20 Proceedings of the Royal Irish Acadcmjj. 



So that Ai is the vcrkx of a right parabolic segment ; half base, 10 ; height, 

 19x19; placed with its vertical axis 2 feet to the left of the middle of the span, 

 and with its base lowered 105 units below the base of the diagram. 



The apexes A, B, C, D, and E of the instantaneous bending-moment 

 diagram, fig. 1, describe the five parabolic loci whose vcrtexcs are A\, A,, A,, Ai, 

 and As shown on fig. 2. These have their five vertical axes placed about the 

 middle of the span in the same way that the five wheels are placed about the 

 centre of gravity of the locomotive, but at half the distance in each case 



thus — 



OSi OS2 OS3 OSi OS-, 



8-5 6 2 - 3 - 6| feet. 



The dotted parabola BA„C standing on the span is the locus of the apex 

 of the triangle which is the instantaneous bending-moment diagram for a 

 single load of 42 tons rolling on the 42-foot span. This triangle is scalene for 

 every position of the single load, except when the single load is at the middle 

 of the span, when the triangle is isosceles, and its height is then 

 OAo = 5 load X span = 21 x 21 = 441 foot-tons. 



We see now that all six parabolic loci have this in common that each 

 parabolic right segment has its height numerically equal to the square of its 

 half base. If a template of the parabolic right segment BAf,C be made, then 

 the other five segments are drawn from portions of that template. If the 

 template be only a quadrant OAoC, it is then called a parabolic-set-square. A 

 parabolic template, such as BAoC, whose height OAo is 441, the square of its 

 half base, which is 00= 21, is said to be of mochdus unity ; and, further, the 

 height of the template at any other point is the product of the two segments 

 of the base BG into which it is divided at that point. Such a parabolic tem- 

 plate or set-square would be far too lofty for actual use ; but any template or 

 set-square of convenient proportions can be used by employing two scales — 

 one for horizontal or feet measurements, and another for vertical or foot-ton 

 measurements. The modulus of the template for drawing the six parabolic loci 

 in fig. 2 is unity, because the ratio of the load to the span is unity. Had the 

 locomotive only weighed half as much, say 21 tons, then the modulus must 

 have been one-half ; still the same template would serve by employing a new 

 vertical scale for foot-tons — in that case twice as coarse. 



We may now consider the point Di on fig. 2 where the fourth and fifth 

 parabolas intersect. At the point F^ the bending-moment will have a common 

 value F^D^ whether the locomotive be moved so as to bring Wt or W^ to that 

 point. If we put Z for BFt in the first ease, the distance from B to G will be 

 (Z- 13) feet, so that the push-up of the abutment at C is {Z- 13) tons and the 

 bending-moment at Fi is {Z- 13) (42 - Z) foot-tons. In the second case the 



