McMri> /f>v T (fnrttell Itniueraitg ffithraqj Strata. Hew fork BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE 1891 MATHEMATICS Cornell University Library QA 403.T94 Tables for facilitating the use of harmo 3 1924 001 121 874 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001121874 TABLES FOR FACILITATING THE USE OF HARMONIC ANALYSIS 2— AS ARRANGED BY H. H. TURNER, D.Sc, F.R.S. SAVILIAN PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF OXFORD Bonbon HUMPHREY MILFORD OXFORD UNIVERSITY PRESS, AMEN CORNER, E.C. I9 X 3 ty*\ b%%% OXFORD : HORACE HART PRINTER TO THE UNIVERSITY TABLES FOR FACILITATING HARMONIC ANALYSIS If we arrange a set of numbers a i> a 2> a i • • • a n> a n+\ • • ■ ■ representing the values of a quantity at equal intervals of time (e. g. the total rainfalls in consecutive months) in series of n terms (either singly or as means of several series), then to find the expression s sin 6 + c cos 8 representing them most closely, where nd = 2-rr, we form S = o + a 2 sin 8 + a. A sin 28 + ... + a n sin (n — 1) 6 C = a Y + a 2 cos# + a. A cos 26 + . . . + a„cos (n — 1) 6 and then s'= 2S/n, c = 2C/n The following tables are arranged to allow the products a r+1 sin v6 and a r+1 cos r6 to be taken out at sight, to two figures, which suffice for much of the exploring work represented by Professor Schuster's periodogram method. Explanation is scarcely necessary. On the left is given the value of a r+] : and on the same line, in the column headed r + 1, is given the product a r+1 sin r 6 or a r+1 cos r 6, as the case may be. If sin r 6 or cos r 6 is negative this is indicated by attaching the minus sign to r + 1 in the heading (or at the foot). Note that, to keep corresponding tables for 11 terms at the same opening, the short tables for 10 and 12 terms are put together. An example of the use of the tables is given at the end. 9 TERMS — SINE 6 1 is zero +2 + 3 + 4 + 5 + 2 + 3 + 4 + 5 Nos. •643 ■985 •866 •342 Nos. •643 •985 ■866 •342 1 1 1 1 51 33 5° 44 17 2 1 2 2 1 52 33 5i 45 18 3 2 3 3 1 53 34 52 146 18 4 3 4 3 1 54 35 53 47 18 5 3 5 4 2 55 35 54 48 19 6 4 6 5 2 56 36 55 48 19 7 5 7 6 2 57 37 56 49 19 8 5 8 7 3 58 37 57 50 20 9 6 9 8 3 59 38 58 51 20 10 6 10 9 3 60 39 59 52 21 11 7 11 10 4 61 39 60 53 21 12 S 12 10 4 62 40 61 54 21 13 8 13 11 4 63 41 62 55 22 14 9 14 12 5 64 41 63 55 22 15 10 15 13 5 65 42 64 56 22 16 10 16 14 5 66 42 65 57 23 17 11 17 15 6 67 43 66 58 23 18 12 18 16 6 68 44 67 59 23 19 12 19 16 6 69 44 68 60 24 20 13 20 17 7 70 45 69 61 24 21 14 21 18 7 71 46 70 61 24 22 14 22 19 8 72 46 71 62 25 23 IS 23 20 8 73 47 72 63 25 24 IS 24 21 8 74 48 73 64 25 25 16 25 22 9 75 48 74 65 26 26 17 26 23 9 76 49 75 66 26 27 17 27 23 9 77 5° 76 67 26 28 18 28 24 10 78 SO 77 68 27 29 19 29 25 10 79 51 78 68 27 30 19 3° 26 10 80 51 79 69 27 31 20 31 27 11 81 52 80 70 28 32 21 32 28 11 82 S3 81 7i 28 33 21 33 29 11 83 53 82 72 28 34 22 33 29 12 84 54 83 73 29 35 23 34 3° 12 85 55 84 74 29 36 23 35 3i 12 86 55 85 74 29 37 24 36 32 13 87 56 86 75 3° 38 24 37 33 13 88 S7 87 76 30 39 25 38 34 13 89 57 88 77 30 40 26 39 35 14 90 58 89 78 3i 41 26 40 36 14 91 59 90 79 3i 42 27 41 36 14 92 59 91 80 31 43 28 42 37 15 93 60 92 81 32 44 28 43 38 15 94 60 93 81 32 45 29 44 39 15 95 61 94 82 32 46 30 45 40 16 96 62 95 83 33 47 30 46 4i 16 97 62 96 84 33 48 31 47 42 16 98 63 97 85 34 49 32 48 42 17 99 64 98 86 34 50 31 49 43 — 7 17 100 . 64 99 87 34 1- — 8 — 6 — 9 — 8 — 7 — 6 9 TERMS — COS 6 1 is 1000 + 2 + 3 — 4 — 5 + 2 + 3 — 4 — 5 Nos. .766 •174 •500 ■940 Nos. ■766 •174 •500 •940 1 1 1 1 51 39 9 26 48 2 2 1 2 52 40 9 26 49 3 2 2 3 53 4i 9 27 50 4 3 2 4 54 41 9 27 5i 5 4 3 5 55 42 10 28 52 6 5 3 6 56 43 10 28 S3 7 5 4 7 57 44 10 29 54 8 6 4 8 58 44 10 29 55 9 7 2 5 8 59 45 10 3° 55 10 8 2 5 9 60 46 10 3° 56 11 8 2 6 10 61 47 11 31 57 12 9 2 6 11 62 47 11 3i 58 13 10 2 7 12 63 48 11 32 59 14 11 2 7 13 64 49 11 32 60 15 11 3 8 14 65 5° 11 33 61 16 12 3 8 15 66 5i 11 33 62 17 13 3 9 16 67 51 12 34 63 18 14 3 9 17 68 52 12 34 64 19 15 3 10 18 69 53 12 35 65 20 15 3 10 19 70 54 12 35 66 21 16 4 11 20 71 54 12 36 67 22 17 4 11 21 72 55 13 36 68 23 18 4 12 22 73 56 13 37 69 24 18 4 12 23 74 57 13 37 70 25 19 4 13 24 75 57 13 38 71 26 20 5 13 24 76 58 13 38 71 27 21 5 14 25 77 59 13 39 72 28 21 5 14 26 78 60 14 39 73 29 22 5 IS 27 79 61 H 40 74 30 23 S 15 28 80 61 14 40 75 31 24 S 16 29 81 62 14 41 76 32 25 6 16 3° 82 63 14 41 77 33 25 6 17 31 83 64 14 42 78 34 26 6 17 32 84 64 15 42 79 35 27 6 18 33 85 65 15 43 80 36 28 6 18 34 86 66 15 43 81 37 28 6 19 35 87 67 15 44 82 38 29 7 19 36 88 6 7 15 44 83 39 30 7 20 37 89 68 15 45 84 40 3i 7 20 38 90 69 16 45 85 41 3i 7 21 39 91 70 16 46 86 42 32 7 21 39 92 70 16 46 86 43 33 7 22 40 93 71 16 47 87 44 34 8 22 4i 94 72 16 47 88 45 34 8 23 42 95 73 17 48 89 46 35 8 23 43 96 74 17 48 90 47 36 8 24 44 97 74 17 49 91 48 37 8 24 45 98 75 17 49 92 49 38 9 25 46 99 76 17 50 93 50 38 9 25 47 100 77 17 50 94 + 9 + 8 — 7 — 6 + 9 + 8 — 7 — 6 10 TER MS — SINES AND COSINES 1 is zero 1 is 1-000 + 6 is zero 6 is 1000 - Sine. Sine. Cos. Cos. + 2 & 5 3&4 + 2& 5 3&4 + 2&10 3& 9 + 2&10 3&9 Nos. ■588 •951 Nos. ■588 •951 Nos. •809 ■309 Nos. •809 ■309 1 1 1 51 i 3° 49 1 1 51 41 16 2 1 2 52 3i 49 2 2 1 52 42 16 3 2 3 53 31 50 3 2 1 53 43 ! 16 4 2 4 54 32 51 4 3 1 54 44 ' 17 5 3 5 55 32 52 5 4 2 55 44 .' 17 6 4 6 56 33 53 6 5 2 56 45 17 7 4 7 57 34 54 7 6 2 57 46 18 8 5 8 58 34 55 8 6 2 58 47 18 9 5 9 59 35 56 9 7 3 59 48 18 10 6 10 60 35 57 10 8 3 60 49 19 11 6 11 61 36 58 11 9 3 61 49 19 12 7 11 62 36 59 12 10 4 62 50 19 13 8 12 63 37 60 13 11 4 63 51 J9 14 8 13 64 38 61 14 11 4 64 52 20 15 9 14 65 38 62 15 12 5 65 53 20 16 9 15 66 39 63 16 13 5 66 53 ' 20 17 10 16 67 39 64 17 14 5 67 54 21 18 11 17 68 40 65 18 15 6 68 55 21 19 11 18 69 4i 66 19 15 6 69 56 21 20 12 19 70 41 6 7 20 16 6 70 57 22 21 12 20 71 42 68 21 17 6 71 57 22 22 13 21 72 42 68 22 18 7 72 58 22 23 14 22 73 43 69 23 19 7 73 59 23 24 14 23 74 44 70 24 19 7 74 60 23 25 15 24 75 44 71 25 20 8 75 61 23 26 15 25 76 45 72 26 21 8 76 61 23 27 16 26 77 45 73 27 22 8 77 62 24 28 16 27 78 46 74 28 23 9 78 63 24 29 17 28 79 46 75 29 23 9 79 64 24 30 18 29 80 47 76 30 24 9 80 65 25 31 18 29 81 48 77 31 25 10 81 66 25 32 19 30 82 48 78 32 26 10 82 66 25 33 19 31 83 49 79 33 27 10 83 67 26 34 20 32 84 49 80 34 28 11 84 68 26 35 21 33 85 50 81 35 28 11 85 69 26 36 21 34 86 51 82 36 29 11 86 70 27 37 22 35 87 5i 83 37 30 11 87 70 27 38 22 36 88 52 84 38 3i 12 88 71 27 ' 39 23 37 89 52 85 j 39 32 12 89 72 28 40 24 33 90 53 86 40 32 12 90 73 28 41 24 39 91 54 87 41 33 13 91 74 28 42 25 40 92 54 87 42 34 13 92 74 28 43 25 41 93 55 88 43 35 13 93 75 29 44 26 42 94 55 89 44 36 14 94 76 29 45 26 43 95 56 90 45 36 14 95 77 29 46 27 44 96 56 9i 46 37 14 96 78 30 47 28 45 97 57 92 47 38 IS 97 78 3° 48 28 46 98 58 93 48 39 15 98 79 30 49 29 47 99 58 94 49 40 *s 99 80 31 50 29 48 100 59 95 50 40 15 100 81 3i 7&10 8& 9 7&10 8&9 — 5&7 4&8 — 5&7 4&8 12 TERMS 1 and 1 are zero 4 and 10 are 1-000 SINES AND COSINES 1 and 7 are 1-000 4 a«,cZ 10 are zero Sine. Sine. Cos. Cos. + 2& 6 3 & 5 + 2 & 6 •500 3&5 + 2&12 3&11 + 2&12 3&11 Nos. •500 ■866 Nos. ■866 Nos. •866 ■500 Nos. •866 •500 1 1 1 51 26 44 1 1 1 51 44 26 2 1 2 52 26 45 2 2 1 52 45 26 : 3 2 3 53 ^7 46 3 3 2 53 46 27 4 2 3 54 27 47 4 3 2 54 47 27 5 3 4 55 28 48 5 4 3 55 48 28 6 3 5 56 28 48 6 5 3 56 48 28 7 4 6 57 29 49 7 6 4 57 49 29 S 4 7 58 29 50 8 7 4 58 5° 29 9 5 8 59 30 5i 9 8 5 59 5i 30 10 5 9 60 30 52 10 9 5 60 52 30 11 6 10 61 31 53 11 10 6 61 53 3i 12 6 10 62 31 54 12 10 6 62 54 31 13 7 11 63 32 55 13 11 7 63 55 32 14 7 12 64 32 55 14 12 7 64 55 32 15 8 13 65 33 56 15 13 8 65 56 33 16 8 14 66 33 57 16 14 8 66 57 33 17 9 15 67 34 58 17 15 9 67 58 34 18 9 16 68 34 59 18 16 9 68 59 34 19 10 16 69 35 60 19 16 10 69 60 35 20 10 17 70 35 61 20 17 10 70 61 35 21 11 18 71 36 61 21 18 11 71 61 36 22 11 19 72 36 62 22 19 11 72 62 36 23 12 20 73 37 63 23 20 12 73 63 37 24 12 21 74 37 . 