where x = the distance between the two lines=N1/N0. These formulae also hold for converting moments of a solid figure with regard to a plane into moments with regard to a parallel plane through the centroid; x being the distance between the two planes. A line through the centroid of a plane figure (drawn in the plane of the figure) is a central line, and a plane through the centroid) of a solid figure is a central plane, of the figure.
The centroid of a rectangle is its centre, i.e. the point of intersection of its diagonals. The first moment of a plane figure with regard to a line in its plane may be regarded as obtained by dividing the area into elementary strips by a series of parallel lines indefinitely close together, and concentrating the area of each strip at its centre. Similarly the first moment of a solid figure may be regarded as obtained by dividing the figure into elementary prisms by two sets of parallel planes, and concentrating the volume of each prism at its centre. This also holds for higher moments, provided that the edges of the elementary strips or prisms are parallel to the line or plane with regard to which the moments are taken.
33. Solids and Surfaces of Revolution.—The solid or surface generated by the revolution of a plane closed figure or a plane continuous line about a straight line in its plane, not intersecting it, is a solid of revolution or surface of revolution, the straight line being its axis. The revolution need not be complete, but may be through any angle.
The section of a solid of revolution by a plane at right angles to the axis is an annulus or a sector of an annulus (fig. 5), or is composed of two or more such figures. If the solid is divided into elements by a series of such planes, and if h is the distance between two consecutive planes making sections such as ABDC in fig. 5, the volume of the element between these planes, when h is very small, is approximately h×AB × arc PQ＝h.AB.OP.θ. The corresponding element of the revolving figure is approximately a rectangle of area h.AB, and OP is the distance of the middle point of either side of the rectangle from the axis. Hence the total volume of the solid is M.θ, where M is the sum of the quantities h.AB.OP, i.e. is the moment of the figure with regard to the axis. The volume is therefore equal to S.ȳ.θ, where S is the area of the revolving figure, and ȳ is the distance of its centroid from the axis.
Similarly a surface of revolution can be divided by planes at right angles to the axis into elements, each of which is approximately a section of the surface of a right circular cone. By unrolling each such element (§ 30) into a sector of a circular annulus, it will be found that the total area of the surface is M′.θ＝L.z̄.θ, where M′ is the moment of the original curve with regard to the axis, L is the total length of the original curve, and z̄ is the distance of the centroid of the curve from the axis. These two theorems may be stated as follows:—
(i) If any plane figure revolves about an external axis in its plane, the volume of the solid generated by the revolution is equal to the product of the area of the figure and the distance travelled by the centroid of the figure.
(ii) If any line in a plane revolves about an external axis in the plane, the area of the curved surface generated by the revolution is equal to the product of the length of the line and the distance travelled by the centroid of the line.
These theorems were discovered by Pappus of Alexandria (c. A.D. 300), and were made generally known by Guldinus (c. A.D. 1640). They are sometimes known as Guldinus’s Theorems, but are more properly described as the Theorems of Pappus. The theorems are of use, not only for finding the volumes or areas of solids or surfaces of revolution, but also, conversely, for finding centroids or centres of gravity. They may be applied, for instance, to finding the centroid of a semicircle or of the arc of a semicircle.
34. Segment of Parabola.—The parabola affords a simple example of the use of infinitesimals. Let AB (fig. 6) be any arc of a parabola; and suppose we require the area of the figure bounded by this arc and the chord AB.
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Fig. 6. |
Draw the tangents at A and B, meeting at T; draw TV parallel to the axis of the parabola, meeting the arc in C and the chord in V; and draw the tangent at C, meeting AT and BT in a and b. Then (see Parabola) TC＝CV, AV＝VB, and ab is parallel to AB, so that aC＝Cb. Hence area of triangle ACB＝twice area of triangle aTb. Repeating the process with the arcs AC and CB, and continuing the repetition indefinitely, we divide up the required area and the remainder of the triangle ATB into corresponding elements, each element of the former being double the corresponding elements of the latter. Hence the required area is double the area of the remainder of the triangle, and therefore it is two-thirds of the area of the triangle.
