BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF 1891 AJBAm ^114 ^ olin.anx The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031363165 MATHEMATICAL AND PHYSICAL TABLES MATHEMATICAL AND PHYSICAL TABLES FOR THE USE OF STUDENTS IN TECHNICAL SCHOOLS AND COLLEGES BY JAMES P. WRAPSON, B.A. (Dublin) CHIEF LECTURER IN PURE AND APPLIED MATHEMATICS AT THE MUNICIPAL TECHNICAL SCHOOL, MANCHESTER AND W. W. HALDANE GEE, B.Sc. (Lond.) CHIEF LECTURER IN PHYSICS AND ELECTRICAL ENGINEERING AT THE MUNICIPAL TECHNICAL SCHOOL, MANCHESTER ; FORMERLY LECTURER OF THE VICTORIA UNIVERSITY MACMILLAN AND CO., Limited NEW YORK: THE MACMILLAN COMPANY 1898 All rights reserved GLASGOW : PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO. PREFACE This set of tables embodies in a compact form useful mathematical tables and the more important formulae and constants required in the teaching of mathematics and physics. The tabulation of formulae and constants gives to the student a more comprehensive and less confused idea of the instruction he receives throughout his two or three years of study, besides providing him with a necessary work of reference for the Physical, Electrical Engineering, and Mechanical Laboratories and the Mathematical Class- room. The tables should therefore become a constant companion to the student of Mathematics, Physics, and Engineering. The student of Chemical Physics will find a number of the tables of especial value. Considerable pains have been taken to obtain correct data, but since a large number of constants have still to be determined, blanks in the tables are unavoidable. These we hope gradually to fill up, either by availing ourselves of the experiments of others or those made in our own laboratories. Although not specially designed for the use of candidates for examinations, we believe that the Tables will be of assistance to students, especially those in the classes of the recently instituted course of PRACTICAL Mathematics of the Science and Art Department. J. P. w. W. W. H. G. November 1898. CONTENTS PAGE Use of Logarithmic Tables, 3 Four Place Logarithms, - - g Four Place Antilogarithms, io Use of Trigonometrical Tables, 12 Natural Sines, 14 Natural Cosines, 16 Natural Tangents, 18 Logarithmic Sines, 20 Logarithmic Cosines, - 22 Logarithmic Tangents, 24 Table of Squares, Square Roots, Cube Roots, and Reciprocals, 26 Table of Weights and Measures, 27 Formulae in Mensuration, 31 Algebraical Formulae, 37 Table of Approximations, 41 Plane Trigonometrical Formulae, 42 Spherical Trigonometrical Formulae, 46 Formulae used in Analytical Geometry, 49 viii CONTENTS PAGE Differential and Integral Calculus, 5^ Formulae in Dynamics, - 62 Table of Moments of Inertia, 82 Formulae in Hydrostatics, 86 Tables of Physical Properties, 93 Vibratory Motion, 143 Formulae in Acoustics, - - 146 Formulae in Optics, 150 Formulae in Heat, - 162 Formulae in Pure and Applied Magnetism and Electricity, 171 Appendix — Conversion Table, - 203 Table of Velocities, - io4 Dimensions of British Coinage, 205 The Morse Code, 206 The Greek Alphabet, 207 Exponential Functions, 208 Logarithms to Base e, 208 Mechanical and Electrical Analogies, 209 Books of Reference, 210 Index,- - - 211 TABLES IN PURE AND APPLIED MATHEMATICS. USE OF THE LOGARITHMIC TABLES. Definition. — The logarithm of a given number to any base is the index of the power to which the base must be raised in order that it may be equal to the given nuniber. The logarithms in the following tables are to base lo, and are called common logarithms, hence The common logarithm of a given number is the index of the power to which lo must be raised in order that it may be equal to the given number. Examples: lo-^^" =2, hence .3010 is the logarithm of 2. J0.1771 -^^ hence .4771 is the logarithm of 3. K^-iTTi - joo, hence 2.4771 is the logarithm of 300. These relations are always written thus : log 2 = .3010, log 3 = .4771. log 300 = 2.4771. A logarithm will in general consist of two parts, an integral part called the characteristic, and a decimal part called the mantissa. Thus log 2oo = 2.3oro; here 2 is the characteristic and .3010 the mantissa. The characteristic is determined by inspection of the given number. The mantissa is found from the tables. To determine the characteristic of the log of a number. (i) If the given number is greater than i, count the number of figures before the decimal point ; the characteristic is one less than this number. Examples : The characteristic of log 302.64 = 2, of log 65.243 = i. 4 MATHEMATICAL AND PHYSICAL TABLES. (2) If the number is less than i, count the number of zeros between the decimal point and the first significant figure; the characteristic is one greater than this number, and is negative, the minus sign being written above the figure. Examples: The characteristic of log .0064 = 3, of log .6051 = i. "To determine the mantissa of the log of a number. Find the first two figures of the number in the left-hand column of the table headed logarithms of numbers ; opposite to this in the column headed by the third figure of the number will be found a number consisting of four figures ; add to this the number obtained from that difference-column which is headed by the fourth figure of the given number, and the result will be the mantissa of the required logarithm. Example : Find the logarithm of 342.6. First, we note that there are three figures before the decimal point, hence the characteristic will be 2. Next, we proceed to find the mantissa. Turn to the table headed logarithms of numbers, and look for 34 in the left-hand column. Opposite to this in the column headed 2 we find 5340. Looking along the same line we find in the difference-column headed 6 the figure 8, which, added to 5340, gives 5348. This is the required mantissa, hence log 342.6 = 2.5348. Find the logarithm of .005172. Here we note that the number is less than i, and there are two zeros between the point and the first significant figure, hence the characteristic is 3. The mantissa, determined as in the above example, is 7137 j hence log .005172 = 3.7137. Note on negative characteristics. It must be carefully borne in mind that the mantissa of a logarithm is always kept positive, though the characteristic may be negative; thus 3.5283 is equiva- lent to -3 + .5283, and must be distinguished from -3.5283, in which both the integral and decimal part are negative. If a logarithm is obtained which is entirely negative, as -2.4717, it must be converted into one in which the characteristic alone is negative; this is done by subtracting i from the characteristic and adding i to the mantissa, thus : -2.4717= -3-Ki-.47i7) = 3-S283. MATHEMATICAL AND PHYSICAL TABLES. Addition, subtraction, and multiplication of such logarithms is performed by the algebraic method for addition, subtraction, and multiplication of negative quantities. Example I. Add together 4.3712 3-1864 3-5971 6.4132 Atis. 1.5679 Example II. From 3-4771 Subtract 2-5015 Arts. 4.9756 Example III. Multiply 2.4963 By 3 Ans. 5. Division of logarithms having negative characteristics requires a special artifice. This consists in subtracting from the charac- teristic such a number as will make it a multiple of the divisor, and adding the same quantity to the mantissa. Example I. : Divide 3.4772 by 4. 3.4772 ^ 4 + 1.4772 4 4 ^1-3693- Example II. : Divide 7.6241 by 3. 7.6241 _ 9 + 2.6241 _ _ 8747. Definition. — The antilogarithm of a given logarithm is the number corresponding to it. Examples: Wehaveseen thatlog2 = . 3010, log3 = . 4771, log 300 = 2.4771, hence antilog .3010= 2, antilog .4771 = 3, antilog 2.4771 = 300. To find the number corresponding to a given logarithm. In the table of antilogarithms find the first two figures of the mantissa in the left-hand column, the third figure at the top of the page, and the fourth figure at the top of the difference-column. The result will be the required number, and the decimal point must be placed as follows : If the characteristic of the given logarithm is positive, point off one figure more, counting from the first figure of the number; if the characteristic is negative, cyphers one less in number than the characteristic must be prefixed, the decimal point being placed before them. Example I. : Find the number whose logarithm is 2.4063. From the table of antilogarithms opposite to .40 and in the column headed 6 we find 2547, and in the difference-column headed 3 we find 2, which must be added to 2547, giving 2549. We then note that the characteristic of the given logarithm is 2, and therefore by the rule given the number required is 254.9. Example II. : Find the number whose logarithm is 2.5194, 6 MATHEMATICAL AND PHYSICAL TABLES. Proceeding as in the above, we find 3307 to be the number corresponding to the mantissa given. The characteristic being 2, we must prefix one cypher, and therefore the required number is -0330^ APPLICATIONS OF LOGARITHMS. Logarithms are used for facilitating numerical calculations where the operations involved are multiplication, division, in- volution, and evolution. The following rules and examples will show how this is done : 1. Multiplication : To multiply numbers together, add their logs and find the antilog of the result. Example : Multiply 1263 by 3.845, log 1263 =3.1014 log 3.845= .5849 Antilog 3. 6863 = 4856 /4 ns. 3- 6863 2. Division : To divide numbers, subtract the log of the divisor from the log of the dividend, and find the antilog of the result. Example: Divide 1263 by 3.845, log 1263 =3.1014 log 3.845= .5849 Antilog 2.5165 = 328.5 Ans. 2.5165 3. Involution : To raise a number to any power, multiply its log by the index of the power, and find the antilog of the result. Example : Find the value of (.3845)3, log .3845 = 7.5849 3 Antilog 2.7547 = .05684 Alls. 2.7547 4. Evolution : To extract any root of a number, divide the log of the number by the index of the root, and find the antilog of the result. Example : Find the value of V.03845, log .03845 = 2.5849 Antilog 1.5283 = .3375 Ans. 5^ = T.5283 MATHEMATICAL AND PHYSICAL TABLES. 7 The following numerical examples will show the method of applying logarithms to laboratory calculations : 1. Find the diameter of a wrought-iron shaft which is subjected to a twisting moment of 10 tons acting at a leverage of 24 inches. 7"= 10x2240 X 24 inch-lbs., /= 9000 lbs., ^7r = 3. 142, c 2240 X 24 X 16 log 10 =1 9000x3.142 ' log 2240 = 3.3502 log 24 =1.3802 log 16 =1.2041 6-9345 log 9000 =3.9542 log 3. 142= .4972 4.4514 s/iox; And antilog .8277 = 6.725, 3 )2.4831 .'. (/= 6.725 inches ^Ki. -8277 2. Calculate the brake horse-power of an Immisch motor from the follow- ing data : — Drum armature, 14 inches long, 7.25 inches diameter ; number of revolutions per minute, 1000. H.P. = 14 X (7.25)2 X 1000 x. 000015. logi4 =1.1461 2 log 7.25 =1.7206 log 1000 = 3.0000 log. 000015 = 5.1761 And antilog 1.0428 = 11.04, .■. H.P. = 11.04 /^«j. 1.0428 Find the value of g from the following data : — Length of pendulum, 230 cm. ; time of oscillation, 1.52 sees. ,- (3-142)^x230 2 log 3. 142= .9944 .?■= (1-52)2 log23o =2.3617 Antilog 2.9925 = 982.8, 3-3S6i 2 log 1.52 = .3636 ■'■ .^=982.8 cm.-sec^. Ans. 2.9925 4. Find Young's modulus by flexure of a steel bar supported at both ends, the following data having been determined : Z = 6i.lcm., /■= 1000x981.4 dynes, ? = . 1075cm., ^=1.95 cm., a?=. 509cm. Y_ 1000 X 981.4 X (61. 1 )' log 1000 =3.0000 4X 1. 95 X (.509)3 X. 1075" log 981.4 = 2.9919 3 log6l.l =5.3580 Antilog 12.3062 = 2.024 X lo'' II -1499 log 4 = .6021 log I ■95 = .2900 3log- 509 = 1.1201 log. 1075 = 1.0315 I -0437 y= 2.024 X 10^2 dynes per sq- cm. Ans. 12.3062 LOGARITHMS. 2 3 4 5 6 7 8 9 Dlfferenoe-Oolumn. 123 456 789 10 11 12 13 14 16 16 17 IS 19 20 21 22 23 24 26 27 2S 29 30 31 32 33 34 36 36 91 38 39 40 41 42 43 44 46 46 47 48 49 50 51 52 63 64 0414 0792 "39 1461 1761 2041 2304 2SS3 2788 3010 3222 3424 3617 3802 3979 4150 4314 4472 4624 4771 4914 5051 5Li8S 5315 5441 "1^563 5682 S7e8 59" 6021 6128 6232 6335 643s 6532 6628 6721 6812 6902 6990 7076 7160 7243 7324 0043 0086 0128 0453 0492 0531 0828 0864 0899 I 173 i2o6 1239 1492 1523 1553 1790 1818 1847 2068 2095 2122 2330 2355 2380 2577 2601 2625 2810 2833 2856 3032 3054 3075 3243 3263 3284 3444 3464 3483 3636 3655 3674 3820 3838 3856 3997 4014 4031 4166 4183 4200 4330 4346 4362 4487 4502 4518 4639 4654 4669 4786 4800 4814 4928 4942 4955 5065 5079 5092 5198 5211 5224 5328 S340 S3S3 5453 5465 5478 5575 5587 5599 5694 5705 5717 5809 5821 5832 5922 5933 5944 6031 6042 6053 6138 6149 6160 6243 6253 6263 6345 635s 6365 6444 6454 6464 6542 6551 6561 6637 6646 6656 6730 6739 6749 6821 6830 6839 691 I 6920 6928 6998 7007 7016 7084 7093 7101 7168 7177 7185 7251 7259 7267 7332 7340 7348 2 3 0170 0212 0253 0569 0607 0645 0934 0969 1004 1271 1303 1335 1584 1614 1644 1875 1903 1931 2148 2175 2201 2405 2430 2455 2648 2672 2695 2878 2900 2923 3096 3118 3139 3304 3324 3345 3502 3522 3541 3692 371 I 3729 3874 3892 3909 4048 4065 4082 4216 4232 4249 4378 4393 4409 4533 4548 45S4 4683 4698 4713 4829 4843 4857 4969 4983 4997 5105 5119 5132 5237 5250 5263 5366 5378 5391 5490 5502 SS14 561 1 5623 5635 5729 5740 5752 5843 5855 5866 5955 5966 5977 6064 607s 6085 6170 6180 6191 6274 6284 6294 6375 6385 639s 6474 6484 6493 6571 6580 6590 6665 6675 6684 6758 6767 6776 6848 6857 6866 6937 6946 695s 7024 7033 7042 7110 7118 7126 7193 7202 7210 7275 7284 7292 7356 7364 7372 4 5 6 0294 0334 0374 0682 0719 0755 1038 1072 1106 1367 1399 1430 1673 1703 1732 1959 1987 2014 2227 2253 2279 2480 2504 2529 2718 2742 2765 2945 2967 2989 3160 3181 3201 3365 3385 3404 3560 3579 3598 3747 3766 3784 3927 3945 3962 4099 41 16 4133 4265 4281 4298 4425 4440 4456 4579 4594 4609 4728 4742 4757 4871 4886 4900 501 I 5024 5038 5145 5159 5172 5276 5289 S302 5403 5416 5428 5527 5539 555" 5647 5658 5670 5763 5775 5786 5877 5888 5899 5988 5999 6010 6096 6107 61 17 6201 6212 6222 6304 6314 6325 6405 6415 6425 6503 6513 6522 6599 6609 6618 6693 6702 6712 6785 6794 6803 6875 6884 6893 6964 6972 6981 7050 7059 7067 7135 7143 7152 7218 7226 7235 7300 7308 7316 7380 738S 7396 7 8 9 4 8 12 8 II 7 10 6 10 6 9 6 8 5 8 5 7 5 7 4 7 4 6 3 4 3 4 3 4 3 4 2 4 2 4 2 3 2 3 2 3 2 3 2 3 2 2 2 2 2 2 12 3 17 21 25 15 19 23 14 17 21 13 16 19 12 15 18 II 14 17 11 13 '6 10 12 IS 9 12 14 9 II 13 8 II 13 8 10 12 8 10 12 7 9 II 7 9 II 7 9 10 8 10 8 9 8 9 7 9 7 9 7 8 7 8 6 8 6 8 6 7 3 4 5 3 4 5 3 4 5 3 4 5 4 5 6 LOGARITHMS. 7404 7482 7559 7634 7709 7782 7853 7924 7993 8062 8129 8195 8261 8325 8388 8451 8513 8573 8633 8692 8751 8808 8865 8921 8976 9031 9085 9138 9191 9243 9294 9345 9395 9445 9494 9542 9590 9638 9685 9731 9777 9823 9868 9912 9956 2 3 7412 7419 7427 7490 7497 7505 7566 7S74 7582 7642 7649 7657 7716 7723 7731 7789 7796 7803 7860 7868 7875 7931 7938 7945 8000 8007 8014 8069 8075 8082 8136 8142 8149 8202 8209 8215 8267 8274 8280 8331 8338 8344 8395 8401 8407 8457 8463 8470 8519 8525 8531 8579 8585 8591 8639 8645 8651 8698 8704 8710 8756 8762 876S 8814 8S20 8825 8871 8876 8882 8927 8932 8938 8982 8987 8993 9036 9042 9047 9090 9096 9101 9143 9149 9154 9196 9201 9206 9248 9253 9258 9299 9304 9309 9350 9355 9360 9400 9405 9410 9450 9455 9460 9499 9504 9509 9547 9552 9557 9595 9600 9605 9643 9647 9652 9689 9694 9699 9736 9741 9745 9782 9786 9791 9827 9832 9836 9872 9877 9881 9917 9921 9926 9961 9965 9969 2 3 4 5 6 7435 7443 745 i 7513 7520 7528 7589 7597 7604 7664 7672 7679 7738 7745 7752 7810 7818 7825 7882 7889 7896 7952 7959 7966 8021 8028 8035 8089 8096 8102 8156 8162 S169 8222 8228 823s 8287 8293 8299 8351 8357 8363 8414 8420 8426 8476 8482 S488 8537 8543 8549 8597 8603 8609 8657 8663 8669 8716 8722 8727 8774 8779 8785 8831 8837 8842 8887 8893 8899 8943 8949 8954 8998 9004 9009 9053 9058 9063 9106 9112 9117 9159 9165 9170 9212 9217 9222 9263 9269 9274 9315 9320 9325 9365 9370 9375 9415 9420 9425 9465 9469 9474 9513 9518 9523 9562 9566 9571 9609 9614 9619 9657 9661 9666 9703 9708 9713 9750 9754 9759 9795 9800 9805 9841 984S 9850 9886 9890 9894 9930 9934 9939 9974 9978 9983 4 5 6 7 8 9 7459 7466 7474 7536 7543''755i 7612 7619 7627 7686 7694 7701 7760 7767 7774 7832 7839 7846 7903 7910 7917 7973 7980 7987 8041 8048 8055 8109 81 16 8122 8176 8182 8189 8241 8248 8254 8306 8312 8319 8370 8376 8382 8432 8439 844s 8494 8500 8506 8555 8561 8567 861S 8621 8627 8675 8681 8686 8733 8739 8745 8791 8797 8802 8848 8854 8859 8904 8910 8915 8960 8965 8971 9015 9020 9025 9069 9074 9079 9122 9128 9133 9175 9180 9186 9227 9232 9238 9279 9284 9289 9330 9335 9340 9380 9385 9390 9430 9435 9440 9479 9484 9489 9528 9533 9538 9576 9581 9586 9624 9628 9633 9671 9675 9680 9717 9722 9727 9763 9768 9773 9809 9814 9818 9854 9859 9863 9899 9903 9908 9943 9948 99'52 9987 9991 9996 7 8 9 Diflferenoe-Oolumn. 123 456 789 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I 2 I I I I I I I I I I I I I I I I I I 12 3 3 4 5 4 4 3 4 3 4 3 4 3 4 234 234 234 234 233 233 233 233 233 233 233 223 223 223 223 223 223 223 223 223 2 2 .3 223 223 223 4 5 6 ANTILOGARITHMS. •00 12 3 4 5 6 7 8 9 Difference-Column. | 1 2 3 4 5 6 7 8 9 1000 1002 1005 1007 1009 1012 1014 1016 1019 1021 I r I I 2 2 2 • •01 •02 •03 •04 •05 1023 1047 1072 1096 1122 1026 1028 1030 1050 1052 1054 1074 1076 1079 1099 1 102 1 104 1125 1127 1130 1033 1035 1038 1057 1059 1062 1081 1084 1086 II07 IIO9 IU2 II32 113s II38 1040 1042 1045 1064 1067 1069 1089 1091 1094 1114 1117 1119 1 140 1 143 1 146 I I I I I I I I I I 2 I 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 •06 •07 •08 •09 •10 1 148 1202 1230 1259 1151 1153 1156 1178 1180 1183 1205 1208 121 1 1233 1236 1239 1262 1265 1268 IIS9 1161 1164 1186 1189 1191 1213 1216 1219 1242 124S 1247 1271 1274 1276 1167 1169 1172 1 194 1 197 1 199 1222 1225 1227 1250 1253 1256 1279 1282 1285 I 2 I 2 1 2 I 2 I 2 2 2 2 2 2 2 2 2 2 2 3 2 3 2 3 •11 •12 •13 •14 •15 1288 1318 1349 1380 1413 1291 1294 1297 1 321 1324 1327 1352 135s 1358 1384 1387 1390 1416 1419 1422 1300 1303 1306 1330 1334 1337 1361 136s 1368 1393 1396 1400 1426 1429 1432 1309 1312 131S 1340 1343 1346 1371 1374 1377 1403 1406 1409 1435 1439 1442 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 3 3 3 3 3 3 3 •16 •17 •18 •19 :20 144S •479 1514 1549 1585 1449 1452 145s 1483 1486 1489 1517 1521 1524 1552 1556 1560 1589 1592 1596 1459 1462 1466 1493 1496 1500 1528 1531 1535 1563 1567 1570 1600 1603 1607 1469 1472 1476 1503 1507 1510 1538 1542 1545 1574 1578 1581 1611 1614 1618 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 •21 •22 •23 •24 •25 1622 1660 1698 1738 1778 1626 1629 1633 1663 1667 1671 1702 1706 1710 1742 1746 1750 1782 1786 1791 1637 1641 1644 167s 1679 1683 1714 1718 1722 1754 1758 1762 1795 1799 1803 1648 1652 1655 1687 1690 1694 1726 1730 1734 1766 1770 1774 1807 181 I 1816 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 3 4 3 4 •26 •27 •28 •29 •30 1820 1862 190S 1950 1995 1824 1828 1832 1866 1871 187s 1910 1914 1919 '954 1959 1963 2000 2004 2009 1837 1841 1845 1879 1884 1888 1923 1928 1932 1968 1972 1977 2014 2018 2023 1849 1854 1858 1892 1897 1901 1936 1941 1945 1982 1986 1991 202S 2032 2037 2 2 2 2 2 2 3 2 3 2 3 2 3 2 3 3 3 3 3 3 3 4 3 4 4 4 4 4 4 4 •31 •32 •33 •34 •35 2042 2089 2138 2188 2239 2046 2051 2056 2094 2099 2104 2143 2148 2153 2193 2198 2203 2244 2249 2254 2061 2065 2070 2109 2113 21 18 2158 2163 2168 2208 2213 2218 2259 2265 2270 2075 2080 2084 2123 2128 2133 2173 2178 2183 2223 2228 2234 2275 2280 2286 I 2 I 2 2 2 2 2 2 2 3 2 3 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 4 5 •36 •37 •38 •39 •40 2291 2344 2399 2455 2512 2296 2301 2307 2350 2355 2360 2404 2410 2415 2460 2466 2472 2518 2523 2529 2312 2317 2323 2366 2371 2377 2421 2427 2432 2477 2483 2489 2535 2541 2547 2328 2333 2339 2382 2388 2393 2438 2443 2449 2495 2500 2506 2553 2559 2564 I 2 I 2 I 2 I 2 I 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 4 5 4 5 5 5 5 5 •41 •42 •43 •44 •46 2570 2630 2692 2754 2818 2576 2582 2588 2636 2642 2649 2698 2704 2710 2761 2767 2773 2825 2831 2838 2594 2600 2606 2655 2661 2667 2716 2723 2729 2780 2786 2793 2844 2851 2858 26J2 2618 2624 2673 2679 2685 2735 2742 2748 2799 2805 2812 2864 2871 2877 I 2 I 2 I 2 I 2 I 2 2 2 3 3 3 3.