Mr. J. M. Heppel on the Theory of Continuous Beams. 449 



Pole had to investigate the case of a much larger work, the Britan- 

 nia Bridge, where he had to deal with some new conditions, which, 

 as far as the writer is aware, were then for the first time successfully 

 treated. 



These were that, besides variation of load on the different spans, 

 their cross sections also varied, and there was imperfect continuity 

 over the centre pier — that is to say, the points of support being 

 supposed to range in a straight line, the beam if relieved from all 

 weight would cease to remain in contact with them all, and would 

 consist of two equal straight portions, forming an angle pointing 

 upwards. The process which, for distinction, may be called that 

 of M. Navier was skilfully extended by Mr. Pole so as to include 

 these new circumstances ; and by its means results were obtained 

 certainly true within a very small limit, and as near the absolute 

 truth as any existing means of treating the subject would produce. 



Mr. Pole's researches on this subject are published in Mr. Edwin 

 Clark's work on the Britannia and Conway Bridges, 1850. Both 

 from the clear and accurate treatment of the case and the record of 

 the numerous and delicate observations by which the theoretical 

 conclusions were continually verified and kept in check, they are 

 most strongly to be recommended to the attention of engineers 

 having to deal with works of this character. 



The sequence of events now compels the writer to advert to some 

 studies of his own. In 1858-59, being then Chief Engineer of the 

 Madras Railway, he had occasion to investigate the conditions of a 

 bridge of five continuous spans over the river Palar. Having in India 

 no books to refer to but those of Moseley and Edwin Clark, he found 

 himself unable to extend the treatment of the cases there given to 

 that of a beam with an increased number of openings and varying 

 loads. After many attempts and failures, the same idea occurred to 

 him which appears to have struck M. Clapeyron nine or ten years 

 before — that if the bending-moments over the supports were known, 

 the whole conditions would become known. 



Following this clue, he was fortunate enough to succeed in at 

 once eliminating the other unknown quantities, which M. Clapeyron 

 had been obliged to retain in his equations for many years after his 

 original discovery of the method, and thus to arrive at an equation 

 precisely identical with that which had been first published in France 

 by M. Bertot in 1856, and was known as the " Theorem of the three 

 Moments." 



This was sufficient for the immediate purpose, as the beams in 

 question were straight and of uniform section throughout, conditions 

 to which this theorem is strictly applicable without any modification 

 whatever. 



As, however, the writer was at this time under the impression 

 that he was using an entirely new mode of analysis, he was natu- 

 rally anxious to check its results by comparison with those obtained 

 in some well-known case by other means. Fortunately he had at 

 hand that of the Britannia Bridge, perhaps the best that could have 

 been selected ; but for this purpose it became necessary to import into 



