GIFT OF 
 

PHYSICAL MEASUREMENTS 
 
 MINOR 
 
 PART I. HEAT, MECHANICS AND 
 
 PROPERTIES OF MATTER 
 
 1916-1917 
 
PHYSICAL MEASUREMENTS 
 
WETZEL BROS. PRINTING CO. 
 
 2110 ADDISON STREET 
 
 BERKELEY, CAL. 
 
PHYSICAL MEASUREMENTS 
 
 A LABORATORY MANUAL IN GENERAL PHYSICS 
 FOR COLLEGES 
 
 BY 
 RALPH S. MINOR, Ph. D. 
 
 Associate Professor of Physics, University of California 
 
 PART I. 
 HEAT, MECHANICS and PROPERTIES of MATTER 
 
 In Collaboration With 
 Wendell P. Roop, A. B. 
 
 and 
 Lloyd T. Jones, Ph. D. 
 
 Instructors in Physics, University of California 
 
 BERKELEY, CALIFORNIA 
 191.6 
 
GO 3 7 
 
 Copyrighted in the year 1916 by 
 Raloh S. Minor 
 
PREFACE 
 
 This manual presents the laboratory exercises of the 
 Courses in General Physics offered at the University of 
 California. It is printed in four parts, each part accom- 
 panied by appropriate tables. Parts I and II contain 
 experiments on Heat, Properties of Matter, and Mechan- 
 ics. Part III deals with Magnetism and Electricity, and 
 Part IV is devoted to Sound and Light. 
 
 While the manual is largely based upon the exercises 
 printed in Physical Measurements in Properties of Matter 
 and Heat-Minor and Elston, and Physical Measurements 
 in Sound, Light and Electricity-Minor, it represents a 
 revision in which the writer has attempted two things. 
 First, to base the college laboratory work squarely upon 
 the foundation afforded by a first course, and second, to 
 omit exercises based upon topics which properly belong 
 to the lecture demonstration and which experience has 
 shown to be adequately treated there. 
 
 The writer's predecessors, Professor Harold Whiting, 
 Professor E. R. Drew, Professor Elmer E. Hall, Dr. T. 
 Sidney Elston, A. C. Alexander, G. K. Burgess, Bruce 
 V. Hill, A. S. King, and T. C. McKay have all influenced 
 the development of these exercises. 
 
 Material has been freely drawn from the literature of 
 the subject whether in magazine or text book form with- 
 out specific credit being always indicated. 
 
 RALPH S. MINOR 
 University of California, 
 August, 1916. 
 
 3G2703 
 
LIST OF EXPERIMENTS 
 
 1. Sensitive Beam Balance - - 12 
 
 2. Density of Air - 16 
 
 3. Relative Density of Carbon Dioxide - 19 
 
 4. Volumenometer - 20 
 
 5. Surface Tension by Jolly's Balance - 23 
 
 6. Capillarity. Rise of Liquids in Tubes - 25 
 
 7. Viscosity. Flow of Liquids in Tubes - 28 
 
 8. Young's Modulus by Stretching - 30 
 
 9. Comparison of Alcohol and Water Thermometers 33 
 
 10. Coefficient of Expansion of a Liquid by Reg- 
 
 nault's Absolute Method - 36 
 
 11. Coefficient of Expansion of a Liquid by Pycnom- 
 
 eter Method - 38 
 
 Plotting of Curves - 41 
 
 12. Coefficient of Expansion of Glass by Weight- 
 
 Thermometer Method - - 42 
 
 Calorimetry - 44 
 
 13. Specific Heat of a Liquid by Method of Heating 46 
 
 14. Specific Heat of a Liquid by Method of Cooling 48 
 
 15. Heat of Fusion of Wood's Metal - 50 
 
 16. Fuel-Value of Alcohol - 53 
 
 17. The Principle of Moments: Statics 55 
 
 18. The Parallelogram Law: Statics - 58 
 
 REFERENCE BOOKS 
 
 Duff: Text-Book of Physics (Fourth Edition). 
 
 Edser : Heat for Advanced Students. 
 
 Gariot: Text-Book of Physics (18th Edition). 
 
 Kimball: College Physics. 
 
 Lodge: Elementary Mechanics. 
 
 Preston: Theory of Heat (Second Edition). 
 
 Watson: Text-Book of Practical Physics. 
 
 Kaye and Laby: Physical and Chemical Tables. 
 
 Landolt and Bernstein: Physical and Chemical Tables. 
 
 Smithsonian Institute: Physical Tables. 
 
INTRODUCTION 
 
 This book is intended to be mainly a manual of direc- 
 tions. It is complete enough, however, so that when used 
 in conjunction with Duff's Physics, the text-book, it 
 will cover the minimum requirement for the year's work. 
 
 Experiments are assigned on the bulletin boards. Note 
 your assignment and find out what you are going to do 
 before you enter the laboratory. The necessary apparatus 
 may be obtained from the store-keeper. In writing re- 
 ports, use the paper and binders sold for the purpose. 
 Take the data in duplicate, and drop the carbon copy in 
 the box in the hall on leaving. Make your report concise. 
 Criticize your results. If they are not satisfactory to 
 you, explain why. 
 
 Within one week after you have finished taking data, 
 drop your completed report in the box in the hall under 
 your recitation instructor's name. In writing the report, 
 follow the general directions printed in the binder. All 
 curves must be plotted on millimeter paper. 
 
 Your report will be returned to you by your recitation 
 instructor. If it is graded, note corrections, and keep it. 
 If it is not graded, make indicated corrections, and return 
 it as soon as possible. 
 
 Errors 
 
 Every measurement is subject to errors. In the simple 
 case of measuring the distance between two points by 
 means of a meter rod, a number of measurements usually 
 give different results, especially if the distance is several 
 meters long and the measurements are made to fractions 
 of a millimeter. The errors here arise from inaccuracy 
 
8 SIGNIFICANT FIGURES 
 
 in setting, inaccuracy in estimating the fraction of a 
 division, parallax in reading, faults in the meter rod, ex- 
 pansion due to change in temperature, and so on. Blun- 
 ders in putting down a wrong reading or in not adding 
 correctly are not classed as errors. 
 
 SIGNIFICANT FIGURES 
 
 It is necessary always, in recording measurements, to 
 put down all the figures that can be trusted, even if some 
 of them are ciphers. For instance, if a distance has been 
 measured to hundredths of a centimeter and found to be 
 50.00 cm., it is not correct to put the distance down as 
 50 cm., for the statement that the distance is 50 cm. implies 
 that the distance has been only roughly measured and is 
 found to be more nearly equal to 50 cm. than it is 49 or 51 
 cm., whereas the statement that the distance is 50.00 cm. 
 implies that the distance lies between 49.99 and 50.01 cm. 
 When the distance is said to be 50 cm., the implication is 
 that the second figure, counting from the left, is the last in 
 which any confidence can be placed; but when the distance 
 is said to be 50.00 cm., the implication is that confidence 
 can be placed in the quantity as far out as the fourth 
 figure from the left. This fact is indicated by saying that 
 in the first case the quantity can be trusted to the second 
 significant figure, and in the other case to the fourth. 
 
 The number of significant figures in a quantity is irre- 
 spective of the decimal point; thus there is one significant 
 figure in 0.01, two in 0.000026, and three in 2.10. A 
 convenient method of writing a number so as to show its 
 precision is to place the point after the first significant 
 figure and multiply by an appropriate positive or negative 
 power of ten. If a mass, for instance, is about 25,000 gm. 
 and the third significant figure is the last in which any 
 
SIGNIFICANT FIGURES 9 
 
 confidence can be placed, this fact is indicated by saying 
 that the mass is 2.50 X 10 4 gm. The power of ten is the 
 characteristic of the logarithm of the number. 
 
 In indirect measurements the result is usually calculated 
 by some formula. To find how many figures should be 
 kept in the result and how accurately the several quantities 
 in the formula should be measured, the student should 
 observe the following rules. 
 
 Sums and Differences. 
 
 In sums and differences no more decimal places should be 
 retained than can be trusted in the quantity having fewest 
 trustworthy decimal places. Also, 
 
 Quantities which are to be added or subtracted should be 
 measured accurately to the same decimal place, irrespective 
 of whether they have the same number of significant figures 
 or not. 
 
 Products and Quotients. 
 
 In products and quotients no more significant figures 
 should be kept than can be trusted in the factor having fewest 
 trustworthy figures. Also, 
 
 Quantities which are to be multiplied or divided should be 
 measured to the same number of significant figures. 
 
 Until the final result is reached, it is often worth while 
 to keep one more figure than the above rules indicate. 
 
 In cases in which a result must be calculated from several 
 different measurements, it is useless to take some of these 
 measurements with high precision when the precision of 
 the result is limited by that of some measurement whose 
 precision is necessarily low. For example, if you are 
 
10 SIGNIFICANT FIGURES 
 
 measuring the area of a rectangle which is ten times as 
 long as it is broad, it is a waste of time to determine the 
 length within a tenth of a millimeter if the breadth can 
 be determined at the best only within a tenth of a milli- 
 meter. For if this were done, the length would be measured 
 to one more significant figure than the breadth, whereas 
 the precision of the calculated area would not exceed 
 that of the breadth. 
 
 Percent Error 
 
 The number of significant figures used in expressing 
 a quantity is only a rough indication of its precision. A 
 more exact statement of the precision is made in terms of 
 percent error. 
 
 The error in a quantity may be estimated by comparing 
 two successive determinations. The deviation of either 
 from the mean indicates the order of magnitude of the 
 probable error. Or, if several determinations are made, 
 the average deviation of the different determinations 
 from their mean may be taken as the probable error of 
 a single observation. This, divided by the number of 
 observations, gives the approximate probable error of 
 the mean value. The probable error may be calculated 
 exactly by means of the method of least squares, which 
 will be taken up in later work. 
 
 The error, as thus determined, is then expressed as a 
 fraction of the quantity measured, in terms of percent. 
 
 Do not estimate the precision of your measurements by 
 comparing your calculated result with a tabular value. 
 
 Do not get the impression that a result apparently in 
 good agreement with a tabular value is preferable to an 
 
PERCENT ERROR 11 
 
 honest reduction of data actually obtained. Frequently 
 the tabular value would be wrong, by reason of the 
 difference in the sample. But even if you have good 
 reason to believe that agreement with tabular results is 
 desirable, you will do better to cultivate the manipulative 
 skill which is necessary to obtain such results rather 
 than to make your data appear better than they are. 
 Manipulative skill is a matter of experience, of which 
 you have little. Correctness of calculations is a matter 
 of arithmetic, but mistakes are common. But when, by 
 a "mistake", poor data give good results, it is a matter of 
 fundamental honesty. A clear scientific conscience is 
 absolutely essential to the validity of any sort of scien- 
 tific work. 
 
 When the greatest errors involved are of the order of 
 0.01%, use five-place logarithms; when they are of the 
 order of 0.1%, use four-place logarithms. Careful work 
 with a 10-inch slide rule will introduce errors not exceeding 
 two or three tenths of a percent. 
 
12 
 
 WEIGHING BY METHOD OF VIBRATIONS 
 
 1. SENSITIVE BEAM BALANCE. DENSITY OF A 
 
 SOLID. 
 
 Weighing by Method of Vibrations. 
 
 When free to swing, the pointer of an equal-arm beam 
 balance oscillates about a mean position called the rest 
 point. This is the position it would occupy if allowed to 
 come to rest. The rest point with no load in either pan 
 is called the zero rest point. The construction of the bal- 
 ance is such that if the two pans contain equal loads 
 the pointer will oscillate about this zero rest point, but 
 if there is a small excess in one pan, the rest point will 
 be shifted slightly in the direction corresponding to a 
 lowering of the heavily loaded pan. 
 
 To determine a rest point, set the beam to swinging 
 freely. Read an odd number of successive turning points 
 (say 5 or 7), on the scale behind the pointer. This scale 
 has zero on the left and ten in the center. The average 
 of the mean of all the left hand and the mean of all the 
 right hand readings is the rest point. 
 
 A rest point determination by the method of Vibrations 
 not only eliminates the friction of the knife edges, but 
 
 Fig. 1. The three turning points shown in this figure 
 indicate 13.3 as the rest point. 
 
1] METHOD OF VIBRATIONS 13 
 
 also avoids loss of time which would elapse in waiting 
 for the pointer to stop. 
 
 In weighing, place the unknown mass in the left pan 
 and load the right pan with known masses until a rest 
 point on the scale can be observed. In making trials 
 proceed systematically, beginning with the large masses. 
 
 THE BALANCE IS A DELICATE INSTRUMENT. 
 HANDLE IT WITH CARE. Keep the door closed, and 
 only open it to change masses. Never change masses 
 except when the beam is clamped. Handle the masses 
 with forceps, to prevent corrosion. 
 
 Use the rider instead of masses of less than 10 mg. The 
 rider weighs 12 mg., but when placed at a point on the 
 beam instead of in the pan, is equivalent to a proportion- 
 ately less amount. 
 
 The sensitiveness of the balance is the shift of the rest- 
 point per mg. excess in one pan. It depends to a certain 
 extent on the load, and must be determined with the 
 pans loaded. Having found the sensitiveness, calculate 
 the amount which would have to be added to or sub- 
 tracted from the mass in the right pan in order to make 
 the rest point come to its zero position. 
 
 An illustrative set of data follows. 
 
 From the data in (ii) and (iii) the sensitiveness is found 
 to be (14.87-8.54) / 2 = 3.2 scale-divisions per mg. From 
 this and the data in (i) and (ii) the approximate mass 
 of the body is found by calculation to be equal to 12.2302 
 Rm. 
 
 The reciprocal of the sensitiveness, called the "sensi- 
 bility" represents the number of milligrams required to 
 shift the rest point one division. From the data given 
 in (ii) and (iii) the sensibility is 2/(14.87 - 8.54) = 0.316 
 mg. per scale division. 
 
14 METHOD OF VIBRATION [1 
 
 (i) Pans empty; 
 
 Left Right 
 
 0.7 15.2 
 
 1.3 14.5 
 
 2.1 13.8 
 
 13.2 
 
 3)4.1 4)56.7 
 
 1.37 14.18 
 
 Zero rest-point, 7.78 
 
 (ii) Body on left pan, 12 + 0.200 + 0.020 + 0.008 
 12.228 gm. on right pan; 
 
 Left Right 
 
 10.9 18.6 
 
 11.4 18.2 
 
 11.7 17.8 
 12.1 
 
 4)46.1 3)54.6 
 
 11.53 18.20 
 
 Rest-point, 14.87 
 
 (iii) Body on left pan, 12 + 0.200 + 0.020 + 0.010 
 12.230 gm. on right pan; 
 
 Left Right 
 
 6.3 11.0 
 
 6.5 10.7 
 
 6.7 10.4 
 
 10.2 
 
 3)19.5 4)42.3 
 
 6.50 10.58 
 
 Rest-point, 8.54 
 
1] DENSITY OF A SOLID 15 
 
 (a) Weigh the solid, by the method of vibrations, as 
 outlined above. Record its number and the number of 
 the set of weights. 
 
 (6) Repeat in detail. 
 
 (c) Read the barometer and the thermometer. 
 Measure the dimensions of the cylinder with a metric 
 scale and calculate its volume. 
 
 (d) From your data in (a) and (b) find the mean 
 value of the apparent mass of the solid. 
 
 The balance only serves to compare forces, and hence 
 indirectly masses. But the forces which you have bal- 
 anced against each other in (a) are not simply the weights 
 of the cylinder on one side and of the marked brass 
 weights on the other. The buoyancy of the air has an 
 appreciable influence. From the approximate volumes 
 of displaced air and the density of air, (see page 62), 
 calculate the bouyant forces and apply the correction 
 to the observed mass to get the true mass. 
 
 (e) Calculate the percent deviation of each determin- 
 ation of the mass from the mean value. 
 
 The balance does not even compare forces, if its arms 
 have not exactly equal lengths, but only force-moments. 
 The error introduced by the inequality of the arms may 
 be eliminated by re-weighing with cylinder and masses 
 interchanged. Show using moments that if this be done, 
 the true mass is given by 
 
 m = / m' m" 
 
 S 
 
 where m' and m" are the results of the two weighings. 
 (/) Calculate the density of the solid. 
 
16 DENSITY OF AIR [2 
 
 2. DENSITY OF AIR. 
 
 To find the density of the air at atmospheric pressure. 
 
 By the density of a body is meant the mass per unit 
 volume of the body, and it is usually found by measuring 
 the mass and the volume and then calculating their 
 quotient. In the case of air the density may be found by 
 an indirect method which does not require that the mass 
 of air filling a given volume shall be known. 
 
 Let a glass bulb of volume V be weighed full of air at 
 atmospheric pressure Pi, and let M be the mass necessary 
 to balance it. Then let some of the air be pumped out 
 until the pressure is P 2 , the mass as determined by weigh- 
 ing now being (M m), where m is the mass of air that 
 has been pumped out between the weighings. Then, if 
 di and d 2 be the densities corresponding to the pressures 
 Pi and P 2 , it follows, from the definition of density and 
 the interpretation of m, that 
 
 (1) Vd l Vd> = m. 
 
 The reciprocal of the density is the volume of unit 
 mass; hence, if Boyle's law is applied to unit mass of the 
 air, the temperature being assumed constant, we have by 
 equating the products of pressure by volume of unit mass 
 before and after the change in pressure, 
 
 (2) Pi/^-P./d,. 
 Eliminating d 2 from (1) and (2), we get 
 
 (3) d l 
 
2] DENSITY OF AIR 17 
 
 In the application of this formula it is essential that the 
 temperature should be the same during the two weighings. 
 (Why?) This condition is approximately satisfied in 
 practice. If the temperature be not the same, the observed 
 pressure in the second case will need to be corrected 
 (through the application of Charles' law), so as to give 
 the pressure that would have existed had the temperature 
 been the same as during the first weighing. The volume 
 V is obtained by weighing the bulb when empty and 
 then when full of water at a known temperature. 
 
 (a) Carefully dry the flask by exhausting it several 
 times and admitting air each time through a calcium 
 chloride drying-tube. Ask an assistant for instructions in 
 regard to manipulating the pump. If moisture is visible 
 inside the flask, it may be necessary to put in a little 
 alcohol, rinse the flask, vaporize the alcohol over a Bunsen 
 burner, and rinse with dry air as before. With the dried 
 flask in connection with the drying tube, admit air at 
 atmospheric pressure. Close the pinch-cock and care- 
 fully weigh the flask. Note the temperature. Read the 
 barometer for the pressure. 
 
 (6) Pump some of the air out until a moderately low 
 pressure is obtained, and weigh again at this reduced 
 pressure. Again note the temperature and record the 
 pressure. The pressure of the air in the partially ex- 
 hausted flask is equal to the difference between the height 
 of the barometer column and the height of the mercury 
 column in the manometer connected with the pump. 
 
 (c) Repeat (a) and (6) before continuing the exper- 
 iment. 
 
 Fill the flask completely with water up to the pinch- 
 cock, taking care to have no water above it. The temper- 
 
18 DENSITY OF AIR [2 
 
 ature of the water should be recorded. Dry the outside 
 of the flask and then weigh upon trip scales or a decigram 
 balance. 
 
 (d) Using the data obtained in (a), (b), and (c), find 
 the density of the air, in grams per cc., at the given tem- 
 perature and atmospheric pressure. From this result 
 the density of dry air under standard conditions (that 
 is, at C. and 76 cm. pressure) may be found through the 
 application of Boyle's and Charles' laws, or a combination 
 of the l!wo. If Pi, di, and 7\ represent the pressure, den- 
 sity and absolute temperature of a given kind of gas at 
 one time, and P 2 , d 2 , and T 2 represent the corresponding 
 values at another time, then it follows, from a combina- 
 tion of the two laws, that 
 
 (4) p l /d 1 r l = p 2 /j 2 r 2 
 
 for the given kind of gas to the degree of approximation 
 with which it observes the given laws. Making use of 
 this relation, calculate the density of dry air under stand- 
 ard conditions of temperature and pressure. Calculate 
 the probable error in your result. Point out the chief 
 sources of error. 
 
 (e) Why is it desirable in (b) to exhaust the flask until 
 the pressure is quite low? 
 
 In order that the method of this experiment may apply, 
 is it necessary, or not, to exhaust the flask completely? 
 To what form would the formula (3) be reduced in this 
 case? 
 
3] RELATIVE DENSITY OF CARBON DIOXIDE 19 
 
 3. RELATIVE DENSITY OF CARBON DIOXIDE. 
 
 The relative density of carbon dioxide compared with 
 air as a standard is to be measured. 
 
 The method employed here is that used in Exp. 2. 
 Using the same symbols as there used, and making the 
 weighings and noting the pressures as there indicated, 
 we have for the air, 
 
 (1) d l 
 
 If the measurements are then repeated for the carbon 
 dioxide, 
 
 (2) df =m f P i f /[V(P i f P 2 ')], 
 
 the symbols having the same meaning as in the case of air. 
 
 From (1) and (2), if D is the relative density of the 
 carbon dioxide, we get, by division, 
 
 (3) D = <*//</, 
 
 = [m'P/ (P, P,)] / [mPi (P/ P/)] ; 
 
 from which we see that a determination of the volume of 
 the flask is unnecessary. 
 
 (a) Read the directions given under Exp. 2. Ask an 
 assistant for instructions in the use of the pump. Care- 
 fully dry the flask, and fill it with dry air admitted through 
 the calcium -chloride tube. Using a sensitive balance, 
 weigh the flask full of air at atmospheric pressure, noting 
 the pressure and temperature. In weighing, follow the 
 method given in Exp. 1. 
 
 (b) Pump some of the air out until a low pressure is 
 obtained, and weigh the flask again at the reduced pres- 
 
20 THE VOLUMENOMETER [3-4 
 
 sure. If the temperature is not the same, within 0.5, 
 the observed pressure should be corrected as in Bxp. 2. 
 
 (r) Fill the flask with dry carbon dioxide at atmos- 
 pheric pressure. This can best be done by pumping out 
 the flask and admitting the gas from the generator several 
 times in succession. Take care not to allow any air to 
 pass through the acid into the generator; and keep the 
 stop-cock closed when not using the generator. When 
 the flask is filled with carbon dioxide at a known pressure 
 and temperature, weigh it as before. 
 
 (d) Pump some of the carbon dioxide out until a low 
 pressure is obtained, as in the case of the air, and weigh again. 
 
 (e) By the use of equation (3), calculate from your 
 data the relative density of carbon dioxide, with respect 
 to air, under the given conditions of temperature and 
 pressure prevailing in the room. 
 
 4. THE VOLUMENOMETER. 
 
 To find the density of an irregular solid by means of the 
 volumenometer and the balance. 
 
 In the volumenometer, A is a glass tube which may be 
 closed at the top by a ground glass plate. It corresponds 
 to the closed tube in the experiment on Boyle's law. As 
 in that experiment, the pressure and volume of air in A 
 are varied by raising or lowering a tube containing mer- 
 cury. The pressure is determined by noting the differ- 
 ence in the levels of the mercury columns and sub- 
 tracting it from the atmospheric pressure. The volume is 
 unknown. The volume of a portion of the tube between 
 the two points M and N, however, is known. 
 
4] 
 
 THE VOLUMENOMETER 
 
 21 
 
 Let the volume between M and N be k, and that above 
 M be V. By determining the pressures when the tube A 
 is full of air at atmospheric pressure and volume V 
 (mercury meniscus at M), and again when the tube con- 
 tains the same mass of air expanded to occupy the volume 
 V+k (mercury meniscus at N), an equation involving 
 Boyle's law may be written containing these two volumes 
 
 Fig, 2. The volumenometer is shown, filled with a certain mass 
 of air of volume V, and then with the same mass of air 
 expanded to occupy volume (V + k). 
 
 and the corresponding pressures. For instance, if Pi and 
 P 2 are the two pressures, we will have, if the temperature 
 is constant, 
 
 P, V = P 2 (V + k). 
 From this equation we get, by solution, 
 
 (1) V = P 2 k/(P l P z ). 
 
 The volume of the air-space in A above M thus becomes 
 known in terms of the two pressures and the volume k. 
 
 If now the solid whose density it is desired to find be 
 placed in A, the volume of the air-space in A above M 
 will be less than before by an amount equal to the volume 
 
22 THE VOLUMENOMETER [4 
 
 of the solid. Let the volume of the air-space now above 
 M be V . By proceeding as before, we get 
 
 (2) V = P 2 'k/(P>' P/), 
 
 where PI and P 2 ' are the two pressures in the enclosed 
 air corresponding to the volumes V and V'+k, the first 
 representing the atmospheric pressure. From (1) and 
 (2) the volume of the solid can be found. The density 
 of the solid can then be calculated from its volume and 
 mass. 
 
 (a) With the tube A uncovered bring the mercury 
 meniscus to M, recording the pressure, evidently just 
 equal to the atmospheric pressure and obtained by 
 reading the barometer. Carefully place the plate 
 on A, so as to insure an air-tight joint. The plate 
 must be clean and have on it only a little grease. By 
 lowering the open tube on the side L cause the mercury 
 meniscus in the tube A to drop from M to N, then note 
 the difference in mercury levels at TV and L and again 
 determine the pressure in the enclosed air. Test for 
 leakage by allowing the tube to remain a minute or more 
 in this position, and make sure that the mercury levels 
 do not change. Note the value of k recorded on the tag 
 attached to the apparatus. 
 
