im Dhonncao mac Cogain connaċtaiġ mec Suibne, im Dhoṁnall Ballaċ mac mec Suibne Girr, agus dá řičit ap dá ċéd immaille Friú, agus Tuatal fém do teċt i ttír ar éigin i nXlbain.— Annals of the Four Masters.

Cia is tusca do rinnead Cruit ina Timpán? Ni featarra γη αργεί φάτο, αγ Capmael. Τo peaounγα, αγ Μαγβάη, agus atber friutsa he. Lánaṁuin bui feaċt naill .1. Macuel mac Miduel, agus Cana cluỏmór a bean; agus tuc a bean Fuat τό, αζύγ το δι ας τειεαό norme an fuo feas αζυγο κάρας, agus buisium na leanṁuin. Xgup lá da ndeaċaid in bean cu τράις mara Camarp, αζup bu oc frubal na τράτα, αζur fuam ρί ταιρι mil mom αγ αν τηάις, αζur ac clum γή του να ξαειτι re féičib in míl móir, agus do ċoduil risin fojur sin. agur τάκια a reap nα διαις, αζαρ το tunc unub term Βροξα για το tuit a suan fuirri, agus téit roime fon bfiġ coillead ba comrocur τό, αζύγ το ζη1 puoba cnuit, αζur cuinear τέασα μέντιο in míl móir innti, agus as í sin céad ċruit do ponad piaṁ.— Book of Lismore,

IV.

Bá fás diu Ere fri re da ched bliadon, ier ndul na ttri noeineaban ac pubnamun empre, co cornecht so plioche τοις dechneabair innti do ridisi ina bfearaib bolg. Do clannaib Neimidh iar mbunadus dóib side, Uair Semeon mac Erglain mie beoain mic Stairn mic Neiṁió, airiġ an treas nonbair do clanoart Πειιό, Σοταγ α henn αγ ττόξαι τum Conann το γιο ζαιγε η necc. Καταγ urbe combatα 10m6a molaou a cclanna, αζur a ccenela. Τα βροηβαιιτ τοι samlad ni po faomsat Greccais a mbeith imaille re noccaib buboem ache ac achτρατ (.1. cucrac) voeine ronna. Τα για med moite scoit seamracha do deanom doib do slebtib clochoa ceannanba la hum sonant ole, 1en na homchup οι gus na maiżnib ina forcongarta agup ina norduiġthe doib a cup. Καταγ pertex τοις commeanmnac το γιός conat comample no rcίορατ εατοηpa buboem eluό αγ αοιιθ εττιαίας α mbaran. Or κατεγιό occo Fo oποιό. Το ζητατ τα om cuγαις αζur caoum eat a o choicnib αζur cabal bol.ccaib iomchuir na huire gur bat eallma ionṁana. Lotar inntib ara haithle do ascnaṁ na hathaire or luidfeat a πηγη. Πι hachprean a numteira fon mum, αέτ nama το riachtatur Erinn an aoin feċtmain.-O'Clery's Book of Invasions,

PURE MATHEMATICS.

Examiner, GEORGE J. ALLMAN, LL.D.

11th October, 1858.--Morning.

1. Find the sum of the cubes of n terms of a series in arithmetical progression.

2. Find the present value of an annuity to commence at the end of p years, and then to continue q years.

3. Approximate to the value of the real root of the equation

x3-6x=100.

4. Prove that the rectangle under the diagonals of a quadrilateral figure inscribed in a circle is equal to the sum of the rectangles under the opposite sides; and hence calculate the side of a regular quindecagon inscribed in a given circle.

5. Prove that a circle contains within a given perimeter the greatest possible area.

6. Describe an ellipse with a given focus, and passing through three given points.

π

7. Prove Machin's formula =4 tan-1 } —tan ̄1‚·

4

8. Find the volume of a parallelepiped in terms of its three edges and the angles between them.

9. Given the obliquity of the ecliptic, and the right ascension and declination of a star, find its latitude and longitude. 10. Separate the real roots in the equation

x1-4x3-3x+23=0.

11. Find the condition that the two conic sections, Ar2+Bry+Cy+Dx+Ey+F=0, ax2+bxy+cy2+dx+ey+f=0, should be similar, though not similarly placed.

12. If in each of a system of parallel chords of a curve of the nth degree there be taken the centre of mean distances of the points where the chord meets the curve, the locus of this centre is a right line.

13. Investigate the equation which proves the existence of three principal diametral planes in all surfaces of the second degree.

14. Form the general equation (1) of surfaces of revolution, (2) of conoidal surfaces, and give an exact definition of the word family, as used in the classification of surfaces.

Afternoon.

1. Prove the theorem of Leibnitz for finding the successive differential coefficients of a function consisting of the product of two functions of the same variable.

