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D > _ 22> Dy bE 22 >> 50> _ B®» 23> >» > >>> DD DW) 2 aE > 22> b> + > D_ DP D> 22> Sy BW; 32 >» >> B+ DdY? >» DD p | 22> > +2 B>DdI?P 2») DD. >. >> Ie > 12 IPF D> > DD 2 ODD Y»> >>> >> yr > » > BBD D>>S> > > F 227 >» DD Y> so > ke > ’ 2 PP) D> D> >_> DDD D>» 35 —P> > > > PRP > > > > 3% >>> > SS LOL > = 2 om > >> » Se5> > eo >> > 2 32 >>> > => > Dw yp 92 >= > 2&3 9D > > > 5 o> Ss a ee 2 Bw > , 3S > > 2322 29> Dw > Dw » 7 22> 2 2 DD) oP D> D>. 22 => =>>> 2 222 29 2> oP Do) eS OS > > > +> DRS 9 > >> > yd i ao > => > +> 2&2 ES §2 >> 22 D>) Im SOS >>» > 2 2 2 >>> > 22>) E> > . » >> > > > 2B >> v> 22> >>> > ep - . 2) yw >, a i. - >> > 2) 5D>>5 >>> >>> => > my, 7” > 2 Be »» > >> +> 22 @ 7 22> > > Ss 22> 3 2 222 Sp » >> 2 > >> SP > >>> O-PS D> » > > >> =. 2 23> Ss. Diz = »> >> Te 0 oe 2% >>" 2 > >> “>> D> 32> 35> 5>>> Sa Se a aD = >> ; < } AP iz > 5 » > = >> S he a , ee? | D> > SID a >> oe EP 2 a SS PROCEEDINGS OF THE VOL. VI. => Se SA ss Sy afer? On the Miaemotic Influence of the ‘Moon, By the Rev. i. Lloyd, D. D. . 28, 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.. On the Structure of the Starch enn obtained from the Potato. By G J. Allman,M.D. . .. . ier On the Method of solving a large ‘saw of Linear Differential guatiins by the application of certain Theorems in the Calculus of Operations. By the Rev. Charles Graves, D.D. - « «© © © © e © apolar On the results of a Series of Experiments on the Decomposition of Water by the Iron Galvanic Battery. By the Rev. Professor Callan. . On a new method of Measuring the Angular Aperture of the Diets of Mi croscopes. By the Rev. T. R. Robinson, D.D. On an ancient terraced Gravel Hill near out pong County of Galway. By D. H. Kelly, Esq. - - - On a remarkable Lunar Halo seen on Feb. 10, 1854, By R. Mallet, Esq, On the British Earthquake of 9th Nov. 1852. By R. Mallet, Esq. . On the Transmutation of Metals. By Geo. J. Knox, Esq. On a new method of determining approximately the Spherical Aberration of a, Combination of Lenses for Microscopic purposes. By Thomas Grubb, Esq. On the Integrations of certain Equations. By Sir. Wm. R. Hamilton. On the Laws of Diurnal Tide at Donaghadee, County of Down, and at Bun- owen, County of Galway. By the Rev. S. Haughton. A notice of some anomalous facts respecting the Tides in fase George s anti Australia. By W.H. Harvey, M.D. : ‘ On Assyrian Weights. By the Rev. Edward Hincks, D. D. ; On the Inscribed Stones in the Isle of Man. By the Rev. J. G. Gusta: a PAGE: 1 3 20 194 ~ 31 31 34 37 38 vi PAGE, On the importance of making a Collection of pi tt, of the Inscribed Stones of Ireland. By J. O. Westwood, Esq. . 3 On the last bright Comet. By Andrew Graham, Esq. Pic On the Comparison of Adjectives in the ancient Irish i ac By the IRevuCnarles' Graves, D.'D.-. ct suas «© eh ace aks we On, Experiments to determine the Velocities of me Bullets. By the Rev. S. ape ntON. =) os se boae (owe. he ae aes Ns The Calculus of Quaternions in connexion with xt taal Theorem of Dupin. By Sir W. R. Hamilton, LL.D. . . On the Barometrical Measurement of the Peak of Teneriffe. By J. Beete Jukes, Esq. . . «= » Biers ays Pease oat ea On the Ancient Name of the City of Dublin. one Charles Haliday, Esq. On the MSS. of John Colgan, ess at St. Isidore’s, Rome. as Charles Mac Donnell, Esq. . . . sits Se . On the Monastic Foundations a the Irish on dia Tee By hate ea Mac Donnell, Esq. On the same. By the Rev. Charles iia D. Dz. On the Properties of Electro- eee (second part): sf the Rey. D. R. Robinson, D.D. . Sayed! GAs On the Personal Peaniton e the Assyrian and palit Language, ‘especialy Hebrew. By the Rev. Edward Hincks, D. D. é On the Principles which regulate interchange of Symbols in certain Symbolic Equations. By the Rev. Charles Graves, D.D. . . . - . « «114 On some Extensions of Quaternions. By Sir W. R. Hamilton, LL.D. . . On Aphanizomenon Flos-aque. By G. J. Allman, M.D. On a species of Peridinea. By G. J. Allman, M. D. On the Meteorology of Ireland. By the Rey. H. Lloyd, D.D. . On the Vine Disease in Ireland. By D. Moore, Esq. On the Chronology of the er ae iris bo “ieniek By the Rey. Edward Hincks, D. D. - 4 On Assyrian Mythology. By the Rev. Edward Hincks, D. D. On the Reflexion of Plane Polarized Light from the surface of Transparent Bodies. By the Rev. Samuel Haughton, . . On the relative quantities of Potash and Soda in the Felpa of the Dublin aad Wicklow Granites. By the Rev. J. A. Galbraith. . . Onithe;same. By the Rey. 8S. Hanehton%.5 2) a wee es oe ones On the Marine Botany of Western Australia. By W. H. Harvey, M. D. On the Dying Gladiator. By Robert Ball, LL.D... . . . ... On the use of the Hygrometer in Barometric Measurements of Heights. By Capt. H. L. Renny, C. E. PO epee eit On the existence of a true Medusoid Structure in the Male Gemme of ee By Gi. Allman, MD assist er clams A Plan for ascertaining the Deviation of Ships’ ee so tei Loaal Attraction, By T: A. Dillon; Esq. “7.055. . @ % nck gia On the Solution of the Equation of nee 's Functions. aa the Rev. Charles Graves, DOB. 2 +). . ey Ge ck On the Draining of the Haarlem Lake. By per iat LL. D. . . . . . . . . 77 80 84 85 86 89 95 95 103 112 113 113 , 144 114 115 118 120 120 125 128 129 134 143 152 152 155 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. On the Solution of the Equation of Laplace’s Functions. By Sir W. R. EfanuitonseilanDeumcevenh cit sOhd iis a tases se bis. ey bars On Ecliptic Catalogues. By Edward J. Cooper, Esq... . - . ws On certain Notes in the Ogham Character on the margin of an ancient MS. of Priscian. By the Rey. Charles Graves, D.D. . . . « «. . . 199, On Laplace’s Equation and the Calculus of Quaternions. By the Rev. Robert AU enICHie ene reese atime tis cc aG te eect a esi a og NEL Brn On the same. By the Rev. Charles Graves, D.D. . .... 2... On an Inscription in the Church of Galway. By the Rev. J. H. Todd, D.D. 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. . . . Address to the Earl of Carlisle, Lord Lieutenant of Ireland. . pammwer. to thayNddresss eit sili cilc paMite,) sepa ye onl UPON Le On the Granites of Leinster. By the Rev. Samuel Haughton. . . . . . On the same. By Richard Grifith,LL.D: . . .......44 On the Molecular Formation of Crystals. By Robt. T. Forster, Esq. On the Precipitate formed in the eae of Alkaline Ee By John Aldridge, M.D. . . . si eee aioe SA eG a) tea On the subject of Dreams. By the Rev. jie Walls DeDro fe ant Os On the Ogham Inscriptions found on a Monument at Bressay, in Shetland. By the Rev. Charles CHESTER TOE IDE pitty ethane ee ia So i nD tes a RS On some Symbolical Extensions of Quaternions, and on a Theory of Associ- ative Quines, and on some Geometrical panne of his Theories. By Sir W. R. Hamilton, LL.D. . . . aan he fe oe be On a peculiar Organic Production forwarded ie W. H. Prey M. D., from Western Australia. By G.J. Allman, M.D. . . Shires wae On the Cover of the Gospels of St. Molaise. By George Petrie, LL. D. 5 On the Revival of Jtalian Literature in the Fourteenth Century. By J. F. Weallers UT, Divi a Uist cae wl 2 ey (cae hemrinerceg jatties On the Bursting of Ordnance when firing Heated Shot. By Robt. Mallet, Esq. On certain Animals mentioned in the Assyrian erp a3 the Rev. Hdward Hincks, D. De .- 2. eh et wk ore On some new Geometrical Applications of Quaternions. By Sir W. R. Ha- MUtON ME Diemer ge Wakes re kc iouei nee) eae iusaee On the early Discovery and extensive Adoption of the Bissextile Intercalation. By the Rey. F. Nolan, LL.D, F.R.S. 2. . 2... 1 On a Babylonian Tablet in the British Museum. e the Rey. Edward Hincks, D.D. . . satiate yao Tee On aremarkable ita wine sia on Lough Bisel e John Tews Esq, On Reciprocal Surfaces. By the Rev. George Salmon. . . : On the Barometric Measurement of Mountain a Rey By the Rey. J cep MCG AIDLALEAY, Ors > coh oS Mat RMM MS cot ee cece : On the Causes of certain Phenomena observed on Lough aie By Henry Hennessy, Esq. . . . . Birch sear On the Luminous Phenomena mown by the dieters of Ruhmkorff g In. duction Apparatus in Vacuo. By the Rev. T. R. Robinson, D. D. 283, 176 181 194 209 216 220 223 225 226 229 230 230 240 244 247 248 250 250 251 251 251 251 260 261 270 271 273 276 279 426 vill On certain Formulz which occur in the Works of Marcellus. By Dr. Jacob Grimm, H.M.R.EA . . - slate water oe wel teense On the same. By the Rev. J. H., Todd, D. D. Beech erd shes Be aetewe 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. . . . - - + © © ee On Meteorology. By Henry Hennessy, Esq. - - - + - A On an Optical Phenomenon observed in July, 1856, while on ets to Havre. By Henry Hennessy, Esq. . . - - - On the Extension of Taylor’s Theorem to Non-Commnniatv Symbols By the Rey. Charles Graves, D. D. ete ric On the Locality where the ancient Gold Fibula, ‘called the Dalriada Brooch, was found. By James Gilmour, Esq.. - - + + + + + + + On the Action of the Wind in quae Waves in a ate Vivarium. By 19:50, Compan, MoD eecaye) te. = htat tobe On the Castle and Manor of Pee, By J. "Hiatiend ‘Smith, Esq. On a new Experimental Hydraulic Formula for finding the pes of Water in Water-Channels. By John Neyille, Esq. . : On a Geometrical Extension of the Calculus of Quaternons. By Sir William R. Hamilton, LL. D. Inaugural Address of the President, hes: J. m" Toad, D. D. On some Properties of Solid Figures revolving on Axes in Sree, at the Surfaces of Fluids. By Gilbert Sanders, Esq. 5 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. On the Solution of Cubic Equations. By Thomas J. Campbell, Esq. On the Depth of the Ocean deducible from Tidal Observations. bes the Rev. Samuel Haughton. On the Introduction and Period . fhe general use of the , Potato in Teenie By W. R. Wilde, Esq. =e Pe aon 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 iti an Tncompresible F laid. By George Boole, LL.D. . - - ab cog : On the same. By the Rey. Charles Becai D. D. nie On the Scandinavian Antiquities of Dublin. By Charles Haliday, Pew 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 ginem of the Be By Henry Hennessy, Esq. On Inscriptions from the Market rcidies at ¢ Campbellton, in pare ia Inverary. By J. Huband Smith, Esq. cate ang On an ancient Missal, described by Dr. O’Conor in his 5 Catalogue of the Stowe MSS. By the Rey. J. H. Todd, D. D. ° On a MS. of Dr. Willoughby’s, written in 1690, ‘‘ On the 2 Climat and Dis- ases of Ireland.” By W. R. Wilde, Esq. ; On a new System of Roots of Unity. By Sir W. R. Hamilton, LL. D. PaGe. ~ 290 291 292 299 301 302 302 303 304 311 311 319 338 347 351 352 354 356 372 375 385 386 386 388 390 393 399 415 An Autograph Letter of the late Dr. sii vib Lira by Edward Bewley, M.D. . . =... ashe, outa ates On some Experiments on the Poisonous Properties of Stryhnin and Nicotine. By the Rev. Samuel Haughton. . . SU GOR hecgce At a tbaee On an Ogham Inscription. By Richard Hitchcock, ‘ihe Sia aalo gee bs On some Roman Coins found near Rathfarnham. 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. a the Rev. W. Reeves, D.D. . . . . Mw OLSh oad On the Irish Abbey of Honau, on the Rhine. By dl the Rev. W. ands 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. . . . . oGthe Abas On the Distribution of Heat over the Island of Great Britain, By iy Hennessy, Esq. . . . «= aOR al tivetitisw: ftallnt biapcetai tics Verda On the General Expression by Quaternions oi Cones of the Third Order. By Sir W. R. Hamilton, LL. D. Re! Ws Fe tr mh CORA Memorial to the Rt. Hon. Lord ‘eaeccetian, Praying the ‘‘ Resolute” to be fitted for another Arctic Exploration. . . . Aes my ton Meta 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. . . . . . . Je saute mies harika eens Letter from the Rey. J. H. Todd, D. D., to the Ci ibeieate proposing to have the Materials collected for the Ordnance Survey deposited i in the Academy. Answer from Lieut.-Col. Larcom. . . . . . 2 © © « meercttertromG@aptain meses 6s bevy se. 8 ee be te ats On the Personal Pronouns in their most ancient form. By the Rev. Edward Pian. DSP RR z 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. Hamilton, BID Sele as) he he . On a Fundamental Theorem respecting Congruences, affecting a class of Com- plex Integers, which involve the Imaginary Cube Roots of Unity. By J. FES iGIAyCs, Bist. awe: sae Bi ) or nearly =f" +f’ + op using which in 2, and setting 2a for x, we have d” — 60a*d' = 120a‘, Equating the value of d’ given by this to its general one, we have Si —- 3) + 60°790a?— 0°63151a' + 0°43461a4 — 0°29979a'+ Ke. A ea F+ 5 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. JS = 6°68. f= 1°35. M = 240. d=5-4l. @ =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 }A4+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 2b the axis minor of the small speculum, Ib. y LEH), AVS =9), fag) Fs fg’ 2-5 (24); pment os 48 bie & The condition of vision is V=u+u, whence oe ya prss) pe, and the expression of x gives F Ped (3 i‘ car) whence f' Pree cde o 2 J at as of get) ht gga) 130° which for F = 36 = 94 gives 2b = 0:2222 = 2°67 inches. f' = 01575 =1:89 ,, M = 240. @ = 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: :: b= a(5 +o -u)s 9. 4 (os ey _ ) F Zi ‘ 28 We thus obtain : - F2 : at Y : Sietths Tae o=5(1 3 = 2b = 0°2308 = 2°77 inches. J’ = 0°2998 = 3°60 ,, M = 240. @ = 13°46. whence ‘«‘ 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., Presipent, 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. ye 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 :— TaBiE I.—Diurnal Variation of the Horizontal Intensity related to the Moon’s Hour-Angle. Yearly Means. Hours. 1841. 1842. 1843. Mean. 0 = 45 ee” + 34 - 16 2 — 130 — 94 - 9 - 78 4 - 57 0 -— 18 ie 6 + 6 + 64 -1l + 19 8 +173 +94 + 55 + 107 10 +116 + 80 + 30 a iG 12 + 80 - 4 - 2 + 25 14 + 62 — 50 — 32 sy, 16 — 100 — 4] — 82 =a 18 = N8D — 32 —14 — 42 20 + 47 — 25 + 55 + 12 22 — 27 + 9 -— 4 ai G, 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 :-— Fase 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 - 61 4 aay i + 27 6 Gee sl + 32 8 + 96 fe 119 10 + Work! + 74 12 + 57 - 6 14 + 9 — 22 16 - 80 | — 69 18 -— 39 — 46 20 + 21 + 4 22 + 27 — 42 ext 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 ofthat 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 iodinized 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 g and yf be any functions whatsoever of x, and mand r any numbers, positive or negative, whole or fractional, the symbolic equation (D+ 9+) y-y (D+ 9+ C2P*) holds good for any subject which we may conceive operated on by its two members. it will be convenient to put te pI) pr © it so that the preceding equation may be written in the form A yr” = PW Ansr- And operating on this again with the symbol {-"( _) ¥", we get PrAn = Amir po" 2. It is easy to show that, for \ and wany functions of 2, (D +X) (D+ w)-(D+p)(D+rA) =H -X. Therefore, if be any function of x, and m any number, AD) (Dey) Aa eE (5), whence AvA(D+ x)={ (D+ y) Art 9-7() bbe Ac 6): If we now suppose that Xx-p=ep’, (1) where ¢ is some constant, this becomes A,A,(D+ x) =(D + x) ArAot {2eprr-r(3) } do whence again, 35 Ay A,A,(D + x) = { (D+ ) Aer tep-r- 2(¥ |} ards +Aby { 2eyr* - (2) } 4. But if we further suppose that k being some constant, the last equation assumes the simpler form, A.A, A.(D + x) =(D+ x) AwArAo+ (8¢-3rk)P7A,Ao. And continuing the same process, we should find generally AnrAn-1r yom Agate (D+ x) = (D+ x) (Ann Atnetyy ...ApAo + {(n+l)e- slash rh) b-" Aim-ayp «Ay Ao Or, since by the theorem in § 1 the variable part of the last term is equal to Any Acn-1)r shay's A, Arp", n(n+1 fee... ... A TAD y)— (Cas lc - ae vr] = (D 3P x) Ac A n-ayr see A,rAas 3. This last formula enables us to effect the solution of the linear differential equation [D+ 9)(D+x) — {es De ee a ar] y— x, @) whenever the conditions (1) and i 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 y= A} Ay... Ay (Dryyt An... AgpA,X. 4. As regards the conditions (1) and (2), it will be ob- served, that the latter limits the nature of the function y, whilst the former makes the difference between e and y to depend upon that same function. ‘ oo D2 36 The solution of the equation ($)-w 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 22" — 2 = x = Differentiating this again, we find zz” -2z2' =0, the integral of which is 2’+a’z=0; therefore, 2e(P' 4 a} 0, and = {(D + at) 0}F 5: To exemplify this theory we may assume yf = 2”; yp a m Wigs yee 5 + _¢ whenee ( gee, teeth m, and @— ¢ =¢, The general formula becomes, therefore, in this case { (D+ 9) (D+ p+a-5)- Caese ee By making c = 0, and writing - m in place of 2, this be- comes {(D+9)(D+ 9+ o) ~ MRD) 92x, 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 vy = (cos x)"; whence (5) =— 2m sec? 2, 7 = - k=- 2m, and 0-g=c,+¢ tan2. Tn this case the general formula (3) becomes [(D+9)(D + ¢+e.4etan2)-(n+1)(c+n) sec? a] y= X. By putting ¢ =c,, and c=0, this is reduced to {(D + ¢) (D+ &)-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., PresipeEnt, 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 optical art. 