on it, the position of the link is readily found, and by repeating the
process for other positions of the eccentrics a diagram of positions
(fig. 32) is drawn for the assigned state of the gear. A line AB
drawn across this diagram in the path of the valve’s travel determines
the displacements of the valve, and enables the oval diagram
to be drawn, which is shown to a larger scale in another part of fig.
32. The example refers to Stephenson’s link-motion in nearly full
forward gear; with obvious modification the same method may be
used in the analysis of Gooch’s or Allan’s motion.
Fig. 32.
The same diagram determines the amount of slotting or sliding motion of the block in the link. In a well-designed gear this sliding is reduced to a minimum for that position of the gear in which the engine runs most usually. In marine engines the suspension-rod is generally connected to the link at the end of the link next the forward eccentric, to reduce this sliding when the engine is in forward gear.
70.—Radial Gears.—Many forms of gear for reversing and for
Varying expansion have been devised with the object of escaping the 
Fig. 33.—Hackworth’s Valve-Gear.
use of two eccentrics, and in some both eccentrics are dispensed with.
Hackworth’s gear, the parent of several others, to which the general
name of radial gears is
applied, has a single eccentric
E (fig. 33) opposite the
crank, with an eccentric-rod
EQ, whose mean position is
perpendicular to the travel of
the valve. The rod ends in
a block Q, which slides on a
fixed inclined guide-bar or
link, and the valve-rod receives
its motion through a
connecting rod from an intermediate
point P of the eccentric-rod,
the locus of which
is an ellipse. To reverse the
gear the guide-bar is tilted
over to the position shown
by the dotted lines, and
intermediate inclinations give
various degrees of expansion
without altering the lead.
The steam distribution is
quite satisfactory, but an
objection to the gear is the
wear of the sliding-block and guide. In Bremme’s or Marshall’s
form this objection is obviated with some loss of symmetry
in the valve’s motion by constraining the motion of the point
Q, not by a sliding-guide, but a suspension-link, which makes
the path of Q a circular arc instead of a straight line;
to reverse the gear the centre of suspension R of this link
is thrown over to the position R′ (fig. 34). In the example
sketched P is beyond Q, but P may be between Q and the crank
(as in fig. 33), in which case the eccentric is set at 180° from the
crank. This gear has been applied in a number of marine
engines. In Joy’s gear, which is extensively used in locomotives,
no eccentric is required; and the rod corresponding to the
eccentric rod in Hackworth’s gear receives its motion from a
point in the connecting rod by the linkage shown in fig. 35, and is
either suspended, as in Marshall’s form, by a rod whose suspension
centre R is thrown over to reverse the motion, or constrained, as
in Hackworth’s, by a slot-guide whose inclination is reversed. Fig.
36 shows Joy’s gear as applied to a locomotive. A slot-guide E is
used, and it is curved to allow for the obliquity of the valve connecting-rod AE.
C is the crank-pin, B the piston path and D a fixed
centre.
A form of radial gear very largely used in locomotives, especially on the continent of Europe, is the Walschaert or Heusinger-Waldegg gear, in which the valve receives its motion in part from the piston cross-head through a reducing lever, and in part from a single eccentric set at right angles to the crank, which actuates a rocking link. Reversing is effected by shifting a sliding block along this rocking link from one side to the other of the centre on which it rocks.
Fig. 34.—Bremme’s or Marshall’s Valve-Gear.
71. Separate Expansion-Valves.—When the distribution of steam is effected by the slide-valve alone the arc of the crank’s motion during which compression occurs is equal to the arc during which expansion occurs, and for this reason the slide-valve would give an excessive amount of compression if it were made to cut off the supply of steam earlier than about half-stroke.
Fig. 35.—Diagram of Joy’s Valve-Gear.
Hence, where an early cut-off is wanted it is necessary either to use an entirely different means of regulating the distribution of steam, or to supplement the slide-valve by another valve—called an expansion-valve, usually driven by a separate eccentric—whose function is to effect the cut-off, the other events being determined as usual by the slide-valve.
Fig. 36.—Joy’s Gear as applied to a Locomotive.
Such expansion-valves belong generally to one or other of two types.
In one the expansion-valve cuts off the supply of steam to the chest
In which the main valve works.
Fig. 37.—Expansion-Valve on back of Main Slide-Valve.
In the other the expansion-valve slides on the back of the main 
Fig. 38.
slide-valve, which is provided with through ports which the expansion-valve
opens and closes. Fig. 37 shows
one form of this type. Here the resultant
relative motion of the expansion-valve and
main-valve has to be considered. If 𝑟𝑎 and
𝑟 (fig. 38) are the eccentrics working the
main and expansion valves respectively, then
CR drawn equal and parallel to ME is the
resultant eccentric which determines the
motion of the expansion-valve relatively
to the main-valve. Cut-off occurs at Q,
when the shaft has turned through an
