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

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î = F(x, dx, d2x,... dx, y, dy, d2y,... dmy, z, dz, d2z,..... dz). Let us first substitute as follows:

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+ Yoy + Y2 d.oy + Y2 d2.dy + ... + Ym dm.dy

+zdz +Z2 d.dz+Zq d2.dz + ... + Z2d*.8z} ; (61)

and reducing these terms by partial integration, we have finally,

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When involves more than three independent variables, the

1

expression for the variation of fr

0

a is of course similar.

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

L = ƒ {x, dx, d2x,... y, dy, d2y,... z, dz, d2z,...} = 0. (63) If L involves only x, y, z, z may be expressed in terms of x and y, and thence dz, d2z,... may be found, and substituted in 2, so that 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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and employing a convenient and abbreviating notation,

OL = εdx+1d.òx + §1⁄2Ã2òx + ....

dL=

+noy+n1d.dy+n2 d2dy + ...

+58z+51d.8z+51⁄2d2dz + ...=0.

(65)

Now since the equation L = 0 must be satisfied for all values of x, y, z which are admissible into the problem, therefore the variation of x, y, z must be subject to the condition 8L = 0, that is, to equation (65); but since

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it is plain that we may add to it the right-hand member of (65) multiplied by an undetermined constant A, without destroying the truth of the expressions; so that

dv = (x+λ§) dx + (x1+λ§1) d.òx + (x2+\§2) d2dx + ...

+ (Υ + λη) δη + (Υ, + λη)d.δy + (Yg + λη,) doy +....

2

(67)

+ (z+λ8)dz + (Z1 +λ51) d.dz + (Z2 +λ52) d2dz +...... Observing now the process by which (62) was deduced from

(61), a result similar to (62) will be deduced from (67), wherein instead of x will be x+λ, instead of x1, X1+λ§1,... instead of y, Y+Aŋ,..., and so on for the others; and as λ is undetermined, consider the variation to involve three independent quan

we may

tities.

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

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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,

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+z(")dz(");

+z8z+z′dz′+z′′dz′′ +...+z(") dz("); (68)

and following a process precisely similar to that of Art. 303, and

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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 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

vôx]'is

the normal displacements only; and that [v da'is the only term

wherein the tangential displacement appears.

If v contains any number of undetermined functions, the vari

ation of

0

v de 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 z 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 A, and adding it

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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

dr

such that in the place of y will be y+λ (a); in the place of Y',

dy

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to the form of a definite integral, whose element-function involves x and two unknown and independent functions of æ.

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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