422 A Resultant System for the [Oct. 



proved beyond doubt from the result of those bridges enumerated in 

 the 20th paragraph, as remodelled on the " Resultant" system. 



22. I will now detail the theory on which the "Resultant" princi- 

 ple is based. 



In Fig. 7, A B C represents the chain of a tension bridge, the 

 centre link of which is above the level of the railing ; abed, the 

 roadway, or suspended platform, (See Fig. 7,) the small portions 

 x x being supported by the abutments. Let 1, 2, 3 ; 3, 2, 1, be the 

 auxiliary oblique rods from the chain, the angle of those at the centre 

 not being less than 25° and those next the standards not greater than 

 45°. It is evident that the platform is entirely upheld by the auxiliaries, 

 and it is to them therefore that our attention is first directed. 



23. The auxiliary rods being by construction attached at equal 

 distances, it is intended that each set shall bear an equal duty or ten- 

 sion, and as the stiffness of the platform to resist the force of gravity 

 is uniform throughout, the whole series of oblique rods benefit equally 

 thereby, and being thus common to all, it may be omitted in consider- 

 ing the strains on the auxiliary rods. (See Fig. 8). 



Suppose the platform to be divided into as many equal parts as there 

 are oblique rods, thus giving to each rod an equal load, the points of 

 attachment of which being the centres of gravity, we have six rods, 

 1, 2, 3, 3, 2, 1, supporting the equal portions of platform having corre- 

 sponding numbers. 



24. The several portions of the platform acting by gravity whilst 

 the sustaining force is oblique, a third force is necessary to preserve 

 the whole in equilibrio. This force is, in the present system, tension in 

 the horizontal line as shown in annexed Fig. 9, and acting from the 

 standard towards the centre. These three forces, viz. vertical, oblique, 

 and horizontal, being in proportion to the radius, cosecant, and co- 

 tangent of the angle of obliquity ; the tensile force being that under 

 consideration, it is necessary to connect the portions of the platform 

 in Fig. 8, in such a manner that the weight or force of gravity shall act 

 freely, whilst the several parts are prevented from separating. Fig. 10, 

 will show the meaning. 



Here we have the tensions on the several portions 1, 2, 3, on one 

 side, or half span, counterbalanced by an equal amount of tension on 

 the portions 3, 2, 1 , of the opposite half, hence the greatest strain is in 



