SECT, vii.] STEAM ENGINES. 231 



a steam engine, is often obtained by this method ; and as the motion is not per- 

 fectly rectilinear, it is desirable to determine the point which renders it most 

 nearly so. 



490. In any regulating apparatus of this kind, it is of considerable importance 

 that the strains on the parts should not change their directions during the stroke ; 

 and this condition being premised, we shall have less difficulty in forming them to 

 act with regularity and certainty. The entire arcs described by B and D, in 

 Plate x. (A) have their equal chords in the same vertical line b d ; and since the 

 distance between the upper extremities and between the lower extremities of these 

 chords is in each case equal to the length of the link B D, it is plain that the 

 distance between the middles of these chords is also equal to the link ; that is, if 

 A B, C D, were both horizontal, we should have a D = the link B D, which evi- 

 dently cannot be the case, as the link is in an oblique position at half stroke. The 

 beam A B, and the radius bar C D, will however be both nearly in a horizontal 

 position at the middle of the stroke ; and if the strain is not to change its direc- 

 tion, the connecting bar B D should not pass a vertical position at either termi- 

 nation of the stroke : and to limit it to this condition, we shall in Plate x. (A) suppose 

 the bar, as shown by the dotted lines, to be exactly vertical, or coinciding with the 

 direction of the piston rod at each end of the stroke. 



Let A B and C D, Fig. 4. Plates x. (A) and (B) be the bars, B D the connecting 

 rod, and E the point to which the piston rod is to be attached ; b d being the direc- 

 tion it is to move in. Put AB = ns, CD = ms, B D = /, and the length of the 

 stroke of the piston rod s, which is equal to the chord of the arc described by the 

 bar A B. Make the versed sine of that arc v, and the versed sine of the arc 

 described by the end D of the radius rod = w. Then a B is the sum of these 



versed sines = v + w ; and v + w.v :: I : BE = - -. But, by the properties of 



the circle, we have s (m V m 2 ) = w, and s (n v/ n 2 ^) = v ; therefore, 



BE = -frL-V ' ~ i)!^ 



(w V m 2 ^) + ( <s/ 2 ) 



^___^^__^ T 1 



But we have very nearly v/ w 2 i = n 5, v/ ? 2 a = m 5 : and there- 



o n o wi 



fore w v/ 2 5 = 5 , m \/ m" i = ^ ; consequently, 



O M O Wl 



BE = OT/ DE = nl 



+ m + n 



.-. BE : DE ::::: CD : AB; 



that is, the segments of the link are inversely proportional to the lengths, or radii, of 

 the beams. 