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 O ¥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 ease. 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.f 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, Z the intensity of light trans- mitted at the incidence @: the element of the hemisphere which transmits this light = 27. sin 0. d@; and therefore, the quantity of light transmitted by the first surface is— For an uncovered object, . . . . 2fZsin 6d6. Foracovered, . .. . . . - 2afJ*sinOd0. 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. 4l First Sur- Aperture. T=1. Uncovered. | Covered. | face curved 0 =75°. ‘, 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. « 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, A.D. 1351, it is stated— ‘William mac Donogh Moyneagh O’Kelly invited all the Irish 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 :— Filid Cipeanh 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— OOFFPO Piond o Oalargs ccc Fd Cipein so 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 Mummnecé O'Kelly, Chief of Hy-Maine, ob. 1307, by his wife, Ournéapa m 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— bids aca aléne a Geile To each other will be known Oatha Podla piofiperve The professors of smooth-landed Fodhla; Ipoarha Alban eaécpa cian And the far-travelled professor of Alba, Cp oceacca a n’apobpus Uil- Coming to William’s noble man- ham. sion. Tiucpad pin na feacc ngndoa Here will come the seven orders Do md veilb an veags Dana Who form the shape of good poe- try; E2 52 Sean diogbala a oceatc apteaé =A charm for misfortune is their coming, Na peaéc bppom-gpada pi- The seven chief orders of the lead. poets. «¢ A little further on he tells us what they were— bed bpeateatham bpeaét noli- There will be the Brehons of legal 510 judgment, bed Opaoice 7 0e1s-PIlId There the Druids and good poets ; bé1H Na pulps GFoaip Cipeat In his mansion will be the au- thors of Erin, luéc cimdais na seaitper The chroniclers of triumphant mean. histories. Cepériileipeatianba anopeam The musicians of Erin in vast numbers, Luct 506 ceipoe HO coiscean The followers of every science in common, Cn cuile dam leat an leat The flood of professors from all quarters, Cn nodal ule 50 haon ceaé. Are all journeying to one house, ‘“« He then, in a succeeding rann, describes the accommo- dation provided for them— Acdc loingiige leabta There are long houses of beds Pa comaip na curoeacta On the smooth ridge of the dry eminence, Cpdpomélad nglan ocealacoce In preparation for the company, Cp n’eagap dedoaé pice. 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 Acdio biiidean tnom 04 Gos Spdio bptigean copp Pa a scomoip Na ngoipe pm puaipe an pgiam Oo hépoaigead le hUitliam. ~Spdio apleit vonluéc reanma Od mbead pe a uét Ippeadma Seanéaide Cineai artle. Ipuomptgadna cpom-ddine. Acdio0 pan mbaile an biidean Cdide ppdio na Seanéaidead Acad ppdio pap ping oile lona lpuilio pion-colge. *60n cplog Pion ap pial cpord- ead Cp cion chap ap cleapaigead beasg pin ag PEacaid a bpuilb Timé1oll catparg wi Ceallarg. Oo ppdionib’ ap mo meadaip 5 plog ai5-F Ip oipbealais Cpé oeangad Pil oppa Rao faip pm se eacoppa. Map acdio lucpe na limb Na ppdio tilce cao’ Oh Hac pnaice pligead lume Réide DeIpse O10puTie. 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 histrionic troop, 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 Hib blade cleat copp Of houses graceful, handsome- ridged, 04 pndite teaé na ciméroll Each street by two ranges is lined, Cp clap cilce 00 Goisead Thick set with houses is the level Opum an aéaid profi-cloipos ~——Of the ridge of habitations, with its white enclosure. cd ap Gl an éldaip cilce At the end of the crowded level OGn ina caip cinlucpe Is a mansion like a capital letter ; Cinlicip cloice aille An illuminated capital, a beau- teous castle; Ofin na plata Pion saille The Dun of a fair-cheeked chief, Oaingean cloé an ofinaid se = And the stronger is the Castle Dun by it, Loé ap chlaib na cloiée. A lake behind the Castle. Realca cloice ap cian popeap A star of a Castle as such long acknowledged Of lini loca na n’Cisear Over the waters of Lough na-n- Eigeas ; G10 dille an élaé von caob call However beauteous the Castle within, CG 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.” 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. “T 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 rmg, 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. Mm. 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. « eDecgieane 78 first, by making as extensively known as possible the simplest means of making rubbings of such kind of monuments; and second, by employing some competent person, for a certain period, in visiting the o/d localities, and making such rubbings ; or by issuing printed instructions to the incumbents of theoutly- ing districts, setting forth the wants of the Academy, and the means of supplying them in this respect. From my own prac- tice Iam inclined to consider, that the old plan of heel-ball and cartridge paper is not the best which can be adopted for the process. Ihave myself found that the powdered black lead used for cleaning grates, &c., used with a hard leather rubber (an old glove stuffed with tow or hay will make an ex- cellent substitute), is more effectual for producing a representa- tion of the rough surface of stones, especially when rubbed upon common cap paper, such as is sold for lapping up gro- cer’s parcels, &c. A still simpler process has been adopted by Mr. O’Neill, and as it can be employed in cases where the black lead and leather ball are not at hand, it is worthy of being recorded. Mr. O’Neill simply uses a ball of grass, which must, however, not be wet nor too juicy, and the paper must not be thin or porous, as the moisture of the grass brought out by the friction in making the rubbings would, of course, rub holes in the paper. ‘* Considering, as I do, a collection of these rubbings of very great value, as enabling us to correct the representations of such stones which have already been published, and as forming an unimpeachable collection of figures of the monuments them- selves, I hope you will use your influence with the Academy in undertaking the commencement ofa systematic collection of these rubbings. I have myself done so with the carved and inscribed stones of Wales to a very considerable extent, and have also, whenever occasion offered, made rubbings of the more deeply carved crosses, although my rubbings show only the highest portion of the sculpture, giving, of course, no idea of the depth of the relieyo. Still, I thus obtain a general idea 79 of the size of the cross, and the distribution of the figures carved on it, which I have also found extremely useful in test- ing and correcting drawings made by the eye, of the objects in question. During the Great Dublin Exhibition, I thus formed rubbings of several of the crosses, and casts of crosses, exhi- bited in the central avenue, and in doing so I had occasion to examine very carefully the sculpture upon the smaller of the two great crosses of Monasterboice, which enables me to cor- rect the description of it given by Mr. J. D. Chambers in the ‘“‘Ecclesiologist” for October, 1848, and at the same time to give an explanation of one of the figures which appears to have been hitherto unnoticed. Our Lord is represented in the centre of the east side of this cross, not seated on His throne of judg- ment, but standing erect with a cross in the left hand, and a sceptre, with a double scroll at the top, in His right hand. On the arm of the cross to His right-hand side is represented a number of figures approaching Him (according to the words, ‘Come ye blessed’)—whilst on His left hand, a number of figures are driven from His presence—(‘ Depart ye wicked’)— by a fiend armed with a three-pronged fork, and by another figure who holds a book, and kneels on one knee, kicking the figure in front of him with the other leg. Between these two — fiends is represented a short, squat figure, with outstretched limbs, which appears to me to be intended as a representation of one of those singular sculptures termed ‘ shéela na gigs,’ to which it bears a perfect resemblance, and which is here repre- sented in its appropriate position, supposing it a personifica- tion of vice.. Beneath the feet of our Saviour is a compart- ment occupied by a representation of St. Michael weighing, in a huge pair of scales, a smaller figure, the balance prepon- derating in his favour; but beneath the scales lies the fiend, who is endeavouring to pull down the lighter scale—a mode of representing the weighing of souls, common in medieval illuminations, which Mr. Chambers has incorrectly described. Immediately beneath this group the Adoration of the Magi is 80 represented, which I here notice, as Mr. Chambers says that amongst the designs on the crosses ‘ there does not appear the ordinary representation of the Blessed Virgin with the Holy Child in her arms, nor, we believe, is it to be found in Ireland.’ On several of the crosses also, as that at Moore Abbey, is the flight into Egypt represented, the Blessed Virgin holding the Child in her arms, as usual. Anothergroup of figures common on the Irish crosses appears to me also to have been mistaken, as to its design, by Mr. Chambers. I allude to that of a figure standing with several animals on each side, from head to foot, ready to tear him. Mr. Chambers considers this to repre- sent our Saviour attacked by fierce wolves or dogs-——(‘ Many dogs came about me, they gape upon me as a roaring lion.’) It appears to me to be rather intended for a representation of Daniel in the lion’s den. «‘T remain, my dear Sir, yours very truly, «« Jno. O. WEstTwoop. ‘* The Rev. Dr. Todd.” Sir W. R. Hamilton communicated extracts from two letters which he had lately received on the subject of the last bright comet, from Mr. Andrew Graham, Astronomical Assis- tant in the Observatory of E. J. Cooper, Esq., at Markree, and discoverer of the planet Metis :— *« Markree Observatory, Collooney, *¢ 28th April, 1854. «* Your observations of the comet have come to hand to- day. They will be to me valuable, as I hope to find, leisure for further researches on the orbit. Last night was cloudy, and, having made some previous preparations, I was tempted to obtain a second approximation to the elements. At half an hour past midnight I obtained a very satisfactory verifica- tion. The observations selected were those made at Markree, March 30th, and April 15th, and at Paris, April 7th, viz. :— 81 Greenwich Mean Time. March 30 -369639 1® 22 27%34 + 19° 37’ 143 April 7 *324534 3. 7 57°46 13 17 3°9 3» lo oe 42 4 13 16°36 5 44 2:4 1854. Appar. a. Appar. 0. The corrections for aberration and parallax were obtained by the aid of my first set of elements, which, as it happens, ap- pear to be unexpectedly near the truth. To save you the trouble of reference, I here place the two sets side by side. First Approx. Greenwich M. T. Second Approx. T March 24-01183 March 24:01376 T CAA aed 213° 50’ 89 R 315 34 50 315 28 16°1 z 82 42 26 82 30 17 °4 log q. 944192 9-449544 Retrograde. Retrograde. “The second set of elements are referred to the mean equinox of April 0:0. The most remarkable circumstance connected with them is, that the observations are precisely re- presented by them: the correction of the calculated middle place, to reduce it to the observed, is + 0”1 in longitude, and + 0’:3 latitude. I do not recollect that in all my calculations I have ever known a parabolic orbit to agree so closely with the observa- tions on which it was founded. “You probably recollect that the first set of elements are founded on the Markree observations of March 30, April 1, and April 3. ‘¢ Tf you think these results worth laying before the Royal Irish Academy, at their next meeting, may I trouble you to do it? I can easily make the communication somewhat longer if you judge it necessary, by entering more into the details: but, perhaps, the whole affair may be too trifling to occupy a moment’s attention. Is not the perfect coincidence 82 of the parabolic hypothesis with three complete observations very remarkable ?” ** Markree Observatory, Collooney, “5th May, 1854. ** Dear Sir,—A set of elements can hardly be regarded complete without the addition of the constants for facilitating the computation of the heliocentric co-ordinates. They are here subjoined,— x=asin(A+yv)sec*l pv y=bsin(B+v) sec*} v z=csin(C+,) sec*} v log. a = 9°2990850 + 20:0 d log. b = 9°3470892 - 12-4d log. c= 9°4044724- 28d A = 198° 56'54"2 - 025d B= 334 54 32°5-0°57d i C= 82 53 25°7+0:28d pecs, a’ | Values of d. | T= March 24:01376 | March 22, -O°77 1:04 log nee 0:8361840 | April 1, +0-27 he | 1-08 it + 1-35 | bg log m=0°7963117 ) al, + 2°54 v = True anomaly. m = Mean daily motion, if Barker’s table be used. q = perihelion distance. T = time of perihelion passage. ‘«« The longitude of the ascending node was diminished 16’, in deducing the constants from the elements: this referred the axis of x nearly to the apparent equinox of March 31; the small equations annexed will reduce precisely to the ap- 83 parent equinox of the date. The coefficients of d after log a, &e., have for unit the seventh decimal place. The obliquity of the ecliptic has been assumed invariable, and equal to 23° 27’ 346. “¢ Tt appeared to us that the diminution of the comet’s light was much more rapid than theory would indicate. On March 30, it shone as a good second magnitude star; on April 15 certainly not more than ninth; at least, such was Mr. Cooper’s impression* as well as my own. A comparison of the distances from the earth and sun, at these two dates, gives the light on April 15, 6-4 times less than that on March 30, which would be perhaps equal to that of a star of fifth or sixth magni- tude. Iam not aware that this circumstance has been no- ticed with regard to the present comet, and therefore venture to direct attention to it as having an important bearing on the physical theory of these remarkable bodies. *¢ One is still disposed to sift the probability ofa collision with the earth or one of her sister planets. Weare certainly out of harm’s way so far as this comet isconcerned. It wasin ascending node on March 1, at 22> 13™, astronomical mean time at Greenwich. Distance from the sun, 66,193,000 miles ; therefore, 3,259,000 miles within the orbit of Venus. It was then nearly 157 millions of miles from us. It was in descend- ing node April 44 22" 46™. Distance from the sun, 43,973,000 miles; nearly 12 millions of miles without the orbit of Mer- cury. It was then nearly 83 millions of miles distance from the earth. The comet was in perigee April 1‘ 8"; distance, 80,600,000 miles. ‘‘The only thing worthy of notice, with regard to the computations, is, that in correcting the elements Laplace’s method failed. The cause is easily explained. For the mid- dle time the angle at the comet, formed by lines drawn to the * Such, Sir W. R. Hamilton stated, was also the impression of his Assis- tant, Mr. Charles Thompson, and his own, on the evening above referred to. 84 earth and sun, was nearly right, so nearly that an increase of one-tenth of a day in the perihelion passage gave an impossi- ble value for the sine of this angle. The orbit and all the cir- cumstances were particularly favourable for Newton’s method ; which was, therefore, applied with considerable modifications. «¢ Without a complete discussion of all the observations the result cannot be regarded as final. It is, therefore, need- less to dwell upon the subject, at present, to any greater length. Part of what is here inserted may be too common- place to bring before the Academy. You will oblige me by pruning as you judge proper. ‘«* The rate of motion at the perihelion was fifty-one miles per second. *¢ With sincere thanks for your kind encouragement, ‘*] am very truly yours, ‘* ANDREW GRAHAM. “ Sir W. R. Hamilton, esr ice. The Rey. Charles Graves made the following communica- tion on the comparison of adjectives in the ancient Irish lan- guage :— ‘*The most eminent Irish grammarians have constantly denied the existence of a superlative form, as distinguished from the comparative. I was, therefore, surprised to find some undoubted instances of the use of a distinct superlative form oc- curring in an ancient Irish tract, in the study of which I was en- gaged more than two years ago; and since then I have continued to collect such other examples of this kind as I met with, in- tending to bring the subject under the notice of the Academy. ‘¢On looking, however, to the ‘Celtic Grammar,’ lately published by Professor Zeuss, I found that I had been antici- pated by that learned and accurate scholar in the statement of this fact. He shows that in the old Welsh language there was a superlative ending in am, of which he adduces hinham, 85 sup. of hen, old, as aninstance. This final m passed into f in the later Welsh, which furnishes several examples of superla- tives in af. In the old Armoric, too, we meet with superla- tives in af or aff, though in the language as spoken at present the termination is reduced to a. ‘‘To the old Irish superlatives enumerated by Zeuss (Gram. Celt.’ p. 287), the following may be added :-— ampam, . . . . sup.ofampa, admirable. anoum, se. 2 3 anoac? bad, annpam, ... - » annpa, difficult. GNoaMg.: ' Hee ecei ts » apd, high. cainem, ... . » cam, delightful. epmeny eee » epion, old. SGM «2. eb. % » Slap, gray. spuloem,. . . - » Spulo, bitter. millpem, . >» milip, sweet. THOOPOMR 2110) 5” ‘s » mon, _ great. pinems ef 50... Sy ee, sy \ Olte PRUIGEM, » [Tpuid, distinguished. GINOEM 5” Fe Wis. sw “- ceann, strong. cpeipim, . . . . >» cpeip, strong. uaiplem } and ROE PES » uapal, noble. uaipluum, i «¢ 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 Rev. 