2 J , ; q Wy j ‘ 42) 2 . a i) Uy, PI fs . A, ¥ P, “* . } > ’ e . \\} i my a + 4 d / id / } Se SF LI VID? Iw SS * + >> D2 TED Dn SS = IY D> i> SS De PIP D> D> D> By) Ses 2k > SS Sess => D- > » > es »>) >_> > Ses SSD 5a I> oy Se >) 2 ee > > D>S> ae 2 > > > > >» ee SD) > > 3) >> ae — > > es ee > eae => La an »> a> »> > Ss ; SS + PDD. ss" aD » — 2 > > >> lp > SS: ee = > > DD > D>? 23> > SP > we Ss SD 3 >> >> Sp a > ow B28 >> 222 >»). SP 2 D> = > Sp ». >e> PP an : » D> Le es > | ro > 22g SS — > as > SDP > > > S > wy >»\ re ay 3p > > Sa a D> 72 >> 2D. > Pp => > yp» > Se. >> ey DD» as =e 2 I> wv > ip. Pe => >a» D p> >> DID Bae D2 2B > >> » 2? ») P_ >> p . > >> > > b) 22> 2» D SS > ie > : > 2 22D _ >: ee 2S SYD Wow SS Ss > > > >>> ie: 22> DPD DD) _—_ SPS PD == So5 P22 D>) Sere 22 22D? I> I SS > 5 3 > Sse >> DP I > > Se > EE os 8 5 Daa D> > I> D> B® > 1 > So 2 SS Ss See Se ie > ee ee ee Sea ee > pS) es es oS ee eee >D) : 2 D>> D > » IDD Pi) = s te > ) 332 > — >> >» PAS >> >»>>> DR > 22> >> 2 D>) > 5 >) >> > > > Se» DD» > [SE > >>) FD OP > > >| 2 Ee >>> o> Rh > w Po > D»s>~ po > Se uw : C 2 Dee > a >, 3 > > 2D 35> >>> > > Ep» 2D >> > 2D >_>», = >>) RD >» BD» D®>D >>>. os = = p> DD > ©: >» DPF >>>. PP LD, > yy D > >>> > ED > 42 RP PDO 2 2D > D» yo lt > 2 PY» 2 1D DW YD » Ee >> > Pi >> D>? V Y vil v nahn vi" it y o> => Sp» v Vv) ry vy ws \ \ vy i \ Vwi Y We aig H iy Vv ¥ wl 1A; Tie) wv vi) w ¥ Wey y¥y We very Ww \ Www > Se SD Ise Dp 23 22> VSS Wy NAVAL ly WW >> > > > >i > > >a 2 $2 Tp Se @. ae siete PROCEEDINGS OF THE ) Ni \) NES DUBLIN: PRINTED BY M. H. GILL, PRINTER TO THE ACADEMY. MDCCCLVIII. CONTENTS. ) VOLUME VI. ———_@———_ 1853-1857. PAGE. On the Solution of Linear Differential sisi hie By the Rev. Charles Graves, D.D. .- . - = + ae kone 1 On the Fresco Painting in the Abbey of eee oy, the i I. H. Todd, D.D. . . - Wien weeny Po ae TNS On the Construction of te Case ciara BY. the Beek dba Robin- son,D.D. ..- - Pa rwe Pere cay ee! On the Magnetic Influence of the Mie By the: Rey. i. Lloyd, D.D. . 28, 194 © On the Meteorology of Ireland (second part), as deduced from the R.I. A. Observations made in 1851. By the Rev. H. Lloyd, D.D.. . . 31 On the Strueture of the Starch Granule obtained from the Potato. By G. J. Alimama De s5 ole POE Mica i! On the Method of solving a large class of L Linear Differential yaa by the application of certain Theorems in the Calculus of Operations. By the RevaCharles Graves yA) saereeete late co Meeel (eh. pills]u on e\hist velo rae On the results of a Series of Experiments on the Decomposition of Water by the Iron Galvanic Battery. By the Rev. Professor Callan... . . . . 37 On a new method of Measuring the Angular Aperture of the Ohientine: of Mi- croscopes. By the Rey. T. R. Robinson, D.D. . . oi de) ees tos On an ancient terraced Gravel Hill near Castle Beer Conny of Calgary: By D. H. Kelly, Esq. . . . « - pmei ‘ Aha ie veoh ate) On a remarkable Lunar Halo seen on Feb. 10, 1854. By R. Mallet, Esq, - 55 On the British Earthquake of 9th Nov. 1852. By R. Mallet, Esq.. . . . 58 On the Transmutation of Metals. By Geo. J. Knox, Esq . . . ... 59 On a new method of determining approximately the Spherical Aberration of a Combination of Lenses for Microscopic purposes, By Thomas Grubb, Esq. 59 On the Integrations of certain Equations. By Sir. Wm. R. Hamilton. . . 62 On the Laws of Diurnal Tide at Donaghadee, County of Down, and at Bun- owen, County of Galway. By the Rev. S. Haughton. . . sig lee NORA Oe A notice of some anomalous facts respecting the Tides in WEE cake s Sound, Australia. By W.H.Harvey,M.D. .. . Sinaat Pease tetetead see On Assyrian Weights. By the Rev. Edward Hincks, IDEIDA BRM 5 Se Serge On the Inscribed Stones in the Isle of Man. By the Rev. J. G. Giana eens a vi PAGE On the importance of making a Collection of Banter of the Inscribed Stones of Ireland. By J. O. Westwood, Esq. . . . ee ec 77 On the last bright Comet. By Andrew Graham, Esq. = Cea nS OF On the Comparison of Adjectives in the ancient Irish Tange By the Rey. Charles Graves, D.D. . . . . aoe PIR ele 2 On, Experiments to determine the Velocities of Rifle Bullets By the Rev. S. ian htone sce car tre eleetangire| ors oe 6 si bje Sehinea ae 85 The Calculus of Quaternions in connexion with the ‘pees Theorem of Dupin. By Sir W. R. Hamilton, LL.D. . . . 52 eo On the Barometrical Measurement of the Peak of Teneriffe “By J. Beete Jukes, Esq. . - - - - 3. tee pe aed mentees s/t felt rane eee On the Ancient Name of the City of Dublin. “By Charles Haliday, Esq. 95 On the MSS. of John ee pane at St. Isidore’s, Rome. By Charles Mac Donnell, Esq. . . Soe c 5% = Pome On the Monastic Foundations of the Irish on iis siCentinaat- By Charles P; MacDonnell, Esq. . . . . - aes 103 On the same. By the Rey. Charles eure D. D. : 112 On the Properties of Electro-Magnets (second Wee BY the Rey. r. R. [Robinsons JAD eae nels one tne - . 113 On the Personal Pronouns of the Assyrian and ohne Teneo especially Hebrew. By the Rev. Edward Hincks, D.D. eH ree 113 On the Principles which regulate interchange of Symbols in certain Symbolic Equations. By the Rey. Charles Graves, D:D wera oh ig | Sipe, Aes On some Extensions of Quaternions. By Sir W. R. Hamilton, LL.D. . . 114 On Aphanizomenon Flos-aque. By G. J. Allman, M. D. Boe a: 85) On a species of Peridinea. By G. J. Allman, M. D. ite ncs<" 118 On the Meteorology of Ireland. By the Rey. H. Lloyd, D.D.. . . » 120 On the Vine Disease in Ireland. By D. Moore, Esq. Apertench aes se 7c On the Chronology of the Nene Beyptian an ee the Rey. Edward Hincks, D. D. : aubeas 3) airs On Assyrian Mythology. By the Rev. Edward Hincks,D.D. . .. . 128 On the Reflexion of Plane Polarized Light from the surface of eanepaeee Bodies. By the Rev. Samuel Haughton. . . . <2 ole wee. On the relative quantities of Potash and Soda in the Pela of the Dublin and Wicklow Granites. By the Rey. J. A.Galbraith,. . . . .. - 134 On the same. Bythe Rey.S. Haughton. . . ...... =. - « 148 On the Marine Botany of Western Australia. By W.H. Harvyey,M.D. . 152 On the Dying Gladiator. By Robert Ball, LL.D... . . . . . = . 152 On the use of the Hygrometer in Barometric Measurements of Heights. By Capt. H. L. Renny,C.E. .. . Cetin eohe Lube tee reac Gat 155 On the existence of a true Medusoid pistes? in the Male Gemmz of Hyde By G.J.Allman,M.D. . . . ... uote 155 A Plan for ascertaining the Deviation of Ships’ Compass arising from ee Attraction. By T. A. Dillon, Esq. » « 157 On the Solution of the Equation of Fence 's Functions. By the Rev. Charles Graves; DD iy. cairn. shee a athGds On the Draining of the Haarlem Lake. By Sarid Dies LL. D. vil PAGE. On the Chemical Composition and Optical Properties of the Mica of the Dub- lin, Wicklow, and Carlow Granites. By the Rev. Samuel Haughton. 176 On the Solution of the Equation of Laplace’s Functions. By Sir W. R. ncn en Meer tiy. he ckyesubd bet be lca te, wat ha solve soo Te On Ecliptic Catalogues. By Edward J. Cooper, Esq... . - - . . 194 On certain Notes in the Ogham Character on the margin of an ancient MS. of Priscian. By the Rev. Charles Graves, D.D. . . . «. «. ~ . 199, 209 On Laplace’s Equation and the Calculus of pines. By the Rev. Robert Carmichael. . Sie aS ESC een Cullis, On the same. By the Rev. Charles ene D. D. : é 220 On an Inscription in the Church of Galway. By the Rev. J. i Toda, Dz. D. 223 On the Inscription found on the base of the Cross which stands in the princi- pal street in Cong. By the Rev. J. H. Todd, D.D. . . . 225 Address to the Earl of Carlisle, Lord Lieutenant of Ireland. . 226 PAHSWOL LO THGPACOTCSA Ny Natl chu c beh Poul auner@tebird ot aki aiari ten, fe 229 On the Granites of Leinster. By the Rev. Samuel Haughton. . . . . . 230 On the same. By Richard Griffith, LL.D. . .. . Metanicerte Moe On the Molecular Formation of Crystals. By Robt. T. en Esq, 240 On the Precipitate formed in the emg of Alkaline shee ca a John Aldridge, M.D. . . . . sith cae - 244 On the subject of Dreams. By the Rev. wi Wills, D. DS free ation te 247 On the Ogham Inscriptions found on a Monument at Bressay, in Shetland. By the Rey. Charles Graves; "Da Dige int. ee ve celatee - 248 On some Symbolical Extensions of Quaternions, and on a Theory of Mages! ative Quines, and on some Geometrical sgl ona of his Theories. By Sir W. R. Hamilton, LL.D. . . . shpat « ‘« 250 On a peculiar Organic Production forwarded iby W. H. Harvey, M. D, from Western Australia. By G. J. Allman, M. D. S “200 On the Cover of the Gospels of St. Molaise. By as Pettie, LL. D. 251 On the Revival of Italian Literature in the Fourteenth Century. By J. FF. Willers Tbe ries ate ies) Sal ee Pst othe RO? te + eb On the Bursting of Ordnance when firing Heated Shot. “By Robt. Mallet, Esq, 251 On certain Animals mentioned in the Assyrian rept se the Rey. Edward Hincks,D.D. . ... . eRe 251 On some new Geometrical Bt oe of Quaternions. By Sir Wl R. Ha. Tatton; Wl Drs) Sos Sic Auer acne: Micke er onc sag tye) Taber Bian OD: On the early Discovery and extensive Adoption of the eee: Intercalation. iBy the Rey. FE: Nolan; LU. Di BOR Sio0.5 soos ats 261 On a Babylonian Tablet in the British Museum. oe the Rey. Edward ineks D5 Ds is sis, 270 On aremarkable Phenomenon eas on inh eel By Jo ohn Peet a 271 On Reciprocal Surfaces. By the Rev. George Salmon. 273 On the Barometric Measurement of Mountain Heights. By ie Rev. J oseph PACK Geil Dralthcciherss int ieect cae eer eae es nes 276 On the Causes of certain Phenomena observed on Lough aide By Henry Hennessy, Esq. . . . . ets 279 On the Luminous Phenomena ete aa the ativdeter of Ruhmkorff gs Te duction Apparatus in Vacuo. By the Rev. T. R. Robinson, D. D. 283, 426 Vill On certain Formule which occur in the Worksof Marcellus. By Dr. Jacob Grimm, H.M.R.L A .. . sate’) Spates, ds Noh tele Melee ame On the same. By the Rev. J. H. Toa, D. D. BOC RT 0a OR eee On the Name said to have been given to St. Patrick when a Captive in Ire- land. By the Rev. J.H. Todd, D.D. . . . . . . «2 ee On Meteorology. By Henry Hennessy, Esq. . ... .- E ie On an Optical Phenomenon observed in ee 1856, while on a voyage to Havre. By Henry Hennessy, Esq. . . Hoe. On the Extension of Taylor’s Theorem to Non-Commsnfetive Symbols By the Rey. Charles Graves, D. D. : ee On the Locality where the ancient Gold Fibula, ‘called the Dalriada Biel was found. By James Gilmour, Esq.. . - - - Been one On the Action of the Wind in Pies Waves in a Glass Vivarium. By D. J. Corrigan, M. D. . Ad ero uenk tees < hiay oe On the Castle and Manor of Baggotvath, By J. Huband Smith, Esq. On a new Experimental Hydraulic Formula for finding the pre, of Water in Water-Channels. By John Neville, Esq. . On a Geometrical Extension of the Calculus of f Quaternions ‘By sir William R. Hamilton, LL. D. Inaugural Address of the President, hee J. H. Todd, D. D. On some Properties of Solid Figures revolving on Axes in Supeoa at the Surfaces of Fluids. By Gilbert Sanders, Esq. On the Identity of the Chronological System of the Egyptian Priests, as ex- plained by Herodotus, and that of Manetho. By Edward Clibborn, Esq. On the Stomach of the Zebu. By John B. Barker, M. B. A On the Solution of Cubic Equations. By Thomas J. Campbell, Esq. On the Depth of the Ocean deducible from Tidal Observations. Ey the Rev. Samuel Haughton. . . . . . . On the Introduction and Period of the Cpa use of the Potate in Ireland. By W. R. Wilde, Esq. . . .- 5 9 On the Effect of the Internal Fluidity of the Earth on the Length of the e Day. By the Rey. J. H. Jellett. On the Solution of the Equation of Contin ati an Tconpeenbls Fluid. By George Boole, LL.D. . . . + . ai ete smile On the same. By the Rev. Charles Graves, D. D. 5 Ee cRrie On the Scandinavian Antiquities of Dublin. By Charles Haliday, Esq. . Letter from F. M. Jennings, Esq., relating to Silver Ornaments collected by _ himself in Morocco, and presented to the Royal Irish Academy. . . On the Influence of the Earth’s Internal Structure on the inci of the Ea By Henry Hennessy, Esq. On Inscriptions from the Market ics at t Campbelion, in ay rE ‘and Inverary. By J. Huband Smith, Esq. é aoe On an ancient Missal, described by Dr. O’Conor in his Catalogue of the Stowe MSS. By the Rey. J. H. Todd, D. D. SiFiay Mal eee TE On a MS. of Dr. Willoughby’s, written in 1690, ‘‘ On the Climate and Dis ases of Ireland.” By W. R. Wilde, Esq. 5 : On a new System of Roots of Unity. By Sir W. R. Hamilton, LL. D. 399 415 1X An Autograph Letter of the late Dr. Charles na ce by Edward Bewley, M. D. : aio On some Experiments on the Poisonous Properties of Styehnin and Nicotine. By the Rev. Samuel Haughton. . . SEC MM ARI RCM Nee 5 ’ . . On an Ogham Inscription. By Richard Hitchcock, hn LEAR neti bce On some Roman Coins found near Rathfarnmham. By George Petrie, LL. D. A Biographical Notice of Colonel Jules Terence O’Reilly. By A. O'Reilly, Esq. On the early System of Abbatial Succession in the Irish Monasteries. ai the Rev. W. Reeves, D.D. . . . naka On the Irish Abbey of Honau, on the Rhine. “By tl the ase W. vee D. D. On the Icosian Calculus. By Sir William R. Hamilton, LL.D. . .. . Address on the Utility of Antiquarian Collections in relation to the Pre- Historic Annals of Europe. By the late J. M. Kemble, Esq. . . . . On Professor Mac Cullagh’s Theorem of the Polar Plane. By A. Cayley, Esq. On a movable Horizontal Sun-dial. By M. Donovan, Esq. . . . .. . On a new and singular Acoustic Phenomenon. By M. Donovan, Esq. On a new Arrangement of Grove’s Galvanic Battery. By G. J. Stoney, Esq. On the Expedition to the Neighbourhoot of the Magnetic Pole about to leave England. By the Rev, Samuel Haughton. On the Distribution of Heat over the Island of Great Britain, By Henry Hennessy, Esq. . . - - Bey fe eure eee On the General Expression by Quatetnions vit Ane of the Third Order. By Sir W. R. Hamilton, LL. D. Si eeee ee WR a Memorial to the Rt. Hon. Lord eta praying the ‘‘ Resolute” to fi fitted for another Arctic Exploration. . . . . «© 2» «© «© « « On the Foundation of Archbishop Marsh’s Library. By J. Huband Smith, Esq. On the General Equation in Quaternions for Cones of the Third Order. By Sir W. R. Hamilton, LL.D. . . . . FM ES MG CaCO Ouse Letter from the Rev. J. H. Todd, D. D., to the a ae proposing to have the Materials collected for the Ordnance Survey deposited in the Academy. Answer from Lieut.-Col. Larcom. . . . - «© © «© «© «© © «© «@ _ Letter from Captain Leach. . . . - » © «© «© © « © @ © On the Personal Pronouns in their most ancient form. By the Rev. Edward Bancks DED yi Src io ies st On a certain Harmonic Property of the Envelope of the Chord connecting two corresponding points of the Hessian of a Cubic Cone. By Sir Wm. R. LEK WD S Dis Gs Oia Ge He hr be Oe Be ON so On a Fundamental Theorem respecting Congruences, affecting a class of Com- plex Integers, which involve the Imaginary Cube Roots of Unity. By J. PES Gra ves Vhs: tims cs “erica! faire cietal. i Rkel cals toate, \'s . . . . . . . . 8 ANTIQUITIES BOUGHT,—pp. 203, 313. 9 EXCHANGED,—p. 155. By-Laws,—pp. 17, 31, 38, 71, 319, 499. Deposrr or ANTIQUITIES,—p. 526. Pace, 416 420 439 441 445 447 452 462 462 481 491 499 499 499 506 506 506 512 512 515 516 516 518 524 525 x Donations of Antiquities,—pp. 19, 48, 58, 71, 88, 130, 131, 160, 161, 171, 172, 179, 250, 275, 277, 300, 419, 424, 451, 511, 512, 525, "528; Books,—pp. 1, 63, 154, 261, 346, 451; MSS. »—pp: 1, 131, 416, 525 ; Maps,—pp. 1, 510; Miscellaneous Articles,—pp. 47, 88, 386, 419, 445, 451, 525, 528; Models, &c.,—pp. 17, 31, 94, 123, 311; Photographs, &c.,—pp. 58, 223, 225, 300, 419, 451, 528; Coins, &c.,—pp. 277, 351, 445, 525; Pictures, §c.,—pp. 1, 345, 416. Exections of Members,—pp. 31, 49, 71, 73, 155, 173, 209, 229, 261, 301, 319, 352, 375, 388, 447, 462, 498, 506, 513, 527; Honorary Members,—p. 64; President and Council »—pp. 69 (1854), 207 (1855), 317 (1856), 497 (1857), 505. Exursirions of Antiquities,—pp. 19, 94, 112, 113, 303, 388, 526; Coins,—p. 224; Miscellaneous Articles,—pp. 180, 311, 388; Photographs, §c.,—pp. 71, 390, "391. Grants for Antiquities,—pp. 95, 155, 251, 304; Books,—p. 804; Catalogue of Museum,—p. 497 ; Payments,—pp. 209, 513; Photographs,—p. 17. GRANTS WITHDRAWN,—p. 518. Oxrituary Norices,—pp. 68, 204, 314, 495, 498. _ PARLIAMENTARY GRANTS, —pp. 261, 279. RECOMMENDATIONS OF CouNcIL,—pp. 70, 133, 209, 261, 374, 420, 425 (447), 513, 526, 527. RESOLUTIONS OF THE ACADEMY,—273, 279, 304, 815, 346, 374, 497, 498, 504, 510, 528. VICE-PRESIDENTS NOMINATED,—pp. 70, 208, 318, 497. APPENDICES. PAGE. I.—Account of the Royal Irish Hey: from 1st ieee 1853, to 31st : March, 1854, . . : Prd II.—Account of the pveal Trish Academy, Hee 1st Api 1854, to 31st March, 1855, _ . 2 ix III.—Catalogue of Books siebentea by Mrs, Thomas Moi pe catret et eee eR ITV.—Account of the Blea Trish ee from 1st eee 1855, to 31st March, 1856, . . a ey) Lia V.—Account of the Royals hish Academy, As, 1st tps 1856, to 31st March, 1857, . . . . « Llxvii INDEX OF CONTRIBUTORS’ NAMES TO PROCEEDINGS. VOLUME VI. Aldridge, 244.—Allman, 31, 115, 118, 155, 250. Ball, R., 152.—Barker, J.'B., 351.—Barton, 271.—Bewley, 416.—Boole, 375. Callan, 37.—Campbell, 352.—Carmichael, 216, Cayley, 481.—Clibborn, 347,— Cooper, 194.—Corrigan, 303.—Cumming, 73. Dillon, 157.—Donovan, 491, 499.—Downing, 173. Forster, 240.4 Galbraith, 134, 276.—Gilmour, 302.—Graham, 80.—Graves, C., 1, 34, 84, 112, 114, 144, 162, 186, 199, 209, 220, 221, 248, 302, 385.—Graves, J. T., 525.— Griffith, 230.—Grimm, 290.—Grubb, 59. Haliday, C., 95, 386.—Hamilton, Sir W. R., 62, 86, 114, 181, 250, 260, 311, 415, 462, 506, 512, 524._Harvey, 72, 152.—Haughton, 72, 85, 129, 143, 176, 230, 354, 420, 499.—Hennessy, 279, 299, 301, 388, 506.—Hincks, 72, 113, 125, 128, 251, 270, 518.—Hitchcock, 439. Jellett, 372.Jennings, 386.—Jukes, 89. Kelly, D. H., 49.—Kemble, 462.—Knox, G. J., 59. Larcom, 516.—Leach, 516.—Lloyd, H., 28, 31, 120, 194. Mac Donnell, C. P., 95, 103.—Mallet, 55, 58, 251.—Moore, D., 120. Neville, 311.—WNolan, 261. O'Reilly, 445. Petrie, 251, 441. Reeves, 447, 452.—Renny, 155.—Robinson, 20, 38, 113, 283, 426. Salmon, 273.—Sanders, 338.—mith, J. H., 304, 390, 512.—Stoney, G. J., 499. Todd, 3, 223, 225, 291, 292, 319, 393, 515. Waller, 251.—Westwood, 77. Wilde, 356, 399.—Wills, 247. THE ROYAL IRISH ACADEMY. Marcu 16, 1855. ——— Patroness. HER MOST SACRED MAJESTY THE QUEEN. Visitor. HIS EXCELLENCY THE LORD LIEUTENANT OF IRELAND. President. REV. THOMAS ROMNEY ROBINSON, D.D. March 16, 1852. March 16, 1853. March 16, 1854. March 16, 1855. Elected. March 16, 1838. March 16, 1840. March 16, 1844. March 16, 1846. 30, 1847. March 15, 1851. March 16, 1855. Nov. VOL. VI. Elected 16th March, 1851. Vice- Presidents. (Nominated by the President.) CotoneL Toomas A. Larcom, R.E., F.R.S. Grorce Petrisz, LL. D. Rev. Cuartes Graves, D.D. Sm Rosert Kang, M.D., F. B.S. COUNCIL. Committee of Science. Rosert Batu, LL. D. Sir Rosert Kane, M. D., F. B.S. Grorce J. Atiman, M.D., F.R.S. Sir Witu1am R. Hamitton, LL. D. Rev. SaMuEL Haveuton, A. M. Rev. Humenrey Luoyp, D.D., F.R.S. Rey. Georce Satmon, A. M. Qa * Committee of Polite Literature. Elected. March 16, 1821. Rey. Witt1am H. Drummonp, D. D. March 16, 1843. Joun Anster, LL. D. Feb. 12,1844. Rev. Cuartes Graves, D. D. March 16, 1844. Rev. Samuex Burcuer, D. D. March 15, 1851. Rev. Joun H. Jevtert, A. M. March 15, 1851. Diesy P. Starxey, Esq. March 16, 1855. Joun F. Watter, LL. D. Committee of Antiquities. March 16, 1829. Grorce Perrin, LL. D. March 16, 1837. Rey. James H. Topp, D. D. March 16, 1849. Aguitia Suit, M. D. March 15, 1851. Ear or Dunraven, F.R.S. March 16, 1852. Cotonen Tuomas A. Larcom, R. E., F.R.S. March 16, 1853. Lorp TaLBot p—E MALAnIDE. March 16, 1855. Wuttram R. Wits, Ese. Officers. Treasurer—Rosert Bau, LL. D. Secretary of the Academy—Rev. Jawes H. Toop, D. D. Secretary of Counctl—Rry. Joun H. Jeter, A. M. Secretary of Foreign Correspondence—Rrv. Samurt Burcurr, D. D. LIubrarian—Rey. Wituram H. Drummonn, D. D. Clerk and Assistant Librarian—EpwarD CLIBBoRN. Aug. 2, 1849. Nov. Nov. Nov. Nov. Oct. Jan. Mar. - Mar. Jan. June 27, 1825. Nov. 30, 1849. June 26, 1837. 25, 1836. Jan. HONORARY MEMBERS. Elected. 25, 1836. 22, 1826. . 30, 1852. . 25, 1836. 30, 1832. 30, 1826. 30, 1850. 30, 1852. 28, 1822. 16, 1841. 16, 1820. June 27, 1825. 23, 1826. His Royat Hicuness Prince ALBERT. SECTION OF SCIENCE. Airy, George Biddell, M. A., F. R.S., &e., Astro- nomer Royal. Greenwich. Babbage, Charles, M. A., F.R.S., &c. London. Bache, Alexander D. Washington, D. C., United States. Beaumont, Elie de. Paris. Brewster, Sir David, K. H., LL. D., F.R.S., &c. St. Andrews. Brisbane, Lieut-General Sir Thomas Mac Dougal, K. C. B., F. R.S8., Pres. R.S.E., &. Kelso. Brown, Robert, D.C. L., F. R.S., &. British Museum, London. Cauchy, Augustin Louis. Paris. Daubeny, Charles Giles Bridle, M. D., F. R. S., &c. Oxford. Dumas, Jean Baptiste. Paris. Dupin, Charles. Paris. , Greville, R.K., LL. D. Edinburgh. Herschel, Sir John Frederick William, Bart. D.C.L., F.R.S., &. Hawkhurst. Hooker, Sir William Jackson, K.H., LL.D., F.R.S. Humboldt, Alexander Von. Berlin. Liebig, Justus. Gressen. Murchison, Sir Roderick Impey, Knt., F.R.5., &e. London. a2 Elected. Nov. Mar. Nov. 29, 1828. 16, 1841. 30, 1852. . 25, 1836. . 25, 1835. 26, 1834. . 16, 1827. . 25, 1836. . 16, 1854. . 16, 1842. . 25, 1836. . 80, 1850. . 30, 1849. . 16, 1835. . 80, 1850. . 30, 1849. . 30, 1849. . 25, 1836. . 80, 1835. . 80, 1850. . 30, 1849. . 30, 1852. 25, 1830. . 30, 1852. . 30, 1849. . 80, 1850. 4 Parry, Sir William Edward, Knt., D.C.L., Rear- Admiral, F.R.S., &c. London. Quetelet, Adolphe Jacques. Brussels. Regnault, Victor. Paris. Rennie, George, Esq., F. R.S., &. London. Sedgwick, Rev. Adam, M.A., F.R.S., &. Cam- bridge. Somerville, Mrs. Mary. Chelsea. South, Sir James, Knt., F.R.8., &c. Kensington. Sykes, Lieut-Col. William Henry, F.R.S., &e. London. Vrolik, Wm. Amsterdam. Wheatstone, Charles, Esq., F. R.S., &e. London. Whewell, Rev. William, D. D., F.R.S., &c., Mas- ter of Trinity College, Cambridge. Cambridge. SEcTION oF PowitE LITERATURE. Boeck, Augustus. Berlin. Bopp, Franz. Berlin. Combe, George, Esq. Edinburgh. Cousin, Victor. Paris. Grimm, Jacob. Berlin. Guizot, Francoise Pierre Guillaume. Paris. Harcourt, Rev. William Vernon, M. A., F.R.S. &e. York. Hobhouse, Right Hon. Henry. Hadspur House, Somersetshire. Irving, Washington. Sunnyside, Dobb’s-Ferry, New York. Lepsius, Richard. Berlin. Macaulay, Right Hon. Thomas B. London. Macloughlin, David, M.D. Paris. Prescott, William H., Esq. United States. Ranke, Leopold. Berlin. Thiers, A. Paris. Nov. Elected. . 80, 1848. . 24, 1826. . 30, 1848. 27, 1833. 15, 1835. . 30, 1832. . 30, 1832. . 16, 1841. . 30, 1832. . 16, 1854. . 30, 1850. . 10, 1827. faa. 1837. 30, 1848. SECTION OF ANTIQUITIES. Botta, P.E. Paris. Brewer, James N., Esq. Bunsen, Chevalier C.C.J. Berlin. Cooper, Charles Purton, LL. D., F. R.S., F.S. A., &c. London. Donop, Baron. Saxe Meiningen. Ellis, Rt. Hon. Sir Henry, K.H., Sec. S.A., F.R.S., &ce. London. Forshall, Rev. Josiah, M.A. F.R.S., F.S.A., &c. London. Halliwell, James Orchard, Esq., F. R.S., F.S. A, &e. Brixton Hill, Surrey. Madden, Sir Frederick, K.H., F.R.S., F.S.A,, &e, London. Mauray, M. Alfred de. Paris. Petit-Radel, L.C.F. Paris. Rafn, C.C. Copenhagen. Smyth, William Henry, Rear-Admiral, F. R.S., D.C.L., F.S.A., &e. St John’s Lodge, near Aylesbury, and Atheneum Club. Thomsen, C. J. Copenhagen. Apr. 10, 1843. Jan. Apr. Mar. May Apr. Apr. May Jan. Apr. Feb. Apr. Jan. Jan. MEMBERS. The Names of Life Members are marked with an Asterisk. Elected. 14, 1839. 8, 1843. 12, 1838. 28, 1828. 16, 1815. 9, 1838. 13, 1846. 11, 1846. 10, 1842. 8, 1850. 10, 1837. 26, 1818. 28, 1822, *Allman, George James, M.D. T.C.D., F.R.S., Professor of Botany, T.C.D. Blackrock. * Andrews, Thomas, M. D., F. R.S., Vice-President and Professor of Chemistry, Q. C., Belfast. *Apjohn, James, M.D. T.C. D., F. R.S., Professor of Chemistry and Mineralogy, T.C.D. 32, Bag- got- street, Lower. *Ashburner, John, M.D. 1%, Hyde Park Place, London. Abeltshauser, Rev. I. George, LL. D., Queen’s Pro- fessor of French and German, T.C. D. 32, Rutland-square, West. Adams, Robert, M.D. T.C.D. 22, Stephen’s-green, North. : Alcorn, Rev. John, A.M. Marifield, Clonmel. Aldridge, John, M.D. 3, Sackville-street, Lower. Andrews, William, Esq. 21, Mountpleasant-square, West. Angeli, Signor Basilio, Queen’s Professor of Italian and Spanish, T.C.D. 17, College. Anster, John, LL. D., Regius Professor of Civil Law, T.C.D. 5, Lower Gloucester-street. Armstrong, William, Esq., C.E. 25, Henry-street. *Baillie, Rev. J. Kennedy, D.D. Stewaristown. *Bald, William, Esq., F. R. 8. E. Elected. Feb. Apr. Jan. Apr. Apr. Oct. Nov. Nov. Jan. Jan. Nov. Apr. Feb. Feb. Nov. Jan. Apr. Jan. 23, 1834. 13, 1840. 10, 1842. 14, 1851. 24, 1809. 22, 1882. 30, 1825. 30, 1836. 8, 1849. 9, 1843. 30, 1836. 12, 1841. 12, 1838. 27, 1832. 12, 1838. 10, 1842. 12, 1847. 9, 1837. 7 *Ball, Robert, LL. D., Director of the Museum, T.C.D.; Secretary to the Queen’s University ; Se- cretary of the Royal Zoological Society of Ireland; President of the Zoological Association, T.C.D.; V. P. Geological Society of Dublin; Local See. Botanical Society of Edinburgh, and of the Ray Society, &c.—TREASURER. 3, Granby-row. *Ball, John, Esq., M. P. 85, Stephen’s green, South. *Banks, John T., M.D. T.C.D., King’s Professor of the Practice of Medicine, T.C.D. 29, Merrion- street, Upper. *Barker, John B., M.B.T.C.D. 48, Waterloo-road, *Bateson, Sir Robert, Bart. Belvoir Park, Belfast. *Beaufort, Sir Francis, Admiral, K.C. B., D.C. L., F.R.S., F.R.A.S., Corr. Inst., France, &c. 11, Gloucester-place, Portman-square, London. *Benson, Charles, A.M., M.D. T.C.D. President, Royal College of Surgeons. 12, Fitzwilliam- square, West. *Bergin, Thomas F., Esq. 49, Westland-row. *Bewglass, Rev. James, LL. D. Wakefield, York- shire. *Blacker, Stewart, Esq. 20, Gardiner’s-place. *Bolton, William Edward, Esq. 3, James’s-terrace, Malahide. *Botfield, Beriah, Esq., F.R.S. 9, Strattan-street, London. *Boyle, Alexander, Esq. Belvue Park, Dalkey. *Brady, Right Hon. Maziere, A. B., Lord High Chancellor of Ireland. Hazelbrook, Roundtown, and 26, Pembroke-street, Upper. *Bruce, Halliday, Esq. Glenageary House, Dalkey. *Butcher, Rev. Samuel, D. D., Regius Professor of Divinity, T.C. D.—SercretTary or ForEIGN Cor- RESPONDENCE. College, and 13, Fitzwilliam- square, West. Baker, A. Whyte, Esq. Ballaghtobin House, Callan. Barker, Francis, M.D.T.C.D. 26, Baggot-st., Lr. Elected. Jan. 25, 1836. Feb. 10, 1851. May 10, 1847. Jan. 9, 1837. Apr. 10, 1848. June 8, 1846. June 24, 1833. Jan. 11, 1841. Jan. 8, 1849. Apr. 13, 1846. Dec. 11, 1843. Jan. 8, 1855. Feb. 11, 1850. Apr. 10, 1854. Apr. 9, 1849. Feb. 14, 1853. Jan. 13, 1851. Apr. 10, 1854. Jan. 10, 1842, Feb. 10, 1840. Jan. 8, 1855. Apr. 11, 1842. 8 Barker, William, M. D. T. C. D., Vice-President, King and Queen’s College. 