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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 CERTAIN DISCONTINUOUS INTEGRALS}

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\centerline{\Largebf CONNECTED WITH THE DEVELOPMENT OF}

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\centerline{\Largebf THE RADICAL WHICH REPRESENTS THE}

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\centerline{\Largebf RECIPROCAL OF THE DISTANCE BETWEEN}

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\centerline{\Largebf TWO POINTS}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Philosophical Magazine, 20 (1842), pp.\ 288--294.)}

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

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\centerline{\largerm 2000}

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\noindent
{\largeit On certain discontinuous Integrals, connected with the
Development of the Radical which represents the Reciprocal of the
Distance between two Points.  By\/}
{\largesc William Rowan Hamilton}, {\largeit LL.D., P.R.I.A.,
Member of several Scientific Societies at Home and Abroad,
Andrews' Professor of Astronomy in the University of Dublin, and
Royal Astronomer of Ireland\/}\footnote*{Communicated by the
Author.}.

\bigbreak

\vskip 12pt

\centerline{[{\it The London, Edinburgh and Dublin Philosophical
Magazine and Journal of Science,}}
\centerline{3rd series, vol.~xx (1842), pp.\ 288--294.]}

\bigskip

1.
It is well known that the radical
$$(1 - 2 xp + x^2)^{-{1 \over 2}},
   \eqno (1.)$$
in which $x$ and $1$ may represent the radii vectores of two
points, while $p$ represents the cosine of the angle between
those radii, and the radical represents therefore the reciprocal
of the distance of the one point from the other, may be developed
in a series of the form
$${\rm P}_0 + {\rm P}_1 x + {\rm P}_2 x^2 + \ldots
       + {\rm P}_n x^n + \ldots;
   \eqno (2.)$$
the coefficients~${\rm P}_n$ being functions of $p$, and
possessing many known properties, among which we shall here
employ the following only,
$${\rm P}_n
   =  [0]^{-n} \left( {d \over dp} \right)^n
         \left( {p^2 - 1 \over 2} \right)^n;
   \eqno (3.)$$
the known notation of factorials being here used, according to
which
$$[0]^{-n}
   =  {1 \over 1} \mathbin{.}
      {1 \over 2} \mathbin{.}
      {1 \over 3} \, \ldots \,
      {1 \over n}.
   \eqno (4.)$$

It is proposed to express the sum of the first $n$ terms of the
development~(2.), which may be thus denoted,
$$\Sigma_{(n) \,}{}_0^{n-1} {\rm P}_n x^n
   =  {\rm P}_0 + {\rm P}_1 x + {\rm P}_2 x^2 + \ldots
       + {\rm P}_{n-1} x^{n-1}.
   \eqno (5.)$$

\bigbreak

2.
In general, by Taylor's theorem,
$$f(p + q)
   =  \Sigma_{(n) \,}{}_0^\infty [0]^{-n} q^n
         \left( {d \over dp} \right)^n f(p);
   \eqno (6.)$$
hence, by the property~(3.), ${\rm P}_n$ is the coefficient of
$q^n$ in the development of
$$\left( {(p + q)^2 - 1 \over 2} \right)^n;
   \eqno (7.)$$
it is therefore also the coefficient of $q^0$ in the development
of
$$\left( {p^2 - 1 \over 2q} + p + {q \over 2} \right)^n
   \eqno (8.)$$
If then we make, for abridgment,
$$\vartheta
   =  p + {p^2 \over 2} \cos \theta
       + \sqrt{-1} \left(1 - {p^2 \over 2} \right) \sin \theta,
   \eqno (9.)$$
we shall have the following expression, which perhaps is new, for
${\rm P}_n$:
$${\rm P}_n
  =  {1 \over 2\pi} \int_{-\pi}^\pi
         \vartheta^n \, d\theta;
   \eqno (10.)$$
and hence, immediately, the required sum (5.) may be expressed as
follows:
$$\Sigma_{(n) \,}{}_0^{n-1} {\rm P}_n x^n
   =  {1 \over 2 \pi} \int_{-\pi}^\pi
         d \theta \, {1 - \vartheta^n x^n \over 1 - \vartheta x};
   \eqno (11.)$$
in which it is to be observed that $x$ may be any quantity, real
or imaginary.

