450 Royal Society : — 



the fundamental equation the conditions of varying sections in the 

 different spans and imperfect continuity. This, however, presented 

 no great difficulty ; and by means of an equation thus modified, he 

 had the satisfaction of reproducing all Mr. Pole's results, and thus 

 convincing himself of the trustworthiness of the method in question. 



The equation thus generalized is absolutely identical with that 

 arrived at by M. Belanger in the work above referred to*. 



It would appear, then, that the theory of this subject was indepen- 

 dently advanced to about the same state of perfection in France and 

 in England, though as regards the development of its application to 

 practice no doubt very much the more has been done in the former 

 country. 



The writer will now advert to some inherent defects of this theory, 

 the cure of which is the principal object of the investigation which 

 follows. 



The chief one, which is admitted by all writers on the subject, 

 is the necessity for supposing the moment of inertia of the section 

 constant throughout each span ; any more general hypothesis, it is 

 said, would render the calculation inextricable. Still it is certain that 

 the conclusions arrived at on the hypothesis of a constant section 

 cease to be true if a variation of section is introduced ; and the amount 

 of error thereby induced, though considered to be probably small, is 

 still a matter of uncertainty. 



The next defect is the assumption of uniformity of load through- 

 out each span ; for although as far as rolling load is concerned no 

 more correct hypothesis could be made, the weight of the bridge 

 itself, if a large one, usually varies considerably in the different parts 

 of the same span. 



The equation given by M. Bresse, as has been stated, provides for 

 certain kinds of variable loads by the use of integrals ; but the writer 

 is not aware that they have been applied, even by that author him- 

 self, to the purposes of calculation, and it seems to him that in most 

 cases the attempt to make such an application would be beset with 

 difficulties. 



It will, however, it is hoped, be seen from what follows, that the 

 dealing with variations of the above elements does not in fact pre- 

 sent any very formidable difficulty, though no doubt the labour of 

 calculation is greater ; but what the writer regards as most satisfac- 

 tory is the very small difference in the principal results in the case of 

 the Britannia (where these variations greatly exceed in amount those 

 usually occurring), whether obtained by the approximate method 

 hitherto followed, or by the more rigorous one to be explained, afford- 

 ing a strong presumption that in all ordinary cases the former method 

 may be confidently employed without risk of any important error. 



Should the following treatment of the case be deemed successful, 

 the author would remark that its success is mainly due to the use 

 of an abbreviated functional notation, by which a great degree of 

 clearness and symmetry is preserved in expressions which would 

 otherwise have become inextricably complex. 



* A paper on this subject bj the writer was published in the Minutes of Pro- 

 ceedings of Inst. C. E. vol. xix. 1850-60. 



