154. General Multiple Integration.-If X', X", X", X", etc., are the successive x-gradients of some function X, and if we start with a knowledge of the lowest of these gradients only, and wish to work upwards to a knowledge of the higher gradients and of X by repeated integration; we find

√xrds = X

dx = X"" + C3

where C, is the constant of integration. Then

This result might perhaps be more clearly understood when expressed as follows:-Xiv may be written (Xiv +0). Then the proposition is that the fourth integral of the known function Xiv is the function X whose fourth x-gradient is Xiv, plus the function (C323+ C2 + C1+ Co),whose fourth x-gradient is 0.

If there were n integrations, there would be (n + 1) terms in the result, one of which would be a constant, and (n-1) of which would be multiples of the first (n - 1) integral powers of x. § 152 illustrates one special example of this general proposition.

The constants are to be determined from the "limiting conditions." The number of limiting conditions, a knowledge of which is necessary to definitely solve the problem, is the same as the number of "arbitrary constants" C appearing in the general solution.

In the above case C2 might be determined from a knowledge of one particular value of X", and then C, from that of one particular gradient X', the remaining C。 being found from one particular value of X being given.

ef xdx

155. General Multiple Integration.—If in § 126 we write / Xdx instead of X, and therefore X instead of X', we obtain

Χα

[{='s Xdx}dx == (Xdx - [EXdx.

If in this formula the x-function E be x itself, so that E' = 1, there results

which enables two possibly easy single integrations to be substituted for one double integration which may be otherwise impracticably difficult.

Conversely, a given function (xX") may be difficult to integrate once, while the part of it X" is recognised as the second x-gradient of a known function X, and then the form

156. Reduction Formulæ.-From § 150 we have

In this substitute X for X', and therefore X' for X", and

and

Again, if in the same formula there be substituted X for X", "x Xdx2 for X, the result appears as

therefore (Xdx for X' and

*III

II

"Xdx = 4x ("Xdx2 - } [" xXdx2.

157. Graphic Diagram of Double Integration.—The meaning of double integration can be very easily represented graphically.

In fig. 5 the slope of the curve is X' and the height of the curve is X, the first integral of X' by dx. Thus (X8x) or the strip of area between two contiguous verticals under the curve is the increment of the second integral of X' by dx. Thus the area under the curve included between two given limiting verticals is their second integral, or

==

This graphic representation will help the student to perceive clearly that this integral is not the sum of a number of terms, each of which is the square of de multiplied by the slope X'. The square of any one de multiplied by the coincident slope X' would be the rectangle of base 8x and height 8X, because X'8x=dX. The sum of the series of such rectangular areas stretching between given limits on the curve is not any definite area, and it can be made as small as desired by taking the dx's sufficiently small. But this small rectangular area (dx.8X) is easily recognised to be the second increment of the area under the curve. The first difference is the area of the whole vertical strip between contiguous verticals. The difference between two such successive narrow strips (each being taken the same width 8x) is the above (8x.SX). Thus as X'da2 is this second difference which equals X'(8x)2, there is nothing illegitimate in considering the symbol da2 in X'dx2 to represent the value of (8x)2 when dr is taken minutely small. 158. Graphic Diagram of Treble Integration. The idea of treble integration may be similarly represented graphically.

nf"x

If the various areas in fig. 5 under the curve measured from any given lower limit up to the various vertical ordinates at the successive values of x, be looked upon as projections or plan-sections of a solid, the successive sections for each x and the following (x+dx) being raised above the paper by the heights x and (x+dx); then this volume is the true graphic representation of "X'de3, because the increment of this volume, or the slice of volume lying between two successive parallel sections dx apart, is the section-area at the

middle of the thickness de multiplied by dx. This section-area we

have seen in § 157 to be "X'da2; and, therefore, the above increment of volume is "X'da. The integral of this is

X'dx3.

If the lower limiting vertical ordinate of the area be at x=0, then two of the side surfaces of the above integral volume are planes normal to the paper of the diagram and passing through the axes of x and X. A third side surface, namely, that passing through the successive X edges (which are the various upper limiting ordinates in the area integrals), is also a plane: it passes through the X-axis and is included at 45° to the diagram paper. The fourth side surface is in general curved. These four side surfaces, three of which are flat, give to the volumetric representation of treble integration the general form of a quadrilateral pyramid. The base of this pyramid is plane parallel to the diagram paper.

As a valuable exercise, the student should endeavour to obtain a clear mental conception of the fact that X'(x)3, the value of which becomes X'da3 when 8x is minutely small, is the third difference in the continuous increase with x of this pyramidal volume.

CHAPTER VIII.

INDEPENDENT VARIABLES.

159. Geometrical Illustration of Two Independent Variables. -Hitherto there have been considered combinations of such functions alone as are mutually dependent on each other. The functions x, X, X, etc., have been such that no one of them can change in size without the others concurrently changing size.

In fig. 1, § 11, we have a vertical plane section of the surface of a piece of undulating land. Suppose it to be a meridional or north and south section. On it each distance measured northwards from a given starting-point corresponds to a definite elevation of the ground. If we take other meridional sections of the same piece of country, this same northward co-ordinate will correspond with other heights in these other sections. Thus, if h be used as a general symbol to mean the height of the surface at any and every point of it, then h depends not only on the northward co-ordinate

or latitude, but also upon the westward co-ordinate or longitude. If there be freedom to move anywhere over the surface, the two co-ordinates of latitude and longitude may be varied independently of each other, that is, a change in one does not necessitate any change in the other.

Under such circumstances the elevation is said to be a function of two independent variables.

160. Equation between Independent Increments. In moving from any point 1 to any other point 2, the elevation rises (or falls) from say h, to h2. Let the latitudes, or northward ordinates, of the two points be n1 and no, an the westward ordinates or longitudes be w1 and w2. Then the same change of elevation would be effected by either of two pairs of motions; namely, first, a motion northwards (n - n1) without change of longitude, followed by a motion (ww) westwards without change of latitude; or, second, a motion (ww) without change of n, followed by a motion (-n) without change of w. This is true whether these motions be large or small. Suppose them to be small, and further suppose that there are no sudden breaks in the ground, that is, that the change of elevation is continuous or gradual over the whole surface. Call the small northward, westward, and vertical movements by the symbols

Then if the meridional northward slope of the ground, just north Əh of point 1, be called

, дп

the rise during the small northward

Əh

movement Sn from 1 will be (oh).on; and if the westward slope

1

of the parallel of latitude through 2, just east of the point 2, be called the rise during the small westward movement Sw which, following the above, completes the motion to 2, will be

Əh .Sw. The sum of these two rises gives the whole of Sh, or

If

Here the two gradients are not gradients at the same point. fig. 28 be a plan and two elevations of the small part of the surface