^4 3 4 3 4 3 4 3 4 4 4 4 4 5 5 5 5 6 5 6 5 6 •46 •47 •48 •49 2884 2951 3020 3090 2891 2897 2904 2958 2965 2972 3027 3034 3041 3097 310S 3112 291 I 2917 2924 2979 2985 2992 3048 3055 3062 3119 3126 3133 2931 2938 2944 2999 3006 3013 3069 3076 3083 3141 3148 3155 I 2 I 2 I 2 I 2 3 3 3 3 3 4 3 4 4 4 4 4 5 5 5 5 5 ^ 5 6 6 6 6 6 12 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 ANTILOGARITHMS. •50 12 3 4 5 6 7 8 9 Difference-Column. | 1 2 3 4 5 6 7 8 9 3162 3170 3177 3184 3192 3199 3206 3214 3221 3228 I I 2 3 4 4 5 6 7 ■61 •52 •53 •54 ■55 3236 3388 3467 3S48 3243 3251 3258 3319 3327 3334 3396 3404 3412 3475 3483 3491 3556 3565 3573 3266 3273 3281 3342 3350 3357 3420 3428 3436 3499 3508 3516 3581 3589 3597 3289 3296 3304 3365 3373 3381 3443 3451 3459 3524 3532 3540 3606 3614 3622 I 2 1 2 I 2 I 2 I 2 2 2 2 2 2 3 4 5 3 4 5 3 4,5 3 4 5 3 4 5 5 6 7 5 6 7 667 667 6 7 7 •56 •57 -58 •59 '60 3631 371S 3802 3890 3981 3639 3648 3656 3724 3733 3741 381 I 3819 3828 3899 3908 3917 3990 3999 4009 3664 3673 3681 3750 3758 3767 3837 3846 3855 3926 3936 3945 4018 4027 4036 3690 3698 3707 3776 3784 3793 3864 3873 3882 3954 3963 3972 4046 4055 4064 I 2 I 2 1 2 I 2 I 2 3 3 3 3 3 3 4 5 3 4 5 4 4 5 4 5 5 456 678 678 678 678 678 •61 •62 ■63 ■64 •65 4074 4169 4266 4365 4467 4083 4093 4102 4178 4188 4198 4276 4285 4295 4375 4385 4395 4477 4487 4498 4111 4121 4130 4207 4217 4227 4305 4315 4325 4406 44:6 4426 4508 4519 4529 4140 4150 4159 4236 4246 4256 4335 4345 4355 4436 4446 4457 4539 4550 4560 I 2 I 2 I 2 I 2 I 2 3 3 3 3 3 456 456 456 456 456 7 8 9 7 8 9 7 8 9 7 8 9 7 8 9 ■66 ■67 ■68 ■69 ■70 4571 4677 4786 4898 5012 4581 4592 4603 4688 4699 4710 4797 4808 4819 4909 4920 4932 5023 5035 5047 4613 4624 4634 4721 4732 4742 4831 4842 4853 4943 4955 4966 5058 5070 5082 4645 4656 4667 4753 4764 4775 4864 4875 4887 4977 4989 50CO 5093 5105 5117 I 2 I 2 I 2 I 2 I 2 3 3 3 3 4 456 4 5 7 467 5 6 7 5 6 7 7 9 10 8 9 10 8 9 10 8 9 10 8 9 II ■71 ■72 ■73 ■74 ■75 5129 5248 5370 5495 5623 5140 5152 5164 5260 5272 5284 5383 5395 5408 5508 S52I 5534 5636 5649 5662 5176 5188 5200 5297 5309 5321 5420 5433 5445 5546 5559 5572 5675 5689 5702 5212 5224 5236 5333 5346 5358 5458 5470 5483 5585 5598 5610 5715 5728 5741 I 2 I 2 I 3 I 3 I 3 4 4 4 4 4 5 6 7 5 6 7 568 568 5 7 8 8 10 II 9 10 II 9 10 11 9 10 12 9 10 12 •76 •77 •78 •79 •80 Pi 6026 6166 6310 5768 5781 5794 5902 5916 5929 6039 6053 6067 6180 6194 6209 6324 6339 6353 5808 5821 5834 5943 5957 597° 6081 6095 6109, 6223 6237 6252 6368 6383 6397 5848 5861 5875 5984 5998 6012 6124 6138 6152 6266 6281 6295 6412 6427 6442 I 3 I 3 I 3 I 3 I 3 4 4 4 4 4 5 7 8 5 7 8 678 679 679 9 II 12 10 II 12 10 II 13 10 II 13 10 12 13 •81 •82 ■83 •84 •85 6457 6607 6761 6918 7079 6471 6486 6501 6622 6637 6653 6776 6792 6808 6934 6950 6966 7096 7112 7129 6516 6531 6544 6668 6683 6699 6823 6839 685s 6982 6998 7015 7145 7161 7178 6561 6577 6592 6714 6730 6745 6871 6887 6902 7031 7047 7063 7194 721 1 7228 2 3 2 3 2 3 2 3 2 3 5 5 5 5 5 689 689 689 6 8 10 7 8 10 II 12 14 II 12 14 II 13 14 11 13 15 12 13 15 •86 ■87 '88 •89 •90 7244 7413 7586 7762 7943 7261 7278 729s 7430 7447 7464 7603 7621 7638 7780 779^7816 7962 798^^998 731 1 7328- 7345 7482 7499 7516 7656 7674 7691 7834 7852 7870 8017 8035 8054 7362 7379 7396 7534 7551 7568 7709 7727 7745 7889 7907 79^ 8072 8091 81 10 2 3 2 3 2 4 2 4 2 4 5 5 5 5 6 7 8 10 7 9 10 7 9 II 7 9 II 7 9 II 12 13 IS 12 14 16 12 14 16 13 14 16 13 15 17 •91 •92 •93 ■94 •95 8128 8318 8511 8710 8913 8147 8166 8185 8337 8356 8375 8531 8551 8570 8730 8750 8770 8933 8954 8974 8204 8222 8241 8395 8414 8433 8590 8610 8630 8790 8810 8831 8995 9016 9036 8260 8279 8299 8453 8472 8492 8650 8670 8690 8851 8872 8892 9057 9078 9099 2 4 2 4 2 4 2 4 2 4 6 6 6 6 6 8 9 II 8 10 12 8 10 12 8 10 12 8 10 12 13 15 17 14 15 17 14 16 18 14 16 18 15 17 19 •96 ■97 •98 •99 9120 9333 9550 9772 9141 9162 9183 9354 9376 9397 9572 9594 9616 9795 9817 9840 9204 9226 9247 9419 9441 9462 9638 9661 9683 9863 9886 9908 9268 9290 931 1 9484 9506 9528 9705 9727 9750 9931 9954 9977 2 4 2 4 2 4 2 5 6 7 7 7 8 II 13 9 II 13 9 II 13 9 " 14 15 17 19 15 17 20 16 18 20 16 18 20 12 3 4 5 6 7 9 8 1 2 3 4 5 6 7 8 9 12 MATHEMATICAL AND PHYSICAL TABLES. USE OF TRIGONOMETRICAL TABLES. To find the natural sine, cosine, or tangent of any angle. Turn to the table of natural sines, cosines, or tangents as the case may- be; find the number of degrees in the left-hand column; pass along to the column headed by the number of minutes next less than the minutes in the given angle; take out the number so obtained; find from the difference-column the difference cor- responding to the remaining minutes in the given angle ; add this to the number just obtained in the case of the sine and tangent, but subtract it in the case of the cosine, and the result will be the required function. Example I. : Find the sine and tangent of 52° 28'. From table of natural sines sin 52° 24' = 7923 difference for 4' = 7 (to be added) sin 52° 28' = 7930 From table of natural tangents tan 52° 24' = I '2985 difference for 4'= 31 (to be added) tan 52° 28' = I ■3016 Example II. : Find the cosine of 24° 15'. From the table of natural cosines cos 24° 12'= -9121 difference for 3' = 4 (to be subtracted) cos 24° 15'= "91 17 To find the logarithmic sine, cosine, or tangent of any angle. Proceed as above, but use the tables of logarithmic sines, cosines, and tangents. It should be borne in mind that the numbers given in these tables are the tabular logarithms, that is to say, they are the true values of log sin, log cos, etc., increased by 10. Example : Find the value of tan 26° 18' x •1563. log tan 26° 18' = 9-6939- io = T'6939 log -1563 =T-I939 Antilog 2 -8878 = -07723 Aus. 2-8878 USE OF TRIGONOMETRICAL TABLES. 13 In the tables of natural tangents, and of logarithmic sines, cosines, and tangents, some of the numbers will be seen to have a line drawn over the first digit ; this indicates that the integral part in the case of a natural tangent, and the characteristic in the case of a logarithmic function, is to be taken from the beginning of the next line lower down. Thus tan 63° 30' — 2.0057, L cos 84° 18' = 8.9970. The following example will further illustrate the method of using the tables : The sides a, i of a triangle ABC are respectively 142 and 128 feet long, and the included angle C is 35° 36' ; find the remaining side and the other two angles to the nearest minute. (i.) ^:^=9o-j =90° -17° 48' =72° 12'. (u.) log tan ^ = log(a-i)-Iog(a + i5)-logtan — = log 14 - log 270 - log tan 17° 48' log log tan 17' 270= ■48'= log 14= =2.4314 = 1.5066 = 1. 1461 1.9380 = 1.2081 ; T.2081 . A-B „ , . . = 9 10 nearly. , A+B,A-B (m.) ^"2 + 2 = 72° 1 2' + 9° 10' = 81° 22'. (iv.) B-^-/-^-^^ = 72° 12' - 9° 10' = 63-2'. (v. ) log f = log a + log sin C - log sin A jog 5;^ 3^0 ^^.^-[^^^^ = log 142 + log sin 35° 36' - log sin 81° 22' ;;^ = 1.9222; logsin8i°22'=T.99Si .'. (r = 83.6. 1-9222 14 NATURAL SINES. 6' 12' 18' 24' 30' 36' 42' 48' 54' Minutes. | 0° 0' orO°-l or0°'2 or0°'3 or0°-4 or0°-5 or 0°-6 or0°*7 or 0°-8 or0°-9 1' 2' 3' 4' 5' oo'oo 0017 003s 0052 0070 0087 0105 0122 0140 0157 3 6 9 12 IS 1 0I7S 0192 0209 0227 0244 0262"- -0279 0297 °3'4 0332 3 f 9 12 IS 2 0349 0366 0384 0401 0419 0436 0454 0471 0488 0S06 3 ! 9 12 15 3 0523 0541 0558 0576 '0593 0610 062^8 0645 0663 0680 3 6 9 12 IS i 0698 0715 0732 0750 0767 0785 d8S2 0819 0837 0854 3 f 9 12 15 5 0871 0889 0906 0924 0941 0958 0976 0993 ion 1028 3 6 9 12 14 6 104s 1063 1080 1097 ins 1132 1 149 1 167 1 184 1201 3 6 9 12 14 7 1219 1236 1253 1271 1288 1305 1323 1340 1357 1374 3 6 9 12 14 8 1392 1409 1426 1444 1461 1478 149s IS?3 1530 1547 3 f 9 12 14 9 1564 1582 1599 1616 1633 1650 1668 i68s 1702 1719 3 6 9 12 14 10 1736 I7S4 1771 1788 1 80s 1822 1840 1857 1874 1891 3 6 9 12 14 11 1908 1925 1942 1959 1977 1994 2011 2028 2045 2062 3 6 9 II 14 12 2079 2096 2113 2130 2147 2164 2:181 2198 2215 2232 3 f 9 II 14 13 2250 2267 2284 2300 2317 2334 2351 2368 2385 2402 ■■5 f 8 II 14 14 2419 2436 2453 2470 2487 2504 2521 2538 2554 2571 3 6 8 II 14 15 2588 2605 2622 2639 26s6 2672 2689 2706 2723 2740 3 6 8 II 14 16 2756 2773 2790 2807 2823 2840 2857 2874 2890 2907 3 6 8 II 14 17 2924 2940 2957 2974 2990 3007 3024 3040 3057 3074 3 6 8 II 14 18 3090 3107 3123 3140 .315S 3173 3190 3206 3223 3239 3 6 8 II 14 19 3256 3272 3289 3305 3322 3338 3355 3371 3387 3404 3 5 8 II 14 20 3420 3437 3453 3469 3486 3502 3518 3535 3551 3567 3 5 8 II 14 21 3584 3600 3616 3633 3649 3665 3681 3697 3714 3730 3 5 8 II 14 - 22 3746 3762 3778 3795 381 1 3827 3843 3859 3875 3891 3 5 8 II 14 23 3907 3923 3939 3955 3971 3987 4003 4019 4035 4051 3 5 8 II 14 2i 4067 4083 4099 4115 4131 4147 4163 4179 4195 4210 3 5 8 n 13 25 4226 4242 4258 4274 4289 4305 4321 4337 4352 4368 3 5 8 II 13 26 4384 4399 4415 4431 4446 4462 4478 4493 4509 4524 3 S 8 10 13 27 4540 4555 4571 4586 4602 4617 4633 4648 4664 4679 3 5 8 10 13 28 469s 4710 4726 4741 4756 4772 4787 4802 4818 4833 3 5 8 10 13 29 4848 4863 4879 4894 4909 4924 4939 4955 4970 4985 3 5 8 10 13 30 5000 5015 5030 5045 5060 5075 5090 5105 5120 S135 3 5 8 10 13 31 5150 5165 5180 5195 5210 5225 5240 5255 5270 5284 2 5 7 10 12 32 5299 5314 5329 5344 5358 5373 5388 5402 5417 5432 2 5 7 10 12 33 5446 5461 5476 S490 5505 5519 5534 5548 5563 5577 2 5 7 10 12 34 SS92 5606 5621 5635 5650 5664 5678 5693 5707 5721 2 5 7 10 12 35 5736 575° 5764 5779 5793 S807 5821 5835 5850 5864 2 5 7 9 12 36 S878 5892 5906 5920 5934 5948 5962 5976 5990 6004 2 5 7 9 12 37 6018 6032 6046 6060 6074 6088 6101 6m5 6129 6143 2 5 7 9 12 38 6157 6170 6184 6198 621 1 622s 6239 6252 6266 6280 2 5 7 9 II 39 6293 6307 6320 6334 6347 6361 6374 6388 6401 6414 2 4 7 9 II 40 6428 6441 6455 6468 6481 6494 6508 6521 6534 6547 2 4 7 9 n 41 6561 6574 6587 6600 6613 6626 6639 6652 666s 6678 2 4 7 9 11 42 6691 6704 6717 6730 6743 6756 6769 6782 6794 6807 2 4 6 9 II 43 6820 6833 6845 6858 6871 6884 6896 6909 6921 6934 2 4 6 8 II 44 6947 6959 6972. 6^84 6997 7009 7022 7034 7046 7059 2 4 6 8 10 0' 6' orO°a 12' or0°'2 18' 01' 0° -3 24' orO°-4 30' or 0°'5 36' or0°-6 42' or 0°'7 48' ox 0°'8 54' or 0°'9 1' 2' 3' 4' 5' Minutes. | NATURAL SINES. 15 6' 12' 18' 24' sc 36' 42' 48' 54' Minutes. | 45° orO°'l or0°-2 or 0°'3 or 0°'4 or 0°'6 or 0°-6 or0°-7 or0°-8 or0°-9 1' 2' 3' 4' 5' 7071 7083 7096 7108 7120 7133 7145 7157 7169 7181 246 8 10 46 7193 7206 7218 7230 7242 7254 7266 7278 7290 7302 246 8 10 47 7314 7325 7337 7349 7361 7373 7385 7396 7408 7420 246 8 10 48 7431 7443 7455 7466 7478 7490 7SOI 7513 7524 7536 246 8 10 49 7547 7558 7570 7581 7593 7604 7615 7627 7638 7649 246 8 9 50 7660 7672 7683 7694 7705 7716 7727 7738 7749 7760 246 7 9 51 7771 7782 7793 7804 7815 7826 7837 7848 7859 7869 2 4 5 7 9 62 7880 7891 7902 7912 7923 7934 7944 7955 7965 7976 2 4 5 7 9 53 7986 7997 8007 8018 8028 8039 8049 8059 8070 8080 2 3 5 7 9 64 8090 8100 8111 8l2I 8131 8141 8151 8161 8171 8181 235 7 8 65 8192 8202 82H 8221 8231 8241 8251 8261 8271 8281 2 3 5 7 8 56 8290 8300 8310 8320 8329 8339 8348 8358 8368 8377 2 3 5 6 8 57 8387 8396 8406 8415 8425 8434 8443 8453 8462 8471 2 3 5 6 8 68 8480 8490 8499 8508 8517 8526 8536 8545 8554 8563 2 3 5 6 8 59 8572 8581 8590 8599 8607 8616 8625 8634 8643 8652 I 3 4 6 7 60 8660 8669 8678 8686 8695 8704 ,8712 8721 8729 8738 I 3 4 6 7 61 8746 8755 8763 8771 8780 8788 8796 8805 8813 8821 I 3 4 6 7 62 8829 8838 8846 8854 8862 8870 8878 8886 8894 8902 I 3 4 5 7 63 8910 8918 8926 8934 8942 8949 8957 8965 8973 8980 I 3 4 5 6 64 8988 8996 9003 901 1 9018 9026 9033 9041 9048 9056 I 3 4 5 f 65 9063 9070 9078 9085 9092 9100 9107 9114 9121 9128 I 2 4 5 6 66 913s 9143 9150 9157 9164 9171 9178 9184 9191 9198 I 2 3 5 ^ 67 9205 9212 9219 9225 9232 9239 9245 9252 9259 9265 I 2 3 4 6 68 9272 9278 9285 9291 9298 9304 93" 9317 9323 9330 I 2 3 4 5 69 9336 9342 9348 9354 9361 9367 9373 9379 9385 9391 I 2 3 4 5 70 9397 9403 9409 9415 9421 9426 9432 9438 9444 9449 I 2 3 4 5 71 9455 9461 9466 9472 9478 9483 9489 9494 9500 9505 I 2 3 4 5 72 95" 9516 9521 9527 9532 9537 9542 9548 9553 9558 I 2 3 4 4 73 9563 9568 9573 9578 9583 9588 9593 9598 9603 9608 122 3 4 74 9613 9617 9622 9627 9632 9636 9641 9646 9650 9655 I 2 2 3 4 75 9659 9664 9668 9673 9677 9681 9686 9690 9694 9699 I I 2 3 4 76 9703 9707 97" 9716 9720 9724 9728 9732 9736 9740 I I 2 3 3 77 9744 9748 9751 9755 9759 9763 9767 9770 9774 9778 I I 2 3 3 78 9781 978s 9789 9792 9796 9799 9803 9806 9810 9813 I I 2 2 3 79 9816 9820 9823 9826 9829 9833 9836 9839 9842 9845 I I 2 2 3 80 9848 9851 9854 9857 9860 9863 9866 9869 9871 9874 I I 2 2 81 9877 9880 9882 9885 9888 9890 9893 9895 9898 .9900 I I 2 2 82 9903 990s 9907 9910 9912 9914 9917 9919 9921 9923 I 1 2 2 83 9925 9928 9930 9932 9934 9936 9938 9940 9942 9943 I I I 2 84 994S 9947 9949 9951 9952 9954 9956 9957 9959 9960 !■ I I 2 85 9962 9963 996S 9966 9968 9969 9971 9972 9973 9974 I I I 86 9976 9977 9978 9979 9980 9981 9982 9983 9984 9985 I 1 I 87 9986 9987 9988 9989 9990 9990 9991 9992 9993 ■9993 000 I 1 88 9994 9995 9995 9996 9996 9997 9997 9997 9998 9998 000 89 9998 9999 9999 9999 9999 9999 9999 9999 9999 9999 000 0' 6' or 0°-l 12' or 0°'2 18' or 0°'3 24' or0°4 3cy or 0°'5 36' or 0°'6 42' or0°-7 48' or0°'8 54' or0°9 1' 2' 3' 4 5' Minutes. | i6 NATURAL COSINES. & 12' 18' 24' SO 36' 42' 48' 54' Minutes. | 0° 0' orO°-l orO°-2 or0°-3 orO°-4 or0°-5 or0°-6 orO°-7 orO°'8 or0°'9 1' 2' 3' 4' 5' 1. 000 9999 9999 9999 9999 9999 9999 9999 9999 9999 1 9998 9998 9998 9997' 9997 9997 9996 9996 9995 9995 2 9994 9993 9993 9992 9991 9990 9990 9989 9988 9987 I I 3 9986 9985 9984 9983 9982 9981 9980 9979 9978 9977 I I 4 9976 9974 9973 9972 9971 9969 9968 9966 9965 9963 I I 5 9962 9960 9959 9957 9956 9954 9952 9951 9949 9947 I I 2 6 9945 9943 9942 9940 9938 9936 9934 9932 9930 9928 I I 2 7 9925 9923 9921 9919 9917 9914 9912 9910 9907 9?2S I 2 2 8 9903 9900 9898 9895 9893 9890 9888 9885 9882 9880 I 2 2 9 9877 9874 9871 9869 9866 9863 9860 9857 9854 985' I 2 2 10 9848 9845 9842 9839 9836 9833 9829 9826 9823 9820 I I 2 2 3 H 9816 9813 9810 9806 9803 9799 9796 9792 9789 9785 I I 2 2 3 12 9781 9778 9774 9770 9767 9763 9759 9755 9751 9748 I I 2 3 3 13 9744 9740 9736 9732 9728 9724 9720 9715 97" 9707 I I. 2 3 3 14 9703 9699 9694 9690 9686 9681 9677 9673 9668 9664 I I 2 3 4 16 9659 9655 965° 9646 9641 9636 9632 9627 9622 9617 I 2 2 3 4 16 9613 9608 9603 9598 9593 9588 9583 957s 9573 9568 I 2 2 3 4 17 9563 9558 9553 9548 9542 9537 9532 9527 9521 9516 I 2 3 4 4 18 95" 9505 9500 9494 9489 9483 9478 9472 9466 9461 I 2 3 4 5 19 9455 9449 9444 9438 9432 9426 9421 9415 9409 9403 I 2 3 4 5 20 9397 9391 9385 9379 9373 9367 9361 9354 9348 9342 I 2 3 4 5 21 9336 9330 9323 9317 93" 9304 9298 9291 928s 9278 I 2 3 4 5 22 9272 9265 9259 9252 924s 9239 9232 9225 9219 9212 I 2 3 4 ^ 23 9205 9198 9191 9184 9178 9171 9164 9157 9150 9143 I 2 3 5 ^ 24 9135 9128 9121 9114 9107 9100 9092 9085 9078 9070 I 2 4 5 ^ 25 9063 9056 9048 9041 9033 9026 9018 901 1 90C3 8996 1 3 4 5 6 26 8988 8980 8973 8965 8957 8949 8942 8934 8926 8918 I 3 4 5 6 27 8910 8902 8894 8886 8878 8870 8862 8854 8846 8838 I 3 4 5 7 28 8829 8821 8813 8805 8796 8788 8780 8771 8763 8755 I 3 4 6 7 29 8746 8738 8729 8721 8712 8704 8695 8686 8678 8669 I 3 4 6 7 30 8660 8652 8643 8634 862s 8616 8607 8599 8590 8581 I 3 4 6 7 31 8572 8563 8554 8545 8536 8526 8517 8508 8499 8490 2 3 5 6 8 32 8480 8471 8462 8453 8443 8434 8425 841 5 8406 8396 2 3 5 6 8 33 8387 8377 8368 8358 8348 8339 8329 8320 8310 8300 2 3 5 6 8 34 8290 8281 8271 8261 8251 8241 8231 8221 821 1 8202 2 3 5 7 8 36 8192 8181 8171 8161 8151 8141 8131 8121 8111 8100 2 3 5 7 8 36 8090 8080 8070 8059 8049 8039 8028 8018 8007 7997 2 3 5 7 9 37 7986 .7976 7965 7955 7944 ,7934 7923 7912 7902 7891 2 4 5 7 9 38 7880 7869 7859 7848 7837 7826 7815 7804 7793 7782 2 4 5 7 9 39 7771 7760 7749 7738 7727 7716 7705 7694 7683 7672 2 4 6 7 9 40 7660 7649 7638 7627 7615 7604 7593 7581 7570 7559 2 4 6 8 9 41 7547 7536 7524 7513 7501 7490 7478 7466 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6129 5990 5850 5707 6252 6115 5976 5835 5693 6239 6101 5962 5821 5678 6225 6088 5948 5807 5664 621 1 6074 5934 5793 5650 6198 6060 5920 5779 5635 6184 6046 5906 5764 5621 6170 6032 5892 5750 5606 2 2 2 2 2 5 5 5 5 5 7 7 7 7 7 9 II 9 12 9 12 9 12 10 12 66 67 68 69 60 S592 5446 5299 5150 5000 5577 5432 5284 5135 4985 5563 5417 5270 5120 4970 5548 5402 525s 5105 4955 lilt 5240 5090 4939 5519 5373 5225 5075 4924 5505 5358 5210 5060 4909 5490 5344 5195 5045 4894 5476 5329 5180 5030 4879 S461 53'4 5165 5015 4863 2 2 2 3 3 5 5 5 5 5 7 7 7 8 8 10 12 10 12 10 12 10 13 10 13 61 62 63 64 66 4848 4695 4540 4384 4226 4833 4679 4524 4368 4210 4818 4664 4509 4352 4195 4802 4648 4493 4337 4179 4787 4633 4478 4321 4163 4772 4617 4462 4305 4147 4756 4602 4446 4289 4131 4741 4586 4431 4274 4115 4726 4571 4415 4258 4099 4710 4555 4399 4242 4083 3 3 3 3 3 S 5 5 5 5 8 8 8 8 8 10 13 10 13 10 13 11 13 II 13 66 67 68 69 70 4067 3907 3746 3584 3420 4051 3891 3730 3567 3404 4035 3875 3714 3551 3387 4019 3859 3697 3535 3371 4003 3843 3681 3518 3355 3987 3827 366S 3502 3338 3971 3811 3649 3486 3322 3955 3795 3633 3469 3305 3939 3778 3616 3453 3289 3923 3762 3600 3437 3272 3 3 3 3 3 5 5 5 5 S 8 8 8 8 8 II 14 II 14 II. 14 II 14 II 14 71 72 73 74 78 3256 3090 2924 2756 2588 3239 3074 2907 2740 2571 3223 3°57 2890 2723 2554 3206 3040 2874 2706 2538 3190 3024 2857 2689 2521 3173 3007 2840 2672 2504 3156 2990 2823 2656 2487 3140 2974 2807 2639 2470 3123 2957 2790 2622 2453 3107 2940 2773 2605 2436 3 3 3 3 3 6 6 6 6 6 8 8 8 8 8 II 14 II 14 II 14 II 14 II 14 76 77 78 79 80 2419 2250 2079 1908 1736 2402 2233 2062 1891 1719 2385 2215 2045 1874 1702 2368 2198 2028 1857 1685 2351 2181 201 1 1840 1668 2334 2164 1994 1822 1650 2317 2147 1977 1805 1633 2300 2130 1959 1788 1616 2284 2113 1942 1771 1599 2267 2096 1925 -1754 1582 3 3 3 3 3 6 6 6 6 6 8 9 9 9 9 II 14 II 14 11 14 12 14 12 14 81 82 83 84 86 1564 1392 1219 1045 0872 1547 1374 1 201 1028 0854 1530 1357 1 184 lOII 0837 1513 1340 1 167 0993 0819 1495 1323 1149 0976 0802 1478 1305 1132 0958 078S 1461 1288 "15 0941 0767 1444 1271 1097 0924 0750 1426 1253 1080 0906 0732 1409 1236 1063 0889 0715 3 3 3 3 3 6 6 6 6 6 9 9 9 9 9 12 14 12 14 12 14 12 14 12 IS 86 87 88 89 0698 0523 0349 OI7S 0680 0506 0332 0157 0663 0488 0314 0140 0645 0471 0297 0122 0628 0454 0279 0105 0610 0436 0262 0087 0593 0419 0244 0070 0576 0401 0227 0052 0558 0384 0209 0035 0541 0366 0192 0017 3 3 3 3 6 6 6 6 9 9 9 9 12 15 12 15 12 15 12 15 a & 12' 18' 24' aa 36' 42' 48' 54' 1' 2' 3' 4' 5'| orO°-l or0°'2 orO°'3 or0°-4 or0°-5 or0°-6 or0°-7 or0°'8 or0°"9 Minutes. | Numbers in difFerence-columns to be subtracted, not added. B NATURAL TANGENTS. 0° 0' 6' orO°-l 12' or0°-2 18 or0°'3 24' or0°-4 30' or0°-6 36' or0°'6 42' orO°-7 48 or0°'8 54' or0°-9 Minutes. { 1' 2' 3' 4' 5' .0000 0017 0035 0052 0070 0087 0105 0122 0140 0157 3 6 9 12 14 1 .0175 0192 0209 0227 0244 0262 0279 0297 0314 0332 3 6 9 12 15 2 •0349 0367 0384 0402 0419 0437 0454 0472 0489 0507 3 6 9 12 15 3 .0524 0542 0559 0577 0594 0612 0629 0647 0664 0682 3 6 9 12 15 i .0699 0717 0734 0752 0769 0787 0805 0822 0840 0857 3 6 9 12 15 5 .0875 0892 0910 0928 0945 0963 0981 0998 ]oi6 1033 3 6 9 12 15 6 .1051 1069 io86 1 104 1122 "39 "57 "75 1 192 IZIO 3 6 9 12 15 7 .1228 1246 1263 1281 1299 1317 1334 1352 1370 1388 3 6 9 12 15 8 .I40S 1423 1441 I4S9 1477 1495 1512 1530 1548 1566 3 6 9 12 15 9 .1584 1602 1620 1638 1655 1673 1691 1709 1727 1745 3 6 9 12 15 10 • 1763 1781 1799 1817 1835 1853 1871 1890 1908 1926 3 6 9 12 IS 11 ■ 1944 .2126' 1962 1980 1998 2016 2035 2053 Z071 2089 2107 3 6 9 12 IS 12 2144 2162 2180 2199 2217 2235 2254 2272 2290 3 6 9 12 15 13 .2309 2327 2345 2364 2382 2401 24.19 2438 2456 2475 3 6 9 12 IS 14 •2493 2512 2530 2549 2568 2586 2605 262^ 2642 2661 3 6 9 12 16 15 .2679 2698 2717 2736 2754 2773 2792 281 1 2830 2849 3 6 9 13 16 16 .2867 2886 2905 2924 2943 2962 2981 3000 3019 3038 3 6 9 13 16 17 •3057 3076 3096 3115 ■3134 3t53 3172 3191 321 1 3230 3 6 10 13 16 18 •3249 3269 3288 3307 3327 3346 3365 3385 3404 3424 3 6 10 13 16 19 •3443 3463 3482 3502 3522 3541 3561 358r 3600 3620 3 6 10 13 17 20 .3640 3659 3679 3699 3719 3739 3759 3779 3799 3819 3 7 10 13 17 21 •3839 3859 3879 3899 3919 3939 3959 3979 4000 4020 3 7 10 13 17 22 .4040 4061 4081 4101 4122 4142 4163 4183 4204 4224 3 7 10 14 17 23 .4245 4265 4286 4307 4327 4348 4369 4390 44" 4431 3 7 10 14 17' 21 .4452 4473 4494 4515 4536 4557 4578 4599 4621 4642 4 7 10 14 18 25 .4663 4684 4706 4727 4748 4770 4791 4813 4834 4856 4 7 II i^ 18 ■ 26 .4877 4899 4921 4942 4964 4986 5008 5029 5051 5073 4 7 II IS 18 27 ■5095 5117 5139 5161 5184 5206 5228 5250 5272 5295 4 II IS 18 28 ■5317 5340 5362 5384 5407 5430 5452 5475 5498 5520 4 8 II 15 19 29 •SS43 5566 5589 5612 5635 5658 5681 5704 5727 5750 4 8 12 IS 19 30 •5774 5797 5820 5844 5867 5890 5914 5938 5961 598s 4 8 12 16 20 31 .6009 6032 6056 6080 6104 6128 6152 6176 6200 6224 4 8 12 16 20 32 .6249 6273 6297 6322 6346 6371 639s 6420 6445 6469 4 8 12 16 20 33 .6494 6519 6544 6569 6594 6619 6644 6669 6694 6720 4 8 13 17 21 31 ■674s 6771 6796 6822 6847 6873 6899 6924 6950 6976 4 9 13 17 21 35 .7002 7028 7054 7080 7107 7133 7159 7186 7212 7239 4 9 13 18 22 36 .7265 7292 7319 7346 7373 7400 7427 7454 7481 7508 5 9 14 18 23 37 ■7536 7563 7590 7618 7646 7673 7701 7729 7757 7785 5 9 14 18 23 38 •7813 7841 7869 7898 7926 7954 7983 8012 8040 8069 S 10 '4 19 24 39 .8098 8127 8156 8185 8214 8243 8273 8302 8332 8361 5 10 IS 20 24- 40 ■8391 8421 8451 8481 8511 8541 8571 8601 8632 8662 5 10 15 20 25 41 .8693 8724 8754 8785 88i6 8847 8878 8910 8941 8972 5 10 16 21 26 42 .9004 9036 9067 9099 9131 9163 9195 9228 9260 9293 5 II 16 21 27 43 •9325 9358 9391 9424 9457 9490 9523 9556 9590 9623 6 II 17 22 28 44 ■9657 9691 9725 9759 9793 9827 9861 9896 9930 9965 6 II 17 23 29 0' 6' orO°-l 12' 18 24' or0°-4 30' or0°"6 36' or0°-6 42' or0°-7 48' or0°-8 54' or0°9 1' 2' 3' 4' 6' or0°-2 or0°'3 Minutes. | NATURAL TANGENTS. 0' 6' orO°l 12' or0°-2 18' or0°-3 24' or0°'4 30' or0°'5 36' or0°'6 42' or0°-7 48' or0°'8 54' or0°-9 Minutes. | 1' 2' 3' 4' 5' 46° I.OOOO 0035 0070 0105 0141 0176 0212 0247 0283 0319 6 12 18 24 30 46 I-03SS 0392 0428 0464 0501 0538 0575 0612 0649 0686 6 12 18 25 31 47 1.0724 0761 0799 0837 0875 0913 0951 0990 1028 1067 6 13 19 25 32 48 1.1106 "45 1 184 1224 1263 1303 1343 1383 1423 1463 7 13 20 26 33 49 1. 1504 1544 1585 1626 1667 1708 1750 1792 1833 1875 7 14 21 28 34 60 1.1918 i960 2002 2045 2088 2131 2174 2218 2261 2305 7 14 22 29 36 51 1-2349 2393 2437 2482 2527 2572 2617 2662 2708 2753 8 15 23 30 38 62 1.2799 2846 2892 2938 2985 3032 3079 3127 3175 3222 8 16 23 31 39 63 1.3270 3319 3367 3416 3465 3514 3564 3613 3663 3713 8 16 25 33 41 54 1-3764 3814 3865 3916 3968 4019 4071 4124 4176 4229 9 17 26 34 43 55 1-1281 4335 4388 4442 4496 4550 4605 4659 4715 4770 9 18 27 36 45 66 1.4826 4882 4938 4994 5051 5108 5166 5224 5282 5340 10 19 29 38 48 67 1-5399 5458 5517 5577 5637 5697 5757 5818 5880 5941 10 20 30 40 50 68 1.6003 6066 6128 6191 6255 6319 6383 6447 6512 6577 II 21 32 43 53 59 1.6643 6709 6775 6842 6909 6977 7045 7113 7182 7251 II 23 34 45 56 60 1.7321 7391 7461 7532 7603 7675 7747 7820 7893 7966 12 24 36 48 60 61 1.8040 8n5 8190 8265 8341 8418 8495 8572 8650 8728 13 26 38 SI 64 62 1.8807 8887 8967 9047 9128 9210 9292 9375 9458 9542 14 27 41 55 68 63 1.9626 9711 9797 9883 99^70 0057 0145 0233 0323 0413 15 29 44 58 73 64 2.0503 0594 0686 0778 0872 0965 1060 "55 1251 1348 16 31 47 63 78 66 2. 