\bigbreak

3.
We have therefore, rigorously, for the sum of the $n$ first
terms of the series
$${\rm P}_0 + {\rm P}_1 + {\rm P}_2 + \ldots,
   \eqno (12.)$$
the expression
$$\Sigma_{(n) \,}{}_0^{n-1} {\rm P}_n
   =  {1 \over 2 \pi} \int_{-\pi}^\pi
         d \theta \, {1 - \vartheta^n \over 1 - \vartheta};
   \eqno (13.)$$
of which we propose to consider now the part independent of $n$,
namely,
$${\rm F}(p)
   =  {1 \over 2\pi} \int_{-\pi}^\pi
         {d \theta \over 1 - \vartheta};
   \eqno (14.)$$
and to examine the form of this function~${\rm F}$ of $p$, at
least between the limits $p = -1$, $p = 1$.

\bigbreak

4.
A little attention shows that the denominator $1 - \vartheta$ may
be decomposed into factors, as follows:
$$1 - \vartheta
   =   - {\textstyle {1 \over 2}}
         (\alpha + e^{\theta \sqrt{-1}})
         (1 - \beta e^{-\theta \sqrt{-1}});
   \eqno (15.)$$
in which,
$$\alpha = 2s (1 - s),\quad
  \beta  = 2s (1 + s),
   \eqno (16.)$$
and
$$p = 1 - 2 s^2;
   \eqno (17.)$$
so that $s$ may be supposed not to exceed the limits $0$ and $1$,
since $p$ is supposed not to exceed the limits $-1$ and $1$.
Hence
$${1 \over 1 - \vartheta}
   = {-2 (\alpha + e^{- \theta \sqrt{-1}})
            (1 - \beta e^{\theta \sqrt{-1}})
         \over (1 + 2 \alpha \cos \theta + \alpha^2)
               (1 - 2 \beta  \cos \theta + \beta^2)};
   \eqno (18.)$$
of which the real part may be put under the form
$${\lambda \over 1 + 2 \alpha \cos \theta + \alpha^2}
       + {\mu \over 1 - 2 \beta \cos \theta + \beta^2},
   \eqno (19.)$$
if $\lambda$ and $\mu$ be so chosen as to satisfy the conditions
$$\lambda (1 + \beta^2) + \mu (1 + \alpha^2)
   =  2 (\beta - \alpha),
   \eqno (20.)$$
$$\lambda \beta - \mu \alpha = 1 - \alpha \beta,
   \eqno (21.)$$
which give
$$\lambda = {1 - \alpha^2 \over \alpha + \beta},\quad
  \mu     = {\beta^2 - 1  \over \alpha + \beta}.
   \eqno (22.)$$

The imaginary part of the expression (18.) changes sign with
$\theta$, and disappears in the integral~(14.); that integral
therefore reduces itself to the sum of the two following:
$${\rm F} (p)
   =  {1 \over 4 s \pi} \int_0^\pi
         { (1 - \alpha^2) \, d \theta
            \over 1 + 2 \alpha \cos \theta + \alpha^2 }
    + {1 \over 4 s \pi} \int_0^\pi
         { (\beta^2 - 1)  \, d \theta
            \over 1 - 2 \beta  \cos \theta + \beta^2 };
   \eqno (23.)$$
in which, by (16.), $\alpha + \beta$ has been changed to $4s$.
But, in general if $a^2 > b^2$,
$$\int_0^\pi {d \theta \over a + b \cos \theta}
   =  {\pi \over \sqrt{a^2 - b^2}},
   \eqno (24.)$$
the radical being a positive quantity if $a$ be such; therefore
in the formula (23.),
$$\int_0^\pi
         { (1 - \alpha^2) \, d \theta
            \over 1 + 2 \alpha \cos \theta + \alpha^2 }
   =  \pi,
   \eqno (25.)$$
because, by (16.), $\alpha$ cannot exceed the limits $0$ and
${1 \over 2}$, $s$ being supposed not to exceed the limits $0$
and $1$, so that $1 - \alpha^2$ is positive.  On the other hand,
$\beta$ varies from $0$ to $4$, while $s$ varies from $0$ to $1$;
and $\beta^2 - 1$ will be positive or negative, according as $s$
is greater or less than the positive root of the equation
$$s^2 + s = {\textstyle {1 \over 2}}.
   \eqno (26.)$$
Hence, in (23.), we must make
$$\int_0^\pi
         { (\beta^2 - 1)  \, d \theta
            \over 1 - 2 \beta  \cos \theta + \beta^2 }
   =  \pi, \hbox{ or } = - \pi,
   \eqno (27.)$$
according as
$$s > \hbox{ or } < {\surd 3 - 1 \over 2};
   \eqno (28.)$$
and thus we find, under the same alternative,
$${\rm F}(p) = {1 \over 4s} (1 \pm 1),
   \eqno (29.)$$
that is,
$${\rm F}(p) = {1 \over 2s}, \hbox{ or } = 0.
   \eqno (30.)$$
But, by (17.),
$$s = \sqrt{1 - p \over 2};
   \eqno (31.)$$
the function~$F(p)$, or the definite integral~(14.), receives
therefore a sudden change of form when $p$, in varying from $-1$
to $1$, passes through the critical value
$$p = \surd 3 - 1;
   \eqno (32.)$$
in such a manner that we have
$${\rm F}(p) = (2 - 2p)^{-{1 \over 2}},
   \quad \hbox{if}\quad p < \surd 3 - 1;
   \eqno (33.)$$
and, on the other hand,
$${\rm F}(p) = 0
   \quad \hbox{if}\quad p > \surd 3 - 1;
   \eqno (34.)$$
For the critical value~(32.) itself, we have
$$s = {\surd 3 - 1 \over 2},\quad
  \alpha = 2 \surd 3 - 3,\quad
  \beta = 1,
   \eqno (35.)$$
and the real part of (18.) becomes
$${1 - \alpha \over 1 + 2 \alpha \cos \theta + \alpha^2};
   \eqno (36.)$$
multiplying therefore by $d \theta$, integrating from
$\theta = 0$ to $\theta = \pi$, and dividing by $\pi$, we find,
by (25.) and (14.), this formula instead of (29.),
$$F(p) = {1 \over 1 + \alpha} = {1 \over 4s},
   \eqno (37.)$$
that is,
$${\rm F}(p) = {\textstyle {1 \over 2}} (2 - 2p)^{-{1 \over 2}},
   \quad \hbox{if}\quad p = \surd 3 - 1;
   \eqno (38.)$$

