If in this expression N is successively replaced by the numbers 1, 2, 3...., the number of balls in the successive layers, beginning at the top, will be ob Whence the sum of the whole number of balls contained in the pile is the most convenient expression for the number of balls in a triangular pile. EXAMPLE. How many balls in a triangular pile, the side of whose base contains 35? (2) A square pile is formed by continued horizontal courses of shot laid one above another, and these courses are squares whose sides decrease by unity from the bottom to the top row, which is also composed simply of one shot; and hence the series of balls composing a square pile is If a side of the base of a quadrangular pile contains 35 balls, how many in the pile ? (3) A rectangular pile is one in which the layers, except the uppermost, are arranged in rectangles. Representing by m+1 the number of balls in the top row, the layer below it must contain 2 rows of m+2 balls, the next layer 3 rows of m+3 balls, and so on, to the Nh, which contains N rows of m+N balls each; and the number in this pile is (m+1)+2(m+2)+3(m+3)+4(m+4)+ .... N(m+N) =m+2m+3m+4m+.... Nm+1o+22+32+42+ . . . . N3 .... (4) The number of balls in a complete triangular or square pile must evidently depend on the number of courses or rows; and the number of balls in a complete rectangular pile depends on the number of courses, and also on the number of shot in the top row, or the amount of shot in the latter pile depends on the length and breadth of the bottom row; for the number of courses is equal to the number of shot in the breadth of the bottom row of the pile. Therefore, the number of shot in a triangular or square pile is a function of N, and the number of shot in a rectangular pile is a function of N and m. The expression for a rectangular pile, may be written N(N+1)(3m+2N+1) _' 'N(N+1) [2(m+N)+m+1]. 6 N(N+1) But m+1 is the number of balls in the top row, N is the number in the smaller side of the base, and m+N the number in the greater side, 2(m+N) the number in the two parallel greater sides; moreover, is the number of balls in the triangular face of each pile; hence we have also this general rule for rectangular or square piles. RULE. 2 Add to the number of balls or shells in the top row the numbers in its two parallels at bottom, and the sum multiplied by one third of the slant end or face gives the number of balls in the pile. EXAMPLES. (1) How many balls are in a triangular pile of 15 courses? Ans. 680. (2) A complete square pile has 14 courses: how many balls are in the pile, and how many remain after the removal of 5 courses? Ans. 609 and 554. (3) In an incomplete rectangular pile, the length and breadth at bottom are respectively 46 and 20, and the length and breadth at top are 35 and 9: how many balls does it contain? Ans. 7190. (4) The number of balls in an incomplete square pile is equal to 6 times the number removed, and the number of courses left is equal to the number of courses taken away: how many balls were in the complete pile ? Ans. 385. (5) Let k and k denote the length and breadth at top of a rectangular truncated pile, and N the number of balls in each of the slanting edges; then, if B be the number of balls in the truncated pile, prove that N -N)+1}. B=~/ { 2N3+3N(h+k)+6hk—3(h+k+N)+1 6 VARIATION. 343. Let a denote a constant quantity, or one which does not change its value, and ≈ a variable which is supposed to increase or diminish. The product of the quantities a and a being denoted by X, if x is increased or diminished, X will be increased or diminished in the same proportion. Thus, if r become r', and, consequently, X become X', we shall have Under these circumstances X is said to vary directly as x. The symbol of variation is ; and the expression X varies directly as x, is indicated by the combination of symbols X x x. 344. If the product of x and y be constant, and x, y both variable, since ry=x'y' =C, 1 1 x:x:y'y::- : .. y y In this case as x varies as the reciprocal of y, x is said to vary inversely as y, and the symbolical expression is τα;· If ry=X and x'y'=X', then X : X' : : xy : x'y'. The variation of X in this case depends on the variation of two quantities which is expressed thus, r and y, Xxxy. 345. If xy=X and x'y'=X', then, x== and r': X = In this case x is said to vary as X directly, and as y inversely. The symbol is that is, if one quantity vary as a second and the second as a third, the first varies as the third. Again, let xx y and z α y .. x x z, or x:x'::z:z', or x:z:: x': z′; But zzy: y', .. x±z: x' ±z'::y:y', i. e., yxxz. Again, since xxy, x:x' :: y: y', and since za y, z:: : z' :: y: y', ... xz : x'z′ : : y2 : y'3, and √xz : √x'z' :: y: y', or y a √xz; that is, if two quantities vary respectively as a third, their sum, difference, or square root of their product, varies as this third quantity. 