the second letter will have one accent, the third two, and the rth three. The num ators of the unknown quantities, u, z, y, x, are found by the rule already given.* 9. We may employ these formulas for the resolution of numerical equations. In doing this, we must compare the terms of the equations proposed with the corresponding terms of the general equations, given in the preceding articles. To resolve, for example, the three equations 7x+5y + 2 z = 79, 8 x + 7y + 9 z = 122, x + 4y + 5 z = 55, compare the terms with those of the equations We have then it is necessary to given in art. 86. α = 2, d = 79, =7,6 = 5, c a" 1, b" 4, c" 5, d" = 55. = = Substituting these values in the general expressions for the unknown quantities x, y, and z, and going through the operations, which are indicated, we find x = 4, y = 9, z=3. It is important to remark, that the same expressions may be employed, even when the proposed equations are not, in all their terms, affected with the sign +, as the general equations, from which these expressions are deduced, appear to require. If we have, for example, in comparing the terms of these equations with the corresponding ones in the general equations, we must attend to the signs, and the result will be We are then to determine by the rules given in art. 31., the sign, * M. Laplace, in the second part of the Mémoires de l'Académie des Sciences for 1772, p. 294, has demonstrated these rules à priori. See also Annales des Mathématiques pures appliquées, by M. Gervol. iv, p. 148. g which each term of the general expressions for x, y, and z, ought to have, according to the signs of the fa ors of which it is co posed. Thus we find, for example, that the first term of the c mon denominator, which is a b' c', becoming + 3 x + 4x changes the sign of the product, and gives 72. If we observe the same method with respect to the other terms, both of the numerators and denominators, taking the sum of those which are positive, and also of those which are negative, we obtain Equations of the First Degree with two or more Unknown X = y = 50. x=- 39.8121 y = 58.5421 (x + 5) (y +7) = (x + 1) (y — 9) + 112 2 x = 3, 2x+10=3y+1 x + y = c b Sax + by = cr Sy = 5. ag-bf" 2 b2-6 a2+d = ah-cf ag-bf 3a2-b2+d x= За ,y= 3 b 3 a + x b+y a (c3 Sbc x = cy 26 - b3) bc 11. 3x+5y= bcf2 (8b-2f)bf b + ƒ + (b + + f ) f y = f2x (b+2f) bf ( bf x + y = 10 ay=7, z = 16. 12. 19 y + z x + y + x = 16, y = 73, z = 5 1. To find a number which added to bges a sum a (1.) Let b = 29, a = 47. 2.) Let b 31, a = 24. 2. Aather is a years, his son years old. In how many years will the son's age be one fourth of his father's? (1.) Let a = 54, b = 0. (2.) Let a 45, b = 3. A cistern, into which water was let by two pipes, will be filled by them both in a number a of hours, and by the first alone, in a number b of hours. In what time will it be filled by the second alone? (1.) Let a = 12, b = 20. (2.) Let = 12, b = ̧10. 4. A person sent to buy oranges which cost a pence each, he should bought those which cost b pence each How many was he sent for? 5. A nd that if he bought those (1) Let a 5, b = 4, c = 24. puued by et a = 4, b= 6, c = 24. all his money, but if he hould have c pence left. which had started from a certain place 10 days, is other boat from the same place and by the same wa The first goes 4 miles every day, the other 9. In how many days will the second overtake the first? 6. Let n equal t mber of days elapsed since the departure of the first boat, a thumber of miles it goes per day, and b the number of miles the second goes per day. 7. What will be the change in the question if n = 10, a = 6, and b = 4? 8. A courier, who goes 31 miles every 5 hours, is sent from a certain place; when he had gone 8 hours another is sent after him, and this one in order to overtake the first must go 22 miles, d overtake the first? every 3 hours. When will the s 9. When all the conditions of the preceding problem remain the same, excepting that the first courier, besides the advantage of starting earlier, has this also, that he travelled from a place 36 miles farther on the road: in how many hours will they come together? 10. Make the problem general. Let the place from which the first starts be situated miles in advance on the road; further, let the number of hours by which he had the start of the other be equal to b; let the speed of the first be such that he goes c miles in d hours, and the speed of the second such that he goes e miles in f hours. In how many hours after the departure of the second courier will they be together? Ans. (ab+bc)f hours. de-cf 11. In how many hours will they come together, when the first courier, instead of starting from a place a miles in advance, starts from one as many miles backwards? What must be done in order to adapt the solution of the preceding problem to this case? 12. Two bodies move in opposite directions; one runs c feet in each second, the other C. The two places from which they start at the same time are distant d feet from one another. When will they meet? 13. In what time will the two bodies come together when that which goes C feet each second runs after the other? Is the problem as it here stated always possible? What is required for it to be possible? What does the expression signify when C c? What does it denote when C<c? C Equations of the Second Degree, having only one Unknown d 90. HITHERTO I have been employed upon equations of the first degree, or such as involve only the first power of the unknown quantities; but were the question proposed, To find a number, which, multiplied by five times itself, will give a product equal to 125; if we designate this number by x, five times the same will be 5 x, and we shall have 5x2125. This is an equation of the second degree, because it contains 2, or the second power of the unknown quantity. If we free this second power from its coefficient 5, we obtain |