Ex. 123. How many men would in 40 minutes raise the anchor in the last example? Ans. 43 men. Ex. 124. Through how many fathoms could 15 men raise the anchor of Ans. 17 nearly.

Ex. 122 in 10 minutes? Ex. 125. If 13 men are required to raise an anchor through 180 fathoms in 20 minutes, what must be the weight of that anchor? Ans. 753) lbs. Ex. 126. A town is situated 25 miles from the mouth of a coal pit, from which coal is taken to the town by a level railway on which the resistance is 10 lbs. per ton; the engine employed is of 15 horse-power and weighs with its tender 10 tons; each truck weighs 3 tons and contains 7 tons of coals; on each journey the engine takes 5 full trucks and returns with 5 empty trucks; supposing no time to be lost at the ends of the journey, how many tons of coals will be taken to the town in 48 hours? How many horses would be required to convey the same quantity of coals in the same time? Ans. (1) 445 tons. (2) 665 horses.

17. Remarks on the work yielded by different agents. -The following remarks upon the preceding tables and examples are worthy of the attention of the reader :

(1) Every agent must be allowed to move at a certain rate in order to do the greatest amount of work it is capable of yielding; thus, a horse walking does considerably more work than a horse trotting, as an inspection of the tables will show. And this is true not of animate agents only, but of inanimate; thus the work yielded by the consumption of a given quantity of coal will be larger in the case of a slow than of a fast engine.

(2) Also, in order that an animate agent may do its greatest amount of work it must not be required to exert more than a certain amount of pressure. This is also plain from an inspection of the table.

(3) It follows from the above considerations that though two agents may be capable of doing the same work in the same time, it may be in practice impossible or disadvantageous to substitute the one for the other. Thus an ox and a horse walking in a whim gin do very nearly the same amount of work; but since the ox moves more slowly, and exerts a greater pressure than the horse, it would generally be disadvantageous to substitute a horse

for an ox in a machine requiring a slow heavy pressure. Again, in cases where great speed is a desideratum, it would generally be impossible by any machinery to make the slow agent perform the labour of the rapid agent; as, for instance, in the case of locomotion.

18. On the Cost of Labour. The chief elements in the cost of labour may be enumerated as follows :

(1) In the case of human labour, the whole cost is the wages paid.

(2) In the case of a horse, the elements of expense are attendance, keep, and the original cost; the last is but a small portion of the expense. Thus, if we suppose a horse to cost 20l. and to continue in working order for ten years, and reckon the value of money at four per cent. per annum, the element of cost would be 2.465l. yearly, or not quite 1s. per week.

(3) In the case of a steam engine, the chief elements are the original cost and subsequent repairs, attendance, and fuel. Of these elements the most important is that of fuel; and accordingly there is a special definition of the power of an engine with reference to the consumption of fuel. The definition is as follows :

Def. The number of units of work yielded by an engine in consequence of the consumption of 1 bushel (i.e. 84 lbs.) of coal, is called the duty of that engine.

The extent to which the economy of fuel may be carried is very remarkably illustrated by the engines employed to drain the mines in Cornwall. In 1815, the average duty of these engines was 20 millions; in 1843, by reason of successive improvements, the average duty had become 60 millions, effecting a saving of 85,000l. per annum;

*

* Bourne on the Steam Engine, p. 171. It may be remarked that this result depends largely on the construction of the boiler; 1 lb. of coal in the Cornish boiler evaporates 11 lbs. of water, while in the waggon-shaped boiler 8.7 is the maximum.-FAIRBAIRN, Useful Information, p. 177.

D

it is stated also, that, in the case of one engine, the duty was raised to 125 millions.

The actual cost of 1,000,000 units of work, when done by different agents, cannot be specified with great precision; but a sufficiently accurate notion of the relative cost of different agents may perhaps be obtained from the annexed table, which has been calculated upon the following suppositions :

(1) The wages of a labourer, 3s. a day.

(2) Keep of a horse, 2s. a day; attendance of 6 horses, 38. a day; cost of each horse, 2d. a day.

(3) Steam engine of 50 horse-power, at an annual cost of 5l. per horse-power; attendance, 12s. a day; coal, 6d. a bushel.*

TABLE X.

COST OF LABOUR.