21, Hatch-street. Barker, William Oliver, M.D. T.C.D. 6, Gar- diner’s-row. Barnes, Edward, Esq., C. E. Ballymurtagh Lodge, Ovoea. Barrington, Sir Matthew, Bart. 50, Stephen’s- green, East. Barry, Michael, Esq., Professor of English Law, Q. C., Cork. 75, Gardiner-street, Lower. Beasley, Thomas John, Esq., A.M. 11, Stephen’s- green, North, and Brighton Cottage, Rathgar. Beatty, Thomas E., M. D., Professor of Midwifery, R.C.8. 18, Merrion-square, North. - Beauchamp, HenryC., M. B. T. C.D. 115, Baggot- street, Lower. Bell, John, Esq., F.S. A. E. Dungannon. Bevan, Philip, M.D. T.C.D., F.R.C.S.L. 21, Baggot- street, Lower. Bewley, Edward, M.D. Leighsbrook, Navan. Blackburne, Right Hon. Francis, LL.D. The Castle, Rathfarnham, and 34, Merrion-square, S. Booth, Sir Robert Gore, Bart., M.P. Lissadill, Co. Sligo. Brady, Cheyne, Esq. 46, Waterloo-road. Brady, D.F., M.D., F.R.C.S.1. 14, Frederick- street, North. Brereton, David, A. M., Fellow K.&Q.C.P., M. D. 12, York-street. Browne, Robert Clayton, Esq. Brown’s-hill, Carlow. Burke, Sir Bernard, Knt., Ulster King-of-Arms. Record Tower, Castle, and 2, Pembroke-place. Burrowes, John, Esq. 1, Herbert-street. Burton, Frederick W., Esq., R.A. Nuremberg. Butcher, Richard G.H., F.R.C.S. 21, Herbert-place. Butler, Very Rev. Richard, Dean of Clonmacnoise. Trim, Elected. Feb. 10, 1838. Mar. 16, 1831. Feb. 12, 1855. July 30, 1821. Feb. 12, 1838. Oct. 27, 1798. Oct. 25, 1819. Dec. 28, 1793. Nov. 30, 1835. Jan. 11, 1841. June 23, 1845. May 13, 1839. Feb. 27, 1832. Nov. 30, 1825. Oct. 28, 1822. Nov. 30, 1835. June 23, 1828. Oct. 27, 18384. Nov. 30, 1833. Mar. 16, 1829. June 13, 1842, Feb. 22, 1836. May 18, 1850. Feb. 12, 1855, 9 *Callwell, Robert, Esq. 25, Herbert-place. *Campbell, W. W., M.D. Portstewart, Coleraine. *Carmichael, Rev. Robert, M.A., F.T.C.D. College. *Carne, Joseph, Esq., F.R.S., F.G.S. Penzance. *Carson, Rev. Jos., D. D., F.T.C.D. 18, Fitzwil- liam-place. *Caulfield, Hon. Henry. Hockley, Armagh. *Chamley, George, Esq. 6, Belvidere-place. *Charlemont, Francis W., Earl of. Charlemont House. *Clarke, Thomas, Esq. 124, Baggot-street, Lower. *Clermont, Thomas, Baron. Ravensdale Park, Flurry Bridge. *Connolly, Daniel, LL.D. 36, Fitzwilliam-place. *Conroy, Sir Edward, Bart. + &e. % A Bis) mee os With F = 9A = 36, this gives a = 0°7115 = 8:54 inches. f = 8-02. f= 1-423. M = 240. d = 6:40. @ = 13°88. d= 31:34. Distance of mirror = 29°60. With F = 8A = 32 we have a = 0°6760 = 8:11 inches. S = 6-68. ff = 1:35. M = 240. d=5-4l. 6 =13°88. d' = 28°30. Distance of mirror = 26°59. In these the field is as large as can be obtained with this power m any telescope without a triple eye-glass. 27 «« The Newtonian does not require the same exactness in its arrangement, but I add the formula for it. The data are, that the image must be outside the tube, distant from the axes 34+ m+o-u; m may be neglected if the tube be iron; the eye must receive the whole pencil; and the eye- stop, placed at the image of the large speculum, must be V distant from the last image of the object. With a single lens, calling 26 the axis minor of the small speculum, Fy LEH), , AVES'-8) Sat aE nif ap? ah rf 2 = 5 (24); nano gee ee uU= The condition of vision is V=ur+4, whence , , F-+ 2f' Ref (rss) pO, and the expression of x gives ; F ics ¢ ~ 6A r) PE A a al whence 154 * 900d?) ~ 120° which for F = 36 = 9.4 gives 2b = 0:2222 = 2°67 inches. f' =0-1575 = 1:89 ,, M = 240. 6 = 8°37. For the Huyghenian the formule given for the Cassegrain apply, changing d’ into F, and a, when it belongs to the small speculum, to A: A tan | 2b - m= 5(F + #-)5 0 (oH Se. RS 28 We thus obtain 2 S32 +22f = = od “F(a + soa) whence 26 = 0°2308 = 2-77 inches. Sf’ =0°2998 = 3°60 ,, M = 240. @ = 13°46. ‘«‘ The field here is nearly the same as in the Cassegrain ; with the single lens it is something larger, which arises from that lens acting differently in the two cases, in that it dimi- nishes the image, and therefore requires an increased value of d’.” DEcEMBER 12TH, 1853. THOMAS ROMNEY ROBINSON, D.D., Presipenr, in the Chair. Tue Rey. H. Luoyp, D. D., read a supplemental note “ on the magnetic influence of the moon.” Ina previous communication* the author had shown, from a discussion of the observations made at the Magnetical Ob- servatory of Dublin, that the magnetic declination was subject to a small periodical variation dependent upon the moon’s hour-angle, the north pole of the magnet deviating twice to the east, and twice to the west, in the course of the lunar day. It was, of course, to be expected that a similar variation would be found to affect the other two magnetic elements. In order to trace its existence, and to determine its law, in the case of the horizontal component of the magnetic intensity, the author * Proceedings, May 9, 1853. has since discussed the two-hourly observations made with the bifilar magnetometer in the years 1841, 1842, 1843, the whole series being re-arranged according to the moon’s hour-angle, in the manner already described in the corresponding inves- tigation relating to the declination. No correction has been applied for temperature, the effect of the diurnal variation of temperature being assumed to be eliminated in this mode of grouping the results. The following Table contains the yearly mean results for the several lunar hours, reckoned from the time of the moon’s upper meridian passage. ‘The numbers are the differences between the horizontal intensity at each hour, and that of the entire day, expressed in millionths of the intensity. The re- sults are given for each year separately, and for the mean of the three :— Tasie I.—Diurnal Variation of the Horizontal Intensity related to the Moon’s Hour-Angle. Yearly Means. Hours. 1841. 1842. 1843. Mean. 0 — 45 cient + 34 - 6 2 — 130 oe Geet rh 4 — 57 0 — 18 — 25 6 + 5 + 64 -—l1l + 19 8 +173 +94 + 55 + 107 10 +116 + 80 + 30 fo) ais 12 | + 80 | - 4 EZR) AROS 14 | + 62 | -50 EV 16 | -100 | -41 —82 | — 74 18 | —- 80 | -—32 —14 | ~ 42 20ie lis A 325 + 55 pene 22 = 27 ap) - 4 is 7 It appears from the foregoing Table that the horizontal component of the magnetic intensity is subject to a periodical variation in the course of the lunar day, analogous to that already established in the case of the declination. The hori- 30 zontal intensity is a minimum at about 2 and 16 (lunar) hours, and a maximum at about 8 and 20 hours. The mean amount of the fluctuation is 86 millionths of the intensity, when the moon is to the east of the meridian, and 185 millionths, when it is to the west. The summer and winter lunations yield analogous results. These are given in the following Table :— TaBrE I].—Diurnal Variation of the Horizontal Intensity related to the Moon’s Hour-Angle, in Summer and in Winter. Hours. Summer. Winter. 0 + 8 = 21 2 — 104 - 1 4 itn + 27 6 + 6 + 32 8 + 96 +119 10 + UE + 74 12 + 57 - 6 “14 + 9 — 22 16 — 80 — 69 18 — 39 — 46 20 + 21 + 4 22 + 27 — 42 If it be assumed that the total intensity undergoes no change,—or, in other words, that the variation above deduced is produced by a change-in the inclination alone,—we can in- fer the latter. Its law will of course be similar to that of the horizontal intensity, the greatest inclination corresponding to the least intensity, and vice versd. The total amount of the change, on this supposition, is 0°22, or about one-fourth of the corresponding change of the declination. The magnitude of the change of direction of the resultant magnetic force in the perpendicular plane (= change of declination x cos inclina- tion) is 0°27. 31 Rey. H. Lloyd, D. D., read the second part of a paper “on the Meteorology of Ireland, as deduced from the observations made in 1851 under the direction of the Royal Irish Aca- demy.” JANUARY 9TH, 1854. THOMAS A. LARCOM, Esa., V.P., F. R.S., &e., in the Chair. J. Thomas Rosborough Colclough, Esq.; and J. Butler Pratt, Esq., were elected Members of the Academy. On the recommendation of the Council it was Resolved :— To insert the following By-Law, between Nos. 6 and 7 of Chap. vu. of the Statutes of the Royal Irish Academy :— ‘* Donations received and acknowledged.” ‘The Secretary presented, from James F. Bland, Esq., a very exact and beautiful model of that remarkable and curious ancient structure called Staig Fort, situated on the property of Mr. Bland, near Kenmare, county Kerry. The model was made of portions of stone selected from the original building, and constructed on the spot by Messrs. Thomas and William Jermyn, the tenants of the farm on which the Fort stands. Professor Allman read a paper on the structure of the starch granule obtained from the potato. The author combated the theory of involution recently proposed by Martin, and modified by Busk; he maintained that the conclusions to which these observers arrived were drawn from incorrect interpretation of the phenomena, and that the appearance of unrolling or unfolding of the granule 32 under the action of hot water or mineral acids was a purely secondary phenomenon, and dependent on a condition in- duced in the granule by the action of these reagents. The immediate effect of this action is a swelling up of the granule, but the latter, not at once responding to the action of the re- agents uniformly over its whole surface, certain portions of the surface are first elevated into ridges or projections, which neces- sarily leave depressions of greater or less depth between them, and the appearances which have been mistaken for an unrolling or unfolding of the original granule are due to the act of formation of these ridges, but especially to the rolling outwards of the intervening depressions when these, in their turn, al- most immediately afterwards, respond to the action of the re- agents. The author believes that there is no difficulty in demon- strating in the most undeniable way the composition of the starch granule out of a series of hollow concentric lamelle. If potato starch previously exposed to the prolonged action of a rather weak alcoholic solution of iodine be treated under the microscope with sulphuric acid diluted with about one- fourth water, the granules will, for the most part, present a beautiful dissection of the lamelle from one another, which will then be plainly seen to consist of a series of hollow con- centric shells. In this experiment a solution was generally employed formed by mixing equal parts of water and the com- mon tincture of iodine; and the granules were exposed to its action for the space of two or three weeks. The author also maintained, that while the various la- mellz are probably all identical in chemical composition—for they present no difference in their behaviour towards iodine,— they possess, nevertheless decided differences of another kind, which appear to be referable to conditions of integration. These differences are beautifully demonstrated by the ac- tion of acetic acid on the granule, previously slightly iodized and treated with sulphuric acid ; when thus operated on, the 33 internal layers will be seen to withdraw themselves from the ex- ternal, in the form first of a wrinkled membrane ; and this, gra- dually contracting towards the centre, the granule will finally appear as a spherical smooth-walled vesicle, with fluid con- tents, and with an irregular nucleus-like body—the altered internal layers—lying upon some part of the inner surface of its walls. The author believes, with Schleiden, that the so-called ‘*nucleus” of Fritsche is a minute cavity in the unaltered gra- nule, becoming greatly enlarged by the action of a high heat, as in roasting. ‘The contents of this cavity are rendered blue by iodine, and assume a granular appearance under the action of acetic acid; they are probably fluid or amorphous amylum. From the appearance frequently presented by the granule under the operation of certain re-agents, and especially during the commencing action of hot water, it would seem to follow that there are definite lines of cleavage in the granule at right angles with the concentric lamelle. In conclusion, the author maintained, that the structure of the starch granule, as advocated in the present paper, was much more in accordance with the centripetal than with the centrifugal theory of its growth; but that, while it is to be viewed as really a laminated vesicle, it cannot be properly included in the category of the true organic cell. Dr. Neligan objected, that if the internal and external coats of the potato starch granule be different in constitution, the chemical test commonly applied to distinguish the differ- ent kinds of starch would be inapplicable. Wheaten starch, when triturated slightly, is not as readily coloured by iodine as the starch produced from potatoes, and this seemed to him to be inconsistent with Dr. Allman’s theory. Mr. L. Moore made some observations in reference to the experiments and observations made by Dr. Allman and others who have studied the structure of the starch granule. VOL. VI. D 34 The Rev. Professor Graves communicated the following method of solving a large class of linear differential equations by the application of certain theorems in the calculus of ope- rations :— 1. If ¢ and yp be any functions whatsoever of x, and mand r any numbers, positive or negative, whole or fractional, the symbolic equation (Di 9+ 7F) yay (D+9+ 9") holds good for any subject which we may conceive operated on by its two members. It will be convenient to put pe ged we wy so that the preceding equation may be written in the form Am yr rT YW" Amer ° And operating on this again with the symbol {"( ) ¥", we get Yo An = Arie Yr. 2. It is easy to show that, for \ and wany functions of a, (D+) (D+ p)-(D+p)(D+A) =u -N. Therefore, ify be any function of x, and m any number, ALD sy @ sy ate e yon (5), whence Ay Ag(D + x) =4 (D+ x) Art x’ 9 (5) } At Ar(x- 9!) If we now suppose that X- ~ = ap", (1) where c is some constant, this becomes A, A,(D+ x) =(D+ x) ArAo+ {2eprr=r(F) } de whence again, 35 A,,A,Ay(D + x) = {(D + x) Aap tep-" — Or (5) } 4rd. +A, { 2c" — r (=) } A, : But if we further suppose that (Fars (2) k being some constant, the last equation assumes the simpler form, A, A, A.(D +x) =(D+ x) AxwArdAot (8¢-3rh)p"A,Ao. And continuing the same process, we should find generally AnrAqa-yr.-»ArAo(D+ yx) =(D+ x) Anr Aner.» Ar Ao + {(n+1)e- so rh} b-" Aig-ijp + AA: Or, since by the theorem in §1 the variable part of the last term is equal to Any Aqn-iyr alae A,A,w', We, As [Au(D+ y) = (at 1) ay = (D ar x) Ape (aay toe AA 3. This last formula enables us to effect the solution of the linear differential equation [(D+4)(D+x)- (+ 1je-* AD ray yr] y=X, @) whenever the conditions (1) and (2) are satisfied; as it fur- nishes us in that case with the means of inverting the operator in the left-hand member. Thus we find yA} Al es Ay (Deyyt An.» «Ag A,X. 4, As regards the conditions (1) and (2), it will be ob- served, that the latter limits the nature of the function x, whilst the former makes the difference between $ and y to depend upon that same function. D2 36 The solution of the equation (a may be obtained by putting y = 2* where s is as yet indeter- ‘minate; thus it is reduced to Ss () = hz", z And if we now determine s by making rs = 2, we get x rk dead fr a 2 Differentiating this again, we find ze’ —- 22 = 0, the integral of which is 2 +a°z=0; therefore, z=(D +a’) 0, and 2 y= (D's ay 0) 5. To exemplify this theory we may assume yf = 2”; Cha Wei oh 7 c whence (7 see r=, h=—m,and@-$ =e ——. The general formula becomes, therefore, in this case c\ (n+1)(e+n) N By making c = 0, and writing - m in place of n, this be- comes {(D+9) (D+ 940) - 21 yex, which is equivalent to a general soluble form which Dr. Har- greave has obtained by an entirely different method.* * Phil. Trans., 1848, p. 35. 37 6. Or we may assume yb = (cos x)"; whence (5 =—2msec?2, r= = k=- 2m, and 0-g¢=c,+¢ tanz. In this case the general formula (3) becomes [(D+9)(D + ¢+e.4+ ¢tan 2)-(n+1)(c+n) sec? a]y= Xx. By putting ¢=c,, and c=0, this is reduced to {(D + &) (D + c)-—n (n+ 1) sec? x} y= X, which is the equation solved by Dr. Hargreave at p. 52, in the Paper already referred to, and from a particular case of which he derives the solution of the equation of Laplace’s functions. Sir Robert Kane read a paper by the Rev. Professor Callan, on the results of a series of experiments on the decompo- sition of water by the iron galvanic battery, with the view of obtaining a brilliant lime light. 38 JANUARY 23RD, 1854. THOMAS ROMNEY ROBINSON, D.D., Presipent, in the Chair. On the recommendation of the Council it was Resolved, that the following By-Law be adopted :— ‘¢ That no Member whose subscription shall be due on the 30th November in each year shall have the privilege of voting on or after that date, until his subscription be paid up.” The President read the following Paper on a new method of measuring the angular aperture of the objectives of micro- scopes. ‘< Shortly after the commencement of those improvements which have made the microscope such a powerful instrument of research, it was observed that complete freedom from aber- ration and high magnifying power are not sufficient alone to give perfect vision to a certain class of objects; but that these essential qualities must be combined with a large angle of aperture. The scales of Lepidoptera and Thysanura, and still more the siliceous valves of certain Diatomacez, are well- known examples of this fact; and to the use of them as tests we mainly owe the astonishing progress which has lately been accomplished in this department of opticalart. Several among us remember the admiration which was excited by the first objectives of 20° or 30° aperture, and which became still more enthusiastic when Ross carried this element to 60°, which was considered a ne plus ultra. But as objectives were improved, more difficult tests were found which stimulated progress, till angles of 170° and upwards have been obtained, by this great optician with us; by Nachet in France; and Spencer in Ame- rica. The combination of science and operative skill which is 39 required to produce such miracles of art cannot be too much praised; and up to a certain point it must be confessed, that the performance of these objectives corresponds fully to their promise. It seems to me, however, that in the extreme cases to which I refer, there is not unfrequently some defect of con- struction which prevents them from being quite as effective as their great aperture might lead us to expect. ‘« The effect of angular aperture is merely an increase of illuminating power* analogous to that of linear aperture in a A Cc B E 0 ¥F telescope. Let O be a point of an object seen by an ob- jective whose anterior surface is AB. This point, in the case of a test object, may be considered as self-luminous and equally so in every direction. Therefore, the light which it sends to the objective is measured by the portion of the hemisphere ECF, which is included by the cone AOB. If all that light came to the eye, the illumination would be mea- sured by 47 sin? 0 being = AOC; but this is never the case. The object is almost invariably covered with a piece of thin glass, both whose surfaces reflect a portion of the light if it be mounted dry, one only if it be in balsam.t A * It does not depend on greater convergence of the rays; when the disked stop, hereafter described, was placed in a quarter 105° aperture, the ring of the objective left free showed a test object just as the same illuminating power of an ordinary one would do. + For objects in balsam no light can escape at a greater incidence than 46°; therefore, notwithstanding the absence of the first reflexion, they will ' be less illuminated than in the other way. It is in fact equivalent to re- ducing the aperture of the objective below 100°, as far as illumination is concerned, though a much larger one may be required to take in the pencil ; and it should not be used when it is desired to see details of the utmost minuteness. 40 similar loss occurs by the reflexion at the first surface of the objective, which, like the cover, is plane; and at all the others which are uncemented. To compute these last would re- quire a plan of the objective’s construction ; but as I only wish to give an approximate estimate of the effect of aperture, and as the incidences there, and consequently the reflexions are comparatively small, it is sufficient to consider the loss of light at the first surface and at the cover alone. Further, as the first lens is dense flint glass, and the cover of the ordinary sort, the loss by the two reflexions may be assumed as equal to that caused by the single one of the lens. Taking for the dense flint, » = 1:67, we can compute, by the help of a well- known formula of Fresnel, J the intensity of light trans- mitted at the incidence #: the element of the hemisphere which transmits this light = 27. sin 9. d0; and therefore, the quantity of light transmitted by the first surface is— For an uncovered object, . . . . 2ajJsin Odé. Foracovered, . .. . . . . 2afJ*sin Odd. As yet, however, I have never seen an objective which, when compensated for uncovered objects, has a very large angle; and in some of them the difference is very great. No.4, when set to the mark “‘ uncovered” (which, I presume, was correctly placed by its maker), gave only 70°. From the short working distance which is inseparable from a large aperture, it is not a desirable mode of'using them, as there is a great chance of the lens being sullied.* *< The following Table gives a few values of these angles, omitting the factor 7. *« This reasoning assumes that the light transmitted through the cover is not less reflexible that it was before transmission. 41 First Sur- Aperture. T=1, Uncovered. | Covered. | face ae 20° 1:74 1:62 1:53 1:52 40° 6°91 6°47 6:07 6:06 60° 15°35 14:37 13°47 13°46 80° 26°81 25°07 23°45 23°46 100° 40:93 38°12 35°52 35°60 120° 57°30 52-90 48°86 49°21 130° 66°16 60°38 55°33 56:14 140° 75°40 67:93 61°89 63.05 145° 80°13 71.25 64:70 66°39 150° 84:93 75°23 67-26 69°69 155° 89°79 78:09 69-70 72°83 160° 94-69 81:68 71:54 75°89 165° 99°63 84:77 73:11 78°73 170° 104:60 86:56 73°97 81:52 175° 10959 88°43 74:33 84:17 180° | 114:59 88-60 74:52 86°84 ‘‘ These numbers (which, from neglecting the remaining reflexions, must give rather too much weight to the larger apertures) show clearly that, especially for covered objects, nothing is gained above 150° at all commensurate to the dif- ficulty of constructing such objectives. But in addition to this, I wish to call attention to the fact, that the whole of these great apertures is not in every case thoroughly effective. “The mode of measuring them which is commonly used is that given by Mr. Lister, in which the microscope is at- tached to the alidad ofa circular instrument, with its objective over the centre, and directed towards a luminary at some dis- tance. Looking into it, the field is filled with light, which, on turning the alidad, is seen to have a circular boundary: if this be brought from each side to the middle of the field, the intercepted arc is the aperture.* For objectives of considerable * When the aperture is very large, there is a sluggishness in the appa- rent movement of the boundary, which makes me have some doubt of the accuracy of the process ; it seems almost stationary in No. 6. 42 focal length, this process is unexceptionable; but does not succeed so well with those of high power and aperture; the boundary is very faint, and sometimes difficult to observe. This is especially the case if the light be not uniformly dif- fused, or if the boundary be irregular; and if there be any false light, it is impossible to distinguish it from that which really contributes to vision. «‘This was strikingly the case with an objective which lately came under my examination ; it was a sixteenth, said by its maker to be 160°, and in which light actually was visible to that extent, though not satisfactorily. When, how- ever, it was tried on the Pleurosigma Fasciola, it could not (even with the most oblique illumination that Amici’s prism can give) be made to show this test better than a twelfth from the same artist, whose aperture is only 129°. This unex- pected result made me seek some mode of measurement which would not only give the angle of aperture, but also show how the light was distributed; and the following seems to fulfil both these requirements. “* As a lucid point in the focus of the objective sends out from the eye-piece rays nearly parallel, so light sent in the op- posite direction through the microscope will converge at that focus, and then diverge in a cone whose angle equals the aperture of the objective. If this cone be intercepted at right angles to its axis by a screen, and the diameter of its section, together with the distance of the screen from the surface of the objective, be carefully measured, they give the aperture. If S be the diameter of the section, D the distance, O the diameter of the objective, and J that of the image of the luminary used which is formed in its focus: A S§+0O O 2), al Dalene O ze 2 tan the upper sign being used if the section is measured within the penumbra, and vice versd. In my practice J was so small 43 that the second factor is = unity, for I directed the light of the sun into the instrument by means of the reflecting part of a solar microscope, and not only got measures with extreme facility, but had at once a beautiful map of the objective’s light-territory. «No.1. As a term of comparison I began with a quarter of known excellence. The section was a neat circle nearly uniformly bright, surrounded by a penumbra also equally well-defined. ‘The measures were taken to 0:005, and they gave (denoting their results by 4’, and those of the ordinary process by A), AGE 80°; 8.) (Sic oo, coe S809; Toye which for such determination may be reckoned identical. ‘¢ No. 2. The sixteenth just mentioned presented a very different appearance. The central part of the section was bright, with a jagged outline formed by dark bands spreading outwards in the direction of radii very far; so far indeed, that I could not trace their end, or see any boundary of the sec- tion. Between these bands were stripes of light, but so mottled and confused as to satisfy me that this part of the illumination could not give distinct vision. Looking directly at. the objective, light could be seen even at an obliquity of 85°. The bright part within the radial shadows was not ex- actly circular; but, taking an average diameter, I found for the part which seemed to me alone likely to give effective vision, AC Ors eee saan eo OOL. ‘< No 3. The twelfth, also referred to, is much better: the section was toothed all round like a wheel, but to no great extent, and a defined boundary was visible. The diameter of the bright part gave Wratpg save tien! | eo & poeP..o; x * The compensation of these lenses was adjusted by the Pleurosigma Angulata. 