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Let the sum of this series be found
= (2 n—1) f + (2 n-3) f + &c. to f .. 2 S = 2 n f + 2 n f t &c. to n terms
2 nfn And S = =n's
2 The usual form of this equation is,
ť X g
, in which t represents the num
2 ber of seconds, and the f or 16,':
(3) (2) t =
9 We have now obtained equations from which either the time of the descent of (3), the space passed over by (2), and the acquired velocity of (1), a falling body may be found. Further, it is seen from the expression a, that space passed over by a falling body in any particular second of its descent, is equal to the product of 16,. by twice the number, in the order of that second with unity deducted; or, the spaces passed over in each succeeding second are to each other as the odd numbers, 1, 3, 5, 7, 9, &c.
3. “Show how it may be determined whether a pillar will stand or fall, when any given pressure is applied obliquely to its summit.”
With the assistance of a scale of equal parts draw a figure of the pillar, whose stability is to be determined, in such a manner, that not only the relative magnitude of the parts will be correctly delineated, but also that the inaccuracy of perpendicular erection, if any, may
be represented. Then upon a horizontal base erect a perpendicular passing through the point which indicates the centre of gravity of the pillar. Next, from the point at which the overturning force is applied, draw a line in the direction of that force, intersecting the perpendicular from the point of intersection, set off the units of force upon this line produced, and from the same point set off the units of weight in the pillar in the same denomination ; complete the parallelogram of forces, and extend the diagonal to the base line. If this diagonal produced cut the horizontal line within the base of the figure, the pillar will stand, and vice
See Tate's Mechanics, page 111. 4. "Show generally, how the traction of a body up an inclined plane, subject to friction, may be determined, and investigate the direction of least traction.”
Chap. VI. of " Moseley's Mechanics Applied to the Arts,” and pages 127-129 of Tate's Mechanics, afford a lucid reply to this question. But, in either work, the knowledge of preceding articles is requisite for a right understanding of the one under consideration, and a mere reference to them would not make our solution more complete. We therefore refer the reader to those authorities, from which our reply could only be a transcript.
1. Mention proofs of the extreme divisibility of matter. 2. What do you understand by density? How are the
densities of different bodies compared ? 3. Explain the use of the terms heat and cold. What
is latent heat? What specific heat ? How does a thermometer enable us to compare the temperatures of bodies ?
SECTION II. 1. Draw a diagram of Bramah's hydrostatic press, and show how to determine the pressure which
be produced by means of it. 2. There is a barge whose section is an equilateral tri
angle, each side being a feet in length, and whose length is 6 feet, how deep will it sink in the water; its weight and that of its lading being together
w lbs. ? 3. Define the centre of pressure of a fluid, and show
that the centre of pressure of a rectangular floodgate is situated at two-thirds the depth.
SECTION III. 1. How does the intensity of the light of a candle vary
with the distance of the illuminated surface from the flame ? Illustrate your meaning by a numeri
cal comparison in some supposed instance. 2. If a ray of light passes from air into water, what
change takes place in its direction, and according
to what law ? Under what circumstances is it
impossible for a ray to pass out of water into air ? 3. Explain the decomposition of light by means of a
prism. 4. Show distinctly how it is that a lense may be used,
first as a burning-glass, and secondly to correct defective sight.
SECTION IV. 1. What solid substance, and what three gases, form
the organic parts of all vegetable and animal sub
stances ? 3. Describe the functions of the leaf in plants by night
and by day. 3. From considerations founded on a knowledge of the
inorganic constituents of plants, account for the fact that one crop will grow where another fails that mixed crops grow well together—that a rota
tion of crops is necessary. 4. What is the chemical constitution of lactic acid ?
What relation does it bear to that of milk of sugar ? In what way does the formation of lactic acid determine the separation of curds in milk? How may these be made to disappear, and in what way does the rennet act in cheese making ?
SECTION I. 1. “Mention proofs of the extreme divisibility of matter."
A grain of gold can be beaten out so as to occupy a surface of 50 square inches; and this leaf can be divided into half a million of visible parts. 124,500 such leaves would be but one inch in thickness. The particles of light are so infinitely small, that though they impinge on our organs of vision with a velocity of 192,000 miles per second, no sensible inconvenience results. A grain
of cochineal dissolved in three gallons of water, imparts to each drop a colouring matter equal to the one hundred and thirty thousandth part of a grain. A block of talc, less than an inch in thickness, has been divided into more than a million distinct laminæ. Other instances will readily suggest themselves.
2. “What do you understand by density ? How are the densities of different bodies compared ?"
Density is a relative term used to imply the proportion between the weights of equal bulks of different bodies. Its synonym is Specific Gravity. Distilled water is the usual standard to which the densities, or relative weights of bodies are referred. A cubic inch of distilled water, at the temperature 62° Fahrenheit, and barometrical column of 30 inches, weighs 252.458 Troy grains; 7000 such grains make a pound Avoirdupoise; and a cubic foot of water, 997.136 ounces, or, in round numbers, 1000 ounces.
A cubic foot of water considered to weigh 1000 ounces is the unit of volume; and the number of times that 1000 ounces is contained in the weight of a cubic foot of any other substance, expresses the density, or specific gravity of that substance. Thus, a cubic foot of hammered platinum weighs 20,336 ounces, in which 1000 is contained 20.336 times, and this number is the specific gravity of platinum.
3. “ Explain the use of the terms heat and cold. What is latent heat ? What specific heat ? How does a thermometer enable us to compare the temperatures of bodies ?"
Heat is a universally diffused agency which becomes sensible to us by our bodily sensations and by the changes it produces on all substances; and cold is a term that indicates the sensation felt on the abstruction of heat.