1879.] On Most General Problems in Continuous Beams. 499 



same, and the loading over both spans is uniform and equal to w per 

 unit length, then the theorem becomes — 



M +4M 1 + M 2 --^!=0. 



When in the two consecutive spans, OQ and QE, there is the same 

 common cross-section, and when there is a uniform load, iv, per unit 

 length over the first span, and w 2 over the second, we get the theorem 

 in M. Clapeyron's form — 



+ 2M 1 (h + Z 2 ) + M 2 Z 3 - - = 0. 



"4 4 



Graphic Determination of the Summations of paragraph 3. 



5. It has hitherto been usual for engineers to assume that the load 

 and section are uniform in each span, because, as we have seen, the 

 calculations in this case are easy, whereas those necessary when 

 attempting the general solution are exceedingly complicated ; in fact, 

 an examination of Mr. Heppel's solution of a comparatively simple 

 case is sufficient to deter engineers from such calculations. But by 

 the following graphic method the solution of the most general case 

 only requires a very elementary mathematical knowledge, and may be 

 completed in a few hours, or in a much shorter time if a Thomson's 

 integrating machine, or even a good planimeter be available. 



We employ a link polygon method for the calculation of m, and it 

 may be expedient to draw the deflected beam by Mohr's method. To 

 employ, however, a link polygon method throughout the whole of the 

 investigation would lead to much waste of time, however interesting it 

 might be from a mathematical point of view ; and as in many other 

 engineering problems, for instance, the finding of centres of gravity 

 and moments of inertia, so in this we see that the use of a little arith- 

 metic, and the actual measurement of lines gives quicker and more 

 accurate integrations. 



Our students in Japan made the necessary summations for spans of 

 300 and 200 feet respectively. Their drawings were exhibited at the 

 meeting, and one of them is shown on a reduced scale in fig. 6. It is 

 advisable to use some well-known scale for the measurement of the 

 lines ; for example, a scale of centimetres. The distance AB, fig. 6, 

 represents the span of 200 feet to the scale — one centimetre stands for 

 3^ feet. The ordinate of the diagram AJEEE, &c, shows the value 



of .J- at every point of the span, and in all cases, except where there 

 EI 



is a sudden change of inclination of the beam over a pier (see the end 



of § 3), any unit of measurement whatever may be employed for JL. 



EI 



6. To find the curve for m and the value of m. — In fig. 1, AB is the 



