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problema inverso dei momenti d'Inerzia" in vol. xxiv. of the 'Politecnico, Giornale dell' Ingegnere architetto civile ed industriale' (Milano, 1876), which contains two lithographic tables, and, as an appendix, a comparison between the numerical and the graphical calculation of a Zorès iron.

II. Retaining the same notation as before, let R be the given moment of resistance of F in the direction A with respect to a given barycentric axis x, i.e. let k2 v

R==F.r, where r= is the radius (or arm) of resistance of F with respect to

v

x in the direction λ.

Let R' and ' 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 numerical calculation, the ratio

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draw through O' a straight line x' parallel to r; take on a'any segment O'X' and on a segment OX=u.O'X' so that X is with respect to O on the same side as X' with respect to O'; and draw the straight lines OÙ' and XX cutting one another in S.

Then transform the figure F' into the homothetical figure F, taking S as centre of similitude (see above). This figure 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 A with respect to the axis x = the given moment R.

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 resistenza,' which will appear in the 'Politecnico, Giornale dell' Ingegnere arch. civ. ed industr.' (Milano, 1876).

Résumé of Researches upon the Graphical 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 graphical methods for calculating the radii of resistance r in an arbitrary direction (and, consequently, the corresponding moments of inertia R=F.r) of the figure F with regard to any barycentric axis x, and I have found several representative curves, viz. in this sense that these curves have for radii vectores the radii of resisSo that, having given an axis x and one of the representative curves (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 representative curve.

tance r.

It is remarkable that when the direction A is conjugate to the direction of the given axis x (i. e. that when the diameter of the central ellipse of F parallel to λ is conjugate to the diameter x), one of the representative curves is the central nucleus (Centralkern)† of the figure F, and we have the following theorem:

* See Rendiconti dell' Istituto Lombardo,' ser. 2, t. ix. 1876, No. xv. Rappresen tazioni grafiche dei momenti resistenti di una sezione piana." No. xvi. "Complemento alla nota precedente."

+ Perhaps it will be useful to recall here rapidly some notions which are, however, well known (see, for example, my memoir "Sui momenti d'Inerzia" in the 'Rendiconti dell' Istituto Lombardo,' 1875).

In the plane of F to every straight line, considered as a neutral axis, corresponds a point X which is the centre of the pressures (tensions) or the 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 ; and the straight line a is the

The radius of resistance with respect to a barycentric axis, in the direction of the conjugate diameter, is equal to the smaller of the two radii vectores 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 difference consists chiefly in this, that each of the radii vectores of the ellipse situated on the diameter y is equal to the radius of gyration of F with regard to the conjugate diameter x; while in general one only (the smaller) of the two radii vectores of the nucleus situated upon y is equal to the radius of resistance of F with regard to the conjugate diameter x.

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 sollicitation of the section F, and the straight line which passes through its barycentre and is conjugate to the axis of sollicitation is the neutral barycentric axis. This being premised, I show that

*

The moment of resistance with respect to a barycentric axis x, in the conjugate direction y, is equal to the resistance specific to the cohesion with respect to the flexure relatively to the axis of sollicitation y.

From which follows a theorem giving the law of variation of the specific resistance of F, when the axis of sollicitation turns round its centre of gravity, viz. :

·

The central nucleus of a given section is the curve of resistances specific to the cohesion with respect to the flexure. A radius of the nucleus (the smaller of the two situated on the barycentric axis considered) multiplied by the area F gives the specific resistance with respect to its direction, considered as axis of sollicitation.

3. Taking still the barycentre of F as pole and for radii vectores segments proportional to the maxima† specific resistances of the section with respect to the flexure and corresponding to each axis of sollicitation, I find the remarkable

theorem :

1

The curve of maxima resistances of F is a transformation by reciprocal radii vectores (the inverse ) of the central nucleus of the section. A radius vector of this inverse curve, multiplied by F gives the specific maximum resistance of F with respect to its direction, considered as axis of sollicitation.

4. Two other theorems are connected with a note of M. Ritter, "Ueber eine neue Festigkeitsformel" (see the 'Civilingenieur,' 1876, Heft iii., iv.). The more important 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)

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antipolar of the point X. If in any one given direction A, d is the 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 axis parallel to a, the distance, measured parallel to A, of the straight line a from its antipole=d+. If a straight line y passes through X, its antipole y lies upon a. The point X, which is the antipole of the straight line & in this reciprocal system (antipolar system), is also the pole, in Poncelet's sense, with respect to the central ellipse of F, of the straight line a' which is symmetrical to a with respect to the point O (barycentre of F and centre of its central ellipse).

