RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 25 



axis of z cut off by the tangent plane. The two surfaces may be denned as recip- 

 rocally polar (in the ordinary sense) with respect to the paraboloid of revolution 



x 2 + y 2 = 2cz (11), 



and the diagrams are the projections on the planes of my and £77 of points and 

 lines on these surfaces. 



If one of the surfaces is a plane-faced polyhedron, the other will also be a 

 plane-faced polyhedron, every face in the one corresponding to a point in the 

 other, and every edge in the one corresponding to the line joining the points 

 corresponding to the faces bounded by the edge. In the projected diagrams 

 every line is perpendicular to the corresponding line, and lines which meet in a 

 point in one figure form a closed polygon in the other. 



These are the conditions of reciprocity mentioned at p. 8, and it now 

 appears that if either of the diagrams is a projection of a plane-faced poly- 

 hedron, the other diagram can be drawn. If the first diagram cannot be a pro- 

 jection of a plane-faced polyhedron, let it be a projection of a polyhedron whose 

 faces are polygons not in one plane. These faces must be conceived to be filled 

 up by surfaces, which are either curved or made up of different plane portions. 

 In the first case the polygon will correspond not to a point, but to a finite por- 

 tion of a surface ; in the second, it will correspond to several points, so that the 

 lines, which correspond to the edges of such a polygon, will terminate in several 

 points, and not in one, as is necessary for reciprocity. 



Second Method of representing Stress in a Body. 

 Let a, b be any two consecutive points in the first diagram, distant s, and a, /3 

 the corresponding points in the second, distant cr, then if the direction cosines 

 of the line a b are /, m, n and those of a (3, X, fi, v 



(12). 



o-X = sl-^r- + sm-^- + sn-f- 

 dx dy dz 



Art drt drt 



oiJb = si-— + sm-r + sn-^r- 



dx dy dz 



At dt dt 



av = sl-^r- + sm-^r- + sn^- 



dx dy dz ' 



Hence 



j(a + ^ +)! ,)=^| +m f +K f + ^ + |) +K <| + f) + ,<| + |)(iB). 



If we put IX + m\L + nv — cos e, where e is the angle between s and cr, and 

 if we take three sets of values of linn, corresponding to three directions at right 

 angles to each other, we find 



<r, 0-0 a. dP dr, d£ d 2 ~F d 2 ~F d 2 ¥ nu 



- 1 - cos e, + — ?- cos e 9 + — *■ cos e 3 = -f- + -~- + ^- = j-s + j-s + -7^2 ( 14 )- 



Sj s 2 s 3 dx dy dz dx/ dy* dz 1 



VOL. XXVI. PART I. G 



