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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
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% Trinity College, 2000.

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\centerline{\Largebf ON THE APPLICATION OF THE}

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\centerline{\Largebf METHOD OF QUATERNIONS TO SOME}

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\centerline{\Largebf DYNAMICAL QUESTIONS}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Proceedings of the Royal Irish Academy,
   3 (1847), Appendix, pp.\ xxxvi--l.)}

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\centerline{\largerm Edited by David R. Wilkins}

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\centerline{\largerm 2000}

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\centerline{{\largeit On the Application of the Method of Quaternions
to some}}

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\centerline{{\largeit Dynamical Questions.}}

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\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}

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\centerline{Communicated July~14 and July~21, 1845.}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~3 (1847), Appendix, pp.\ xxxvi--l.]}

\bigskip

The following is the substance of the communications made to the
Academy by Sir William Hamilton, on the application of the method
of Quaternions to some dynamical questions:

The author stated that, during a visit which he had lately made
to England, Sir John Herschel suggested to him that the internal
character (if it may be so called) of the method of quaternions,
or of vectors, as applied to algebraic geometry,---that character
by which it is independent of any foreign and arbitrary axes of
coordinates,---might make it useful in researches respecting the
attractions of a system of bodies.  A beginning of such a
research had been made by Sir William Hamilton in October, 1844,
which went so far, but only so far, as the deducing of the
constancy of the plane of an orbit, and the equable description
of areas, under one common formula, namely, the following:
$$    \rho {{\rm d} \rho \over {\rm d} t}
       - {{\rm d} \rho \over {\rm d} t} \rho
   =  \hbox{const.},$$
from the general expression of a central force, namely, from the
equation
$$    \rho {{\rm d}^2 \rho \over {\rm d} t^2}
       - {{\rm d}^2 \rho \over {\rm d} t^2} \rho
   =  0,$$
which asserts merely the {\it coaxality\/} of the vector~$\rho$
and the force
$\displaystyle {{\rm d}^2 \rho \over {\rm d} t^2}$,
or the existence of one common line along which this vector and
this force are (similarly or oppositely) directed.

Since the suggestion above acknowledged was made, Sir William
Hamilton has proposed to himself to express by an equation, on
the principles of the method of vectors, the problem of any
number of bodies attracting according to Newton's law: and has
arrived at the formula
$${{\rm d}^2 \alpha \over {\rm d} t^2}
   =  \Sigma
      {m + \Delta m
         \over - \Delta \alpha \surd (- \Delta \alpha^2 )};
   \eqno ({\sc a})$$
which may also be thus written,
$${{\rm d}^2 \alpha \over {\rm d} t^2}
   =  \Sigma
      {m' \over (\alpha - \alpha') \surd \{ - (\alpha - \alpha')^2 \}};
   \eqno ({\sc b})$$
and from which he has deduced anew the known laws of the centre
of gravity, of areas, and of the {\it vis viva}, under the forms:
$$\left( {{\rm d} \over {\rm d} t} \right)^2
      \Sigma \mathbin{.} m \alpha = 0;
   \eqno ({\sc c})$$
$${{\rm d} \over {\rm d} t} \Sigma \mathbin{.} m
      \left(
         \alpha {{\rm d} \alpha \over {\rm d} t}
       - {{\rm d} \alpha \over {\rm d} t} \alpha
      \right)
   =  0;
   \eqno ({\sc d})$$
$${{\rm d} \over {\rm d} t}
      \Sigma \mathbin{.} {m \over 2}
      \left( {{\rm d} \alpha \over {\rm d} t} \right)^2
    + {{\rm d} \over {\rm d} t}
      \Sigma \mathbin{.}
      {m m' \over \surd \{ - (\alpha - \alpha')^2 \}}
   =  0.
   \eqno ({\sc e})$$
$\alpha$ is the vector and $m$ the mass of one body; $\alpha'$
and $m'$ of another; $\Sigma$ sums for the system; $t$ is the
time, ${\rm d}$ the characteristic of differentiation; $\Delta$
(where used) is the mark of finite differencing.

