Dr. Rankine on Mr. HeppePs Theory of Continuous Beams, 457 



If the variation of section alone be considered, the load being 

 taken at its mean value, 



^ a 46382, 2 =34465, Y=4'52. 



Tt therefore appears that the amount of variation in the section 

 and load which occurs in each span of the Britannia Bridge, when 

 taken strictly into account, produces scarcely any effect on the 

 values of the bending-moments and deflections, which are practically 

 the same as those resulting from their mean values considered as 

 constant ; and it may be considered 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 ap- 

 plied 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 de- 

 monstration. 



" Remarks on Mr. HeppePs Theory of Continuous Beams. " By 

 W. J. Macquorn Rankine, C.E., LL.D., F.R.S. 



1. Condensed form of stating the Theory. — The advantages pos- 

 sessed 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 arithmetical 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 princi- 

 ples 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(a?=0, y=0) and (x=l, y = 0)be the coordinates of two adjacent 

 points of support of a continuous beam, x being horizontal. Lety 

 and the vertical forces be positive downwards. 



At a given point x in the span between those points let \x be the 

 load per unit of span, and El the stiffness of the cross section, each 

 of which functions may be uniform or variable, continuous or dis- 

 continuous. 



In each of the following double and quadruple definite integrals, 



Phil. Mag. S. 4. Vol. 40. No. 269. Dec. 1870. 2 H 



