308 Messrs. H. J. Tomlinson and K. Pearson : 



for the solution of Mr. Wilson's problem. It is necessary 

 to remind the reader that the original drawing has been 

 photographed down to nearly £ its size, and that the scales 

 attached are those of the original drawing, or about \ in. 

 in the diagram stands for 1 in. of the original drawing. 



(3) At the top of the drawing A J gives the three spans 

 with bending-moment curves for each span (in this case 60 ft., 

 100 ft., and 40 ft. loaded uniformly with 3 tons per foot run) on 

 the hypothesis that the spans are crossed by discontinuous 

 simple beams. In the particular example these curve3 are 

 all parabolas. These parabolas then have their vertical 

 ordinates altered in the ratio of the constant flexural rigidity, 

 that at D, to the flexural rigidity at each section. The 

 moment of inertia is 2100/144 ft. 4 at A, changes uniformly 

 to 3300/144 at D, then uniformly to 3000/144 at G, and 

 finally to 2200/144 at J. This is represented by the line 

 marked (i Curve of Moment of Inertia/' and the process is 

 the same whatever be the form of this curve. Any ordinate 

 like ob was then altered graphically as follows : — 



Let the ordinate meet the constant moment of inertia line 

 at c and the curve of inertia at d. Rotate ob down to the 

 horizontal as at od f , draw cc' to meet the horizontal in c', 

 parallel to dd'\ then rotate oc f up to oa; it gives the point on 

 the altered bending-moment curve. The outer curves were 

 thus constructed. 



The next step is to construct the vertical through the 

 centroid of each one of these curves. This was done by 

 drawing the so-called centroid-curves, the areas of which 

 give the first moments of the primitive curves about chosen 

 verticals. This is the process usually attributed to Collignon, 

 but as the method of " equivalent areas " it has had probably a 

 much longer drawing-office existence. The areas were as usual 

 read off by a planimeter. After the centroid-verticals had 

 been inserted, the so-called " third-lines," or lines at third and 

 two-third distance of each span were drawn, and also the 

 " inverted third-lines/' or lines obtained by drawing a vertical 

 between the right-hand third-line of one span and the left- 

 hand third-line of the adjacent span, dividing the distance 

 between them inversely as the spans. The third step is to 

 draw the " cross-lines.''' From the centroid- vertical of each 

 span is taken off a distance equal to the constant flexural 

 rigidity, i. e. in our case 33,000,000 sq. ft. tons, not on the 

 scale on which areas of the bending-moment curves will 

 afterwards be plotted, but on some reduced scale, which 

 marks by its inverse the exaggeration of the deflexions as 

 given in the elastic line. Without such an exaggeration the 



