The chief purpose of the paper was to indicate the rise of transformations of the type where the integral in the first member of the equation is taken throughout a closed surface, and that in the second member over the surface, a being the angle made with the axis of x by the normal to the element dS drawn outwards. It is on this transformation the analytical proof of Green's theorem depends, and it was shown to have been employed in various forms by Poisson, Duhamel, Gauss, and others, before Green's essay was generally known on the Continent. It may be observed that the essay was published at Nottingham in 1828, and seems to have been unknown to continental mathematicians till its reprint in Crelle's Journal, vols. 39 (1850), 44 (1852), and 47 (1854).