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% School of Mathematics, Trinity College, Dublin 2, Ireland
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% Trinity College, 1999.
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\centerline{\Largebf ON THE APPLICATION OF THE CALCULUS OF}
\vskip12pt
\centerline{\Largebf QUATERNIONS TO THE THEORY OF THE MOON}
\vskip24pt
\centerline{\Largebf By}
\vskip24pt
\centerline{\Largebf William Rowan Hamilton}
\vskip24pt
\centerline{\largerm (Proceedings of the Royal Irish Academy,
3 (1847), pp.\ 507--521.)}
\vskip36pt
\vfill
\centerline{\largerm Edited by David R. Wilkins}
\vskip 12pt
\centerline{\largerm 1999}
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\null\vskip36pt
\centerline{{\largeit On the Application of the Calculus of
Quaternions to the Theory}}
\vskip 3pt
\centerline{{\largeit of the Moon.}}
\vskip 6pt
\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}
\bigskip
\centerline{Communicated June~14, 1847.}
\bigskip
\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~3 (1847), pp.\ 507--521.]}
\bigskip
Sir William Rowan Hamilton made a communication respecting the
application of the Calculus of Quaternions to the Theory of the
Moon.
\bigbreak
I.
At two Meetings of the Royal Irish Academy, in the month of July,
1845, Sir William Rowan Hamilton had exhibited and illustrated
the following general equation of motion of a system of bodies,
with masses $m, m',\ldots$, and with vectors
$\alpha, \alpha',\ldots$, and attracting each other according to
Newton's law:
$${{\rm d}^2 \alpha \over {\rm d} t^2}
= \Sigma {m'
\over (\alpha - \alpha') \surd \{ - (\alpha - \alpha')^2 \}}.
\eqno (1)$$
He had, at that time, deduced from this equation the known laws
of the centre of gravity, of areas, and of living force, for any
such multiple system; and had shown that the corresponding, but
less general, equation of relative motion of a binary system,
which (by changing $\alpha - \alpha'$ to $\alpha$, and $m + m'$
to ${\sc m}$) becomes
$${{\rm d}^2 \alpha \over {\rm d} t^2}
= {{\sc m} \over \alpha \surd ( - \alpha^2 )},
\eqno (2)$$
can be rigorously integrated by the processes of his new calculus
of quaternions, so as to conduct, with facility, when the
principles and plan have been caught, to the known laws of
elliptic, parabolic, or hyperbolic motion of one of the two
attracting bodies about the other. (See the Proceedings of July
14th and 21st, 1845, Appendix to Volume III., pp.~xxxvii., \&c.)
At a subsequent Meeting of the Academy, in December, 1845, Sir W.
Hamilton has shown that the general differential equation~(1)
might be put under this other form:
$$0 = {\textstyle {1 \over 2}} \Sigma \mathbin{.} m
\left(
\delta \alpha \, {{\rm d}^2 \alpha \over {\rm d} t^2}
+ {{\rm d}^2 \alpha \over {\rm d} t^2} \, \delta \alpha
\right)
+ \delta \Sigma
{m m' \over \surd \{ - (\alpha - \alpha')^2 \}};
\eqno (3)$$
and that it might, theoretically, be integrated by an adaptation
of that ``General Method in Dynamics'' which he had previously
published in the Philosophical Transactions of the Royal Society
of London, for the years 1834 and 1835; and which depended on a
peculiar combination of the principles of variations and partial
differentials, already illustrated by him, in earlier years, for
the case of mathematical optics, in the Transactions of this
Academy. (See Proceedings of December 8th, 1845, Appendix
already cited, pp.~lii., \&c.)
At the same meeting of December, 1845, Sir W. Hamilton assigned
the two following rigorous differential equations for the
internal motions of a system of {\it three\/} bodies, with masses
$m$,~$m'$,~$m''$, and with vectors $\alpha$, $\beta + \alpha$,
$\gamma + \alpha$,---that is, for the motions of the two latter of
these three bodies (regarded as points) about the former,---as
consequences of the general equation~(1):
$$\eqalignno{
{{\rm d}^2 \beta \over {\rm d} t^2}
&= {m + m' \over \beta \surd ( - \beta^2 )}
+ m''
\left\{
{(\beta - \gamma)^{-1}
\over \surd \{ - (\beta - \gamma)^2 \}}
+ {\gamma^{-1} \over \surd ( - \gamma^2 )}
\right\};
&(4)\cr
{{\rm d}^2 \gamma \over {\rm d} t^2}
&= {m + m'' \over \gamma \surd ( - \gamma^2 )}
+ m'
\left\{
{(\gamma - \beta)^{-1}
\over \surd \{ - (\gamma - \beta)^2 \}}
+ {\beta^{-1} \over \surd ( - \beta^2 )}
\right\}.