64 24 21 12 74 64 37 25 13 22 75 38 65 25 22 13 75 65 38 26 13 23 76 38 66 26 23 13 76 66 38 27 14 23 77 39 67 27 23 14 77 67 39 28 14 24 78 , 39 68 28 24 H 78 68 39 29 IS 25 79 40 68 29 25 15 79 68 40 30 15 26 80 40 69 30 26 15 80 69 40 31 16 27 81 41 70 31 27 16 81 70 41 32 16 28 82 41 71 32 28 16 82 71 4i 33 17 29 83 42 72 33 29 17 83 72 42 34 17 29 84 42 73 34 29 17 84 73 42 35 18 30 85 43 74- 35 30 18 85 74 43 36 18 31 86 43 74 36 3i 18 86 74 43 37 19 32 87 44 75 37 32 19 87 75 44 38 19 33 88 44 76 38 33 19 88 76 44 39 20 34 89 45 77 39 34 20 89 77 45 40 20 35 90 45 78 40 35 20 90 78 45 41 21 36 91 46 79 41 36 21 91 79 46 42 21 36 92 46 80 42 36 21 92 80 46 43 22 37 93 47 81 43 37 22 93 81 47 44 22 38 94 47 81 44 38 22 94 81 47 45 23 39 95 48 82 45 39 23 95 82 48 46 23 40 96 48 83 46 40 23 96 83 48 47 24 4i 97 49 84 47 41 24 97 84 49 48 24 42 98 49 85 48 42 24 98 85 49 49 25 42 99 5° 86 49 42 25 99 86 50 50 25 43 100 50 87 50 43 25 100 87 5° , — 8&12 9&11 — 8&12 9&11 — 6&8 5&9 — 6&8 5&9 * - 9 11 TERMS — SINE 6 1 is zero + Nos. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 •541 i i 2 2 3 3 4 4 5 5 6 6 7 9 9 io io ii ii 12 12 13 14 14 15 15 16 16 17 17 18 18 19 19 20 21 21 22 22 23 23 24 24 25 25 26 27 27 11 •910 1 2 3 4 5 5 6 7 8 9 10 11 12 13 14 15 15 16 17 18 19 20 21 22 23 24 25 25 26 27 28 29 30 31 32 33 34 35 35 36 37 38 39 40 41 42 43 44 45 46 10 • 99 o 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3° 31 32 33 34 35 36 37 33 39 40 41 42 43 44 45 46 47 48 49 5° •756 1 2 2 3 4 5 5 6 7 9 10 11 11 12 13 14 14 15 16 17 17 18 19 20 20 21 22 23 23 24 25 26 26 27 28 29 29 3° 3i 32 33 33 34 35 36 36 37 38 o 1 1 1 1 2 2 2 3 3 3 3 4 4 4 5 5 5 5 9 IO IO IO IO II II II 12 12 12 12 13 13 13 14 14 14 + Nos. 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 2 •54i 28 28 29 29 3° 30 31 31 32 32 33 34 34 35 35 36 36 37 37 38 38 39 39 40 41 41 42 42 43 43 44 44 45 45 46 47 47 48 48 49 49 50 5° 5i 5i 52 52 53 54 54 11 ■910 46 47 48 49 50 5i 52 53 54 SS 56 56 57 58 59 60 61 62 63 64 65 66 66 67 68 69 70 7i 72 73 74 75 76 76 77 78 79 80 Si 82 83 84 85 86 86 87 90 91 10 ■990 50 5i 52 S3 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 37 88 89 90 91 92 93 94 95 96 97 98 756 39 39 40 41 42 42 43 44 45 45 46 47 48 48 49 5° 51 51 52 53 54 54 55 56 57 57 58 59 60 60 61 62 63 64 64 65 66 67 67 68 69 70 70 71 72 73 73 74 75 7 6 10 11 TERMS — COS 6 1 . is L-000 + 2 + 3 -4 -5 -6 + 2 + 3 -4 — 5 -6 Nos. •841 ■416 ■142 •655 •960 Nos. •841 •416 •142 •655 •960 1 1 1 1 51 43 21 7 33 49 2 2 1 1 2 52 44 22 7 34 50 3 3 1 2 3 53 45 22 8 35 51 4 3 2 3 4 54 45 22 8 35 52 5 4 2 3 5 55 46 23 8 36 S3 6 5 2 4 6 56 47 23 8 37 54 7 6 3 5 7 57 48 24 8 37 55 8 7 3 S 8 58 49 24 8 38 56 9 8 4 6 9 59 50 25 8 39 57 10 8 4 7 10 60 50 25 9 39 58 11 9 5 2 7 11 61 5i 25 9 40 59 12 10 5 2 8 12 62 52 26 9 4i 60 13 11 5 2 9 12 63 53 26 9 4i 60 14 12 6 2 9 13 64 54 27 9 42 61 15 13 6 2 10 14 65 55 27 9 43 62 16 13 7 2 10 15 66 56 27 9 43 63 17 H 7 2 11 16 67 56 28 10 44 64 18 15 7 3 12 17 68 57 28 10 45 65 19 16 8 3 12 18 69 58 29 10 45 66 20 17 8 3 13 19 70 59 29 10 46 67 21 18 9 3 14 20 71 60 30 10 47 68 22 19 9 3 14 21 72 61 3° 10 47 69 23 19 10 3 15 22 73 61 30 10 48 70 24 20 10 3 16 23 74 62 3i 11 48 71 25 21 10 4 16 24 75 63 3i 11 49 72 26 22 11 4 17 25 76 64 32 11 50 73 27 23 11 4 18 26 77 65 32 11 So 74 28 24 12 4 18 27 78 66 32 11 Si 75 29 24 12 4 19 28 79 66 33 11 52 76 30 25 12 4 20 29 80 67 33 11 52 77 31 26 13 4 20 30 81 68 34 12 53 78 32 27 13 5 21 31 82 69 34 12 54 79 33 28 14 5 22 32 83 70 35 12 54 80 34 29 14 5 22 33 84 71 35 12 55 81 35 29 15 5 23 34 85 71 35 12 56 82 36 30 15 5 24 35 86 72 36 12 56 83 37 3i 15 5 24 36 87 73 36 12 57 84 38 32 16 5 25 36 88 74 37 12 58 84 39 33 16 6 26 37 89 75 37 13 58 85 40 34 17 6 26 38 90 76 37 13 59 86 41 34 17 6 27 39 91 77 38 13 60 87 42 35 17 6 28 40 92 77 38 13 60 88 43 36 18 6 28 41 93 78 39 13 61 89 44 37 18 6 29 42 94 79 39 13 62 90 45 38 19 6 29 43 95 80 40 13 62 91 46 39 19 7 30 44 96 81 40 14 63 92 47 40 20 7 31 45 97 82 40 14 64 93 48 40 20 7 3i 46 98 82 4i 14 64 94 49 4i 20 7 32 47 99 83 4i 14 65 95 50 42 21 7 33 48 100 84 42 14 66 96 + 11 + 10 -9 -8 — 7 + 11 + 10 — 9 — 8 -7 11 13 TERMS — SINE 6 1 is zero + 2 3 4 5 6 7 + 2 3 4 5 6 '7 ,' Nos. •465 •823 •993 •935 •663 •239 Nos. ■465 ■823 •993 ■935 ■663 •239 1 1 1 1 1 51 24 42 5i 48 34 12 j 2 1 2 2 2 1 52 24 43 52 49 34 12 3 1 2 3 3 2 1 53 25 44 53 50 35 13 4 2 3 4 4 3 1 54 25 44 54 50 36 13 5 2 4 5 5 3 1 55 ' 26 45 . 55 51 36 13 6 3 5 6 6 4 1 56 26 46 56 52 37 13 7 3 6 7 7 5 2 57 27 47 57 53 38 14 8 4 7 8 7 5 2 58 27 48 58 54 38 14 9 4 7 9 8 6 2 59 27 49 59 55 39 14 10 5 8 10 9 7 2 60 28 49 60 56 40 14 11 5 9 11 10 7 3 61 28 5° 61 57 40 15 12 6 10 12 11 8 3 62 29 5i 62 58 4i 15 13 6 11 13 12 9 3 63 29 52 63 59 42 15 14 7 12 14 13 9 3 64 30 S3 64 60 42 15 • 15 7 12 15 14 10 4 65 30 53 65 61 43 16 16 7 13 16 15 11 4 66 31 54 66 62 44 16 17 8 14 17 16 11 4 67 3i 55 67 63 44 16 18 8 IS 18 17 12 4 68 32 56 68 64 45 16 : 19 9 16 19 18 13 5 69 32 57 69 65 46 16 20 9 16 20 19 13 5 70 33 58 70 65 46 17 - 21 10 17 21 20 14 5 71 33 58 71 66 47 17 22 10 18 22 21 15 5 72 33 59 71 67 48 17 23 11 19 23 22 15 5 73 34 60 72 68 48 17 24 11 20 24 22 16 6 74 34 61 73 69 49 18 25 12 21 25 23 17 6 75 35 62 74 70 50 18 . 26 12 21 26 24 17 6 76 35 63 75 71 5° 18 27 13 22 27 25 18 6 77 36 63 76 72 5i 18 28 13 23 28 26 19 7 78 36 64 77 73 52 19 29 13 24 29 27 19 7 79 37 65 78 74 52 19 ! 30 14 25 3° 28 20 7 80 37 66 79 75 53 19 ■ 31 14 26 3i 29 21 7 81 38 67 80 76 54 19 32 15 26 32 30 21 8 82 38 67 81 77 54 20 33 IS 27 33 31 22 8 83 39 68 82 78 55 20 ' 34 16 28 34 32 23 ! 8 84 39 69 83 79 56 20 35 16 29 35 33 23 8 85 40 70 84 ■ 79 56 20 : 36 17 30 36 34 24 9 86 40 71 85 80 57 21 37 17 30 37 35 25 9 87 40 72 86 81 58 21 38 18 3i 38 36 25 9 88 41 72 87 82 58 21 39 18 32 39 36 26 9 89 4i 73 88 83 59 21 40 19 33 40 37 27 10 90 42 74 89 84 60 22 41 19 34 4i 38 27 10 91 42 75 90 85 60 22 42 20 35 42 39 28 10 92 43 76 91 86 61 22 43 20 35 43 40 29 10 93 43 77 92 87 62 22 44 20 36 44 41 29 11 94 44 77 93 88 62 22 45 21 37 45 42- 30 11 95 44 78 94 89 63 23 46 21 38 46 43 30 11 96 45 79 95 90 64 23 47 22 39 47 44 3i 11 97 45 80 96 91 64 23 48 22 40 48 45 32 11 98 46 81 97 92 65 23 49 23 40 49 46 32 12 99 46 81 98 93 66 24 50 23 41 50 47 33 12 100 46 82 99 94 66 24 — 13 12 11 10 9 8 — 13 12 11 10 9 8 12 13 TERMS — COS 6 1 is 1-000 + 2 + 3 + 4 — 5 — 6 — 7 + 2 + 3 + 4 — 5 — 6 -7 Nos. •886 •568 •120 •355 748 ■97i Nos. •886 •568 •120 ■355 •748 •971 1 i I 1 1 51 45 29 6 18 38 5° 2 2 I 1 1 2 52 46 3° 6 18 39 5° 3 3 2 1 2 3 53 47 3° 6 19 40 5i 4 4 2 1 3 4 54 48 3i 6 19 40 52 5 4 3 2 4 5 55 49 3i 7 20 41 53 6 5 3 2 4 6 56 5° 32 7 20 42 54 7 6 4 2 ' 5 7 57 5i 32 7 20 43 55 8 7 5 3 6 8 58 5i 33 7 21 43 56 9 8 5 3 7 9 59 52 34 7 21 44 57 10 9 6 4 7 10 60 53 34 7 21 45 58 11 IO 6 4 8 11 61 54 35 7 22 46 59 12 ii 7 4 9 12 62 55 35 7 22 46 60 13 12 7 z. 5 10 13 63 56 36 8 22 47 61 14 12 8 4 5 10 14 64 57 36 8 23 48 62 15 13 9 2 5 11 15 65 58 37 8 23 49 63 16 14 9 2 6 12 16 66 58 37 8 23 49 64 17 IS 10 2 6 13 17 67 59 38 8 24 50 65 18 16 10 2 6 13 17 68 60 39 8 24 5i 66 19 17 11 2 7 14 18 69 61 39 8 24 52 67 20 18 11 2 7 15 19 70 62 40 8 25 52 68 21 19 12 3 7 16 20 71 63 40 9 25 53 69 22 19 12 3 8 16 21 72 64 41 9 26 54 70 23 20 13 3 8 17 22 73 65 41 9 26 55 71 24 21 H 3 9 18 23 74 66 42 9 26 55 72 25 22 14 3 9 19 24 75 66 43 9 27 56 73 26 23 15 3 9 19 25 76 67 43 9 27 57 74 27 24 15 3 10 20 26 77 68 44 9 27 58 75 28 25 16 3 10 21 27 78 69 44 9 28 58 76 29 26 16 3 10 22 28 79 70 45 9 28 59 77 30 27 17 4 11 22 29 80 71 45 IO 28 60 78 31 28 18 4 11 23 30 81 72 46 IO 29 61 79 32 28 18 4 11 24 3i 82 73 47 IO 29 61 80 33 29 19 4 12 25 32 83 74 47 IO 29 62 81 34 30 19 4 12 25 33 84 74 48 IO 30 63 82 35 31 20 4 12 26 34 85 75 48 IO 30 64 83 36 32 20 4 13 27 35 86 76 49 IO 3i 64 84 37 33 21 4 13 28 36 87 77 49 IO 3i 65 84 38 34 22 5 13 28 37 88 78 SO II 3i 66 85 39 35 22 5 14 29 38 89 79 51 II 32 67 86 40 35 23 5 14 30 39 90 80 5i II 32 67 87 41 36 23 5 15 3i 40 91 81 52 II 32 68 88 42 37 24 5 15 31 4i 92 82 52 II 33 69 89 43 38 24 5 15 32 42 93 82 53 II 33 70 90 44 39 25 5 16 33 43 94 83 53 II 33 70 91 45 40 26 5 16 34 44 95 84 54 II 34 7i 92 46 41 26 6 16 34 45 96 85 55 12 34 72 93 47 42 27 6 17 35 46 97 86 55 12 34 73 94 48 43 27 6 17 36 47 98 87 56 12 35 73 95 49 43 28 6 17 37 48 99 88 56 12 35 74 96 50 44 28 6 18 37 49 100 89 57 12 36 75 97 1 + 13 + 12 + 11 — 10 — 9 — 8 1 + 13 + 12 + 11 — 10 — 9 — 8 13 14 TERMS — SINE 6 1 and 8 are zero + Nos. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2 & 7 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 •434 o i i 2 2 3 3 3 4 4 5 S 6 6 7 7 7 20 9 21 9 22 IO 23 10 24 IO 25 II 26 II 27 12 28 12 29 13 3 & 6 13 13 14 14 IS 15 16 16 16 17 17 18 18 19 19 20 20 20 21 21 22 •782 I 2 2 3 4 5 5 6 7 8 9 9 10 11 12 13 13 14 15 16 16 17 18 19 20 20 21 22 23 23 24 25 26 27 27 28 29 3° 30 3i 32 33 34 34 35 36 37 38 38 39 4 & 5 •975 9&14 10&13 11&12 10 11 12 13 14 15 16 17 18 19 20 20 21 22 23 24 25 26 27 28 29 3° 3 1 3 2 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 + Nos. 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 2 & 7 •434 22 23 23 23 24 24 25 25 26 26 26 27 27 28 28 29 29 30 30 3° 31 31 32 32 33 33 33 34 34 35 35 36 36 36 37 37 38 38 39 39 39 40 40 41 4i 42 42 43 43 43 3 & 6 ■782 40 41 41 42 43 44 45 45 46 47 48 48 49 50 5i 52 52 S3 54 55 56 56 57 58 59 59 60 61 62 63 63 64 65 66 66 67 68 69 70 70 71 72 73 74 74 75 76 77 77 78 4 & 5 •975 9&14 10&13 50 51 52 53 54' 55 56 57 58 59 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 90 91 92 93 94 95 96 97 11&12 14 1 is + 1-000 14 TERMS — COS 8 is - 1-000 + 2&14 3&13 4&12 + 2&14 3&13 4&12 Nos. •901 •623 •223 Nos. ■901 •623 •223 1 1 1 51 46 32 11 2 2 1 52 47 32 12 3 3 2 1 53 48 33 12 4 4 2 1 54 49 34 12 5 5 3 1 55 50 34 12 6 5 4 1 56 50 35 12 7 6 4 2 57 51 36 13 8 7 5 2 58 52 36 13 9 8 6 2 59 53 37 13 10 9 6 2 60 54 37 13 11 10 7 2 61 55 38 14 12 II 7 3 62 56 39 14 13 12 8 3 63 57 39 14 14 13 9 3 64 58 40 H 15 14 9 3 65 59 40 14 16 14 10 4 66 59 41 i5 17 15 11 4 67 60 42 *5 IS 16 11 4 68 61 42 15 19 17 12 4 69 62 43 15 20 18 12 4 70 63 44 16 21 19 13 5 71 64 44 16 22 20 14 5 72 65 45 16 23 21 14 5 73 66 45 16 24 22 IS 5 74 67 46 17 25 23 16 6 75 68 47 17 26 23 16 6 76 68 47 17 27 24 17 6 77 69 48 17 : 28 25 17 6 78 70 49 17 29 26 18 6 79 71 49 18 30 27 19 7 80 72 50 18 31 28 19 7 81 73 50 18 32 29 20 7 82 74 5i 18 33 30 21 7 83 75 52 19 34 31 21 8 84 76 52 19 35 3 2 22 8 85 77 53 19 36 3 2 22 8 86 77 54 19 37 33 23 8 87 78 54 19 38 34 24 8 88 79 55 20 39 35 24 9 89 80 55 20 40 36 25 9 90 81 56 20 \ 41 37 26 9 91 82 57 20 42 38 26 9 92 83 57 21 43 39 27 10 93 84 58 21 44 40 27 10 94 85 59 21 45 4 1 28 10 95 86 59 21 46 41 29 10 96 86 60 21 47 42 29 10 97 98 87 88 60 61 22 22 48 43 30 1 1 49 44 3 1 11 99 89 62 22 50 45 31 11 100 90 62 22 — 7 & 9 6&10 5&11 - 7 & 9 6&10 5&11 15 15 TERMS — SINE 6 1 is zero + 2 3 4 5 6 7 8 + 2 3 4 5 6 7 8 j Nos. ■407 ■743 ■951 ■995 •866 ■588 •208 Nos. ■407 743 ■951 ■995 •866 ■588 •208 '■ 1 1 1 1 1 1 51 21 38 49 5i 44 3° 11 2 1 1 j. 2 2 1 52 21 39 49 52 45 31 11 3 1 2 3 3 3 2 53 22 39 5o 53 46 3i 11 4 2 3 4 4 3 2 54 22 40 51 54 47 32 11 5 2 4 5 5 4 3 55 22 41 52 55 48 32 11 6 2 4 6 6 5 4 56 23 42 53 56 48 33 12 7 3 5 7 7 6 4 57 23 42 54 57 49 34 12 8 3 6 8 8 7 5 2 58 24 43 55 58 5° 34 12 9 4 7 9 9 8 5 2 59 24 44 56 59 5i 35 12 10 4 7 10 10 9 6 2 60 24 45 57 60 52 35 12 11 4 8 10 11 10 6 2 61 25 45 58 61 53 36 13 12 5 9 11 12 10 7 2 62 25 46 59 62 54 36 13 13 5 10 12 13 11 8 3 63 26 47 60 63 55 37 13 14 6 10 13 14 12 8 3 64 26 48 61 64 55 38 13 15 6 11 14 15 13 9 3 65 26 48 62 65 56 38 14 16 7 12 15 16 14 9 3 66 27 49 63 66 57 39 H 17 7 13 16 17 15 10 4 67 27 5o 64 67 58 39 14 18 7 13 17 18 16 11 4 68 28 5i 65 68 59 40 14 19 8 14 18 19 16 11 4 69 28 5i 66 69 60 41 14 20 8 15 19 20 17 12 4 70 28 52 67 70 61 41 15 21 9 16 20 21 18 12 4 71 29 53 68 7i 61 42 15 22 9 16 21 22 19 13 S 72 29 53 68 72 62 42 15 23 9 17 22 23 20 14 5 73 3° 54 69 73 63 43 15 24 10 18 23 24 21 14 5 74 30 55 70 74 64 44 15 25 10 19 24 25 22 15 5 75 31 56 71 75 65 44 16 26 11 19 25 26 23 15 5 76 3i 56 72 76 66 45 16 27 11 20 26 27 23 16 6 77 3i 57 73 77 67 45 16 28 11 21 27 28 24 16 6 78 32 58 74 78 68 46 16 29 12 22 28 29 25 17 6 79 32 59 75 79 68 46 16 30 12 22 29 30 26 18 6 80 33 59 76 80 69 47 17 31 13 23 29 3i 27 18 6 81 33 60 77 81 70 48 17 32 13 24 30 32 28 19 7 82 33 61 78 82 7i 48 17 33 13 2 5 31 33 29 19 7 83 34 62 79 83 72 49 17 ' 34 14 25 32 34 29 20 7 84 34 62 80 84 73 49 17 35 14 26 33 35 3° 21 7 85 35 63 81 85 74 5° 18 36 15 27 34 36 31 21 7 86 35 64 82 86 74 51 18 37 15 27 35 37 32 22 8 87 35 65 83 87 75 51 18 38 IS 28 36 38 33 22 8 88 36 65 84 88 76 52 18 39 16 29 37 39 34 23 8 89 3<5 66 85 89 77 52 19 40 16 3° 38 40 35 24 8 90 3-7 67 86 90 78 53 19 41 17 30 39 41 36 24 9 91 37 68 87 91 79 54 19 42 17 31 40 42 36 25 9 92 37 68 87 92 80 54 19 43 18 32 41 43 37 25 9 93 38 69 88 93 81 55 19 44 18 33 42 44 38 26 9 94 38 70 89 94 81 55 20 45 18 33 43 45 39 26 9 95 39 71 90 95 82 56 20 46 19 34 44 46 40 27 ' 10 96 39 7i 91 96 83 56 20 47 19 35 45 47 41 28 10 97 39 72 92 97 84 57 20 48 20 36 46 48 42 28 10 98 40 73 93 ; 98 85 58 20 49 io 36 47 49 42 29 10 99 40 74 94 j 99 86 58 21 50 20 37 48 50 43 29 10 100 4i 74 95 : 100 87 59 10 21 ~ 15 14 13 12 11 10 9 | - 15 14 13 12 11 9 16 15 TERMS— COS 6 1 is + 1-000 + 2 + 3 + 4 — 5 — 6 — 7 — 8 + 2 + 3 + 4 -5 — 6 — 7 —8 Nos. •914 •669 ■309 ■los •500 •809 ■978 Nos. •914 •669 •309 ■105 •500 •809 •978 1 1 1 1 1 1 51 47 34 16 5 26 4i 5° 2 2 1 1 1 2 2 52 48 35 16 5 26 42 51 3 3 2 1 2 1 3 53 48 35 16 6 27 43 52 4 4 3 1 2 3 4 54 49 36 17 6 27 44 53 5 5 3 2 3 4 5 55 50 37 17 6 28 44 54 6 5 4 2 3 5 6 56 5i 37 17 6 28 45 SS 7 6 S 2 4 6 7 57 52 38 18 6 29 46 56 8 7 5 2 4 6 8 58 53 39 18 6 29 47 57 9 8 6 3 5 7 9 59 54 39 18 6 3° 48 58 10 9 7 3 5 8 10 60 55 40 19 6 30 49 59 11 10 7 3 6 9 11 61 56 41 19 6 31 49 66 12 11 8 4 6 10 12 62 57 41 19 7 31 50 6i 13 12 9 4 7 11 13 63 58 42 19 7 3 2 51 62 14 13 9 4 7 11 14 64 58 43 20 7 32 52 63 15 14 10 S 2 8 12 15 65 59 43 20 7 33 53 64 16 15 11 5 2 8 13 16 66 60 44 20 7 33 53 65 17 16 11 S 2 9 14 17 67 61 45 21 7 34 54 66 18 16 12 6 2 9 IS 18 68 62 45 21 7 34 55 67 19 17 13 6 2 10 IS 19 69 63 46 21 7 35 56 67 20 IS 13 6 2 10 16 20 70 64 47 22 7 35 57 68 21 19 14 6 2 11 17 21 71 65 47 22 7 36 57 69 22 20 15 7 2 11 18 22 72 66 48 22 8 36 58 70 23 21 15 7 3 12 19 22 73 67 49 23 8 37 59 71 24 22 16 7 3 12 19 23 74 68 SO 23 8 37 Co 72 25 23 17 8 3 13 20 24 75 69 50 23 8 38 61 73 26 24 17 S 3 13 21 25 76 69 51 23 8 38 61 74 27 25 18 8 3 14 22 26 77 70 52 24 8 39 62 75 28 26 19 9 3 14 23 27 78 7 1 52 24 8 39 63 76 29 27 19 9 3 15 23 28 79 72 53 24 8 40 64 77 30 27 20 9 3 IS 24 29 80 73 54 25 8 40 65 7 8 31 28 21 10 3 16 25 30 81 74 54 25 9 41 66 79 32 29 21 10 3 16 26 3i 82 75 55 25 9 41 66 80 33 30 22 10 3 17 27 32 83 76 56 26 9 42 67 81 34 31 23 11 4 17 28 33 84 77 56 26 9 42 68 82 35 32 23 11 4 18 28 34 85 78 57 26 9 43 69 83 36 33 24 11 4 18 29 35 86 79 58 27 9 43 70 84 37 34 25 11 4 19 30 36 87 80 58 27 9 44 70 85 38 35 25 12 4 19 31 37 88 80 59 27 9 44 71 86 39 36 26 12 4 20 32 38 89 81 60 28 9 45 72 87 40 37 27 12 4 20 32 39 90 82 60 28 9 45 73 88 41 37 27 13 4 21 33 40 91 83 61 28 10 46 74 89 42 38 28 13 4 21 34 41 92 84 62 28 10 46 74 90 43 39 29 13 5 22 35 