A siz do cúm neaṁ is Lás,

Agus tug slán na sé sluaig,

Fóiption leat an ealtan éan,

Leantar an tréan go m-bad truaiz.'

'a bráične,' ap Fionnguala, 'cpeidig an fír-Ŏia forórda na fírinne do cúm neaṁ go n-a néallaib, agus talaṁ go n-a tortaib, agus an fairrge go n-a h-iongantaib, agus do geabċaoi cabair agus cómfustaċt o'n g-Combe.' Cpeidmío,' as ad; agus creidimsi lib,' ap Fionnuala, 'do'n fír-Dia foipfe, píp-eolaċ.' Agus do creidiodas ar an uair ċóir, agus fuaradar cabain agus cobranao 6'n g-Combe da éis sin, agus níos cuir doinionn ná doisbšíon orra 6 sin amaċ.

Agus ro poillsigeað dó gurab iad Clanna Lis do sinne é. Agus iar d-teaċt na maidne ar n-a ṁápač, gluaiseas Mocaoṁóg go loċ na h-Éanlaiċe agus do connaise na h-éin uada ap an loċ; agus do ċuaid go h-oirear an cuain mar a b-facaid iad, agus do fiafpaiz díob: 'An sib Clann lip,' ap sé. 'Is sinn go deimin,' as iadsan. Do beisim a buide sin lé Dia,' ap Moċaoṁ6, 6ir is ar bur son tangusa čum na h-innse-si, tas gaċ n-innsi eile a n-Éirinn; agus tigíd a d-tís, agus tabraid taob liomsa, 6ir is annso atá a g-cinnead Ŏíb deagoibreaca do déanaṁ, agus dealúgað né bun b-peacċaib.'

Tánzadas a d-tír iar sin, agus tugadar taob leis an g-cléireaċ; agus do pug leis da adbuid féin iad; agus do bídís ag déanaṁ tráć, agus ag éisteaċt aifrinn a b-foċair an cléiriz. Agus tug Moċaomog céard mai cuige, agus d'fupáil air slabrada airgid aoingil do déanaṁ dóib; agus do ċuip slabraid idis Aod agus Fionnguala, agus slabraid idir Čonn agus Piaċra; agus do bídís 'n-a g-ceaċrar ag urgáirdiúgað intinne, agus ag méadúgað meɑnmna ag an g-cléireaċ.

:

5. Identify Abainn Chapa, bealach Conglais, bearnán eile, buas, bun Suaiṁne.

IDIOMS.

6. Translate into Irish :

He bid us drink water out of a cup (give two translations, using in one d'orduig sé; in the other, d'aithin sé).

7. Translate into English:

Cupio mearapdacht duine a fearg ar cáirde; ní críonna an duine do chuirfead a leas ar cáirde á ló go lá; is mór an fear cum foğluma do chur ar agaid é; cuirfead fa deara air fonn oile do cantain. Ná múch an lion as a beuil deatach; an baraṁuil diriseal atá againn asainn féin.

III.—MATHEMATICAL SCIENCE.

MATHEMATICS.

FIRST PAPER.

ALGEBRA.

PROFESSOR GIBNEY.

1. Establish, stating clearly the definitions on which you rely, that (a) is equivalent to + a.

2. If a + B+ Y

=

0 prove that the value of

a(l2y — n2a) (m'a — l2ß) + B(m3a − l2B) (n*B − m2y) + y(n2ß — m2y) (l2y — n2a) ̧

where A, B, . . . A', B', . . . are functions of a, ß, y find the values of A and A'.

4. Solve the equation

(26 + x)3 + (7 − x)3 = 3.

5. Find the values of x, y, %, w, from

and the resulting expressions multiplied together, prove that the product is rational in y, and does not contain a higher power of y than y”.

8. If log, n and log, n' be each given to p places of decimals, show how to find log, n', and how to determine to how many places of decimals the value is certainly correct.

9. Find the number of ways in which ten similar marbles may be distributed amongst ten boys, if no boy is to get more than two.

Find the number of ways in which Ρ similar marbles may be distributed amongst p boys, there being no limitation on the number each may receive.

10. If x1, x2 x, are n real positive quantities, prove

...

that the value of

SECOND PAPER.

[Full credit will be given for answering FIVE-SIXTHS of this Paper.]

PROFESSOR DIXON.

ANALYTICAL GEOMETRY AND CONIC SECTIONS.

1. A, B are two fixed points whose coordinates are (f. g), (h, k). P is a variable point, and squares APQR, BPST are described on AP, BP, each being on the side opposite to the triangle APB. Find the coordinates of the middle point of RT, and prove that it is fixed, so long as P keeps to one side of the line AB.

2. Prove that the points (x, y) and

are always inverse to each other with respect to a fixed circle, and find its equation.

3. Find the length of the least chord of the circle

x2 + y2 + 2gx + 2fy + c =

which passes through an internal point (x', y').

Two equal circles are given and a point P within both moves so that the sum or difference of the least chords through P of the two circles has a constant value. Find the locus of P.

4. The normal at P to a parabola meets the axis in G and the curve again in Q. Prove that the abscissæ of P, G, Q are in geometrical progression.

5. Two ellipses

x2

a2

have the same foci.

A point (x, y) on the first, and a point

(x', y') on the second, are said to correspond when

prove that the distance between any two points, one on each ellipse, is equal to the distance between the two corresponding points.

Prove also that if four points on the one ellipse lie on one circle, the corresponding points on the other also lie on one circle.

6. The focus and directrix of an unknown conic are known, and the polar of one known point passes through another known point. Construct this polar geometrically and find a point on the curve.

7. Prove that the locus of the intersection of two straight lines in given directions, each of which passes through the pole of the other with respect to a given ellipse, is a hyperbola having its asymptotes in the given directions.

8. Given the foci and vertices of a hyperbola, construct the tangents from any point, and their points of contact, without drawing the curve.

Examine the case when the tangents are to be drawn from the centre.

9. A conic whose focus is S touches the sides of a triangle ABC. If D is the point of contact of BC, prove that the angles ASB, CSD are equal or supplementary.

10. Prove that the area of any segment of a conic is bisected by the diameter which bisects the chord of the segment.

The segments cut off by two chords PQ, QR are equal. Show that PR is parallel to the tangent at Q.

THIRD PAPER.

[Full credit was given for answering THREE-FOURTHS of this

Paper.]

PURE GEOMETRY.

PROFESSOR BROMWICH.

1. If DEF be the middle points of the sides BC, CA, AB, respectively, show that a triangle A'B'C' can be drawn with its sides equal and parallel to AD, BE, CF, and that its