ces is 8; their sum is 41; and the sum of their squares 699. What are the numbers ? Let :=the second number, and y=the difference of the second and least ; .::x-y, x, and x+y+8 are the numbers, and their sum=3x+8=41; ... 3x=33, and x=11; :::(11—y) +121+(19+y)'=669, or y?+8y=48 ; ... completing the square, and extracting the root, y+4=+8, and y=4, or-12, both which values answer the conditions ; and the numbers are 7, 11, and 23. Prob. 6. What number is that, which being divided by the product of its two digits, the quotient is 2; and if 27 be added to it, the digits will be inverted ? Aos. 36. Prob. 7. There are three numbers, the difference of whose differences is 5; their sum is 44 ; and contioual produd is 1950. What are the numbers ? Ang. 6, 13, and 25. Prob. 8. A farmer received 71. 48. for a certain quantity of wheat, and an equal sum at a price less by Is. 6d. per bushel, for a quantity of barley, which exceeded the quantity of wheat by 16 bushels. How many bushels were there of each? Ans. 32 bushels of wheat, and 48 of barley. Prob. 9. A poulterer bought 15 ducks and !2 turkeys for live guineas. He had two ducks more for 18 shillings, than he had of turkeys for 20 shillings. What was the price of each ?-Ans. the price of a duck was 3s. and of a turkey 5s. Prob. 10. There are three numbers, the difference of .whose differences is 3; their sum is 21; and the sum of the squares of the greatest and least is 137. Required the numbers. Ans. 4,6, and 11. Prob. 11. There is a number consisting of 2 digits, which, when divided by the sum of its digits, gives a quotient greater by 2 than the first digit. But if the digits be inverted, and then divided by a number greater by unity than the sum of the digits, the quotient is greater by 2 than the preceding quotient? Required the number. Ans. 24. Prob. 12. What two numbers are those, whose product is 24, and whose sum added to the sum of their squares is 62 ? Ans. 4, and E. Prob. 13. A grocer sold 80 pounds of mace, and 100 pounds of cloves, for 651. ; but he sold 60 pouods more of cloves for 201. tban he did of mace for 101. What was the price of a pound of each? Ans. the mace cost los, and the cloves 5s. per pound. Prob. 14. To divide the number 134 into three such parts, that once the first, twice the second, and three times the third, added together may be equal to 278; and that the sum of the squares of the three parts may be equal to 6036. Ans. 40, 41, and 50, respectively. Prob. 15. Find two numbers, such, that the square of the greater ininus the square of the lesser, may be 56 ; and the square of the lesser plus one-third their product may be 40. Ans. 9, and 5. Prob. 16. There are two numbers, such, that three times the square of the greater plus twice the square of the less is 110 ; and half their product, plus the square of the lesser, is 4. What are the numbers ? Ans. 6, and 1. Prob. 17. What number is that, the sum of whose digits is 15; and if 31 be added to their product, the digits will be inverted ? Ans. 78. Prob. 18. There are two numbers such, that, if the lesser be taken from three times the greater, the remainder will be 35; and if four times the greater be divided by three times the lesser plus one, the quotient will be equal to the lesser number. What are the numbers ? Ads. 13 and 4. Prob. 19. To find two numbers, the first of which, plus 2, multiplied into the second minus 3, may produce 110; and the first minus 3, multiplied by the second plus 2, may produce 80. Ans. 8, and 14. Prob. 20. Two persons, A and B, comparing their wages, observe, that if A had received per day, in addition to what he does receive, a sum equal to one-tourth of what B received per week, and had worked as many days as B received shillings per day, he would have received 488. ; and had B received 2 shillings a day more than A did, and worked for a number of days equal lo half the number of shillings he received per week, he would have received 41. 185. What were their daily wages ? Ans. A's 5 shillings, and B's 4. Prob. 21. Bacchus caught Silenus asleep by the side of a full cask, and seized the opportunity of drinking, which he continued for two-thirds of the time that Silenus would have taken to empty the whole cask. After that Silenus awoke, and drank what Bacchus had left. Had they drunk both together, it would have been emptied two hours sooner, and Bacchus would hare drunk only half what he left Sileous. Required the time in which they could have emptied the cask separately. Ans. Silenus in 3 hours, and Bacchus in 6. Prob. 22. Two persons, A aad B, talking of their money, A says to B, if I had as many dollars at 55. 