The books on the subject, repeating over a constant which was first, I believe, given by Brunel in testimony before a parliamentary commission, have held that one four-hundredth of a span is the proper ratio of flexure. This may have been a very good rule to give to the parliamentary commission, but it is hardly the practical method of limitation for a matter of engineering construction, because the flexure of a loaded beam is in the form of a curve, and therefore its law is that of a curvilinear function, and not of a straight line. I have examined a great number of precedents of good construction in this connection, and for mill use have deduced the formula for deflection in inches, _d_ = 0.0012 L², in which L is the length of span in feet. It will be readily recognized that the true constant of deflection of span is measured by the radius of curvature which will give a uniform and allowable distortion to the floor in either direction to the limit of the radius upon which this formula is based, which is 1,250 feet. I do not propose to offer to you on this occasion any remarks in regard to the treatment of the mathematics of the problem of applied mechanics concerned in the questions of transverse stress, knowing that you have certainly received instruction upon these subjects. But referring to the questions of mill floors, I would state that Southern pine beams of solid timber twelve by fourteen up to fourteen by sixteen inches are used; and instead of attempting the use of one piece of timber, it is preferable to use two pieces of the same depth and of half the breadth. These should be bolted together, with a space of an inch or so between them left by placing small vertical pieces of wood between the timbers when they are bolted together. In this manner one is more sure of sound timber, and in the process of seasoning there is less liability of dry rot in the interior, or of injurious checking, warping, or twisting. The end of the beams should rest upon iron plates in the masonry, and should be secured by means of a tongue upon the plate entering a groove across the lower side of the beam. It is not feasible to make this groove to a close fit with the tongue; but it is cut a great deal larger, and the whole brought to a firm bearing by means of pairs of wedges or quoins driven into the groove each side of the iron tongue. The outer end of the plate contains ribs or tongues reaching down into the brickwork. In this manner the timber is securely fastened to the brickwork; and yet in time of accident or of fire the falling of the beam in the middle of the mill will raise it up sufficiently so that it will clear the tongue and fall without tearing the wall down, which is the case whenever the beams are secured by bolts entering the end of the beam from the face of the wall.