Multiply half the height of the segment by the area of the base, and the cube of the height by .5236 and add the two products. Or, from three times the diameter of the sphere subtract twice the height of the segment; multiply the difference by the square of the height and by .5236. Or, to three times the square of the radius of the base of the segment add the square of its height, and multiply the sum by the height and by .5236. Spheroid or ellipsoid.-When the revolution of the spheroid is about the transverse diameter it is prolate, and when about the conjugate it is oblate. Convex surface of a segment of a spheroid.-Square the diameters of the spheroid, and take the square root of half their sum; then, as the diameter from which the segment is cut is to this root so is the height of the segment to the proportionate height of the segment to the mean diameter. Multiply the product of the other diameter and 3.1416 by the proportionate height. Convex surface of a frustum or zone of a spheroid.-Proceed as by previous rule for the surface of a segment, and obtain the proportionate height of the frustum. Multiply the product of the diameter parallel to the base of the frustum and 3.1416 by the proportionate height of the frustum. Volume of a spheroid is equal to the product of the square of the revolving axis by the fixed axis and by .5236. The volume of a spheroid is two thirds of that of the circumscribing cylinder. Volume of a segment of a spheroid.-1. When the base is parallel to the revolving axis, multiply the difference between three times the fixed axis and twice the height of the segment, by the square of the height and by 5236. Multiply the product by the square of the revolving axis, and divide by the square of the fixed axis. 2. When the base is perpendicular to the revolving axis, multiply the difference between three times the revolving axis and twice the height of the segment by the square of the height and by 5236. Multiply the product by the length of the fixed axis, and divide by the length of the revolving axis. Volume of the middle frustum of a spheroid.-1. When the ends are circular, or parallel to the revolving axis: To twice the square of the middle diameter add the square of the diameter of one end; multiply the sum by the length of the frustum and by .2618. 2. When the ends are elliptical, or perpendicular to the revolving axis: To twice the product of the transverse and conjugate diameters of the middle section add the product of the transverse and conjugate diameters of one end; multiply the sum by the length of the frustum and by .2618. Spindles.-Figures generated by the revolution of a plane area, when the curve is revolved about a chord perpendicular to its axis, or about its double ordinate. They are designated by the name of the arc or curve from which they are generated, as Circular, Elliptic, Parabolic, etc., etc. Convex surface of a circular spindle, zone, or segment of it-Rule: Multiply the length by the radius of the revolving arc; multiply this are by the central distance, or distance between the centre of the spindle and centre of the revolving arc; subtract this product from the former, double the remainder, and multiply it by 3.1416. Volume of a circular spindle.-Multiply the central distance by half the area of the revolving segment; subtract the product from one third of the cube of half the length, and multiply the remainder by 12.5664. Volume of frustum or zone of a circular spindle.-From the square of half the length of the whole spindle take one third of the square of half the length of the frustum, and multiply the remainder by the said half length of the frustum; multiply the central distance by the revolving area which generates the frustum; subtract this product from the former, and multiply the remainder by 6 2832. Volume of a segment of a circular spindle.-Subtract the length of the segment from the half length of the spindle; double the remainder and ascertain the volume of a middle frustum of this length; subtract the result from the volume of the whole spindle and halve the remainder. Volume of a cycloidal spindle = five eighths of the volume of the circumscribing cylinder.-Multiply the product of the square of twice the diameter of the generating circle and 3.927 by its circumference, and divide this product by 8. Parabolic conoid.-Volume of a parabolic conoid (generated by the revolution of a parabola on its axis).-Multiply the area of the base by half the height. Or multiply the square of the diameter of the base by the height and by .3927. Volume of a frustum of a parabolic conoid.