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6. Prove that the complete algebraic solution of the cubic equation
ax3 + 36x2 + 3cx + d=0 may be presented in the form
ax + b = VP+
VP show that this radical expression has three, and only three, distinct values, and obtain the values of p and m in terms of the coefficients.
7. Calculate the value, in terms of the coefficients, of the symmetric function
(a – B)2 (a – y)2 + (B - y)2 (B – a)2 + (y – a)? (0 - 3)2 of the roots of the cubic equation in the preceding question.
8. Form the equation with rational coefficients whose roots are all the possible values of the expression
-I = 0.
VI + m +n + VI + wm + w’n + Vi+w2m + wn,
+ where w is an imaginary root of the equation x3 9. Form the biquadratic equation whose roots are
a + 2a4, a? + 2a, 23 + 2a?, at + 2a,
where a is an imaginary root of the equation 25 – 1 = 0.
10. Prove that when the expression
ax? + by2 + cz2 + 2lyz + 2mzx + 2nxy
consists of two factors of the form 2x + xy + vz, x'x + u'y + u'z, the following determinant relation must exist among the coefficients :
1.4.3 +2.9.4 + 3.16.5+ + n(n + 1)2 (n + 2). 12. If pn and an be the numerator and denominator of the nth convergent to the continued fraction
I 1 ai +
&c. ... a2 + a3 +
prove the relation
Pn9n-1 – Pn-19n = (-1)".
MR. F. PURSER.
1. State and prove De Moivre's theorem for all indices. Write down in a trigonometrical form (1 + V-1)}.
2. Prove that
7 3. On examining the table of differences of log sin 0, log cos 6 for consecutive minutes, it is found that their product is q. p. constant. Explain the cause of this.
4. Show that the area of the circle inscribed in a triangle bears to that of the triangle the ratio
A cot | cot .
6. The altitudes of a meteor as seen from three stations A, B, C are 0, 0,4; show that if the point vertically under the meteor lie within the triangle A, B, C, the height h of the meteor will be given by
cotxy + cot20 — 2 cot y cot 8 cos B + B'
c2 cot20 + cot-$— 2 cot 8 cot p cos C + C'' a, b, c, A, B, C being the sides and angles of the triangle A, B, C, A', B', C', the angles of a triangle whose sides are proportional to a cot , b coto, c coty.
7. The opposite sides AB, CD of a spherical quadrilateral ABCD are produced to meet in (); prove that, denoting the angle which they form at O by ,
cos AD cos BC-COS AC cos BD = cos sin AB sin CD.
where @ <
2 9. Prove that for a spherical triangle right-angled at C,
sin c cos B = sin a cos b.
(a) Hence prove that for any spherical triangle,
10. Find the relation subsisting between the latitudes and longitudes of three places on the globe which lie in a great circle. 11. Trove Napier's analogies for a spherical triangle, viz.,
cos } (A - B) tan } (a + b) =
tan cos (A + B)
12. From any point P on the circumference of a small circle of the sphere an arc of a great circle is drawn, meeting the circle again in 0, and the tangent at the opposite extremity P' of the diameters through Pin 0 ; prove that
tan PO' tan } PO = tan R tan 2R, R being spherical radius of small circle.
1. The angles of a triangle are A = 47° 46', B= 65° 12', C=67° 2', and the radius of inscribed circle is to feet; determine the sides.
2. From a vessel which is pursuing a due easterly course, two lighthouses A, B are observed, the former bearing due north, the latter 14° east of north. After running a certain distance, B bears due north, A 23° west of north. The distance between the lighthouses being known to be 30 miles, find the length run by the vessel in the interval between the observations.
3. Calculate the height of a meteor by the method indicated in (6) of preceding Paper ; the triangle of observation being equilateral, each side being 20 miles, and the altitudes 0, 0, y being cot-1\, cot-), cot-*1.
