842 Transactions of the American Institute. 



the quadrant preceding, exactly at the middle point, where <p = 135°. 

 The point of equal division of the whole combined will be not far 

 from that in which <p = 126°. 



In working steam without cut-off, the moving parts being supposed 

 to be without inertia, the points of equal division of the work in the 

 successive quadrants are those in which 9 = 60°, and 9 = 120°. 



The ])receding figure shows these several points. It may be 

 remarked finally in respect to this matter, that, inasmuch as the 

 logarithmic theory, applied to the pressure of steam working 

 expansively, is too favorable to the low pressures in the second half 

 stroke ; therefore, in practice, considerably more mass may with pro- 

 priety be given to the reciprocating parts of the engine than the 

 theoretical determinations here reached would justify ; and the effect 

 of such increase of weight will be to advance the point of equal divi- 

 sion of the work of the second quadrant still nearer to the middle 

 point where 9 = 135°. 



One further particular deserves a moment's attention here: It will 

 be seen, by a careful comparison of the last diagram and the accom- 

 panying table, that the effective force exerted tangentially on the 

 crank, for points only 10° from the centers, exceeds one-fourth the mean 

 pressure on the piston, or that which, exerted uniformly, would do the 

 same work per revolution, while the mean tangential effect of such 

 uniform pressure would be about five-eighths ; also, that the effective 

 force runs from these points rapidly up. It is only, therefore, for 

 about one-ninth part of the semi-revolution that the driving force 

 falls below, say, two-fifths of the mean ; and this, with 120 revolu- 

 tions to the minute, gives less than the thirty-sixth part of a second 

 to the duration of each interval in which this force is largely 

 deficient. To this fact may be ascribed, in large measure, the facility 

 with which the crank passes the centers, so that this facility is mani- 

 festly promoted by running at high speed. 



The conclusions to which the foregoing discussion leads, may be 

 summed up briefly thus : 



High velocities and considerable weight in the reciprocating parts 

 of cylinder steam-engines, working expansively, are advantageous in 

 the following particulars : First. In distributing the work done with 

 a near approach to uniformity over the circumference described by 

 the crank in its revolution ; and, secondly, in greatly reducing the 

 irregularities of strain experienced by the working parts of the 

 engine, especially as it respects the torsion of the shaft. 



The advantage derivable from the use of heavy pistons increases 



