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SECTION 5.- Investigation of the critical values of a double

definite integral. 361.] It only remains for us now to investigate the conditions of the critical values of a definite double integral, of which the variation has been calculated in Art. 311. On referring to equation (95), Art. 311, it appears that the expression for du consists of three parts; viz., two partially integrated terms whose value depends on the values which w and its derived-functions have at the limits which are assigned by the given limiting equation; and the third term, which is wholly unintegrated, and cannot be reduced unless w receives a determinate value. Now let

dr dz,, d27" daz' daz
2- dx - + dva + dedy + dyz =

m = n., (145) Then as du = 0, by reason of u having a critical value, it follows that e=0; and from this differential equation the required function is to be determined. (145) is plainly a partial differential equation of the fourth order ; the general integral of which is in most cases beyond the present powers of the integral calculus : we can in many cases however deduce from it some geometrical property which is sufficient to define the required surface.

362.] To find the surface the portion of which enclosed by a given curve has a minimum area.

In this problem the limits of integration are given by the given curve: and

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and on comparing this with (94), Art. 413, Vol. I, it is seen that the two are identical; and therefore the geometrical interpretation is, “The surface of minimum area is such that the sum of the reciprocals of its principal radii of curvature at every point vanishes :" bence we infer that the principal radii of curvature at every point are equal and of opposite signs.

363.] Let the problem be “ To determine the form of the surface which being of given extent, and terminated by a given curve, includes the greatest volume between it, the plane of (x, y), and the right cylinder whose director is the projection of the given curve on the plane of (x, y):" in this case

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and thus the equation n = 0 becomes

dz dz,

" = 0;

- dx dy whence by development, as in the preceding example, we have

dazu, dx2)

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and to interpret this geometrically ; let P, and p2 be the principal radii of curvature at any point on a surface; then by equation (27), Art. 399, Vol. I, we have 1 1 U? (K + L) + v2(L+1) +w2 (H + K).

(147) Pi P2

(u2 + 12 + wa) i and if these symbols are expressed in terms of the derived-functions of z, it will be seen by comparison with (146) that

1 1 1

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and therefore the surface which under a given superficial area contains the greatest volume is such, that the sum of its prin. cipal curvatures at every point is constant: and this result is usually expressed as “ The mean curvature is the same at every point of the surface.”

The equations (146) and (147) have never yet been directly integrated, but Mr. Jellet has in Liouville's Journal* shewn indirectly that the sphere is the only surface which satisfies them.

* Tome XVIII, p. 163, 1853.








SECTION 1.-General considerations on Differential Equations.

364.] EXPRESSIONS or equations which involve differentials, and the variables of which they are the differentials, are called differential expressions, and are distinguished from finite expressions, inasmuch as the latter contain only finite quantities. In the preceding parts of this volume differential expressions have been the subject-matter, but in that restricted form wherein the element-function of an integral is the function of only one vari. able for that integration. A more general case however is that in which a differential expression involves many variables, and the determination of the corresponding finite expression from that given differential expression is the problem. The process by which such differential expressions are derived from the finite equation has been explained, and fully illustrated in Section 7, Chap. III, Vol. I; and it is the inverse process of that of that section which we have here to develop. The formation or the genesis of differential equations, containing both total and partial differential coefficients, has been so largely explained, that it is unnecessary to say more on the subject; but the student is recommended to study that section, that he may ascertain not only the conditions


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of the problem but also the form of the solution in the most general case.

It is convenient to classify differential equations. They may contain either total differentials or derived-functions; or they may contain partial derived-functions. In the former case they will be of either of the following forms;

F(x, y, ... dx, dy, ... dar, dy, ... dmx, d"y, ... ) = 0, (1)


r(x, y, die eine ) = 0; (2) when they are called total differential equations. Or they may be of the forms, »{,9, 2, (), (), ale), le in), (... } = 0, (3) = {, ......... m), le ...} = 0; (4)

in which case they are called partial differential equations. On referring to Vol. I, Chap. III, Section 7, it will be observed that equations of the former class arise from the elimination of constants and determinate functions; whereas the latter arise from the elimination of arbitrary and undetermined functions. Much more however will be said on this subject hereafter.

These equations also require to be classified on other principles: (1) on the order of the highest differential or derived-function which is involved; and (2) on the degree or index to which the highest differential or derived-function is raised ; thus order is predicated of a differential equation as to the former, degree as to the latter; and if a differential equation contains x, y, dx, dy, dèx, dạy, or x, y, ay, it is said to be of the second order; and if the highest differentials or derived-functions enter in only linear forms, or to the first power, such an expression is said to

dy 3 be of the first degree; but an equation containing x, y,

d2v C ) is of the second order and of the second degree; and so of other similar expressions.

365.] As in Chap. III, Vol. I, the subject is considered from only an analytical point of view, let us also examine the geometrical aspect of it. And to fix our thoughts let us first take a

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total differential equation of the first order and first degree, and suppose it to be in the form

- f (x, y);

(5) let x and y be the rectangular coordinates of a plane curve; and let , be the angle between the axis of x and the tangent to the curve at the point (x, y): so that tan r = = f(x,y). As x and y are general in (5), let Xo, yo be particular, although arbitrarily chosen, values of x and y; and let to be the corresponding value of r: so that

tanto = f (x, yo); and through the point (xo, Yo) let a line be drawn cutting the axis of x at the angle To On this line let there be taken a point (x1, y1) contiguous to (xo, Y.), and through it let a line be drawn cutting the axis of x at an angle 11, so that

tan T, = f(x1, ); on this line let there be taken a point (X2, Yz) contiguous to (oxy, y1), and through (X2, yz) let a line be drawn making an angle T, with the axis of x, where

tan Tg = f (x2, Y2): and let a similar process be repeated n times, until at last we arrive at the point (Xn, Yn); hereby we shall have formed a series of short lines inclined to each other at different angles, and abutting at the points (%o, yo) and (x, yn). Let now every two successive points be infinitesimally near to each other, and also let the number of times that the process is repeated be infinite; then the distance between the extreme points is still finite, and the broken line which joins them becomes a continuous curve, and the distances between each two successive points become arc-elements of the curve: and hereby the curve between the two points will have been constructed from the given differential equation. Now from the process thus conducted it is manifest that the position of each point of the curve depends on that of the immediately preceding point, the law of dependence being given by the differential equation (5); the nature of the curve therefore is given by the differential equation : but it is also equally manifest that the position of every point, and so of the curve, depends on that of the first assumed point, viz., on (xo, Y.), and the position of this point is arbitrary: although therefore the

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