44 This explained the equal performance of these objectives, by showing that in fact the effective portions of their apertures were nearly the same. “No. 4. A similar twelfth, but not quite so good, gave for the bright part, and for the limit of the radial stripes of light, Al = 102°, 0.06 35 Ae 122 in a Ae «No. 5. A tenth, said to be of 156°, had the same jagged edge of shadows fading to darkness without any definite ter- mination; the light stripes were fainter than usual. A= 146; oes, ss A= Lobe In the usual method light could be seen through 159°, but was unbroken only through 115°. “No.6. A twelfth, said to be 170°, and in which light was seen through 169°, but for much of that arc so streaky, that I was prepared to find its performance not quite conformable to its aperture, though its workmanship is of the highest order. It presented in the solar apparatus the same appearance as No. 2, with this exception, that the bright central part was not uniform, two-thirds of it being brighter than the rest, and the change rather abrupt; three or four of the radial shadows were also much broader and blacker than the rest. For the clear diameter, AP 399° Gre A acrme. Ran LPT Gea On looking at this objective, light (and pretty bright too) could be seen even at 90°; which induced me to establish a second microscope, power 41, and fitted with a shade- glass, to examine the image of the sun which was formed in the focus of the objective. This image, whose diameter was 0:0007, was seen through a considerable range as a well-de- fined circle; but at last a bright curve, probably the edge of one of the cells, was seen to approach it o}) When it came into contact with this it became deformed, and increasing the 45 angle, it was drawn out into a brilliant line accompanied by several other images, which caused much confusion, but it was quite visible as faras 170°. All, I think, after it touched the image of the cell, were useless.* «¢ This objective was tried on that most difficult test, the Grammatophora Subtilissima, which it did show, but not nearly so well as a sixth of 132°, whose whole aperture was found to be quite perfect : the image which it gave was ofa shadowy character, contrasting very unfavourably with the sharp definition of its companion; a result quite obvious if it be considered that the good part of its aperture is only 123°, and that the effect of all the rest must be actually injurious. This was verified by introducing behind the objective a stop * This mode of examining the aperture may, in many instances, be useful to opticians, as it can be performed with a camphine lamp. If the ex- amining microscope be fixed on the alidad of the instrument, and that to be examined on a radial slide, light being transmitted through its eye-piece, and both be slided till their focal points be in the axis of rotation (which is ascertained by the image of the flame not moving when the alidad is shifted), it will be found that the image will continue sharp and distinct if the aper- ture be good, till:—1. It begins to decrease in brightness; 2. Its edge dis- appears; and 3. Lastly, it vanishes entirely. Let & be the are between the points where any of these facts are ob- served on each side, and a the aperture of ex. microscope’s objective: se (R + a) O Wee be tan 3 4 = tan ~—3— a x. (R -a) O De pan rate hae Se ntaTe gCh Lhgh art we (R-a) O Dy ae MEE gr Pm BS eS ei The first of these is the least accurate. If the objective have the defect just described, the image retains its brightness, but becomes deformed after a certain angle. In this way I esti- mated by the formula 2, the good part of No. 6 to be = 122°, 75, not far from the result given in the text. Four other objectives of large aperture gave by it measures according with those of Mr. Lister’s method; though, among them also, one of 105 was indistinct for a few degrees. 46 of 0:125 diameter, which cuts off the irregular part. With this the Angulatum was seen in the most satisfactory man- ner; but when it was replaced by one carrying a central disc of 0°125, which stops out the good part, it would scarcely show even a trace of lines on the coarsest Pleuro- sigmata. «¢ These facts, I trust, are sufficient to show the necessity of attending not merely to the amount of aperture, but also to its quality. The objectives which I have examined are all of first-rate excellence in the good part of their apertures : I have not named their makers, nor the friends to whose kindness I am indebted for the power of examining so pre- cious a collection, because it might lead some to unjust con- clusions respecting the merits of the former. It is from no wish to depreciate the debt of gratitude which we owe them, or to undervalue the wonderful skill which they have shown in correcting aberrations so perfectly as they have done in these very objectives, that I make this communication, but from a wish to point out to them a defect which they will be able easily to remedy, and which at present occasion- ally interferes with their complete success. What its cause is can only be determined by one familiar with the construc- tion of the objectives, but it probably arises from some of the lenses being so small that their edges meet the lumi- nous pencil and reflect false light. I am induced to suspect this from observing that when in No. 6 the first lens was separated from the others by a revolution of the compensating screw beyond its proper adjustment (in which state, however, it would have no definition), the radial shadows disappear altogether, and A’ = 110°,3. Reducing the separation by a fourth of the revolution, they are seen, but of no great length, and enveloped in a blue zone, which surrounds the bright centre, the latter giving A’= 139°,9. ‘This increase of dis- tance has the effect of stopping out the margins of the other 47 - lenses. The disturbance, however, may also be owing to the brass of the cells, but if so the remedy is the same, namely, increasing a little the diameter of the posterior lenses. It is true that this involves an increase of their thickness, and of course a considerable change in the combination. I would also suggest another alteration, in case it be thought de- sirable still to make objectives of these extreme apertures ; that the anterior surface be concave instead of plane. I do not suppose this would much increase the difficulty of the work of correction, and in fact No.5 was so constructed. To show how much illumination would be gained by this, I have computed the fourth column of the Table, supposing the curvature such that the final incidence is 75°, and the object covered; from which it will be seen, that the difference at the limits is nearly the full power of an objective of 60° aper- ture, and that it almost compensates for the loss of light due to the cover.” The Rey. Dr. Todd read a Letter which he had received from William H. Harvey, M.D., written in Ceylon in No- vember last, giving an account of his botanical and zoological collections. He also exhibited a restoration by Joseph H. Smith, LL.D., made from a rubbing of an ancient Irish in- scribed grave-stone, with an inscription. Dr. Petrie made some remarks upon the inscription. The Rev. Dr. Todd presented a Walloon tobacco-box, with several Flemish inscriptions, from the Rev. William Thomp- son, said to have been found on the person of a soldier slain in the Battle of the Boyne. ——— 48 Sir W. R. Hamilton having arrived after the period for reading papers, handed in a notice of his being ready to read his paper on the geometrical interpretation of biquaternions. Dr. Petrie presented an ancient brass cauldron found at Sallow Glin, near Newtown, county of Kerry, from William Sandes, Esq., forwarded by Henry Stokes, Esq. 49 Fesruary 137TH, 1854. HUMPHREY LLOYD, D.D., Vice-Presipent, in the Chair. Cuar.es Domvit_e, Ese., Rev. Robert Ferguson, LL.D., and J. R. O’Flanagan, Esq., were elected Members of the Academy. D. H. Kelly, Esq., read a paper on an ancient terraced gravel hill, near Castle Blakeney, county of Galway :— « This remarkable object is a gravel esker, near the vil- lage of Castle Blakeney, in the county of Galway, and is situated close to the remains of the old Castle of Gallagh, the seat of a once-powerful family of the O’Kellys of Hy-Many, and which furnished several chiefs to that ancient toparchy. ‘* The present appearance of this esker is very remark- able, as may be seen by the rough sketch sent herewith. *« A length of 355 yards is cut off by two deep trenches at each end from the centre of the gravel ridge, and the part thus isolated is carefully levelled on the summit and the sides, artificially cut into terraces, like the mountain vineyards of the south of Europe, and the East. *«'The summit is carefully levelled into a terrace, 36 feet wide; on its southern side are five other terraces, 16 feet wide each; and on the north are three terraces of the same width still existing, but there may have been originally more, as a large fence now skirts the hill on that side, whose con- struction may have obliterated others. “¢ At the eastern end the terraced portion is cut off by a deep trench, or roadway, 20 feet wide, from an uneven mound, the remains of the debris of the ancient Castle of Gallagh, out of the ruins of which was constructed the mansion of Gallagh, VOL. VI. E 50 which, together with the surrounding property, became for- feited in 1641, and then passed from the O’Kellies of Gal- lagh, now worthily represented by Connor J. O’Kelly, late of Tycooly, but who now resides on a purchased estate, to which he has given the name of Gallagh, after the home of his an- cestors; it then passed to the Blakeneys, of Abbert, now represented by J. H. Blakeney, Esq., a family always distin- guished in the military service of their country, and of which the gallant defender of Minorca, in the days of George IL., and in our own day, Sir Edward Blakeney, K.C.B., Com- mander of the Forces in Ireland, have been the most illus- trious ornaments. ‘Of this rebuilt pile, but one solitary chimney now re- mains to tell of its former splendour, but the present occupant of the farm assures me, that, when he got possession of it some twenty years ago, sixteen such chimneys were then standing, and a pile of building that gave evidence of very considerable extent. «To the western end of the terraces the gravel esker appears to have been left completely in a state of nature, nor do I believe that it ever has been disturbed by any agricul- tural process. “In the Annals of Clonmacnois, 4.p. 1351, it is stated — * William mac Donogh Moyneagh O’Kelly invited all the Trish poets, brehons, bards, harpers, gamesters, or com- mon kearrachs, jesters, and others of their kind in Ireland, to his house upon Christmas, this year, where every one of them was well used during Christmas holydays. And he satisfied each of them with presents at their departure, so as that every one of them was well pleased, and extolled William for his bounty, one of which assembly made certain Irish verses in commendation of William and his house, which began thus :— Pld Eipean so h’aon ceaé.’ 51 ‘‘ We are enabled to ascertain the author of this poem by one of Dean Lyons’ tracings, now in the library of the Royal Irish Academy, and which was taken from a MS. in the College of St. Isidore, at Rome, from which it appears to have been the composition of Geoffrey Fionn O’Daly. ‘¢ This is the tracing— OOPFPod Prono o Oalargs ccc Fd Efpein 50 h’aén ceaé. *T am in possession of a very beautiful copy of this ancient poem, transcribed and literally translated for me by my valued friend, Eugene Curry, to whom Ireland is deeply indebted for rescuing most valuable portions of her history from oblivion and misrepresentation. «‘ But before comparing this remarkable vestige of the olden time with the description given in this ancient poem, it may be well to observe that William O’Kelly was the son of Donogh Muimneaé O'Kelly, Chief of Hy-Maine, ob. 1307, by his wife, Ourhéara ni Conéobaip (Duveassee O’Conor), daugh-~ ter of the King of Connaught. He will be found No. 27 in the Tabular Genealogy of the O’Kellies, in ‘ The Tribes and Customs of Hy-Maine,’ edited by my learned friend, Professor O'Donovan, for the Irish Archeological Society. “ This poem, after extolling William’s liberality, and de- scribing how, by means of his invitation, the other districts of Erin will that year be bereft of their bards, proceeds— bidid aca altne a éeile To each other will be known Oatha Podla pionperoe The professors of smooth-landed Fodhla; Ipoatha Alban eaécpa cian And the far-travelled professor of Alba, Cp oceacta a n’apobpug Uil- Coming to William’s noble man- liam. sion. GiucRpad pin na peact nsndoa Here will come the seven orders Do mo veilb an veas dana Who form the shape of good poe- try; E2 52 Sean v10sbala a oceaéc apteac A charm for misfortune is their Na feacc bpmom-spada Fi- lead. coming, The seven chief orders of the poets. ‘¢ A little further on he tells us what they were— bé10 bpeateatham bpeas noli- $10 bE Opaoize 7 0e1§-pIhd bé1d Na pulps soaip Cipean Luéct cimoaig na sceaitpei mean. Cep citil Cipeat anba anopeam Luct gaé ceipove H0 coiccean Cn cuile dam leac an leat Cn nodal uile 50 haon ceaé. There will be the Brehons of legal judgment, There the Druids and good poets; In his mansion will be the au- thors of Erin, The chroniclers of triumphant histories. The musicians of Erin in vast numbers, The followers of every science in common, The flood of professors from all quarters, Are all journeying to one house, *‘ He then, in a succeeding rann, describes the accommo- dation provided for them— Acdc loingtige leabta Fa comaip na curoeacca Cp dpomélad nglan ocealacoze Cp n’eagap dedoaé fice. There are long houses of beds On the smooth ridge of the dry eminence, In preparation for the company, Well furnished with woven cloth. « This exactly—‘ the smooth ridge of the dry eminence’— describes the terraced esker on which were erected wattle houses, covered with cloths like tents, and the poet then goes on to describe the respective streets, laid out for the accom- modation of each class, and if we suppose a terrace allotted to each, we exactly coincide with the locality. 53 Achid biidean ctpom 04 Gos Spmio bptigean copp Fa a scomoin Na ngoipe pn puaipe an psiaih Oo hénoargead le hUitiam. Spdio apleit oonluéc peanma Od mbead pe a uéc ppeadma Seancaide Eineani ailte. lpuompaésgadna ctnom-ddime. Ccdi0 fan mbaile an bii1dean Cdide ppdio na Seancaidead Acd ppdio pain ping oile Tona lpuilio pion-corse. ?60n cplos Pion ap pial cpord- ead Cp cion chiap ap cleapaiseao beasg pin as Péacald a bpuil Timéioll catpars ui Ceallarg. Oo ppdioi’ ap mo meadain 5g flog aig-pip ofpbealais Cipé veapgad pil oppa Raom faip pin se eacoppa. Map acdio lizpe na limb Na ppdio tilce caoib ois Hac pndice pligead lume Réide DeIpPse O10s5uTie. A numerous company approaches the mansion ; A street of well-formed houses awaits them; Near unto these, joyous its fea- tures, Has been ordered by William— A separate street for the musi- clans To be ready before him; The historians of beauteous Erin, And the heavy throng of their associates. In the town is the multitude, Where is the street of the Shena- chies; There is another extensive street, In which are fair houses. With free hearted hosts, For receiving the histrionictroop, Trifling are these, seeing all that are Around the Dun of O’Kelly. Of streets of greater merriment, With generous hosts of manly aspect; The manner of their situation is With wide passages between them. As letters are placed in line, So are those straight, intersected streets ; Every line of every street exact, Smooth, unobstructed, pleasant. 54 Oo GigIb blade cleat copp Of houses graceful, handsome- ridged, O04 pndite teaé na ciméroll Each street by two ranges is lined, Cp clap cilze 00 Go1sead Thick set with houses is the level Opum an acaid Pron-cloipos ~——Of the ridge of habitations, with its white enclosure. Aca ap Chl an cldaip cilce At the end of the crowded level OGn ina cafp cinlicpe Is a mansion like a capital letter; Cinlicip cloice alle An illuminated capital, a beau- teous castle; On na plata pion saille The Dun of a fair-cheeked chief, Oainsean cloé an ofnaid de And the stronger is the Castle Dun by it, Loc ap cilaib na cloiée. A lake behind the Castle. Realca cloice ap cian popeap A star of a Castle as such long acknowledged Of lif loca na n’Cigear Over the waters of Lough na-n- Eigeas; O10 dille an élaé von caob Gall However beauteous the Castle within, G caom amaé map theampam. Its outer surface was like vellum. « After carefully examining the locality, I feel no doubt on my own mind that this very remarkable vestige of the days of yore is the scene of the entertainment immortalized by O’Daly, and that the Castle of Gallagh, which then adjoined it, was the ‘illuminated capital letter of a Dun’ described by the poet as dominating over the straight lines of streets, as may be beheld in any of our old MSS., and a very apt simile it was for the relative positions of the locality. It is true the lake no longer forms part of the scene, but any one who takes his stand on the mound where once the Castle stood will perceive, just behind it, an extensive morass, which, before these days of drainage, may well have been Loch na-n-Eigeas.” rh 55 Robert Mallet, Esq., read an account ofa remarkable lunar halo on the night between the 10th and 11th instant :— «¢ At 12 o'clock, night, between the 10th and 11th Febru- ary, 1854, looking from a southern window of my house (Delville, Glasnevin), I observed the nearly round disc of the moon, then thirteen days old, riding high in an almost clear heaven, and surrounded by a very large and perfectly circular halo. ‘The size of the circle was such as to convey an idea of great grandeur, almost of awe, and the great comparative diminution of the apparent magnitude of the moon’s disc within it, which really seemed as though it could be covered by a shilling, was equally striking. ‘The annexed dia- gram is intended to repre- sent the general appear- ance of the halo. ‘7 had no means of directly measuring the an- gle subtended by the inner edge of the ring, which, however, was so large as to be with difficulty kept within the field of vision § with the eye fixed. “The inner edge of ™™ the halo was well defined, and slightly tinted with prismatic red, passing outwards into orange and yellow, and the whole becoming evanescent into white vapoury mist or light cloud. The prevailing colour of the mass of the ring was that of white bright moon-illuminated cloud. The width of the distin- guishable annulus (which was almost uniform all round) was from one-eighth to one-seventh of the internal diameter. To- wards the lower part, the shading off into vapour assumed a streaky appearance, like scirrus cloud, and below the ring were larger surfaces of very light fleecy scirrus and cumulo stratus, 56 clouds. One star of the third magnitude was clearly visible in the deep blue-gray surface inside the ring, which was free from any trace of vapour or cloud,—the other stars were lost in the moon’s light. ‘There was neither corona nor paraselene. “‘ The night was cold and frosty, the air dry and crisp, and pleasant to the feelings at the surface of the earth. The day had been remarkable fine and clear; the preceding night a clear one, with hard frost. ‘« There was little or no wind stirring at 12 o’clock, night, 10th-11th February. The barometer had been high for several days, and at 8 o’clock a. m. on the morning of the 11th February was 30°52 inches ; thermometer on a northern exposure, 35° Fahr. ‘¢The front of the house whence I looked is due south, the face ranging therefore E.and W. Looking out of window, and directly upwards. with my eye as nearly as possible plumb under the edge of the stone cornice about twelve feet above me, I perceived that the interior of the upper limb of the ring almost exactly reached the zenith, and formed a tangent to the line of cornice. «The moon’s place being known, therefore, it is easy to find the apparent diameter of the ring. H Thus let HH be the horizon; O the place of the observer ; Z the zenith ; M, moon’s place; ZR, apparent diameter of the halo. 57 ‘¢ For the position of the moon I am indebted to our Pre- -sident, Dr. Robinson. «‘ The place of observation may be taken as— Latitude, 53° 22’ 29” Longitude, 0° 25™ 12s Then at 12, night, of 10th-11th February, 1854,— Zenith distance of moon’s centre = 42° 38’ 29” Parallax, + 1 Sond Refraction, = ane Apparent zenith distance of moon’scentre,. . . . . . 43° 14° 33° Semi-diameter, i 14’ 45” This would give the apparent internal diameter of the ring about 864 degrees, which would bring that of the densest and most highly illuminated part, at or near the yellow light, to about 90 degrees. ‘‘ This halo may, therefore (as suggested by Dr. Robinson), have been one of those formed by two refractions. ‘The ab- sence of any inner ring, however, throws some doubt upon this. ‘‘ The determination of the precise angles subtended by these halos is of interest. Dr. Young having shown that they depend upon refraction and reflexion from minute crys- tals of suspended ice, and Arago’s experiment having proved that the light which passes has been polarized by refraction, it follows that the angle of the primary form of ice being as- sumed = 60° or 30°—measurements of this character afford the means of obtaining the possible secondary crystalline forms of ice crystals suspended in the higher regions of the atmos- phere, and there produced under conditions likely to extend our knowledge, as yet so very limited, of some of the forces upon which secondary crystalline forms depend for their pro- duction. . 58 ‘‘It may be remarked, that Mr. Howard has recorded his observation that halos of this sort occurring in spring are usually succeeded by very hot weather.” Mr. Mallet also read a paper containing notices of the facts of the British earthquake of 9th November, 1852. On the part of Mrs. Lambert, of Lough Scur, Cashcarri- gan, Rev. Dr. Todd presented a stone mould for casting bronze celts. He also presented, from William Wakeman, Esq., a rubbing from an inscribed stone in the island of Inis no Coill, in Lough Corrib, shown by Dr. Petrie to be the grave- stone of Lugnat, son of Liemania, sister of St. Patrick. 59 Fresruary 27TH, 1854. THOMAS ROMNEY ROBINSON, D.D., PresipEnt, in the Chair. Tue Secretary read a paper, by George James Knox, Esq., on the transmutation of metals. Mr. Thomas Grubb read the following account of a new method of determining, approximately, the spherical aberration of a combination of lenses for microscopic purposes :— «‘ The methods hitherto at our option for investigating the spherical aberrations of a system of lenses, having spherical surfaces, are—firstly, the purely mathematical, involving (where the thickness of the lenses is required to be included, and more especially where the angle of aperture is consider- able) such intricacy in the calculations as renders the process nearly useless to practical persons; and, secondly, the more practical method of constructing diagrams of large size, in which two or more rays, at different distances from the axis of the compound, are geometrically traced, according to the laws of optics. «‘ Much of my leisure time has, for several years, been devoted to inquiries including, necessarily, the construction of such diagrams; and I cannot speak too highly of their value in giving to the experimenter in optics a thorough practical insight into the effects of the various forms and combinations which will be suggested, more especially when the improve- ment of the compound objective of the microscope is under consideration. ‘Tt may be desirable here to state, that these diagrams were usually drawn on a scale ten times that intended for use, the radii of the arcs, from which the sines were measured 60 (and which were constant for all the surfaces of a combina- tion), were usually three and sometimes six inches ; the arcs were struck and the sines measured at both sides of the centre (to avoid errors of eccentricity), and the measurement of the sines taken with a scale of fiftieths of an inch, using a magni- fier, and estimating to tenths of divisions, or say 1-500th of an inch. i «¢ By tracing—with these precautions, and with care—two rays, one passing through nearly the margin of the combina- tion, and the other at one-half that distance from the axis, an approximation to the state of the spherical aberration of the compound, adequate for such combinations as are used in the construction of the objective of the microscope, may be obtained. ‘‘Some practice in delicate manipulation is, however, required to arrive at the precision indicated, and even for approximate investigations a more accurate method would be desirable. ‘¢ When, however, combinations of small aperture, com- pared with their foci (for example, such proportions of these as are adapted to the object glass of the telescope), are under examination, the method of diagram fails entirely in the ac- curacy requisite to estimate the correction of the spherical aberration. ‘Such difficulties in the way of practical research in- duced me to consider how far the objections, arising out of the intricacy of the one mode of investigation, and the inade- quacy of the other, might be obviated by adopting a mixed method of proceeding. In short, it appeared that the diagrams not only included those conditions arising out of the thickness of the lenses, but that they would serve to furnish with suffi- cient accuracy for all cases that point in each surface where the ray impinges, and also the angle of its incidence. Little more thus remained to be done than to find a simple expres- sion for the aberration of each surface for the ray, and this, 61 after some trials, I obtained in a very convenient form, as follows :— ‘«‘ J) being the deviation of the ray (not from one lens into another, but always) either from air into a lens, or from a lens into air, a is as (sin? D). Then, calling the aberration of the entire combination A, and a, a,, a3, &c., the aberrations (calculated from the above formula) at the Ist, 2nd, 3rd, &e., surfaces of the compound, 4 is equal to the sum of a, a, a;, &e. (respect being paid to their signs), these being plus when the ray is deflected towards the axis, and minus when the contrary. «¢ When the spherical aberration of the compound under examination is corrected, the sums of the + and — aberrations will be equal, and A will evidently be zero, while in any other case, the + and — quantities will exhibit a fraction showing their relative proportion, and will indicate to the practical operator the means of a closer approximation to perfect correction. «¢ The entire process may be shortly described as follows:— «A diagram of the combination to be examined being made with care, and of a size as large as circumstances permit, —one single ray must be traced (in the usual manner) through the diagram; the most suitable distance for this ray from the axis of the combination will depend upon the angular aperture. For combinations of small angular aperture I prefer 3-4ths to 4-5ths of the semi-aperture for this distance. In addition to this (should the combination have any two sur- faces in contact), the direction of the ray from these into air must be projected. “The diagram is now fitted for obtaining the sine of D for each surface; these, being measured on a scale of equal parts, and tabulated, and afterwards squared, Give di, G2, As, &e., which being separated, as directed, into +-and -, give, in their sums, two- quantities, representing respectively the positive and negative aberration of the whole combination. * 62 ‘¢ Having examined diagrams made some time ago, and the calculations connected with same, with a view to ascertain the limits of error of this process, it appears that, allowing an error equal to 2-10ths of a division of the scale used in estimating the sines, the probable limits of error are only 1-50th part of the numerical sum of the aberrations, a quantity which may be considered insensible in practice, and probably not one-half of that error, or departure from the true spherical figure of a surface, which takes place (and with contrary signs) during the polishing, according as the lens, or polisher, is upper during the process.” Rev. Dr. Graves read a note from Sir W. R. Hamilton, in which he stated that he had lately arrived at a variety of results respecting the integrations of certain equations, which might not be unworthy of the acceptance of the Academy, and the investigation of which had been suggested to him by Mr. Carmichael’s printed Paper, and by a manuscript which he had lent Sir W. Hamilton, who writes,—‘‘ In our conclu- sions we do not quite agree, but I am happy to acknowledge my obligations to his writings for the suggestions above alluded to, as I shall hereafter more fully express. <¢ So long ago as 1846, I communicated to the Royal Irish Academy a transformation which may be written thus (see the Proceedings for the July of that year) : D,?