Character of Agent

Cost per Million
Units of Work

(1) Labourer carrying weights up a ladder
(2) Labourer raising weights by rope and pulley

(3) Labourer turning a winch

28.80

(4) Labourer turning a capstan

(5) Horse in a whim gin trotting

(6) Horse in a whim gin walking

(7) Horse walking in a cart

(8) Steam engine, duty 20 millions

(9) Steam engine, duty 90 millions

* In Weale's Contractor's Price Book for 1859 the prices of various steam engines are estimated to be from 25l. to 35l. per horse-power, boilers and fittings included; as the nominal horse-power (which is determined by measurement) is considerably less than the working horse-power the estimate in the text is very ample; that estimate assumes 50l. the cost of a horse-power, and assumes that 10 per cent. will represent interest on capital, repairs, and restitution. It may interest the reader to consider the following statement taken from Mr. R. Stephenson's paper on Railway Economy which forms an appendix to Mr. Smiles's Life of George Stephenson. In 1854 there were in the United Kingdom 5000 locomotive engines costing from Ex. 127. How many bushels of coal must be expended in a day of 24 hours in raising 150 cubic feet of water per minute from a depth of 100 fathoms; the duty of the engine being 60 millions? Ans. 135 bushels.

Ex. 128. Determine the number of horses working in whim gins required to do the work of the last example. Determine also the weekly saving effected by employing steam power, supposing the total weekly expense of the engine to be double the price of coals consumed; the coals costing 10s. a ton; and each horse 20s. a week.

Ans. (1) 960 horses. (2) 924l. 11s. Od. weekly saving.

Ex. 129. There are three distinct levels to be pumped in a mine; the first 100 fathoms deep, the second 120, the third 150; 30 cubic feet of water are to come from the first, 40 from the second, and 60 from the third per minute; the duty of the engine is 70 millions. Determine its working horse-power and the consumption of coal per hour.

Ans. (1) 191 H.-P. (2) 5.4 bushels.

Ex. 130. In the last example suppose there is another level of 160 fathoms to be pumped, that the engine does as much work as before for the other levels, and that the utmost power of the engine is 275 H.-P. Find the greatest number of cubic feet of water that can be raised from the fourth level. Ans. 464 cub. ft.

Ex. 131. An engine raises every minute a cubic feet of water from a depth of a fathoms, в cubic feet of water from a depth of b fathoms, and c cubic feet of water from a depth of c fathoms. The diameter of the piston of the steam engine is d in., the length of the stroke I ft., it makes n strokes per minute; also it consumes o bushels of coal in twenty-four hours, and has a modulus m. Determine (1) the pressure per square inch upon the piston; (2) the horse-power of the engine (as measured by pressure of steam on piston); (3) its duty.

Ex. 132. Water is to be raised from three levels of 20, 30 and 40 fathoms respectively; 10 cubic feet of water are to be taken per minute from the first, 20 from the second, and 40 from the third. The engine consumes 15 bushels of coal in a day. The diameter of the piston is 4 ft., it makes 10 strokes of 6 ft. each per minute. The modulus of the engine is 0.65.

2000l. to 2500l. apiece, and consuming annually 13 million tons of coke, made from 20 million tons of coal. It appears moreover that if a railway company start with 100 new engines about 20 or 25 will need repair at the end of four years, and after that there will always be about 25 in the workshop.

Find the pressure per square inch on the piston, the horse-power (as measured by pressure of steam) and the duty of the engine.

Ans. (1) 12.75 lbs. (2) (nearly) 42 H.-P. (3) 133,000,000 duty. Ex. 133. In Ex. 126 suppose the engine and trucks on the one hand and the horses and carts on the other to want renewal every ten years; suppose also that each horse and cart costs 40l., that one man attends to every six horses and is paid 3s. a day, that each horse's keep is 1s. 6d. a day, that there are two turnpikes on the road at each of which there is a toll of 6d.; determine the cost of transporting 445 tons of coals. Next suppose the engine and tender to cost 1000l., each truck 120l. (15 trucks are required to prevent loss of time); that there are three drivers and three stokers each at 6s. a day; that money is worth 5 per cent. and that each mile of road cost 10,000l. to make and 365l. a year to keep in repair; determine in this case the cost of transporting 445 tons of coals. Also if coal cost 3s. a ton at the pit mouth what will it cost in the town according to each method of transport, neglecting profit?

Ans. (1) 2141. (2) 1231. (3) 12s. 6d. a ton by cart.

(4) 8s. 6d. a ton by rail.

[Interest on the cost price of engine, trucks, horses and carts can be neglected.]

SECTION II.

19. On the Work done by a Variable Force. There are two important questions in the subject of work which we shall treat in the present section: they are (1) the work done by a variable force, when exerted through a certain space; (2) the total amount of work done in raising a number of weights through different heights.

As an introduction to the theorem which follows, it may be remarked, that if a constant force of P lbs. act through a space of s feet, and if a rectangle ABCD be drawn, of which the base a B represents the s feet on scale, and the perpendicular AD represents the P lbs. on the same scale: then, since the area of ABCD contains Ps square units on the same scale, that area will correctly represent the work done by P.