III. The log of half-span = log of sum of half-chain and sag plus log of their difference, plus log of the difference of these two logarithms plus colog of sag plus 0.0612.

IV. Radius divided by half-chain measures the “batter" at the point selected; that is, measures the horizontal falling back for every unit of vertical ascent in a straight line tangent at that point.

136. How strong a horizontal pull on a chain, weighing half a pound to the yard, is required to make the lowest part curve with an 18-in. radius? with a 6-ft. radius? 137. A 1-in. rope, weighing of a pound to the yard, is fastened at one end to a staple, and near the other end, on the same level, runs over a pulley, and has a 25-lb. weight hung to it. What is the radius of its curvature at the middle?

138. A shower wets the rope of Ex. 137, and increases its weight 40%; what does its radius now become? 139. A steam-tug, in attempting to move a ship, straightened her hawser until the radius of the lowest point was 1980 ft. The rope was wet, and weighed 3 lbs. With what force was it stretched?

to the yard.

140. A chain 31 ft. long hangs between points on a level, and sags 4 ft.? What is the radius at the lowest point? 141. The whole chain, in Ex. 140, weighs 18 lbs. What is the horizontal tension? What is the distance of the points apart? What is the slant, or batter, of the end of the chain?

142. A chain weighing 1kg to the meter is suspended from points on a level; the length of chain is 31m, and it sags 1.3. Find all the conditions, and find how much it falls below a level at 10cm from each end. 143. A chain 100m long, weighing 14 oz. to the foot, is suspended from points on a level 80m apart. What is the sag, the batter at the ends, and the horizontal tension?

144. Suppose the points of suspension in Ex. 143 to remain unchanged, and the chain to be shortened 5m. What does the tension become?

145. How long a rope is required between points 100 ft. apart to sag 30 ft.? 20 ft.? 10 ft.?

146. If 4 cu. in. of iron weigh 1 lb. avoirdupois, what is the weight of 1 cu. in. in grains? What is the specific gravity of the iron?

147. If 4 cu. in. of iron weigh 1 lb., what is the diameter of a 6-lb. ball? of a 32-lb. ball?

148. If a block of iron with all its corners square measures 17.36 in. by 8.7 in. by 1.76 in., what does it weigh at lb. to the cubic inch? and what would be the diameter if cast into a ball, if 11% is allowed for waste?

149. Answer the same questions as in the last example, for a block of which the dimensions are 71.4 in. by 83 in. by 33 in.

150. What is the diameter of a cylinder 11 in. long that holds 2 gals.

151. What is the diameter of a cylinder 9 in. long that holds 2 gals.?

152. What is the diameter of a cylinder 30cm long that holds 101?

153. What is the circumference of a globe if the square centimeters of its surface are three times the cubic centimeters of its volume?

154. Find the diameter of a circle of which the number of inches in its circumference is equal to the square feet of its area?

155. How many times does a carriage-wheel 3 ft. 2 in. in diameter turn in going a mile on a smooth road?

156. The top of a wheel is at each instant moving with twice the velocity of the carriage, and is moving in a curve whose centre, at the instant, is as far below ground as the point is above ground. What, then, is the force exerted to separate the mud from the top of a wheel 3 ft. 2 in. in diameter, when the carriage is moving at the rate of 10 mi. an hour?

157. A point in the tire, as a spike-head, moves, while the wheel rolls over once, just four times the diameter of the wheel. How far does the point travel while the wheel, 3 ft. 2 in. in diameter, travels 1 mi.?

158. An oil-can is formed of two cylinders connected by a frustum of a cone. The upper cylinder, or neck, is 6cm in diameter, and 75mm high; the lower cylinder is 13cm in diameter, and 153mm high; the total length of the can is 30cm. Find its capacity in liters. 159. A common tunnel is formed of a frustum of a cone

terminated with a cylinder. The height of the frustum is 14cm, and the diameters of the two bases are 175mm and 16mm respectively. The cylinder is 8cm long. Find the capacity of the tunnel in liters. 160. A pan is in the form of a frustum of a cone. The interior measurement is 10cm deep, 12cm across the

bottom, and 23cm across the top. Find the capacity of the pan in liters.

161. A stove-pipe is 4m long, 26cm in diameter, and 1mm thick. Find how many square centimeters of sheetiron it has taken to make it, if the edges lap one centimeter; and give the weight of the pipe, if the specific gravity of the sheet-iron is 7.8.

162. A spherical bomb is 32cm in diameter, and the sides 38mm thick. The specific gravity of the metal of which it is made is 7.2. Find its weight and interior capacity.

163. The diameters of a lamp-shade are 25cm and 7em; its slant height is 134mm. Give its curved surface in square centimeters.

164. A niche is formed like a half-cylinder surmounted by a quarter of a sphere. The height of the cylinder is 1.2m, the diameter .8m. Find the volume of the niche, and the area of its interior surface.

165. A steam-boiler is formed of a cylinder terminated at each end by a hemispherical cap of the same diameter. The length of the cylinder is 3.4m, interior diameter .8m. Find the number of hektoliters of water required to fill the boiler half full.

458. In the average state of the atmosphere, the distance at which an object is visible at sea is found by the following formula:

The square of the distance in English miles is sevenfourths the height of the object in English feet; or,

log miles = 0.1215 + log feet,

log feet 2log miles -0.2431.

=

166. A hill 482 ft. high is 8 mi. from the shore. How many miles out at sea is it visible?

167. A sailor at the topmast 80 ft. above the sea can just see a sailor at the topmast of a similar ship. How many miles apart are the vessels?

168. A vessel approaching Valparaiso at day-break just makes out the peak of Aconcagua, 23,000 ft. high

and 140 mi. back from the coast.

How far is the vessel from land if the eye of the observer is 30 ft. above the water?

169. If Mount Washington is 6240 ft. high and 76 mi. in

an air-line from Cape Elizabeth, how far out from the Cape will its peak be visible in the ordinary state of the atmosphere?

170. How many acres of water can a man see, standing on a raft with his eyes just 6 ft. above water, and no land in sight?

171. How far would a mountain 30,000 ft. high be visible? one of 5000 ft. high? one of 1000 ft. high?

172. How high must a mountain be in order to be visible at sea-level 50 mi.? 100 mi.? 150 mi.?

459. When the distance is given in kilometers, and the height in meters,

The square of the distance in kilometers is 15 times the height in meters; or,

log kilometers = 0.5880+ log meters,

log meters

173. How far is a mountain 1000m high visible? 2000m high?

174. How far can a man see from the shore, if he stands on a bluff that raises his eye 11m above the sea?

175. If in steaming away from a mountainous island a sailor estimates his distance at 171km, when the island dis

appears beneath the wave, how high shall he estimate the mountains?

460. Sound travels in still air at 57° F. 1114 ft. a second, and 1 ft. a second faster for every degree above 57°; so that at 63° it goes 1120 ft. a second; at 53°, 1110 ft. a second.

176. The flash of a gun is seen 7 sec. before the report of the gun is heard; there is no wind, and the thermometer is 73° F. How far off was the gun? 177. A meteor was seen to burst; the report followed in 4 min. 17 sec. What was its distance if the average temperature of the intervening air was 50° F.?