Coincidentally, Murphy was presented with another opportunity to gain admission to a university when he met Mr. McCarthy, a junior fellow at Gonville and Caius College, Cambridge, while he was on vacation in Cork [Creedon 2001]. Impressed, Mr. McCarthy brought some of Murphy’s work back to Gonville and Caius and presented it to Professor Robert Woodhouse (1773-1827). Woodhouse saw potential in Murphy’s work and offered him admission to Cambridge as a result [Creedon 2001]. Several people in Mallow, led by Mulcahy, raised the tuition required to send Murphy to Gonville and Caius in October 1825 [Barry 1986]. Unfortunately, little is known about Murphy’s undergraduate studies at Cambridge; however, Venn  reported that Murphy received a “1st. math prize” in 1826. He graduated in 1829, earning a B.A. with a rank of third wrangler. A student was awarded the title of wrangler if he gained first-class honors on the Mathematical Tripos [Barry 1999]. Murphy scored the third highest on the Mathematical Tripos examination.
Figure 2. Murphy’s entry in the Gonville and Caius Matriculation Book, which was made when he arrived in 1825. (Source: Permission granted by the Master and Fellows of Gonville and Caius College, Cambridge.) The entry reads:
Murphy Robert, son of John of Mallow, Ireland, born there: educated at Mallow for three years under Mr. Armstrong; age 18; admitted on 7 July as a pensioner of this College, tutor within the same, and released by death.
Kate McQuillian, Archivist at Gonville and Caius College, provided the above translation of the excerpt from the Gonville and Caius College Matriculation Book. (See Figure 2.) Ms. McQuillian stated that the abrupt ending of the excerpt means that Murphy remained a member of the College until his death.
After graduating, Murphy was appointed a fellow of Gonville and Caius College. This fellowship gave him a small salary, and the opportunity to teach classes – he held a position as a Hebrew and Greek lecturer [Venn 2009] – and tutor students for additional pay. On June 4, 1831, he was ordained a deacon (Chichester) [Venn 2009]. Shortly after, Murphy was elected dean in May 1831. According to Barry , this position paid him handsomely. As dean, Murphy ensured that the rules of the church were enforced. During this time, Murphy began publishing papers in physics and mathematics, which we discuss in the next three sections.
Figure 3. Gonville and Caius College, Cambridge University, England, in 2011 (Source: Public domain)
Murphy received his M.A. in 1832 [Venn 2009]; however, due to a lifestyle that consisted largely of drinking and gambling, he left Cambridge in 1832 when creditors seized his property. He then traveled to West Cork, Ireland, and became a shoemaker; however, his heavy drinking continued [Barry 1986]. Murphy returned to Mallow in 1835 and, with the aid of his friends, he stopped drinking [Barry 1986]. In 1836 he was ready to return to England and restart his career in academia.
While Murphy was in Ireland, he wrote several papers on physics. He earned royalties from his book Elementary Principles of the Theories of Electricity, Heat and Molecular Action: On Electricity , which aided him in paying off a large portion of his debts. As a result of his numerous publications and growing reputation, Murphy resumed his life as a teacher and writer when he moved to London in 1836. He was granted a position as examiner of mathematics and natural philosophy at the University of London in October 1838, which he held until his death.
The next three sections are devoted to Murphy’s contributions to mathematics. Allaire  determined that Murphy’s works fall into three categories: Algebraic Equations, Integral Equations, and Operator Calculus. We adopt these classifications in this paper.