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Monday, 21st April 1851. 

 Dr CHRISTISON, Vice-President, in the Chair. 

 The following Communications were read : — 

 1. On the Economy of Single-acting Expansive Steam En- 

 gines, and Expansive Machines generally; being Supple- 

 ments to a Paper on the Mechanical Action of Heat. 

 By W. J. M. Rankine, Esq., C.E. 

 The author, in the first place, states the equations, which, when 

 used in conjunction with the Tables in the Appendix to the original 

 paper referred to, serve to compute the action of Cornish pumping 

 engines. They are similar in form to those of M. de Pambour, 

 but differ in the expressions for the pressure and volume of steam, 

 and for its expansive action, which the author in the original paper 

 deduced from theory. 



Let A denote the area of the piston. 



/, the length of the stroke. 



«, the number of double strokes in unity of time. 



c, the fraction of the whole bulk of steam above the piston at 

 the end of a down stroke, which is employed in filling the 

 valve-boxes and the clearance of the cylinder. 



Z', the length of stroke performed, when the steam is cut off. 



5, the ratio of expansion of the steam, so that 



1_ 

 1 ,. .V V J ^ 



Let W be the weight of steam expended in unity of time. 



Pj, the pressure at which it enters the cylinder. 



Vj, the corresponding volume of unity of weight of steam, 

 which may be found by means of Table I., already re- 

 ferred to. 



F, the resistance per unit of area of piston not depending on 

 the useful load. 



R, the resistance per unit of area of piston arising from the 

 useful load. 



Z, the ratio of the total action of the steam at the expansion s, 

 to its action at full pressure ; which may be found from 

 Table II. 



E, the useful effect in unity of time. 