 (b) Remove the plate, place inside the volumenometer 
 one of the bodies whose density is to be determined, and 
 repeat (a). Weigh the body. 
 
 (c) Repeat for at least one other body. 
 
 (d) By applying Boyle's law find the volume of the 
 solids used and then calculate their densities. 
 
 What are the advantages and disadvantages of this 
 method of determining density? 
 
5] SURFACE TENSION BY JOLLY'S BALANCE 23 
 
 5. SURFACE TENSION BY JOLLY'S BALANCE. 
 
 To obtain a direct measure of the surface tension of a 
 liquid by balancing it against the tension in a stretched spring. 
 
 A wire rectangle is hung from the spring of a Jolly's 
 balance and allowed to dip in a soap solution which forms 
 a film across the rectangle. When equilibrium is estab- 
 lished, the force due to surface tension in the two surfaces 
 of the film must just balance the tension in the spring. 
 By knowing the force which will stretch the spring the same 
 amount, we have a measure of the total force due to sur- 
 face tension. If T is the value of the surface tension in 
 dynes per centimeter width of the film, / the width in 
 centimeters of the rectangle along the surface of the 
 liquid, and F the force in dynes exerted by the spring, 
 then 
 
 F - 2 1 r. 
 
 The force F in dynes is equal to 980 m, where m is the 
 mass in grams whose weight will stretch the spring the 
 given amount and 980 is approximately the number of 
 dynes of force which the earth exerts on 1 gm. Know- 
 ing m and /, the value of T can be calculated. 
 
 Bare wire, pincers, thread and a piece of emery cloth 
 are provided for the construction of the rectangles. The 
 greatest care must be taken to have the beaker and 
 rectangles clean. Do not touch with the fingers the inside 
 of the beaker, the liquid, or the part of the rectangle on 
 which the film is formed, for a slight trace of grease will 
 very greatly decrease the surface tension of water. 
 
 (a) Having first cleaned the pincers and the wire 
 with -emery cloth construct three rectangles about 2, 
 4 and 6 cm. wide respectively. Make the rectangles to 
 
24 SURFACE TENSION BY JOLLY'S BALANCE [5 
 
 approximate size and determine their exact width after 
 you have finished using them. These should have the 
 form of a staple with square, corners and legs from 3 to 5 
 cm. long. Suspend a rectangle, 2 cm. wide, from the 
 spring, and let it be partially immersed in a beaker of 
 soap-solution. Read the extension of the spring when 
 there is no film in the rectangle, and again with a film 
 across it. 
 
 Repeat until consistent results are obtained, and 
 average. 
 
 Repeat these measurements, using rectangles 4 cm. 
 and 6 cm. wide. 
 
 Determine by trial whether the force exerted by the 
 film depends on the length of the film, measured parallel 
 to the direction of the force. 
 
 (6) Calibrate the balance by observing the extension 
 produced by known weights. Use extensions equal to 
 or somewhat larger than those obtained with the film. 
 
 (c) Make a new rectangle, 4 cm. wide; clean the beaker 
 thoroughly; and repeat (a) with water fresh from the 
 tap. As a film of no appreciable height will form with 
 water, take the reading of the balance without the film 
 when the under side of the upper wire of the rectangle 
 is just above the surface of the water and not in contact 
 with it; and again, after immersing the upper wire of the 
 rectangle so as to wet it, take a reading when it breaks 
 away from the surface. 
 
 Repeat until consistent results are obtained, and 
 average. 
 
 (d) Repeat (c), using water at 50 C. or higher. 
 
 (e) Repeat (c), using alcohol. 
 
5-6] CAPILLARITY 25 
 
 (/") From the data taken in (a), state how the total 
 tension in the film varies with its width. Calculate the 
 surface tension, T, in dynes per cm., for the liquids used 
 in (a), (c), (d), and (<?). Present your results in tabular 
 form. 
 
 6. CAPILLARITY. RISE OF LIQUIDS IN TUBES. 
 
 To determine the surface tension of water and of alcohol 
 from the rise of these liquids in capillary tubes. 
 
 When the inner surface of a tube is wet by a liquid, the 
 surface tension of the latter may be considered as acting 
 upward at all points around the circumference of the 
 tube. The total vertical component of this force is 
 2wrT cos a, where r is the radius of the tube in cm., T the 
 surface tension in dynes per cm., and a is the angle of con- 
 tact between the liquid and the tube. If the tube is of 
 small bore, the liquid will rise inside the tube, equilibrium 
 being established when the weight of the column of liquid 
 within the tube (provided the bore is uniform) equals the 
 vertical component of the force due to surface tension. 
 If d is the density of the liquid in gm. per cc., h the height 
 in cm. of the liquid column above the general surface of 
 the liquid outside the tube, and g is the acceleration due 
 to gravity (approximately 980 cm. per sec 2 .), it follows that 
 
 TT r 2 h d g = 2 TT r T cos a. 
 
 From this equation the value of T, the surface tension 
 in dynes per cm., can be found. 
 
 (a) Capillary tubes of different sizes are provided. 
 These may be thermometer-tubes or larger glass tubing 
 drawn out to a fine bore. In either case every precaution 
 must be taken to have the tubes perfectly clean and free 
 from all traces of grease. They should be cleaned with 
 caustic potash solution, rinsed with tap water and then 
 
26 
 
 CAPILLARITY 
 
 [6 
 
 B 
 
 Fig. 3. Showing the capillary action between water and glass in 
 tubes of various shapes and sizes. 
 
 with the liquid to be experimented with (in this case 
 water). With a rubber band fasten the tubes side by 
 side to a glass scale, and place the scale and tubes verti- 
 cally in a small dish of distilled water. Lower the tubes 
 first to the bottom of the dish so as to wet the inside for 
 some distance above the point to which the water will 
 rise. Then clamp them with the ends below the surface, 
 and note on the scale the point to which the water rises 
 in each tube. To obtain the reading for the water-sur- 
 face in the dish, a wire hook is provided which should be 
 brought up so that the point is just even with the surface. 
 Then read the height of this point on the glass scale. 
 
 (b) Measure the inside diameter of the tube (if its 
 bore is uniform) with a micrometer microscope or by 
 means of a thread of mercury drawn into the tube. In 
 case the latter method is used, the length and mass of 
 
6-7] CAPILLARITY 27 
 
 the thread and the density of mercury are all the data 
 needed for calculating the diameter. If the tube is not 
 uniform in bore, it will be necessary to find the diam- 
 eter at the point to which the water rises; this can be 
 done by breaking the tube by scratching it with a file at the 
 desired point. 
 
 (c) In the same way find the surface tension of alcohol. 
 For alcohol and glass the angle of contact is practically 
 zero. 
 
 (d) Calculate the surface tension of water from the 
 data for each tube, and take the average. For pure water 
 and clean glass the angle of contact is practically zero. 
 Would the water or alcohol rise as high in the tubes if 
 the experiment were performed in a vacuum? Explain. 
 
 The height at which a liquid will be held by capillarity 
 in a tube depends upon the diameter of the tube only at 
 the upper end of the liquid column where the surface 
 film is attached to the tube, being independent of the 
 shape and size of the tube everywhere else. Explain. 
 
 7. VISCOSITY. FLOW OF LIQUIDS IN TUBES. 
 
 To determine the coefficient of -viscosity of a liquid. 
 
 When a liquid flows through a tube, its motion is uni- 
 form under the action of two opposing forces, that due 
 to the head, or pressure difference between the two ends, 
 urging it forward, and that of viscosity, opposing the 
 motion. The latter force is proportional to the rate at 
 which the liquid is forced through, to the length of the 
 
28 VISCOSITY [7 
 
 tube, and inversely proportional to the square of the 
 radius, (if the tube has a circular cross-section). 
 
 V I 
 
 F = x X constant, 
 t r* 
 
 If the tube is of uniform cross-section, the force urging 
 the liquid forward equals the pressure difference between 
 the two ends multiplied by this cross-section. We there- 
 fore have 
 
 F' = p TT r 2 . 
 
 If the liquid moves with unaccelerated motion, F = F f , 
 whence 
 
 ~ p TT r* t 
 
 Constant = 
 
 V I 
 
 In this expression, p is the pressure difference between the 
 ends of the tube, V/t is the volume flowing through per 
 second, / is the length of the tube, r its radius. If all 
 these quantities are measured in C. G. S. units (pressure 
 in dynes per cm. 2 ), the constant, when divided by 8, is 
 called the coefficient of viscosity of the liquid in C. G. S. 
 measure. The factor 8 arises in the fact that the coefficient 
 of viscosity was not originally defined in this way. 
 
 If the coefficient of viscosity is measured by determin- 
 ing experimentally all the other quantities occurring in 
 the equation and is found to be constant even though the 
 conditions of the experiment are varied, you will have 
 succeeded in verifying the statements made above. Cap- 
 illary glass tubes are used, bent in the form of a syphon. 
 
 (a) Fill one of the syphons with tap water and arrange 
 it so as to transfer the liquid from one beaker to another 
 at a lower level. The difference in level of the surfaces 
 
7] VISCOSITY 29 
 
 of water in the two beakers gives the required pressure- 
 difference, provided that the end of the tube is below the 
 surface in the lower beaker as well as in the upper one. 
 If there is an appreciable change in level in the beakers 
 during the experiment, take the mean value of the dif- 
 ference. Use a head of water of not over 3 cm. 
 
 Allow the syphon to run for an observed length of 
 time, and determine the volume of liquid which runs 
 through by weighing the beakers and contents at the 
 beginning and end of the run. Note the temperature of 
 the water. Weigh both beakers, and take the mean of 
 the two determinations, in case it is desirable to increase 
 the accuracy of the determination of V. In this way 
 make one determination with each of three tubes of 
 different length and cross-section. 
 
 Note. The tube must be very clean. Also the method 
 is only accurate if the fluid is run through very slowly. 
 
 (b) I may be measured with a paper scale. Secure 
 data for calculation of the radii of the tubes as follows. 
 Fill a part of the tube with clean mercury. Measure 
 the length of the thread in several different positions in 
 the tube, distributed over its whole length. Run the 
 mercury into a small beaker or a watchglass and weigh 
 carefully on a sensitive balance. 
 
 (c) Make a determination of the coefficient of viscosity 
 of tap water at about 60 C. 
 
 The effect of evaporation may be eliminated by aver- 
 aging loss in one beaker with gain in the other, provided 
 the water is at the same temperature in both beakers. 
 
 (d) Calculate the radii of the tubes from your data. 
 The density of mercury at 18 C. is 13.55. Calculate 
 
30 VISCOSITY [7-8 
 
 the coefficient of viscosity for the various cases under 
 test, expressing your results in C. G. S. Units and present- 
 ing them in tabular form. 
 
 (e) Calculate the probable error of your mean value 
 for the coefficient of viscosity for water at room tempera- 
 ture. 
 
 (/) State clearly the way in which the rate of flow of 
 a liquid depends on the length and diameter of the tube 
 and the character of the liquid. 
 
 Since the water in the upper beaker is at rest, accelera- 
 tion must be imparted to it in order to cause it to flow 
 through the siphon. Our equation is therefore only 
 approximately true. Will the observed value of the 
 constant be larger or smaller than the true value on 
 this account? 
 
 8. YOUNG'S MODULUS BY STRETCHING. 
 
 References. Duff, pp. 119, 121; Kimball, 159. 
 
 Hooke's Law states that in elastic bodies, within their 
 elastic limits, the strain or deformation produced is pro- 
 portional to the change in the stress or distorting force. 
 In particular it states that if different forces be applied to a 
 wire, e. g., by suspending it and hanging masses from it, 
 the amount of stretching will be (within certain limits) 
 proportional to the applied force. For a wire of any given 
 material the ratio of the change in stress per unit area of 
 cross-section to the increase in length per unit length is 
 known as Young's modulus. If f* is the additional force 
 applied to a wire of length L and cross-section a, and / is 
 the elongation produced, the value of the ratio is 
 
 / / P 
 
 1/1 - /. or 
 r I ~ a L al 
 
8] YOUNG'S MODULUS BY STRETCHING 31 
 
 It is approximately a constant for any given material, 
 thus verifying Hooke's law. The constant has widely 
 varying values, however, for different materials. 
 
 In determining the stretch / of the wire a spirit level, C , 
 carried in a hinged frame is used. The frame is sup- 
 ported by two wiresA^If one of these wires is stretched 
 the micrometer sere Wp. supporting one end of the level 
 must be turned up before the level will read as before. 
 The amount of stretch can thus be directly measured. 
 
 * . 
 
 (a) Hang first a large enough mass on the wire to 
 
 insure that it is straight at the beginning. Make a setting J 
 with the screw and read it to 0.01 mm. or less. Then / 
 increase the load by adding 500 gm. at a time (each time L 
 reading the screw), until the wire carries a load of about 
 3000 gm., or until the elastic limit is approached. 
 
 Take the masses off, 500 gm. at a time, reading the screw 
 each time. 
 
 (b) Repeat (a) at least once. Record the observa- 
 tions in tabular form. 
 
 (c) Measure the length of the wire between the clamps, 
 and also the diameter of the wire. In measuring the latter, 
 apply the caliper to four or five different points along the 
 wire. 
 
 (d) From the data in (a) and (6) calculate separately 
 for each series the mean elongation produced by a stretch- 
 ing force of 500 grams- weight, as follows: 
 
 Take the difference between each reading and every / 
 other reading of the series, making as many combina- ' 
 tions as possible. Divide the sum of these differences 
 by the sum of the number of 500 gm. weights which 
 produced them. For example, with an initial reading 
 
32 YOUNG'S MODULUS BY STRETCHING [8 
 
 (called 0) and four readings (called 1, 2, 3, 4) taken after 
 adding one, two, three, and four weights respectively 
 we have, 
 
 Reading Difference No. of Difference No. of 
 
 Weights Weights 
 
 3.62 - 1 = 0.45 1 1 - 3 = 0.90 2 
 
 1 4.07 - 2 = 0.92 2 1 - 4 = 1.35 3 
 
 2 4.54 0-3 =.1.35 3 2-3 = 0.43 1 
 
 3 4.97 - 4 = 1.80 4 2 - 4 = 0.88 2 
 
 4 5.42 1 - 2 = 0.47 1 3 - 4 = 0.48 1 
 
 Sum of differences = 9.03. Sum of weights = 20. 
 
 Average difference per 500 gm. weight = 0.452 mm. 
 
 , ^ 
 
 Expressing the force in dynes and the lengths in cm., | 
 calculate Young's modulus. ( ; 
 
 (e) Should the result be independent of the diameter 
 of the wire? 
 
 What evidence does the experiment give for the veri- 
 fication of Hooke's law? If there is any variation from 
 the law, assign a reason if you can. 
 
 The Optical Lever 
 
 In another form of apparatus the elongation / is deter- 
 mined using an optical lever and a telescope and scale. 
 The optical lever stands upon three pegs with spherical 
 ends, ABC, two of these, A, B, resting in a hole and 
 groove respectively, form the fulcrum of the lever, while 
 the third, C, rests on the flat horizontal surface of a chuck 
 H attached to the wire under test. The lever carries a 
 mirror-which is adjusted to lie in a plane through A B. 
 
 From the figure it is seen that the stretch / is given by 
 the relation / = r sin A. r may be determined from the 
 distances between the centers of the spheres A, B, C, 
 
8-9] 
 
 ALCOHOL AND WATER THERMOMETERS 
 
 33 
 
 T-0* 
 
 Fig. 4. 
 
 from the relation r = /a 2 -c 2 . Where a is the average of 
 the distances AC, BC and c half the distance AB. The 
 double angle 2a is determined from the radius\R (distance 
 from the mirror to the curved scale) and the length of arc 
 (the difference between successive readings^ in the tele- 
 scope) by means of the relation Angle (in radians) x radius 
 = length of arc. 
 
 If you used the optical lever, prove from the law of 
 reflection that a beam of light reflected by a mirror 
 turns through twice the angle through which the mirror 
 turns. 
 
 9. COMPARISON OF ALCOHOL AND WATER 
 THERMOMETERS. 
 
 In this experiment the relative expansions of water and 
 alcohol are to be studied, and the behavior of these liquids 
 when used in thermometers is to be observed. 
 
 (a) Two thermometer-bulbs are to be filled, one with 
 water and the other with ethyl alcohol, by the aid of the 
 reservoir-tube. The reservoir is fitted on the end of the 
 
34 ALCOHOL AND WATER THERMOMETERS [9 
 
 thermometer-stem, filled with the liquid and warmed. 
 (In the case of the water-thermometer, the water should 
 first be boiled to drive out the oxygen held in solution, 
 before filling the reservoir with it.) The liquid is then 
 introduced into the thermometer-bulb by alternately 
 heating the bulb to drive out the air and allowing it to 
 cool to admit the liquid. When all but a tiny bubble of 
 air has been removed, place the bulb in ice-water and 
 force the liquid to dissolve the air. If this does not suc- 
 ceed, ask for assistance. Take care not to ignite the 
 alcohol. The liquid in each thermometer should stand 
 1 or 2 cm. above the lower end of the stem when the bulb 
 is in melting ice; for this purpose it will be found neces- 
 sary to remove liquid from the stem by means of a fine 
 wire. 
 
 (6) Glue or otherwise fasten a strip of stiff paper along 
 the back of each stem, and use it as a scale. Then place 
 the thermometers in clamps with their bulbs in shaved 
 ice or in a mixture of water and ice. When^the readings 
 become steady, indicate the position of the meniscus of 
 each by a sharp line on the card. Mark the line zero. 
 This is the first fixed point of the thermometer. 
 
 (c) To determine a second fixed point, place the bulbs 
 in a beaker of methyl alcohol which is itself placed on a 
 support in a bath of water. Heat the water-bath slowly 
 until the methyl alcohol begins to boil. Be very careful 
 not to bring the alcohol itself to the flame, and avoid 
 inhaling the fumes. When the readings become steady, 
 again indicate the position of the meniscus on each stem 
 by a sharp line. Mark this point 66, which is the boiling 
 point of methyl alcohol on the centigrade scale. 
 
 (d) Lay the stems of the thermometers on a flat sur- 
 face, measure the distance on each between the two fixed 
 
9] ALCOHOL AND WATER THERMOMETERS 35 
 
 points, then by means of the customary geometrical 
 construction divide this distance into 66 equal parts and 
 call each part a degree. Put the marks for each degree 
 on the scale and number every tenth one. 
 
 (e) Place the two arbitrarily calibrated thermometers 
 in a water-bath at 0, as recorded by each thermometer. 
 Gradually raise the temperature of the water-hath and 
 note the readings of the two thermometers, at first at 
 short intervals, then at longer intervals, until the upper 
 fixed point is reached. Each time that the temperature is 
 raised, it should be kept at its new value for three or four 
 minutes, so as to give the bulbs time to assume the tem- 
 perature of the bath. 
 
 (/) Plot the series of points on co-ordinate paper, hav- 
 ing as abscissae the temperatures by the alcohol-ther- 
 mometer, and as ordinates the corresponding temperatures 
 by the water-thermometer. Draw a smooth curve through 
 the points. This curve gives the relation between the 
 temperatures as recorded by the two thermometers. 
 For the general method of plotting data, see the direc- 
 tions, page 41. 
 
 What inferences can you draw from the curve? If 
 alcohol be taken as the standard substance, what can 
 you say of the uniformity of the expansion of the water? 
 If water be assumed as the standard, what of the expan- 
 sion of the alcohol? Which would be the better sub- 
 stance to use in a practical thermometer, and why? 
 
36 EXPANSION OF LIQUID -REGNAULT'S METHOD [10 
 
 10. COEFFICIENT OF EXPANSION OF A LIQUID 
 BY REGNAULT'S ABSOLUTE METHOD. 
 
 To determine the coefficient of expansion of a sample of oil. 
 
 If the expansion coefficient of a liquid is determined by 
 measuring its volume at different temperatures, correc- 
 tion must be made for the expansion of the vessel used 
 in the volume measurements. The present method was 
 devised by Dulong and Petit for the purpose of eliminating 
 that correction. It was later improved and made practical 
 by Regnault. 
 
 The apparatus consists essentially of a metal U-tube. 
 One arm is surrounded by a steam-jacket, the other by 
 an ice-pack. Each arm terminates in a glass tube, so 
 that the heights of the liquid surfaces may be read off. 
 
 The oil in the hot side is less dense than that on the 
 cold side, and hence stands higher. Since the pressure at 
 the bottom of the two sides is the same, we have 
 
 h d g = h t d t g, (1) 
 
 where h is the total height of the cold column, d the 
 density of the cold oil, h t and d t , the values of the same 
 quantities on the hot side, and g is the number of dynes 
 in a gram. The last factor is inserted in order to give 
 the pressures in C. G. S. units. 
 
 Now if V is the volume of a given mass of oil at C., 
 and V t its volume at t C., we have 
 
 V, = V. (1 4- j8<), (2) 
 
 j8 being the cubical expansion coefficient of the oil. Since 
 the densities are inversely as the volumes, this gives 
 
 d I d t = 1 + # (3) 
 
10] EXPANSION OF LIQUID REGNAULTS METHOD 37 
 
 Combining this with the first equation above gives 
 
 (A, - *.) / * fc. - ft (4) 
 
 From this, /3 may readily be determined by observation 
 of h t , h , and t. 
 
 (a) With the two arms -of the manometer at the same 
 temperature, read the level of the oil in the glass tubes. 
 If the two levels are not the same, it may be due either 
 to the instrument not being level, or to the presence of 
 foreign material in the oil (air, water). Adjust until 
 the two levels show the same reading before proceeding. 
 
 (6) The manometer should contain oil enough so that 
 when the temperature difference is established, the level 
 on the cold side can be conveniently observed. The 
 column projecting above the ice-pack on one side and 
 the steam jacket on the other should be as short as 
 possible, as the formula is based on the assumption that 
 the whole of each arm is at one single temperature. 
 
 Pack the ice jacket full of shaved ice and keep it full. 
 Let the water run off below and keep the ice well packed 
 down. There is a tendency for the lower part of the 
 cold side to warm up, by conduction from the hot side. 
 This is partly prevented by the insertion of a non-con- 
 ducting segment in the cross-tube. But if the ice is not 
 packed down, it will melt out below and introduce error. 
 
 Pass steam through the steam jacket, from above 
 downward. The steam should escape freely at the lower 
 opening, and not all pass out at the small opening around 
 the glass tube. While heating, slide the cap up so as to 
 cover the oil column completely. 
 
 (c) In reading, slide the cap on the steam jacket 
 down below the meniscus far enough to read conveniently. 
 
38 EXPANSION OF A LIQUID. PYCNOMETER [10-11 
 
 Your first reading may be in error if time has not 
 been allowed for temperatures to become steady. A 
 point will be reached, however, beyond which further 
 variations are irregular, no uniform increase being 
 observed. Now let out a little oil by means of the stop 
 cock at the bottom of the U-tube and read both columns 
 again. Repeat this process at least twice. 
 
 (d) Calculate the value of ft. 
 
 11. COEFFICIENT OF EXPANSION OF A LIQUID 
 BY PYCNOMETER METHOD. 
 
 To determine the coefficient of cubical expansion of alcohol. 
 
 The method consists in finding the mass of alcohol 
 filling the pycnometer at each of several temperatures. 
 
 Let Mo represent the mass of alcohol which would 
 fill the pycnometer at C., 
 
 V , the volume of M , and hence of the pycnometer 
 at C., 
 
 m, the difference between the masses contained in the 
 pycnometer at C. and at t C., 
 
 ft, y, the coefficients of cubical expansion respectively 
 of alcohol and glass. 
 
 Now 
 
 Fo (1 + fti) = the volume of the mass M at t C, and 
 V (1 + yt) = the volume of the pycnometer at t C, and 
 
11] EXPANSION OF A LIQUID. PYCNOMETER 39 
 
 V (1 + fit) - V (1 + 7 i) = V (/3 - 7) t = the volume 
 of the mass m at t. If d and d t represent the densities 
 of liquid at and t, respectively, then 
 
 (1) d = Mo/Fo, 
 
 (2) d t = m/V (0 - y)t, 
 
 (3) d = d t (l + ftt) (See Exp. 10). 
 
 By eliminating d and d t from equations (1), (2), and (3), 
 we get 
 
 (4) Mo (0-7)* = w(l + 0*). 
 
 Besides containing the known masses, M Q and m, this 
 equation contains the three quantities, 0, 7, and t. Any 
 two of these three quantities being known, the third will 
 be given by the equation. 
 
 Solving this equation for we have 
 
 = m I (M - m) t + Mo 7 / (M - m). 
 
