as the formula for interpolation, which coincides with the one obtained before, 81, 82, 83.. being the first, second, and third differences of the functions, as is evident from the manner in which they have been assumed above. Let us apply it to a table in the Nautical Almanac, which gives the moon's latitude at noon and midnight for every day in the year. EXAMPLE. Let it be required to find the moon's latitude for August 4, 1842, at 16h 18 mean time at Greenwich, that is, at 4.3 hours after midnight. m/m -40.463 minutes; d=+11′′.4, —1——0.642, 1 (−1)8,: =-1′′.31. m n Therefore, y=-0° 8′ 35′′.87, which, without the sign correct latitude south at the time for which it was required. Second differences will ordinarily insure sufficient accuracy. Third and fourth differences are rarely used. INEQUATIONS. 237. In discussing algebraical problems, it is frequently necessary to introduce inequations, that is, expressions connected by the sign >. Generally speaking, the principles already detailed for the transformation of equations are applicable to inequations also. There are, however, some important exceptions which it is necessary to notice, in order that the student may guard against falling into error in employing the sign of inequality. These exceptions will be readily understood by considering the different transformations in succession. I. If we add the same quantity to, or subtract it from, the two members of any inequation, the resulting inequation will always hold good, in the same sense as the original inequation; that is, if Thus, if So, also, if a>b, then a+a'>b+a', and a—a'>b—a'. we have still 8+5>3+5, and 8-5>3-5. -3-2, we have still -3+6<−2+6, and -3-6-2—6.† *The moon's latitude is marked + when north, when south. The negative quantity of greater numerical value is always considered less than the negative quantity of less numerical value. The truth of this proposition is evident from what has been said with reference to equations. This principle enables us, as in equations, to transpose any term from one member of an inequation to the other by changing its sign. II. If we add together the corresponding members of two or more inequations which hold good in the same sense, the resulting inequation will always hold good in the same sense as the original individual inequations; that is, if then a>b, c>d, e>f, a+c+e>b+d+f. III. But if we subtract the corresponding members of two or more inequations which hold good in the same sense, the resulting inequation WILL NOT ALWAYS hold good in the same sense as the original inequations. Take the inequations 4 <7, 2<3, we have still 4—2<7—3, or 2<4. But take 9<10 and 6<8, the result is 9-6> (not <) 10-8, or 3>2. We must, therefore, avoid as much as possible making use of a transformation of this nature, unless we can assure ourselves of the sense in which the resulting inequality will subsist. IV. If we multiply or divide the two members of an inequation by a positive quantity, the resulting inequation will hold good in the same sense as the original inequation. Thus, if This principle will enable us to clear an inequation of fractions. V. If we multiply or divide the two members of an inequation by a negative quantity, the resulting inequation will hold in a sense opposite to that of the original inequation. Thus, if we take the inequation 8>7, multiplying both members by -3, we have the opposite inequation, -24<-21. VI. We can not change the signs of both members of an inequation unless we reverse the sense of the inequation, for this transformation is manifestly the same thing as multiplying both members by -1. VII. If both members of an inequation be positive numbers, we can raise them to any power without altering the sense of the inequation; that is, if Thus, from But, a>b, then a">b". 5>3 we have (5)o>(3)2, or 25>9. (a+b)>c, we have (a+b)>c2. VIII. If both members of an inequation be not positive numbers, we can not determine, a priori, the sense in which the resulting inequation will hold good, unless the power to which they are raised be of an uneven degree. IX. We can extract any root of both members of an inequation without altering the sense of the inequation; that is, if a>b, then Va>Vb. If the root be of an even degree, both members of the inequation must necessarily be positive, otherwise we should be obliged to introduce imaginary quantities, which can not be compared with each other. EXAMPLES IN INEQUATIONS. (1) The double of a number, diminished by 6, is greater than 24; and triple the number, diminished by 6, is less than double the number increased by 10. Required a number which will fulfill the conditions. Let x represent a number fulfilling the conditions of the question; then, in the language of inequations, we have 2x-6>24, and 3r—6<2x+10. From the former of these inequations we have and from the latter we get 2x30, or x>15; 3x−2x<10+6, or x<16; therefore 15 and 16 are the limits, and any number between these limits will satisfy the conditions of the question. Thus, if we take the number 15.9, we In the second example, 12 5' or 23, is an inferior limit of the values of x. 82 In the second, -11, and, in the third, or 91, are superior limits of the 9' value of x. If the second and fourth of the above inequalities must be verified simultaneously by the values of x, these values must be comprised between 23 and 91. If the third and fourth, it is sufficient that it be less than -11. Finally, there is no value which will verify at the same time the 2o and 3°. 3x-2y>5, 5x+3y>16; (5) We can attribute to y any value whatever, and for each arbitrary value of y we can give to r all the values greater than the greatest of the two quantities 5+2y 16-3y We determine, also, from the proposed inequalities, of x there should be admitted for y but values comprised between the two limits above. GENERAL THEORY OF EQUATIONS. THE NATURE AND COMPOSITION OF EQUATIONS. 238. The valuable improvements recently made in the process for the determination of the roots of equations of all degrees, render it indispensably necessary to present to the student a view of the present state of this interesting department of analytical investigation. The beautiful theorem of M. Sturm for the complete separation of the real and imaginary roots, and for discovering their initial figures, combined with the admirable method of continuous approximation as improved by Horner, has given a fresh impulse to this branch of scientific research, entirely changed the state of the subject, and completed the theory and numerical solution of equations of all degrees. We recapitulate here two or three DEFINITIONS. 1. An equation is an algebraical expression of equality between two quantities. 2. A root of an equation is that number, or quantity, which, when substituted for the unknown quantity in the equation, verifies that equation. 3. A function of a quantity is any expression involving that quantity; thus, ar+b ar+b, ar+er+d, a are all functions of x; and also ar—by3, cx+d' 2x+3y √4x—5y, 3x-2y' y2+yx+x2+a2+b+2, are all functions of x and y. These functions are usually written f(x), and ƒ(x, y). 4. To express that two members of an equation are identical or true for every value of x, the sign is sometimes used. when divided by x-a, will leave a remainder, which is the same function of a that the given polynomial is of x. ..... Let f(x)=x"+px"¬1+qx"¬3+ ; and, dividing f(x) by x-a, let Q denote the quotient thus obtained, and R the remainder which does not involve r; hence, by the nature of division, we have f(x)±Q(x−a)+R. Now this equation must be true for every value of r, because its truth depends upon a principle of division which is independent of the particular values of the letters; hence, if x=a, we have *ƒ(a)=0+R; and, therefore, the remainder R is the same function of a that the proposed polynomial is of x. EXAMPLES. (1) What is the remainder of r2-6x+7, divided by x-2, without actually performing the operation? The student will recollect that f(r) stands for "+px"1+, &c., and that, therefore, (a) will stand for a"+pa"-1+qa"-2+, &c. |