[180] H. COX, ESQ., ON THE DEFLECTION OF IMPERFECTLY ELASTIC BEAMS 

 where 



6 = - d (/3 + SaV 2 ) (1 + aly-t)-\ 



c = ! -(l+aJ 7 -J)- 3 (/3' + ^a5 7 -«), 

 5 a 



P 3 

 k = - -rr-(l +a^^) 8 . 

 ju era 



The constants in the equation to the curve are determined by the conditions that it passes 



through the origin and that — = 0, when x = a. 



ax 



If according to the ordinary hypotheses we put a = 7, and neglect terms depending on 



/3, )8', 8, & (or put those coefficients equal to zero) 6 and c become zero. 



The equation to the curve then coincides with ordinary equation to the elastic curve given 

 by Poisson, (Traite de Mecanique, 324) and others. 



If in the equation to the elastic curve here investigated, w be put = a, the value of the 

 deflection at the centre of the beam is 



K a 3 6«V (26 2 -c)« 2 a 5 



+ , 



3 4 5 



whence it may be seen that the deflection is greater than it would be if the elasticity were 



perfect. 



All these formulae however lead to very complicated results when applied to investigations 



respecting the deflection of beams. The cubic and quadratic formulas have moreover the 



serious inconvenience that from the expression for the weight in terms of the extension, to 



find conversely the extension in terms of the weight involves the difficulty of solving in one 



case a cubic, in the other, a quadratic equation with large numerical coefficients. 



Hyperbolic Law of Elasticity. 



The formula about to be proposed* is far more accurate than the formula (l), and has 

 greatly the advantage in point of simplicity of computation from it. It possesses the practical 

 advantage of great facility in calculating either the extension in terms of the weight or the 

 weight in terms of the extension. When applied to the theory of beams it leads to an ex- 

 pression, similar in form to itself, from which with the utmost readiness the deflection can be 

 computed from the transverse pressure or the transverse pressure from the deflection. 



If e be the extension of a rod produced by a stretching weight w, it will be found by 



• A notice of this formula was read by Prof. Stokes on I for the Advancement of Science, 1850, and is printed with the 

 behalf of the author at the meeting of the British Association | last Annual Report of the Association. 



