104 REPORT — 1869. 



With these observations I have now to refer to the drawings of the furnaces 

 and apparatus which I have attached in illustration as an Appendix. In 

 conclusion I may state that, looking at this new process and its further develop- 

 ment as a step in advance of what has already been done by Bessemer and 

 others, we may reasonably look forward to a new and important epoch in the 

 history of metaUurgic science. 



Before entering upon the experiments, it will be necessary to repeat the 

 formula of reduction as given in my previous Report of 1867. This appears 

 to be the more requisite, as it may be inconvenient to refer to the Transactions 

 of 1867, where it was originally introduced. 



FoKMULiE OF EeDUCTION. 



For the reduction of the Expervinents on Transverse Strain. — When a bar 

 is supported at the extremities and loaded in the middle, 



E = 4W^' (1) 



where I is the distance between the supports, K the area of the section of the 

 bar, cl its depth, w the weight laid on added to |- of the weight of the bar, 

 S the corresponding deflection, and E the modulus of elasticity. 



^=m^ (2) 



when the sectioi" of the bar is a square. 



These formulae show that the deflection, taken within the elastic Hmit, for 



. 2 

 unity of pressure is a constant, that is, — =D, a constant. 



Let —,—,..., -p be a series of values of D, determined by experiment 

 in a given bar, then 



I>4(^. + |+ ■■•+!) (3) 



which gives the mean value of this constant for a given bar. 

 Now, for the same material and length, 



^'°^^^^- (4) 



and when the section of the bar is a square, 



Tu'^'^'^I^ (5) 



If D, be put for the value of D when d=l, then 

 D^=DcZ^ 



which expresses the mean value of the deflection for unity of pressure and 

 section. This mean value, therefore, may be taken as the measxire of the 

 flexihilitif of the bar, or as the modidus of flexure, since it measures the 

 amount of deflection produced by a unit of pressure for a unit of section. 

 Substituting this value in equation (2), we get 



^=W,' 0) 