If a variable straight line envelops the contour of F without cutting it, its antipole X describes a closed curve which is the central nucleus (Centralkern) of the figure F (see Culmann, Die graphische Statik,' 2nd edition, t. i. 3ter Abschnitt, Zurich, 1875).

* It is the moment of resistance of the section for which, the axis of sollicitation of the 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.

That is, the maximum unit tensions (or pressure) on the hypothesis that the moment of the exterior forces which produces the flexure in the sections of the cylinder is = 1. See, for example, Hirst, "Inversione quadratica" (Annali di Matematica,' Roma. 1st series); Darboux, Sur une classe remarquable de courbes et de surfaces algébriques,' Paris, 1873.

6

of the resultant of the forces which act normally upon F, C' the point in which OC is met by the (unknown) neutral axis, A and B the points in which OC meets the contour of the central nucleus, A' and B' the points in which OC meets respectively the antipolars of A and B *, then this last theorem can be enunciated thus: The point C' is conjugate to C in the involution AA', BB'.

Consequently, if C is given, we have C' linearly, and we construct the neutral axis by drawing through C' a straight line parallel to the conjugate direction of OC.

On a new Construction for the Central Nucleus of a Plane Section.
By Professor GIUSEPPE JUNG (Milan).

I have the honour to communicate to Section A a new and very easy method of representing the radii of gyration of a given plane (figure F), which appears to be more simple than the known methods of Poinsot, Reye, and Mohr.

From this representation I deduce a new construction for the central nucleus of F, independent of that of the central ellipse of the figure. This I regard as interesting, because of the importance of the central nucleus in the study of the stability of constructions, on account of its remarkable properties with regard to the moment of resistance of the section &c. (See Culmann, Die graphische Statik,' and the memoirs of which a résumé has just been given.)

1. Let O be the centre of gravity of F; AA and BB its principle axes of inertia, i. e. the axes of the central ellipse E of F; f and f' (upon AA) the two foci of E; C the circle which has for diameter the major axis AA. Then the radius of gyration of F (in the normal direction), with respect to any barycentric axis x, is the segment MM' of the perpendicular drawn to x from one of the points ff' included between the axis x and the circle C. In fact the circle C is the locus of the feet of the perpendiculars let fall from the foci ff' upon the tangents to the ellipse E.

Thus the circle C represents the radii of gyration of F (in the normal direction) with respect to all the barycentric axes. If from M' we draw the straight line m parallel to r, the segment NN' of any straight line A, included between a and m, is equal to the radius of gyration with respect to r in the arbitrary direction A; that is to say, if we take the angle Ar=90-w, we have the radius of gyration, in the MM' direction X, = =NN'. We can dispense with the perpendiculars. It is sufficient to construct, besides the circle C, the circle r on Of as diameter: if x meets the circle r in the point M, and Mf be drawn cutting the circle C in the point M', the segment MM' will be the required radius of gyration.

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2. Let G be a circle passing through O, and of arbitrary radius t. If through the points A we draw two parallels to BB, and through the points B two parallels to AA, the diagonals of the rectangle so produced meet G in two points a and a', and the straight lines AA, BB meet the same circle in ẞ and B'. Let U be the point of intersection of the chords a a' and ßß'.

By means of this point U we construct the barycentric axis y, conjugate to any given barycentric aris x. It is only necessary to observe that if a cuts G in the point X, and XU cuts G in the point Y, the straight line OU is the axis y required. This is, in fact, merely the construction for the radius y conjugate to a in the involution of the straight lines O(A B. aa'); but these latter are two pairs of conjugate diameters of the central ellipse of F, whence &c.

3. Construction for the central nucleus. Draw any suitable number of straight lines enveloping the contour of F without cutting it. Let be one of these lines, i. e. a tangent which does not cut elsewhere the contour of F (unless it be convex). Draw through O the axis a parallel to 1, and through f the perpendicular to 1, which meets 1, x, and the circle C in the points V, M, and M' respectively. With centre M and radius MM' describe a circle, intercepting on a the distance MK'

*These antipolars are tangents to the contour of F and parallel to the conjugate direction of OC; and we know that the barycentre O is situated on each of the finite segments AA' and BB'.

We might take G coincident with r; then ẞ coincides with fand B' with O, and U is the point of intersection of ax' and AA'.

1876.

3

(=radius of gyration, normal with respect to ; see No. 1), and through K' draw the perpendicular to K'V meeting MM' in the point K. The straight line passing through K and parallel to cuts the axis y, conjugate to a (see the construction for it in No. 2), in the point L, antipole of 1*; consequently L is a point on the

central nucleus.

