U,_2Z’,F SON a aU GA yp Oy Pe OY Wh pA 1 Nagy ee gk = €Y,,U,. 42 ZF + (XZ, XY, UU, 4 4¥ +XZ°_,XZ',.F—XZ', ZF )y,_>. Introducing the conditions Vii-2R.=0, V,_3Vn-2R,=0, we find R, — (jae Soca SM or PE oie GRY 4g U 2 2 — De RN GO eng ZOE =fZe ok Vn 3U gM peek Yn oA aE +2 2X Vn 3Un_3Zn XZ) F + (Zi!) 2Zin Zak —Z', of XY ,_.U,_.Z', F—Z',_oXY,3U,,_-3Z', 1Z2F SPL oO gig Un Fie RY SU pe Fie k 4+Z', oX¥,_-3U,_3Z_1XZ',F +20 _,XZ',_ XY,,_2U,_.Z’,F + Zi-gXZ)_)XZ',,F—Zo_ XZ", DOF )yy_2 428 Mr. W. H.L. Russell on the Calculus of Symbols. Now a? ' 0 70 qin n—2Lin1Lnk — Zi, -2L, XY n_2QU yp 2Z',F — Zi n_2XV,_3Un_3Z, sZAF +Z', 2X Vp -3Un_sZ'n_1 XY n_2U,_20',,F | + Zi, 2X Y,3U n-sZn1XZ',F + Zh 9 XZ", XV n_2Uy20'F + Z)_oXZn_:XZ',F —Z)_.XZ',_ ZhP) = XY, 10, 4Z'n_2Zn 1 ZF —XY,,_ 0,2 p22, s1XY,2U, 20, F —XY,,_,U, 12, 2XY,_3U,,_3Z', ,Z,F +XY,,_,U,,4Z',_2XY,,_3U,,-3Z', 1X Y,,2Uy_2Z'.F +XY,,4U,,4Z',, 2X Y,,_3U,,_sZ)_.XZ',.F +XY, U, 2) XZ, XY, U..78 +XY,,_,U,,_,Z20_.XZ7_,XZ'F —XY,,_,U,,_, 2 2XZ', s ZF + (XZ, oF), ZF —XZ', Zi XY, .U,_.Z',F —XZ',,_.XY,,3U,-3Z',aZF +XZ',_2XY,_3U,_34', 1XY,_2U,,_24',F +XZ', oXY,3U,sZ,1XZ',F +XZi_-2XZ', :XY,,2U, 20", F +XZ0_,XZ°_,XZ',F—X2Z_,X7', ,22F)y,-s (45 2, 54,0 Zi, 0, sk, ee — Zin 2X ¥n_sUy_3Z!naZ¥ + Z',_2X Yn_3U ng Z'n1& Vn-2U p24 F + Zn 2X VY, -3U yn _3Zn1XZ',F +2) XZ, XV n-2U yf nk +2_,XU2:KU/,B—2, XU, OR) y yo We thus find R3= Zio, ZF — ZZ, XY »2U no, F — Zi, 2&V,-3Un-3Z', 1ZnF + Z,_2XY,3U,_3Z/,1X VY, 2U no EF + Zi XY, 3Un 34, 1X2, —XY,_,U,,_ 4, 2A gee +XY,,,U,_,4', 24 XY, 2U,,_ 22", F +XY,,_,0,_,4'n2XYn_3U,3L'n ZF — XY ,,1,U, 2 2X, U2 RV BU neo —XY,,_,U,_,4/n 2X Yn3U,_sZ,_ XZ), F —XY,,_,U,_,27-2X0', aXY,_2U, 20) EF —XY,_,U,_,27_.X 2) _,XZ' EF =F DG Gc, Ula, nary OA pa Ze + (XZ",_oZ) sXY,2U,_2Z', F—XZ',_.Z) sZ0F “eh, SON ee, 52, —XZ', oY, sUy_sf'n1XV,2U p22 ,F : — XZ’, 2X V¥,_3U,-s4,1¥4Z',F —XZ)_ XZ! XY, _2U p20), F + XZ) XZ XZ', B+ XZ) XZ ‘nn LB Yn—s) : Mr. W.H. L. Russell on the Calculus of Symbols. 429 Let us now assume ; R,=M,+N,yn-> Then M, is formed according to the following rule :—Form the term ON Wie Ae 6, Ay ao RN gery eee pas eaiviveive DEY ea ee VD’ E. Z',, may in any place be changed into Z',; but in this case either the preceding XY,, .U,,-. must be omitted, or the succeeding XY,,_,U,,_1 changed into X. The signs of the terms follow this law. A term not containing X introduced in place of XYU is positive if Z' occurs in it an even number of times, negative in the contrary case. But every X in- troduced in place of XYU occasions a change of sign. The aggregate of all the terms thus formed will give M.. We form N thus: construct the term et ONG Ur Unt p iy see a VIA and a precisely similar rule holds good. R, is subject to the condition Ve Oe 2” hu == 0. Let us now investigate the criterion that F may be divisible by dl” d ait P+ where P and Q are functions of () and (y). Proceeding as before, we have a? d P ie Wie aa PS == Q)U, 2 r= (z P) (Y,,-2+Yn—1Vn—2) Un_2Z' nF + QU, 20’, F = (XY,-2+ PY; 2+ Q)U, 2ZnF + ynaA(X+ PZ F+ yn ZF. The form of this equation gives us the following rule to ascertain the successive remainders. Construct the remainder in the last ease as before, and substitute at pleasure Q in any place where XY is found, P in any place where X is found. The aggregate of the term thus formed will give the remainder in this case. We now investigate the condition that ie may be an external factor of F. We put, as before, F=Z)F+y,Z’,F, where Z',F must contain neither Yn—1 NOY Y,_2, Which gives the conditions ViF=0, ViaVAL=0, V,,-2Vr,H=0. Now we have a? a? beth ZF) = dix® (U,,-4' nF) dic” @? , If =F Yn 24 nt ar Yn—h iby) = XY ,,-3U,_34',F oe ¥ nad Z a te 2y n-1 XZ H sf Yn aes (¥), —3 a Yn— oVa =) U,_34 ful 430 Mr. W. H. L. Russell on the Calculus of Symbols. And we consequently obtain R= Zlk— KYU, Se Fy JEU 9y, F. Introducing the conditions Ven 0; Veo hi — 0; Vee and expanding in terms of y,_,, we have B= (22_2F— 20 XY, 5U,22',F) + (Zi Pgh fy i XY 523 U nisl Ya 1 Ena Yn Yn nL, F— 2y,_sZ>_ 1 XZ" EB. As the coefficient of y,_, in this cannot contain Yn—2, We may write this expression, R,=(Z) sa — Be XY, 30,34, Yn ot 4 (Zn, ZF —Z)_ LZ", ~X7Y, 30,32", F 22) _2Ln_ 1 XZ! nb) Yn Let us now assume Rn=L,+ MiYn—-m—1+ NnYn—ms where R,, is the mth remainder, and N,, does not contain ¥,,_ 1 OT Yn_m—2 Hence, expanding in terms of y,_,,, we have RS Za ae Ze ls, a5 (2) _,,.M.,, + Una ‘nm ML) Yn—m-1 ae (Zire Ne Vil Nn Un =) Aaery Lites ay Spain| Da ae gl Skee ey Arman, Ui eS AO DI Now B} = : Us CAE APN Op == Vipera pee) t aaa a fay —m—s s+ (Speeaietir | io i Urs ST a: ie pla Le ol aio, Nowe 22 KW) OU) 5(Z) o's oi at Zi ig ie ee == RA iat pees te Di iphy Dea ND - Yn—m—2 DR TE NDGA GP AD ae a INA eae BELG step Lg = Lit oN gc Hence (R= (Biome Mncm—sU fa, nolan, UG in (fe eames Fa SL ga ma at gare egg Ny SDR! eh SM Le My + 2X2) Zin Nn )Yn—m=i* Now consider for a moment the equations Lnu=G,L, + HM, + K,Niw a Gi lin == H’,,M., == I< Ns Nru=G",L,,+ HH", M,+tK,Niw Mr. W. H. U. Russell on the Calculus of Symbols. 431 and suppose that Dinas = ee Sal iM,,i+ N41 =) Lot poM 2+ v2N,_2= &e. = Ee Ee Ne —— ee. Then we find dy = Gn.Gnr—1 == EG: a= KG A, — G36, 1G. 5 te HB, GaGa. =i K rns et Ogee 1G be Got A. iG yo K, EE oe Gee +G,,K,,1G@", 2+ H,K' ni G' not K,,.K",-1G"'m 7. G.G.,_.G, .G,.,+ H, GG, Gs See RG iG oa, Gb Ge GC... eee 5G Gs Kk HG G's eG, kG) Gee Ke GG. SP Ee yO tat Og 9 Ae gy Oe EN yg oF ass ee GE Ge aK Gr on Gs + GH EG ee-+ HW ng Bak oG ses SP a, 2G ae EK, LH 2G ies eG. Red Ee |G BW KRY Ae Ges Stes Or Crea Ke Gy ample, one ns 11 ONS id SGA 8 sre SM Coe i WEN read GS eters SN Ee es ier curt ED AR ecg ake pee Ke Ge ee WA, le a oss ripe, Mer pea Rigo sass Hence we obtain the following rule for the determination of X,.:— Write down the term G,,,G,,_:G,,......G,_,. Wemay substitute H and K at pleasure for G anywhere except in the last factor, which is always G. Whenever we put H for G, the succeeding letter is to receive a single accent ; whenever K for G, the succeeding letter receives a double accent. The aggregate of all the terms thus formed will be A,, and we may of course obtain similar expressions for p,, &c. Now if we put -Q? G,,, = ae (22-4-y2) sin—1 —_" __ Vere $, (@® +9"), — (20° + 2ay’ iE ABA: Mr. W. H. 1. Russell on the Calculus of Symbols. This equation may be written ad a : 2 2 d ad 2 2\2 44 (05 Vas) 20 +Y) (#5, age) + HIE, ile iO Saas yb ft bo es 5 tae 1% Vg MOTI)? Gy ag which may be treated as before. In order to find the most general form of equation to which the symbols @ dy d d\” d d Ne ae : : Ge —y a) a UA zy +y a and z°—y’ likewise combine according to the laws of algebraical symbols, we shall take (oF +y = y"to avoid the a Bart, : A fe ae and x°+y” give rise, we must determine the expansion of negative sign. Now the expansion of (es +y <\ will consist of all the terms of the (+a) (va) (a) (a) C464 .0. bebe onan, <, and dy for 7 g —, where it is to be understood that dz and dy do not apply to the aha Moreover we shall use, as in the third memoir, a—l a—2 a—r+l] ; form in which We shall write 6, for a, for @.—5—* B+ +s 7 Then we shall have, if a+a=7, d\& d \@ gq Goa gd’ —2 ests, — ) mre ye od Cae ues xe? (3) (vz) OY ye dga WN" Gat dae) ia +a,0% 0, y% yd ee Again, a+a+6=n, (yaa) I Te ( dy (vz, a dn aot an a+b ce Ore digas TO Oe a gatia t a Oa Te gato $5, y2t? Fae Oo" dy* datto—3 ** Mr. W.H. 1. Russell on the Calculus of Symbols. 435 +a,xry? 6 aaa wee +a,b, y? 3 xb, y% ie aol 1 ] dy*- yet b« att y dy*—deute—" —3 +4, b, y? Py vt by" d” dy*— 1 dayats—2 qd—2 qd-3 6b 2 7,0 b HY 0x6 v4 dy*— 2 yat 6 a oe by On U2 0y ge dy* 2 yet o— 1 Bde ss q2-3 PON OS aps dear t + &e. And again, if a+a+6+ (=n, we shall have yay a2) ( =) (9-55 ( dy Te =yttb pee et +6, y2*? x8 a fe an} dy**B dete dy*tB daets—1 n—2 PR gah aes TRO ye a 0 qd—3 dy +B yats—3 Hits g?-) : a qd”-2 a, vatBy? dy dy2+B—1 agate + a,b, xB y? dna Oy Y dyttB-X dageto=1 qd»—-3 9/0 + 0,6, xP y On wo Oyu" Ger dee +... n—2 d™—3 ae eee b ~a+B e2 + ay ¢ y dy*+b—2 daets—i qdn—-3 dy*+B—8 datt 6 claerele =F ay? wotB O°, y@ q”-) ‘ x dn—-2 + Bert Poy pate deat TBE eye at Tarai gat i= qd2—3 +B, 6,0 byy2t? Orn, 0 dypP) dato qn—2 qd’—-3 + fa, wtb by, y?dyy% dyrtP—2 qcats + a,b, 8, byy? d,* 0 yy" dy? *B-2qlpetb-1 2-3 + 2,3, 2 t8 0,4? Oy y" dy*tb—3 datto—] Op tet 436 Mr. W. H. L. Russell on the Calculus of Symbols. “eee 4.5, 8, 08 5,ce%82, 248 ee dy* +82 duets 101M: yd dyB=2 qyato-1 q”2-3 dye BS dogo +024 3%, ye? + a, 3, oyy? yo yy Heo: +B, x8 tB§, yate ee Dy ee dxete We are consequently able to see that the bis term of (2 =) (vz) e a (a) is qy> q! 8 xp! got ais By Cr Byte By Spee OLY? Py. .0L 08 SP la pai igo where pte +o. -=7, 0 . 6. 8 Pte + sey vw is 2 es 5 and r+s Me ig 2 es d NP i 8 Hence the general term of (« iy +y aE ) will be q2-m > PP ae S Ont st* Bg Gy one OLY? RYO, OO ho 7 ea ee under the ‘ohteLEE (1), (2), (8), and also a+64+...+ta+6+...=n. Calling the expanded form of (95+ y za): An, it is easily seen that we can resolve all linear partial differential equations of the form S(e—y)Anit+f,(a?—y)An-1 Uu+f,(e—y)An-1u+ &. =F (a, y). The same property is possessed by a great number of other symbols. Let us examine the condition that , (ax + by +c) *. —(a'ae+b'y+c') = and Az’ +2Bay+ Cy’+ 2Kv+2Fy+H may combine according to the algebraical law. The condition is easily seen to be (Av+ By+ E) (z+ by+¢)—(Cy+ Be+F) (e+ o'y+c')=0, from whence we have Aa —Ba'=0, Bod—Cd'= Ab +Ba =Ca'+ Bo’, Be +Ed =Cc!+ Fd’, Ea +Ac =Fa'+Be', and Ec =Fe’. ha pile teat Fa Mr. W. H.L. Russell on the Calculus of Symbols. 437 We may consider B=1, which gives the following conditions: a'/b=a0', a=b'. Also ! A= ee C=] —— =-_, a a And the symbols may be written d 1, & b(ax + by +c) ae meee | tyere ay and. a+ 2ab xy + by? + 2acle+ 2cby+H. It fallow: hence me in order to find the form of the differential equa- tions to which these symbols give rise, we must know the expansion of d d\” : (x are +Y a) , where X and Y are functions of # and y. The expanded form will be a series of terms like d \B d \? d \* d\* (rq) Ga) ("a) (a). We must consequently find an expression for (x =) in powers of Le It must be remembered that X is a function of x and y, in which (y) during the present process is considered as constant, and therefore X may be looked upon as a function of (x) only. Now we shall find after a few differentiations, that d 5 ae a’ (x dz} = dx’? + (X6X* + X°5X3 + X°SK24 X*9X) = + (XdXOX? + XSX7SX?+ X*IXOX? + Xo + (XdXOX6X?+ XOXdX*5X + Bs a8 = + Xoxexaxex 2, dx Now let x xy ee ae ( Z) = =iG ee da" nase di"? ee Then X=, X*OXFOXY... where there are 7 0’s, and a+(B+y+....=n. Hence we shall have CHa CH Cs 438 Mr. W. H. L. Russell on the Calculus of Symbols. dp dB = ( ) 1% eget ae as grat} Le p-2 be ee Oo A= Stag pee See 8 4. ee 4 aye dy*— —2 eae (2) d4—2 “teh Foy XO Fh oe it : obvious that the general expression for the expansion of (x< <+¥ =) will depend upon principles not materially differing from oo one considered. The symbols we have already considered are only of the first order of differentiation ; we shall show that there exist symbols of the second order which combine with certain algebraical quantities as if they were themselves algebraic. Let us take the symbols ' a a? d’ ,@ , a oe 5 Aen” dor dy ay a ap + 2a rn +26 a +e, and Az’? + 2Bay+Cy?+ 2A'r+ 2Bly+H. Proceeding as before, we arrive at the following conditions : Ab +Be =0,0 0.0." Bé +Ce-=0; 2 2 ee A+ Ble=0) 4. he Ac@!+- Bb! = 0, yer wc yuk ie 2 ee ar Be! +Co'=0,.. . 6 Aa +Bé o ay 't oo Deke) Bas = sg be oh Ala+ een oo Ss Se DA ABB 42004 doll 4 40H c= 0 . 0. ee Whence we have, putting B=1, A=— and with the following conditions, ac=—6*?, ac—bd'=0 4a! A'+ 40'B/+e=0 the condition a!e— 6d! may be otherwise written a'6—ab'=0, in consequence of the equation ac=0’. It will be observed that several of the nine above equations are not inde- pendent of the rest ; so that the result is much simplified. p) C=— Mr. W. H. L. Russell on the Calculus of Symbols. 