&(5)\cr}$$
It was remarked, that by regarding $m$,~$m'$,~$m''$, as
representing respectively the masses of the earth, moon, and sun,
$\beta$ and $\gamma$ become the geocentric vectors of the two
latter bodies; and that thus the laws of the disturbed motion of
our satellite are contained in the two equations (4) and
(5)---but especially in the first of those equations (the second
serving chiefly to express the laws of the sun's relative
motion).
The part of this equation~(4), which is independent of the sun's
mass~$m''$, is of the form~(2), and contains the laws of the
undisturbed elliptic motion of the moon; the remainder is the
disturbing part of the equation, and contains the laws of the
chief lunar perturbations. A commencement was made of the
development of this disturbing part, according to ascending
powers of the vector of the moon, and descending powers of the
vector of the sun; and an approximate expression was thereby
obtained, which may be written thus:
$$m'' \left\{
{(\beta - \gamma)^{-1}
\over \surd \{ - (\beta - \gamma)^2 \}}
+ {\gamma^{-1} \over \surd ( - \gamma^2 )}
\right\}
= m'' { (\beta + 3 \gamma^{-1} \beta \gamma)
\over 2 ( - \gamma^2)^{3 \over 2} }.
\eqno (6)$$
There was also given a geometrical interpretation of this result,
corresponding to a certain decomposition of the sun's disturbing
force into two others, of which the greater is triple of the
less, while the angle between them is bisected by the geocentric
vector of the sun; and the lesser of these two component forces
is in the direction of the moon's geocentric vector prolonged, so
that it is an ablatitious force, which was shown to be one of
nearly constant amount.
Although the foregoing formul{\ae} may be found in the Appendix
already cited, to the Proceedings of the above-mentioned dates,
yet it is hoped that, in consideration of the importance and
difficulty of the subject, and the novelty of the processes
employed, the Academy will not be displeased at having had this
brief recapitulation laid before them, as preparatory to a sketch
of some additional developments and applications of the same
general view, which have since been made by the author. It may,
for the same reason, be not improper here to state again, what
was stated on former occasions, that all expressions involving
{\it vectors} $\alpha,$~$\alpha'$, \&c., such as are considered
in this new sort of algebraical geometry, and enter into the
foregoing equations, admit of being translated into others, which
shall involve, instead of those vectors, three times as many
rectangular {\it co-ordinates},
$x$,~$y$,~$z$, $x'$,~$y'$,~$z'$, \&c.,
by means of relations of the forms
$$\alpha = i x + j y + k z,\quad
\alpha' = i x' + j y' + k z',\quad\hbox{\&c.};
\eqno (7)$$
where $i$~$j$~$k$ are the three original and coordinate vector
units of Sir William Hamilton's theory of quaternions, and
satisfy the fundamental equations
$$\left. \eqalign{
i^2 &= j^2 = k^2 = -1;\cr
ij &= k,\quad
jk = i,\quad
ki = j;\cr
ji &= -k;\quad
kj = -i;\quad
ik = -j;\cr}
\right\}
\eqno (8)$$
which were communicated to the Royal Irish Academy at the Meeting
of the 13th November, 1843. (See the Proceedings of that date,
and the author's First Series of Researches respecting
Quaternions, which Series has lately been printed in the
Transactions of the Academy, Vol.~XXI.\ Part~2.)
\bigbreak
II.
It is evident, from inspection of the equations above
recapitulated, that every transformation of the {\it vector
function},
$$\phi(\alpha) = \alpha^{-1} ( - \alpha^2 )^{- {1 \over 2}}
\eqno (9)$$
which represents, in direction and amount, the attraction exerted
by one mass-unit, situated at the beginning of the
vector~$\alpha$, on another mass-unit situated at the end of that
vector, must be important in the theory of the Moon; and
generally in the investigation, by quaternions, of the
mathematical consequences of the Newtonian Law of Attraction.