 (a) Fill the pycnometer with alcohol and set it on a 
 platform in a kettle of water, so that the water comes well 
 up to the neck of the pycnometer. Hang a 50 thermome- 
 ter in the bath alongside the pycnometer, and keep the 
 bath well stirred for about five minutes. The tempera- 
 ture of the bath, which must be a little above that of the 
 room, should remain constant within 0.1 during this 
 time, and at the end of it the alcohol will have the same 
 temperature within 0.1. Take the pycnometer out of 
 the bath, wipe the outside dry, and weigh to an ac- 
 curacy of 1 mg. 
 
 (b) In this way take at least four readings, at tem- 
 peratures ranging up to 50 C. Keep the temperature 
 steady in each case for five or ten minutes by holding the 
 flame under the kettle for a few seconds occasionally. 
 
40 EXPANSION OF A LIQUID. PYCNOMETER [11 
 
 Before reheating after each weighing, refill the pycnometer. 
 This will prevent the formation of a bubble under the 
 stopper. 
 
 (c) Empty the alcohol into the bottle from which it 
 was taken, dry the pycnometer, and weigh. 
 
 (d) Determine the mass of the alcohol filling the 
 pycnometer at each temperature. Plot the results, with 
 temperatures, starting from 0, as abscissae, and masses 
 as ordinates. Assuming that the expansion is uniform, 
 draw the straight line which best represents the plotted 
 points, and by extending it find the mass filling the pyc- 
 nometer at 0C, and at 50C, and then calculate the 
 coefficient of expansion of alcohol. 
 
 (e) Error in this experiment may be estimated by 
 carrying out the calculation of /3 separately for each pair 
 of values of m and t, and comparing the results. 
 
PLOTTING OF CURVES 41 
 
 PLOTTING OF CURVES. 
 
 It is very frequently advantageous to represent the 
 way in which one quantity depends on another by means 
 of a plot, or graph. The values of the quantity regarded 
 as independently variable are laid off on the horizontal 
 axis, and are called abscissa. The corresponding values 
 of the dependent variable are called ordinates, and are 
 laid off on the vertical axis. 
 
 Choose the scale of abscissas so that the largest value 
 reaches nearly across the page. Choose the scale of 
 ordinates so that the largest value reaches nearly the 
 vertical distance assigned to the curve. Always use 
 scales such that each space of the co-ordinate paper 
 represents a convenient number of units, or a convenient 
 fractional part of a unit. The two scales need not be alike. 
 
 As a rule, take for origin of co-ordinates the point in 
 the lower left-hand corner of the sheet of plotting paper, 
 and for co-ordinate axes the two perpendicular lines 
 which intersect at this point. Along these axes indicate 
 the scales used. 
 
 Each pair of values plotted should be plainly indicated 
 by a point at the center of a small circle, or by a cross 
 intersecting at the point plotted. 
 
 After the values are plotted, draw the smooth curve 
 that best fits all the points plotted. This curve will not, 
 in general, pass through all the points. Deviation of 
 points from this curve usually indicates errors of obser- 
 vation. 
 
 Give each curve a title. 
 
 All curves should be drawn on the best 18 x 25 mm. 
 co-ordinate paper. Both curves and lettering should be 
 done in ink. 
 
42 EXPANSION OF GLASS. WEIGHT THERMOMETER [12 
 
 12. COEFFICIENT OF EXPANSION OF GLASS 
 BY WEIGHT-THERMOMETER METHOD. 
 
 To determine the coefficient of cubical expansion of glass 
 by means of the weight-thermometer. 
 
 The weight-thermometer consists of a glass tube closed 
 at one end and ending in a curved capillary at the other 
 end. It is filled with mercury at 0C., and the mass of 
 the mercury measured. When later placed in a bath of 
 higher temperature, some mercury overflows, since mer- 
 cury expands more rapidly when heated than does glass. 
 The mass of this overflow is measured. 
 
 Equation 4, as developed in the previous experiment, 
 applies here also, but M and m are measured directly. 
 13 and t are to be observed and y calculated. 
 
 (a) Weigh the empty weight-thermometer to an 
 accuracy of 10 mg. Then fill it with mercury. In doing 
 so it should be held by a clamp, or suspended in a gauze 
 jacket, and heated by a flame held in the hand, care being 
 taken to keep from heating too rapidly and from apply- 
 ing the flame too long at any point of the empty bulb. 
 
 The end of the capillary dips under the surface of mer- 
 cury in a porcelain dish. The mercury in this dish should 
 first be heated, and then the weight-thermometer gently 
 heated until the air bubbles out through the mercury. 
 On allowing the bulb to cool, some mercury will run into 
 it. The process is then repeated. When considerable 
 mercury is in the bulb, heat it until it boils vigorously, 
 but be careful not to heat too hot that portion of the glass 
 where there is no mercury. Keep the mercury hot in the 
 porcelain dish, otherwise the glass is apt to crack when the 
 cooler mercury rushes in. The tube must be completely 
 filled with mercury to the end of the capillary, the last 
 
12] EXPANSION OF GLASS. WEIGHT THERMOMETER 43 
 
 bubble of air being expelled. To accomplish this it will 
 be found helpful to turn the weight-thermometer so as to 
 give the capillary above the air-bubble an upward slant. 
 A gentle tapping with a light splinter or pencil will then 
 probably cause the bubble to work its way along the tube 
 far enough to be easily expelled by further heating. 
 
 (b) Keeping the end of the capillary in the dish of 
 mercury, allow the weight-thermometer to cool in the 
 air sufficiently so that you can bear your hand on it, and 
 then surround it with shaved ice and leave it long enough 
 to contract as much as it will. Assume that its tempera- 
 ture is now 0C. Carefully remove the dish and brush 
 the mercury off the end of the capillary. Place a watch- 
 glass under the end to catch the mercury as it begins to 
 expand and flow out. Now remove the ice-bath and 
 warm the bulb with the hand until its temperature is 
 raised to the temperature of the room. 
 
 (c) Place the weight-thermometer in the boiler pro- 
 vided. Heat it to the boiling point of water by passing 
 steam over it until no more mercury comes out. Read 
 the barometer and calculate the temperature of the 
 steam. Very carefully weigh the mercury in the watch- 
 glass to an accuracy of 1 mg. Weigh the weight-ther- 
 mometer and contained mercury to an accuracy of 10 mg. 
 
 (d) From the data obtained in (a) and (c) determine 
 M and m Using your values of M and m, and taking 
 the coefficient of expansion of mercury as found in the 
 Tables, calculate the coefficient of cubical expansion of 
 glass. 
 
 What additional measurements would you need to 
 have made in order to measure the room temperature 
 with your weight- thermometer? 
 
44 CALORIMETRY [12 
 
 GALORIMETRY. 
 
 Calorimetry is that branch of physical measurements 
 which has to do with quantities of heat. In such changes 
 as temperature-rise, fusion, vaporization, combustion, 
 and solution, quantities of heat are either absorbed, or 
 evolved, or transferred from one set of substances to 
 another. The unit employed in their measurement is 
 the calorie. 
 
 Measure of heat. In the case of the rise in tem- 
 perature of a body, the quantity of heat which passes 
 into the body is measured by the product ms(t 2 ti), 
 where m is the mass, 5 the specific heat, and (t* t^ the 
 rise in temperature. This is not true if the body changes 
 state during the heating. In such a case the quantity of 
 heat absorbed during the change of state alone is meas- 
 ured by the product m L, where m is the mass and L the 
 heat absorbed per gram. L is called the heat of the change 
 of state (e. g., fusion, vaporization). 
 
 In every experiment performed with a calorimeter 
 the quantity desired is obtained by solving an equation 
 which expresses the fact that the total amount of heat 
 lost by those bodies which lose heat equals the total 
 amount of heat received by those bodies which receive 
 heat. 
 
 The Calorimeter. The vessel in which the transfer 
 of heat takes place, in experiments involving the meas- 
 urement of heat, is called a calorimeter. It consists of 
 an inner chamber, insulated as completely as possible, 
 so as to prevent the transfer of heat either to or from the 
 material in the chamber. Transfers of heat by conduction 
 are eliminated by dead air or vacuum jackets. Radiation 
 is reduced by polished surfaces. In the Dewar flask, 
 and the thermos bottle, convection is also eliminated. 
 
12] CALORIMETRY 45 
 
 In precision work with a calorimeter, allowance must 
 be made for the heat absorbed by the calorimeter itself, 
 with its accessories, thermometer, stirrer, etc. It is 
 convenient to calculate this correction in terms of water- 
 equivalent. A body's water-equivalent is the mass of 
 water whose heat capacity is equal to that of the body. 
 It may be found either by multiplying the body's mass 
 by its specific heat, or by experiment. 
 
 The water equivalent of a thermos bottle or Dewar 
 flask may be determined as follows: Weigh the thermos 
 bottle when empty, add about 25 grams of hot water; 
 determining the exact amount by re-weighing after the 
 water has been put into the bottle. Insert the stopper 
 with the thermometer and move the bottle about so 
 that the water will come into contact with all parts of 
 the inside. In particular be sure to invert the bottle 
 and move it about for some time in this position so that 
 all parts will come to a uniform temperature. Do not 
 shake the bottle. 
 
 Next pour in from 40 to 50 grams of water at a known 
 temperature near that of the room. Determining the 
 exact amount by weighing after pouring it in. Move the 
 flask about as before and note the constant temperature 
 finally reached. Equating the heat lost by the bottle, 
 hot water, and thermometer to that gained by the cold 
 water will give the water equivalent of the bottle and 
 thermometer. 
 
 If a heating coil is to be used with the thermos bottle 
 its water equivalent may be included in the determination 
 by keeping it in the bottle throughout the preceedure 
 just described. 
 
 In the case of a thermometer, which is part glass and 
 part mercury, the water-equivalent may be determined 
 
46 SPECIFIC HEAT. METHOD OF HEATING [12-13 
 
 by finding the volume of the immersed part of the ther- 
 mometer and then calculating the water-equivalent of 
 this volume of mercury. This is possible since equal 
 volumes of glass and mercury have practically the same 
 heat capacity, and hence the thermometer may be treated 
 as though it were made entirely of mercury. The student 
 will find that 0.45 is approximately the factor by which 
 the volume in cc. of the immersed part should be multi- 
 plied to give the water-equivalent of the thermometer. 
 
 13. SPECIFIC HEAT OF A LIQUID BY METHOD 
 OF HEATING. 
 
 To find the specific heat of a given liquid. 
 
 In this experiment a heating coil, composed of high 
 resistance metal through which an electric current is 
 passed, is immersed for a given time, first in water and 
 then in the liquid whose specific heat it is desired to find. 
 If -the same current passes through the coil in the two 
 cases, equal quantities of heat should be generated in 
 equal times. Noting in each case the mass of the liquid 
 and the rise in temperature, the two quantities of heat 
 may be equated and the specific heat of the liquid calcu- 
 lated, if that of water is known. 
 
 (a) Place the bottle, containing the second liquid, in 
 a vessel of ice-water to cool for part (b). Put enough 
 water (chilled to 6 or 7 below room temperature) to 
 cover the heating coil to a depth of about 2 cm., deter- 
 mining the exact amount by weighing the bottle both 
 before and after the water is poured in. Move the 
 bottle about so that the water comes in contact with 
 all parts of the inside holding it inverted part of the 
 time. Note the temperature after it becomes constant. 
 Turn on the current at a noted time and let it run for 
 
13] SPECIFIC HEAT. METHOD OF HEATING 47 
 
 exactly two minutes. Equalize the temperature as 
 before and record its value. Repeat for a series of two 
 minute intervals until the temperature of the water is 
 as far above room temperature as it started below it. 
 
 (b) Repeat (a), using the second liquid, instead of 
 water, in the calorimeter cup. It should be chilled to a 
 lower temperature than the water, as it will warm up 
 faster than the water did. Heat for approximately 
 the same length of time as in (a). Use a considerably 
 larger quantity of the second liquid than you did of the 
 water. It will be necessary to find the water-equivalent 
 of the calorimeter cup, stirrer, and thermometer. For 
 this purpose see the paragraph on Water-Equivalent on 
 page 45. 
 
 (c) Repeat (a) and (b) in 'detail. 
 
 (d) Plot on one sheet of co-ordinate paper the results 
 of (a) and (b), using temperatures as ordinates and times 
 as abscissae. Erect two perpendiculars to the time axis 
 which will include between them as wide segments of the 
 two curves as is consistent with accuracy. From these 
 intersections obtain the range of temperatures passed 
 through by the water and the other liquid in equal times. 
 
 Write the equation expressing the fact that the same 
 amount of heat is absorbed in the two cases. 
 
 Plot (c) also. The two plots may be drawn on the 
 two halves of the same sheet of co-ordinate paper. From 
 the data obtained calculate the value of the specific heat. 
 
 (e) By comparing the two determinations, estimate 
 the probable error of your result. 
 
 Explain why no correction is necessary for the amount 
 of heat lost by radiation. 
 
48 SPECIFIC HEAT. METHOD OF COOLING [14 
 
 14. SPECIFIC HEAT OF A LIQUID BY METHOD 
 OF COOLING. 
 
 To find the specific heat of a given liquid. 
 
 The method consists in a comparison of the quantities 
 of heat lost by two liquids, one of which is water, when 
 equal volumes are allowed to cool under exactly the same 
 conditions through the same range of temperature. The 
 conditions of radiation being the same for both, if water 
 of mass mi and specific heat Si cools through a certain 
 temperature-range A in TI seconds, and a second liquid of 
 mass m z and specific heat s 2 requires T 2 seconds for the 
 same temperature-change, then the quantities of heat lost 
 will be proportional to the times, i. e., Qi/Qz = Ti/Tz. 
 For the similarity of conditions make the rate of losing 
 heat the same in the two cases. If w denote the water- 
 equivalent of the containing vessel, thermometer, and 
 stirrer, the above relation becomes 
 
 (mi Si + w Si) A J/(m 2 s 2 + w Si) A t = 
 from which the unknown specific heat can be determined. 
 
 (a) Set up the calorimeter, with the inner cup sup- 
 ported from the screws on the lower side of the wooden 
 cover. The small nickeled brass bottle is to be supported 
 by the thermometer, which fits tightly in the cork of the 
 bottle. A second cork on the thermometer fits trie hole 
 in the wooden cover, and carries the weight of the bottle 
 and contents. The bottle is thus suspended in air, and 
 cools primarily by radiation. The space outside the cup 
 should be filled with water. /vs# 
 
 Fill the bottle, heat nearly to 100C., in a water-bath, 
 handling the bottle by means of the thermometer. Trans- 
 
14] 
 
 SPECIFIC HEAT. METHOD OF COOLING 
 
 49 
 
 fer to the cooling chamber, and read the temperature at 
 one minute intervals until it has fallen through a sufficient 
 range to give a smooth curve covering an interval of 
 20 to 30C. A smooth curve can only > be " obtained if 
 the liquid is continuously and uniformly stirred. 
 
 Make the necessary weighings. The water-equivalent 
 may be found as explained under Calorimetry. 
 
 (b) Repeat, using the other liquid. The water in the 
 jacket should have the same initial temperature as in 
 the first case. 
 
 10 20 JO 40 50 
 
 Fig. 5. Cooling curves of water and turpentine. 
 
 (c) Plot the cooling curves of oil and water on co- 
 ordinate paper, using temperatures as ordinates and 
 corresponding times as abscissae. (See Fig. 5.) Since 
 the temperature range must be the same in the two cases, 
 draw two lines parallel to the axis of abscissae, marking 
 
50 HEAT OF FUSION BY METHOD OF MIXTURES [14-15 
 
 off equal temperature intervals on the two curves. Make 
 this temperature interval as long as possible, consistant 
 with accuracy. The intersections of these lines with the 
 curves will give >the times required. Calculate the specific 
 heat of the oil. 
 
 If the water had not been changed between the two 
 sets of observations, in what way would the value for the 
 specific heat of the oil have been affected? What source 
 of error still remains even if the water in the jacket is 
 changed before the second measurement? Suggest a 
 way to avoid this uncertainty. 
 
 15. HEAT OF FUSION BY METHOD OF 
 MIXTURES. 
 
 To determine the heat of fusion of Wood's Alloy by the 
 method of mixtures. 
 
 The melted alloy, with container, is plunged into water 
 in a calorimeter cup. The following changes occur 
 wherein heat is given out: (1) The alloy cools as a liquid 
 from the temperature of the hot water-bath down to the 
 freezing point of the alloy, (2) the alloy changes from a 
 liquid to a solid without change of temperature, (3) the 
 alloy cools as a solid from its freezing point down to the 
 final temperature of the mixture in the calorimeter, (4) 
 meanwhile the nickle crucible and the copper cage cool 
 from the temperature of the hot water-bath to the final 
 temperature of the mixture. The changes wherein heat is 
 absorbed are those accompanying the rise in temperature 
 of the nickeled brass calorimeter cup and contents form 
 the initial temperature of the cold water up to the final 
 temperature of the mixture. From these data, if the 
 
15] HEAT OF FUSION. WOOD'S ALLOY 51 
 
 specific heat of the metal in the solid and in the liquid 
 state be known, the heat of fusion may be found. 
 
 (a) Make the necessary weighings. Determine the 
 melting point by melting the alloy and noting its freezing 
 point on cooling. The heating is conveniently done in 
 a water bath. The thermometer must be placed in the 
 water, and not in the alloy, as the liquid alloy wets the 
 glass. 
 
 Set up the calorimeter, using the sensitive thermometer. 
 The procedure outlined under (d) gives the desired 
 result with accuracy only in case the initial temperature 
 of the water is the same as that of the room. 
 
 (b) Bring the water to the boiling point, taking care 
 that no water gets inside the crucible containing the 
 alloy. Remove the crucible, and quickly lower it into 
 the calorimeter, right side up. Note the time when the 
 crucible is immersed in the cold water. Then, at the end 
 of every 5 minutes for 15 minutes, read the temperature 
 of the mixture, keeping it well stirred meanwhile. 
 
 Do not leave the gas burning near the calorimeter. 
 
 (c) Repeat (a) and (b) to check your work. 
 
 (d) Let M be the mass of the alloy; m, the mass of 
 the nickle crucible; W, the mass of the water in the 
 calorimeter cup plus the water-equivalent of the cup, 
 thermometer, and stirrer; si, the specific heat of the 
 liquid alloy; s 2 , that of the solid alloy; s 3 , that of the 
 nickel crucible; s 4 , that of the copper cage; T, the melt- 
 ing point of the alloy; ti, the initial temperature of the 
 alloy and crucible; to, the initial temperature of the 
 calorimeter and contents; /, the final temperature; and 
 L, the heat of fusion per gram of the alloy. Write the 
 
52 
 
 HEAT OF FUSION. WOOD'S ALLOY 
 
 [15 
 
 proper equation representing the transfer of heat in the 
 above process, using the symbols indicated, and solve 
 the equation for L. Take the value of the specific heat 
 of substances involved from the table on page 65. 
 
 From each set of data calculate the heat of fusion 
 of the alloy, and take the average. In order to obtain 
 for each set the value of t, the temperature of the mixture 
 after the immersion of the alloy, proceed as follows: 
 Plot the times as abscissae and the temperature as or- 
 dinates, and draw the straight line which most nearly 
 represents the plotted points. This straight line gives the 
 rate of cooling of the mixture, and will, when prolonged 
 backward to intersection with the axis of ordinates, give 
 
 20* 
 
 i 
 
 S 10 IS 
 
 Fig. 6. Showing the method of finding the radiation correction. 
 
15-16] FUEL VALUE OF ALCOHOL 53 
 
 the temperature which the mixture would have had if 
 its temperature had been made uniform the instant that 
 the alloy entered it. 
 
 (e) Calculate the probable error of your result. 
 Point out the principal sources of error in the experiment. 
 Examine the equation set up in (d) and state why only 
 a rough determination of the melting point of the alloy 
 is necessary? 
 
 16. FUEL VALUE OF ALCOHOL. 
 
 To find the amount of heat generated by the combustion 
 of unit mass of the sample furnished. 
 
 The Junker calorimeter is used (Fig. 7). This consists 
 of a double-walled copper vessel provided with an inner 
 cavity C for running water and with a number of tubes 
 or flues FF for the heated gases from the flame. The 
 heat given up by the gases during their passage through 
 these tubes passes by conduction into the stream of 
 water which flows continuously through the calorimeter 
 from the inlet / to the outlet O. A thermometer T 2 
 placed at I registers the temperature of the water entering 
 the calorimeter and a second thermometer T\ at registers 
 the temperature of the water leaving it. A steady flow 
 of water is maintained by means of an overflow reservoir 
 placed above the level 6f the calorimeter and kept filled 
 with faucet-water. The warm water, as it issues from 
 the outlet O, is caught in a beaker and weighed. 
 
 The heat gained by the water is measured by the pro- 
 duct m s t, where ra is the mass of water caught in a given 
 time, s its specific heat, and t the rise in temperature be- 
 tween I and O. This is only an approximate measure of 
 
54 
 
 FUEL VALUE OF ALCOHOL 
 
 [16 
 
 the heat liberated by the combustion of the gas since 
 some heat is lost by conduction and radiation while the 
 water is flowing through the calorimeter. 
 
 (a) Arrange the calorimeter as shown in the figure. 
 Place the overflow reservoir in position so that, when 
 connected with the orifice I, a steady flow of water through 
 the calorimeter will be established. For this purpose it 
 will be found necessary to keep the level of the water 
 constant. This can be done by adjusting the flow of 
 
 O 
 
 Fig. 7. Junker's Calorimeter. 
 
 water from the faucet to the reservoir in such a way as 
 to keep the water-level constant. The rate of flow is 
 conveniently controlled by a pinch-cock. 
 
 Weigh the alcohol lamp, after filling it, light it at a 
 noted time and place it in position. 
 
 When the conditions have become steady, as shown by 
 the constancy of the temperatures registered by the 
 
16-17] PRINCIPLE OF MOMENTS 55 
 
 thermometers Ti and T 2 , place a beaker in position to 
 catch the water as it flows out at 0, and allow the flow to 
 continue for a measured length of time until 200 gm. or 
 more of water have passed. Record the time and the 
 temperature at I and 0. 
 
 Extinguish the lamp at a noted time and weigh. 
 
 (6) Make two more determinations with the rate of 
 flow of water different each time. 
 
 (c) From the data in (a) and (b) calculate the heat 
 liberated per gram of alcohol, assuming that the rate of 
 consumption of alcohol is uniform. 
 
 (d) Calculate the probable error of your result. To 
 what sources of error is the method of this experiment 
 subject? 
 
 To what sources of error is the method of this experi- 
 ment subject? 
 
 The calorimeter may also be used to determine the 
 fuel value of illuminating gas. In this case, in place of 
 the weight of alcohol consumed, the quantities to be 
 measured are volume of gas, corresponding time, baro- 
 metric pressure and gas pressure. 
 
 17. THE PRINCIPLE OF MOMENTS: STATICS. 
 
 To verify the proposition that the sum of the moments 
 about any axis, of forces in equilibrium^ is zero. 
 
 When forces act on an extended body, they are not 
 necessarily concurrent. The condition of translational 
 equilibrium, however, is just the same as though they 
 were. On the other hand, in order that a body be in 
 
56 THE PRINCIPLE OF MOMENTS [17 
 
 rotational equilibrium, the forces must also have zero, 
 net, rotating effect. The rotating effect, or moment of 
 a force, is measured by the product of the force by the 
 normal distance to the axis about which rotation is 
 considered. Since a body in equilibrium is in rotation 
 about no axis, the forces must have zero rotating effect 
 about any axis. As you will see in this experiment, 
 when forces have zero moment about one axis, their 
 moment about any other axis is also zero. 
 
 The apparatus used to test the principle consists of a 
 circular table with a movable disk resting on bicycle balls. 
 The disk may be pivoted in the center if desired. To pegs, 
 placed at will in the disk, cords are attached which pass 
 over pulleys clamped at different points around the cir- 
 cular table. From the ends of the cords are suspended 
 known masses whose weight produces the forces required. 
 
 (a) Place the disk on four bicycle balls, widely sepa- 
 rated, and level up the table so that the disk will not 
 tend to move in any one direction in preference to another. 
 Pivot the disk in the center and place a sheet of manila 
 paper upon it. Attach cords to it at three different 
 points chosen at random; and, placing the pulleys at any 
 convenient points, add masses until two of the forces 
 have large values. Adjust the third force, both as to 
 magnitude and direction, until no motion results on 
 removing the peg. See that the disk is free to move 
 on the bicycle balls and that the cords all lie in a plane 
 close to and parallel to the top of the disk; then mark 
 points or lines on the paper to indicate the directions of 
 the forces. Note the magnitude of each force, including 
 the weight of the hanger. 
 
 (b) Repeat (a), using four forces instead of three. 
 
17] THE PRINCIPLE OF MOMENTS 57 
 
 (c) On the papers used, trace the lines of direction 
 of the forces, and make the necessary measurements to 
 determine their moments about a point chosen at random. 
 Find the sum of these moments, taking those as positive 
 which tend to produce a counter-clockwise rotation about 
 the given axis, and those as negative which tend to 
 produce a clockwise rotation. Select in turn as centers 
 of moments three points as widely separated as possible. 
 Find the sum of the moments for each of these centers 
 and do this for both (a) and (b). 
 