439 I now proceed to apply the calculus of symbols to the solution of func- tional equations. daz i =X. Let Then the following formule are known: d MODS =f Xe +1) Gone (EY) =x +2)}- &e. (cP a) -f0) =fyUy(@) +1)}- These formule may ee thus expressed in the notation of the calculus of symbols: if p= a ms m=, § a functional symbol acting on f(7) in such a manner as to convert f(r) into fy—'(yr +1); then pf (=) =Ofr .p, a general law of symbolical combination due to Professor Boole. We will now consider two cases of internal and external division in which the symbols combine according to this law. The results, as will appear afterward, will be found useful in the solution of functional equations. And, first, for internal division. We shall determine the condition that pv, (7) +,(7) may divide p”$,(7) +p” dn—1(7) +. oe es The process will be, mutatis mutandis, the same as in my former memoir. The symbolical quotient is p2-} ont n—2 Qn-17 Wot p Ont . of +p oe a Soe ee: and the required condition is cea by equating the symbolical final re- mainder to zero, and we have | QT bx Yor pom Wor ve 9 Pat — o,7—wW,70 We +W,78 Wis ee te —&.twv, Wee 0 : Pee 6 affecting every part of the term which succeeds it. I shall now give the corresponding condition for an external factor. The symbolical quotient is . Vie ae oe ar} n—2 4 2Q—(m—-1) 1—2 —(1=2) 4, -A—(—2)_—_ =! . yr z! { oe eee yr ae Ga TR a The required condition is found by equating the final remainder to zero; we have Wor Wom WoT py Wor Por=0 oT — Gut +9 ~ Be. + gg HoT g~1 YOR g—1 YO g~1 YoR= 9 As ae Ee ame aaah 6-1 in each term affecting ever ything which comes after it. I conclude with some examples of functional equations. Let the functional equation be f(0) of Fs FO): VOL. XIII. 2 4.40 Mr. W..H. L. Russell on the Calculus of Symbols. this may be written © d f(a) —ae* fe) =F (2), or K(2)= see hh 1 —qede-? Afi fo gy bil ={1 oo ieee Re a C2) =F a ee (2) +aF 7 ate OTe hE To make this solution ee we must add a complementary function, and we have C C, (Va) (— Va)” & +F eta aE ot Ch. eee Ore V1+ ne i poe es As an tpi of te put F(2)=a, and the series becomes Oa an Bee 2 et * Tiga * Vicar 2 vise = ae Mg e (lee ae oe : NV x 0 _ 22 3) @seedp REN a é 1— ae—2202 As a second example, we will take the equation f(=<5) ar ( a) tYO=FO). This equation may be written g,4_ ee Fe die 2h d — (e Be aL ye) Fa (= 22) =y@). and the functional equation resolves itself into the two, (54) -FO=x Now let and. x (se) -x@=FC). which are known forms. Mr, W. H. L. since onthe Calculus of are Add As a last example, we will take the equation ~ 32—2 v+3a—1 .2a—1: es Ue i on otetfe)= F(a); 4 or putting Seis 2 : (e—1) is = 0, wv=n, d ze = hel : ; ¢—— we can write the equation (since p=e *—*) 2n*—7r—3 \ep (==) 4 +m (741) \ u=F(2). Applying the method of divisors, we see that if the symbolical portion of the first member admit of an internal factor, it must be either p—* or p—(r-+1). Now in this case oft )=f(— )p. ll ) Wherefore the meres ¢ Po™ — oer i ~ + bor io ae —&e. +70 = becomes, if we A the factor p—7, and put Yea], syn —m, Cea @(=— 5) - ae ery Hence afte) =f (= Wor Wet Var _ ie aati an identical equation if we put for the symbol 0 its equivalent as given above. . Hence p—z is an internal factor of the symbolical portion of the first member. Effecting the internal division, we have (o—@+1))(p—a flo) =F). (e i f(z) = x(x), and the equation resolves itself into the two, (ep—(7+ 1))x(@) =F@) Let and r (p—m) f(x) = x(2) ; x (= = F(z) and AAA )-A@ =x 2K 2 442 My. Warren De la Rue—Comparison of De la Rue’s {| Recess, forms which I have considered in my memoir on the Calculus of Functions published in the Philosophical Transactions for 1862, in which the general solution of the equations | at+be go) —x(w)g | Eb =I), where ¢ is the unknown function, has been obtained. COMMUNICATIONS RECEIVED SINCE THE END OF THE SESSTON. I. “ Comparison of Mr. De la Rue’s and Padre Secchi’s Hclipse Photographs.” By Warren De ta Ruz, F.R.S. Received August 8, 1864. T have stated, in the Bakerian Lecture read at the Royal Society on April 10, 1862, that the boomerang (prominence E)* was not depicted on Senor Aguilar’s photographs. This is true of the prints which came into my hands in England. A visit to Rome in November 1862, however, afforded an opportunity for the examination of the first prints which had been taken in Spain on the day of the eclipse, previous to those printed off for general distribution by Senor Aguilar. I was agreeably surprised to find that the photograph of the first phase of totality showed not only this prominence very distinctly, but also other details, presently to be described, which were quite invisible in Sefior Aguilar’s copies. I had in fact experi- enced some difficulty in comparing measurements of my photographs with those of Senor Aguilar’s, on account of the indistinctness (woolliness) of the latter, which I have attributed to Padre Secchi’s telescope not having fol- lowed the sun’s motion perfectly. A careful examination of the prints in Padre Secchi’s possession has, however, convinced me that this was not the case during the period of exposure of the first negative ; for I have been able to identify with a magnifier many minute forms which could only have been depicted by the most perfect following of the sun’s apparent motion. For instance, my statement that the prominence H (the fallen tree) was not seen from having been mixed up with the prominence G, is not applicable to Padre Secchi’s copy of the first phase of totality, for in it every detail of the fallen tree can be made out. 7 On expressing to Professor Secchi my surprise at the great discordance between the copy of the first phase of totality sent to me by Seftor Aguilar and that of the same phase in his possession, I was informed that after a few positive prints had been taken from the then unvarnished negative, it was strengthened by the usual photographic process with nitrate of silver. This I look upon as an unfortunate mistake, as the images of the promi- nences were increased and their details hidden, and the beauty of the negative for ever lost. It occurred to Padre Secchi and myself that although there was no hope * See Index Map, Plate XV. Phil. Trans, Part I. 1862. 1864.] and Secchi’s Kelipse Photographs. 443 of procuring more satisfactory prints from the original negative of the first - phase of totality, yet some advantage would arise from taking an enlarged negative from the positive print in his possession, although it could not be expected to yield as perfect an impression as might have been obtained by enlarging from the original photograph. The enlargement has been successfully accomplished in my presence ; and although Professor Secchi will take such means as he may think proper to make known the results of comparisons he may make between my photographs and his own, it will not be out of place for me to add a few remarks by way of appendix to my paper. Taking the prominences in the order in my index map, Plate XV. :— Prominence A (the cauliflower or wheatsheaf) has the same form in Padre Secchi’s photograph as in mine. It extends considerably less in height above the moon’s edge in this copy than in that printed off from the strengthened negative (Senor Aguilar’s copy); the bright points of the two branching streams which issue from the summit towards the North are well depicted in the Secchi photograph, but not the fainter parts. ’ There exists a faint indication of the minute prominence B in the §, photograph. The convolutions of the prominence C (the floating cloud) are seen in the S. photograph, and its form coincides absolutely with that of mine; it is a little nearer the moon’s edge at the point c, probably because the telescope was uncovered relatively a little later than at Rivabellosa. The prominence D cannot be clearly traced in the S. photograph. The boomerang HE is distinctly visible in the 8. photograph ; the point e is apparently prolonged ; but this I attribute to an accidental photographic stain, for the bright part ¢’ can be well made out. The long prominence F cannot be made out in the 8S. photograph, pro- bably from the cause explained in reference to C. The fallen tree (Hi in the S. photograph) corresponds in its minutest details with its picture in my own. The articulated extremity f, the round points h' h”, the point 4”, and the connecting branch joining it with the stem are clearly seen. The prominence G from g_to g’ corresponds precisely in the S. photo- graph with its image in my own, and a dark marking near g also is seen ; the narrow portion of this prominence, from g to the point immediately below A, is not seen in the S. photograph. The prominence I (the mitre) agrees in form in the S. photograph with its image in my own, even the faint point zis there seen. This prominence in the 8. photograph extends further from the edge of the moon than in mine; and whereas in my photograph the convex boundary next the moon is cut off by the moon’s limb, in Padre Secchi’s the convex boundary is complete, and hence in all probability the prominence I presented another case of a floating cloud. About midway between G and I there is a small round prominence visi- AAAs ~~ Prof. Guthrie on Drops. [Recess, ble in the S. photograph not seen in mine, which may be accounted for from our different positions in respect to the central line of the eclipse. Between I and K, at a distance from I equal to about two-thirds the angular interval, there is in the S. photograph a prominence consisting of two round dots, which extend beyond the moon’s limb to precisely the same extent as the prominence K protrudes in Professor Secchi’s photo- graph beyond the moon’s limb in excess of what it does in my own. The prominence K has precisely the same form in every respect in the S. photograph as in mine, so far as mine shows it ; but on account of parallax, more of it is seen in the S. photograph than in mine. Beyond K is another prominence, visible in the S. photograph about 17° distant from K, a small round prominence which could not have been visible from my station. | Of the remaining prominences, L, M,N, O, P, Q R, none were visible at the epoch of the photograph. In conclusion, the photographic images of the prominences, so far as they are common to the two photographs taken at Miranda and Desierto de las. Palmas, accord in their most minute details. The photographs must, from the difference of position of the two stations, have been made at an abso- lute interval of about seven minutes; and this fact, while it strongly sup- ports the conclusion that the protuberances belong to the sun, at the same time shows that there is no change in their form during an interval much. greater than the whole duration of an eclipse. I. “On Drops.” By Freprrick Gururiz, Esq., Professor of Che- mistry and Physics at the Royal College, Mauritius. Communi- cated by Professor Stoxss, Sec. R.S. Received July 15, 1864, In the following investigation, the word drop is used in a rather more definite sense than that which is usually attached to it. In common speech a drop signifies any mass of liquid matter whose form, is visibly influenced towards the spherical by the attraction of its parts, and whose sensible motion or tendency is towards the earth. This definition both includes drops with which we are not here concerned, and excludes others which we shall have to consider; for we shall; have to measure the size of drops; and it can only be of avail to measure the size of ss drops as are formed under fixed and determinable conditions. How many drops, according to the usual scope of the term, are formed under indefinite conditions. For instance, a rain-drop depends for its size’ upon such circumstances as the quantity of vapour at the time and place of its formation, the tranquillity and electrical condition of the air, its rate of motion, the number and size of the drops it meets with in its course, &c., all of which are fortuitous, or, at least, immeasurable conditions. With such drops we have here nothing to do, but only with those which are. formed under fixed circumstances. On the other hand, we e 1864.] Prof. Guthrie on Drops. AAS shall have to consider drops which move upwards*. “Drops of this kind are so seldom. met with that no distinguishing name has been given to them. We shall find it convenient to include them in the Benet term drop, though it may appear at first inapplicable to them. — Without attempting to give an exhaustive definition, it will be sufficient to define a drop as a mass of liquid collected and held together by the attraction of its parts and separated from other matter by the attraction of gravitation. This definition will exclude such drops as those of mist or rain, and will include the upward-moving drops mentioned above. It follows that the size of a drop may depend upon and be influenced by variation in— (1) The self-attraction and cohesion of the drop-generating liquid ; (2) Its adhesion to the matter upon which the drop is formed ; (3) The shape of the matter from which the drop moves ; (4) The physical relation of the medium through which the drop moves, on the one hand, to the liquid of which the drop is formed, and on the other, to the matter on which it is formed ; (5) The attraction of the earth, or gravitation, upon the drop-forming liquid and upon the medium, as influenced by their respective and relative densities, and by variation in the attracting power of the earth. In order to study systematically the influence which each of these factors exerts, each must be varied in succession while the others remain con- stant. Denoting the three states of matter, solid, liquid, and gaseous, by the symbols 8, L, G respectively, and considerimg the symbols in the order in which they are written to denote respectively the matter from which the dropping takes place, the drop and the medium, we get a convenient notation. As we are speaking at present exclusively of liquid drops, L must always hold the middle place in the symbol. Of the eight symbolically possible variations, Bid Baten Q) (2) (3) (4) (5) (6) (7) (8) SLS, SLL, SLG, LLS, LLL, LLG, GLS, GLG, (1), (4), and (7) are physically impossible on account of the superior cohesion of solids over liquids, (6) and (8) are physically impossible on account of the superior density of liquids over gases. SLL, SLG, and LLL are therefore the only cases we have to consider. That is, * Owing to the numerical preponderance of downward-moving drops, we are prone to associate the ideas of “drop” and “down.” How far I may be justified philologically in using the expression ‘‘ drop up,”’ must depend upon the relative primitiveness of the noun and verb “drop.” Once for all, I beg permission to use the term drop in this more extended sense. Of course, in the absence of positive levity, an upward drop can only be caused by the downward motion of the medium in which the drop moves. 