4. On a circular base whose radius is 20 yards, is constructed a dome of spherical form having a surface 3000 square yards. Determine whether the dome will be less or greater than a hemisphere, and the radius of the sphere of which it forms a part.
5. Given in a spherical triangle a=32°, b= 47°, A= 40°; determine whether the triangle is a possible one, and if so, calculate the angles B, C. 6. Determine the error in the approximate formula
13 log 2 = 4 + log 9 - log 11, and hence calculate log 2 to seven decimal places, assuming modulus = .4342944.
CLASSICAL SIZARSHIP EXAMINATION.
HOMER, iii. 127-147. 2. Beginning, ΧΟ. “Ο μέγας όλβος & τ' αρετά, κ. τ. λ. Ending, πατρώων παθέων αμοιβών.
EURIPIDES, Or., 807-843. 3. Beginning, έπει δ' αφείθη, πυρσός ώς, Τυρσηνικής, κ. τ. λ. Ending, στρατός δ' ανηλάλαξε Δαναϊδων άπας.
ID., Ph., 1377-1395. 4. Beginning, Ταυτί τα ψηφίσματ’, ώ άνδρες Αθηναίοι, κ. τ.λ. Ending, ουχί του τη πόλει συμφέροντος έσεσθαι.
MR. L. C. PURSER.
Translate into English 1. (a) Beginning, Quum dictator, stipatus agmine patriciorum, . Ending, primae tribus dicerent, tum Camillus ... inquit.
Livy, vi. 38. (6) Beginning, Haec omnia sagulo gregali amictus, Ending, ne ducem circumire hostes notarent, perlustravit.
Ibid., vii. 34. (c) Beginning, Bina in Latino iugera ita ut dodrante, Ending, etiam pro longinquitate adiectis.
Ibid., viii. 11. (d) Beginning, Duo exercitus erant: scuta alterius auro, Ending, militibus versicolores, argentatis linteae candidae.
Livy, ix. 40. (e) Beginning, Eodem anno Cn. Flavius Cn. F. scriba, Ending, nocturno altero, altero coloniae deducendae.
Ibid., ix. 46. (f) Beginning, Eodem anno Cn. et Q. Ogulnii aediles, Ending, ludi facti pateraeque aureae ad Cereris positae.
Ibid., X. 23. 2. (a) Beginning, Itaque, excussis vocibus, et ad te et ad praedes, Ending, Tam bonus gladiator rudem tam cito accepisti ?
CICERO, Philipp., ii. 29. (6) Beginning, Quaero igitur, si Lysiades citatus iudex, Ending, si ullam speciem reipublicae cogitavisset ?
Ibid., v. 5. 3. (a) Beginning, Ch. Homo sum : humani nihil a me
Ending, Rectumst, ego ut faciam : non est te ut deterream.
(6) Beginning, Patrem novisti, ad has res quam sit perspicax :...
Ending, Gemitus, screatus, tunis, visus abstine.
(c) Beginning, Sy. Lectulos in sole ilignis pedibus faciundos, Ending, et cyathos sorbilans paulatim hunc producam diem.
TERENCE, Adelph., iv. 2. 46.
(d) Beginning, Heus, proxumus sum egomet mihi.
ID., Andria, iy, i. 11.
1. Beginning, Ergo negatum vincor ut credam miser, Ending, Quae finis aut quod me manet stipendium ?
HORACE, Carm., xvii. 27–36. Beginning, Quo teneam voltus mutantem Protea nodo ? .... Ending, Diruit, aedificat, mutat quadrata rotundis ?
Ibid., lib. i. Epist. 1. 3. Beginning, Si proprium est, quod quis libra mercatur et aere, Ending, Sit proprium quidquam.
Ibid., lib. ii. Epist. 2. 4. Beginning, Thy. Pastores, hedera crescentem ornate poetam, Ending, Si qua tui Corydonis habet te cura, venito.
Virgil, Bucol. Ecl., vii. 25-40.