To illustrate the method of treating equations of such forms as
these, let us consider briefly the problem of {\it two\/} bodies,
or of {\it one\/} body, as it presents itself, in the method of
quaternions, with Newton's law of attraction, coordinates being
not employed.  The differential equation may be thus written,
$${{\rm d}^2 \alpha \over {\rm d} t^2}
   =  {{\sc m} \over \alpha \surd (- \alpha^2)};
   \eqno (1)$$
$\alpha$ being the vector of the attracted body, drawn from the
attracting one; $t$ the time, ${\rm d}$ the mark of
differentiation; and ${\sc m}$ the attracting mass, or the sum of
the two such masses.  This equation gives
$$    \alpha {{\rm d}^2 \alpha \over {\rm d} t^2}
    - {{\rm d}^2 \alpha \over {\rm d} t^2} \alpha
   =  0;
   \eqno (2)$$
which expresses merely that the force is central; and gives by
integration a result already alluded to (as independent of that
function of the distance which enters into the law of
attraction), namely,
$$       {\alpha \over 2} {{\rm d} \alpha \over {\rm d} t}
       - {{\rm d} \alpha \over {\rm d} t} {\alpha \over 2}
   = \beta;\quad
  {\rm d} \beta = 0;
   \eqno (3)$$
the constant~$\beta$ being a new vector, perpendicular in
direction to the plane of the orbit, and in magnitude
representing the double of the areal velocity, which velocity is
thus seen to be constant, as also is the plane.  For we have at
once, by (3),
$$\alpha \beta + \beta \alpha = 0,
   \eqno (4)$$
implying that the variable vector~$\alpha$ is perpendicular to
the constant vector~$\beta$; and also
$$\int ( \alpha \mathbin{.} {\rm d} \alpha
       - {\rm d} \alpha \mathbin{.} \alpha )
   =  2 \beta (t - t_0),
   \eqno (5)$$
if $t_0$ be the value of $t$ at the commencement of the integral.

Make now, to distinguish between the length and direction of the
vector,
$$\alpha = r \iota,\quad
  r = \surd (- \alpha^2),\quad
  \iota^2 = -1;
   \eqno (6)$$
we shall have
$${\rm d} \alpha
   =  r \mathbin{.} {\rm d} \iota
       + {\rm d} r \mathbin{.} \iota,
   \eqno (7)$$
and because $r$ and ${\rm d} r$ are scalar (or real) quantities,
$$\alpha \mathbin{.} {\rm d} \alpha
   =  r^2 \iota \mathbin{.} {\rm d} \iota
       - r \mathbin{.} {\rm d} r,\quad
  {\rm d} \alpha \mathbin{.} \alpha
   =  r^2 \, {\rm d} \iota \mathbin{.} \iota
       - r \mathbin{.} {\rm d} r;
   \eqno (8)$$
therefore
$$\beta \mathbin{.} {\rm d} t
   =  {\textstyle {1 \over 2}}
      (  \alpha \mathbin{.} {\rm d} \alpha
       - {\rm d} \alpha \mathbin{.} \alpha )
   =  {r^2 \over 2}
      (  \iota \mathbin{.} {\rm d} \iota
       - {\rm d} \iota \mathbin{.} \iota )
   =  r^2 \iota \mathbin{.} {\rm d} \iota,
   \eqno (9)$$
observing that the equation
$$\iota^2 = -1
   \quad\hbox{gives}\quad
  \iota \mathbin{.} {\rm d} \iota
       + {\rm d} \iota \mathbin{.} \iota
   =  0.
   \eqno (10)$$