1445 1543 1642 1742 1842 1943 2045 2148 2251 235i 17 34 51 68 85 66 2.2460 2566 2673 2781 2889 2998 3109 3220 3332 3445 18 37 55 74 92 67 2-3559 3673 3789 3906 4023 4142 4262 4383 4504 4627 20 40 60 79 99 68 2.4751 4876 5002 5129 5257 5386 5517 5649 5782 5916 22 43 65 87 108 69 2.6051 6187 6325 6464 6605 6746 6889 7034 7179 7326 24 47 71 95 118 70 2-7475 7625 7776 7929 8083 8239 8397 8556 8715 8878 26 52 78 104 130 71 2.9042 9208 9375 9544 9714 9887 6061 0237 6415 0595 29 58 87 115 144 72 3-0777 0961 1 146 1334 1524 1716 1910 2106 2305 2506 32 64 96 129 161 73 3-2709 2914 3122 3332 3544 3759 3977 4197 4420 4646 36 72 108 144 180 74 3-4874 510S 5339 5576 5816 6059 6305 6554 6806 7062 41 81 122 162 203 75 3-7321 7583 7848 8118 8391 8667 8947 9232 9520 9812 46 93 139 186232 76 4.0108 0408 0713 1022 1335 1653 1976 2303 2635 2972 53 107 160 213 267 77 4-3315 3662 4015 4374 4737 5107 5483 5864 6252 6646 62 124 186 248 310 78 4.7046 7453 7867 8288 8716 9152 9594 0045 0504 6970 73 146 219 292 36s 79 5-1446 1929 2422 2924 3435 3955 4486 5026 5578 6140 87 175 262 350 437 80 S-6713 7297 7894 8502 9124 9758 0405 1066 1742 2432 81 6-3138 3859 4596 535° 6122 6912 7920 8548 9395 0264 1' 2' 3' 4' 5' 82 83 7.1154 8-1443 2066 2636 3002 3863 3962 5126 4947 6427 5958 7769 6996 8062 0579 9158 2052 0285 3572 Minutes. | 9152 84 9-5144 9-677 9.845 10.02 10.20 10.39 10.58 10.78 10.99 11.20 85 11-43 11.66 11.91 12.16 12.43 12.71 13.00 13-30 13.62 13-95 - 86 87 14.30 19.08 14.67 19-74 15-06 20.45 15.46 21.20 15.89 22.02 16.35 22.90 J6.83 23.86 17-34 24.90 17-89 26.03 18.46 27.27 Difference maybeiK and the taken to est tentl columns ;glected, tangent he near- 1 of a 88 89 28.64 57-29 30.14 63.66 31.82 71.62 33-69 81.85 35-80 95-49 38.19 1 14.6 40.92 143.2 44.07 191.0 47-74 286.5 52.08 573-0 degree. 0' & 12' 18' 24' 30' 36' 42' 48' 54' orO°'l or0°-2 or0°-3 or0°4 or0°-6 orO°-6 orO°-7 or0°'8 or0°-9 LOGARITHMIC SINES. 6' 12' 18' 24' 30' 36' 42' 48' 54' Minutes. | 0' orO°-l or0°-2 or0°-3 or 0° -4 or 0° "5 or0°'6 orO°-7 or 0°-8 or0°-9 1' 2' 3' 4' 5' 0° Inf. Neg. 7.2419 5429 7190 8439 9408 0200 0870 1450 1 96 1 1 8.2419 2832 3210 3558 3880 4179 4459 4723 4971 5206 2 8.5428 564Q 5842 6035 6220 6397 6567 6731 6889 7041 3 8.7188 7330 7468 7602 7731 7857 7979 8098 8213 8326 21 41 62 82 103 i 8.8436 8543 8647 8749 8849 8946 9042 9135 9226 9315 16 32 48 64 80 5 8.9403 9489 9573 9655 9736 9816 9894 9970 0046 0120 13 26 39 52 65 6 9.0192 0264 0334 0403 0472 0539 0605 0670 0734 0797 II 22 33 44 55 7 9.0859 0920 0981 1040 1099 1 157 1214 1271 1326 1381 10 19 29 38 48 8 9-1436 1489 1542 1594 1646 1697 1747 1797 1847 189s 8 17 25 34 42 9 9- 1943 1991 2038 Z085 2131 2176 2221 2266 2310 2353 8 15 23 30 38 10 9-2397 2439 2482 2524 2565 2606 2647 2687 2727 2767 7 14 20 27 34 11 9.2806 2845 2883 2921. 2959 2997 3034 3070 3107 3't3 6 12 19 25 31 12 9-3179 3214 3250 3284 .3319 3353 3387 3421 3455 3488 6 II 'I 23 28 13 9-3S2I 3554 3586 3618 3650 3682 3713 3/45 3775 3806 5 II 16 21 26 11 9-3837 3867 3897 3927 3957 3986 4015 4044 4073 4102 S 10 15 20 24 15 9-4130 4158 4186 4214 4242 4269 4296 4323 4350 4377 5 9 14 18 23 16 9-4403 4430 4456 4482 4508 4533 4559 4584 4609 4634 4 9 13 17 21 17 9.4659 4684 4709 4733 4757 4781 4805 4829 4853 4876 4 8 12 16 20 18 9.4900 4923 4946 4969 4992 5015 5037 5060 5082 5104 4 8 II 15 19 19 9.5126 5148 5170 5192 5213 5235 5256 5278 5299 5320 4 7 II 14 18 20 9-S34I 5361 5382 5402 5423 5443 5463 5484 5504 5523 3 7 10 14 17 21 9-5S43 5563 5583 5602 5621 5641 5660 5679 5698 5717 3 6 10 13 16 22 9-5736 S754 5773 5792 5810 S828 5847 5865 5883 5901 3 6 9 12 15 23 9-5919 5937 5954 5972 5990 6007 6024 6042 6059 6076 3 6 9 12 •5 21 9.6093 6110 6127 6144 6161 6177 6194 6210 6227 6243 3 6 8 II 14 25 9.6259 6276 6292 6308 6324 6340 6356 6371 6387 6403 3 5 8 II 13 26 9.6418 6434 6449 6465 6480 6495 6510 6526 6541 6556 3 5 8 10 13 27 9.6570 6585 6600 6615 6629 6644 6659 6673 6687 6702 2 5 7 10 12 28 9.6716 6730 6744 6759 6773 6787 6801 6814 6828 684Z 2 S 7 .9 12 29 9.6856 6869 6883 6896 6910 6923 6937 6950 6963 6977 2 4 7 9 II 30 9.6990 7003 7016 7029 7042 7055 7068 7080 7093 7106 2 4 6 9 II 31 9.7118 7131 7144 7156 7168 7181 7193 7205 7218 7230 2 4 6 8 10 32 9.7242 7254 7266 7278 7290 7302 7314 7326 7338 7349 2 4 6 8 10 33 9-7361 7373 7384 7396 7407 7419 7430 7442 7453 7464 2 4 6 8 10 31 9.7476 7487 7498 7509 7520 7531 7542 7553 7564 7575 2 4 6 7 9 35 9-7586 7597 7607 7618 7629 7640 7650 7661 7671 7682 2 4 5 7 9 36 9.7692 7703 7713 7723 7734 7744 7754 7764 7774 7785 2 3 5 7 9 37 9-7795 7805 7815 7825 7835 7844 7854 7864 7874 7884 2 3 S 7 8 38 9-7893 7903 7913 7922 7932 7941 7951 7960 7970 7979 2 3 5 6 8 39 9.7989 7998 8007 8017 8026 8035 8044 8053 8063 8072 2 3 5 6 8 40 9.8081 8090 8099 8108 8117 8125 8134 8143 8152 8i6i I 3 4 6 7 41 9.8169 8178 8187 8195 8204 8213 8221 8230 8238 8247 I 3 4 6 7 42 9-S255 8264 8272 8280 8289 8297 8305 8313 8322 8330 I 3 4 6 7 43 9-8338 8346 8354 8362 8370 8378 8386 8394 8402 8410 I 3 4 5 7 44 9.8418 8426 8433 8441 8449 8457 8464 8472 8480 8487 I 3 4 5 6 a 6' orO°-l 12' or0°-2 18' or0°'3 24' or0°-4 30' or0°-6 36' or 0°'6 42' or0°-7 48' or 0°-8 54' orO°-9 1' 2' 3' 4' 5' Minutes. | LOGARITHMIC SINES. 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' Minutes. | orO°-l or 0°-2 or 0°'3 or0°-4 or 0°'5 or 0°'6 or0°-7 or0°-8 or0°-9 1' 2' 3' 4 5' 45° 9.8495 8502 8510 8517 8525 8532 8540 8547 8555 8562 I 2 4 5 6 46 9.8569 8577 8584 8591 8598 8606 8613 8620 8627 S634 2 4 5 6 47 9.8641 8648 8655 8662 8669 8676 8683 8690 8697 8704 2 3 5 6 4S 9.871 1 8718 8724 8731 8738 8745 8751 8758 8765 8771 2 3 4 6 49 9.8778 8784 8791 8797 8804 8810 8817 8823 8830 8836 2 3 4 5 60 9.8843 8849 8855 8862 8868 8874 8880 8887 8893 8899 2 3 4 5 51 9.890s 8911 8917 8923 8929 8935 8941 8947 8953 8959 2 3 4 5 52 9.896s 8971 8977 8983 8989 8995 9000 9006 9012 goi8 2 3 4 5 53 9.9023 9029 9035 9041 9046 9052 9057 9063 9069 9074 2 3 4 5 64 9.9080 9085 9091 9096 9101 9107 9112 9118 9123 9128 2 3 4 5 55 99134 9139 9144 9149 9155 9160 9165 9170 917s 9181 2 3 3 4 66 9.9186 9191 9196 9201 9206 921 1 9216 9221 9226 9231 2 3 3 4 67 9.9236 9241 9246 9251 9255 9260 9265 9270 927s 9279 2 2 3 4 5S 9.9284 9289 9294 9298 9303 9308 9312 9317 9322 9326 2 2 3 4 59 99331 9335 9340 9344 9349 9353 9358 9362 9367 9371 I 2 3 4 60 9-9375 9380 9384 9388 9393 9397 9401 9406 9410 9414 I 2 3 4 61 9.9418 9422 9427 9431 9435 9439 9443 9447 9451 9455 I 2 3 3 62 9-9459 9463 9467 9471 9475 9479 9483 9487 9491 9495 I 2 3 3 63 9.9499 9503 9507 9510 9514 9518 9522 9525 9529 9533 I 2 3 3 64 9-9537 9540 9544 9548 9551 9555 9558 9562 9566 9569 I 2 2 3 65 9-9573 9576 9580 9583 9587 9590 9594 9597 9601 9604 I 2 2 3 66 9.9607 961 1 9614 9617 9621 9624 9627 9631 9634 9637 I 2 2 3 67 9.9640 9643 9647 9650 9653 9656 9659 9662 9666 9669 I 2 2 3 68 9.9672 9675 9678 9681 9684 9687 9690 9693 9696 9699 I I 2 2 69 9.9702 9704 9707 9710 9713 9716 9719 9722 9724 9727 1 I 2 2 70 9-9730 9733 9735 9738 9741 9743 9746 9749 9751 9754 I I 2 2 71 9-9757 9759 9762 9764 9767 9770 9772 9775 9777 9780 I I 2 2 72 9.9782 9785 9787 9789 9792 9794 9797 9799 9801 9804 I I 2 2 73 9.9806 9808 981 1 9813 9815 9817 9820 9822 9824 9826 I I 2 2 74 9.9828 9831 9833 9835 9837 9839 9841 9843 9845 9847 I I I 2 75 9.9849 9851 9853 985s 9857 9859 9861 9863 9865 9867 I I I 2 76 9.9869 9871 9873 9875 9876 9878 9880 9S82 9884 9885 I I I 2 77 9.9887 9889 9891 9892 9894 9896 9897 9899 9901 9902 I I 78 9.9904 9906 9907 9909 9910 9912 9913 991 5 9916 9918 I I 79 9.9919 9921 9922 9924 9925 9927 9928 9929 9931 9932 I 80 9-9934 9935 9936 9937 9939 9940 9941 9943 9944 9945 I 81 9.9946 9947 9949 9950 9951 9952 9953 9954 9955 9956 I 82 9-9957 9959 9960 9961 9962 9964 9965 9966 9967 .0 I 83 9.9968 9968 9969 9970 9971 9972 9973 9974 9975 9975 ,0 84 9.9976 9977 9978 9978 9979 9980 9981 9981 9982 9983 I 85 9-9983 9984 9985 9985 9986 9987 9987 9988 9988 9989 86 9.9989 9990 9990 9991 9991 9992 9992 9993 9993 9994 87 9.9994 9994 9995 9995 9996 9996 9996 9996 9997 9997 88 9.9997 9998 9998 9998 9998 9999 9999 9999 9999 9999 89 9.9999 9999 0000 0000 0000 0000 0000 0000 onno 0000 54' or0°'9 C & orO°-l 12' or 0"'-2 18' orO°'3 24' or 0°-4 30' or 0°*5 36' or 0°'6 42' orO°-7 48' or0°'8 1' 2' 3' 4' 5' Minutes. | LOGARITHMIC COSINES. 6' 12' 18' 24' 30' 36' 42' 48' 54' Minutes. | 0° 0' orO°-l or0°-2 or0°-3 orO°-4 or0°'5 or0°-6 or0°'7 or0°'8 orO°'9 1' 2' 3' 4' 5' lO.OOOO 0000 0000 nnnn 0000 0000 0000 oonn 0000 9-9999 000 1 9-9999 9999 9999 9999 9999 9999 9998 9998 9998 9998 000 2 9-9997 9997 9997 9996 9996 9996 9996 9995 9995 9994 000 3 9-9994 9994 9993 9993 9992 9992 9991 9991 9990 9990 000 4 9.9989 9989 9988 9988 9987 9987 9986 9985 9985 9984 000 5 9-9983 9983 9982 9981 9981 9980 9979 9978 9978 9977 000 1 6 9.9976 9975 9975 9974 9973 9972 9971 9970 9969 9968 000 7 9.9968 9967 9966 9965 9964 9963 9962 9961 9960 9959 I 8 9-9957 9956 9955 9954 9953 9952 9951 9950 9949 9947 I 9 9.9946 9945 9944 9943 9941 9940 9939 9937 9936 9935 I 10 9-9934 9932 9931 9929 9928 9927 9925 9924 9922 9921 I 11 9-9919 9918 9916 9915 9913 9912 9910 9909 9907 995^ I I 12 9.9904 9902 9901 9899 9897 9896 9894 9892 9891 9889 oil 13 9.9887 9885 9884 9882 9880 9878 9876 9875 9873 9871 oil I 2 14 9.9869 9867 9865 9863 9861 9859 9857 985s 9853 9851 I I I 2 16 9.9849 9847 9845 9843 9841 9839 9837 9835 9833 9831 I I I 2 16 9.9828 9826 9824 9822 9820 9817 981S 9813 981 1 9808 oil 2 2 17 9.9806 9804 9801 9799 9797 9794 9792 9789 9787 9785 oil 2 2 18 9.9782 9780 9777 9775 9772 9770 9767 9764 9762 9759 1 I 2 2 19 9-9757 9754 9751 9749 9746 9743 9741 9738 9735 9733 oil 2 2 20 9-9730 9727 9724 9722 9719 9716 9713 9710 9707 9704 oil 2 2 21 9.9702 9699 9696 9693 9690 9687 9684 9681 9678 9675 oil 2 2 22 9.9672 9669 9666 9662 9659 9656 9653 9650 9647 9643 112 2 3 23 9.9640 9637 9634 9631 9627 9624 9621 9617 9614 961 1 112 2 3 24 9.9607 9604 9601 9597 9594 9590 9587 9583 9580 9576 112 2 3 2S 9-9573 9569 9566 9562 9558 9555 9551 9548 9544 9540 112 2 3 26 9-9537 9533 9529 9525 9522 9518 9514 9510 9507 9503 112 3 3 27 9-9499 9495 9491 9487 9483 9479 9475 9471 9467 9463 112 3 3 28 9-9459 9455 9451 9447 9443 9439 9435 9431 9427 9422 112 3 3 29 9.9418 9414 9410 9406 9401 9397 9393 9388 9384 9380 I I 2 3 4 30 9-9375 9371 9367 9362 9358 9353 9349 9344 9340 9335 I I 2 3 4 31 9-9331 9326 9322 9317 9312 9308 9303 9298 9294 9289 12 2 3 4 32 9.9284 9279 9275 9270 9265 9260 9255 9251 9246 9241 I 2 2 3 4 33 9.9236 9231 9226 9221 9216 921 1 9206 9201 9196 9191 I 2 3 3 4 34 9.9186 9181 917s 9170 916S 9160 9155 9149 9144 9139 I 2 3 3 4 35 9-9134 9128 9123 9118 9112 9107 9101 9096 9091 9085 1 2 3 4 5 36 9.9080 9074 9069 9063 9057 9052 9046 9041 9035 9029 I 2 3 4 5 37 9.9023 9018 9012 9006 9000 8995 8989 8983 8977 8971 I 2 3 4 5 38 9.8965 8959 8953 8947 8941 8935 8929 8923 8917 891 1 I 2 3 4 5 39 9.8905 8899 8893 8887 8880 8874 8868 8862 88SS 8849 I 2 3 4 5 40 9.8843 8836 8830 8823 8817 S810 8804 8797 8791 8784 I 2 3 4 5 41 9.8778 8771 8765 8758 8751 8745 8738 8731 8724 8718 I 2 3 5 6 42 9.871 1 8704 8697 8690 8683 8676 8669 8662 8655 8648 I 2 3 5 6 43 9.8641 8634 8627 8620 8613 8606 8598 8591 8584 8577 I 2 4 5 6 44 9.8569 8562 8555 8547 8540 8532 8525 8517 8510 8502 I 2 4 5 6 0' 6' orO°-l 12' or0°-2 18' orO'-S 24' or0°-4 30' or0°-5 36' or0°'6 42' or0°-7 48' orO°'8 54' orO°-9 1' 2' 3' 4' 5' Minutes. Numbers in minutes-columns to be subtracted, not added. LOGARITHMIC COSINES. 45° 0' 6' orO°l 12' or0°-2 18' orO°'3 24' or0°-4 30' orO°-6 36' or0°'6 42' orO°-7 48' or0°'8 54' 01 0° -9 Minutes. | 1' 2' 3' 4' 5' 9.8495 8487 8480 8472 8464 8457 8449 8441 8433 8426 I 3 4 S 6 46 9.8418 8410 8402 8394 8386 8378 8370 8362 8354 8346 I 3 4 5 7 47 9-8338 8330 8322 8313 830s 8297 8289 8280 8272 8264 I 3 4 6 7 48 9.8255 8247 8238 8230 8221 8213 8204 819s 8187 8178 I 3 4 6 7 49 9.8169 8161 8152 8143 8134 8125 8117 8108 8099 8090 I 3 4 6 7 50 9.8081 8072 8063 8053 8044 8035 8026 8017 8007 7998 2 3 5 6 8 61 9.7989 7979 7970 7960 7951 7941 7932 7922 7913 7903 2 3 5 6 8 S2 9-7893 7884 7874 7864 7854 7844 783s 7825 7815 7805 2 3 5 7 8 63 9-7795 7785 7774 7764 7754 7744 7734 7723 7713 7703 2 3 5 7 9 54 9.7692 7682 7671 7661 7650 7640 7629 7618 7607 7597 2 4 5 7 9 55 9-7586 7575 7564 7553 7542 7531 7520 7509 7498 7487 2 4 6 7 9 56 9.7476 7464 7453 7442 7430 7419 7407 7396 7384 7373 2 4 6 8 10 57 9-7361 7349 7338 7326 7314 7302 7290 7278 7266 7254 2 4 6 8 10 58 9.7242 7230 7218 7205 7193 7181 7168 7156 7144 7131 2 4 6 8 10 59 9.7118 7106 7093 7080 7068 7055 7042 7029 7016 7003 2 4 6 9 II 60 9.6990 6977 6963 6950 6937 6923 6910 6896 6883 6869 2 4 7 9 II 61 9.6856 6842 6828 6814 6801 6787 6773 6759 6744 6730 2 5 7 9 12 62 9.6716 6702 6687 6673 6659 6644 6629 6615 66co 658s 2 5 7 10 12 63 9.6570 6556 6541 6526 6510 6495 6480 6465 6449 6434 3 5 8 10 13 64 9.6418 6403 6387 6371 6356 6340 6324 6308 6292 6276 3 5 8 II 13 65 9.6259 6243 6227 6210 6194 6177 6161 6144 6127 6110 3 6 8 II 14 66 9.6093 6076 6059 6042 6024 6007 5990 5972 5954 5937 3 6 9 12 15 67 9-5919 5901 5883 5865 5847 5828 5810 5792 5773 5754 3 6 9 12 15 68 9-5736 5717 5698 5679 5660 5641 5621 5602 5583 5563 3 6 10 13 16 69 9-5543 5523 5504 5484 5463 5443 5423 5402 5382 5361 3 7 10 14 17 70 9-5341 5320 5299 5278 5256 523s 5213 5192 5170 5148 4 7 II 14 18 71 9.5126 5:04 5082 5060 5037 5015 4992 4969 4946 4923 4 8 II 15 19 72 9.4900 4876 4853 4829 4805 4781 4757 4733 4709 4684 4 8 12 16 20 73 .9.4659 4634 4609 4584 4SS9 4533 4508 4482 4456 4430 4 9 13 17 21 74 9.4403 4377 4350 4323 4296 4269 4242 4214 4186 "^If 5 9 14 18 23 75 9-4130 4102 4073 4044 4015 3986 3957 3927 3897 3867 5 lo 15 20 24 76 9-3837 3806 3775 3745 3713 3682 3650 3618 3586 3554 1 " 16 21 26 77 9-3521 3488 345S 3421 3387 3353 3319 3284 3250 3214 6 11 17 23 28 78 9-3179 3143 3107 3070 3034 2997 2959 2921 2883 2845 6 12 19 25 31 79 9.2806 2767 2727 2687 2647 2606 2565 2524 2482 2439 7 14 20 27 34 80 9-2397 2353 2310 2266 2221 2176 2131 2085 2038 1991 8 IS 23 30 38 81 9-1943 1895 1847 1797 1747 1697 1646 1594 1542 1489 8 17 25 34 42 82 9-1436 1381 1326 127 1 1214 1157 1099 1040 0981 0920 10 19 29 38 48 83 9.0859 0797 0734 0670 0605 0539 0472 0403 0334 0264 II 22 33 44 55 84 9.0192 0120 0046 9970 9894 9816 9736 9655 9573 9489 13 26 39 52 65 86 8.9403 9315 9226 9135 9042 8946 8849 8749 8647 8543 i6 32 48 64 80 86 8.8436 8326 8213 8098 7979 7857 7731 7602 7468 7330 21 41 62 82 103 87 8.7188 7041 6889 6731 6567 6397 6220 6035 S842 5640 88 8.5428 5206 4971 4723 4459 4179 3880 3558 3210 2832 89 8.2419 1961 1450 0870 0200 9408 8439 7190 5429 2419 0' & orO°l 12' or0°'2 18' orO°-3 24' orO°-4 30' or0°-6 36' or0°'6 42' orO°-7 48' or0°'8 54' or0°-9 1' 2' M 3' 4' 5' inutes. | Nu mber.s in minutes columns to be SI btracted not ad( ied. 24 LOGARITHMIC TANGENTS. 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' Minutes. | 0° orO°-l orO°-2 orO°-3 or0°-4 or0°-5 orO°-6 orO°-7 or0°-8 or0°-9 1' 2' 3' 4' 5' Inf. Neg. 7.2419 5429 7190 8439 9409 0200 0870 T450 T962 1 8.2419 2833 321 1 3559 3881 4181 4461 4725 4973 5208 2 8-S43I 5643 5845 6038 6223 6401 6571 6736 6894 7046 29 58 87 116 145 3 8.7194 7337 7475 7609 7739 7865 7988 8107 8223 8336 21 41 62 83103 4 8.8446 8554 8659 8762 8862 8960 9056 9150 9241 9331 16 32 48 64 81 5 8.9420 9506 9591 9674 9756 9836 9915 9992 0068 0143 13 26 40 53 66 6 9.0216 0289 0360 0430 0499 0567 0633 0699 0764 0828 II 2234 45 56 7 9.0891 0954 1015 1076 "35 1 194 1252 1310 1367 1423 10 20 29 39 49 8 9.1478 IS33 1587 1640 1693 1745 1797 1848 1898 1948 91726 35 43 9 91997 2046 2094 2142 2189 2236 2282 2328 2374 2419 8 1623 31 39 10 9.2463 2507 2551 2594 2637 2680 2722 2764 2805 2846 7 14 21 28 35 11 9.2887 2927 2967 3006 3046 3085 3123 3162 3200 3237 613 19 26 32 12 9-3275 3312 3349 3385 3422 3458 3493 3529 3564 3599 6 12 18 24 30 13 9-3634 3668 3702 3736 3770 3804 3837 3870 3903 3935 6 II 17 22 28 11 9.3968 4000 4032 4064 4095 4127 4158 4189 4220 4250 5 10 16 21 26 16 9.4281 43" 4341 4371 4400 4430 4459 4488 4517 4546 5 10 15 20 25 16 9-457S 4603 4632 4660 4688 4716 4744 4771 4799 4826 5 9 14 19 23 17 9-4853 4880 4907 4934 4961 4987 5014 5040 5066 5092 4 913 18 22 18 9.5118 5143 5169 5195 5220 5245 5270 5295 5320 5345 4 813 17 21 19 9-5370 5394 5419 5443 5467 5491 5516 5539 5563 5587 4 8"I2 16 20 20 9. 561 1 5634 5658 5681 5704 5727 5750 5773 5796 5819 4 8 12 15 19 21 9.5842 5864 5887 5909 5932 5954 5976 5998 6020 6042 4 7 II 15 19 22 9.6064 6086 6108 6129 6151 6172 6194 6215 6236 6257 4 7" 14 18 23 9.6279 6300 6321 6341 6362 6383 6404 6424 6445 6465 3 7 10 14 17 21 9.6486 6506 6527 6547 6567 6587 6607 6627 6647 6667 3 7 10 13 17 26 9.6687 6706 6726 6746 6765 6785 6804 6824 6843 6S63 3 7 10 13 16 26 9.6882 6901 6920 6939 6958 6977 6996 7015 7034 7053 369 13 16 27 9.7072 7090 7109 7128 7146 7165 7183 7202 7220 7238 369 12 15 28 9-7257 7275 7293 73" 7330 7348 7366 7384 7402 7420 369 12 15 29 9-7438 7455 7473 7491 7509 75?6 7544 7562 7579 7597 369 12 15 30 9.7614 7632 7649 7667 7684 7701 7719 7736 7753 7771 369 12 14 31 9.7788 7805 7822 7839 7856 7873 7890 7907 7924 7941 369 II 14 32 9-7958 7975 7992 8008 8025 8042 8059 8075 8092 8109 368 II 14 33 9-8125 8142 8158 8175 8191 8208 8224 8241 8257 8274 3 5 8 II 14 34 9.8290 8306 8323 8339 8355 8371 8388 8404 8420 8436 3 5 8 II 14 36 9.8452 8468 8484 8501 8517 8533 8549 8565 8581 8597 3 5 8 II 13 36 9-8613 8629 8644 8660 8676 8692 8708 8724 8740 8755 3 5 8 II 13 37 9.8771 8787 8803 8818 8834 8850 8865 8881 8897 8912 3 5 8 10 13 38 9.8928 8944 8959 8975 8990 9006 9022 9037 9053 9068 3 5 8 10 13 39 9.9084 9099 9115 9130 9146 9161 9176 9192 9207 9223 3 5 8 10 13 40 9.9238 9254 9269 9284 9300 9315 9330 9346 9361 9376 3 5 8 10 13 41 9-9392 9407 9422 9438 9453 9468 9483 9499 9514 9529 3 5 8 10 13 42 9-9544 9560 9575 9590 9605 9621 9636 9651 9666 9681 3 5 8 10 13 43 9.9697 9712 9727 9742 9757 9773 9788 9803 9818 9833 3 5 8 10 13 44 9.9848 9864 9879 9894 9909 9924 9939 9955 9970 9985 3 5 8 10 13 0' 6' orC-l 12' or0°-2 18' or0°'3 24' orO°-4 30' or0°'6 36' or0°-6 42' orO°-7 48' or0°-8 54' orO°-9 1' 2' 3' 4' 5' Minutes. | LOGARITHMIC TANGENTS. 45° 0' 6' orO°l 12' or0°-2 18' or0°-3 24' or0°-4 30' or0°'6 36' or0°'6 42' or0°'7 48' or0°-8 54' or0°-9 Minutes. 1 1' 2' 3' 4' 5' lO.OOOO 0015 0030 0045 0061 0076 0091 0106 0121 0136 3 5 8 10 13 46 10.0152 0167 0182 0197 0212 0228 0243 0258 0273 0288 3 5 8 10 13 47 10.0303 0319 0334 0349 0364 0379 0395 0410 0425 0440 3 S 8 10 13 48 10.0456 0471 0486 0501 0517 0532 0547 0562 0578 0593 3 5 8 10 13 49 10.0608 0624 0639 0654 0670 0685 0700 0716 0731 074b 3 5 8 10 13 50 10.0762 0777 0793 0808 0824 0839 0854 0870 0885 0901 3 5 8 10 13 51 10.0916 0932 0947 0963 0978 0994 lOIO 1025 1041 1056 3 5 8 10 13 52 10. 1072 1088 1 103 II19 1135 1 150 1 166 1182 1 197 1213 3 5 8 10 13 53 10. 1229 1245 1260 1276 1292 1308 1324 1340 1356 1371 3 5 8 :i 13 54 10.1387 1403 1419 1435 1451 1467 1483 1499 1516 1532 3 5 8 II 13 55 10.1548 1564 1580 1596 1612 1629 1645 1661 1677 1694 3 5 8 II 14 56 IO.I7IO 1726 1743 1759 1776 1792 1809 1825 1842 1858 3 5 8 II 14 57 10. 1875 1891 1908 1925 1941 1958 1975 1992 2008 2025 368 II 14 58 10.2042 2059 2076 2093 2110 2127 2144 2161 2178 2195 3 6 9 II 14 59 10.2212 2229 2247 2264 2281 2299 2316 2333 2351 2368 369 12 14 60 10.2386 2403 2421 2438 2456 2474 2491 2509 2527 2545 369 12 15 61 10.2562 2580 2598 2616 2634 2652 2670 2689 2707 2725 369 12 15 62 10.2743 2762 2780 2798 2817 2835 2854 2872 2891 2910 369 12 15 63 10. 2928 2947 2966 298s 3004 3023 3042 3061 3080 3099 369 13 16 64 I0.3II8 3137 3157 3176 3196 3215 323s 3254 3274 3294 3 610 13 16 65 10.3313 3333 3353 3373 3393 3413 3433 3453 3473 3494 3 710 13 17 66 10.3514 3535 3555 3576 3596 3617 3638 3659 3679 3700 3 710 14 17 67 10.3721 3743 3764 3785 3806 3828 3849 3871 3892 3914 4 711 14 18 68 10.3936 3958 3980 4002 4024 4046 4068 409: 41 13 4136 4 711 15 19 69 10.4158 4181 4204 4227 4250 4273 4296 4319 4342 4366 4 812 15 19 70 10.4389 4413 4437 4461 4484 4509 4533 4557 4581 4606 4 812 16 20 71 10.4630 4655 4680 4705 4730 4755 4780 4805 4831 4857 4 813 17 21 72 10.4882 4908 4934 4960 49S6 5013 5039 5066 S093 5120 4 913 18 22 73 10.5147 5174 5201 5229 5256 5284 5312 5340 5368 5397 5 914 19 23 74 10.5425 5454 5483 SS12 SS4I 5570 5600 5629 5659 5689 5 10 15 20 25 75 10.5719 5750 5780 5811 5842 5873 5905 5936 5968 6000 5 10 16 21 26 76 10.6032 6065 6097 6130 6163 6196 6230 6264 6298 6332 6 II 17 22 28 77 10.6366 6401 6436 6471 6507 6542 6578 6615 6651 6688 6 12 18 24 30 78 10.6725 6763 6800 6838 6877 6915 6954 6994 7033 7073 6 13 19 26 32 79 10.71 13 7154 7195 7236 7278 7320 7363 7406 7449 7493 7 14 21 28 35 80 10.7537 7581 7626 7672 7718 7764 781 1 7858 7906 7954 81623 31 39 81 10.8003 8052 8ro2 8152 8203 825s 8307 8360 8413 8467 91726 35 43 82 10.8522 8577 8633 8690 8748 8806 8865 8924 8985 9046 TO 20 29 39 49 83 10.9109 9172 9236 9301 9367 9433 9501 9570 9640 9711 II 1234 45 56 84 10.9784 9857 9932 0008 0085 0164 0244 0326 0409 0494 132640 53 66 85 11.0580 0669 0759 0850 0944 1040 1138 1238 1341 1446 163248 64 81 86 "■ISS4 1664 1777 1893 2012 2135 2261 2391 2525 2663 20 41 62 83103 87 11.2806 2954 3106 3264 3429 3599 3777 3962 4155 4357 295887 116 144 88 11.4569 4792 5027 5275 5539 5819 6119 6441 6789 7167 89 11.7581 8038 8550 9130 9800 0591 1561 2810 4571 7581 0' 6' orO°-l 12' or0°-2 18' or0°'3 24' or0°-4 30' or0°-5 36' or0°'6 42' or0°-7 48' or0°-8 54' orO°-9 V 2' 3' 4' 5' Minutes. | 26 TABLES OF SQUARES, CUBES, SQUARE ROOTS, CUBE ROOTS, AND RECIPROCALS. n ffi «5 x/» Vn I n n n? »3 v/» 3/- I n 1 I I I I I 61 2601 I3265I 7. 