The integration of the equation of motion~(2) of a binary system
was deduced, in the communication of July, 1845, from a
transformation of that vector function, which may now be written
thus:
$$\alpha^{-1} ( - \alpha^2 )^{- {1 \over 2}}
= {2 {\rm d} \mathbin{.} \alpha ( - \alpha^2 )^{-{1 \over 2}}
\over \alpha \, {\rm d} \alpha - {\rm d} \alpha \, \alpha};
\eqno (10)$$
where ${\rm d}$ is, as in former equations, the characteristic of
differentiation. And the hodographic theory of the motion of a
system of bodies, attracting each other according to the same
Newtonian law, so far as it was symbolically stated to the
Academy, at the meeting of the 14th of December, 1846, depends
essentially on the same transformation. In fact, if we make
$$d \alpha = \tau \, {\rm d} t,\quad
\alpha = \int \tau \, {\rm d} t;
\eqno (11)$$
and if, by the use of notations explained in former
communications, we employ the letters ${\sc u}$ and ${\sc v}$ as
the characteristics of the operations of taking the versor and
the vector of a quaternion, writing, therefore,
$${\sc u} (\alpha) = \alpha (- \alpha^2)^{-{1 \over 2}};\quad
{\sc v} \mathbin{.} \alpha \tau
= - {\sc v} \mathbin{.} \tau \alpha
= {\textstyle {1 \over 2}} (\alpha \tau - \tau \alpha);
\eqno (12)$$
the equation~(2) of the internal motion of a binary system
becomes
$${\rm d} \tau
= {\displaystyle
- {\sc m} \, {\rm d} {\sc u}
\left( \int \tau \, {\rm d} t \right)
\over \displaystyle
{\sc v} \left( \tau \int \tau \, {\rm d} t \right)};
\eqno (13)$$
where the denominator in the second member is constant, by the
law of the equable description of areas. Hence, this second
member, like the first, is an exact differential; and an
immediate integration, introducing an arbitrary, but constant
vector~$\epsilon$, coplanar with $\alpha$ and $\tau$, gives the
{\it law of the circular hodograph}, under the symbolical form
$$\tau
= {\displaystyle
{\sc m}
\left( \epsilon - {\sc u} \int \tau \, {\rm d} t \right)
\over \displaystyle
{\sc v} \mathbin{.} \tau \int \tau \, {\rm d} t }:
\eqno (14)$$
the constant part of this expression~(14) for the vector of the
velocity,~$\tau$, being the vector of the centre of the
hodograph, drawn from that one of the two bodies which is
regarded as the centre of force; while the variable part of the
same expression for $\tau$ represents the variable radius of the
same hodographic circle, or the vector of a point on its
circumference, drawn from its own centre of figure as the origin.
Multiplying this integral equation~(14) by
$\displaystyle \int \tau \, {\rm d} t$,
taking the vector part of the product, dividing by ${\sc m}$, and
multiplying both members of the result into the constant
denominator of the second member of (13) or of (14), we find, by
the rules of the present calculus,
$${\displaystyle
- \left( {\sc v} \mathbin{.} \tau
\int \tau \, {\rm d} t \right)^2
\over {\sc m}}
= {\sc s} \mathbin{.} \epsilon \int \tau \, {\rm d} t
+ {\sc t} \mathbin{.} \int \tau \, {\rm d} t;
\eqno (15)$$
where ${\sc s}$ and ${\sc t}$ are the characteristics of the
operations of taking respectively the scalar and tensor of a
quaternion, so that, as applied to the present question, they
give the results,
$${\sc t} \mathbin{.} \int \tau \, {\rm d} t
= {\sc t} \alpha = \surd ( - \alpha^2 ) = r;
\eqno (16)$$
and
$${\sc s} \mathbin{.} \epsilon \int \tau \, {\rm d} t
= {\textstyle {1 \over 2}}
(\epsilon \alpha + \alpha \epsilon)
= e r \cos v;
\eqno (17)$$
where
$$e = {\sc t} \epsilon = \surd ( - \epsilon^2 )
= \hbox{const.};
\eqno (18)$$
while $v$ is the {\it angle\/} (of true anomaly) which the
variable {\it vector\/}~$\alpha$ of the orbit makes with the
fixed vector $- \epsilon$ in the plane of that orbit; and $r$
denotes the {\it length\/} of $\alpha$, or what is usually called
(and may still in this theory be named) the {\it radius\/} vector
of the relative orbit. The first member of the equation~(15) is
a positive and constant number, representing the quotient which
is obtained when the square of the double areal velocity in the
relative orbit is divided by the sum of the two masses; if then
we denote, as usual, this constant quotient (or semiparameter) by
$p$, and observe that the constant~$e$ is also numerical
(expressing, as usual, the eccentricity of the orbit), we shall
obtain again, by this process, as by that of July, 1845, the polar
equation of the orbit, under the well-known form,
$$r = {p \over 1 + e \cos v}.
