« ElőzőTovább »
12. Give Newton's proof that the difference of forces required to cause two bodies to revolve equally—the one in a fixed orbit, and the other in the same orbit revolving-varies inversely as the third power of the common distance from the centre of force.
1. Find the envelope of a right line of given length, the extremities of which move along two given right lines.
2. Find the equation of the evolute of the cubical parabola y3 =p2x, and show how to determine its singular points.
3. Given the radii of the inscribed and circumscribing circle of a spherical triangle, compute the distance between their poles.
4. Find the equation of the locus of a point from which if tangents be drawn to a parabola they will have a given ratio to one another.
5. If U, V, W represent three surfaces of the second degree
(a). Find the locus of the vertices of all cones which can be represented by 1 U + 4 V + v W.
(6). Find the locus of points whose polar planes with respect to the three surfaces have a common line of intersection.
6. (a). If tangent planes be drawn to a quadric through the six chords joining any four points on a plane section, prove that the twelve points of contact lie three each on one of the eight generators at the four points.
(6). Hence deduce Sir Wm. Hamilton's theorem that the twelve umbilics lie in threes on eight imaginary right lines.
7. The normal planes to the two lines of curvature at any point on an ellipsoid intercept from the centre on the axis major portions whose rectangle is constant.
8. If o denote a homogeneous function of the differences of the roots of the equation
aozn + naian-1 + &c. =0, homogeneously expressed in the binomial coefficients ac, ai, az, &c.,
dar where & stands for the operator
+ 221 + dai
d + nanli
MR. M. ROBERTS.
9. If yı, Yz are the particular integrals of the equation
y=0, daca I-ex dx
I Yi Y2
- Ce-, I yi y2
I yi" ya" the accents denoting successive differentiations with respect to the independent variable. 10. Integrate the partial differential equation of the first order,
f(p) 0 (x) + fi (9) + (y)=F(P) + F(@). 11. Find the centre of gravity of a lamina whose form is that of an elliptic quadrant, the density at any point varying inversely as the difference of the squares of the semiaxes of the ellipse and hyperbola passing through the point and confocal with the elliptic circumference. 12. Dr. Boole presents the integral of the partial differential equation
ZV1 + p2 + q* under the form
() dc Show from geometrical considerations the identity of this solution with the solution ordinarily given, namely,
I t-a)2 + (-a)2 +22= 1,
1. A heavy beam rests with its extremities on two rough cylinders whose axes are at right angles to the beam, and which rest on a rough horizontal plane. If the cylinders be slowly drawn away from each other till equilibrium is broken, determine the nature of the initial motion.
2. The extremities of a heavy rod are attached by cords of equal length o a horizontal beam. If one of the cords be cut, find the initial tension on the other, and determine the angle at which these cords should intersect the beam in order that the tension may be a maximum.
3. In the motion of a point on a surface, if the osculating plane to the trajectory pass through the direction of the acting force, the trajectory is either that which would be described freely, or a geodetic line?
4. A heavy particle is attached by an elastic string to one extremity of a rod whose weight is the same as that of the particle, the attachment being so arranged, that when the string has its natural length the particle shall be situated at that extremity. If, the rod being laid on a smooth horizontal plane, the particle be drawn to the other extremity and let go, find the motion.
5. If the ordinary expression for the velocity of a liquid issuing from a small orifice, namely,
= V/2gh, were true at the orifice, show that the pressure at the opposite point would be greater than the statical pressure, and determine the ratio.
a. Given that the pressure is equal to the statical pressure, find the area of the vena contracta.
Moderatorships in Classics.
CHARLES GRAVES, D. D.
Translate the following passages into English: I. Beginning, οι δ' ες βήσσαν ίκανον έπακτήρες προ δ' άρ' αυτών, κ.τ.λ. Ending, λικριφίς αίξας, ουδ' οστέον ίκετο φωτός.
HOMER, Odyssey, xix. 435-5Ι. 2. Beginning, Moίσα, και παρ Δεινομένει κελαδήσαι, κ. τ.λ. Ending, Ελλάδ' εξέλκων βαρείας δoυλίας.
PINDAR, Pythia, i. 58-75. 3. Beginning, όρα, όρα μάλ' αύ, κ. τ.
τ.λ. Ending, δελτογράφω δε πάντ' επωπά φρενί.
Æschylus, Eumen., 255-275. 4. Beginning, ώ χθονία βροτοίσι φάμα, κ. τ.λ. Ending, σοφά τ' αρίστα τε παίς κεκλήσθαι.