141 3-708 .01961 2 4 8 1-414 1.260 •50000 62 2704' 140608 7.211 3-733 .01923 , 3 9 27 1.732 1.442 .33333 63 2809 148877 7.280 3-756 .01887 4 i6 64 2.000 1.587 .25000 64 2916 157464 7-348 3.780 .01852 6 25 125 2.236 1. 710 .20000 66 3025 166375 7.416 3.803 .01818 6 36 216 2.449 1. 817 .16667 56 3136 I75616 7.4S3 3.826 .01786 ■7 49 343 2.646 1-913 .14286 67 3249 I85I93 7-550 3-849 -OI7S4 8 64 512 2.828 2.000 .12500 68 3364 I95II2 7-616 3-871 .01724 9 8i 729 3.000 2.080 .mil 69 3481 205379 7-681 3-893 .01695 10 100 1000 3.162 .2.154 .10000 60 3600 216000 7.746 3-915 .01667 11 121 i33> 3-317 2.224 .09091 61 3721 226981 7.810 3-936 -01639 12 144 ■ 1728 3-464 2.289 -08333 62 3844 238328 7.874 3-958 .01613 13 169 2197 3-606 2- 351 .07692 63 -396^ 250047 7-937 3-979 .01587 :. 11 196 2744 3-742 2.410 -07143 64 4096 262144 8.000 4.000 .01563 15 225 3375 3-873 2.466 .06667 65 4225 274625 8.062 4.021 -01538 16 256 4096 4.000 2.520 .06250 66 4356 287496 8.124 4.041 .01515 -^ 17 289 4913 4-123 2-571 .05882 67 4489 300763 8.185 4.C62 •01493 18 324 S832 4-243 2.621 -05556 68 4624 314432 8.246 4.082 .01471 19 361 6859 4-359 2.668 .05263 69 4761 328509 8.307 4.102 .01449 20 400 8000 4.472 2.714 .05000 70 4900 343000 8.367 4.121 .01429 21 441 9261 4-583 2.759 .04762 71 5041 3579" 8.426 4. 141 .01408 22 484 10648 4.690 2.802 -04545 72 5184 373248 8.48s 4.160 .01389 23 529 12167 4.796 2.844 .04348 73 5329 389017 8.544 4-179 .01370 21 576 13824 4-899 2.884 .04167 74 5476 405224 8.602 4.198 .01351 25 62s 15625 5.000 2.924 .04000 75 5625 421875 8.660 4.217 •01333 26 '676 17576 5.099 2.962 .03846 76 5776 438976 8.718 4-236 .01316 27 729 19683 5-196 3.000 .03704 77 5929 456533 8.775 4-254 .01299 28 784 21952 5.291 3-037 -03571 78 6084 474552 8.832 4-273 .01282 29 841 24389 5- 38s 3-072 .03448 79 6241 493039 8.888 4.291 .01266 30 900 27000 5-477 3-107 -03333 80 6400 512000 8-944 4-309 .01250 31 961 29791 5-568 3- 141 .03226 81 6561 531441 9.000 4-327 .01235 32 1024 32768 5-657 3- 175 .03125 82 6724 551368 9.C5S 4-344 .01220 33 1089 35937 5-745 3.208 .03030 83 6889 571787 9. 1 10 4-362 .01205 31 1156 39304 5-831 ■3.240 .02941 84 7056 592704 9.165 4-380 .01191 35 1225 42875 5-916 3-271 .02857 85 7225 614125 9.220 4-397 .01177 36 1296 46656 6.000 3-302 .02778 86 7396 636056 9.274 4.414 .01163 37 1369 50653 6.083 3-332 -02703 87 7569 658503 9.327 4-431 .01149 38 1444 54872 6.164 3-362 .02632 88 7744 681472 9-381 4-448 .01136 39 1521 59319 6.245- 3-391 .02564 89 7921 704969 9-434 4-465 .01124 40 1600 6.325 3-420 .02500 90 8100 729000 9-487 4.481 .01111 41 1681 68921 6.403 3448 .02439 91 8281 753S7I 9-539 4.498 .01099 42 1764 74088 6.481 3-476 .02381 92 8464 778688 9.592 4-5H .01087 43 1849 79507 6- 557 3-503 .02326 93 8649 804357 9-644 4- 531 -01075- 44 1936 85184 6-633 3-530 .02273 94 8836 830584 9-695 4-547 .01064 45 2025 91125 6.708 3-557 .02222 96 9025 85737s 9-747 4-563 -01053 46 2116 97336 6.782 3-583 .02174 96 9216 884736 9.798 4.579 .01042 47 2209 103823 6.856 3-609 .02128 97 9409 912673 9:849 4-595 .01031 48 2304 I 10592 6.928 3-634 .02083 98 9604 941192 9-899 4.610 .01020 49 2401 I 17649 7.000 3-659 .02041 99 9801 970299 9.950 4.626 .01010 60 2500 125000 7.071 3.684 .02000 100 lOOOO lOOOOOO 10.000 4.642 .01000 n «2 »» ^n Vn I n n «2 »s ^/« V^ . I » 27 TABLES OF BRITISH WEIGHTS^ AND MEASURES. AVOIRDUPOIS WEIGHT. (for general merchandise.) Hundred- Metric weights. Equivalents. I Ton 20 2240 35840 573440 15680000 1015 kilogrammes. I Hundredweight (cwt.) = I 112 1792 28672 784000 50-7 kilogrammes. I Pound (lb.) .008928 I 16 256. 7000 453-6 grammes. I Ounce (oz.) = — .0625 I 16 437-5 28.35 grammes. I Dram (dr.) — .003906 .0625 I 27-35 1.772 grammes. I Grain (gr.) = — .0001428 . .002286 -03657 I .0648 grammes. TROY WEIGHT. (for silver, gold, and precious stones.) Ounces. i Pennyweights. Grains. Grammes. I Pound (lb.) I Ounce (oz. ) - = I Pennyweight (dwt.) = I Grain (gr. ) - = 12, I -OS .002084 240 20 .04168 5760 480 24 I 373-2 31.10 I-5S5 .0648 I carat = 4 grains when used as a weight ; used with reference to gold alloys it signifies ^jth pure gold, thus 18 carat gold contains J|ths of its weight of pure gold. APOTHECARIES WEIGHT. Ounces. Drachms. Scruples. Grains. Grammes. - I Pound (lb.) I Ounce (S) I Drachm (3) I Scruple O) 12 I .125 .04166 96 8 I -3333 288 24 3 I 5760 480 60 20 373-2 31.10 3.888 1.296 28 IMPERIAL FLUID MEASURE. Quarts. Pints. Fluid Ounces. Fluid Drachms. Minims. Cubic Inches. Cuhic Centimetres. I Gallon (gal.) - 4 8 160 1280 76800 277-3 4544 I Quart (qt.) = 1 2 40 320 19200 69.32 1 136 I Pint (pt.) - = ■5 I 20 i6o 9600 34.66 567.9 I Fluid Ounce .025 ■OS I 8 480 1-733 28.39 I Fluid Diachm = .003125 .00625 .125 I 60 .2166 3-548 I Minim .00005 .0001 .00021 .0167 I .00361 -059 I cubic inch = 16.39 cubic centimetres. I cubic centimetre = .06101 cubic inch. LENGTH. Miles. Furlongs. Chains. Rods. Yards. Feet. Inches. Centimetres. I Mile I 8 80 320 1760 5280 63360 160931.49 I Furlong = •125 I 10 40 220 660 7920 201 16.4 I .Chain = .0125 .1 I 4 22 66 792 20II.6 1 Rod .003125 .025 .25 I 5-5 16.S 198 502.9 I Yard = .0005682 .004545 -04545 .1818 I 3 36 91.44 I Foot = .0001894 .001515 .01515 .06061 •3333 I 12 30.48 ■I Inch .00001578 .0001263 .001263 .005051 .02778 ■08333 I 2.54 Gunter's Chain : i link = 7.92 inches. 100 links = i chain. SURFACE. Square Square Square Square Square Square Metric Miles. Chains. 6400 Rods. Yards. Feet. Inches. Equivalents. I Sq. Mile = I 640 102400 3097600 27878400 4014489600 2.5899 sq. kilometres. I Acre .0016 I 10 100 4840 43560 6272640 4046.7 sq. metres. I Sq. Chain = .0002 .1 r 10 484 4356 627264 404.67 sq. metres. I Sq. Rod = — .0062 .0625 , I 30.25 272.25 39ZO4 25.29 sq. metres. I Sq. Yard = — .0002 .0021 •033 I 9 1296 .836 sq. metre. I Sq. Foot = — — .0002 .0037 .1111 I 144 929 sq. centimetres. I Sq. Inch = .0007 .007 1 6.451 sq. centimetres. FORMULAE IN PURE AND APPLIED MATHEMATICS. FORMULAE IN MENSURATION. 31 FORMULAE IN MENSURATION. I. LENGTHS OF CURVES. Circumference of Circle =27r/-, „ „ = ird, where 7r = -^y?- or 3.1416. Circumference of Ellipse = 27r w , approximately. j' Arc of Parabola = 2a n'^ + ^^ — Arc {BAC) of Hyperbola 32 LENGTHS OF CURVES. ..- I ■ Rule I. Arc (/) of Circle = — ^ ' i8o 6 being in degrees. Rule II. Arc (/) of Circle = = .0175/-^, U-a Arc {ASCf of Cycloid = 8/-, where r is the radius of the generating circle. 2. AREAS OF PLANE FIGURES. Area of Triangle _6h 2 ' Area of Triangle a =\bcsva.A. Area of Triangle =Js{s-a){s-b){s-c), ^ \ ■'■ , a + 6 + t: 6 *o where 5= ^!— . Area of Parallelogram = M. Area of Parallelogram = ad sin A. Area of Parallelogram = ^dd-^ sin 6, Area of Quadrilateral ^c-*-B FORMULAE IN MENSURATION. z:i AREAS OF REGULAR POLYGONS. Pentagon. Square the side, and multiply by 1. 721 Hexagon. ,, ,, ,, 2.598 Heptagon. ,, ,, ,, 3-634 Octagon. ,, ,, ,, 4.828 Nonagon. ,, ,, ,, 6.182 Decagon. ,, ,, ,, 7.694 Undecagon. ,, ,, ,, 9.366 Dodecagon. ,, ,, ,, II. 196 Area of Circle = i^r^. = .7854^2. Rule I. Area of Sector of Circle = —r- 360 = .00873^^5, ^ 6 being in degrees. RULK II. Ir Area of Sector of Circle = — , 2 where /= length of arc. Area of Segnnent of Circle _Tr'^e y^sin(9 360 2 Q being in degrees. c 34 AREAS OF PLANE FIGURES, Area of Parabola = ^xy. Area of Ellipse = ^a^. ^ff^^i 1 Area {BAC) of Hyperbola I^: i X 75 « Area of Cycloid = ^^ — Simpson's Rule for finding Areas : Divide AB into any odd number of equal parts and erect ordinates. Add together the two end ordinates, twice the sum of the even ordinates, and four times ...the sum of the odd ordinates, multiply, by the distance between any two ordinates, and divide by three. FORMULAE IN MENSURATION. 35 Thus, if AD and BC are the end ordinates, ^, = the sum of the even ordinates, 2, 4, 6, etc., ^»= „ „ odd „ I, 3, 5, etc.,' ^ = area of ABCD; then A = {AD+BC+2S,^-^S^^'~- 3. SURFACES AND VOLUMES OF SOLIDS. \h Curved Surface of Cylinder = zirrh. Volume of Cylinder = ■Kr''h. Curved Surface of Cone = irrJr'^ + /i'. Volume of Cone 3 = 1.047/-^,^. Surface of Sphere =4Trr^ = 12.57^2- Volume of Sphere = |7r;-3 = 4. 189^^. 36 SURFACES AND VOLUMES OF SOLIDS. Curved Surface of Spherical Segment = 217 ah. Rule I. Volume of Spherical Segment Rule II. Volume of Spherical Segment = \-^h\ir-h) = \.o/^'jh\ir-h), where r is the radius of the sphere. Volume of Pyramid = Area of base x \ height. FORMULAE IN ALGEBRA. 37 FORMULAE IN ALGEBRA. USEFUL FACTORS. {a + b)^ = «3 + la'-b + lab"- + -J^. (a - bf = aS _ 3«2^ + T^alP- - b^. a^-^ = (a-b){a-hb). a«~b^ = {a-b)(a^ + ab + b'i). a^ + b^={a + b)(a^ -ab + b^). a" - /5" = (a - b){a"-^ + a"-^b+...+ b"-'') for all values of n. a'- - ^» = (a + /;)(a»-i -a'-^+...- b"''), if n be ^z;««. a" + (5" = (fl + b)(a"-'' -a"-^+...+ 3""'), if « be odd. a* + aW + ^* = (flS + a* + ^2)(a2 _ ^^ + ^2)_ (a + ^ + c)- = a^ + ^2 + ^ + 2ai5 + 2a^ '^ 'x^ +etc. 4° BINOMIAL SERIES. (\-x)-^=-i. + ^+ y^-ir oj2 2 + V^ 1 00 - 1 2v/2 >/s+i 4 \/lO + 2Vs 4 4 ViO + 2n/s 4 Vs-zVs ■JlO-Z-y/s \/5 + 2^5 RELATIONS BETWEEN THE FUNCTIONS. 43 RELATIONS BETWEEN THE FUNCTIONS ^vs\A= -, cosec/4=-^ — -, cosec A sin A cos A = :, sec A = ~, sec A cos A I sin ^ .1 cos A ta.aA= _, = -, cotA = 2 a=~ — j' cot A cos A tan A sin ^ sin ^ = V I - cos^y^, cosA = Ji -sin^^, secA= Jtzn^A + i, tan ^ = Jsec'A - 1 , cosec ^ = jJcot^A + I , cot ^ = Vcosec^yi - i . sin (90 — ^)= cos^, cos (90-^)= sin^, tan (90-^)= cot^, sin(i8o-^)= sin^, cos(i8o -^)=-cos^, tan(i8o -^)=-tan^, sin(-y^) =-sin^4, cos(-^) = cos^, tan(-^) =-tan^, sin (90 + ^)= cos^, cos (90 + ^)=- sin y^, tan (90 + ^; =- cot^, sin(i8o + ^) = -sin^, cos(i8o + yi)=-cos^, tan(i8o + ^)=- tan^. FUNCTIONS OF SUMS OF ANGLES. sin (^ ±B)= sin A cos B ± cos A sin B, cos{A ±B) = cos A cos j5 + sin ^ sin B, , . , _, tany4±tan^ tan (A±B) = -. 5. ^ I + tan ^ tan ^ FUNCTIONS OF MULTIPLE ANGLES, sin 2 A = 2 sin /4 cos A, cos 2 A = co?,^A — sw?-A = 2 cos^A — 1 = 1-2 sin^^, . 2 tan A sin 3^ = 3 sin ^ - 4 sin^^, cos3/4 = 4COS^^- 3cos^, ^ -itanA- ta.n^A tan %A = 5-; — •' 1-3 tan^/i PRODUCTS OF FUNCTIONS. 2 sin A cos B = sin (sum) + sin (difference), 2 cos ^ sin ^ = sin (sum) - sin (difference), 2 cos A cos B = cos (sum) + cos (difference), 2 sin A smB = cos (difference) - cos (sum). 44 PLANE TRIGONOMETRY. SUMS AND DIFFERENCES OF FUNCTIONS. Sum of two sines = 2 sin (half-sum) x cos (half-difference), difference of two sines = 2 cos (half-sum) x sin (half-difference), sum of two cosines = 2 cos (half-sum) x cos (half-difference), difference of two cosines = 2 sin (half-sum) x sin (half-difference reversed). RELATIONS BETWEEN THE SIDES AND ANGLES OF ANY TRIANGLE. sin A sin B sin C 7 ^ . d = — ; — = , fl! = ^cos CH-^cos^, a c ^2 ^ ^ _ ^2 a'' = b''' -\- fi - 2bc cos A, cosA-- 2bc 2 a-Vb 2' """""iJf^ ^ A-B a-b ^C ■ , 2 , ■ '""-^=;^'=°'T' ^^nA = j.Js(s-a)(s-b){s-c), sind = JlZ^HlzA, cos^=>J£ffl, tan^ = >-/)(---). 2 V be 2 \ be 2 V j(j-a) SOLUTION OF TRIANGLES. Case i. Given the three sides {a, b, c), J. (i.) L tan — = A {log(j - b)+\og(s -c)-\ogs- \og{s - a)} -1- 10, which gives A. (ii.) Z tan - = l{\og{s -a) + \og{s -c)-\ogs- \og{s - b)} + 10, which gives £. (iii.) C= 180 - (^ + B), which gives C. Case 2. Given two angles and a side {A, B, a), (i. ) C = 1 80° - (^ + ^), which gives C. (ii. ) log ^ = log a H- Z sin ^ - Z sin ^, which gives b. (iii.) log tf = log a -f Z sin C - Z sin A, which gives c. Case 3. Given two sides and the included angle {a, b, C), /,\ A + B C u- , • A^-B (i.) = go , which gives — — . 22 2 A-B C (ii. ) Z tan = log(a -b)- log(a ■{■b) + L cot — , A — B which gives .. SOLUTIO.V OF TRIANGLES. 45 /•■•\ A A + £ A-B , • . . (111.) A = 1 , which gives A. 2 2 (iv.) B== , which gives ^. 22 (v.) log^=loga + Zsin C-Zsin^, which gives f. Case 4. Given two sides and the angle opposite one of them (A, a, b), (i.) Z sin ^ = log * + Z sin ^ - log a, which gives B. (ii. ) C = 1 80 - (^ + ^), which gives C. (iii.) logf=loga + Zsin C-Zsin^, which gives ^. Note. — If ^ > a we obtain two values of B to each of which (ii.) and (iii.) apply. TRIANGLES AND CIRCLES. Radius of Circumcirck: R=^ ^ 2 sin ^ 2 sin B 2 sin C Radius of In-circle : r=—, where A = area of triangle. Radius of Ex-circles : A _ A _ A ''"^s-a' ''"-J^re "'"-T^Tc INSCRIBED AND CIRCUMSCRIBED POLYGONS. 1. Polygon of n sides inscribed in circle of radius r. Perimeter = 2nr sin -, n Area = ^w/^sin — . 2. Polygon of n sides circumscribed about circle of radius r. Perimeter — 2nr tan -, n Area= nr^ ta.n-. 46 SPHERICAL TRIGONOMETRY. SPHERICAL TRIGONOMETRY. RELATIONS BETWEEN THE SIDES AND ANGLES OF ANY SPHERICAL TRIANGLE. sin^ sin ^_ sin C sin a sin 3 sin;;' cos a — cos b cos c-\- sin b sin c cos A, cos a — cos b cos c cos A = • sin b sin r sin^ = VI - cos^fl - cos^^ - cosV + 2 cos a cos b cos f siniJsinf cot a sin iJ = cot ^ sin C + cos ^ cos C. A /sin (j - b) sin (j - c) cos tan sin b sin f sin J sin (j- - fl) sin bw\c ' Vsin{s - b)s'in(s — c) sin s sin (^ - a) cos tan a_ I cosScos{S-A) 2 \ sin ^ sin C a_ \coi{S^B)co%{S - C) 2 ~ V sin ^ sin C f=V- cos6-cos(.S'-^) cos(6'-^)cos(.S'-6y sin -^ = ^j^^j^jj^s/sin ^ sin (y - a) sin (^ - b) sin ( j - 2 A+B 2 2 ■ A+B 2 cos sin sin C= 2 sin A sin ^ Case 5. Given two sides and the angle opposite one of them (a, b, A), — ^ - b cos . „ sin^sin^ ^ C 2 ^A + B smB = -. , tan- = jcot — , sin « 2 a + b 2 cos 2 A+B cos c 2 ^ a+b tan-= =tan- 2 A-B 2 cos Case 6. Given two angles and the side opposite one of them (A, B, a), a-b cos . , sm ^ sin a C 2 ^A + B sin 3 = -. — 3 — , tan = rcot , sin ^ 2 a + b 2 cos 2 A + B cos c 2 ^ a+b tan - = -. T, tan 2 A-B 2 cos AREA OF A SPHERICAL TRIANGLE. where r is the radius of the sphere. ANALYTICAL GEOMETRY. THE POINT. Distance (8) between two points. Rectangular coordinates, 8= ± 'Jl^^^T+{y^^^^ Oblique coordinates, angle between axes, fc), 8=±V(jr2-jri)2 + (j,2_j/j)2+2(^2-;iri)(;|/2-J'i)coso). Polar coordinates, 8= + 'Jr{^+ri--irjr^cos(,0i-6^). Example : Find the length of the straight line joining the points (2, 30°) and (4, 120°). Here ri=2, ^2=4, ^1 = 3°°, ^2=120°; 8=\/4+ 16—16 cos 90° = 4.472. Area of a triangle. Rectangular coordinates, A=i{jriJ)'2-^2j'l + ^2j'3 Example : Find the area of the triangle whose angular points are (i, 3), (2, 5), and (4,4). A=i{s-6+8-2oH-i2-4} = —2.5 (the negative sign is due to the order in which the points are taken). 5° ANALYTICAL GEOMETRY. Polar coordinates, A=i{V2sin(^2-^i) + rj^gsinC^a - B^) + nr-^Anifi-^ - 63)} Example : The area of the triangle whose angular points are (I, 30°), (2, 60°), (3, 9o°)> is -701 units, the units being square centi- metres if the radii vectores are in centimetres. THE STRAIGHT LINE. Equation in terms of inclination, rect. axes, j/ = mx+c. Thus the equation F=.'i6lV+2.s is the equation to a straight line cutting the axis of y or P" at a height of 2.5 and inclined to the horizontal at an angle whose tangent is .36. Equation to line through (x^, yj), rect. axes, j/-y^ = m{x-Xi). Example : The equation to the line through the point (j, 6) and inclined at an angle of 60° to the axis of X is jf-6='>/3{x-s). THE STRAIGHT LINE. SI Equation to line through (-niJ'l)(^2,^2), rect. and oblique axes, Example : The equation to the line through the points (3, 4) and (S, 6)is^=;ir+i. Equation in terms of intercepts on axes, a Example : The equation to the line cutting intercepts of 2 and 3 from the • ■*■ y , ■, axes is - +- = I ; also the equation X y - + 7= I is the equation to a line cut- 3 4 ting off intercepts of 3 and 4 from the axes. Note that in this form the right- hand side of the equation must be unity. Equation in terms of perpen- dicular from origin, ;f cos d+ysm Q=p. Example : The perpendicular from the origin upon a straight line is inclined to the axis of x at an angle of 30°, its length being 6 ; the equation to the line will therefore be W3-l-j»'= 12 52 ANALYTICAL GEOMETRY. General equation of straight line, Ax+By+C=o. Length of perpendicular from (Xi,yi) upon Ax+By+C=o, A£i±By^±£ /=- Example : The length of the per- pendicular from the point (4, 5) upon the straight line 2x+^y—lo=o is 4.4. Length of perpendicular from (^ijJi) upon jTcos d+y sin d=P, l=x^cos ^+j'isin 6— p. Note that the equation to a line is of the form ;rcos d+y sin 0=p when the sum of the squares of the coefficients oix and j' is unity. Example : The length of the perpendicular from the point (5, 6) upon the line ^^ +- = 5 is 2.33. Equation to straight line, axes oblique, y = -T—. .X + c. sm (o) — a) Example : The angle between the axes being 60°, find the equation to a line inclined at 30° and cutting off an intercept of 3 from the axis of _j'. Ans. y=x+3. THE STRAIGHT LINE. 53 Polar equation to straight line, rcos(^-a)=/. THE CIRCLE. Equation to circle, centre at {a, b). Example : The equation to the circle whose centre is at the point (S, 5), and radius lo is x^+y^— lojr— i2y=39. Equation to circle, origin at centre, x^+y^ = c\ Equation to tangent at {xi,y-^, xx-^+yy-^=t?. Equation to normal at {x^y^, Polar equation to circle, pole on circumference, r=2^C0s(^— a). Example : The diameter through the pole makes an angle of t- with the initial line, its length being 5. What is the equation to the circle ? Ans. /-=5cosf ^--j j. 54 ANALYTICAL GEOMETRY. Polar equation to circle, centre at (/, a), ^2 = ^2+/2_2;./cos(^-a). THE PARABOLA. Equation to parabola, origin at vertex, 7^ = 4^:^. Equation to tangent at x-^, j/j^ ^/i=2a(jr+jri). Equation to normal at jtj, ^j, J'->'i=-|i(^-A-.). Polar equation to parabola, pole at focus, - = I - cos 6. THE PARABOLA. 55 Polar equation to parabola, pole at vertex, _\a cos Q This equation is also written /■=4acot ^cosecft THE ELLIPSE. V y ^ :::^ / 1 h / ^' \^\ 1 « X, J-- '' 1 ^^-^ L a — / A J ^>^ Equation to ellipse, origin at centre, Equation to tangent at (^i, Ji), Equation to normal at (^i,J'i), 56 ANALYTICAL GEOMETRY. Polar equation to ellipse, pole at focus, -=i — «cos Q, where e<\. Polar equation to ellipse, pole at centre. THE HYPERBOLA. \ \ n Y\ Equation to hyperbola, origin at centre. Equation to tangent at (ix-^,y^. Equation to normal at (^i.j'i); THE HYPERBOLA. 57 Polar equation to hyperbola, pole at focus : I a - = i— ecos a, where e>i. Polar equation to hyperbola, pole at centre : „ am aH\nW~b^cosW GENERAL EQUATION OF THE SECOND DEGREE. The general equation of the second degree is ax^+2hxj/+by^+2gx+2fy+c=o. This equation represents An Ellipse, when h^ab; A Circle, „ ii = b, and /i=o ; Two Straight Lines,t „ abc+'2.fgh-af'-b^-ch^=o. * Which is rectangular, when a + i = o. t Which are parallel, when X' = ab. S8 DIFFERENTIAL AND INTEGRAL CALCULUS. DIFFERENTIAL AND INTEGRAL CALCULUS. RULES FOR DIFFERENTIATION. Rule i. A constant connected by a plus or minus sign disappears on differentiation. Rule 2. A constant multiplier or divisor remains as such after differentiation. Rule 3. To find the differential coefficient of any power of x, multiply together the index and x with its index diminished by unity. Example: y=x^, :. -^ = nx"-^ ; y=x^, .". -^ = 6x^, etc. ax ax Rule 4. To find the differential coefficient of the product of two functions of x, multiply each function by the D.C. of the other and add the results. Example: y=ax%c+x^), J-=ax^X2x^ + 2ax{c+x^)=^2acx+sax*. Rule 5. To find the differential coefficient of a fraction, multiply the D.C. of numerator by the denominator, subtract the D.c. of the denominator multiplied by the numerator, and divide by the square of the denominator. _ , 4x^ dy I2X^{$ + X^)-2XX4X^ 36;ir^ + 4jr* Example : ^ = ^-j:^,, ^= ^^^^^^ = (3 + x^y ' Rule 6. To find the differential coefficient of an exponentiaF function, multiply together the function itself, the D.C. of the exponent, and the hyperbolic log of the base. Example : J/ = a"*, -^ = ndr\o%^a; y = e^, j^ = «e~logee=«£~- RULE 7. To find the differential coefficient of a logarithmic function, multiply the D.C. of the function by the log of « to the given base, and divide by the function itself. Example : J/ =loga«j:, -2=^_2l£f=ilog„e ; y=\og^nx, -^=~. dx nx x ■' dx X Rule 8. The differential coefficient of the sine of an angle is equal to the cosine of the angle multiplied by the D.c of the angle. Example : j = sin mx, ^ = (cos mx) y.m=m cos mx. Rule 9. The differential coefficient of the cosine of an angle is equal to minus the sine of the angle multiplied by the D.C. of the angle. Example : y—za% mx, -j- = —in sin mx. RULES FOR DIFFERENTIATION. 59 Rule 10. The differential coefficient of the tangent of an anglais equal to the square of the secant of the angle multiplied by the D.c. of the angle. Example: _)' = tanw/x, -~ = rm&:?mx. Rule i i. The differential coefficients of the inverse functions are obtained as in the following examples : . dy I dv in j=sm 'jr, -T-j= r ^ i jc=sm-'?«a-, *) v,- I r, J y= sill in^. ,,, — , — 5) _, dv —I dy —m y=cos X, -T-_= .- — _; ^=cos-'7«j.-, ;r;^ '" Ji-x' -^ ' dx Ji-m^x^ » ^1 dy I ^ . dy in J'=tan 'x, -j- = -; y = ta.n-hnx, f-=— — j— ,. EULER'S THEOREM. If w = Ax'y^ + Bx^'^y^ + etc., where p + f =p^ + fi = etc. = «, then 3a . 3« LEIBNITZ'S THEOREM. If u and z/ are both functions of x, then d''{uv) _d"u .^r'^—H ^'^ «r ^"~'" ^4. rfjr»-i dx^ ^dx' d^ dx''-' ' lix" ' "' ' "dx' TAYLOR'S THEOREM. f(;r+y5) = f(;r)+>4fX:r)+|-f» + ^f,»+ ... +^f(nlW+ ■.