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, * i d d=! Frat, a +h a A ~ and had pointed out a theorem, differing only slightly in its expression from the following : Vi.aV. By=yS.aB-BPS.ay; which may also be thus written, V.a(V. By)=S8.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, te V.a(V.dv)=S.ad.v-d8.av; wee, 2: V.a(V. Bv) =S.B4.v-BS. dv; where >){7¥Q. On a cylinder of Esarhaddon in the Museum two goddesses are mentioned : 129 the 15 of Nineveh, and the 15 of Arbela. The former of these must be Ishtar, always called the mistress of Nineveh; and the latter is presumed to be the goddess named on the tablet, who presided over childbirth. Of the goddesses named on the obelisk, the first is stated to have been named Biltu and Jar- panitu. She was the wife of Bil; and is called “mother of the gods,” as Bil is called their father. She is presumed to have been connected with the planet Venus, as Ishtar is cer- tainly the moon. ‘The name of Ishtar occurs last on the obelisk ;- she was regarded as the wife of Assur, and mistress of Nineveh. Her name is used as an appellative for “ god- dess.” On the figure of a lion in the British Museum is a long inscription in honour of the goddess of war, to whom it is dedicated. She may have been the same as Jarpanitu. Mr. Haughton communicated an account of some experi- ments made by him during the last summer, on the reflexion of plane polarized light from the surface of transparent bodies. These experiments were made with sunlight, and repeatedly verified. The new laws established a the experiments are the following :— First Law.—TIf light polarized in any azimuth be incident on a transparent surface, and the angle of incidence be in- creased from 0° to 90°, the reflected elliptically polarized light has a minimum ratio of axes at the principal incidence, and is plane polarized at 0° and 90°; or the ratio of axes is infinity. Second Law.—That as the azimuth of the incident po- larization approaches a certain limit, which Mr. Haughton calls the circular limit, the minimum ratio of axes diminishes. Third Law.—That when the azimuth of the incident po- larization is equal to the circular limit, the ratio of axes of reflected light is unity ; or the reflected light is circularly po- larized. 130 Fourth Law.—That when the azimuth of incident polari- zation exceeds the circular limit, and recedes from it, the ratio of axes again increases from unity. Fifth Law.—That when the azimuth is less than the cir- cular limit, the major axis moves always in the same direction, passing through the plane of incidence, when the incidence is. the principal incidence. Sixth Law.—That when the azimuth is greater than the circular limit, the axis major moves towards the plane of inci- dence, reaches a minimum, and turns back,—passing through the plane perpendicular to plane of incidence at principal inci- dence ; and as the angle of incidence increases, the major axis describes a minimum on the other side of perpendicular, and returns to a position at 90°, making an equal angle with that at which it set out, and at the other side. Rey. Mr. Jellett made some observations on this commu- _ nication, pointing out the great value of Mr. Haughton’s ex- periments. The following antiquities were presented by Joseph Hu- band Smith, Esq. :— 1. Stone urn, found in a tumulus at Dunadry, county of Antrim, in 1837. 2. Two fragments of rings, one of glass, the other of jet, found in the same place. 3. Portion of large stone hammer, found in Island Magee, county of Antrim. 4. Ball of green glass, found in a bog near Clogher, county of Tyrone. 5. Small globular glass bottle, found at Slievegullion, county of Armagh. 6. Fourteen ancient glass beads, found in the county of Antrim. 7. Double glass bead, purple. 8. Silver globular bead, ornamented. ee Layee 131 9. Lozenge-shaped silver reliquary, ornamented in niello, containing a ring of plaited rush and a piece of linen cloth, found in a small artificial cave at Straidcalye, near Glenarm, county of Antrim, in 1839. 10. Bronze spear-head, highly ornamented. 11. Bronze pin, with lateral ornamented head, found near Clogher, county of Tyrone. 12. Circular ecclesiastical seal. -13. Small leaden bulla, with the figure of the Virgin on the one side, a crucifix on the other, and the date 1728. 14. Stone bowl of basalt, found near Gorey, in the county of Wexford. 15. Square stone, with heraldic shield, containing the ar- morial bearings of the Cheevers family, from the church of Cheeverstown, in the county of Meath. 16. Triangular monumental stone of John Joel Josse, Kettle-drummer of Charles II., containing the arms of Ireland (three harps, two and one), and the date 1678, found in the graveyard of St. Andrew’s church, Dublin. 17. MS. containing Irish Fenian tales, from Fintona, in the county of Tyrone. 18. Two specimens of melted stone, from a vitrified fort near Banagher, in the county of Derry. The Secretary presented, on the part of William Ander- son, Esq., a wooden candlestick, found in a bog at a conside- rable depth by Mr. John Byrne, on the lands of Lower Lyrane, county of Kerry, the property of the Marquess of Lansdowne. 132 NovemBer 30TH, 1854. (Stated Meeting.) LIEUT.-COL. LARCOM, F.R.S., Vicse-Presipent, in the Chair. Tue Secretary of Council read the following recommenda- tions of the Council, adopted at their meeting held on 26th June last :— ‘¢1, That one Member of each Committee shall go off each year, in addition to any vacancies which may be caused by death, resignation, or non-attendance. «62, That the senior Members of each Committee, with the exception of the Secretary of the Academy, the Secretary of the Council, the Treasurer, the Librarian, and the Vice- Presidents, go off in each year. «©3, That in preparing the list before each election, the names of the Members who go off shall be printed in Italics, and the names of those whom the Council recommend to fill their places be printed in capitals.” After some discussion, it was moved by John Anster, LL. D., and seconded by George Petrie, LL.D.,— «© That the opinion of counsel be taken, whether the pro- posed Resolutions are consistent with the Charter.” A division having been called for, it was found that 19 Members voted for the proposition, and 31 against it ; where- upon the Chairman put the first of the above Resolutions recommended by the Council. It was moved by Sir Robert Kane, and seconded by Aquilla Smith, M. D., and Reso.tvep,—That this recommendation be amended as follows :— « That it is expedient that one Member of each Committee be removed in each year, in addition to any vacancies which may be caused by death, resignation, or non-attendance.” 133 The second recommendation of the Council having been read by the Chairman, it was moved by F. J. Sidney, LL.D., and seconded by W. T. Mulvany, Esq., and Reso.vep,—** That this recommendation be amended as follows :—‘'That the Members so to be removed should be - the senior Members of each Committee, with the exception of the Secretary of the Academy, the Secretary of the Coun- cil, the Treasurer, the Librarian, and the Vice-Presidents.’ ” The third recommendation of the Council having been read by the Chairman, it was moved by J. M. Neligan, M.D., and seconded by H. C. Beauchamp, M. D.,— «That this recommendation be amended by omitting the words, ‘and the names of those whom the Council recommend to fill their places be printed in capitals.’” The Chairman having declared that this amendment was negatived, it was proposed by F. J. Sidney, LL.D., and seconded by Aquilla Smith, M. D., and Resotvep,—“ That in the third recommendation, the words ‘who go off’ be omitted, and the words ‘so to be re- moved’ be substituted in their place.” The recommendations of the Council as amended will stand thus :— ‘1, That it is expedient that one Member of each Com- mittee be removed in each year, in addition to any vacancies which may be caused by death, resignation, or non-attendance. «2. That the Members so to be removed should be the senior Member of each Committee, with the exception of the Secretary of the Academy, the Secretary of the Council, the Treasurer, the Librarian, and the Vice-Presidents. : «3. That in preparing the list before each election, the names of the Members so to be removed shall be printed in Italics, and the names of those whom the Council recommend to fill their places be printed in capitals.” . VOL. VI. N 134 Monpay, DecempBer 11TH, 1854. LIEUT.-COL. LARCOM, F.R.S., Vicse-Presipent, in the Chair. Tue following account ofa series of analyses, instituted to determine the relative quantities of potash and soda in the felspar of the Dublin and Wicklow granite, was read by the Rev. Joseph A. Galbraith, Erasmus Smith’s Professor of Natural and Experimental Philosophy in the University of Dublin. *‘ Granite is a rock which generally consists of three con- stituents, namely, quartz, mica, and felspar, in varying propor- tions. The first of these, quartz or silex, is a mineral of in- variable composition ; the second, mica, is a mineral of com- plex constitution, admitting of considerable varieties both in its chemical composition and in its physical properties. In some granites it is replaced either wholly or in part by horn- blende; as, for instance, in the syenite of Egypt, and the granites of the county of Galway. In some by tale, as in the Protogene of Mont Blanc, and in parts of Cornwall ; and occa- sionally by the remarkable hydrated mineral chlorite, as in the granite rocks in the vicinity of Liskeard. The third con- stituent, felspar, is a mineral of very definite composition, ad- mitting of no important chemical variety, except in its alkaline constituents. The nature and extent of this variety will be best understood by comparing together the following theo- ~ retical analyses, which are extreme cases :— Composition of Typical Orthose, Composition of Typical Albite, or Potash Felspar. or Soda-Felspar. Silex ie Rs GSE Silex, | “SS O87 Pee Alumina; .. <2 abel'S:38 io Alimina,oe esa iep eee oO Potashe si... ra EAB Soda,” -* ..'\. See 100-00 | 100-00 ¢ 135 One is calculated on the hypothesis that the mineral contains no other alkali than potash, the other on the hypothesis that it contains soda only. “The fact is, that we rarely, if ever, find a felspar whose composition coincides altogether with either of these calculated analyses, both the alkalies being, we may say, always present, but in almost every case (with a few exceptions specified in the standard works on mineralogy) one of them is found to preponderate greatly over the other, and accordingly the name orthose, or potash felspar, has been generally assigned to those specimens whose chemical character is defined by a great ex- cess of potash, while the name albite, or soda felspar, has been given to those in which soda exists in great excess. There are other distinctions founded on crystallographic form, which at present I purposely abstain from dwelling upon, my present object being to lay before the Academy the results of the chemical examinations I have made of this important consti- tuent of our granite rocks. ‘The ordinary felspar occurring in granite is orthose or potash felspar, and in those cases in which this mineral is re- placed or accompanied by albite, the granite is designated albitic. In Ireland we have a very interesting case of this description in the granite of the Mourne mountains. And accordingly, we frequently find it cited both by English and foreign writers on geology as a typical locality. The granite district which stretches in a south-westerly direction from Williamstown, in the county of Dublin, through the county of Wicklow, to Poull Mounty, in the neighbourhood of New Ross, has been, up to a very late period, supposed to furnish that variety of felspar only which is called orthose or potash- felspar. Any statement, therefore, to the contrary, if made on competent authority, must naturally attract attention; and any amount of pains or trouble may be said to be well expended if we can arrive at the determination of a question of such high interest to the geological inquirer. 136 «¢ A statement of this nature, made by a Member of this Academy about two years ago, first drew my attention to the subject. Many circumstances have contributed to prevent me from carrying on the investigation as rapidly as I could have wished. Nor have I even yet completed the series which I originally projected, as I had determined not to rest satis- fied until I had fairly examined the whole range of the Dublin and Wicklow mountains from Killiney to New Ross. Al- though I have not completed my series, I have, notwithstand- ing, examined that portion of the district which extends from the quarries of Dalkey to the mountain of Lug-na-quilla, and which, so far as Sir Robert Kane’s statement is concerned, is the most important; and accordingly, I do not hesitate to lay before the Academy, as a first contribution towards a com- plete chemical investigation of this subject, the results of seven analyses of felspars taken from different localities in these mountains, and of founding upon them my dissent from the statement made by Sir Robert Kane, viz., ‘that the felspar of our Dublin granites is upon the whole a soda or albitie felspar. «¢ These words occur in the concluding statement of a com- munication made to the Academy in the month of January, 1853, in which he ‘ brought under the notice of the Academy the results of the analysis of the waters of the streams which descend from the side of the Dublin mountains, such as the Three Rock Mountain, with a view to illustrate the process of decomposition of the granite masses of these rocks, and the conversion of the felspathic elements into clays adapted to ceramic manufactures.’ ‘In this paper we find two analyses, one that of a water taken from a stream running down the side of the Three Rock Mountain at a place called Ticknock; the other of a water collected from a hole in a quarry, excavated for the purpose of cutting out the substance so well known in this city as free- stone. From the residues of both these waters after evapo- ration, the soda greatly preponderates over the potash, in the Sw ae oe ee, we 137 former in the ratio of 6: 1; in the latter in theratio of 10:1. From these facts he drawn an inference which, from its im- portance, I shall quote from the Proceedings in his own words :— «<« The analysis of these waters have placed in view an- other fact of much interest in regard to the geognostic cha- racter of the granite of the Dublin mountains. In the water there were found both potash and soda, but the latter in very great excess. This indicates that the felspar of our Dublin granite is upon the whole a soda or albitic felspar, although in particular spots orthose, or potash felspar, may be found. ‘The fact has been also verified by a great’ number of analyses of specimens of granites from various parts of the great mass which extends from Dublin into the county of Wicklow. In all the analyses made, which included both ordinary granites and elvan or granite porphyries, both potash and soda were present, and the latter almost always so pre- ponderant as to lead to the conclusion that the potash should in most cases be considered to belong to the mica 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.’