10. A vessel whose shape is a truncated square pyramid is filled with water; given the particulars of the vessel, determine the pressure and centre of pressure of the water against any face. 11. In a hemispherical vessel filled with water, determine the total pressure on the zone of its surface intercepted between any two horizontal planes. 12. State the principle of "constancy of work" in the equilibrium of machines; and show that it holds when the power equilibrates the resistance by propagated pressure through an incompressible fluid. SIZARSHIP EXAMINATION. Science Sizarship. GEOMETRY. MR. TOWNSEND. 1. Solve the following problems in maxima and minima : a. To a given triangle inscribe another of given species and minimum area. b. To a given triangle exscribe another of given species and maximum area. 2. For a figure of any nature, variable in magnitude and position, but invariable in form, prove that a. If three of its points move on fixed lines, all its points move on fixed lines. b. If three of its lines turn round fixed points, all its lines turn round fixed points. 3. For a fixed arc of a fixed circle, determine a. The envelope of the connectors of all pairs of points dividing it harmonically. b. The locus of the intersections of all pairs of tangents dividing it harmonically. 4. For two fixed arcs of the same fixed circle, determine a. The envelope of the connectors of all pairs of points dividing them equianharmonically. b. The locus of the intersections of all pairs of tangents dividing them equianharmonically. 5. For two fixed circles intersecting at right angles, determine a. The envelope of a variable line intersecting them in an harmonic system of points. b. The locus of a variable point subtending them in an harmonic system of tangents. 6. Prove the following cases of coaxal circles : a. Every three circles coaxal each with two of three others, and orthogonal to the third, are coaxal with each other. b. Every three circles coaxal each with two of three others, and orthogonal to a fourth, are coaxal with each other. 7. Prove the following properties of poles and polars with respect to triangles: :- a. If a variable point describe a fixed circle, its polar with respect to the sides of any fixed inscribed triangle turns round a fixed point. b. If a variable line envelope a fixed circle, its pole with respect to the vertices of any fixed exscribed triangle moves on a fixed line. 8. Prove the following properties of poles and polars with respect to circles a. The polar of a point with respect to any system of circles is the same as if each circle were removed, and replaced by the polar of the point with respect to itself. b. The pole of a line with respect to any system of circles is the same as if each circle were removed, and replaced by the pole of the line with respect to itself. 9. Prove the following properties of poles and polars with respect to homographic systems: a. If a fixed line intersect with the axes of three homographic systems of points in a triad of homologous points, its variable pole with respect to the general triad moves on a fixed line. b. If a fixed point connect with the vertices of three homographic systems of rays by a triad of homologous rays, its variable polar with respect to the general triad turns round a fixed point. 10. Prove the following properties of triangles in perspective :— a. When two triangles inscribed to the same circle are in perspective, every point on the circle connects with the vertices of either by three lines which intersect collinearly with the corresponding sides of the other. b. When two triangles exscribed to the same circle are in perspective, every tangent to the circle intersects with the sides of either at three points which connect concurrently with the corresponding vertices of the other. 11. For two triangles either inscribed or exscribed to the same circle and in perspective with each other, prove that 1 a. Every line through the centre of perspective intersects with their three pairs of corresponding sides at three pairs of points in involution. b. Every point on the axis of perspective connects with their three pairs of corresponding vertices by three pairs of rays in involution. 12. What do the following properties become by inversion to an arbitrary circle: : a. If a variable line turn round a fixed point, its pole with respect to any fixed circle moves on a fixed line. b. If a variable point move on a fixed line, its polar with respect to any fixed circle turns round a fixed point. 2. If TRIGONOMETRY. MR. LESLIE. 1. Prove geometrically that in a plane triangle, a = tan } (A-B), a-b sin (4 – B) с (4+ B) * - b sin = sin 20 2 cos (a + 20) 3. From a point 0 outside a circle whose centre is C, a secant OPP' is drawn, prove that , prove that 0 = nπ. 3 cos (a + 30)' tan OCP.tan OCP' = constant. 4. If sin 0=n sin e', and 8 cos 20 n2 1, prove that (n2 − 1)1 (9 − n2) 3 8n3 7. If tan 0: sin (30'- 0) = 5. If da, db, dC be the errors in two sides and the included angle of a plane triangle, prove that the error in the computed value of the third side is dc = cos Boa+cos Adb+ a sin BoC. 6. Prove that the area of the triangle formed by joining the points of contact of the inscribed circle is to the area of the original triangle as the radius of the inscribed to the diameter of the circumscribed circle. x sin a , prove that IX COS a sin 2a + X3 x cos 0+ y sin 0 = a, x y = b. sin 3a + &c. 2 3 8. From the series for the sine in terms of the arc, deduce, by reversing the series, the series for the arc in terms of the sine. 9. Eliminate from ALGEBRA. 10. State and prove the rule by which the angle corresponding to a given logarithmic sine not found in the Tables is determined. MR. WILLIAMSON. 1. Find the coefficient of 14 in the expansion of (ax + bx3 + cx3+ &c.)¤. 2. Rationalize the expression x + √ x2 − 1 + ✩ x2 + 1 = 0. 3. a. Find the sum of n terms of the series 2.5x+3.7x2+4.9x3+ &c. b. Also find the limit to the sum ad infinitum, when x is less than unity. 4. Prove that the roots of the equation x3 − (a2 + b2 + c2) x − 2abc= 0 are all real; and solve it when two of the quantities, a, b, c, become equal. 5. Prove that the determinant a, b, с a is divisible by a + wb+w2c, where w is a cube root of unity. 6. Determine the conditions among the coefficients of an equation of the fifth degree, that it should have a root of the form √a + √-B. 7. If a be a commensurable root of the equation f(x)= o, prove f(z) that must be a whole number for all integral values of z; the coeffi a-2 cients of the equation being all integers. 8. Find the value of (a2 + ẞ2 − 2y2). (a2 + y2 − 2ß2) · (B2 + y2 − za2), where a, ẞ, y are the roots of the cubic 9. Find a root of the equation 1. From the formula ax3 +3bx2 + 3cx + d=0. where c2 bd. 10. If ax + ẞy, a'x + B'y be substituted for x and y in the expressions ax2+2bxy + cy2, a'x2+2b'xy + c'y2, prove that the quantity ac' + ca' — 2bb' remains unaltered; provided aß' - ßa' = ± 1. 11. If a, b, c, &c., be the roots of an equation of the nth degree; find the value of (1 + a2) (1 + b2) ( 1 + c2) &c., in terms of the coefficients. 12. Prove that the reducing cubic for the equation ax1+4bx3+6cx2 + 4dx+e=0 admits of being written in the form of the determinant, a, b, 30-y b, 2 y, d, π ax3 + 3bx2 + 3cx+d=0, d I 4 5 calculate to five places of decimals. 4 tan-1. e TRIGONOMETRY. MR. LESLIE. = 0. I I tan-1 +tan-1 ; 99 b and -= a =√3-1; find A and B with 2. If in a plane triangle C= 30°, out the Tables. 3. The diagonals of a quadrilateral enclosure are 17.21, and 24.32 chains, and the angle between them is 39° 14'; find the area. 4. Solve the equation x3 = 3x+ 1 by the Tables. |