Substituting these in formula (1) and reduc- annexed figure. The top layer contains ing, we have, (m + 1) balls, the second layer contains n (n + 1) (2n + 1) 2 (m + 2), the third, 3(m + 3), and so on. S= 1.2.3 10. To find a formula for the number of balls in a The rectangular pile is formed as in the complete rectangular pile, we have the series 1.(m+1), 2(6+2), 3(m+3), 4(m+4)&c. Add to the number of balls in the longest 1st ord. diff., m+3, m+5, m+7, &c. side of the base the number in the parulle side 2d " " 2 2 &c. opposite, and also the number in the parallel 3d " si 0 &c. top row; multiply this sum by one-third of the Hence, number of balls in the triangular face of the a=m+1, d, =(m+3), d,=2, d,=0,0,=0, &c. prle, and the res d=0cc pile, and the result will be the number of balls Substituting in formula (1) and reducing, we in the pile. have, | This rule is easy to remember, and is S n (n + 1)(1 +2n + 3m) equally applicable to each of the three forms; 1.2.3 ...(4). it is often called the workman's rule. In order to find the number of balls in an Where space is an object, the rectangular pile is preferable to either of the others, and incomplete pile, compute the number that the the longer the pile, the greater the number of pile would contain if complete, and the num-! balls that can be piled upon a given area, ber required to complete it; the difference of these two numbers is the number of balls in | having a given breadth. One long pile is the pile. more economical of space than two or more Formulas (2), (3) and (4), may be written short ones. The square pile occupies most space for the number of balls contained in it. thus, In(n+1) Triangular, S=3 PiNT. A unit of measure of capacity, 2 (n equivalent to one-eighth of a gallon, or about 1 n(n+1). 39 cubic inches. See Measures. Square, S= 2: 0 (n+n+1)... (3), PLAN. In Descriptive Geometry and SurRectangu'r, S= 3: veying, a representation of the horizontal 2(n+m)+(n+ projection of a body. The plan of an object +(m+1)). . (4). is the same as its horizontal projection. The Now, since term is particularly applied to architectural n (n + 1) drawings. PLĀNE. [L. planus, even, flat). A suris the number of balls in the triangular face face such that, if any two points be taken at of each pile, and the next factor in each case pleasure and joined by a straight line, that denotes the number of balls in the longest line will lie wholly in the surface. A plane side of the base, plus the number in the sides is supposed to extend indefinitely in all of the base opposite, plus the number in the directions. A plane may be generated by a top parallel row, we have the following prac- straight line moving in such a manner as to tical rule for finding the number of balls in touch a given straight line, and continue any pile. parallel to its first position. A plane may also be generated by revolving one straight line about another straight line, perpendicular to it, as an axis of revolution. The rectilineal equation of a plane may be reduced to the form, z = ex + dy -r g, in which x, y and z, arc the co-ordinates of every point of the surface, and c, d and g, constants. The plane in given when c, d and g are known, and may be constructed by points, as follows: Assume any two values for x and y, and substitute them in the equation; there will result a corresponding value for :, which with the assumed value of x and y will be the co-ordinates of a point that may be constructed by known principles. In like manner, any number of points may be found and constructed; a surface passed through them will be the surface required ; three points arc sufficient to fix the position of a plane provided they are not in the same straight line. Planes arc generally constructed by finding the points in which they cut the co-ordinate axes, and then passing a plane through these three points. Planes are given in Descriptive Geometry by their traces, that is, by their intersections with the planes of projections; they may, in like manner, be determined analytically. To find the equation of the trace of a plane upon the planes XV, XZ and YZ respectively : make in the equation of the plane, :, y and x, respectively equal to 0 in the equation; the resulting equation* will be the equations of the required traces. Two traces will be sufficient to fix the position of a plane. If wc take two planes, whose equations are t = fj- + rfy + iT, and z = c'x + d'y+g', they will be parallel when e = d and d — d'. They will be perpendicular to each oilier, when 1 + cc' + dd' - 0. In general, the an<;lc which they make with each other, may be determined by means of the formula, 1 + cc + dd' natc one variable; the resulting equation will be the equation of the projection of their intersection on the plane of the other two. Combine the equations again, eliminating a second variable, and the resulting equation will be that of the projection of the line of intersection upon a second plane, and these will be sufficient to determine the line of intersection. If the equation of a plane is z = ex + dy + g, and the equations of a straight line arc x = az + a, and y = bz + (3, the line and plane will be parallel when 1 - ac - bd = 0; thry will be at right angles when a = — c and b = — d; and in general, the angle which they make with each other is given by the formula 1 - ac - bd sin