that this state of equilibrium is produced proves conclusively that the rule above described and thus applied, although possibly it may be considered empirical, results in the correct solution of the question, and that the stresses shown are actually those which the girder would have to sustain under the given position of the live load.  Figs. 2 to 10 inclusive show stresses arrived at in this manner for every position of the live load.  An inspection of these diagrams shows:  a.  That there is no single instance of compression in a vertical member of the bowstring girder, b.  That every one of the diagonals is subjected to compression at some point or other in the passage of the live load over the bridge, c.  That the maximum horizontal component of the stresses in each of the diagonals is a constant quantity, not only for tension and compression, but for all the diagonals.  The diagrams also show the following facts, which are, however, recognized in the common formulae:  d.  The maximum stress in any vertical is equal to the sum of the amounts of the live and dead loads per bay of the girder. e.  The maximum horizontal component of the stresses in any bay of the top flange is the same for each bay, and is equal to the maximum stress in the bottom flange.  Having taken out the stresses in several forms of bowstring girders, differing from each other in the proportion of depth to span, the number of bays in the girder, and the amounts and ratios of the live and dead loads, similar results were invariably found, and a consideration of the various sets of calculations resulted in the following empirical rule for the stresses in the diagonals:  “The horizontal component of the greatest stress in any diagonal, which will be both compressive and tensile, and is the same for every diagonal brace in the girder, is equal to the amount of the live load per bay multiplied by the span of the girder, and divided by sixteen times the depth of girder at center.”  The following formulae will give all the stresses in the bowstring girder, without the necessity of any diagrams, or basing any calculations on the assumed action of any of the members of the girders: 

Let S = span of girder. 
    D = depth at center. 
    B = length of one bay. 
    N = number of bays. 
    L = length of any bay of top flange.
    l = length of any diagonal.
    w = dead load per bay of girder.
    w¹= live load per bay of girder. 
    W = total load per bay of girder = w + w¹.

Then:  S/B = N.

Bottom Flange.  WNS/8D = maximum stress throughout. (1)

Top Flange.—­In any bay the maximum stress =

+ WNS/8D x L/B = + WLN squared/8D (2)

Verticals.—­The maximum stress = -W. (3)

Diagonals.—­The maximum stress is

+- w¹lS/16DB = +- w¹lN/16D (4)

These results show that the method generally adopted in the construction of bowstring girders is erroneous; and one consequence of the method is the observed looseness and rattling of the long embraced ties referred to at the commencement of the article during the passage of the live load; the fact being that they have at such times to sustain a compressive stress, which slightly buckles them, and sets them vibrating when they recover their original position.