382. Examples. (1) The length of a cask is 40 inches, the bung diameter 32, and the head diameter 24. 39 x 32 x 32 = 39936, 25 × 24 × 24=14400, 26 × 32 × 24=19968, 39936+14400+19968=74304, 74304 × 40 x '000031473=93'54279... Thus the volume of the cask is about 93 gallons. (2) The length of a cask is 20 inches, the bung diameter 16, and the head diameter 12. 39 × 16 × 16=9984, 25 × 12 × 12=3600, 26 × 16 × 12=4992, 9984+3600+4992=18576, 18576 x 20 x 000031473=11.6928... Thus the volume of the cask is about 11.7 gallons. 383. It is sometimes necessary to know the quantity of liquid in a vessel which is only partly filled. The word ullage means strictly the portion of a partly filled cask which is not occupied by the liquid: but the word is now applied to the occupied portion as well as to the unoccupied portion, the former being called the wet ullage and the latter the dry ullage. 384. Two cases may occur; namely that of a standing cask, and that of a lying cask. We will consider the former case. The depth of the liquid is called the wet inches; the difference between the wet inches and the length of the cask is called the dry inches. 385. To estimate the wet ullage of a standing cask which is less than half full. RULE. Multiply the square of the dry inches by the difference between the bung diameter and the head diameter; and divide the product by the square of the length: subtract this from the bung diameter, and the result may be taken as the mean diameter of the occupied portion of the cask. Then proceed as in finding the volume of a cylinder: Multiply the square of the mean diameter by the wet inches, and the product by 0028326; the result may be taken for the number of gallons in the wet ullage. 386. To estimate the wet ullage of a standing cask which is more than half full. Apply the method of Art. 385, using wet inches instead of dry inches, and dry inches instead of wet inches; we thus obtain the dry ullage: subtract the dry ullage from the volume of the cask, and the remainder is the wet ullage. 387. Examples. (1) The length of a cask is 40 inches, the bung diameter 32, and the head diameter 24; the number of wet inches is 10: find the wet ullage. The number of dry inches is 30; the difference between the bung diameter and the head diameter is 8. (2) The length of a cask is 20 inches, the bung diameter 16, and the head diameter 12; the number of wet inches is 15: find the wet ullage. Here we first find the dry ullage. Subtract this from 11'6928, the volume of the whole cask, by Art. 382; the remainder is 90151. Thus the wet ullage is about 9 gallons. 388. No satisfactory rule can be given for estimating the ullage of any lying cask. A rule proposed by Hutton amounts to assuming the cask to be a cylinder; the rule is substantially this: find the area of the segment of a circle which is obtained by the section of the fluid with a plane perpendicular to the length of the cask; multiply this area by the length of the cask, and divide the product by 277 274; the result may be taken for the number of gallons in the wet ullage. 389. The business of gauging is practically performed by excisemen with the aid of instruments called the gauging or diagonal rod, and the gauging or sliding rule. These instruments however are recommended not so much on account of the accuracy of the results to which they lead, as of the expedition with which these results are obtained. The construction and mode of using these instruments can be far more readily understood by actual experience than by any description. EXAMPLES. XLIII. Find in gallons the volumes of casks having the following dimensions: 1. Length 50-2, bung diameter 31-5, head diameter 22-7. 2. Length 47-5, bung diameter 28-5, head diameter 21.4. 3. Length 42-5, bung diameter 32-5, head diameter 26-5. 4. Length 305, bung diameter 26-5, head diameter 23. 5. Length 468, bung diameter 30-5, head diameter 26. 6. Length 345, bung diameter 32-3, head diameter 27·6. 7. Length 46-9, bung diameter 31-2, head diameter 26·1. [The dimensions in the above seven examples are stated in Lubbock's tract on Cask-gaging, 1834, to be the average dimensions respectively of a Port-pipe, a Madeira-pipe, a Sherry-butt, a Sherry hogshead, a Bourdeaux brandy-puncheon, a Rum-puncheon, a Brandy-piece.] Find in gallons the wet ullage in the following cases of standing casks: 8. Length 60, bung diameter 36, head diameter 30, wet inches 12. 9. Length 50, bung diameter 32, head diameter 27, wet inches 10. 10. Length 30, bung diameter 27, head diameter 23, wet inches 9. 11. If the length of the cask in Example 1 is increased by '1 of an inch, shew that the volume will be increased by about 22 of a gallon. 12. If the head diameter in Example 2 is increased by 1 of an inch, shew that the volume will be increased by about 27 of a gallon. 13. If the bung diameter in Example 3 is increased by 1 of an inch, shew that the volume will be increased by about 43 of a gallon. 14. If all the dimensions in Example 1 are increased by 1 of an inch, shew that the volume will be increased by about a gallon. 237 SEVENTH SECTION. LAND XLIV. USE OF THE CHAIN. 390. A VERY important application of some of the rules of mensuration is furnished in Land Surveying; and to this we now proceed. 391. Land is measured with a chain, called Gunter's chain, which is 4 poles, that is, 22 yards long; the chain consists of 100 equal links, so that each link is yard long, that is 7.92 inches. of a 392. A picket is a rod stuck into the ground to mark a certain position. 393. A field-book is a book in which the results obtained by measurement are recorded. 394. We will now explain how a straight line is measured with the chain. We will suppose that the straight line which is to be measured is the distance between two points each marked by a picket. Ten small arrows are provided which may be stuck in the ground. Two persons engage in the work, one of whom is called the leader; both place themselves at one of the pickets. The leader takes in his hand the ten arrows and one end of the chain, and walks towards the second picket; the follower keeps the other end of the chain at the first picket. When the leader has stretched the chain to its full length he puts an arrow in the ground to mark the spot to which the chain reached; he then walks towards the second picket carrying with him his end of the chain as before. The follower now comes up to the arrow, and holds his end of the chain at it until the leader |