42 93 85 62 29 10 47 75 91 44 40 29 14 5 22 • 36 43 94 86 63 29 10 47 76 92 45 4 1 3° 14 5 23 36 44 95 87 64 29 10 48 77 93 46 42 31 14 5 23 37 45 96 88 64 30 10 48 78 94; 47 43 3i 15 S 24 38 46 97 89 65 30 10 49 78 95 48 44 32 15 5 24 39 47 98 90 66 30 10 49 79 96 49 45 33 15 5 25 40 48 99 90 66 31 10 SO 80 97 50 46 33 15 5 25 40 49 100 91 67 31 11 So 81 98 + 15 + 14 + 13 -12 -11 -10 — 9 + 15 + 14 + 13 — 12 — 11 — 10 — 9 17 1 and 9 are zero 16 TERMS — SINE 6 5 and 13 are 1000 + 2 & 8 3 & 7 4 & 6 + 2 & 8 3 & 7 4 & 6 Nos. •383 •707 •924 Nos. ■383 ■707 ■924 1 : 1 51 20 36 47 2 1 I 2 52 20 37 48 3 1 2 3 53 20 37 49 4 2 3 4 54 21 38 5° 5 2 4 5 55 21 39 51 6 2 4 6 56 21 40 52 7 3 5 6 57 22 40 53 8 3 6 7 58 22 4i 54 9 3 6 8 59 23 42 55 10 4 7 9 60 23 42 55 11 4 8 10 61 23 43 56 12 5 8 11 62 24 44 57 13 5 9 12 63 24 45 58 14 5 10 13 64 25 45 59 15 6 II H 65 25 46 60 ■ 16 6 II 15 66 25 47 61 17 7 12 16 67 26 47 62 18 7 13 17 68 26 48 63 i 19 7 13 18 69 26 49 64 20 8 14 18 70 27 49 65 : 21 8 15 19 71 27 50 66 I 22 8 16 20 72 28 51 67 I 23 9 16 21 73 28 52 67 : 24 9 17 22 74 28 52 68 ' 25 10 18 23 75 29 53 69 26 10 18 24 76 29 54 70 27 10 19 25 77 29 54 71 ' 28 11 20 26 78 30 55 72 29 II 21 27 79 3° 56 73 30 11 21 28 80 31 57 74 31 12 22 29 81 3i 57 75 32 12 23 30 82 31 58 76 33 13 23 3° 83 32 59 77 . 34 13 24 3' 84 32 59 78 1 35 13 25 32 85 33 60 79 I 36 14 25 33 86 33 61 79 37 H 26 34 87 33 62 80 ; 38 15 27 35 88 34 62 81 39 IS 28 36 89 34 63 82 40 IS 28 37 90 34 64 83 41 16 29 38 91 35 64 84 42 16 3D 39 92 35 65 85 : 43 16 30 40 93 36 66 86 44 17 31 41 94 36 66 87 45 17 32 42 95 36 67 88 46 18 33 43 96 37 68 89 47 18 33 43 97 37 69 90 48 18 34 44 98 38 69 91 49 19 35 45 99 38 70 91 50 19 35 4*3 100 38 7i 92 — 10&16 11&15 12&14 — 10&16 11&15 12&14 18 1 and 9 zre 1000 16 TERMS — COS 6 5 and 13 are zero + 2 A 16 3&15 4& 14 + 2&16 3&15 4&14 Nos. ■924 7°7 ■383 Nos. •924 •707 ■383 1 1 1 51 47 3f> 20 2 2 1 1 52 48 37 20 3 3 2 1 53 49 37 20 4 4 3 2 54 5° 38 21 5 5 4 2 55 51 39 21 6 6 4 2 56 52 40 21 7 6 5 3 57 53 40 22 8 7 6 3 58 54 41 22 9 8 6 3 59 55 42 23 10 9 7 4 60 55 42 23 11 10 8 4 61 56 43 23 12 11 8 5 62 57 44 24 13 12 9 5 63 58 45 24 14 13 10 5 64 59 45 25 15 14 11 6 65 60 46 25 16 15 11 6 66 61 47 25 17 16 12 7 67 62 47 26 18 17 13 7 68 63 48 26 19 18 13 7 69 64 49 26 20 18 14 8 70 (35 49 27 21 19 15 8 71 66 50 27 22 20 16 8 72 67 5i 28 23 21 16 9 73 67 52 28 24 22 17 9 74 68 52 28 25 23 18 10 75 69 S3 29 26 24 18 10 76 70 54 29 27 25 19 10 77 71 54 29 28 26 20 11 78 72 55 3° 29 27 21 11 1 79 73 56 3D 30 28 21 ii 80 74 57 31 31 29 22 12 81 75 57 31 32 30 23 12 82 76 58 31 33 30 23 13 83 77 59 32 34 31 24 13 84 78 59 32 35 32 25 13 85 79 60 33 36 33 25 H 1 86 79 61 33 37 34 26 14 87 80 62 33 38 35 27 15 88 81 62 34 39 36 28 IS 89 82 63 34 40 37 28 15 90 83 64 34 41 38 29 16 91 84 64 35 42 39 30 16 92 85 65 35 43 40 3° 16 93 86 66 36 44 4i 31 17 94 87 66 36 45 42 32 17 95 88 67 36 46 43 33 18 96 89 68 37 47 43 33 18 97 90 69 37 48 44 34 18 98 9i 69 38 49 45 35 19 99 91 70 38 50 46 35 19 100 92 71 38 - Il0&8 11 & 7 12 & 6 — 10 & 8 11 & 7 12 & 6 19 17 TERMS — SINE 6 1 is zero + 2 3 4 5 6 7 8 9 Nos. •36i •674 ■895 •996 •962 •798 •527 •184 1 1 1 1 1 1 1 2 1 1 2 2 2 2 1 3 1 2 3 3 3 2 2 4 1 3 4 4 4 3 2 5 2 3 4 5 5 4 3 6 2 4 5 6 6 5 3 7 3 5 6 7 7 6 4 8 3 5 7 8 8 6 4 9 3 6 8 9 9 7 5 2 10 4 7 9 10 10 8 5 2 11 4 7 10 II II 9 6 2 12 4 8 11 12 12 10 6 2 13 5 9 12 13 13 10 7 2 14 S 9 13 14 13 11 7 3 15 5 10 13 15 14 12 8 3 16 6 11 14 16 15 13 8 3 17 6 11 IS 17 16 14 9 3 18 6 12 16 18 17 14 9 3 19 7 13 17 19 18 15 10 3 20 7 13 18 20 19 16 11 4 21 8 14 !9 21 20 17 11 4 22 8 15 20 22 21 18 12 4 23 8 16 21 23 22 18 12 4 24 9 16 21 24 23 19 13 4 25 9 17 22 25 24 20 13 5 26 9 18 23 26 25 21 14 5 27 10 18 24 27 26 22 14 5 28 10 19 25 28 27 22 15 5 29 10 20 26 29 28 23 IS 5 30 11 20 27 30 29 24 16 6 31 11 21 28 31 30 25 16 6 32 12 22 29 32 31 26 17 6 33 12 22 30 33 32 26 17 6 34 12 23 30 34 33 27 18 6 35 13 24 31 35 34 28 18 6 36 13 24 32 36 35 29 19 7 37 13 25 33 37 36 30 19 7 38 14 26 34 38 37 30 20 7 39 14 26 35 39 38 3i 21 7 40 14 27 36 40 38 32 21 7 41 15 28 37 41 39 33 22 8 42 IS 28 38 42 40 34 22 8 43 16 29 38 43 41 34 23 8 44 16 30 39 44 42 35 23 8 45 16 30 40 45 43 36 24 8 46 17 31 41 46 44 37 24 8 47 17 32 42 47 45 38 25 9 48 17 32 43 48 46 38 25 9 49 18 33 44 49 47 39 26 9 50 18 34 45 So 48 40 26 9 — 17 16 15 14 13 12 11 10 20 17 TERMS — SINE 6 1 is zero + 2 3 4 5 6 7 8 9 Nos. •361 •674 •895 •996 ■962 ■798 •527 •184 51 18 34 46 5i 49 4i 27 9 52 19 35 47 52 5° 4i 27 IO 53 19 36 47 53 5i 42 28 10 54 '9 36 48 54 52 43 28 10 55 20 37 49 55 53 44 29 10 56 20 38 SO 56 54 45 30 10 57 21 38 5i 57 55 45 30 10 58 21 39 52 58 56 46 3i 11 59 21 40 53 59 57 47 3i 11 60 22 40 54 60 58 48 32 11 61 22 41 55 61 59 49 32 11 62 22 42 55 62 60 49 33 11 63 23 42 56 63 61 5° 33 12 64 23 43 57 64 62 51 34 12 65 23 44 58 65 63 52 34 12 66 24 44 59 66 63 53 35 12 67 24 45 60 ■ 67 64 53 35 12 68 24 46 61 68 65 54 36 *3 69 25 47 62 69 66 55 36 13 70 25 47 63 70 67 56 37 13 71 26 48 64 7i 68 57 37 13 72 26 49 64 72 69 57 38 13 73 26 49 65 73 70 58 38 13 74 27 5° 66 74 7i 59 39 14 75 27 5i 67 ■ 75 72 60 40 H 76 27 51 68 76 73 61 40 14 77 28 52 69 77 74 61 41 14 78 28 53 70 78 75 62 41 14 79 28 53 71 79 76 63 42 15 80 29 54 72 , 80 77 64 42 15 81 29 55 72 81 78 65 43 15 82 30 55 73 82 79 65 43 15 \ 83 30 56 74 83 80 66 44 15 84 3° 57 75 84 81 67 44 IS 85 31 57 76 85 82 68 45 16 86 31 58 77 ; 86 83 69 45 16 87 31 59 78 87 84 69 46 16 88 32 59 79 ■ 88 85 70 46 16 89 32 60 80 89 86 7i 47 16 90 32 61 81 : 90 87 72 47 17 91 33 61 81 ; 9I 88 73 48 17 92 33 62 82 92 89 73 48 17 93 33 63 83 93 89 74 49 17 94 34 63 84 94 90 75 50 17 95 34 64 85 95 91 7 6 5° 17 96 35 65 86 96 92 77 51 18 97 35 65 87 97 93 77 5i 18 98 35 66 88 98 94 78 52 18 99 36 67 89 99 95 79 52 18 100 36 67 90 100 96 80 53 18 10 — 17 16 15 14 13 12 11 21 17 TERMS — COS 1 is 1-000 + 2 + 3 + 4 + 5 — 6 -7 — 8 — 9 Nos. 1 ■933 739 •445 •092 •274 •602 ■850 ■983 i 1 1 1 1 2 2 1 1 1 1 2 2 3 3 2 1 1 2 3 3 4 4 3 2 1 2 3 4 5 5 4 2 1 3 4 5 6 6 4 3 2 4 5 6 7 7 5 3 2 4 6 7 8 7 6 4 2 5 7 8 9 8 7 4 2 5 8 9 10 9 7 4 3 6 9 10 11 10 8 5 3 7 9 11 12 ii 9 5 3 7 10 12 13 12 10 6 4 8 11 13 14 13 10 6 4 8 12 H 15 14 11 7 4 9 13 IS 16 IS 12 7 4 10 14 16 17 16 13 8 2 5 10 14 17 18 17 13 8 2 5 II 15 18 19 18 H 8 2 5 11 16 19 20 19 15 9 2 5 12 17 20 21 20 16 9 2 6 13 18 21 22 21 16 10 2 6 13 19 22 23 21 17 10 2 6 14 20 23 24 22 18 11 2 7 14 20 24 25 23 18 11 -> 7 15 21 25 26 24 19 12 2 7 16 22 26 27 25 20 12 2 7 16 23 27 28 26 21 12 3 8 17 24 28 29 27 21 13 3 8 17 25 29 30 28 22 13 3 8 18 26 29 31 29 23 14 3 8 19 26 30 32 30 24 14 3 9 '9 27 3i 33 31 24 15 3 9 20 28 32 34 32 25 15 3 9 20 29 33 35 33 26 16 3 10 21 30 34 36 34 27 16 3 10 22 , 3i 35 37 35 27 16 3 10 22 3i 36 38 35 28 17 3 10 23 32 37 39 36 29 17 4 11 23 33 38 40 37 30 18 4 11 2 4 34 39 41 38 3° 18 4 11 25 35 40 42 39 31 19 4 12 25 36 41 43 40 32 19 4 12 26 37 42 44 4i 33 20 4 12 ; 26 37 43 45 42 33 20 4 12 27 38 44 46 43 34 20 4 13 28 39 45 47 44 35 21 4 13 28 40 46 48 45 35 21 4 13 29 41 47 49 46 36 22 5 13 29 42 48 50 47 37 22 5 14 30 43 49 + 17 + 16 + 15 + 14 — 13 — 12 -11 — 10 22 17 TERMS — COS 6 1 is 1000 + 2 + 3 + 4 •445 + 5 — 6 -7 — 8 — 9 Nos. •933 •739 ■092 •274 •602 ■850 43 •983 51 48 3 s 23 5 '4 31 50 52 49 3« 23 5 H 31 44 5i 53 49 39 24 5 15 32 45 52 54 50 40 24 5 15 33 46 53 55 5i 4i 24 5 15 33 47 54 56 52 4i 25 5 15 34 48 55 57 S3 42 25 5 16 34 48 56 58 54 43 26 5 16 35 49 57 59 55 44 26 5 16 36 5° 58 60 56 44 27 6 16 36 5i 59 61 57 45 27 6 17 37 52 60 62 58 46 28 6 17 37 53 61 63 59 47 28 6 17 38 54 62 64 60 47 28 6 18 39 54 63 65 61 48 29 6 18 39 55 64 66 62 49 29 6 18 40 56 65 67 63 50 30 6 18 40 57 66 68 63 50 30 6 19 41 58 67 69 64 5i 31 6 19 42 59 68 70 65 