SECT, vii.] STEAM ENGINES. 229 



may be any where in the lune represented by the dotted lines, and drawn from 

 the centres A and B, with radii each equal to the length of the connecting rod. If 

 we sum up the forces acting in the circle, we find them exactly equal to the mecha- 

 nical power in the straight line, the additional friction excepted. 



The following table is calculated for an uniform force acting in a straight line ; 

 the moving force in the straight line is supposed to be unity, and the table shows 

 the pressure it produces in the direction of a tangent to the circle, at the quarters 

 and at every thirty degrees of its path. It will enable the reader to judge of the 

 effect of a variable force ; and where the acting point describes a curve, it is less 

 regular, but not so different nor so important as to require investigation : the last 

 column is added to show the additional stress on the axis above that which would 

 take place if the axis were turned by toothed wheels. 1 



1 When the moving force reciprocates in the straight line A B, and the end D of a connecting 

 rod is moved round in a circle ; let c be the angle D C O which the connecting rod forms with the 

 direction A B of the motion, and a the angle D O E or arc described from E. The force in the 

 direction of the connecting rod is P sec. c, where P is the force when the rod is in the position 

 A E. Also, since C F is parallel to O D, the L O C F = L D O E = a, and consequently the 

 Z.DCF = c + a; therefore the force in the direction F D of the tangent to the circle = P sec. c 



sin. (c + a) = P sec. c (sin. c cos. a + cos. c sin. a) = P [ !_' cos. a + sin. a ). But, when the 



\cos. c / 



connecting rod is n times the length of the crank, sin. a = sin. c, sin. c = Sln ' a , and cos. c 



' -; therefore, 



[ J = the circular force at D for any 



n 



cos. a 



n 2 sin. 2 a 

 The additional stress on the axis or shaft, and consequently the friction, is as 



P sec. c cos. (c + a) or P ( cos. a sin. 2 a \ 



v // 2 sin. 2 a / 



The additional friction (being about one-eighth of the pressure) is therefore never greater than 



where a is the diameter of the shaft, and r the radius of the crank, both in inches. 



Pa 

 167' 



By construction of the figure the values may be found from a scale of equal parts ; for if C G 

 be the pressure, F D will be the force in the circle, and C F the stress on the axis. 



