58 Mr. J. M. Heppel on the Theory of Continuous Beams. 



monstrates the theorem of the three moments, at the knowledge of which 

 he had himself arrived from M. Clapeyron's investigations, independently of 

 M. Bertot. He then goes on to the investigation of an equation of much 

 greater generality, in which what is termed by English writers e ' imperfect 

 continuity " is taken into account, being, however, there replaced by the 

 precisely equivalent notion of original differences of level in the supports, 

 the beam being always supposed primitively straight; besides this the loads, 

 instead of being taken as uniform for each span, are considered as distributed 

 in any given manner. 



Having obtained this fundamental equation, M. Bresse proceeds to 

 investigate the nature of the curves, which are the envelopes of the greatest 

 bending moments produced at each point, by the most unfavourable 

 distribution of the load in reference to it, and finally gives tables for the 

 ready calculation of results in a great variety of cases, comprising most of 

 those likely to occur in practice. 



During the time that M. Bresse was engaged in these researches, an 

 Imperial Commission was formed, of which he was a member, for the 

 purpose of devising rules applicable to practice, and the results of his 

 labours have been the basis of legislative enactments equivalent to our 

 Board of Trade regulations prescribing the methods to be followed in 

 determining the stresses in the various parts of the structure. 



About the same time that M. Bresse turned his attention to this sub- 

 ject, it appears also to have engaged that of M. Belanger, who in his work 

 entitled ' Theorie de la Resistance et de la Flexion Plane des Solides &c, 

 Paris, 1862,' gives a very complete demonstration — resulting in an equation 

 which in one point of view is slightly more general than that of M. Bresse, 

 as it takes in variation of the moment of inertia of the section from one 

 span to another. In another point of view its generality is slightly less, 

 as it deals only with loads distributed over each separate span uniformly, 

 whereas M. Bresse replaces the simple algebraical terms expressing these 

 by definite integrals expressing the load as a function of the distance from 

 one of the points of support. 



As far as the writer is informed, little has been done in France to 

 advance this theory beyond the point to which it was brought by the 

 writers last mentioned, and especially by M. Bresse ; but valuable contri- 

 butions to its development in reference to application to practice are to be 

 found in the work of MM. Molinos and Pronnier above referred to, as well 

 as in various papers by MM. Renaudot, Albaret, Colignon, Piarron de 

 Mondesir, &c. 



In England little or no attention appears to have been paid to this sub- 

 ject by writers on mechanics till 1843, when the Rev. Henry Moseley, 

 Professor of Natural Philosophy and Astronomy at King's College, London, 

 published his work on 'The Mechanical Principles of Engineering and 

 xirchitecture.' In part 5 of this work, which treats of the strength of 

 materials, four cases of continuous beams are fully investigated, and the 



