nth part of what remains after the portion of the eldest and 2a have been subtracted; to the third he leaves a sum 3a, together with the nth part of what remains after the portions of the two other sons and 3a have been subtracted. The property is found to be entirely disposed of by this arrangement. Required the amount of the property." Ans. It is clear that 3a is the property received by the third son, since there is no remainder; and as the second son received 2a and an nth part of the remainder, if we repre n 1 sent the remainder by x, we shall clearly have n (2n+1)a is the prop n-1 Again, x+2a: 3na = + 2a = (5n - 2)a is the remain der of the father's property after the 1 first son received his share, which by the question must be the the remainder after he received the portion a; consequently, if y denotes the remainder, we shall get (n − 1)2 (n − 1)2 1)a is the portion of the eldest son; and it is (5n2 — 2n)a + a = presses the father's property. 45. The same things being supposed as in the preceding question, with the addition that the father has n-1 children instead of three, and that the eldest receives a and an nth part of the remainder, the next 2a and an nth part of the remainder, and so on to the last, who receives (n-1)a; then it is proposed to show that (n - 1)a is the portion of each child, and that the father's estate equals (n − 1)3a. Ans. We have (as before) — 1 consequently, + (n − 2)a = a + (n = 2)a = (n − 1)a= the n portion of the child whose number is n-2. Also, + (n − 2)a = 2(n − 1)a gives the equation n 1 n or y = 2na; consequently, Y + (n−3)a= 2a + (n − 3)a = n In -1)a is the portion of the child whose number is n-3; and in like manner it may be shown that the child whose number is n-4 receives (n-1)a for his portion, and so on; consequently, (n − 1)2a = the father's estate. Remarks. We have extracted this and the preceding question from Davies' Translation of Bourdon's Algebra. [See p. 79, etc., of the work; published by Wiley & Long in 1835.] We have given the questions, and a slight sketch of the methods to be used in their solution; because the methods used by Bourdon appear to us to be unnecessarily long and embarrassing. 46. Two market-women, A and B, had (each) the same number of eggs; A sold hers in parcels consisting of 9 eggs each, and received 8 cents for each parcel sold; also, B sold hers in parcels consisting of 13 eggs each, and received 12 cents for each parcel sold; and the money received by A and B together was $10.40. A sold more eggs than B, and received less money by twice as many cents as A and B had eggs that remained unsold; moreover, B had three times as many eggs remaining as A had. It is proposed to find the number of eggs which A and B had, and what they each. received for the eggs sold, and the number of eggs that remained to each. Ans. A and B each had 578 eggs; A received $5.12, and B received $5.28; A had two eggs left, and B had six. 47. "Two artillerymen together filled 1000 cartridges, and used the same quantity of powder. One says to the other, If I had filled as many cartridges as you have, I should have made use of 18 cwt. of powder.' Then the other answers him, 'Had I filled as many cartridges as you have, I should have used only 8 cwt. of powder.' How many cartridges did each fill? and how much powder did each use? "Ans. The first filled 400, the second 600 cartridges, and each used 12 cwt. of powder." APPENDIX TO (SIMPLE) EQUATIONS. Since in finding the limits of numbers or quantities it is frequently necessary to use those which are not equal to each other, we here propose to give the general principles of rea-soning which are applicable to INEQUALITIES OR INEQUATIONS. 1. If A and B stand for two numbers or quantities of the same kind (which may consist of one or more terms), such that A is greater than B, then, according to 18, of Sec. I., we signify that A is greater than B, or B less than A, by writing A>B, or B<A, which are called inequalities or inequa tions, noticing that A and B are called the first and second members of the first inequality, while B and A are those of the second. The first inequality is read by saying that A is greater than B, or (which is often preferable) by saying that A is not less than B; also, the second inequality is read by saying that B is less than A, or B is not greater than A. Inequalities whose first members are all greater or less than their second members, are said to exist in the same sense; but inequalities whose first members are not all greater or less than their second members, are not all said to exist in the same sense. Thus, 3>2 and 7>4 exist in the same sense; also, 5<6 and 10<14 exist in the same sense; but 8<15 and 9>6 exist in a contrary sense. 