Royal Society. 4A7 



these equations a second set of unknown quantities (" inconnues auxi- 

 liaires"), being the inclinations of the deflection-curve at the points 

 of support, and, not having arrived at a general method of eliminating 

 these latter, was obliged to operate in each case on a number of 

 equations equal to twice the number of spans. M. Clapeyron does 

 not appear, as yet, to have made any formal publication of his method, 

 but to have used it in his own practice, and communicated it freely 

 to those with whom he came into contact. 



In 1856, M. Bertot, Ingenieur Civil, appears to have found the 

 means of eliminating this second set of unknown quantities n-\-l in 

 number for a bridge of n spans, and thus reducing the number of 

 equations to n — 1. 



Each of these equations involved as unknown quantities the bend- 

 ing-moments over three consecutive supports, and was considered, 

 from its remarkable symmetry and simplicity, to merit a distinctive 

 name, that of " The Theorem of the three Moments." 



The method, however, to which this theorem is the key, is still 

 everywhere called that of M. Clapeyron, and, as it appears to the 

 writer, justly so, as it was an immediate and simple result from his 

 investigations, with which M. Bertot was well acquainted. 



The next important advance was made in 1861, when M. Bresse, 

 Professeur de Mecanique appliquee a l'Ecole Imperiale des Ponts et 

 Chaussees, completed the matter of the third volume of his course, 

 which is exclusively devoted to this subject*. M. Bresse explains 

 and demonstrates the theorem of the three moments, at the know- 

 ledge of which he had himself arrived from M. Clapeyron' s investi- 

 gations, independently of M. Bertot. He then goes on to the inves- 

 tigation of an equation of much greater generality, in which what is 

 termed by English writers "imperfect continuity" is taken into ac- 

 count, 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 un- 

 favourable 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 me- 

 thods 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 



* This was communicated to the Academy of Sciences in 1SG2, though the 

 Tolume was not published till 1865. 



