5. Shew that in the case when m and n are positive integers and m is greater a" ; a" = a"-". thann = 7. Solve the equations: (a) 5x + 2y— 1 = 3x – y + 14 = x + 197 +6, (25x® – 9y* = 675, (3y + 5x = 45. 8. A train travelling from A to C direct at a uniform rate of 54 miles an hour accomplishes the distance in the same time as a 'train which travels from A to a station B between A and C at the uniform rate of 60 miles an hour and without stopping at B proceeds from B to C at the uniform rate of 50 miles an hour. If the distance between B and C be 3 miles greater than that between A and B, find the number of miles between each pair of stations. 9. Find the sides of a rectangle whose area is unaltered if its length be increased by 4 feet while its breadth is diminished by 3 feet, and which loses one third of its area if its length be increased by 16 feet while its breadth is diminished by 10 feet. 10. Shew that a ratio of greater inequality is diminished by adding the same quantity to its antecedent and its consequent. If X – 2 : Y-2=x* :y", shew that 11. Shew how to insert two geometrical means between 2com and yon. If xy, yé, be in arithmetical progression, shew that y, 2, 2y – « are in geometrical progression. 12. Sum to 10 terms each of the series : (a) 64 -34 +14 – - ANSWERS. SECOND GENERAL EXAMINATION. WEDNESDAY, November 26, 1884. STATICS. (A.) (Page 140.) 1. 7 ft. 4 in. and 5 ft. 4 in. 3. 45 in. from the top of the cross-bar. 7. 1 ft. 5 in. and 1 ft. 4 in. 10. 3 ft. HYDROSTATICS AND HEAT. (B.) (Page 143.) 3. 2 and 4. 4. 311 lbs. 5. 9 inches. 9. 200 cubic inches. 10. 79°C. 7. 8R. FRIDAY, November 28, 1884. = ALGEBRA. (A.) (Page 149.) 1. (i) x = 1; ) (ii) x= 4; (iii) x = 15, y=6; = (iv) x = 2 or - . 2. 9 and 10. 3. 10 of each kind. a? 12 4. (i) x = }, y=1; (ii) 2 (iii) x = b y 6. £9900; £90 and £110. 9. 12s. 10. 12 and 11: 11. (i) 27; ) (ii) - 1705; (ii) 9. 12. 5 and 45. 3 or }; a ALGEBRA. (B.) (Page 150.) 1. (i) x=21; ) (ii) x= 3; (iii) x=6, y=15; (iv) x = } or – 2. 2. 8 and 9. 3. 10 of each kind. a? 62 4. (i) x = 1, y=$; (ii) x=4 or (iii) x = or b, y or a b 6. £9975; £95 and £105. 9. 8s. 10. 16. 11. (i) 29; (ii) - 1023; - 12. .3 and 75. = 1; a PREVIOUS EXAMINATION. FRIDAY, December 5, 1884. 121–3. ALGEBRA. ADDITIONAL. (A) (Page 154.) 3. (1) 480; (2) n; (3) 17955; (4) 32. 4. 5, 6, 7. 5. 5, 20, 80, or -9, - 41, -21. 6. 40 men; 80 days. 7. 4; 398. 8. a + (r – 1) (B –a). 9. 2:5105450. ALGEBRA. ADDITIONAL. (B.) (Page 155.) 3. (1) 483; (2) m; (3) 17955; (4) 27.' 4. 4, 5, 6. 6 5. 7, 21, 63. 6. 40 men ; 80 days. 7. 4; 3858 B-a 8. at 9. 2.8115750. r-1. FRIDAY, December 12, 1884. 9–111. . ARITHMETIC (A) · (Page 190.) . ( 1. 660539. 2. (1) 362 p. 26 gal. 2 qts. 1 pt. (2) 15 a. 3 r. 10} p. 3. 850. 4. (@) 14. (B) 1-259525. 5. £1. 12s. 10% d. 7. £82. 0s. 337d. 8. 10 months. 9. 93 yds. 2 ft. 6,3 in. 10. £630. 11. 1663 ARITHMETIC. (B) (Page 191.) 1. 582216. 2. (1) 1843 p. 5 yds. 1 ft. 3 in. (2) 36 a. 2 r. 17 p. 4. (a) , (B) +50381. 5. 16s. 35d. 7. £184. 10s. 7491d. 9. 83 yds. 1 ft. 3 in. 10. £1155. 11. 9. 3. 1700. 8. £30. 2 24 FRIDAY, December 12, 1884. 