Analytical dynamics, being a synopsis of leading topics in the analytical theory of dynamics with numerous examples and selections from Newton's Principia and other sources, by Arthur S. Hathaway.

sponding positive units OL,.. OM,....O'..., i. e. such that OB=
V.OL, OH=v.OM, OH'-=v'.OM'. This gives a curve LMM'
of unit radius, whose arc measures from L as origin are, arc
LM = —, arc LM'=-'.
These are the radian measures of the angles through which the
tangent to the path has turned, in the motion of its point of contact from A to P and to P'.
The curve LMM' (on a sphere of. unit radius and center 0),
may be called the angular directrix of the path, and ~, the angular
direction of the path at P.
The angular velocity of the direction of motion PS, is the
velocity of the corresponding point M on its angular directrix; and
is represented at time t by a tangent MN at M, whose numerical
measure is dq/dt, the angular speed. (Remember that the treatment of velocity and speed on this curve is the same as on any
other, p being the arc distance of the moving point at time t.)
Draw on fig. 1, HH" parallel to MM' to intersect OH' in H".
Then,HHT' is the resultant of HH" and H"H', i.e. of v.MM' and
(v'-r) OM'. Hence, dividing by t'-t, to obtain HT' and its components, and thence, their limits,HT, and vMN, Dv.OM, the latter
must be KT, HK, respectively, or,
HK=-Dv.OMl, KT=vMN.
Remembering that the measure of OM is unity, and of MN,
d4/dt, we have:
(2) Tangential acceleration=f:=Dv-dv/dt.
(3) Normal acceleration =vdc/dt =v2do/ds
-v2/R, where R-ds/dc, the radius of curvature (PC) of the
path. (Calc. Art. 83.)
Note. The above is differentiation of the velocity OH, D being
a special differential symbol whose multiplying factor is n=1/(t' —t), so that Dtlim.n (t'-t) 1.
Thus let, OM —u, and denote resultant addition and subtraction by + and -.
Then,
D (vu) — [T=.n (v'u-' vu) =i —m.n (v'u' —vu' +vu' ---vu)
-lim. [n (v'-v).u'+nv (u'-u) ] — Dv.u+vDu.
and Du-=MN, the angular velocity.
MOTIONS IRRESPIECTIVE OF PATH
The character of a motion, irrespective of path, is determined
by the function which s is of t. This function may be given explicitly, or implicitly by sufficient equations between s and t and
other variables for the explicit solution of s in terms of t. From

5