446 Prof. Guthrie on Drops. [ Recess, SLL, from a solid a liquid drops through a liquid. SLG, from a solid a liquid drops through a gas. LLL, from a liquid a liquid drops through a liquid. Of these three cases, two, SLL and LLL, may be inverted ; that is, the motion of the drop may be towards or from the earth. ‘The gravitation of the drop may be greater than and overcome the gravitation of the medium, the drop descends; or the gravitation of the medium may over- come that of the drop, the drop ascends. ‘The case SLG cannot be in- verted, because, at all events, at the same pressure every known gas is lighter than every known liquid. It will be convenient to consider the case SLG first, because instances of it come more frequently under our notice than of the other two, and because it will be convenient to consider together those cases which are capable of inversion. ' As we are considering the physical aspect of the question, we will only discuss those cases where no chemical action takes place between the terms, and where either no solution takes place, or where it is so small as to be negligible, or of such a kind as to admit of experimental elimination. This limitation of course excludes a vast number of combinations, but it must be made in order to study the purely physical and definite influences which determine the size of a drop. SLG. From a Solid a se drops through a Gas. The variable factors are 1. The self-attraction and aneaod of the liquid : A. Dependent on its purely chemical constitution. - B. Dependent on the proportion and physical relation between its heterogeneous parts, when a mixture. C. Dependent on temperature. 2. The adhesion between the solid and the liquid: A, B, C asin 1. D. Dependent upon the shape of the solid. 3. The adhesion of the gas to the solid. 4. The adhesion of the gas to the liquid. The factors 3 and 4 may be neglected, as we shall at present only con- sider the case where the gaseous medium is air at the ordinary barometric pressure. One of these factors, namely temperature, though varying in different cases, may be supposed in the same case to be the same for the different kinds of matter present. Another factor in the same predicament is the locally constant gravitation at the place where the dropping occurs. Lastly, — a condition of great influence is the length of the time-interval between the successive drops. This interval we shall call, for brevity, the growth-time, and denote by gt. If the above conditions are exhaustive, we may assert that a drop of 1864.] Prof. Guthrie on Drops. 447 liquid will always be of the same size, if it is formed of the same liquid substance and falls from a solid of the same substance, size, and shape, provided that the temperature remain the same, and the growth-time be constant. The size of the drops may be most conveniently determined by weighing a noted number of them. We are concerned rather with the relative than with the absolute sizes of the drops. The sizes of drops formed of the same liquid are proportional to their weights; of different liquids, to those weights divided by the specific gravities of the liquids. In the first series of experiments the apparatus, fig. 1, was employed. The globe A, full of the liquid under experiment, is inverted into the cylinder B, containing the same. The mouth of A is supported just in contact with the surface of the liquid in B, by means of the tripod stand D. A and B are carried on a table, which may be raised or lowered at pleasure. A siphon, E, leads from the reservoir B, and is firmly held by the clamp F, The longer limb of E, from which the liquid flows, is ee 448 Prof, Guthrie on Drops. [Recess, turned up at the end, and touches a plug of cotton wool at G. The sphere H, from which the dropping takes place, is hung by three thin wires from the ring of a retort-stand. The upper half of the sphere is clothed in cotton wool, which reaches up to the plug at G. The whole arrangement is placed upon a separate table from that which supports the balance, so as to avoid the vibration caused by opening and shutting the balance case. The drops which fall from H enter the funnel L, whose lower end is somewhat bent, so that the drops are thrown out of the ver- tical, and all upward splashing avoided. The rapidity of the flow through the siphon, and consequent dropping from H, is regulated by raising or lowering the table C. The vessel A acts as a regulator for keeping the level of the liquid in B at a constant height. The first series of experiments was made with the double object of determining how far the rapidity of dropping influenced the size of the drops, and to establish the uniformity of the size of the drops which drop at equal intervals of time. In these experiments cocoa-nut oil was taken as the liquid, an ivory sphere as the solid, and atmospheric air as the gas. The ivory sphere was washed in hydrochloric acid, so as to deaden its surface. Immediately before and after each batch of drops, the same number of drops were counted, and their time of fallmg compared with the time which elapsed in the actual experiment. In no case, however, was there a difference between the two of a single second, so that g¢ may be considered in each case to be exactly given. TasiE 1.—Cocoa-nut oil. T =28°:5 C. ge— Al Radius of ivory sphere = 22:1 millims. Numb tae gt. Weight of drops. Mu gramme. 60 1 3-9817 60 1 3984] ; 60 1 3-9784 | 60 1 3°9742 | bee alge 3:9730 | | 60 1 3°9735 60 1 3-9682 gt. | Mean weight of single drop. a” 0:066279 Preliminary experiments having shown that the size of a drop is greatly affected by the rate at which the dropping takes place, that is, by the time occupied by the drop in its formation, the following experiments were per- formed to establish the connexion between the two. It may be here remarked that with some liquids, of which cocoa-nut oil 1864.] Prof. Guthrie on Drops. 449 is one, a continuous stream of liquid by no means implies a faster delivery of it than may be achieved by a succession of drops. On the contrary, just as by walking more rapid progress may be made than by running, so may dropping deliver more liquid than passes ina stream. A uniformly rapid series of drops may be converted into a stream, and reconverted into drops under certain restrictions, at pleasure, without altering the quatiniy of liquid delivered. We shall return to this point. TaBLE IT.—Cocoa-nut oil. devas Soy ad Radius of ivory sphere =22°1 millims. | Time between fall : | Time between fall ‘ Te ber of first but one and Weight Number of first but one and Weight of drops. last drop. of drops. || of drops. last drop, of drops. . gramme. || 7 gramme. 60 26 45212 || 60 38 4:3678 60° 26 45173 60 38 43628 60 26 45265 || 60 38 43682 60 26 45316 120 76 87403 60 | 30 43676 || 60 38 43646 60 30 43668 || 60 42 4-23.42 60 30 | 43593 60 | 42 4:2357 60 30 43665 || 60 42 42362 60 34 4°4827 60 42 4:2368 60 34 44731 60.6. 42 4-2330 60 34 44643 60 | 42 4-9378 60 34 44779 60 46 41487 60 | 34 44681 || 60 | 46 4-1438 60 | 34 44752 60 | 46 4:1499 60 | 38 43778 | 60 | 46 4147) ! From this Table is constructed the followmg Table III., which shows g# in seconds and the corresponding drop-weights in grammes, the latter values being the mean of the resultsin Table II. g# is got by dividing the time-lapses of Table II. by the number of drops. TaBLeE III.—Cocoa-nut oil. ‘E=28?*7 Radius of ivory sphere = 22°] millims. ‘ Mean weight of y single drop. j | grm. 0-433 | 0:07540 0-500 0:07275 | 0-567 0°07456 | 0-633 0:07281 | 0-700 0-07059 0°767 0:06912 | 1-000 0:06628* * Table I. T=2875 C. 450 Prof, Guthrie on Drops. [ Recess, Hence it appears that, within the above limits, on the whole, the weight or size of a drop diminishes as its growth-time increases. Further, it seems that between the rates gi =°433 and gt=°567 a minimum occurs, that is, instead of there being a continuous diminution in the weight as the growth- time increases, there is at first a diminution, then an increase, and finally a continuous diminution, so that drops of the rate gf=*500 have sensibly the same size as those of the rate gé="633. In order to establish more precisely the position of this minimum and the general relation between rate and size, the observations must be both more minute and more extended. For this purpose a fresh sample of oil was taken, and the time-intervals extended from 25" per 60 drops to 240" per 20 drops; as before, four experiments were made at each time-interval. The mean results are given in Table IV.*, in which the values of gé are obtained by dividing the time-intervals by the number of drops. The mean weights of the single drops are got as in Table III. The weights of oil passing in one second are found by dividing the terms of column 2 by those of column 1, which correspond to them. TaBLe 1V.—Cocoa-nut oil (specific gravity =0°9195). T=29° C.—29°4 C. Radius of ivory sphere=22°1 millims. 1 | Mean weight Weight of oil es of single drop. |passing per second. hy grm. erm. (0°333) (009264) (0:27792) 0°417 0:08265 0°19837 0:433 0:08074 0°186381 0-450 0:08185 0:18189 0467 0-07918 0:16968 0°483 0:07932 0-16412 0:500 0:08017 0:16035 0°517 0:08017 0:15518 0533 0-:07961 0:14927 0:550 0:07698 0:13985 0-567 0:07664 0:138524 0-583 0:07558 0°12957 0600 0:07334 0:12221 0-617 0 07820 0:11871 0633 0:07821 0°11560 0-667 007260 0:10891] 0-750 0:07102 0:09469 0°833 0:06902 0:08283 1-000 0:06605 0-06605 1:500 006215 004144 2-000 0:05986 0:02993 3000 0:05710 0-01903 4°000 0:05561 0:01432 5000 0:05469 0:01094 12-000 0 05201 0:00433 * A Table exhibiting the details is given in the MS., which is preserved for reference in the Archives. x 1864. } Prof, Guthrie on Drops. 451 It was found impossible to arrest an exact number of drops when the rate was faster than 60 drops in 25". A few rather discordant results, got at the rate of 60 drops in 20", gave a mean of 0°09264 grm. as the weight of a single drop; this tends to show that at this high rate the drops were considerably larger than at any lower rate. Towards the end of the Table, at the slower rates, the error of time be- comes so exaggerated (the least alteration in the adjustment of the instru- ment makes so sensible a change in the entire time-lapse) that it is nearly impossible to avoid an error of about 0”*5 in the whole time of several minutes. Although the time-error thus becomes palpable, it nevertheless remains, relatively to the whole time-lapse, as immaterially small as the inappreciable errors of the swifter rates of dropping. The numbers of Table IV. present us with several interesting and im- portant facts. From gf='333 to gé= 433 there is diminution. ss 9 == 433.5. 5,—= “490 5, __,, mMerease. wees a AD, oo ag CIINTL IOI, 33 oe) =°*467 3) ae °900 3 -, Icrease. Be a "00 7g. 55. 