Centroids, and their Application to some Mechanical Problems.
By Professor A. B. W. KENNEDY.

Elementary Demonstration of a Fundamental Principle of the Theory of Functions. By PAUL MANSION, Professor in the University of Ghent.

M. Thomae ('Abriss einer Theorie der complexen Functionen,' 2te Auflage, Halle, 1873, pp. 11-13) first demonstrated rigorously the theorem that " a function y=Fr, whose differential coefficient, both in the positive and in the negative direction, is zero for every value of x, from 2 to X, is constant in this interval." This important proposition can be demonstrated in an elementary manner by the following method, which seems capable also of other applications.

I. If the differential coefficient of a function y=Fr in the positive direction is the same as in the negative direction, this differential coefficient is equal, for a F(x2)-F(x1) system of values (x, y), to the limit of the ratio X2-X1

,, and a, converging

towards the intermediate value x.

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and, e, and being multiplied by proper fractions, since x is intermediate to z

and

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II. Let xo, 19 values yo, Y1, We have

....

X2-X1

....1, X be increasing values of x, to which correspond the Yn-1, Y of the function y=Fr.

....

Y-Yo_(Y1 — Yo) + (Y2 — Y1) · · · · (Y — Yn−1)
X-x (x1-x)+(x12-x1).... (X—xn−1)'

It results from this equation that Y-yo has a value intermediate to the greatest

X-xo

and least of the ratios -1, unless they are all equal. Thus :-Unless all the

Ay,

Δη

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's are equal in the interval (x, X), there is at least one of them greater than and

one of them less than yo

Y-Yo
X-x

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III. If the differential coefficients of a function y=Fæ are the same in the positive direction as in the negative direction, from x, to X, then either all these differential * In fact if OL-y meets l in L', and if k is the radius of gyration with respect to r in the conjugate direction y, we have, by construction, L'L=L'O+; but the distance, in the direction y, of the straight line 7 from its antipole has exactly this value (see note to my Résumé of Researches upon the Graphical Representation' &c.); therefore L is the antipole of l

coefficients are equal, or there is at least one of them greater than and one less than

Y-Yo

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Subdivide the interval X-x, into n parts: the

Ay
Ax

corresponding to one of them,

Y-
X-x

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Δε

--1, will be greater than (No. II.). Operate in the same manner with --1, and so on. We shall thus have an indefinitely increasing series of Y-Yo greater than X-xo

(No. II.), and having for limit the differential coefficient y' of Fr for a certain value of x (No. I.). There is, then, a differential coefficient y greater than Y-yo. In the same way we can show that there is one smaller. We

X-x

must, however, except the case of y constant, which arises when y=ax+b.

Δ.Τ

IV. If the differential coefficient of a function, supposed the same in the positive and negative directions, is equal to a constant a, from x to X, the function is linear and of the form ax+b.

Necessarily, x and x, being any two values included in the interval (x, X),

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whatever x and x, may be. For, were it otherwise, there would be between a and z, a differential coefficient greater than a, and one smaller than a.

y=ax+(y1—ax1).

Corollary.—If a=0, y=constant. Q.E.D.

Thus:

On Convergents. By THOMAS MUIR, M.A., F.R.S.E.

In Lagrange's additions to Euler's Algebra (2nd Eng. ed. vol. ii. p. 279), he sets himself the problem,-A fraction expressed by a great number of figures being given, to find all the fractions, in less terms, which approach so near the truth that it is impossible to approach nearer without employing greater ones; and for solution he gives in effect the following rule :-Transform the given fraction into a continued fraction with unit numerators and positive integral partial denominators, and the so-called convergents of this continued fraction will be the fractions required. In this he is in error, the fractions found being some of the fractions required, but not all. Thus, taking as the given fractional form, he transforms it into

3+

1 1 1
7+ 15+ I+...

3 22 333 355

the so-called convergents of which are 17 106 1139

.; and in regard to them

3

he says:-"So that we may be assured that the fraction approaches nearer the truth than any other fraction whose denominator is less than 7; also the fraction 22 approaches nearer the truth than any other fraction whose denominator is less than 106; and so of others."

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The statement here made in reference to is easily seen to be incorrect by com13 16 paring the difference of from with that of or the former being, of 4, 5” course, 14159..., and the three latter 10840..., ·05840..., 02507...; and the incorrectness extends to what is said of the other convergents. The true solution lies in the fact that not only is 3+ one of the required fractions, but so also

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