52 3i 6 19 42 60 69 71 66 52 32 7 19 43 60 70 72 67 53 32 7 20 43 61 71 73 68 54 32 7 20 44 62 72 74 69 55 33 7 20 45 63 73 75 70 55 33 7 21 45 64 74 76 71 56 34 7 21 46 65 75 77 72 57 34 7 21 46 65 76 78 73 5S 35 7 21 47 66 77 79 74 58 35 7 22 48 67 78 80 75 59 36 7 22 48 68 79 81 76 60 36 7 22 49 69 80 82 77 61 36 8 22 49 7° 81 83 77 61 37 8 23 5° 7i 82 84 78 62 37 8 23 51 71 83 85 79 63 38 8 23 5i 72 84 86 80 64 38 8 24 52 73 85 87 81 64 39 8 24 52 74 86 88 82 65 39 8 24 53 75 87 89 83 66 40 8 24 54 76 87 90 84 67 40 8 25 54 77 88 91 85 67 40 8 25 55 77 89 92 86 68 41 8 25 55 78 90 93 87 69 41 9 25 56 79 91 94 88 69 42 9 26 57 80 92 95 89 70 42 9 26 57 81 93 96 90 71 43 9 26 58 82 94 97 91 72 43 9 27 58 82 95 98 91 72 44 9 27 59 83 96 99 92 73 44 9 27 60 84 97 100 93 74 45 9 27 60 85 98 + 17 + 16 + 15 + 14 -13 — 12 -11 -10 23 18 TERMS — SINE 1 and 10 are zero + 2 & 9 3 & 8 4 & 7 5 & 6 + 2 & 9 3 & 8 4 & 7 5 & 6 Nos. •342 •643 1 •866 ■985 Nos. •342 •643 •866 •985 1 o 1 1 51 V 33 44 5° 2 t 1 2 2 52 18 33 45 5i 3 i 2 3 3 53 18 34 46 52 4 i 3 3 4 54 18 35 47 53 5 2 3 4 5 55 19 35 48 54 6 2 4 5 6 56 19 36 48 55 7 2 5 6 7 57 19 37 49 56 8 3 5 7 8 58 20 37 50 57 9 3 6 8 9 59 20 38 51 58 10 3 6 9 10 60 21 39 52 59 11 4 7 10 1 1 61 21 39 53 60 12 4 8 10 12 62 - 21 40 54 61 13 4 8 11 13 63 22 4i 55 62 14 5 9 12 14 64 22 4i 55 63 15 5 10 13 15 65 22 42 56 64 16 5 10 14 16 66 23 42 57 65 . 17 6 11 15 17 67 23 43 58 66 18 6 12 16 18 68 23 44 59 67 19 6 12 16 19 69 24 44 60 68 20 7 13 17 20 70 24 45 61 69 21 7 14 18 21 71 24 46 61 70 22 8 14 19 22 72 25 46 62 71 23 8 15 20 23 73 25 47 63 72 24 8 15 21 24 74 25 48 64 73 25 9 16 22 25 75 26 48 65 74 26 9 17 23 26 76 26 49 66 75 27 9 17 23 27 77 26 50 67 76 28 to 18 24 28 78 27 50 68 77 29 IO 19 25 29 79 27 5i 68 78 30 10 19 26 30 80 27 5i 69 79 31 II 20 27 3i 81 28 52 7° 80 32 II 21 28 32 82 28 53 71 81 33 II 21 29 33 83 28 53 72 82 34 12 22 29 33 84 29 54 73 83 35 12 23 30 34 85 29 55 74 84 36 12 23 3i 35 86 29 55 74 85 37 13 24 32 36 87 3° 56 75 86 38 13 24 33 37 88 30 57 76 87 39 13 25 34 38 89 30 57 77 88 40 14 26 35 39 90 3i 58 78 89 41 14 26 36 40 91 3i 59 79 90 42 H 27 36 41 92 31 59 80 9i 43 IS 28 37 42 93 32 60 81 92 44 IS 28 38 43 94 32 60 81 93 45 15 29 39 44 95 32 61 82 94 46 16 30 4° 45 96 33 62 83 95 47 16 30 41 46 97 33 62 84 96 48 16 31 42 47 98 34 63 85 97 49 17 32 42 48 99 34 64 86 98 50 17 32 43 49 100 34 64 87 99 — 11&18 12&17 13&16 14&15 — 11&18 12&17 13&16 14&15 24 18 TERMS — COS 6 1 and 10 are 1-000 ■•■* + 2&18 3&17 4& 16 5&15 + 2&18 3& 17 4& 16 5* 15 Nos. •940 766 •500 •174 Nos. •940 •766 ■500 ■174 1 1 1 1 51 48 39 26 9 2 2 2 1 52 49 40 26 9 3 3 2 2 53 50 41 27 9 4 4 3 2 54 51 41 27 9 5 5 4 3 55 52 42 28 10 6 6 5 3 56 53 43 28 10 7 7 5 4 57 54 44 29 10 8 8 6 4 58 55 44 29 10 9 8 7 5 2 59 55 45 30 10 10 9 8 5 2 60 56 46 30 10 11 10 8 6 2 61 57 47 31 11 12 11 9 6 2 62 58 47 31 11 13 12 10 7 2 63 59 48 32 11 .44 13 11 7 2 64 60 49 32 11 15 14 11 8 3 65 61 50 33 1 1 16 15 12 8 3 66 62 5i 33 11 17 16 13 9 3 67 63 5i 34 12 18 17 14 9 3 68 64 52 34 12 19 18 15 10 3 69 65 53 35 12 20 19 15 10 3 70 66 54 35 12 21 20 16 11 4 71 67 54 36 12 22 21 17 11 4 72 68 55 36 13 23 22 18 12 4 73 69 56 37 13 24 23 18 12 4 74 70 57 37 13 25 24 19 13 4 75 71 57 38 13 26 24 20 13 5 76 7 1 58 38 13 27 25 21 14 S 77 72 59 39 13 28 26 21 14 5 78 73 60 39 H 29 27 22 15 5 79 74 61 40 14 30 28 23 15 5 80 75 61 40 14 31 29 24 16 5 81 76 62 4i 14 32 30 25 16 6 82 77 63 41 14 33 31 25 17 6 83 78 64 42 14 34 32 26 17 6 84 79 64 42 15 35 33 27 18 6 85 80 65 43 15 36 34 28 18 6 86 81 66 43 15 ' 37 35 28 19 6 87 82 67 44 15 38 36 29 19 7 88 83 67 44 15 39 37 30 20 7 89 84 68 45 15 40 38 3i 20 7 90 85 69 45 16 41 39 31 21 7 91 86 70 46 16 42 39 32 21 7 92 86 70 46 16 43 40 33 22 7 93 87 71 47 16 44 41 34 22 8 94 88 72 47 16 45 42 34 23 8 95 89 73 48 17 46 43 35 23 8 96 90 74 48 17 47 44 36 24 8 97 9i 74 49 V 48 45 37 24 8 98 92 75 49 17 49 46 38 25 9 : 99 93 76 50 17 50 47 38 25 9 ! 100 94 77 50 17 — 9&11 8&12 7&13 6&14 — 9& 11 8&12 7&13 6&14 25 19 TERMS — S1JME 1 li zero 1 + 2 3 4 5 6 7 8 9 10 Nos. •325 ■614 •837 •969 • 997 •916 •736 •476 •164 1 1 1 1 1 r [ 2 1 1 2 2 2 2 1 1 3 1 2 3 3 3 3 2 1 4 1 2 3 4 4 4 3 2 5 2 3 4 5 5 5 4 2 6 2 4 S 6 6 5 4 3 7 2 4 6 7 7 6 5 3 8 3 5 7 8 8 7 6 4 9 3 6 8 9 9 8 7 4 10 3 6 8 10 10 9 7 5 2 11 4 7 9 11 11 10 8 5 2 12 4 7 10 12 12 11 9 6 2 13 4 8 11 13 13 12 10 6 2 14 5 9 12 14 14 13 10 7 2 15 5 . 9 13 15 15 14 11 7 2 16 5 10 13 16 16 15 12 8 3 17 6 10 H 16 17 16 13 8 3 IS 6 11 IS 17 18 16 13 9 3 19 6 12 16 18 19 17 14 9 3 20 7 12 17 19 20 18 15 10 3 21 7 13 18 20 21 19 15 10 3 22 7 H 18 21 22 20 16 10 4 23 7 14 19 22 23 21 17 1 1 4 24 8 IS 20 23 24 22 18 1 r 4 25 8 IS 21 24 25 23 18 ] 2 4 26 8 16 22 25 26 24 19 12 4 27 9 17 23 26 27 25 20 13 4 28 9 17 23 27 28 26 21 13 5 29 9 18 24 28 29 27 21 14 5 30 10 18 25 29 30 27 22 H 5 31 10 J 9 26 30 31 28 23 15 S 32 10 20 27 31 32 29 24 IS 5 33 11 20 28 32 33 30 24 16 S 34 11 21 28 33 34 3i 25 16 , 6 35 11 21 29 34 35 32 26 17 6 36 12 22 30 35 36 33 26 17 6 37 12 23 31 36 37 34 27 18 6 38 12 23 32 37 38 35 28 18 6 39 13 24 33 38 39 36 29 19 6 40 13 25 33 39 40 37 29 19 7 41 13 25 34 40 41 38 30 20 7 42 14 26 35 41 42 38 31 20 7 43 H 26 36 42 43 39 32 20 7 44 14 27 37 43 44 40 32 21 7 45 15 28 38 44 45 4i 33 ■ 21 7 46 IS 28 39 45 46 42 34 22 8 47 15 29 39 46 47 43 35 22 8 48 16 29 40 47 48 44 35 23 8 49 16 30 4i 47 49 45 36 23 8 50 16 31 42 48 50 46 37 24 8 — 19 18 17 16 15 14 13 12 11 26 19 TERMS — SINE 1 is zero + 2 •325 3 4 5 6 7 8 9 10 Nos. • ■614 ■837 ■969 •997 •916 736 ■476 •164 51 "7 3i 43 49 5i 47 38 24 8 52 17 32 44 50 52 48 38 25 9 53 17 33 44 5i 53 49 39 25 9 54 18 33 45 52 54 49 40 26 9 55 18 34 46 53 55 50 40 26 9 56 18 34 47 54 56 51 41 27 9 [ 57 19 35 48 55 57 52 42 27 9 58 19 36 49 56 58 53 43 28 10 59 19 36 49 57 59 54 43 28 10 60 20 37 50 58 60 55 44 29 10 61 20 37 5i 59 61 56 45 29 10 62 20 38 52 60 62 57 46 3° 10 63 20 39 53 61 63 58 46 30 10 64 21 39 54 62 64 59 47 3° 10 65 21 40 54 63 65 60 48 31 11 66 21 41 55 64 66 60 49 3i 1 1 67 22 41 56 65 67 61 49 32 11 68 22 42 57 66 68 62 5° 32 11 69 22 42 58 67 69 63 5i 33 11 70 23 -43 59 68 70 64 52 33 11 71 23 44 59 69 71 65 52 34 12 72 23 44 60 70 72 66 53 34 12 73 24 45 61 7i 73 67 54 35 12 74 24 45 62 72 74 68 54 35 12 75 24 46 63 73 75 69 55 36 12 76 25 47 64 74 76 70 56 36 12 77 25 47 64 75 77 7i 57 37 '3 78 25 48 65 76 78 71 57 37 13 79 26 49 66 77 79 72 58 38 '3 80 26 49 67 78 80 73 59 38 13 81 26 50 68 78 81 74 60 39 13 82 27 50 69 79 82 75 60 39 13 i 83 27 5i 69 80 83 76 61 40 14 84 27 52 70 81 84 77 62 40 14 85 28 52 7i 82 85 78 63 40 14 86 28 53 72 83 86 79 63 41 14 87 28 53 73 84 87 80 64 41 14 88 29 54 74 85 88 81 65 42 14 89 29 55 74 86 89 82 66 42 15 90 29 55 75 87 90 82 66 43 15 91 30 56 76 88 9i 83 67 43 15 92 30 56 77 89 92 84 68 44 «5 93 3° 57 78 90 93 85 68 44 ! 5 94 3i 58 79 9i 94 86 69 45 15 95 3i 58 80 92 95 87 70 45 16 96 31 59 80 93 96 88 71 46 16 97 32 60 81 : 94 97 89 71 46 16 98 32 60 82 95 98 90 72 47 16 99 32 61 83 : 96 99 91 73 47 16 100 33 61 84 i 97 100 92 74 48 16 - 19 18 17 i ! 16 : 15 14 13 12 11 27 + 1000 19 TERMS — COS 6 + 2 + 3 + 4 + 5 — 6 — 7 — 8 — 9 — 10 Nos. •946 •789 •547 •246 •082 ■401 ■677 •879 •986 1 1 1 1 1 1 1 2 2 2 1 1 1 2 2 3 3 2 2 1 1 2 3 3 4 4 3 2 1 2 3 4 4 5 5 4 3 1 2 3 4 5 6 6 5 3 1 2 4 5 6 7 7 6 4 2 3 5 6 7 8 8 6 4 2 3 5 7 8 9 9 7 5 2 4 6 8 9 10 9 8 5 2 4 7 9 10 11 10 9 6 3 4 7 10 11 12 n 9 7 3 5 8 11 12 13 12 10 7 3 5 9 11 13 14 13 11 8 3 6 9 12 14 15 14 12 8 4 6 10 13 15 16 15 '3 9 4 6 11 H 16 17 16 13 9 4 7 12 15 17 18 17 14 10 4 7 12 16 18 19 18 15 10 5 2 8 13 17 19 20 19 16 11 5 2 8 14 18 20 21 20 17 1 1 5 2 8 14 18 21 22 21 17 12 5 2 9 15 19 22 23 22 18 13 6 2 9 16 20 23 24 23 19 13 6 2 10 16 21 24 25 24 20 14 6 2 10 17 22 25 26 25 21 14 6 2 10 18 23 26 27 26 21 15 7 2 11 18 24 27 28 26 22 15 7 2 11 19 25 28 29 27 23 16 7 2 12 20 25 29 30 28 24 16 7 2 12 20 26 3° 31 29 24 17 8 3 12 21 27 3i 32 30 25 18 8 3 13 22 28 32 33 31 26 18 8 3 13 22 29 33 34 32 27 19 8 3 14 23 3° 34 35 33 28 19 9 3 H 24 31 35 36 34 28 20 9 3 14 24 32 35 37 35 29 20 9 3 15 25 33 36 ! 