\eqno (19)$$
This sketch of a process for employing the general
transformation (10) in the theory of a binary system, may make it
easier, than it would otherwise be, to understand how the
following equation for the motion of a multiple system,
$${\rm d} \tau
= \Sigma
{\displaystyle
(m + \Delta m) \, {\rm d} {\sc u}
\left( \int \Delta \tau \, {\rm d} t \right)
\over \displaystyle
{\sc v} \left( \Delta \tau \mathbin{.}
\int \Delta \tau \, {\rm d} t \right)},
\eqno (20)$$
(where $m + \Delta m$, $\tau + \Delta \tau$, are the mass and the
vector of velocity of an attracting body, as $m$,~$\tau$ are
those of an attracted one, which is analogous to, and includes,
the equation~(13) for the motion of a binary one, and which
agrees with a formula communicated to the Academy in December,
1846), was obtained by the present author; and how it may
hereafter be applied.
\bigbreak
III.
The vector function $\phi(\alpha)$ in (9) may be called the
{\sc tractor} corresponding to the vector of position~$\alpha$,
or simply the tractor of $\alpha$; and another general
transformation of this tractor, which is more intimately
connected than the foregoing with the problem of
{\it perturbation}, may be obtained by supposing the
vector~$\alpha$ to receive any small but finite
increment~$\beta$, representing a new but shorter vector, which
begins, or is conceived to be drawn, in any arbitrary direction,
from the point of space where the vector~$\alpha$ ends; and, by
then {\it developing}, in conformity with the rules of
quaternions, the {\it new tractor\/} $\phi(\beta + \alpha)$,
(answering to the new vector $\beta + \alpha$, which is drawn
from the beginning of $\alpha$ to the end of $\beta$), according
to the ascending powers of this added vector~$\beta$. In this
manner we find
$$\eqalignno{
\phi(\beta + \alpha)
&= \{ - (\beta + \alpha)^2 \}^{-{1 \over 2}}
(\beta + \alpha)^{-1} \cr
&= \{ - \alpha^2 (1 + \alpha^{-1} \beta)
(1 + \beta \alpha^{-1}) \}^{-{1 \over 2}}
\{ \alpha (1 + \alpha^{-1} \beta) \}^{-1} \cr
&= (1 + \beta \alpha^{-1})^{-{1 \over 2}}
(1 + \alpha^{-1} \beta)^{-{3 \over 2}}
\alpha^{-1} (- \alpha^2)^{-{1 \over 2}};
&(21)\cr}$$
that is,
$$\phi(\beta + \alpha) = \Sigma_{n,n'} \phi_{n,n'}
\eqno (22)$$
if we make, for abridgment,
$$\phi_{n,n'}
= m_{n,n'} (\beta \alpha)^n (\alpha \beta)^{n'}
\alpha^{-1} (- \alpha^2)^{-{1 \over 2} - n - n'}
\eqno (23)$$
where
$$m_{n,n'}
= {1 \mathbin{.} 3 \, \ldots \, (2n - 1)
\over 2 \mathbin{.} 4 \, \ldots \, (2n)}
\times
{3 \mathbin{.} 5 \, \ldots \, (2n' + 1)
\over 2 \mathbin{.} 4 \, \ldots \, (2n')}.
