586 



REPORT — 1893. 



When the correspondence exists in two figures they are said to be 

 reciprocal to each other, but the case given is only a special one of a more 

 general theorem, hereafter considered. The diagram mai-ked I. is alluded 



Fig. 



to as the ' first derived diagram,' and that marked II. as the ' second derived 

 diagram ' — terms which might with advantage be respectively substituted 

 for the present ones of ' force ' polygon and ' cord,' ' rope,' or ' funicular ' 

 polygon, since the figures are themselves perfectly general in their appli- 

 cations.' 



If the segments to be combined do not all lie in one plane, the projections 

 on two planes must be obtained and combined in each, just as for the 

 case of given segments in a plane ; but the points from which the first 

 lines and the derived polygon are drawn must be the projection of the 

 same points in space. The two resultants so obtained are the projections 

 of the true resultant of the given segments.^ The method of projection 

 on planes enables the properties of segments in space to be dealt with 

 in the same way as with segments in planes. The figures so obtained 

 are reciprocal, but they are the projection of soUd bodies, the bounding 

 planes of which intersect in the given segments, and the solids themselves 

 are also said to be reciprocal. 



In fig. 7 the point O would be the vertex of a pyramid, but cases in 

 which there are polyhedra of other forms of eveiy possible kind also fall 

 under the same laws relating to reciprocal figures. 



A large number of theorems have been discovered as a result of 

 combining segments in this way. Thus, suppose two segments are to be 

 combined, the magnitudes of which are constant, but which take every 

 possible direction, the inclination of both making the same angles on 

 opposite sides of the vertical. The locus of the resultant, having the 

 fixed point for its centre, is an ellipse, the major and minor axes of 

 which are respectively a + b, and a — b, a and b being the respective 

 segments. If this theorem is extended to three dimensions of space the 

 locus of the surface described by the extremity of the resultant is an 

 ellipsoid. Rankine's ellipse and ellipsoid of stress, also the ellipse and 

 ellipsoid of strain, follow from this proposition. Again, if the two seg- 

 ments are always in the same direction, and, though not constant in 



' Professor Ritter, of Zurich, in a conver.sation with the writer of this report 

 expressed his approval of the terms above suggested, and stated that they were 

 already used in a treatise by Herr M. Nehls, Director of Waterworks, Hamburg. 



- It is to be noted that this applies to segments as such. 



