28 M E N M E N c 2 c 2 or (since v.i 2 . = - , TTS: = - , &c. )
a formula more convenient in practice, as it is sometimes more easy to measure equidistant circumferences than equidistant radii. J. Theorems of Pappus. 110. The following general propositions concerning surfaces and solids of revolution, usually called Guldin s theorems, are worth the reader s attention. If any plane curve revolve about any external axis situated in its plane, then (a) the surface of the solid which is thereby generated is equal to the product of the perimeter of the revolving curve and the length of the path described by the centre of gravity of that peri meter ; (j8) the volume of the solid is equal to the product of the area of the revolving curve and the length of the path described by the centre of gravity of the revolving area. We content ourselves with an example or two of the application of these theorems, referring to the article INFINITESIMAL CALCULUS for the proofs. Example 1. To find the surface and volume of a circular ring. Let a be the distance of the centre of the generating curve, in this case a circle, from the axis of rotation, and r the radius of the circle, then perimeter of generating curve = 27ir, area of generating curve = irr 2 , and path described by the centre of gravity either of the perimeter or area = 2ira ; hence surface of ring = 2ir?" x 2ira = 47r 2 ra, and volume of ring = ?rr 2 x 2ira = 2ir 2 r 2 a . Example 2. To find the volume swept out by an ellipse whose axes are 2 and 2&, revolving about an axis in its own plane whose distance from the centre of the ellipse is c. Here area of generating curve = irab, and path described by centre of gravity of area = lire ; hence volume generated = irab x 2irc=2ir 2 abc. Example 3. A circle of r inches radius, with an inscribed regular hexagon, revolves about an axis a inches distant from its centre, and parallel to a side of the hexagon ; to find the difference in area of the generated surfaces and volumes. Here perimeter of circle = 2irr , and perimeter of hexagon = 12 x rsin30 ( 17) -6r; also area of circle = Tir 2 , and area of hexagon = 3r 2 sin 60 ( 18, j8) = fV3> 2 ; hence difference of surfaces generated = 47r 2 r - 12-n-ar = 4irar(ir - 3) ; and difference of volumes generated PART III. GAUGING. 111. By gauging is meant the art of measuring the volume of a cask, or any portion of it. The subject is one of great interest and practical importance, but space will only permit us to discuss it very briefly. If the cask whose capacity we wish to determine be a solid of revolution, then its volume can at once be computed, either exactly or approximately, by the methods already described. It is usual to divide casks into the following four classes according to the nature of the revolving curve : (a) the middle frustum of a spheroid ; (/3) the middle frustum of a parabolic spindle ; (7) two equal frusta of a paraboloid, united at their bases ; (5) two equal frusta of a cone, united at their bases. Casks of the second, third, and fourth variety are rarely met with in practice, and we shall accordingly confine our attention to the first kind, which is considered the true or model form of cask. Let ABCD (fig, 56) be a section of the cask, and assume it to be the middle frustum of a prolate spheroid, then its volume = ^ir( 2b- + bl)k , where b = OY, ^ - AX, and k = XX ( 99). YY is called the bung diameter, and AB or CD the head diameter. An imperial gallon contains 277 274 cubic inches, and therefore the number of gallons in the above cask 3x277-274 831 82 X X whence we have the rule : to the square of the head diameter add twice the square of the bung diameter, multiply the sum by the length and divide the result by 1059 - 1, and the answer is the con tent in imperial gallons. Casks as ordinarily met with are not true spheroidal frusta, but it is better to consider them as such, calculate their capacity on this assumption, and then make allowance for the departure from the spheroidal form. The de termination of the proper allowance to be made in each case is a matter depending on the skill and ex perience of the ganger, and pro ficiency in the art can only be attained by considerable practice. 112. If the cask be very little curved, we obtain an approxima tion to its capacity by considering it as made up of two equal frusta of a cone, united at their bases. Hence from 83 we have volume of cask = |7r/t(rJ + r^ + ?;) nearly. Here we neglect the small volumes generated by APY, YSD, BQY , and Y HC ; and therefore the volume is too small. If we put r 1 r 3 =r" l we obtain volume = sirh(2rl + r~ s ) , which is a little too large, and therefore the true volume lies between these two limits, and a very close approximation to it is said to be given by the formula JirA{2r+r;-a(r?-r*)}. 113. Ullage of a Cask. The quantity of liquor contained in a cask partially filled and the capacity of the portion which is empty are termed respectively the wet and dry ullage. (a) Ullage of a standing cask. By means of the method applied in 105, the following rule is deduced : Add the square of the diameter at the surface, the square of the diameter at the nearest end, and the square of double the diameter half-way between ; multiply the sum by the length between the surface and the nearest end, and by 000472. The product will be the wet or dry ullage according as the lesser portion of the cask is filled or empty. (#) Ullage of a lying cask. The ullage in this case is found approximately on the assumption that it is proportional to the seg ment of the bung circle cut off by the surface of the liquor. The rule adopted in practice is ullage = | content x segmental area. (W. T.*) MENTAL DISEASES. See INSANITY. MENTON (Ital., Mentone), a cantonal capital in the department of Alpes-Maritimes, France, situated 15 miles north-east of Nice, on the shores of the Mediterranean. The town, which has a population of about 8000, rises like an amphitheatre on a promontory by which its semi circular bay (5 miles wide at its entrance, and bounded on the W. by Cape Martin and on the E. by the cliffs of La Murtola) is divided. It is composed of two very distinct portions : below, along the sea-shore, is the town of hotels and of foreigners, which alone is accessible to wheeled vehicles; above is that of the native Mentonese, with steep, narrow, and dark streets, spread over and clinging to the mountain, around the strong castle which was once its protection against the attacks of pirates. Facing the south-east, and sheltered on the north and west by high mountains, the Bay of Menton enjoys a delicious climate, and is on this account much frequented by invalids re quiring a mild and equable temperature. The mean for
the year is 61 Fahr., exceeding that of Rome or of