■ to inf. MACLAURIN'S THEOREM. f(;r)=f(o)+;rf,(o) + ^''f„(o)+^f„Xo)+ ... + Jf(")(°)+ - to inf. EQUATION TO TANGENT AND NORMAL. (i.) Curve, y = i{x) ; tangent, I'-J'=(^-^) Ji "°™^>' (^-^) + (J'-^) J=o. (ii.) Curve, f(;f,/)=o; tangent, (.V-x)|^+( K_^)|=o ; normal, ^=-^. 3a- 3y 6o DIFFERENTIAL AND INTEGRAL CALCULUS. FORMULAE FOR % %. % AND J ds dd RADIUS OF CURVATURE, p = - ^5 /> = ^^AdQl ' dd' FUNDAMENTAL FORMULAE OF INTEGRATION. / ax'^dx=- 2+1' X j-=log^; / cos Jrfl&r=sin;ir; /sec*-^jr=logtan ( ~+- )' / cot xdx=\a% sin ;f, I cosec''jr«&-= - cot jr ; J log.fl / sin xdx= — cos j,-, / cosec xdx=\a% tan / tan xdx=\o% sec jt, / sec^jr^;ir=tan a". / sin inxdx= cos mx, I cos mxdx= — sin mx ; r — = log tan X. J sin:ircos;ir C dx . 1 ■*■ / , ==sin~^-, C dx I ,*■ I -2-, — »=-tan-'-, J d'+x' a a f dx i_ , x-a }l?^^^~2a°^ x + a' Lfd' -dx ,x = COS"' -, -dx I -5-; — 5=-cot ''+x' a dx I , ajrx ia "° a — x ^2__jj „„ a FUNDAMENTAL FORMULAE OF INTEGRATION. 6i — dx _ I -7==^= vers-'-, ; ,~ ^ - , = covers-'-; ^2ax-x^ a J ^2ax-x^ a f dx X r -dx ,x I / „ = vers ' -, / - . = covers-' - J ^iax — x' a J / f / .---lV ^ xjx- + a- a- x+JJF+a? U'x-+a-dx=-^ h— log ^ , J 2 2 " a \4^^^dX=^^^^^"^-^ log "^+-1^^^' ; J 2 2 " a f r^i 3 J xja" -j~ a^ . .X / ^a- -a^dx=-^ 1 — sin-' -. J 2 1a LENGTH OF CURVE. Rectangular Coordinates. Polar Coordinates. AREA OF CURVE. A={ydx. A=-\{r^de. SURFACE OF SOLID OF REVOLUTION. S= \2iryds. VOLUME OF S0L;D OF REVOLUTION. V= [iTfdx. 62 FORMULAE IN DYNAMICS. FORMULAE IN DYNAMICS, , I. RESULTANTS. Two forces at a point, R^ = P^^-(X'-\-2PQ_zo%e. Ij Any number of forces at a point, where X='2Fcos0, y=2Psin6l. Q... A / Two //ie parallel forces, F.AC=Q.BC. Q Two tinlike parallel forces, Ii = P-Q, P.AC=Q.BC. The resultant in this case is of the same sign as the greater force, and the point C is in AB pro- duced or inBA produced. RESULTANTS. 63 Any number of parallel forces, 2. CENTRES OF GRAVITY. vG-' Parallelogram. Intersection of diagonals. This point is also the inter- section of the lines join- ing middle points of opposite sides. Triangular lamina, where D is the middle point of 64 FORMULAE IN DYNAMICS. .a -------J i> II il II i! II Trapezoid : £G = c a + 2b 3 a + b' I' Circular sector ; 0G = '-.^. Semicircle : (9G = — = 0.424;-; 3T Circular segment : AG- _ 12a' where a = area of segment. (See Mensuration.) SOL/DS. 65 SOLIDS. Pyramid : where g is the C. of G. of base. Cone : Gg=kVg, where g is the C. of G. of base. Hemisphere : OG=^r. Spherical Segment : F-t- W ^„= FRICTION. /*= coefficient of friction. i^=foroe of friction. ^=normal reaction. 66 RESULTANTS OF VELOCITIES. RESULTANTS OF VELOCITIES. I. Two velocities on a particle : r^2 _ 2^2 ^ j,2 ^ 22JV^ — 2g/i. Greatest height attained : {/2 V Time to greatest height : — . g' Velocity due to fallin g from rest through a height h ■ -ligh. MOTION DOWN AN INCLINED PLA^fE. B ^= velocity of projec- tion. ^= acceleration due to gravity. Plane smooth. Acceleration down"! the plane, / Velocity acquired'] in sliding from \ BtoA, j Time of sliding"! from Bto A, j g'smz 'J'2gABl 4, zAB g sin i Plane rough. ^(sinz — /icosz) iJigABiiva. i-ficosi) '^(sinz — /ncosz) z'= inclination of plane. g = acceleration due to gravity. /x = coefficient of friction. PROJECTILES. y A D B Velocity after t seconds : v='J V^cos'^a + {Vs\aa-gt)K Direction of motion after / seconds : tan 6= — ,. ^• Kcosa Time to highest point C : i Total time of flight : T= Fsini 2 F sin a 68 PROJECTILES. Greatest height attained : CD- J^sin^a Range on horizontal plane : AB-- 2,r B Range on inclined plane : AB-- 2 F^cosasin(a-;8) Total time of flight up inclined plane : — _ 2^^sin(a-j6) SYSTEMS OF TWO HEAVY PARTICLES. Acceleration _ »Zisin Q — m^svci^ mi + m^ m2 Tension of string _»«i»22(sin ^4- sin <^) m-^ + m^ Acceleration _m2 — m^sm 6 mi + m^ Tension of string vf- _ ;«;»Z2(i+sin 6) Acceleration m^ + m^ Tension of string= — 1— ^£ Acceleration OTj + WZj' Tension of string = } — -g. IMPACT. IMPACT. 69 I. Impact of a sphere on a fixed plane : v=u Vsin^a + ^^ cos^a. cot 6=ecota. 2. Direct impact of two spheres : _ mu + ni^u^ — etn-iiu — «,) jn + «i ' OTZ/ + in,u, +em(u — u-,) m + ;«] 3. Oblique impact of two spheres : z/sin d=usma, (i) z/,sin^ = aisin/3, (2) „ mucosa+m-tU,cos B-em,(ucosa — UjCOs 6) , ~. ■ = ' . ^ ^^• •(4) »j and »«i= masses. u and «i=veIocities before impact. V and »!= velocities after impact. a and /3=angles of incidence. and 0=angles of reflection or rebound. «=coefllicient of restitution. (see over. 7° IMPACT. 1. To find 9, divide (i) by (3), which gives tan d. 2. To find <^, divide (2) by (4), which gives tan <^. 3. To find V, square (i) and (3), add the results, and extract sq. root, 4. To find Vi, square (2) and (4), add the results, and extract sq. root. THE SIMPLE MACHINES. LEVERS. First Order of Levers. («) Lever without weight : Py.AC= Wy.BC, R = P+W. (d) Weight of lever = w : PxAC+ivxGC= W-x.BC, P = P+ W+w. Second Order of Levers. (a) Lever without weight : PxAC=WxBC, R=W-P. {6) Weight of lever ='Z£/ ; PxAC=WxBC+wxGC, R=W+w-P. Tc Third Order of Levers. {a) Lever without weight : PxAC=WxBC, R=P-. W. (fi) Weight of lever = w/ : PxAC=WxBC+wxGCy R=p- W-w. LEVERS. 71 Bent Lever. Py.AC=lVxBC, Safety Valve. {a) Weight of lever and valve neglected : W><.L=Px.A-x.l. P = pressure of steam in lbs. per sq. in. above atmo- sphere. A = Area of valve in sq. in. (J)) Weight of lever=w; weight of valve =7/ : PAl=WL-\-'wd-^v. WHEEL AND AXLE. [a) Thickness of rope neglected : PxOA= Wy.OB. (t>) Thickness of rope = / : p[0A+^^=lv[0B + ^). 72 THE SIMPLE MACHINES. DIFFERENTIAL WHEEL AND AXLE {a) Thickness of rope neglected : PY.OC=~{pB-OA). (J)) Thickness of rope =/: P^OC+'-^^^iOB-OA). PULLEYS. Single Movable Pulley. (a) Pulley without weight : W=2P cos-. 2 a = angle between parts of string. {b) Weight of pulley ='Z£': W+ w = iP cos -. First System of Pulleys. (a) Pulleys without weight : IV= 2"P. n = number of pulleys. {fi) Weight of each pulley = 'Z£/ : f^=2"/'-'Ze'(2"-l). Note. — The fixed pulley does not count as part of the system. PULLEYS. 73 Second System of Pulleys. (a) Pulleys without weight : JV= nP. n = number of strings at lower block. ((5) Weight of lower block = «/ : lV=nP-w. Third System of Pulleys, (a) Pulleys without weight : W= (2" - I ) P. n= number of pulleys. {b) Weight of each pulley = z£' ■ i^=(2"-l)/'+(2"-?Z-l)w. THE INCLINED PLANE. B Pull Parallel to Plane. Plane smooth : P PC n ri7 • ^^^, or R=IV cos i. Plane rough, coefficient of friction /i : (a) Body about to move up the plane, P — W{iva. i + fi cos t). (b) Body about to move down the plane, P= IV {sm i- fi cos z). 74 THE SIMPLE MACHINES. ^ >-P ^ '<\ >'w A c Pull Horizontal. Plane smooth : P BC W R AC or P= Wx^ni r — ^-^ or P=IVseci IV- AC Plane rough, coefficient of friction /x : {a) Body about to move up the plane, _ „, sinz'+ucosz' P=W : — ' i i- C0S2 — fisin? (b) Body about to move down the plane, ,sinz'— /icos/ /'=»' cosz + Msmz Pull Inclined at an Angle d to Plane. Plane smooth : P=^W R=W smz. cos ff cos(^+z") COS d Plane rough, coefficient of friction \l : (a) Body about to move up the plane, „ _ j^ sin z'+ /i cos i cos ^+j«.sin 0' (^) Body about to move down the plane, „ ,,, sinz — licosz' P=yy a—^ — -■ — a- cos p— jusin t) THE SCREW. Screw smooth P=lVx a = angle of thread. pitch circumf of circle desc. hyP 2Trl Screw rough, coefficient of fric- tion fj, : (a) Screwabout to move downwards^ p_ -„ r(sin g — /x cos a) /(cosa-f-/i sin a) (i) Screw about to move upwards, p_ ,,, r{sa\a+fico% a) /(cos a — /x sin a)' r= radius of screw. HUNTER'S DIFFERENTIAL SCREW. HUNTER'S DIFFERENTIAL SCREW. 75 P=lVx difference of pitches circumf. of circle desc. hyP 2nrl WORK AND HORSE-POWER. U=Ps, s U H.P.= P= Ps 33000' H.P. X 33000 s _ H.P. X 33000 '~ ? • H.P. of an engine ^ PLAN 33000 ■ „_ H.P.x 33000 LAN ' ^ _ H.P. X 33000 PAAT ' „_ H.P. X 33000 PLA ' A = H.P. X 33000 PLN If working expansively APNa(i+\os H.P.=- t/=work in ft.-lbs. or ft.- poundals. P=iorcQ in lbs. or poundals. j= space moved through in feet. H. P. = horse-power. P=iorc& in lbs. 5 = number of ft. moved through per minute. P=mesLn pressure per sq. in. /-= length of stroke in feet. A = a.rea of piston in sq. ins. = .7854^^. jV= number of strokes per min. 33000 where a = distance travelled by- piston in feet to point of cut-off. 76 TOOTHED WHEELS. TOOTHED WHEELS. If « = number of teeth, d= diameter of pitch circle, p = pitch of teeth, rf= px.n dxi Proportions of Teeth. T=A^P, S=.P, D=.TlP, breadth of face = 2 to 3/". />= pitch. Z= thickness of tooth. 5=breadth of space. 0=length outside pitch circle. /= length inside pitch circle. Z)=0 + /=total depth of tooth. CIRCULAR MOTION. If a particle describe a circle of radius r, with linear velocity v, and if -=radius of wire. T= fi' 7"= value of torsional couple- called into operation by twist through unit angle. /=moment of inertia of vibrator. /=tinie of vibration. Twisting Angle of Shafts. Twisting angle in degrees -F= force in lbs. -weight. 7?= leverage in inches. /= length of shaft in inches. Z>= diameter of shaft in inches. ^= modulus of elasticity in lbs. per sq. in. TABLE OF COEFFICIENTS OF ELASTICITY. 8l >< H I— I o (—1 H CO < O Cfi Cd O O Pd •o «.s O Ta"" s 1? X \r\ \r\ °^ 0^ cu i; o in CO t-^ On r^ a a M N "a •o o tn "bo £ ^ 2 B H t> 1 k X 8 8 o VO * i O S r^ ro ON rr> K p. >H « M h^ M « ^■^ o !^ " !: = : ^< i" X m tN. 8, CO N VO Ov m ti & & CO lA O d tN. vd >ts "2 *M '^ ■§. SE ■b „ „ „ « ^ E " " " " " en X (4 w m m VO On c< CO =Ss CO VO ^ VO CO CO N PO t^ r>. m Tf ■fe ^.s X " ' " " S 1^ VO 00 Ml ^ - O >, Ai; « t^ ■^ vq t^ r^ S ? D. d lO vn ^ d CO r^ N N 3 S « o g •• - " »• « r. a X 1 i^ d in 8. 00 VO ^ N '^s n o u^ VO O •a- o lo 2" Ov N " • • • ■ ■ ■ ■ o 1 1 ■ • 1 1 a> cd (J B h « • ' ■ A • ;i "S I ■ ■ 3 W ^ .^ -—^ OJ •3 I '& % g c §: .12 S. 4J 7} 2 o Q O m * w U. fa fa O U m H W n o H at 212° F. I joule = - ' I watt second. .000000278 k.w. hour. .102 kg.m. 0.00094 heat units. I 7-3 ft- lbs. I lb. water raised from 32° F. and evaporated 212° F = r 0.33 k.w. hour. 0.44 h.p. hour. 1,148 heat units. 124,200 kg.m. 1,219,000 joules. 887,800 ft. lbs. '^ 0.076 lb. of coal oxidised. ift.lb.=N 1.36 joules. 1,383 kg.m. .000000377 k.w. hours. 0.00129 heat units. . 0.0000005 h.p. hour. TABLES IN SOUND. TABLES IN SOUND. THE DIATONIC SCALE. Names of Notes. C. Do. D. Re. E. Mi. F. Fa. G. Sol. A. La. B. Si. c Do,. Proportional "1 Number of I Vibrations, J or or I I 24 9 1. 125 27 5 4 I.2S 30 4 3 1-3 32 3 "3" 1-5 36 1.6 40 15 ~s" 1.87s 45 2 2 48 Consecutive \ Intervals, -J 9 10 16 9 10 9 10 8 "IT" TT B" ~TS 8 TT Complete ^ Number of Vibrations for Middle Octave (Society of Arts) Pitch, ) 264 256 297 330 352 396 440 49S 528 Physical Pitch \ of Chladni, J 288 320 340 3S4 428 480 SI2 TABLES IN LIGHT. WAVE LENGTHS OF FRAUNHOFER'S LINES (Van dkr Willingen). (In Air in Millionths of a Millimetre.) A 760.92 F 486.40 a 718.95 d 467.03 B 687.13 e 438.56 C 656.56 f- 434.26 D, 589.84 G 431-14 D, 589-23 g 422.88 E 527.20 h 410.40 b, 517.52 H 397-15 c 495-98 K 393-87 IVAVE LENGTHS OF IMPORTANT SPECTRAL LINES. 103 WAVE LENGTHS OF IMPORTANT SPECTRAL LINES. Millionths of a millimetre. Designation according to Boisbaudran. Order of intensity — order of Greek alphabet. Element. Name of Line. How Produced. Wave Length. Sodium - ai Flame 5S9-5 588.9 Lithium a a |3 Flame Spark 670.7 610.2 670.6 Potassium a Group a KCl in flame K2SO4 fused with spark 769.7-766.3 583-1-578.3 Caesium a Flame 456 459-7 Rubidium a /3 Flame 420.2 421.6 Thallium a Flame 534-9 Barium - a s a /3 BaClj in flame BaClj with spark 553-6 588.1 604.4 549-2 524.2 51.3-6 Strontium a /3 7 a band 3 SrCla, HCl in flame SrClj, HCl'with spark 3) 635 659.8 672.9 613-600 460.7 Calcium ^ a 7 7 5 CaCIa, HCl in flame CaClj, HCl with spark 620.2 &618.1 593-3 606.8-604.4 593-3 422.6 Zinc a ZnClo with spark 481.2 636.1 Cadmium a CdClg with spark 508.5 479-9 Lead a Metal with spark 405.6 500.3 Copper - "2 Metal with spark 521.8 510.6 Hydrogen = C of solar spectrum 7=/ Vacuum tube 656.2 486.1 434- 410. 1 104 TABLES IN LIGHT. a < o o < Q o >- O < O O w Oh 8 --»~ >S •^l, ^ 'J a* D* 00 D^ . ^ cr tx CO CT^ iH « ►1 '^ '^ ly-l CO u^ o 8 CO CO OO OS 00 d o u^ CO O 00 g ^ + iJ5 + ^i t.%-^ ^ r^ t^ \r\ \s\ « vo On OS 1- CO 5- vd " -i- CO CO vO 1^ •*- d* Og- 00 • ^ \r\ OO • 0\ »o r^ OS OS IS : I 1 • O O "^ ■■ ■■ ^ ■■ a^ « li^ CO u-» 1^ o O O '^ M N M M M M 6 . " § + + 1 1 + + + + + + P- ■M c « (U o o ro O o^ O d " «■ hJ 1-1 W U u iz; > < tn 1-1 H U 0- d I |WK o C S S tdU fn Sen 5 a S ■" O a ^ * e bB o. *j ■a y . -s u : : < 3 o ■O 1 « g ^ ■a O o 3 - 3 ho. !■ • • • ■ o . . . 7" o . . g . • t^o" , d °o " a a a None CI, Solut Na uoo ^11 "— .— ' /— *"- •a U .3 > 3 u u : = = ' o O en o s SI 2 5 1/3, "o +J " 5 'go Sen oy -.2 tuO 5 „ i^-i^i _ _,SP4 SO tn O 3 ^'*..2.2w-.«£MXWO •2.2 gggSS 11.^5 O O ro ii-ii5 IT •- « O 0< 3 Ou en D3si O^, oig „su" C N B I N i "3 e ?: «- Q Buns Bich (F Q cS 1 3 'O *^ oj rt 4J rt i-lJOhJ o !l= -- s en •!3 m «_i3 us PhOW GALVANIC CELLS. "S a u u < > < -— ^, ^^> \o t^ .S 1.072 at 15° p. coefft.-o.oooi 1.025 1.0 1.434 at 15° p. coefft. -0.0007 1.072 .S •1 Si S'rt .11 CO VO • s 1 1 s « ' ... • ;'»' •c •s ^ oS > Of:; V E S 1 -e rf g "^ ^ , Daniell (gi Weston, Von Helm Latimer C Fleming, 1 Faure, etc, Regnier (l „ (2 ii6 PURE COPPER WIRE. TABLE OF DIMENSIONS AND RESISTANCES OF PURE COPPER WIRE AT IS°.S C. IN B.A. OHMS. Area in Sq. Ins. Ohms per 1000 r eet. Ohms i4 S ^ Area in Sq. Ins. Ohms per 1000 Feet. Ohms 0000000 0.500 • •1963s 0.040216 .0U0053175 26 .0180 .000254460 31-031 31-659 000000 .464 .169093 0.046699 .000071700 27 .0164 .000211241 37-381 45-943 00000 ■432 .146574 0.053886 .000095468 28 .0148 .000172034 45-900 69.270 0000 .400 . 125664 0.062837 .00012982 29 .0136 .000145267 54-358 97.148 000 •372 .1086868 0.072653 .00017355 30 .0124 .000120763 65.388 140.57 00 ■348 ■324 .0951150 .0824481 0.083020 0.095774 .00022661 .00030158 31 32 .0116 .0108 .000105683 .000091609 74.718 86. 197 183.55 244.29 1 .300 .0706860 0. II 171 .00041030 33 .0100 .000078540 ICO. 541 332.34 2 .276 .0598286 0.13199 .00057274 34 .0092 .000066476 118.79 463.91 3 .252 .0498760 0.15832 .00082411 35 .0084 .000055418 142.49 667.53 4 6 .232 .212 ■0422733 .0352990 0.18680 0.22370 .0011472 .0016453 36 37 .0076 .0068 -000045365 . 000036316 174-07 216.94 996.18 1554.4 6 .192 .0289529 0.27273 .0024455 38 .0060 .000028274 279.28 2564-4 7 .176 .0243285 0.32458 .0034637 39 .0052 .000021237 371.82 4545-4 8 .160 .0201062 0.39273 .0050711 40 .0048 .000018095 436-37 6260.8 9 10 .144 .128 .0162860 .0128679 0.48486 0.61365 .0077293 .012381 41 42 .0044 .0040 .000015205 .000012566 5i9^32 628.37 8867.0 12982.5 11 .116 .0105683 0.74718 •018355 43 .0036 .000010179 775^77 19786.9 12 .104 .00849488 0.92955 .028409 44 .0032 .0000080425 981.83 31694.9 13 .092 .00664762 1. 1878 .046391 45 .0028 .0000061575 1282.4 54116.1 14 15 .080 .072 .00502656 .00407151 1-5709 1-9394 .081139 .123670 46 47 .0024 .0020 .0000045239 .0000031416 1745^5 2513^5 100171.5 207719 16 .064 .00321699 2.4546 .19890 48 .0016 .0000020106 3927-3 507010 17 .056 .00246301 3.2060 •33794 49 .0012 .0000011310 6982.0 1602799 18 .048 .00180956 4-3637 . 62607 50 .0010 .0000007854 10054.15 3323456 19 .040 .00125664 6-2837 1.2982 20 .036 .00101787 7-7577 1.9787 21 • 032 .00080425 9.8184 3-1695 Th ; resistance of pure copper wirf 5 increases 22 23 24 .028 .024 .022 .00061575 •00452390 .00038013 12.S24 17-455 20.773 5.4116 10.017 14.187 as its temperature cent, for each degre rises about e C. 0.38 per 25 .020 ,00031416 25-135 20.772 Note. — To convert inches into millimetres, sq. inches ,, sq. millimetres, ohms per 1000 ft. , , ohms per metre, ohms per lb. ohms per kilogramme, multiply the figures in column 2 by 25 4. 3 by 6451. 4 by 00328. 6 by 2-2046. Note. — For a revised table in legal ohms, compiled by Cheesman, see Electrician, p. 446 and p. 666, Vol. XL. (1898). PROPERTIES OF ELEMENTS. 117 •aouBisisa^ IBDUjoaia joj jaapiyaoQ ajtUBjauuiajL 0^ ON w^ M 00 1 §■ 1 : : : ^ : : 1 : : ^ ; ; : ; : 1 aouejsisa^ IB3lJ53aia •^tjiAponpuo^ c POO '• '• ' '■ : :« 1 : ^ '■ '• : : : u •luiOtj 3ui[ioa : B : : : S : '^S : : : :^ • : ■ - ■jiiioj Suijjaj^ u 600 440 Sublimes 1300 900 266 Electric Arc -7-3 27 700 Electric Arc 700 - 102 above 2000 1500 1050 "luajj oyioadg i-i m cxj m rt mcx) u-i 00-3 -d-woi-t a- NOo : :u-io«^H'acio«'-<'- :o ' " " * ' * "^ * ' ' " '0 •uoisu^dxg jeauii JO juaiDgyad^ i '^t-^vO c^ HH .^00 Mr^ N-o : : : >- : : f^ : :-50: •>- -- 000 ■ : -o . -o ■ -nO- • =0 ^o ■iCjisuaQ ']Qa{BAmb3 IE3imaq3 -oijDaia ■* ro M CXJ -juaiBAinba l-Eiiiuaq^ ■Adu31b\ P. III., IV. III., V. III.,. V. II. II. III., V. III. I. II. I. II. IV. IV., III. I. II. -VI. II., p. III. V. II., p. I. 'jqSia^Diiuoiv 00 ^o rn ON*^*-* t-«oo "^'>;^^ooo rn I^O^Orhr^ONOO w N M N i-J»^N06fOf-O •pqiuXg o o m M •quioj 3aiipj\[ u H^ t^ 1- 070 CO 00 m fO 1 ns^ 00 o ^ i^ ■* « ■jBajj Dyioadg O ; ° = o"S S' = ? ro -* in « OS M ^H •uoTSUEdxa „ ■^ t-, « OS t^ b jo'3iiai3[ys63 ^- o : t \ 8 5 : gig: OS W •jtqisusQ OS ii-> ro - ro 00 00 1- •^ m 00 « VD vo in IT) ON • t^ ^ N I-^ vo M l-l IN. CO 00 ^ ^ vo PO •juajBAinba O 00 ro PI t^ -^ VO IN •* 00 m a HH t>. a» 00 ; f^ Os OS tN. tN r^ M 00 rn o^ -OJIDSI'J l-H \q q M fO ^11" q q "H^ N q ^ r^ t^ M VO m N in •luaiBAinbg lEDiiiaqo m u^ O m Th 00 m ■* M "^ u lo cyi 00 in _) vn >-< vo W M ON « •* Ml N ^ l-l G ^ 2 '-' > G '-' >^ n •AousiBA p^ : > J • - f^ o > 1^ S "^^ "-5 ^ > ^1 ? > > ^ '"' Oh" '"' l-H — 00 vo \o w t^ t^ 00 OS '^ < 'IH^raM 3IUI0JV \0 On O "^ N vri « M ro VO fO VO 00 vo tN Tj- in R ^ \0 "-I r^ in I^ OS 4 •-I « OS u-> m § « ^ ■^ ■^ i-i iH M >-) ■[oquiXg ClJ O 3 s ^ « ^[5i^ fi -^^ S S K S ' ' r. R ' ' ' ' ' ■ . s" ■ - S * KO 1 1 ■4uiO(i Sanppi u m 10 :8 :'§8 : 858^ '"^S : :C-8 •JBSJI oyiDadg 00 i^ »ni-t :•-< ;';^t.*^n^-<"^lnO POt^ ;io ;vo ; ;0000 .w -MtHOCtMOMOM .0 -O . ■O"-' •noTsuBdxg * JESUIT JO luaiDggao;;) i :o . :o :ooSo :o :o : :E'00 •Xiisua(j • r^o .0 .inooinoo- roq;;for^ « - « w « ^u^|EAmba IB3iuiaq3 -0413313 ■^ ■*>hoo ON00-* r^t^r^ win .0 .OOOW'-'VO'^u^.OOTl-OO . .OsN ;0 :'1-'-00rnO0O :roOOcn; ;Or-- •inaiBAinbg lEOimaqo ■.N«roOO ■* \OtN.iHiooO i^rr»\o cooorovoc»i>»'*M .vo-^ro.Ot^.cooQT^h "-|fOlrJ^O^-.»J^»HO :i-iWii^WtN. lOrnro •-iMTh'-'m, vot-ivo »0'-' ON0t>-0"^iO«N Hio "-ivOOOvOvO t^enenkovdci'^oo :O\eiN0i>.rA.d\«N 0«^'-«(^V00»^ li^i-iOvvO^-'iOMrow •Xdusiea I. I. II. II., IV. V, II., IV, I. IV. II., IV. IV. II., VI. IV., VI., VIII. III., V. III. III. II. IV. •jqSi3M.=W01V O^Omooo « Tf-t-itH m \nrnm rC.fOt-^« M lo'^poc^odco '^hO *^ Pod^lnd MOO tooo « povo >-" ^00 TfmrN.oovo on ,-1 mmMN>-'^ >-iN i-H oqiu/Cg ■^'IcScoE^HHeS ;^P&tD>^^t5t5 1 s '.al^lllil '.flflli '1 s 1 g I. II l -g 1 „- s II 1 1 -1 - 1 PROPERTIES OF ELEMENTS. NOTES ON THE TABLE OF THE PROPERTIES OF ELEMENTS. A. The Atomic Weights are taken partly from the data furnished by Van der Plaats, supplemented by Clarke's recalculation of atomic weights. B. The Valencies quoted are from Karl von Buchka's Physikalisch- Ckemiscke Tabellen ; where P. I. or P. III. occurs it signifies that the element is found as a pseudo-monad or pseudo- triad. C. The Chemical Equivalent quoted is that derived from the first- mentioned valency in column B. D. The Electro-Chemical Equivalent is derived from the experi- ments of Lord Rayleigh, which make the electro-chemical equivalent of silver 1.118 milligrammes per coulomb, hence T T T S that of hydrogen is — ^ or .01036. That of the other 107.93 elements has been obtained by multiplying .01036 by the equivalent in column C. E. The Densities quoted are average values, and represent the mass of I c.c. of the substance at 1 5° C. F. The Linear Coefficients given are average values between 0° C. and 100° C. for 1° C. G. The Specific Heats given are average values between 0° C. and 100° C. H. The Melting Points are estimated from the best data obtainable ; many of the higher melting points have not been determined with accuracy. K. The Boiling Points are for a pressure of 76 cms. of mercury ; in many cases they are only approximations. L. The Thermal Conductivities are expressed in C.G.S. units. M. The Electrical Resistances are expressed in microhms per c.c. N. The Temperature Coefficient for electrical resistance is expressed as the change of resistance per degree centigrade per unit. ELEMENTS ARRANGED TO PERIODIC ^KS7 '^AT. > 1 00 00 1 ^8 1 1 CO J:: ji OS- 1 1 > ^1 vO pq ON 0-, 1 1 1 1 > ol to to (UOO c/3od On ■2; ON 1 NO 1 01 > Z °- VO « q^ "7 CO "^ On NO 1 H~ CO 1 > CO HOC 1) CO ON 00 1 -1 » «d si 1^ (U ON en m '4- cd q> 0$ 00 vO C <0 PS' 1 " b/3 ^ n! On CO c - ts) 10 nD 1^ l- is NO 1 «l 1 " ■^E CO CO U CO 4 U CO 1 < ON 1 PROPERTIES OF ALLOYS. 123 ^^^H ' — ' — < •aouBisiss^ ^ * ?o * Tf Tj- N . w Q JOJ juapmao;) 8 8 ■ § § § § ' § ■ ■ ■uiD DiqnD .§ =0 * S, jsd suii[o 7 00 00 N 00 * \r\ y}^ : fc^ ^^■ ON 1 00 M ' 8 IBOU303ia ogpads N N M P. m « — ^— ^ :2"3 -* . -3^ •lESfi ogpadg 0^ On 0\ 000 VO ■* q q ■apBjSnua^ "-) S33aS3(J • CTi •]mo3 Suniaj^ VO m VO -uoisacdx^ 0^ ON 00 00 jeaajT; : 8 8 8 ^ JO 1113105^303 8 § q ■A]ipi3i-a 3[duitg 2 PI en W -sninpoM _ xi-1 ^ ■* s,3uno^ >-l •tUD •bs Jad saiiAp « CO i^ */t:;lDBU3X '^ 1^ 00 tn vO r^ VO •.C]isua(i : 00 00 00 00 Ov 00 1 — N .— — ^— '■ «-**— ^ ' .