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their greatest generality let us suppose all the variables and differentials to receive variations. Let

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When a involves more than three independent variables, the expression for the variation of / 2 is of course similar.

307.] Suppose however that an equation of relation is given between the variables and their differentials which are involved in n; and, to fix our thoughts, let us take the case of three variables X, y, ; and suppose the equation to be

l=f{x,dx, dx, ... y, dy, dạy, ... 2, dz, dz, ...} = 0. (63) If l involves only x, y, z, z may be expressed in terms of x and y, and thence dz, dz, ... may be found, and substituted in , so that a will become a function of only two variables, x and y: but as l involves the differentials of the variables, such an elimination is generally impossible, and we are obliged to have recourse to the following process. Take the variation of l, viz.

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(61), a result similar to (62) will be deduced from (67), wherein instead of x will be x + 1€, instead of x,,X, +1&v ... instead of y, Y+in, ..., and so on for the others; and as a is undetermined, we may consider the variation to involve three independent quantities.

The variation will also be found in a similar manner if the original element-function involves more variables, and if these are related to each other by many equations of condition.

308.] Suppose however that the element involves three variables x, y, z; that x is equicrescent, and that y and z are two unknown functions of x, and independent of each other, and that the quantity whose variation is to be calculated is

ri
u= / v dx,

dy dạy dmy dz daz d"z) wherev=P(x, y, da dr ,... dam, z, Ta' d. 22).. dxn), f being a known function.

To give v the most general variation, let us suppose that not only x, y, z, but that also the derived functions of y and z vary: then, adopting the following substitutions,

da y el

9 = y(); daca

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+32 - dz + -...(-)"0."}«, dx ; (69) an expression consisting of two parts; of which one involves the values of the variables and their variations at the limits only; and the other involves a process of integration, and which cannot be performed unless the relations between x and y and z are given. These several parts admit of explanation similar to that which has been given of (56) in Art. 304.

Also it admits of a geometrical explanation similar to that of Art. 304; the relations between x and y and between x and 2 represent two cylinders which are perpendicular to the planes of (x, y), and of (x,z) respectively; and these by their intersection define a curve in space.

Let us consider the general displacement of a point on this curve to be due, (1) to two displacements perpendicular to each other in the normal plane, and (2) to one along the tangent line; now by a process exactly analogous to that of Art. 304 it may be

shewn that the quantities under the signs of integration involve the normal displacements only; and that vox is the only term wherein the tangential displacement appears.

If v contains any number of undetermined functions, the variation of v dx will be 'calculated in a similar manner, and will consist of a series of terms and quantities similar to those of equation (69).

309.] In the last article y and are considered to be independent of each other; if a relation is given connecting them and their derived-functions and x, and of the form

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multiplying which by an undetermined constant à, and adding it to dv, we have

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comparing which expression with that of (68), and noticing the process by which (69) is deduced from (68), it is palpable that (72) will lead to a result of the form given in (69), and with quantities such that in the place of y will be y+ 1); in the place of Y', Y+ (),...; in the place of 2, 2+1 (45); in the place of 2', z' + c ),...; and so on : and thus the variation will be reduced to the form of a definite integral, whose element-function involves x and two unknown and independent functions of x.

310.] Certain processes in the sequel will require the calculation of the variation of a variation, that is, of the second variation of a definite integral. As the principles and the method are the same as those explained and applied in the preceding articles, I

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