38 36 3° 21 9 3 15 26 33 37 39 37 31 21 10 3 16 26 34 38 ' 40 38 32 22 10 3 16 27 35 39 41 39 32 22 10 3 16 28 36 40 42 40 33 23 10 3 17 28 37 41 43 4i 34 24 11 4 17 29 38 42 44 42 35 24 11 4 18 30 39 43 45 43 36 25 11 4 18 30 40 44 46 44 36 25 11 4 18 31 40 45 47 44 37 26 12 4 19 32 41 46 48 45 38 26 12 4 19 32 42 47 49 46 39 27 12 4 20 33 43 48 50 47 39 27 12 4 20 34 44 49 + 19 + 18 + 17 + 16 — 15 — 14 — 13 — 12 — 11 28 19 TERMS — COS 6 1 is + 1000 + 2 + 3 + 4 + 5 -6 — 7 — 8 ■677 — 9 — 10 Nos. ■946 •789 •547 •246 •082 •401 ■879 •986 51 48 40 28 13 4 20 35 45 50 52 49 41 28 13 4 21 35 46 51 53 50 42 29 13 4 21 36 47 52 54 Si 43 30 13 4 22 37 47 53 55 5^ 43 30 H 5 22 37 48 54 56 53 44 3i 14 5 22 38 49 55 57 54 45 3i H 5 23 39 50 56 58 55 46 32 14 5 23 39 51 57 59 56 47 32 15 5 24 40 52 58 60 57 47 33 15 5 24 4i 53 59 61 58 48 33 15 5 24 41 54 60 62 59 49 34 15 5 25 42 54 61 63 60 50 34 15 5 25 43 55 62 64 61 50 35 16 5 26 43 56 63 65 61 51 36 16 5 26 44 57 64 66 62 52 36 16 5 26 45 58 65 67 63 53 37 16 5 27 45 59 66 68 64 54 37 17 6 27 46 60 67 69 65 54 38 17 6 28 47 61 68 70 66 55 38 17 6 28 47 62 69 71 67 56 39 17 6 28 48 62 70 72 68 57 39 18 6 29 49 63 7i 73 69 58 40 18 6 29 49 64 72 74 ' 70 58 40 18 6 30 50 65 73 75 71 59 41 18 6 30 51 66 74 76 72 60 42 19 6 3° 51 67 75 77 73 61 42 19 6 31 52 68 76 78 74 62 43 19 6 31 53 6 9 77 79 75 62 43 19 6 32 53 69 78 80 76 63 44 20 7 32 54 70 79 81 77 64 44 20 7 32 55 71 80 I 82 78 65 45 20 7 33 56 72 81 83 79 65 45 20 7 33 56 73 82 84 79 66 46 21 7 34 57 74 83 85 80 67 46 21 7 34 58 75 84 86 8r 68 47 21 7 34 58 7<5 85 87 82 69 48 21 7 35 59 76 86 88 83 69 48 22 7 35 60 77 87 89 84 70 49 22 7 36 60 78 88 90 85 71 49 22 7 36 61 79 89 91 86 72 50 22 7 36 62 80 90 92 87 73 50 23 8 37 62 81 91 93 88 73 5i 23 8 37 63 82 ■ 92 94 89 74 5i 23 8 38 64 83 93 95 90 75 52 23 8 38 64 84 94 96 91 76 53 24 8 38 65 84 95 97 92 77 53 24 8 39 66 85 96 98 93 77 54 24 8 39 66 86 97 99 94 78 54 24 8 40 67 87 98 100 95 79 55 25 8 40 68 88 99 + 19 + 18 + 17 + 16 -15 -14 — 13 -12 — 11 29 1 and 11 are zero 6 and 16 are 1-000 20 TERMS — SINE 6 \ + 2& 10 3 & 9 4 & 8 5 & 7 + 2&10 3 & 9 4 & 8 5 & 7 Nos. •309 •588 •809 ■951 Nos. •309 • •588 •809 •951 1 1 1 1 51 16 30 4i 49 2 1 1 2 2 52 16 31 42 49 3 1 2 2 3 53 16 3i 43 50 4 1 2 3 4 54 17 32 44 Si 5 2 3 4 S 55 17 32 44 52 6 2 4 5 6 56 17 33 45 S3 7 2 4 6 7 57 18 34 46 54 8 2 S 6 8 58 18 34 47 55 9 3 5 7 9 59 18 35 48 56 10 3 6 8 10 60 19 35 49 57 11 3 6 9 11 61 19 36 49 58 12 4 7 10 11 62 19 36 SO 59 13 4 8 11 12 63 19 37 5i 60 14 4 8 11 13 64 20 38 52 61 15 5 9 12 14 65 20 38 53 62 16 5 9 13 15 66 20 39 53 63 ! 17 5 10 14 16 67 21 39 54 64 18 6 11 15 17 68 21 40 55 65 19 6 11 15 18 69 21 4i 56 66 20 6 12 16 19 70 22 4i 57 67 21 6 12 17 20 71 22 42 57 68 22 7 13 18 21 72 22 42 58 68 23 7 14 19 22 73 23 43 59 69 24 7 14 19 23 74 23 44 60 70 25 8 IS 20 24 75 23 44 61 71 26 8 is 21 25 76 23 45 61 72 27 8 16 22 26 77 24 45 62 73 28 9 16 23 27 78 24 46 63 74 29 9 17 23 28 79 24 46 64 7S 30 9 18 24 29 80 25 47 65 76 31 10 18 25 29 81 25 48 66 77 32 10 19 26 30 82 25 48 66 78 33 10 19 27 31 83 26 49 67 79 34 11 4 20 28 32 84 26 49 68 80 35 11 21 28 33 85 26 5° 69 81 36 11 21 29 34 86 27 Si 70 82 37 11 22 3° 35 87 27 5i 70 83 38 12 22 3i 36 88 27 52 71 84 39 12 23 32 37 89 28 52 72 85 40 12 24 32 38 90 28 53 73 86 41 13 24 33 39 91 28 54 74 87 42 13 25 34 40 92 28 54 74 87 43 13 25 35 4i 93 29 55 75 88 44 H 26 36 42 94 29 55 76 89 45 H 26 36 43 95 29 56 77 90 46 14 27 37 44 96 30 56 78 91 47 15 28 38 45 97 30 57 78 92 48 IS 28 39 46 98 30 58 79 93 49 15 29 40 47 99 3i 58 80 94 50 IS 29 40 48 100 31 59 81 95 — 12&20 13&19 14 & 18 15&17 — 12 & 20 13&19 14&18 15&17 30 1 and 11 are 1000 6 and 16 are zero 20 TERMS — COS 6 + 2&20 3&19 4&18 5&17 + 2&20 3& 19 4&18 5&17 Nos. •951 •809 •588 ■309 Nos. •951 •809 ■588 •309 1 i 1 1 51 49 41 30 16 2 2 2 1 52 49 42 3t 16 3 3 2 2 53 5° 43 3i 16 4 4 3 2 54 51 44 32 : 7 5 s 4 3 2 55 52 44 32 17 6 6 5 4 2 56 53 45 33 •7 7 7 6 4 2 57 54 46 34 18 8 8 6 5 2 58 55 47 34 IS 9 9 7 5 3 59 56 48 35 18 10 IO 8 6 3 60 57 49 35 19 11 ii 9 6 3 61 58 49 36 19 12 ii 10 7 4 62 59 5° 36 19 13 12 11 8 4 63 60 5i 37 19 14 13 11 8 4 64 61 52 38 20 15 14 12 9 5 65 62 53 38 20 16 IS 13 9 5 66 63 53 39 20 17 16 14 10 5 67 64 54 39 21 18 17 15 1 1 6 68 65 55 40 21 19 18 15 11 6 69 66 56 4i 21 20 19 16 12 6 70 67 57 41 22 21 JO 17 12 6 71 68 57 42 22 22 21 18 13 7- 72 68 58 42 22 23 22 19 14 7 73 69 59 43 23 24 23 19 H 7 74 70 60 44 23 25 24 20 15 8 75 71 61 44 2 3 26 -5 21 15 8 76 72 61 45 23 27 26 22 16 8 77 73 62 45 24 28 ^7 23 16 9 78 74 63 46 24 29 28 23 17 9 79 75 64 46 24 30 29 24 18 9 80 76 65 47 25 31 29 25 18 10 81 77 66 48 25 32 30 26 19 10 82 78 66 48 25 33 3 1 27 19 10 83 79 67 49 26 34 32 28 20 11 84 80 68 49 26 35 33 28 21 11 85 81 69 50 26 36 34 29 21 . 11 86 82 70 51 27 37 3S 30 22 11 87 83 70 5i 27 38 36 31 22 12 88 84 71 52 27 39 37 32 23 12 89 85 72 52 28 40 38 32 24 12 90 86 73 53 28 41 39 33 24 13 91 87 74 54 28 42 40 34 25 13 92 87 74 54 28 43 41 35 25 13 93 88 75 55 29 44 42 36 26 14 94 89 76 55 29 45 43 36 26 U 95 90 77 56 29 46 44 37 27 14 96 91 78 56 3° 47 45 38 28 15 97 92 78 57 30 48 46 39 28 15 98 93 79 5 8 ' 30 49 47 40 29 15 99 94 80 58 31 50 48 40 29 15 100 95 Si 59. 31 — 12&10 13 & 9 14 & 8 15 & 7 — 12&10 13 & 9 14 & 8 15 & 7 31 21 TERMS — SINE 6 1 is zero + 2 3 4 782 5 6 7 8 9 10 11 Nos. ■295 •563 ■931 • 997 ■975 •866 ■680 ■435 •149 1 1 1 1 1 1 1 1 2 i 1 2 2 2 2 2 1 1 3 i 2 2 3 3 3 3 2 1 4 i 2 3 4 4 4 3 3 2 5 i 3 4 5 5 5 4 3 2 6 2 3 5 6 6 6 5 4 3 7 2 4 5 7 7 7 6 5 3 8 2 5 6 7 8 8 7 5 3 1 J 9 3 5 7 8 9 9 8 6 4 10 3 , 6 8 9 IO IO 9 7 4 11 3 6 9 IO 11 11 IO 7 5 2 12 4 7 9 11 12 12 IO 8 5 2 13 4 7 IO 12 13 13 11 9 6 2 14 4 8 11 13 H 14 12 IO 6 2 15 4 8 12 14 15 15 13 IO 7 2 16 5 9 13 15 16 16 14 11 7 2 17 5 10 13 16 17 17 15 12 7 3 18 5 IO 14 17 18 18 16 12 8 3 19 6 11 15 18 19 19 16 13 8 3 20 6 11 16 19 20 20 17 14 9 3 21 6 12 16 20 21 20 18 14 9 3 22 6 12 17 20 22 21 19 15 IO 3 23 7 13 18 21 23 22 20 16 10 3 24 7 14 19 22 24 23 21 16 IO 4 25 7 14 20 23 25 24 22 17 11 4 26 8 IS 20 24 26 25 23 18 1 1 4 27 8 15 21 25 27 26 23 18 12 4 28 8 16 22 26 28 27 24 19 12 4 29 9 16 23 27 29 28 25 20 13 4 ; 30 9 17 23 28 3° 29 26 20 13 4 31 9 17 24 29 31 3° 27 21 13 5 32 9 18 25 30 32 31 28 22 14 5 33 IO 19 26 3i 33 32 29 22 14 5 34 IO 19 27 32 34 33 29 23 15 5 35 IO 20 27 33 35 34 3° 24 15 5 36 ii 20 28 34 36 35 31 24 16 5 37 ii 21 29 34 37 36 32 25 16 6 38 ii 21 30 35 38 37 33 26 16 6 39 12 22 3° 36 39 38 34 27 17 6 40 12 23 3i 37 40 39 35 27 17 6 41 12 23 32 38 41 40 36 28 18 6 42 12 24 33 39 42 41 36 29 18 6 43 13 24 34 40 43 42 37 29 19 6 44 13 25 34 4i 44 43 38 30 19 7 45 13 25 35 42 45 44 39 3i 20 7 46 14 26 36 43, 46 45 40 3i 20 7 47 14 26 37 44 47 46 4' 32 20 7 48 14 27 38 45. 48 47 42 33 21 7 49 H 28 38 46 49 48 42 33 21 7 50 15 28 39 47 5° 49 43 34 22 7 — 21 20 19 18 17 16 15 14 13 12 32 21 TERMS — SINE 6 ,. 1 is zero + 2 3 4 5 6 7 8 9 10 11 Nos. •295 •563 ■782 •93i •997 •975 •866 ■680 •435 •149 51 15 29 40 47 5i 50 44 35 22 8 52 15 29 41 48 52 5i 45 35 23 8 53 16 30 4i 49 53 52 46 36 23 8 54 16 30 42 50 54 53 47 37 23 8 55 16 3i 43 5i 55 54 48 37 24 8 56 17 32 44 52 56 55 48 38 24 8 57 17 32 45 .53 57 56 49 39 25 8 58 17 33 45 54 58 57 5° 39 25 9 59 17 33 46 55 59 58 51 40 26 9 60 18 34 47 56 60 59 52 41 26 9 61 IS 34 48 57 61 59 53 41 27 9 62 18 35 48 58 62 60 54 42 27 9 63 19 35 49 59 63 61 55 43 27 9 64 19 36 5° 60 64 62 55 44 28 10 ! 