 Put your work in neat tabular form. Give in each 
 case the percent deviation from the mean between the 
 sum of the positive moments and the sum of the negative 
 moments. 
 
 (d) Prolong two of the lines in (a) until they inter- 
 sect. At what distance from the intersection is the 
 third line at its nearest point? Assuming the principle 
 of moments to be true, give a reason why three forces 
 in equilibrium must be concurrent. Does the same 
 reason apply to four forces? Do the four forces in (6) 
 intersect in a single point? 
 
 A ladder leaning against a smooth vertical wall is 
 prevented from sliding by the reaction of the ground. 
 Assuming that the force which the wall exerts against 
 the ladder is horizontal, find, by construction, the force 
 exerted on the ladder by the ground. 
 
58 THE PARALLELOGRAM LAW [18 
 
 18. THE PARALLELOGRAM LAW: STATICS 
 
 To verify the parallelogram law by means of the force- 
 table. 
 
 The force-table is a device for applying known forces 
 at a common point so as to make known angles with each 
 other. When the forces are in equilibrium, the point on 
 which they act will not change position when released. 
 
 The law will be verified in two different ways, for two 
 different force-groups. 
 
 (1) In the graphical method, the forces are repre- 
 sented by lines. Every force is characterized by a mag- 
 nitude and a direction. The line representing the force 
 may exhibit the magnitude of the force by its length, 
 and the direction by its direction. It should be drawn 
 with a dot at one end and an arrowhead at the other. 
 A diagram in which each force is represented by such 
 a line is called a force-diagram. Unknown forces may 
 be determined by direct measurement on such a diagram. 
 
 (2) In the method of components, use is made of the 
 proposition that when a point is in equilibrium, the sum 
 of the components of forces acting on it taken parallel 
 to any direction is zero. 
 
 (a) Set up the force-table, and adjust three large 
 forces, not equal, and not forming right-angles with 
 each other. Two of the forces should be set with the 
 pin in place, and the third adjusted, both as to direction 
 and magnitude, so as to secure equilibrium when the 
 pin is removed. Record the circular scale readings and 
 the forces. Include the weight of the hangers with that 
 of the weights. 
 
18] THE PARALLELOGRAM LAW 59 
 
 (b) Repeat (a), using four forces instead of three. 
 
 (c) Draw the force- diagram for each of the two cases 
 accurately to scale, with the protractor and dividers. 
 In the first case, construct a parallelogram on any two 
 of the forces, and determine their equilibrant. Record 
 the deviation between this and the third force, both as 
 to magnitude and direction. In the second case, lay off 
 three of the forces in succession to form a force-polygon, 
 and compare the vector required to close the polygon 
 with the fourth force. 
 
 Also determine, in each of the two cases, the component 
 of each of the forces parallel to some line not coinciding 
 in direction with any of the forces. No diagram is 
 necessary, but you may find it convenient. The com- 
 ponent is calculated simply by multiplying the magnitude 
 of the force by the cosine of the angle through which it 
 is projected. To estimate error, express the percent 
 difference from the mean .of the algebraic sums of the 
 components. 
 
 Tabulate all results neatly. 
 
 Solve the following problem: Three men stand in a 
 row on top of a wall and pull a heavy load up the wall 
 by means of ropes attached to the load. The angles 
 at which the ropes act on the load are +22, -12, and 
 -25, measured from the vertical. Each man pulls with 
 the same force. What fraction of the load does each 
 man lift? Calling the load W and the force each man 
 exerts F, write the equation for vertical components. 
 Solve this equation for F, and then find the vertical 
 component of the force exerted by each man. Check 
 your solution by adding vertical components to see if 
 they equal the total load. 
 
60 PHYSICAL TABLES 
 
 USEFUL NUMERICAL RELATIONS. 
 Mensuration. 
 
 Circle: circumference = 27rr; area = 7rr z . 
 Sphere: area = 47Tr 2 ; volume = ^TTr 3 . 
 Cylinder: volume = 7T> 2 /. 
 
 Angle. 
 
 1 radian = 57.2958 = 3437'.75. 
 1 degree = 0.017453 radian. 
 
 Length. 
 
 1 centimeter (cm.) = 0.3937 in. 1 inch (in.) = 2.540 cm. 
 
 1 meter (m.) = 3.281 ft. 1 foot (ft.) = 0.3048 m. 
 
 1 kilometer (km.) = 0.6214 mi. 1 mile (mi.) = 1.609 km. 
 
 1 micron (M) = 0.001 mm. 1 mil = 0.001 in. 
 
 Area. 
 
 1 sq, cm. =* 0.1550 sq. in. 1 sq. in. = 6.451 sq. cm. 
 
 1 sq. m. - 10.674 sq. ft. 1 sq. ft. = 0.09290 sq. m. 
 
 Volume. 
 
 1 cc. = 0.06103 cu. in. 1 cu. in = 16.386 cc. 
 
 1 cu. in. = 35.317 cu. ft. 1 cu. ft. = 0.02832 cu. m. 
 
 1 liter (1000 cc.) = 1.7608 pints. 1 quart = 1.1359 liters. 
 
 Mass. 
 
 1 gram (gm.) = 15.43 gr. 1 grain (gr.) = 0.06480 gm. 
 
 1 kilogram (kg.) = 2.2046 Ib. 1 pound (Ib.) = 0.45359 kg. 
 
 Density. 
 
 1 gm. per cc. = 62.425 Ib. per cu. ft. 
 1 Ib. per cu. ft. = 0.01602 gm. per cc. 
 
 Thermometric Scales. 
 
 C=5 (F 32)/9 F = (9C/5) 4 32 
 
 (C centigrade temperature; F = Fahrenheit temperature) 
 
PHYSICAL TABLES 61 
 
 USEFUL NUMERICAL RELATIONS. 
 
 / Force. 
 
 1 gram's weight (gm. wt.) = 980.6 dynes (g, = 980.6 cm./sec.'.) 
 1 pound's weight (Ib. wt.) = 0.4448 megadynes (g = 980.6.) 
 
 (The "gm. wt.' is here defined as the force of gravity acting on 
 a gram of matter at sea -level and 45 north latitude. The "Ib. wt." is 
 similarly defined.) 
 
 Pressure and Stress 
 
 1 cm. of mercury at 0C. 1 in of mercury at 0C. 
 
 = 13.596 gm. wt. per sq. cm. = 34.533 gm. wt. per sq. cm. 
 
 = 0.19338 Ib. wt. per sq. in. = 0.49118 Ib. wt. per sq. in. 
 
 Work and Energy. 
 
 1 kilogram-meter (kg. m.) = 7.233 ft. Ib. 
 
 1 foot-pound (ft. Ib.) = 0.13826 kg. m. 
 
 1 joule = 10 7 ergs. 
 
 1 foot-pound = 1.3557 X 10 7 ergs. (g = 980.6 cm./sec. s .) 
 
 1 foot-pound = 1.3557 joules (g = 980.6.) 
 
 1 joule = 0.7376 ft. Ib (g = 980.6.) 
 
 Power (or Activity). 
 
 1 horse-power (H. P.) = 33000 ft. Ib. per min. 
 
 1 watt = 1 joule per sec. = 10 7 ergs per sec. 
 
 1 horse-power = 745.64 watts (g = 980.6 cm./sec. 1 ) 
 
 1 watt = 44.28 ft. Ib. per min (g = 980.6) 
 
 Mechanical Equivalent. 
 
 I gro -calorie = 4.187 X 10 7 ergs. 
 
 = 0.4269 kg. m. (g, = 980.6 cm./sec. 2 .) 
 = 3.088 ft. Ib. (g = 980.6.) 
 
62 
 
 PHYSICAL TABLES 
 
 DENSITY OF DRY AIR. 
 
 (Values are given in gms. per cc.) 
 
 Temp. 
 
 Barometric Pressure (Centimeters of Mercury) 
 
 C. 
 
 72 
 
 73 
 
 74 
 
 75 
 
 
 76 
 
 77 
 
 
 
 .001225 
 
 .001242 
 
 .001259 
 
 .001276 
 
 .001293 
 
 .001310 
 
 1 
 
 220 
 
 237 
 
 
 254 
 
 
 271 
 
 288 
 
 305 
 
 2 
 
 216 
 
 233 
 
 
 250 
 
 
 267 
 
 283 
 
 300 
 
 3 
 
 212 
 
 228 
 
 
 245 
 
 
 262 
 
 279 
 
 296 
 
 4 
 
 207 
 
 224 
 
 
 241 
 
 
 257 
 
 274 
 
 290 
 
 5 
 
 .001203 
 
 .001219 
 
 .001236 
 
 .001253 
 
 .001270 
 
 .001286 
 
 6 
 
 198 
 
 215 
 
 
 232 
 
 
 248 
 
 265 
 
 282 
 
 7 
 
 194 
 
 211 
 
 
 227 
 
 
 244 
 
 260 
 
 277 
 
 8 
 
 190 
 
 206 
 
 
 223 
 
 
 239 
 
 256 
 
 272 
 
 9 
 
 186 
 
 202 
 
 
 219 
 
 
 235 
 
 251 
 
 268 
 
 10 
 
 .001181 
 
 .001198 
 
 .001214 
 
 .001231 
 
 .001247 
 
 .001263 
 
 11 
 
 177 
 
 194 
 
 
 210 
 
 
 226 
 
 243 
 
 259 
 
 12 
 
 173 
 
 189 
 
 
 206 
 
 
 222 
 
 238 
 
 255 
 
 13 
 
 169 
 
 185 
 
 
 202 
 
 
 218 
 
 234 
 
 250 
 
 14 
 
 165 
 
 181 
 
 
 197 
 
 
 214 
 
 230 
 
 246 
 
 15 
 
 .001161 
 
 .001177 
 
 .001193 
 
 .001209 
 
 .001225 
 
 .001242 
 
 16 
 
 157 
 
 173 
 
 
 189 
 
 
 205 
 
 221 
 
 237 
 
 17 
 
 153 
 
 169 
 
 
 185 
 
 
 201 
 
 217 
 
 233 
 
 18 
 
 149 
 
 165 
 
 
 181 
 
 
 197 
 
 213 
 
 229 
 
 19 
 
 145 
 
 161 
 
 
 177 
 
 
 193 
 
 209 
 
 224 
 
 20 
 
 .001141 
 
 .001157 
 
 .001173 
 
 .001189 
 
 .001204 
 
 .001220 
 
 21 
 
 137 
 
 153 
 
 
 169 
 
 
 185 
 
 200 
 
 216 
 
 22 
 
 133 
 
 149 
 
 
 165 
 
 4 
 
 181 
 
 196 
 
 212 
 
 23 
 
 130 
 
 145 
 
 
 161 
 
 ^177 
 
 192 
 
 208 
 
 24 
 
 126 
 
 141 
 
 
 157 
 
 
 173 
 
 188 
 
 204 
 
 25 
 
 .001122 
 
 .001138 
 
 .001153 
 
 .001169 
 
 .001184 
 
 .001200 
 
 26 
 
 118 
 
 134 
 
 
 149 
 
 
 165 
 
 180 
 
 196 
 
 27 
 
 114 
 
 130 
 
 
 145 
 
 
 161 
 
 176 
 
 192 
 
 28 
 
 110 
 
 126 
 
 
 142 
 
 
 157 
 
 172 
 
 188 
 
 29 
 
 107 
 
 122 
 
 
 138 
 
 
 153 
 
 169 
 
 184 
 
 30 
 
 .001103 
 
 .001119 
 
 .001134 
 
 .001149 
 
 . 01165 
 
 .001180 
 
 Corrections for Moisture in the 
 
 Atmosphere 
 
 Dew-point Subtract Dew-point 
 
 Subtract 
 
 Dew-point 
 
 Subtract 
 
 10 .000001 + 2 
 
 .000003 
 
 -4-14 
 
 .000007 
 
 8 2 +4 
 
 4 
 
 + 16 
 
 8 
 
 6 2 +6 
 
 4 
 
 + 18 
 
 9 
 
 4 2 +8 
 
 5 
 
 + 20 
 
 .000010 
 
 2 3 +10 
 
 6 
 
 + 24 
 
 13 
 
 3 +12 
 
 6 
 
 + 28 
 
 16 
 
PHYSICAL TABLES 
 
 63 
 
 DENSITIES AND THERMAL PROPERTIES OF GASES. 
 
 (The densities are given at 0C. and 76 cm. pressure, and the 
 specific heats at ordinary temperatures. The coefficients of cubical 
 expansion (at constant pressure) of the gases listed below are not 
 given in this Table; they are about the same for all the permanent 
 gases, being approximately 1/273 or 0.003663, if referred in each 
 case to the volume of the gas at 0C. The specific heats at constant 
 pressure and at constant volume are represented by the symbols 
 Sp. and Sv) . 
 
 Gas or Vapor 
 
 Formula 
 
 Density 
 (gms. per cc.) 
 
 Molecular 
 Weight 
 
 Sp ~5~ Sv 
 
 (cafs. 
 per gm.) 
 
 Air 
 
 
 n nm OQQ 
 
 
 1/11 
 
 OOQ7 
 
 Ammonia 
 Carbon dioxide 
 Carbon monoxide 
 Chlorine 
 Hydrochloric acid 
 Hydrogen 
 Hydrogen sulphide 
 Nitrogen, pure 
 
 NH 3 
 C0 2 
 CO 
 Cli 
 
 HC1 
 H 2 
 H 2 S 
 
 N 2 
 
 0.000770 
 0.001974 
 0.001234 
 0.003133 
 0.001616 
 0.0000896 
 0.001476 
 0.001254 
 n 001 9^7 
 
 17.06 
 44.00 
 28.00 
 70.90 
 36.46 
 2.016 
 34.08 
 28.08 
 
 .41 
 
 1.33 
 1.29 
 1.40 
 1.32 
 1.40 
 .41 
 .34 
 .41 
 
 .Zot 
 
 .530 
 .203 
 .243 
 .124 
 .194 
 3.410 
 .245 
 .244 
 
 Oxygen 
 Steam (100C.) 
 Sulphur dioxide 
 
 2 
 H 2 O 
 
 S0 2 
 
 0.001430 
 0.000581 
 0.002785 
 
 32.00 
 18.02 
 64.06 
 
 .41 
 .28 
 .26 
 
 .218 
 .421 
 .154 
 
 DENSITY AND SPECIFIC VOLUME OF WATER. 
 
 Temp. 
 C. 
 
 Density 
 (gms. per cc.) 
 
 Specific 
 Volume 
 (cc. per gm.) 
 
 Temp. 
 C. 
 
 Density 
 (gms. per cc.) 
 
 Specific 
 Volume 
 
 (cc. per gm.) 
 
 
 
 0.999868 
 
 1.000132 
 
 20 
 
 0.99823 
 
 1.00177 
 
 1 
 
 927 
 
 073 
 
 25 
 
 777 
 
 294 
 
 2 
 
 968 
 
 032 
 
 30 
 
 567 
 
 435 
 
 3 
 
 992 
 
 008 
 
 35 
 
 406 
 
 598 
 
 3.98 
 
 1.000000 
 
 000 
 
 40 
 
 224 
 
 782 
 
 5 
 
 .999992 
 
 008 
 
 50 
 
 .98807 
 
 1.01207 
 
 6 
 
 968 
 
 032 
 
 60 
 
 324 
 
 705 
 
 7 
 
 929 
 
 071 
 
 70 
 
 .97781 
 
 1.02270 
 
 8 
 
 876 
 
 124 
 
 80 
 
 183 
 
 902 
 
 9 
 
 808 
 
 192 
 
 90 
 
 .96534 
 
 1.03590 
 
 10 
 
 727 
 
 273 
 
 100 
 
 .95838 
 
 1.04343 
 
 15 
 
 126 
 
 874 
 
 102 
 
 693 
 
 501 
 
64 
 
 PHYSICAL TABLES 
 
 DENSITIES AND THERMAL PROPERTIES OF LIQUIDS. 
 
 (The values given in this Table are mostly for pure specimens 
 of the liquids listed. The student should not expect the properties 
 of the average laboratory specimen to correspond exactly in value 
 with them. With a few exceptions the densities are given for ordi- 
 nary atmospheric temperature and pressure. The specific heats and 
 coefficients of expansion are in most cases the average values be- 
 tween and 100C. The boiling points are given for atmospheric 
 pressure, and the heats of vaporization are given at these boiling 
 points.) 
 
 
 * 
 
 o 
 
 ill 
 
 ba 
 
 c 
 o 
 
 - 
 
 MM 
 
 5 ~rt 
 
 *7| 'Q co 
 
 C 
 
 O rt 
 
 
 C 
 
 W o> 
 
 Jfi ^ d 
 
 ^ "o 
 
 +- 1 .i 
 
 Liquid 
 
 Q 
 
 K 
 
 o?i 
 
 A< 
 
 d >- 
 
 (LI O 
 
 
 
 (calories 
 
 
 
 > 
 
 
 
 per gm. 
 
 (per degree 
 
 (degrees 
 
 (calories 
 
 
 (gms. per cc.) 
 
 per deg.) 
 
 C.) 
 
 C.) 
 
 per gm.) 
 
 Alcohol (ethyl) 
 
 0.794 
 
 .58 
 
 .00111 
 
 78 
 
 205* 
 
 Alcohol (methyl) 
 
 .796 
 
 .60 
 
 .00143 
 
 66 
 
 262f 
 
 Benzene 
 
 .880 
 
 .42 
 
 .00123 
 
 80 
 
 93.2 
 
 Carbon bisulphide 
 
 1.29 
 
 .24 
 
 .00120 
 
 466 
 
 84 
 
 Cotton seed oil 
 
 .925 
 
 .47 
 
 .00077 
 
 
 
 Ether 
 
 .74 (0C) 
 
 .55 
 
 .00162 
 
 35 
 
 90 
 
 Glycerine 
 
 1.26 
 
 .576 
 
 .000534 
 
 
 
 Hydrochloric acid 
 
 1.27 
 
 .75 
 
 .000455 
 
 110 
 
 
 Mercury 
 
 13.596 (0) 
 
 .033 
 
 .0001815 
 
 357 
 
 67 
 
 Olive oil 
 
 .918 
 
 .47 
 
 .000721 
 
 
 
 Nitric acid 
 
 1.56 
 
 .66 
 
 .00125 
 
 86 
 
 115 
 
 Sea-water 
 
 1.025 
 
 .938 
 
 
 
 
 Sulphuric acid 
 
 1.85 
 
 .33 
 
 .00056 
 
 338 
 
 122 
 
 Turpentine 
 
 .873 
 
 .47 
 
 .00105 
 
 159 
 
 70 
 
 * The heat of vaporization of ethyl alcohol at 0C. is 236.5. 
 t The heat of vaporization of methyl alcohol at 0C. is 289.2. 
 
PHYSICAL TABLES 
 
 65 
 
 DENSITIES AND THERMAL PROPERTIES OF SOLIDS. 
 
 (The values given in this Table are mostly for pure specimens 
 of the substances listed. The student should not expect the prop- 
 erties of the average laboratory specimen to correspond exactly 
 in value with them. As a rule the densities are given for ordi- 
 nary atmospheric temperature. The specific heats and coefficients 
 of expansion are in most cases the average values between and 
 100 C. The melting points and heats of fusion are given for 
 atmospheric pressure.) 
 
 The coefficient of cubical expansion of solids is approximately 
 three times the linear coefficient. 
 
 
 
 
 *j U fl 
 
 
 
 
 ^v 
 
 o 
 
 S rt o 
 
 C) bfl 
 
 <+- 1 
 
 
 +J 
 
 !+ 
 
 - 2 '55 
 
 
 O g 
 
 
 
 Ti d 
 
 o c 
 
 
 
 
 g 
 
 CJ a> 
 
 K 
 
 *3 a 
 
 |2 
 
 2 3 
 
 Solid. 
 
 Q 
 
 CO 
 
 CJ^W 
 
 
 ffi^ 
 
 
 
 cals. per 
 
 
 degrees 
 
 cals. per 
 
 gms. per cc. 
 
 gm. 
 
 per degree C. 
 
 C. 
 
 gm. 
 
 Acetamide 1.56 
 
 
 82 
 
 
 Aluminum 
 
 2.70 
 
 0.219 
 
 .0000231 
 
 658 
 
 
 Brass, cast 
 
 8.44 
 
 .092 
 
 .0000188 
 
 
 
 " drawn 
 
 8.70 
 
 .092 
 
 .0000193 
 
 
 
 Copper 
 
 8.92 
 
 .094 
 
 .0000172 
 
 1090 
 
 43.0 
 
 German-silver 
 
 8.62 
 
 .0946 
 
 .000018 
 
 860 
 
 
 Glass, common tube 
 
 2.46 
 
 .186 
 
 .0000086 
 
 
 
 " . flint 
 
 3.9 
 
 .117 
 
 .0000079 
 
 
 
 Gold 
 
 19.3 
 
 .0316 
 
 .0000144 
 
 1065 
 
 
 Hyposul. of soda 
 
 1.73 
 
 .445 
 
 
 48 
 
 
 Ice 
 
 .918 
 
 .502 
 
 .000051 
 
 
 
 80. 
 
 Iron, cast 
 
 7.4 
 
 .113 i. 0000106 
 
 1100 
 
 23-33 
 
 " wrought 
 
 7.8 
 
 .115 .000012 
 
 1600 
 
 
 Lead 
 
 11.3 
 
 .0315 .000029 
 
 326 
 
 5.4 
 
 Mercury 
 
 13.596 
 
 .0319 
 
 
 39 
 
 2.8 
 
 Nickel 
 
 8.90 
 
 .109 
 
 .0000128 ' 1480 
 
 4.6 
 
 Paraffin, wax 
 
 .90 
 
 .560 
 
 .000008-23 i 52 
 
 35.1 
 
 liquid 
 
 
 .710 
 
 
 
 
 Platinum 
 
 21.50 
 
 .0324 
 
 .0000090 
 
 1760 
 
 27.2 
 
 Rubber, hard 
 
 1.22 
 
 .331 
 
 .000064 
 
 
 
 Silver 
 
 10.53 
 
 .056 
 
 .0000193 
 
 960 
 
 21.1 
 
 Sodium chloride 
 
 2.17 
 
 .214 
 
 .000040 
 
 800 
 
 
 Steel 
 
 7.8 
 
 .118 
 
 .000011 
 
 1375 
 
 
 Wood's alloy, solid 
 
 9.78 
 
 .0352 
 
 
 75.5 
 
 8.40 
 
 " , liquid 
 
 
 .0426 
 
 
 
 
66 
 
 PHYSICAL TABLES 
 
 VISCOSITY OF AQUEOUS SOLUTIONS OF SUGAR. 
 
 % Sugar 
 
 Coeff. at 20 C. 
 (C. G. S. Units.) 
 
 Coeff. at 30C. 
 (C. G. S. Units.) 
 
 
 5 
 10 
 20 
 40 
 
 0.0100 
 .0117 
 .0132 
 .0191 
 .0600 
 
 0.0080 
 .0089 
 .0104 
 .0145 
 .0423 
 
 COEFFICIENTS OF FRICTION. 
 
 Substances. 
 
 Static Coefficient. 
 
 Kinetic Coefficient 
 
 Metals on metals (dry) 
 " (wet) 
 (oiled) 
 Wood on wood (dry) 
 (a) direction of fiber 
 (b) normal to fiber 
 Leather belt on wood pulley 
 " " iron " 
 
 fro 
 
 m 0.2 1 
 0.15 
 0.15 
 
 0.5 
 0.4 
 0.45 
 0.25 
 
 :o 0.4 
 0.3 
 0.2 
 
 0.7 
 0.6 
 0.6 
 0.35 
 
 fro 
 
 m 0.18 1 
 0.14 
 0.14 
 
 0.2 
 0.18 
 0.3 
 0.2 
 
 .0 0.35 
 0.28 
 0.18 
 
 0.3 
 0.3 
 0.5 
 0.3 
 
 ELASTIC CONSTANTS OF SOLIDS. 
 (Approximate Values.) 
 
 Substance 
 
 Bulk-Modulus. 
 (C.G.S. Units.) 
 
 Simple Rigidity 
 (C.G.S.Units.) 
 
 Young's Modulus 
 (C.G.S.Units.) 
 
 Aluminum 
 Brass, drawn 
 Copper 
 
 5.5 x 10 11 
 10.8 x " 
 16.8 x " 
 
 2.5 x 10 11 
 3.7 x 
 4.5 x 
 
 4K Y 
 
 6.5 x 10 n 
 10.8 x 
 12.3 x 
 
 10 Q Y 
 
 Place 
 
 
 24. Y 
 
 7 * 
 
 Iron, wrought 
 Steel 
 
 14.6 x " 
 18.4 x " 
 
 7.7 x 
 8.2 x 
 
 19.6 x 
 21.4 x 
 
PHYSICAL TABLES 
 
 67 
 
 SURFACE TENSION OF PURE WATER IN CONTACT 
 WITH AIR. 
 
 (The value of the surface tension of a liquid is dependent only 
 upon the character and temperature of the liquid and upon the 
 nature of the gas above the surface of the liquid. It is independent of 
 the curvature of the surface film and of the material of the con- 
 taining vessel.) 
 