12-000- ,, ,5,continual dimination. The most prominent fact is that, on the whole, the drops undergo a continuous diminution in weight or size as gé increases. To such an extent is this the case, that the most rapidly falling drops of the above Table are nearly twice as heavy as the most slowly falling ones. The cause of this is probably to be sought for in the circumstance that when the flowing to the solid is more slow, the latter is covered with a thinner film of liquid, so that, as the drop parts, the solid reclaims by adhesion more of the root of the drop than is the case when the adhesion of the solid to the liquid can satisfy itself from the thicker film which surrounds the drop ‘in the case of a more rapid flow. The influence of rate is seen to extend even to the exceedingly slow rate of gf=12". This connexion between rate and weight (or quantity) should not be lost sight of by prescribers and dispensers of medicine. A pharmacist who administers 100 drops of a liquid drug at the rate of three drops per second, may give half as much again, as one who measures the same number at the rate of one drop in two seconds, and so on. For our present purpose the effect of rate upon the size of a drop is of great moment, because it proves that there is no such thing as a drop of normal size. At no degree of slowness of dropping do drops assume a size unaffected by even a slight change in the rate of their sequence. Hence, whenever a comparison has to be made between the sizes of different drops, we shall have to eliminate this source of difference by taking drops which follow at exactly the same rate. About the rate at which the diminution of size takes place for equal increments of gf, the Table gives us little information beyond the fact that, 452 Prof. Guthrie on Drops. [Recess, on the whole, the sizes of the drops at the slower rates are less influenced by equal increments of g¢ than are those of the quicker rates. This, how- ever, ouly appears distinctly at and below the rate of about gé=1"-00. _ If the connexion between gt and the drop-size be represented by a curve (fig. 2, A), the abscissee being the values of gt, and the ordinates the corresponding drop-weights, there is apparently no asymptote parallel to the axis of X. The curve presents, however, in its course two cecmnane maxima and minima: ‘Secondary maxima. Secondary minima. (1) “gt “= "450" ot="435a" = 500! | (2) at} Le pit , gt= "467" Although at these minima the drops are less than at the immediately succeeding rates, yet the quantity of liquid passed in a given time is, at every rate of dropping, greater than the quantity passed in the same time at every slower rate. The decrease of rate more than counterbalances the temporary increase in the drop-size. This is seen on comparing the num- bers of column 3, Table IV., with one another. They are found to de- crease continually, though by no méans uniformly, as the rate of dropping decreases.. ‘The same fact is shown graphically in fig. 2, B. The second maximum (at gf ='500 and gt = 517) is in remarkable connexion with the rate at which a series of drops may be converted into a continuous stream. At all rates of dropping, from gt=°333 to gt='517 inclusive, the drops may be converted into a permanent stream by pouring a little additional oil upon the sphere as the drops are falling from it. A stream is thus established which remains for any length of time, if it be protected from all currents of air and vibration. At the rate gf=+519 the stream may be established by the same means for a few seconds (about 30"), but the continuous part inevitably begins to palpitate, becoming alternately longer and shorter, thinner and thicker, until at last it draws up and is converted into a succession of drops. At the immediately slower rates of dropping the same effect follows, but in each case in a shorter time, so that the slowest rate of dropping, which may be converted into permanent running, coincides with the rate which gives the second maxi- mum size of drops (gé=°500 and gt=°517). ‘The appearance of a drop- convertible stream is peculiar, the narrowing which it undergoes on leaving the solid beg remarkably sudden. iad #In many liquids such secondary maxima are entirely wanting. They — appear in liquids of the physical nature of oils, whether those oils be che- mically fatty (adipic salt’of glycerine), or whether they be miscible with water, as syrups, glycerine itself, &c. In order to avoid the influence of variations in rate, we shall for the future take the same rate of dropping in all cases, and, unless the contrary be stated, the rate adopted will be gt=2". 1864. | Prof. Guthrie on Drops. 453. _ The factor, the influence of whose variation on the size of the drop we have next to consider, is the constitution of the liquid.of which the drop is formed. For the foregoing experiments concerning the influence of rate, cocoa-nut oil was employed on account of its non-volatility. On allowing a quantity of it, having an exposed surface of about two square inches, to stand for 70 hours, it was found to have increased about 2 milli- grammes in weight, probably in consequence of oxidation, Its fixedness, therefore, and its perfect liquidness at the temperature of 28°-30° C., make it well adapted for this special purpose. Chemically. and physically, how- ever, it is of little interest for our immediate purpose, because it is a mix- ture of several substances, the proportion between which is indefinite. The constitution of a ‘liquid may vary in two ways. A liquid may be a mixture of two or more simple liquids, or a solution of one or more solids in a single or mixed liquid; or secondly, the liquid being single, may vary in the sense of its chemical constitution. It would be clearly impossible to exhaust experimentally the countless variations which might thus arise. We must be satisfied with taking a few simple examples of the two cases. : 2 With the more mobile liquids the apparatus, fig. 1, fails to give a strictly uniform flow. As the liquid descends in B, it adheres by capillary action to the lip of A for some time after the level of B is below the lip. The air at last separates the two, enters the flask A, displaces the liquid there, and restores the level to B, so that although the average height of B is constant, yet it undergoes a series of slight but ceaseless variations. As even such slight irregularities sensibly affect the rate of flow through the siphon, and consequently the rate of dropping from the sphere, the apparatus is slightly modified as follows, fig. 3. Between the reservoir, B, fig. 1, and the dripping sphere, a second reservoir, M, is placed. This is kept in a state of continual overflow. The overflow is regulated by means of a few filaments of cotton wool hanging over the edge of the overflowing vessel, and so fashioned that the end in the overflowing vessel tapers to a point. Finally, the rate of flow is in many instances so sensitive, that it is impossible to procure exactly a predetermined rate by the ordinary screw-adjustment of the holder which carries the siphon. For the final adjustment, it is convenient to depend upon the elasticity of the siphon. A heavy ring is passed over the siphon, which is then firmly fixed so: as to deliver the liquid at nearly the required rate. The ring slipped back- wards and forwards, bends the siphon more or less, and regulates the flow through it. Solution of Chloride of Calcium in water.—A. solution of chloride of calcium, nearly saturated at 28° C., was taken as the starting-point or solution of maximum saline contents. Half of this solution was mixed with an equal volume of water (solution 2). Half of solution 2 was mixed with its own volume of water, giving solution 3, and so on. In this manner, without knowing the absolute strength of solution 1, we know. 45 4: Prof. Guthrie on Drops. : [ Recess, that the successive strengths of the saline solutions, whether there be loss S 8 Pee ey These numbers give exactly the relative quantity of solid matter in a unit of volume of the liquid. As, however, solution 1 on dilution evolves heat and therefore probably contracts, the sizes of the drops cannot be derived directly from their weights. The specific gravity of each solution has to be determined experimentally. TasLe V.—Solutions of Ca Cl. of volume owing to chemical union or not, areas s, j6=—2 Radius of ivory sphere=22°1 millims. T= 28°. Solution; Mean weight | Specific | Relative size of CaCl.| of single drop*.| gravity. | of single drop. Wateror; stm. 2 0185166 | 1-0000| 0-18517 = 0:168137 | 1-0039 016750 = 0172907 | 1.0084 0-17147 5 0:172593 | 1-0172 0-16967 = 0:167222 | 1.0383 0-16105 - 0:191008 | 1-0720 0:17817 : 0195839 11721 0-16742 2 0-211396 1:2786 0-16533 S 0:225558 | 1-4939 015098 The column of the relative sizes of the single drops (which is got by dividing the mean weights by the corresponding specific gravities) shows that, under like conditions, a drop of water is larger than a drop of solution of chloride of calcium of any strength whatever. The comparatively small quantity of solid matter in me causes the drop to diminish about 3th of its volume. We must bear in mind that the successive increments of solid matter may affect the size of the drop in opposite directions,—by affecting the * The first number from six, the following numbers from four determinations of the weight of 30 drops. 1864..] Prof. Guthrie on Drops. 455 cohesion of the water, by asserting its own cohesion, by increasing the gravity of the liquid and thereby determining an earlier separation of the drop, and, in this particular case, by the chemical affinity of the solid to the liquid, and the probable formation of hydrates. It is seen that these influences cause an irregularity in the diminution of the size of the drop as it acquires more solid matter. In fact, it is only when the liquid has the considerable strength of 3 that the diminution in drop-sizé becomes continuous. In fig. 2, C shows graphically the relation between drop-size and strength. The abscissee represent the strengths of the solution progressing in geometric ratio; the ordinates show the corresponding comparative drop-sizes. It may be remarked that the curve C bears a striking resemblance to the curve A, as though increase in solid constituent produced a similar effect upon the drop-size as increase in the time-interval on the drops of a homogeneous liquid. We may also notice the great difference in size between a drop of water and a drop of oil under the same conditions. From Table IV. we find that a drop of oil of specific gravity 9195 has the weight 05986 when gt=2". Hence the comparative sizes of the two are,— — } Radius of T Comparative ui: sphere. i sizes. Water 2! 22°1 mm. 237... 0°18517 Oil oe 22°] mm. 29°-29°°4 C. 0:06510 Or a drop of water is nearly three times as large as the drop of oil, the only difference in the circumstances being that the oil was 1°-1%4 C, warmer. We shall have to study this point more especially hereafter. On account of the chemical union which takes place on dissolving Ca Cl in water, it would be useless to give the absolute strengths of the various solutions. VOL. XIII. 21 a a 456 . Prof. Guthrie on Drops. [Reces, Solution of Nitrate of Potash in water.—Nitrate of potash was the next solid examined, on account of the probable non-existence of hydrates. Seven solutions of nitrate of potash were made of the following eye by weight :— : (1) 22 of water to 1 of nitrate of potash. (2) 99 9 29 2 29 29 9 - (3) cy) De DS 3 93 9 oy) (4) oy) 29 29 4 239 29 29 (5) ry) oy) 23 5) 9 29 ” (6) 29 99 99 6 29 39 23 (7) 99 99 9 7 29 99 33 These solutions were made to drop from the ivory sphere at the rate of gt=2", Ineach instance four batches of drops, of 30 each, were weighed. In the following Table the mean results only are given. TaBLe VI.—Solutions of Nitrate of Potash. Gi—2 T =28° C. Radius of sphere =22°1 millims. 1. 2. 2 4, 5. a Mean weight | Specific gravity | Relative size | Weight of KNO, of single drop. | by experiment. | of single drop. in a drop. KNO, : Water = 0:18517 1-0000 0:18517 -00000 = 0-18613 1-0164 0:18314 00846 = 0:17908 10341 0:17318 01628 = 0717714 10511 016853 02411 = 0-16917 1-0680 0-15840 03075 — 0:17805 1:0832 0-16439 04047 22 018254 1:0987 0:16618 04978 6 22 0-18611 1/1130 0°16723 05921 7 Hence it appears that on the addition of the first quantities of nitre 22 22 22 22 22 22 22 (=, elec =) the size of the drop is diminished. Afterwards (= 5” rae =) 1864. | Prof. Guthrie on Drops. 