65 19 37 5i 61 65 63 56 44 28 10 66 19 37 52 61 66 64 57 45 29 10 67 20 38 52 62 67 65 58 46 29 10 68 20 38 53 63 68 66 59 46 3° 10 69 20 39 54 64 69 67 60 47 3° 10 ! 70 21 39 55 65 70 68 61 48 3° 10 71 21 40 56 66 71 69 61 48 3' 11 72 21 41 56 67 72 70 62 49 3i 11 ! 73 22 41 57 68 73 71 63 50 32 11 i 74 22 42 58 69 74 72 64 5° 32 11 i 75 22 42 59 70 75 73 65 51 33 11 ! 76 22 43 59 7i 76 74 66 52 33 11 i 77 23 43 60 72 77 75 67 52 33 11 i 78 23 44 61 73 78 76 68 53 34 12 79 23 44 62 74 79 77 68 54 34 12 ! 80 24 45 63 74 80 78 69 54 35 12 1 81 24 46 63 75 81 79 70 55 35 12 82 24 46 64 76 82 80 71 5<5 36 12 83 24 47 65 77 83 81 72 56 36 12 84 25 47 66 78 84 82 73 57 37 13 85 25 48 66 79 85 83 74 58 37 13 86 25 48 67 80 86 84 74 58 37 13 87 26 49 68 81 87 85 75 59 38 13 88 26 5° 69 82 88 86 76 60 38 13 89 26 50 70 83 89 87 77 61 39 13 90 ! 2 7 5i 70 84 90 88 78 61 39 J 3 i 91 27 5i 71 85 91 89 79 62 40 14 92 27 52 72 86 92 90 80 63 40 14 93 27 52 73 87 93 91 81 63 40 H 94 28 53 74 88 94 92 81 64 41 14 95 28 53 74 88 95 93 82 65 41 14 96 28 54 75 89 96 94 83 65 42 14 97 29 55 76 90 97 95 84 66 42 14 98 29 55 77 91 98 96 85 67 43 15 99 29 56 77 92 99 97 86 67 43 15 100 3° 56 78 93 100 98 87 68 44 15 — 21 20 19 18 17 16 15 14 13 12 21 TERMS — COS 6 1 :.s + i-ooo + 2 + 3 + 4 + 5 + 6 — 7 — 8 — 9 — 10 — 11 Nos. •956 ■826 •623 •364 ■07 s ■222 •500 •733 ■901 ' ■989 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 2 2 3 i 3 2 2 1 1 2 7 3 3 4 4 3 2 1 1 2 3 4 4 5 S 4 3 2 1 3 4 5 5 6 6 5 4 2 1 3 '4 5 6 7 i 7 6 4 3 2 4 5 6 7 8 8 7 5 3 2 4 6 7 8 9 9 7 6 3 2 5 7 8 9 10 10 8 6 4 2 5 7 9 10 11 11 9 7 4 2 6 8 10 11 12 11 10 7 4 3 6 9 11 12 13 12 11 8 5 3 7 IO 12 13 14 13 12 9 5 3 7 10 13 x 4 ; 15 14 12 ,9 5 3 8 II 14 15 16 IS 13 10 6 4 8 12 14 16 17 16 14 11 6 4 9 12 15 17 18 17 15 11 7 4 9 13 16 18 19 18 16 12 7 4 10 H 17 19 20 19 17 12 7 2 4 10 15 18 20 21 20 17 13 8 2 5 11 15 19 21 22 21 18 14 8 z 5 1 1 16 20 22 23 22 19 14 8 2 5 12 17 21 23 24 23 ! 20 15 9 2 5 12 18 22 24 25 24 21 16 9 2 6 13 18 23 2 5 26 25 21 16 9 2 6 13 •9 23 26 27 26 22 "7 10 2 6 14 20 24 27 28 27 23 17 10 2 6 14 21 25 28 29 28 24 18 1 1 2 6 15 21 26 29 30 29 25 19 11 2 7 15 22 27 30 31 30 26 19 1 1 2 7 16 23 28 31 32 31 26 20 12 2 7 16 23 29 3~ 33 32 27 21 12 2 7 17 24 30 33 34 33 28 21 12 3 8 17 25 3i 34 35 33 29 22 13 3 8 18 26 32 35 36 34 3° 22 13 3 8 18 26 32 36 37 35 31 23 13 3 8 '9 27 33 37 38 36 31 24 H 3 8 19 28 34 38 39 37 32 24 14 3 9 20 29 35 39 40 38 33 25 15 3 9 20 29 36 40 41 39 34 26 15 3 9 21 3° 37 41 1 42 40 35 26 15 3 9 21 31 38 42 43 4i 36 2 7 16 3 10 22 32 39 43 44 42 36 27 16 3 10 22 32 40 44 45 43 37 28 16 3 10 23 33 41 45 46 44 38 29 17 3 10 23 34 41 45 47 45 39 29 17 4 10 24 34 42 46 48 46 40 30 17 4 1 1 24 35 43 47 49 47 40 3i 18 4 1 1 25 36 44 48 50 48 4i 31 18 4 11 25 37 45 49 + 21 + 20 + 19 + 18 + 17 — 16 — 15 — 14 — 13 -12 34 21 TERMS - -COS 5 1 is + 1-000 + 2 ! +3 + 4 + 5 + 6 -7 — 8 — 9 — 10 -11 Nos. ■956 | -826 •623 •364 •075 4 ■222 •500 733 •901 •989 51 52 49 4-! 32 19 II 26 37 46 50 5° 43 32 19 4 12 26 38 47 51 53 54 55 5i 44 33 19 4 12 27 39 48 52 5 2 45 34 20 4 12 27 40 49 53 5 3 45 34 20 4 12 28 40 50 54 56 54 46 35 20 4 12 28 41 50 55 57 54 47 36 21 4 13 29 42 51 56 58 55 48 36 21 4 13 29 43 52 57 59 56 49 37 21 4 13 3° 43 53 58 60 57 5° 37 22 5 13 30 44 54 59 61 5S 50 38 22 5 14 31 45 55 60 62 59 51 39 23 5 14 3i 45 56 61 63 60 52 39 23 5 H 32 46 57 62 64 6r 53 40 23 5 14 32 47 58 63 65 62 54 40 24 5 14 33 48 59 64 66 63 55 41 24 5 15 33 48 59 65 67 64 55 42 24 5 15 34 49 60 66 68 65 56 42 25 5 15 34 5° 61 67 69 66 57 43 25 5 15 35 51 62 68 70 67 58 44 25 5 16 35 5i 63 69 71 68 59 44 26 5 16 36 52 64 70 72 69 59 45 26 5 16 36 53 65 71 73 7° 60 45 27 5 16 37 54 66 72 74 /i 61 46 27 6 16 37 54 67 73 75 7~ 62 47 27 6 V 38 55 68 74 76 73 63 47 28 6 17 38 56 68 75 77 74 64 48 28 6 17 39 56 69 76 78 75 64 49 28 6 17 39 57 70 77 79 76 65 49 29 6 18 40 58 71 78 80 76 66 SO 29 6 18 40 59 72 79 81 77 67 5° 29 6 18 41 59 73 80 82 78 68 51 3° 6 18 41 60 74 81 83 79 69 52 30 6 18 42 61 75 82 84 80 69 52 31 6 19 42 62 76 83 85 81 70 53 31 6 19 43 62 77 84 86 82 71 54 31 6 19 43 63 77 85 87 83 72 54 32 7 19 44 64 78 86 88 «4 73 55 32 7 20 44 65 79 87 89 85 74 55 32 7 20 45 65 80 88 90 86 74 56 33 7 20 45 66 81 89 91 87 75 57 33 7 20 46 67 82 90 92 88 76 57 33 7 20 46 67 83 9i 93 89 77 58 34 7 21 47 68 84 92 94 90 78 59 34 7 21 47 69 85 93 95 91 78 59 35 7 21 48 70 86 94 96 92 79 60 35 7 21 48 70 86 95 97 93 80 60 35 7 22 49 71 87 96 98 94 81 61 36 7 22 49 72 88 97 99 95 82 62 36 7 ■ 22 5° 73 89 98 100 96 83 62 36 8 22 50 73 90 99 + 21 + 20 + 19 + 18 + 17 -16 — 15 — 14 -13 — 12 35 TABLE FOR FINDING V«a + b 2 Let b be the smaller of the two : then Va 2 4- b % = (a 4- i b) + following tabic. a\ 10 20 30 40 50 60 70 80 90 100 100 -2-8 -47 -5-6 -5-6 -4-9 -3-4 — 1-2 4-1-4 + 4-5 + 7-9 99 -2-8 -47 -5-6 -5-6 -4-8 -3-2 — I'O 4-1-6 + 4-7 + 8-4 98 -2-8 -47 -5-5 -5-5 -47 -3-0 -o-8 4-1-8 + 5-o + 87 97 -2-8 -47 -5'5 -5-4 -4-6 -2-9 -o-6 4-2-0 + 5-3 + 9-o 96 -2-8 -4'6 -5'4 -5-3 -4'5 -2-8 -o-4 + 2-2 + 5-6 + 9-3 95 -2-8 -4-6 -5-4 -5-2 -4-4 -2-6 -o-3 + 2-5 + 5-9 + 9-6 94 -2-8 -4-6 -5-3 -5-i -4'3 -2-4 — o-i + 27 4- 6-1 + 9-9 93 -2-8 -4-6 -5'3 -5-o -4-i -2-3 4-0-1 4-2-9 + 6-4 + IO-2 92 -2-8 -4-6 -5'2 -5-o -4-0 — 2-1 +o-3 + 3-2 + 6-7 4-IO-6 91 90 -2-8 -27 -4-5 -4-5 -5-2 -5-i -4-9 -4-8 -3-9 -37 -1-9 -i-8 +0-5 +07 + 3-4 + 37 + 7-o + 7-3 4-10-9 -f 11 -2 89 -27 -4-5 -5-i -47 -3-6 -1-6 +o-9 + 3'9 + 7-6 88 -27 -4-5 -5-0 -4-6 -3'5 -1-4 + i-i 4-4-2 + 7-9 87 -27 -4"4 -5-o -4-5 -3-3 -1-3 + i-3 +4-5 4- 8-2 86 —2-7 —4-4 -4'9 -4-4 -3-2 — 1-2 + i-5 +4-8 + 8-5 85 -27 -4-4 -4'9 -4-3 -3'i — I-O 4-1-8 + 5-o 4- 8-8 84 -27 -4-4 -4-8 -4-3 — 2-9 -o-8 + 2-0 + 5-2 + 9-1 83 -27 -4-3 -4-8 -4-2 -2-8 -o-6 +2-3 + 5-5 + 9-4 82 -27 -4'3 -47 -4-i -27 -0-4 +2-5 + 5-8 + 97 81 80 -27 -27 -4-3 -4-2 -47 -4-6 -4-o -3-9 -2-6 -2-4 — 0-2 o-o + 27 + 3-o 4-6-1 +6-4 4-io-o 4-10-4 79 -27 -4-2 -4-6 -3-8 -2-3 + 0-2 + 3'3 +67 78 -27 -4-2 -4-5 -37 — 2-1 4-0-4 + 3-5 +7-o 77 -27 -4-1 -4'4 -3-5 -1-9 4-0-6 + 3-8 +7-3 76 —2-7 -4-i -4-3 -3-4 — 1-8 4-0-8 4-4-0 +7-6 75 -27 -4-i -4-2 -3-3 -1-6 + i-o +4-3 +7-9 74 -27 -4-0 -4-2 -3'2 -i-5 + 1-2 +4-6 +8-4 73 -27 -4-o -4-1 -3-o -i-3 + i-5 +4'8 +87 72 -27 -4-o -4-o — 2-9 — i-i + 17 + 5-1 4-9-0 71 70 -2-7 -2-6 -3-9 -3-9 -3-9 -3-8 -2-8 -27 -0-9 -07 + 1-9 + 2-2 + 5-4 + 57 +9-3 + 9-6 69 -2-6 -3-9 -3-8 -2-6 -0-5 4-2-4 4-6-o 68 -2-6 -3-8 -37 -2-4 -0-3 + 27 +6-3 67 -2-6 -3-8 -3-6 -2-3 — o-i + 2-9 +6-6 66 -2-6 -37 -3-5 — 2-2 4-0-1 4-3-2 +6-9 65 -2-6 -37 -3-4 — 2-0 +0-3 + 3-5 4-7-2 64 -2-6 -3-6 -3'3 -1-9 +0-5 + 3-8 + 7-6 63 -2-6 -3-6 -3-2 -17 4-o-8 4-4-0 + 7'9 62 -2-6 -3-5 -3'i -i-5 + i-o +4-3 4-8-2 61 60 -2-6 -2-5 -3-5 -3-4 -3-0 -2-9 -1-4 — 1-2 + 1-2 + '•4 +4-6 +4-9 +8-5 +8-9 59 -2-5 -3-4 -2-8 — I'l + 17 + 5-2 58 -2'5 -3-3 -27 -o-9 + 1-9 + 5-5 57 -2-5 -3-3 -2-6 -07 + 2-2 + 5-8 56 -2-s -3-2 -2-4 — o-5 +2-4 +6-i 55 -2-5 -3-2 -2-3 -o-3 + 2-6 +6-4 54 -2-5 -3-i — 2-2 — o-i 4-2-8 4-6-8 53 -2-5 -3-i — 2-1 +o-i +3-1 4-7-1 52 -2-5 -3-o — 2-0 +o-3 + 3-4 +7-4 51 50 -2-5 -2-3 -2-9 -2-8 -1-9 -17 +o-5 +07 + 37 +4-0 + 77 +8-i Example (1). If a = yy and 6 = 73 a+ J 6 = 101-3 corr" = 4- 4-8 Vo 2 4- b' = 106-1 This is correct to o-i, though the tables are not intended to be used beyond an accuracy of i-o. Example (2). If a = 43, 6 = 29 a + ib= 52-7 corr >= — o-8 5i-9 The correction is obtained by doubling o and b and then halving the correction given by the table. 36 To find q from sin q = 6/Va* + V, where b < a. Since q is always less than 45°, only the part of the table to the left of the thick broken line is for direct use : the rest of the table gives merely the proportional parts for interpolation. V 10 20 30 40 50 60 70 8 9 10 O O O O O O O O 100 57 us I7-? 