Page:Somerville Mechanism of the heavens.djvu/121

or, eliminating ${\displaystyle dt}$ by means of the preceding integral, and making

it becomes

The integral of this equation will give ${\displaystyle u}$ in functions of ${\displaystyle x}$, and when substituted in

it will furnish a new equation of the first order between ${\displaystyle z,x,}$ and ${\displaystyle t}$, which will be the differential equation of the trajectory.

If the resistance of the medium be zero, ${\displaystyle h=0}$, and the preceding equation gives

and substituting ${\displaystyle {\frac {dz}{dx}}}$ for ${\displaystyle u}$, and integrating again

${\displaystyle b}$ and ${\displaystyle b'}$ being arbitrary constant quantities. This is the equation to a parabola whose axis is vertical, which is the curve a projectile would describe in vacuo. When

and as the second differential of the preceding integral gives

therefore

If the particle begins to move from the origin of the co-ordinates, the time as well as ${\displaystyle x,y,z}$, are estimated from that point; hence ${\displaystyle b'}$ and ${\displaystyle a'}$ are zero, and the two equations of motion become

${\displaystyle z=ax^{2}+bx}$; and ${\displaystyle t=x{\sqrt {\frac {2a}{g}}}}$;

whence

${\displaystyle z=g{\frac {t^{2}}{2}}+tb{\sqrt {\frac {g}{2a}}}}$.