The value of the discontinuous function~${\rm F}$ is therefore,
in this case, equal to the semisum of the two different values
which that function receives, immediately before and after the
variable~$p$ attains its critical value, as usually happens in
other similar cases of discontinuity.

\bigbreak

5.
As verifications of the results (33.), (34.), we may consider the
particular values $p = 0$, $p = 1$, which ought to give
$${\rm F}(0) = 2^{-{1 \over 2}},\quad
  {\rm F}(1) = 0.
   \eqno (39.)$$
Accordingly, when $p = 0$, the definitions (9.) and (14.) give
$$\vartheta = \sqrt{-1} \sin \theta,
   \eqno (40.)$$
$${\rm F}(0)
   =  {1 \over 2 \pi} \int_{-\pi}^\pi
         {d \theta \over 1 - \sqrt{-1} \sin \theta}
   =  {1 \over \pi} \int_0^\pi
         {d \theta \over 1 + \sin \theta^2};
   \eqno (41.)$$
which easily gives, by (24.),
$${\rm F}(0)
   =  {2 \over \pi} \int_0^\pi
         {d \theta \over 3 - \cos 2 \theta}
   =  {1 \over \pi} \int_0^{2\pi}
         {d \theta \over 3 - \cos \theta}
   =  2^{-{1 \over 2}}.
   \eqno (42.)$$

And when $p = 1$, we have
$$1 - \vartheta
   =  - {\textstyle {1 \over 2}}
         (\cos \theta + \sqrt{-1} \sin \theta),
   \eqno (43.)$$
$${1 \over 2 \pi} {d \theta \over 1 - \vartheta}
   =  - \pi^{-1} (\cos \theta - \sqrt{-1} \sin \theta) \, d \theta,
   \eqno (44.)$$
of which the integral, taken from $\theta = - \pi$ to
$\theta = \pi$, is ${\rm F}(1) = 0$.

\bigbreak

6.
Let us consider now this other integral,
$${\rm G}(p)
   =  {1 \over 2 \pi} \int_{-\pi}^\pi
         {\vartheta^n \, d \theta \over \vartheta - 1}.
   \eqno (45.)$$
The expression (13.) gives
$$\Sigma_{(n) \,}{}_0^{n-1} {\rm P}_n
   = {\rm F}(p) + {\rm G}(p);
   \eqno (46.)$$
therefore, by (34.), we shall have
$${\rm G}(p) = \Sigma_{(n) \,}{}_0^{n-1} {\rm P}_n,
   \quad \hbox{if}\quad p > \surd 3 - 1.
   \eqno (47.)$$
For instance, let $p = 1$; then multiplying the expression~(44.) by
$$- \vartheta^n
   =  - (1 + {\textstyle {1 \over 2}} e^{\theta \sqrt{-1}})^n,
   \eqno (48.)$$
the only term which does not vanish when integrated is
${1 \over 2} n \pi^{-1} \, d \theta$, and this term gives the result
$${\rm G}(1) = n,
   \eqno (49.)$$
which evidently agrees with the formula~(47.), because it is well
known that
$${\rm P}_n = 1,
   \quad\hbox{when}\quad p = 1,
   \eqno (50.)$$
the series~(2.) becoming then the development of $(1 - x)^{-1}$.

