Substituting this value of u in equation (623), and

by which expression the variation of the section of the chain of uniform strength is determined.

du

dx c

substituting for its value from equation (624).

dx

=

in respect to x, and

Substituting for u' its value from equation (626),

ing between the limits b and y, observing that when y=b,

dy

de=0,

Now let it be observed, that the value of 7, being in all practical cases exceedingly great as compared with the values of, and m, the value of a (equation 625) is exceedingly small; so that we may, without sensible error, assume those terms of the series 2(-b) which involve powers of 2aly-b) above the first, to vanish as compared with unity

*Church's Int. Cal. Art. 140.

This supposition being made, we have 2(-b)-1=2a(y—b), whence, by substitution and reduction,

Extracting the square root of both sides, transposing, and integrating,

the equation to a parabola whose vertex is in D, and its axis vertical.*

The values a and' H of x and y at the points of suspension being substituted in this equation, and it being solved in respect to c, we obtain

21126) a..... (629);

by which expression the tension e upon the lowest point of the curve is determined, and thence the length y of the sus pending rod at any given distance a from the centre of the span, by equation (628), and the section K of the chain at that point by equation (627), which last equation gives by a reduction similar to the above

405. The section of the chains being of uniform dimensions, as in the common suspension bridge, it is required to determine the conditions of the equilibrium.+

The weight of the suspending rods being neglected, and the same notation being adopted as in the preceding arti cles, except that , is taken to represent the weight of one foot in the length of the chains instead of a bar one square inch in section, we have by equation (614), since K is here constant,

This problem appears first to have been investigated by Mr. Hodgkinson in the fifth volume of the Manchester Transactions; his investigation extenda to the case in which the influence of the weights of the suspending rods is included.

Differentiating this equation in respect to , and observing

+ d) * = (1 + 2)* (equation 615), and that

The former of these equations may be rationalised by assuming (c+u2)+= c + zu, and the latter by assuming (c+u'); there will thus be obtained by reduction

x=2c

(1+z2)dz

!

zdz

(1—2°) { (μ, +μ2) + (μ, —μ2)1⁄23} y=

The latter equation may be placed under the form

which expression being integrated and its value substituted for 2, we obtain

The method of rational fractions (Church's Integ. Calc. Art. 135) being applied to the function under the integral sign in the former equation, it becomes

The integral in the first term in this expression is repre

according as , is greater or less than ",, or according as the weight of each foot in the length of the chains is greater or less than the weight of each foot in the length of the road

way.

Substituting for 2 its value, we obtain, therefore, in the

If the given values, a and H, of x and y at the points of suspension, be substituted in equations (633) and (632), equations will be obtained, whence the value of the constant c and of u at the points of suspension may be determined by approximation. A series of values of u, diminishing from the value thus found to zero, being substituted in equations (633) and (632), as many corresponding values of and y will then become known. The curve of the chains may thus be laid down with any required degree of accuracy.

This common method of construction, which assigns a uniform section to the chains, is evidently false in principle; the strength of a bridge, the section of whose chains varied according to the law established in Art. 401. (equation 619), would be far greater, the same quantity of iron being employed in its construction.

*

RUPTURE BY COMPRESSION.

406. It results from the experiments of Mr. Eaton Hodgkinson, on the compression of short columns of different heights but of equal sections, first, that after a certain height is passed the crushing pressure remains the same, as the

* Seventh Report of the British Association of Science.

*

So that

heights are increased, until another height is attained, when they begin to break; not as they have done before, by the sliding of one portion upon a subjacent portion, but by bending. Secondly, that the plane of rupture is always inclined at the same constant angle to the base of the column, when its height is between these limits. These two facts explain one another; for if K represent the transverse section of the column in square inches, and a the constant inclination of the plane of rupture to the base, then will K sec. a represent the area of the plane of rupture. ify represent the resistance opposed, by the coherence of the material, to the sliding of one square inch upon the surface of another, then will K sec. a represent the resistance which is overcome in the rupture of the column, so long as its height lies between the supposed limits; which resistance being constant, the pressure applied upon the summit of the column to overcome it must evidently be constant. Let this pressure be represented by P, and let CD be the plane of rupture. Now it is evident that the inclination of the direction of P to the perpendicular QR to the surface of the plane, or its equal, the inclination a of CD to the base of the column, must be greater than the limiting angle of resistance of the surfaces; if it were not, then would no pressure applied in the direction of P be sufficient to cause the one surface to slide upon the other, even if a separation of the surfaces were produced along that plane..

R

α

Let P be resolved into two other pressures, whose directions are perpendicular and parallel to the plane of rupture; the former will be represented by P cos. a, and the friction resulting from it by P cos. a tan. ; and the latter, represented by P sin. a, will, when rupture is about to take place, be precisely equal to the coherence Ky sec. a of the plane of rupture increased by its friction P cos, a tan. 9, or P sin. a=Ky sec. a+P cos. a tan. 9, whence by reduction

It is evident from this expression that if the coherence of the material were the same in all directions, or if the unit of

The force necessary to overcome a resistance, such as that here spoken of, has been appropriately called by Mr. Hodgkinson the force necessary C shear it across.