The fundamental equation~(1) of the problem becomes, by (6) and
(9),
$${{\rm d} \over {\rm d} t} {{\rm d} \alpha \over {\rm d} t}
   =  {{\sc m} \over r^2 \iota}
   =  {{\rm d} \iota \over dt} {{\sc m} \over \beta},
   \eqno (11)$$
(in which last member the order of the factors is not
indifferent), and therefore gives, by integration, since $\beta$
as well as ${\sc m}$ is constant,
$${{\rm d} \alpha \over {\rm d} t}
       - \iota {{\sc m} \over \beta}
   =  \hbox{const.};
   \eqno (12)$$
or, as we may also write it,
$$\iota - {{\rm d} \alpha \over {\rm d} t} {\beta \over {\sc m}}
   =  \epsilon,\quad
  {\rm d} \epsilon = 0.
   \eqno (13)$$

We have, consequently, by (6) and (4),
$$\alpha \epsilon
   =  - r
       - \alpha {{\rm d} \alpha \over {\rm d} t}
         {\beta \over {\sc m}},\quad
  \epsilon \alpha
   =  - r
       + {{\rm d} \alpha \over {\rm d} t} \alpha
         {\beta \over {\sc m}};
   \eqno (14)$$
and finally, by (3),
$$\alpha \epsilon + \epsilon \alpha + 2r
   =  2 p,
   \eqno (15)$$
if
$$p = {- \beta^2 \over {\sc m}},
   \eqno (16)$$
the constant~$p$ being here not only a scalar by an essentially
positive quantity, because the force is supposed to be
attractive, or ${\sc m} > 0$, while $\beta^2 < 0$.  The
equation~(15) thus obtained, contains the law of elliptic,
parabolic, or hyperbolic motion.  For if we make (by way of
comparison with known results),
$$\surd (- \epsilon^2) = e,
   \eqno (17)$$
and
$$(\alpha, - \epsilon) = v,
   \eqno (18)$$
$(\alpha, - \epsilon)$ denoting here the angle between the
directions of $\alpha$ and $\epsilon$, we have (by the
formula~(a) of the abstract of last November),
$$\alpha \epsilon + \epsilon \alpha
   = 2 e r \cos v;
   \eqno (19)$$
and therefore, by (15),
$$r = {p \over 1 + e \cos v},
   \eqno (20)$$
which is the known equation of a conic section referred to a
focus.  The Greek letters, throughout, represent vectors: and the
Italics, scalar quantities.

Suppose that we had no previous knowledge of the properties of
cosines or of conics, we might have proceeded thus to investigate
the nature of the locus represented by the equation~(15).  This
locus is a {\it surface of revolution\/} round the
line~$\epsilon$; because the differential of its equation being
$${\rm d} \alpha \mathbin{.} \epsilon
      + \epsilon \mathbin{.} {\rm d} \alpha
      + 2 \, {\rm d} r
   =  0
   \eqno (21)$$
if we cut it by a series of concentric spheres round the origin
of vectors, the sections are contained in a series of planes
perpendicular to $\epsilon$; since
$${\rm d} r = 0,
   \eqno (22)$$
which is the differential equation of the first series, gives, by
(21),
$${\rm d} \alpha \, \epsilon + \epsilon \, {\rm d} \alpha
   =  0,
   \eqno (23)$$
which is the differential equation of the second series.  To
study more closely this surface of revolution~(15), make
$$\alpha = \gamma + \alpha',
   \eqno (24)$$
$\gamma$ being an arbitrary constant, and $\alpha'$ a variable
vector; and since it must evidently give simpler and more
symmetric results to suppose the vector~$\gamma$ co-axal with
$\epsilon$, than to make the contrary supposition, since we
shall thus place the origin of the new vectors~$\alpha'$ upon the
axis of revolution of the surface, let
$$\epsilon \gamma - \gamma \epsilon = 0,
   \quad\hbox{or}\quad
  \gamma = g \epsilon,
   \eqno (25)$$
$g$ being an arbitrary scalar, to be disposed of according to
convenience.  Equations (24) and (25), combined with (6) and
(17), will give, for every point of space,
$$- \alpha'^2 = - (\alpha - \gamma)^2
   =  r^2 + g^2 \epsilon^2
       + g (\alpha \epsilon + \epsilon \alpha);
   \eqno (26)$$
and therefore, for every point of the locus (15),
$$- \alpha'^2 = r^2 - 2gr + g^2 \epsilon^2 + 2gp.
   \eqno (27)$$

