22 



Proceedings of the Koi/al Irish Academy. 



from d on fig. ?>, dropped on the vertical line through the middle of the span, 

 is the base of a small part of the locus whose vertex is At, which can be drawn 

 with the template. In the same way perpendiculars on the middle vertical 

 line from a, b, r, and e give bases" to guide the template to draw parts of the 

 other four loci. Note that at the common height of c and d it is only neces- 

 sary to add the square of the half of the small bases, thus : 252 + 4 = 256, 

 and 252 + 9 = 261 are the heights of A3 and Ai respectively. 



The height at F^, the junction of the fourth and fifth fields, to the dotted 

 parabola BAaC, fig. 2, is the product BF^ x CPt = 33 x 9. In a preceding 

 paragraph we had F^D^ = (33 - 13) x 9 ; hence the height from A to the dotted 

 parabola is (9 x 13), that is, the load on the wheel JFj = 9 tons multiplied by its 

 distance from 6, the centre of gravity of the locomotive. In the same way it is 

 found that DiAn is the sum of the moments of the two weights W^ and Wt 



about G. Generally the depths of the junctions of the parabolic arcs in pairs 

 below the dotted parabola are given by the moments of the weights of the wheels 

 about their common centre of gravity. The numerical value for D, is 5 x 17, for 

 A it is greater by 5 x 12, for D3 it is still greater by 11 x 4, while for A it is 

 lesser by 12 x 6, calculating them in order from the left end. Depths from 

 the dotted parabola BAoO, due to the 42 tons rolling on one wheel, to the 

 junction of the arcs of parabolas giving the locus due to the 42-ton locomotive 

 crossing the 42-foot span, are : 



At J), ih A A 



85 145 189 117 foot-tons. 



On fig. 4 are shown the two parabolic right segments BA^C standing on 

 the span, and c'Aib', the arc A^iA of which is the locus of the maximum 

 bending-moments for the fourth field commanded by TF*. Let us suppose 