-Multiply half the sum of the areas of the two ends by the height. Volume of a parabolic spindle (generated by the revolution of a parabola on its base).-Multiply the square of the middle diameter by the length and by .4189. The volume of a parabolic spindle is to that of a cylinder of the same height and diameter as 8 to 15. Volume of the middle frustum of a parabolic spindle.-Add together 8 times the square of the maximum diameter, 3 times the square of the end diameter, and 4 times the product of the diameters. Multiply the sum by the length of the frustum and by .05236. This rule is applicable for calculating the content of casks of parabolic form. Casks.-To find the volume of a cask of any form.-Add together 39 times the square of the bung diameter, 25 times the square of the head diameter, and 26 times the product of the diameters. Multiply the sum by the length, and divide by 31,773 for the content in Imperial gallons, or by 26,470 for U. S. gallons. This rule was framed by Dr. Hutton, on the supposition that the middle third of the length of the cask was a frustum of a parabolic spindle, and each outer third was a frustum of a cone. To find the ullage of a cask, the quantity of liquor in it when it is not full. 1. For a lying cask: Divide the number of wet or dry inches by the bung diameter in inches. If the quotient is less than 5, deduct from it one fourth part of what it wants of .5. If it exceeds .5, add to it one fourth part of the excess above .5. Multiply the remainder or the sum by the whole content of the cask. The product is the quantity of liquor in the cask, in gallons, when the dividend is wet inches; or the empty space, if dry inches. 2. For a standing cask: Divide the number of wet or dry inches by the length of the cask. If the quotient exceeds .5, add to it one tenth of its excess above .5; if less than .5, subtract from it one tenth of what it wants of .5. Multiply the sum or the remainder by the whole content of the cask. The product is the quantity of liquor in the cask, when the dividend is wet inches; or the empty space, if dry inches. = Volume of cask (approximate) U. S. gallons square of mean diam. X length in inches x .0034. Mean diam. half the sum of the bung and head diams. Volume of an irregular solid.—Suppose it divided into parts, resembling prisms or other bodies measurable by preceding rules. Find the content of each part; the sum of the contents is the cubic contents of the solid. The content of a small part is found nearly by multiplying half the sum of the areas of each end by the perpendicular distance between them. The contents of small irregular solids may sometimes be found by immersing them under water in a prismatic or cylindrical vessel, and observ ing the amount by which the level of the water descends when the solid is withdrawn. The sectional area of the vessel being multiplied by the descent of the level gives the cubic contents. Or, weigh the solid in air and in water; the difference is the weight of water it displaces. Divide the weight in pounds by 62.4 to obtain volume in cubic feet, or multiply it by 27.7 to obtain the volume in cubic inches. When the solid is very large and a great degree of accuracy is not requisite, measure its length, breadth, and depth in several different places, and take the mean of the measurement for each dimension, and multiply the three means together. When the surface of the solid is very extensive it is better to divide it into triangles, to find the area of each triangle, and to multiply it by the mean depth of the triangle for the contents of each triangular portion; the contents of the triangular sections are to be added together. The mean depth of a triangular section is obtained by measuring the depth at each angle, adding together the three measurements, and taking one third of the sum. PLANE TRIGONOMETRY. Trigonometrical Functions. Every triangle has six parts-three angles and three sides. When any three of these parts are given, provided one of them is a side, the other parts may be determined. By the solution of a triangle is meant the deternination of the unknown parts of a triangle when certain parts are given. The complement of an angle or arc is what remains after subtracting the angle or arc from 90°. In general, if we represent any arc by A, its complement is 90° — A. Hence the complement of an arc that exceeds 90° is negative. Since the two acute angles of a right-angled triangle are together equal to a right