Mr. J. M. Heppel on the Theory of Continuous Beams. 57 



Messrs. Molinos and Pronnier, in their work entitled < Traite The'orique et 

 Pratique de la construction des Ponts Metalliques,' describe this process 

 fully, and show that for a bridge of n openings, the solution must be effected 

 of 3n-\- 1 equations, involving as many unknown quantities, these equations 

 being themselves of a complex character ; and they observe, " Thus to find 

 the curve of the moments of rupture for a bridge of 6 spans 1 9 equations 

 must be operated on ; such calculations would be repulsive, and when the 

 number of spans is at all considerable this method must be abandoned." 

 " The method of M. Navier, however, remained the only one available 

 till about 1849, when M. Clapeyron, Ingenieur des Mines, and Member of 

 the Academy of Sciences, being charged with the construction of the Pont 

 d'Asnieres, a bridge of five continuous spans over the Seine, near Paris, 

 applied himself to seek some more manageable process. He appears to 

 have perceived (and so far as the writer is informed, to have been the first 

 to perceive) that if the bending moments over the supports at the ends of 

 any span were known as well as the amount and distribution of the load, the 

 entire mechanical condition of this portion of the beam would become known 

 just as if it were an independent beam. Upon this M. Clapeyron proceeded 

 to form a set of equations involving as unknown quantities the bending 

 moments over the supports, with a view to their determination. He found 

 himself, however, obliged to introduce into these equations a second set of 

 unknown quantities (" inconnues auxiliaires"), 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 in contact. 



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

 of eliminating this second set of unknown quantities n + 1 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 bending 

 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 every- 

 where 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 Me'canique 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 de- 



* This was communicated to the Academy of Sciences in 1862, though the volume 

 was not published till 1865. 



