1. Picking up Gunter Weiss's helpful suggestion about H P Hudson's 'Cremona transformations', I consulted its bibliography which lists three works by Hirst on the subject:
'On the quadric inversion of plane curves', Proc Roy Soc London 14 (1865), 91-106 (This also appeared in Italian, in Rome -- Ann di M 7 (1865), 49-65; and in Naples -- Battaglini G 4 (1866), 278-293)
'Sur la transformation quadrique', Nouv. Ann (2) 5 (1866), 213-218
'On quadric transformation', Quart J 17 (1881), 301-311 (Hudson remarks that an abstract of this paper appeared in the British Association report for 1865 (Birmingham))
Gunter Weiss's citation of the 1864 (Bath) meeting of the British Association refers to another short report by Hirst, 'On a generalization of the method of geometrical inversion', which is not included in Hudson's bibliography.
2. H P Hudson is Hilda Hudson (1881-1965). For people who don't know it, her book is a fine achievement, whose final chapter (pp. 388-395) is a history of Cremona transformations from Apollonius onwards. Her discussion of Hirst (this is getting towards answering Clark Kimberling's query) is contained in the following sentence (which illustrates her compact historiographical style):
"Inversion, as a transformation of points, is treated by Plucker, 1831-4, Bellavitis, who extends it to space and generalizes it into conical inversion, Thomson, who was led to it in his electro-static researches, Liouville, who elaborates Thomson's method and introduces the phrase 'reciprocal radii', Mobius, whose Kreisverwandtschaft became widely known, Hirst, after whom this form of involution is often named, and many other writers of the period."
3. Gunter made a very helpful and interesting final remark about the value and possible use of Hirst inversion in high school classes. If any school-teachers would take up the challenge and share their lesson plans for such a classroom discussion, it would be a great service to those concerned about the uses of history in the mathematics classroom and would be worth sharing more widely for those concerned about geometry education.