A V 1 + a' -t- b' • 1 + c" To find the point in which a line pierces a plane, combine their equations and find the corresponding values of the variables; these i will be the co-ordinates of the required point Plank Angle. A portion of a plane lying between two straight lines, meeting at a point. The lines are called sides of the angle, and their common point is the vertex. See Angle. Plane Chart. A chart constructed so that the parallels of latitude and longitude are represented by straight lines parallel to each other, and at the same distance from each other, in every latitude. Plane Curve. A curve all of whose points He in the same plane. Plane Director. A plane parallel to every element of a warped surface of the first class. See Warped Surface. Plank Fiourk. A portion of a plane limited by lines either straight or curved. When the bounding lines arc straight, the figure is rectilinear, and is called a polygon. When they arc curved, the figuie is eumlincar. Plank Geometry. That part of Geometry which treats of the relations and properties of plane figures. Horizontal Plank A plane parallel to the surface of still water, or, parallel to a under side and screw firmly into the plates. The object of these plates is to secure the paper on which the drawing is to be made. By loosening the screws, and pushing up the plates, the paper may be introduced; then, by turning the screws back again, the plates are drawn down, and the paper is held tightly. The paper might bo slightly moistened, which would secure a smoother surface when it is dried. A ruler accompanies the table, with two eights like compass sights, or sometimes with a telescope, in their stead. One edge of the ruler is beveled, and this edge is so placed that it is in the plane of the openings through the sights. The sights arc constructed so as to fold down for convenience in carriage. A compass is sometimes attached for determining the bearings of lines. The plane table, as described, is used for two distinct purposes; 1st. To measure horizontal angles; and 2d, For determining the shorter lines of a survey, both in extent and position. To measure a horizontal ancle. Place, by means of a plumb line, the centre of the table exactly over the angular point; then level the table and clamp the limb; after which place the ruler with the sights raised so that the beveled edge shall rest against the steel pin at the centre; direct the sights to the left hand object, and read the reading at each end of the ruler, and take a mean of the results for the first reading. Then direct the sights to the right hand object, and take the second reading in like manner. If the ruler has not passed over the 03 point of the limb, the excess of the second reading over the first is the value of the required angle. If the ruler has passed theO° point, the first reading must be subtracted from ISO" and the difference added to the second reading; the sum will be the value of the angle required. To determine lines in extent and position. Having fastened a sheet of paper on the table, examine the lines and objects which arc to be determined in iH>sition and select for a base a convenient line which is connected with some point of the triangulation, taking care that as many prominent objects as possible may he seen from its two extremities. Then place the plane table over one extremity of the base, so that the point on the paper corresponding may be exactly over the extreme point of the base. Clamp the limb and make it truly hor] izontal. Mark the point corresponding to the end of the base by a needle, and pressing the ruler against the needle, direct the sights to the other extremity of the base. With a finepointed pencil, draw a straight line along the beveled edge of the ruler, and lay off on it, : from a scale of equal parts, the length of the base, and mark the second point; then direct ! the sights, in succession, to all the principal objects that arc visible from the first station, and draw pencil lines along the edge of the ruler. Next plant the plane table so that the second end of the plotted base line shall bo over the second end of the base line in the field; level the instrument, and having placed the needle at the second end of the base line,, bring the beveled edge of the ruler to coincide with the plotted base, and then turn the limb of the instrument till the sights are directed to the first end of the base; clamp the limb and direct the sights, in succession, to every object sighted from the first station, marking the points of intersection of these lines with the corresponding ones from the first station. To illustrate: let it be required to determine the relative position of several houses. From station A, and on one of the line of the triangulation. as AB measure the base line AB, which we will suppose equal to 300 yards. Place the plane table over A, and sight to the corners of the houses, and mark the lines 1,2, 3, 4, &c. Then more the table to B, place the plotted line AB in the direction from B to A, and sight to the same corners as before, and draw the lines as in the figure; the points at which they intersect the corresponding lines before drawn, determine the plot of the comers of the houses; the front lines of the houses may then be drawn on the paper, and upon these the plots of the houses themselves may be con |
