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?

The question now is, what number is x, when x(365x30) is greatest. The number 5 is a common multiplier, therefore we have to consider x73 - x26 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 7 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?

8. 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 Green wich 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.

Both are expressed in terms of L per T; but, in the case of speed, L denotes a length merely; whereas, in the case of velocity, it denotes a vector (Art. 69). When our attention is restricted to motion along a definite path, it is not necessary to specify the direction of the velocity; it is sufficient to state whether it is backwards or forwards.

The idea which is reciprocal to that of speed is slowness. It is expressed in terms of T per L.

ART. 107. British Units. According to the common usage of this country, we may have any of the units of length for L, and any of the units of time for T; for example, miles per hour, miles per minute, miles per second, yards per minute, feet per second, etc. The choice of each unit depends on the magnitude of the quantities of that kind which come into consideration. In the case of the motion of trains the distances coming into consideration are great, and so are the times occupied; hence the speed is commonly expressed in miles per hour. In the motion of a projectile the distance coming into consideration is not great, and the time occupied is small; hence foot per second is a more convenient unit.

Calculation, however, is usually facilitated by choosing one set of fundamental units. Hence in the British system of absolute units, the F.P.S. system, foot per second is the primary unit of speed.

ART. 108.-Metric and C.G.S. Units. The primary unit of speed in the French system is naturally the metre per second. However, as the first founders of the system departed from the metre in taking the centimetre to define the unit of mass (Art. 127), ́ the founders of the C.G.S. system have adopted the centimetre throughout as the primary unit of length, and accordingly adopt the centimetre per second as the primary unit of speed.

As the foot per second and the centimetre per second involve

the same unit of time, the conversion of the former into the latter is the same as the conversion of the foot into the centimetre. The mean solar units being the same in all countries, the only conversions to which they give rise are those due to the relations of the several denominations of mean solar time to one another.

EXAMPLES.

Ex. 1. Express a speed of 60 miles per hour in terms of kilometres per second.

Ex. 2. Find the average speed of a lamplighter who spends 10 seconds at each lamp, and walks to the next, 25 yards off, at the rate of 5 miles an hour.

Ex. 3. A person standing on a railway platform noticed that a train took 21 seconds to pass completely through the station, which was 88 yards long, and that it took 9 seconds to pass himself. How long was the train, and at what rate per hour was it travelling?

1. Express a speed of 1 mile per hour in terms of feet per second; and of 1 foot per second in terms of mile per hour.

2. Express a speed of 60 miles per hour in terms of feet per second.

3. The speed with which light travels is 186,000 miles per second; express it in metres per second.

4. Reduce 1 kilometre per hour to centimetres per second.

5. Römer found that a ray of light took 16m. 36s. to cross the diameter of the earth's orbit. The mean distance of the sun from the earth is 92.39 million miles. Deduce the speed with which light travels, given that it travels uniformly.