Second part. Let ABC be the triangle, D the point of bisection of the base, then or Similarly, BD2 = BA2 + AD2 – 2BA.AD cos▲, a2=c2 + AD2-2c. AD cos A. a = b2 + AD2 — 2b. AD cos§A; therefore by subtraction we have b2 – c2 — 2 (b − c) AD cos§A = 0; therefore bc must 0 or b=c, since b+c is obviously not equal to 2AD costa. 9. Find the area of a triangle in terms of the sides. 10. Find the radius of the circle which touches one side of a triangle and the two other sides produced. If ABC and DEF be two triangles with the same perimeter, prove that the radius of the escribed circle touching the side BC of the first is to the radius of the escribed circle touching the side EF' of the second as tan 4: tan D. Tod. Trig., Arts. 250, 251. 11. Given two sides b and c, and the included angle A of a triangle, find a formula for determining the without the previous determination of the angles B and C, adapted to logarithmic computation. side Tod. Trig., Art. 231. Apply to the case of b=354 yards, c=426 yards, A = 49° 16'. 2L sin 0 = log4+logb+logc − 2 log (b + c) + 2L cos §A therefore = 19.9134055; 1 0 = 64° 50′ 18". Now therefore loga = log (b+c) + cos 0 - 10 a = 331.6356 yards. 12. A building on a square base ABCD has two of its sides AB and CD parallel to the bank of a river. An observer standing on the river's bank in the same straight line with DA finds that the side AB subtends, at his eye, an angle of 45°. Having walked a yards along the bank, he finds that the side DA subtends an angle whose sine is .. Prove that the length of each side of the building is a √/2 yards. Let E (fig. 8) be the first position of the observer, then, since angle BEA = 45°, AE= AB=x say. Let F be the second position of the observer. Let AFD=0, EAF=6; therefore sin; there therefore tan + tan2 tan0 = 2 tan – 2 tan 0, PURE MATHEMATICS (1). THURSDAY, 7TH DECEMBER, 1876. 10 A.M. TO 1 P.M. 1. If two parallel planes be cut by another plane, their common sections with it are parallel. Euclid, Book XI., Prop. 16. If a straight line CD be parallel to a plane, prove that the intersection of any plane passing through CD with the given plane will lie in a straight line parallel to CD. Through CD (fig. 9) drawn a plane parallel to the given plane EF. Then we have the plane CDGH intersecting two parallel planes, and therefore the common sections; that is, CD and GH are parallel. 2. Define a solid angle. When are solids said to be similar? If a solid angle be contained by any number of plane angles, these together shall be less than four right angles. Euclid, Book XI., Defs. 9, 11, Prop. 21. Is this proposition true if any of the planes which contain the solid angle make re-entering angles with one another? If the planes make re-entering angles with each other, the sum of the plane angles may have any magnitude. 3. Define a prism and a pyramid, and shew that the volume of a pyramid is a third part of a prism which has the same base and altitude. Euclid, Book XI., Defs. 12, 13. The base of a right pyramid is a regular hexagon inscribed in a circle whose radius is 10 inches; find the ratio of the contents of the pyramid and of a cone having the circle for its base, the altitudes both of the cone and pyramid being 12 inches. Vol. of the cone = 1256.64 cubic inches, pyramid=1039-2 4. In any plane triangle, of which A, B, C are the angles, and a, b, c the sides subtending them, prove (1) c = (a+b)2 sin2 C+ (a - b) cos* C. b2 sin C+c2 sin B (2) √(bc sin B sin C) = b+c To prove (1) we have in any triangle c2=a+b2-2ab cos C = = · (a2 + b2) (cos2 1⁄2 C + sin2 1 C) – 2ab (cos2 1 C – sin2 1 C) · (a + b)2 sin2 1⁄2 C + (a − b)2 cos2 1 C. To prove (2) we have √(bc sin B sin C)=√(b2 sin2 C) = b sin C, 1 y=coso +√(−1) sind and cosp√(−1) sinø. y cos me+√(−1) sinmė, cosm✪ — √(− 1) sinm¤; m =2 cosmo, y"= cos no +√(-1) sinno; n therefore "y" = cos (me + np) + √(−1) sin (m0 + np). Similarly, 1 m n xy therefore = cos (me + np) − √(− 1) sin (m✪ +np); =2 cos (mo+no). 6. Investigate the expression sine = 0 (1–2) (1 - 0) (1 – 307), &c. In the expression just obtained put =π, and we obtain |