457 the size of the drop is partially recovered. There is a stage of dilution when the specific gravity is 1-0680, where the drop-size is a minimum. _ Further, it is seen from column 5 that the quantity of nitre in a drop in- creases continually as the strength of the solution increases, although both the weight and the volume of the drop vary. Inversely, the regularity of the variation of drop-size, in the case of nitre, points to the absence of hydrates of that body. It would be delusive to endeavour to construct a formula connecting the specific gravity with the drop-size or drop-weight of the solution, but, as before, a graphic representation serves to show the connexion between the variables. In curve D, fig. 2, the abscissee represent the quantity of nitrate of potash in solution, the ordinates show the corresponding drop- sizes. As with chloride of calcium, it is seen that the drop-size of water is larger than that of any solution of nitre. Curve H, fig. 2, having the same abscissz as D, has ordinates which represent the drop-weights. It is confessedly a matter of great interest, and still greater difficulty, to determine exactly the relation which exists between a dissolved solid and its solvent; that is, to find out whether or when a solid should be viewed as being in combination with a portion of the liquid in which it is dissolved. Such questions may perhaps receive additional light from experiments similar to the above, but more extensive, and performed with this special object in view. Comparing the curves C and D, for instance, there can be little doubt that the secondary maxima and minima of C are owing to the existence of hydrates of chloride of calcium in solution. The only known hydrates of chloride of calcium are Ca Cl, 2HO and Ca Cl, 6 HO, the latter of which contains 50-7 per cent. of CaCl. Solution S contains about 42°5 per cent. It is noteworthy that, while the six-water chloride in the solid state absorbs heat on solution, the solution S evolves heat on dilution, as already mentioned. In the case of nitre we have in the drop-sizes evidence only of the opposite efforts of two cohesions, that of the water and that of the nitre. By pursuing this direction of experimental inquiry, evidence may probably be got concerning the truth of Berthollet’s hypothesis of reciprocal recomposition in the case of the mixture of the solutions of two salts, AX and BY, where AY and BX are also soluble in water. frm ie Peasy: - ® a » » + a see So ae “sere Se 1864. ] Prof. Guthrie on Drops. 457 the size of the drop is partially recovered. There is a stage of dilution when the specific gravity is 1:0680, where the drop-size is a minimum. Further, it is seen from column 5 that the quantity of nitre in a drop in- creases continually as the strength of the solution increases, although both the weight and the volume of the drop vary. Inversely, the regularity of the variation of drop-size, in the case of nitre, points to the absence of hydrates of that body. It would be delusive to endeavour to construct a formula connecting the specific gravity with the drop-size or drop-weight of the solution; but, as before, a graphic representation serves to show the connexion between the variables. In curve D, fig. 2, the abscissee represent the quantity of nitrate of potash in solution, the ordinates show the corresponding drop- sizes. As with chloride of calcium, it is seen that the drop-size of water is larger than that of any solution of nitre. Curve H, fig. 2, having the same abscissze as D, has ordinates which represent the drop-weights. It is confessedly a matter of great interest, and still greater difficulty, to determine exactly the relation which exists between a dissolved solid and its solvent —that is, to fnd out whether or when a solid should be viewed as being in combination with a portion of the liquid in which it is dissolved. Such questions may perhaps receive additional light from experiments similar to the above, but more extensive, and performed with this special object in view. Comparing the curves C and D, for instance, there can be little doubt that the secondary maxima and minima of C are owing to the existence of hydrates of chloride of calcium in solution. The only known hydrates of chloride of calcium are Ca Cl, 2HO and Ca Cl, 6 HO, the latter of which contains 50°7 per cent. of CaCl. Solution S contains about 42°5 per cent. It is noteworthy that, while the six-water chloride in the solid state absorbs heat on solution, the solution S evolves heat on dilution, as already mentioned. In the case of nitre we have in the drop-sizes evidence only of the opposite efforts of two cohesions, that of the water and that of the nitre. By pursuing this direction of experimental inquiry, evidence may probably be got concerning the truth of Berthollet’s hypothesis of reciprocal recomposition in the case of the mixture of the solutions of two salts, AX and BY, where AY and BX are also soluble in water. III. “On Drops.’—Part If. By Frepverick Guturiz, Esq., Pro- fessor of Chemistry and Physics at the Royal College, Mauritius. Communicated by Professor Stokes, Sec.R.S. Received October 17, 1864. We have next to consider the influence which variation in the chemical nature of the drop-forming liquid may exercise upon the drop-size in the case SLG. The liquids which were selected for this purpose were chosen as being VOL, XIII, 2M 458. Prof. Guthrie on ‘Drops. [ Recess, typical of extensive classes, rather than as being connected mee one another in immediate chemical relation. They were— Water. | Oil of turpentine (turpentol). Alcohol. ~~. Benzol. Acetic acid.’ Glycerine. Acetic ether. Mercury. Butyric acid. These several liquids were allowed to drop under the same conditions, from the bottom of a hemispherical platinum cup. The arrangement of the apparatus was quite similar to that described in Part I., the ivory ball being replaced by the platinum cup, and the overflow of the cup being deter- mined by strips of paper bent over its edge. The case of mercury is the only one which requires some explanation. A few years ago I noticed the fact that mercury which holds even a very little sodium in solution has the power of “ wetting” platinum in a very remarkable manner. The appearance of the platinum is quite similar to that presented by amalga- mable metals in contact with mercury. But the platinum is in no wise attacked. Further, the amalgam may be washed off by clean mercury, and the latter will also continue to adhere equally closely to the platinum. All the phenomena of capillarity are presented between the two. The surface of the mercury ina platinum cup so prepared is quite concave; and a basin of mercury may be emptied if a few strips of similarly prepared platinum foil be laid over its edge—just as a basin of water may be emptied. by strips of paper or cloth, and under the same condition, namely that the _ external limb of such capillary siphon be longer than the internal one. I generally use this curious property of sodium-amalgam for cleaning platinum vessels. It enables us now to examine the size of drops of mer- cury under conditions similar to those which obtain in the case of other liquids*. After the cup had been used for the other liquids, its surface * In regard to the above-mentioned property of sodium, the following observations may be of interest. At first the explanation naturally suggests itself, that the effect wrought by the sodium may be due to an absorption of oxygen, in consequence of the oxidation of the sodium, the consequent diminution of the gaseous film between the two metals, and the resulting excess in the superior pressure of the air. This, however, cannot be the true explanation, because it is found that the perfect contact between the two, or “ wetting,” takes place equally well in an atmosphere of nitrogen, carbonic acid, or in vacuo. Hence, if I may venture upon a guess, unsupported by experimental evi- dence, I should be rather disposed to assign the phenomenon to the reducing action of nascent hydrogen derived from the contact of sodium with traces of water. Perhaps — even the least oxidizable metals are covered with a thin film of oxide, which is reduced by the nascent hydrogen at the same moment that the mercury is presented to the re- duced metal. It is found that iron, copper, bismuth, and antimony are also wetted by mercury if their surfaces are first touched with sodium amalgam. Not only do -the latter metals lose this power on being heated (as we might expect, in consequence of their superficial oxidation), but platinum, from which the adhering mercury film has been wiped by the cleanest cloth, or from which it has been driven by heat, also loses the power. It is true that the surface of clean platinum is supposed to condense a 1864.] | Prof. Guthrie on Drops. 459: was rubbed with sodium-amalgam and washed with clean mercury. A few strips of similarly prepared platinum foil being bent over the edge and pressed close to the sides of the cup: the mercury could be handled simi- gee to the other liquids. The following Table VII. shows,— 1. The liquids examined. 2. The number of drops which were weighed. 3. The weights found. 4. The mean weights of single drops. 5. The observed specific gravity at the given temperature. 6. The relative sizes of single drops. Tasxe VII. 2 0, gt=2" Radius of curvature of platinum cup=11°4 millims. 1. 2, 3. 4, b. 6. Name and formula] Number | Weight | Mean weight of | Specific | Relative size of of liquid. of drops.| of drops.| single drop. | gravity. | single drop. erm. grm 20. | 2:9703 20 | 2:9923 ‘AGE Sh a Ae ae ne 20 | 29472 0:14828 1:0000 0°14828 HO. 20 | 29603 20 | 29533 20 | 2:5496 Glycerine ......... 20 | 2:5576 | 0:12804 1:2452 0:10280 C, H, O,. 10 | 1-2877 20 | 1:1616 Butyric acid ...... 20 | 1°1630 | 0:05813 1:0017 005803 , H, O,. 20 | 11634 20 | 7:9655 Meroury ....+sseses. Br Noeeee i. Overos. |1eare8. |) won7ee 8 20 | 7:8197 20 | 0°9514 Benzol ...sesssee a een +| 04778 | 08645 | 0-05527 ei | 579 12 “"6" 20 | 0-9644. 20 |0°8675 é Turpentol ......... 20 | 0:8656 | 0:04331 0:8634 0:05016 ia 20 | 0:8653 20 |0°7890 J G5) 10) ree 20 |0°7910 | 003949 08163 0:04960 6, H,.0, 20 | 07896 20 | 08214 Acetic-ether .....: 20 | 0:8300 | 0:04149 08930 - 0:04647 C,H, 0,C,H,0,.) [20 |0-8384 30 | 1:3636 Acetic acid ....:.... 20 |0°9055 | 0:04540 1:0552 0:04302 HO C,H, 0,. 20 | 0:9095 | film of oxygen ; and the removal of this might alter the adhesion between the mercury and platinum ; but such a film could scarcely exist 7 vacuo or in another gas, ved 2m 2 460 Prof. Guthrie on Drops. [ Recess, The experimental numbers obtained are given without omission. The liquids are arranged in the order of magnitude of their drop-sizes. It appears from column 5 (of the specific gravities) that some of the liquids employed were not perfectly pure. This, however, is quite immaterial in the present direction of examination, provided that in all cases where the liquids named are in future employed and compared with those of Table VII., identically the same liquids are meant. The numbers of column 6, with which we are now exclusively concerned, present several points of great interest. In the first place, it appears that the specific gravity of a liquid is not by any means the most powerful de- terminant of the drop-size. Thus-butyric acid, which has sensibly the same specific gravity as water, gives rise to a drop less than half the size of the water-drop ; while mercury, of singular specific gravity, has no ex- ceptional drop-size. Lastly, it may be observed how that remarkable body water asserts here again its preeminence. ‘The first impression which these numbers make is, that there are three groups of magnitude, n, 2 n, 3n. But it is possible that a change in the nature of the solid might throw these drop-sizes into a different order of magnitude; and certainly until a very much greater number of bodies is examined in this sense, it would be premature to attempt to establish anything like a law. It is sufficient for the present to poimt out that the drop-size is not directly dependent upon either the specific gravity or boiling-point ; nor does it stand in any obvious relation to what is sometimes called the liqui- dity, mobility, or thinness of a liquid. For we find that glycerine and (from former experiments) cocoa-nut oil both form smaller drops than water, the one being heavier and the other lighter than that body, and both being viscid or sluggish. On the other hand, alcohol and acetic acid, both perfectly mobile liquids, give rise to drops about half as large as those of glycerine*. Hence it is clear that we are still ignorant of that property of a liquid upon which its drop-size mainly depends. We are not yet in a position to connect the drop-size with any of the known physical or chemical pro- perties of liquids. We approach the solution of the problem by studying the effects of change in some others of the variables. The adhesion between the liquid which drops and the solid from which it drops is also affected by the curvature and general geometric distribution of the solid at and about its lowest point. And the variation in the adhe- sion between the solid and liquid, caused by the variation in the geometric distribution of the solid, may and does in its turn affect the size of the drop. Yd From this aspect, one of the simplest kinds of variation is that offered * The evaporation of the more volatile of these liquids is a source of slight error ; not so much on account of the direct loss in weight of the drop in falling, as by reason of the cooling which it causes, and the consequent variation in density and adhesion. Such source of variation we shall examine in the sequel, and find insignificant. 