middle of 8. The integral, or total, moment exerted upon this section by the part of the load lying between l1 and l1⁄2 is

= whole load on (2-7) multiplied by the distance of the middle of the same length from the given section.

It must be noted that this is the moment exerted by the load alone independently of that exerted by the forces supporting the beam.

70. Volume of Sphere.-Passing now to volumetric integrals, we may consider a very small sectorial part of the volume of a sphere as an equal-sided cone of very small vertical angle placed at the centre of the sphere, and with a very small spherical base nearly coinciding with the flat surface of small area touching the sphere. The volume of the small cone with the flat base is known to be the product of its base area by its height. The height here is r, the radius of the sphere. This is true whatever be the shape of the cross section of the cone. Now the whole volume of the sphere is made up of a very large number of such small-angled cones with spherical bases, these cones fitting close together so as to fill up the whole space. They would not fit close together if their cross sections were, say, circular; but the argument does not depend on the shape of the cross section, and this is to be taken such as will make the cones fit close together. In all these small conic volumes, the common factor r appears as a constant: each is r.SA, if SA represent the area of the small base. Thus the sum of the volumes is greater than any one of them in the same ratio as the sum of the areas of the bases is greater than the basearea of that one.

Thus if A be the sum of the bases, or (dA=A,

we have the sum of the volumes equal to 3rA. For any sectorial portion of the volume of the sphere, the sum of the areas of the flat tangent bases approximates to the area of the corresponding portion of the spherical surface pari passu with the approxima tion of the sum of the flat-based conical volumes to the sum of the round-based conical volumes, which latter is the true spherical volume. Thus, if A be the area of the spherical surface, the volume subtended by it at the centre is rA. If A be taken as the complete spherical surface, then rA is the total spherical volume. This integration is in form identical with that of fig. 9. It differs from that in kind, inasmuch as the differential SA is an

area, while in fig. 9 the differential dp is a line. The mathematical process is the same in both cases; but the legitimacy of the application of this process depends in the one case upon the physical relations between certain curved and straight lines, while in the other case it depends on the physical relations between certain curved and flat surfaces.

When it is known that the ratio of the surface-area of a sphere to the square of its radius is 47, the above integration proves the complete spherical volume to be r3 (see § 76 below).

71. Volume of Expanding Sphere.-Consider now the spherical volume as swept through by the surface of a gradually expanding sphere. If the radius be r1 at one stage of the expansion, and at another, the volume swept through between these two stages is (3-3). During any small increase of size dr from the radius (~ − 1dr) to (r+18), the volume swept out is the normal distance Sr between the smaller and larger spherical surfaces multiplied by the mean area of the spherical surface during the motion, viz., 4m2. That is, the increment of volume is

Thus 42 is the r-gradient of (+C).

If x were used to represent the radius, and X the volume, and X' the x-gradient of X and the constant factor 4π be written k: we would here have

Expressed in words, the radius-gradient of the spherical volume is the spherical surface.

72. Volume of Expanding Pyramid.-Consider a rectangularbased pyramid of height x, and the two sides of whose base are ma and nx. The area of the base is mna2, and, therefore, the pyramidal volume is mnx. Now, suppose the size of this pyramid to be gradually increased, keeping its shape unaltered, by extending

*See Classified List, III. A. 2.

its sides in the same planes, and moving the base away from the vertex while keeping the base always parallel to its original position. As the height x increases, the sides of the rectangular base both increase in the same ratio so as to remain always mx and nx; and, therefore, the increasing volume is always equal to mnx3. As the base moves a distance Sx away from the vertex from the height (x-6x) to (x+dx), the increase of volume thus added to the pyramid is the mean area of the base during this motion, viz., mna, multiplied by the normal distance Sx between the old and the new bases. The increment of volume is thus mnx2.8x. The definite integral volume taken between the limit X1 and X2 is

If the constant factor mn be written k, this result would be thus expressed, taking the indefinite form of the integral :—

which is formally or symbolically identical with the last result obtained. The difference between the two in kind is perhaps best recognised by comparing the word-expression of the last result with the following similar statement of our present one :—

The height-gradient of the volume of a pyramid of given shape is the area of the base of the pyramid.

In this last statement of the result no reference is made to the special shape of the cross section of the pyramid, and it is readily perceived that the reasoning employed above did not depend in any degree upon the rectangularity of the base.

73. Stress Bending Moment on Beam.-Take as another example of this formula leading from the second power in the gradient to the third power in the integral, the calculation of the stressbending moment of a rectangular beam section exposed to pure bending of such degree as produces only stresses within the elastic limit. Under this condition the normal stress on the section increases uniformly with the distance from the neutral axis, which in this case is at the middle of the depth. Thus, if the whole depth of the section be called H, and the intensity of stress at the top edge (at distance from neutral axis) be called k; then the

H 2 intensity of stress at any distance h from the axis is k

h

2k

=

h.

H

H

If the width of the section be B, the area of a small cross strip of it, of depth Sh, is B8h. If h mean the height to the middle of Sh, then the whole normal stress on this strip is

2kB
H

2kB

H

•hôh, and the

moment of this round the neutral axis is h2. Sh, because h is

the leverage. The sum of these moments over the half of the section lying above the axis is the integral of this between the limits h=0 and h=1H, or

An equal sum of moments of like sign is exerted by the stresses on the lower half of the section, and thus the

Total Stress Bending Moment = BH2.

N

S

74. Angle Gradients of Sine and Cosine and Integration of Sine and Cosine.-In fig. 13 the angle a is supposed measured in radians, that is, in circular measure, the unit of which is the angle whose arc equals the radius. Radians, sines, cosines, tangents, etc., are pure numbers, or ratios between certain lengths and the radius of a circle; but if the radius be taken as unity, as in fig. 13, then these ratios are properly represented by lengths of lines, this graphic representation being to an artificial scale just as to other artificial scales, velocities, moments, weights, etc., can be graphically represented by line-lengths. In fig. 13 the angle a is measured to such a scale by the length of the arc Na, while to the same scale sin a is measured by as and cos a by ac. small angle da, and mark off from N the two angles (a - da) and (a+da). The horizontal and vertical projections of da (parallel to as and ac) are evidently the increments of the sine and cosine

Take a very

FIG. 13.

for the angle increment da. The horizontal projection is a positive increment of the sine; the cosine decreases as a increases, so that the vertical projection is the decrement or negative increment of the cosine. If Sa be taken small enough to justify the short arc being taken as a straight line, da and its two projections form a small right-angled triangle of the same shape as Oas. We have, therefore, Increment of sina = 8(sina) = Horizontal projection of da

Integrating these increments between any limits a and 02:

results are

the

The student should carefully follow out this integration on the diagram through all four quadrants of the complete circle, paying attention to the changes of sign.

Written as indefinite integrals these results are

Expressed in words, this is, the angle-gradient of the sine of an angle is its cosine, and that of its cosine is its sine taken negatively. 75. Integration through 90°.-Since sin 0°=0 and cos 0° = 1, while sin 90° 1 and cos 90° =0, we find, integrating between the limits 0° and 90°,

90°