Page:Somerville Mechanism of the heavens.djvu/91

consequently,

$\scriptstyle {\left\{\left({\frac {\delta r}{\delta x}}\right)^{2}+\left({\frac {\delta r}{\delta y}}\right)^{2}+\left({\frac {\delta r}{\delta z}}\right)^{2}\right\}=1,}$

on account of which, the equation

$\scriptstyle {\delta r={\text{N}}\delta u}{\text{ gives }}{\text{N}}^{2}\left\{\left({\frac {\delta u}{\delta r}}\right)^{2}+\left({\frac {\delta u}{\delta y}}\right)^{2}+\left({\frac {\delta u}{\delta z}}\right)^{2}\right\}=1,$

or

$\scriptstyle {{\text{N.}}={\frac {1}{\sqrt {\left({\frac {\delta u}{\delta x}}\right)^{2}+\left({\frac {\delta u}{\delta y}}\right)^{2}+\left({\frac {\delta u}{\delta z}}\right)^{2}}}},}$

for u is a function of x, y, z; hence,

${\begin{aligned}\scriptstyle {{\text{R}}_{'}\delta r}&\scriptstyle {={\frac {{\text{R}}_{'}\delta u}{\sqrt {\left({\frac {\delta u}{\delta x}}\right)^{2}+\left({\frac {\delta u}{\delta y}}\right)+\left({\frac {\delta u}{\delta z}}\right)^{2}}}}{\text{;}}}&\scriptstyle {\text{ and if}}\\\scriptstyle {\lambda }&\scriptstyle {={\frac {{\text{R}}_{'}}{\sqrt {\left({\frac {\delta u}{\delta x}}\right)^{2}+\left({\frac {\delta u}{\delta y}}\right)^{2}+\left({\frac {\delta u}{\delta z}}\right)^{2}}}},}\end{aligned}}$

then $\scriptstyle {{\text{R}}_{'}\delta r}$ becomes λδu, and the equation of the equilibrium of a particle m, on a curved line or surface, is

Xδx+Yδy+Zδz+λδu=0(4),

where δu is a function of the elements δx, δy, δz: and as this equation exists whatever these elements may be, each of them may be made zero, which will divide it into three equations; but they will be reduced to two by the elimination of λ. And these two, with the equation of the surface u=0, will suffice to determine x, y, z, the co-ordinates of m in its position of equilibrium. These found, N and consequently λ become known. And since $\scriptstyle {{\text{R}}_{'}}$ is the resistance

$\scriptstyle {{\text{R}}_{'}=\lambda {\sqrt {\left({\frac {\delta u}{\delta x}}\right)^{2}+\left({\frac {\delta u}{\delta y}}\right)^{2}+\left({\frac {\delta u}{\delta z}}\right)^{2}}}}$

is the pressure, which is equal and contrary to the resistance, and is therefore determined.

50. Thus if a particle of matter, either free or obliged to remain on a curved line or surface, be urged by any number of forces, it will continue in equilibrio, if the sum of the products of each force by the element of its direction be zero.