152 Cambridge Philosophical Society. 



of the latter quantities expressed by a converging series of their 

 ascending integral powers, the terms after the third may in general be 

 neglected as of inconsiderable magnitude. If then e be the exten- 

 sion of a uniform rod of a unit of length and a unit of sectional area, 

 the longitudinal force producing that extension is 



ae + /3<? 2 + /3'e 3 , 

 where a, /3, /3' are empirical constants. 



Similarly, if c be the compression of a similar rod, the force pro- 

 ducing that compression is 



yc + 8c°- + $'c\ 

 where y, 8, <T are three other empirical constants. 



These formula? are to be applied to a uniform beam of rectangular 

 section, resting on horizontal supports and slightly deflected at its 

 centre. For this purpose, the compression and extension of every 

 filament of the beam are expressed in terms of the radius of curva- 

 ture and the distance from the neutral axis. Analytical expressions 

 are thus obtained for the elastic forces developed in any transverse 

 section of the beam ; and the position of the neutral axis is obtained 

 from the integrals of these expressions by the principle, that the sum 

 of all the horizontal forces above is equal to the sum of all the hori- 

 zontal forces below the neutral axis. 



Next, the sums of the moments of the elastic forces about the neu- 

 tral axis are obtained ; and the sums are equated to the moment 

 about that axis of the pressure (P) of the fulcrum, the latter moment 

 being the product of half the deflecting weight by the distances (x) 

 of the fulcrum from the point of the neutral axis here considered. 

 This equation involves the radius of curvature, and is solved with 

 respect to the reciprocal of that quantity. It is to be observed, that 

 this equation, and also the preceding one determining the neutral 

 axis, are not of such a form as to admit of direct solution, and are 

 therefore solved by an ordinary method of approximation. 



The reciprocal of the radius of curvature of a point (x, y) of a 

 curve is equal to (the second differential of y with respect to <r)-r- 

 (a quantity which becomes equal to unity when, as here, the incli- 

 nation to the axis of x of the tangent at any point of the curve is 

 comparatively very small). 



Making the substitution indicated, and integrating twice the equa- 

 tion last obtained, we obtained finally for the equation to the neutral 

 line of a rectangular beam of vertical depth d, and horizontal breadth 

 p, and length 2a, 



_ kx? _ bxT-x* (2b <i —c)K 3 x b _ /xa g _ bxW (26 2 — c)a 4 x 3 \ 

 y ~273 3.4 4.5 \2 IT A )' 



where 



6=|<?(/3 + ^aY- J2 X 1 +a*y^ 4 )- 2 



c= | *1(1 +a*y-4)-3(/3' + a'aS y -S) 

 5 a 



jw, a 3 a 



