dy dx 7. Among the values of a which cause to vanish, to determine those which give maximum values of y and those which give minimum values? 8. Find the least parabola which can circumscribe a given circle. 10. In the ellipse the radius of curvature varies as the cabe of the normal? 11. Three forces acting on a point balance each other when their directions make angles 105°, 120°, 135° with each other ;-find the relation of the forces to each other. 12. A beam 30 feet long balances upon a prop 10 feet from one end; but when a weight of 10 lbs. is suspended from the thin end the prop has to be moved 2 feet to preserve the equilibrium; required the weight of the beam. 13. A given body is supported on an inclined plane, first by a power parallel to the base, and then by a power parallel to the plane ; compare the pressures on the plane in the two cases. 14. The centre of gravity of the semicircle? 15. The theory of the pulley? H. GREEN. CLARE (OR IST YEAR'S) SCHOLARS. ALGEBRA AND GEOMETRY. 1. Q. Prove the principle of indeterminate co-efficients, and apply it to find the log (1+x). A. If there be two series, and if these two series be always equal to one another, whetever value we give to the unknown quantity, the co-efficients of like powers of the unknown quantity must be equal. 2 Take a + bx + c x2 + &c. a+ b' x2 + c2x2 + &c.; let x = 0 in this equation, then we shall have a = a'; take off a and a' from both sides of the equation, and divide everywhere by r; take x = 0, we shall have b b' and in like manner c= = Take log (1+x) = = c'. A+ B x + С x' Dx3+ &c.; take x = this equation, we get log 1 = A, but log 10, .. A = 0; .. log (1x)=Bx+Cx+ Dx + Ex+ &c. ; o in but the logarithm of the square of (1+x) = 2 B x + 2 C x2 +2 Dx + 2 Ex+ &c.; 3 2 4 x2+ log (1 + x)2 = 2 B x + 2 Cx2 + Dx2+2 Ex1+ &c. 2 3 2 3 or log (1+2x+x2)=2 Bx+2Cx2+2Dx + 2 Ex+&e. Take 2 x + x as one term, and substitute it for x in the equation, (1) log (1 + 2x + x2) = B (2 x + x2) + C (2 x + x2) + D (2 x + x 2) + &c. 2x+Bx2+4 Cx2+Cx'+4 Cx2+8D3+ 12 Dæ +16 Ex+ &c. 4 3 2 Now by equating the co-efficients of like powers of x, By substituting the values of B, C, D, E, &c. in the equation (1) 2. Q. Show that 5 log X = 2 M { (+ + 3 ) + 1 (¦ + 3 ) + + (1 + x)2 + &c.} and apply this to find the Napierian logarithm of the common base. By subtracting the 2nd equation from that of the 1st 3 log (1+x) — log (1 − x) = B (2 x + ÷ x 3 + 7 x 3 + ‡ x2 + &c.) but the difference of the logarithm of the two numbers is equal to the quotient of those two numbers; ... log (1 + 2) = = 2 B (x + 2 + + 3 5 x2+&c.) B is called modulus, which is represented by M. -! Take = x..x = X; substitute these values in the above equation, x+ 3 1 5 log X = 2 M {(x + 1) + } ( X − 1)2 + } (x + 1) + &c. } (1 + 2 z)2 + &c. ; ) 1 1 1 in the Napierian system of logarithms M = ..log(1+)-log x=2 (1 + 2x + 3 (1 + 2x)2 By transposition, log 1 + z=log x + 2 (1 + 2 z + 3 (1 + 2 1 1 5 1 ; + 5 This is the formula for calculating the Napierian number: 1 (1 + 2x) 5(1+2x) (1 + 22)2 + &c.) 3. Q. Prove that the sum of the cubes of n natural numbers is By subtracting the 1st equation from that of the 2nd, (n + 1)3 = 4 A n3 + 6 A n2 + 4 A n + A +3 B+3 B n + B n3 + 3 n2 + 3 n + 1 + 2 Cn + C + D 4 A 1 ; 6 A+ 3 B By equating the co-efficients of like powers of n, 3 ; 4 A + 3 B + 2 C = 3 By substituting the values of A, B, C, and D in the equation, n1 3 n2 n1 + n2 + 2 n3 S + + = this is equal to the square of their sum. 4 +1 4. Q. Prove, without Trigonometry, that the area of a triangle is given by the formula area = S (8-a) (8-b) (8 — c) Where a, b, &c. are the sides and s = a + b + c 2 A. Let A B C be a triangle; and draw C D perpendicular on A B; bisect the base A B in E; b Now b2 a2 = A D2 - D B2 = (A D + D B) (AD — D B) a2 = 2 c. Ꭼ Ꭰ = ED 24 2 2 с addc on both sides. =AD; now, by the property of a right angled triangle, |