Page:Somerville Mechanism of the heavens.djvu/100

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Chap II.]

VARIABLE MOTION.

24

by δs. And since the single force $\scriptstyle {{\text{R}}_{'}}$ is resolved into X', Y', Z', we must have

X'δx + Y'δy + Z'δz = $\scriptstyle {{\text{R}}_{'}}$ δs;

so that the preceding equation becomes

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

+ $\scriptstyle {{\text{R}}_{'}}$ δs - λδu(8)

and this is true whatever λ may be.

But λ being thus left arbitrary, we are at liberty to determine it by any convenient condition. Let this condition be $\scriptstyle {{\text{R}}_{'}}$ δs — λδu = 0, or λ = $\scriptstyle {{\text{R}}_{'}}$ . δs/δu, which reduces equation (8) to equation (6). So when X, Y, Z, are the only acting forces explicitly given, this equation still suffices to resolve the problem, provided it be taken in conjunction with the equation δu = 0, or, which is the same thing,

pδx + qδy + rδz = 0.

which establishes a relation between δx, δy, δz,

Now let the condition λ = s . δs/δu be considered which determines λ.

Since $\scriptstyle {{\text{R}}_{'}}$ is the resultant of the forces X', Y', Z', its magnitude must be represented by $\scriptstyle {\sqrt {{\text{X}}'^{2}+{\text{Y}}'^{2}+{\text{Z}}'^{2}}}$ by article 37, and since $\scriptstyle {{\text{R}}_{'}}$ = λδu, or

X'δx + Y'δy + Z'δz = λdu/dxδx + λdu/dyδy + λdu/dzδz,

therefore, in order that dx, dy, dz, may remain arbitrary, we must have

X' = λdu/dxδx; Y' = λdu/dyδy; Z' = λdu/dzδz;

and consequently

$\scriptstyle {\text{R}}'={\sqrt {{\text{X}}'^{2}+{\text{Y}}'^{2}+{\text{Z}}'^{2}}}=\lambda .{\sqrt {\left({\frac {du}{dx}}\right)^{2}+\left({\frac {du}{dy}}\right)^{2}+\left({\frac {du}{dz}}\right)^{2}}}$ (9)

and

$\scriptstyle \lambda ={\frac {{\text{R}}'}{\sqrt {\left({\frac {du}{dx}}\right)^{2}+\left({\frac {du}{dy}}\right)^{2}+\left({\frac {du}{dz}}\right)^{2}}}}$

and if to abridge

$\scriptstyle {\frac {1}{\sqrt {\left({\frac {du}{dx}}\right)^{2}+\left({\frac {du}{dy}}\right)^{2}+\left({\frac {du}{dz}}\right)^{2}}}}={\text{K}};$

then if α, β, γ, be the angles that the normal to the curve or surface makes with the co-ordinates,

Kdu/dx = cos α, Kdu/dyδy = cos β, Kdu/dzδz = cos γ,

and

X’ = $\scriptstyle {{\text{R}}_{'}}$ . cos α, Y' = $\scriptstyle {{\text{R}}_{'}}$ . cos β, Z' = $\scriptstyle {{\text{R}}_{'}}$ . cos γ.

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