(92) A cylindrical boiler, 16 ft. long, and 2 ft. in diameter, with hemispherical ends, in addition to the above length, has to be covered with felt; what will it cost at 1 d. per square foot? Ans. 14s. 1 d.

(93) It is required to make a cistern, 3.2 ft. long, and 2.6 ft. wide, that shall contain 216 gallons; how deep must the cistern be, if 2774 cubic inches make 1 gallon? Ans. 4.33...ft.

(94) A rectangular mass of earth is 9:45 yds. long, 3 yds. broad, and 16 yds. thick; find the edge of a cubical mass of equal volume. Ans. 3.15 yds.

(95) Two cylindrical cups of the same height will hold 9 and 16 pints respectively; what is the content of another of the same height the diameter of whose base is equal to the sum of the diameters of the former two?

Ans. 49 pints.

(96) Not only the capacity, but the form also, of the Imperial Bushel, is defined by Act of Parliament. Explain the necessity for this enactment.

(97) The Act of Parliament directs, that the Imperial Bushel used for heaped measure shall be an upright cylinder, the diameter of whose base is not less than twice the height, and that the height of the conical heap shall be at least three-fourths of the depth of the bushel, the boundary of its base being the outside of the measure. State fully the effect of not complying with this regulation.

(98) Required the content of a tub, in the form of a frustum of a cone, whose greatest diameter is 60 in., diagonal 66 in., and slant side 30 in.

(99) A gentleman wishes to raise his garden 1 foot higher throughout by means of earth dug out of a moat to be formed 8 feet wide round two adjacent sides of it; the garden is 300 feet long and 200 feet broad, and is rectangular. How deep must he dig the moat, supposing it uniform and rectangular? Ans. 14 ft.

(100) Of what diameter must the bore of a cannon be cast for a ball of 24 lbs. weight, so that it may be onetenth of an inch more than that of the ball? pendix.)

(See ApAns. 5-61 in.

(101) A railway which for some distance has been laid in a straight line at a certain point takes a circular bend for 476 yards, and then proceeds again in a straight line, which deviates from the former by an angle of 12°38′; find the radius of the curve.

Ans. 1 mile 397 yards.

(102) In levelling for canals and railways engineers allow a depression of 8 inches per mile for the curvature of the earth. What is the earth's diameter, supposing this to be correct?

Ans. 7920 miles.

(103) A cubic foot of copper is to be drawn into a wire of th of an inch diameter; what will be the length of the wire ? Ans. 55 miles nearly.

An excellent collection of easy examples, by the Rev. W. N. Griffin, is published by the National Society, and sold at the Depository, Sanctuary, Westminster, at the low price of 1d. It is exactly adapted to this work, and junior students will find it of great service.

NOTE.

AMSLER'S PLANIMETER.

(See p. 314.)

For the following popular explanation of the principle of this instrument, I am indebted to Dr. Alfred Day, of Clifton, near Bristol; though I have not adhered precisely to that gentleman's own words:

Suppose a rod of wood, or brass, carried a wheel attached to it revolving at right angles to its length, like the rolling Parallel-Ruler deprived of one of its wheels, it would, on being moved parallel to itself, describe an area which would obviously be measured, as a rectangle, by the product of the distance moved over by the wheel into the length of the rod, or ruler. And if, in addition to this parallel motion, the ruler were made to rotate, or deviate from its first direction, we could resolve, or separate, the area traced out into two parts, one due to the advance of the ruler parallel to itself, and the other wholly due to rotation, backwards or forwards, that is, diminishing or increasing the previously described area. By means of these two motions, and a third in the direction of the ruler's length, (which latter will affect neither of the two former, nor add any thing to the area), we can make the end of the ruler trace out any continuous curve or irregular line we please. If then, at the same time, we note the track of the other extremity of the ruler, and complete the figure by two straight lines coinciding with the edge of the ruler in its first and last position; and if we see further, that for every partial rotation of the ruler round itself in one part of its course, an equal and opposite partial rotation takes place in the other direction, so that at last the ruler is exactly parallel to its first position, then, in this case, it is plain, that we should at once have a measure of the area above described, viz. the product of the ruler's length into the rotation of the wheel.

It will be obvious, that, in practice, the operation here described would be of very limited utility, because in most cases, when we had a given irregular figure to measure, while we made one end of the ruler trace out the given boundary, the other end would be tracing a

boundary with which we were in no way concerned, besides the difficulty connected with the two parallel straight The Planimeter at once obviates this inconve

lines. nience.

Let ECDF be the instrument; EC the arm fixed at E; and jointed at C to the tracer-rod CF; F the tracing point, and D the position of the wheel on CF; FGF'K the area to be measured. Beginning with the tracerpoint at F, it is plain that, in passing from F to F', along the boundary FGF', the arm CF will, by means of the three sorts of mo

tion before mentioned, trace out the area CFGF'C', while CE will trace the sector CEC'. Accordingly, the wheel will register all the advance of the arm CF due to parallel motion, together with that which has resulted from rotation round its own axis, when it gets to the extent of its positive progress; certain quantities of rotation, positive and negative, having balanced themselves wholly, and left no record of their existence. Then, as the tracer moves along the boundary F'KF, not only will all the sector described by EC' be, as it were, wiped out, but all the area FKF'C'C, and all the rotation of the tracerarm round itself is also counter-balanced by a negative rotation, when the tracer has returned to its first position F. In going back, therefore, the wheel will register the actual parallel motion of the tracer-arm, and leave all the rotations balanced.

Hence the area FGF'K=CFGF'C' – FKF'C'C,

=rotation of the wheel × the arm which carries it.

[Dr Day has further discussed the various details connected with this beautiful instrument, and has proved the truth of its determinations, with admirable skill and

success, in every possible case. But the above will suffice for our purpose here, that is, to give the ordinary student a notion, at least, of the principle of the instrument.]

For the advanced reader I am permitted to give the following elegant proof of the principle by Professor Adams, of Cambridge, the celebrated astronomer :—

N

M

F

E

Let E be the fixed point, F the tracer, N the projection of the hinge upon the plane of the paper, D the point in which the plane of the wheel meets NF, M the middle point of NF.

Also let

NF-a, EN=b, DM=c. If the boundary of any closed figure be traced out by F, the area of the figure is equal to the algebraical sum of the elementary areas described by the lines EN, NF in passing from any position to a consecutive position, considering an elementary area to be positive when it passes from the left to the right side of the lines, and negative when it passes in the opposite direction.

In any position of the tracer, let 4, v, be the angles which NF, EN, make with their respective initial positions; and lets be the arc through which the wheel has turned in the same time.

For a consecutive position of the tracer, let ø, Y, and s, become p+dp, +d, and s+ds, respectively. Then ds the resolved part of the motion of the point D perpendicular to the line NF.

=

Hence the resolved part of the motion of M perpendicular to the same line=ds+cdp; and therefore the elementary area described by NF-a (ds+cop). Also the elementary area described by EN={b3d↓.

Hence the algebraical sum of the elementary areas described by EN, NF, in passing from their initial positions to any other positions

= as+acp+b3Y.