convenient to use one standard of time throughout Great Britain, and the standard naturally adopted is the time for the meridian of Greenwich. A ship reckons time from its own varying mean noon. Recently the problem ART. 103.-American Standard Time. of how best to arrange the origin of the civil day was brought into prominence in North America owing to the construction of the Pacific Railways, which cross many degrees of longitude. The problem has been solved as follows:-The whole of North America has been divided into five broad belts running north and south, each extending over 15 degrees of longitude. In each belt one standard time will be maintained, the difference of one hour existing between two contiguous belts. ART. 104.-Rate of Working; Resistance, Facility. In the solution of problems on the time required to accomplish a piece of work, several assumptions are made. It is assumed that for each workman or kind of workman there is a definite rate of working which is independent of the number of hands engaged, and is also uniform, though the length of the working day varies. Without constant or approximately constant rates of working to reason from, nothing can be concluded. Suppose that the piece of work is reaping a field of grain, each hand working independently. Suppose that a man could do it in p hours, working uniformly, though not necessarily continuously. Then the resistance which the reaping of the field offers to a man is Ρ hours = field of grain. Here we use "field of grain" as an expression for the temporary unit “reaping the field of grain in question.” His facility of working, which is the reciprocal of the resistance of the field, is 1 ART. 105.-Collective Facility, Resulting Resistance. Let the resistance of the field of grain to a woman, boy, girl be respectively q, r, s hours field of grain. Suppose a men, b women, c boys, d girls set to work simultaneously, then the collective facility will be and the resulting resistance, which is the reciprocal, will be b c + +-+ hours = field of grain. p q 2° S When water flows at constant rates into or out of a cistern, we have the same kind of ideas; only if a current inwards is reckoned a positive facility towards filling the cistern, a simultaneous current outwards is a negative facility. Similar ideas are encountered when we consider the flow of electricity. EXAMPLES. Ex. 1. Washington Time is 5h. 8m. 12s. later than Greenwich Time; what is the longitude of Washington relatively to Greenwich? 15 unit of angle = corresponding unit of time, i.e., 77° 3' W. Ex. 2. A besieged garrison has sufficient provisions to last it for 23 weeks at the rate of 18 oz. per man per diem, but receiving a reinforcement of 40 per cent. upon its original number, the allowance is reduced to 15 oz. per man per diem; how many Ex. 3. A can perform a piece of work in 12 days, B in 15 days, and C in 18 days, when working separately. Find the time in which they could perform it when working together. Ex. 4. Suppose that every hour per day that a student works requires 30 days of rest during the year; how many hours per day must he read so as to do the greatest amount possible? hours of work day of work; x30 days of rest = year, (1) 365-230 days of work = year. (2) Therefore from (1) and (2), x(365 – x30) hours of work = year. The question now is, what number is x, when x(365 – x30) is greatest. The number 5 is a common multiplier, therefore we have to consider 273-226 only. From which it appears that 6 is the number. Hence, 6 hours per day. EXERCISE XV. 1. Given the length of the sidereal day 23h. 56m. 4'09s., and the length of the mean solar day 24h.; find the length of the year. 2. A person, on being asked what time it was, answered that the time past noon was three-fifths of the time till midnight. What was the time? 3. What time is it at Pekin, Calcutta, Rome, Washington, Sydney, when it is 6:30 p.m. at London (Greenwich) ? 4. A fortress was originally provisioned for 60 days, but after 20 days 15,000 additional troops were driven to take shelter in it; in consequence of which the provisions held out for only 10 days subsequent thereto. What was the number of the original garrison? 5. If 24 boys or 15 men can do three quarters of a piece of work in 71⁄2 hours, in what time will 10 men and 12 boys do the remainder? 6. Two men A and B working together can do a piece of work in 10 days; but if A stops working after 4 days, B can finish the work in 14 days more. Compare their rates of working. 7. A cistern is supplied from two taps, by one of which it can be filled in 39 minutes, and by the other in 52 minutes. In what time will it be filled by both together? S. What time would 36. men, working 10 hours a day, require to build a wall which 24 men, working 93 hours a day, can build in 9 days? 9. A cistern is fitted with three pipes, one of which will fill it in 48 minutes, the second in an hour, and the third in half-an-hour: how long will it take to fill the cistern when all three pipes are open together? 10. Assume that 6 men can do as much work in an hour as 7 women, and 8 women as much as 11 boys, and that 5 men can do a certain piece of work in 10 hours how long will it take 1 man, 2 women, and 3 boys together to do the same piece of work? : 11. If B and C working together take p days to a piece of work, for which C and A together take q days, and A and B together take r days; find how long each would take by himself. 12. Assume that 4 English navvies can do as much work in a day as 5 French navvies, that 4 French navvies can do as much work as 7 negroes, and that 13 English and 12 French do a piece of work in 3 days: how long will it take 10 negroes to do that piece of work? 13. Compare the time of a place 7° 30′ 15′′ west of Greenwich with Greenwich time. 14. Find the successive convergents to the difference between 365 days and the true solar year. SECTION XVI.-SPEED. ART. 106.-Speed and Velocity distinguished. It is important to distinguish between speed and velocity, or at least to discriminate between two different ideas, which these words may be used to fix. Velocity may be defined as rate of change of position with respect to time; while speed may be defined as the rate, with respect to time, of change of distance measured along a specified path. The elaborating of this distinction is due to Tait (MECHANICS, Ency. Brit., vol. xv., p. 681). Speed thus defined does not involve direction in its conception, while velocity does. |