—Proceedings, vol. v. p. 351, January 10, 1853. «¢ A statement so precise as this, and at the same time so novel, naturally called forth discussion; and accordingly I find that Dr. Apjohn, who was present when the communi- cation was made, expressed surprise at the fact, which he then learned for the first time, namely, that potash was either absent from the Dublin felspar, or, at most, only casually present, and then only in insignificant quantities; that Sir Robert Kane’s communication stated as a principal fact that which was wholly contrary to hisexperience. Dr. Apjohn considered this question one of such importance that he made it the sub- ject of a communication, which he shortly after laid before the Academy, in which he brought forward a number of facts 138 which appeared to him irreconcilable with the conclusion arrived at by Sir Robert Kane, namely, that in the granite of the Dublin and Wicklow granites soda existed in great ex- cess, and that only in some cases they contain a small quantity of replacing potash; and in which he stated his opinion that we could not rest such a conclusion on an argument which appeared to him too circuitous, especially when the investi- gation might be conducted in a simpler and more direct man- ner than that pursued by Sir Robert Kane, namely, the analysis of surface water, and the examination of rock speci- mens taken from the district ; and that he did not think that mineralogists would feel themselves safe in adopting Sir Robert Kane’s conclusions, until it was supported by the re- sults of experiments made directly on the felspars themselves. ‘‘ Feeling the importance of this suggestion, it occurred to me that good service might be done to the sciences, both of mineralogy and geology, if I were to undertake the che- mical examination of the minerals whose character was called in question, in order, if possible, to arrive at the determination of such an interesting question. And that I might not fall on these ‘ particular spots where orthose or potash-felspar was to be found,’ I cast my eye over the map of Dublin and its adjacent county, and selected a number of localities pretty evenly distributed over the range commencing at Dalkey and extending as far as Glenmalur, situated on the flanks of Lug- na-quilla, which is about thirty-six miles from Dublin. ‘The results of these analyses I beg to lay before you. The distances of the localities from which the specimens were taken may be stated as follows :— From Dalkey to Three Rock Mountain, . . . . 5 miles. 55 Three Rock Mountain to Lough Bray, .5 ,, »» ~duough Bray to Lough Dan, . . 2. 25. es »» Lough Dan to Glenmacanass, ...... Rivss s» Glenmacanass to Glendalough, ..... Ges, », Glendalough to Glenmalur, ....... 2 39 Per Cent. Atomic Quotients. Proportionals. Silex, 64:00 . 1-412 4:01 Alumina, : Wil... 0°352 1:00 Magnesia, O57 .. 0:028 Lime, - » Trace, . . . 0000 0:396 ie Potash, . . 1273... 0271 (99 | Soda, . . 3:00 0:097 Loss by ignition, 0°55 98:96 Sp. gr.=2°540. THREE-ROCK MOUNTAIN. Per Cent. Atomic Quotients. Proportionals. Se lke lal GSAO se 1444 . ... 4:19 Alumin, .... 17-71 . US. . Saree 1:00 Measnesia, . . ¢ ie. 1°77 0-088 Mamie, . ... «.-« Trace, 0-000 0-420 1-22 Mash, . . so 10°68 0°227 2 3:26 . 0:105 Loss by ignition, 0°69 99:51 Sp. gr. = 2'562. LOUGH BRAY. Per Cent. Atomic Quotients. Proportionals. Silex, . : 65°44 . - 44... . 404 Alumina, . 1836 . 0357 1:00 Magnesia, - 0:00 - 0:000 iime, . . sa POOR . 0:038 Potash, és 12°34 - 0:262 ouns 108 Soda, A eda” NG 0:088 Loss by ignition, 0°52 100°19 Sp. gr. = 2'554. ey 139 DALKEY QUARRIES. Silex, Alumina, . Magnesia, Lime, Potash, Soda, : Loss by ignition, Silex, Alumina, Magnesia, Lime, Potash, 4 S101 ba Ags ae Re a Loss by ignition, Silex, Alumina, Magnesia, Lime, Potash, Soda, : Loss by ignition, " LOUGH DAN. Sp. gr. = 2°453. Per Cent. Atomic Quotients. 65:05 1:436 17-72 0°345 . “Brace, 0-000 0°23 0-008 13-42 o2ss (938! 2-75 0-088 0°36 99°53 Sp. gr. =2°559. GLENMACANASS. Per Cent. Atomic Quotients. 64:19 1-417 18°39 0°358 0°34 0017 0:70 0:025 11°39 0-242 (9979 2°95 0:095 0°58 98°54 Sp. gr. = 2°553. GLENDALOUGH. Per Cent. Atomic Quotients. 63°60 oa e404 5. 18°84 0°366 0-40 0:020 } . Trace, 0-000 . 0-385 14°33 0-305 | 1-92 0-060 4 0:60 99°69 Proportionals, 4:16 1:00 1:10 Proportionals, 3°96 1-00 1:05 Proportionals. 3°83 1:00 1:05 141 GLENMALUR. Per Cent. Atomic Quotients. Proportionals. emer 2 OI GEAR OED ey BBE ig 8S SIG O4 es? of ODIO ia. EO Mapnesia, . . . . 2102... 0051 oro. 5) ais hrace,. .. 7 0000 ae LOZ... epagpmeiaien Scena ae BE ee we | OE. ww oO 085 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 3} 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. o 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 diffi- 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 Rey. 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=mpta, (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 ) = 12 = (OTT, + My or (2) pT =7p +a; the same as before. From this it follows, that in any symbo- lical equation, p (7, p)=9, (3) which has been directly deduced from the fundamental equa- es 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 as at once to form the correlative equation, ¢ (ep, —7) = 9; (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. Forasymbolical 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 7 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=7p+1, pm’ = 1pt + 7, =7(zp+1)+7; =7’p +27. Again, pt = mom + 27”, =m" (rp +1) + 27’, =mp+3r’. And for x any positive integer we get, pm" = 1"p + nmr"-*. 146 Again, operating on the equation pr=m7p+l, w( )a, with we get pw =r p-7?*; from which we deduce pr =7"p-nr". So that the equation pr = "9 +27", holds good for any integer value of n. From this again we infer that p Yr= dr ptr, (5) where Ym represents any function of integral powers of 7. And from (5), finally, we can ascend to the more general theorem, , ‘it 1 "4 4 op da = Ur op tyr ppt Tam op tke. (6) Changing 7 into p, and p into —7, in the last two equations, we obtain the correlative ones, 7 $p= 9p ™- Pp; (7) U 1 " a vr gp=9p Yr-9'p a+ > op Yr Ke. (8) In the theorems here given the reader will recognise an ex- tension to the symbols 7 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= a oD+Pa ¢D+ a vx 9’ D+ &e., (9) ea =) mp as 147 px ¢D=9D ya-g¢D Wu+ a ¢’D W'x-&e., (10) separately, Dr. Hargreave observed that the latter might be deduced from the former by changing z into D, and D into —w; 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 (afx) = x'af + fr, or, detaching the subject fx from the operations effected on it, we find that Daz=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 2 and D combine. And as in this equation we may change x into D, and D into -2; 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 Sp=Jp ps and adding these equations together, we get p(w +Jp) = (a+ 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 7+fp. Similar reasoning will show that in symbolical formule obtained in the same way, we may change p into p+fz. As particular cases of this we may observe that in any symbolical equation involving z and D, we are at liberty to change z into 2+/D, or D into D+ fe. Again, if we operate on (5) with (fz), it becomes (Uap dam da (Yr) p 41, inasmuch as any two functions of 7 are commutative. Now this again is an equation of the form pr=7ptl, (Wr) p being put for p, and ym for z. It follows then that in any deduction from (1) we may change p into (pr) "p, and (13) w into Wr. In like manner, if we operate on (7) with (¢’p)', it be- comes gp (pp) *7=(~'p)*rop +1; showing that in any deduction from (1) we may change p into ¢p, and (14) m into (¢'p)' 7. Writing « for 7, and D for p, we learn from (13) that it is legitimate in symbolical formule to change z into yz, and D - | 149 into (yz)1D. 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 zx into (¢'D)'z; js 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 PHP +t; when P and II can‘be 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). Asa particular case of formula (5), we have pelt = el" + ety, therefore, p tpn =evrper" 5 whence we conclude, that Sle + ¥n)=e "foe. (15) That is to say, the symbol, evr( jew; operating on any function of p will change it into the corre- sponding function of p + Wr. Changing 7 into p, and p into — 7, we should find Sf (w+ Yip) = evefr es (16) which shows that the symbol, evr ( ) ev, 150 operating on any function of 7 will change it into the corre- sponding function of 7+ wp. The substitution of w for 7, and D for p, in (16), leads to a result which is of considerable value, viz., that eVDfirevD =f (x+ YD). If in this symbolical equation we suppose the subject to be unity, we shall have eWPfre V1 =f(a+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- mulz 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 d end d dir dp’ either of which denotes the operation of taking the deriveée, in an algebraic sense, of any function of the symbol involved in it. : ‘ eee According to this definition — must operate on 7 only where Tv d it appears explicitly ; and so for a p ss : : : , Hence i inoperative on p, or any function of it, and is Ett. = ad , ‘ ‘ commutative with p. So also dp is commutative with 7, or d, commutative with one another; but they combine respectively any function of it. The two symbols and are plainly ate i ¥ 151 with + and p, in conformity with the law expressed by the equations, d d Bi gods d d 1 dae dps e The formule (6) and (8) may now be expressed in the sym- bolical form, ae op Yr=E™ be op, (18) dd Yr go=e ™ gp Yr. (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 Yr gp and ¢p pz, which immediately fol- low them, and not allowed to affect the subject operated on by these terms. a. : : pas As di is commutative with 7, and ra with p, we may write the formule (18) and (19) in the form, go dead (w+7) op tx 9p=9(p-< \um, and, for the same reason, are at liberty to develope urea) ; . es d shire according to ascending powers of either 7 or a A similar . p observation applies to the development of 152 ? (r- =). 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 7 and p- Thus, if pr=mp+l1, and fim =71p,+ 1, the symbols being otherwise mutually commutative, we shall have a fae A. F (Ps pr) $ (ym) = PF Cr, 1) F(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 ~ denotes a function of 7 and m, d; d t(p fF s, pit =) = ev (p, pi) ev. And, analogous to (16), f(a oe mish) = etf(ns myer; y denoting in this case a function of p and p,. Writing x 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.e. 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-strg. 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. et tee! ale Oe 155 Monpay, January 8TH, 1855. THOMAS ROMNEY ROBINSON, D.D., Presivenr, in the Chair. Tue Right Hon. Francis Blackburne; Richard G. Butcher, Hsq.; 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 dise 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- & 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. <¢ Str,—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 4 gone wrong, even after every scientific precaution had been i taken to preserve and compensate them in the most perfect - VOL. VI. P q 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. ‘© 2nd. 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 iron 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. Zhe 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., Vicse-PresipEnt, 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, PV PV @V de dy? + de =U, or (1) D}V + DV + DjV=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,+jD2+hD,) (Di-jD.-kD;) V=0. Its complete solution is, therefore, the sum of those of the two equations of the first order, (D, +jD, Se kD;) V= 0, (D, -jD, ae kDs) V= 0 3 and these latter solutions have been presented to us by Mr. Carmichael in the symbolic forms, ae alata oh (Y, Z), (2) Vu Fail eet oF 7 (Y, 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 formule. He has made a 163 GmIDytP af, (Y, 2) =f, (y — jt, z- ke), (4) . gr IPukD? f(y, z) =f, (yt ja, 2+ har). (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 gidvikDs _ giDa gkDs, This last equivalence does noé subsist, because the symbols 7 and # are not commutative. ‘< Indeed, the consideration of a simple case might lead to the suspicion that the formula (4) was incorrect. Suppose that Si (y, z2)=yz: it becomes at once a question, what is the meaning of fi(y—jx, z-ha)? Is it (y—ju)(z-kax), or (z-khx)(y—jx)? 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 —jw) (z— zx), nor (z —kx)(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, GME s, 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. x y™=(y4+ jr), and 7 2°=(2+ka)". 2. 7 yz=yz+juz + kay. 3. mt PZ=Y2 + Wayet hay? — 22 — tha’. 4.07 PB=yr2z? + Ways + Skay?z? — x25 — 3a°y?z — kasz* — Qja®yz — katy? + avtz + thas. 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+ kz. ‘* But the second example shows that its effect upon yz is to change it, not into either (y +jx) (z+ ka), or (z+ha) (y+ja), but into half the sum of these different expressions. For 4{(y+jau) (2 + kz) + (2+ ke) (y+ jx)| = yz + juz + hay, in virtue of the equation jk + 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+ ja) (z+khe) (yt+jx), and (z+ha) (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, pa-l, ‘B=-1, je+hj=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+je)? (2+ kx)’, (y+jx) (+h) (ytju) (z+ ka)’, (yt+jx) (z+ha)? (y+ju) (2+ he), (y+ ju) (2+hx) (y+ je), (z+hx) (y+ju) (z+ ke), (z+hax) (yt+jx) (z+ha) (y+ja) (2 +h), (2+hx) (yt+ja) (z+ka)? (yt+je), (z+hzx)? (y+ jx)? (2+ ke), (z+hu)? (yt jx) (2+ kv) (y+je), (e+ha) (y+ jz)’, which arise as the differently arranged products of the five factors, of which two are equal to y +jz, and three to 2+ Aa. ‘‘ Following up the analogy, we are led to expect that the effect of + upon ¥” 2” will be to change it into the oe Ti ee ee ee eT 165 (m+n)! m!n! m!n! ! {fe Bias . part of the sum of all the different expressions which arise as the differently arranged products of the m+n factors of which m are equal to y+jz, and n to z+kz. ‘“‘ The following reasoning demonstrates the truth of the proposition just stated. «« Let C be the coefficient of ay” z”” in the develop- ment of 7 y"z". Then C will be equal to the coefficient of av DD} in the development of EMID ADs), multiplied by m(m—1t)i...'. (m-p+l)n(n-1)..... (n-v +1). But, in the development of the exponential, DD; occurs only in the term at” (7D, +kD;3)"_ (uty)! ” and there has for its coefficient = (u, v) 5 (u+v)!? 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 min! SE a ye A : a (m—)! (n-v)! (ut v)! (u, ») “ But again, C, the coefficient of x” y”™ 2"” in the (m+n)! m!n! {or st \ part of the sum of the differently arranged products of the m+ factors, of which m are equal to y+jz, and n to z+ kx, will be equal to m!\n! (m+n)! 166 where S denotes the sum of all the differently arranged pro- ducts of m+n factors, of which m— are ys, n—v are 28, w are js, and vy ares. Now the number of these arrangements in S is (m+n)! (m—p)! (n—v)l plow? and § itself will obviously be of the form NZ (nu, v), N being some numerical coefficient depending upon m, n, pw, and v. But as the number of differently arranged terms in & (yu, v) is (u - v) ! piv! ? it is plain that we shall have (m+n)! Nan)! @—) Gey)? and consequently, min! C,= - = (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! th } part of the sum of all the differently arranged products of the m+n factors, of which m are equal to y+ jr, and z to z+ ke. «¢The statement of this 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+ ke). We may now proceed to interpret the expression 167 gr DarhDs) t (Ys z), in which f(y, z) is supposed to be of the form S Aya, m and n, as before, being positive integers. ‘«< 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™2". Consequently, gIPwkDs Fy, zZ)= > AM (y+ j2, 2 i kz), and, with an interpretation suggested by what has been already said, we may write finally, | Er IPakDd F(y, 2) = Mf (y+ jx, 2+ 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, aj DztkD. we are enabled, in general, to solve the equation, DiV + DjV + DV = U, (6) in which U denotes any function expressed by means of posi- tive and integer powers of 2, y, and z. The solution depends upon our being able to invert the operations denoted by D, +jD,+ kD;, and D, ~jD, —kD,; 3 168 and as these inverse operations are respectively, Z gts DerkD3) J) e™ jDytkD3) or 72D: Tr and e jDkD3) D3 eu SEED): or 7 Dy Ww, we shall have Ve cD? Dy a U+ Mf, (y —jar, z-he) + Mf. (y+ jx, +h), the two latter terms being the solution of (1). This com- plete solution, when developed, appears, in general, in the form, Fi, + jP,+kF, F,, F., and F;, being different functions of x, y, z, which singly satisfy the proposed equation. ‘«¢ For instance, we have seen above that GIP Ds) y2z28 = ye — 2 (2 ES 3y°2) + xz, + 27 (ayz® + xyz), +k (3ay?z? — vz? — ay? + t2°), 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. ‘«¢ The conclusions already obtained may be further genera- lized. For the equation, GV av gl ee sh Aa dw da dy dz’ in which U is a function of w, «, y, and z, may be reduced to the symbolic form, (D + iD, +jD, + kDs) (D -~ iD, -jD, = kDs) V= U, the solution of which depends on the inversion of the opera- tors, D+tD,+jD,+ kD;, and D-iD,-jD,-kD,. Putting iD, +jD,+kD;=P, a notation employed by Sir William Hamilton, we shall have ie a iS i i a ae tan a eel ee 169 (D + Db) ee per Ds PP (D it 5) la eve D ewe. Consequently, V= evb D7 eee P—- eur U + er? fi (a, ys Z)+e°>fe (2 Y> 2), (7) and to interpret this we must ascertain what is the effect of the symbol wb a) upon any function of 2, y, and z. ‘* Reasoning as before, we should find that its effect upon a term a’y™ 2", 1, m, and n being positive integers, will be to change it into the (c+m+n)! (l+m+n)! Ll! m!n! I! m! n! a part of the sum of all the differently arranged products of the 7+m-+~2 factors, of which _lare 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 l m n M (2+iw, y+jw, z+ kw): and, more generally, the effect of oe upon f(a, y, z), any function consisting of positive and integer powers of x, y, and z, will be to change it into Mf (a + tw, y+jw, z+ kw). We are thus enabled to interpret the formula (7), when U, Ji, 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, ky we have (ta +jb + ke)? = (— 1)" (a2 + B+ 0?) se, (8) Now, the coefficient of a?