-5-? s , ' , ■ — ' — ' '-'~' 5 « 11 :Uc< "Uc^nuSn i^A CJ3 B : :c3lg<'dt^'dl M 00 VO ^ r^ ON rj- rtN-£"^"^0"^"^^^0 -^N-^^mi^o n n 1 ^J ir> W HH PM^=>0 " I^H^^'^a «• « u-i 00 w VO for^ ro 1 o-| , , , 1 1 1 , , ^ >» CU ■ ' ^ ^■l ■ F h f a" - ' i i tT ^ 3 1 1 s ^ 2 Brass, Cast, „ Hard Bronze, 1 S 1 Platinoid, - Fusible Alloy : Phosphor Bro Silicium Bron Manganin, - m S '.5 B < B 124 PROPERTrES OF VARIOUS LIQUIDS. Q in O S < > o in I— I H Pi H O Pi ■USi) ro ^^ ro vo CO ro r^ I^ HH CO ^ ro ^ •XjioBde^ aAnonpuj oypsdg ci N CO CO « ■£ ON ro 'ch ON CO T- CO fO o\ CO ■XiiA;jonpuo3 j^sh § 8 § 8 8 00 3 8 8 u ■(,oe) jnode^ jo sanss-ssij; ON 00 5 ? ■uou n-. at vo n -BSUOdE^ JO JB3H 1U31BT M e» •* 00 ON r^ ■(■uiD 9Z) luioj Suiitog u VD N I^ VT) ON in vo " vo GO W ro CO ■quioj; Suizsaij ■^ 1 1 1 1 1 n-. 10 ro J, ■(„£i) i^H ogpsdg VO "I" q 00 10 ° ■. M in ro Cv cc •uoiireoduio^ a X u & ro fa K ' ■ ' ■ ■ ■ ' ' OJ -T^ TS ^ -a 'h3 s 0" sf n 1 (U 1 s .g r 1 )-< ^ g ON** : - : :0> : : :« . .0 : : : ojo •auinio^ lUEjsuo^ — leafj Dyiaadg « •aanssajjmejsuo-) 00 u^ -o rn ro ■saiaqdsounv "1 vO r->. On _ 0^ On _ t^ . " ^^ O* "^ ON M 10 1~^ : r-s. 10 ro ; : ^oo -oo roi-^u^fooododiA ro'O "-« ■* r>. f*^oo m 1000 IN u-it^ j>. Tj- "~.oo r-. On •3 . 'ajniEjadma j^ "oroovo .-iiH__(n 0.^^ -^"O 00 in 1 1 1 1 1 1 i -IU3 9^ ■3,'U0UBSU3pUO3 JO sitijEisdiuax N ". "-H* . !>* . ro ^ M»S Q 0.00 OOONOOn-*- .irih-. COOO vooOi-ifO- ■CO'-i-. -ro-i-ONt-ti^OOM-t II III II 1 1 1 1 1 I JO 3jn}iii3dui3x r<-i fO N ^ 00 ..10 Vt^OO....«.ON..ONO.^0O * " 1 1 i 1 ! " ■ * ■ "J* ■ 1 * ■ 1 1 ' 1 1 uoisuedx^ JO juapiaaoo «o i>*vo VO r--\o VO CN rn: : : :roro: : : : :fo: : :romfo:ro: 8 ■ • • • 8 8 8 ■ • • 8 8 8 • 8 ■ •3 ,0 3mn[OA JO 3DU3T^IS3'^ 10" X •999 .992 •999 1. 001 .■988 •999 .984 ■uaSojpXH 01 pajiajaj ^tjiABJQ Dypadg ■^ rood « '■^h lovo "-) t^ ■«*od 1-1 Q 00 u-> N ^J-vo t^ M 0» PI M '^00 Ni-ifO ,_rD-^'-< ww^HtHMtr) 1 Si i 1 Air, Alcohol, Ethyl, - Benzene, ... - Bromine, .... Carbon Dioxide, - Carbonic Oxide, - Chlorine, .... Coal Gas, - Ethane, .... Ether, Ethylene, .... Hydrochloric Acid Gas, Hydrogen, . - . - Mercury, .... Methane (Marsh Gas), Nitric Oxide, Nitrous Oxide, Nitrogen, .... Oxygen, . . . - Sulphuretted Hydrogen, Sulphur Dioxide, Water, .... 126 PROPERTIES OF MISCELLANEOUS SOLIDS. PROPERTIES OF MISCELLANEOUS SOLIDS. Caoutchouc, A = .92-.99, cr=.48. p=45oxioil Cotton Wool, - ,4 =.00043. Gutta-percha, - A = .97, a= 1. 98x10-*, p = 45oxio'8. Ebonite, A-iir r^ = .ooo37, a=.84Xlo-*, ^-'•15, V=.33, p=3oxio«. Glass, A = 2.4 to 4.5, yj=.ooos to .0023, a=(.o5 to .09) X IQ-*. Porcelain, A=2.3to2.5, a^.o4.3Xio- ^=j;,l'°"' '^^'^''''^^ Mica, A=3.6 to 3.2, p=84 X 10". Marble, A=2.5 to2.8, (7-=. 21. Paper, A=.7toi.i5, ^=.0001. Slate, A=2.6to2.7, ^=.004? Resin, A =1.07. Wood, A=.35 to 1.16, a=(.o5 to .34)x 10-*, {^^"^'ogg Wax, Bees', A=.96to.97, iI//'=52.4, il = 42.3, 0=2.3x10-* „ Paraffin, A = .87to.9..^P = 6..8, {J=4|3, ^^^ A = density. -l q d q en 00 d d q VO 10 ^q • '5 a m (3 00 ij~i lr^ n <^ CO 00 &cJ i^ t^ M CO CO rt- ^ "* q N rn Tt- LO vO I^ n«> ° ^ *« 2 't; *:C -3 ir^ u-i 0" (S ro ^ ir\ vO I^ sz P i£ S o* <1 ^^_ ^^^ ^^_ ^^^ Oi u O ^^ M ft o •| O .2 'a; W i-H (J 00 fXCO 5 rt 6 hi ^ ffi c ca lO >i f^ u + 3 7i X 15 ca; ^; 3 d a, CO rt > o o (n V 1) n 3 H) rr u. T3 n! c> 00 \q i-H* "-J o od H-i ro oo ■^ OO CJ o o - O t^ o S b o (U X ■a « ^ vO C^ =° U ^ II oo O o >.~ o (I) C/J UC (y ^ ,^ G o e o ■ >i o PROPERTIES OF AMMONIA. 129 O < o W I— « H Pi W Oh o pi! &: >i H > 1'% \T\ Tt- t<^ •* D Q 1 U ci CO 00 - to 0\ \D vO N u-1 u u Pi < •-) D U s s. s p.i u III »-< 5 8 § I i^ CO i-i m rt c ■ •3E n-. (li So • ir% vO I "^ bOOo vO 1^ CO ^ • 1 CO ._= t^ 'o P2 rt « M vO Ov \r\ crv P^ m rt >.* n -* " vO ■* .-* ■C*^ OS Ov >-< u^ t-^ vO •* ^ V IV u^ Tt- rl 00 00 OS qs Cr. o\ CO 00 00 " ■3 1 *j M 3 gKctn "15 3 Ln 1^ ITS \0 -. M CO CO CO 8 S ft S & < K in + w t»i K 2; ?^ » q- fl 1 rn X d CO r) + (T\ ffi ^ ffi rt g; 'S q So ^ rt en u c tn .y > 3 m o> 3 a) ffi w H M o w W PL, O td ^ 1.^ ir» so ■* 00 t-< I-* r^ 0\ „ sO IpIi CO en to ro i-H N so O 0^ 00 > 3 bo q q q q q q q q q q u a ^ < Vl o S E-i i -a- ■+ o 00 m r^ s oo hj ". "1 \n \q -; CO 00 to q> N N lA 00 ci *-H lA d\ 00 IjTS ds W M CO ■* VO 00 Ox OS Ph -^ .•§ ^ 00 P* O C U DO u " HI Index fractio bout I : : £S . . . . 00 • • • • a m en <« 13 c o ^*w O c . Qsli^>?,^ ■* cng-;jt^ . . r„^^ = = = • § o y 'IS. "3 (4 C . -3 E N ^ g \jr\ ^ vO •* en M M Os OV 00 ^^ 0-. 0-. 00 00 00 00 00 00 1^ r-^ t^ JS tN 'o CQ rt u X « ro Ln vO 00 m •J 0. 0. q q q> : : : ; : vq « M * * * • 'o u a CO rt >^0 •+ ON t^ ■* •* ■* ■=!- « •* 00 en ■--5l " ■* in m Os On rs. -^ o ■* so c°o o oo ^^ VO •* N Q 00 Os OS O* 0^ 00 vO en o qx qv Ov 00 00_ 00 n ^ "Sq 1 s^ i \r\ 8 en ^ m so t^ §> a 8 p. s & CJ M ^ bo Q d) •a • g .« o s Tl m s • 1-1 o o u ffi C3 u K rt o ^ tA c "^ -M ,n fn (!) U ^ I 1 o "3 < J UI ix; \0 Tj- 00 ir^ N VD ^ ■^ CO vO -* n 4 "* "^t- -* \0 ^0 t-^ I-^ t^ !>. t^ CO CO CO CO CO CO CO CO . vo xq "^ \r\ m M* t^ c^ ■* -+ p< CO "^ i^ ro CO en CO >g 2" « 00 i-c CO -^ i-O rt M N ■* 1^ VD ^0 tN. rs. IS. u-^ >> 000 q q K) l-l M M M H-( HI M M M s " B ^0 "s ■§ -.= ■3 u^ •^ 8 l-H "^ " P) CO •^ \r\ \D tN 00 ON (S § & < -t-> ra o 3 c o o K 3 nj . O ^ °o -Si u^ U .5E V V J°o" CO ■O o " : B "?; ^ O (U ^o '-^ t^l t^ ho & *; (u -c .S p P o 2 s .S -^ g pq O " U o ■- — ' " .- (U ,-^ S ^ c/] C to.s 8 w ^ . • §■» " tr°- 132 PROPERTIES OF SULPHURIC ACID. u^ VO CO I^ M CO ro CO -^ -^ rt -^ CO n VO O ir^ tri ON lO ^ VO . ON t) CO * vO CO OOOOOOOOOQ 134 PROPERTIES OF HYDROCHLORIC ACID SOLUTIONS. I? O H 1-1 O m o <; u o H-1 o o Q >< h O W Pi W Plh O P< Oh ^ (M ■^^ ■'^" ^3 °S ° s sS • J .llJ^e ro M l-l o li^ K N lA rj C) M CO vO Q Is-'Or N \r\ 00 >-< ^ t^ > o U s K D 3o 1 ja u si o \rv q q CO o q o u S = 'rt 8 rt.S " c 00 i^ CO r^ 00 N M N N C<1 ; q q q q q -J — « n O « y m 4> . tS o 5 e Oo* o VO i^ "S M q vO *-" t^ ; ; m-v^ 1^ tx vd lA -* p. 1 u (£ °= lO ■* T^^s 6 M ro lO vD t^ o<5 Mrt g o O O O 1 ^^ *"* •^ *" '"' ^ ^ t-H ^ .s 'o Ph . 00 5 q "^ qv , ro 1-^ 2" !>. fe rt ffi -a- Ov LTl _ Ov ss 0^ 00 00 CO t^ Q< M t! vO CO CI M ri bH m tv ut 0^ o •s. q q c^ g" 1-.' H » ■ n C u. .2 ° = Per cent. NaCl in neousSol irl tn o ^r\ VO 00 w W CM CM cr 1 < ^^ ^^ o K o + u 2; ■S f^ 13 O QJ N ^ PC S o ;i II '^ o o « I J C< r<5 J3 _ o II •^ OG O " S U s •3 o o OS 136 PROPERTIES OF CALCIUM CHLORIDE. III vO »v "^ ■* Ov vO i^ t^ -* -.a . 3 VO CO C3V t^ t^ vO 0-3 E N M -■ 6 d d d t gS& J m D. •-] m i" ; N -=t- vO 00 ov E^° ' ^ g'S owering 0I 'ressure at p. of Boili ter at 76 c 00 in irv u fO 0< vO ^ 2" N tn hJ^ EJ" « 1^ Pk « 0) 3 l1^ TJ^ q q : : s. '^ 3 n « « r^ en ^ ov . . Tt- • ^ : « a C 8 ^ i-i )-i i-i ,_ ^ VD Ov ■* 00 »-4 1-4 ^ >8 l-H I-I ■ « t^ pa ^ CH w 00 vO -* : •1 q> Ov qv . \ \ U3 rt t^ ■+ t-t 1^ On t^ N k- 00 N CO S-d )M - U * "o dj ho O o 2 -^ U « a H 'o . ,n! irv U ov 3 u H Pi PROPERTIES OF COPPER SULPHATE. 137 < Oh Pi w o u o S Pi W &< o g >. > :!'% - 00 ■* Ov u^ N VO H lr^ -!t 00 N t^ r^ i-n \0 00 s - N N •=J- ^ vO Ov > s oK K < .J D U '^ U> 1 ir% q q 8 § i % CO « a. >-;S.iS W 0— s « ratu cien r rica anct I^ W t^ en en Tempe Coeffi fo; Elect Resist n M N M 9 Q q q — 41 C trica stanc ihms b.ce en -^ CO M rn qs U^ CO N N a rt OS a\ CO ■5^ 00 00 &o - Ti- ir^ m J n \o o\ en t^ •|So q q q '^ "^ N Sj "^ (w .S ■" ^■3 1 16' \ tn iri irt K. >n ""* " " M o» en a< S CP t < 8 S S « y ^ II c 3 . >^ ^11 r CO rt -a • OS M O O '" 2 c o " ^>, ~r *+J ■5 O "o C W in « 3 I- •^ U£ ■^ C -5 won ^ 3 „ ■" o + o S 9 1- w ^ -a ii u3 00 'o "^ B « II ^ « '2 5* "" " 1 H >> O O tlH U I, o ^ I" c 2 1=^ o "o ft- 'S w ^ 5 I o W § § £ (1^ 138 PROPERTIES OF ZINC SULPHATE. W <: U g H W Ph o Plh liii ^^^ ^"^^ § S K E "g. ■«■ 1^ I-; -i- 6 "H N ^''W " M ■«• NO S o S3 S ■ s ^ M iri w N en ■^ e"3S Ul ■" - >. < I-^ M \0 C> N M in m ti u 00 -^ -^ CO 00 C C? vO p II « M CO ^ vO OS -d bjo M §K s> u op ■ '^ a ^ >^ Q JJ * J" ci C/3 A 00 vg- 00 PL, ■- III CO CO CO (x, ^s rt C/5 s-i K w 0^ 00 vO Xi^ -* CO Ov CO 00 1 q> qv qv CT> qv qv 00 00 I^ 'A w O, ^ 0) (J „ ^ 02; PL, t^ \r\ ro ^ ^^ n OV OV N Ov ui t/5 rt § N ro CO •* 00 n t^ CO 00 vj-i r^ fl >* >^ 3 q t-i HI M M CO CO ^ . £ ao 140 PROPERTIES OF WATER. PROPERTIES OF WATER. (Also see previous Tables.) Absorption of Gases by Water. Bulk Modulus (Compressibility). "C. Volume of Gas at 0° and 76 cm., which unit volume of water at 76 cm. and the temperature named absorbs. °C. Grms per sq. cm. X 106. Pounds per sq. in. X io5. 206 2.93 CO2 N. 0. Air. NH3. 10 220 232 3-13 3-30 1.80 .024 .049 .025 1 174 30 243.2 346 IS 1. 19 .0192 •039 .018 7S6 40 2S3-I 3.60 40 •SI .0120 .023 80 260.8 3-71 100 .24 .010 .017 100 252.4 3-S9 SuRFACB Tension. Viscosity. Index of Refraction. 10° C. °C. Dynes per sq. cm. °C. C.G.S. Units. Line of Spectrum. Index. 75.6 .0178 10 74.2 10 .0131 c 1.3318 IS 73- S IS .0113 D I..3336 20 72.8 20 .0100 F 1-3377 SO 68.6 2S .0089 Hy 1.3409 80 64-3 40 .0066 H 1.3442 100 61.S So .0056 Conductivity for Heat. Specific Electrical Resistance. "C. C.G.S. Units. ■c. Ohms per cm. Critical Temperature, 307° C. ■0012 Liquid, 3.234x10= Temperature of Disassoci- 9-iS 4 .00136 .00129 Solid, - 10 39870 210900 ation, 1200° C. Latent Heat of Vapour, 535.9 calories. r8 .00124 -17 S3S40O Latent Heat of Fusion, 30 .00157 79 calories. Density of Ice at 0°, .918. Specific Heat of Ice, "5. FORMULAE IN PURE AND APPLIED PHYSICS. VIBRATORY MOTION. HARMONIC MOTION. f] IB ^ XX ^ y \ A'l C B' /e \ / Definitions : Let a point P move uniformly in a circle, and draw the perpendiculars PX and P Y. Then the points X and Y execute Simple Harmonic Motions. The circle ABA'B' is called the auxiliary circle : length CA or CA' ( = a) is the amplitude : time of movement from A to A' is the period ( = J'). The following formulae are of importance : jir=acos 9, (i) (2) j/=asm C, where ;ir=C;r, j/=CF, e=angle PCX. If 6 is measured from a fixed radius CJS, x=acos (0 — e), j/=«sin (9- e), where £ is the angle £CA (the epocK). ITTt .(3) .(4) e = - •(5) where t is the time of describing 9. (iTTt \ where x=a cos (nt—e), n=-^, .(6) •(7) .(8) 144 VIBRATORY MOTION. In (7) X and t are variables ; denote them by y and x respectively ; then y = a cos {nx — i) (9) This is the equation of the simple harmonic curve. It is the curve which would be traced out by a point making simple harmonic vibrations upon a sheet of paper travelling with uniform velocity in a direction at right angles to the line of vibra*:ion, j)/=acos (^-«)' (1°) where A (called the wave-lengtti) is the distance the paper advances in the period of one complete vibration. COMPOSITION OF VIBRATIONS. I. Two simple harmonic motions in the same straight line, and of the same period. x-^=a-^Q.o% (6 — €1), x^=aiCos{Q-e.^ compound to Xi+x^^A cos (d-E), (II) where A = s/ai^+a2^+2a^a2COS (ej-ea), (12) , i E- «! sin £, + ao sine, , . and tan^ = ^^ ^ — ^ % (13) aicosei+a2cose2 hence the amplitude and phase of the resultant vibration is known. If 5=ei-£2, then A'=ai^ + a^^+2aia2Cos S. Case I. If 8=0 or 2mr, that is, one wave retarded on the other by any even number of wave lengths, then Case 2. l{ 8= IT or (2«+i)x, that is, one wave retarded on the other by any odd number of wave lengths, then If ai=a2, then A=o. Case 3. If 8=- or f «ir-l--j, or one wave retarded on the other a quarter wave length, then ^2 = aj2 + a^2_ COMPOSITION OF VIBRATIONS. 145 II. Two simple harmonic motions of the same period at right angles, diifering by a quarter period in epoch, x = a cos 0, y = b cos (6±~\—+bsmd compound to (fy+©'='=°''^+'^"'^='' ('4) which is an ellipse of semi-axes a and 6. III. Composition of any two simple harmonic motions at right angles. x=a cos 6, y=a cos (d-S) compound to (^^__cosSj =^ sm^S, an equation of the second degree, which in particular cases gives an ellipse or a straight line. (XoTE. The student is advised to see "Vibratory Motion and Sound," by J. D. Everett; "Alternate Current Working," by Alfred Hay; "The Alternate Current Transformer," by J. A. Fleming; and "The Theory of Light," by T. Preston. These works treat the subject from different points of view. In con- sidering electric current waves the sine form of equation is generally used, and the curve of the figure, page 143, is called a. sine curve or sinusoid.) 146 FORMULAE IN ACOUSTICS. FORMULAE IN ACOUSTICS. VELOCITY OF SOUND. Newtoris Formula: F= velocity, jE' = coefficient of elasticity of medium, /)= density of medium. For liquids, £■= resilience of volume, or bulk modulus. For solids in the form of rods, wires, or tubes, £■= Young's modulus. For gases, E=P, the pressure, and the formula becomes : Laplac^s Formula: where y = ratio of specific heat at constant pressure to specific heat at constant volume (=1.414). For dry air, F=3324ov'i +.00366/, or approximately K= 33240 + 60/, where K= velocity in cms. per sec. and /= temperature centigrade. If the air be considered as half saturated with aqueous vapour, we have approximately V= 33240 \/i+.co4A For the value of D in general see the formulae on Heat. DOPPLER'S PRINCIPLE. ., = «(i±^). «i = number ot vibrations which reach the ear, « = number emitted by the source of sound, 7/ = velocity of source, F= velocity of sound. + for approaching source, - for receding source. BIOT'S FORMULA. L L Z= length of tube, such as iron water-pipe, F= velocity of sound in air, Fi= velocity of sound in material of tube, «= number of seconds which elapse between hearing sound through material and through air. VIBRATING STRINGS. 147 VIBRATING STRINGS. Longitudinal Vibrations ; String stretched and fixed at both ends. V=2nL. «=iVS- F= velocity of propagation of longitudinal vibrations in cms. per sec, «= number of longitudinal vibrations per sec, Z= length of string in cms., E and Z)=Young's Modulus and Density. Unaffected by tension, harmonics follow the series of natural numbers. Transverse Vibrations: V=inL. V= V m f^= velocity with which transverse vibrations travel along string in cms. per sec, «=number of transverse vibrations per sec, Z = length of string sounding fundamental note, /"= tension of string in grammes weight, _^= acceleration of gravity =98 1 nearly, « = mass of unit length of string, r= radius of string, A = density of string. The harmonics follow the series of natural numbers. ORGAN PIPES. BernouilU's Laws : Closed Pipes, A = 2wz/, V—4nX., Open Pipes, \=i,ml, V=2nk. /=length of pipe, A=wave length, m=i, 3, 5, etc., for a closed pipe, and i, 2, 3, 4, etc., for an open pipe, ;^ = frequency. Corrected Formulae: Closed Pipes, V= \n{l-\-o.^r\ Open Pipes. V=2n{J-\-\.\r). F'= velocity of sound, r= radius of pipe if circular, =.^b^ab for rectangular section with sides a and b. RESONATING COLUMN. A = 4(/+o.8;^). A,=wave length of sound, /=length of column, r=radius of column. 148 FORMULAE IN ACOUSTICS. BEATS. tx=t:{x+\). t and / = times of vibration of two notes, ;r=number of vibrations of one note coinciding with x-\-\ of the other. f From the above ^~7Zr>' and therefore , is the interval between the instants of time at t-i which the vibrations oppose each other, and is therefore the period of beats. Let the faster gain one vibration in x sees. Then _ tt is the period of beats. INTENSITY OF SOUND. In free air L-'ll I, /j intensities of sound at distances d and d^ from the source. I=kDa\ /= intensity of sound produced when air is of density D, and a the amplitude of vibration, k\s a. constant depending on the units used. P = 2irWd^n^V. /'=quantity of energy per second, «=frequency, F=velocity, /> = density, and a = amplitude. VIBRATIONS OF PLATES. « = number of vibrations of fundamental, A = a.reiL of plate, ^= thick- ness of plate, E and /?= Young's modulus and density, k = a constant. This formula applies to round or square plates fixed at the centre. VIBRATIONS OF TUNING-FORKS. Frequency, n=o.i64j^\^jy i^= number of vibrations of fundamental, /= length of one prong, /=thickness in plane of vibration. E and Z* as before. VIBRATIONS OF PLATES. I ir^ Energy of Vibration, W=-^m-r;{2'nf. o t" 149 W^=work stored in one vibrating prong, »z = mass of prong, /=time of vibration, •)} = amplitude of vibration. Harmonics follow the series of odd numbers. VIBRATIONS OF RODS. kt Ir Free at both ends, "~'M\^'n' (^=1.0279.) Fixed at one end, ^~'N'\~n' (^=0.28.) «= number of vibrations per second of fundamental note, /= thick- ness of rod, /= length of rod, i?= modulus of rigidity of material, Z)= density of material. A rod free at both ends can vibrate with two, three, or more nodes, but not with one. (Note. "Lessons on Sound,'' by W. H. Stone; "Practical Acoustics," by C. L. Barnes; "Sound," by Catchpool ; and Deschanel's "Physics," Part IV., should be consulted. ) ISO FORMULAE IN OPTICS. FORMULAE IN OPTICS. ILLUMINATION OF SURFACES. Law of Inverse Squares. I and /i intensities of light at distances OA and OB from luminous point O. 2. Law of Cosines. 7A /i=/cosa. /= intensity of light on normal sur- face. A = intensity of light when surface at angle a. REFLECTION OF LIGHT. I. Plane Surface. \ /" ^ \. y "N /? " A / N If MN be a plane reflecting sur- face, 10 the incident ray, OR the reflected ray, OP the normal to MN, then LP 01= LP OR =a, lI0M=lR0N=^. Also, if /' be the image of /, IA=I'A. REFLECTION OF LIGHT. 2. Angular Rotation of Plane Mirror. 151 y=LNON', through which mirror is turned. 01, incident ray. OP, OP', normals to mirror in the two positions. OR, ORi, reflected ray in the two positions of mirror. a=LROR-i, through which re- flected ray moves. a=2'y. 3. Hadle^s Sextant. S = 2a. 0= angle between two mirrors A and B. S = deviation of hght after two reflec- 3 '\ tions. 4. Multiple Reflection from Parallel Mirrors. P, Q, rr—c- O ia»»-i» Q, Pa OP'M or 0Qin=2nc, 0P^.n=2{nc+a), 0Q^+i=2{nc+b). IS2 FORMULAE IN OPTICS. S. Multiple Reflection from Inclined Mirrors. If AOB is an exact submultiple of 360°, then number of images _36o° J n° 6. Reflection from Curved Surfaces, Convention of Signs : Suppose the light always proceeding from the right to the left ; all lengths measured from the surface of mirror in the opposite direction to that from which the light comes are called POSITIVE ; lengths measured from the surface of the mirror in the same direction as that in which the light is pro- ceeding are called Negative. General Formula for Spherical Mirrors. £ I_2_I_ u V r f u = AQ, v=Aq, r=radius of surface, /=-=focal length. Special Values of « and v. When «= Then v= Remarks. 00 r 2 Incident rays parallel, q at prin- cipal focus. r r Image falls on object. r 2 cx> Reflected rays parallel. . >> ■c ■" ni bo S ^ & 3 < ^ 4C *« + 1 + WJ m V- + + 1 bO m 0) CU > y. > § t 3 c a c u U -l-J ^_, ^ C _: g bo '0 s 5 r ■ 1 ■q 5 • u / % /// ^/fc < '\ < ^ a^ < '■' \''^ IS4 FORMULAE IN OPTICS. "^ ?! VA ■a o u S 1-1 8 '="'S O o R ° ."2 "■ ° ni (n o ;h (U 13 O o (U Tl .a a 3 hn (1) ■a .-3 1 4i I 3 u ^ 1 o o rt -t-J . >2 0> CJ ni fj 3 n u rt TJ aj ■M f > X. CI) bo* + + + > 3 O o til T3 W 'o J? o > 3 o O O i3 Ah, ;t REFLECTION OF LIGHT. 7. Magnification. Linear magnification k _ size of image i ~size of object _0q_ OQ v — r r — ii 155 REFRACTION. SnelPs Law. AB, boundary of media. POPi, normal to surface. 10, incident ray. OR, refracted ray. smPOI Index of refraction or /x= sinPiOT?' sinz' sin/ 2. Total Reflection. smgo _ I . sin r sin r ' smr=— , ■©■ Light passing through several Media. ai^r smz "sin/ 0^= sinr sin ri I sin ri . smz Vacuum ^^= =2^ af'y IS6 FORMULAE IN OPTICS. ij/io= index of refraction from medium a to medium )8 |3/*.y= !> )J " P " 7* al^y~ " " " " " y' 4. Direction of Pencil after refraction at a Plane Surface. JR v=[M, when R near A. 5. Direct refraction through a Plate. For a small pencil per- pendicular to plate, v=u — t- • For glass, fj,—-; hence Refraction through a Curved Surface. fj,_ I _fi — i V u r 7. Refraction through Two Curved Surfaces, thickness negligible. General Formula : Signs all +. f V u '^ \r sJ Bl lA O, ? O Q u=AQ, v=Ag, r=AO, s=BO^. This applies to all cases with proper signs. REFRACTION. LENSES. 157 Case. II. III. IV. V. VI. Nanie of Lens. Section Converging Meniscus. Double Convex. Piano- Convex. Diverging Coucavo- Convex. Double Concave. Piano- Concave. + + 8. Combinations of Lenses. I. Lenses close together, i^= focal length of combination, 2 ( -^ ) = sum of powers of the several lenses. 2. Two lenses with distance a between, F /2 '/,-f« 9. Prisms. For minimum deviation, sinK^+^) '^ sin \A A=a.ng\e of prism. i3= angle of deviation. ju,=index of refraction for mono- chromatic light. 158 FORMULAE IN OPTICS. When ^=60°, /;t = 2 sin J^6o + U). When A is small, or D = {[i,-\)A. Dispersive power = — ^= — =' ^ _ • Dr, Dv, D minimum deviation of refraction produced by the extreme red, violet, and mean rays. Hr, fi,,, [i indices of refractive of these rays. LAWS RELATING TO INDEX OF REFRACTION. Cauchys Formula. ft = index of refraction for wave length X. A, B, C= constants for the medium, when X = oo, then p,=A. Gladstone and Dale's Law. iiZl-i A -^- A = density of a liquid. A = a. refractive constant. /4l«l + ^2^22 — ^ (-^1 + '^2)- /^i, ^2 refractive constants of two liquids mixed in the proportion of «i to «2 j ^ tl^6 refractive constant of the mixture. Law of Lorenz and Lorentz. A /x2 + 2 More accurate than above, except for mixtures. Applies both for liquids and gases. LandoWs Law. »2^ = «]»Z]/6l + «2''22^2 + '^3''%^3+ wz = molecular weight of a compound. W2i, ;«2) ^3= atomic weights of constituents, j^i, ^2> ^23 = number of atoms of each kind. k\, ^2) '^3= refractive constants of constituents. k= „ constant of compound. mk = molecular refractive power, nk, ttc.