Also in 1843 MacCullagh published his most important work on geometry, namely On surfaces of the second order which described how surfaces such as the ellipsoid could be generated.
Din sa 1843 MacCullagh-publish na ang kanyang pinaka-mahalagang trabaho sa geometry, namely Sa ibabaw ng mga ikalawang-order na inilarawan kung paano ibabaw tulad ng ellipsoid ay maaaring nalikha.