\b*c” in the left-hand member of this equation is 3 (2A, 2u, 2v), in conformity with the notation explained in p. 165: and the same coefficient in the right-hand member is plainly (A+pt+yv)! Alply! ° Consequently, we have the theorem I. (- pee (A+ m+ v)! Aluly! ° ‘“¢ Multiplying both sides of the equation (8) by ta +jb+ ke, we get (ia +jb + he) ?e = (— 14 (ta + 7b + ke) (a? + B+ che, The coefficient of abc” in the left-hand member is = (2 +1, Qu, 2v); and the same coefficient in the right-hand member is = (2X, 2u, 2v) = (- LD i ga a ie aes 7 ae i We have, therefore, II., S(2X+1, Qu, 2v) = (- 1) See i, and similar expressions for j and /. « Again, multiplying (8) by (ta +jb + ke)? =— (a? + b+ c*) we get (ia + Jb: + fee) eee — (1) Mao (a? + 0? + ch) eet, The coefficient of a" 6°" c” in the left-hand member is =(2A+1, 24+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, III, 171 = (2A +1, 2n+1, 2v)=0, If\=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 jen 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, Fresruary 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 rivers and 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, . 15°9,, : 167 °,, «' 22:6>.,, -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— Das a ei bse a te. ce | COG UOC Benes Natural lands," . 2)... +, LOU = PORMCLEy se a ee RE tee 305,000 acres. 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 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 Rey. 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,)+ 2(R.0;, SiO;) in which formula, in the potash micas m=1 n = 2, 3, or 4. «‘ The last two cases, = 3 and = 4, being the Muscovite of mineralogists, and the first, n = 2, or —- 177 being known as Margarodite. ROSIO, + 2(R,0,Si0,) + kHO : , «‘ The Micas of the Dublin, Wicklow, and Carlow granites, analyzed by me, belong to the Margarodite genus, and contain two atoms of water, corresponding to k = 2. “The following are the analyses of three micas selected from three distant localities of the granite chain of the south. east of Ireland. Silica, . . . 43°47 Alumina, . . 3142 Peroxide ofiron, 4°79 fiime, -. . ..- 138 Magnesia, . . 1°13 Potash, . . . 1071 Boge, . . . . see Loss by ignition, 5°43 . 99°77 Meuica, . . . 44°71 Alumina, .. 31°13 _ Peroxideofiron, 4°69 Lime, ... 1:09 Magnesia, . . 0:90 Berotash,. . . 9°91 memogs,. . . . 1°27 _ Loss by ignition, 6-22 99°92 q gray, transparent. Atomic Quotients. 0959 . . . 2°863 0611 0:059 0:049) 0:056 | 0:228 0:046 0603. . . 1:800 } 0670 2000 Atomic Quotients. 0987 . . . 2973 0:606 ‘ 4 * pose | 0064 21000 0-035 0.045 211 [0832 1-000 0-041 J 0691 . . . 2083 a 1131 e «6 No. 1. Mica from the Three-Rock Mountain, county of Dublin; gray, transparent, containing specks or flakes of ‘a bronze-coloured or black mica. Per Cent. no ww _ a “ No. 2. Mica from Glendalough valley, county of Wick- _ low, gray, transparent, with specific gravity = 2-793. Per Cent. * No. 3. Mica from Mount Leinster, county of Carlow, 178 Per Cent. Atomic Quotients. Giles ea AOS. OO RR. . CSE ee Aloming, cn.) 3018) 25, «00587 . ; Peroxide ofiron, 6°35 . . 0:079 } SSS. (Ca Sane PINE, ? riser O00). oe JOMOO Magnesia, . . O72 . . 0036\ Paragh. «0, Sag: I aoe (9298: ee sodas, it 4% teirace vt +), 0°000 Loss byignition, S32... . OS91 ... Ite we 99°61 «Tf we take the mean of these analyses we find— Average Mica of Dublin, Wicklow, and Carlow Granite. Per Cent. Atomic Quotients. Sidi wa eS AAT 2200. SRD «2... Ga Alomina,) Sis:.4.. << S0SL S72), ..".2-O60L : z 0°667. 2 Peroxide ofiron,. . 527 . . . . 0-066 } 7 Tana pee oe = DBS OIPSP So Marnena, =. OOS See, 3. Oa | : Pou... se « LIRR)... 4 Se eda ! Sata Pt 6 SSO GEM 2) CO Loss by ignition,. . 566 . . .. 0629 .... 2 99°76 «“ There can be little doubt, from the foregoing analyses, that the transparent gray Mica ofthe district under considera- tion is a Margarodite, having the rational formula ROSiO, + 2(R.0,, SiO;) + 2HO. The mica which deviates most from this formula is the mica of the Three Rock Mountain, and this deviation may possibly be due to the presence of grains of black mica, which also occurs in the mass of the granite, and of which I was unable to obtain a sufficient quantity for chemical or optical examination ; the quantity of protoxides in the Three Rock mica is somewhat in excess of that required by the formula. ‘«‘ The angles between the optic axes of these micas were carefully determined, and found to be as follows :— 179 Angles between Optic Axes of Micas. 1. Three Rock mica, ong SAT ake aaaee OTe 2. Glendalough mica, .. . a BET Pe: 3. Mount Leinster mica, . . . . .- 72 18 4. Lough Dan mica, - . + ++ + 70 0 5. Glenmalure mica, ..... . 67 Ii <¢T have added to the determination of angles of the micas analyzed the optic angles of two other specimens of gray, trans- - parent mica from Lough Dan and Glenmalure, county of ~ Wicklow. _ « The four micas which were free from any intermixture of black mica have a high angle, while the angle between the optic axes of the mica from the Three Rock Mountain, which contained flakes of black mica, is nearly 20° less than _ that of the pure transparent gray micas. E «I think that the uniformity of the preceding analyses is _ sufficient to show that Margarodite is entitled to be considered "asa distinct species of hydrated mica, and that it is not merely an altered form of Muscovite, an opinion advocated by Mr. 999 Dana, in the last edition of his ‘System of Mineralogy. | James Apjohn, M. D., made some remarks on Mr. Haugh- q ton’s communication. The Secretary read a letter from Mr. Hornsby, Secretary _ to the Board of Works, presenting to the Academy Museum _ the following articles :— 1. A small wooden bucket or situla, bound with bronze hoops, having perforated lozenge patterns. 2. Ornamented bead, composed of blue glass, with bands of red and white glass. 3. Bone bead or ornament. 4. Fragments of bone comb. 5. Neck and handles of an earthen jar. 180 . Head ofa fibula, ornamented with a curious mosaic. . Bronze hoop. . Bronze spear. . Three dorsal vertebre. 10. Os pelvis. 11. Boar’s tusk. All found in the river bed, between Aunagassin and Castle Bellingham. Nom? os Mer) 12. Bottom of an ancient bronze vessel, curiously orna- mented with raised bands. 13. Large elk’s head and horns, found in the bed of the river Dee, between Nobber and Whitewood, in the county of Meath. 14. Small brass pin or bodkin, found in the river Glyde, about a mile below Castle Bellingham. Dr. Ball made some remarks on the animal remains; and Mr. Wilde drew the attention of the Academy to the bucket, which resembled two others in the Museum. Dr. Tuffnell exhibited a curious wooden staff, with Turkish inscriptions, which he procured at Silistria, and read transla- tions of the inscriptions made by Mr. Benmohel. 181 Monpay, Fesruary 26TH, 1855. THOMAS ROMNEY ROBINSON, D.D., Presipen’, . in the Chair. Rev. Proressor Graves communicated the following extract } from a letter addressed to him (under date of January 26th, 1855) by Sir William R. Hamilton :— «My pear Graves,—Y ou may like, perhaps, to see a way in which I have to-day, for my own satisfaction, confirmed (not _ that they required confirmation) some of the results announced by you to the Academy on Monday evening last. ‘«¢ Let us then consider the function (suggested by you), S ajnn” = (1, m, n) vk” 5 (1) where /, m, n are positive and integer exponents (0 included) ; _ the summation & refers to all the possible arrangements of the 1+m+n factors, whereof the number is (l+m+n)!_ Ni, =, 9= Tata (2) q each of these N arrangements gives (by the rules of 7h) a pro- duct =+1. aymk"; and the sum of all these positive or nega- tive unit-coefficients, + 1, thus obtained, is the numerical co- efficient denoted by (/, m, 2). ___ * Since each arrangement must have 7 or 7 ork to the left, we may write, Sym he = tS MR” 4 PSTN + hVUymh (3) _ and it is easy to see that the coefficient (/, m, m), or the sum _ = 1), vanishes, if more than one of the exponents, J, m, n, be _ odd. Assume, therefore, as a new notation, 7 (2A, 2u, 2v) =(A, pw v}3 (4) =) VOL. Vi. R . 182 which will give, by (3), and by the principle last mentioned respecting odd exponents, (2X +1, 2, 2v) = {A; bs Me (2-1, 4p, 2») ={A—1) wy v)- We shall then have, by the mere notation, > pk =(r, py vp PPR” 5 (6) and, by treating this equation on the plan of (3), {A, ws v}=(A-1, py vj + (A, w—1,v}+ {Asp v1}. (7) By a precisely similar reasoning, attending only to j and &, or making \ = 0, we have an expression of the form, 3 jel = (uy 9) Pa 8) where the coefficients {y, v} must satisfy the analogous equa- tion in differences, 6) {u, vj={u—-1, v}+{uwv-1}, (9) together with the initial conditions, (a, 0}=1, (0, v)=1. (10) Hence, it is easy to infer that heal (ut+v)!. [Hs vj = leila (11) one way of obtaining which result is, to observe that the ge- nerating function has the form, S {u,v} u've’ = (1 -u—-v)". (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, sv} = (ps V}> {A, 0, vy} ={A, v}, {A, uw, O}={A,u}, (13) when the second members are interpreted as in (11), we find that the (slightly) more complex generating function sought is, = (A, wy, vjPute=(1-t-u-v)7; (14) and therefore that the required form of the coefficient is, a 183 el Al ply! as, I have no doubt, you had determined it to be. {A, Ia v}= «« With the same signification of { }, we have, by (2), ° Ni, m, n= {2 m, nj; (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°y2#k, the following : : ig um whe M dA j2H fe = (20, 2u, 2v} er) to 5 (17) or, substituting for { } its value (15), and writing for abridg- ment K=Atuty, (18) : aN oD shes ae SCE | eT ee eT Ne aw | (19) In oe manner, 2A+1 neh te yb: ef (2X + Wye (2u) ! (2v) ! ae De ee eek (2«+1)! AL pikencarlod eo ‘The whole theory of what you call the mean values, of products of positive and integer powers of ijk, being con- tained in the foregoing remarks, let us next apply it to the determination of the mean value of a function of w + iw, y + jw, _ 2+kw; or, in other words, let us investigate the equivalent for your Mf (a2+iw, ytjw, zt+kw): (21) by developing this function f according to ascending powers of w, and by substituting, for every product of powers of ijk, its mean value determined as above. Writing, as you - propose, ; d d d ——— —_ = op dw > dx Dy dy ‘we are to calculate and to sum the general term of (21), namely, d 29 — D;, (22) R2 184 MM phe" x i DPD: SF (&s Ys 2): (23) 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 = 2A, m= 2u, r= 2v, (24) the corresponding part of the general term of Wf namely, the part independent of zk, is by (15), (18), (19), (— *)* (2x) whereof the sum, relatively to A, u, v, when their sum xk is given, is, {r; My v} DOE D? f(z. Y 2) 3 (25) ‘Got (Di + D3 + Dif (@, y, 2) = & = hi if my signification of < be adopted, so that 4=1D,+jD.,+kDs; (27) and another summation, performed on (26), with respect to x, gives, for the part of M/fwhich is independent of ¢j/, the ex- pression, 1 (6% +64) f(a, y, 2). (28) ‘* Again, by supposing, in (23), 12 ONS 1, WS Qu, BSD; (29) and by attending to (20), we obtain the term, wiD, (—w*)*, Gee Tyh Doro vi DEDEDE f(a, 9.2) (80) Adding the two other general terms correspondent, in which iD, is replaced by jP, and by kD,, we change iD, to q; and obtain, by a first summation, the term (wa) (2«+1)! and, by a second summation, we obtain I (45 Ys Z) 5 (31) 1 185 a (ert —e"4) f(x, ys 2), (32) as the part of the mean function Mf, which involves expressly yk. 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 («+ tw, y+ jw, 2+ kz) =e4 f (&; Ys 2) (33) _ In like manner, Mo (a - tw, y—jw, 2—hz) =e" (a, 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*)70; (35) that is, the partial differential equation, (D? + Di +D} + D3) V=0, (36) has its general integral, with two arbitrary functions, fand ¢, - expressible as follows : a V= Mf (2+ iw, y+ jw, 2 + kw) + Mp (2 - iw, y - jw, 2 - kw; (37) _ 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, ews D1 ew 4 D1 evs U7, (38) Tn fact, D+qz=ersrDerst, D-q=ersDers, (39) and therefore, CDS ayo Aye = eu? errs Dane (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, «© Witt1aM Rowan Hamitton. -“ The Rev. Charles Graves, D.D.” — 186 The Rev. 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 (7 Dy+kD. é J+ 2, 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 x be positive integers, into the ((m+n)!\* m!n! (m+n)! m!n! part of the sum of all the differently arranged products, of which each contains m factors equal to y+ jx, and 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 » are positive integers: and this has been accomplished by the help of mean products of 7s, 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 in 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. «I will now proceed briefly to describe the manner in which I investigate the effect of 7 upon yz", when m and 2 are negative or fractional. «© As before, I commence with particular and simple cases, expecting that the results will guide us, by the observation of analogies, to a general conclusion. «Let us first calculate r y%z'. Putting p in place of jD.+kD;, we have . Db ay) = —jy*z" ats ky iz, pty st=- 9! foe = 91 Pz, D3 ytzt=3! jytet4 2) hye? 4 Ql jyrets 3 lhyizt, piytzt=4lyizt+2.212lytzt+4l yz, pi ytet=-5! jytzt—4lhytz?- 2.312! jyte® —~2.2!3! hkytet—4ljy?z-5!l hy'z*, De yizt=_6! yz 3.412! y%29-3.214 1 y%2*-6 lye, _ Hence, 2 2 T mee =yz) —2 Gy?z* me ky 27) ue = (2 ! yz Lig 2 lye) : 1 ai (3! jytet + 2! ky sz? 42! jy2z? + 3! ky te) x +g Alytet+2.21alytzts 4tyz”) ee f ‘ = (6 liytet + A lhytet 42.3121 jy te +2.2'3!kyiz*+4 lyytze +5 | kytz-*) 6 — 7 (6!yte" +3 412 Raye 84.05 2 LAL yates G !ytz?) . (A) 188 4 ‘«* Now let us compare this with 3 {(y +juy? (2+ hay? + (+ he) (yt jx)"}, 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+ joy? (2+ ha ay et —a (jy shy zt) ~ 2 (y? ord 2 ufZt +y) Zz) +23 (jyte? + kysz? +jy%z? + ky z+) +24 (y 82) —iytz? + ytz3 iy? zt t+ yz’) —-2° ( ym af hy®z* a ys ay hye ae jyZ b (B) co + hy z*) — (yiz) a iy *z* ae yz = iy tz 4 yz 2 ty *z® + yp z7 ) Ea ical lus nay J And (z+ a)" (y+ja) differs from this only in the signs of the terms containing 7. Consequently, the development of 4 {y+ jay (2+ha)? 4+ (e+hey (yt+jzy} differs from the series just given only by the omission of these terms. But this omission will not make it agree with the ex- pression already found for 7 yz". «‘ The discrepancy first shows itselfin the numerical coeffi- cients of the terms sy%2, sy%z*, and sty%z°. In the former development (A) these coefficients are all =4. In the latter (B) to unity. ‘* Again, the coefficients of a°y*z*, ay%z4, a8y?z*, 2y*z”, asy®z*, and 2*y*z* are all equal in (A) to }, 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“z= and a*y?z*; it must be observed that in (B) these terms have 189 respectively the imaginary coefficients & and j; or, more ex- actly, after the restoration of the powers of j and & suppressed in virtue of the equations 7?=h?