The static head against which the pumping-engine operates is the vertical height or elevation between the water surfaces in the pump-well and the reservoir. The head which represents the resistances of the passages through the pump and force-main, when added to the sum of the static head and the head due to the velocity in the force-main, gives what is called the dynamic head; it represents the total head against which the pump acts. If _h_ represents the static head, _hʹ_ the head due to all the resistances, and _h″_ the head due to the velocity in the force-main, then the dynamic head will be _l v² v² v²_ _H_ = _h + hʹ + h″_ = _h_ + _f_ --- ---- + _n_---- + ----, _d_ 2_g_ 2_g_ 2_g_ in which _f_ has a value of about .015 and _n_ is a coefficient which when multiplied by the velocity head will represent the loss of head incurred by the water in passing through the pump-cylinder and valves. The latter quantity is variable in value; but it is seldom more than a few feet. =197. Duty of Pumping-engines.=—It is thus seen that the collective machines and force-main forming the pumping system afford opportunity for a number of serious losses of energy found chiefly in the boiler, the engine, and the pump. The excellence of a pumping-plant, including the boilers, may obviously be measured by the amount of useful work performed by a standard quantity, as 100 pounds of coal. Sixty or more years ago, in the days of the old Cornish pumping-engine, the standard of excellence or “duty” was the number of foot-pounds of work, i.e., the number of pounds lifted one foot high, performed by one bushel of coal. As early as 1843 the Cornish pumping-engine reached a duty, per bushel of coal, of 107,500,000 foot-pounds. These pumping-engines were single-acting, the steam raising a weight the descent of which forced the water up the delivery-pipe. At a later date and until about ten years ago the usual standard or criterion applied to pumping-engines for city water-works was the amount of work performed in lifting water for each 100 pounds of coal consumed; this result was also called the “duty” of the engine. In order to determine the duty of a pumping-engine it was thus only necessary to observe carefully for a given period of time, i.e., twenty-four hours or some other arbitrary period, the amount of coal consumed, the condition of the furnace-fires at the beginning and end of the test being as nearly the same as possible, and measure at the same time the total amount of water discharged into the reservoir. The total weight of water raised multiplied by the total number of feet of elevation from the water surface in the pump-well to that in the reservoir would give the total number of foot-pounds of useful work performed. This quantity divided by the number of hundred pounds of coal consumed would then give what is called the “duty” of the pumping-engine.