 Temp. 
 C. 
 
 Tension 
 (dynes pr. cm) 
 
 Temp. 
 C. 
 
 Tension 
 (dynes pr. cm) 
 
 Temp. 
 
 c. 
 
 Tension 
 (dynes pr.cm) 
 
 
 5 
 10 
 15 
 20 
 25 
 
 75.5 
 
 74.8 
 74.0 
 73.3 
 72.5 
 
 71.8 
 
 30 
 35 
 40 
 45 
 50 
 55 
 
 71.0 
 70.3 
 69.5 
 68.6 
 67.8 
 66.9 
 
 60 
 65 
 70 
 80 
 100 
 
 crit. temp 
 
 66.0 
 65.1 
 64.2 
 62.3 
 56.0 
 0.0 
 
 SURFACE TENSIONS OF SOME LIQUIDS IN CONTACT 
 WITH AIR. 
 
 (The angle of contact between pure water and clean glass vessels 
 of all sizes is 0; the angle of contact between pure water and clean 
 steel or silver is about 90; the angle of contact between mercury 
 and glass is about 132. See the note to Table VIII.) 
 
 
 Dynes 
 per cm. 
 
 
 Dynes 
 per cm. 
 
 Alcohol (ethyl) at 20 
 Alcohol (methyl) at 20 
 Benzene at 15 
 Glycerine at 18 
 
 22-24 
 22-24 
 28-30 
 63-65 
 
 Mercury at 20 
 Olive oil at 20 
 Petroleum at 20 
 Water (pure) at 20 
 
 470-500 
 32-36 
 24- 26 
 
 72- 74 
 
 VISCOSITY OF WATER. 
 
 Temp. 
 C. 
 
 Coeff. of Vise. 
 (C.G.S.Units) 
 
 Temp. 
 C. 
 
 Coeff. of Vise. 
 (C.G.S.Units) 
 
 Temp. 
 C. 
 
 Coeff. of Vise. 
 (C.G.S.Units) 
 
 
 5 
 10 
 15 
 20 
 
 0.0178 
 .0151 
 .0131 
 .0113 
 .0100 
 
 25 
 30 
 35 
 40 
 50 
 
 0.0089 
 .0080 
 .0072 
 .0066 
 .0055 
 
 60 
 70 
 80 
 90 
 100 
 
 0.0047 
 .0041 
 .0036 
 .0032 
 .0028 
 
68 
 
 PHYSICAL TABLES 
 
 (a) BOILING POINT OF WATER AT DIFFERENT BARO- 
 
 METRIC PRESSURES. 
 
 (b) VAPOR-PRESSURE OF SATURATED WATER-VAPOR. 
 
 (This table may be used either (a) to find the boiling point / of 
 water under the barometric pressure P, or (b) to find the vapor- 
 pressure P of water-vapor saturated at the temperature /, the dew- 
 point.) 
 
 t 
 c. 
 
 P 
 cm. 
 
 D 
 
 gm./cc. 
 
 t 
 C. 
 
 P 
 cm. 
 
 D 
 gm./cc. 
 
 t 
 C. 
 
 P 
 cm. 
 
 D 
 gm. /cc. 
 
 -10 
 
 .22 
 
 2.3xlO- 6 
 
 30 
 
 3.15 
 
 30.1x10- 
 
 88.5 49.62 
 
 
 - 9 
 
 .23 
 
 2.5x " 
 
 35 ! 4.18 
 
 39.3x" 
 
 89 
 
 50.58 
 
 
 - 8 
 
 .25 
 
 2.7x" 
 
 40 5.49 
 
 50.9x" 
 
 89.5 
 
 51.55 
 
 
 - 7 
 
 .27 
 
 2.9x" 
 
 45 7.14 
 
 65.3x " 
 
 90 
 
 52.54 
 
 428.4xlO- 6 
 
 - 6 
 
 .29 
 
 3.2x " 
 
 50 
 
 9.20 
 
 83.0x " 
 
 90.5 
 
 53.55 
 
 
 - 5 
 
 .32 
 
 3.4x " 
 
 55 
 
 11.75 
 
 104.6x " 
 
 91 
 
 54.57 
 
 
 - 4 
 
 .34 
 
 3.7x " 
 
 60 14.88 
 
 130.7x " 
 
 91.5 
 
 55.61 
 
 
 - 3 
 
 .37 
 
 4.0x " 
 
 65 
 
 18.70 
 
 162.1x" 
 
 92 
 
 56.67 
 
 
 - 2 
 
 .39 
 
 4.2x " 
 
 70 
 
 23.31 
 
 199.5x " 
 
 92.5 
 
 57.74 
 
 
 - 1 
 
 .42 
 
 4.5x " 
 
 71 
 
 24.36 
 
 
 93 
 
 58.83 
 
 
 
 
 .46 
 
 4.9x " 
 
 72 
 
 25.43 
 
 
 93.5 
 
 59.96 
 
 
 1 
 
 .49 
 
 5.2x " 
 
 73 
 
 26.54 
 
 
 94 
 
 61.06 
 
 
 2 
 
 .53 
 
 5.6x " 
 
 74 
 
 27.69 
 
 
 94.5 
 
 62.20 
 
 
 3 
 
 .57 
 
 6.0x " 
 
 75 
 
 28.88 
 
 243.7 " 
 
 95 
 
 63.36 
 
 511.1" 
 
 4 
 
 .61 
 
 6.4x " 
 
 75.5 |29.49 
 
 
 95.5 
 
 64.54 
 
 
 5 
 
 .65 
 
 6.8x " 
 
 76 130.11 
 
 
 96 
 
 65.74 
 
 
 6 
 
 .70 
 
 7.3x " 
 
 76.5 J30.74 
 
 
 96.5 
 
 66.95 
 
 
 7 
 
 .75 
 
 7.7x " 
 
 77 
 
 31.38 
 
 
 97 
 
 68.18 
 
 
 8 
 
 .80 
 
 8.2x " 
 
 77.5 32.04 
 
 
 97.5 
 
 69.42 
 
 
 9 
 
 .85 
 
 8.7x " 
 
 78 132.71 
 
 
 98 
 
 70.71 
 
 
 10 
 
 .91 
 
 9.3x " 
 
 78.5133.38 
 
 
 98.2 
 
 71.23 
 
 
 11 
 
 .98 
 
 lO.Ox " 
 
 79 34.07 
 
 
 98.4 
 
 71.74 
 
 
 12 
 
 1.04 
 
 10.6x " 
 
 79.5 34.77 
 
 
 98.6 
 
 72.26 
 
 
 13 
 
 1.11 
 
 11.2x" 
 
 80 35.49 
 
 295.9 " 
 
 98.8 
 
 72.79 
 
 
 14 
 
 1.19 
 
 12.0x " 
 
 80.5 :36.21 
 
 
 99 
 
 73.32 
 
 
 15 
 
 .27 
 
 12.8x " 
 
 81 36.95 
 
 
 99.2 
 
 73.85 
 
 
 16 
 
 .35 
 
 13.5x " 
 
 81.5 37.70 
 
 
 99.4 
 
 74.38 
 
 
 17 
 
 .44 
 
 14.4x " 
 
 82 38.46 
 
 
 99.6J 74.92 
 
 
 18 
 
 .53 
 
 15.2x " 
 
 82.5 39.24 
 
 
 99.8 
 
 75.47 
 
 
 19 
 
 .63 
 
 16.2x " 
 
 83 J40.03 
 
 
 100 
 
 76.00 
 
 606.2 " 
 
 20 
 
 .74 
 
 17.2x " 
 
 83.5 '40.83 
 
 
 100.2 
 
 76.55 
 
 
 21 
 
 .85 
 
 18.2x " 
 
 84 
 
 41.65 
 
 
 100.4 
 
 77.10 
 
 
 22 
 
 1.96 
 
 19.3x " 
 
 84.5 
 
 42.47 
 
 
 100.6 
 
 77.65 
 
 
 23 
 
 2.09 
 
 20.4x " 
 
 85 
 
 43.32 
 
 357.1 " 
 
 100.8 
 
 78.21 
 
 
 24 
 
 2.22 
 
 21.6x" 
 
 85.5144.17 
 
 
 101 
 
 78.77 
 
 
 25 
 
 2.35i22.9x " 
 
 86 45.05 
 
 
 102 
 
 81.60 
 
 
 26 
 
 2.50|24.2x " 
 
 86.5i45.93 
 
 
 103 
 
 84.53 
 
 
 27 
 
 2.65 
 
 25.6x " 
 
 87 46.83 
 
 
 105 
 
 90.64 
 
 715.4 " 
 
 28 
 
 2.81 
 
 27.0x " 
 
 87.5 J47.74 
 
 
 107 
 
 97.11 
 
 
 29 
 
 2.9728.5x" 
 
 88 : 48.68 
 
 
 110 
 
 107.54 
 
 840.1 " 
 
PHYSICAL TABLES 
 
 69 
 
 THE WET- AND DRY- BULB HYGROMETER. DEW-POINT. 
 
 (This Table gives the vapor-pressure, in mercurial centimeters, of 
 the water-vapor in the atmosphere corresponding to the dry-bulb 
 reading /C. (first column) and the difference (first row) between 
 the dry-bulb and wet-bulb readings of the hygrometer. Having 
 obtained from this Table the value of the vapor-pressure for a given 
 case, the dew-point can be found by consulting Table XIV. The data 
 given below are calculated for a barometric pressure equal to 76 cm.) 
 
 jo/ ^j Difference between Dry-bulb and Wet-bulb Readings. 
 
 
 1 
 
 Oo 
 
 3' 4 
 
 5 
 
 6 
 
 7 8 9 10 
 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 10 
 
 .92 
 
 .81 
 
 .70 
 
 .60 
 
 .50 
 
 .40 
 
 .31 
 
 .22 
 
 .13 
 
 
 
 11 
 
 .98 
 
 .87 
 
 .76 
 
 .65 
 
 .55 
 
 .45 
 
 .35 
 
 .26 
 
 .17 
 
 
 
 12 
 
 .105 
 
 .93 
 
 .82 
 
 .71 
 
 .60 
 
 .50 
 
 .40 
 
 .30 
 
 .21 
 
 .12 
 
 .03 
 
 13 
 
 1.12 
 
 .00 
 
 .89 
 
 .76 
 
 .65 
 
 .55 
 
 .45 
 
 .35 
 
 .25 
 
 .16 
 
 .07 
 
 14 
 
 1.19 .07 
 
 .94 
 
 .83 
 
 71 
 
 .61 
 
 .50 
 
 .40 
 
 .30 
 
 .20 
 
 .11 
 
 15 
 
 1.27! .14 
 
 .01 
 
 .90 
 
 .78 
 
 .66 
 
 .55 
 
 .45 .34 
 
 .25 
 
 .15 
 
 16 
 
 1.35) .22 
 
 .09 
 
 .97 
 
 .84 
 
 .73 
 
 .60 
 
 .50 .40 
 
 .30 
 
 .19 
 
 17 
 
 1.44 
 
 .30 
 
 .17 1.04 
 
 .91 
 
 .80 
 
 .67 
 
 .56 
 
 .45 
 
 .35 
 
 .24 
 
 18 
 
 1.54 
 
 .39 
 
 .25 
 
 1.12 
 
 .99 
 
 .86 
 
 .74 
 
 .63 .51 
 
 .40 
 
 .30 
 
 19 
 
 1.63 
 
 .49 
 
 .34 
 
 1.20 1.07 
 
 .94 
 
 .81 
 
 .69 .57 .46 
 
 .35 
 
 20 
 
 1.74 
 
 .59 
 
 .43 
 
 1.29 1.15 
 
 1.02 
 
 .88 
 
 .76 .64 .52 
 
 .41 
 
 21 
 
 1.85 
 
 .69 
 
 .53 
 
 1.38 1.24 
 
 1.10 
 
 .96 
 
 .84 .71| .59 
 
 .47 
 
 22 
 
 1.97 
 
 .80 
 
 .64 
 
 1.48 1.33 
 
 1.19 
 
 1.05 
 
 .91 
 
 .78 ! .66 
 
 .54 
 
 23 
 
 2.09 
 
 1.92 
 
 .75 
 
 1.59 1.43 
 
 1.28 
 
 1.13 
 
 .00 .86; .73 
 
 .61 
 
 24 
 
 2.22 
 
 2.04 
 
 .86 
 
 1.70 1.53 
 
 1.38 
 
 1.23 
 
 .09 .94| .81 
 
 .68 
 
 25 
 
 2.35 2.17 
 
 .99 
 
 1.81 1.64 
 
 1.48 
 
 1.33 
 
 .18 1.03J .90 
 
 .76 
 
 26 
 
 2.50 
 
 2.31 
 
 2.11 
 
 1.94 1.76 
 
 1.59 
 
 1.43 
 
 .28 1.13 .98 .84 
 
 27 
 
 2.65 2.45 
 
 2.25 
 
 2.07 1.88 
 
 1.71 
 
 1.54 
 
 .38 1.23 1.08 
 
 .93 
 
 28 
 
 2.81J 2.60 
 
 2.40 
 
 2.20 2.01 
 
 1.83 
 
 1.66 
 
 .49 1.33 1.18 
 
 1.02 
 
 29 
 
 2.98 2.76 
 
 2.55 
 
 2.35 2.15 
 
 1.96 
 
 1.78 
 
 .61 1.44 1.28 
 
 1.12 
 
 30 
 
 3.15 2.93 
 
 2 71 
 
 2.50 2.29 
 
 2.10 
 
 1.91 
 
 .73 1.55 1.39 1.23 
 
 Miscellaneous. 
 
 (1.) Heat of Neutralization. 
 
 Any strong acid with any strong alkali evolves (+) about 
 761 calories for every gm. of water formed. 
 
 (2.) Heat of Solution in water. 
 
 For Calcium oxide (CaO), -f 327 cals. per gm. 
 
 " Sodium chloride (NaCl), - 21 " " " 
 
 hydroxide (NaOH), + 248 " " " 
 
 hyposulphite (Na 2 S 2 O 3 +5H 2 O), - 44.8 " " " 
 
 (3.) Fuel value of illuminating gas is 5500 to 6500 calories per liter, 
 its density is .00058 gm. per cc. at 0C and 76 cm. pressure. 
 
 Fuel value of ethyl alcohol is 7400, of methyl alcohol 5700, 
 calories per gram. 
 
70 
 
 NATURAL SINES. 
 
 
 0' 
 
 6 
 
 12 
 
 18 
 
 24' 
 
 30' 
 
 36 
 
 42' 
 
 48' 
 
 54' 
 
 123 
 
 4 5 
 
 
 
 0000 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 oi57 
 
 3 6 9 
 
 12 15 
 
 1 
 2 
 3 
 
 0175 
 0349 
 0523 
 
 0192 
 0366 
 0541 
 
 0209 0227 
 0384 0401 
 0558 0576 
 
 0244 
 0419 
 0593 
 
 0262 
 0436 
 0610 
 
 0279 
 
 0454 
 0628 
 
 0297 
 0471 
 0645 
 
 0314 
 0488 
 0663 
 
 0332 
 0506 
 0680 
 
 3 6 9 
 3 6 9 
 3 6 9 
 
 12 15 
 12 15 
 
 12 15 
 
 4 
 5 
 6 
 
 0698 
 0872 
 1045 
 
 0715107320750 
 0889 0906 092^ 
 1063! 1080 1097 
 
 0767 
 0941 
 IH5 
 
 075 
 0958 
 1132 
 
 0802 
 0976 
 1149 
 
 O8ig 
 
 993 
 1167 
 
 0837 
 ion 
 1184 
 
 0854 
 1028 
 
 1201 
 
 3 6 9 
 369 
 3 6 9 
 
 12 15 
 12 14 
 12 14 
 
 7 
 8 
 9 
 
 1219 
 1392 
 1564 
 
 1236 
 1409 
 1582 
 
 1253 1271 
 1426 1444 
 I599ji6i6 
 
 1288 
 1461 
 1633 
 
 1305 
 1478 
 1650 
 
 i323 
 1495 
 1668 
 
 1340 
 
 1513 
 1685 
 
 1357 
 1530 
 1702 
 
 1374 
 1547 
 1719 
 
 3 6 9 
 3 6 9 
 3 6 9 
 
 12 14 
 12 14 
 
 12 14 
 
 10 
 
 1736 
 
 1754 
 
 1771 
 
 1788 
 
 1805 
 
 1822 
 
 1840 
 
 1857 
 
 1874 
 
 1891 
 
 369 
 
 12 14 
 
 11 
 12 
 13 
 
 1908 
 2079 
 2250 
 
 925 
 2096 
 2267 
 
 1942 
 2113 
 2284 
 
 '959 
 2130 
 2300 
 
 1977 
 2147 
 2317 
 
 1994 
 2164 
 2334 
 
 201 1 
 
 2181 
 
 2351 
 
 2028 
 2198 
 2368 
 
 2045 
 2215 
 
 2385 
 
 2062 
 2232 
 24O2 
 
 369 
 369 
 368 
 
 II 14 
 II 14 
 II 14 
 
 14 
 15 
 16 
 
 24*19 
 
 2588 
 2756 
 
 2436 
 2605 
 2773 
 
 2453 
 2622 
 2790 
 
 2470 
 2639 
 2807 
 
 2487 
 2656 
 2823 
 
 2504 
 2672 
 2840 
 
 2521 
 2689 
 
 2857 
 
 2538 
 2706 
 2874 
 
 2554 
 2723 
 2890 
 
 2571 
 2740 
 2907 
 
 368 
 368 
 368 
 
 II 14 
 II 14 
 II 14 
 
 17 
 18 
 19 
 
 2924 
 3090 
 3256 
 
 2940 
 3107 
 3272 
 
 3437 
 
 2957 
 3123 
 3289 
 
 3453 
 
 2974 
 3140 
 3305 
 
 2990 
 3156 
 3322 
 3486 
 
 3007 
 3173 
 3338 
 
 3O2^ 
 3190 
 
 3355 
 
 3040 
 3206 
 3371 
 
 3057 
 3223 
 3387 
 
 3074 
 3239 
 3404 
 
 368 
 368 
 
 3 5 8 
 
 II 14 
 II 14 
 II 14 
 
 20 
 
 3420 
 
 3469 
 
 3502 
 
 35i8 
 
 3535 
 
 355i 
 
 3567 
 
 3 5 8 
 
 II 14 
 
 21 
 22 
 23 
 
 3584 
 3746 
 397 
 
 3600 
 3762 
 3923 
 
 3616 
 3778 
 3939 
 
 3633 
 3795 
 3955 
 
 3649 
 3811 
 3971 
 
 3665 
 3827 
 3987 
 
 3681 
 
 3843 
 4003 
 
 3697 
 3859 
 ^019 
 
 3714 
 3875 
 4035 
 
 3730 
 3891 
 4051 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 II 14 
 II 14 
 II 14 
 
 24 
 25 
 26 
 
 4067 
 4226 
 4384 
 
 4083 
 4242 
 4399 
 
 4099 
 4258 
 4415 
 
 4"5 
 4274 
 
 4431 
 
 4131 
 4289 
 4446 
 
 4M7 
 4305 
 4462 
 
 4163 
 4321 
 
 4478 
 
 4179 
 4337 
 4493 
 
 4195 
 4352 
 4509 
 
 4210 
 
 4368 
 4524 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 II 13 
 II 13 
 
 10 13 
 
 27 
 28 
 
 29 
 
 4540 
 4695 
 
 4848 
 
 4555 
 4710 
 4863 
 
 4571 
 4726 
 4879 
 
 4586 
 474i 
 4894 
 
 4602 
 4756 
 1909 
 
 4617 
 4772 
 4924 
 
 4633 
 4787 
 4939 
 
 4648 
 4802 
 4955 
 
 4664 
 4818 
 4970 
 
 4679 
 4833 
 4985 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 10 13 
 10 13 
 10 13 
 
 30 
 
 5000 
 
 5015 
 
 5030 
 
 5045 
 
 5060 
 
 5075 
 
 5090 
 
 5105 
 
 5120 
 
 5135 
 
 3 5 8 
 
 10 13 
 
 31 
 32 
 33 
 
 5i5o 
 5299 
 5446 
 
 5165 
 53M 
 546i 
 
 5180 
 5329 
 5476 
 
 5195 
 5344 
 5490 
 
 5210 
 
 5358 
 5505 
 
 5225 
 5373 
 5519 
 
 5240 
 
 5388 
 5534 
 
 5255 
 5402 
 
 5548 
 
 5270 
 5417 
 5563 
 
 5284 
 5432 
 
 5577 
 
 257 
 
 2 5 7 
 2 5 7 
 
 10 12 
 IO 12 
 10 12 
 
 34 
 35 
 36 
 
 5592 
 5736 
 5878 
 
 5606 
 5750 
 5892 
 
 5621 
 5764 
 5906 
 
 5635 
 5779 
 5920 
 
 5650 
 5793 
 5934 
 
 5664 
 5807 
 5948 
 
 5678 
 5821 
 5962 
 
 5693 
 5835 
 5976 
 
 5707 
 5850 
 
 599 
 
 5721 
 
 5864 
 6004 
 
 257 
 2 5 7 
 2 5 7 
 
 IO 12 
 10 12 
 
 9 12 
 
 37 
 38 
 39 
 
 6018 
 
 6i57 
 6293 
 
 6032 
 6170 
 6307 
 
 6046 
 6184 
 6320 
 
 6060 
 6198 
 6334 
 
 6074 
 6211 
 6347 
 
 6088 
 6225 
 6361 
 
 6101 
 6239 
 6374 
 
 6115 
 6252 
 6388 
 
 6129 
 6266 
 6401 
 
 6143 
 6280 
 6414 
 
 257 
 257 
 247 
 
 9 12 
 
 9 " 
 9 ii 
 
 40 
 
 6428 
 
 6441 
 
 6455 
 
 6468 
 
 6481 
 
 6494 
 
 6508 
 
 6521 
 
 6534 
 
 6547 
 
 247 
 
 9 ii 
 
 41 
 
 42 
 43 
 
 6561 
 6691 
 6820 
 
 6574 
 6704 
 
 6833 
 
 6587 
 6717 
 
 6845 
 
 6600 
 6730 
 
 6858 
 
 6613 
 6743 
 6871 
 
 6626 
 6756 
 6884 
 
 6639 
 6769 
 6896 
 
 6652 
 6782 
 6909 
 
 6665 
 6794 
 6921 
 
 6678 
 6807 
 6934 
 
 247 
 246 
 246 
 
 9 ii 
 9 ii 
 8 ii 
 
 44 
 
 6947 
 
 6959 
 
 6972 
 
 6984 
 
 6997 
 
 7009 
 
 7022 
 
 034 
 
 7046 
 
 7059 
 
 246 
 
 8 10 
 
NATURAL SINES. 
 