=-1; the imaginary coefh- cients —7?k and —k?j7. Now, by the theorems in p. 170, the mean value of the product of two js and one & is, 2!1 3, 2!1 1 3 (2, Dray I)k=—gk; _ and the mean value of the product of two &s and one 7 is — 4). So that, so far as concerns the terms 2*°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. «The next discrepancy noticeable is in the coefficient of z'y%z3, In (B) this is /*h, if the suppressed powers of 7 and : k be restored. Now the mean value of the product of two js and two &s, by the formule of p. 170, is PAN pi ak | Acta a Whee Here again we find a mean product of js and #s in (A), cor- responding to an ordinary product in (B). ‘The next discrepancy occurs in the case of the coeffi- cients of 2°y4z* and a*°y*z*. In (B) these are —7*h?, and — 7?h, if we restore the suppressed powers of jandk. Now the mean value of the product of three j s and two fs is, See edly Bre ein Be” and the mean value of the product of two js and three /s is a 1k. Here again, therefore, we find mean values of products _ of js and ks in (A), corresponding to ordinary products m in (B). ‘* Let us next consider the coefficients of 5y*z” and a*°y*z°, _ In (B) they are —j*k and — jk‘: but the mean values of pro- 190 ducts of one j and four £s, or four js and one &, are respec- tively ifs! 78 nd = aS. or gj and 5 k, so that here likewise we find mean products in (A) standing in place of ordinary products in (B). ‘‘ Lastly, the coefficients of x°y*z*, and x'y*z* in (B) are j*k? and 7*k*. Now the mean value of the produgt of four js and two ks, or of two js and four /s, is = — 3, which is the coefficient belonging to a’y%z? and a°y*z* in (A). ‘¢ It is, moreover, to be observed that all the terms which disappear out of (B) have coefficients like jk*, the exponents of both j and & being odd numbers. Now the mean value of a product containing odd numbers both of js and /s has been proved equal to 0. ‘It is also deserving of remark, that where the developments coincide, the mean values and the ordinary products are equal. Th fact, these coincidences occur in the case of the first and last terms in each group of terms multiplied by the same power of x; and in their coeflicients js and /s are not combined. ‘¢ So far, then, as our examination has extended, the discre- pancy between the developments (A) and (B) consists in this, that mean values of products of js and /s stand in the former in place of ordinary products occurring in the latter. ‘«‘ The careful examination of this one example led me to suspect, that when m and n are integers, the difference between the expression 7 y™z”" and the ordinary algebraic develop- ment of (y + ju)™ (z+hka)”, effected without any regard to the properties of 7 and &, consists merely in this, that mean pro- ducts of js and ks take the place in the former of ordinary products occurring in the latter. To test this hypothesis let us try another simple example, in which y and z are not symmetrically involved. Let us calculate 7 yz1. We shall have then 191 D> yzi=j21-kyz, Db? yz? =-2lyz, De yzt=— 2) 92343! hye, Dt yzt=4! y2°, De yzt=4! j2°-5! kyz, b§ yz'=-6! yz", Hence, we find 3 w yz =ye)+ a (jet — kyz*) - ayz? - a (2! jz%-3! hyz*) . A) 5 ( + avtyz + A (41 j25-5! kyz*) - ayz" — &e. Now let us compare this with the development of (y+ja) (z+kn). Expanding by the binomial theorem, and preserving the powers of j and 2, when both appear in the same coefficient, we have (y+ je) (2+ hey? = ye + a (ja? — hye) — a" (jhe + yz") + 08 (jh?23 + hyz*) — at (jh824 — yz) +25 (phi — ky2*) — a8 (jh’e*+yz7) .. (®B) 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 a°z*, atz+, a'z*, &c., do not appear in (A). In (B) they have the coefficients jh, jh, jh®, &c. But the mean values of such products of 7 and & are equal to zero. *«‘ Again, the mean value of the product of one j and 2» : 1 = ie ks is, a1 (— 1)”. Hence the coefficients in the two develop- Vv ments of a°z*, wz, &c., differ just in this: that in (A) they are mean products, in (B) ordinary products of js and ks. 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) (2 +2)", we shall produce the development of 7 yz". *¢ Without stopping to consider the case where m or n is 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 yz" in the development of a y™z", is equal to the coefticient of «,*,DD; in the develop- ment of e%/2*"», multiplied by m(m—1)....(m—p+1)n(m-1)....(n-v+1). But, in the development of the exponential, Ds D; occurs only in the term a” (7D, +k.D;)e (u =F v) ! and there has for its coefficient = (us, v) (u + v) Ly Consequently, the coefficient sought is ra (m= Us hei ih oe St ohee eee “‘ But again, if we develop (y+ jx)” (z+ kx)" in the manner already mentioned, that is to say, preserving all the powers of jand k, and afterwards substituting mean products for ordinary products of these imaginaries; the coefficient of gevyme 2" is plainly . men): - ons) eles as Oe ae pp! v! or, since M(j, k) = TE: A 1 = (us v); to 1a (me Tie ot Nails We» 1 owt l) ae (u+v)! Thus, we have demonstrated generally that the expression 193 Pars f. (y; z), is to be interpreted as follows :— “ Substitute y+ jx for y, and z+ kw 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 gi0iDyjDyskDy) upon any function whatsoever of 2, 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 2, 7, and & 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 limiiation of the nature of the arbitrary functions, provided we modify, 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 that given 2% p. 166, in the case where m and n are positive integers: and it includes the cases where m and n are negative or fractional, to which the original defini- tion of a mean product is inapplicable. ‘¢ Tf it should prove that the solution of Laplace’s equation now attained to, viz. : V= Mf, (y + ja, z+ ke) + Mf, (y -j2, z-ke), 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. Rey. Samuel Haughton made some observations on the Rev. Dr. Graves’ paper. Rev. 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 ‘¢ Kcliptic 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 ‘Kcliptic Stars, one which we considered the most likely to oad a 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- 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 j _ membered, that it originates in a comparison of previously de- 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 4 its greatest freedom from rain, he would find it low in the order of productiveness; or November as the least variable, still the 196 Taste L. Montu. 1848, | Be ati) 1850. | 1851, c. | x. | D al & || ee, ade co. | x. | p. |rotan|| c | x July, . J... .]. .|} 353] 41] 38] 432] s6|- 31 7} 96|| 874) 28 August, . .| 949| 49) 67 1065! 5| 19| 34] 58|| 183). 18] 16] 217|/1205] 168 September, .| 734] ~64| 47 g44l'1i51| 921 168\1411|1076 98| 20111375/1137| 165 October, . .{1009] 198) 18011387) 864| 101] 95|1060| 637| 83| 62| 782)| 597| 120 November, .| 470] 54} 56) 580) 209, 34, 11) 254) 230, 35] 23) 288) 914) 143 December, {1534 222) 7/1763/| 260) 31| 80] 321/1181) 117) 461294| 912) 132 January, . 959| 164, 20/1143| 676| 101| 60| 837/1520 199 February, a 47| 7| 2) 56)| 886) 91 March, ; 502, 69 87| 6081340, 214) 3911593\| 602) 77 April, « : | 0 | 0!0/0 | 249) 33] 8 m0 850| 77 May, . 5 .| 0 | 0 0 | 0 | 172) 26 9] 207) 404) 18 June, . ‘ | i : 3 4 55} 0 | 010 | 0 1s 8 ite aes | | TaB.e II. Taste III. Months in the Order of abso- Months in the Order of least lute Productiveness. Inches. Stars. | Average. Rain. September, averaging. .| 1215 |] May: x a gauionts eee tosae April, + abe bee Ki 7 Aa March, ~ sf) * 5.2) MR O8o March, = came 1112 | September, . . . . . |” 2383 December, os cheeses (i 11) Din, AIpIalS ets Je” ool we tle ae January, e Pins 898 | | February, «).< « =: «)|@20862 October, + Sa 85 818 789 PNVCG, cy se) is fev, +: oe eee November, fi hae 699 December, . . .. .| 3.266 February, 3 car 201848 i Sy etbiay si ape Oo . ae - “AR hdr Pome te aarti ie Vol. 115 page: (8,4 a. 4 Ba Sie gh MUR 4 ea eee 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. Rey. 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. 15, Seeeigo in ci PAIS Aen ES ay, 20) cotit nee 1d Wie o 1% .Y md , . €.1 Tir Alt Sky ee oh i eo i hes by abate ves oA oe ‘ ».s 201 Marcu 16TH, 1855. (Stated Meeting.) THOMAS ROMNEY ROBINSON, D.D., Present, 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 ;” 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 and from Pro- 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 > with several Downing, on ‘‘ The Draining of the Haarlem Lake;’ 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 , _—— | : 4 ; 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 ; - abronze 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, Jonny Epwarp Butter, Esq.; elected 11th April,1853: died 23rd July, 1854. 2. C. T. WeszBer, 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 Rem; elected 12th April, 1841: died 12th Sep- tember, 1854. , 5. Oxtver Sprovte, Esq.; elected 10th May, 1841: died 3rd September, 1854. 6. Rosert Forster, Esq. ; elected 13th January, 1845: died 2nd June, 1854. 7. Witt1am Epineton, Esq.; elected 10th March, 1835: died 13th October, 1854. 8. Wynpuam Goo.p, M.P.; elected 9th June, 1845: died 27th November, 1854. 9. The Eart or Lerrrm; elected 7th July, 1802: died 2nd January, 1855. 205 10. Mavricr Coxts, Esq.; elected 12th February, 1849: died 15th January, 1855. 11. Bryvon Boop, Esq, ; elected 16th March, 1802: died 27th January, 1855. _ Iv was Resotvep,—That the Report of the Council be adopted, and printed in the Proceedings. The Rey. 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., Rey. 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.—Rev. Thomas R. Robinson, D. D. Treasurer.—Robert Ball, LL. D. Secretary to the Academy.—Rev. J. H. Todd. Secretary to the Council_—Reyv. 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 —Kdward 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. Rey. 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., Vicz-Presipenr, 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. Resoivep,—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 on the 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 of the MS. On this lattter pot, 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 etghth 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 5ann membpum (the vellum is scanty); Ip cana an oub (the ink is thin); or complains of his health, as, uch mo chliab a norb mgen (alas, my chest, O holy Virgin!). But the following memoranda are important as they furnish means to determine the date of the MS. “1, Aglossatp. 157: Hucusque Calvus Patricii depinait, 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, signed 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. «2. At p. 194, marg. inf., we find, vo mip Mabdvoc dain «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- tpic 7 Opis. ap Maelbpigcae namba olcc a menma ppimm:::: Pepibunod 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, A.D. 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 saith mnochc, Fa puapna paipeae pinvpolc. Ni dsop perme mona minn Oond laechparo 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 Ruatom avert, 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 Ruaidhri, called by the Welsh Rodric Mawr, son of Mervyn the Freckled, was 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 ninth 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. «¢ Having now established the date of the Oghams, let us proceed to describe them. ** They are as follows :— «T, Page 50, marg. inf.— FERIA 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. “JI. Page 70, marg. sup.—Fel MARTAIN—1. 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. «JIT. 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.—CO9GRT —i.e. Corrige. The word is not to be found in the diction- aries; but there can be no doubt as to its meaning. We have at p. 90, ol apcogapti as a gloss on the Latin guod sit emen- dandum. ' “VII. Page 195, marg. sup.— ASCOHGRT INSO.—i. e. Hoe est corrigendum. Compare the gloss just referred to: also, m1 aeopapchi mpo (gl. minime hoc est adhibendum) ; also a gloss in the Wurtzburgh MS. of St. Paul’s Epistles, cio apoéna ppt pincna (gl.quid ergo fratres ? i.e. quid faciendum in hoc ergo). "F ie 214 The scribe has used the character called eamhancoll to stand for the letters pe in apeoganc. According to the Uraicept it is properly used to denote 2, which is equivalent to es: 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. “VIII. Page 204, marg. sup.—lLatheirt.—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 cepzia hopa, 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 lachoipz, so little differing in ortho- graphy that it may be equivalent to the one before us. “ Laithoipc «1. Laith opt .1. laich po n-opcz .1. ol conmae. “ Laichoipe, i. e. from laith, champion, and opt, 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. VIL., a point is used for this purpose between the words cosgapc and po. There is also a point at the end of No.I. In Ogham inscriptions occurring 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 poppeavha, 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 opGécpolup deapsporp. Oo cpil pigpaige ap buan blag Sluaig pan pidbaile ppuidglan. Acdic uaiple Clomoe Cumo Fan pelig lecaio, leapgéuino. Snaidm n6é cnaeb 6p 5a¢ colaimno Acar amm éaem Geant 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 bochz [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 with a 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. ‘‘ Early 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+ Di U=0, the solution of which was known to be U=@ (a+ ty) + ¥ (a-y), where i?=-1, it seemed probable that the solution of the higher equation, D.V+ Di V+ DiV =0, should be susceptible of deduction by the employment of two imaginaries 7 and j, governed by the laws P=-1, fa=-l,. G=-ji The integral thus deduced appeared to be V=0 (a+iz, ytjz)t ¥ (w-iz, y—J): Unable to interpret this form, and impressed with the convic- tion that, to render the solution, iftrue, of any value, such in- aS Ses ee : i a : i & . ; Hy a 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 ¢ cos a+jsina, which, as_referrmg to an unit circle is planar, whereas it would be desirable that the explicit vector-unit should be 2 cOS at+jcos (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. © ‘“ (fit afr). This transformation suggests an elementary process, by means of which the solution of Laplace’s function, in the form of a series arranged according to ascending powers of 2, may be obtained without recourse to imaginaries. Let the equation, (Di + _D} + D2) V=0, be integrated twice with respect to 2; $, and g,, two arbitrary functions of y and z, being successively introduced in the in- tegration ; it will then assume the form { 1 3h i Oh (D; ae D3)} V=29, ate pi. _ Hence we shall have, V=({14+ Dy? (D} + D3)}7 (#24 o:)- 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 arepresentation of an adze and square, or rule, the emblems _ of the trade of coopers, to which the brothers O’Tiernagh be- longed. b) 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 or1ERNAGH in the inscription are the words ‘*].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-PresipEn7, 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 Irish 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 :— @rx’ do nichol ag’ do gille bert o dubthaich ra bin abaiddeact Cunga. Which he reads thus: OROIT DO Nichol agus Do sillebert oO OubThaich ra bIN abalohedcT CUNSa. f ‘ VOL. VI. Y Ne 242 grant molecules are spheres, each having six poles on its sur- face, and it is owing to the different degrees of force possessed by these poles that the different cleavages are due.’ In this he is clearly in error, for no matter with what cleavage we divide a cube, we will eventually separate every pole from every other. ‘‘ J will, in this paper, confine my observations to the first system, but it will be readily perceived that they apply (mu- tatis mutandis) word for word to the other systems. ** The cleavages are of three kinds, cubical, octahedral, and dodecahedral ; the cubical we have already explained ; the oc- tahedral formation will arise if each spherical molecule have twelve poles on its surface, whose position is given by the in- tersections of four great circles, having the same inclination and relative position as the faces of the tetrahedron; the form which they will assume will be the same as Wollaston pro- pounded. *« Tf each layer of molecules be deficient by one row, planes will appear on the edges, which planes belong to the cube; we have thus got a compound form consisting of the cube and tetrahedron, or in other words, we have a cube with half its corners replaced by planes. By other decrements at the edges, or corners, we can explain all forms of the first system, except hemihedral forms with parallel faces, which never combine with forms without parallel faces. ‘*¢ Thirdly, the dodecahedral formation will result if each molecule be a sphere with eight poles on its surface, situated with respect to each other in the same positions as the angles of a cube; as can be very easily demonstrated. *«* Any crystal will of course cleave in whatever direction the least resistance is met with. Thus in the cubical forma- tion the cleavage is cubical because by such division we sepa- rate each molecule from one adjacent molecule only, whereas if we divided it in any other direction, we would have to sepa- rate each molecule from two or three others. In the octahedral 243 formation the cleavage is octahedral, for such cleavage over- comes the attraction of each molecule to three others, while any other would have to overcome the attraction of at least four. And in the dodecahedral the cleavage is in like manner parallel to the faces, as this cleavage destroys the cohesion of two poles, while the two other possible cleavages would destroy the cohesion of three and four respectively. «‘ With respect to the cause of decrements, they will evi- ‘dently result from the loss of polarity in the external mole- cules, and this loss is what we would have every reason to an- ticipate ; for if we consider the attraction of the particles to arise from the attraction of a fluid contained in them, and that this fluid is only held in its place by a certain coercive power of the molecule itself, as is the case with a magnet, it will fol- low, that as the crystal increases in size the combined attrac- tion of the whole fluid will draw from the extreme molecules such fluid as they may contain, and thus they will lose their polarity. «* Thus, if the time which elapses while a layer of molecules is being laid on a cube, is equal to that which elapses while the polarities of the corner molecules is being removed, octa- hedral faces will appear; and in a similar manner we can ex- plain all the other cases. ‘‘ Tn this theory we must of course suppose the particles of the fluid to be infinitely smaller than those of the crystal.” 