AEDC. The area of the semicircle ADB = 2r2 × .78539, and of the whole circle 4r2 x .78539: D2 x .78539. The sum of the series within the brackets (.78539) is the area of a quadrant whose radius is 1, and hence it is the area of a circle whose diameter is 1. NOTE.-The above series converges very slowly, requiring some twelve or fifteen terms to obtain the above result; but, with the exception of the part of the denominator of each term which arises from the exponent of x, each term can be obtained from the preceding one with comparatively little labor. Example 5.—To find the solidity of a cylinder, having the diame ter of the base and the altitude given. Here the area of the circle EHF is y, or the generating quantity, and it is known definitely, being equal to the base AGB. Put AD a, AB—b, AE F area AGB B x, area EHF—y (putting p.78539) pb2 = y'. Then dx the solidity of the cylinder = Sy'dæ = § pb'dx = pb2x (when x = a) pb'a the area of the base multiplied by the altitude. Example 6.-To find the solidity of a cone, having the diameter of the base and the altitude given. Here the area of the circle EIF is y, the generating quantity. Put AD a, the diameter BC=b, AG X x; then a:b::x:EF b2x2 PEF-p- y'. Hence the solidity of the cone= (when x= a) pb2 × 1 a the area of the base multiplied by one-third of the altitude, or one-third of the cylinder of the same base and altitude. Example 7.-To find the solidity of the paraboloid when the base and height are given. Here the area of the circle EIFL is the generating quantity, or y, beginning at the base CHDO, and moving parallel to itself up to A, generating the paraboloid ACHDO. Put AB a, the diameter CD=b, AG=x; then (by Prob. I. of the "Parabola," p. 253) a:x:: b2: b2x pb2x a:x::b2: Now, y = area of circle EIF A I E G F H D B y'. Hence the solidity of the paraboloid=y'dx = EF2 a PEF2 a the area of the base CHD multiplied by half the altitude AB half the cylinder of the same base and altitude. Example 8.—To find the solidity of a hemisphere, and thence of the sphere, having the radius given. Here the area of the circle EIFL is the generating quantity, or y, beginning at the base AHBO, and moving parallel to itself up to D, generating the hemisphere DAHBO. Put radius CA or CE-r, CG÷ x, p=4 ▲ X .78539 3.1416; then EG p2 — x2, and area EIFpEG2= p(r2 — x2) — y'. 2 Hence the solidity of the hemisphere = Sy'da = §p(r2 — x2)dx = the area of a great circle of the sphere multiplied by one-third of the radius. (See Legendre's Geometry, Book VIII., Proposition XIV. and corollaries.) Example 9.-To find the solidity of a spheroid; that is, of the solid described by an ellipse revolved about either of its axes. A C F I G E H D B Case 1.-Let the revolution of the ellipse be about the transverse axis AB, which gives the prolate spheroid; then the ordinate EF will describe a circle, which is the generating circle, y', supposed to move from and parallel to COD to B. The axis about which the revolution is made is called the fixed axe. The equation of the ellipse (see Prob. I. of "Ellipse," p. 253) is a2y2 + b2x2 = a2b2, where a= OB, b=OC, x= OE GF, y=EF=OG. Put p3.1416, the area of a circle whose radius is 1. Now, the area of the generating circle FEH=EF2 × b2 p = y2 × p = — (a2 — x2)p = y'. Hence the solidity Sy'dx = ab'p the solidity of the semi-spheroid CODB. The whole spheroid ADBC=ab'p. Now, putting the major axis A ABA, and the minor axis CDB, we have a= and b 2' B 2 substituting these values for a and b in the equation ab2p, it becomes 3.1416 A B2 X × 3.1416 A x B2 × A× B2 × .5236—the 2 4 solidity of the whole prolate spheroid ADBC, where the ellipse is revolved about AB. Case 2.-When the revolution is about the conjugate axis CD, then the ordinate GF describes the circle IGF, which is the generating circle, moving from and parallel to the circle AOB to C. Hence, as x is the moving line and y the space moved through, the solidity in this case will be fa'dy. The circle IF = GF2 × p= x2p=(b2-y3)p=a', and Sa'dy= 2 S' "(b2 — y23)pdy = Sa2pdy — fa2py'dy — a2py — the solidity of the semi-spheroid A2 B ACBO. The whole spheroid ADBC4a2bp = 4 × X 4 3.1416 3.1416 = A2 × B × solidity of the whole spheroid equals the square of the revolving axe multiplied by the fixed axe, and the product multiplied by .5236, which is one-sixth of the area of a circle whose radius is 1. A2 × B × .5236. In either case the SOME ELEMENTARY PROBLEMS IN ANALYTICAL GEOMETRY. INTRODUCTORY REMARKS.-In many mathematical investigations the operations are much simplified by designating a point, line, or surface by an equation, referring its position to two lines called axes, crossing each other at right angles. This method of representing points, lines, and surfaces by algebraic equations was first introduced by Des Cartes in his Geometry, published in the year 1637; but the principle has been extended and amplified by Lagrange, Delambre, Laplace, and many others, and it has now become one of the most powerful instruments in mathematical research. It is proposed to give here an idea of the principle, accompanied by a few practical problems as illustrations. B A 21 E 314 C D Y' Let XX', YY', in the adjacent figure, represent the lines crossing each other at right angles in the point O, and let these lines be taken as axes, to which it is proposed to refer points and lines on the same plane. XX' is called the horizontal axis, YY' the vertical axis, and their point of intersection, O, the origin of the axes, or simply the origin. The perpendicular distance, either way, from the vertical axis, measured on the horizontal axis, or on a line parallel thereto, is called an abscissa; and such distance is generally represented by the letter x. The perpendicular distance, either way, from the horizontal axis, (240) |