SECT, vii.] STEAM ENGINES. 233 



commonly made to divide the versed sine of the arc, described by the end of the 

 beam, into two equal parts, as in Plate x. (B.) l 

 Let, 



b = the length of the beam from the centre of motion, A F. 



c = the length of the parallel bar D G. 



r = the length of the radius bar C D. 



s = half the length of the stroke. 



v = the versed sine of the angle described by the radius bar. 



a = half the angle described by the beam. 



Assume that the radius bar is horizontal when the beam is horizontal ; this cannot 

 be strictly true, except when the vibration is bisected, but is generally very nearly so. 

 Then, (b c) sin. = V 2 r v 1>* = half the chord of the arc described by the 

 end D of the radius bar. But v = c (1 cos. a) ; and substituting this value of v 

 in the equation and squaring, it becomes 



(b c)" sin. a = 2 r c (1 cos. a) e 2 (1 cos. a)* ; 

 and by reduction, observing that sin. 2 a = (1 cos. a) (1 + cos. a), 



b(b- 2c) . 

 r = -ig- - '- (1 + cos. a) + c, 



which is a convenient formula for the radius bar when the angle is fixed; but 

 when it is not, we have 



= (.- 



the length of the radius bar. 



If the centre C of the radius rod should be fixed by necessity or convenience 

 as in steam packet engines, let h = C G its horizontal distance from the line 

 traversed by the piston ; then c = r h, which substituted in the above, and 

 again solved for r, we get, 



j ft* (1 + cos, a) i&(6 y 



A + b (I + cos. a) H - A + 6 + v/6 e_J - 



in which the former term expresses the parallel bar G D ; and h must be considered 



1 Tredgold has adopted the former construction in Plates xi. xix. and xx. : there can be 

 no doubt, however, that almost every circumstance is in favour of the latter mode of ad- 

 justment, and that manufacturers have shown good judgment in adopting it. The bisection 

 of the vibration certainly causes the strain to change its direction, but this is of little con- 

 sequence, compared with the important advantage of rendering it of the least possible amount. 

 Another advantage in this construction is, that the beam and radius bar are parallel to the horizon 

 at half stroke ; and that the radius bar, like the beam, works equally on both sides of it. ED. 



2 G 



