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 II III [" 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. 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. The As a valuable exercise, the student should endeavour to obtain a clear mental conception of the fact that X'(x), the value of which becomes X'da3 when Sx 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 ha. Let the latitudes, or northward ordinates, of the two points be n1 and n2, anl 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 (n2 - 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 of point 1, be called , movement Sn from 1 will be the rise during the small northward (On).on; and if the westward slope дп 1 of the parallel of latitude through 2, just east of the point 2, be the rise during the small westward movement Sw called Əh which, following the above, completes the motion to 2, will be ().u. The sum of these two rises gives the whole of ô, or Here the two gradients are not gradients at the same point. If fig. 28 be a plan and two elevations of the small part of the surface considered, they are the northward and westward gradients at v1 and V2 at the middle points of 1N and N2 in the plan. If now the passage from 1 to 2 be effected by passing through ward slopes of the ground at w, and w2; then the same change of elevation may be calculated thus, where the Sw and the Sn are also the same lengths as before, the quadrilateral 1N2W being a parallelogram. Əh and би Əh дго 1 are the westward slopes on opposite sides of this parallelogram; they are the slopes of N2 and 1W in the "North Elevation." and (3) are the northward slopes upon the 1 2 other pair of opposite sides; they are the slopes of IN and 2W in the "West Elevation." Adding these two equations and dividing each side by 2; and, further, calling the means between the gradients on the opposite дп sides of the parallelogram by the symbols and ; we have дп дп ди On a continuous surface such as is here supposed, the above arithmetic means are, with great accuracy, equal to the actual gradients along the centre lines w12 and v1w of the small rectangle ; W1V2 V1W2 that is, the gradients at the centre of the short straight line 1 2. 161. Equation between Independent Gradients. If the short length 12 be called ds, so that Sn and Sw are the northward and westward projections or components of ds; then we have, as general truths, by dividing successively by dn, dw, and ds, where the restrictive symbol (), indicates a ratio of increments occurring concurrently along the special path s over the surface, an element of which path is 1 2 or ds; while the ratios of increments not marked with this symbol are pure northward and eastward dh gradients. does not need to be marked, as its terms indicate ds plainly that it means the actual whole gradient of the ground. along the path s. are different measures of the direction of the path s in plan; the first two are the tangents of the inclination of this path from the west and from the north respectively; the last two are the sines of the same inclinations. These measures of its direction parдп дп ticularise the special path to which the equations apply. and дп dw have no connection with, and are quite independent of, the direction of this paths: they are the due north and due west gradients at a point of the path, and depend upon the position of this point in the field, but not upon the direction of the path at such point. The gradients and are called the "partial" differential Әһ Əh бо coefficients or gradients of h with respect to n and w. dh is the ratio of rise to northward progress in travelling dn/s along the paths, and depends upon the direction of this path.` It is quite different from Əh дп dh In fig. 28 it, (d), equals the tangent of inclination of the line 12 to the horizontal base in the "West Elevation"; while equals the tangent of inclination of line IN Əh дп to same base also in the "West Elevation." 'dh Similarly, (d) equals the tangent of inclination of 12 to the Əh ди horizontal base in the "North Elevation," while is the tangent of inclination of line 1W to same base also in the "North Elevation.' 162. Constraining Relation between Three Variables.-We have above considered the ordinates n and w to any point of the surface as mutually independent of each other, and has dependent upon both n and w. But we may equally well consider w a function dependent on both n and h, while looking on n and h as mutually independent of each other. Generally between the three functions n, w, and h there is only one restrictive relational law established, leaving one degree of freedom of variation. among the three. If a second restrictive law be imposed upon the relations between the three, this means that we are restricted to some particular path, such as s, over the surface, and are no longer free to take points all over the surface. 163. Equation of Contours.-The meridional section is such a restricted path; the restriction being Sw=0. The parallel of latitude is another such restricted path; the restriction in this case being in= 0. A level contour line is a third example of such a restricted path, the restriction being 80. Therefore, if the path s be a contour line, we have dh ds = 0, and thus one form of |