6d. each, as ! have shillings, I should have as much money as you ; but, if the number of my shillings were squared, I should have twice as much as you, and 12 shillings more. What had each ? Ads. A had 12, and B 66 shillings. Prob. 23. It is required to find two numbers, such, that if their product be added to their sum it shall make 62; and if their sum be taken from the sum of their squares it shall leave 86. Ans. 8, and 6. Prob. 24. It is required to find two numbers, such, that their difference shall be 98, and the difference of their cube roots 2. Ans. 125, and 27. Prob. 25. There is a number consisting of two digits. The left-hand digit is equal to 3 tiines the right-hand digit; and if 12 be subtracted from the number itself, the remainder will be equal to the square of the left-hand digit. What is the number? Ans. 93. Prob. 26. A person bought a quantity of cloth of two sorts for 71. 18 shillings. For every yard of the better sort he gave as many shillings as he had yards in all; and for every yard of the worse as many shillings as there were yards of the better sort more than of the worse, And the whole price of the better sort was to the whole price of the worse as 72 to 7. How many yards had he of each? · Ans. I yards of the better, and 7 of the worse, Prob. 27. There are four towns in the order of the letters, A, B, C, D. The difference between the distances, from A to B, and from B to C, is greater by four miles than the distance from B to D. Also the number of miles between B and D is equal to two-thirds of the number between A to C. And the oumber between A and B is to the number between C and D as seven times the number between B and C: 26. Required the respective distances. Ans. AB=42, BC=6, and CD=26 miles. CHAPTER XII. ON THE BINOMIAL THEOREM. 434. Previous to the investigation of the Binomial Theorem, it is necessary to observe, that any two algebraic expressions are said to be identical, when they are of the same value, for all values of the letters of which they are composed. Thus, *~1. =x-1, is an identical equation : and shows that x is indeterminate; or that the equation will be satisfied by substituting, for X, any quantity whatever. Also, (x+2) X(X -- a) and 2-a?, are identical expressions ; that is, (x+a) x(x-a)=x-a%; whatever numeral values may be given to the quantities represented by c and a. 435. When the two members of any identity consist of the same successive powers of some indefinite quantity x, the coefficient of all the like powers of x, in that identity, will be equal to each other. For, let the proposed identity consist of an indefinite number of terms, as, a+bx+cre +d2+&c.=d'+bx+c'x + du +-&c. Then since it will hold good, whatever may be the value of %, let x=0, and we shall have, from the vanishing of the rest of the terms, a=a'. Whence, suppressing these two terms, as being equal to each other, there will arise the new identity bx+cx®+dx+ &c. =b+c'x?+dox+&c. which, by dividing each of its terms hy x, becomes b+cx+dx+&c. = 6'+c'x+d22+&c. And, consequently, if this be treated in the same manner as the former, by taking x=0, we shall have b=b', and so on ; the same mode of reasoning giving c=c', d=d', &c., as was to be shown. § 1. INVESTIGATION OF THE BINOMIAL THEOREB. 436. Newton, as is well known, left no demonstration of this celebrated theorem bat appears, as has already been observed, (Art. 163), to have deduced it merely from an induction of particular cases, and though no doubt can be entertained of its truth from its having been found to succeed in all the instances in which it has been applied, yet, agreeably to the rigour that ought to be observed in the establishment of every mathematical theory, and especially in a fundamental proposition of such general use and application, it is necessary that as regular and strict a proof should be given of it as the nature of the subject, and the state of analysis will admit. 437. In order to avoid entering into a too prolix investigation of the simple and well-known elements, upon which the general formula depends, it will be sufficient to observe, that it can be easily shown, from some of the first and most common rules of Algebra, that whatever may be the operations which the index (m) directs to be performed upon the expression (a +x)m, whether of elevation, division, or extraction of roots, the terms of the resulting series will necessarily arise, by the regular integral powers of x; and that the first two terms of this series will always be amtmamx; so that the entire expansion of it may be represented under the form am+nam-x+Bam-?x? +Ca-37-Dam+&c. Where B, C, D, &c. are certain numerical coefficients, that are independent of the values of a and x; which two latter may be considered as denoting any quantities whatever. 438. For supposing the index în to be an integer, and taking a=1, wbich will render the following part of the investigation more simple, and equally answer the purpose intended; it is plain that we shall have, according to what has been shown (Art. 289), (1+x)=l+mz+bx+cm+dx' + &c. 439. And if the index in, of the given binomial, be negative, it will be found by division, that (1+x)-m, or the equivalent expression 1 1 1. 6'22-23_&c. (1+x)m 1+mx+bx+cx? &c. where the law of the terms, in each of these cases is similar to that above mentioned. 440. Again, let there be taken the binomial (1+2), bar (1, ing the fractional index ; where m and n are whole positive numbers. m n |