SOPHOCLES, Electra, 1066-1089. 5. Beginning, ά δ' εν δόμοις έδρασε θαυμάσει κλύων, κ. τ.λ. Ending, κακόν μεθίστη χρωτός ευειδή φύσιν.
EURIPIDES, Alcestis, 151-174. 6. Beginning, ΑΙ. συ δ', ώ θεοίσιν εχθρέ, ποιά γ' εστίν, άττ' επoίεις; Ending, ΛΙ. φημί κάγω.
ARISTOPHANES, Ranæ, 967-985.
κ. τ. λ.
Translate the following passages into English 1. Beginning, Πείθωμεν τοίνυν αυτόν πράως, κ. τ.λ. Ending, Και όρθώς γ' έφη.
Plato, De Republ., ix. 12. 2. Beginning, Tέλος δε ο Περίανδρος κήρυγμα εποιήσατο, κ. τ.λ. Ending, έστ' αν πυνθάνηται περιεόντα τον πατέρα.
HERODOTUS, lib. iii. c. 52, 53. 3. Beginning, Και εί τω άρα παρέστηκε, κ τ.λ. Ending, ου γαρ έργω ίσον, ώσπερ τα δικαιώματι, εστίν.
THUCYDIDES, lib. vi. c. 78-80. 4. Beginning, Λοιπόν δ' έστιν ημίν περί τε της νομιζομένης, κ. τ.λ. Ending, και ότι ου πορρω αύται αλλήλων, φανερόν.
ARISTOTLE, Politic., lib. iv. c. 6.
Translate the following passages into English : 1. Beginning, Pallada nunc pueri, teneræque ornate puellæ...... Ending, Quique facis docta mollia saxa manu.
Ovid, Fasti, lib. iii. 818-832. 2. Beginning, Omnia Romanæ cedent miraculæ terræ. ..... Ending, Spes et venturæ conjugis aptus amor.
PROPERTIUS, lib. iii. Eleg. xxii. 17-42. 3. Beginning, Quadringenta tibi si quis Deus aut similis Dis...... Ending, Ante illum uxoris, post quem nil amplius edit.
JUVENAL, Sat. v. 132-148. 4. Beginning, Scis etenim justum gemina suspendere lance. Ending, Sed præcedenti spectatur mantica tergo.
PERSIUS, Sat. iv. 10-24. 5. Beginning, P. Nequeo te exorare, ut maneas triduum hoc?...... Ending, A. Mihin' domi'st ? immo, quod aiunt, Auribus teneo lupum.
TERENCE, Phormio, act. III. sc. ii. 4-21. 6. Beginning, Denique, ubi in medio nobis equus acer obhæsit.... Ending, Navigia, aplustris fractis, obnitier undis.
LUCRETIUS, De Rerum Nat., lib. iv. 422–439.
Translate the following passages into English Prose :1. Beginning, Cetera item funebria, quibus luctus augetur,... Ending, aliud habitum esse sepelire, et urere.
CICERO, De Leg., lib. ii. c. 24. 2. Beginning, Mandatum tuum curabo diligenter : Ending, ut, quamprimum te videre possimus. Vale.
CICERO, Epist. Famil., lib. vii. Epist. 2. 3. Beginning, Forte illis diebus Cæsar inspecto omatu, ... Ending, quod insitus et adoptivus per injurias matris exerceret.
Tacitus, Annal. lib. xiii. C. 13. 4. Beginning, Negatis quemquam præstare quæ loquitur, ..... Ending, suos spectatores conspuerent.
SENECA, De Vita Beata, c. 19. 5. Beginning, Et ulmus detruncata media in ramorum .. Ending, arborem aut circa proxumam cælibem.
PLINY, Hist. Nat., lib. xvii. C. 23.
Translate the following passage into Greek Prose : – Beginning, Igitur Thrasullus iisdem rupibus inductus, . Ending, et antiqua ætas et nostro tulerit.
TACITUS, Annal. vi. 21, 22.
Translate the following passage into Greek Trimeter Iambics :Beginning, Pan. Discover, noble queen, to us thy handmaidens, . Ending, Hel. . . . . Disperses back to their abysses, or subdues.
CARLYLE from GOETHE
Translate the following passage into Latin Prose :Beginning, To speak now of the true temper of empire: Ending, and yet not to endure the means.
Bacon, Essays of Empir.
Translate the following passages into Latin Verse :1. Beginning, Fired at the sound, my genius spreads her wing, .. Ending, And learns to venerate himself as man.
GOLDSMITH, Traveller. 2. Beginning, Vain, very vain, my weary search to find..... Ending, Leave reason, faith, and conscience, all our own.