68 Dr. W. J. M. Bankine on Mr. HeppeV, 



considered as demonstrated that, for most ordinary cases of large bridges, 

 calculations founded on equation (26) may be confidently relied on. It 

 need scarcely be remarked that these are much more simple and easy than 

 those founded on the more exact but complex equations above given. 



In smaller bridges, however, the error of the approximate process will 

 be more considerable, and the process above given may be applied with 

 advantage to its correction. 



In concluding this paper, the author desires to record his thanks to his 

 young friend, Mr. Henry Reilly, for the patience and skill with which he 

 made, in detail, all the intricate calculations of the numerical values of the 

 various functions involved in the above demonstration. 



" Remarks on Mr. Heppel's Theory of Continuous Beams." By W. 

 J. Macquorn Bankine, C.E., LL.D., F.B.S. Received De- 

 cember 22, 1869*. 



1 . Condensed form of stating the Theory. — The advantages possessed 

 by Mr. Heppel's method of treating the mathematical problem of the state 

 of stress in a continuous beam will probably cause it to be used both in 

 practice and in scientific study. 



The manner in which the theory is set forth in Mr. Heppel's paper is 

 remarkably clear and satisfactory, especially as the several steps of the 

 algebraical investigation correspond closely with the steps of the arithme- 

 tical calculations which will have to be performed in applying the method 

 to practice. 



Still it appears to me that, for the scientific study of the principles of 

 the method, and for the instruction of students in engineering science, it 

 may be desirable to have those principles expressed in a condensed form ; 

 and with that view I have drawn up the following statement of them, which is 

 virtually not a new investigation, but Mr. Heppel's investigation abridged. 



Let (#=0, y=0) and (x=l, y=0) be the coordinates of two adjacent 

 points of support of a continuous beam, x being horizontal. Let y and the 

 vertical forces be positive downwards. 



At a given point x in the span between those points let /u be the load 

 per unit of span, and EI the stiffness of the cross-section, each of which 

 functions may be uniform or variable, continuous or discontinuous. 



In each of the following double and quadruple definite integrals, let the 

 lower limits be x=0. 



When the integrations extend over the whole span I, that will be denoted 

 by affixing 1 ; for example, n lf &c. 



* Read January 27, 1870. See vol. xviii. p. 178. 



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