TRANSACTIONS OF THE SECTIONS. 21 



The two values coincide if 



sin' C(4 cos-' A— 3) = (cos A +3 cos B cos C +2 cos A cos= C)=, 

 or 



(cotB+3co8«cotC) (cotB+cos rr cot 0)4-3 sill- rt = 0, 



t]ie biaugiilar equation to a conic. 



That is, the locus of the double focus when the cusps are coUinear, or the boimd- 

 ini? curve, on either side of which equihai-monic values are or are not possible, is a 

 splierical ellipse, whose cyclic arcs, veal or imaginary, are perpendicular to the line 

 connecting the single focus and satellite. 



In piano the bounding line is tlius denoted : — 



cotB+3 cot = 0, or cosO = — jr^-. 



The double focus is in a line, wliich cuts orthogonally the connector of the single 

 focus and the satellite. 



6. On harmonic cubics with double foci. 

 The invariant T = 0. 



cos A (9-8 cos= A)k3 + ;! {0-4 cos- A-9 cos= B-f 3 cos= -8 cos" A cos- 



-12cosAcosBcosO}K--h3sin^C(cosA-|-3cosBcosC-f2cosAcos2C)K+sin''C=0- 



For every position of the foci and satellite there is at least one value of the para- 

 meter which yields a harmonic cubic. 



7. On the discriminant of the cubic. 



Equate « to zero in the point-equation (§3); besides the point of contact 

 (/3— /cy = 0), three tangential points are determined by the aggregate : — 



Ky(/3--f 2/3y cos A+y-)— /3(/3- sin" B+2/37 sin B sin cos «+y- sin- 0). 



Since the anharmonic ratio of the lines connecting the tangential points depends 



T- 



upon the function 64——, the discriminant of the ternary cubic is simply found 



from this binary cubic : — 



{9k sin- B + {2k cos A - sin- C) (/c— 2 sin B sin cos a) }* 



+4{ 3 sin- B(2k cos A - sin' C ) -f (k— 2 sin B sin cos of} 



X {3k(k -2 sin B sin cos a) - (2k cos A - siu^ 0)'} =0. 



8. By dualising, this investigation is equally applicable to order-cubics v\irh 

 double cyclic arcs and double foci. 



Resume of Researches on the Inverse Problems of Moments of Inertia and of 

 Moments of Resistance. Bij Professor Giuseppe Juxg (Milan). 



Tn the study of the resistance of materials and the stability of constructions, the 

 two following problems continually present themselves : — 



I. To construct a plane iigure (for example, the cross-section of a cylinder loaded 

 in a given manner) of which we may suppose given the orientation, the form, the 

 centre of gravity, and also tlie moment of inertia with respect to a given neutral 

 axis. 



II. To construct a plane section, given the orientation, the form, the centre of 

 gravity, and also the moment of resistance * with respect to a given neutral axis. 



* If e is an element of a plane section F, and ;/ be the distance of the barycentre of e 

 fi'0)u an axis .r measured in the direction \ (\ being a straight line making any angle with 

 the axis x), then the moment of inertia of F with respect to .r in the direction \ is 

 — Si?y- = J, the i; extending over the contour of F. 



If, besides, v is the distance (ineasurcc' parallel to X) of the barycentre O of F from the 

 tangent to F parallel to x and furthest removed from O, then the moment of resistance 



of P with respect to a barycentric axis .r, in the direction X, is defined to be the ratio — 

 i. « ^. 



In fact, if we multiply this ratio by a certain constant v,'c have the ordinary moment of 

 resistance of tl)e section F. 



