3 1.2.3.4 = n2 (n+3) 2"-4. (n − 4)3 + &c. The series is of the coefficient of 2" in (e*+ 1)" + (e" − 1)", 2 If n>3, x3 appears only in the first of these two terms, n2 (n+3) 2n−3 and its coefficient is and therefore the required sum is n2 (n+3) 21−4. 2. Prove that there are only five kinds of regular polyhedrons. If one of each kind be inscribed in the same sphere, prove that their edges will be in the ratio of 2 √√(2) : 2 : √(6) : √(5) − 1 : √[§ {5 −√√(5)}]. If m be the number of sides in each face of a regular polyhedron, n the number of plane angles in each solid angle, a the length of an edge, and D the diameter of the cir cumscribing sphere α Ꭰ In the (i) tetrahedron m=3, n=3, therefore√(3), α D = = (5 — √√(5) - √(5) 10 =2√(2): 2 : √(6): √(5) 1 : √[§ {5-√(5)}]. 3. Find the equation of the chord joining the points of contact of two tangents drawn to the parabola y=4ax from the point (h, k). If p(x, y) = (ax+ By)*+2gx+2fy+c=0 be the equation of a parabola, prove that the equation of its axis is The direction-cosines of a diameter are proportional to a and 8, and of the polar of (xy) to do аф аф If (xy) is on the axis these lines are at right angles, and iv. Find the length of the straight line drawn from the point (a, ß, y) parallel to the straight line the plane la+my+nz=p. Prove that the four planes, х y 2 ν λ = μ my+nz=0, nz+lx=0, lx + my=0, lx+my+nz=p, form a tetrahedron whose volume is If the first three planes meet the coordinate planes of yz, zx, xy in BC, CA, AB respectively (fig. 59), and the fourth plane meet the coordinate planes in bc, ca, ab; then BC passes through a and is parallel to bc, CA passes through b and is parallel to ca, and AB passes through c and is parallel to ab. Hence the triangle abc is the triangle ABC. 5. Find the equations of the tangent plane and the normal at a point of the surface If the plane lx+my+nz=p cut this surface in a parabola, prove that l'a+mb2 = n°c, and the coordinates of the vertex of the parabola satisfy the equation The equations of the line conjugate to the plane lx+my+nz=p are and if the curve of section be a parabola, this line must be parallel to its conjugate plane, therefore l'a2 + m*b* - n°c2 = 0. If (ay) be the coordinates of the vertex, the equations of the diameter are 6. Shew how to determine the maximum and minimum values of a function of two or more variables connected by a given equation. A framework crossed or uncrossed is formed of two unequal rods joined together at their ends by two equal rods; prove that the distance between the middle points of either pair of rods is a maximum when the unequal rods are parallel and a minimum when the equal rods are parallel; unless the two unequal rods are together less than the two equal rods, in which case the unequal rods are parallel in both the maximum and minimum positions. Let ACDB be the framework; AB=a, AC=BD=b, CD=c, 4 CAB=0, LDBA=0; and let x be the distance L between the middle points of the unequal rods. This is to be a maximum or minimum subject to the condition c2 = a2-2ab (cos + cos p) +263 (1+cos (0+)}. Hence, 2ab sin 0-26 sin (0+ 4) = 2ab sin 4 – 263 sin (0 + $), = +¢ sin sin and 0; for += would require the quadrilateral to be a parallellogram. If ac, has its greatest numerical value when BDC is a straight line, and when ACD is a straight line. If a +c>2b, 0 and can vanish, and it is possible for the equal rods to become parallel, the unequal rods being crossed. If a + c <2b, 0 has its least numerical value when BCD is a straight line, and 6 when ADC is a straight line, and it is impossible for the equal rods to become parallel. From the figures 60 and 61 we see that is a maximum when the unequal rods are parallel; and if a+c>2b, x2 is a minimum, and zero when the equal rods are parallel; but if a+c<2b, x is a minimum when the unequal rods are parallel and the equal rods crossed. If y be the distance between the middle points of the equal rods, vii. Prove that there are two ways of generating the same hypocycloid by the trace of a point on a circle rolling |