121-3. For candidates under the old regulations. ELEMENTARY ALGEBRA, (A.) (Page 192.) 1. a+c; 216. atc 2. 20% – 4y + 5y*. 3. (a) a (x - 12y) (x – 7y), (B) 9 (x – 4y) (x + 4y) (x – 2y) (x + 2y), (w) (1 – 4xy°) (1 + 4xy’ + 16x+y). x2 + 2y 1 4. (a) (B) 5. ** – 5xy + 7y. 6. 2x? – 7xy + 1ly. a + 8y2' 1-a' 7. a. 8. (a) x = 7, (B) x = 9, or – 63. 9. (a) x=1, y=6, 1 (B) x = 10, y = 9. 10. X:y :: 5 : 4. x . = 12. y=+3 = ELEMENTARY ALGEBRA. (B.) (Page 193.) 1. a + 6c; -76. 2. 2x2 + 9xy – 4y. 3. (a) x (x – 12yo) (x - 5y), (B) (x – 3y) (x + 3y) (5x – y) (5x + y), (W) (1 – 3xy') (1 + 3xy* + 9x+y). y * 202 + 2y 2.cc + 1 4. (a) (B) 5 5. – 7xy +9y. 6. 2* — 9xy + 7yo. ma + 9ya' 7. 8. (a) x=9, (B) x = 2, or – 111. 9. (a) x = 6, 1 (B) x=6, y = 5. 10. 12. y = =t +1. a. y=1; FRIDAY, December 12, 1884. 121-3. For candidates under the new regulations. 2cm + 2y 1. a +c; 216. 2. 2x® – 4xy + 5y". 3. (a) a + 8ye? 1 (B) 4. x – 5xy + 7y. 5. a. 6. (a) x = 7 1 - x (B) x = 9, or -63. 7. (a) x= 1, y=6, (B) x = 10, y = 9. 8. The distance between A and B is 131 miles. 9. The length and breadth of the rectangle are 9 inches and 8 inches respectively. 12. (a) 8.27 (B) – 24971. 256) ELEMENTARY ALGEBRA. (B.) (Page 196.) una + 2y* 1. a + 6c; -76. 2. 2.x2 + 9xy – 4y?. 3. (a) x +9y' 2x + 1 (B) 4. X* – 7 xy +9y?. 5. 6. (a) x = 9, (B) x = 2, or -14. 7. (a) x=6, y=1, (B) x= 6, 8. The distance between A and B is 12 miles, 9. The length and breadth of the rectangle are 16 feet and 15 fect respectively. 12. (a) 4162 (B) – 5511. a. XC +1 = y = 5. MATHEMATICAL TRIPOS. PART III. MONDAY, January 5, 1885. 9 to 12. GROUP B. n n.n 1-2 1. PROVE that the Zonal Surface Harmonic P, satisfies the difference equation nP (2n – 1) uPm-1 - (n − 1) Pn-z. Shew how the Solid Zonal Harmonics are associated with the Surface Harmonics. X', is the Solid Zonal Harmonic of positive order n having the axis of , for its axis and the origin of coordinates for its origin; Xm is the Solid Zonal Harmonic of positive order m having the same axis and a point distant a from the origin for its origin; prove that 1 1.2 The corresponding Solid Zonal Harmonic of negative order being denoted by Y', prove that, for points included within any sphere whose radius is less than å and whose centre is the new origin, 1 n +1! X, n + 2! X n + 3! X Y'. + n! 2!n! a 3!n! a Obtain the expression for Y', for points outside any sphere whose radius is greater than a and whose centre is the new origin in the form n+1! n+3! a Y.., t. 3! n! 1 [1 2 anti a n a Y n+1 n+2 +3 e 2. Explain Kirchhoff's kinetic analogue to the statics of a bent wire. A uniform wire in the shape of a helix is pushed into a uniform tube in the shape of a different helix and of the same length, which the wire just fits; investigate the shape of the new helix formed by the combination, and point out the kinetic analogue. 3. Establish the equations in the Planetary Theory da 2 dR do 6 dR dt na' de Prove that, if the disturbance is due only to a resisting medium producing retardation kum/pl and if u denote the excentric anomaly, then da - (m-1) - P (1 – e®) cos u (1 +e cos ujtim-1) (1 – e cos u) (m-1)=p -}(m-1)-p na de' m-p m - 2 n -1)-9, |