= atomic „ „ When the substance is dissolved in water, {i^ + m)k=i?ipk-i^ + mx. ZAIVS RELATING TO INDEX OF REFRACTION. IS9 /= number of molecules of salt in one molecule of water. i8 = molecular weight of water. m= „ „ salt. i, ^1= refractive constants of the solutions and of water.' x= „ ,, of the dissolved substance. WAVE THEORY OF LIGHT. Velocity of Waves and Index of Refraction. _ V-^ sin i ly V2 sinr 4 Fj and V2 velocities in air and dense medium, /j and 4 length waves travels in unit time in the media. /j=jLi/2 = optical length or reduced to equivalent in air. Interference of Waves. Diffraction. {a) Knife Edge. O Luminous source. PM Screen. OA=a. AM=b. PM=d. mX = d^ — —7. a + b m = 2n or even for dark bands = 2n+i or odd for bright bands. {b) Narrow Slit. AB=s. BM=l. PM=d. , s(d-\s) m even for bright bands. „ odd „ dark „ (c) Transparent Grating, (i) Light Normal. sin Qn A=- nN S„= angle of deviation of the spectral line of «"■ order. A'^= number of lines ruled per unit length on the grating. (2) Position of Minimum Deviation. \=2%m^jnN. ^„= position of minimum deviation of the «* image. i6o FORMULAE IN OPTICS. Colours of Thin Plates. 2fjie cos r—nX for extinction. 2/i«cosr=«AH — for maximum intensity e= thickness of film. r= angle of refraction. Newton's Rings. Radii of dark rings = V/f sec<^;iA. Radii of bright rings =Vi?sec<^(»+i)A. .ff= radius of curvature. ^= angle of incidence of light. POLARISATION. Brewster's Law. tan«=/*: smz sin?' sm;'=cos2, 2'+^ =90°. z = polarizing angle. ?'= angle of refraction. FresnePs Equation. (a^ — i^)ta.nr= c^ tan i. a, b and c amplitudes of the ether vibrations of the incident reflected and refracted waves. Rotation of Plane of Polarisation. Specific rotatory power, [a], a = observed angle. \a\=jj- /= thickness of layer. ^= density of layer. Molecular rotatory power, »?[a], OT = molecular weight. r -. m[a\ ^. , im] = — i^^ = practical unit. For solutions and mixtures, w= aV r T 100 a [m'} = kdl' F=volume of liquid in cm., and p the number of grams dissolved. ^=number of grams of substance in 100 grams of solution. t,ld coo = rotation for water of length /q and d(j. r= specific rotation. Mial^n M /3 = molecular rotation. i8u},)id 1 8 ■ iW= molecular weight of substance. ^= rotation produced by difference of d=wV. magnetic potential V. w= Verdet's constant. ELECTROMAGNETIC WAVES. Electrical Oscillations. ^~2ir\KL « = frequency of oscillations per second. ■^ Ar= capacity of condenser in farads. i^I? Z = inductance of circuit in henries. .ff=resistance of circuit in ohms. Maxwell's Equatio7is. K= velocity of electro-magnetic propa- _ gation. K=i/V/fe/i. /&= specific inductive capacity of medium. /*= permeability of medium. k=\^^. ^= specific inductive capacity. /i= index of refraction. (Note. See "The Theory of Light," by T. Preston; "Light," by W. T. A. Emtage; Ostwald's "OutUnes of General Chemistry," translated by J. Walker; "Treatise on Optics,"by S. Parkinson ; "Physical Optics," by R. T. Glazebrook ; "Light, Visible and Invisible," by S. P. Thompson ; "Light," by P. G. Tait.) 1 62 FORMULAE IN HEAT. FORMULAE IN HEAT. CONVERSION OF THERMOMETER SCALES. C=IR. R=kC. i^= degrees Fahrenheit. C= degrees Centigrade. 7? = degrees R&umur. C+273=7', where 7" is the absolute temperature Centigrade. EXPANSION OF BODIES. Approximate Formulae — -Linear: Lt=L^{\-'raf). Superficial : St = S^{l->rZa£). Cubical : F, = Fq ( i + 3a/). Z-oi A = lengths at 0° and f. 5o, 5t = surfaces at 0° and f. a = coefficient of linear expansion. Vfj, Fj= volumes at 0° and f. More complete Formula: £>t=B(,(,i+af+/3fi + yfi). 250= dimensions at 0°. a, |8, y are constants. i?«= dimensions at t" APPARENT EXPANSION. ^ = absolute coefficient of expansion of substance. yi= apparent coefficient of expansion of substance. G = cubical coefficient of expansion of glass or envelope. WEIGHT THERMOMETER. {M-m)t A = apparent coefficient of expansion of mercury. ?;z = mass of mercury expelled at t°. j1/=mass of mercury filling thermometer at 0°. ABSOLUTE EXPANSION OF MERCURY. 163 ABSOLUTE EXPANSION OF MERCURY. (DULONG AND PETITS' METHOD.) Ao/i=A,/r, (Formula i) ■\Fo=A,!/, = A,Fo(H-,^4 (Formula 2) hence k= — r-— . ht Ao, A, = densities of mercury at 0° and i°. //, A''= heights of columns at o" and t°. Vo, F; = volumes of unit mass at 0° and i°. >J = absolute coefficient of expansion. BAROMETRIC CORRECTIONS. Hq, /r,=heights of barometer at 0° and i°. i, /8= coefficients of expansion of mercury and scale. For brass scale, (yJ — /8) = .oooi6i ; for graduations on glass of barometer, (^ — y8) = .oooi7i5. Also see Hydrostatics. LAW OF CHARLES. V,= Vo{l + ai) (I) ^0, f^ = volume at 0° and i(° a = .003665 = 2^3 nearly. V, ^ i+g/ ^ 273 + / ^r V,^ l+ai^ 273 + i'i T{ V,, K(, =volumes at t° and t°. T, 7'i= absolute temperatures corresponding to t and /j. BOYLE'S LAW. K/'=constant. See Mechanics. CHARLES' AND BOYLE'S LAW COMBINED. ^ = constant =i?, or VP = RT. where V, P, and T axe the volume, pressure, and absolute tempera- ture of any given mass of gas. Note. — When V is kept constant, we have PjTi=P/T, which is the formula for Jolly's Air Thermometer. 1 64 FORMULAE IN HEAT. DEPARTURE FROM BOYLE'S LAW. Regnault. m=VJVj^. A and B constants for different gases. For CO2 signs negative, and log .4 = 3.9310, log 5=6.8625. Van der Waals. a and b constants. An equation of the form V^-qV'^ + rV-s=o. The value of the three equal roots (<^) of this equation are given by or I'^^b+RTIP, 34>'^=alP, <^^=ablP ; hence <^ = 3^=critical volume, /' = a/27i5^ = critical pressure, T=8a/27i?i5= critical temperature. SPECIFIC HEAT. METHOD OF MIXTURES. mi — ■-'2— ~^' CHANGE OF FREEZING POINT DUE TO PRESSURE. , pe{T+27z) Jld ' — /= lowering of the F.P. for an additional pressure of/ dynes per sq. cm. «=increase of unit volume of liquid when frozen. /= latent heat of liquefaction. 2"= freezing temperature of water. ^j?= density of liquid. For water, ^ = .087, /=7g.25, T=o, d=i, -/=.oo7i4 for a megadyne. 170 FORMULAE IN HEAT. FREEZING POINT OF SOLUTIONS. RaouWs Law. . n rp ri) g mg i:^g A = depression of the freezing point of the solution. « = number of molecular weights of the substance dissolved in g grms. of the solvent. r= constant depending on the nature of the solvent, and which may- be found by dissolving substances of known m in the hquid. ^ = number of grms. of substance dissolved. Varit Hoff's Law. w ' 7'= absolute temperature of fusion of the substance. w= latent heat of fusion. (Note. The following books should be consulted by the student : ' ' Elementary Treatise on Heat," by Balfour Stewart; "Heat and the Principles of Thermo- dynamics," by C. H. Draper; "Heat," by P. G. Tait ; "Heat and Light." by R. T. Glazebrook; " Theory of Heat," by J. Clerk Maxwell; Deschanel's " Natural Philosophy," Part II.— Heat— translated by J. D. Everett; "The Theory of Heat," by Thomas Preston.) FUNDAMENTAL LAW. 171 FORMULAE IN MAGNETISM. FUNDAMENTAL LAW. mV-.-d- m, F=+- d^ i^= force of attraction or repulsion. m, «! = strength of magnetic poles. rf= distance between poles. Use + for North Magnetism and - for South Magnetism. DEFINITION OF H. F=mH. F={orce acting on a magnetic pole. ;, »z = strength of pole. :' .^= strength of magnetic field. When m = i, then F=H. TRIANGLE OF MAGNETIC FORCES. 7^2 = ^2+ j72^ F=.^tanz, F= T sin z, Ii=T cos i. 7"= total magnetic force. jy= horizontal magnetic force. 1^= vertical magnetic force. z = angle of inclination or dip. 172 FORMULAE IN MAGNETISM. COMPASS NEEDLE IN MAGNETIC FIELD. F=mlHiva.a. =MH sin a. i?= moment of couple tending to bring magnetic needle back into the magnetic meridian NS. angle of deflection of needle from meridian. /= length of needle m and H as above. M=ml =moiaent of magnet. Fi=imJ7ta,na. Fi = one of forces of a couple acting' at right angles to the meridian and causing the magnet to remain at angle a to meridian, the strength of each of the earth's forces being mH. <-,.. --21 > ^ d--— k- --- >; iMd M , 3 ^=(^q:75?=2^ (appr"^-)- i^= magnetic force of a magnet end on to a magnetic pole I— half length of magnet. H- 2d ^^^ -\ --" II ^3 = --tana (approx.). K a- — >■ 2 il/= magnetic moment of a magnet end on to a needle, a = deflection of needle. /^= strength of the field in which the needle is placed. M 3 ■^=((/2 + /^)*tana=rf'tana (approx). Symbols as before, but broadside^ on position. H --21- VIBRATING MAGNET. t=1T7 4\ P K Mir /=time of double or complete vibra- tion of magnetic pendulum NS. j^= moment of inertia about AB. il/= magnetic moment of NS. S jy= horizontal strength of field in which NS vibrates. MAGNETIC POTENTIAL. +?«)" ■/a V,. ..=m{ ). Fot= magnetic potential at point P. ^1 and ^2 distances of P from the poles + m and —m. 174 FORMULAE IN MAGNETISM. If the magnet is short, then „ _ Mcos 6 il/= moment of magnet. 6= angle FOB, O being the centre of the magnet. r=OP. M Also, i^=^(i + 3cos2^). i^= resultant of the magnetic force at the point p. F is maximum when ^=o° or i8o°, and least when 6=(p° or 270°. A O B TEMPERATURE EFFECT. Ars= moment of a magnet at temperature t. Mf,= „ „ „ o°C. a = temperature coefficient. (Note. Books of Reference: "Treatise on Magnetism," by H. Lloyd; "Treatise on Magnetism," by G. B. Airy; "Practical Physics," Vol. 11., by Stewart and Gee. FUNDAMENTAL FORMULAE. 175 FORMULAE IN ELECTROSTATICS. FUNDAMENTAL FORMULAE. Fundamental Law. 9' F=+^. F={oTce of attraction or repulsion between two small electrified spheres in air. f, j'l = quantities of electricity. rf= distance between centres. Note. — If the dielectric have a specific inductive capacity . C= strength of current in absolute units. ^ j^ r= distances of pole from wire. i is FUNDAMENTAL FORMULAE. Force due to Circular Current. 185 /=2,r;«Cy/(.r2+/)*. /= force on m units of magnetism at P. j:= distance from coil of radius j carrying current of C absolute units. TANGENT GALVANOMETER. F Cx 27rrxi = HtanS. C=-^HtanS. 5 l-KH C=-^tan8. C=^tanS. C= current in absolute units passing round n coils of radius r producing deflection 8. ^ 2Trn . . , Or= =pnncipal constant. ^= working constant. SINE GALVANOMETER. -*H C=Ji'sin8. S angle between meridian and position in which the needle is in the plane of coils. Note. — When the current is expressed in amperes divide the value of C by 10 ; all other quantities are in C.G.S. units. IMPORTANT EXPRESSIONS. N=niunber of lines proceeding from m units of magnetism. FORMULAE IN ELECTROMAGNETISM. o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o B Work, W, done in moving a current C absolute units across N lines of force. f=C\M. ° ° / force exerted by field H on current C of length /. N= number of lines. B = number of lines per square centi- metre in iron. ^ ^=area in square cms. Ai = B/H. /!= coefficient of permeability. ._M__my.l ^ I = intensity of magnetization. M—mome.ni of magnet = wx/. F'= volume of magnet=ax/. »2 = strength of pole. /= length of magnet. <2=area of polar surface. ^= coefficient of susceptibility. B = H + 4irl. An important connection between the above quantities. Hysteresis. ^ Energy spent in carrying iron round the magnetic cycle is, per unit of volume, I I^H = (the amount represented by the shaded area) 41- J B^H. E Energy dissipated in ergs per cubic centimetre dliring a complete cycle of doubly reversed strong magnetisation is for very soft iron about 9.300, for glass-hard steel wire 76.000 or about 28 ft.-lbs. per cubic ft. for soft iron. If there be 100 cycles per second=i8 H.P. per ton of soft iron. The rise of temperature (° C.) per cycle per c.c. = 2.84 X 10-^ X ergs per c.c. PRINCIPLE OF THE MAGNETIC CIRCUIT. 187 PRINCIPLE OF THE MAGNETIC CIRCUIT N = M/R. N = number of lines, or magnetic flux. IV1= magnetomotive force. R = magnetic resistance or reluctance. ^=11 Ap.. /= length of iron .<4=area of cross section. jii = permeability. M=47rCi'/io = 1.257 X ampere turns. M= magnetomotive force. C= current in amperes. 5= number of spirals. H=4jrC5/io/ = 1.257 ampere turns per centimetre length. (H = strength of field, or number of lines per sq. cm.) /= length of solenoid. If the helix contains iron, then B=/xH, where B is the number of lines per square cm. of iron. ANCHOR RING. N = M 1.257C5 "R" HA\i. ' Z = mean length of lines. ELECTROMAGNET WITH AIR GAP. 1.257C5 N = A +- 4 - + - 2/0 ^i/xj A^i'.i A^ /i = length of path in magnet ol area A^ and permeability \i^. armature A^ each path in air 1 88 FORMULAE IN ELECTROMAGNETISM. Ampere-turns required for N lines in armature= Nx^J— -=-1.257. » .. » ,, gaps = Nx ^ vN „ core =vNx ' Hence Total ampere-turns required = N ( -^-S^ -(- -— -^ +-^ 1 1.257. 1.257. 1-257, where v is the coefficient of leakage, which varies from 1.2 to 1.6. LAW OF MAGNETIC TRACTION. Stt /'=pull in dynes. ^=area of cross section of iron in sq. cms. FIELD MAGNETS. Winding of Bobbins. Let i/= diameter of bobbin. D= „ flange. /= length of bobbin between flanges. Z= length of wire on bobbin with very close winding, if = thickness of wire, including the insulation. 7z = number of turns of wire in a cross-section. N= „ „ on bobbin. D-d n- ■zt N= {D-d)l 2fl ■ Temperature of Bobbin. (i) Each sq. cm, of surface warmed 1° C. above surrounding air dissipates 0.0029 watt. If 50° C. above the air be taken as a safe Hmit of rise of tem- perature, then C=.-s%JaIR. C=highest permissible current in amperes. .4= number of sq. cms. of surface. i? = resistance of bobbin hot in ohms. (2) Probable rise of temperature in degrees F. Watts expended in coil x 100 ~ surface in square inches FIELD MAGNETS. 189 C^sldAjlooR. 0=rise of temperature. C= highest permissible current in amperes. .4= number of sq. inches of surface. /?= resistance of bobbin hot in ohms. If 60° F. rise be allowed = 2 sq. inches per watt as a rough i-ule. SIZE OF WIRE ON FIELD MAGNET. A fixed E.M.F.=£ at the terminals given, and a given number of ampere-turns ( = CS). Each turn of outer layer of wire will have a length of irD, each turn of inner layer ird. Mean length of a tum=-(i? + ^ = /. where r is the resistance of the mean length of a turn, or _E_ ''~CS' From tables the gauge of the wire of length / and resistance r can be ascertained. E.M.F. OF TWO-POLE DYNAMO. «= speed of armature revolutions per second. 5=number of wires or conductors in series on the armature counting all round. N= number of magnetic lines passing through the armature core. .£:=E.M.F. in volts produced in armature. OUTPUT OF TWO-POLE DYNAMOS AND MOTORS. SneWs Rules. /= length of armature in inches. i/= diameter „ „ « = number of revolutions per minute. Dynamos. Motors. Drum, .oi'^liPn watts. .00001 5/(^« B.H.P. Cylinder, .01 l(Pn „ .00001 Wn „ igo FORMULAE IN ELECTROMAGNETISM. Definitions of Efficiency of Dynamos. Gross efficiency or efficiency of conversion _ total electrical energy developed total mechanical energy supplied' Nett efficiency or commercial efficiency useful electrical energy developed ~ total mechanical energy supplied Electrical efficiency or economic coefficient _ useful electrical energy total electrical energy MECHANICS OF DYNAMO AND MOTOR. Torque. T=fr. T= torque or angular force. /= force. r= radius of pulley. P=f.v =fr.- = T(D = Tiirn. /*= power supplied or produced by pulley. 2/= linear velocity. (0= angular velocity. « = revolutions per second. /'H.=£'C=27r«rx 1.356. /',r= power in watts supplied by motor. £■= number of volts generated by armature. C=the number of amperes flowing through the armature. 7^= torque in pound-feet. £'=«5N-=-io^ by previous formula. Drag on Conductor / (dynes) = C/H-=- 10. C= current in conductor. /= length of conductor. H = intensity of magnetic field. . H.P. X 33,000 / (lbs. average drag per conductor) = ^^ ^^^ min. x 5 " THE DIRECT CURRENT DYNAMO. 191 THE DIRECT CURRENT DYNAMO. Series Machine. £'„ = total induced E.M.F. -£',=E.M.F. at terminals. C>=current in circuit. i?<,=resistance of armature. R-m— „ magnet coils, vj pg g g eju) BBf t j nj j .) R,= „ external circuit. ij= economic coefficient. E R, C« £a = (R. + Ra + R,„)Ce. R, ' aHR;,+R,+R„) E R,+R,+R^ R„ should not be greater than 7?„, and never more than two-thirds its value. S^nl Machine. ■ R, ODPOOPDpopappOQVO 1- = E.M.F. at terminals. = resistance of „ = current in armature. „ shunt. „ external circuit. And above symbols. c,= E,= V= T] is i £,/R.. JS-CJ C;'R,-\-C^R„ I maximum when R='Jrar,. C, should not be more than 5 % of C«, often is not more than 2 or 3 %. R, must be at least 324 times Ra to obtain 90 % efficiency. COMPOUND DYNAMO. This may be either short or long shunt (see figs.). For the former we have C = C.-|-C .OQOOOOOOOOOBOO. '' C^R,JrC.^R.+ C^R,+ C,^R, Short Shun . igz FORMULAE IN ELECTROMAGNETISM. (Tf^rrnirinmnnnnnnri ,^<5yjw^,- Long Shunt. For the latter, C?Re If i?™ is small, there is but little differ- ence between the two types. ELECTRO-MOTORS (Two-Pole Direct Current). „_.£-/ ^~ R ■ CE = C^R +Ce = watts converted into heat + watts converted into mechanical power. £=E.M.F. at brushes of motor. ^=back E.M.F. of the armature. C= current passing into the armature. /?= resistance of armature. Ce is a maximum when e=El2. Efficiency = -^7^= -p- 50 % efficiency when maximum work is done. E.H.P.=4=^^^. 746 33.000 T=7.o^Cln. E. H. P. = electrical horse-power. 2"= torque in ft.-lbs. ?z = number of revolutions per minute. e = Snfi\o~^ (see Dynamo Formulae), where n here is number of revolutions per second. FARADAY'S LAW. 193 ELECTRO-MAGNETIC INDUCTION AND ALTERNATING CURRENTS. (Including Formulae of Importance in Electrical Engineering.) FARADAY'S LAW. _ N1-N2 ^N ^=^T- ""' -di- £'=total induced E.M.F. Ni=number of magnetic lines passing- through the closed circuit, as at I- ^^r^-^-T- ^- . |S N ^ jilll^ ,' — j — »- N2 = number passing through the circuit, as at B. f=a. short interval of time in which the change of the number of lines takes place. If lines of magnetic force perforate a circuit positively, as in the above figure, a diminution in the number of lines makes a positively directed E.M.F., i.e. a current in a clockwise direction as seen from N, hence negative rate of change of the number of lines creates a positively directed induced E.M.F., or E=—dNldt. Divide by lo^ to obtain volts. IDEAL DYNAMO. A u — b — > B • • • m • % C D Let AB and CD be rails of copper of negligible resistance, horizontal and parallel ; let j- be a metal slider, then : ^ N Hab Hav. C= Had tR' .£'=total induced E.M.F. when N lines are cut out of the circuit in time /, the shder moving with uniform velocity v. H= strength of magnetic field supposed perpendicular to the paper. ab—Xhe. area swept over when the slider moves from S to S'. J?=resistance of circuit. C= current produced when the slider moves. N 194 INDUCTION AND ALTERNATING CURRENTS. c A ) B C D ROTATING RECTANGLE. ABCD conducting rectangle rotating with uniform velocity about the axis OP, plane per- pendicular to lines of force. E= Ncose=H^cosft dH cos dt = -N sin Q-72- dt N=Hnes offeree passing through ABCD. H = strength of field. .<4=area of rectangle. 0= angle through which the rectangle is displaced from the position of maximum lines to the position where Nj lines pass through it. ^= instant of time arriving at position 6. £=E.M.F. acting round the circuit at the time t. SIMPLE ALTERNATOR. E= N5« X 27r sin iirnt-^ lo*. 5=number of spirals. «=revolutions per second. t=o, %m.iirnt=o, and E.M.F.=o. t=\, sm2irni=i, and E.M.F. = maximum, If If hence £'=±2ir«N5-M0« expresses limits of variation of E.M.F. COMMERCIAL ALTERNATOR. £'=average E.M.F. in volts. « = number of cycles or periods per second equal to number of revolutions in case of two-pole machines. 5= number of coils in series. /= pairs of poles. A'=a constant depending on the relative breadth and shape of pole pieces, varying from 0.29 to i.l. SINE CURVE ORDINATES. SINE CURVE ORDINATES. 195 2 £a<,-Eu^. X - = 0.6369 iimat E,=E„^ X -= = 0.707 ^„a=c. ^r — -ff on = .07 Emai. -£'mai. = maximum value of ordinate corresponding to the maximum value of E.M.F. in alternator. .£'ar= average value of ordinate or of E.M.F. £■,= square root of mean squares of ordinate corresponding to the virtual E.M.F. DEFINITION OF SELF-INDUCTION. U=L^C. Zj= coefficient of self-induction or inductance defined as the ratio of the counter E.M.F. in any circuit to the time rate of variation of the current. ^2 = inductance defined as the ratio of the total number of lines passing through a circuit to the current producing the lines. Z3=inductance defined as that coefficient by which half the square of the current must be multiplied to obtain the electro-kinetic energy (7") of the circuit at that instant. In circuits not containing iron L-^=L^=L^. INDUCTANCE OF HELIX. N=47rC5/R. N= number of lines passing through the helix when a current of C absolute units is started. 5= number of spirals. R = magnetic reluctance. Total cutting of lines = N5=ZC Z = inductance = 4?r57R = henries x 10*. If the current varies at the rate dC\dt, E=-LdCldt, ■or volts =henriesx amperes per second. 196 INDUCTION AND ALTERNATING CURRENTS. MUTUAL INDUCTION. Suppose two helices, one inside the other, the inner called the primary of S-^ spirals, the outer tho secondary of ^"2 spirals. Lines produced by primary=4a-C5i/R = N. Suppose all these lines to be enclosed by the secondary. Total cutting of lines by secondary =6'2N=J/C. il/=mutual inductance = 4x5i52/R=henriesx 10'. If the current in the primary varies at the rate dCjdf, then Ei=-MdCldt, or volts in secondary=henriesx amperes per second. CIRCUIT WITH SELF-INDUCTION. E=RC+L~. at ^=impressed E.M.F. /?C= resistance x current = effective E.M.F. Z Ni=t0tal amount of cutting of magnetic lines of the Sj spirals of the primary of length /, sectional area A, when a current of C absolute units is passed through the coil. N2= total amount of cutting by the 52 spirals of the secondary coil when current C is passing through the primary. ^ = constant /^irAjl. Now let the same current C pass through the secondary instead of the primary. 