2. A shaft is 20 feet long, and has to transmit a power of 4500 lbs., acting with a leverage of 2 feet; what must be its diameter so that the angle of flexure may be 11°? Ans. 6.8 inches.

131. The resistance to a crushing force cannot be determined with even an approximation to the truth, when the length is small. The theory of the subject is in a very imperfect state; and the experiments regarding it are equally deficient, either in accuracy or on account of the smallness of the specimens. The following are some of the results obtained by Rennie, showing the force necessary to crush different kinds of materials*:

A cube of cast-iron inch,
A cube of cast-iron inch,
A piece inch square and
A piece inch square and I inch long,

inch long,

A cube of elm of 1 inch,

A cube of deal of 1 inch,

A cube of English oak 1 inch,

A cube of Craigleith stone of 14 inch, crushed
in direction of strata,

The same crushed across the strata,
Cornish granite, of same size,
Peterhead granite, do.
Aberdeen blue granite, do.

132. PROBLEM XVII.-To find the weight that could be safely supported by a column of cast-iron of considerable length resting on a horizontal plane.

The rules for this problem would be very tedious, if expressed in common language; they are, therefore, given algebraically.t

Let W and d

the weight in pounds, 6 the greater side, the less side both in inches, and the length

15300 bd3 W =

in feet; then

d2+1872

when the weight acts along the axis.

*Tredgold's Essay on the Strength of Cast-Iron, art. 64; and Leslie's Natural Philosophy, p. 221.

+ See Tredgold's Essay, articles 240-247.

When the weight acts at the distance a inches from the 15300 bd3 axis, then d2 + 6 ad + ·18 72°

W =

For a cylinder, the force acting in the direction of its axis, 9562 d1 d2+1812°

W =

When the direction of the force is a inches from the axis, 9562 d1 W = d+6ad+187

W =

for a rectangular column of cast-iron,

It is considered to be probable that the weight generally acts at the edge which is nearest to the axis, that is, at the distanced; hence, a = d, and then 15300 Zd3 4 d2+18 12 17800 ld3 4d2+16 12 3960 ld3 4d2+5 12

for a rectangular column of malleable iron, W =

for a rectangular column of oak,

for a cylinder of cast-iron,

9562 d1 4 d2+1812

11125 d1

for a cylinder of malleable iron,

4 ď2 + ·16 l2
2470 d1
4 d2 + ·5 12

for a cylinder of oak,

133. As the pressure has the greatest effect in bending a pillar when it acts farthest distant from the centre, that is, at the edge, it is safest in practice to compute the strength by the last six formulas.

W=

W =

W =

W =

In practice, for safety, a beam ought not to be subjected to a pressure exceeding or of its strength, as determined by computation.

EXERCISES.

1. Find the weight that can be safely supported by a square pillar 5 inches in the side and 61 feet long, the direction of the force being along the axis. Ans. 298537.

2. What weight can a rectangular column of malleable iron 6 inches broad, 5 thick, and 12 feet long support, supposing the pressure to be at the edge nearest the axis? Ans. 106800 lbs.

3. What weight can a cylindrical pillar of oak 10 feet long and 10 inches in diameter support, the pressure being at the edge of the top? Ans. 54889.

4. Find the pressure that can be sustained by the pistonrod of a steam-engine, its length being 4 feet, and its diameter 4 inches. Ans. 42789 lbs.

mmmmm

GAUGING.

134. Gauging is the art of measuring the dimensions, and computing the capacity, of any vessel or any portion of it.

The vessels usually gauged are casks, tuns, stills, and ships. The dimensions of the three former kinds are generally taken in inches, as the object is to determine the number of gallons of liquid contained in them.

When the capacity of a vessel is known in cubic inches, the number of gallons contained in it could then be easily found, by dividing the capacity by 277-274, the number of cubic inches in an imperial gallon. The capacities of vessels can be found by means of the preceding rules in the Mensuration of Solids (Part I.), but they can be found more readily by means of certain numbers called divisors, multipliers, and gauge points.

PRINCIPLES AND DEFINITIONS OF TERMS.

For Rectilineal Figures.

135. The number of cubic inches in the measure of сараcity or quantity of any vessel or solid, is called the divisor for that body.

The number of cubic inches in the capacity being divided by the divisor, will give the capacity or quantity in the required denomination. Thus, the number of cubic inches in the capacity of a vessel being divided by 277-274, gives the number of imperial gallons; by 2218-192, gives the

number of imperial bushels. So the number of cubic inches contained in a quantity of dry starch being divided by 403, will give the number of pounds, for 40.3 is the number of cubic inches in a pound of starch; the pound in this case is the measure of quantity.

136. The reciprocals of the divisors are the multipliers. It is evident that, if instead of dividing by the preceding divisors, we multiply by their reciprocals, the results will be the same. These multipliers will, therefore, be found by dividing 1 by the preceding divisors.

137. The square roots of the divisors are called gauge points.

The gauge points are just the sides of squares, of which the content at one inch deep is the measure of capacity or of quantity, that is, 1 gallon, 1 bushel, or 1 pound of starch, soap, tallow, or glass.

138. By the content of any given surface at one inch deep, is meant the content in cubic inches of a right prism or vessel, whose height or depth is 1 inch, and base the given surface.

Thus, the content of a circular area is the content of a cylinder 1 inch high, whose base is the circle; the content of a square is the content of a parallelopiped 1 inch high, whose base is the given square.

Let the volume of a vessel or solid in cubic inches, c = the capacity of it in the required denomination, the number of cubic inches in the measure of capacity, as in 1 gallon, 1 pound, &c.,

m

n
g

the multiplier,

the gauge point,

v

1

c = =nv, for m=

m

n

g2 x 1=m, and g√m.

v

m

For Circular Areas.

139. If the number of cubic inches in the measure of capacity or quantity is divided by the number 785398 or 7854, the quotients are the divisors.

then

also

Hence also,

c=

=

v

92

Let ՊՈՆ 1

then

the divisor and d the diameter of the area, v d2 v=785398 d2, and hence c = =

m

for

m

m1 = 785398

140. If the number 785398 is divided by the number of cubic inches in the measure of capacity or quantity, the quotients are the multipliers.

It is evident that the multiplier n, is the reciprocal of m1; hence c = n, d2.

141. The square roots of the divisors are the gauge points.

The gauge points are the diameters of circles, of which the content at one inch deep is the number of cubic inches in the measure of capacity or quantity.

d2

d2

Since c = when c = 1,

mi

or d=√m1 =91 1

m

m1

1

= 1, therefore d2 =m1,

=

Polygonal Areas.

142. If the number of cubic inches in the measure of capacity is divided by the tabular areas of polygons (in 271, Part I.), the quotients are the polygonal divisors.

Thus, if a the area of any regular polygon and s its

side,

a1 = the area of a similar regular polygon whose side is 1,

s2a1

$2

a

m 2

a = s2a1, c =

ՊՈՆ 2

a1

m2 = the polygonal divisor, then m2 = m m2a1 143. The reciprocals of the divisors are the multipliers. If n the multiplier, then n2 = and hence

1

ՂՈՆ Ջ

c = ns2.

144. The square roots of the divisors are the gauge points. The gauge points are the sides of regular polygons whose areas are equal to the ber of cubic inches in the measure of capacity.