\eqno (24)$$
The attraction $\phi(\beta + \alpha)$ which a mass-unit, situated
at the beginning of the vector $\beta + \alpha$, exerts on
another mass-unit situated situated at the end of that vector, is
thus decomposed into an infinite but convergent series of other
forces,~$\phi_{n,n'}$, of which the {\it intensities\/} are
determined by the {\it tensors}, and of which the
{\it directions\/} are determined by the {\it versors}, of the
expressions included in the formula~(23); or by the following
expressions, which are derived from it by the rules of the
calculus of quaternions:
$${\sc t} \phi_{n,n'}
= m_{n,n'}
\left( {\sc t} {\beta \over \alpha} \right)^{n+n'}
( {\sc t} \alpha )^{-2};
\eqno (25)$$
$${\sc u} \phi_{n,n'}
= ( {\sc u} \mathbin{.} \beta \alpha )^{n - n'}
( {\sc u} \alpha )^{-1}
= \left( {\sc u} {\beta \over - \alpha} \right)^{n - n'}
{\sc u} (- \alpha).
\eqno (26)$$
Let $a$, $b$, be the lengths (or tensors) of the vectors
$\alpha$,~$\beta$, and let ${\sc c}$ be the angle between them,
which angle we may conceive to express the amount of the positive
rotation, in their common plane, from the direction of $- \alpha$
to the direction of $+ \beta$; then the positive or negative
rotation in the same plane, from the same direction of
$- \alpha$, to the direction of the component
force~$\phi_{n,n'}$, is measured as follows:
$$\hbox{{\it angle}, from $-\alpha$ to force $\phi_{n,n'}$}
= (n - n') {\sc c};
\eqno (27)$$
and
$$\hbox{{\it intensity\/} of same component force}
= m_{n,n'} \left( {b \over a} \right)^{n+n'} a^{-2}.
\eqno (28)$$
The case $n = 0$, $n' = 0$, answers to the old tractor
$\phi(\alpha)$, or to a force of which the intensity is
represented by $a^{-2}$, while its direction is the same as that
of $- \alpha$.
\bigbreak
IV.
Thus, if the vector~$\alpha$ be conceived to being at a
point~${\sc b}$, and to end at a point~${\sc c}$, while the
vector~$\beta$ shall be conceived to begin at ${\sc c}$ and to
end at ${\sc a}$; and if we conceive a unit-mass at ${\sc b}$ to
attract two other masses, regarded as collected into points, and
as situated respectively at ${\sc c}$ and at ${\sc a}$; this
attraction of ${\sc b}$ will {\it disturb\/} the relative motion
of ${\sc a}$ about ${\sc c}$, if ${\sc a}$ be supposed to be
nearer than ${\sc b}$ is to ${\sc c}$, by producing a {\it series
of groups of smaller and smaller forces}, of which groups it may
be sufficient here to consider the two following.
The first and principal group consists of the two disturbing
forces $\phi_{1,0}$ and $\phi_{0,1}$, and of these the first is
purely {\it ablatitious}, or is directed along the prolongation
of the side of the triangle ${\sc a} {\sc b} {\sc c}$, which is
drawn from ${\sc c}$ to ${\sc a}$, and it has its intensity
denoted by the expression ${1 \over 2} b a^{-3}$, since we have
for this force, and for its tensor and versor, the expressions
$$\phi_{1,0}
= {\textstyle {1 \over 2}} \beta (- \alpha^2)^{-{3 \over 2}};\quad
{\sc t} \phi_{1,0}
= {\textstyle {1 \over 2}} b a^{-3};\quad
{\sc u} \phi_{1,0}
= {\sc u} \beta.
\eqno (29)$$
The second disturbing force, of this last group, has for
expression
$$\phi_{1,0}
= {\textstyle {3 \over 2}} \alpha \beta \alpha^{-1}
(- \alpha^2)^{-{3 \over 2}}
= {\textstyle {3 \over 2}} \alpha \beta \alpha^{-1} a^{-3};
\eqno (30)$$
it intensity is {\it exactly triple\/} of that of the former
force, being represented by ${3 \over 2} b a^{-3}$; and its
direction is the same as that of a straight line drawn from
${\sc c}$ to ${\sc a}'$, if ${\sc a}'$ be a point such that the
line ${\sc a} {\sc a}'$ is perpendicularly bisected by the line
${\sc b} {\sc c}$ (prolongued through ${\sc c}$ if necessary).