2. Transform the equation-

dez

d2z

dez

=x

dz

dz

+y dy

y -2xy
dx2 dxdy dy dx

by assuming x=p cos 0, y=p sin 0.

3. Prove the principal properties of the evolute of any curve, and show how to form its equation from that of the primitive.

4. Prove that all the envelopes of the different families of surfaces represented by the equation F{x, y, z, a, (a)}=0 (in which a is a parameter, and ø a variable function), admit of a generatrix of a common species.

5. Prove that the sum of the curvatures of any two normal sections of a surface at right to each other is constant. 6. Find the following integrals :

୮

(1+x)* i

7. Prove that an integral of the form fdam (a+bx2) a

dx

may be made to depend on any one of the four following forms:

P

S dxxm±n-1 (a+bxn)7

m-1

Sdxx-(a+bx" )?

8. Separate the variables in the equation—

(y−x) (1+x2)*dy=n(1+y3)*dr.

9. Give any method of integrating the equation

+ A1 (a + bx)n-1 dan-1+ &c. + Any=ƒ (x). Apply your method to integrate the equation

10. Find the curve such that each of its tangents cuts two fixed ordinates (produced if necessary) so that the product of the parts of these ordinates comprised between the tangent and axis of abscissae shall be constant.

11. Investigate a method for the integration of the partial du du

du

differential equation P. + Q +R
dx dy dz

and S, are any functions of x, y, z, and u.

děz
+2xy; + y2
dx2 dxdy dy

13. Being given the velocity with which a body moves at any three points of a given orbit described by it under the action of forces tending to a common centre, find that centre.

14. Determine the orbit described under the action of a force whose absolute quantity is given, and which varies inversely as the square of the distance, the body being projected from a given point, with a given velocity, and in a given direction.

MIXED MATHEMATICS.

Examiner, JOHNn Stevelly, ll.D.

6th October, 1858.--Morning.

1. Show that any number of couples acting upon a rigid body either produce equilibrium, or are reducible to a single couple. Find the conditions if they equilibrate; if not, find the equivalent couple.

2. Explain what is meant by "the central axis" of a system of forces which act on the several points of a rigid body; and find its equations.

3. Prove that if a right cylinder be described round the central axis of such a system of forces, the principal moment retains the same value for any origin taken on the surface of this cylinder.

4. Prove that the volume of a cylinder included between two plane sections, one of which is perpendicular to its axis, is equal to the area of that plane section, multiplied by the perpendicular distance of the centre of gravity of the other plane section from it.

5. Prove that the volume of a cylinder included between any two plane sections is equal to the area of a section of the cylinder formed by a plane perpendicular to its axis, multiplied by the line which joins the centres of gravity of its plane ends.

6. Investigate the force of attraction of a rectilinear prism or cylinder whose length is 7, and whose cross section is indefinitely small on a particle placed anywhere outside it, the force varying as the product of the masses divided by the square of the distances.

7. With the same law of force find the point at which, if a material particle be placed which is attracted by the three material lines of indefinitely small cross section, which form the sides of a triangle, the attracted particle is in equilibrio.

8. Investigate the position of the axis of suspension of a given body forming a pendulum, with respect to which the small oscillations will be most rapid.

9. Investigate the motion of a cylinder which rolls without any sliding down an inclined plane.

10. Prove that if an equal velocity in any given direction be combined with the actual motions of each of the bodies of any system their relative motions will not be changed.

11. A point describes a circle with a uniform velocity, find the expression for--(1), the actual velocity of the orthographic projection of that point on any plane inclined to the plane of the circle about the centre; (2), for the angular velocity of the same projection about that centre.

12. A heavy particle is allowed to glide down a helical guide whose axis is vertical, investigate the pressure each point of the curve has to sustain in turn.

13. A body descends from rest from an infinite distance towards a centre, attracting with a force inversely as the square of the distance. When it reaches the distance R, from that centre it meets a perfectly elastic surface, inclined at an angle to the direction in which it is then moving; investigate its motion after impact on that surface.

Afternoon.

1. Show how, when the altitude and azimuth of a heavenly body have been observed at a place whose latitude is known, its declination and right ascension can be computed.

2. Explain what angle the astronomer calls "refraction," its effect on the place of an object; and that for objects near the zenith it varies nearly as the tangent of the apparent zenith distance.

3. Show how the parallax of the sun or moon, or one of the planets can be deduced from two observations of its angular distances from the same fixed star, made at two distant places on the Earth which have the same (or nearly the same) meridian.

4. Give the investigation of the differential equation of motion of a body describing an orbit round a fixed centre of force the co-ordinates being polar; explain its terms, and prove what you assert respecting the constant h, which occurs in it.