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 A B = n s, CD = ms, BD = /, 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 :: l: BE = —— . But, by the properties of 
the circle, we have s (m — >/ = w, and s (n — \/ if — = v; therefore, 
B E = (” ~y ~ 1) 1 
(m — s/ m? — A) + (» — s/ n~ — ±) 
But we have very nearly \/ rd — \ = n — \/ m z — | = m — ~ ; and there 
fore n — \/ rd — J m — \/ inf — ^ ; consequently, 
BE=- m? DE=. . 
m + n m + n 
.*. BE : DE :: m : n :: CD : AB; 
that is, the segments of the link are inversely proportional to the lengths, or radii, of 
the beams.