These two principal disturbing forces evidently correspond to
those which were considered for the case of our own satellite in
a communication above alluded to; the second force being the one
which was described in that former communication as being
directed to what was there called the ``fictitious moon,'' and
was conceived to be as far from the sun in the heavens on one
side, as the actual moon is on the other side, but in the same
great circle.
If now we extend that mode of speaking so far as to conceive a
similar {\it reflexion\/} of the sun with respect to the moon,
and to call the point in the heavens so found the ``fictitious
sun,'' the moon being thus imagined to be seen midway among the
stars between the actual and the fictitious sun: and if we
farther imagine a ``second fictitious sun,'' so placed that the
actual sun shall appear to be midway between this and the first
fictitious sun; we shall then be able to describe in words the
directions of the three disturbing forces of the second group,
and to say that they tend respectively, for the case of our own
satellite, to these three (real or fictitious) suns. For these
three {\it forces\/} will have, for their respective expressions,
the three corresponding {\it terms\/} of the development of the
{\it tractor\/} (22), namely, the following:
$$\phi_{2,0}
= {\textstyle {3 \over 8}} \beta \alpha \beta
(- \alpha^2)^{-{5 \over 2}};\quad
\phi_{1,1}
= {\textstyle {3 \over 4}} \beta^2 \alpha
(- \alpha^2)^{-{5 \over 2}};\quad
\phi_{0,2}
= {\textstyle {15 \over 8}}
\alpha \beta \alpha \beta \alpha^{-1}
(- \alpha^2)^{-{5 \over 2}};
\eqno (31)$$
of which the {\it intensities\/} are respectively
$${\textstyle {3 \over 8}} b^2 a^{-4};\quad
{\textstyle {3 \over 4}} b^2 a^{-4};\quad
{\textstyle {15 \over 8}} b^2 a^{-4};
\eqno (32)$$
so that they are {\it exactly proportional to the three whole
numbers}, $1$,~$2$,~$5$; while they are {\it directed},
respectively, to the first fictitious sun, the actual sun, and
the second fictitious sun. The {\it disturbing force of a
superior planet}, exerted on an inferior one, may be developed or
decomposed into a series of groups of lesser disturbing forces,
the intensities of the several forces in each group being
constantly proportional to whole numbers, in an exactly similar
way; nor does the application of the principle and method of
development thus employed terminate here. In the applications to
the lunar theory, $a$ and $b$, in the recent expressions, are to
be regarded as denoting the variable distances of the sun and
moon from the earth; and the expressions for the forces are to be
multiplied by the mass of the sun. Nothing depends, so far, on
any smallness of eccentricities or inclinations.
\bigbreak
V.
The lunar theory is, very approximately, contained in the
differential equation~(4), provided that we regard $\gamma$ as
the elliptic vector of the sun, drawn from the common centre of
gravity of the earth and moon; and the laws of the sun's relative
elliptic motion, with respect to that centre of gravity, are then
contained in the following differential equation, which takes the
place of the equation~(5):
$${{\rm d}^2 \gamma \over {\rm d} t^2}
= {m + m' + m'' \over \gamma \surd (- \gamma^2)}.
\eqno (33)$$
Indeed, when we come to consider the small disturbing forces
which belong to the second group, and which depend on the inverse
fourth power of the sun's distance, the corresponding terms of
the development of the first member of the formula~(6) are then,
for greater accuracy, to be multiplied by the fraction
$\displaystyle {m - m' \over m + m'}$,
which expresses the ratio of the difference to the sum of the
masses of the earth and moon. But if we neglect, for the
present, those small disturbing terms, we may regard that
formula~(6) as accurate, without as yet neglecting anything on
account of smallness of eccentricities or of inclinations; and
even without assuming any knowledge of the smallness of the
moon's mass, as compared with the mass of the earth; $\gamma$
still denoting, as just stated, the elliptic vector of the sun.
