contained in it may be precisely equal in weight to the material of the suspending rods. It is evident that the conditions of the equilibrium will, on this hypothesis, be very nearly the same as in the actual case. Let μ, represent the

re

weight of each square foot of this plate, then will μydz present the weight of that portion of it which is suspended from the portion DP of the chain, and the whole load u upon that portion of the chain will be represented by

It may be shown, as before (Art. 400.), that

dy น KT=m(c2+u2)1
dx-c'
с

.... •

(623),

SKds="f(c2+u2)dr. Substituting in equation (622),

TC

A linear equation in u2, the integration of which by a wellknown method (Hall's Diff. Cal. p. 397.) gives

Assuming the length of the shortest connecting rod DC to be represented by b, integrating between the limits b and y, and observing that when y=b, u=0,

Substituting this value of u2 in equation (623), and re

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

Substituting for u2 its value from equation (626),

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

Now let it be observed, that the value of, 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 2a(y-b) above the first, to vanish as compared with unity. This supposition being made, we have 2(y—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

by which expression the tension c upon the lowest point of the curve is determined, and thence the length y of the suspending rod at any given distance r 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 articles, except that μ, 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,

u =μ‚s +μqx.. ... (631). Differentiating this equation in respect to x, and observing

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

ds

that da=(1+dy) = (1+23) (equation 615), and that

dx

The former of these equations may be rationalised by assuming (c2+u2)3=c+zu, and the latter by assuming (c2+u2)=z; there will thus be obtained by reduction

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

The method of rational fractions (Hymer's Integ. Calc. § 2.) 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

sented by & log., (12), and that of the second term by

according as is greater or less than 2, 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 roadway. Substituting for z its value, we obtain therefore, in the two

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 e 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 x 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 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

* Seventh Report of the British Association of Science.