Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

138 FIGURE OF THE EARTH
portional to the length of the included column multiplied
by the accelerating force that acts on it, and may therefore
be represented by Pp. cp*. Let u be the whole pressure;
since, upon increasing the distance by Îp, the pressure is
diminished by Pp. p, we have
u =-Pp;. u = C- --
dp 2
PbBut the pressure at the surface =o, or C - o;
P. = (b2 - 2);
Pb2
hence, the pressure at the center -. Similarly the
pressure at the center, produced by the equatoreal column,
Q-r r rr r
* The accelerating force is represented by the acceleration which it would
cause in the matter upon which it acts, if that matter were allowed to move freely
under its action: it is measured by the velocity in feet per second which its action
during one second would cause: and cannot be correctly represented in any other
way.
The pressure caused by a quantity of fluid or other matter under the action
of an accelerating force is a pressure of the same kind as that which we commonly
call weight. It is connected with the accelerating force by means of the third law
of motion, which teaches that the pressure corresponding to a certain acceleration
of a certain quantity of matter is proportional to the product of the acceleration
by the quantity of matter.
If we estimate the quantity of matter by the weight in pounds of that matter
at a certain point of the Earth's surface, under the action of gravity at that place,
and if we put g for the acceleration which gravity would produce in 1" at the
same place; then we mean by this that the pressure W estimated in pounds
corresponds to the accelerating force g acting on the mass TW. Substituting
these in the general theorem,
pressure = C x acceleration x quantity of matter,
we have W=Cxgx g W,
and therefore C = -. Hence in any other instance of accelerating force,
accelerating force
pressure in pounds = x weight in pounds,
where the weight in pounds and the value of g are both referred to the same
locality.
We shall frequently omit the divisor g in mercly representing by a proportional
quantity the pressure which accelerating forces cause.