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therefore neglecting infinitesimals of the higher orders,

d.8x, d.dy



d.8y, cos a = a + dy

cos ß




dz And as all the quantities in the right-hand members of these last three equations are infinitesimals of an order higher than that of those retained in (31), it is evident that a, b, and y are approximately right angles; and that the volume-element in its displaced position is approximately a rectangular parallelepipedon.

· Hence it appears that if ox, oy, öz are functions of only x, y, z respectively, we shall have rigorously cos a = cos ß = cos y = 0; so that the volume-element is a rectangular parallelepipedon in both its original and displaced states. Thus, the variation of the volume-element is, to the first order of infinitesimals, as general in this restricted case of the variations of the coordinates, as it is in the general case where 8x, dy, dz are arbitrary functions of i, y, and z.

300.] Finally, let us take the general case, and investigate the variation of a product of differentials of the form dx, dx, dxg... dxm, X1, X2, Xg, ... In being n variables independent of each other.

Let &1, $2, &g, ... be the values of X1, X2, Xg, ... in their varied states; so that $1 = x1 +8X1,

$2 = , +8X2,
. . . . . .

En = xo +8Xn; where 8X1, 8x2, ... are functions of X1, X3, Xg, ... : then we have

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so that, by the method of Art. 211, and neglecting infinitesimals of the higher orders, we have

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2.002 t..

.:: 8.dx, dx, d&g ... = ?

dx, dx, dxg... (40)

(dxy dxn and this gives the variation of the multiple infinitesimal element.

301.] I come now to the investigation of the variation of definite integrals, in which the element-functions involve finite variables, their differentials, and differential coefficients. Here the processes will be longer ; but by a judicious employment of integration, the final results assume a practicable form.

Now the value of a definite integral may be varied in many ways; (1) by a change of form in the element-function; this is a variation which falls entirely within the scope of this Calculus : (2) by a change in the limits; and this may occur in two ways; either the limits may be assigned by certain determinate functions; in which case a change of them involves a passage from one value to another in accordance with certain given conditions, as along a given curve or a given surface; and although the variations of the variables at these limits will have to be determined, yet at those points they will not be arbitrary, but of the nature of differentials and not of variations; or the functions which assign the limits may be undetermined, and thus subject to variation; and thus there will be variations of these limits due to a change of form in the functions which assign them. In all these several cases the total variation will be the sum of the several partial variations which are due to the changes as they occur in particular problems; and these partial variations will be estimated separately.

In these problems it appears to me most desirable to maintain as far as possible symmetry of notation and symmetry of expression; for a large amount of labour, both of brain and hand, is thereby saved. With this object in view I have at first considered all the variables to be equally dependent and equally subject to variation, and have made none equicrescent; the rationale of this process being that each variable is supposed to

be a function of another variable which is not involved in the expressions either explicitly or implicitly. The process is applicable with especial advantage, as the sequel will shew, when the number of variables is small. In many cases however it leads to long expressions which are capable of abbreviation, if some hypothesis is made as to the character of one or more of the variables or of their variations, as to equicrescence and variation, and the generality of the problem is not thereby abridged. Thus, suppose the element-function of the definite integral to contain two variables x and y, the functional relation between which is not fixed; then, for the variation of the element-function apart from its values at the limits, so long as the variation of one of the variables, say y, is arbitrary and indeterminate, the generality of the whole variation is not abridged if x is not subject to variation, or if a particular value, or a series of particular values, is given to dx ; apart from the limits, I say; because at the limits a relation may be given between x and y, and this relation taken in connection with the general functional relation may absolutely determine the values of 8x and of dy at the limits. This is also evident geometrically. Each point in a plane curve may be displaced in a direction parallel to the axis of y, and thus Ox =0. Thus in fig. 47, P may be shifted to R; in which case if all the points do not move through equal spaces, but through spaces which are functions of the coordinates of the point in its original position, the form of the equation to the curve will change, although the point has the same abscissa in both its positions. If however the extreme points P, and P, are constrained to move on given curves, at the limits generally x and y must both vary, and consistently with the equations to the limiting curves. In many cases in the sequel I shall take one or more of the variables to be equicrescent, or assume that they have no variation, or take the variation of a variable to be a function of that variable only, or make some other hypothesis, as far as it is applicable, which will shorten the operations, and not abridge the generality.

302.] In the first place let us consider a function of two variables x and y, which are connected with each other by a functional symbol, which is not known; and suppose the element to be

F(x, dx, dạx, ...d"x, y, dy, dy, ... dmy), where F expresses a known function; let u represent the sum of these elements, between the limits X1, y, and xo, Yo; so that

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the order of the symbols of operation having been changed in accordance with the commutative law established in Art. 295. But

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and similar results are of course true for the integration of the Y's; therefore, substituting in (49),

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