William Price (1) was on the List of Retired Navy Surgeons in 1841 and 1858.
Leeds School of Medicine
He was one of the first lecturers of the Leeds School of Medicine -
In June 1831 Mr. Price senior, formerly a naval surgeon, was invited to join a group of Leeds doctors who had decided that it was high time the dominance of London over medical training was ended. Previously, any aspiring English physician or surgeon had had to attend the courses of the Society of Apothecaries or the Royal College of Surgeons in London, and the expense and inconvenience of having to study so far from home were compounded, in the opinion of many, by the moral dangers to which innocent young men from the North were exposed in the wicked metropolis.  To Prove I'm not Forgot: Living and dying in a Victorian City - Sylvia M. Barnard
In the 1861 UK Census William Price was 75 years old and lived with his wife Catherine age 73 and their son William Nicholson Price age 34, daughter Maria E age 31 (unmarried) Mary F age 29 (unmarried). Also residing with them is William F. Watts a medical student age 18, and servant/cook Emily Hodgson. Both William Price and his son William Nicholson Price give their occupations as surgeon. Their address is the Hospital for women and children, Leeds.
Death of William Price
William Price died at Leeds on 20 September 1867. 
Death of William Nicholson Price in 1888
British Medical Journal Obituary of William Nicholson Price who died in 1888........ By the death of Mr. William Nicholson Price, which occurred on June 25th, Leeds loses one of its most prominent medical practitioners, and the profession of Leeds a much valued colleague. The deceased gentleman, who was 62 years of age at the time of his death, was the second son of the late William Price, at one time a naval surgeon, but who for many years practised in Leeds, being one of the first lecturers at the Leeds School of Medicine.