22 . REPORT— 1876. 



These two problems liave recently engaged my attention. 



It is well known that engineers resolve these questions by tentative methods 

 which sometimes require long calculations, and which besides are incapable of 

 performance when tlie section is quite irregular; and this is wliy it is necessary to 

 lixtj-pes (such as a Zores iron, T's, I's, &c.) which, being decomposHble into parts 

 whose moments of inertia or moments of resistance can be determined analytically, 

 are calculable. 



By the simple and uniform graphical method which I have proposed, we can 

 treat in the same manner as the simple sections (triangles, rectangles, &c.) the most 

 complicated forms (such as a Zores iron or even figures with arbitrarj^ and irregular 

 contours, whose equations would not. admit of expression) ; so that, in order to 

 render possible the solution of these important problems, we need sacrifice nothing, 

 either from the economical or the aesthetic point of view. 



I. Let F be the unknown figure that we wish to construct, J its moment of 

 inertia in a direction X with respect to a given baryceutric axis .f *, F' a figure 

 homothetical to F, its centre of gravity, and x' a sti-aight line parallel to .r and 

 passing through 0'. 



Let, besides, k be the (unknown) radius of gyration of F, in the direction X with 

 respect to x, so that J = k-F ; and let J' and A' be analogous quantities to J and Jc, 

 relating to F'. 



Finallj', let us suppose two orthogonal axes it, to drawn anywhere, on which we 

 have respectively the segments UA = WA = 1, A being the point of intersection of 

 u and 10. 



Solution : (a) We find directly J' the moment of inertia of F', either by the inte- 

 grometer or graphical (e. </. by Cidmaun's) method. 



(b) On the axis ti take two segments AB, AB' respectivelj' proportional to J and 

 J', and describe two semicircles on the diameters UB, UB' wliich intercept on the 

 axis of to the segments AC, AC respectively proportional to n/J and \/ J' ; and two 

 semicircles upon the diameters WC, WC which intercept upon tc the two segments 

 AD and AD' respectively proportional to 4/j and //J'. 



(c) From 0' draw a straight line x' parallel to .(', and take upon x' and x the 

 segments O'X ', OX respectively equal or proportional to AD' and AD, so that X is 

 with respect to on the same side as X' with respect to 0'. Draw the straight 

 lines 0' and XX' meeting at the point S. 



(d) Finally, transform the figure F' into the homothetical figure F, taking S as 

 centre of similitude ; that is to say, draw through S a series of lines cutting the 

 contour of F in the points M', and the coiTesponding points M of the required con- 

 toiu- F are formed Ijv constructing the intersections of the radii Si\l' with the straight 

 lines 051 parallel to O'M'. 



This figure F is evidently the secticm required, that is to say, a figure which has 

 given the centre of gravity, the orientation, the form, and also the moment of inertia, 

 in the direction X, with respect to .r, equal to the given quantity J. In fact the 

 ratio of similitude of the two figures F and F' is = A^J : !y/J'- 



Nofel, When J' has been found, we can calculate directly the number 



/J 



ve.)=. 



and then we should take O'X' upon ,r' arbitrarily, and on OX we should take 

 OX=^i . O'X ' ; we should then continue the procedure as above. 



N'ote 2. If the position of x and the magnitude of J are not given absolutely, i. e. 

 if the inverse proolem is to be resolved several times supposing ,r and J successively 

 variable, and if for the determination of J' (see («)) we emj^loy the graphical 

 method, it is convenient to use the central ellipse of F'. 



On this point, and for more details, see three notes tliat I have published in 

 vol. ix. of the ' Eendiconti dell' Istituto Lombardo/ 1870, or my memoir, ">Sul 



* We may restrict oursehes to consider the axes whicli pass tln-ough the centre of 

 gravity O ot'F, on account of the well-known relation between the moments of inertia 

 which" have reference to these axes, and those which have reference to any parallel axes. 