 71 
 
 
 0' 
 
 6' 
 
 7083 
 
 12' 
 
 7096 
 
 18 
 
 24 
 
 30 
 
 36 
 
 7M5 
 
 42' 
 
 7157 
 
 48' 
 
 7169 
 
 54 
 
 7181 
 
 123 
 
 4 5 
 
 45 
 
 7071 
 
 7108 
 
 7120 
 
 7133 
 
 246 
 
 8 10 
 
 46 
 47 
 48 
 
 7193 
 73M 
 743i 
 
 7206 
 7325 
 7443 
 
 7218 
 7337 
 
 7455 
 
 7230 
 7349 
 7466 
 
 7242 
 736i 
 7478 
 
 7254 
 7373 
 7490 
 
 7266 
 7385 
 75oi 
 
 7278 
 7396 
 7513 
 
 7290 
 7408 
 7524 
 
 7302 
 7420 
 7536 
 
 246 
 246 
 246 
 
 8 10 
 8 10 
 8 10 
 
 49 
 50 
 51 
 
 7547 
 7660 
 
 777i 
 
 7558 
 7672 
 7782 
 
 7570 
 7683 
 7793 
 
 758i 
 769-4 
 7804 
 
 7593 
 7705 
 7815 
 
 7604 
 7716 
 7826 
 
 7615 
 7727 
 7837 
 
 7627 
 7738 
 7848 
 
 7638 
 7749 
 7859 
 
 7649 
 7760 
 7869 
 
 246 
 246 
 245 
 
 8 9 
 7 9 
 7 9 
 
 52 
 53 
 54 
 
 7880 
 7986 
 8090 
 
 7891 
 
 7997 
 8100 
 
 7902 
 8007 
 8m 
 
 7912 
 8018 
 8121 
 
 7923 
 8028 
 8131 
 
 7934 
 8039 
 8141 
 
 7944 
 8049 
 8151 
 
 7955 
 8059 
 8161 
 
 79^5 
 8070 
 8171 
 
 7976 
 8080 
 8181 
 
 245 
 2 3 5 
 235 
 
 7 9 
 7 9 
 
 7 8 
 
 55 
 
 8192 
 
 8202 
 
 8211 
 
 8221 
 
 8231 
 
 $241 
 
 8339 
 8434 
 8526 
 
 8251 
 
 8261 
 
 8271 
 
 8281 
 
 2 3 5 
 
 7 8 
 
 56 
 57 
 58 
 
 8290 
 
 8387 
 8480 
 
 8572 
 8660 
 8746 
 
 &300 
 8396 
 8490 
 
 8310 
 8406 
 8499 
 
 8320 
 8415 
 8508 
 
 8329 
 8425 
 8517 
 
 8348 
 8443 
 8536 
 
 8358 
 8453 
 S_545 
 8634 
 8721 
 8805 
 
 8368 
 8462 
 8554 
 
 8377 
 8471 
 8563 
 
 235 
 2 3 5 
 2 3 5 
 
 6 8 
 6 8 
 6 8 
 
 59 
 60 
 61 
 
 S 5 8i 
 8669 
 8755 
 8838 
 8918 
 8996 
 
 8590 
 8678 
 8763 
 
 8599 
 8686 
 
 8771 
 
 8607 
 8695 
 8780 
 
 8616 
 8704 
 8788 
 
 8625 
 8712 
 8796 
 
 8643 
 8729 
 8813 
 
 8652 
 8738 
 8821 
 
 3 4 
 3 4 
 3 4 
 
 6 7 
 6 7 
 6 7 
 
 62 
 
 63 
 64 
 
 8829 
 8910 
 
 8988 
 
 8846 
 8926 
 9Q3 
 
 8854 
 8934 
 9011 
 
 8862 
 8942 
 9018 
 
 8870 
 8949 
 9026 
 
 8878 
 8957 
 9033 
 
 8886 
 8965 
 9041 
 
 8894 
 8973 
 9048 
 
 8902 
 8980 
 9056 
 
 3 4 
 3 4 
 3 4 
 
 5 7 
 5 6 
 5 6 
 
 65 
 
 9063 
 
 9070 
 
 9078 
 
 9150 
 9219 
 9285 
 
 9085 
 
 9092 
 9164 
 9232 
 9298 
 
 9100 
 9171 
 9239 
 9304 
 
 9107 
 9178 
 9245 
 93U 
 
 9114 
 9184 
 9252 
 9317 
 
 9121 
 
 9128 
 
 2 4 
 
 5 ^ 
 
 66 
 67 
 68 
 69" 
 70 
 71 
 
 9135 
 9205 
 9272 
 
 9M3 
 9212 
 9278 
 
 9157 
 9225 
 9291 
 
 9191 
 9259 
 9323 
 
 gigh 
 9265 
 9330 
 
 2 3 
 2 3 
 2 3 
 
 5 6 
 4 6 
 4 5 
 
 9336 
 9397 
 9455 
 
 9342 
 9403 
 9461 
 
 9348 
 9409 
 9466 
 
 9354 
 9415 
 9472 
 
 9361 
 9421 
 9478 
 
 9367 
 9426 
 
 9483 
 
 9373 
 9432 
 9489 
 
 9379 
 9438 
 9494 
 
 9385 
 9444 
 
 9391 
 9449 
 9505 
 
 2 3 
 2 3 
 2 3 
 
 4 5 
 4 5 
 4 5 
 
 9500 
 
 72 
 73 
 74 
 ~75~ 
 
 95U 
 9563 
 9613 
 
 95!6 
 9568 
 b6i 7 
 
 9521 
 9573 
 9622 
 
 9527 
 9578 
 9627 
 
 9532 
 9583 
 9632 
 
 9537 
 9588 
 9636 
 
 9542 
 9593 
 9641 
 
 9548 
 9598 
 9646 
 
 9553 
 9603 
 9650 
 
 9558 
 9608 
 9655 
 9699 
 
 2 3 
 
 2 2 
 2 2 
 
 4 4 
 3 4 
 3 4 
 
 9659 
 
 9664 
 
 9668 
 
 9673 
 
 9677 
 
 9681 
 
 9686 
 
 9690 
 
 9694 
 
 I 2 
 
 3 4 
 
 76 
 77 
 78 
 
 9703 
 9744 
 9781 
 
 9707 
 9748 
 9785 
 
 9711 
 9751 
 9789 
 
 9715 
 9755 
 9792 
 
 9720 
 9759 
 9796 
 9829 
 9860 
 9888 
 
 9724 
 9763 
 9799 
 
 9728 
 9767 
 9803 
 
 9732 
 9770 
 9806 
 
 9736 
 9774 
 9810 
 
 9740 
 9778 
 9813 
 
 2 
 2 
 2 
 
 3 3 
 3 3 
 2 3 
 
 79 
 80 
 81 
 
 9816 
 9848 
 9877 
 
 9820 
 
 9851 
 9880 
 
 9823 
 9854 
 9882 
 
 9826 
 
 9857 
 9885 
 
 9833 
 9863 
 9890 
 
 9836 
 9866 
 9893 
 
 9839 
 9869 
 
 9895 
 
 9842 
 9871 
 9898 
 
 9845 
 9874 
 9900 
 
 I 2 
 
 
 
 2 3 
 
 2 2 
 2 2 
 
 82 
 83 
 84 
 85 
 
 9903 
 9925 
 9945 
 
 9905 
 9928 
 
 9947 
 
 9907 
 9930 
 9949 
 9965 
 
 9910 
 9932 
 995i 
 9966 
 
 9912 
 9934 
 9952 
 
 9914 
 9936 
 9954 
 
 9917 
 9938 
 9956 
 
 9919 
 9940 
 9957 
 
 9921 
 9942 
 9959 
 
 99 2 3 
 9943 
 9960 
 
 O 
 
 
 
 2 
 2 
 I 
 
 9962 
 9976 
 9986 
 9994 
 
 9963 
 
 9968 
 
 9969 
 
 9971 
 
 9972 
 
 9973 
 
 9974 
 
 O O I 
 
 I 
 
 86 
 87 
 88 
 
 9977 
 9987 
 9995 
 
 9999 
 
 9978 
 9988 
 9995 
 
 9979 
 9989 
 9996 
 
 9980 
 9990 
 9996 
 
 9981 
 9990 
 9997 
 
 9982 
 999 i 
 9997 
 
 9983 
 9992 
 9997 
 
 9984 
 9993 
 9998 
 
 9985 
 9993 
 9998 
 
 O O I 
 O 
 O 
 
 I 
 
 I 
 
 
 89 
 
 9998 
 
 9999 
 
 9999 
 
 9999 
 
 1.000 
 nearly 
 
 1. 000 
 nearly 
 
 1.000 
 
 nearlv 
 
 1. 000 
 
 nearly 
 
 I.OOO 
 nearly 
 
 o o a 
 
 
 
72 
 
 NATURAL TANGENTS. 
 
 
 
 
 6 
 
 12' 
 
 18 
 
 24' 
 
 30' 
 
 36 
 
 42 
 
 48 
 
 54' 
 
 123 
 
 4 5 
 
 
 
 1 
 
 2 
 3 
 
 .0000 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 0157 
 
 369 
 
 12 14- 
 
 .0175 
 
 0349 
 .0524 
 
 0192 
 0367 
 0542 
 
 0209 
 0384 
 0559 
 
 0227 
 0402 
 0577 
 
 0244 
 0419 
 0594 
 
 0262 
 
 0437 
 0612 
 
 0279 
 
 0454 
 0629 
 
 0297 
 0472 
 0647 
 
 0314 
 0489 
 0664 
 
 0332 
 0507 
 0682 
 
 369 
 369 
 369 
 
 12 1 5 
 12 15 
 12 1 5 
 
 4 
 5 
 6 
 
 .0699 
 
 .0875 
 .1051 
 
 0717 
 0892 
 1069 
 
 0734 
 0910 
 1086 
 
 0752 
 0928 
 1104 
 
 0769 
 0945 
 
 1122 
 
 0787 
 0963 
 "39 
 
 0805 
 0981 
 H57 
 
 0822 
 0998 
 "75 
 
 0840 
 1016 
 1192 
 
 0857 
 1033 
 
 1210 
 
 369 
 
 369 
 369 
 
 12 1 5 
 12 1 5 
 12 I 5 
 
 7 
 8 
 9 
 
 .1228 
 .1405 
 
 .1584 
 
 1246 
 
 1423 
 1602 
 
 1263 
 1441 
 1620 
 
 1281 
 
 1459 
 1638 
 
 1299 
 1477 
 1655 
 
 1317 
 1495 
 1673 
 
 1334 
 1512 
 1691 
 
 1352 
 1530 
 1709 
 
 1370 
 1548 
 1727 
 
 1388 
 I 5 66 
 1745 
 
 369 
 369 
 369 
 
 12 I 5 
 12 15 
 12 I 5 
 
 10 
 
 1763 
 
 1781 
 
 1799 
 
 1817 
 
 1835 
 
 1853 
 
 1871 
 
 1890 
 
 1908 
 
 1926 
 
 369 
 
 12 15 
 
 11 
 12 
 13 
 
 .1944 
 
 .2126 
 
 .2309 
 
 1962 
 2144 
 2327 
 2512 
 2698 
 2886 
 
 1980 
 2162 
 
 2345 
 
 1998 
 2180 
 2364 
 
 2016 
 2199 
 
 2382 
 
 2035 
 2217 
 2401 
 
 2053 
 2235 
 2419 
 
 2071 
 2254 
 2438 
 
 2089 
 2272 
 2456 
 
 2107 
 2290 
 2475 
 2661 
 2849 
 3038 
 
 369 
 
 3 6 9 
 369 
 
 12 15 
 
 12 15 
 12 15 
 
 14 
 15 
 16 
 
 2493 
 .2679 
 .2867 
 
 2530 
 2717 
 2905 
 
 2549 
 2736 
 2924 
 
 2568 
 2754 
 2943 
 
 2586 
 
 2773 
 2962 
 
 2605 
 2792 
 2981 
 
 2623 
 28ll 
 3000 
 
 2642 
 2830 
 3019 
 
 369 
 369 
 3 6 9 
 
 12 l6 
 
 13 16 
 13 16 
 
 17 
 18 
 19 
 
 .3057 
 3249 
 
 3443 
 
 3076 
 3269 
 3463 
 
 3096 
 
 3288 
 3482 
 
 3"5 
 3307 
 3502 
 
 3134 
 3327 
 3522 
 
 3153 
 3346 
 3541 
 
 3172 
 3365 
 356i 
 
 3191 
 
 3385 
 3581 
 
 3211 
 3404 
 3600 
 
 3230 
 3424 
 3620 
 
 3 6 10 
 3 6 10 
 3 6 10 
 
 13 16 
 13 16 
 13 17 
 
 20 
 
 3640 
 
 3659 
 
 3679 
 
 3699 
 
 3719 
 
 3739 
 
 3759 
 
 3779 
 
 3799 
 
 3819 
 
 3 7 '0 
 
 '3 17 
 
 21 
 22 
 23 
 
 3839 
 .4040 
 
 4245 
 
 3859 
 4061 
 4265 
 
 3879 
 4081 
 4286 
 
 3899 
 4101 
 
 4307 
 
 3919 
 4122 
 
 4327 
 
 3939 
 4142 
 
 4348 
 
 3959 
 4163 
 4369 
 
 3978 
 4183 
 4390 
 
 4000 
 4204 
 44" 
 
 4020 
 4224 
 4431 
 
 3 7 10 
 3 7 '0 
 3 7 10 
 
 13 17 
 
 4 '7 
 14 17 
 
 24 
 25 
 26 
 
 4452 
 .4663 
 
 .4877 
 
 4473 
 4684 
 4899 
 
 4494 
 4706 
 4921 
 
 4515 
 4727 
 4942 
 
 4536 
 4748 
 4964 
 
 4557 
 4770 
 4986 
 
 4578 
 479i 
 5008 
 
 4599 
 4813 
 5029 
 
 4621 
 4834 
 5051 
 
 4642 
 4856 
 5073 
 
 4 7 10 
 4 7 ii 
 4 7 ii 
 
 14 18 
 14 18 
 15 18 
 
 27 
 28 
 29 
 
 5095 
 5317 
 5543 
 
 5H7 
 5340 
 5566 
 
 5139 
 5362 
 5589 
 
 5161 
 
 5384 
 5612 
 
 5184 
 5407 
 5635 
 
 5206 
 5430 
 5658 
 
 5228 
 5452 
 5681 
 
 5250 
 5475 
 5704 
 
 5272 
 5498 
 5727 
 
 5295 
 5520 
 5750 
 5985 
 
 4 7 M 
 8 ii 
 
 8 12 
 
 15 18 
 "5 19 
 '5 19 
 
 30 
 
 5774 
 .6009 
 .6249 
 .6494 
 
 5797 
 
 58205844 
 
 5867 
 
 5890 
 
 59M 
 
 5938 
 
 59 6 . 1 
 
 8 12 
 
 l6 20 
 
 31 
 32 
 33 
 
 6032 
 6273 
 6519 
 
 6056 
 6297 
 6544 
 
 6080 
 6322 
 6569 
 
 6lO4 
 6346 
 6594 
 
 6128 
 
 6371 
 6619 
 
 6152 
 
 6395 
 6644 
 
 6176 
 6420 
 6669 
 
 6200 
 6445 
 6694 
 
 6224 
 6469 
 6720 
 
 8 12 
 8 12 
 
 S 13 
 
 1 6 20 
 
 16 20 
 
 17 21 
 
 34 
 35 
 36 
 
 6745 
 .7002 
 7265 
 
 6771 
 7028 
 7292 
 
 6796 
 7054 
 7319 
 
 6822 
 7080 
 7346 
 
 6847 
 7107 
 
 7373 
 
 6873 
 7133 
 7400 
 
 6899 
 7159 
 7427 
 
 6924 
 7186 
 7454 
 
 6950 
 7212 
 7481 
 
 6976 
 
 7239 
 7508 
 
 9 '3 
 9 '3 
 5 9 "4 
 
 17 21 
 
 l8 22 
 l8 23 
 
 37 
 38 
 39 
 
 7536 
 7813 
 .8098 
 
 7563 
 7841 
 8127 
 
 7590 
 7869 
 8156 
 
 7618 
 7898 
 8185 
 
 7646 
 7926 
 8214 
 
 7673 
 7954 
 8243 
 
 7701 
 
 7983 
 8273 
 
 7729 
 8012 
 8302 
 
 7757 
 8040 
 8332 
 
 7785 
 8069 
 8361 
 
 5 9 '4 
 5 10 '4 
 5 i '5 
 
 18 23 
 19 24 
 
 20 24 
 
 40 
 
 .8391 
 
 8421 
 
 845i 
 
 8481 
 
 8511 
 
 8541 
 
 8571 
 
 8601 
 8910 
 9228 
 9556 
 
 8632 
 
 8662 
 
 5 '0 15 
 
 20 25 
 
 41 
 42 
 43 
 
 .8693 
 .9004 
 9325 
 
 8724 
 9036 
 9358 
 
 8754 
 9067 
 
 9391 
 
 8785 
 9099 
 
 9424 
 
 8816 
 9131 
 9457 
 
 8847 
 9163 
 9490 
 
 8878 
 9195 
 9523 
 
 8941 
 9260 
 9590 
 
 8972 
 9 2 93 
 9623 
 
 5 10 16 
 5 ii 16 
 6 ii 17 
 
 21 26 
 21 2 7 
 22 28 
 
 44 
 
 9657 
 
 9691 
 
 9725 
 
 9759 
 
 9793 
 
 9827 
 
 9861 
 
 9896 
 
 9930 
 
 99 6 5 
 
 6 ii 17 
 
 2 3 2 9 
 
NATURAL TANGENTS. 
 
 73 
 
 45 
 
 
 
 6' 
 
 12 
 
 18 
 
 24' 
 
 30 
 
 36 
 
 42' 
 
 48 
 
 54 
 
 123 
 
 4 5 
 
 24 30 
 
 1. 0000 
 
 0035 
 
 0070 
 
 0105 
 
 0141 
 
 0176 
 
 O2I2 
 
 057? 
 0951 
 1343 
 
 0247 
 
 0283 
 
 0319 
 
 6 12 18 
 
 46 
 47 
 48 
 
 1-0355 
 1.0724 
 1.1106 
 
 0392 
 0761 
 H45 
 
 0428 
 0799 
 1184 
 
 0464 
 0837 
 1224 
 
 0501 
 0875 
 1263 
 
 0538 
 0913 
 1303 
 
 0612 
 0990 
 383 
 
 0649 
 1028 
 1423 
 
 0686 
 1067 
 1463 
 
 6 12 18 
 6 43 19 
 7 13 20 
 
 25 3' 
 25 32 
 26 33 
 
 49 
 50 
 51 
 52 
 53 
 54 
 "55 
 
 1.1504 
 1.1918 
 1-2349 
 
 1544 
 1960 
 
 2393 
 2846 
 3319 
 3814 
 
 1585 
 
 2OO2 
 
 2437 
 
 1626 
 2045 
 2482 
 
 1667 
 2088 
 2527 
 
 1708 
 2131 
 
 2572 
 
 I75C 
 2174 
 2617 
 
 792 
 2218 
 2662 
 
 1833 
 2261 
 2708 
 
 1875 
 2305 
 2753 
 
 7 '4 21 
 7 4 
 
 8 15 23 
 
 28 3-1 
 29 36 
 30 38 
 
 1.2799 
 1.3270 
 1.3764 
 
 2892 
 3367 
 
 3865 
 
 2938 
 34i6 
 3916 
 
 2985 
 3465 
 3968 
 
 3032 
 35H 
 4019 
 
 3079 
 3564 
 4071 
 
 3127 
 36i3 
 4124 
 
 3175 
 3663 
 4176 
 
 3222 
 
 3713 
 4229 
 
 8 16 23 
 8 16 25 
 9 '7 26 
 
 3' 39 
 33 4' 
 34 43 
 
 1.4281 
 
 4335 
 
 4388 
 
 4442 
 
 4496 
 
 4550 
 
 4605 
 
 4659 
 
 4715 
 
 4770 
 
 9 18 27 
 
 36 45 
 
 56 
 57 
 58 
 
 1.4826 
 
 1-5399 
 1.6003 
 
 4882 
 545B 
 6066 
 
 4938 
 5517 
 6128 
 
 4994 
 5577 
 6191 
 
 5051 
 5637 
 6255 
 
 5108 
 
 5697 
 6319 
 
 5166 
 5757 
 6383 
 
 5224 
 5818 
 6447 
 
 5282 
 5880 
 6512 
 
 5340 
 594i 
 6577 
 
 o 19 29 
 
 10 20 30 
 II 21 32 
 
 38 48 
 40 50 
 43 53 
 
 59 
 60 
 61 
 62 
 63 
 64 
 
 1.6643 
 1.7321 
 1.8040 
 
 6709 
 739 1 
 
 8115 
 
 6775 
 7461 
 
 8190 
 
 6842 
 7532 
 8265 
 
 6909 
 7603 
 8341 
 
 6977 
 7675 
 8418 
 
 7045 
 7747 
 8495 
 
 7"3 
 
 7820 
 8572 
 
 7182 
 
 7893 
 8650 
 
 7251 
 7966 
 8728 
 
 ii 23 34 
 
 12 24 36 
 13 26 3 8 
 
 45 56 
 48 60 
 51 64 
 
 1.8807 
 1.9626 
 2.0503 
 
 8887 
 9711 
 0594 
 
 8967 
 
 9797 
 0686 
 
 9047 
 
 9883 
 0778 
 
 9128 
 9970 
 0872 
 
 9210 
 6057 
 0965 
 
 5292 
 0145 
 1060 
 
 9375 
 6233 
 
 "55 
 
 9458 
 0323 
 1251 
 
 9542 
 0413 
 
 1348 
 
 I 4 2 7 41 
 
 S 29 44 
 
 16 31 47 
 
 55 68 
 58 73 
 63 78 
 
 65 
 66 
 67 
 68 
 69 
 70 
 71 
 
 2.1445 
 
 1543 
 
 1642 
 
 1742 
 
 1842 
 
 1943 
 
 2045 
 
 2148 
 
 2251 
 
 2355 
 
 '7 34 5' 
 
 68 85 
 
 2.2460 
 2-3559 
 2.4751 
 
 2566 
 3673 
 4876 
 
 2673 
 3789 
 5002 
 
 2781 
 3906 
 5129 
 
 2889 
 4023 
 5257 
 
 2998 
 4142 
 
 5386 
 
 3109 
 4262 
 5517 
 
 3220 
 4383 
 5649 
 
 3332 
 4504 
 5782 
 
 3445 
 4627 
 59i6 
 
 18 37 55 
 
 20 40 60 
 
 22 43 65 
 
 74 9 2 
 79 99 
 87 108 
 
 2.6051 
 
 2.7475 
 2.9042 
 
 6187 
 7625 
 9208 
 
 6325 
 7776 
 9375 
 
 6464 
 7929 
 9544 
 
 6605 
 8083 
 9714 
 
 6746 
 8239 
 9887 
 
 6889 
 8397 
 6061 
 
 7034 
 8556 
 0237 
 
 7179 
 8716 
 
 0415 
 
 7326 
 8878 
 5595 
 
 24 47 7 1 
 26 52 78 
 29 58 87 
 
 95 "8 
 104 130 
 115 M4 
 
 72 
 73 
 74 
 
 3-0777 
 32709 
 
 3.4874 
 
 0961 
 2914 
 5105 
 
 1146 
 3122 
 5339 
 
 1334 
 3332 
 5576 
 
 1524 
 3544 
 5816 
 
 1716 
 
 3759 
 6059 
 
 1910 
 3977 
 6305 
 
 2106 
 4197 
 6554 
 
 2305 
 4420 
 6806 
 
 2506 
 4646 
 7062 
 
 32 64 96 
 36 72 108 
 
 41 82 122 
 
 129 161 
 144 k8o 
 162 203 
 
 75 
 76 
 77 
 78 
 
 3-7321 
 
 7583 
 
 7848 
 
 8118 
 
 8391 
 
 8667 
 
 8947 
 
 9232 
 
 9520 
 
 981 
 
 46 94 139 
 
 186 232 
 
 4.0108 
 
 4-3315 
 4.7046 
 
 0408 
 3662 
 7453 
 
 0713 
 4015 
 7867 
 
 IO22 
 
 4374 
 8288 
 
 1335 
 4737 
 8716 
 
 1653 
 5107 
 9152 
 
 1976 
 5483 
 9594 
 
 2303 
 5864 
 0045 
 
 2635 
 6252 
 050^ 
 
 297 
 6646 
 0970 
 
 53 107 i6c 
 62 124 186 
 73 146 219 
 
 214 267 
 248 310 
 292 365 
 
 79 
 80 
 81 
 
 5.1446 
 5-67I3 
 6.313^ 
 
 1929 
 7297 
 3859 
 
 2422 
 7894 
 4596 
 
 292^ 
 8502 
 5350 
 
 3435 
 9124 
 6122 
 
 3955 
 9758 
 6912 
 
 4486 
 640 
 7920 
 
 5026 
 1066 
 
 8548 
 
 5578 
 1742 
 9395 
 
 6140 
 
 o 1? 
 
 87 175 262 
 
 35<> 437 
 
 026 
 
 
 
 82 
 83 
 84 
 
 7.H54 
 8.144. 
 9-5I44 
 
 2066 
 2636 
 9-677 
 
 3002 
 3863 
 9-845 
 
 3962 
 5126 
 
 10.0 
 
 4947 
 6427 
 
 10.20 
 
 5958 
 7769 
 10.39 
 
 6996 
 
 9*5 
 10.5 
 
 8062 
 
 0579 
 10.78 
 
 9158 
 2052 
 10.99 
 
 028 
 
 357 
 
 11.20 
 
 Difference - col- 
 umns cease to be 
 useful, owing to 
 the rapidity with 
 which the value 
 of the tangent 
 changes. 
 
 85 
 86 
 87 
 88 
 
 n-43 
 
 11.66 
 
 11.9 
 
 I2.I 
 
 1243 
 
 12.7 
 
 13-00 
 
 13-3 
 
 13-62 
 
 13-9 
 
 14.30 
 19.08 
 28.64 
 
 14.67 
 19.74 
 30.14 
 
 15-06 
 20.45 
 31-82 
 
 ?5-4 
 21.2 
 
 33-6 
 
 15.89 
 
 22.02 
 35-80 
 
 16.35 
 22.90 
 38.19 
 
 16.8 
 23-8 
 40.9 
 
 17-34 
 24.90 
 44-o 
 
 17-89 
 26.03 
 47-74 
 
 I8. 4 
 27.2 
 52.0 
 
 89 
 
 57-29 
 
 63.66 
 
 71.62 
 
 81.8 
 
 95-49 
 
 114.6 
 
 143- 
 
 I 9 I.C 
 
 286. 
 
 573- 
 
74 
 
 ANTI-LOGARITHMS 
 
 Mats. 
 
 01234 
 
 
 PROPORTIONAL PARTS. 
 
