ON A CONSTANT INDICATOR FOR STEAM-ENGINES. 315 



with the surface of the cone, from the apex of the cone, measured along its 

 side, is represented by (* r"- P + F) A,. Whence it follows that the radius of 

 the circle, which this point is in the act of describing by reason of the re- 

 volution of the cone, is represented in inches by (tt r- F~-f F) \ sin i. Also 

 the space described by the cord, which passes round the pulley N, and there- 

 fore by the circumference of that pulley, whilst the number A N is registered, 



12 

 is represented in inches by — A L ; and this pulley carries I'ound with it the 



cone, so that the space in inches described by a point in the cone at distance 

 unity from its axis, whilst the number A N is registered, is represented by 



12 



— =- A L. Now let the pressure P upon the piston of the engine be conceived 



m R 



to remain unchanged, whilst the exceedingly small space A L is. in the act of 



being described by it, the piston-rod of the indicator will then remain at rest 



during that time ; and the point of contact of the integrating wheel with the 



surface of the cone will, during that time, describe the arc of a circle whose 



radius is represented by (■n' r- P + F) X sin i, and which subtends at its centre 



12 

 an angle measured (at distance unity) by — ^ A L. The length of this arc 



described by the point of contact of the cone with the integrating wheel, and 

 therefore by the circumference of the latter, whilst A N is registered, is re- 

 presented therefore, on the above supposition, by 



J^r . A L . (ir 7-"- P + F) A sin i. 

 mR 



But since the circumference of the integrating wheel is represented by 2 tf p, 

 its circumference describes a space represented by 2 ff f A N, whilst A N re- 

 volutions, or parts of a revolution, are made by it : 



.-. J4 • A L . (tt 7-2 P + F) A sin » = 2 ff /> . A N 



...P.AL = _^i^l-.AN + -l.AL 



6 r* A sm < ~ vr^ 



... 2 P . A L = -^'"f . N + Jl . L, 



6 r'^ A sm J ~ ir r"^ 



where S P . A L represents the sum of all the different pressures P upon 

 a square inch of the piston, when situated at as many different points of the 

 length L of its strolic, each such pressure in pounds being first multiplied by a 

 space A L in feet, which the piston is supposed to describe under that pres- 

 sure remaining constant. Now this equation obtains, however small the 

 spaces may be taken which are represented by A L, through each of which 

 spaces the pressure upon the piston is supposed to remain constant ; it obtains 

 therefore if the spaces be taken infinitely small, or if each pressure be sup- 

 posed to remain constant only through an infinitely small space, or, which is 

 the same thing, if the pressure constantly vary, which is the actual case ; and 

 in this case the sum of all such products of the pounds in the pressure upon 

 a square inch of the piston by the feet in the space described under that pres- 

 sure, is evidently the effective work done by the steam upon a square inch of 

 the piston through tiie space L, estimated in pounds one foot high. Let this 

 number of units of work be represented by U, 



.•.U= ^'""P -N + JL.L, (1.) 



6 r- A sm i -^ it r- 



which formula determines (in pounds one foot high) the work U done by the 

 steam upon each square inch of the piston of the engine whilst the space L 