1864. | Prof. Guthrie on Drops. 46] by a system of spheres of various radii, but made of the same material. And this case is an important one, because it undoubtedly offers the key to all drop-size variation arising from a similar cause. ‘To study this point we may make use of any one convenient liquid, such as water, and cause it to drop at a fixed rate from spheres of various radii, including the ex- treme case of a horizontal plane. This extreme case, however, presents certain practical difficulties. From a plane it is almost impossible to get a series of drops uniform in growth-time and in position. A ripe drop hanging from a horizontal plane will seek the edge thereof. Several drops may form upon and fall from the same plate at the same time and inde- pendently of one another. It is only by employing a plate not absolutely flat, that an approximation to the required conditions can be made. Taking ry for the radius of curvature, the first numbers for 7 =o can therefore be considered only as an approximation. The arrangements for the other cases were quite similar to that described in Part I., fig. 3. No. 1. A glass plate, fastened to and held by a vertical rod. Nos. 2, 3, 4. Selected globular glass flasks. Nos. 5, 6, &c. Perfectly spherical glass spheres. TaBLe VIII.—Water. ge 2) T—22°5 ©. ae | 9, 3, 4, F Number} Radius of | Weight Mean weight and relative size of drops.| curvature. | of drops. of single drop. ——— erm. 5.229 1. {50} a { ae \ 026549 ° 20 mm. 49296 . 2, 130} 1131 { oie \ 0:24808 3. { a \ 70-1 | 45018 1) 022619 4 | {So} |] 472 |{a5o4q}| 021257 20 | £35055 . Be { 50} 175 | { 34733 \ 0-17497 6. {50} 15-1 {33600 | 0-16765 m7. {30 | 115 { 3.0206 \ 0:15122 8. { 5 \ 11-2 { ae \ 0:14896 20 Py | (9-8665 a. 9. { 50 10:0 | 5-8619 | 0-14321 20 29-6765 So 10. | 30} TD { 26660 \ 0:13356 . 7) 1, {*o} ral 111001 | 0:12877 It appears, therefore, that the drop increases in size according as the radius 462 Prof. Guthrie on Drops. [Recess, of the sphere increases from which the drop falls, and, further, that the difference of drop-size brought about by this cause alone may easily amount to half the largest drop-size. For dispensers of medicine this fact is as im- portant as that pointed out in Part I., where it was shown that the growth- time so materially influenced the drop-size. The lip of a bottle from which a drop falls is usually annuloid. The amount of solid in contact with the dropping liquid is determined by the size of two diameters, one measuring the width of the rim of the neck, the other the thickness of that rim. In most cases the curvature and massing of the solid at the point whence the liquid drops is so irregular as not to admit of any mathematical expression. The reason why drops which fall from surfaces of greater curvature are larger than those which fall from surfaces of less curvature is surely this :—In the case of a surface of greater curvature the base of the drop has more nearly its maximum size; the centre of gravity of the liquid film from which the drop hangs is nearer to the centre of gravity of the hang- ing drop ; the contact between the two is more extensive and intimate ; so that the drop is held for a longer time and therefore grows more. On comparing columns 3 and 5 of Table VIII., there does not appear to be any obvious law of connexion between the two; nor indeed can the numbers of column 4 pretend to such a degree of accuracy as would justify us in attempting to establish one. This is seen on comparing znter se the numbers of column 4. Especially with the spheres of longer radu, there is so much difficulty in getting a uniform wetting of the surface whence the drop falls, and this so materially influences the drop-size, that the numbers found are seen to vary considerably. Greater accord is obtained with spheres of less radii. As we might expect, the same absolute increase in length of radius takes less effect upon the drop-size in the case of longer than in that of shorter radii. The infinite, or at least indefinitely great difference between the radii 1 and 2 produces about the same effect upon the drop-size as the difference of 43 millims. between the radii 2 and 3, and so on. | The following Table of first differences shows this more strikingly :— Trti—Tn Wyti1—Wn. | (oe) 0-01854 43° 0:02189 22'9 0:01862 | 29-7 0:03760 2-4 0:00732 36 0:01643 0:3 0:00226 1:2 0:00575 2°5 0:00965 0-4 0:00479 The relation exhibited in this Table supports the supposition that the size of the drop varies inversely as the contents of a figure bounded below by a circular horizontal plane of constant diameter (less than that of the 1864.] - . Prof. Guthrie on Drops. 468 sphere) tangent to the sphere, laterally by a cylinder of vertical axis stand- ing on the tangent plane and cutting the sphere, and above by es convex surface of the sphere itself (Plate IV. fig. 4). As the diameter of the sphere still fabithiee diminishes, the size of the drop is limited by the possible size of its base, until finally the sphere is completely included in the drop. It would be interesting, but it would take us too far, to consider the vari- ous cases of liquids dropping from cones, edges, solid angles, cylinders, rings, &c. We must content ourselves in this direction with the fact that the size of a drop is greater the more nearly plane is the surface from which the dropping takes place. If it were possible for a drop to fall from a concave surface, we should anticipate a still further increase in its size. The relation between drop-size and curvature may be more strikingly shown by arranging the spheres one above the other in the order of mag- nitude. Plate IV. fig. 5.—Each sphere receives the drops from the higher one. The quantity of water which drops in a given time, from every sphere, is the same. Hence in all cases the number of drops is inversely as the drop- TABLE [X.—Water. er TJ GC, E, 2. 3. 4, Radius | Number| Weight Mean weight i nd relat: of disk. | of drops.| of drops. a ees in. orm. (20 | 33682 5 | 90 | 31193 | 5 90 | 325934 | 0-16895 20 20 | 3:3256 | 20 | 32594) ; 20 | 2-9693 | 90 | 29-9854 0 20 a O'14915 20 | 30031 : 20 | 1-9333 90 | 1-9244 0 20 A 009666 90 | 1-9248 : 20 | 1-4618 | 20 | 1-4672 0 20 | 1-4688 007332 | 20 | 1-4682 | : 20 | 0:8250 | 20 | 0:8212 me | 50 90 | 08208 0-04107 20 | 0:8190 size; so that by counting the number of drops which fall from any two spheres in the same time, we get at once the relative sizes of the respective nnn nen a 464: Prof. Guthrie on Drops. [ Recess, drops. For several reasons, this plan of comparison is not sufficiently ac- curate to measure drop-sizes ; but it offers a method of making the differ- ence of drop-size visible to any number of persons at once. The only other variation in the geometrical relation between the solid and the liquid, which we shall consider, is the variation in the size of a cir- cular horizontal plane from which drops fall. Five disks of copper foil were cut of the radii 35, 35, 3 o> soth of an inch respectively. These were fastened horizontally to vertical wires, and, having been thoroughly cleaned by momentary immersion in nitric acid and washing, water was made to drop from them at the rate gf= 2". Table IX. shows the influence of this kind of variation upon drop-size. The want of accord in the numbers of the largest disk is owing to a pecu- liar tremor which the drops exhibit at the moment of delivery. The same phenomenon was noticed, but to a less extent, with the next smaller disk. With the remainder it was not noticed. The curvature and shape of the solid, and its consequent massing towards the liquid, is intimately connected with the next phase of variation which we shall consider, to wit, the variation in the chemical composition of the solid from which the drop falls. The influence of this kind of variation is to be studied by examining the size of drops formed under like circum- stances, from spheres of the same size, but made of different material. Since in this case the liquid remains the same, we must limit the solids examined to such as the liquid completely wets. In this case, variation in the drop- size implies a variation in the thickness of the liquid film covering the solid. The latter must be caused by variation in the adhesion between the solid and liquid. Finally, such adhesion can only vary through one or both of two causes—namely, variation in the density of the solid, or in its spe- cific adhesion dependent upon its chemical nature. The first qualitative experiment was made upon three equal spheres of brass, glass, and cork. They were hung one above the other im the manner before described, so that the drop from one sphere fell upon the lower one*, It was found that, in whatever order the spheres were arranged, when the flow was uniform and not quicker than gi= 2", the dropping from the cork took place with the greatest rapidity, that from the glass next, and that from the brass most slowly—showing that the brass gives rise to the largest, the glass to the next largest, and the cork to the least drops. From this it would seem that the drops are in the same order as to size as are the solids as todensity. We shall find, however, that this is not always the case, and that some other property as well as density is at work to in- fluence the drop-size. The quantitative experiment, the results of which are given in Table X., confirms the result of the qualitative experiment given above, but shows, at the same time, that the joint influences of den- * Jn this kind of experiment there should be a considerable mass of cotton wool on each sphere to receive the drops from the higher one, and, by acting as a reservoir, to regulate the flow. 1864. Prof. Guthrie on Drops. 4.65, sity and chemical diversity of the solid have only a small effect upon the drop-size. The conditions of the experiment were similar to those pre- viously described. TABLE X.—/VVater. GEE ae 7=7'1 millims. Number of drops=20. Weight of Mean weicht Substance. 20 ga rere: of single drop. erms. grm. 72-4846 e 4848 cohecaas aale 2 oe 012418 2: 2821 ‘A877 + 5930) 2:5985 | j 25980 (CC ROMA Oreo 92-5949 ( 25953 ( 2-5900 26295 9-629 9-626 oe eee ee 0-13118 92-6296 | 26116 0712975 When a liquid drops from a solid it is not always that the adhesion be- _ tween the solid and liquid is overcome. The phenomenon of “ wetting ”’ implies a superiority of the adhesion between the solid and liquid over the cohesion of the liquid ; and in all cases where a liquid drops from a solid which it wets, the act of separation is a disruption of the liquid, and not a separation of the liquid from the solid ; that is, the separation of the drop is a failure of cohesion and not of adhesion. We are not, however, justified on this account in anticipating that the size of a drop is unaffected by the chemical nature of the solid from which it drops, even in those cases where the adhesion between the solid and liquid is greater than the cohesion of the liquid (that is, where the liquid completely wets the solid), because, although it is the liquid which is broken, yet the size of the broken-off part, or drop, depends in great measure upon the thickness of the residual film, as we have seen in examining the influence of the growth-time (in Part I.) and of the radius of curvature. Adhesion may also exist between a solid and a liquid which does not wet it, as when a drop of mercury hangs from a glass sphere. But the cohe- sion of the liquid in such a case, by its effort to bring the liquid to the spherical form, and the weight of the drop so modify the adhesion between 466 Prof. Guthrie on Drops. [Recess, the solid and liquid, by altering the size of the surface of contact between the two, that the size of the drop gives no direct clue to the cohesion of the liquid. - We may now examine a. few cases in which, the size of the sphere re- maining the same, and its density in some instances nearly so, the matter of the solid varies, but the liquid wets it in all cases. This will show whether the differences of Table X. are due wholly to differences of density of the solid, or also or wholly to differences of chemical constitution. TaBLe XI.