360 Mb ROHRS, on THE MOTION OP BEAMS 



subject of vibrating railway girders. There is one case, however, and that the most common 

 one, in which the deflection of the girder is very slight, and the pressure of the load so nearly 

 constant as to be capable of being so assumed, at least for a first approximation ; in this 

 case — whether the mass of the girder be neglected or not — the problem admits of an easy 

 solution by the aid of Fourier's Functions; and the central deflection can be exhibited in the 

 form of a rapidly convergent series, and in a shape very convenient for discussion. Pro- 

 fessor Stokes has also considered this case, but he has solved it by a quite difierent method 

 from the one I have employed. 



And now to proceed to the analysis. 



Let AO, UA' be a section of a slender elastic prismatic quadrangular rod made by the 

 plane of the paper, which is supposed parallel to either of the narrow sides of the rod. 



Let the length of the rod = a, 



thickness = A, 



breadth = k, 



FT, Pi'Fi an element length = Ss, 



OX tangent at the axis of x, 



OP = fi, y the ordinate at P. 



Then the element Pi'P is kept at rest by the tensions and thrusts arising from the elasticity 

 of the rod and by reactions R, B + SR acting along PP", P/A. We shall assume that the 

 "elements of the rod are incapable of sliding over each other, and that the thickness remains 

 uniform, so that all the laminae of the rod will always have the same centre of curvature at 

 the same time. We shall also suppose the rod to be so slightly disturbed that the longitudi- 

 nal motion of its elements may be neglected; consequently the neutral axis will be in the middle 

 of the rod. Also if we suppose m to be the mass of an unit's length and breadth of the 

 rod, and V a pressure arising from other than molecular forces extending over an unit's 

 length and breadth of the rod at point (y, tv), we shall have 



RSs^S('-!^^).k 

 d? \ds) " ~ . li' 



eP (dy\ 



mk — & 

 12 



where C is a constant, depending on the elasticity. 