+ Dj + D?=-(iDr+jDy + kD:z)?; (1) and which was obviously connected with the celebrated equa- tion of Laplace. “But it had quite escaped my notice that the principles of quaternions allow also this other transformation, which Mr. Carmichael was the first to point out: D?+D,?+ Dy =(Dz-tDz—jDy) (Dz+iDz+jDy). (2) And therefore I had, of course, not seen, what Mr. Carmichael has since shown, that the integration of Laplace’s equation of 63 the second order may be made to depend on the integrations of two linear and conjugate equations, of which one is (D,-iDz,-jDy,)V=0. (3) ‘ Ase os » annpa, difficult. QPoGtipe: sees hart seis GTLOS high. caimems) 59.29. » cam, delightful. epmemsy 7.270. » epion, old. SIGIR Su ieees » lap, gray. OIOGIN een) ia 3 Sputo, bitter. millpem, ... . >» milip, sweet. WIOOPOUNy G of 5 Hy mon, great. minemiy | Sep es 3 Pell, yold PURGE, 5's » [Ppuitd, distinguished. CINOEM ee wie oe re “ ceann, strong. cTpeipim, . . . . >» cpeip, strong. uaiplem } and ise Ds » uapal, noble. uaiplium, f ‘¢ Amongst these will be observed two in im, and two in um, terminations of which Professor Zeuss seems to have met with no examples.” The Rey. Samuel Haughton read an account of some ex- periments made to determine the velocities of the rifle bullets commonly used. The Chairman remarked, that such papers as that read by Mr. Haughton, combining both the highest science and the most exact practical knowledge, were of the greatest national importance. 86 Sir W. R. Hamilton, having been lately induced to con- sider, in connexion with the Calculus of Quaternions, the celebrated theorem of Dupin, respecting the character of the intersection lines of three systems of orthogonal surfaces, as lines of curvature thereon, stated that he had thus been led to perceive some symbolical results which he supposed to be new, and which seemed to him to be of sufficient interest to be submitted to the Academy. As long ago as 1846, he had proposed the notation,* id tere d Saiaancd 7 +k ae and had pointed out a theorem,} differing only slightly in its expression from the following : V.aV. By=yS.aB—-BS. ay; which may also be thus written, V.a(V. By)=8.aB.y-BS.ay, V.a(V.By)=S.Ba.y-BS. ay. The recent results just referred to have a remarkable sym- bolical resemblance to those comparatively old ones, since they admit of being written thus : or thus, Teese V.a(V.dv)=S.ad.v-d8.a0; PPD: 2: V.A(V.Bv) =8.B4.v-BS. dv; where >. 4.9500 Mapnesiay: . oa) O34) |< 0:017 Lime, 9 0:70 0:025 Potash, ; . 11:39 0:242 eee ek ie SOdR ids 0) eink & M2590 0:095 Loss by ignition, 0°58 98°54 Sp. gr. = 2553. GLENDALOUGH. Per Cent. Atomic Quotients. Proportionals. Silex, 63°60 1-404 3:83 Alumina, 18:84 0°366 1:00 Magnesia, 0-40 0:020 } Lime, . Trace, 0:000 ee 108 Potash, 14:33 0:305 | : Soda, Ae 1-92 0-060 J Loss by ignition, 0°60 99°69 Sp. gr. = 2°453. 141 GLENMALUR. Per Cent. Atomic Quotients. Proportionals. Sens rat ST HOA AB Ss PORAB SU”. het ato B84 Ainmimaes: . . 2 19:04 6. 2 O:370).96 .e 1:00 Meemesia,at 0!) S102) ~ 2. 0051 Hamers wht els. v4 Drace,. ») .. :0:000 Fe i be 1MOCTA,. «:..< k MIBI dene Soda, Pr ies el OA 3 ee O8S Loss by ignition, 0°78 98°70 ; Sp. gr. = 2:560. ‘¢On examining these analyses, it will be found that not even in a single instance do they deviate from the well-known composition of orthose or potash felspar. The average ratio of potash to soda from these analyses is 9:2; the greatest amount of soda present being 34 per cent., which is found in the specimen taken from the Three Rock Mountain, which quantity, although relatively great, is less than one-third of the potash in the same specimen. ‘We find, from the extract which I have read, that Sir Robert Kane was unwilling to found a statement as to the nature of our granites so important as this on an inference drawn from the analysis of a water taken from the locality ; for he affirms that his conclusion is verified by a number of analyses of granite specimens taken from various parts of the range, and that in all these specimens so great was the excess of the soda over the potash, that he felt himself warranted in drawing the conclusion that the potash should, in most cases, be considered as belonging to the mica, and that the felspar was almost exclusively an albitic or soda felspar, containing only in some cases a small quantity of replacing potash. ‘On referring to the analyses which I now lay before you, I think I am warranted in the statement, that the felspars of this district are orthose or potash felspar, containing only small quantities of replacing soda. . VOL. VI. ) 142 ‘¢On the occasion when Dr. Apjohn made his communi- cation, I find that Sir Robert Kane repeated his statement as to the nature of these granites, together with the grounds on which he rested it. ‘The idea,’ viz., that the predominant characters of the granite district of Dublin and Wicklow was the presence of soda felspars, ‘ was founded not merely on the results of the analysis of the waters read at the last meeting, and which in itself he did not consider very important, but was the result of a widely-spread series of observations, which, on another occasion, he hoped to be able to bring before the Academy.’— Vol. v. p. 382, February 28, 1853. ‘“¢ Feeling, as I do, a great interest in this question, I con- fess I have for a long time looked forward with some anxiety for the production of these analyses; and I hope I may be allowed, on the part of the Academy, to express the hope, that at no distant day Sir Robert Kane will lay before the Academy what I am sure will be esteemed a most valuable communication. Independently of my wish to arrive at the settlement of a scientific question, I feel some anxiety on the subject for another reason, namely, my utter inability to re- concile Sir Robert Kane’s statement, with regard to the results of his rock analyses, with what (at least to my own satisfac- tion) I think I have succeeded in proving. The granite of Dublin and Wicklow, as I have said, consists of three ele- ments, quartz, mica, and felspar. Of these quartz, as is well known, contains no alkali, and may be therefore dismissed from our consideration. Mica contains both alkalies, but the potash always in great excess. Felspar, the third element, con- tains, as I have shown, both potash and soda, but the former always in great excess, in the ratio, namely, of 9:2. This being the case, I cannot in any way see how a rock, which is a mix- ture of these three constituents, should on analysis yield ‘both potash and soda; but the latter, viz., soda, almost always so preponderant, as to lead to the conclusion that the potash should, in most cases, be considered to belong to the mica 143 which the granite contained, and that the felspar was almost exclusively an albitic or soda felspar, containing only in some cases a small quantity of replacing potash.” Rey. Mr. Haughton stated, with referenee to Mr. Gal- braith’s communication, that he had himself made a chemical examination of some specimens of mica from the Wicklow and Dublin granites, with reference to some remarkable differences in their optical properties, and that he had found in these micas, as Sir Robert Kane himself had observed, a prepon- derance of potash over soda, on an average of about 15 to 2. This result seemed to Mr. Haughton to render very difi- cult of satisfactory explanation the result of Sir R. Kane’s unpublished analyses of granites, in which he found a great preponderance of soda; for if the micas contain potash to soda in the proportion of 15 to 2, and the felspar, as Mr. Galbraith had just demonstrated, contain potash to soda in the proportion of 9 to 2, Mr. Haughton confessed himself unable to understand how a rock made up of such minerals could contain a great preponderance of soda. He quite concurred in Mr. Galbraith’s wish, that Sir R. Kane would favour the Academy with his promised analyses. Dr. Apjohn remarked, that in his opinion Mr. Galbraith’s analyses were conclusive as to the relative quantities of potash and soda in the Dublin and Wicklow felspars, and were quite coincident with the opinions which he had himself previously expressed upon this subject. He further observed, that he objected to the use of the terms orthose and albite, as distin- guishing between potash and soda felspars, as both alkalies might and did occur in both minerals in varying proportions. The true distinction, in his opinion, between these minerals was crystallographic, the former always occurring in the fifth, and the latter in the sixth system. Mr. Mallet mentioned a fact which had come under his observation, which confirmed in a remarkable manner the.re- 02 144 sults of Mr. Galbraith’s analyses. Having had occasion to analyze the water of the river Liffey above King’s Bridge, in order to ascertain the quantity of alkalies contained in it, he found distinct evidence of the presence of potash, and none whatever of the presence of soda. And as this river takes its rise in the granite platform of the Wicklow hills, and might be said to contain the washings of that district, the presence of potash strongly confirms the opinion maintained by Mr. Galbraith, that the felspar of the Dublin and Wicklow hills was potash felspar. Mr. Galbraith explained, that he had used the terms orthose and albite in the sense in which Sir R. Kane had used them, although he did not consider it quite exact, as his object was to confine himself exclusively to the consideration of the rela- tive numerical quantities of potash and soda in the Wicklow felspars. The Rev. Professor Graves continued the reading of his Paper on the principles which regulate the interchange of symbols in certain symbolic equations. Let 7 and p be two distributive symbols of operation, which combine according to the law expressed by the equation, pw =7p+a, (1) a being a constant, or at least a symbol of distributive opera- tion commutative with both 7 and p. In this fundamental equation, if we change 7 into p and p into —7, it becomes —Tp=—pT+a, or (2) pT =T7P Ta, the same as before. From this it follows, that in any symbo- lical equation, p (x; p)=9, (3) which has been directly deduced from the fundamental equa- 145 tion (1), without any further assumption as to the nature of the operations denoted by 7 and p, we may change 7 into p, and p into — 7; so asatonce to form the correlative equation, $ (p, — 7) = 05 (4) for this latter will be deducible from the primitive, in the form (2), by the same processes, whatever they are, which conduct us from (1) to (3). The value of this principle must depend upon the extent of its application ; and this will be found much wider than might at first sight be supposed. For asymbolical equation of the form (3), which is verified for any subject whatever ope- rated on by its left-hand member, if it be not, in its existing state, identically true, must hold good in consequence of our being able to transform it into one that is identically true by means of the fundamental equation (1), which connects 7 and p- ‘Thus we may regard all useful equations of the form (3) as deductions from the single primitive (1). Of the general nature of the results which may be deduced from this one very simple equation, and that without the in- troduction of any fresh hypothesis as to the operation of and p, the following example will give a sufficient idea. Making a=1, which does not much diminish the gene- rality of our conclusions, we have pr=m7p+1, pm = 1pT + 7, =7(7p+1)+7, =7’p+2r. Again, pm =m’ om + 27°, = (rp +1) + 27’, =7'p+3r’. And for n any positive integer we get, pm" = 7p + nr}, 146 Again, operating on the equation pr=7pt1, mC )ar-, 71 al, _ 2 pw = p-7 with we get from which we deduce pr" =7"o—nr™}. So that the equation pm” = 70 + na" holds good for any integer value of n. From this again we infer that p or=dr p+ pr, (5) where yr represents any function of integral powers of z. And from (5), finally, we can ascend to the more general theorem, ’ oo 1 4 4 gp dw= dr gotym ppt oPm o'p + ke. (6) Changing 7 into p, and p into - 7, in the last two equations, we obtain the correlative ones, 7 ¢p=$p 7- $s (7) , 1 “ 4 ot gp=$p da-gp wrt > ¢'p bn - ke. (8) In the theorems here given the reader will recognise an ex- tension to the symbols w and p of the theorems respecting x and D, stated by Dr. Hargreave at the commencement of his remarkable paper on the Solution of Differential Equations, printed in the Transactions of the Royal Society for 1848. Having obtained the theorems, oD pr= ya Dia (D+ 7¥'x Deas (9) 147 ya oD=4D fa-g¢D Parr aa ¢'D p'x-&e., (10) separately, Dr. Hargreave observed that the latter might be deduced from the former by changing 2 into D, and D into —wx; and on this observed fact he founded the conclusion that, in expressions capable of being reduced to the form (9) or (10), we are at liberty to effect the above-mentioned inter- change of symbols. The preceding investigation enables us to account for the fact just referred to, and to establish on what seems to be its real foundation the validity of the proposed method of de- riving formule one from the other. If we take f (x) any func- tion of x, we shall have D (afz) = «'af + fr, or, detaching the subject fz from the operations effected on it, we find that Da=2D+1 (11) is a symbolical equation which holds good whatever subject be operated on by each of its terms. It is, in fact, the fundamen- tal equation which defines the law according to which w and D combine. And as in this equation we may change into D, and D into —z; we may do the same in (9), or in any other equation derived from it. From this one equation (11) the principal symbolic for- mule of the Differential Calculus can be deduced ; and we may, therefore, regard a great part of it as included in that single branch of the Calculus of Operations which refers to the pro- perties of symbols connected by the fundamental equation with which this Paper commences. But there are other changes of symbols which may be made in formule deduced from the equation, pr=mp+1. 148 Since p is distributive, we shall have p Se =Jp p> and adding these equations together, we get p(w +Jp)=(7+fp)p +), (12) an equation still of the same form as (1). And, therefore, in any symbolical equation deduced from (1) merely in virtue of its form, we are at liberty to change 7 into r+fp. Similar reasoning will show that in symbolical formule obtained in the same way, we may change p into p+/fm. As particular cases of this we may observe that in any symbolical equation involving # and D, we are at liberty to change « into #+fD, or D into D+ fe. Again, if we operate on (5) with (f’7)-1, it becomes (Wr) pdr=da (Yr) *p +1, inasmuch as any two functions of 7 are commutative. Now this again is an equation of the form pr =7p+1, (Wr) 1p being put for p, and pr for 7. Tt follows then that in any deduction from (1) we may change p into (p'r) *p, and (13) w into dr. In like manner, if we operate on (7) with (¢’p)}, it be- comes ge (pp) *7=(9'p)*rHp +1; showing that in any deduction from (1) we may change p into ¢p, and (14) m into (¢'p) 7. Writing 2 for 7, and D for p, we learn from (13) that it is legitimate in symbolical formulz to change z into Ya, and D - 149 into ('x)D. This, in fact, reproduces the known rule for the change of the independent variable. From (14) we conclude that the change of D into oD, and z into (¢'D)'7, is a legitimate one. The validity of this change has not, we believe, been noticed before. It is unnecessary to adduce any more particular instances of the general law of interchange of symbols which may be established, viz.: If from (1) we can deduce any equation of the form PIT=1IP +1; when P and II canbe expressed in terms of p and 7, then, in any symbolical equation derived from (1), we are at liberty to change p into P, and z into II. Some very important deductions may be made from the equation (1). As a particular case of formula (5), we have pe’ = eo + evar, therefore, ptpr= eY™eV" 5 whence we conclude, that S(p + ¥n)=eMfper. (15) That is to say, the symbol, CA ens operating on any function of p will change it into the corre- sponding function of p + wz. Changing 7 into p, and p into — 7, we should find F (n+ ¥p)=erefi ev, (16) which shows that the symbol, eve ( ) ev, 150 operating on any function of 7 will change it into the corre- sponding function of 7 +p. The substitution of « for 7, and D for p, in (16), leads to a result which is of considerable value, viz., that eVfre) =f(a+ yD). If in this symbolical equation we suppose the subject to be unity, we shall have eV fre] = f (m+ YD) 1. (17) This is a remarkable extension of Taylor’s theorem, when stated in the symbolical form; and will be found useful in the interpretation of symbolical expressions which are met with in the solution of differential equations. In the development of the right-hand member of formule (15) and (16), the terms involving D may be all brought by means of the theorems (6) and (8) to the right or left hand at pleasure. The for- mule thus obtained will be found of considerable use. In the deduction and statement of theorems involving 7 and p, we shall find it convenient to employ the symbols either of which denotes the operation of taking the derivée, in an algebraic sense, of any function of the symbol involved in it. d — must operate on 7 only where T According to this definition 7 ‘ ay. d it appears explicitly ; and so for —. dp Gnas : : : F Hence as inoperative on p, or any function of it, and is betes ad. : : commutative with p. So also a is commutative with 7, or any function of it. The two symbols £ and © are plainly 7 dp commutative with one another; but they combine respectively 151 with 7 and p, in conformity with the law expressed by the equations, d d pee gis 1 dato dpe e The formule (6) and (8) may now be expressed in the sym- bolical form, ae op Wr=e* Fhe gp, (18) dd Ur gp=e "pp dr. (19) But it must be observed that they are no longer as gene- ral as they were in their original form. The equations (6) and (8) would hold good, whatever expression involving 7 and p was written to the right of each of their terms. Whilst the operation of the exponentials in (18) and (19) must be re- stricted to the terms Yz gp and gp yz, which immediately fol- low them, and not allowed to affect the subject operated on by these terms. Gis é : ae As ap is commutative with 7, and ae with p, we may write the formule (18) and (19) in the form, d ge Yr=y i ae $ps tn yp=9(p-< dm, and, for the same reason, are at liberty to develope seed ; : : d aie according to ascending powers of either 7 or az A similar P observation applies to the development of 152 : d p (6 dz < The very general theorems already stated may be ex- tended to any number of systems of variables connected by equations, such as define the mutual action of + and p. Thus, if pr=7pt1 and Pim = 7101+ 1, the symbols being otherwise mutually commutative, we shall have aa pl a I (p» pr) ) (7, 71) = fm dt dm dy ? (7, 7) S(p, P1)s and so on for any number of pairs of symbols. Again, as a generalization of the formula (15), we shall. find, if y denotes a function of 7 and 7, d d And, analogous to (16), t(= +e m+3) = eVf (n,m) e; y denoting in this case a function of p and pi. Writing 2 and y for 7 and 7, and = and for p and p, in these latter formule, we obtain results of considerable importance, the statement and discussion of which is reserved for the conclud- ing part of this Paper. The Secretary read a paper by W. H. Harvey, M.D., on the Marine Botany of Western Australia. Robert Ball, LL.D., drew the attention of the Academy to the fact, that in the celebrated statue, known as the Dying 153 Gladiator, the figure was represented as lying on two trum- pets, and had on its neck a torque. Dr. Ball exhibited a remarkably well-executed statuette copy of the figure, kindly lent him by Mrs. Hutton, and pointed out that the trumpets were in form and dimension excellent representations of the larger part of the fine trumpet found at Banbridge, and now in the Museum of the Academy. In the statue they appear connected, the larger end of one to the smaller end of the other by a strap or thong. This induced him to believe, that in the original statue, not two trumpets, but two pieces, iden- tical with those in the Academy, were represented, and that in the restoration of the right arm, &c., which took place when the statue was found, the artist had added a bell- mouth to the curved cylindrical tube, believing that two trum- pets should have been figured, not knowing that these instru- ments were made in two or more pieces: it is not probable that two exactly similar trumpets were carried. The mode of attachment of the strap points out its use, i. é. to hold the trumpet in its proper position when put together; and in the figures on the column of Trajan, several men, probably Da- cians, are represented blowing trumpets resembling the Irish specimen, and with the strap attached and strained in the manner of a bow-string. The date of the statue in question has been much disputed. Byron, it does not appear on what authority, in his beautiful notice of the Dying Gladiator, assumes that he was a Dacian, and the figures on Trajan’s column seem to support this view : however, learned artists, for the greater part, give it a higher antiquity. With respect to the torque on the neck, it appears to be a very fair representation of the general form of the torques found in Ireland. It may be observed, similar ornaments are said to be in use in the interior of Africa,—a fact known long since, it appears, by a fine old picture, the property of Dr. 154 O’Ferrall, in which an African slave is represented as having a heavy golden torque on his neck. A copy of the London Obsequies, found by the Hon. C. Neville at Little Wilbraham, was presented by the Secretary, on the part of Sir John Young, Bart. The thanks of the Academy were voted for this valuable donation. 155 Monpay, JANUARY 8TH, 1855. THOMAS ROMNEY ROBINSON, D.D., Prestvenr, in the Chair. Tue Right Hon. Francis Blackburne; Richard G. Butcher, Esq.; James H. Owen, Esq.; and Edward Senior, Esq., were elected Members of the Academy. On the recommendation of the Council it was Resolved :— I. To place the sum of £50 at the disposal of the Com- mittee of Antiquities, for the purchase of antiquities. II. That permission be given to the Council to exchange with Mr. Anketell a modern representation of the Crucifixion, made of ivory, and not Irish, now in the Academy’s collec- tion, for one made of bronze, and probably of great antiquity, and of native manufacture. Rey. Joseph Galbraith read a Paper by Capt. H. L. Renny, C.E., on the Use of the Hygrometer in the Barometric Mea- surements of Heights. Dr. Allman read a Paper on the existence of a true me- dusoid structure in the male gemme of Hydra. In this communication the author endeavoured to demon- strate that the peculiar spermatophorous tubercles which are developed upon the body of Hydra at the commencement of winter, possess a true medusoid type of structure, and thus form no exception to the general law which he had already enunciated, that the fixed reproductive capsules of the hydroid zoophytes are constructed on the medusoid type, and that for 156 true sexual reproduction in these animals the existence of such a type is a necessary condition. In Hydra fusca the organs in question consist, in their early stage, of minute depressed tubercles, attached by the whole of their broad base to the body of the Hydra. In their completely developed condition they present the appearance of more or less spherical capsules, attached by a contracted base to the Hydra, and slightly prolonged at the opposite point into a small mamilla, which is alternately perforated by an aperture for the egress of the contents. Into the interior of these bodies the gastric cavity of the Hydra is continued as a blind sac, which occupies the axis of the capsule and gives off from its sides numerous ramified tubes, which extend themselves in the interval between the central sac and the walls of the capsule. In these tubes, whose cavity, however, does not seem to communicate with that of the central sac, the spermatozoa appear to be developed, and sub- sequently escape into the cavity of the capsule to be finally expelled through the aperture in its summit. We have here, as in Cordylophora and the marine Hy- droida, the walls of the capsule representing the disc of a Me- dusa, the central sac homologous with the stomach, and the spermatophorous tubes with the radiating gastro-vascular canals. The spermatozoa possess a conical body with a long caudal filament attached to the larger end. They are produced in the interior of vesicles of evolution, a single spermatozoon being formed in each vesicle, and the vesicles themselves are produced in the interior of mother-cells. These mother-cells may ge- nerally be seen to contain two or three spermatozoa, which ap- pear to be often free on the mother-cell, but which are also doubtless frequently contained each in its own vesicle of evolu- tion, though the extreme delicacy of the latter renders it in- visible through the walls of the surrounding cell. Very fre- quently the bodies of the spermatozoa may be seen still in- 4 157 cluded within the mother-cell, while the tails have become disengaged and vibrate freely in the external fluid. In the mother-cells, besides the spermatozoa in their vesi- cles of evolution, a large, well-defined, nucleus-like body, with considerable refractive powers, seems to be invariably pre- sent. The motions of the free spermatozoa are peculiar, and con- sist in a succession of sudden jerks. By permission of the Academy, Mr. T. A. Dillon ex- plained a plan of his for ascertaining the deviation of ships’ compasses arising from local attraction. «¢ Sir,—I have the honour to submit, for the consideration of the Royal Irish Academy, a plan which I hope will tend in some degree towards reassuring the proprietors and com- manders of iron ships, whose uneasiness in consequence of Dr. Scoresby’s communication, ‘ On the Influence of Iron Ships over their own Compasses,’ has been, and still continues to be, of the most serious character. ‘¢ What Dr. Scoresby says is this :—‘ So soon as a vessel made of iron puts to sea and undergoes the tossing and strain- ing of the waters, she becomes an immense magnet, as it were, something in the same way that a poker is transformed into a magnet by striking it repeatedly with a hammer.’ And this distinguished philosopher goes on to state, that the loss of the Tayleur, and of many other iron vessels, can be assigned to-no other cause than to the very startling one above mentioned ; for the proximity of such an enormous magnet to the delicate compass needles disturbs and overrules these instruments as a matter of course. The ship goes astray, and all hands perish. ““Now, it little matters whether Professor Scoresby’s magne- tic theory be correct in every particular or not. We know that the most admirably constructed instruments have mysteriously gone wrong, even after every scientific precaution had been taken to preserve and compensate them in the most perfect VOL. VI. P 158 © manner. Whatever be the true theory, we know that ves- sels have been lost, and the only plan as yet proposed for obviating this danger is a recommendation by Dr. Scoresby himself, to place a compass at the mast-head for reference in case of suspicion, which is merely tantamount to saying, that the disease baffles, for the present at least, all his sci- entific skill. Professor Scoresby is wise enough to know that the motion of a compass situated on the top of a mast, even when masts are forthcoming, which sometimes is not the case, would cause the most perfect gimbals to sulk or give way. Again, a steam-ship’s funnel is made of iron, which fact con- fuses matters more than ever. «¢ We make out plainly enough, however, that the more dis- tant a compass can be placed from the seat of danger, the more trustworthy it becomes in the eyes of the ship’s commander. Consequently if, regardless of masts or internal attraction, we can have a reference compass always at hand, the binnacle in- struments may try any vagaries they please without disturbing the ship’s running in the least. Day by day we remark their deviation, and make allowances accordingly. «< My plan is meant as a simple mode of discovering the error to which the binnacle compasses are liable in consequence of the ship’s local attraction. ‘‘1st. Let it be granted that a line may be drawn along the keel of the vessel, and prolonged indefinitely astern. “Ond. That there exists astern of the ship a point on this line where, if we place a compass, the needle does not suffer from the ship’s influence, but exhibits the true po- larity. «¢ Now the difference between the binnacle compass and the compass alluded to must be the error we so anxiously wish to discover.’ As magnetic attraction decreases as the square of the distance, I should say that a compass one hundred yards or so astern would not be influenced by the iron of the ship: consequently :— 159 «‘ See that the jolly-boat, towing under the counter, drops astern a hundred yards or so, and thatit contains a careful officer and good compass, seeing of course that the boat is free from tron ballast, rings, &c. Pay out the tow-line until the boat, as described, is a hundred yards astern. One officer is now in the boat ; another stands by the binnacle compass ; a third takes his position at the taffrail, looking towards the boat. The officer in the boat, and the officer at the binnacle, each keeps one eye, as it were, upon his respective instrument, and one upona little flag which the third officer, who stands at the taffrail, holds in his hand. The moment this look-out officer finds the jolly- boat’s head and stern to be on a line with the keel of the ship, he lowers the flag. ‘The other two officers check their respec- tive compasses as the flag falls. The difference between the ship's compass and the boat's compass is the error sought after. ‘< Because, if the ship’s course was ‘ due east,’ the course of the boat was ‘due east’ likewise, since her head and stern were in a line with the ship’s head and stern by observation. If the course of the ship and boat were ‘due east,’ the fleur de lis on binnacle and boat should stand. at right angles to the bows of ship and boat. But the local attraction of the ship interferes with this result, and as the boat’s compass is the correct one, the difference between the instruments is the in- fluence of local attraction on board the ship. *« Asa first step, this is the simplest mode of escaping from the threatened overturning of our faith in the mariner’s com- pass. The other plans of ‘floating compasses’ are merely alterations in the form of the experiment, which do not inter- fere with the principle just detailed in the slightest degree. I give the following as the best I have devised. “Take a long plank of timber; cut it sharp at either end ; make a hole in the middle, insert a compass therein, having a spring so arranged that by chucking a trigger the needle is instantly arrested. at any point. Place a shining mark at the head and stern of this plank. Now float it astern at the end P2 160 of a log-line, and sight the two shining points alluded to from the ship’s taffrail. Chuck the line and the thing is done; for on hauling the plank alongside and examining its compass (which, it is needless to remark, is impervious to water), the difference between the plank’s head and stern and the ship’s head and stern, as taken by observation and the position of their cards at the same moment, indicates, as in the previous experiment, the precise amount of derangement caused to the ship’s needle by local attraction. «In concluding these observations, will you permit me to state that I am not satisfied with Dr. Scoresby’s theory? at the same time I cannot presume to deny it as yet, at least. You will notice that my plan is a mechanical test for determining the error of a ship’s compass, whether that error arises from imperfect compensation originally, shifting of guns, &c., acci- dental presence of iron or steel in the immediate neighbour- hood of the instrument, or from the percussion and straining of the waters, as Dr. Scoresby has ascertained.” W.R. Wilde, Esq., on the part of George O'Flaherty, Esq., of Lemonfield, presented a curious oval wooden bowl or vessel with handles, and carved out of timber, found in the turf-bog near the old church in the Demesne of Lemonfield, and about four and a half feet from the surface. Two others were found at the same time, but in the haste of the workmen to examine their contents they were broken and lost; the one presented was saved by a gentleman who happened to pass at the time of the discovery. The three vessels contained neither coins nor other antiquities. Mr. Wilde also presented an iron spear-head, a dagger, and swivel gun constructed with a chambered breech, found in 1853 by George Warder, at Inishdauwee, an island in Lough Corrib, near Oughterard. These articles were discovered in consequence of attention being drawn to the spot by a fragment of iron projecting above the*surface. 161 Rev. Dr. Reeves presented a small conical brass box, found in a bog near Cullybacky, county of Antrim ; also, for James R. Hutchinson, Esq., a box of the same kind, but larger and older, found in the Demesne of Stranocum, by workmen levelling on the bank of the River Bush, about twenty feet below the surface. 162 Monpay, JANUARY 22ND, 1855. LIEUT.-COL. LARCOM, F.R.S., Vicze-PReEsipENT, in the Chair. Rey. Proressor Graves, D.D., read a Paper on the solution of the equation of Laplace’s functions. ‘¢ Mr. Carmichael was the first to observe that the partial differential equation of the second order, dV av a&V da dy? d@’ or (1) DiV+ D}V + D3V =0, known as the equation of Laplace’s functions, may be reduced, by means of Sir William Hamilton’s imaginaries, to the sym- _ bolic form, (D,+jD,+kD;) (D:-jD:-kD;)V=0. Its complete solution is, therefore, the sum of those of the two equations of the first order, (D,+jD,+ kD: V=0, (D, —jD, os kD;) V= 0 3 and these latter solutions have been presented to us by Mr. Carmichael in the symbolic forms, We ETE D fF (Y, Z), (2) Va at Df, (Ys Z)s (3) in which f, and f; stand for quite arbitrary functions. Follow- ing, however, too closely the analogies of ordinary algebra, Mr. Carmichael has fallen into an error in interpreting the right-hand members of these formula. He has made 163 er IPek PDF, (Yy, 2) =fily — jes z- ha), @) griDakDe f, (y, 2) = Kt (y + Jes 2+ kz). (5) Sir William Hamilton at once perceived the inaccuracy of these results, and referred it to its origin, which was the erro- neous supposition, that gh Dak Ds = gre os, This last equivalence does zot subsist, because the symbols j and 4 are not commutative. ‘Indeed, the consideration of a simple case might lead to the suspicion that the formula (4) was incorrect. Suppose that Ay, 2)=yz: it becomes at once a question, what is the meaning of fi(y—jx, z-hxv)? Is it (y-ju)(z-hx), or (z—hx)(y—jzx)? for these expressions have different values. Thus, in the first instance, it is apparent that the assigned re- sult is ambiguous. - But from what follows it will appear that neither (y —jx) (z— kx), nor (z —hwx)(y —jx), is equivalent to eID ykDs) yz, ‘‘ The question relative to the interpretation of the sym- bolic formule (2) and (3) being in this state, I have endea- voured to resolve it in the present Paper. «* As a first step in our investigation, let us inquire what is the effect of the symbol, graDykDsy, or 7, as it will be more convenient to denote it, upon the term y" 2", m and n being positive integers. *‘ Beginning with simple cases, we shall find by actual ex- pansion of the exponential symbol, lL. wr y™=(yt jx), and 7 2°=(2+ha)”. 2. 7 yZ=yz+jaz t+ kay. 3.7 pe=y2t Qjay2 + kay? — xz - Tha. 4.7 yYea=yP2? + Bayz + dkay?z? — v2° — 3a°y?2 — kasz? — Qju®yz — katy? + vz + Lhe. 164 The first of these results shows that the effect of 7 upon any function of y alone is to change it into the like function of y+jx; and the effect of the same symbol upon any function of z alone is to change it into the like function of z+ ka. ‘‘ But the second example shows that its effect upon yz is to change it, not into either (y + jx) (z+ ka), or (z+hx) (y+ja), but into half the sum of these different expressions. For 3{(y+ja) (e+ kz) + (2+ ke) (y+jx)} = yz + juz + hay, in virtue of the equation jh + kj=0. ‘‘ Again, the third example shows that the effect of 7 upon yz is to change it, not into any one of the three different ex- pressions, (y+ ja)? (z+hz), (y+ je) (z+he) (y+jz), and (z+khx) (y+jx), but into the third part of the sum of the three. It is easy to see that this result follows from the equations, poH-l, ‘R=-1, jk+kj=0. ‘«« Pursuing the same course we shall find that the effect of 7 upon y*z* is to change it into the tenth part of the sum of the ten expressions, (y+ju) (2+ ka)’, (y+jx) (e+hkx) (ytjau) (2 + kx)’, (y+ja) (2+ha)? (y+ju) (2+ ha); (y+ju) (2+ hu) (y+ je), (z+hx) (y+ju) (2+ ke)’, (z+hx) (y+ju) (z+hu) (ytjax) (2+ha), (2 +ha) (y +jo) (2+ ke)? (y + je), (z+khz)? (y+ ju)? (2+ ke), (z+khx)? (yt+jx) (2+ kha) (y+ja), (etka) (y+ ja)’, which arise as the differently arranged products of the five factors, of which two are equal to y +jx, and three to z + ha. “‘ Following up the analogy, we are led to expect that the effect of 7 upon y” 2” will be to change it into the 165 = n!n! {@ - ”) \" part of the sum of all the m!n! different expressions which arise as the differently arranged products of the m+n factors of which m are equal to y +z, and n to z+ka. “The following reasoning demonstrates the truth of the proposition just stated. “« Let C be the coefficient of ay" z"” in the develop- ment of 7y™2". Then C will be equal to the coefficient of avy DD" in the development of gtIDgrkDs), multiplied by m(m—1)c.... (m-p+1)n(n-1)..... (n-v+1). But, in the development of the exponential, DyD: occurs only in the term ah” (7D, + kD;)"_ (u+v)! and there has for its coefficient = (uv) | (ut+y)l? the numerator = (u, v) denoting the sum of all the variously arranged products, into each of which enter wjs, and v ks. Consequently, we have m!n! ~ (m=)! (n—-v)! (ut+v)! “‘ But again, C, the coefficient of x” y" 2" in the ~ (u, v). (m+n)! m! n! ! 1S = ”) He part of the sum of the m!n! differently arranged products of the m+n factors, of which m are equal to y+jz, and n to z+ ka, will be equal to m!n! 166 where S denotes the sum of all the differently arranged pro- ducts of m+n factors, of which m-j are ys, n—v are Z8, are js, and vy are ks. Now the number of these arrangements in S§ is (m+n)! (m—p)! a=v) lv? and. § itself will obviously be of the form NZ (u, v), N being some numerical coefficient depending upon m, n, m, and v. But as the number of differently arranged terms in 3 (p, v) is (u+v)! piv! ? it is plain that we shall have (m+n)! N= (m—p)!(n—-v)! (u+v)? and consequently, aalval C,= aon = : (m=)! (n=v)! (ut v)! (us ») «Thus, we have found that C =C,, and as this is true for the numerical coefficient of every term in the development of 7 y™2", we are warranted in concluding that this latter ex- pression is equal to the ((m+n)!)* m!n! (m+n)! m! n! part of the sum of all the differently arranged products of the m+n factors, of which m are equal to y+ ja, and n to 2+ ke. ‘¢'The statement ofthis theorem, and of other similar ones, may be rendered simpler and more elegant by our assigning a name and symbol to the last-mentioned expression. I propose to call.it the mean value of the product of the factors combined in different orders: and for the present to denote it by the symbol M(y+ju, 2+ kn). We may now proceed to interpret the expression ee a rt 167 gts DerkDs) t (Ys z); in which f(y, z) is supposed to be of the form = A y™ 2", m and n, as before, being positive integers. ‘«s The exponental symbol being distributive in its nature, - we shall have the proposed expression equal to the sum of the mean values of the products corresponding to the several terms such as Ay”z”". Consequently, grIPukDd Fly, Z) = AM (y + Ye ae kz), and, with an interpretation suggested by what has been already said, we may write finally, Er IP2kDD f(y, z) = Mf (y+ ja, z+ ka). «¢ The boundaries of algebra having been of late extended so as to include symbols which are not commutative with each other, it becomes absolutely necessary to have the means of denoting certain standard and constantly occurring combina- tions in brief and unambiguous ways. The symbol M, pro- posed in this paper, may perhaps be a useful contribution to mathematical language. It has the recommendation of having been already used in a similar, though less extensive, meaning by M. Cauchy. It may also be regarded as an extension of Sir William Hamilton’s notation of §(a, 3), which stands in the Calculus of Quaternions for $ (a + Ba). ‘«¢ Knowing how to interpret the expression, (jj DetkD. é (7 Dz 35 we are enabled, in general, to solve the equation, DiV + D;V + D3V = U, (6) in which U denotes any function expressed by means of posi- tive and integer powers of x, y, and z. The solution depends upon our being able to invert the operations denoted by D,+jD.+kD;, and D,-jD.-kD;; 168 and as these inverse operations are respectively, P P y EPID DS DT) 71 gXIDYkDy) oy 1D) a and erIDykDs) DP) -1 peo). or 7 Dy Tr, we shall have V= wD aw? Dy 2 U+ M, ] (y — jkr, a2 ke) + Mf, (y Tiley Zo+ kr), the two latter terms being the solution of (1). This com- plete solution, when developed, appears, in general, in the form, F,+jF,+kFr, F,, F,, and F;, being different functions of x, y, z, which singly satisfy the proposed equation. ‘¢ For instance, we have seen above that EID HDS) y223 = yz — a (2° + 3y?z) + atz, + 27 (ayz? + x°yz), +k (Bayz? — v2? — ay? + 145), It will be found on trial, that each line in the right-hand mem- ber will by itself satisfy the equation of Laplace’s functions. “ G's upon any function of x, y, and z. ‘* Reasoning as before, we should find that its effect upon a term 2! y™z", 1, m, and n being positive integers, will be to change it into the ‘Sear part of the sum of all the a differently arranged products of the 7+m-+n factors, of which Jare equal to x+iw, m to y+jw, and nto z+kw. In other words, to change it into the mean value of this product, that is, into M (2+iw, y + jv, z+ hw): and, more generally, the effect of (pile upon f(x, y, z), any function consisting of positive and integer powers of w, y, and z, will be to change it into Mf (a+ iw, y+jw, z+ kw). We are thus enabled to interpret the formula (7), when U, JA» and f;, are functions of positive and integer powers. To that case the demonstrations given in this paper are essentially limited. I hope to be able to lay before the Academy the investigation of the cases where /, m, and n, are negative or fractional. So far as I have yet discussed them, they seem to present results which it is more difficult to express with elegance. ; ‘In conclusion, I may be permitted to state some theorems at which I arrived whilst discussing the subject of the present 170 paper. In fact, I at first imagined that the proof of them was necessary to my purpose. They are obtained as follows :— _* In virtue of the laws of combination of the imaginaries t, j, k, we have (ta ae jb a1 ke pee — (- Pye (a? a3 b+ ey rs (8) Now, the coefficient of a?4#c” in the left-hand member of this equation is = (2A, 2u, 2v), in conformity with the notation explained in p. 165: and the same coefficient in the right-hand member is plainly ies (r tpt v) ! Tillie Tp Ge Consequently, we have the theorem I. sie A+ ut v)! = (2A, 2u, 2v) = (- 1)* e ae ‘‘ Multiplying both sides of the equation (8) by 2a +jb+ ke, we get (ia +jb + he)? = (— 1)" (ta +b + ke) (a? + B+ ye, The coefficient of a*10*c” in the left-hand member is = (2A +1, 2u, 2v); and the same coefficient in the right-hand member is (- 1pm CD We have, therefore, I1., S(2A+1, Qu, 2v) =(- ae ee i, and similar expressions for 7 and £. « Again, multiplying (8) by (éa+jb + ke)? =— (a? + B* +c). we get (ia + jb + ke) — (— 1) sl (9? + B+ 2) se The coefficient of a6" c” in the left-hand member is =(2A+1, 2u+1, 2v); whilst in the development of the right-hand member no such term appears, as all the exponents of a, b, c, must be even numbers. We have, therefore, II. — ee 171 3 (tA, Qu41, 2v)=0, IfX=0, we must put 1 in place of A! in the preceding for- mule.” Sir William R. Hamilton made some remarks on Professor Graves’ Paper. Dr. Aquilla Smith presented the following donation from the Archdeacon of Clonmacnoise : 1. Coloured window glass, seventeen fragments :—one fragment in lead fitting; two fragments of lead fittings for glass: from the Abbey of St. Peter and St. Paul, Newtown, Trim. 2. Oval silver reliquary and medal. 3. Ten Italian medals in brass and copper. 4. Two bulle; a figure representing St. Patrick ; a weight stamped with a heart and the letters J. B.; a square ingot or weight ; and a small cup: all made of lead. 5. Two silver, one gold, and one brass brooch ; one bead made of white glass, with blue stripes. 6. Three silver and four brass finger rings; four rings, iron and brass. 7. One silver and nine brass buckles, of different patterns. 8. Three very small spoons, and two fragments of large spoons. 9. A coat button with glass centre. 10. Two ornaments and harp pin, made of brass. 11. Two pins and portions of a comb, made of bone. 12. One brass and seven iron keys. 13. One old matchlock and fragment of iron chain. 14. One small padlock and stopper for key-hole of safe. 15. One pommel ofsword and arrow-head ; two knives and one large fork ; made of iron. 172 16. One spur, with silver ornaments, and large rowel of another : made of iron. 17. One ring and small cross, made of stone. All these _ were found at Trim or Newtown. 18. One large brass thimble, found at Bective Abbey. Dr. Aquilla Smith, on the part of F. Higgins, Esq., pre- sented a small hammer, made of porphyry, and found at Hig- ginsbrook, near Trim. 173 Monpay, Fesruary 12TH, 1855. JOHN ANSTER, LL.D., in the Chair. Tue Rev. Robert Carmichael, F.T.C.D.; Alexander Carte, M.D.; and the Rev. Ogle William Moore, were elected Mem- bers of the Academy. Professor Downing read a Paper on the draining of the Haarlem Lake. The lake of Haarlem, situated in North Holland, con- tains 44,500 acres, which, previous to its drainage, was covered with a depth of thirteen feet of water, the surface of which was under the mean tide level of the sea; it is now completely dry and under cultivation. To have an adequate idea of the difficulties encountered in bringing this work to a suceessful termination, it is necessary to consider the peculiar physical and artificial circumstances of the Netherlands. The greater portion of the surface is at or below the level of the sea, and only protected from being again covered with water by immense dykes, which guard it alike from the riversand the sea. Along the greater portion of its western boundary, it is, however, in a great degree protected by the dunes or sand-hills which form the coast line. The rise of tide along the coasts of the Zuyder Zee is only two feet, and upon the west, in the German Ocean, it is six feet, the mean level being very nearly the same. The annual rain-fall, as deduced from observations continued for nearly one hundred years, is on the average 25-15 inches; the mean annual evaporation is22°6 inches, distributed, however, very unequally in the winter and summer seasons, thus : Summer. Winter. Total. Fall of rain, . 105in. . 1465 in. . 25°15 in. . Evaporation, . 159, . 167 ,, .° 226 .,, -5-4in. . +795in. . =2-55in. VOL. VI. Q 174 As to the artificial features of Holland, we find that from the very earliest times it has been divided into districts of greater or less extent, placed under the control of a govern- ing body ( Waterschappen), which we may call the Hydraulic Administration; the-boundaries of these administrations (which are not conterminous with those of the provinces, or any fiscal or municipal districts) are formed by large and lofty dykes, in which are placed self-acting sluices for the discharge of the waters within the boundary dyke, and closing against the admis- sion of any of the external waters. Lake Haarlem is situated in the administration of the Rynland, which has discharging sluices into the German Ocean at Katwyck, into the Zuyder Zee at Sparndam and Halfwege, and into the river Yssell at Gouda. Within every Hydraulic Administration are three divisions of surface, called the Natural Lands, the Basin, and the Pol- ders. ‘The basin is the total area of water-surface within the boundary dyke; the natural lands area little above the level of the basin, and discharge the rain-fall off their surface naturally ; the polders are lands below the level of the basin, at various depths, from a few inches to twenty feet, and from which, con- sequently, the water must be raised mechanically, by windmills generally, and latterly by steam-power. The Rynland con- tains— : Basin,. . .... . . . 656,000 acres. Natural lands, 2... 4 10, O00 Poldergs,. sweets ee ro O00 2. Lake Haarlem, which had been part of the basin, is now added to the polders, so that, instead of 56000 and 173000 acres, we now have 11500 acres of basin, and 217500 acres of polders, in this Administration. Regarding, then, the basin in its most important duty, that, namely, of a receptacle of the rain-fall when the self-acting sluices may happen to be closed against the external waters, we see how greatly its powers of a ee 175 storage are now reduced. To obviate this disadvantage it was necessary to put up engines of 200 and 100 horse-power at Sparndam and Halfwege, and widen the channel leading to the Katwyck sluice. Another work preliminary to the drain- age was the navigable canal (Ringvart), adapted to vessels drawing 8 feet of water, which previously traversed the lake; this canal had a total length of 36 miles, and width of 146 feet, the inner bank being in fact the dyke surrounding the lake, and cutting off the waters which otherwise would flow in during and after the laying dry of the bed. All preliminary works being thus completed, the raising of the waters up to the level of the sea was effected by three engines of 350 horse-power each, on the Cornish principle, constructed by Harvey and Co., at Hayle foundry, after designs by Messrs. Gibbs and Deane; the cylinders were 12 feet diameter and 10 feet stroke. From numerous and unforeseen causes of delay they were thirty-nine months in raising the water; and instead of 800000000 of tons of water, the computed contents of the lake, they actually raised 1100000000 tons. These engines will be required for all time to keep dry the land they may be said to have created, not, however, by that continuous working by which the first operation has been performed, but by throwing off extraordinary rain-falls before they have injuriously affected the land. Eight inches of rain-fall and infiltration per month is ‘the maximum quantity that long-continued observations lead them to expect, and this can be raised in about twenty-five working days by the 1150 horse-power of the three engines. The original estimated cost of all the works of the drainage was £687500; the actual expenditure, £827200. The sale of the land has realized about £400000, and the land tax, 7s. 4d. per acre, being capitalized, would yield a like sum; nor must we omit the saving of £5000 per annum, formerly expended in guarding the banks of the lake from destruction during storms, but which now of course ceases. ——_ 176 The Rev. Samuel Haughton, M. A., Professor of Geology in the University of Dublin, read a Paper on the chemical composition and optical properties of the Mica of the Dublin, Wicklow, and Carlow granites. “‘The minerals included in the mica family may be divided into various groups, founded on their chemical composition and optical properties. “¢ Chemically considered, they are divided by Rammelsberg and others into three families :— 1. Potash mica. 2. Lithia mica. 3. Magnesia mica. “‘ Optically considered, they are divided into three groups also :— 1. Biaxial micas, the angle between the optic axes being from 44° to 75°. 