 
 123 
 
 456 
 
 789 
 
 .00 
 .0 
 .0 
 .03 
 .04 
 
 1000 1002 1005 1007 1009 
 1023 1026 1028 1030 1033 
 1047 1050 1052 1054 1057 
 1072 1074 1076 1079 1081 
 1096 1099 1102 1104 1107 
 
 1012 1014 1016 1019 1021 
 1035 1038 1040 1042 1045 
 1059 1062 1064 1067 1069 
 1084 1086 1089 1091 1094 
 1109 1112 1114 1117 1119 
 
 
 
 
 
 1 
 
 1 1 
 1 1 
 1 1 
 1 1 
 112 
 
 222 
 222 
 222 
 222 
 232 
 
 .0 
 .06 
 .07 
 .08 
 .09 
 
 1122 1126 1127 1130 1132 
 1148 1151 1153 1156 1159 
 1175 1178 1180 1183 1186 
 1202 1205 1208 1211 1213 
 1230 1233 1236 1239 1242 
 
 1135 1138 1140 1143 1146 
 1161 1164 1167 1169 1172 
 1189 1191 1194 1197 1199 
 1216 1219 1222 1225 1227 
 1245 1247 1250 1253 1256 
 
 1 
 1 
 1 
 1 
 1 
 
 112 
 112 
 112 
 112 
 112 
 
 222 
 222 
 222 
 223 
 223 
 
 .1 
 
 .11 
 .12 
 
 :12 
 
 1259 1262 1265 1268 1271 
 1288 1291 1294 1297 1300 
 1318 1321 1324 1327 1330 
 1349 1352 1355 1358 1361 
 1380 1384 1387 1390 1393 
 
 1274 1276 1279 1282 1285 
 1303 1306 1309 1312 1315 
 1334 1337 1340 1343 1346 
 1365 1368 1371 1374 1377 
 1396 1400 1403 1406 1409 
 
 1 
 1 
 1 
 1 
 1 
 
 112 
 122 
 122 
 122 
 182 
 
 223 
 2 3 
 2 3 
 2 3 
 2 3 
 
 .15 
 .16 
 .17 
 .18 
 .19 
 
 1418 1416 1419 1422 1426 
 1445 1449 1452 1455 1459 
 1479 1483 1486 1489 1493 
 1514 1517 1521 1524 1528 
 1649 1552 1556 1560 1563 
 
 1429 1432 1435 1439 1442 
 1462 1466 1469 1472 1476 
 1496 1500 1503 1507 1510 
 1531 1535 1538 1542 1545 
 1567 1570 1574 1578 1581 
 
 1 
 1 
 1 
 1 
 
 1 2 2 
 122 
 1 2 2 
 122 
 
 2 3 
 2 3 
 2 3 
 2 3 
 3 3 
 
 
 
 20 
 
 .21 
 .22 
 .23 
 
 1585 1589 1592 1596 1600 
 1622 1626 1629 1633 1637 
 1660, 1663 1667 1671 1675 
 l^ 1702 1706 1710 1714 
 1738 1742 1746 1750 1754 
 
 1603 1607 1611 1614 1618 
 1641 1644 1648 1652 1656 
 1679 1683 1687 1690 1694 
 1718 1722 1726 1730 1734 
 1758 1762 1766 1770 1774 
 
 1 
 1 
 1 
 1 
 1 
 
 122 
 222 
 222 
 222 
 
 8 3 
 3 3 
 3 3 
 3 4 
 
 
 
 
 .25 
 .26 
 
 .27 
 .28 
 .29 
 
 1778 1782 1786 1791 1795 
 1820 1824 1828 1832 1837 
 1862 1866 1871 1875 1879 
 1905 1910 1914 1919 1923 
 1950 1954 1959 1963 1968 
 
 1799 1803 1807 1811 1816 
 1841 1845 1849 1854 1858 
 1884 1888 1892 1897 1901 
 1928 1932 1936 1941 1945 
 1972 1977 1982 1986 1991 
 
 1 
 1 
 1 
 1 
 1 
 
 222 
 223 
 223 
 223 
 223 
 
 3 
 3 
 3 
 3 
 3 
 
 .30 
 .31 
 .32 
 .63 
 .84 
 
 1995 2000 2004 2009 2014 
 2042 2046 2051 2056 2061 
 2089 2094 2099 2104 2109 
 2138 2143 2148 2153 2158 
 2188 2193 2198 2203 2208 
 
 2018 2023 2028 2032 2037 
 2065 2070 2075 2080 2084 
 2113 2118 2123 2128 2133 
 2163 2168 2173 2178 2183 
 2213 2218 2223 2228 2234 
 
 1 
 1 
 1 
 1 
 
 1 1 
 
 223 
 223 
 223 
 
 3 4 
 3 4 
 3 4 
 
 
 
 .35 
 .86 
 .87 
 .88 
 .39 
 
 2239 2244 2249 2254 2259 
 2291 2296 2301 2307 2312 
 2344 2350 2355 2360 2366 
 2399 2404 2410 2415 2421 
 2455 2460 2466 2472 2477 
 
 2265 2270 2275 2280 2286 
 2317 2323 2328 2333 2339 
 2371 2377 2382 2388 2393 
 2427 2432 2438 2443 2449 
 2483 2489 2495 2500 2506 
 
 119 
 
 
 
 1 1 2 
 112 
 
 233 
 
 445 
 
 
 
 
 .40 
 .41 
 .42 
 .43 
 .44 
 
 2512 2518 2523 2529 2535 
 2570 2576 2582 2588 2594 
 2630 2636 2642 2649 2655 
 2692 2698 2704 2710 2716 
 2754 2761 2767 2773 2780 
 
 2541 2547 2553 2559 2564 
 2600 2606 2612 2618 2624 
 2661 2667 2673 2679 2685 
 2723 2729 2735 2742 2748 
 2786 2793 2799 2805 2812 
 
 112 
 112 
 112 
 112 
 112 
 
 234 
 
 455 
 
 234 
 
 456 
 
 
 
 45 
 
 .46 
 .47 
 .48 
 .49 
 
 2818 2825 2831 2838 2844 
 2884 2891 2897 2904 2911 
 2951 2958 2965 2972 2979 
 3020 3027 3034 3041 3048 
 090 3097 3105 3112 3119 
 
 2851 2858 2864 2871 2877 
 2917 2924 2931 2938 2944 
 2985 2992 2999 3006 3013 
 3055 3062 3069 3076 3083 
 3126 3133 3141 3148 3155 
 
 112 
 1 1 2 
 112 
 112 
 112 
 
 834 
 334 
 384 
 344 
 344 
 
 556 
 556 
 566 
 666 
 666 
 
ANTI-LOGARITHMS 
 
 75 
 
 Hants. 
 
 01234 
 
 56789 
 
 PROW 
 
 >RTIONAL F 
 
 ARTS. 
 
 
 
 
 J. 2 3 
 
 456 
 
 789 
 
 .50 
 
 .51 
 52 
 
 3162 3170 3177 3184 3192 
 3236 3243 3251 3258 3266 
 3311 3319 3327 3334 3342 
 
 3199 3206 3214 3221 3228 
 3273 3281 3289 3296 3304 
 3350 3357 3365 3373 3381 
 
 112 
 122 
 122 
 
 
 
 .53 
 .54 
 
 3388 3396 3404 3412 3420 
 3467 3475 3483 3491 3499 
 
 3428 3436 3443 3451 3459 
 3508 3516 3524 3532 3540 
 
 122 
 
 
 
 .55 
 
 3548 3556 3565 3573 3581 
 
 3589 3597 3606 3614 3622 
 
 
 
 
 .56 
 .57 
 .58 
 
 3631 3639 3648 3656 3664 
 3715 3724 3733 3741 3750 
 3802 3811 3819 3828 3837 
 
 3673 3681 3690 3698 3707 
 3758 3767 3776 3784 3793 
 3846 3855 3864 3873 3882 
 
 123 
 1 2 3 
 
 445 
 
 678 
 
 
 
 
 
 
 
 .60 
 
 .61 
 
 3981 3990 3999 4009 4018 
 4074 4083 4093 4102 4111 
 
 4027 4036 4046 4055 4064 
 4121 4130 4140 4150 4159 
 
 1 2 
 
 
 
 .62 
 .63 
 .64 
 
 4266 4276 4285 4295 4305 
 4365 4375 4385 4395 4406 
 
 4315 4325 4335 4345 4355 
 4416 4426 4436 4446 4457 
 
 1 2 
 1 2 
 
 
 
 
 
 
 
 
 
 .66 
 .67 
 
 4571 4581 4592 4603 4613 
 4677 4688 4699 4710 4721 
 
 4624 4634 4645 4656 4667 
 4732 4742 4753 4764 4775 
 
 1 2 
 1 2 
 
 456 
 
 7 9 10 
 
 .68 
 .69 
 
 4786 4797 4808 4819 4831 
 4898 4909 4920 4932 4943 
 
 4842 4853 4864 4875 4887 
 4955 4966 4977 4989 5000 
 
 1 2 
 1 2 
 
 567 
 
 8 9 10 
 
 
 
 
 
 
 
 .71 
 
 5129 5140 5152 5164 5176 
 
 5188 5200 5212 5224 5236 
 
 1 2 
 
 567 
 
 8 10 11 
 
 .73 
 
 5370 6383 5395 5408 6420 
 
 5433 5445 5458 5470 5483 
 
 1 3 
 
 568 
 
 9 10 11 
 
 
 
 
 
 
 
 .75 
 
 .76 
 
 5623 5636 5649 5662 5675 
 5754 5768 5781 5794 5808 
 
 5689 6702 6716 6728 6741 
 6821 5834 5848 5861 5876 
 
 
 
 9 10 IS 
 9 11 12 
 
 .78 
 .79 
 
 6026 6039 6053 6067 6081 
 6166 6180 6194 6209 6223 
 
 6095 6109 6124 6138 6152 
 6237 6252 6266 6281 6295 
 
 
 
 10 11 13 
 10 11 13 
 
 .80 
 .81 
 .82 
 
 6310 6324 6339 6353 6368 
 6457 6471 6486 6501 6516 
 6607 6622 6637 6653 6668 
 
 6383 6397 6412 8427 6442 
 6531 6546 6561 6577 6592 
 6683 6699 6714 6730 6745 
 
 134 
 
 235 
 
 6 7 9 
 689 
 
 10 12 13 
 11 12 14 
 11 12 14 
 
 .83 
 .84 
 
 6761 6776 6792 6808 6823 
 6918 6934 6950 6966 6982 
 
 6839 6855 6871 6887 6902 
 6998 7015 7031 7047 7063 
 
 235 
 
 6 8 10 
 
 11 13 14 
 11 13 15 
 
 .85 
 
 7079 7096 7112 7129 7146 
 
 7161 7178 7194 7211 7228 
 
 236 
 
 7 8 10 
 
 12 13 15 
 
 .87 
 .88 
 .89 
 
 7413 7430 7447 7464 7482 
 7586 7603 7621 7638 7656 
 7762 7780 7798 7816 7834 
 
 7499 7516 7534 7551 7568 
 7674 *S91 7709 7727 7745 
 7852 7870 7889 7907 7926 
 
 236 
 245 
 245 
 
 7 9 10 
 7 9 11 
 7 911 
 
 12 14 16 
 12 14 16 
 18 14 16 
 
 
 
 
 
 
 
 .90 
 
 .91 
 .92 
 
 .93 
 .94 
 
 7943 7962 7980 7998 8017 
 8128 8147 8166 8185 8204 
 8318 8337 8356 8375 8395 
 8511 8531 8651 8570 8590 
 8710 8730 8750 8770 8790 
 
 8222 8241 8260 8279 8299 
 8414 8433 8453 8472 8492 
 8610 8630 8650 8670 8690 
 8810 8831 8851 8872 8892 
 
 246 
 246 
 246 
 246 
 
 8 9 11 
 8 10 12 
 8 10 12 
 8 10 12 
 
 13 15 17 
 14 15 17 
 14 16 18 
 14 16 18 
 
 .95 
 
 .96 
 .97 
 .98 
 .99 
 
 8913 8933 8954 8974 8995 
 9120 9141 9162 9183 9204 
 9333 9354 9376 9397 9419 
 9550 9572 9594 9616 9638 
 9772 9795 9817 9840 9863 
 
 9016 9036 9057 9078 9099 
 9226 9247 9268 9290 9311 
 9441 9462 9484 9506 9528 
 9681 9683 9705 9727 9750 
 9886 9908 9931 9954 9977 
 
 246 
 246 
 247 
 247 
 267 
 
 8 10 12 
 8 11 13 
 9 11 13 
 9 11 13 
 9 11 14 
 
 15 17 19 
 15 17 19 
 15 17 20 
 16*18 20 
 16 1820 
 
76 
 
 LOGARITHMS. 
 
 10 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 123 
 
 456 
 
 789 
 
 oooo 
 
 0043 
 
 0086 
 
 OI2S 
 
 0170 
 
 0212 
 
 0253 
 
 0294 
 
 0334 
 
 0374 
 
 Use Table on p. 58 
 
 11 
 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 19 
 
 0414 
 0792 
 
 "39 
 
 0453 
 0828 
 
 "73 
 
 0492 
 0864 
 1206 
 
 0531 
 
 0899 
 1239 
 
 0569 
 0934 
 1271 
 
 0607 
 0969 
 1303 
 
 0645 
 1 004 
 1335 
 
 0682 
 1038 
 1367 
 
 6719 
 1072 
 1399 
 
 0755 
 1106 
 T43Q 
 
 4 8 ii 
 3 7 10 
 3 6 10 
 
 15 19 23 
 
 14 17 21 
 
 13 16 19 
 
 26 30 34 
 24 28 31 
 23 26 29 
 
 1461 
 1761 
 2041 
 
 1492 
 1790 
 2068 
 
 1523 
 1818 
 2095 
 
 1553 
 1847 
 
 2122 
 
 1584 
 
 1875 
 2148 
 
 1614 
 1903 
 
 2175 
 
 1644 
 !93i 
 
 2201 
 
 1673 
 
 1959 
 2227 
 
 i7<>3 
 1967 
 
 2253 
 
 1732 
 
 20 1. } 
 
 2279 
 
 369 
 368 
 5 8 
 
 12 15 18 
 
 II 14 17 
 ii 13 16 
 
 21 24 27 
 
 20 22 25 
 
 18 21 24 
 
 2304 
 
 2553 
 2788 
 
 2330 
 2577 
 2810 
 
 2355 
 2601 
 
 2833 
 
 2380 
 2625 
 2856 
 
 2405 
 2648 
 2878 
 
 2430 
 2672 
 29OO 
 
 2455 
 2695 
 2923 
 
 2480 
 2718 
 2945 
 
 2504 
 2742 
 2967 
 
 2529 
 
 2765 
 2989 
 
 5 7 
 5 7 
 4 7 
 
 10 12 15 
 
 9 12 14 
 9 i3 
 
 17 2O 22 
 
 16 19 21 
 
 16 18 20 
 
 20 
 21 
 22 
 23 
 
 3010 
 
 3032 
 
 3054 
 
 3075 
 
 3096 
 
 3118 
 
 3139 
 
 3160 
 
 3181 
 
 3201 
 
 4 6 
 
 8 ii 13 
 
 15 17 *9 
 
 3222 
 
 3424 
 3617 
 
 3243 
 3444 
 3636 
 
 3263 
 3464 
 3655 
 
 3284 
 3483 
 3^74 
 
 3304 
 3502 
 3692 
 
 3324 
 3522 
 3711 
 
 3345 
 3541 
 3729 
 
 3365 
 356o 
 3747 
 
 3385 
 3579 
 3766 
 
 3404 
 
 3598 
 3784 
 
 4 6 
 4 6 
 
 4 6 
 
 8 10 12 
 8 IO 12 
 
 79" 
 
 14 16 18 
 14 15 1.7 
 13 "5 *7 
 
 24 
 26 
 26 
 
 3802 
 3979 
 4!*> 
 
 3820 
 
 3997 
 4166 
 
 3838 
 4014 
 4183 
 
 3856 
 4031 
 4200 
 
 3874 
 4048 
 4216 
 
 3892 
 4065 
 4232 
 
 3909 
 4082 
 
 4249 
 
 3927 
 4099 
 4265 
 
 3945 
 4116 
 4281 
 
 3962 
 
 4133 
 4298 
 
 4 5 
 3 5 
 3 5 
 
 7 9 ii 
 
 7 9 1 
 7 8 .10 
 
 12 14 16 
 
 12 14 15 
 II 13 15 
 
 27 
 28 
 
 29 
 
 43M 
 4472 
 4624 
 
 4330 
 1487 
 4639 
 
 4346 
 4502 
 4654 
 
 4362 
 4518 
 4669 
 
 4378 
 4533 
 4683 
 
 4393 
 
 4548 
 4698 
 
 4409 
 4564 
 4713 
 
 4425 
 4579 
 4728 
 
 4440 
 4594 
 4742 
 
 4456 
 
 4609 
 4757 
 
 3 5 
 3 5 
 3 4 
 
 689 
 689 
 679 
 
 II 13 14 
 II 12 14 
 10 12 13 
 
 30 
 31 
 32 
 33 
 34 
 35 
 36 
 
 4771 
 
 4786 
 
 4800 
 
 4814 
 
 4829 
 
 4843 
 
 4857 
 
 4871 
 
 4886 
 
 4900 
 
 3 4 
 
 679 
 
 10 ii 13 
 
 4914 
 5051 
 
 5185 
 
 4928 
 5065 
 5198 
 
 4942 
 
 5079 
 5211 
 
 4955 
 5092 
 5224 
 
 4969 
 5105 
 
 5237 
 
 4983 
 5"9 
 5250 
 
 4997 
 5132 
 5263 
 
 5011 
 
 5145 
 5276 
 
 5024 
 
 5159 
 5289 
 
 5038 
 5172 
 5302 
 
 3 4 
 3 4 
 3 4 
 
 678 
 5 7 8 
 5 6 8 
 
 10 II 12 
 
 9 ii 12 
 9 10 12 
 
 5315 
 5441 
 5563 
 
 5328 
 5453 
 5575 
 
 5340 
 5465 
 
 5587 
 
 5353 
 5478 
 5599 
 5717 
 5832 
 5944 
 
 5366 
 5490 
 5611 
 
 5378 
 5502 
 5623 
 
 539 1 
 5514 
 5635 
 
 5403 
 5527 
 5647 
 
 54i6 
 
 5539 
 5658 
 
 5428 
 
 5551 
 5670 
 
 3 4 
 4 
 
 4 
 
 5 6 8 
 5 6 7 
 5 6 7 
 
 9 10 ii 
 9 jo u 
 8 10 ii 
 
 37 
 38 
 39 
 
 5682 
 5798 
 59 TI 
 
 5694 
 S&og 
 5922 
 
 5705 
 5821 
 5933 
 
 5729 
 5843 
 5955 
 
 5740 
 5855 
 5966 
 
 5752 
 5866 
 5977 
 
 5763 
 
 5877 
 5988 
 
 5775 
 5888 
 
 5999 
 
 5786 
 
 5899 
 6010 
 
 3 
 3 
 3 
 
 5 6 7 
 5 6 7 
 457 
 
 8 9 10 
 8 9 jo 
 
 8 9 10 
 
 40 
 
 6021 
 
 6031 
 
 6042 
 
 6053 
 
 6064 
 
 6075 
 
 6085 
 
 6096 
 
 6107 
 
 6117 
 
 3 
 
 4 5 6 
 
 8 9 10 
 
 41 
 42 
 43 
 
 6128 
 6232 
 6335 
 
 6138 
 6243 
 6345 
 6444 
 6542 
 6637 
 
 6149 
 6253 
 6355 
 
 6160 
 6263 
 6365 
 
 6170 
 6274 
 6375 
 
 6180 
 6284 
 6385 
 
 6191 
 6294 
 6395 
 
 6201 
 6304 
 6405 
 
 6212 
 6314 
 6415 
 
 6222 
 6325 
 6425 
 
 3 
 3 
 3 
 
 456 
 4 5 6 
 4 5 6 
 
 7 9 
 7 8 9 
 7 8 q 
 
 44 
 45 
 46 
 "47 
 48 
 49 
 
 6435 
 6532 
 6628 
 
 6454 
 655i 
 6646 
 
 6464 
 6561 
 6656 
 
 6474 
 6571 
 6665 
 
 6484 
 6580 
 6675 
 
 6 493 
 6590 
 6684 
 
 6503 
 
 6599 
 6693 
 
 6513 
 6609 
 6702 
 
 6522 
 6618 
 6712 
 
 3 
 3 
 3 
 
 5 6 
 5 6 
 5 6 
 
 7 8 9 
 7 8 g 
 
 7 7 8 
 
 6721 
 6812 
 6902 
 
 6730 
 6821 
 6911 
 
 6739 
 6830 
 6920 
 
 6749 
 6839 
 6928 
 
 6758 
 6848 
 6937 
 
 6767 
 
 6857 
 6946 
 
 6776 
 6866 
 6955 
 
 6785 
 6875 
 6964 
 
 6794 
 6884 
 6972 
 
 6803 
 6893 
 6981 
 
 3 
 3 
 3 
 
 5 5 
 4 5 
 445 
 
 6 7 8 
 6 7 fe 
 6 7 8 
 
 50 
 
 51 
 52 
 53 
 
 6990 
 
 6998 
 
 7007 
 
 7016 
 
 7024 
 
 7033 
 
 7042 
 
 7050 
 
 7059 
 
 7067 
 
 3 
 
 345 
 
 6 7 8 
 
 7076 
 7160 
 7243 
 
 7084 
 7168 
 7251 
 
 7093 
 7177 
 
 7259 
 
 7101 
 
 7185 
 7267 
 
 7110 
 7193 
 
 7275 
 
 7118 
 7202 
 7284 
 
 7126 
 7210 
 7292 
 
 7135 
 7218 
 7300 
 
 7M3 
 7226 
 7308 
 
 7152 
 7235 
 7316 
 
 3 
 
 2 
 2 
 
 345 
 345 
 345 
 
 6 7 8 
 6 7 7 
 667 
 
 667 
 
 54 
 
 7324 
 
 7332 
 
 7340 
 
 7348 
 
 7356 
 
 7364 
 
 7372 
 
 738o 
 
 7388 
 
 7396 
 
 122 
 
 345 
 
LOGARITHMS 
 
 77 
 
 55 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 123 
 
 456 
 
 789 
 
 7404 
 
 7412 
 
 7.419 
 
 7427 
 
 7435 
 
 7443 
 
 745i 
 
 7459 
 
 7466 
 
 7474 
 
 122 
 
 3 4 5 
 
 5 6 7 
 
 56 
 57 
 58 
 
 7482 
 7559 
 7634 
 
 7490 
 7566 
 7642 
 
 7497 
 7574 
 7649 
 
 7505 
 
 7582 
 7657 
 
 7513 
 7589 
 7664 
 
 7520 
 
 7597 
 7672 
 
 7528 
 7604 
 7679 
 
 7536 
 7612 
 7686 
 
 7543 
 7619 
 7694 
 
 7551 
 7627 
 7701 
 
 122 
 I 2 2 
 I I 2 
 
 345 
 345 
 
 344 
 
 5 6 7 
 5 6 7 
 5 6 7 
 
 59 
 60 
 61 
 
 7709 
 
 7782 
 7853 
 
 77i6 
 7789 
 7860 
 
 7723 
 7796 
 7868 
 
 773i 
 7803 
 7871 
 
 7738, 
 7810 
 7882 
 
 7745 
 7818 
 7889 
 
 7752 
 7825 
 7896 
 
 7760 
 7832 
 7903 
 
 7767 
 
 7839 
 7910 
 
 7774 
 7846 
 79^7 
 
 I I 2 
 112 
 112 
 
 344 
 344 
 344 
 
 5 6 7 
 
 5 6 C 
 5 6 6 
 
 62 
 63 
 64 
 
 7924 
 
 7993 
 8062 
 
 793i 
 Sooo 
 8069 
 
 7938 
 8007 
 8075 
 
 7945 
 8014 
 8082 
 
 7952 
 8021 
 8089 
 
 7959 
 8028 
 8096 
 
 7966 
 8035 
 8102 
 
 7973 
 8041 
 8109 
 
 7980 
 
 8048 
 8116 
 
 7987 
 805^ 
 8122 
 
 I 1 2 
 112 
 
 I I 2 
 
 112 
 
 334 
 334 
 334 
 
 5 6 6 
 
 5 S 6 
 
 s s e 
 
 65 
 66 
 67 
 68 
 
 8129 
 
 8i95 
 8261 
 
 8325 
 
 8136 
 
 8202 
 8267 
 8331 
 
 8142 
 
 8149 
 
 8156 
 
 8162 
 
 8169 
 
 8176 
 
 8182 
 
 8189 
 
 3 3 4 5 5 C 
 
 8209 
 
 8274 
 8338 
 
 8215 
 8280 
 8344 
 
 8222 
 8287 
 8351 
 
 8228 
 8293 
 8357 
 
 8235 
 8299 
 8363 
 
 8241 
 8306 
 8370 
 
 8248 
 8312 
 8376 
 
 8254 
 8319 
 8382 
 
 112 
 112 
 I I 2 
 
 334 
 334 
 334 
 
 5 5 6 
 5 5 6 
 4 5 6 
 
 69 
 70 
 71 
 
 8388 
 8451 
 8513 
 
 8395 
 8457 
 8519 
 
 8401 
 8463 
 8525 
 
 8407 
 8470 
 8531 
 
 8414 
 8476 
 8537 
 
 8420 
 8482 
 8543 
 
 8426 
 8488 
 8549 
 
 8432 
 8494 
 
 8555 
 
 8439 
 8500 
 8561 
 
 8445 
 8506 
 8567 
 
 1 I 2 
 I I 2 
 112 
 
 234 
 234 
 
 234 
 
 4 5 6 
 4 5 (' 
 455 
 
 72 
 73 
 74 
 
 8573 
 8633 
 8692 
 
 8579 
 8639 
 8698 
 
 8585 
 8645 
 8704 
 
 859J 
 8651 
 8710 
 
 8597 
 8657 
 8716 
 
 8603 
 8663 
 8722 
 
 8609 
 
 8669 
 8727 
 
 8615 
 8675 
 8733 
 
 8621 
 8681 
 8739 
 
 8627 
 8686 
 8745 
 
 112 
 J I 2 
 112 
 
 234 
 
 2 3 4 
 2 34 
 
 455 
 455 
 455 
 
 75 
 
 875i 
 
 8756 
 
 8762 
 
 8768 
 
 8774 
 
 8779 
 
 8785 
 
 8791 
 
 8797 
 
 8802 
 
 I I 2 
 
 233 
 
 4 5 5 
 
 76 
 77 
 78 
 79 
 80 
 81 
 
 8808 
 8865 
 8921 
 
 8814 
 8871 
 8927 
 8982 
 9036 
 9090 
 
 8820 
 8876 
 8932 
 
 8825 
 8882 
 8938 
 
 8831 
 8887 
 8943 
 
 8837 
 8893 
 8949 
 9004 
 9058 
 9112 
 
 8842 
 8899 
 8954 
 
 8848 
 8904 
 8960 
 
 9015 
 9069 
 9122 
 
 8854 
 8910 
 8965 
 
 S85S) 
 8915 
 897] 
 
