the number and horizontal reach of diagonals, while the vertical reach is as the depth of truss, it follows that the stress of chords is directly as the length and inversely as the depth of truss, other conditions being the same.

Hence, if the depth of truss be so reduced as to make the ratio of length to depth indefinitely large, the stress and required material of chords, become indefinitely large. On the contrary, if the depth be indefinitely great, although the stress of chords be ever so small the length and required material for diagonals and verticals must be indefinitely large. It is manifest, then, that between these two extremes there is a practical optimum,— a certain ratio of length to depth of truss, which, though it may vary somewhat with circumstances, will give the best possible results as to economy of material in the truss. This matter will be taken into consideration hereafter, and is referred to here, to show the expediency of generally increasing the depth, with increase of lengths in the truss.

Now, in trusses of considerable length, and, consequently, depth, it becomes expedient, in order to avoid too great a width of panel (horizontally), or an inclination of diagonals too steep for economy of material in those members, to extend them horizontally across two or more panels, or spaces between consecutive nodes of the chords. In such cases, the truss may be called double or treble cancelated, according as the diagonals cross two or three panels.

LVI. To estimate the stresses of the members of double cancelated trusses with vertical members, a slight modification of the process already described, [XL, &c.], is required, as follows:

Having placed the numbers 1, 2, 3, &c., over the nodes of the upper chord, as seen in Fig. 18, place under each odd number, the sum of all the odd numbers in the first series, up to and including the one under which the sum is placed; and the same with respect to the even numbers. Then, the second series of figures may be used in precisely the same manner as that explained with reference to Fig. 12, to determine the weight sustained by, and the maximum stress produced upon, each diagonal and vertical, by equal weights upon all or any of the nodes of either chord.

For example; supposing the truss to have tension diagonals and thrust verticals; take the diagonal havits lower end under 5 (upper series), and its upper end under 7. This diagonal may be represented by 5/7, while 57 may indicate its antagonist, and so of other diagonals. Then, as we see 9 (the sum of 1+3+5), in the second series, over the lower end of 5/7, and, as the diagram represents a truss of 16 panels, we know that the diagonal in question is liable to a maximum weight of w, =9w". This amount is to be diminished, of course, by the weight due from weight of structure to the counter diagonal.

Again, the diagonal 9/11 sustains as a maximum from variable load, 25w"; which will require to be increased on account of weight of structure, since the latter, in this case, acts upon the main, and not upon the counter diagonal, as in case of 5/7.

Now, to obtain the effects of weight of structure and uniform load, the truss having even panels, we place under the centre node of the lower chord, because half of the weight w', which is supposed to be concentiated at that point, tends to act on each of the dia gonals rising from that point.

At the next node from the centre, each way, the figure 1 is set, because, of the weights (w'), concentrated at those points, each bears upon its nearest abutment (the truss being uniformly loaded), through the diagonals running upward and outward from those points. If this be not so, each must transmit a part of its amount past the centre, through the antagonistic diagonals 7/9 and 79, which is contrary to statical law.

Then we put 11, 21, 31, etc., under alternate nodes from the centre, and 1, 2, 3, etc., under alternates beginning at the first on each side of the centre; as shown in diagram Fig. 18.

These figures form the coefficients of w', to indicate the weights acting, or tending to act, upon the diagonals running upward and outward from these numbers respectively, arising from weight of structure, and also, the co-efficients for (w+w'), to express the load tending to act on diagonals, arising from both superstructure and movable weight, when the truss is fully loaded. For illustration;

the diagonal 5/7 we have seen to be liable to a maximum stress of 9w" from variable load, and, as we have the figure 1 at the foot of 57, it shows that the weight due to the latter on account of structure is lw', which must be subtracted from 9w" to obtain the actual maximum to which 5/7 is liable; which is 9w'-w'.

If w' be equal to or greater than 9w", then 5/7 is subject to no action, and may be dispensed with. As to the advantage of introducing counter diagonals, merely for the purpose of stiffening the truss, the results of my investigations will be given in a subsequent part of this work.

The maximum weight sustained by any thrust upright, is manifestly equal to the greatest weight borne by either diagonal connected with it at the upper end, since any weight borne by 3/5, for instance, being transferred to the antagonist of 57, thereby diminishes by a like amount, the maximum action of the latter. Whence the upright at 5, can receive no more load from the two diagonals, than the maximum load of one, and this relation holds in general.

The reason of adding alternate figures to form the second series over the diagram, will be obvious, when it is observed that there are two independent systems of uprights and diagonals; one of which includes the uprights under even numbers in the upper series, and the diagonals connecting therewith, and the other, the remaining uprights and diagonals. Now weight applied at the nodes of either of these systems, can only act upon members of that same system; that is, weight applied at nodes indicated by even numbers in the upper series can only act upon the first above named system of uprights and diagonals, and vice versa.

The main end braces are acted upon by both systems; so that to obtain the weight sustained by them, we must add the numbers 56 and 64 (and corresponding numbers in other cases), making in this case 120w" equal to 7w.

The uprights under 1 and 15, sustain each a tension equal to w, for variable load, and to w+w, for weight of variable load and superstructure together; which obviously gives their greatest strain.

Having thus determined the greatest weights to which the several verticals and diagonal members are liable, we proceed as in former cases, to multiply those weights by lengths of diagonals, and divide the products by lengths of verticals, to obtain the stresses of diagonals; remembering to take into account the difference in length between those having a horizontal reach of only one panel, at and near the ends of the truss, and those that reach across two panels.

h

The mode of estimating the stresses upon the different portions of the chords, depending upon the horizontal action of diagonals, has been sufficiently explained. It is only necessary to observe that the end braces produce compression upon the upper, and tension upon the lower chord, through their whole lengths, equal to (w+w'), multiplied by the number of nodes. of the lower chord, and that product multiplied by and that each pair of intermediate diagonals analogously situated with respect to the ends of the truss, whether acting by thrust or tension, produce tension and thrust in like manner, upon the portions of the lower and upper chords, between their points of connection with the chords. Thus is generated a progresBIV and determinate increase of action upon succeeding