The second member of this last equation may be made an exact
square, by assuming
$$g^2 e^2 + 2gp = g^2,
   \quad\hbox{that is,}\quad
  g = {2p \over 1 - e^2} = 2 a;
   \eqno (28)$$
the scalar quotient
$${p \over 1 - e^2} = a,
   \quad\hbox{or the transformation}\quad
  p = a (1 - e^2),
   \eqno (29)$$
being thus suggested to our attention; and with this value of $g$
we shall have, by (27),
$$- \alpha'^2 = (2a - r)^2,
   \eqno (30)$$
that is,
$$2a = \surd (- \alpha^2) \pm \surd (- \alpha'^2);
   \eqno (31)$$
so that either the sum or the difference of the distances of any
point on the locus~(15) from the two {\it foci\/} of which the
vectors are respectively $0$ and $2a \epsilon$, is equal to the
constant~$2a$.  It is not difficult to prove that the upper or
the lower sign is to be taken, in the formula~(31), according as
$e^2$ is $<$ or $> 1$.  For the case $e^2 = 1$, the recent
transformation fails.

Again, to find whether the locus has a {\it centre}, we may make
$$\alpha = \gamma' + \delta + g' \epsilon + \delta,
   \eqno (32)$$
$g'$ being a new disposable scalar, and $\delta$ a new variable
vector; and, after having cleared the equation~(15) of the
radical~$r$, or $\surd (- \alpha^2)$, by writing it as follows,
$$\alpha^2
       + \left(
            p - {\alpha \epsilon + \epsilon \alpha \over 2}
         \right)^2
   =  0,
   \eqno (33)$$
we get
$$\eqalignno{
0  &= (g' \epsilon + \delta)^2
       + \left(
            p + g' e^2
              - {\delta \epsilon + \epsilon \delta \over 2}
         \right)^2 \cr
   &= \delta^2
       + \left(
            {\delta \epsilon + \epsilon \delta \over 2}
         \right)^2
       + g'' (\delta \epsilon + \epsilon \delta)
       + (p + g' e^2)^2 - g'^2 e^2;
   &(34)\cr}$$
if we make for abridgment
$$g'' = g' - (p + g' e^2).
   \eqno (35)$$

If $\gamma'$ or $g' \epsilon$ is to be the constant vector of the
centre of the locus, it is necessary that to every variable
vector,~$\delta$, which satisfies the equation~(34), should
correspond another vector~$- \delta$, equal in length but
opposite in direction, and satisfying the same equation;
therefore the terms $g'' (\delta \epsilon + \epsilon \delta)$
must disappear, and we we must have
$$g'' = 0,\quad
  g' = {p \over 1 - e^2} = a,
   \eqno (36)$$
the constant~$a$ being thus suggested by the search after a
centre, as well as by the search after a second focus.  Making
then $g' = a$ in (34), we find the following equation of the
surface, when referred to its centre,
$$0 = \delta^2
       + \left(
            {\delta \epsilon + \epsilon \delta \over 2}
         \right)^2
       + ap;
   \eqno (37)$$
in which
$$ap = a^2 (1 - e^2) = a^2 (1 + \epsilon^2).
   \eqno (38)$$

And because in general, for any two vectors $\delta$,~$\epsilon$,
the following relation holds good,
$$    \left(
         {\delta \epsilon + \epsilon \delta \over 2}
      \right)^2
   =  \left(
         {\delta \epsilon - \epsilon \delta \over 2}
      \right)^2
       + \delta^2 \epsilon^2,
   \eqno (39)$$
we may write the equation (37) under the form
$$0 = (1 + \epsilon^2) (\delta^2 + a^2)
       + \left(
            {\delta \epsilon - \epsilon \delta \over 2}
         \right)^2.
   \eqno (40)$$