2. If C is added to the members of the preceding inequalities, they will clearly become A C>B± C, or B±C<A±C, which exist in the same sense as the given inequalities. Thus, if 5 is added to the members of a5>2, it becomes >7; and subtracting 3 from the members of 39, it becomes a<6. Hence, any term in either member of an inequality (as in an equation) may be transposed into the other member by changing its sign, without altering the sense of the inequality; also, any term which is common to both members, and has the same sign in each, may be erased or stricken out of the inequality, without affecting its sense. Thus, taking A ± C >B±C or B±C<A ± C, and erasing C, the inequalities become A>B or B<A; also, transposing A and B, the inequality A > B or B < A will be changed to - B> A or AB; consequently, A > B and - A < – B exist in a contrary sense. Hence, when the signs of both members of any inequality are changed (or multiplied by -1), the sense of the inequality will be changed. 3. Adding the corresponding members of A>B or B<A, and C> D or DC, the sums clearly give A+ C>B+D or B + D <A + C. Consequently, the sum of any number of inequalities which exist in the same sense will exist in the same sense as the inequalities. Thus, from the addition of x + 3 > 5 and y-37 there results + y> 12. If from 8> 5 or 58 we take 64 or 46, we get 2 > 1 or 1 < 2, which exist in the same sense as the given inequalities. But if from 8 > 5 or 5 < 8 we take 7 > 3 or 37, we get 1 <2 or 2> 1, which do not exist in the same sense as the given inequalities. Hence, the difference of two inequalities which exist in the same sense is not always an inequality which exists in the same sense as the given inequalities. and reducing its terms to a common denominator, and then rejecting the denominator, there results the inequality 318 2x+6 or 2x + 63x-18, which gives a 24 or 24 <x. x > Hence, considering and treating the denominators of the fractional terms of any inequality as being positive, we free the inequality from fractions in the same way that we free an equation from fractions, and the sense of the inequality will remain unchanged. Also, any positive factor which is common to the terms of an inequality may be erased, and vice versa, and that without altering the sense of the inequality. Thus, from 8> 6 and <1 we get 4 > 3 and 1 < 2. 1 2 5. If we have the inequalities A > B or BA and C>D or DC, such that their members are essentially positive, then by taking the product of their corresponding members, we shall plainly have AC > BD or BD < AC. Hence, the product of any number of inequalities which exist in the same sense, and whose members are essentially positive, is an inequality which exists in the same sense as the given inequalities. Hence, too, if the members of such an inequality are raised to any power whose index is a positive integer, the result will be an inequality which exists in the same sense as the given inequality; but if the index of the power is a negative integer, it is plain that the result will be an inequality which exists in a sense contrary to that of the given inequality. Thus, from > 3 and y> 2 we get xy > 6; by BDF > > 2 we get x < 27 and (y ̄1⁄2)-2< ACE. - 6. If we take the product of the (corresponding) members of the inequalities- A <- B or B>- A and C < - D or — D > - C, we get AC > BD or BD < AC, an inequality which exists in a sense contrary to that of the given inequalities. If we multiply AC > BD or BD <AC EF or FE, the result will be ACE BDF or Hence, the product of an even number of inequalities which exist in the same sense, and whose members are essentially negative, is an inequality which exists in a sense contrary to that of the given inequalities; and the product of an odd number of such inequalities is an inequality which exists in the same sense as the given inequalities. < Hence, also, any even positive integral power of such an inequality will be an inequality which will exist in a sense contrary to that of the given inequality; and if we take the index of the power with the negative sign, the power will be an inequality which will exist in the same sense as the given inequality; but if the members of such an inequality are raised to any odd positive integral power, the result will be an inequality which will exist in the same sense as the given inequality, while, if the index of the power is taken with the negative sign, the result will be an inequality which will exist in a sense contrary to that of the given inequality. Thus, from 4 <3 and 52 we get 20 > 6; and from 2 > − 3, − 4 > 5, and 6>- 7 we have - — |