Euler considers general linear continua, where the body in question is an unknown curve but is described with respect to a known reference curve. In this case, Euler interprets the integral as being an integration along the reference curve. Together with E410, this work combines the elastica and catenary into one conceptual framework. (Based on Clifford Truesdell's An idiot's fugitive essays on science: methods, criticisms, training, circumstances.)
Original Source Citation
Novi Commentarii academiae scientiarum Petropolitanae, Volume 20, pp. 286-303.
Opera Omnia Citation
Series 2, Volume 11, pp.180-193.