The line TCV is parallel to the axis of the parabola. If we draw a line at right angles to TCV, meeting TCV produced in M and parallels through and B in K and L, the area of the triangle ATB is 12KL.TV＝KL.CV; and therefore the area of the figure bounded by AK, BL, KL and the arc AB, is
KL.12(AK+BL)+23KL{CM−12(AK+BL)}
＝16KL(AK+4CM+BL).
Similarly, for a corresponding figure K′L′BA outside the parabola, the area is
16K′L′(K′A+4M′C+L′B).
35. The Ellipse and the Ellipsoid.—For elementary mensuration the ellipse is to be regarded as obtained by projection of the circle, and the ellipsoid by projection of the sphere. Hence the area of an ellipse whose axes are 2a and 2b is πab; and the volume of an ellipsoid whose axes are 2a, 2b and 2c is 43πabc. The area of a strip of an ellipse between two lines parallel to an axis, or the volume of the portion (frustum) of an ellipsoid between two planes parallel to a principal section, may be found in the same way.
36. Examples of Applications.—The formulae of § 24 for the area of a trapezoid are of special importance in land-surveying. The measurements of a polygonal field or other area are usually taken as in § 25 (ii); a diagonal AE is taken as the base-line, and for the points B, C, D, . . . there are entered the distances AN, AP, AQ, . . . along the base-line, and the lengths and directions of the offsets NB, PC, QD, . . . The area is then given by the formula of §25 (ii).
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Fig. 7. |
37. The mensuration of earthwork involves consideration of quadrilaterals whose dimensions are given by special data, and of prismoids whose sections are such quadrilaterals. In the ordinary case three of the four lateral surfaces of the prismoid are at right angles to the two ends. In special cases two of these three lateral surfaces are equally inclined to the third.
(i) In fig. 7 let base BC＝2a, and let h be the distance, measured at right angles to BC, from the middle point of BC to AD. Also, let angle ABC＝π−θ angle BCD＝π−φ, angle between BC and AD＝ψ. Then (as the difference of two triangles)
area ABCD＝(h cot ψ+a)22(cot ψ−cot φ) − (h cot ψ−a)22(cot ψ+cot θ) ⋅
(ii) If φ＝θ, this becomes
area＝tan θtan2 θ−tan2 ψ(h + a tan θ)2 − a2 tan θ.
(iii) If ψ =0, so that AD is parallel to BC, it becomes
area＝2ah+12(cot θ + cot φ)h2.
(iv) To find the volume of a prismoidal cutting with vertical ends, and with sides equally inclined to the vertical, so that φ＝θ, let the values of h, ψ for the two ends be h1, ψ1, and h2, ψ2, and write
m1≡cot ψ1cot ψ1−cot θ (a + h1 cot θ), n1≡ cot ψ1cot ψ1+cot θ (a + h1 cot θ),
m2≡cot ψ2cot ψ2−cot θ (a + h2 cot θ), n2≡ cot ψ2cot ψ2+cot θ (a + h2 cot θ),
Then volume of prismoid＝length × 13 {m1n1 + m2n2+ 12(m1n2 + m2n1)−3a2} tan θ.
mensuration of graphs
38. (A) Preliminary.—In § 23 the area of a right trapezium has been expressed in terms of the base and the two sides; and in § 34 the area of a somewhat similar figure, the top having been replaced by an arc of a parabola, has been expressed in terms of its base and of three lengths which may be regarded as the sides of two separate figures of which it is composed. We have now to consider the extension of formulae of this kind to other figures, and their application to the calculation of moments and volumes.
39. The plane figures with which we are concerned come mainly under the description of graphs of continuous variation. Let E and F be two magnitudes so related that whenever F has any value (within certain limits) E has a definite corresponding value. Let u and x be the numerical expressions of the magnitudes of E and F. On any line OX take a length ON equal to xG, and from N draw NP at right angles to OX and equal to uH; G and H being convenient units of length. Then we may, ignoring the units G and H, speak of ON and NP as being equal to x and u respectively. Let KA and LB be the positions of NP corresponding to the extreme values of x. Then the different positions of NP will (if x may have any value from OK to OL) trace out a figure on base KL, and extending from KA to LB; this is called the graph of E in respect of F. The term is also sometimes applied to the line AB along which the point P moves as N moves from K to L.
To illustrate the importance of the mensuration of graphs, suppose that we require the average value of u with regard to x. It may be shown that this is the same thing as the mean distance
