a n-ais dóib go h-Éirinn arís, do fearad an díle dóib ag Tuaig-Inbir, gur báiċiod iad: Capa, Laigne, agus Luasad, a n-anmanna. As dóib so canad an pann :Capa, Laigne is Luasad grinn, bádor bliadain pia n-dílinn, For inis Banba na m-báż, Adeistior, trá, gur ab í Cearain ingion bheaca mic Noe, táinic innte pé n-dílinn, gonad dó do pónað an pann: Ceapaip ingion bheata buain, Mad áil, iomorro, a fios d'fagáil créd tug go h-épinn í biot do cuip teaċta zo Noe, d'pios an b-fuiġbiod féin agus a ingion Ceapaip ionad isin áire d'a g-caomna ar dílinn; ráidis Noe naċ fuiġbidís. Foċtais Fionntain an g-cévna, agus po páiò Noe naċ fuiġbiod. Téid bioċ, Fionntain, Ladra, agus an infion Ceapaip a g-comairle iapaṁ. 'Deuntop mo coṁairlesi lib,' as Ceasair. 'Do jeuntos,' ol síad. 'Maread,' ol rise, tabruiò láiṁ-dia čugaib, agus adsaid dó, agus tréigid Dia Noe.' 2. Ní tuigim cionnas fuapados na seanċuide sgeula na n-drong a deirit do teaċt a n-Éipinn pia n-dílinn, aċt munab iad na deaṁain aieurda do bioò 'n-a leannánaib síte aca pe linn a m-beit págánta tug dóib iad; no munab a leacaib cloc puaissiod sgriobċa iad iar d-trázad dílinne, dá mad fíor an sgél; óis ni h-ionráid gur ab é an Fionntain úd do baoi sés an dílinn do maipfiod dia h-éis, do bríz go b-fuil an Scrioptúir 'n-a ajaid, map a n-abais naċ deaċaid don druing daonna gan bátað, aċt oċtop na h-áirce abáin, agus as follus nap díob sin éision. As neimėíor an suidiugad atá ag druing do seancuidib ap Phionntain do maptain se linn dílinne, map a n-abpuid gos maissiod ceatror a g-ceitre h-áirdib an domain pe linn na dílinne, map atá, Fionntain, Fearon, Foss, agus Andóid. idead, a leuztóir, ná meas gur ab í so ceudfaió na muinntire as uzdarba san seancus. Uime sin cuirid ugdar dáisiċe an ní po pomainn a laoid, da foillsíofað naċ tig sé le fírinne an ċreidiṁ a rád go maisfiod Fionntain, no neaċtar don triar oile iar n-dórtad dílinne agus roimpe. Ag so an laoi8 : Anmann ceatrais ceart so činn, Fionntain, Feapon, Fors caom cóir, Agus Andoid mac Catoir. Fors a n-oistior ¿oir do dliġ ; KEATING, Forus Feasa ar Erinn. 3. Adeis Stanihurst gur ab í an Mide fá cuid sonna do Šláinze, mac Deala, mic loić; gidead ní fíor dó sin. Óir do réir an leabair Gabála, ní raibe de mide ann in aimsir Sláinge, aċt an aon-tuait fearainn atá láiṁ se huisneac, go haimsir Tuatail Teaċtṁais: agus mar adeir gur ab 6 Šláinge adeirtean baile Sláinge, agur, d'á réir sin, gur ab í an Mide an ṁír ponna páinic é 6 n-a bráiċrib, níos cópa a meas gur ab í an Mide páinic map ṁíp ponna dó, 'ná a meas gur ab í cúigeaŎ laigean fá mír sonna dó, agus gur ab uaid ainmnifċear Inbear Sláinge snigeas tré lár laigean go loċ-Garman, agus fós gur uaid ainmniġċear Dúṁa Sláinge ré' sáidtear Dionn-ríog as bruaċ Bearba, idir Čeaċaplaċ agus leitglinn do'n leit tiap do'n bearba. 4. Ni hiongantac gan fios na neite-so do beit ag Stanihusst, agus naċ faca seanċus Éireann siaṁ, as a mbiad fios sean-dál na hÉireann aige, agus is 6 ṁeasaim naċ mór an losg do bí aige orra, mar go bfuil sé com ainbfiosaċ a's sin ar dálaib Éireann go n-abair gur ab 'san Muṁain atá Ros-mic-Triuin; agus go n-abair gur ab cúigeaỏ nó 'Prouincia' an mide, in azaid Cambrens féin, nac áipmeann an Mibe 'na cúigead, agus in agaid leabair Gabála Éireann. 