AND THE HYPERBOLIC LAW OF ELASTICITY. [179 



possible error, have been laid down by Gauss in the Theoria Combinationis Observationum, by 

 Legendre, Laplace, Poisson, Ivory, and others. 



Mr J. C. Adams, Fellow of St John's College, has by the method of Least Squares com- 

 puttd for the author of the present paper numerical formulae from the experiments here 

 considered, and has shewn that when the cube of the extension is included, the coincidence 

 of the formulae with the results of experiments on extension may be made almost absolutely 

 exact. 



According to such a formula, if e be the extension of a uniform rod of a unit of length, 

 and a unit of sectional area, the longitudinal force producing that extension is 



ae + fie* + /3V, 

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



Similarly, if c be the compression of a similar rod, the force producing that compression is 



7c + $c 2 + $V, 

 where y, 5, 6 are three other empirical constants. 



The following is a brief abstract of the method of applying these formulae to compute 

 small deflections of a uniform beam of rectangular section, resting horizontally on supports at 

 its extremities and deflected by a weight midway between them. First, the compression and 

 extension of every filament of the beam are expressed in terms of its radius of curvature and 

 distance from the neutral axis. From these expressions the sum of the forces of compression 

 above, and of extension below, the neutral axis, are obtained by integration. But the conditions 

 of equilibrium require that the horizontal elastic forces developed in any section of the beam, 

 above the neutral axis, be equal and opposite to those below the axis. Therefore by equating 

 the two integrals just referred to, the position of the neutral axis is obtained. 



Next the sum of the moments of the elastic forces about the neutral axis are obtained, and 

 the sums are equated to the moment about that axis of the pressure (P) of the nearest 

 fulcrum, the latter moment being the product of half the deflecting weight by the distance (x) 

 of the fulcrum from the axis about which moments are taken. This equation involves 

 the radius of curvature and is solved, by approximation, with respect to the reciprocal of 

 that quantity. 



Let the fulcrum be the origin of co-ordinates ; x, as above defined, the horizontal, and 

 y the vertical co-ordinate of a point in the elastic curve. The reciprocal of the radius of 

 curvature at the point (a?, y) is equal to (the second differential coefficient of y with respect 

 to x) -5- (a quantity which becomes equal to unity, when, as here, the horizontal inclination of 

 the tangent at any point of the curve is very small). 



It has not been considered necessary to give here the steps of the investigation, as the 

 analysis though tedious and involved is of an ordinary kind. Making the substitution indi- 

 cated, and integrating twice the equation last obtained ; it is found finally that, taking the 

 origin at the fulcrum, the equation to the elastic curve or neutral line of a rectangular beam 

 of depth d, horizontal breadth /x and length 2a is 



kw 3 fc,cV (26 2 -c)kV Ua? 6kV (26 2 -c)aVl 

 2.3 3.4 4.5 [2 3 4 J 



47—2 



