A.—­To find how far the piston is from the end of its stroke when the steam that is propelling it by expansion is allowed to escape to the atmosphere or condenser—­to the lap on the steam side add the lead; divide the sum by half the stroke of the valve, and find the arc whose sine is equal to the quotient; find the arc whose sine is equal to the lap on the eduction side, divided by half the stroke of the valve; add these two arcs together and subtract 90 deg.; find the cosine of the residue, subtract it from 1, and multiply the remainder by half the stroke of the piston; the product is the distance of the piston from the end of its stroke when the steam that is propelling it is allowed to escape into the atmosphere or condenser.  In using these rules, all the dimensions are to be taken in inches, and the answers will be found in inches also.

193. Q.—­Is it a benefit or a detriment to open the eduction passage before the end of the stroke?

A.—­In engines working at a high rate of speed, such as locomotive engines, it is very important to open the exhaust passage for the escape of the steam before the end of the stroke, as an injurious amount of back pressure is thus prevented.  In the earlier locomotives a great loss of effect was produced from inattention to this condition; and when lap was applied to the valves to enable the steam to be worked expansively, it was found that a still greater benefit was collaterally obtained by the earlier escape of the steam from the eduction passages, and which was incidental to the application of lap to the valves.  The average consumption of coke per mile was reduced by Mr. Woods from 40 lbs. per mile to 15 lbs. per mile, chiefly by giving a free outlet to the escaping steam.

194. Q.—­To what extent can expansion be carried beneficially by means of lap upon the valve?

A.—­To about one-third of the stroke; that is, the valve may be made with so much lap, that the steam will be cut off when two thirds of the stroke have been performed, leaving the residue to be accomplished by the agency of the expanding steam; but if more lap be put on than answers to this amount of expansion, a very distorted action of the valve will be produced, which may impair the efficiency of the engine.  If a further amount of expansion than this is wanted, it may be accomplished by wire drawing the steam, or by so contracting the steam passage that the pressure within the cylinder must decline when the speed of the piston is accelerated, as it is about the middle of the stroke.

195. Q.—­Will you explain how this result ensues?