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



The coefficients A and B would be found from experiment. That B must be affected by 

 the negative sign appears from the consideration that dividing by e we have 



- = A - Be (2) 



And as the ratio w : e decreases as e increases the second term on the second side of equation 

 (2) must be negative. 



From the experiments recorded in the Report (1849) of the Royal Commission " appointed 

 to inquire into the Application of Iron to Railway Structures," it is manifest that the formula (1) 

 there adopted, is subject to unavoidable inaccuracies. A large number of results from it are 

 compared in the Report with the results of experiments. The bars were subjected to extension 

 by weights regularly increased up to the breaking weights. The differences between the 

 results of experiment and formulae are not + and — promiscuously, but their signs observe a 

 certain order ; that is, they are negative for several terms together, and positive for several 

 terms together. 



Eight formulae are given for extension of different kinds of cast iron, and in every case, 

 without exception, at least one half of all the results of each set come together in the middle of 

 the series, with errors in excess, and are preceded and followed by results in which the errors 

 are in defect. The formulas cannot however be considered satisfactory until the errors be 

 affected by the + and — signs promiscuously and without regular sequence. For while the 

 errors observe any general law, they are themselves capable of being represented by an 

 algebraical expression, which may be added to the original formula by way of correction. 



The general character of the errors is this : they are at first negative, then positive, and 

 increasing up to some term near the middle of the series; they then decrease till they 

 become negative again. Now since the sign of the error undergoes in general two changes, first 

 from — to +, secondly from + to — , the algebraical expression for it passes through zero 

 at each of these changes. Therefore if c be the correction which is zero when e, the extension, 

 is zero, and changes sign when e — a, and e = b, the equation 



c = A'e(a — e) (6 — e), 

 where A is constant, would express these characteristics of c. 



Effecting the multiplication of the quantities in the brackets, and adding the resulting 

 expression for the correction to the original formula we obtain a formula involving the first 

 three powers of the extension, which if correctly applied will be found to be much more 

 accurate than the formula involving the first and second powers only. 



In the Report above referred to, the weight producing an assigned extension of iron is in 

 one case computed by a formula involving the first four integral powers of the extension. 

 But in this biquadratic formula, as well as in the quadratic formulae, the numerical coefficients 

 are obtained by substitution in experimental results selected at random, and by taking the 

 mean of the coefficients so computed. This process being immethodic, is not likely to produce 

 the most accurate results. The mathematical laws of combination of observations, in which 

 several results are to be represented by a formula which on the whole shall give the least 