This last equation shows that
$$\hbox{when}\quad
  \delta \epsilon - \epsilon \delta = 0,
   \quad\hbox{then}\quad
  \delta^2 + a^2 = 0;
   \eqno (41)$$
that is to say, when $\delta$ is co-axal with, or parallel to
$\epsilon$, or, in other words, when the vector from the centre
coincides (in either direction) with the axis of revolution of
the surface, its length is $= \pm a$, according as $a$ is $>$
or $< 0$.

The equation~(37) shows that
$$\hbox{when}\quad
  \delta \epsilon + \epsilon \delta = 0,
   \quad\hbox{then}\quad
  \delta^2 + a^2 (1 + \epsilon^2) = 0;
   \eqno (42)$$
if therefore $\epsilon^2$ be $> -1$, that is, if $e^2 < 1$, the
length of every vector drawn from the centre perpendicularly to
the axis of revolution will be
$$\surd (- \delta^2)
   =  \alpha \surd (1 - e^2) = b,
   \eqno (43)$$
$b$ being a new scalar quantity; but if $e^2 > 1$,
$\epsilon^2 < -1$, $1 + \epsilon^2 < 0$, then we shall have, by
(42), {\it the absurd result of a\/} {\sc vector}~$\delta$
{\it appearing to have a\/} {\sc positive square}: whereas it is
a first principle of the present method of calculation, {\it that
the square of every vector\/} is to be regarded as a
{\it negative number: which symbolical contradiction indicates
the\/} {\sc geometrical impossibility} {\it of drawing from the
centre to any point of the locus, a straight line which shall be
perpendicular to the axis of revolution, in the case where\/}
$e^2 > 1$.  The locus has, in this case, two infinite branches
enclosed within the two branches of the {\it asymptotic cone\/}
which has for its equation
$$\delta^2
       + \left(
            {\delta \epsilon + \epsilon \delta \over 2}
         \right)^2
   =  0;
   \eqno (44)$$
and nowhere penetrates within that {\it inscribed spheric
surface}, which has for its equation
$$\delta^2 + a^2 = 0,
   \eqno (45)$$
though it touches this last surface at the two points where it
meets the axis of revolution.  On the other hand, when $e^2 < 1$,
the locus is entirely contained {\it within\/} the spheric
surface~(45), touching it, however, in like manner in two points
upon the axis of revolution.  A {\it finite\/} surface of
revolution (the {\it ellipsoid\/}) might thus have been
discovered, of which each point has a constant {\it sum\/} of
distances from two fixed foci; and an {\it infinite\/} surface
(the {\it hyperboloid\/}), with two separate sheets, of which
each point has a constant {\it difference\/} of distances from
two such foci: and all the other properties of these two surfaces
of revolution might have been found, and may be proved anew, by
pursuing this sort of analysis.  A third distinct surface of the
same class, but infinite in {\it one\/} direction only (the
{\it paraboloid\/}), might have been suggested by the observation
that the reduction to a centre fails in the case $e^2 = 1$,
$\epsilon^2 = -1$.  Its equation may be put under the form
$$(\epsilon \alpha'' - \alpha'' \epsilon)^2
   =  4 p (\epsilon \alpha'' + \alpha'' \epsilon),
   \eqno (46)$$
by making
$$\alpha = \alpha'' - {p \epsilon \over 2},
   \eqno (47)$$
so that $\alpha''$ is the vector from the vertex: and it lies
entirely on one side of the plane which touches it at the vertex,
namely, the plane
$$\epsilon \alpha'' + \alpha'' \epsilon = 0.
   \eqno (48)$$
In general whatever $e$ or $\epsilon$ may be, and therefore for
all the three surfaces, the length of the focal vector
perpendicular to the axis is $p$; for, by (33), if we make
$$\epsilon \alpha + \alpha \epsilon = 0,
   \eqno (49)$$
we get
$$\alpha^2 + p^2 = 0.
   \eqno (50)$$
Indeed (15) then gives $r = p$.