2. Biaxial micas, with angle between optic axes from 5° to 20°. 3. Biaxial and uniaxial micas, with angle between optic axes from 5° to 0°. ‘“‘ The first optical group includes the potash and lithia micas of the chemical division, while the third chemical group, or Magnesia mica, is divided between the second and third optical groups; the second group being usually designated Phlogopite, and the third Biotite. The potash and lithia micas are considered by Rammelsberg as represented by the follow- ing general formula— “‘m(RO, SiO.) + n(R.O., SiO;) in which formula, in the potash micas m=1- n= 2, 3, or 4. ‘“‘ The last two cases, = 3 and n = 4, being the Muscovite of mineralogists, and the first, x = 2, or 177 ROSiO, + 2(R,0,810;) + kKHO being known as Margarodite. « amRn Be ida ymhn + Samper aL heap mes c (3) and it is easy to see that the coefficient (2, m, n), or the sum = (+ 1), vanishes, if more than one of the exponents, J, m, n, be odd. Assume, therefore, as a new notation, (2A, 2u, 2v)={X; pe v}s (4) VOL. VI. R 182 which will give, by (3), and by the principle last mentioned respecting odd exponents, (2X + Ls 2, 2v) ={A, ps Dis (2A - 1, Quy 2v) ={A- 1, [Ms v}. We shall then have, by the mere notation, DTN, Te PE” 5 (6) and, by treating this equation on the plan of (3), {Ay ps v}=(A-I, pm, v}+{A,u—-1,v}4+{A,m,v-1}. (7) By a precisely similar reasoning, attending only to j and &, or 6) making \ = 0, we have an expression of the form, 3 fr = (us v) pk, 8) where the coefficients {y, v} must satisfy the analogous equa- tion in differences, (mw vj=(u—-1, v}+{u,v—-1}, (9) together with the initial conditions, (us O}=1, (0, v}=1. (10) Hence, it is easy to infer that Ges oe 1 es Se a”) one way of obtaining which result is, to observe that the ge- nerating function has the form, = {u, vj ute’ = (1 -u-v)t. (12) In like manner, if we combine the equation in differences (7), with the initial conditions derived from the foregoing solution of a less complex problem, namely, with (0, p,v}=(u vj}, {A, 0, v}=[A, v}, {A, ws OJ=[A, nw}, (13) when the second members are interpreted as in (11), we find that the (slightly) more complex generating function sought is, Dr, wy vj Puro = (1-t-u-v)7; (14) and therefore that the required form of the coefficient is, 183 ere Al ply! ee) [A, Ms v j= as, I have no doubt, you had determined it to be. «¢ With the same signification of { }, we have, by (2), Ura ro Vode ni; (16) therefore, dividing = by N, or the sum by the number, we ob- tain, as an expression for what you happily call the MEAN VALUE of the product 2°2#k”, the following : “on {A; Ms v| "4 Wyp2 fr2v 2Ay2u [r2v « M Pp Ke =P, Qn 20] PNP: (17) or, substituting for { } its value (15), and writing for abridg- ment k=A+pty, (18) aujov . LS «! (2A)! (Qu)! ee) M rye ‘Coa ee (19) In like manner, t(-1)* «! (2A +1)! (Qu)! (2v) ! F2A+1 72, ve Me = (2e+1)! A! pe! vl- (20) «‘ The whole theory of what you call the mean values, of products of positive and integer powers of ijk, beimg con- tained in the foregoing remarks, let us next apply it to the determination of the mean value of a function of x + ww, y + jw, z+kw; or, in other words, let us investigate the equivalent for your Mf (2+tw, yt+jw, 2+hw): (21) by developing this function f according to ascending powers of w, and by substituting, for every product of powers of wk, its mean value determined as above. Writing, as you” propose, d d d d ee WP» Bye dz we are to calculate and to sum the general term of (21), namely, SB (22) R 2 184 yim Mime x = DD" Dt f(a, y, 2). (23) m\n! One only of the exponents, 7, m, n, can ‘usefully be odd, by properties of the mean function, which have been already stated. If all be even, and if we make 1-2. — an, — 2y, (24) the corresponding part of the general term of Mf namely, the part independent of 7k, is by (15), (18), (19), (2x) ! whereof the sum, relatively to A, u, v, when theer sum x is given, is, (Ay ps vj} DX DY D3 f(@, Ys 2) 3 (25) Gat (Di + Dz + D3)" f (& Ys Z) = cares (2; > z), (26) if my signification of < be adopted, so that 4 =tD,+jD,+kD;; (27) and another summation, performed on (26), with respect to x, gives, for the part of Mf which is independent of 7h, the ex- pression, (e044 64) (0, y, 2). (28) ‘«¢ Again, by supposing, in (23), l=2A +1, m=2u, n=2p, (29) and by attending to (20), we obtain the term, wiD, (—w*): “Gesiyt ety) DPDEDY Say). (0) Adding the two other general terms correspondent, in which iD, is replaced by jD, and by £D;, we change iD, to q; and obtain, by a first summation, the term (wa jee (2e+1)! and, by a second summation, we obtain AERO ELA © (31) ; 185 Beer ie 4) F(a) yore); (32) as the part of the mean function Mf, which involves expressly wk. Adding the two parts, (28) and (32), we are conducted _ finally to the very simple and remarkable transformation of the MeAN FuNcTION Mf of which the discovery is due to you: Mf (x + tw, y+ jo, 2+ kz) =e" f (&,Y; 2)- (33) In like manner, Mo (a - iw, y-jw, z—hz) =e4 @ (2, y; 2). (34) Each of these two means of arbitrary functions, and therefore also their sum, is thus a value of the expression (D?- 4*)0; (35) that is, the partial differential equation, (D? + Di + D} + D3) V=0, (36) has its general integral, with two arbitrary functions, f and ¢, expressible as follows : V= Mf (a+ tw, y + jw, 2+ hw) + Mo (a - iw, y - jw, z— hw; (87) which is another of your important results. You remarked that if the second member of the equation (36) had been U, the expression for V would contain the additional term, eas D vere dD teh U7, (38) In fact, D+asersDerst, D-q=esDe4, (39) and therefore, (D2 ay? (D> dyer ens Des. (40) <‘ Most of this letter is merely a repetition of your remarks, but the analysis employed may perhaps not be in all respects identical with yours: a point on which I shall be glad to be informed. ‘“‘T remain faithfully yours, « WituiaM Rowan Hami.ron. ‘The Rev. Charles Graves, D.D.” —_-— 186 The Rey. Professor Graves, D.D., read the second part of his Paper on the solution of the equation of Laplace’s func- tions. ‘* In the former part of this Paper I showed that the sym- bol gu sDakD;). or 7, as we have denoted it for the sake of brevity, when ope- rating upon y” 2", has the effect of changing it, if m and n be positive integers, into the oh (oes oe m'!n ue + By pat of the sum of all the differently arranged products, of which each contains m factors equal to y+jz, and n equal to z+. But I reserved the con- sideration of the cases in which m and n were negative or frac- tional. In fact, I had ascertained by trial that the theorem just announced must undergo some modification in its state- ment before it could be extended to the case where m or n was negative ; and I was at a loss to conceive what modification could render it applicable in the case where either of the ex- ponents was fractional: the rule given for the formation of a mean product seeming of necessity to presume that the expo- nents were at least integer, if not positive numbers. In the present communication I desire to lay before the Academy the discussion of the reserved cases. In dealing with them I have been led so to modify my definition of a mean product as to make it apply where m and n are negative or fractional; at the same time that it coincides with my previous definition in the case where m and n are positive integers: and this has been accomplished by the help of mean products of is, js, and ks, the fundamental theorems respecting which were stated at the end of my former Paper, p.170. Thus it will be found that we are im possession of a complete and perfectly simple solution of the equation of Laplace’s functions :—com- plete, as involving two arbitrary functions ; and simple, as it is 187 disincumbered from all signs of differentiation or integration to be effected upon them. In this latter respect my solution seems to possess an advantage over those which have been given by Drs. Hargreave and Boole. ‘ ye = —jy?z" ky z, Db? = Dp yet — 9 |: en D3 ytzt=3! jyt2142! kytz?+ 2! jy?23+ 3! kyle, Bigeye +2. 212 by sehad bytes, Be ytzt=-5! jy’z1-4! hyez*- 2.312! jytes —~2.213!hkytzt- 4) jy%2z5-5 | ky iz, BS yizt=-6! yi -3.4!2!y%23-3.2!4! y8z*-6! yz, a yiztaylet— a (jy2 + hye) - a! lasz eae) 3 By _ (3! jytz3 + 2) ky 22+ 2) jy?2? +3! ky te) 4 a a (4 Dopben tee DD I yze+ 4! yz?) 5 = 55 (5! jyte + 4 hy te? + 2.3! 2) jy tz EO) | 3) ky?zt+4! ljy?ee+ 5! hy iz) Fy Glytet+ 3.41 21244 3,214 ye + 6 lytzt) fe { | 188 ‘* Now let us compare this with 2 ((y + ju)? (2+ hay + (2 + he) (y+ jay}, to which the analogy of example (2) in my former Paper, p- 163, might lead us to expect to find it equal. ** Developing by the binomial theorem, we have (y + ja)? (2+ he) =y 27 — @ (jy?21 + Rhy 2) ~ 2 (yez7 fe gem +y> Zz) +23 (jytet + hyz* + jy%2 + ky tz) + at (y oz) —iyte? + yz iy? et + ye) — x ( Jy=) a hy?z? 4 jyiz ae hy?z4 ae Wee L (B) ——— = —__] + ky z°) — 76 Grice ra iy*z” 4 pre is iy te 4 oz? ta Were ai ae ) PSEC sihe Yea |e J And (z+ a)" (y+ jx)" differs from this only in the signs of the terms containing 7. Consequently, the development of aly+ jay? (2+ha)*+(2+he) (y+ jay} differs from the series just given only by the omission of these terms. But this omission willnot make it agree with the ex- pression already found for 7 yz". “‘ The discrepancy first shows itselfin the numerical coeffi- cients of the terms ey 2, xyz, and xty%zs. In the former development (A) these coefficients are all =3. In the latter (B) to unity. «« Again, the coefficients of a®yz*, ay%z4, aty?z', ay oz, ay 2%, and 2*y*z* are all equal in (A) to 3, in (B) to unity. It is needless to proceed further in the comparison of the two developments. ‘* As regards the first instance of disagreement, viz. that between the coefficients in the two series of the terms 2*y*z2 and «y*z*; it must be observed that in (B) these terms have 189 respectively the imaginary coefficients A and 7; or, more ex- actly, after the restoration of the powers of 7 and £ suppressed in virtue of the equations 7?=h?=—1; the imaginary coeffi- cients —j*k and — kj. Now, by the theorems in p. 170, the mean value of the product of two js and one & is, 2 l 1 ae peer =. 3 I)k=-5h and the mean value of the product of two 4s and one j is - 4). So that, so far as concerns the terms w*y*z* and a*y*z*, the difference between the two developments consists in this: that in (B) these terms are multiplied by ordinary products, but in (A) by mean products of js and ks. ‘ yzi=jz1—kyz*, D2 yz? =- 2! yz, Ds yzt=—2!j23+3! hyzt, Dt yzt=4! yz, Di yzt=4! jz5-5! kyz, * ee e @ =e Yeitie aete Hence, we find 3 mw yz =ye + xv (jer — kyz*) — xyz? — a (2! j23-3! kyz*) 5 + wtyz?+ - (4! jz - 51 kyz*) — a®yz" — &e. Now let us compare this with the development of (y+jz) (2+kx). Expanding by the binomial theorem, and preserving the powers of j and &, when both appear in the same coefficient, we have y+ je) (2+ hay =yo? + x (jet — hye) — a (ghz? + yz J J J y + 28 (jk228 + hyz*) — a4 (jh8z4 — yz) B) + a5 (jhte’ — hyz*)— 2° (jheet+ yz") . . & The discrepancies between the developments (A) and (B) are numerous, but all of them are of the same kind. In the first place the terms 2%z, xtz4, a°z*, &e., do not appear in (A). In (B) they have the coefficients jh, jh®, jh°, &e. But the mean values of such products of j and & are equal to zero. ‘¢ Again, the mean value of the product of one j and 2» 1 2v+1 ments of a°z*, wz, &c., differ just in this: that in (A) they are mean products, in (B) ordinary products of js and és. k 8 18, (-1). Hence the coefficients in the two develop- Thus it appears, as we anticipated, that if we substitute mean products of js and ’s for ordinary products throughout the 192 entire development of (y + jx) (z + kx), we shall produce the development of 7 yz. ‘¢ Without stopping to consider the case where m or nis fractional, we may now proceed to establish the mode of inter- preting 7 y”2", whatever be the nature of m and n. ‘* The coefficient of a” y™*z"” in the development of 7 y™z", is equal to the coefficient of a *,DyD; in the develop- ment of e*/?:*", multiplied by m(m—-1)....(m-pt+1l)n(m—-1)....(m-v+1). But, in the development of the exponential, D:D; occurs only in the term ah ( 7D, ip k.Ds)" (u +yv)! and there has for its coefficient = (mv) (u as v) ha Consequently, the coefficient sought is m(m—1)....(m—ywt+l1)n(n-1)....(n-v +1) data tela ge “‘ But again, if we develop (y+ jx)” (z+ kx)" in the manner already mentioned, that is to say, preserving all the powers of j and k, and afterwards substituting mean products for ordinary products of these imaginaries; the coefficient of gievy™P 2 is plainly f m (m—1) no Seah 1)....(a-v+4 Dub, Bev. 2 or, since Boy op il Mikag tiene he ©) ) Grpice”? to m (oe TY oe slowed Dias Dev ela) i Thus, we have demonstrated generally that the expression 193 et iDakDs) if (y, 2), is to be interpreted as follows :— “ Substitute y+ jz for y, and z+ ke forzin f(y, 2); taking care to leave all powers of j and & in evidence, and then re- place all the products of js and 4s, obtained in this way by mean products of those imaginaries. ‘¢ Reasoning and processes in all respects similar lead to the conclusion that the effect of the symbol Pe +i DytkD3) upon any function whatsoever of x, y, and z will be to change it into the same function, in its mean state, of w+ iw, y+ jw, and z+kw. By this it is to be understood, that after this change of the variables has been made, and the development effected as if 7, 7, and k were ordinary algebraic quantities, mean values of products of the imaginaries are to be substi- tuted for ordinary ones. ‘«¢ Reverting now to the solutions of the differential equa- tions noticed in the first pait of this Paper, p. 168, we see that they hold good. without any liniiation of the nature of the arbitrary functions, provided we mocify, or rather perfect, our conception of the mean sia e of a function in the manner just described. “¢ Our new definition of a mean product, ov of a mean func- tion, coincides with thai given ep. 166, in the case where m and m are positive integers: and it includes the cases where m and z are negative or fractional, to which the original defini- tion of a mean product is inapplicable. ‘* If it should prove that the solution of Laplace’s equation now attained to, viz. : V= Mf, (y + ja, z+ kx) + Mf, (y -ja, z-kz), is something more than a mathematical curiosity, and answers the demands of physical inquiry, we shall have reason to re- joice not only in the fruits of that particular discovery, but 194 also in the anticipation that other important steps in mathe- matics may be made by the help of Sir William Hamilton’s imaginaries. I hope before long to be able to’ furnish the Academy with some reply to the questions here suggested. Rev. Samuel Haughton made some observations on the Rev. Dr. Graves’ paper. Rey. Humphrey Lloyd, D.D., read a further communi- cation ‘on the magnetic influence of the Moon.” _ The President and Rev. Samuel Haughton made some re- marks, eliciting explanations from Dr. Lloyd as to the analogy of the magnetic phenomena described by him, to correspond- ing phenomena connected with the tides. In the absence of Edward J. Cooper, Esq., his Paper on ‘‘ Keliptic Catalogues” was read by the Secretary :— _ « Having completed the catalogues of ecliptic stars obser- ved here during six years, it occurred to me to employ a few holidays, which I gave myself after the publication of our third volume, in examining some of the general results deducible from them, and comparing these results with concurrent me- teorological phenomena. My object was to ascertain the sound- ness of a preconceived opinion, that the records of the state of the weather are useless as a guide in estimating the most fa- vourable periods of the year for astronomical observations. To the investigation I added a search for any striking facts that might appear during the course of the work, in which Mr. Graham has been the principal performer, in the capacity of an indefatigable observer. “‘'The mode of proceeding which we originally adopted was, as is stated in the Introduction to the first volume of ‘Keliptic Stars, one which we considered the most likely to 195 possess the two essential conditions of rapidity and sufficient accuracy. It would require some little labour and time to as- certain from the published volumes the effectual manner in which the former of these conditions has been carried out. This, however, is easily established from the nightly noted observa- tions. On six occasions the nightly number of stars observed approached 500; on three occasions it exceeded 500, exclu- sively of one, when it approached 900; another 1200; and at last, on January 17th of the present year, no fewer than 1387 were secured. The two last, and one of the first cited, are not yet published. «‘ With reference to the second condition, which, primd facie, might appear unfavourably affected by the success of the first, it will be seen, by reference to the Introduction already alluded to, that the probable error of an observation in right ascension of 0*:288, and in declination of 4”-27, was the result deduced from 1345 known stars. But this statement does not fairly represent the accuracy of the observations; for, be it re- membered, that it originates in a comparison of previously de- termined stars, as observed and used by us for the reduction of the places of those published in our volumes, and their cata- logued places, and with the result is mixed up the errors of those catalogues. Were this not the case, our probable error in right ascension must necessarily have appeared considerably less than in declination: the right ascension being obtained from the mean of two observations ; the declination from their difference. It is fair, then, to assume, that the second condi- tion has been fulfilled. « The Tables on pp.196, 197, will show, as expected, that meteorological records wouldfurnish no data from which a prac- tical astronomer could select a time for his visit to this western station with any certainty of securing clear nights for observa- tion. Were he to choose the month of May in consequence of its greatest freedom from rain, he would find it low in the order of productiveness; or November as the least variable, still the Monta. 1848. |. Supeetaeion ! 1850. 1851. c. | K. | D. |Total.|| Cc. | K D. |Total.|| ©: K. | D. |Total,|| C- | K. | D. ‘Total. uly sleen vel| eo eel eed |e OS 4a eS St 3S 6 3 7| 96 374) 28) 135) 537 August, . .| 949) 49) 67/1065 5) 19) 34) 58) 183). 18) 16) 217)/1205} 168) 169.1542) September, .| 734) 64) 47) 8441151! 92) 168]1411)1076) 98} 201/1375)/1137] 165) 223 1525) October, . .j1009) 198) 180)1387|| 864) 101) 95)/1060|| 637) 83] 62] 782)| 597) 120) 170) 887 ; November, .} 470} 54] 56} 580)| 209} 34) 11] 254/| 230) 85) 23) 288|| 914) 143] 48,1105 December, ./1534) 222) 7/1763)| 260) 31) 30] 821)/1131) 117] 46/1294)! 912) 132) 97/1141) January,. .J. «|. «|. «|. .|| 959] 164) 20/1143] 676] 101] 60] 837|1520| 199 118|1837] February, .j- -|. -|. -|. -|/ 0 | 0 | 0 | 0 |] 47] 7 2] 5é6l| 886] 91) 49\1026) March, . .|. -|. -|. «|. «|| 502} 69/ 37] 608/|1340| 214) 39/1593]! 602) 77] 31] 710) April,. . J. «|. .J. ./. .|1 0 | © | 0 | © || 249] 33) 8] 290/| 850) 77] sel1013ai May, . . «|. -|- «|. -/. -||-0 | 0 | O | © |] 172} 26) 9) 207/| 404) 18] 25) 447) June,. . .. «|. «|. «|. «|| 48) 38 4) 55] 0 | O | O | O |] 132) 8] 95) 445 Tas_e II. Taste III. Months in the Order of abso- Months in the Order of least Gross dif-| lute Productiveness. Stars. | Average. Rain. PPETES ference. | September, averaging. .| 1215 |] WESA oe oy oe 1:983 2014 | April, oa Cutagy | ealalierale i Marchys 0)"; vee 3 | 2089 1-747 March, cf aes 1112 | September, Seid, eo. |e eek 1:002 December, 6 - .{ 1003 Apres ee tee | aco) 1°802 © January, oF Bs 898 | February, . 2. 2. . | 2852 1:468 October, 99 Ares 818 789 DUNES asi sep femurs 70 el MOREL 2°530 November, cf Swi 699 December, . . . . .| 3.266 4:114 | February, a anate 645 DUlys eee er be. ls ipsa) NERO 0-914 © May, Me hcg 614 November, . . . . .| 3941 0904 August, ns ee 608 October, . . . . . . | 4:258 1:041 July, A ohne) 544 August, oe el well eA 1:851 | June, niet aes 141 January,. . . . . .| 4573 | 22060 197 Tas LE 1.—continued. | | Montr. 1852. 1853. 1854. Role) fomual c. | x. | v. |rotaii| o. | x | D. (rotat.|| c | = | Dv. |rotal| July, - . .| 720) 65) 91) 876)) 85) 32 6| 123|) 855) 224) 120/1199|) 3263 544 August, . .| 318} 62) 52) 482) 197) 49) 86) 332). .|. .|. «|. . 3646 608 September, .| 766) 120) 123)1009) 761) 189) 225 LD Elie oa a lea a ea 7289 1215 October, .~.| 451) 83) 62) 596] 141) 28) 26) 195). .|}. -|. .}. . 4907 818 November, .|.605| 96) 54| 755/| 926] 142] 142/1210). .|. .]. .|. . 4192 699 December, .| 819) 111) 64] 994/| 397) 65) 45 BO lea wat sige alray se) aad 6020 1003 January, . .| 580) 82) 45) 707|) 651) 85) 66) 802) 33/17} 10) 60) 5386 | 898 | | February, .| 985) 189) 44/1218)|1081) 221) 157 1439 1078 188] 661332)| 5071 | 845 | March, . .|1370) 194) 82)1646) 468) 75} 80) 623 1180| 221) 921493| 6673 | 1112 | | April,. + .|1853) 232) 208/2293)) 988 166) 172)1326|)1544| 242) 3202106) 7028 | 1171 May, . . -| 254) 20, 22) 296/1489) 294). 88)1871|| 727) 118) 60, 905)) 3726 | 621 > | | June, . . .| 0 | 0 | O | O |] 322) 108) 71) 504|| 110) 30) 6) 146)/ 847 | 141 58048 | 9675 } \ Tasie IV. TABLE V. | | | | Hours dark enough for Observations. ee era os noel Average. | meecembern jo Ok 8 St 258 December, . . . . 39 PMMA ets ie gt bis ar DAO CEI a abe a ee 3°6 MEVEMIDED hel oe a a ck ee ee 28S November. srehike. 3:0 BRONCOS Ss t5 i aS Sta vy hte toa DOW. October, . . . . = . 39 PEEUMMOEV cl Sh vs ees le sw ts BOO Ul RebruaryA avai s me, 42 March, See taiarte. eee: See, aaploaT! Mamrchipeeinns Seated sn ire 61 5-8 13) GLUT ETS oa nc an 1) September, . . - . « 76 one BE Dy et OE 199 ribs 2) 2) see L . BBCI A ey Sg SN eke ea oe ie Ma AUPTUS iiyers re) oe se) omnis oRa: So eae arith Petar May,.- .- +... 8:3 D1 y . SGA SLE Teer, Sonn eR July, . . . + - 99 LINCS “cE TORS Me Re AD aisle MM ie ReaD 1) June, . a ; 49 Jj 198 numbers of stars obtained is below the average of the year. Lastly, were he to select December as the month producing the greatest number of hours dark enough for his purpose, he would again be foiled, the number of stars obtained per hour of darkness in this month being considerably below the average of the year. It will be perceived that September has proved to be the most productive absolutely, although three other months stand higher in horary results. At this season there is little to contend with, excepting clouds. The nights are long enough for the strength of the observer, and there is no frost to coat the object-glass. The latter is a serious obstacle in the colder months, and especially in this Observatory, where the large equatorial is completely exposed ; and the object-glass too heavy to be frequently removed with prudence. “Tn my general examination of the zones, planets weresought — for, but only two detected :—Neptune, in vol. 1., page 196, in right ascension, 22" 27™ 26°; and Melpomene, in vol. m1., page 177, in right ascension, 8" 6" 34°. The paucity of ob- servations of planets found in the zones may be thus accounted for. The telescope is almost invariably used near the meridian, and, the time ofa planet’s passage being known, the zone-ob- serving was frequently abandoned to obtain the most accurate place of the planet with the meridian circle. Facts indicative of an unknown planet were always noted in forming the cata- logue, and many places have been subsequently examined with the meridian circle. Hitherto the only results have been notes of two or three missing stars, on which we cannot at this mo- ment lay our hands. A comparison by Mr. Graham of a por- tion of his maps, in course of execution, with the heavens, gave _ the following stars as missing :— Vol. 1. page 73,. . . 4*18™ 24s jee RAO Tc. se 20.18) 9 29 » 182,. . .20 3 42 45 Ppa ied yee eA ay il Vol. 11.,,page. 8, .) «ye 4 54 9 55 >. 92,0 +» 206. 42 199 The President observed that the stars contemplated in Mr. Cooper’s Paper were many of them below the tenth magnitude, and that, therefore, the conditions of clearness, &c., of atmo- sphere required for such observations would not be applicable to the mere ordinary work of observatories. Rev. Charles Graves, D.D., read a Paper containing an account of certain notes in the Ogham character, occurring in the margin of an ancient manuscript of Priscian, in the Li- brary of St. Gall. = Pd Vea FT Gibwine pris iar © 201 Marcu 167TH, 1855. (Stated Meeting.) THOMAS ROMNEY ROBINSON, D.D., Presrpent, in the Chair. Tue Secretary of the Council read the following Report from the Council :— REPORT. In presenting their Annual Report for the Session of 1854-5, the Council have to congratulate the Academy on increased activity in the publication of their Transactions. The Fifth Part (Science) of the Twenty-Second Volume has been recently published, and is now ready for distribution among the Members. : The Sixth Part (Polite Literature) is in a very advanced state, and the Council hope to be able to announce its publication in a very short time. The First Part (Science) of the Twenty-Third Volume is also partially printed, and the Second Part (Polite Literature) has been commenced. Several interesting papers have been, during the past year, com- municated to the Academy. In Pure and Mixed Mathematics we have had communications from Sir William R. Hamilton on ‘‘ Quaternions ;” and from Pro- fessor Graves on “ The Principles which regulate the Interchange of Symbols,” and on ‘‘ The Equation of Laplace’s Functions ;” and from Mr. Rennie on “ The Use of the Hygrometer in the Barome- tric Measurement of Heights.” The President has given us a paper on ‘‘ The Electro Magnet.” In the Sciences of Observation and Experiment, Professor Haughton has given us a paper on ‘“ The Reflexion of Plane Pola- rized Light from the Surface of Transparent Bodies,” and on “ The Optical Properties of the Micas contained in the Dublin, Wick- low, and Carlow Granites.’ We have also had communications from Dr. Lloyd on ‘“‘ The Magnetic Influence of the Moon;” from VOL. VI. T 202 Mr. Galbraith on ‘‘ The Chemical Composition of the Felspar found in the Dublin and Wicklow Granites ;” from Dr. Allman on “ The Existence of a true Medusoid Structure in the Male Gemme of Hy- dra,” and on “‘ Aphanizomenon Flos-Aque ;” from Dr. Harvey on «The Marine Botany of Western Australia ;”’ and from Professor Downing, on ‘‘ The Draining of the Haarlem Lake;” with several others. In Polite Literature, Dr. Hincks has contributed a paper on “The Chronology of the Twenty-Sixth Egyptian Dynasty and of the Commencement of the Twenty-Seventh.”’ The same author has also given us a paper on “ Assyrian Mythology.” Professor Graves has made a communication on “ The Comparison of Adjec- tives in the Ancient Irish Language.” In Antiquities, Mr. Charles Mac Donnell has made a communi- cation relative to the MSS. of the celebrated John Colgan, preserved at St. Isidore’s, Rome. We have also received a communication from Dr. Ball, on ‘‘An Attempt to prove the Age of the large Bronze Trumpets in the Museum.” The Proceedings of the Academy have appeared with great re- gularity. The Council regret that the progress in the preparation of a Ca- talogue of the Museum has not been as rapid as could be desired. A Report from the Committee appointed to superintend this work is here subjoined. During the past year some important organic changes in the constitution of the Council have been carried into effect. The duties heretofore discharged by the Committee of Publica- tion have been delegated to the Council at large, who are also em- powered to obtain the assistance of any other Members of the Aca- demy, whose acquaintance with the subject of any one or more of the papers submitted for publication may enable them to give to the Council valuable suggestions in the discharge of this important task. This change will probably be considered to be calculated to add to the efficiency of the publishing department. At the Stated Meeting held in March last, the Academy expressed their opinion that it would be desirable to secure a greater amount of rotation among the Members of the Council than exists at pre- sent. In accordance with this opinion, the Council submitted to the , 203 Academy, in November last, a By-Law, by which it was declared to be expedient that, with the exception of the Officers of the Aca- demy, the senior Members of each Committee should he removed in each year. This By-Law, which, in conformity with the Charter, is expressed merely as a recommendation to the Members in the disposal of their votes, was passed by the Academy, and comes into operation on this evening. Should the Members act in conformity with it, a minimum of three vacancies in each year will be secured. An important addition has been made to our Museum during the past year, by the purchase of Mr. Murray’s Collection of Anti- quities. The following purchases of Antiquities have also been made by the Council out of the sum placed at their disposal by the Academy for that purpose :— 1. A highly ornamented spear-head: from Mr. M. Daly. 2. Several bronze and iron antiquities found in the neighbour- hood of Strokestown: from Mrs. Conry. 3. An earthen vessel; a bronze cross; a handle of ancient skil- let, highly ornamented ; a prick spur; a collection of small fibule ; a bronze mace and large ring: from Mr. J. Underwood. 4, A large earthen jar or jug: from the Rev. Mr. Archbold. 5. Two episcopal seals, one belonging to the late President of the Academy, Right Rev. Dr. Brinkley: from Mrs. Flood. ' 6. A miscellaneous collection of Antiquities: from Mr. P. Fegan. 7. A large silver fibula: bought by Mr. J. F. Jones at Mr. Gardner’s auction. 8. A peculiarly beautiful silver fibula: from Mr. Bindon. 9. A remarkably large and perfect bronze trumpet: from Mr. George Du Noyer. 10. A silver pommel of a sword, handsomely inlaid with fine wire filigree: from Mr. Murray. 11. A peculiarly fine bronze pin, with a cone-shaped head; a small bronze cross or ornament; a peculiarly shaped celt, and a pri- vate seal, found in the excavations in College-green, have also been bought from different individuals. The Council has also purchased a collection of plaster casts taken from the ancient stone crosses in the Isle of Man, from the Rey. J. Cumming. 204 A subscription has been opened for the purchase of a portion of the collection of gold ornaments recently discovered in the county of Clare, and. exhibited at the Meeting of the Academy held on the 26th June. During the past year, ten new Members have been added to the Academy. Their names are:— Cheyne Brady, Esq. James Higgins Owen, Esq. Sir Bernard Burke. Edward Senior, Esq. Parke Neville, Esq. Alexander Carte, M.D. Rt. Hon. Francis Blackburne. Rev. Ogle William Moore. Richard G. H. Butcher, Esq. Rev. Robt. Carmichael, F.T.C.D. During the same period two Honorary Members and eleven or- dinary Members have been lost to us by death. The names of the Honorary Members are :— Cart Friepricu Gauss, and Ropert JAMESON, Esq. The names of the ordinary Members deceased within the same period are :— 1, Joun Epwarp Butter, Esq.; elected 11th April, 1853: died 23rd July, 1854. 2. C. T. WepzBER, Esq.; elected 9th April, 1838: died 25th March, 1854. 3. Roger C. Waker, Esq. ; elected 24th June, 1833: died 6th September, 1854. 4, Rev. James Rei; elected 12th April, 1841: died 12th Sep- tember, 1854. 5. Otiver Sprou.e, Esq.; elected 10th May, 1841: died 3rd September, 1854. 6. Rosert Forster, Esq. ; elected 13th January, 1845: died 2nd June, 1854. 7. Wittiam Epineton, Esq.; elected 10th March, 1835: died 13th October, 1854. 8. Wynpuam Gootp, M.P.; elected 9th June, 1845: died 27th November, 1854. 9. The Earu or Lerrrm; elected 7th July, 1802: died 2nd January, 1855. 205 10. Maurice Cottis, Esq.; elected 12th February, 1849: died 15th January, 1855. 11. Biynon Buoop, Esq, ; elected 16th March, 1802: died 27th January, 1855. Ir was Resotvep,—That the Report of the Council be adopted, and printed in the Proceedings. The Rev. Dr. Todd read the following Report from the Committee appointed by the Council to prepare a Catalogue of the Museum :— REPORT. The Committee was appointed on the 18th March, 1852, by the following Resolution of Council :— “‘ That in consequence of the resignation of Dr. Petrie as proved by his Letter addressed to the Council, the following Gentlemen, The Earl of Dunraven, Aquilla Smith, M. D., Rev. J. H. Todd, D.D., Major T. A. Larcom, R. E., be appointed a Committee to carry out the plan already agreed upon by the Academy for the preparation and publication of the Cata- logue.” On the 19th April, 1852, the Committee presented to the Coun- cil the following Report :— “The Resolution of the Council, directing a Catalogue of the Museum to be prepared, distinctly specifies three objects to be kept in view :— ** Ist. An accurate list of everything in the Museum. “2nd. Such a list as may be a guarantee for the safety of the articles. * 3rd. A Descriptive Catalogue, for the use of visiters. “Tt is the opinion of the Committee, that the first of these ob- jects would be attained by continuing the Register commenced some years ago in connexion with the Pictorial Catalogue, which was, how- ever, discontinued about two years since. “« They would, therefore, recommend that this list be continued, and that the Pictorial Catalogue be also completed up to the present time, by which the second of the foregoing objects would be in a 206 great measure attained, especially if weights and measurements were added to each object portrayed. ‘“‘ For the purpose it will be necessary that a sum not exceeding £50 be placed at the disposal of the Committee. “Whilst the Register and Pictorial Catalogue are in course of completion, as above recommended, the Committee will take the ne- cessary steps for a permanent classification of the Museum, with a view to the preparation of the Descriptive Catalogue.” On the 7th of July, 1852, the Committee held a meeting, at which Mr. Clibborn reported that 5373 articles had been already entered in the Number Book or Register of the Museum, and that about 500 articles still remained to be entered, which it was then thought could have been done before the ensuing Session of the Academy. At the same meeting scales and weights were ordered to be pur- chased, and the whole of the gold ornaments were afterwards accu- rately weighed by Dr. Aquilla Smith and Mr. Clibborn, previous to their being exhibited at the Dublin Exhibition. Whilst the Museum remained at the Exhibition nothing could of course be done, and since its return the first care was to place it in the new room now prepared for it; which necessarily took up much of Mr. Clibborn’s time and attention, and prevented his com- pleting the Register as was at first proposed. After the return of the Museum the Committee inspected the numerical Register, and instructed Mr. Clibborn to have the columns headed weights, and where procured, filled up. Much difficulty, however, has been found in completing these entries. Mr. Clibborn reports that many of the numbers formerly pasted on the articles have come off, owing to the dampness of the new rooms, and that this accident has necessarily occasioned much difficulty and delay. The weights of all the gold and most of the silver articles, how- ever, have been inserted in the Register. The numbers have also been attached to the drawings of all the gold articles in the Pictorial Catalogue, referring to the correspond- ing numbers in the Register. The Committee must, therefore, report to the Council, that it is now impossible to make any further progress in the Descriptive 207 Catalogue until the Museum has been more exactly arranged, and the articles fixed in permanent places. The crypt under the Library, which is intended for the reception of the larger and coarser arti- cles, has been but just completed, and the glass cases, which are to stand on the railing round the gallery, have not yet been received. The permanent arrangement, therefore, cannot as yet be satisfacto- rily commenced; but the Committee recommend the immediate at- tention of the Council to this subject. With respect to the Pictorial Catalogue, the Committee after much consideration resolved, that the recent improvement in pho- tography, especially the collodion process, offered the most advan- tageous and economical means of obtaining accurate representations of the principal articles of the Museum. An excellent apparatus was therefore ordered, and several beautiful photographs were taken by Mr. Tennison, who kindly offered his services to carry out this object. This work, however, has also been stopped by the want of a suitable glass chamber for conducting the practical operations of the photographic processes. This chamber, it was hoped, might have been erected at a small expense, but on obtaining estimates it was found that the cost would be much greater than was at first anticipated. The Committee, however, have reason to suppose that this deficiency will shortly be supplied. The Ballot for the annual election having closed, the fol- lowing gentlemen were declared to have been elected Officers and Council for the ensuing year :— President.—Rey. Thomas R. Robinson, D. D. Treasurer.— Robert Ball, LL. D. Secretary to the Academy.—Rev. J. H. Todd. Secretary to the Council.—Rey. J. H. Jellett, A. M. Secretary of Foreign Correspondence.—Rev. S. Butcher, D.D. Librarian. Rey. William H. Drummond, D. D. Clerk and Assistant Librarian.—Edward Clibborn. 208 Committee of Science. Robert Ball, LL. D.; Sir Robert Kane, M. D.; George J. Allman, M. D.; Sir W. R. Hamilton, LL. D.; Rev. Sa- muel Haughton, A. M.; Rev. Humphrey Lloyd, D.D.; Rev. George Salmon, A. M. Committee of Polite Literature. Rev. W. H. Drummond, D. D.; Rev. Charles Graves, D.D.; John Anster, LL. D.; Rev. S. Butcher, D.D.; D. P. Starkey, Esq., A.M.; Rev. J. H. Jellett, A.M.; John F. Waller, LL. D. Committee of Antiquities. George Petrie, LL. D.; Rev. James H. Todd, D.D.; Aquilla Smith, M. D.; Earl of Dunraven; Colonel T. A. Larcom; Lord Talbot de Malahide; William R. Wilde, Esq. The President nominated, under his hand and seal, the fol- lowing Vice-Presidents :—Lieut-Col. Larcom, R. E.; George Petrie, LL. D.; Rev. Charles Graves, D.D.; Sir Robert Kane, M. D. 209 Monpay, APRIL 9TH, 1855. LIEUT-COL. LARCOM, F.R.S., Vice-Presipent, in the Chair. Joun T.Gitpert, Esq.; John Edward Walsh, LL. D.; and John Ringland, Esq., were elected Members of the Academy. On the recommendation of the Council, it was I. Resotvep,—That the Resolutions adopted by the Aca- demy on the 30th of November, 1854, are not intended to limit the right of the Members of the Academy to vote for any name appearing on the Balloting List prepared by the Council ; but to record the deliberate opinion of the Academy, that it is expedient that a Member of each Committee should be removed annually, in the manner which those Resolutions recommend. II. That the Treasurer be authorized to sell out of the Funds a sum not exceeding £500 of Stock, in order to meet the expenditure of moving into the present house. Professor Graves, D.D., read the second part of his Paper on the Ogham Notes in the St. Gall MS. of Priscian. ‘‘ In the Library of St. Gall in Switzerland is preserved a manuscript of Priscian, written in an Irish hand and full of glosses, both interlinear and marginal, in the Irish language. Several of the marginal glosses are in the Ogham character, and on account of their great antiquity deserve a special notice. I had been for some time aware of their existence, and my curiosity respecting them had been excited by seeing a fauc- simile of one of them in Dr. Keller’s Memoir onthe Irish Manu- scripts extant in Swiss Libraries. But it was not till I had been put in possession of trustworthy copies of them, made by Dr. Todd, that I thought it worth my while to attempt to VOL. VI. U 210 decipher them, or to speculate concerning their age. For the conduct of this inquiry I found abundant materials in the Grammatica Celtica of Professor Zeuss. In his introduction he has described the MS. minutely ; in the body of his work he has quoted a vast number of the glosses; and in an Ap- pendix he has exhibited a considerable portion of the text of the MS., together with all the interlinear glosses pertaining to it. He has, moreover, given the readings of the simpler Oghams, and to some extent discussed the question relative to the date ofthe MS. On this lattter point, however, he has not arrived at definite conclusions. He seems rather to incline to the notion, that the MS. was written by an Irish scribe on the Continent; and he does not dispute the dictum of Haenel, who refers it to the eighth century. ‘¢ Before I proceed to read and translate the Oghams themselves, I must state that they form a part of the general body of the glosses; they were written at the same time, by the same person, and with the same objects. The glosso- grapher, as was usual with Irish scribes, made occasional me- moranda in the margin, noting how his work progressed, and occasionally referring to circumstances which occurred as he was actually writing. A few of these relate to the nature of the text, as, Sude qui legat difficilis est ista pagina. Many are ejaculations, or prayers for a blessing on the scribe’s work, as Fave Brigita; Sancta Brigita adjuva scriptorem istius artis (the Ars Grammatica of Priscian); In nomine Almi Patricii; In nomine Sancti Diormitii. The saints in- voked in this way are only Irish ones. Occasionally the scribe complains of his writing materials. Thus, Ip sann membpum (the vellum is scanty); Ip tana an oub (the ink is thin); or complains of his health, as, uch mo chliab a noib ingen (alas, my chest, O holy Virgin!). But the following memoranda are important as they furnish means to determine the date of the MS. “1, A glossatp. 157: Hucusque Calvus Patricii depinatt, 211 showing that the name of one of the scribes employed on the work was Maelpatraic, of which Calvus Patricii, the ton- sured (servant) of Patrick, is a literal translation. In like manner, Maelsuthain, the spiritual adviser of Brian Boroimhe, sioned his name as Calvus Perennis in the Book of Armagh. The Irish Aunals mention several persons of this name, which was common in the ninth and tenth centuries; and two of them are actually said to have been scribes. Maelpatraic, son of Finnchu, bishop, scribe, and anchorite, and abbot elect of Ar- magh, died a.p. 861. Another Maelpatraic, scribe, wise man, and abbot of Treoit (Trevet in the county of Meath), died A.D. 885. Persons of the same name, abbots of Monaster- boice, Clonmacnoise, and Slane, are also mentioned as having died in the years 875, 883, and 886. “9, At p. 194, marg. inf., we find, 00 imp Mavbvoc oan «1. meippe 7 Choipbbpe, i. e. of Inis Maedhoc are we, i.e. myself and Coirpre. This gloss has caused some perplexity to Pro- fessor Zeuss, who was not aware that the island here spoken of was named after the celebrated Irish saint, Maedhoc of Ferns. It is in the lake of Templeport, in the county of Leitrim, and retains its name to the present day. We learn from this gloss the exact district in Ireland from which the writer came. The Coirpre here mentioned may possibly have been Coirpre Crom, Bishop of Clonmacnoise, who died a. p. 889. But the name was so common a one that little weight can be attached to the conjecture. «© 3, There is another marginal note unfortunately imper- fect, of which just enough remains to show that this transcript of Priscian was made under the superintendence of a person named Maelbriget. This note is as follows :—::: aithap Pa- Tic 7 bMS. ap Maelbpigcae namba olcc a menma ppimm:::: pepibund po pepibad in oul po. The sentence is incomplete, but it plainly amounts to a prayer addressed to St. Patrick and St. Bridget, that Maelbridget may not be dissatisfied with the scribe’s performance. Maelbrighde was a very com- u 2 212 mon name amongst ecclesiastics in the ninth, tenth, and ele- venth centuries. M. Bishop of Slane, died a.p. 874. M. Abbot of Clonmacnoise, 4.p. 888. M.Comorb of Patrick, a.p. 889. M. son of Tornan, became Bishop of Armagh, a. p. 885, and died at an advanced age, a. D. 926. “<4, At p. 112, the following quatrain is written in the margin :— Ip achep in gach innochz, Fa fuapna faipcae pinopolc. Ni G5op perme mona minn Oond laechpaio lamn oa Lochlino. Of which the following is a translation : Bitter is the wind to night, To ruffle the white crest of the sea. Long and smooth voyages are not accomplished By the fierce warriors from Lochlinn. «The mind of the writer was evidently full of the ravages of the Danes who plundered all the great ecclesiastical estab- lishments of Ireland during the ninth century. The Annals record that Clonmacnoise, and many other ecclesiastical estab- lishments, were plundered and burned by the Danes under Turgesius in the year 843; Armagh experienced a like fate in the years 831, 839, 850, 867, 893. “© 5. The gloss Ruaiopi avez, p. » probably furnishes us with the means of fixing the actual year in which the MS. was written. For we learn from the Annals of the Four Masters, at the year 874, that Ruaidhri, son of Morminn, King of Britain (i.e. Wales), came to Ireland to shun the Danes. Such an event was very likely to be noticed by ascribe, parti- cularly if it happened that the monarch came to the place where he was. ** According to the Welsh Chronicles, this Ruaidhyi, called by the Welsh Rodric Mawr, son of Mervyn the Freckled, was De 213 killed by the Saxons in the year 876. The Annals of Ulster record this event at the year 877. «« From what has been said we may safely conclude that the MS. was written in Ireland towards the close of the nmth century; and the style of the writing as compared with that of the Book of Armagh, written in the beginning of the same century, fully confirms this inference. ‘«s Having now established the date of the Oghams, let us proceed to describe them. ‘‘ They are as follows :— «TI, Page 50, marg. inf.—rerta cal HODIE.—This marked the 4th of October, which, as we learn from the Martyrology of Marianus Gormanus, was kept in the Irish as well as in other Churches, as the anniversary of the Caius, or Gaius, and Crispus, mentioned 1 Cor. i. 14. “TI. Page 70, marg. sup—fel MaRTaIN—i. e. the Fes- tival of St. Martin of Tours, kept on the 11th of November. St. Martin, as the supposed uncle of St. Patrick, was specially honoured in Ireland. Churches were dedicated to him, and the name is preserved in those of parishes and townlands to this day. ‘TTI. Page 170, marg. sup.—mINchasc—i. e. Pascha minor, or Low Sunday. The word is still in vernacular use. This Ogham has been slightly mutilated in the binding of the MS., but enough remains to make the reading certain. “IV. V. VI. Pages 193, 194, 195, marg. sup.—CO§aRT —i.e. Corrige. The word is not to be found in the diction- aries; but there can be no doubt as toits meaning. We have at p. 90, ol apcogapci as a gloss on the Latin quod sit emen- dandum. “ VIT. Page 195, marg. sup.—_ASCOHdRT INSO,—i. e. Hoe est corrigendum. Compare the gloss just referred to: also, m aeopapch mpo (gl. minime hoc est adhibendum) ; also a gloss in the Wurtzburgh MS. of St. Paul’s Epistles, clo apoéna pm rm cna (gl. quid ergo fratres ? i. e. quid faciendum in hoc ergo), 214 The scribe has used the character called eamhancoll to stand for the letters pc in apeoganc. According to the Uraicept it is properly used to denote x, which is equivalent to cs: but Irish scribes sometimes put se for the Latin 2, e. g. ascella for axilla; Mascimin for Maximin. The present mode of writing is thus easily explained. “ VITI. Page 204, marg. sup.—Latheirt.—The same word occurs in the ordinary character at p. 189. I cannot pro- nounce any positive opinion as to its signification. Professor Zeuss understands it to mean at the third hour, and refers to a gloss cepcia hona, at the bottom of p.212. But this explanation leaves the aspiration of the t unaccounted for. In Cormac’s Glossary we find a word lachoine, so little differmg in ortho- graphy that it may be equivalent to the one before us. “ Laichoipc .1. Lath ops .1. laith po n-onc .1. ol conmae. “ Uaichoipe, i.e. from laich, champion, and opc, it over- comes, i. e. drinking ale. ‘“‘ It seems unlikely that this is the true interpretation of the Ogham word, though it might possibly be a gloss on some such word as ebrietas or crapula. *¢ At the commencement of each of these Ogham notes the following mark occurs: —>. It is used in the Books of Leinster, Lecan, and Ballymote; and generally in Irish MSS., where specimens of Ogham writing are introduced. On a large silver brooch in the Museum of the Royal Dublin So- ciety, it is used both to mark the beginning of each line of Ogham writing, and to separate names from one another. In the Ogham, No. VILI., a point is used for this purpose between the words cosgapc and inpo. There is also a point at the end of No. I. In Ogham inscriptions occurrmg on monuments I have met with indubitable instances of stops employed to se- parate words. But the difficulty of distinguishing between natural and artificial marks ought to make us careful not to pronounce too positively in cases of this kind. “‘ It is to be observed that the diphthongs occurring in these 215 Oghams are written in full, instead of being represented by the poppeaoha, which are said to have been invented for the pur- pose. “‘ In concluding my notice of these Oghams, I must remark, that they furnish an unanswerable proof that the Ogham cha- racter was in use amongst Irish ecclesiastics in the Middle Ages. That Clonmacnoise was a distinguished seat of Ogham — lore is proved by the following stanzas, occurring at the be- ginning of an ancient poem on the families buried in that cemetery :-— Cataip Ciapan Cluain mic noip, baile opGccpolup veapsporp. Do pil pigpaise ap buan blag Sluaig pan piébaile ppwmdslan. Acdic uaiple Clomoe Cuno Pan pelig lecaro, leapgouino. Snaidm no cnaeb 6p 5a¢é colamo Acar amm éaeth Ceapt ogaim. That is, Clonmacnoise is the city of Ciaran, A place of bright dews and red roses. Of the race of kings of lasting fame There is a host beneath the peaceful sacred place. The nobles of the Clann Cuinn lie Beneath the flagged, brown, sloping cemetery, A knot or branch (craobh) over each body, And a correct Ogham name. ‘‘ My attention was pointed to this poem by Mr. Eugene Curry, who found it in a MS. in the Bodleian Library at Ox- ford, marked Rawlinson, 406, at fol. 7. “«‘ The truth of the statement here made is confirmed by the discovery, at Clonmacnoise, of a tombstone bearing the name 216 Colman in the Irish character, with the word boche [poor] written under it in Ogham. I doubt whether this tombstone is still to be found. My information respecting it is derived from Dr. Petrie, who furnished me witha drawing of the mo- nument made by him several years ago. Since then many of the monuments have been broken, buried, or removed to other ‘ churchyards in the neighbourhood.” Rey. Robert Carmichael, F.T.C.D., read a Paper on La- place’s Equation and the Calculus of Quaternions. “‘ Karly in the year 1852 it accidentally suggested itself that the celebrated Equation of Laplace’s Functions, which had hitherto, for all practical purposes, baffled the powers of ordinary analysis, might possibly be solved with simplicity, and in a form admitting of useful application, by the new method of analysis discovered by Sir William Hamilton. The results of the investigation thus set on foot were published in the ‘ Cambridge and Dublin Mathematical Journal,’ February, 1852. “To one starting with the simpler equation, D,U+ DP? U=0, the solution of which was known to be U=@(a+iy)+¥(a-wy), where ??