Page:Somerville Mechanism of the heavens.djvu/99

u is a function of x, y, z, X, Y, Z, and t. Therefore the equation u = 0; establishes the existence of a relation

δu = pδx + qδy + rδz = 0

between the variations δx, δy, δz, which can no longer be regarded as arbitrary; but the equation (6) subsists whether they be so or not, and may therefore be used simultaneously with δu = 0 to eliminate one; after which the other two being really arbitrary, their co-efficients must be separately zero.

In the second case; if we do not regard the forces arising from the conditions of constraint as involved in X, Y, Z, let δu = 0 be that condition, and let X', Y', Z', be the unknown forces brought into action by that condition, by which the action of X, Y, Z, is modified; then will the whole forces acting on m be X + X', Y + Y', Z + Z', and under the influence of these the particle will move as a free particle; and therefore δx, δy, δz, being any variations

$\scriptstyle 0=\left({{\text{X + X}}'-{\frac {{d^{2}}{x}}{d{t^{2}}}}}\right)\delta {x}+\left({{\text{Y + Y}}'-{\frac {{d^{2}}{y}}{d{t^{2}}}}}\right)\delta {y}+\left({{\text{Z + Z}}'-{\frac {{d^{2}}{z}}{d{t^{2}}}}}\right)\delta {z}$, (6),

or,

$\scriptstyle 0=\left({{\text{X}}-{\frac {{d^{2}}{x}}{d{t^{2}}}}}\right)\delta {x}+\left({{\text{Y}}-{\frac {{d^{2}}{y}}{d{t^{2}}}}}\right)\delta {y}+\left({{\text{Z}}-{\frac {{d^{2}}{z}}{d{t^{2}}}}}\right)\delta {z}$ (7),

+ X'δx + Y'δy + Z’δz;

and this equation is independent of any particular relation between δx, δy, δz, and holds good whether they subsist or not. But the condition δx = 0 establishes a relation of the form pδx + qδy + rδz = 0,

$\scriptstyle p=\left({\frac {d{u}}{d{x}}}\right),\left({\frac {d{u}}{d{y}}}\right),\left({\frac {d{u}}{d{z}}}\right)$

and since this is true, it is so when multiplied by any arbitrary quantity λ; therefore,

λ (pδx + qδy + rδz) = 0, or λδu = 0;

because

δu = pδx + qδy + rδz = 0.

If this be added to equation (7), it becomes

$\scriptstyle 0=\left({{\text{X}}-{\frac {{d^{2}}{x}}{d{t^{2}}}}}\right)\delta {x}+\left({{\text{Y}}-{\frac {{d^{2}}{y}}{d{t^{2}}}}}\right)\delta {y}+\left({{\text{Z}}-{\frac {{d^{2}}{z}}{d{t^{2}}}}}\right)\delta {z}$ (7),

+ X'δx + Y'δy + Z’δz - λδu,

which is true whatever δx, δy, δz, or λ may be.

Now since X', Y', Z', are forces acting in the direction x, y, z, (though unknown) they may be compounded into one resultant $\scriptstyle {{\text{R}}_{'}}$, which must have one direction, whose element may be represented