Since
$$\alpha^2 + r^2 = 0,\quad
  \alpha \mathbin{.} {\rm d} \alpha
       + {\rm d} \alpha \mathbin{.} \alpha
       + 2r \, {\rm d} r
   =  0,
   \eqno (51)$$
the differential equation (21) of the locus (15) may be put under
the form
$$(r \epsilon - \alpha) \, {\rm d} \alpha
       + {\rm d} \alpha \, (r \epsilon - \alpha)
   =  0;
   \eqno (52)$$
thus shewing that the vector $r \epsilon - \alpha$ is
perpendicular to the differential~${\rm d}$ of the focal
vector~$\alpha$, or that it is parallel to the normal to the
locus, at the extremity of that focal vector.  That normal,
therefore, intersects the axis of revolution in a point, of which
the focal vector is $r \epsilon$; the position of the normal is,
therefore, entirely known, and every thing that depends upon it
may be found, for the particular surfaces of revolution which
have been here considered.  For example, in the ellipsoid, the
vector of the second focus, drawn from the first, has been seen
to be $2a \epsilon$; if, then, we make
$$2a - r = r',
   \eqno (53)$$
so that $r'$ denotes the length of the second focal vector, drawn
to the same point as the first focal vector, of which the length
is $r$, we have $- r' \epsilon$ for the second focal vector of
the intersection of the normal with this axis; the normal,
therefore, cuts (internally) the interval between the two foci,
into segments proportional to the two conterminous focal
distances of the point upon the ellipsoid, and consequently
bisects the angle between those focal distances.  Again, if we
divide the expression $r \epsilon - \alpha$ by the scalar
quantity~$r$, and multiply the quotient by $a$, we find that
$\epsilon - \iota$ and $a \epsilon - a \iota$ are also
expressions for vectors in the normal direction; and because
$a \epsilon$ is the focal vector of the centre, while $- a \iota$
is a radius of the circumscribed sphere, opposite in direction to
the focal vector of the point upon the ellipsoid, we see that if
the focal vector of the extremity of this radius of the sphere be
prolonged through the focus, it will cut perpendicularly the
tangent plane to the ellipsoid.  Again, the expression
$$\tau = a ( \epsilon + \iota ) - \alpha
   =  (a - r) \iota + a \epsilon,
   \eqno (54)$$
is easily seen to denote here a vector perpendicular to
$\alpha - r \epsilon$, and therefore to the normal, because
$$\alpha \tau + \tau \alpha
   =  r (\varepsilon \tau + \tau \varepsilon)
   =  2 (ap - r r');
   \eqno (55)$$
but $\tau$ is also in the same plane with $\alpha$ and
$\epsilon$, and therefore is a vector parallel to the tangent to
the elliptic section of the locus made by a plane passing through
the axis of revolution; $a \iota$ is therefore the central vector
of a point upon this tangent, because $\alpha - a \epsilon$ is
the central vector of the point of contact; and the central
vector of the second focus being $a \epsilon$, we have
$a \iota - a \epsilon$ as an expression for the second focal
vector of the same point upon the tangent; this second focal
vector is therefore parallel to the normal, because
$\iota - \epsilon$ is parallel thereto, and, consequently, it is
the perpendicular let fall from the second focus on the tangent
line or plane: and the foot of this perpendicular is thus seen to
be at the extremity of that radius of the circumscribed circle or
sphere, which is drawn in a direction similar (and not, as
lately, opposite) to the direction of the first focal vector of
the point on the ellipse or ellipsoid.  We see, at the same time,
that $- \tau$ is a symbol for the projection of the second focal
vector upon the tangent line or plane; from which we may infer,
by (55), that the product of the lengths of the two projections
of the two focal vectors on the tangent is $= r r' - ap$, and
therefore that it is less than the product~$r r'$ of the lengths
of those two vectors by the constant quantity~$ap$, or $b^2$,
which constant must thus be equal to the product of the lengths
of the projections of the same two vectors on the normal, so that
we may write the equation
$${\sc p} {\sc p}' = ap = b^2,
   \eqno (56)$$
if ${\sc p}$ and ${\sc p}'$ denote the lengths of the
perpendiculars let fall from the two foci on the tangent, while
$b$ is the axis minor of the ellipse.  Analogous reasoning may be
applied to the hyperbola, or to the surface formed by its
revolution round its transverse axis.  Most of the foregoing
geometrical results are well known, and probably all of them are
so: but it may be considered worth while to have briefly
indicated the manner in which they reproduce themselves in these
new processes of calculation.