angle, each of them is the complement of the other. The supplement of an angle or arc is what remains after subtracting the angle or arc from 180°. If A is an arc its supplement is 180° - A. The supplement of an arc that exceeds 180° is negative. The sum of the three angles of a triangle is equal to 180°. Either angle is the supplement of the other two. In a right-angled triangle, the right angle being equal to 90°, each of the acute angles is the complement of the other. In all right-angled triangles having the same acute angle, the sides have to each other the same ratio. These ratios have received special names, as follows: If A is one of the acute angles, a the opposite side, b the adjacent side, and c the hypothenuse. The sine of the angle A is the quotient of the opposite side divided by the hypothenuse. Sin. A= α The tangent of the angle A is the quotient of the opposite side divided by с a b The secant of the angle A is the quotient of the hypothenuse divided by the adjacent side. Sec. A =¿ The cosine, cotangent, and cosecant of an angle are respectively the sine, tangent, and secant of the complement of that angle. The terms sine, cosine, etc., are called trigonometrical functions. In a circle whose radius is unity, the sine of an arc, or of the angle at the centre measured by that arc, is the perpendicular let fall from one extremity of the arc upon the diameter passing through the other extremity. The tangent of an arc is the line which touches the circle at one extremity of the arc, and is limited by the diameter (produced) passing through the other extremity. The secant of an arc is that part of the produced diameter which is intercepted between the centre and the tangent. The versed sine of an arc is that part of the diameter intercepted betireen the extremity of the arc and the foot of the sine. In a circle whose radius is not unity, the trigonometric functions of an arc will be equal to the lines here defined, divided by the radius of the circle. If I CÀ (Fig. 70) is an angle in the first quadrant, and C F = radius, If radius is 1, then Rad. in the denominator is omitted, and sine = F G, etc. The sine of an arc = half the chord of twice the arc. The sine of the supplement of the arc is the same M as that of the arc itself. Sine of arc BDF FG = sin arc FA. The tangent of the supplement is equal to the tangent of the a a contrary sign. Tang. BDF The secant of the supplement is equal to the secant of the ard contrary sign. Sec. BDF- CM. Signs of the functions in the four quadran divide a circle into four quadrants by a vertical and a horizo ter, the upper right-hand quadrant is called the first, the upper ond, the lower left the third, and the lower right the fourth. T the functions in the four quadrants are as follows: The values of the functions are as follows for the angles specifi TRIGONOMETRICAL FORMULE. The following relations are deduced from the properties of sim angles (Radius The sum of the square of the sine of an arc and the square of its equals unity. Sin2 A+ cos2 A= 1. Also, 1+ tan2 A = sec2 A; 1+cot A cosec A. Functions of the sum and difference of two angles Let the two angles be denoted by A and B, their sum A+B=0 their difference A B by D. sin (A + B) = sin A cos B + cos A sin B; From these four formulæ by addition and subtraction we obtain sin (A+B) + sin (4 – B) = 2 sin A cos B; If we put 4+ B = C, and A - B = – B) = 2 cos A cos B; + B) = 2 sin A sin B. (8) D, then A = (C+D) and B = (C (C + D) cos D); (9) sin C+ sin D = 2 sin (C sin C – sin D = 2 cos 1⁄2(C + D) sin 1⁄2(C — D); cos C+ cos D = 2 cos 1⁄2(C + D) cos 1⁄2(C — D); cos D - cos C = 2 sin (C + D) sin (C — D). Equation (9) may be enunciated thus: The sum of the sines of any two angles is equal to twice the sine of half the sum of the angles multiplied by the cosine of half their difference. These formulæ enable us to transform a sum or difference into a product. The sum of the sines of two angles is to their difference as the tangent of half the sum of those angles is to the tangent of half their difference. 2 sin (A+B) cos 1⁄2(A — B) tan (A+B) sin Asin B (13) sin A — sin B 2 cos (A + B) sin (4 – B) tan (4B)* The sum of the cosines of two angles is to their difference as the cotangent of half the sum of those angles is to the tangent of half their difference. cos A+ cos B 2 cos (A+B) cos 1⁄41⁄2(A – B) cot (A + B) (A+B) sin (A – B) (14) The sine of the sum of two angles is to the sine of their difference as the sum of the tangents of those angles is to the difference of the tangents. |