\bigbreak

7.
On the other hand, let $p$ be $< \surd 3 - 1$; then, observing
that, by (33.),
$${\rm F}(p)
   =  (2 - 2p)^{-{1 \over 2}}
   =  \Sigma_{(n) \,}{}_0^\infty {\rm P}_n,
   \eqno (51.)$$
we find, by the relation~(46.) between the functions ${\rm F}$
and ${\rm G}$,
$${\rm G}(p)
   =  - \Sigma_{(n) \,}{}_n^\infty {\rm P}_n
   =  - ( {\rm P}_n + {\rm P}_{n+1} + {\rm P}_{n+2} + \ldots ).
   \eqno (52.)$$
For instance, let $p = 0$; then, by (40.) and (45.),
$${\rm G}(0)
   =  {- (\sqrt{-1})^n \over 2 \pi} \int_{-\pi}^\pi
         {d\theta \, (\sin \theta)^n
            \over 1 - \sqrt{-1} \sin \theta};
   \eqno (53.)$$
that is
$${\rm G}(0)
   =  {(-1)^{i+1} \over \pi} \int_0^\pi
         \int_0^\pi
         {d \theta \, \sin \theta^{2i}
            \over 1 + \sin \theta^2};
   \eqno (54.)$$
if $n$ be either $= 2i - 1$, or $= 2i$.  Now, when $p = 0$,
${\rm P}_n$ is the coefficient of $x^n$ in the development of
$(1 + x^2)^{-{1 \over 2}}$; therefore,
$${\rm P}_{2i-1} = 0,
   \quad\hbox{when}\quad p = 0,
   \eqno (55.)$$
and, in the notation of factorials,
$${\rm P}_{2i}
   =  [0]^{-i} [{-\textstyle {1 \over 2}}]^i
   =  (-1)^i \pi^{-1} \int_0^\pi d \theta \, \sin \theta^{2i};
   \eqno (56.)$$
so that, by (54.),
$${\rm G}(0)
   =  - ({\rm P}_{2i} + {\rm P}_{2i+2} + \ldots ),
   \eqno (57.)$$
when $p = 0$, and when $n$ is either $2i$ or $2i - 1$.

\bigbreak

8.
For the critical value $p = \surd 3 - 1$, we have, by (38.),
$${\rm F}(p)
   =  {\textstyle {1 \over 2}} \Sigma_{(n) \,}{}_0^\infty {\rm P}_n;
   \eqno (58.)$$
therefore, for the same value of $p$, by (46.),
$$\eqalignno{
{\rm G}(p)
   &= {\textstyle {1 \over 2}} \Sigma_{(n) \,}{}_0^{n-1} {\rm P}_n
    - {\textstyle {1 \over 2}} \Sigma_{(n) \,}{}_n^\infty {\rm P}_n \cr
   &= {\textstyle {1 \over 2}} (
         {\rm P}_0 + {\rm P}_1 + \ldots + {\rm P}_{n-1}
       - {\rm P}_n - {\rm P}_{n+1} - \ldots );
   &(59.)\cr}$$
so that the discontinuous function~${\rm G}$, like ${\rm F}$,
acquires, for the critical value of $p$, a value which is the
semisum of those which it receives immediately before and
afterwards.