1^2 = -j 02X02^^02 C ^/.2^ ! r,^i =45^5^ X S^=kS^S^C^ M^C. Nji and N2' being now the amount of the total cutting of the primary and secondary, hence Assume a periodic current through the primary E2 = '2.'KnMC-^. £■2= maximum E.M.F. in secondary; Ci = maximum current in primary ; «= frequency of the wave current. P" \T S* -^='Vt^ = t^=^= transformation ratio. This assumes no magnetic leakage and negligible resistance in the primary. .E2 = 27r«ZiC]. £■1 = maximum E.M.F. in the primary, the circuit of the secondary being open. (2) Commercial Transformers "With Iron. Design. — When the transformer contains iron L and M are not constant, the following formulae, however, Ex = ^%irS^^ -=- 10', Ei = kEx, are useful in design. El and .ffj^ virtual E.M.F.'s of primary and secondary in volts. N= maximum induction. TJiANSFORMERS. 199 Efficiency (ij) CEi Co^aH- watts lost in iron + watts lost in copper _ output indicated by voltmeter and ammeter on non-inductive circuit input measured by a wattmeter _ output indicated by voltmeter and ammeter on non-inductive circuit output -1-4. 2 amount of heat in calories produced per second Hysteresis Loss. — Empirical formula for a good sample of iron Watts per lb.=-^«Bi=5. «=frequency. B maximum value of induction in C.G.S. lines per square inch. Eddy Current Loss. Watts per \\>.=^A^^\ /= thickness of plates in inches, n and B as before. BALLISTIC GALVANOMETER. g= quantity of induced electricity. G= principal constant of galvanometer (see tangent galvanometer). 7'=time of a single vibration of the magnet. a=angle through which the magnet is deflected by the cm-rent of short duration. EARTH INDUCTOR. iHA iHtnrr'^ Q=- R R Q = quantity of electricity induced per half-revolution. A = total area of 'vaA\x.tot=mrr^ for a circular coil where « = number of turns and r the radius of the hoop. i?= electrical resistance of the inductor. zoo LIST OF BOOKS OF REFERENCE. ADDITIONAL LIST OF BOOKS OF REFERENCE ON PURE AND APPLIED ELECTRICITY. Magnetic Induction in Iron and other Metals. J. A. Ewing. Lessons in Electricity and Magnetism. S. P. Thompson. Practical Electricity. W. E. Ayrton. Lessons in Electricity and Magnetism. Eric Gerard, translated by Duncan. Elements of Mathematical Theory of Electricity and Magnetism. J. J. Thomson. Mathematical Theory of Electricity and Magnetism. Watson and Burbuiy. Mathematical Theory of Electricity and Magnetism. W. T. A. Emtage. Electro-Magnet and Electro-Magnetic Mechanism. S. P. Xhompsofi. Dynamo-Electric Machinery. S. P. Thompson. Electro- Magnetism and the Construction of DjTiamos. Jackson. Electric Motive Power. Albion T. Snell. Electric Transmission of Energy. Gisbert Kapp. Absolute Measurements in Electricity and Magnetism, 2 vols. Andrew Gray. Alternate Current Transformer in Theory and Practice, 2 vols. J. A. Fleming. Alternating Currents of Electricity. T. H. Blakesley. Alternating Currents of Electricity. Gisbert Kapp. Alternating Currents. Bedell and Crehore. The above are selected as representative standard works. The bibliography of the subject is rapidly increasing. A more complete list will be found in the Electrical Traded Directory. APPENDIX. CONVERSION TABLE. CONVERSION TABLE. 203 I. LENGTH. To reduce Multiply by To reduce Multiply by Kilometres to miles, .6214 Miles to kilometres, 1.609 Metres to miles, .0006214 Miles to metres, - 1609 Metres to yards, - 1.094 Yards to metres, .9144 Metres to feet, - 3.281 Feet to metres. .3048 Centimetres to inches^ - .W^7 Inches to centimetres, Inches to millimetres. 2.540 25.40 Millimetres to inches, - -03937 n. SURFACE. Sq. kilometres to sq. miles, .3861 Sq. miles to sq. kilometres, - 2.590 Sq. metres to sq. yards, - 1. 196 Sq. yards to sq. metres, - .8361 Sq. metres to sq. feet. 10.76 Sq. feet to sq. metres; .09292 Sq. centimetres to sq. inches - -1550 Sq. inches to sq. centimetres,- 6.452 Sq. millimetres to sq. inches, - -ooiSS Sq. inches' to sq. millimetres, - 645.2 III. VOLUME. Cu. metres to cu. yards,- 1.308 Cu. yards to cu. metres,- .7645 Cu. metres to cu. feet. 35-32 Cu. feet to cu. metres, .02832 Cu. centimetres to cu. inches , .06103 Cu. inches to cu. centimetres, 16.38 Litres to cu. feet, -03532 Cu. feet to litres. 28.32 Litres to gallons, - .2642 Gallons to litres, 3-785 Litres of water to lbs.. - 2.205 Lbs. of water to litres, - -4536 IV. WEIGHT. Kilogrammes to tons, .0009843 Tons to kilogrammes. ioi6 Kilogrammes to lbs.. 2.205 Lbs. to kilogrammes. •4536 Grammes to ounces, - -03527 Ounces to grammes, 28.35 Grammes to grains. 15-43 Grains to grammes. .06479 V. FORCE, WORK, Dynes to poundals, .00007232 Dynes to Ibs.-weight, .000002246 Kilogrammetres to foot-pounds, 7.233 Force de cheval to horse-power, Kilogrammes per sq. metre") to pounds per sq. foot, -/ Grammes per sq. cm. to Ibs.l per sq. inch, - - - j' 4 - .9863 .2048 AND PRESSURE, ETC. Poundals to dynes, 13830 Lbs. -weight to dynes, - 445300 Foot-pounds to kilogrammetres, .1382 Horse-power to force de cheval, 1.014 Pounds per sq. foot to kilo-\ grammes per sq. metre, - j Pounds persq.inchtogrammes\ per sq. cm., - 1 4,883 70.31 VI. MISCELLANEOUS. Common logs to Napierian, Radians to degrees, Calories to Ib.-degrees, Fahr., Calories to joules, - - - Joules to foot-pounds, Grammes per litre to ounces"! per gal., - -/ Gallons of water to lbs.. Gallons to cubic feet, Cubic feet of water to lbs.. Tons per sq. foot to lbs. per sq. in., 15.55 Lbs. per sq. in. to atmospheres, .06803 Tons per sq. foot to atmospheres, i .06 Feet per sec. to miles per hour, .68 19 Cm. per sec. to miles per hour, .02238 2.303 57-30 3.968 4.158 -7374 -1336 10 .1606 62.28 ':} Napierian logs to common Degrees to radians, Lb.-degrees F. to calories, Joules to calories, - Foot-pounds to joules, - Ounces per gallon to gms per litre. Lbs. of water to gallons, Cu. feet to gallons,- Lbs. of water to cubic feet, - .01606 Lbs.persq.in.totonspersq.foot, .06431 Atmospheres to lbs. per sq. inch, 14.70 Atmospheres to tons per sq. foot, .94 Miles per hour to feet per sec, 1.467 Miles per hour to cm. per sec, 44.70 •4343 01745 .2520 ■24 1-356 7-491 .1000 6.228 ao4 APPENDIX. ; TABLES OF VELOCITIES. Cms. Feet Kilometres Miles per second. per second. per second. per second. Light and Electromag- netic Propagation, 3 X iqI" 984x108 300000 186420 Electric Signals : Submarine Cable, ■ 4X lO^ 1 3. 1 x-io' 4000 2486 Air Line, 36 X 10* ii8x 108 36000 22374 Earth in orbit. 30 X 10^ 98.4 X 10' 30 18.6 Point on Equator, 465 X 10^ 1525.2 .465 .289 Dry air at o° C, 33.2 X 10' 1089 -332 .2063 Airat 15° C. (average humidity), - 34.96x108 1 147 •3496 .2225 Hydrogen at 0° C, - 1,27 X lo' 4166 1.27 -7892 Oxygen 31.7 X lb' 1040 .317 .1970 Carbon dioxide „ 26.2 X le? 859.4 .262 .1629 Water at 8° C, 143.5 >< 1°' 4707 . 1-435 .8917 Oak, 344 X 10' 1 1280 3-44 2.138 Brass, 339.7 X io3 1 1 140 3-397 2.107 Glass, 450 X 10' 14760 4.5 2.796 Iron and Steel, 510X 10' 16730 5-1 3.169 Cms. Feet Kilometres Miles ^ per second. per second. per hour. per hour. Snail, - .16 .005249 .005759 .00358 Fly (flying slowly). 160 5.249 5-759- 3-58 Hunted Fly, 970 31-83 34-92 21.7 Eagle, 3170 104 1 14. 1 70.92 Carrier Pigeon, 1800 59.06 64.8 40.27 Horse trotting. 1 301 42.68 46.83 29.10 Race-horse, 1670 54-79 60.12 37-36 Man walking, 420 13-78 15-13 9.401 „ running. 630.7 20.89 22.93 14.25 „ skating. 894 29-33 32.19 20.00 „ swimming, - 102.7 3-369 3-697 2.297 Safety Bicycle, 1382 45-35 49-75 30.92 Tandem, 1363 44.74 49.06 30.50 Triplet, 1338 4392 47-88 29.95 Tricycle, - 1 167 38-31 41.76 26.12 Tandem Tricycle, II4I 37-44 41.04 25-53 Sailing Vessel, 720.6 23-63 25-94 16.12 Racing Yacht, 772.1 25-33 27.80 17.28 Man-o'-war, 926.6 30.40 33-36 20.73 Atlantic Liner, I09I 35.80 39.28 24.41 Torpedo Boat Destroyer 162I 53.19 58.37 36.27 Express Train, - 41370 1 26. 1 138.4 86.00 Thrown Stone,' - 1700 55-78 61.19 38.02 Rifle Bullet, 57910 1900 2084 1295 Cannon Ball, 91440 , 3000, 3292 2046 Rivers, 300 9-843 10.80 6.71 1 Mountain Streams, - 600 19.69 21.60 13.42 Winds (average). 550 18.04 19.80 12.30 Hurricane, - 4000 131.2 144.0 89.48 Note. — Many of the velocities can only be regarded as approximate, as new records are constantly being made. The miles per hour are usually calculated from the raie for a short distance. DIMENSIONS OF BRITISH COINAGE. 20S DIMENSIONS OF BRITISH COINAGE. COMPILED PROM INFORMATION SUPPLIED BY THE BOYAL MINT, SEPTEMBER, 1896. Coin. Diameter. Thickness (Approximate). Weight. , ^Chemical ;:omposition. Inches. Millimetres Inches. Millimetres Grains. Grammes. Per cent. Farthing, Halfpenny, - Penny, 0.8 I.O 1.2 20.3 25-4 30.5 0.045 0.048 0.054 I.I4 1.22 1-37 43-75 87.50 145-83 2.835 5.670 9.450 CU95 Sn 4 Zn I Threepence, - Sixpence, Shilling, Florin, Half-crown, - Double Florin, Crown, - 0.64 0.76 0.92 1.12 1.26 1.42 1.52 16.3 19-3 23-4 28.4 32.0 36.1 38.6 0.026 0.036 0.049 0.063 0.065 0.084 0.091 0.66 0.91 1.24 1.60 1.6s 2.13 2.31 21.82 43-^4 87-27 174-55 218.18 349-09 436.36 1.414 2.828 5-655 11.310 14-137 22.619 28.274 Ag92.5 Cu 7-5 Half-sovereign Sovereign, - 0.75 0.86 19-5 21.8 0.037 0.045 0.94 1.14 61.64 123.27 3-994 7.987 JAU91.6 lAg.8.3 j^i = 25 francs French = 20 marks German Empire = 4 dollars 80 cents United States. One franc weighs 5 grammes, and is divided into 100 centimes. The I centime weighs i gramme, and measures 15 mm. diameter. „ S .. 5 .. " ^S ,, 10 ,, 10 ,, ,, 30 •• The German mark = 100 pfennige, weighs 85.7 grains = 5.55 grms. The American dollar = 100 cents, weighs 412.5 grains = 26.7 grms. - - ■ -r France, Belgium, Italy, Greece, and Switzerland have coins of the same weight and fine- ness, although the names of the coins may differ. They constitute the Latin Union. It is useful to remember, that the diameter of a halfpenny is i incf.. And that approximately One sovereign or two half sovereigns weighs H oz. Troy. Half-a-crown, or two shillings and sixpence, weighs % oz. Troy. Crown, or five shillings, weighs i oz. Troy. 2o6 APPENDIX. THE MORSE CODE. 1-4 SIGNALS. e • ■ i • • . s • • a • h t — — m _ _ — o • — a — - — u — • • — — u, German. — — * • ••—V ^.^ . — • r — • • — • — a, German. — • — - • - - y - - . 5 SIGNALS. • n • g . d • z . b -k . c — X . 1 . o -q - ch I. . 9. • 2. . . 8. - • ■ 3- ... 7- • • a 4- .... - 6. - • a • • 5. o. h. . . _ . . French. 6 SIGNALS. Period or fall stop, Repetition or ? - - - Stroke or Divisional bar of a fraction, Hyphen, Apostrophe, Comma, ----- THE GREEK' ALPHABET. 207 OQ < X < w o w DC "S ^ V >» buo ^ « ^ •§ •^ 1 8 H g a. IS :z; >^ ^ fxl '^ a "2 1 « -^i bo •^ •M N ■*4 >«e ^ S ni d rt E n* o iS S<-2 < pq O O W O rt 1 3 tf oa. ?. CO V •v^ sr- •a i: --i S, -l w Q < s < o w OS H U W W CA >>'^ o II '^ II y c -a to ho e 6 o ■^ .2 .S 3 ^J U +j ■*-; ™ > OJ +^ +J *" c bo 3 o o w ex " S o 3 >. 5 bo tn 3 W 02. <1 3 ■2t> ^ II E t; S E^ > s 2 o fe bo Qi -^ ■ ^ XT U ■ — I -^ „ « ^3 ■» 6 I' i; It, ri (U Tl ^' t 1 n^ a t I P, II s s p < < s 'S o >, TO a; 55 i > ^ O o 2 g m o 3 -S, cr (U ^f 3 4 ■" *"■ j1 -^ ^s T u „ ■ .3 ,3 i^S II "^ 'N -a ;3 + P* 3 q &"| S « 3 2 " K "rt "rt II +-" ■4-' o o H H > I .S ■" : ■« Q ^ « I I •- lU ■K. > 3 3 ni cv- "fin 0) II II 0) (/I Si 3 C) 3 a 01 3 Tl S Si S. Si n bo 3 2 S;? 3 S E bo •C o O '3 I*, -B ^ « -S + "! fe II ■a -3 II 4-1 -4-1 o o H H APPENDIX. LIST OF BOOKS OF REFERENCE. Physikalisch-Chemische Tabellen der Anorganischen Chemie. Karl von Buchka, 1895. Physikalisch-Chemische Tabellen. Landolt and Bornstein, 1883. Physical Mahipulation, Part II. Pickering, 1876. Physical Measurements. Kohlrausch, trans, by Waller and Procter. Physical Measurement, Part ill. Whiting, 1891. Numerical Tables and Constants. Lupton, 1884. Units and Physical Constants. Everett. Manipulations de Physique. Buignet, 1877. Smithsonian Physical Tables. Gray, 1896. Smithsonian Meteorological Tables. Guyot, 1884. Constants of Nature, Smithsonian Collections. F. W. Clarke, 1876. Useful Rules and Tables for Engineers and others. Rankine. Pocket-book of Useful Formulae and Memoranda. Molesworth. Pocket-book of Electrical Rules and Tables. Munro and Jameson, 1898. Electrical Engineers' Pocket-book. Kempe. Manual of Rules, Tables, and Data for Mechanical Engineers. D. K. Clark. Engineers' Year Book. Kempe, 1896. Dictionary of Metric and other Useful Measures. Latimer Clark. Tables of Squares, Cubes, etc. Barlow. Mechanics. Nystrom. Pocket-book for Miners and Metallurgists. Power. Electrical Engineering Formulae. Geipel and Kilgour. Formulaire de I'Electricien. Hospitaller. Chemiker Kalender. Biedermann. Constants Physico-Chemiques. D. Sidersky. Unit& et Etalons. Guillaume. Polarisation et Sacchariraetrie. D. Sidersky. INDEX. PAGE Accelerated velocity 66 Acetic acid, properties of 131 Air and the atmosphere 96 Algebra, formulae in 37 Alloys, properties of 123 Alternating currents 193 Alternator, commercial - 194 simple 194 Ammonia, properties of 129 Analogies, mechanical and electrical - 209 Analytical geometry, formulae in 49 Anchor ring - 187 Annuities 39 Antilogarithms 10, II Applications of logarithms 6 Approximations 41 Areas of plane figures 32>6i Arithmetical progression 38 Atomic heat - 165 Ballistic galvanometer 199 Barometer 90 Barometric corrections 90, 163 Batteries in parallel 180 Beams, deflection of 7S Beats 148 Binomial series 39 Blot's formula 146 Boiling point 99 Boyle's law 91, 163 departure from 164 Bursting of shells 92 Calcium chloride, properties of 136 Calculus, differential and in- tegral - 58 Capacity, electrostatic 176 PAGE Capillarity 92 Capillary depression of mer- cury, - 95 Capillary tube, motion of liquid in - 91 Caustic potash, properties of - 128 soda, properties of. 127 Cells, primary 114 secondary - "5 standard 115 Centres of gravity - 63 Charles, law of 163 Circle, the S3 Circles, triangles and 45 Circuit, branching - 179 magnetic 187 shunted 180 simple 179 with self-induction 196 Circular motion 76 Circumscribed polygons - 45 Coinage, dimensions of - 205 Combinations 40 Compound interest 39 pendulum - 77 Conduction of heat 168 Conical pendulum - 77 Contact differences no Conversion table 203 Cooling, laws of 167 Copper sulphate, properties of 137 Cosines, logarithmic 22, 23 natural l6, 17 Cube roots, table of 26 Cubes, table of 26 Current electricity, formulje in 178 Curvature, radius of 60 Curved mirrors 153 INDEX. PAGE Curves, lengths of- 31 Density and specific gravity 86 Diatonic scale 102 Differential and integral cal- culus 58 screw 75 wheel and axle 72 Differentiation, rules for 58 Doppler's principle 146 Dulong and Petit's law - 165 Dynamics, formulas in 62 Dynamo, compound 191 direct current 191 E.M.F. of two-pole 189 ideal - 193 Dynamos and motors, mechan- ics of 190 output of 189 Earth inductor 199 Elasticity 78 table of coefficients of 81 Electricity, current, formulae in 178 Electro-chemistry, formulae in 181 Electromagnetic waves - 161 Electromagnetism, formulae in 184 Electromagnet with air gap 1 87 Electrometers 1 76 Electro-motors 192 Electrostatics, formulae in 175 Elements, properties of - 117 Ellipse, the 55 Ethyl alcohol, properties of 130 Euler's theorem - 59 Expansion, absolute, of mercury 163 apparent - 162 of bodies 1 62 woric done during 91 Exponential functions, table of 208 Friction 65 Functions, numerical values of 42 of multiple angles 43 of sums of angles 43 products of 43 relations between 43 sums and differences of 44 Galvanometer, ballistic - 199 sine 185 tangent 185 Gases and vapours, properties of - - - - 125 General equation of second degree 57 Geometrical progression 38 Gravity, centres of 63 values of ^ 66 Greek alphabet 207 H, definition of 171 Harmonic motion - 143 Heat, atomic 165 conduction of 168 of combination 100 power, and energy, table lOI specific 164 Heights by barometer 90 Helix, inductance of 195 Horse-power, work and 75 Hydrochloric acid, properties of - - 134 Hydrostatics, formulae in 86 Hygrometer - 98 Hygrometry - 167 Hyperbola, the 56 Hyperbolic logarithms - 208 Hysteresis 186 series 40 Illumination of surfeces - Impact - 150 69 Factors, useful 37 Inclined plane 73 Field magnets 188 motion down 67 size of wire on _ 189 Index of refraction 158 Forces, resultants of . 62 Indices, laws of 37 Fraunhofer's lines, wave lengths Inductance of helix 195 of - 102 Induction, electro-magnetic 193 Freezing point, change of 169 mutual 196 of solutions 170 Inductor, earth 199 INDEX. 213 Inertia, moments of Inscribed and circumscribed polygons Integral calculus Integration, formula of - Intensity of light, units of Intensity of sound - Interest, simple and compound Iron, magnetic properties of tractive force of - wrought, magnetic proper- ties of Joule's law KirchhofTs laws Latent heat Leibnitz's theorem Length of curves - Lenses - Levers - Light, intensity of reflection of wave theory of - Liquids, properties of Logarithmic cosines series sines - tables, use of tangents Logarithms applications of Loss of weight, correction for ^Maclaurin's theorem Magnet, temperature effect on vibrating Magnetic circuit elements, table of field, needle in forces, triangle of potential - properties of iron traction, law of - M^netism, formulas in - Mensuration, formulee in Mercury, vapour pressure of volume and density of PAGE 82 45 58 60 104 148 39 107 107 io5 180 179 166 59 31. 61 70 104 150 159 124 22, 23 40 20, 21 3 24, 25 8. 9, 39 6 or 95 59 on 174 173 187 106 172 171 173 06, 107 IS8 171 31 99 Metacentre 89 Mirrors, curved 153 Molecular theory of gases 16S Morse code 206 Motors - 192 Multiple angles, functions of 43 Napierian logarithms 208 Napier's analogies - 46 Natural cosines 16, 17 numbers 38 sines- 14, 15 tangents 18, 19 Nitric acid, properties of 133 Numbers, natural - 38 Numerical values of functions 42 Ohm's law 178 Organ pipes - 147 Parabola, the 54 Pendulums 77 Periodic current, power of 197 system 122 Permutations 40 Pipes, motion of water in gi Plane figures, areas of 32 trigonometry - 42 Plates, vibrations of 148 Point, the 49 Polarisation 160 Polygons 45 Potential, electrostatic 175 magnetic - 173, 184 Practical units 112 Pressure, centres of, etc. 88 Products of functions 43 Progression, arithmetical 38 geometrical 38 Projectiles 67 Properties of acetic acid 1 3 1 alloys - 123 ammonia 1 29 calcium chloride 136 cane sugar 139 caustic potash 128 caustic soda - - 127 copper sulphate - - 137 elements - 117 2 14 INDEX. Properties of ethyl alcohol 130 gases and vapours 125 hydrochloric acid 134 liquids . 124 miscellaneous solids 126 nitric acid - 133 sodium chloride - 135 sulphuric acid 132 water - 140 zinc sulphate 138 Pulleys - 72 Quadratic equation 37 Radius of curvature 60 Ratios, trigonometrical 42 Reciprocals, table of 26 Reflection of light - ISO Refraction - iSS laws relating to - 158 Relations between ftmctions 43 sides and angles 44 Resistance - 178 conversion table "3 of copper wire 116 Resonating column 147 Resultants of forces 62 velocities - 66 Right-angled spherical triangles 47 Rigidity, simple 80 Rods, vibrations of 149 Rotatory power of crystals 104 specific - 103 Screw, the 74 Self-induction 195 circuit with 196 simple periodic current with 196 Series, binomial 39 exponential 40 logarithmic 40 Sides and angles, relations between 44 Simple interest 39 pendulum - - 77 Sine curve ordinates 195 Sines, logarithmic - 20, 21 natural 14, 15 Sodium chloride, properties of 135 PAGE Solids, surfaces and volumes of - - 35, 61 Solution of triangles 44 Sound, intensity of 148 velocity of- 146 Specific gravity 86 heat - - 165 inductive capacities 108 rotatory power - 103 Spherical triangles 46, 47 area of 48 Spherical trigonometry - 46 Square roots, table of 26 Squares, table of 26 Steam, latent heat of 166 total heat of 166 Straight line, the 50 Striking distance in air 1 10 Strings, vibration of 147 Sugar, properties of 139 Sulphuric acid, properties of 132 Sums and differences of func- tions - - 44 Sums of angles, functions of - 44 Surfaces of solids 3S> 6r Systems of particles, motion of 68 Tangent and normal, equation to 59 Tangents, logarithmic 24, 25 natural 18, 19 Taylor's theorem 59 Temperature effect on magnet 174 Temperatures, important 100 Tension of shells 92 Thermodynamics, laws of 168 Thermoelectricity, formulae in 183 Thermoelectric values 109 Thermometer scales, conver- sion of - 162 Toothed wheels 76 Toricelli's theorem 92 Traction, magnetic, law of 188 Tractive force of iron 107 Transformers 198 Triangles and circles 45 Triangles, solution of 44 Trigonometrical ratios 42 tables, use of - - 12 INDEX. 2IS Tjigonometry, plane 42 spherical 46 Tuning forks, vibrations of 148 Twisting angle of shafts 80 Units and dimensions - no Use of logarithmic tables 3 trigonometrical tables 12 Velocities, resultants of - 66 table of 204 Velocity, of sound 146 uniform 66 uniformly accelerated - 66 Vertical motion under gravity 67 Vibrations, composition of 144 of magnets 173 of plates 148 of rods 149 of strings 147 FAGS Vibrations of tuning forks 148 Vibratory motion - 143 Volumes of solids - 35, 61 Water, latent heat of 166 properties of 140 specific heat of - 99 vapour pressure of 97 volume and density of 97 Wave lengths, Fraunhofer's lines 102 spectral lines 105 Wave theory of light 159 Waves, electromagnetic 161 Weight thermometer 162 We^hts and measures 27 Wheel and axle - 71 Work and horse-power 75 done by expanding fluid gi Zinc sulphate, properties of 138 GLASGOW ; PRINTED AT THE UNIVERSITY TRESS I3Y ROBERT MACLEHOSE AND CO, ■00mmlm'