the weight at f being equal to 2w, it follows that f'g makes twice the deflection from e'f' that the latter makes from d'e', that is, equal to 2D in the horizontal distance of lv, or 1, or 10D (= .5173), in the distance a C, or 5. Hence, f'g' produced, cuts the vertical at a, twice as high as e'f' cuts it, or, at a point 1.0346 above a; being just as high as the point f'; except a small difference resulting probably from omitted fractions. This shows that f'g' is horizontal, and tangent to the curve at its vertex.

It follows that all the weight at f', and at the left of that point, is brought to bear at a, aud all that at g', and on the right thereof, bears at k. This affords a check upon our work thus far, as we already knew that the bearing at a was equal to 6w, and we now see that this is made up of 1w at each of the four points b', c', d', e', and 2w at f'. If f'g' were not horizontal the arch could not be in equilibric under the assumed condition of load.

Now, as we manifestly have for the 4 remaining segments, a vertical reach for each, as the weights they respectively sustain; i. e., equal respectively to 2D,4D, 6D, and 8D; making 20D (=Cf'); altogether, we have only to subtract these quantities successively from Cƒ' (=1.0346), to obtain the lengths of verticals at h'. ', '; as follows:

The differences between these lengins of verticals, and those of the normal curve at the same points, show

the aberrations vertically, of the distorted, from the nor mal curve, as below.

LXXV. From this exhibit, we perceive that the greateet vertical aberration externally for the condition of load here assumed, is at hh', and equals .091v, and the greatest internally, at cc' (or a little to the right of these points in both cases), equal to .071v, traversing a zone equal in width to .162v. nearly of the versed sine of the normal curve.

Now, we have seen that the horizontal thrust of the arch for & gross load of 14w, equals 19.23w, with the assumed proportion of versed sine to span, as 1 to 10, whether upon the normal or the distorted curve; and, the thrust being evidently as the gross load, other things the same, it follows that, with the full gross ¡oad of 18w, or 2w at each angle, the thrust would be to 19.33w as 9 to 7. Hence the load, as above assumed, produces, or 77 per cent of the maximum thrust

inder the full uniform load.

The uniform load being supposed to act equally upon the outer and inner members of the rib, the action of 50 per cent is due to each; and, in order that neither

member, at the nearest approach to the equilibrated curve, may be subjected to greater stress than under the greatest uniform load, the web should be so wide. that (assuming the outward and inward aberrations to be each equal to the mean of .081v, and putting x= width of web), x: x+.081v:: 77.7: 50. Whence 50x =38.88x+77.7x.081v; and x=.557v.

But this value of x being equal to the distance vertically across the web between c and d, or between h and i, is greater than the distance square across, about in the ratio of distance from a to f, to the line aC, in this case as 26: 5. The actual width of web, therefore, is only .545v, still considerably more than half the versed sine Cf.

The condition of load here supposed, may or may not be the one requiring the greatest distortion of the equilibrated curve. The case has been assumed to illustrate this discussion, as it seemed likely to be near the condition requiring the greatest width of web; and I leave this part of the subject, without attempting a more general and determinate solution of the question.

LXXVI. The movable load has been taken as only equal to the weight of superstructure, upon the supposition that this style of bridging would seldom be adopted, except for very considerable lengths of span, where the weight of superstructure is relatively greater than in case of short spans.

This double arch, as here under consideration, con sisting of an outer and an inner curved member, connected by a web, in order to act most efficiently should be so adjusted that the outer and inner members may be subjected to equal action under a full maximum,

uniform load. Hence, the normal and equilibrated curves, representing the line of the resultant of forece acting upon the arch, have been assumed as terminating at each end, at points centrally between the extremities of the outer and inner curved members.

It might seem possible that the distorted curve adapted to the above assumed condition of load, might so fall as to recross the normal between the points of greatest departure and the ends, and thus diminish the extent of aberration, and the necessary width of web. If the curve a, b', c', etc., be turned upon its centre, by raising the end at a, by 3rds of the greatest departure, that is, by x.081v,=.054v, the aberration half way between a and f, where it is at or near its maximum point, would be reduced by .027v, and become .054r just the same as at the end. The other end would drop to the same extent, and would reduce the outward aberration in the same degree. This, of conrse, would be the least possible extent of aberration; and if we could rely upon the resultant stress following this curve in such a position, it would enable us to diminish the width of web to .364v.

But there seems to be no obvious reason why we should assume the equilibrated curve to take the posi tion just described, rather than one with the le end below a, and the other above k, thus increasing instead of diminishing the aberration. Hence, in the case of an arch ribbed bridge, liable to a movable load equal tc the weight of structure, foot for foot, upon the whole or any part of its length, if the web of the ribs be less than 36-100th, of the versed sine (Cf Fig 22), certainly. and if less than 54-100ths probably, the material in the principal members is liable to greater strain in some parts, under a partial, than rnder the extreme load

which would be decidedly an unfavorable condition, with regard to economy.

LXXVII. The operation of the web in distributing the action upon the outer and inner curved members of the rib, and transferring it from one to the other, may be understood by the diagram Fig. 23, exhibiting said curved members, connected by a web consisting of a simple system of diagonals, capable of acting by thrust or tension as may be required.

The normal curve is represented parallel with, and midway between the curved members; and the equili

FIG. 23

а

brated curve is represented as crossing the normal near f, meeting it again at a and k, at the ends; and having its greatest aberrations at c and h. It is manifest that the action of the outer member at i, is to that of the inner one at j, as jh to ih (inversely as their distances from the distorted curve), and that the action upon the outer diminishes, while that of the inner one. increases each way from i and j, until the action upon the two becomes equal at the meeting of the curves at k, and at the crossing point near f. Hence the diagonals leaning toward the point i must act by thrust. while those leaning from j, act by tension. On the contrary at d, where the greatest compression is upon the inner member, and diminishes each way, the diagonals leaning from e, act by thrust, while those lean