244 Monpay, May 287, 1855. THOMAS ROMNEY ROBINSON, D.D., Presinent, in the Chair. Dr. Auprince read a paper on the nature of the precipitate which occurs in the preparation of alkaline phosphates. « * pe’, and as the point P also became P’ by expan- sion (equal ratio with c’), the balance is maintained; but water expands more than metals by equal increments of heat. Let u represent the point to which the water expanded, and the triangle pur its volume, which is greater than pxz, but only equal to it in weight; the rarefied volume of water pxz, acting on px’, will not balance the weight acting at P’; but, to keep the semicylinder in equilibrium during changes of temperatnre, the breadth must be so much longer than the 343 radius, as will, by its expansion, present a surface equal to the difference between the triangles, that is, the breadth must be to the radius as (the difference between the triangles divided by the expanded radius) is to the expansion of the radius. In balancing the semicylinders, less or more weight may be employed to produce the same effect, provided the centre of gravity is further from the axis, or nearer to it; for instance, if the balancing for a homogeneously balanced semicylinder be placed at its centre of gravity, 0-4244 from the axis, it will be half the specific gravity of water; but if the balance be placed at half that distance, the whole weight will equal that of an equal bulk of water, and if at one quarter the distance, it will be double the specific gravity of water. Thus, the same effect is produced by bodies whose absolute weights are so different, that is, they will sink by the withdrawal of water below the surface level, or rise on any addition being made, though one may be much lighter than water, one equal to it, and one double its density. But if the weighting be made to act with a force greater than half the specific gravity of water, the power of such excess of weight acts as the whole weight, that is, as the sine of the angle of rotation ; and ifthe whole of the float be elevated to the fluid level, by the withdrawal of a quantity of the fluid, the float will commence to descend, and, in doing so, actually raise the level of the fluid surface, producing the paradox of raising the height of fluid in a vessel by withdrawing a part; but the fluid will continue to rise only while the float is descending through the first quadrant, for, as soon as the point s, in Fig. 1, falls on the line of sur- face MN, the level will fall, and continue to fall during the further descent of the float through the second quadrant ; the converse of this is also true. The form of the float ought to be that of a figure gene- yated by a plane revolving on its axis. If otherwise, let AEFC represent a parallelopiped, equal in weight to the semi- 344 cylinder asc, Fig. 3, and having their centres of gravity coin- AD} teat Fic. 3. - eiding. The pressure of the water on pc balances the weight of either of them, considered separately, but the pressure of the water on Fc tends to make the parallelopiped rotate in the same direction as that on pc, which tendency not being in the same ratio for the angular motion as that exerted on pc, the two figures cannot act in the same manner, nor can any other figure, where the action on the periphery produces a rotating effect, except on the one given part pc. Buta part, or parts, might be excentric in one direction, provided they be coun- teracted by others in an opposite direction. The semi-cylindrical float described above is capable of being applied to many useful purposes; the delicacy of its action, when properly balanced, is such, that a solid body ca- pable of raising the surface level in the cistern, in which the compensator may be placed, only 1-2000th ofan inch, it will, on being gently plunged into the water, cause an elevation of the float quite visible, and indicating a movement, perhaps, equal to the bulk of the immersed solid. It is, therefore, applicable to the measurement of complicated structures, such as groups of crystals, or masses of other matter, and by it also specific gravities, expansion of solids, &c., may be ascertained with ; 345 great exactitude. Extremely small additions of fluid will be measured, as in the instance with the solid already mentioned, and, therefore, its utility as a rain gauge. I also find that, by plunging one end of an open tube, bent at right angles, into the water in the cistern, allowing the wind to act upon the other end,—the surface of the water in the rest of the cistern, and the float, being protected from the influence of the wind,— the float will ascend in exact proportion to the force of the wind, depressing the water in the tube. I have one of these instruments so sensitive that mere breathing or speaking in front of the open end of the tube will act on the float. It has also occurred to us, that a solid cylinder of iron placed in a properly formed cistern of a barometer, would keep the level of the mercury in the cistern constant during the ascent and descent of the mercury in the tube, and would save much trouble in determining the true difference of height of a co- lumn in the barometer, which could always be read off at once on the scale without any allowance for difference of level; the surface level of the cistern being maintained to the 1-2500th of an inch by the action of the float. Since I became acquainted with the properties of the ro- tating float, my friend, Mr. Richard E. Donovan, who first in- troduced it to my notice, has informed me that he has recently heard that a similar float had been proposed many years ago as a method for maintaining the oil level inalamp. However, if that be true, the valuable properties it possesses as a hydro- static balance could not have been investigated, otherwise it would not have been forgotten. I must here acknowledge the obligations I am under to Mr. Donovan, for the part he took in carrying on the experiments and the calculations ne- cessary for this paper. The President presented to the Academy, on the part of © the present Earl of Charleville, a portrait of his grandfather, 346 William Bury Earl of Charleville, who was President of the Academy from 22nd June, 1812, to 16th March, 1822. The President was requested to convey to the Earl of Charleville the special thanks of the Academy for his hand- some gift. Rev. J. H. Jellett, Secretary of the Council, presented the first ten volumes of ‘* Liouville’s Mathematical Journal” to the Library of the Academy. The thanks of the Academy were voted to the Secretary of the Council for his valuable gift. 347 Monpay, Aprit 28TH, 1856. JAMES HENTHORN TODD, D.D., Presipent, in the Chair. Mr. E. Curssory, by permission of the Academy, read a paper on the identity of the chronological system of the priests of Memphis, as explained to Herodotus, and the suc- cession of the kings of Egypt, attributed to Manetho. The object of the author was, in the first place, to show that a critique on Herodotus in the ‘ Royal Irish Academy Trans- actions,” vol. xxii. Pol. Lit. p. 49, was not applicable either to the chronology of the reign of Sabacon, or Anysis, which appears to precede it, and of Sethos, who was a contemporary of Sabacon ; and, in the second place, to prove that the royal chronology of the Egyptian priests at Memphis and Heliopo- lis, and especially at the latter place, as it was explained by a reference to a series of images there of 345 statues of Pironeses, was, in theory, identical with Manetho’s system of chronology to the end of the twenty-sixth dynasty, or the beginning of the Persian dynasty, in whose time both Hecatceus and Hero- dotus visited Egypt. It was explained, that according to the corrected lists of Manetho's dynasties, the actual number of reigns of all the kings of Egyptian and of foreign origin, including Sabacon and two other Ethiopian kings, and one queen’s reign, in Egypt, was 346 only to the Persian Conquest; and thus, the total numbers of reigns of kings, of Pironeses at Heliopolis, and of priests at Memphis, as explained to Herodotus, were the same up to the Persian dominion, as the number of reigns stated by Manetho. It was also explained, that Bunsen, in the exposition in his * Keypt’s Place,” &c., vol. i. p. 105, of what he erroneously calls Herodotus’s view of the chronology of Egypt, had fallen 348 into a great mistake in supposing that the stone statue of Sethos (Smintheos) in the Temple of Vulcan, at Memphis, was to be counted with the 341 (or 345 ?) wooden images in the storehouse there, and that the count of the number of reigns of kings, of priests, and of generations of men in Egypt from Menes, was to be carried down only to the time of Sethos, the contemporary of Sennacherib, and not to Amasis of the twenty-sixth dynasty. A statement preserved by Herodotus, but overlooked by Bunsen and all other modern writers, as to the interval of time between the construction of the mound of Anysis, and its discovery by Amyrtzeus, was adduced to prove that an inter- val of at least 300 years existed between the reign of the king called Anysis, and Sethos, or Sabacon; so that the events of the reign of Anysis are not to be considered as immediately preceding the reign of Sabacon, as is generally supposed by modern writers, who have overlooked the fact of an interval of 700 years between Anysis and Amyrteus. Thus, it was proved that Herodotus’s statements concerning Anysis do not immediately precede Sabacon, who is mistaken in his present text for another king of Cuthean, but not of African origin, whose name or title was Saba, identical with that of the enemy of ‘* the son of Anosh,” of the Arabs, who may be identified with Anysis of Herodotus. It was shown that the two Ethiopian kings—who, with or after Sabacon, ruled in Egypt,—the shepherds or Hyecsos, and the Shethites, or blue-eyed kings of the monuments,—belong to the group of seventeen kings, called altogether, with Saba- con, by the Egyptian priests, eighteen Ethiopians, in the text of Herodotus—to a great extent, fill up the gap of 300 years be- tween Anysis and Sabacon; whose reign, probably in his own country, may have subtended the reigns of his son and grandson, Sabacus and Tirhaka, in Egypt: and thus, the author main- tained, we might reconcile the statements in Herodotus with matters of fact which followed Sabacon’s actual rule in Egypt. 349 The appointment of Sethos, probably in the place of Boc- chonis, to the chief rule in Lower Egypt, at Sais, was con- ' sidered to have been an act of the Ethiopic king or kings of the day. It was denied that Sethos could have been a priest of Vulcan at all; and that, where he is so called in the present text of Herodotus, the words are redundant, and altogether contradict facts stated elsewhere in Herodotus; and hence we are obliged to reject the title of priest ‘‘ of Vulcan,” and call Sethos simply a priest or a prophet of a “god,” or “gods,” whose proper name, if known, would not have been men- tioned by Pagan priests, who, by calling the prophet, Sethos, attributed his gifts to Typhon, or Seth, the evil genius of the neighbourhood of Pelusion, according to Egyptian superstition. The analogy in the description, by Herodotus, of the vision of Sethos with one of the two visions of the prophet Isaiah, led the author to identify the party called Sethos by the priests of Vulcan, at Memphis, by the diabolical title Sethos, with the Jewish prophet Isaiah, or E-Sais ; and adopt Herodotus’s statements as explanatory of the means adopted by Provi- dence for the fulfilment of the prophecy, that the Assyrians should not shoot an arrow or raise a shield at Jerusalem. It was also shown, that the statements in Herodotus, taken in connexion with the Biblical notices of Sennacherib’s defeat, and a quotation from Berosus preserved by Josephus, that the mode of the first discomfiture of Sennacherib at Pelusion was exactly the same in kind with that of the Midianites and their allies,—the different nations, Arabs, Assyrians, Medes, and Persians, composing Sennacherib’s army, having quarrelled and fought with each other with their swords, and without shields. To this battle Herodotus refers, when he notices the bones of the people he saw at Pelusion who fell on the occasion of thefdiscoveryjof the depredations committed by the mice on their bow-strings and shield-handles, during the night after the arrival of the army before Pelusion. 350 These explanations were offered to save Herodotus’s repu- tation from the critique quoted from the Transactions, so far as it related to the real period of Sabacon, and of the reign which appears to precede it, and that which was, at least in part, contemporary with it and after it; and to prove the gene- ral accuracy of Herodotus as a reporter of statements made to him by the Egyptian pagan priests, and others; and as sug- gestive of the omission of a few words in his text, which appear to have been introduced into it by some Arab critic who un- derstood Greek, but who had no knowledge of chronology, or of Manetho’s dynasties; and the identity in duration of the chronology of the priests at Heliopolis and Memphis, with that of the priest of Sebennytis in the time of Ptolemy IT. In conclusion, it was shown, that the particular facts, said, in the critique referred to, to be undeserving of the slightest credit, were on the contrary, worthy of the special notice of Biblical scholars, as being supplementary to facts recorded in the Scriptures, which in themselves are insufficient to realize the historical identities of the ‘son of Anosh,” as a king, both before and after his retreat to the Ausitis. The few words in Herodotus relating to this prince just supply the desiderata which give him an historical reality in time and place, and indicate his position in the monuments as the foster- brother of Horus, or the Hawk, of the eighteenth dynasty of Manetho, and not as the immediate predecessor of Sethos. The notices of Sethos, in the text of Herodotus, are inva- luable, as they supply everything that is necessary to explain the facts of Sennacherib’s discomfiture, and how it was that the prophecies relating to that event were all of them completely fulfilled, ‘here’ at Pelusion, where Isaiah or Sethos was, and ‘ there” at Jerusalem, where Hezekiah was when he sent the embassy to Isaiah. By the identification of Sethos and Isaiah, the notices of this prince in Herodotus at once open the way to the grouping together of a number of other fragments 351 of the personal history of Isaiah, to be found elsewhere; and as these are numerous, extending backwards to his infancy, and are quite consistent with other facts and references in the writings of Isaiah, we thus become indirectly indebted to He- rodotus for a biography of the chief of the prophets, about whose personal history so little has been preserved by the Jewish writers, though he really appears to have been one of the most extraordinary characters who ever appeared in East- ern history. Dr. John B. Barker read a paper on the stomach of the zebu. The Secretary of the Academy announced a donation of two copies of a large medal in copper and silver, struck by the order of George V., King of Hanover, in honour of Carl F. Gauss, and presented to the Academy by order of His Majesty, through the Royal Society of Gottingen. 352 Monpay, May 127u, 1856. JAMES HENTHORN TODD, D.D., Presipent, in the Chair. Tuomas H. Lepwicu, Esq., and John H. Otway, Esq., were elected Members of the Academy. The Rev. John H. Jellett read a Paper, by Mr. Thomas J. Campbell, on the solution of cubic equations. ‘<'To resolve the cubic equation, + az?+ba+c=0, put x=2' + 2, and the equation becomes a3 4+ (3z+ a) w+ (327+ 2az+b) x x (22 + a2* + bz +c) =0, which may be proved by development, for 8 = 43 + 32a"? + 3 22a' + 2° ax? = ax? + Qaza' + az? bz= ba’ + bz c= Cc 08+ an?+ bz+e=0'3+ (32+a)x?+(32°+2az+b)xt+ 24+ az+bz+e. Call the member on the right and left of this equation fx and fx respectively : oft = 23 + (32+ a) a+ (327+ 2az+ bd) a'+ (234+ a2? 