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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 INTEGRATIONS OF}

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\centerline{\Largebf CERTAIN EQUATIONS}

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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,
   6 (1858), pp.\ 62--63.)}

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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 Integrations of certain Equations.}

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\centerline{Sir William Rowan Hamilton.}

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\centerline{Communicated February 27th, 1854.}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
   vol.~vi (1858), pp.\ 62--63.]}

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Rev.\ Dr.\ Graves read a note from Sir W.~R. Hamilton, in which
he stated that he had lately arrived at a variety of results
respecting the integrations of certain equations, which might not
be unworthy of the acceptance of the Academy, and the
investigation of which had been suggested to him by
Mr.~Carmichael's printed Paper, and by a manuscript which he had
lent Sir W.~Hamilton, who writes,---``In our considerations we do
not quite agree, but I am happy to acknowledge my obligations to
his writings for the suggestions above alluded to, as I shall
hereafter more fully express.

``So long ago as 1846, I communicated to the Royal Irish Academy
a transformation which may be written thus (see Proceedings for
the July of that year):
$$D_x^2 + D_y^2 + D_z^2 = - (i D_x + j D_y + k D_z)^2;
   \eqno (1)$$
and which was obviously connected with the celebrated equation of
Laplace.

``But it had quite escaped my notice that the principles of
quaternions allow also this other transformation, which
Mr.~Carmichael was the first to point out:
$$D_x^2 + D_y^2 + D_z^2
   = (D_z - i D_x - j D_y) (D_z + i D_x + j D_y).
   \eqno (2)$$
And therefore I had, of course, not seen, what Mr.~Carmichael has
since shown, that the integration of Laplace's equation of the
{\it second\/} order may be made to depend on the integrations of
{\it two linear\/} and conjugate equations, of which one is
$$(D_z - i D_x - j D_y) V = 0.
   \eqno (3)$$

``I am disposed, for the sake of reference, to call this
`{\it Carmichael's Equation\/};' and have had the pleasure of
recently finding its integral, under a form, or rather forms, so
general as to extend even to {\it bi\/}quaternions.

``One of these forms is the following:\footnote*{``{\it Note,
added during printing}.---Since writing the above, I have
convinced myself that Mr.~Carmichael had been in full possession
of the exponential form of the integral, and probably also of my
chief transformations thereof; although he seems to have chosen
to put forward more prominently certain other forms, to which I
have found objections, arising out of the non-commutative
character of the symbols $i \, j \, k$ as factors, and on which
forms I believe that he does not now insist.---W.~R.~H.''}
$$V_{xyz} = e^{z (i D_x + j D_y)} V_{xy0}.
   \eqno (4)$$
``Another is
$$V_{xyz}
   =  (D_z + i D_x + j D_y)
         \int_0^z \cos \{ z (D_x^2 + D_y^2)^{1 \over 2} \}
            V_{xy0} \, dz;
   \eqno (5)$$
where $V_{xy0}$ is generally an {\it initial biquaternion\/}; and
where the {\it single\/} definite integral admits of being
usefully put under the form of a {\it double definite integral},
exactly analogous to, and (when we proceed to Laplace's equation)
reproducing, a well known expression of Poisson's, to which
Mr.~Carmichael has referred.

``These specimens may serve to show to the Academy that I have
been aiming to collect materials for future communications to
their Transactions.''

\bye

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