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.

OBLIQUE AND RECTANGULAR COORDINATES
Choose any convenient axes OX, OY, OZ, in space, and let
(x,y,z) be the coordinates of the moving point in its position P,
at time t, and (x',y',z') its coordinates in the position P' at time t'.
Thus, x,y,z, will be given functions of t, and x',y',z', will be the
same functions of t'.
If we make the velocity PS the diagonal of a parallelopiped
whose edges are parallel to the axes, those edges, from P as initial
point, are called the components of PS, the one parallel to OX
being the x-component, etc. These components are expressed by
the positive or negative numbers which measure them (positive
when in the directions of the axes). Conversely, given the components of a velocity, construct them in their lengths and directions
parallel to the axes, from the point of application P, and complete
the parallelopiped with the constructed lines as edges; then its
diagonal, PS, represents the velocity determined by them.
With rectangular axes the magnitude of the velocity is the
square root of the sum of the squares of its components.
The x-component of the resultant of two velocities is the sum
of the x-components of the velocities, and similarly for the other
components.
If two velocities are parallel, then corresponding components
are proportional, and in the ratio of the velocities (opposite
velocities in a negative ratio).
These are simply geometrical laws of displacements, and hold
for all quantities which are properly represented by displacements,
as velocities, accelerations, etc. A line representing any such
quantities is called a. vector.
We proceed to find the components of the velocity and acceleration of a moving point, in terms of the functions, x, y, z,
which fix its position at any time t, from their definitions as limits
of average values.
Since [x,y,z] are the components of the displacement
OP, and [x',y',z], of OP', the components of PP' will be
[x'-x, y'-y, z'-z], and the components of average velocity PS',
if 1/(t' —t)-n, will be [n(x'-x),n(y'-y), n(z'-z)]. Hence:
(5) The components of the velocity at time t are [Dx, Dy, Dz].
These components are functions of t which we denote for the
moment by [u,v,w], and by [u',v',w'], at time t', (respectively the
components of OH, OH', fig. 1). Thus, the components of HIF
are [u'-u, v'-v, w'-w], and those of the average acceleration
lIT', are [n(u'-u), n(v-v, n(w'-vw) ], where n=l/(t'-t);
and consequently their limits are [Du,Dv,Dw] the components

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