4+ bz+c)=0. My object is to reduce /’x to the form of 23+ Ad’? +1 4'a' +50, where x” =/f"x (or another function of f’x), and thus to find 2” by completing the cube, for a similar reason as we complete the square in equations of the second degree. «But to effect this important relation of the coefficients 353 of f'z, so as to reduce it to the form I have mentioned, we will divide both sides by (2° + az? + bz +c) x’, and thus we will have a new form of the equation in these terms, viz. :— , 327+2az+b \ , 3z+a ; aw 8 + | —————_—__ Jv-?. +. | ———_______ ]q""1 23+az*+bz+e 23+az*+bz4+¢ + (@4ae'4+bz40) which is an equation the roots of which are the reciprocals of those of fz. Now, let me compare the model equation e+ Av?+iA'a'+e to the above, and we see that 3274+ 2az+6b . ~ 4az4¢bz+0e. also 3z+a ie Ss es 23+az*+bz+¢€ 3 or 2-3 32+a@ _( 32 4+ 2az+6 ae 3z+a ‘ —“\estaz?tbzte) \etaz*+bz+e) ~\2+a2+bz+0e)’ or, by development and clearing of fractions, we get the fol- lowing function of z, viz. :— (327+ 2az +b)? =3 (33+ a)(2°+ a2? + bz +0) 9z! + 12az*+ (6b+4a*)z?) (924+ 12az° + (9b + 3a*)2*) + 4abz+ 0? J (+ (9e+ 3ab)z+ 3ac J - Hence, by contraction or transposition, we get the following - quadratic :— (3b— a) 27+ (9e- ab) z+ (Sac - 5") = 0. 2L2 354 Hence _—(8b-a@) + y [(36- a’) — 4(3ae - b°)(3d — a*)), of 2(3b- a’) but in the equation e+ At? st Aas e¢é=0, 2"=-14+4(4-27c)s, by completing the cube, and transposing c. Therefore, also, asa z- or x , 32+ 2az+b ae 32° + 2az+b \3 27, i! ~ S234 az2+bere 3 \\e+az2+bz+c) 2+azt+bz+e} . gy 3) 322+ 2a2+6 ql 327+ 2az+b \3 27 ee —"( B+az2+be+0¢ 2t+az+bze+e}) 2+az7+bz+e) but x + z2=4a, by the hypothesis in the original equation : °".t=2 (32? + 2az + b) ,af2 2+ 2az+b \s 27 _ BP +ae2+bere) 7 2+az*+bz+e 24 az2?+bz+e consequently, substitute the foregoing value of z into this last formula, and it gives the value of the unknown quantity in any cubic of the form that I proposed.” Rey. Professor Haughton made a communication ‘On the Depth of the Sea deducible from Tidal Observations,” of which the following is an abstract. He stated that, in consequence of his having succeeded in separating the effects of the sun and moon in the diurnal tide, he was enabled to make calculations of the depth of the sea in which the tidal wave was produced, which he believed to be worthy of the greatest attention. The depth of the sea may be inferred from three distinct observations, viz., of heights of diurnal tide; of solitidal and lunitidal intervals; and of the age of the lunar tide compared with the lunitidal interval. 355 The calculations founded on the last two methods gave respectively 11°9 miles and 11°3 miles, each result being a mean of those derived from eight stations, agreeing remarka- bly well with each other. The calculation founded on heights gave as a result a depth of 5:12 miles, agreeing with a result deduced by La- place from the long series of Brest observations on the semi- _ diurnal tide, viz., 5-07 miles. Mr. Haughton was of opinion that the first result, viz., 11°6 miles, is the depth of the central channel of the South Atlantic, up which the tide waves advance from the Antarctic Ocean ; while the second result, viz., 5 miles, deduced from heights, represents the mean depth of the whole Atlantic Ocean, including the shallow water of the soundings in the eastern portion, near the British Islands. This view he considered to be confirmed by the great age of the diurnal tide, viz., five to _ six days,—a circumstance which shows the great distance from the coast at which the tide wave is formed, which gives the character to the diurnal tides of the Irish coasts. 356 Monpay, May 26rTu, 1856. JAMES HENTHORN TODD, D.D., Presipenr, in the Chair. Mr. W. R. Wide read a Paper on the introduction and the time of the general use of the Potato in Ireland,—and its various failures since that period; with some notice of the substance called Bog-butter. ‘¢ Some few years ago, having turned my attention -to the subject of the ‘ Food of the Irish,’ especially in early times, and written some essays upon it in the ‘Dublin University Maga- zine,’ (see Numbers for January and March, 1854), the potato came, in due course and chronological order, under considera- tion. Having looked into the authorities which bore upon the subject of the early introduction of the potato into Ireland, I then arrived at the conclusion that it became an article of gene- ral food, and consequently, as such, was the means of influencing —as far as the mode of producing food, and the constituents and character of that food could be the means of influencing—the moral, physical, social, political, and commercial condition of . the people about the middle of the seventeenth century. My at- tention was again called to the subject by the publication of Mr. Macaulay’s ‘History of England,’ in which he mentions the potato as influencing the feelings and character of the people during the period over which his third and fourth volumes extend. He has twice mentioned the potato (vol. iii. p. 158, and vol. iv. p. 110), and in one instance under very peculiar circumstances—at the siege of Limerick. The beleaguered city, having stood out to the last, capitulated, and then a me- morable scene took place—a scene well worthy the attention of the painter and the poet,—on each side of the gate stood the generals of the respective armies, with their attendants; out 357 marched the soldiers of the garrison to choose their destiny ;— __ and Mr. Macaulay, in describing this scene, took occasion to 3 state—among the various circumstances that influenced the __ minds of the men who were then either to expatriate them- ____ selves, or to remain under what they considered a foreign yoke ; _. —the remembrance of their homes, their potato garden, and ‘ their clamp of turf, with other attractions of a like nature, which still sway the Irish peasantry. «¢ Recently Dr. John Davy wrote me a letter, in which he questioned this early use of the potato as the general food of the people, on account of the statement in the ‘ Great Geographical Dictionary,’ published in 1694, that, ‘in hard times, they [the Irish] lived on water-cresses, roots, mush- rooms, shamrocks, oatmeal, milk, and such other slender diet.’ I have again looked into some authorities to see whether the views of Dr. Davy are supported, or those which I my- self had expressed in the Dublin periodical alluded to, and in which I stated, that in Munster especially the potato formed the staple food of the Irish about the middle of the seventeenth century. The writer in the ‘Geographical Dictionary’ probably took Spencer and Campion, who wrote more than a century before, as his authorities. «Some difficulty has attended the investigation of this subject, from the circumstance of inquirers not distinguishing between the true potato, Solanum tuberosum, and the ‘sweet potato,’ Convolvulus batata, or, as it is sometimes called by old writers, the Spanish potato. *« It is generally believed that Sir Walter Raleigh intro- duced the potato into Ireland. Sir Joseph Banks came to the conclusion when he wrote his Essay (being an attempt to discover the time in which the potato was introduced into the British isles) that it was brought by Raleigh into England, and from England into Ireland about the year 1600. “It must have been at least before the year 1602, because the estates of Raleigh then passed into the Boyle family, and his connexion with Ireland ceased. | E | | : j : ; 358 ‘Clusius, the botanist of Leyden, who wrote in 1586, says the potato was cultivated in Italy prior to that date; and Cuvier denied that Europe derived the potato from Virginia. The researches of Banks also favour this conclusion, and he states that Coccius, in his Chronicle, printed in 1553, mentions potatoes under the term of papas. Herriott, who accompanied Raleigh’s expedition to Virginia, described them under the name of openawk. In Irish they are variously styled potatee, pratea, or phottie, mere Hibernicisms of the English word ‘potato.’ Sir Robert Southwell, President of the Royal Society, stated, at one of its meetings in 1693, that potatoes had been introduced into Ireland by his grandfather, who first had them from Sir W. Raleigh. «¢ T would now ask, what had the people to live on in Ireland before Raleigh introduced the potato? While most other nations have had their history transmitted from the days of the hunter and the fisher, clothed in skins, and using weapons either for the chase, their own preservation, or the production of food, and so rising in the scale of civilization from barbarism to the highest amount of cultivation, in which the arts were made subservient to the food as well as to the ornament and education of man—we find this curious fact, that there is no record of such a state of existence in Ireland. The Irish had mills and ‘pure white wheat,’ and a coexistent state of civili- zation of which that was but a small portion; because, to raise and to grind corn, and to bake it into bread, was compa- ratively an advanced state of society. We had in Ireland at that time a social state very different from that alluded to, as being the character of other nations in similar phases of de- velopment, and which serves to confirm the idea that we are in all probability descended from a colony previously civilized, which had settled in this country. «The people lived, in early times, upon corn and milk, and also upon the flesh of oxen and swine—the latter is shown by the details of feasts and royal banquets, descriptions of 5 ‘, a 359 which were favourite themes for the recitals of the early bards. Subsequently sheep appear to have been introduced; goats were likewise domesticated, and the remains of domestic fowl have been discovered in early tumuli—a circumstance which upon a former occasion I brought under the notice of the Academy. Corn, peas, beans, and possibly parsnips, with cabbages and onions, formed the vegetable food of the peo- ple, prior to the introduction of the potato. ‘* Gerard, the English herbalist of 1597, is one of the first authors who alludes to the potato, and after him Richard Bradley, F.R.S., in his ‘ Planting and Gardening,’ published in 1634. Atameeting of the Royal Society, in March, 1662, a letter was read, containing a proposal for preventing famine, by dispersing potatoes throughout all parts of England ;—this subject is alluded to in Evelyn’s ‘ Sylva.’ Threlkeld, the Irish botanist, described the plant in 1726, and says we had it through Thomas Herriott. The late Crofton Croker, in the introductory matter to his ‘Popular Songs of Ireland,’ has given some very interesting references to the early authorities respecting the introduction of the potato into Ireland, and Mr. MacAdam, of Belfast, has likewise written a valuable treatise on the subject in the ‘Quarterly Journal of Agriculture,’ for June, 1834-5. ‘That potatoes were ordinary food in the south of Ireland,’ writes Mr. Croker, ‘before the time of the Commonwealth, is shown by “An Account of an Trish Quarter,” printed in 1654, in a volume entitled ‘« Songs and Poems of Love and Drollery,” by T. W. The writer and his friend visited Coolfin, in the county of Waterford, the seat of Mr. Poer, where at supper they were treated with codded onions, and in the van— ‘Was a salted tail of salmon, And in the rear some rank potatoes came on.’ ” # although sown in gardens as a rarity, and used at supperas a delicacy, we have no authority for believing that pp Y y g 360 the potato had become the general or principal food of the Trish peasantry until the middle of the century. That, how- ever, the cultivation of the plant was making rapid progress, may be learned by reference to Cole’s ‘ Adam in Eden, or the Paradise of Plants,’—published in London in 1657, which says :—‘ The potatoes which we call Spanish [not the sweet potato], because they were first brought up to us out of Spain, grew originally in the Indies, where they, or at least some of this kind, serve for bread, and have been planted in many of our gardens [in England], where they decay rather than in- crease; but the soyle of Ireland doth so well agree with them, that they grow there so plentifully that there be whole fieldes overrun with them, as I have been informed by divers soul- diers which came from thence.’* ‘The soldiers alluded to by Cole were those of the Parliamentary forces engaged in Ire- land from 1649 to 1653, during a period when Sir William Petty calculated that 616,000 of the Irish and the English in Ireland died by the sword, famine, and pestilence. ‘‘ Ina paper published in the ‘ Philosophical Transactions’ in 1672, and believed to have been written by Dr. Beale, concerning a strange frost which occurred in England in that year, we read that—in 1629 or 1630 there was a dearth in England; and ‘much talk there was then that in London that the had a way to knead and ferment boyled turnips, with a small quantity of meal;’ and then he goes on to say, ‘potadoes were a relief to Ireland in their last famine; they yield meat and drink.’ This famine was evidently that alluded to by Petty in the foregoing reference. «¢ From the researches which I have made it would appear that the cultivation of the potato was very irregular through- out the country; some localities, especially in Ulster, having only adopted it generally within the memory of the past generation. M‘Skimmin, in his ‘ History of Carrickfergus,’ * T am indebted to our Treasurer, R. Ball, Esq., for Cole’s rare book. 361 ~ asserts that not more than two generations back potatoes were seldom used after harvest. ‘In 1663 Mr. Boyle exhibited some specimens to the Royal Society of London, and read before that body a letter from his gardener at Youghal (the cradle of the potato), in which he describes this esculent as ‘ very good to pickle for winter salads, and also to preserve. They are to be gathered in September, before the frost doth take them;’ and, after describing the best mode of culture, he continues—‘I could speak in the praise of the root, what a good and profitable thing it is, and might be to a commonwealth, could it generally be experienced, as the inhabitants of your town can manifest the truth of it.’ One would think from this passage that the potato had not then become an article of common food amongst the Irish, beyond the locality where it was first cultivated. Sir William Petty, in his ‘ Political Anatomy of Ireland,’ written in 1672, although not published until 1691, enumerates among the articles of food, ‘ potatoes from August till May ; muscles, cockles, and oysters near the sea; eggs, and butter made very rancid by keeping in bogs;’ and in another place he asserts—‘ that six out of every eight of all the Irish feed chiefly upon milk and potatoes.’ ‘Certainly the present great historian of England has ample authority for the statement that the potato was cul- tivated in Ireland to such an extent as to influence the cha- racter and feelings of the people, so early as 1689; for, in ad- dition to those authorities already referred to, it is stated in Durfey’s ‘Irish Hudibras,’ published in the May of that year, and in which the esculent is frequently referred to, that after the arrival of William III., the natives are said to have been prevented enjoying their ‘ Banni-clabber [thick milk] and pottados.’ John Dunton, likewise, in his ‘ Conversation in Ireland,’ published in 1699, describes the Irish cabin in his day as having behind it ‘the garden, a piece of ground, some- times of half an acre or an acre, and in this is the turf-stack, 362 their corn, perhaps two or three hundred sheaves of oats, and as much peas; the rest of the ground is full of their dearly-beloved potatoes, and a few cabbages.’ And again, describing the habits of the people generally from Galway to Kilkenny, he says, ‘ Bonny-Clabber and Mulahaan, alias sowre milk and choak-cheese, with a dish of potatoes boiled, is their general entertainment; also in the ‘keens’ of that day, allusion is made to the ‘pigs and potato garden.’ Moreover, John Haughton, who published his ‘ Husbandry and Trade Im- proved’ in 1699, when describing the growth of the potato in Ireland, says, it has ‘thrived very well and to good purpose, for in their succeeding wars, when all the corn above ground was destroyed, this supported them; for the soldiers, unless they had dug up all the ground where they grew, and almost sifted it, could not extirpate them.’ ‘‘As experience has proved the potato to be one of the most fickle of vegetables cultivated to the same extent, the most likely to suffer from atmospheric vicissitudes, and the most liable to disease—one would think that if it had been cultivated in Ireland to such an extent as to constitute the most material portion of the food of the people, its failures would have been noticed in history, contemporaneously with those other losses of food which have been recorded. It is possible, however, that in the earlier years of its general intro- duction, this crop was not so liable to disease as in later times. 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