—Water. Gt=2, A leas dst By Radius of curvature==7 millims. iL 2. 3. 4. Weight of | Mean weight | Specific gravity Bubshance: 20 drops. | of single drop. of solid. germs. m. 011984 6°80 012021 2-00 0°12246 6°86 012264 11-44 Phosphorus 0°12274 2:08 Bismuth......... 0°12285 990° SarG 012425 7-29 Equal spheres of the substances were made by casting them in the same ‘bullet-mould. The surfaces of the metals were roughened by momentary immersion in acid; tin and antimony in hydrochloric, the rest in nitric acid. Without this precaution a metallic surface is apt to be wetted only locally, the base edge of the drop is irregular and inconstant, and the drop- , 1864.] Prof. Guthrie on Drops. 467 weight varies. Indeed with some metals, such as tin, a smooth and bright surface is scarcely wetted by water. As the bodies examined have different coefficients of expansion by heat, and one of them expands on solidification, it was necessary to test the equality of their size and remedy any inequality. ‘This was done by ar- ranging three of them, one at each angle of a small equilateral triangle drawn on a large piece of plate glass. Another piece of plate glass was then placed upon the spheres so as to rest on them all three, and slightly loaded. On passing a gauge between the plates, at their edges, the slightest inequality of the spheres could be detected, because the gauge lifted the plate off the smallest of the three balls, which could then be moved. The larger spheres were then reduced im size by brisk agitation in acid. The sulphur and phosphorus were, for the same purpose, washed in ether. Although there is only a slight difference between the consecutive terms of column 3, yet between the extremes of antimony and tin a well-marked difference exists. | This Table shows that the drop-size stands in no simple relation either to the equivalent density or chemical character of the solid, and establishes the existence of a specific adhesion independent of these. Although the differences of Table IX. may be partly owing to the differences of density of the solids cork, glass, and brass, yet we see from Table X. that there is about half as great a difference between the sizes of drops from antimony and tin as between those from cork and brass, although the difference of density between the first two is small compared with that between the last. Again, sulphur gives rise to drops intermediate between those of antimony and cadmium. Without, therefore, venturing to assert that density is with- out influence on drop-size, it is clearly proved that it does not exert the most powerful influence. We have finally to examine the direction and extent of variation in drop- size caused by change of temperature. By altering the density of the liquid, a change in its temperature may affect the drop-weight without altering the drop-size. It may further alter the drop-size by altering the size and therefore the curvature of the solid. Any error introduced by the first of these sources is eliminated by dividing the observed weight by the specific gravity at the proper temperature, as in the case of different liquids at the same temperature. Errors from the second source may be certainly safely neglected, being far within the errors of observation. In the place where these experiments were made, the range of natural atmospheric temperature is very small. From the coldest to the hottest season the difference scarcely exceeds 10°C. This circumstance made an extended and minute study of the influence of temperature impossible, by preventing more than one observation at each temperature being made. The liquid taken was water, and the solid was glass. The water was heated to the boiling-poimt and placed in the apparatus (Part I. fig. 3). The sphere from which the water fell was the bulb of the thermometer which 468 Prof. Guthrie on Drops. [ Recess, measured the temperature. Fully the upper half of the sphere was covered with cotton-wool, so that the whole of the sphere was kept wet. The con- siderable mass of mercury in the bulb of the dropping sphere or thermo- meter itself served to make more uniform the temperature of the drops; while the actual contact between the drops and the spherical bulb ensured a tolerably close approximation between the actual temperature of the drops and that indicated on the stem of the instrument. Although, therefore, the temperatures observed cannot pretend to any even approximate positive accuracy, yet they are certainly in the actual order of magnitude. The arrangement is seen in Plate IV. fig. 6. : TasLe XII.—Water. oe r=7°A millims. Number of drops=20. Relative mean Temperature,| Weight of | Weight of | size of single drop Centigrade. | 20 drops. | single drop. (corrected for temperature). ie germs. erm. ao 2:5564 0:12782 40°35 40 25795 012897 0°12985 37° 25826 012913 35° 26083 0:13041 339 26105 0138052 32°6 26161 0:13080 31:2 25960 0:12980 30°64 306 26065 0:13032 0-13066 29: 26044 0-13022 | 28-2 25983 0:12992 | 28° 26078 0°13029 (27-5 2:6032 0:13016 ) 20°4 20-4 26480 0-13240 0:13262 In the above Table the temperatures are so grouped together that the means of the groups differ from one another by about 10°C. The single drop-weights are correspondingly grouped, and the mean of each group is then divided by the specific gravity of water (0°=1) at the mean tempe- rature of the group. It appears then that, for a range of 20° Centigrade, or 36° F., the dif- ference in drop-size effected by change of temperature in the liquid is in- appreciably small, not being more than 0:00277, a quantity almost within the limits of experimental error ; for on referring to Table X. we find that the greatest difference between the numbers for glass, which should be equal, amounts to 0°00044 grm., or a sixth of the greatest difference due to variation in temperature. On the whole, then, we may conclude that the temperature has very little influence on the drop-size in the case of water between the above limits. No doubt, near the point of solidification, where liquids have an incipient 1864. ] Prof. Guthrie on Drops. 469 structure, the drop-size would be subject to sudden changes of magnitude. A few experiments with other liquids, namely turpentol, acetic acid, and alcohol, showed that with them the drop-size was almost equally insensible to change of temperature; and in all cases, as with water, the lower the temperature, on the whole, the larger the drop. We have now examined seriatim all the chief causes upon which the drop-size depends in the case SLG. They are, J. Rate of delivery; 2. Solids held in solution; 3. Chemical nature of liquid; 4. Geometric relation between solid and liquid; 5. Density and chemical nature of solid ; 6. ‘Temperature. Our data, however, are still insufficient for us to predict, under all cir- cumstances, the relative sizes of the drops of liquids under known external conditions. Clearly the missing term is closely related to the specific co- hesion of the liquid. But what is cohesion ? and how can it be measured ? It lies perhaps in the nature of things—it seems at least inevitable—that the nomenclature of elementary properties should be vague and unsatisfac- tory. The properties of solids—hard, soft, brittle, tough, tenacious, elastic, malleable—do not stand in any definite relation to one another. Even the hardness which resists abrasion, the hardness which resists penetration, the hardness which resists crushing are by no means identical; so that one body may possess more of the one sort of hardness than a second body does, while the second body exceeds the first in another sort of hardness. Nor do any of the above-mentioned properties of solids stand in any simple relation to that resistance to the separation of the contiguous parts which is called cohesion. Thus, by no attribution of this single property of co- hesion could we define ice or shell-lac, bodies which are at the same time tough, brittle, elastic, and soft. We are forced to the conception of two distinct kinds of cohesion—séub- born and persistent. These may coexist, but are not identical. The one is strong to assert, the other pertinacious to maintain. The four following substances may serve to illustrate the possession of these two cohesions in various quantity. Talc has little stubborn and little persistent cohesion. Glass has much stubborn and little persistent cohesion. Gold has little stubborn and much persistent cohesion. Tron has much stubborn and much persistent cohesion. The necessity for such a discrimination exists in a yet higher degree in liquids. If we conceive two liquids of different nature dropping from the same substance which they both wet, and if there be only one kind of co- hesion, the one which has the greatest cohesion will tend most strongly to assume the spherical form ; and this would tend to cause it to drop sooner, or have a smaller drop-size than the other. Ona the other hand, the liquid of stronger cohesion will cling most strongly to the film of liquid adhering to the solid ; this will keep it longer from falling, and thereby increase its drop-size.- Hence an increase of cohesion tends to produce two contrary 4.70. Prof. Guthrie on Drops. | Recess, effects. But if there be a similar distinction between the two kinds of co- hesion of liquids, as above pointed out in the case of solids, we have the following consequence. It is the persistent cohesion which causes the as- sumption of the spherical form, the stubborn which resists the separation of the drop. The former tends to diminish, the latter to increase its size. As one or other predominates, the size of the drop varies. Accordingly the drop-size is by no means a measure of what is generally called the cohesion of the liquid, but rather a measure of the difference between the two cohesions, stubborn and persistent ; and the law is, that the drop-size varies versely as the persistent, and directly as the iilicin cohesion of the liquid. In mercury, water, and glycerine the stubborn cohesion is greater in proportion to the persistent cohesion than in the other liquids examined ; but it by no means follows that persistent cohesion is wanting in mercury or stubborn in alcohol. When a drop is in the act of falling its stubborn cohesion is in equili- brium with the resultant of two forces—the one, the persistent cohesion, tending to produce a spherical form, the other the weight of the drop. Since the former of these component forces is, for the same liquid, constant, it seems as though the weight of the drop might be taken as a measure and expression of the stubborn cchesion. But such is not the case, because we have no ground for supposing that the diameter of the drop where the separation occurs is of constant size; on the contrary, it must be conceded. that in larger drops this hypothetical surface of stubborn cohesion is larger — than in smaller drops. Further, unless we know the exact shape of a drop in all cases, we are not in a position to deduce the size of the surface of cohesion from the drop-size or drop-weight. _ In the cases where it has been tried, it has not been found that the nature of the gaseous medium in the case of SLG exerts-any appreciable or defi- nite influence upon the drop-size. Taking glass for the solid and water for the liquid, the medium was changed from air to nitrogen, hydrogen, and carbonic acid. The exceedingly slight variation wrought in the drop-size by this change may probably have been due to the different solu- bility of the gases in water, and the consequent alteration in the cohesion of that liquid. Having now traced the effect of variation in the conditions which deter- mine the size of a drop in the general case SLG (or where from a solid a liquid drops through a gas), we come to the case SLL (that is, where from a solid a liquid drops through a liquid). As in the cases of SLG, we must here also take the three terms of such chemical nature as to be without action upon one another. SLL. From a Solid a Liquid drops through a Liquid. A preliminary quantitative experiment was made under the following conditions :—Water was made to drop from a glass sphere at the rate 1864.] _ Prof, Guthrie on Drops. 