348. If ray and m be a constant quantity, integer or fractional, since ry:: x' : y', .. x : y :: mx' : my' (Art. 127), i. e., x x my; that is, if one quantity vary as another, it varies as any multiple or part of this other. When xxy, and, consequently, ramy, so that x:x::my: my' or г:my ::r': my', then, if x=my, x' will be equal to my' in all cases; whence, if r vary as y, x is equal to y multiplied by some constant quantity. 349. If X and Y are two corresponding values of x, y, from which it follows that, when two corresponding values of x, y are known, the constant m may be found. ... xxym; 350. Let xxy · · . x : x' : : y : y' ··· x11 :x′m :: yTM : y'm m being any exponent integer or fractional. Whence, if one quantity vary as another, any power or root of the first quantity will vary as the same power or root of the second quantity. 351. Let xxy, and let t be another quantity, either variable or constant, and of which t, t' are either equal or different values. Then, since that is, if one quantity vary as another, and if each of them be multiplied or divided by any quantity, variable or constant, the products or quotients will vary as each other. that is, if the product of two quantities vary as a third quantity, each of the two quantities varies as the third directly, and as the other inversely. that is, if the product of two variable quantities be constant, these quantities vary inversely as each other. 354. Let a be a constant, and x, y, z variables, and let a:x::y:z, a: x':: y': z', &c.; ... az=xy, az'=x'y', &c.; ... az: az' :: xy: x'y', or z: z' :: xy: x'y' ..zxxy; that is, if four quantities are always proportional, and one or two of them are constant, the others being variable, it can be found how the latter vary. 355. Let x, y, z be three quantities, of which, xay when z is constant, and raz when y is constant; it is required to determine the variation of r when y, z are both variable. Suppose, first, that r is made to vary as y, and that when y becomes y', r becomes r'. Next, that r' (varied from r by the variation of y) is made further to vary as z, and that when z becomes z', x' becomes r". Then, since or x:x::yy', and x': x'':z: z' x:x":: yz: y'z'; i. e., xxyz. Therefore, if x vary as y when z is constant, and as z when y is constant, when y, z are both variable, x varies as the product yz. Similarly, it can be proved, that if t vary as v, x, y, z separately, the others being constant when v, x, y, z are all variable, t varies as the product vxyz. SYMMETRICAL FUNCTIONS OF THE ROOTS OF AN EQUATION. 356. THERE are certain functions of the roots of an equation which may be expressed, in a general manner, by means of the coefficients of that equation, without the equation itself being resolved. These functions, which form a very extensive class, are termed rational and symmetric functions, or simply symmetric functions. They are called rational, because the roots do not enter into them under the radical sign, nor with fractional exponents; the roots are combined only by addition, subtraction, multiplication, and division. These functions are called symmetric, because the roots are combined in such a way that any two of them may be interchanged without altering the value of the function. For example, the expressions ab ac bc ac+be+ab, a+b2+c2, 2c2+262+2a2* -3abc are rational and symmetric functions of a, b, c. All the coefficients of an equation are symmetric functions of its roots, as may be seen in the expressions for the coefficients in Art. 245; for, in these expressions, if a, were written in every place where a2 occurs, instead of and a, in every place where a, occurs, instead of a1, or if any other two of the roots were interchanged, the values of the expressions would not be altered. Several quantities, a, b, c, &c., being given, if we arrange them two and two, in every possible way, and if in each arrangement, e. g., ab, we give the exponent a to the first factor and the exponent ẞ to the second, we have a series of products such as aab, whose sum is evidently a symmetric function of the quantities a, b, c, &c. This function is called a double function, because each term contains two of the given quantities; it is represented, abridged, by S(aab3), the letter S being here employed to denote the word sum. In like manner, triple, quadruple, &c., symmetric functions are represented by S(a b3c), S(a b c d°), &c. In accordance with this notation, simple symmetric functions, as aa+ba |
