TRANSACTIONS OF THE SECTIONS. 23 



problema inverso dei momenti d'Inerzia " in vol. xxiv. of tlic ' Politecnico, Gior- 

 nale dell' lui^egnere arcliitetto civile ed industriale' (Milano, 1870), which contains 

 two lithographic tables, and, as an appendix, a comparison between the numerical 

 and the graphical calculation of a Zores iron. 



II. Retaining the same notation as before, let II be the given moment of resist- 

 ance of F in the direction \ with respect to a given bar3'centric axis x, i. c. let 



J k- 



11= - =F . ?•, where r=— is the radius (or arm) of resistance of F with respect to 



V V 



X in the direction X. 



Let R' and r' be analogous quantities to R and r for the figure F'. 



Solution. Find directly R' {for example, by Culmann's graphical method); 

 determine, either graphically or by a niimerical calculation, the ratio 



\/r' = '^' 



draw through 0' a straight line x' parallel to x ; talce on x' any segment O'X' and 

 on X a segment OX = jx . O'X' so that X is with respect to O on the same side as 

 X' with respect to 0'; and draw the straight lines 00' and XX cutting one another 

 in S. 



Then transform the figure F' into the liomothetical figure F, taking S as centre 

 of similitude (see above). This figvu-e F is evidently the required section ; that is 

 to say, a figure which has the given barycentre, orientation, and form, and also the 

 moment of resistance in the direction X with respect to the axis x = the given mo- 

 ment II. 



For further details see the notes already cited in the ' Rendiconti dell' Istituto 

 Lombardo,' 1876, and also the memoir "Sul problema inverso dei momenti di re- 

 sistenza,' which will appear in the ' Politecnico, Giornale dell' Ingeguere ai'ch. civ. 

 ed industr. ' (Milano, 187(3). 



Resume of Researches upon the Orapliical Representation of the Moments of 

 Resistance of Plane Figures. By Professor Giuseppe Jung (Milan). 



Continuing the investigation upon the moments of resistance of a given plane 

 figure F, I have communicated to the Istituto Lombardo * some results which I 

 have obtained, and of which I here give a short account. 



1. Retaining the same notation as in the last paper, I have given several gra- 

 phical methods for calculating the radii of resistance r in an arbitrary direction X 

 (and, consequently, the corresponding moments of inertia R = F.r) of the figure 

 F with regard to any barj'ceutric axis x, and I ha^e found several representative 

 curves, viz. in this sense that these curves have for radii vectores the radii of resis- 

 tance r. So that, having given an axis x and one of the representative cur\es 

 (which I show how to construct), we have the corresponding moment of resistance 

 by multiplying by the area F of the section a certain radius vector of the represen- 

 tative curve. 



It is remarkable that when the direction X is conjugate to the direction of the 

 given axis x (i. e. that when the diameter of the central ellipse of F parallel to X is 

 conjugate to the diameter x), one of the representative curves is the central nucleus 

 (Centralkeru) t of the figure F, and we have the following theorem :^ 



* See 'Rendiconti dell' Istituto Lombardo,' ser. 2, t. ix. 1876, No. sv. "Rapprcseu- 

 tazioni grafiche dei momenti resistenti di una sezione piiina." _ Ko. xvi. " Coniplemento 

 alia nota precedente." 



■f Perhaps it wiU be usefid to recall here rapidly some notions which are, however, well 

 known (see, for example, my memoir " Sui momenti d'lucrzia " iu the ' Rendiconti dell' 

 Istituto Lombardo,' 1875). 



In tlie plane of P to every straight line, considered as a neutral axis, corresponds a point 

 X which is tlie centre of the pressures (tt-nsious) oi" tlie centre of the second degree or the 

 point of application of the resultant of the normal forces acting on the section F. This 

 point X is also called the antipole of the straight line x; and the straight line .v is the 



