and find the length of the curve, the equation to which is ay2 = x3. 81. If ax2+2bxy + cy2 = d be the equation to an ellipse, 82. Find the length of the curve, the equation to which is 8a3y = x2 + 6a2x2, and the equation to its involute. 84. Prove that between the limits x = a, x = b, Judx = h (U ̧_a + Ux=a+h + • . h2 du du Uz_b_h) + 1.2 dx da'ath dxb-k 85. Prove that a dx (a + bx”), may be obtained in finite form Zdz, where Z is a rational function of z. m being an odd integer, and n an indefinitely small quantity; prove that the latter integral is discontinuous. 88. Integrate the following equations : 1834 π (a + b sin2 ødø = 75 ( 1 + c,) (1 + €2) · · · · · (1 + ©n) 89. c2 sin2 o (1 − √T — c,2) ÷ (1 + ✓1c,2); and the process is carried on till c, does not differ sensibly from zero. Prove this, and thence find the first two terms of the developement, in a series of cosines of multiple angles, of (a2 2aa' cos o + a22) in the case where a' is not much less than a. 91. A plane curve, referred to polar coordinates, is defined by the equation r = (a2 - ·62). sin 0. cos 0 ; prove a2 sin2 0 + b2 cos2 0 a Shew that the curve represented by the equations - + X b y 1, + c = 1, is an hyperbola; when the coordinate axes are inclined to each other at given oblique angles, determine the position of its centre, the magnitude and direction of its axes. 93. Prove that every rational proper fraction can be resolved into the sum of a series of fractions of all or some of the forms 94. Explain what is meant by the process of integration; 1835 95. In the spiral, whose equation is r = a sec included by the curve, the asymptotes and the tangent at the apse = 4a2. 96. Find the differential coefficient (of the volume of a solid of revolution, and determine the volume generated by the revolution round the axis of x of the area of the curve whose equation is ay = x √ a2 x2. 1836 98. The area of the segment included between the arc PP' and chord of an ellipse or hyperbola, is equal to the area of the segment included between the conjugate arc DD', and chord of the ellipse or opposite hyperbola. 99. The frustum of a cone, the radii of whose ends are R, r, and altitude h, is cut by a plane, which touches the circumference of each end; find the axes of the section, and shew that the frustum is divided into parts, which are to each other as R2: r. 3 3 100. Express coso in terms of the cosines of multiples of 0, and find /cos de between the limits = 0 and 0 = 2་ 101. Integrate the following differential coefficients : 102. Make fdx xm (a + bxn) depend upon fdx xm-n (a + bx). Shew for what relation of the coefficients the method fails, and how in that case the integration is to be effected, Find fdxe-ax sin mx between the limits x = 103. Integrate the expressions O and ∞. 2. The curve which is expressed by the particular solution of a differential equation of the first order, is the locus of the intersections of the curves which arise from giving every possible value to the constant in the general solution. 3. Find the equation of the curve traced out by the extremities of the perpendiculars upon the tangents of a circle, drawn from a point in its circumference; and find its greatest ordinate. the variables x, y and z, is reduced to the integration of equations of two variables, when any one of the equations 5. If a ladder slides down a perpendicular wall, shew that each stave describes a quadrant of an ellipse, except the middle one, which describes a quadrant of a circle. 6. Given, in the equation + u = 0, u = a sin v + b. d2u dv2 cos v, to solve the equation d2u + u + II = 0, by the Varia tion of the Parameters. 1822 8. Investigate the equation to the curve, in which the area has the same ratio to the square of the ordinate, that the ordinate has to the abscissa. 11. Required the curve, which within its own arc, its evolute and radius of curvature shall contain the least area. 12. Find the relation of x and y in the equation d2y dx2 + (y a) x2 = 0. dx 1824 13. Integrate ydydx = (x + a) dy2 + adx2. 14. Find the relation between x and y in the equation |