471. gt=5". The drops were collected in a tube bearing an arbitrary mark. The number of drops required to fill the tube up to this mark was noted. Then the sphere was surrounded by turpentol, and the rate having been brought again* to gt=5", the number of drops of water necessary to fill the tube up to the same mark was counted. The turpentol being replaced by benzol, the same operation was performed. The entire arrangement of the Stalagmometer+ is seen in Plate V. fig. 7. X, Y¥ are contrivances described in Part I. for giving a uniform flow of water. _ The siphon A rests upon the cotton-wool covering half of the dropping sphere and thermometer-bulb G. The sphere is held by its stem B in the clamp H. C is half a globular 1-lb. flask, supported by the filter-stand K.. Through the neck of C passes the tube D. C and D are joined liquid- tight by the caoutchouc collar L. A few arbitrary marks are made at E. The lip of C is turned down to a beak at M above the vessel F. In adjusting the instrument, to get the required value of gé, the holder K is slipped along the table so that the drops from G fall between C and D, and not into D. When the required rate is obtained, it is slipped back again. When such liquids as turpentol are used as media, a little water is poured between D and C to protect the caoutchouc. In all cases where a liquid medium is employed C is filled till it runs over. In the first experiment, of which the results are given in the following Table XIII., the numbers are subject to two sources of error. The volume filled is rather small, and no allowance is made for meniscus. In this, as in all cases of SLL, great care must be taken not to shake the instrument. Tas.Le XIII.—Water. Cue" D227 €: Radius of glass sphere=7°4 millims. i, 2. 3. a. Number of drops of) aroan of Relative size of Medium. water required to | Gojimn 2. single dro fill a given volume. (through air=1). DS cole ee ee ; { oe \ 57-0 1-0 28 Purpentol ......... - | 26°7 2:14 26 13-2440) By aoe aS { é \ 7:0 8:14 There is therefore a ereater difference between the drop-sizes of water in benzol and turpentol than between those in turpentol and air. The tur- * A diminution of g¢ is observed. Sita T Sradaypos, a drop. 472 Prof. Guthrie on Drops. [Recess, pentol and benzol here employed had the specific gravities of 0-863 and 0:864 respectively ; they may therefore be considered of equal density. Hence variation in the liquid medium, independent of variation in its den- sity, produces an enormous effect upon drop-size. We shall have occasion to return to this case. . The influence which the liquid medium exerts on the drop-size, and the share of that influence due to the specific gravity of the medium, will be well seen on comparing the drop-sizes of mercury which falls through various liquid media. The arrangement of the apparatus for this purpose is seen in Plate V. fig. 8. As far as A it is similar to fig. 7. The siphon A, fig. 8, is a capillary tube; its lower end, which is turned vertically downwards, rests upon a sphere of bees: R, which has been washed with nitric acid and sodium- amalgam, and allowed to soak for some days under mercury. Mercury adheres perfectly to such a sphere. In every case the sphere was immersed just halfway in the liquid. A small capsule S is supported in the liquid on a stand T about half an inch lower than the bottom of the sphere. As soon as gé becomes constantly =5", the vessel V is moved so that S comes under R. Five drops of mercury having been caught, the cup is moved horizontally as before, taken out and replaced by a fresh one, and so on. The batches of five drops are washed, dried, and weighed. The results are given in Table XV. We may, however, previously notice here with advantage a phenomenon which attends the separation of drops under several circumstances, but which can be watched most narrowly in the cases of SLL, because in a liquid the separation of a drop is less abrupt than in a gas. When water falls from glass through air, immediately after the drop separates, a very minute drop is frequently projected upwards from the upper surface of the drop*. I have not traced the conditions under which this supplementary drop is formed; indeed it is sometimes formed, and sometimes not, under apparently similar circumstances. No doubt the proximate cause is that the drop at the instant of separation is not sphe- rical; the persistent or retentive cohesion, which brings it almost imme- diately to its normal shape, does not allow time for its more excentric parts to collect to the main mass ; they are therefore by the motion of the main drop flung off and projected upwards. The same phenomenon is seen much more distinctly when water drops at this rate (gé=5'') through benzol or turpentol. In these cases the per- sistent cohesion of the liquid medium comes also into play. But the most striking example of supplementary drops is seen when * The secondary drop may be well shown by holding a plate containing anhydrous cupric sulphate about two inches below the dropping solid. The white salt is smoothened by pressure under a plate, and its surface, being porots, absorbs the water-drops instantly and without splashing. The blue spots of hydrated sulphate show where the water has fallen, ~ re, Lith. SL 2 . iC Proc. Roy. Soc. Vol AM FU LV. UA) AOS0T 7 Proc. Rey-Soe. VLXIIELVI 1864..] Prof. Guthrie on Drops. 473 glycerine forms the medium through which mercury drops. In this case, when gé=5", there are always two supplementary drops of mercury formed. It is impossible to determine whether they both have their origin at the same moment and from the same drop. The probability, however, is that they have not, but that one is first separated from the main drop, and the second from the first; for there is always a great disparity between the sizes of the two supplementary drops, whereas, if they were both formed at the same time and for the same reason, we should be justified in expect- ing greater equality. The drops soon separate in falling, in consequence of the difference of their surfaces. The relative sizes of the main and sup- plementary drops in the case of mercury falling from copper through gly- cerine were determined as follows:—A number of porcelain cups (fig. 9) were arranged at the bottom of a shallow dish full of glycerine ; when the rate of dropping was uniform at gé=4", the dish was shifted horizontally so that every drop with its two supplements was caught in a separate cup. The globules of mercury in each cup were removed by a little scoop of copper foil. Ten of each kind were collected. After washing and drying, they were weighed, with the following result :— Taste XIV.—Mercury. gt 4." P=2103,C. Radius of sphere= 12:8 millims. grms. 10 principal drops weighed ............ 6°3447 10 first supplementary drops weighed .... 0°1242 10 second do. do. G0; te. Or0229 10 complete drops weighed ........ 6°4918 In all cases of SLL the supplementary drop or drops were collected and weighed or measured with the main drop. In Table XV.— Column 1 shows the medium through which the mercury dropped. Column 2. The number of drops weighed. Column 3. The weight of the drops. The weight of every batch of drops is given, in order that the approximation between the figures for each liquid may be compared with that between the separate liquids. In two cases only, marked by an asterisk, are the numbers probably erroneous. They are not reckoned in taking the mean. Column 4. Mean weight of single drop, from column 3. Column 5. Specific gravity of medium. Column 6 shows the weight of the drop of mercury in the liquid. Since the falling of the drop is determined in part by its weight, and since the weight depends not only upon the size of the drop, but also upon the den- sity of the medium in which it ig formed, it is interesting to see how the VOL. XIII, 2N A7 4 Prof. Guthrie on Drops. , [Recess, size of the drop is affected by the diminution in its we = by the density of the medium. If W,=weight of drop of mercury in air, W,=required weight of drop of mercury in liquid, A=specific gravity of liquid, | B=specific gravity of mercury ; then W,=W —$W, The values of W, form column 6. The liquid media are arranged according to the order of rinecetiGllle of the numbers of column 4. The salient points of Table XV. are chiefly these :-— 1. The drop-size of a liquid which drops under like conditions through various media does not depend wholly upon the density of the medium and consequent variation in the weight, in the medium, of the dropping liquid. Thus glycerine, whose density is above that of all the other liquids ex- amined, does not, as a medium, cause the mercurial drop to assume either its minimum or maximum size. 2. The liquids in Table XV. are in the same order as in Table VII. In other words, if there be two liquids, A and B, which drop under like conditions through air, and the drop-size of the one, A, be greater than that of the other, B; then of a third liquid, C, be made to drop through A and through B, the drop-size of C through A is greater than the drop-size of C through B. 3. Further, on comparing Tables XIII. and XV. it appears that, whether water or mercury drops through turpentol and benzol, the drop through benzol is greater than the drop through turpentol. This we shall after- wards find confirmed in other instances into the law, If the drop-size of A through B be greater than the drop-size of A through C, then the drop- size of D through B is also greater than the drop-size of D through C. It is further observed that, while mercury exhibits its largest drop when falling through air, water assumes its smallest drop-size under this condition. This method of the examination of liquids by drop-size in the case SLL, which brings so prominently forward a comparatively slight difference be- tween similar liquids, may be used, not only to detect commercial adulte- rations of one liquid by another, but perhaps to distinguish between those remarkably-related isomeric liquid bodies (the number of which is quickly increasing) between whose terms the difference has until lately escaped detection. Of these bodies perhaps the first most remarkable instance was furnished by the two amylic alcohols ; but the greatest number at present known is amongst the hydrocarbons. We may take an example illustrating the use of the stad osiniottielee in approximately measuring the proportion, in a mixture, of its two chemically and physically similar, but not isomeric constituents. Suppose we had a liquid which we knew to consist wholly of a mixture of benzol and turpentol, and we wished to find the propor in which 1864.] . Prof. Guthrie on Drops. 475 TaBLE XV.—WMercury. ge— a P21 sc. Radius of sphere=12°8 millims. i, 2. 3. 4, 5. 6. Medium Mean weight in| Specific | Weight of through which pies Weight air and gravity | single drop in the mercury of drops. | relative size of of respective dropped. single drop. _|medium. medium. ———— = | — | drops. germs. m. grm. 0°76545 0-00 0°76545 3°5047 0°69750 1:00 0:64619 35066) Glycerine ...... 3-4088 0:61508 1245 0:55793 2 — (sy) (J) Or Ean oe 29637 059822 0-364 | 056014 Turpentol ...... 043497 0°863 040715 | pene eysrerenv anor | i) = & \— 476 Prof. Guthrie on Drops. [ Recess, these two ingredients were present. We could scarcely approach to an answer by any of the means hitherto employed. The specifie gravities of the two liquids are so close (864, °863) that the density of the mixture would give us no substantial aid. Though there is a considerable differ- ence (80° C.) in their boiling-points, no one who is familiar with the diffi- culties of fractional distillation would place any reliance upon a quantita- tive separation based upon volatility. heir refractive indices are nearly the same*. Their vapour-densities, 2°77, 4°76, though comparatively different, are not absolutely very wide apart. They are active and passive towards most of the same chemical reagents, and interfere with one another’s reactions. If we have recourse to chemical analysis (C,, H,, C,, H,,), a very small experimental error would point to a great difference in the proportion of the two. To find how far the stalagmometer (Plate V. fig. 7) is applicable in this case, it was filled with five liquids in succession :— Ist, with benzol: ioe. wal east eee ae o's aft ey =e 2nd, with two volumes benzol and one of turpentol =B,T. 3rd, with one volume benzol and one of turpentol.. = BT. 4th, with one volume benzol and two of turpentol =BT,. Sth, with, turpentol