5. Parse, and write grammatical comments on, the underlined words as fully as you can. 6. Translate into Irish : I caught him by the hand and ears. They took him by the hand and lead him to Dublin. He took the bread and blessed it. What man among you would not lay hold of him and lift him up. 7. Translate into Irish : Thou shalt make me hear of joy; thou hast made the land to shake. When the ways of man please the Lord he makes even his enemies to be at peace with him. You have caused men to ride over our heads. MATHEMATICS. FIRST PAPER. PROFESSOR DIXON. ALGEBRA. 1. Multiply 1+4x+9x2+ +n2x2-1 by (1 − x)3. ... 2. Resolve into their simplest factors and abcd (a2 + b2 + c2 + d2) − (b2c2d2 + c2d2a2 + d2a2b2 + a2b2c2) a2 + b2 + c2 + d1- 2b2c2 - 2c2a2 - 2a2b2 - 2a2d2 - 3. (a) Solve the equations - - 2b2d2 - 2c2d2 + 8abcd. x + y + z = 0, ax + by + cz = 1, a2x + b2y + c2z = a + b + c. (b) Find u, x, y, z from the equations ux + yz 25, uy + 8 = 35, uz+xy = 53, x+y+z+ u = 19. 4. In a certain recurring decimal fraction containing two digits, of which the first does not recur, the effect of interchanging the digits is to multiply the fraction by 4. Prove that there is only one fraction with this property and find it. 5. Express concisely the product (1 + x)(1 + x2)(1 + x1)(1 + x3) ... to n factors, and when possible to infinity. 6. Write out the coefficient of x in the expansion of (1 + ax + bx2)", n, r being positive and integral. 7. What finite values of x satisfy the equation 8. The square root of a certain number is wanted, correct to 2n + 1 places. Prove that when the first n + 1 places have been found by the ordinary rule, the next n can generally be found by mere division. Show also that there are cases in which it is not possible to decide on the nearest figure for the (2n+1) place, when the original number is given to the nearest figure in the mth place, however great m may be. SECOND PAPER. GEOMETRY. [Full credit was given for answering THREE-FOURTHS of this Paper.] PROFESSOR BROMWICH. 1. Two triangles ABC, PQR are such that AB, BC are equal to PQ, QR respectively, and the angle at A is equal to the angle at P; prove that the triangles are equal in all respects if BC is greater than AB, or if the angle at C is a right angle. 2. A, B are fixed points and P is variable; find the locus of P, given that (i) the angle APB is constant, (ii) the ratio AP: PB is constant. Prove that the two loci cut at right angles. 3. The sides BC, CA, AB of a triangle ABC are divided internally at D, E, F, so that m. BD = n. CD, n. CE=1. AE, 1.AF=m. BF where l, m, n are given numbers. Prove that AD, BE, CF meet in a point 0, and that if P is any other point, 1. AP2+m. BP2 + n. CP2 == - (Z +m+n) OP +1.40+m.BỘ +n.CO mn. BC2 + nl . CA2 + lm . AB2 = (l + m + n) OP2 + l + m + n When is it 4. Prove that, in general, one and only one circle can be drawn to cut three given circles orthogonally. possible to draw more than one such circle? SECTION B. PROFESSOR GIBNEY. 5. Given a line and two points on opposite sides of it, show how to find the point in the line, the difference of whose distances from the two given points shall be the greatest possible. 6. Taking Euclid's definition of duplicate ratio, prove that the lengths of the perpendiculars from the vertex of a triangle on the tangents at the extremities of the base to the circumcircle of the triangle have to one another the duplicate of the ratio of the sides of the triangle. 7. ABC is a triangle and P is a given point in the base BC: show how to find in the side CA a point whose distance from P, and whose perpendicular distance from AB shall be in the ratio of two given finite lines. How many solutions are there? |