24 REPORT — 1876. 



The radius of resistance with respect to a harycentric axis, in the direction of the 

 conjugate diameter, is equal to the smaller of the tico radii vectorcs of the central 

 nucleus situated on the latter diameter. 



We thus .see that the central nucleus stands in nearly the same relation to 

 the radii of resistance that the central ellipse does to the radii of gyration of the 

 figure F. In fact the diiference consists chiefly in this, that each of the radii 

 yectores of the ellipse situated on the diameter y is equal to the radius of gyration 

 of F with regard to the conjugate diameter .r; while in general one only (the 

 smaller) of the two radii ^ectores of the nucleus situated upon y is equal "to the 

 radius of resistance of F v/ith regard to the conjugate diameter ;r. 



2. Suppose that F is a cross-section of a cylinder upon which are acting forces 

 situated in a plane passing through its axis, the intersection of this plane with the 

 plane of F is the axis of sollieitation of the section F, and the straight line which 

 passes through its barycentre and is conjugate to the axis of pollicitation is the 

 neutral barycentric axis. This being premised, I show that 



The moment of resistance with resj}cct to a harycentric axis x, in the conjuyatc 

 direction y, is equal to the resistance specific * to the cohesion tcith respect to the 

 flexure relatively to the axis of sollieitation y. 



From which follows a theorem giving the law of variation of the specific resist- 

 ance of F, when the axis of sollieitation turns round its centre of gravity, viz. :— 



The central nucleus of a yiven section is the curve of resistances specific to the cohe- 

 sion with respect to the flexure. A radius of the nucleus (tlie smaller of the tivo 

 situated on the harycentric axis considered) multiplied hy the area F yii'cs the specific 

 resistance with respect to its direction, considered as axis of sollieitation. 



?>. Taking still the barycentre of F as pole and for radii vectores segments pro- 

 portional to the maxima t specific resistances of the section with respect to the 

 ilexure and corresponding to each axis of sollieitation, I find the remarkable 

 theorem : — 



The curve of maxima resistances of F /*■ a transformation hy rcciproccd radii vectores 

 (the inverse X) of the central nvclcus of the section. A radius vector of this inverse 



curve, midtiplied hy p, ffives the specific maximum resistance of F with resjiect to its 



direction, considered as axis of sollieitation. 



4. Two other theorems are connected with a note of M. Ritter, "Ueber cine neue 

 Festigkeitsformel " (seethe ' Civilingenieur,' 1876, Heft iii., iv.). The more im- 

 portant is that which gives a simple solution of the following question : — Given 

 the point of application of the resultant of the forces which act normally on the 

 section F and also the central nucleus, but not the central ellipse, of F, find the 

 neutral axis corresponding to this point. 



If O is the centre of gravity of F, C the point of application (in the plane of F) 



antipolar of the point X. If in any one given direction \, d is tlie distance of the centre 

 of gravity of F from the straight line x, and k is the radius of gyration of F with respect 

 to the barycentric asis parallel to .r, the distance, measured parallel to X, of the straight 



line X from its antipole = rf+ -j- If a straight line y passes through X, its antipole y lies 



Tipon x\ The point X, which is the antipole of the straight line x in this reciprocal system 

 (antipolar system), is also the pole, in Poucelet's sense, with respect to the central ellipse 

 of F, of the straight lino x' wliieh is symnietrical to x with resjicct to the point O (bary- 

 centre of F and centre of its eentrnl ellipse). 



If a variable straight line envelops tlie contour of F williout cutting it, its antipole 

 X describes a closed curve which is the central nucleus (Centralkcrn) of the figure F (see 

 Cnlmann, 'Die graphi^che Statik,' 2nd edition, t. i. .3t''i" Abschnitt, Zurich, 1875). 



* It is the moment of resistance of the section for whicli, the axis of sollieitation of tlie 

 forces being given, the unit of tension (or of pressure) is produced in the most distant 

 fibre of the neutral axis upon the unit of area of this fibre. 



t That is, the nuixinmm unit tensions (or pressure) on the hypothesis that the moment 

 of the exterior forces which produces the flexure in the sections of tlie cylinder is =1. 



X See, for example, Hirst, "luversione quadi-atica" (• Annali di Matematicn.' Eoraa. 

 1st series) ; Darbo.ix, ' Sur nne clnsse remarquable de coxu-bes et dc surfaces algt'briqnes,' 

 Pari^^. 1873. 