\bigbreak

9.
We have seen that the sum of these two discontinuous integrals,
${\rm F}$ and ${\rm G}$, is always equal to the sum of the first
$n$ terms of the series (12.), so that
$${\rm F}(p) + {\rm G}(p)
   =  {\rm P}_0 + {\rm P}_1 + \ldots + {\rm P}_{n-1};
   \eqno (60.)$$
and it may not be irrelevant to remark that this sum may be
developed under this other form:
$${1 \over 2\pi} \int_{-\pi}^\pi d \theta \,
         {\vartheta^n - 1 \over \vartheta - 1}
   =  \Sigma_{(k) \,}{}_1^n [n]^k [0]^{-k} {\rm Q}_{k-1};
   \eqno (61.)$$
in which the factorial expression $[n]^k [0]^{-k}$ denotes the
coefficient of $x^k$ in the development of $(1 + x)^n$; and
$${\rm Q}_k
   =  {1 \over 2 \pi} \int_{-\pi}^\pi d \theta \,
         (\vartheta - 1)^k.
   \eqno (62.)$$
Thus
$$\left. \eqalign{
   & {\rm P}_0
      =  {\rm Q}_0; \cr
   & {\rm P}_0 + {\rm P}_1
      =  2 {\rm Q}_0 + {\rm Q}_1; \cr
   & {\rm P}_0 + {\rm P}_1 + {\rm P}_2
      =  3 {\rm Q}_0 + 3 {\rm Q}_1 + {\rm Q}_2; \cr
   & \quad \hbox{\&c.} \cr}
   \right\}
   \eqno (63.)$$
and consequently
$$\left. \eqalign{
{\rm P}_0 &= {\rm Q}_0; \cr
{\rm P}_1 &= {\rm Q}_0 + {\rm Q}_1; \cr
{\rm P}_2 &= {\rm Q}_0 + 2 {\rm Q}_1 + {\rm Q}_2; \cr
   & \quad \hbox{\&c.} \cr}
   \right\}
   \eqno (64.)$$
which last expressions, indeed, follow immediately from the
formula (10.).

\bigbreak

10.
With respect to the calculation of ${\rm Q}_0$, ${\rm Q}_1$,
\&c.\ as functions of $p$, it may be noted, in conclusion, that,
by (15.) and (62.), ${\rm Q}_k$ is the term independent of
$\theta$ in the development of
$$2^{-k} (\alpha + e^{\theta \sqrt{-1}})^k
         (1 - \beta e^{- \theta \sqrt{-1}})^k;
   \eqno (65.)$$
thus
$$\left. \eqalign{
{\rm Q}_0 &= 1,\cr
{\rm Q}_1 &= 2^{-1} (\alpha - \beta),\cr
{\rm Q}_2 &= 2^{-2} (\alpha^2 - 4 \alpha \beta + \beta^2),\cr
{\rm Q}_3 &= 2^{-3} (\alpha^3 - 9 \alpha^2 \beta
               + 9 \alpha \beta^2 - \beta^3),\cr
   & \quad \hbox{\&c.} \cr}
   \right\}
   \eqno (66.)$$
in which the law of formation is evident.  It remains to
substitute for $\alpha$,~$\beta$ their values~(16.) as functions
of $s$, and then to eliminate $s^2$ by (17.); and thus we find,
for example,
$$\left. \eqalign{
{\rm Q}_1 &= p - 1,\cr
{\rm Q}_2 &= {\textstyle {1 \over 2}} (p - 1) (3p - 1);\cr
{\rm Q}_3 &= {\textstyle {1 \over 2}} (p - 1)^2 (5p + 1);\cr
{\rm Q}_4 &= {\textstyle {1 \over 8}} (p - 1)^2
               (35 p^2 - 10 p - 13).\cr}
   \right\}
   \eqno (67.)$$

This, then, is at least one way, though perhaps not the easiest,
of computing the initial values of the successive differences of
the function~${\rm P}_n$, that is, the quantities
$$\left. \eqalign{
{\rm Q}_0 &= \Delta^0 {\rm P}_0
   =  {\rm P}_0,\cr
{\rm Q}_1 &= \Delta^1 {\rm P}_0
   =  {\rm P}_1 - {\rm P}_0,\cr
{\rm Q}_2 &= \Delta^2 {\rm P}_0
   =  {\rm P}_2 - 2 {\rm P}_1 + {\rm P}_0,\cr
   & \quad \hbox{\&c.} \cr}
   \right\}
   \eqno (68.)$$
And we see that it is permitted to express generally those
differences, as follows:
$$\Delta^k {\rm P}_0
   =  s^k \Sigma_{(i) \,}{}_0^k (-1)^i ([k]^i [0]^{-i})^2
         (1 + s)^i (1 - s)^{k-i};
   \eqno (69.)$$
in which
$$s^2 = {\textstyle {1 \over 2}} (1 - p).
   \eqno (70.)$$

\bigbreak

Observatory of Trinity College, Dublin,\par
\qquad Feb.~12, 1842.

\bye