Simplify 5. Shew that in the case when m and n are positive integers and m is greater 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. 11. Shew how to insert two geometrical means between x and y. If xy, y2, z2 be in arithmetical progression, shew that y, z, 2y-x geometrical progression. 12. Sum to 10 terms each of the series: (a) 64-34 +14 – (B) 9-51-19} are in (B) (x-3y) (x + 3y) (5x − y) (5x+y), (y) (1-3xy3) (1 +3xy3 + 9x2y*). 5. x2-7xy+9y3. 1. a6c; -76. 8. (a) x=9, (B) x = 2, or 14. (B) x=9, or - 63. 4. x2-5xy + 7y3. 5. a3. 6. (a) x=7, 7. (a) x=1, y=6, (B) x = 10, 8. The distance between A and B is 13 miles. breadth of the rectangle are 9 inches and 8 inches respectively. (B) — 24971. y = 9. 9. 8. The distance between A and B is 12 miles. MATHEMATICAL TRIPOS. PART III. MONDAY, January 5, 1885. 9 to 12. GROUP B. 1. PROVE that the Zonal Surface Harmonic P satisfies the difference equation 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 z for its axis and the origin of coordinates for its origin; X 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 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 a and whose centre is the new origin, 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 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 Prove that, if the disturbance is due only to a resisting medium producing retardaand if u denote the excentric anomaly, then tion kvm e da du du do du == n (1 − e2) cos u (1 + e cos m-p−1 „m−2 (1 − e2)* sin u (1 + e cos u)1(m−1) (1 − e cos u) −1(m −1)—p |