The values of x and y in both systems, it is evident, will be real, since the quantity placed under the radical is essentially positive. In the first system the values of x and y are essentially negative; in the second the value of x, it is easy to see, is necessarily positive, but the value of y may be either positive or neg $2 ative; in order that it may be positive, we must have 92 3. Let a = b, and therefore a2. 620. On this hypothesis, we have for the first system of values for x and y Returning to the equations of the proposed in order to interpret these last we obtain for x and y on the present hypothesis х PROBLEMS FOR SOLUTION AND DISCUSSION. 1. There are two numbers, whose sum is a and the sum of whose second powers is b. Required the numbers. Putting x and y for the numbers, we have What conditions are necessary in order that the values of x and y may be real? When will the values of x both be positive? Can both be negative? When will one of them be positive and the other negative, and to what question does the negative value belong? 2. To find two numbers such, that the sum of their products by the numbers a and b respectively may be equal to 2 S, and their product equal to p. Putting x and y for the numbers, we have What conditions are necessary in order that the values of x and y may be real? What is the greatest value of which p admits? Can either of the values of x or y be negative? 3. To find two numbers such, that the sum of their products by the numbers a and b respectively may be equal to a given numbers, and the sum of their squares equal to another given number q. Putting x and y for the numbers respectively, we have x= What conditions are necessary in order that the values of x and y may be real? Within what limits must q be comprised in order that both values of x may be positive? Within what limits must q be comprised in order that both values of y may be positive? In the second system of values for x and y, when will the value of x be positive, and that of y negative, and what is the analogous problem, to which this system belongs? 4. To find a number such, that its square may be to the product of the differences between this number and two other numbers a and b in the ratio of Չ to p. Putting x for the number sought, we have x= q (a + b) ±√ q2 ( a − b )2 + 4 p q a b Let this formula be examined on the different hypotheses q<P, q=P, q>p. SECTION XIV. MAXIMA AND MINIMA. 122. In several of the preceding questions, the given things, we have seen, are so connected among themselves, that one is determined by the others to be comprised within certain limits, or to have a greatest or least possible value. A quantity, the value of which may be made to vary, is called a variable quantity; the greatest value of which is called a maximum and the least a minimum. Questions frequently occur, in which it is required to determine under what circumstances the result of certain arithmetical operations performed upon numbers will be the greatest or least possible. We shall resolve a few questions of this kind, the solutions of which depend upon equations of the second degree. 1. To divide a number, 2 a, into two parts such, that the product of these parts may be a maximum. Let x be one of the parts, then 2 a -x will be the other, and their product will be x (2 a —x). By assigning different values to x, the product x (2 a-x) will vary in magnitude, and the question is to assign to x a value such, that this product may be the greatest possible. Let m be the maximum sought, we have by the question Regarding for the moment m as known, and deducing from this equation the value of x, we have From this result it appears, that in order that x may be real, m must not exceed a2; the greatest value of m will therefore be a2, in which case we have x=a. Thus to obtain the greatest possible product, the proposed must be divided into two equal parts, and the maximum obtained will be equal to the square of one of these parts. In the equation x (2 a-x)=m, the expression x (2 a-x) is called a function of x, This function is itself a variable, the value of which depends upon that given to the first variable or x. 2. To divide a number, 2 a, into two parts such, that the sum of the square roots of these parts may be a maximum. Let x2 be one of the parts, then 2 a- x2 will be the other, and the sum of the square roots will be x + 2a — x2. Let m be the maximum sought, we have by the question In order that the values of x may be real, the value of m2 must not exceed 4a; 2 a is therefore the greatest value, which m can receive: Let us put m 2 a x2 2a, we have xa and x2=a, whence =Ɑ. Thus, the proposed must be divided into two equal parts in order that the sum of the square roots of the parts may be a maximum. This maximum moreover will be equal to twice the square root of one of the parts. 3. Let it be proposed next to find for x in the expression p2x2+q2 (p2 — q2) x Putting a value such as to render this expression a minimum. from which we obtain x= 1 (p2 — q2) m 2 p2√ (p2 — q2)2 m2 — 4 p3 q2 In order that the values of x may be real (p2-q3)2 m2 must at least be equal to 4 p2 q2, and by consequence m must at least be equal to 2 p q p2 - q2 2pq in the expression Putting m = p2 -- q2 for x, the radical disappears, and we have x= 2 p2 The least value of the proposed is therefore 2 p q p2 — q2, and the value of x, which will render the proposed a minimum, is From what has been done, the following rule for the solution of questions of the kind, which we are here considering will readily be inferred, viz. Having formed the algebraic expression of the quantity susceptible of becoming a maximum or minimum, make this expression equal to any quantity m. If the equation thus made is of the second degree in x, x designating the variable quantity, which enters into the algebraic expression, resolve this equation in relation to x; make next the quantity under the radical equal to zero, and deduce from this last equation the value of m; this will be the maximum or minimum sought. Substituting finally the value of m in the expression for x, we obtain the value of x proper to satisfy the enunciation proposed. If the quantity placed under the radical remains essentially positive, whatever the value of m, we infer that the expression proposed may be of any assignable magnitude whatever, or in other words, that it will have infinity for a maximum and zero for a minimum. Thus let there be proposed the expression 4x2 + 4x 3 6 (2x+1) to determine whether this expression is susceptible of a maximum or minimum. Putting have x= 4x2+4x-3 = m and deducing the value of x we 6 (2 x + 1) 3 m 1 1 2 ± √9 m2 +4. Here, whatever value we give to m, the quantity placed under the radical will be positive; the proposed therefore may be of any magnitude what ever. EXAMPLES FOR PRACTICE. 1. To divide a given number a into two factors, the sum of which shall be a minimum. Ans. The two factors should be equal. 2. Let d be the difference between two numbers; required |