The vector drawn from the focus first considered to any arbitrary
point upon the normal, may be represented by the expression
$$\nu = (1 - n) \alpha + n r \epsilon,
   \eqno (57)$$
in which $n$ is an arbitrary scalar; and if this normal intersect
another normal infinitely near it, then we may write, as the
expression of this relation,
$$0 = {\rm d} \nu
   =  (1 - n) {\rm d} \alpha + n \epsilon \, {\rm d} r
       + (r \epsilon - \alpha) \, {\rm d} n:
   \eqno (58)$$
comparing which differential equation with the forms (52) and
(21) of the differential equation of the surface of
revolution~(15), we can eliminate the scalar differential
${\rm d} n$, and deduce for $n$ itself the expression
$$n = {{\rm d} \alpha^2 \over {\rm d} r^2 + {\rm d} \alpha^2}.
   \eqno (59)$$
One way of satisfying these conditions is to suppose
$$n = 1,\quad
  {\rm d} r = 0,\quad
  \nu = r \epsilon;
   \eqno (60)$$
which comes to considering the intersection of the given normal
with the axis, and therefore with the other normal from points of
the same generating circle of the surface of revolution: and this
intersection is accordingly one centre of curvature of the
surface.  The only other way of obtaining an intersection of two
normals infinitely near, is to suppose, by (58), the
element~${\rm d} \alpha$ coplanar with $\alpha$ and $\epsilon$,
or to pass to consecutive normals contained in the same plane
drawn through the axis; that is to say, the other centre of
curvature of the surface is the centre of curvature of its
meridian.  The length of the element of this meridian, that is
the length of ${\rm d} \alpha$, is denoted by the radical
$\surd ( - {\rm d} \alpha^2)$, because the
differential~${\rm d} \alpha$ is a vector; and the length of the
projection of this element on the focal vector is
$\pm dr = \surd ( + {\rm d} r^2 )$,
because $dr$ is a scalar differential: therefore the length of
the projection of the same element on a line perpendicular to the
focal vector, and drawn in the plane through the axis, is denoted
by this other radical,
$\surd ( - {\rm d} \alpha^2 - {\rm d} r^2 )$;
but the length of this last projection is evidently to the length
of the element itself, as the length~${\sc p}$ of the perpendicular let
fall from the focus on the tangent is to the length~$r$ of the
focal vector of the point of contact; such, therefore, is, by
(59), the ratio of $n^{-{1 \over 2}}$ to $1$, if the scalar~$n$,
in the {\it equation of the normal\/}~(57); receive the value
which corresponds to the centre of curvature of the meridian;
therefore we have
$${{\sc r} \over {\sc n}} = {\nu - \alpha \over r \epsilon - \alpha}
   = n = {r^2 \over {\sc p}^2} = {r r' \over {\sc p} {\sc p}'}
   = {r r' \over p a},
   \eqno (61)$$
${\sc n}$ denoting the length of the portion of the normal which is
comprised between the meridian and the axis, and ${\sc r}$ denoting the
length of the radius of curvature of the meridian.  The
projection of this radius on the focal vector is evidently the
focal half chord of curvature, of which half chord the length may
be here denoted by ${\sc c}$; we see then that if we again project this
half chord on the normal, the result is the normal itself, that
is the portion~${\sc n}$, because this double process of projection
multiplies ${\sc r}$ twice successively by $n^{-{1 \over 2}}$; and if,
once more, the normal be projected on the focal vector, the third
projection so obtained is equal in length to the
semiparameter~$p$, because, by (15) and (16),
$$\iota (r \epsilon - \alpha) + (r \epsilon - \alpha) \iota
   =  2p;
   \eqno (62)$$
hence
$$\surd ({\sc r}{\sc n}) = {\sc c} = np = {r r' \over a}
   = {2r r' \over r + r'},
   \eqno (63)$$
that is, for any conic section, the geometrical mean between the
radius of curvature and the normal is equal to the harmonic mean
between the two focal distances; of which distances the second,
namely $r'$, is to be regarded as negative for the hyperbola, and
infinite for the parabola, and the harmonic mean determined
accordingly.  We have also, for every conic section (if
$r' a^{-1}$ be suitably interpreted),
$$\surd n = {{\sc r} \over {\sc c}} = {{\sc c} \over {\sc n}} = {{\sc n} \over p}
   = \sqrt{ \vphantom{\biggr\{} }
      \left( {r r' \over pa} \right),
   \eqno (64)$$
so that the semiparameter, the normal, the focal half chord of
curvature, and the radius of curvature, are in continued
geometrical progression: and the analysis may be verified, by
calculating directly, on the same principles, the length of the
normal, as follows:
$${\sc n} = \surd \{ - (r \epsilon - \alpha)^2 \}
    = \surd \{ r^2 (e^2 + 1) + 2r (p - r) \}
    = \sqrt{ \vphantom{\biggr\{} }
      \left( {p r r' \over a} \right).
   \eqno (65)$$