And thus, if the moon's geocentric vector~$\beta$ be changed to
the sum $\beta + \delta \beta$, where the term $\delta \beta$ is
supposed to depend on the disturbing force, and to give a product
which may be neglected when it is multiplied by or into the
expression for that force, we shall have the following
approximate differential equation, by developing the disturbed or
{\it altered tractor\/} $\phi(\beta + \delta \beta)$, and
confining ourselves to the first power of $\delta \beta$:
$${{\rm d}^2 \delta \beta \over {\rm d} t^2}
+ {{\rm d}^2 \beta \over {\rm d} t^2}
- {m + m' \over \beta \surd (- \beta^2)}
= {m + m' \over 2 (- \beta^2)^{3 \over 2}}
(\delta \beta + 3 \beta^{-1} \, \delta \beta \, \beta)
+ {m'' \over 2 (- \gamma^2)^{3 \over 2}}
(\beta + 3 \gamma^{-1} \beta \gamma).
\eqno (34)$$
The {\it disturbance\/} $\delta \beta$ of the moon's geocentric
{\it vector\/} is thus exhibited as giving rise to an
{\it alteration\/} $\delta \phi(\beta)$ in the corresponding
{\it tractor\/} $\phi(\beta)$, which alteration is {\it analogous
to a disturbing force}, and occasions the presence of the first
of the two parts of the second member of the equation~(34): which
equation will be found to contain a considerable portion of the
theory of the moon.
\bigbreak
VI.
The author will only mention here two very simple applications,
which he has made of this equation (34), one to the Lunar
Variation, and the other to the Regression of the Node. Treating
{\it here\/} the sun's relative orbit as exactly circular, and
the moon's as approximately such, neglecting the inclination,
taking for units of their kinds the sum of the masses of the
earth and moon, and the moon's mean distance and mean angular
velocity, and employing, as usual, the letter~$m$ to denote (not
now the earth's mass, but) the ratio of the sun's mean angular
motion to the corresponding motion of the moon, the differential
equation~(34) becomes:
$${{\rm d}^2 \delta \beta \over {\rm d} t^2}
= {\textstyle {1 \over 2}}
( \delta \beta + 3 \beta^{-1} \, \delta \beta \, \beta )
+ {m^2 \over 2} (\beta + 3 \gamma^{-1} \beta \gamma);
\eqno (35)$$
in which the laws of the circular revolutions of the vectors
$\beta$ and $\gamma$ give
$${{\rm d}^2 \beta \over {\rm d} t^2}
= -\beta;\quad
{{\rm d}^2 \gamma \over {\rm d} t^2}
= - m^2 \gamma.
\eqno (36)$$
Assuming, from some general indications of this theory, an
expression for the perturbation of the moon's vector, which shall
be of the form
$$\delta \beta
= m^2 ( A \beta + B \gamma^{-1} \beta \gamma
+ C \beta^{-1} \gamma^{-1} \beta \gamma \beta ),
\eqno (37)$$
and neglecting all powers of $m$ above the square, we find
$${{\rm d}^2 \delta \beta \over {\rm d} t^2}
= - m^2 ( A \beta + B \gamma^{-1} \beta \gamma
+ 3^2 C \beta^{-1} \gamma^{-1} \beta \gamma \beta );
\eqno (38)$$
$$\beta^{-1} \, \delta \beta \mathbin{.} \beta
= m^2 ( A \beta + C \gamma^{-1} \beta \gamma
+ B \beta^{-1} \gamma^{-1} \beta \gamma \beta );
\eqno (39)$$
so that the three numerical coefficients, $A$, $B$, $C$, must
satisfy the three following equations of condition:
$$-A = 2A + {\textstyle {1 \over 2}};
\quad\hbox{giving}\quad
A = - {\textstyle {1 \over 6}};
\eqno (40)$$
and
$$-B = {\textstyle {1 \over 2}} (B + 3 C)
+ {\textstyle {3 \over 2}};\quad
-9C = {\textstyle {1 \over 2}} (C + 3B);
\eqno (41)$$
giving
$$B = - {\textstyle {19 \over 16}};\quad
C = + {\textstyle {3 \over 16}}.