23 '(i 30-0 36-9 44'5 4-8 5-4 6-0 99 98 5-8 5 V u-6 n-8 17-6 17-8 23-8 24-0 30-3 3°7 37'3 377 45-0 4-8 4'9 5-5 5-5 6-i 6-1 4-3 97 SV 1 1 -9 l8-o 24-3 31-0 38-2 4-3 S-o 5-6 6-2 96 6-o I2'0 18-2 24-6 3-'4 387 4-4 5-0 57 6-3 95 6-o 12*2 l8'4 24-9 31-8 39-2 4'5 5'i 5-8 6-4 94 6-1 12-3 1 8 -6 25-2 32-2 397 4-5 5'i S-8 6-4 93 6-3 12-4 18-8 25-5 32-6 40-2 4 -6 5-2 5-9 6-s 92 6-2 12-5 19-0 . 25'8 33-o 407 4-6 5-3 5'9 6-6 91 6-3 127 19-2 26-1 33-4 41-2 47 5-4 6-o 67 90 6-4 12-8 -9'5 26-4 33'8 41-8 47 5-4 fro 67 89 6-5 13-0 197 267 34'2 42-4 4-8 5*5 6-i 6-8 88 6-5 ■3-i 19-9 27-0 34'6 43 -o 4-8 5'5 6-2 6-9 87 6-6 13-2 20-2 27-4 35-i 43-6 4-9 5-6 6-3 7-0 86 67 -3'4 20-5 277 3.V5 44-2 5-0 57 6-4 7-1 85 84 6-8 6-8 13-6 13-8 207 21-0 28-0 28-4 36-0 36-5 44-9 5-0 5"' 5-8 5-8 6-5 6-6 7-2 7-3 4'4 83 6-9 14-0 21-3 28-8 37-o 4-4 5-2 5-9 67 7'4 82 7-0 14-1 21-6 29-2 37'5 4'5 5"3 6-o 6-8 7-5 81 7-1 -4-3 21-9 29-6 38-1 4-6 5" 3 6-1 6-9 7-6 80 7-2 14-5 22'2 30-0 387 4-6 5 '4 6-2 7-0 77 79 7S 147 22-5 3°'5 39-3 47 5'5 fi'3 7-1 7-9 78 7-4 14-9 22-8 30-9 39-9 4-8 5-6 C--4 7-2 8-o 77 7'S 15-0 23-0 3--3 40-5 4-9 57 <>•-, 7'3 8-i 76 7-6 -5-3 23-3 31-8 41-1 4-9 57 6-6 7"4 8-2 75 7'7 * y$ 23-6 32-2 41-8 S-o 5-8 67 7-6 8.4 74 7-8 -57 23-9 327 42-5 5-1 5'9 r>-s 77 8"5 73 7-9 -5 '9 24-2 33-2 43-2 .-■•■ (Vo 6-9 7-8 8-6 72 71 8-o 8-i 16-1 16-3 24-6 25-0 337 34-3 44-0 5'2 5-3 6-1 (r2 7-0 7-1 7-9 8-0 8-8 8-9 4-4 70 8-2 1 6-6 25-4 34-8 4-5 .V4 6-3 7-2 8-1 9-0 69 8-3 16-8 2-,-S 35-4 4-6 5 '3 r>- 4 7'3 8-2 9-1 68 8-5 17-1 26-2 36-0 4-6 5-5 6-4 7-4 8-3 9-2 67 8-6 17-3 26-6 36-6 4-6 'vf> 05 7-4 8-4 9-3 66 87 17-6 27-0 37'3 47 5-6 6-6 7'5 8-5 9-4 65 8-8 17-9 27-5 38-0 4-8 57 67 7-6 8-5 9-5 64 9-0 18-2 27-9 387 ! 4-8 5'8 67 77 8-6 9-6 63 9-1 18-5 28-4 39-5 4-8 5-8 6-8 7-8 87 97 62 9-3 18-8 28-9 40-2 4-9 5-9 6-9 7-8 8-8 9-8 61 9-4 19-1 29-4 41-0 5-0 6-0 6-9 7-9 8-9 9'9 60 9-6 19-5 30-0 41-8 5-0 6-o 7-0 8-0 9-0 IO-0 59 9-8 19-9 30-6 42-6 s-i 6-1 7-1 8-2 9-2 IO'2 58 9-9 20-2 3i-2 43-5 5'2 6-2 7-3 8-3 9-4 IO"4 io-6 io-8 57 56 IO-I 10-3 20-6 21 -O 31-8 32-4 44-5 S'3 5'4 6-4 6-5 7'4 7-6 8-5 8-6 9-5 97 4-3 55 10-5 21-4 33'° 4-4 5-5 6-6 77 8-8 9/9 11-0 54 10-7 22 -O 337 4'5 5-6 67 7-8 9-0 IO-I II'2 53 10-9 22"4 34'4 4-6 57 6-8 8-o 9-1 10-3 n-4 52 1 1 -i 22-8 35-2 4-6 5-8 7-0 8-i 9-3 10-4 1 1 -6 51 11 -3 23-2 36-0 47 5-9 7'i 8'3 9-4 io-6 1 1 '8 50 1 1 -5 23-6 36-9 4-8 6-o 7-2 8-4 9-6 io-8 12-0 Example: If V l^y\ 120° 240° 0° 120° 240° 0° 120" 240° 0° 41 EXAMPLE OF THE USE OF THE TABLES remain and manifest themselves by an accordance of phase which may be even striking, but is not significant of reality. It merely means that the larger terms have been removed. We proceed, therefore, to the exposition of this sounder method, which had not been reached until the foregoing was in type. DEFINITIVE PROCEDURE Suppose the whole available period of observation divided into exact submultiples. In the present case we have observations (of azimuth of the Cape Transit Circle) for the years 1856-1904, i. e. 49 years or 588 months. We need not write down the whole series of submultiples : those which concern us in the example under consideration are near 15 months, and it will suffice to consider the Divisors 36 37 38 39 40 41 4 2 43 Submultiples 16-33 15-89 15-48 15-08 14-70 14-34 14-00 13-68 months The coefficients of sin 6 and cos 6 should be determined for these periods, and for these only : and the information published in this form. It is not absolutely necessary to calculate all the above in connexion with the series in 15 months : the first two are nearer 16 months, and the last three nearer 14. But it will be a useful check on the work to include these overlaps. The reasons for selecting these definite periods cannot be fully given here ; but briefly they may be thus summarized : (a) The investigation of all these periods is necessary. Fourier showed that the coefficients are all independent of one another : hence no one can be inferred from the others. (b) It is also sufficient : it can be shown that the coefficients for any intermediate period can be deduced from these standard constituents. To these essential reasons may be added : (c) That the procedure is instructive : for instance, we see at once why the elimination of one such periodicity (as 13-9 months in the above example) is liable to reveal the existence of others. (d) Finally, the procedure will be found convenient. For instance, it avoids the uncertainty of that above sketched. We will now apply it to the above example. Suppose we are to deal with a true periodicity of 16 months exactly. Since we have cut up the series into lengths of 15 months, successive maxima will fall one month later, until after sixteen periods of 15 months each the maximum returns to the same place : the phase thus changes by 36o°/i6 each period, or by 5 x 36o°/i6 in five periods. Generally, if the period to be dealt with is 15 + x months, the change of phase is 5 a; x 3607(15 + x) 42 EXAMPLE OF THE USE OF THE TABLES which for the periods above enumerated has the values Divisor 36 37 38 39 40 41 42 43 Submultiplo 16-33 15^9 15-48 15-08 14-70 14-34 14-00 13-68 Advance + 146° + 100° + 56° +10° — 37 — 83 -129 —174° The calculation of the coefficients may now be arranged as follows : 1 Add Sum = U, = p sin Q 2 a, = p cos Q.i U9° - 73° + 44 + 42 - 5o° 69° + 57 + 21 266 -219 47 86 + 63 + 59 -150 116 + 77 - 38 164 - 5 159 63 + 23 -59 — 250 274 - 63 + 4 71 — 151 280 103 — IOI + 18 -350 81 + 102 + 16 240 -297 303 158 -133 4-86 - 90 150 + 79 -137 116 - «3 33 25 + 14 + 21 — 190 286 - 24 + 7 300 -229 7i 28 Mean + 26 + 9 — 290 10 + 5 + 28 - 9 + 25 + 33 - 14 In the first column q is repeated from p. 39 : in the second successive multiples of 146 (which is the advance for divisor 36) are written down, beginning, however, with 14672 to reduce from the mean of the first group to the beginning of it. [It will clearly be convenient to reduce all our terms to the same epoch, and the time of the first observation is hereby adopted as that epoch.] In the third column Q 1 is formed as the sum of the first two : then p is repeated from p. 39 : and p sin Q v p cos Qi are formed in the next two columns. The remainder of the columns give the formation of p sin Q 2 and p cos Q 2 in the same way, using now ioo°, the advance for divisor 37. The whole set of values works out as follows : Divisor 36 37 38 39 40 41 42 43 Period i6-33 15-89 15-48 15-08 14-70 14-34 14-00 13-68 S - 9 +33 — 8 + 5 — 10 - 9 +53 -48 a +25 -14 + 8 -25 + 11 + 15 -32 + 7 V& Vi +v 27 36 11 26 15 18 62 49 Before considering the significance of these figures we may notice two sources of inaccuracy. Firstly, these terms should each run through an exact number of periods, and if we had used the whole series of observa- tions this would be the case. But if we stop at the end of the last complete set of 15 months, and all the more if we stop at the end of the last complete set of 5 x 15 = 75 months, some observations are not used. In the present example we only use 7 x 75 = 525 months out of the whole 588 : hence the coefficients deduced by this summary process will be to that extent inaccurate. But since the process is essentially one of exploration, to be supplemented by more exact methods (e.g. the use of 3 or more figures instead of 2) where necessary, this defect is not serious ; and there are several tolerably obvious remedies, if it is desired to apply one. 43 EXAMPLE OF THE USE OF THE TABLES Secondly, grouping five periods together depresses the coefficient of a periodicity which differs from 15 months exactly. Suppose that the change of phase is per period, or 5 in 5 periods. Then a set of 5 periods will give approximately the terms : q cos (6 a - 2 0) + q cos (0 o - 0) + q cos 6 + q cos (0 +