The general relation of a conic section to a {\it directrix\/} is
an immediate geometrical consequence of the equation~(15), which
has been here (in part) discussed, and may be regarded as its
simplest interpretation.  Some of the foregoing symbolical
results respecting such a section admit of dynamical
interpretation also; and, in particular, the expression
$a \iota - a \epsilon$, which has been seen to represent, both in
length and in direction, the perpendicular let fall from the
second focus on the tangent, may suggest, by its composition,
what is, however, a more immediate consequence of the
equation~(12), that in the undisturbed motion of a planet or
comet about the sun, {\it the whole varying tangential velocity
may be decomposed into two partial velocities, of which both are
constant in magnitude, while one of them is constant in direction
also}.  The component velocity, which is constant in magnitude,
but not in direction, is always in the plane of the orbit, and is
perpendicular to the heliocentric radius vector of the body; the
other component, which is constant in both magnitude and
direction is parallel to the velocity at perihelion; and the
magnitude of this fixed component is to the magnitude of the
revolving one in the ratio of the excentricity~$e$ to unity.  The
author supposes that this theorem respecting a decomposition of
the velocity in an excentric orbit is known, though he does not
remember having met with it; but conceived that it might properly
be mentioned here, as being a very easy and immediate consequence
of the present analysis: respecting the general principles of
which analysis, the reader is requested to consult the Abstract,
already referred to, of the communication of November, 1844,
printed at the commencement of the present volume of the
Proceedings of the Academy.  There is no difficulty in deducing,
on the same principles, from formul{\ae} of the present paper,
the known differential equation,
$$\left( {{\rm d} r \over {\rm d} t} \right)^2
   =  {\sc m} \left( {2 \over r} - {1 \over a}
             - {p \over r^2} \right),
   \eqno (66)$$
which connects the radial component of velocity with the
heliocentric distance and the time, and may be integrated by the
usual processes.

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