An Introduction to Celestial Mechanics

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1914 edition. Excerpt: ...by the second of (36), and this value of t0 becomes the date of the second observation when the successive observations are equally distant from one another. With this choice of t0 the errors in X' and X" are of the second order, while X is known up to the third order. 117. The Approximations when there are Four Observations. When there are four observations the equations which correspond to the last three of (28) are f co + C1T1 + c2Tl2 + C8Tl' = Xl--C4Ti4-I, C4T24-I, c4r34 H, C3r44 +. (37) The determinant of the coefficients of c0, Ci, c2, and c3 is which is not zero since the dates of the observations are distinct. The errors of lowest order in C0, Ci, c2 and C3 are determined from (37); when only the first terms in the right members are known they contain c4 as a factor. Let these errors be represented by Ac0, ACi, Ac2, and Ac3; their orders in the T, are required. The expression for Ac0 is When the factors n, T2, T3, and n are removed from this determinant it is identical with S except the columns are permuted. Three permutations of columns bring it to the form of 5; hence (38) Ac0 = + C4T1TJT8T4. The expression for ACi is If 72 is put equal to Ti in this determinant it vanishes because then two lines become the same. Therefore it is divisible by r2--n. Similarly, it is divisible by T3--n, n--n, T3--r2, u--T2, and T4--T3; that is, it is divisible by 5. All the elements of each column are of the same degree; and since every term of the expansion of a determinant has a factor from each column, the terms of the expansion are all of the same degree. The degree of this determinant is nine, because this is the sum of the degrees of its columns. Hence ACi is of the third degree because 5 is of the sixth degree. Moreover, it is...