\eqno (42)$$
Thus, if we neglect eccentricities and inclination, and confine
ourselves to the first power of the disturbing force, or to the
second power of $m$, the perturbation of the moon's vector,
produced by the sun's attraction, is composed of the three
following terms:
$$\delta \beta
= - {m^2 \over 6} \beta
- {19 m^2 \over 16} \gamma^{-1} \beta \gamma
+ {3 m^2 \over 16} \beta^{-1} \gamma^{-1} \beta \gamma \beta.
\eqno (43)$$
The first of these three terms expresses that the sun's
ablatitious force, by partially counteracting the earth's
attractive force on the moon, allows our satellite to revolve in
a somewhat smaller orbit than would otherwise be consistent with
the observed periodic time: the ratio of the diminished to the
undiminished radius of the orbit being that of
$\displaystyle 1 - {m^2 \over 6}$ to $1$. The second term
expresses a displacement of the moon, through perturbation, from
its diminished circular orbit, of which displacement the constant
magnitude or length bears to the radius of the undiminished orbit
the ratio of $\displaystyle {19 m^2 \over 16}$ to unity; while
the direction of this displacement is always {\it from\/} that
fictitious moon, {\it to\/} which it has been seen that one of
the two principal components of the sun's disturbing force is
directed: an opposition of sign which may at first surprise, but
which is exactly analogous to the {\it contraction\/} of the
orbit produced by the {\it ablatitious\/} force (when the
periodic time is given), and is to be explained upon similar
principles. Finally, the third term of the formula (43) for
$\delta \beta$, expresses that with the two foregoing
displacements a third is to be combined, which is, like them, of
constant amount, being equal to ${3 \over 19}$ths of the second
displacement, or bearing to the radius of the moon's orbit the
ratio of $\displaystyle {3 m^2 \over 16}$ to unity; but being
always directed {\it to\/} what, by an extension of a recently
employed phaseology, might be called the second fictitious moon,
being so placed that the actual moon is midway in the heavens
between {\it this\/} fictitious moon and the one which was before
considered. These two latter terms of (43) contain the chief laws
of the {\it Lunar Variation\/}: and are easily shown to give the
known terms in the expressions of the moon's parallax and
longitude,
$$\eqalignno{
\delta {1 \over r}
&= m^2 \cos 2 (\moon - \sun);
&(44)\cr
\delta \theta
&= {11 m^2 \over 8} \sin 2 (\moon - \sun).
&(45)\cr}$$
It may assist some readers to observe here, that when the
inclination of the orbit is neglected, the longitudes of the
first and second fictitious moons are, respectively,
$$2 \sun - \moon,
\quad\hbox{and}\quad
3 \moon - 2 \sun;
\eqno (46)$$
while those of the first and second fictitious suns, mentioned in
a former section of this abstract, are, under the same condition,
$$2 \moon - \sun,
\quad\hbox{and}\quad
3 \sun - 2 \moon.
\eqno (47)$$
\bigbreak
VII.
The law and quantity of the regression of the Moon's Node may also
be calculated on principles of the kind above stated, but we must
content ourselves with writing here the formula for the angular
velocity of a planet's node generally, considered as depending on
the variable {\it vector of position\/} $\alpha$, the {\it vector
of velocity\/}
$\displaystyle {{\rm d} \alpha \over {\rm d} t}$,
and the {\it vector of acceleration\/}
$\displaystyle {{\rm d}^2 \alpha \over {\rm d} t^2}$,
and also on a vector unit~$\lambda$, supposed to be directed
towards the north pole of a fixed ecliptic. The formula thus
referred to is the following:
$$d \Omega
= {{\sc s} \mathbin{.} \alpha \lambda \mathbin{.}
{\sc s} \mathbin{.} {\rm d}^2 \alpha \,
{\rm d} \alpha \, \alpha
\over
( {\sc v} \mathbin{.} \lambda \, {\sc v} \mathbin{.}
\alpha \, {\rm d} \alpha )^2},
\eqno (48)$$
where ${\sc s}$ and ${\sc v}$ are, as before, the characteristics
of the operations of taking the scalar and vector of a
quaternion. The author proposes to give a fuller account of his
investigations on this class of dynamical questions, when the
Third Series of his Researches respecting Quaternions shall come
to be printed in the Transactions of the Academy: the Second
Series being devoted to subjects more purely geometrical; as the
First Series (already printed) relates chiefly to others which
are of a more algebraical character.
\bye