=-1, it seemed probable that the solution of the higher equation, DV + Di V+ DiV =0, should be susceptible of deduction by the employment of two imaginaries 7 and j, governed by the laws Gobo ige ley — ae The integral thus deduced appeared to be V=®@ (a+iz, y+jz)+ ¥ (a@-72, y—-jz). Unable to interpret this form, and impressed with the convic- tion that, to render the solution, iftrue, of any value, such in- 217 terpretation was absolutely necessary, I took the liberty of so- liciting the attention of mathematicians to this point. «‘ Having been honoured with communications from Eng- land and France in connexion with this paper, I resumed the subject in the early part of the year 1853, and entered into correspondence with Sir William Hamilton. With his valu- able assistance I hoped to be able to overcome two difficulties which seemed to lie in the way of interpretation. It appeared desirable that the form of solution should be rendered more purely symmetrical by the introduction of the third imaginary unit &, and that by the aid of the same new element the cha- racter of the solution might be rendered more purely spatial. In one sense this form is undoubtedly spatial. If, however, we extract from it the explicit vector-unit, we get ¢cosa+j sina, which, as_referrmg to an unit circle is planar, whereas it would be desirable that the explicit vector-unit should be 2 cos a+j cos 3 +h cos y, referred to the unit sphere. “In the month of January, 1854, Sir William Hamilton pointed out the necessity of introducing some modification in the form of the solution as stated, arising out of the non-com- mutative character of the terms x + iz, y+jz,and x — iz, y—jz. ‘‘ In the early part of the present year this modification was supplied by Professor Graves, but the same objections lie against the modified form : V= MO (a+ iz, y+ jz) + MY (#- iz, y -jz). In the first place this form is not purely symmetrical ; and in the second place, its character is not purely spatial. For these reasons it seems, I would say with all due respect, improba- ble that any interpretation of this form can be devised which will meet the requirements of physical research. 218 ** Now the symbolic form of Laplace’s equation which was integrated was (D,-iD, -jD,) (D,+iD,+jD,) .V=0, which is obviously unsymmetrical. It appears then possible, that in order to have arrived at a solution susceptible of useful application, we should not have taken this form, but one purely symmetrical, and such was pointed out nearly nine years since by Sir William Hamilton, namely, (iD, + jDy + kD;)*.V = 0. Now, if we confine our regard to this latter form, and substi- tute for the imaginary symbols real quantities a, b, c, it can readily be shown that the solution of the equation, (aD, + bD,+cD,)?.V=0 is Lye EON ys EE lw Le Sees iE tied: Ke cenite 1 Up (6%, &, 6°) + (67, 65, e), where # and are arbitrary homogenous functions, of the order zero, of the quantities respectively under them, or DEY WON. sl Yah eae etter csi Pe a= =- = — er at) — = Shs mee Pe abst ee oS Gees 6 O28 a sae ce) ad bf Ifnow, in the right-hand member of these equations, we replace the real quantities a, 6, c, by the imaginary symbols 3, j, &, respectively, we get i 1D EN Seo eg ( +o4 i) Uy (67, @, e*) + vy (e?, 6, e*), and zy ,?\—(¥_22 22 9) gfy 22 2a y G4+z)o(§ PROT a: ae Bk? i 7 and modifications of these analogous to that established so con- clusively by Professor Graves, for the previous form, will give, I think, solutions of Laplace’s equation, which will sa- tisfy the conditions required. 219 ‘‘Hyom the peculiar nature of the symbols it is evident that the expressions last stated may be written in the forms, aie Uo (ée;, e", e*) Eee (e, é", é*) and p-P(n-S, ¢-&, E-n)+¥(n-¢, c-&, E-n), where &, n, Z, are the co-ordinates of the point 2, y, Z, in mag- nitude and direction, or, if I may presume to invent the phrase, the components of the points «, y, 2, and p the vector of this same point. ‘‘ There is a peculiarity about this expression to which it may be well to solicit attention. It is known to all physicists that, in the lunar theory, and in that of the perturbed motion of pendulums, there occur equations of the form, w= sin (n+) + Acos (n0+ B)+ &., implying that the value of u is not simply periodic, but admits of indefinite increase. ‘‘ Similarly, in the above expression, we observe in the first term the vector p outside the arbitrary function, a circum- stance likely to add considerably to the interest of the physical interpretation of the solution. ‘‘ That the modified form of this expression will satisfy the requirements of physical research, appears probable from the considerations, that it must be perfectly symmetrical, that it must be spatial, and that even the notation exhibits a semi- physical character. “‘ The exact nature of the requisite modification I am not at present prepared to state to the Academy, but with the existence of such I am strongly impressed, and as the subject has recently attracted much attention, these remarks have been submitted in the hope of contributing to the production ofa result which possesses much interest both for the mathemati- cian and the physicist. “Tt may be well to add, that the two other forms in which ~ 990 the solution of Laplace’s equation were presented in the paper of February, 1852,—namely, 4 [ SAe™=*"2" (cos 4/ (mi + m2) z +2, . sin y (m3 + m3) z} 1 | SBemzmy {cos ¥ (mi + m3) z—7,. sin / (m? +m?) 2} } b) with its duplicate SA erin?) { cos (mx + my) + tp . Sin (mye + may) } ) + : = Bem») (cos (mx + may) — t, . sin (mz + my) | where tt =CoSat+/sin a; and [J® (m1, mz) em™Y . cosy (mi + m3) z . dm,dm,z] + b] LJ (ami, mz) em . sin ¥/ (m? + m2) z .dm,dm,} with its duplicate ak (7m, Mz) Cos (1n,@ + my) e222, dm,dm, | AP 3 LJ” (aa, m,) sin (mya + may) es)? , dm,dm, | the limits of the integrals in both cases being supposed inde- pendent of the quantities x, y, and z,—stand unaffected.” Professor Graves observed, with reference to Mr. Carmi- chael’s paper, that he entertained great hopes that Mr. Carmi- chael would succeed in discovering the requisite modification of the symmetrical expression now exhibited by him to the Academy, so as to make it actually satisfy Laplace’s equation. Professor Graves stated that he had pursued the same track of investigation himself; but he had abandoned it in conse- quence of his finding that the expression in which Y denotes an arbitrary function, is not a true solu- tion of Laplace’s equation. This becomes at once apparent on trying the case in which the function just given reduces to 2 (4 = <\, or — (y’ + 27). 221 _ Professor Graves, D.D., read a Paper on the solution of the equation of Laplace’s functions. ‘It is not my design, in the present communication, to dis- cuss the results obtained by giving particular forms to the ar- bitrary functions f, and f,, which enter into the expression, V= Mf (y+ ja, 2+ kx) + Mf, (y—ja, z- ke), which I lately presented to the Academy as the complete so- lution of the equation EV eV dV ia) wy ae ‘¢ But I propose to give some development to the general formula, in order more plainly to exhibit its nature and the mutual relation of its parts. For this purpose let us take Mf (y + jx, 2+kx), and after developing it by Taylor’s The- orem, let us substitute mean products of js, and 4s for the or- dinary products, according to the formule of p.170. It will then assume the form P+ jh, + kFs, Cf aN. a4 ask d'f eel ae ni(qet we) a & 4 2 ade a) oo B= ie Hla a) silent? ae sail ue) Ke. where “ly 3!\dy? dydz*) 5!\ dy? ~ dysdz? * dydz ee Bape GEN ama fe af sf #y= eae a ° da 8 51 apa ; 7) ws (ayia: 2 apde 7) i J being used for brevity to denote f(y, z). It is very easy to ascertain the law according to which the coefficients of the different powers of x are formed. In F, the coefficient of (- 1)" { d? @& n (Qn) lay az 7 In F, the coefficient of 2?"*} is, Glial, sf d2s ye Qn+Ildy [ap def 7? a is, 222 and in F; the coefficient of 2?" is WR hice ibaa a (2n +1)! dz | dy? + ef ie “« With respect to these expressions Fj, F,, Fs, the follow- ing circumstances deserve notice :— ‘1. They are entirely freed from imaginaries. *©2. Any one of them is a solution of Laplace’s equation. ««3. They are connected together by the relations dF, dh ay dB, dhy_ dF. dz dy’ dx dz’ dy dx’ in virtue of which, the expression Fidzx+ F,dy + Fydz is an exact differential. “¢4, From the fact that F, is a solution of Laplace’s equa- tion, it follows, that F, and F, are likewise solutions. For as F-| DF. (Di+ Dis Di F=[' D, (Dis Di+ Dd Fix 0, and a similar proof applies in the case of F;. “5, Writing jf in place of ie in F,, or # in FP, we see that F,=axf,- Pf, Pf # (df. af df nies y = aM © CD ia) ae will be a solution of Laplace’s Equation, whatever function of y and z is denoted by /f.. “°6, If we add this value of F, to F,, we obtain a solution involving two arbitrary functions. It is exactly in this form that Lagrange has presented the solution of Laplace’s Equa- tion in his ‘‘ Mecanique Analytique,” p. 520. «7, It appears, then, that we are able to deduce a complete solution of Laplace’s Equation from one of the arbitrary func- 223 tions Mf,, Vf,: and this arises from the mixture of real and imaginary quantities in them. «8, The solution just mentioned, viz., #,+ F,, might be written in the form {1+D,? (D? + D3)}> (fit ahr). This transformation suggests an elementary process, by means of which the solution of Laplace’s function, in the form ofa series arranged according to ascending powers of z, may be obtained without recourse to imaginaries. Let the equation, (Di + D} + D?) V=0, be integrated twice with respect to x; ¢, and ¢,, two arbitrary functions of y and 2, being successively introduced in the in- tegration ; it will then assume the form {1+.D,? (D? + D3)} V=a¢2 + or Hence we shall have, V=(1+D,? (Dj + D3)}7 (#2 + di): The development of the operations here indicated will actually produce a result equivalent to Lagrange’s. So long ago as in February, 1848, I had suggested this mode of treating diffe- rential equations; but I had then little notion of the possibi- lity of applying it with any success in the case of an equation so intractable as that of Laplace’s coefficients.” Dr. Todd presented a rubbing made by him from an in- scribed tombstone in the north transept of the church of Gal- way. It bears the following inscription :— - HIR - LIETH - THE - BODI- OF - ON - MORIZRTAH - OTIER- NAGH - AND - HIS: WIF - KATERINA - NIGONOHW - AND - HIS - BROTHER - TEIGE - OG- CVPERS - AN°: DNI: 1580 - The stone is elaborately ornamented, and bears on it also a representation of an adze and square, or rule, the emblems of the trade of coopers, to which the brothers O’Tiernagh be- longed. 224 Dr. Todd drew attention to the form of the wife’s name, “« Katerina ni-Gonow,” the ni being the correct form of the Irish patronymic for a female, corresponding to the O’ for a male. Under the word orreRNacH in the inscription are the words ‘7. TEIG” in a smaller character, but of the same date, i.e. “and” or “et Teig,” alluding to the younger brother Teig og, who was interred in the same tomb. Dr. Ball read extracts from a letter from Professor Har- vey, dated Melbourne, 10th January last, in which he stated that he had just packed up his Victoria collection of Algz, and had examined and named all the new species; and that, includ- ing the West Australian collection, his list shows 556 species. He enclosed for the ‘“‘Annals of Natural History” a description of some of his new genera: as Bellotia, named in honour of Lieut. Bellot, who perished in search of Sir John Franklin ; Apjohnia, called in remembrance of Dr. and Mrs. Apjohn; &e., &e. A specimen of the Apjohnia was exhibited. Dr. Harvey was about to sail for Van Dieman’s Land, and ex- pected to proceed to Sydney in May. Dr. John Barker exhibited some bracteate coins, said to have been found in a place called the Giant’s Grave, within six miles of Belfast. 225 Monpay, APRIL 23RD, 1855. LIEUT.-COL. LARCOM, F.R.S., Vice-Presiwent, in the Chair. Rev. Dr. Topp presented a rubbing of the ancient inscription on the base of the cross which stands in the principal street of the village of Cong. This inscription does not appear to have been ever published, nor do the names it contains occur in the Trish Annals. It is written in the black letter text of the fourteenth century, and not in Irish characters, although it is in the Irish language. It is very much effaced, and by no means easy to read, but Dr. Todd, with the assistance of Dr. Petrie, who had copied it many years ago, has succeeded in deciphering it as follows :— @r’ do nichol ag’ do gille bert o dubthaich ra bin abaiddeact Cunga. Which he reads thus: OROIT DO Nichol asus Do Hillebert O OubThaich ra bIN abalohedact CuNnga. ‘« A prayer for Nichol and for Gillebert O’Dubhthaigh [O’Duffy], who were Abbots of Cong.” Dr. Wilde gave it as his opinion, that the base or steps on which the cross stands were of an earlier date than the present shaft of the cross; and he mentioned a report current in the town, that many years ago the ancient cross was carried away by some soldiers and thrown over the bridge into the river. It is said by some that it was recovered and replaced, but others think that this was not so, and that the present shaft belongs to a cross that formerly stood in the old abbey burial- ground. Dr. Todd presented an accurate drawing of a chalice made some time ago by Mr. M‘Carthy ; and stated that when VOL. VI. : x 226 such antiquities could not be themselves procured and pre- served in the Museum, it was very important to have correct drawings of them made to scale, as this was. The chalice was of silver, and stood 53 inches high; the bowl was 3} inches in diameter, and the foot, which was octagonal, with eight seg- ments of circles, as the bases of each of the triangles of the oc- tagon, was 5 inches from point to point, and 4 inches from hollow to hollow. Round the foot ran the following inscription :— ths. Conosus PMMaguir vex fermanach me fi. fe. m°. cecee®. rxtx’, Dr. Todd showed that this was the Cuchonacht Maguire, chief of Fermanagh, who was murdered by his own relatives in 1537. At that year the Four Masters record his death, and give him a high character for virtue, piety, and devotion to the Church,—so that it was quite in accordance with his cha- racter to find his name on a sacramental chalice. He was buried at first in an island in Loch Erne, where there was a small monastic establishment, but his remains were afterwards removed to the Abbey of Donegal by the Franciscan friars of that monastery, and interred there with great solemnity. The following Address to His Excellency the Lord Lieu- tenant was adopted by the Academy :— “ To His Excellency George William Frederick Earl of Car- lisle, Lord Lieutenant General and General Governor of Ireland, &c. «© May IT PLEASE YOUR EXCELLENCY, ** We, the President, Council, and Members of the Royal Irish Academy, beg to tender to your Excellency our respect- ful congratulations on your return to this country in the high office of Her Majesty’s Representative. ‘In approaching your Excellency as the Visitor of the Academy, appointed by our Charter, it is with pleasure we re- 227 cognise in you, not only the Representative of our gracious Sovereign, but also the man of letters, whose cultivated mind and varied and eminent attainments point him out as indivi- dually qualified for such a position, independently of any civil or official rank. «Your Excellency is already well acquainted with the literary and scientific institutions of Ireland, and you are therefore aware that the Academy was founded by the Char- ter of His Majesty King George III. for promoting the study of the Abstract and Physical Sciences, Polite Litera- ture, and Antiquities. <¢ It may be necessary to say, that since the period of your Excellency’s former residence in Dublin, we have obtained, from the liberality of Government, a large and convenient residence, much better adapted to our purposes than that which was formerly occupied by the Academy, and provided with ample accommodation for our Library, as well as for our Mu- * seum, which can now be displayed and arranged as its impor- tance and national character deserves. «* The Museum, which is confined to Irish Antiquities, has of late years received great and valuable additions, chiefly from the liberal contributions of the zealous friends of Irish history and antiquities, amongst whom it is gratifying to us to be able to reckon your Excellency, and to thank you for a contribution to this department of our labours, transmitted to us immediately on your arrival here, and before you had pub- licly entered upon your Viceregal duties. *« It is not for us to speak of the merit of the literary efforts of our members, which have been published in our Transac- tions and Proceedings: we would only remark, that we have endeavoured to cultivate the branches of learning to which our Academy is devoted, in such a manner as to hold out to every one, with strict impartiality, that meed of praise and of honour which his exertions and attainments have earned. ‘* Treland is now, we trust, recovering from the late cala- x2 228 mitous difficulties which depressed her trade and impoverished her people; and we earnestly hope that your Excellency’s ad- ministration, in giving force to the impetus which she has already received, will be crowned with that successful promo- tion of all useful arts, whether literary or practical, which we know it will be your Excellency’s most anxious wish to culti- vate, and with which we believe the true peace, prosperity, and happiness of Ireland will always be identified.” 229 Monpay, May 141x, 1855. GEORGE PETRIE, LL. D., Vicse-PresipEent, in the Chair. Epmunp Wim Davy, Esq., M. B., was elected a Mem- ber of the Academy. The Secretary of the Academy read the following Answer of His Excellency the Lord Lieutenant to the Address adopted by the Academy at the last meeting :-— «¢ GENTLEMEN, ‘**T request the President, Council, and Members of the Royal Irish Academy, to accept my sincere thanks for the obliging expressions which they have addressed to me per- sonally, as well as officially, upon my arrival in Ireland. « T rejoice to find, that since the period of my former ac- quaintance with your proceedings, you have obtained a more commodious area for their future development; I doubt not that your zeal and spirit will keep pace with your enlarged local proportions :— ‘Dignis invitant Pallada templis.’ ‘ 7: 255 7200 horses and mares.’ It can mean nothing else. The name consists of the same determinative sign as before, fol- lowed by characters which I read susim, or susiw. The read- ing of the last syllable is uncertain; but I connect the word with the Egyptian sesem, which agrees both in its signification and in its radical letters. The relation of the word to the Hebrew dap, sis, appears to me doubtful, though an attempt has been made to connect them. The last radical of the Assy- rian and Egyptian words is wanting in the Hebrew; the latter language does not assign to sis a feminine signification ; and I am by no means certain that the Hebrew Samekh is ever represented by the Assyrian s. ‘* In two passages on Bellino’s cylinder, we have four ani- mals in sequence, the first, second, and fourth of which are the horse, the mare, and the camel: all whose names are pre- ceded by the determinative already spoken of; while the third is represented by this determinative alone. In the pas- sage in the sixteenth line, where 7200 horses and mares are said to have been taken, 11,112 of this third animal and 5230 camels were taken. I cannot think of any animal but the ass, which this ideograph can represent. This was the beast of burden first used, and, therefore, likely to have been symboli- cally represented, and to have had its symbol prefixed to the names of other beasts of burden; and it could scarcely have been omitted from a list of captured animals, which, however, it would have been if this ideograph did not express it. I re- gard the value of the character as denoting ‘an ass’ nearly certain ; but I am uncertain how it was pronounced. A fourth word, to which this character is prefixed as a determinative, occurs in an inscription of Tiglath-pileser II., the contempo- rary of Menahem and Rezin (B. M. 68. 2). The word is ana- qatin, which occurs in the second Targum of Esther, i. 2, with the meaning, ‘she camels.’ There can be no doubt that this is the meaning here, as the determinative of females precedes that of beasts of burden, and as the word habba, < camels,’ 256 stands next before it. A fifth word, preceded by the same determinative, follows, which necessarily signifies either a foal generally, or the foal of a camel in particular. The second character in the word is different in two different copies of the inscription, and neither copy appears to be correct. ‘This ren- ders doubtful the second consonant in the word. I read it zukkari, taking it for a pubal form of the root zakar, ‘ to beget ;’ a signification of the root which has hitherto been considered doubtful, but which the word before us seems to confirm. The passage in the inscription terminates a sentence in this manner :—‘ Oxen and sheep, camels, she camels with their foals, I received.’ ‘¢ A seventh word to which this determinative is prefixed is parrin, which occurs in the seventh line of the inscription on Bellino’s cylinder, where the booty abandoned by the Baby- lonians, and captured by the Assyrians on the actual field of battle, is mentioned. It consisted of chariots, waggons, horses, mares, asses, camels and parrif, which I take to be put for pardin, and to signify ‘mules.’ The Assyrians assimilated adjoining consonants to a very great extent ; and when we find anna for anka, ‘tin’ or ‘lead,’ and gallu for gadlu, ‘ great,’ we cannot be surprised at parrii for pardin. Mules are fre- quently represented in Assyrian sculptures. ‘* An eighth word occurs in a fragment of the Annals of Tiglath-pileser II. (B. M. 52, 2nd part, 11). We have here ‘horses, pari, oxen,’ &c., enumerated as a tribute or spoil. It is possible that this may be a modification of the preceding word; but I rather think it means ‘asses,’ corresponding to another Hebrew word pere, 87D. ‘In the sculptures from the North-west Palace at Nimriid the Assyrian king is represented hunting and killing wild bulls and lions. In the pavement inscription from the same palace, where he relates his exploits, he says that he killed forty of one kind of animals, and took eight alive; ofanother he killed twenty, and took twenty alive. The name of the first kind 257 begins with a character, which, when it stands alone, is very commonly used in the inscriptions. It occurs as part of the booty taken from the different conquered nations, following horses, when these are mentioned, which it always exceeds in number; while it is taken in much smaller numbers than another animal which follows it. I cannot doubt that this character signifies ‘an ox ; and it seems to be prefixed as a de- terminative to the following character av, which may signify ‘a buffalo,’ or ‘wild bull. A similar word, determined by the figure of an ox, occurs in the Egyptian inscriptions. Its precise signification has not been ascertained; Mr. Birch, in the vocabulary given in Bunsen’s work, interprets it as ‘a cow; but there was another word in common use, of which this was certainly the signification. ** An obelisk which has been lately brought to the British Museum belonged, I believe, to the same king, the builder of the North-west Palace. I have only seen a rubbing of one of its sides. In it, the word which I have translated ‘ buffaloes’ occurs without the determinative sign. The king says that he killed all those on Harajiq, and on the sides of Lebanon, that were of larger size, carrying away their calves, and (if I be right in interpreting the next word according to the Arabic) keeping their females in confinement. Though I think there is good reason to think that this animal wasa buffalo or bison, it is never used to express any of the stone objects which the Assyrian kings erected at their gates. The word for ‘ bull’ must have been different. ‘The determinative of this word, which, when it stands alone, signifies oxen generically, is followed by different groups, all of which must signify ‘sheep, goats,’ or these two kinds of animals in common. Sometimes a word stands alone, which is evidently the Hebrew yx, ‘a flock,’ though the vowels are different from those expressed by the Masoretic punctuation. Tothis a determinative isgenerally, thoughnotalways prefixed ; namely, the character which has the phonetic value lu. I take VOL. VI. 2A 258 this word, with or without the determinative, to signify ‘ sheep and goats,’ including both species. The determinative, how- ever it was read, when it stands alone, seems to signify sheep 5 and another word, the plural of which is often joined with its plural, probably signifies ‘goats.’ In the pavement inscrip- tion at Nimrid, this pair of words seems used as equivalent to chi-int, | SZ. ‘«¢ The word which I interpret lion consists of three charac- ters, whose ordinary values are mal, sir, and khu. It may be considered as certain that all these are not their values in this group; but I have no idea how the Assyrian group should be read. It is probable, however, that some among the tablets of interpretation that are in the British Museum may solve this enigma. ‘¢ I believe that ‘bull’ is denoted by a group of which the ordinary values of the characters would be wr and makh. ‘The first of these seems to be a determinative, and the second has yery probably some other value in this combination. ‘«¢ Other animals are mentioned, on which I do not feel that I can throw any light. I confine myself to two, which it re- quires more knowledge of natural history than I possess to identify. ‘The first is named the nakhir, which, according to the He- brew value of the root, would signify ‘the snorter.” It is mentioned three times in the Assyrian inscriptions that I have seen, and in three different connexions; but they are all in inscriptions of the same king, the builder of the North-west palace at Nimrfid. At the end of the tribute of the maritime cities of Syria, Tyre, Sidon, Gubal, &c., he mentions ‘ teeth or tusks of the nakhirii, the produce of the sea.’ He says, ‘that he embarked in ships of the Arvadites, and killed the nakhir in the great sea. The noun is here in the accusative singular definite, which would seem to imply great rarity. He speaks of the nakhir much in the same way as the Americans speak of ‘the sea serpent.’ Lastly, he speaks of setting up 259 at the gates of a palace (at Kaleh Shergat, as I believe) two Nakhirin, along with six bulls of stone, and other objects of different kinds of stone. It would seem from this that it was not stone representations of these sea monsters which he set up, but the skeletons of the actual animals; for it is scarcely credible that their skins were preserved. Now what cetaceous animal, or fish, could be found in the Levant, which would satisfy these statements ? ‘¢ The other animal to which I wish to draw attention is one which is very frequently mentioned ; and I am inclined to identify it with the reem of the Bible, and to suppose that it represents a species which does not now exist. At any rate, I think it cannot have been any of the animals which are now living in Syria. ‘