The following works fall within Murphy’s “Integral Equations” research area:

- “On the General Properties of Definite Integrals” [1830],
- “First Memoir on the Inverse Method of Definite Integrals, with Physical Applications” [1832],
- Elementary Principles of the Theories of Electricity, Heat, and Molecular Actions. Part I on Electricity [1833],
- “On the Mathematical Laws of Electrical Influence” [1833],
- “Second Memoir on the Inverse Method of Definite Integrals, with Physical Applications” [1833],
- “Third Memoir on the Inverse Method of Definite Integrals, with Physical Applications” [1835],
- “On a New Theorem in Analysis” [1837],
- “On the Composition of Two Rectangular Forces Acting on a Point” [1837],
- “On Atmospheric Refraction” [1842].

Much of Murphy’s work on integral equations was centered around integral transformations. Using Murphy’s notation, a function \(\varphi(x)\) that is equal to an integral of a function \(f(t)\) with respect to some kernel function, initially \({t^x},\) would be denoted as:\[\phi(x)={\int^1_0}{f(t)}\,{t^x}\,dt\]
[Cross 1985, p. 123]. (More generally, the kernel function, \(k(u,t),\) is a function of variables \(u\) and \(t\) that allows us to perform an integral transform from a function \(f(t)\) to a function \(g(u)\).) Murphy’s goal was to determine the function \(f.\) He adopted the limits \(0\) and \(1\) from Gauss’ *Methodus nova integralium valores per approximationem inveniendi* [1815].
Murphy referred to this problem as “an inverse method of Definite Integrals, by which we may re-ascend from the known integral, to the unknown function under the sign of definite integration” [1832b, p. 353].

Furthermore, Murphy considered both continuous and discontinuous functions in his theory of integral equations. According to Murphy, it was necessary to examine discontinuous functions because “the phænomena presented by nature are mostly of that kind” [Murphy 1832b, p. 355]. Essentially, Murphy’s examination of discontinuous functions was motivated by their applications in areas of physics, such as electrostatics, gravity, and heat [Murphy 1833a].

Figure 5. Portrait of Pierre-Simon Laplace (Source: Public domain) |

In addition to his work in mathematics, Murphy contributed to several areas of physics, such as the
subject of electricity. In his book,
*Elementary Principles of the Theo
ries of Electricity, Heat, and Molecular Actions. Part I on Electric
ity,* Murphy presented the theory of electricity. According to Cross [1985], Murphy’s book was intended to be a text for students studying
at Cambridge. At the start of his book, Murphy acknowledged that Volume III of
the *Mécanique Céleste* [1803] of Pierre-Simon Laplace (1749-1827) was “indispensable in
investigations respecting electricity”
[Murphy 1833a, p. v]. Consequently,
he introduced Laplace’s work in a section titled “Preliminary Propositions.” In general, Murphy discussed
*Laplace functions,* and this was the
first appearance of this term in English [Miller 2010]. Murphy expanded
on Laplace’s work and arrived at a
more general class of functions which
are of importance in the study of
the theory of latent electricity. According to Murphy himself, he “obtained several new and remarkable theorems with respect to Laplace’s functions” [Murphy 1833a, p. vi]. Interestingly, Murphy did not highlight what was “new” in his work because “that will easily be recognized by those who are already acquainted with the subject” [Murphy 1833a, p. vi]. An in-depth analysis of this work, to include revealing results original to Murphy, could serve as an interesting research project or potential thesis topic.

It is clear that Murphy made significant contributions to the fields of mathematics and physics, as demonstrated by his work on integral equations. Readers interested in a more in-depth analysis of Murphy’s work on integral equations and contributions to physics can refer to Cross [1985], Grattan-Guinness [1985], and Wilson [1985].

We end this section by noting a key citation that the authors previously were unaware existed. Murphy mentioned “functions of operation of the distributive kind (that is, such whose action on the whole, is the sum of the actions on the parts)” [1832b, p. 354] and referenced the work of François-Joseph Servois (1767-1847) on the calculus [Servois 1814]. Previously, there was no solid evidence that Murphy read Servois’ contributions to calculus, because Murphy made no direct reference to Servois in his “First Memoir on the Theory of Analytic Operations” [1837b], a paper in which Murphy used and expanded on many of Servois’ ideas. In the next section we provide an exposition of Murphy’s “First Memoir.”