the former of which is value corresponding to equation given, and therefore two values of A are 26° 33′ 54′′, 243° 26' 6". 6. If A, B, C are the angles of a plane triangle, a, b, c the sides subtending them, and s half the perimeter, prove (1) costA= (s ss-a bc Todhunter's Trigonometry, Art. 217. (2) c2 = (a - b)2 cos2 1⁄2 C + (a + b)2 sin* 1⁄2 C. Shew how to adapt (2) to logarithmic computation by introducing a subsidiary angle. We have ca"+b2-2ab cos C = = (a2 + b2) (cos3 1⁄2 C + sin3 C) — 2ab (cos2 1⁄2 C – sin2† C ) = (a − b)2 cos2 1⁄2 C + (a + b)2 sin31⁄2 C. formula which are adapted for logarithmic computation. 7. Find an equation that expresses the relation between the circular measure of an angle and the measure of the same angle in degrees. Todhunter's Trigonometry, Art. 22. Find the circular measure of an angle of a regular quindecagon. 8. If A be the area of a triangle, whose three sides are a, b, c, prove that the radius of the circle circum Todhunter's Trigonometry, Art. 252. In the ambiguous case for the solution of a triangle shew that the radii of the circles circumscribing each of the triangles to which the data apply are equal to each other. Let ABC and AB'C be two triangles, to which the data b, A, a apply. Then the radius of a circle circumscribing either is the same for both triangles. being = a 2 sin A 9. The three sides of a triangle are 320, 440, 560 feet respectively, find its greatest angle or its area. Greatest angle = 93° 35'. Area = 70262-4 square feet. 10. From the intersection of two straight paths which are inclined at an angle of 37°, two pedestrians A and B start at the same instant to walk along the paths, A at the rate of 5 miles an hour, and B at a uniform rate also; after 3 hours they are found to be 9 miles apart; N shew that there are two rates at which B may walk to fulfil the condition, and find either of those rates. Let AB' and AC (fig. 30) be the directions of the paths of B and A respectively. Suppose B the position of B at the end of the three hours, we shall find the length AB. We have L sin Blogb - loga + L sin A = 9.9778307, B=71° 50′ 53′′ or 108° 9' 7", and there will be two positions such as B and B' which will satisfy CB=9.5. The two values of AB corresponding to these two positions will be found to be 14.9392 and 9.01991, and thus the rates at which B may walk are 4.9797 and 3-00664 miles per hour. 11. AB and DE are two chords of a circle at right angles to each other, intersecting in C. AC=40 feet, DC=30 feet, and the radius of the circle is 100 feet. Find the sides of the quadrilateral ADBE, and determine two of its opposite angles. Let CE and CB (fig. 30) be = 4x and 3x respectively. Let be the centre of the circle, and OF and OG perpendiculars on DE and AB. = whence the sides CB and CE 30 (15) and 40 (15). The sides of the quadrilateral are therefore 50, 160, 50√(15) and 120 feet respectively. And the angles 112° 23′ 33′′, 67° 36′ 27′′, 128° 39′ 9′′, 51° 20′ 51′′. PURE MATHEMATICS (1). FRIDAY, 22ND JUNE, 1877. 10 A.M. TO 1P.M. 1. Prove De Moivre's Theorem for a positive integer, and hence prove that is one of the values of {cos + √(−1) sin0}~. What are the other values? Todhunter's Trigonometry, Arts. 267, 268. 2. Prove that cose + √(-1) sine = ev(1). = sin cos {√(-1)} + cos sin {Þ √(−1)} (31 — 3 ̄1) — § (3a — 3 ̄†) + † (3a — 3 ̃†) — &c. = ¿π. Expression given is equal to 3* — § (31)3 + † (31)3 — &c. — {3 ̄1 — § (3 ̃ ̄1)3 + † (3 ̃ ̄1)3 — &c.} = tan 13-tan ̄13 3-3 1+3.3 √(3) 휴. 4. Find the equation of the straight line passing through the two points whose coordinates are x,y,, XY 2° Todhunter's Conic Sections, Art. 35. Find the equation of the straight line joining the origin and the point of intersection of the straight lines 2x + 3y a = 0, 3x+2y+a= 0. The point of intersection of these two lines is (−a, a), and the equation to the line required is y = −x. 5. Find the equation of the circle referred to any pair of rectangular axes. Todhunter's Conic Sections, Art. 88. Find the lengths of the chords cut off from the axes by the circle whose equation is x2+ y2-8x-y-20=0. Putting x = 0, we obtain y3-y-20=0 as the quadratic to determine the lengths of the intercepts on the |