 I I 2 
 
 233 
 
 455 
 
 I I 2 
 
 233 
 
 445 
 
 8976 
 9031 
 9085 
 
 8987 
 9042 
 9096 
 
 8993 
 9047 
 9101 
 
 8998 
 
 953 
 9106 
 
 9009 
 9063 
 9117 
 
 9020 
 9074 
 9128 
 
 9025 
 9079 
 9133 
 
 1 I 2 
 1 I 2 
 
 233 
 233 
 
 445 
 
 445 
 
 82 
 83 
 84 
 
 9138 
 9191 
 9243 
 
 9 T 43 
 9196 
 9248 
 
 9149 
 9201 
 9 2 53 
 
 9154 
 9206 
 9258 
 
 9 r 59 
 9212 
 9263 
 
 9165 
 9217 
 9269 
 
 9170 
 9222 
 9274 
 
 9175 
 9227 
 9279 
 
 9180 
 9232 
 9284 
 
 9186 
 9238 
 9289 
 
 112 
 
 2 3 3: 4 4 5 
 
 112 
 
 233 
 
 445 
 
 85 
 
 9294 
 
 9299 
 
 9304 
 
 9309 
 
 9315 
 
 9320 
 
 9325 
 
 9330 
 
 9335 
 
 9340 
 
 
 233 
 
 
 86 
 87 
 88 
 
 9345 
 9395 
 9445 
 
 9350 
 9400 
 9450 
 
 9355 
 9405 
 9455 
 
 936o 
 9410 
 9460 
 
 9365 
 9415 
 9465 
 
 9370 
 9420 
 9469 
 
 9375 
 9425 
 9474 
 
 938o 
 9430 
 9479 
 
 9385 
 9435 
 9484 
 
 9390 
 9440 
 9489 
 
 
 
 
 I I 
 
 223 
 
 3 4 -1 
 
 89 
 90 
 91 
 
 9494 
 9542 
 959 
 
 9499 
 9547 
 9595 
 
 9504 
 9552 
 9600 
 
 9509 
 9557 
 9605 
 
 9513 
 9562 
 9609 
 
 95i8 
 9566 
 9614 
 
 9523 
 957' 
 9619 
 
 9528 
 9576 
 9624 
 
 9533 
 958i 
 9628 
 
 9538 
 9580 
 9 6 33 
 
 Oil 
 Oil 
 
 223 
 223 
 
 344 
 344 
 
 92 
 93 
 94 
 
 9638 
 9685 
 9731 
 
 9f>43 
 9689 
 9736 
 
 9647 
 9694 
 9741 
 
 9652 
 9699 
 9745 
 979' 
 
 9657 
 9703 
 9750 
 
 9661 
 9708 
 9754 
 
 9666 
 97*3 
 9759 
 
 9671 
 9717 
 9763 
 
 9675 
 9722 
 9768 
 
 9680 
 9727 
 9773 
 
 Oil 
 Oil 
 I I 
 
 223 
 223 
 2-23 
 
 344 
 344 
 3 4 4 
 
 95 
 
 9777 
 
 9782 
 
 9786 
 
 9795 
 
 9800 
 
 9805 
 
 9809 
 
 9814 
 
 9818 
 
 Oil 
 
 2 2 3 
 
 344 
 
 96 
 97 
 98 
 
 9823 
 9868 
 9912 
 
 9827 
 9872 
 9917 
 
 9832 
 9877 
 9921 
 
 9836 
 9881 
 9926 
 
 9841 
 9886 
 9930 
 
 9845 
 9890 
 
 9934 
 
 9850 
 9894 
 9939 
 
 9854 
 9899 
 
 9943 
 
 9859 
 9903 
 9948 
 
 9863 
 9908 
 9952 
 
 Oil 
 
 I I 
 Oil 
 
 223 
 
 223 
 2 2 3 
 
 344 
 
 344 
 344 
 
 99 
 
 9956 
 
 9961 
 
 9965 
 
 9969 
 
 9974 
 
 9978 
 
 9983 
 
 9987 
 
 999 i 
 
 9996 
 
 O I 1 
 
 223 
 
 334 
 
78 
 
 LOGARITHMS. 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 100 
 
 ooooo 
 
 043 
 
 087 
 
 130 
 
 173 
 
 217 
 
 260 
 
 303 
 
 346 
 
 389 
 
 101 
 102 
 103 
 
 432 
 860 
 
 oi 284 
 
 475 
 903 
 326 
 
 5i8 
 945 
 368 
 
 56i 
 988 
 410 
 
 604 
 030 
 452 
 
 647 
 072 
 494 
 
 689 
 H5 
 536 
 
 732 
 157 
 
 578 
 
 775 
 199 
 620 
 
 817 
 242 
 
 662 
 
 104 
 105 
 106 
 
 703 
 
 O2 IIQ 
 
 531 
 
 745 
 160 
 572 
 
 787 
 202 
 612 
 
 828 
 243 
 653 
 
 870 
 284 
 694 
 
 912 
 325 
 
 735 
 
 953 
 366 
 776 
 
 995 
 407 
 816 
 
 036 
 
 449 
 
 857 
 
 078 
 490 
 898 
 
 107 
 108 
 109 
 
 938 
 03342 
 
 743 
 
 979 
 
 383 
 782 
 
 019 
 
 423 
 822 
 
 060 
 
 463 
 862 
 
 IOO 
 
 503 
 902 
 
 141 
 
 543 
 941 
 
 181 
 
 583 
 981 
 
 222 
 623 
 O2 1 
 
 262 
 663 
 060 
 
 302 
 703 
 
 IOO 
 
 To find the logarithm of a number: First, locate in the 
 table the mantissa which lies in line with the first two figures of the 
 number and underneath the third figure, then increase this mantissa 
 by an amount depending upon the fourth figure of the number and 
 found by means of the interpolation columns at the right; secondly, 
 determine the characteristic, or the exponent of that integer power 
 of 10 which lies next in value below the number; for example, 
 log 600- 0.7782 f 2.; log 73.46=0.8661 + 1.; 
 
 log .006=0.7782-3.; log .7346=0.8661-1.; 
 
 log 6.003= 0.7784 + 0. ; log 7349= 0.8662 + 3. 
 
 The logarithm of a product of two or more numbers is the sum of 
 the logarithms of its factors; for example, 
 
 log. (.0821 X 463.2)= (0.9143 -2.) + (0.6658 + 2.) = 0.5801 + 1. 
 The logarithm of a quotient is the difference between the logar- 
 ithms of the dividend and divisor; for example, 
 
 log. (.5321 -916)= (0.7260-1.) - (0.9619 + 2.) = 0.7641-4. 
 The logarithm of a power or root of a number is the exponent 
 times the logarithm of the number; for example, 
 
 log V.863) 3 =3/2 X (0.9360 1.) = 0.9040 -1. 
 
 To find the number from its logarithm: Locate in the table 
 the mantissa next less than the given mantissa, then join the figure 
 standing above it at the top of the table to the two figures at the 
 extreme left on the same line as the mantissa, and finally to these 
 three join the figure at the top of the interpolation column which 
 contains the difference between the two mantissae. In the four- 
 figure number thus found, so place the decimal point that the 
 number shall be the product of some number, that lies between 
 1 and 10, by a power of 10 whose exponent is the characteristic 
 of the logarithm. For example, 
 
 antilog (0.6440 + 3) = 4405; 
 antilog (0.3069 -2) = .02027. 
 
 Caution. In adding and subtracting logarithms it is well to 
 remember that the mantissa is always essentially positive and may 
 or may not therefore, have the same sign as its characteristic. 
 
INDEX 
 
 Absolute, expansion of 
 
 liquid - - - - 36 
 
 Accuracy of measurement 8 
 
 Air, buoyancy, correction 
 
 for in weighing - - 15 
 
 Air, density of - 16 
 
 Alcohol, fuel value of - 53 
 
 Apparent expansion - 36 
 
 of alcohol - - 33 
 
 Archimedes' principle - 15 
 
 Balance, sensitive - - 12 
 Boyle's Law, Applied 16, 19, 21 
 
 Calorimetry - 44 
 
 Calorimeter - - - 44 
 
 Capillarity, rise of liquids 
 
 in tubes - - - 25 
 
 Carbon dioxide, relative 
 
 density of - - 19 
 
 Charles' Law - - * - 18 
 
 Coefficient of Expansion 
 of liquid by Regnault's 
 method - 36 
 by pycnometer method - 38 
 of glass by weight-ther- 
 mometer method - 42 
 
 Contact, angle of - - 25 
 Cooling curve - -49 
 
 Cooling, law of - 48 
 
 Data sheets 7 
 
 Density, of air, determi- 
 nation of, - 16 
 of carbon dioxide, deter- 
 mination of - - 19 
 of a cylindrical solid - 12 
 
 Double Weighing - - 15 
 
 Errors - - 7 
 
 Expansion, 
 
 absolute, of liquid - 36 
 
 Experiments, list of - 6 
 
 Figures, significant - 8 
 
 Force Table 58 
 
 Fuel value of alcohol - 53 
 
 Glass, coefficient of expan- 
 sion - - - - 42 
 
 Heat of Fusion, Wood's 
 
 Alloy - 
 Hooke's Law - 
 
 - 50 
 
 - 31 
 
 - 54 
 
 Junker's Calorimeter 
 
 Logarithms, table of 
 
 4 place - - 76 
 
 Measurements, precision 
 
 of - 8 
 
 Method of cooling - 48 
 
 of Components - . - 58 
 
 of heating - - 46 
 
 of mixtures - - - 50 
 
 of vibrations - - - 12 
 
 pycnometer - - 38 
 
 weight-thermometer - 42 
 
 Parallelogram Law - 58 
 
 Percent error - - 10 
 
80 
 
 INDEX 
 
 Plotting of Curves - - 41 
 
 Physical Tables 60 
 
 Principle of Moments - 55 
 
 Pycnometer, expansion of 
 
 liquid by - 38 
 Radiation, correction for 
 
 rate of - 52 
 
 Reference books - - 6 
 
 Rest point, of balance - 12 
 
 Rider, use of the - - 13 
 
 Sensitiveness of balance 
 
 defined - ^ - - 13 
 
 Significant Figures - 8 
 
 Specific heat of a liquid 
 
 by method of heating 46 
 
 by method of cooling 48 
 
 Surface tension by Jolly's 
 
 balance - 23 
 
 in capillary tubes - 25 
 
 Tension, surface, by di- 
 rect measurement - 23 
 
 Thermometer, comparison 
 of alcohol and water - 33 
 
 Vibration, method of 12 
 
 Viscosity, coefficient of - 28 
 Volumenometer, the - 20 
 
 Water, equivalent of a 
 body - - 45 
 
 equivalent of a ther- 
 mometer - - 45 
 equivalent of a glass 
 calorimeter - 45 
 
 Weighing, method of 
 
 double - 15 
 
 by method of vibrations 12 
 
 Weight thermometer 
 
 42 
 
 Young's Modulus, by 
 
 stretching - - 31 
 

 
 
 PHYSICAL MEASUREMENTS 
 
 MINOR 
 
 PART II. HEAT, MECHANICS AND 
 
 PROPERTIES OF MATTER 
 
 1917 
 

 
 
 
 
 
 
 
PHYSICAL MEASUREMENTS 
 
 A LABORATORY MANUAL IN GENERAL PHYSICS 
 FOR COLLEGES 
 
 In Four Parts 
 BY 
 
 RALPH S. MINOR, Ph. D. 
 
 Associate Professor of Physics, University of California 
 
 PART II 
 HEAT, MECHANICS and PROPERTIES of MATTER 
 
 In collaboration with 
 Wendell P. Roop, A. B. 
 
 and 
 Lloyd T. Jones, Ph, D. 
 
 Instructors in Physics, University of California 
 
 BERKELEY, CALIFORNIA 
 1917 
 
Copyrighted in the year 1917 by 
 Ralph S. Minor 
 
 Wetzel Bros. Printing Co. 
 
 2110 Addison Street 
 Berkeley, Calif. 
 
LIST OF EXPERIMENTS 
 
 Page 
 
 26. Uniformly Accelerated Motion - 5 
 
 27. Centripetal Force - 8 
 
 28. The Simple Pendulum - 10 
 
 29. Momentum - - 14 
 
 30. Moment of Inertia - - 15 
 
 31. Mechanical Equivalent of Heat by Calendar's 
 Method - - 18 
 
 32. Efflux of Gases - 31 
 
 33. Calibration of a set of weights - 23 
 
 34. Absolute Calibration of a Thermometer - - 25 
 
 35. Heat of Solution - - - - 31 
 
 36. Heat of Neutralization - 33 
 
 37. Variation of Boiling Point with Pressure - - 35 
 
 38. Constant Volume Air-Thermometer - 37 
 
 39. Vapor Pressure and Temperature - - 39 
 
 40. Hygrometry - - 41 
 
 41. Ratio of the Two Specific Heats of Air - - 44 
 
 42. The Heat of Fusion of Tin - - - 46 
 
REFERENCE BOOKS 
 
 Duff: Text-Book of Physics (Fourth Edition). 
 
 Edser: Heat for Advanced Students. 
 
 Ganot: Text-Book of Physics (18th Edition). 
 
 Hastings and Beach: General Physics. 
 
 Kimball: College Physics. 
 
 Preston: Theory of Heat (Second Edition). 
 
 Reed and Guthe: College Physics. 
 
 Spinney: Text-Book of Physics. 
 
 Watson: Text-Book of Physics (Fourth Edition, 1903). 
 
 Kaye and Laby: Physical and Chemical Constants. 
 Landolt and Bornstein: Physical and Chemical Tables. 
 Smithsonian Institute: Physical Tables. 
 
HEAT, 
 MECHANICS and PROPERTIES of MATTER 
 
 26. UNIFORMLY ACCELERATED MOTION. 
 
 To determine the ratio of the weight unit of force to the 
 
 absolute unit of force. 
 
 When the body is acted upon by a constant force, 
 its motion is uniformly accelerated. The force required 
 to produce a given acceleration is directly proportional 
 to the mass on which it acts and to the acceleration 
 produced in the mass. 
 
 When a body moves under the influence of its own 
 weight, it moves with a certain characteristic accelera- 
 tion, which is the same for all bodies in any given locality. 
 If large bodies are acted on by proportionally large 
 forces, their large mass operates to prevent the increased 
 acceleration which would otherwise occur. Weight- 
 force is rigorously proportional to mass. 
 
 When a body moves under the influence of some force 
 other than weight, the acceleration produced is propor- 
 tional to the force acting. In weight units, the force 
 exerted on unit mass by the earth is called unit force. 
 In unit mass, it produces an acceleration equal to the 
 characteristic acceleration produced by weight. In 
 absolute units, the force producing unit acceleration in 
 unit mass is called unit force, (the dyne). The ratio 
 of the weight unit to the absolute unit is numerically 
 equal to the acceleration produced by weight-force. 
 
 In this experiment, the acceleration produced in a 
 given body by a given force will be directly measured, 
 and the force producing the acceleration calculated in 
 
6 UNIFORMLY ACCELERATED MOTION [26 
 
 absolute units. Knowing the accelerating force in weight 
 units, the ratio of the units may be calculated. 
 
 The acceleration is to be measured by a device by 
 means of which the body, as it moves, leaves a trace 
 which enables us to infer the position of the body at any 
 instant during its motion. Two forms of apparatus are 
 used. In one, the moving body is a tuning-fork which 
 traces a wavy line on a fixed whitened glass plate. In 
 the other, the fork is stationary and the plate moves. 
 In both, the body moves vertically, between guides. 
 The body may fall freely, under the influence of its 
 weight alone; or, by means of a pulley and counter- 
 weight, the force acting may be reduced to any desired 
 value. 
 
 (a) Paint the plate with a thin coat of corn-starch, 
 mixed with alcohol. The alcohol will evaporate and 
 leave a surface on which the trace may be taken. Level 
 the apparatus carefully. If the body does not move 
 vertically, friction in the guides will cause error. 
 
 Adjust the stylus on the fork so that it presses very 
 lightly against the plate. If the stylus does not press 
 tightly enough, parts of the trace may be missing. But 
 if it presses too hard, friction will interfere with the 
 proper motion of the stylus. 
 
 The traces to be taken are five in number. If con- 
 ditions are favorable, all may be obtained on one plate, 
 and in case this is possible, postpone the measurement 
 of the traces until all of them are ready. Otherwise, 
 the measurements must be taken before the plate is 
 prepared for new traces. 
 
 First obtain a trace with the body falling freely. To 
 be usable, the trace must be distinct over a distance of 
 at least 30 to 40 cm. 
 
26] UNIFORMLY ACCELERATED MOTION 7 
 
 For the second trace use a counterweight weighing 
 300 grams less than the falling fork or plate; then, in 
 succession, others giving accelerating forces of 200 and 
 100 grams. 
 
 Finally, by using all the counterweights, reduce the 
 acceperating weight-force to zero. Give the moving 
 system a push, and take a fifth trace showing the negative 
 acceleration due to friction. If the friction were not 
 acting, all the observed accelerations would be greater. 
 
 (b) In measuring the traces, proceed as follows. 
 Mark the crests of a convenient odd number of consecutive 
 waves near the beginning and also near the end of the 
 trace. Lay the meter stick on the trace and read off the 
 positions of the crests which were marked. Count the 
 number of unmarked crests between those whose posi- 
 tions are noted. Record the vibration frequency marked 
 on the fork. 
 
 (c) Obtain the mean velocity for the intervals at the 
 beginning and end of each trace by dividing the length 
 of the interval by the corresponding time. This time 
 interval is calculated from the number of wave-lengths 
 in the interval and the frequency of the fork. 
 
 Determine the number of wave-lengths lying between 
 the centers of the two intervals. The change of velocity 
 during the time interval between these two middle 
 points, divided by the time interval, gives the change 
 in velocity per second, or the acceleration. 
 
 (d) Make out a table, containing the following 
 columns: (1) acceleration; (2) acceleration, corrected for 
 
8 CENTRIPETAL FORCE [26-27 
 
 effect of friction; (3) mass of body accelerated; (4) force 
 acting on body, in absolute units; (5) force acting on 
 body, in weight units 
 
 (e) By comparing your different values for the ratio 
 of the two units with their mean, estimate the probable 
 error of your work. 
 
 27. CENTRIPETAL FORGE. 
 
 References. Duff, pp. 23, 24, 25; Kimball, pp. 75, 76. 
 
 To determine the force necessary to keep a body of given 
 mass in a circle of given radius, while it moves with con- 
 stant speed. 
 
 Experience shows that a body in motion will continue 
 to move with the same speed in the same straight line, 
 unless acted upon by some outside force. An outside 
 force, if acting in the direction of the motion, will cause 
 a change in speed; if acting at right angles to the direction 
 of motion, it will cause no change in speed, but will 
 cause a change in the direction of the motion. A body 
 in motion always moves in a straight line, unless there 
 is a force applied causing it to leave the straight line. If 
 the force perpendicular to the line of the motion be momen- 
 tarily supplied, the direction of the motion is changed, 
 but the body afterward moves in a straight line at an 
 angle with its former direction. If the force be continu- 
 ously applied, the body moves in a curved path. If 
 the body be kept in a circular path, a force of definite 
 magnitude must be continuously applied to the body, 
 the direction of the force being always perpendicular to 
 the instantaneous direction of the motion. Since the 
 instantaneous direction of motion is along the tangent, 
 the force perpendicular to the direction of motion must be 
 
27] CENTRIPETAL FORCE 9 
 
 along the radius of the circle. If the force ceases to be 
 applied, the body ceases to leave the straight line and 
 hence continues to move in the tangent to the circle at 
 the position occupied by the body at the instant the 
 force ceased to act. 
 
 The force, /, required to maintain motion in a circle 
 of radius, r, of a mass, w, moving with rim speed, v, and 
 angular speed, co, is 
 
 f = k m v 2 / r, or since co r = v, f = k m co 2 r. 
 
 If the quantities m, v, and r are expressed in C. G. S. units 
 the factor k will be unity and the force will be given in 
 dynes. 
 
 To a rotator is attached the "centripetal force" appar- 
 atus. Two masses, nti and w 2 , are arranged to slide along 
 the horizontal guides. One of these, m\, is attached, by 
 means of cords passing over pulleys, to a large mass M, 
 which can slide up and down along the vertical rod. As 
 the speed of rotation is increased, more and more force 
 must be supplied in order to hold it to a circular path. 
 Finally, when the speed passes a certain value, the 
 force necessary to keep the mass moving in its circu- 
 lar path is greater than the weight of M can supply, 
 so the mass M is lifted. Its weight, Mg dynes, represents 
 the normal or centripetal force it supplies to the mass m\. 
 
 (a) To determine the radius for m\, measure the dis- 
 tance from the center of rotation to the center of the 
 mass wi when the mass M is against the lower stop and 
 again when it is against the upper stop and take the 
 average of these distances as the radius r\. Fix m 2 at a 
 distance r\ from the center of rotation. Note the value 
 of the masses. 
 
 The speed is to be calculated from a determination of 
 the number of revolutions in a measured time. This 
 
10 THE PRINCIPLE OF MOMENTS [27-28 
 
 may be most accurately determined by measuring the 
 time required for some even number of hundreds of revo- 
 lutions, starting and stopping the stop-clock while the 
 apparatus is in motion. To eliminate friction the speed 
 should be so regulated that the mass M slowly rises and 
 then falls between the stops continuously during a trial, 
 which should occupy from three to five minutes. 
 
 Make two more determinations of the speed with the 
 masses at the same distance. 
 
 (b) Make several additional trials, selecting each time 
 a different value of the mass M, or of the distance of mi 
 and w 2 from the axis. 
 
 (c) For each trial, test the equality of the weight, 
 (Mg), and. the calculated centripetal force required, and 
 determine the percentage difference from the weight. 
 
 (d) Point out the principal sources of error in the 
 experiment. 
 
 28. THE SIMPLE PENDULUM. 
 
 Reference. Duff, pp. 78, 82. 
 
 To determine the acceleration due to gravity from the 
 observed period and length of a simple pendulum. 
 
 If a conical pendulum and a simple pendulum have the 
 same length, /, and each is oscillating through a small 
 amplitude (2 or less) the periods are equal. The projec- 
 tion of the motion of the conical pendulum on any 
 diameter corresponds to that of the simple pendulum. 
 If r is the radius of the circle the conical pendulum 
 describes, the force toward the center is constant and 
 
28] 
 
 THE SIMPLE PENDULUM 
 
 11 
 
 Fig. 1. Showing a conical pendulum (a) and a simple pendulum (b) 
 of the same length. 
 
 its value is m v* / r or m co 2 r, where v is its linear and co 
 its angular velocity. In t seconds after the conical 
 pendulum has passed one end of the diameter shown in 
 (a) the angle turned through is co/ and the component 
 of the centripetal force along this diameter will be 
 m co 2 r cos co t. Let x be the projection of r on the 
 diameter at this instant. Then cos co t = x / r, and the 
 component of the force along the diameter becomes 
 m co 2 x. 
 
 When the simple pendulum is an equal distance, x, 
 from its rest position the force urging it toward its rest 
 position is / = m g x / I. If the two pendulums have 
 the same mass, m, these forces must be equal. Then 
 
 m 
 
 g x / /, or co 2 = g / I. 
 
 In any circular motion of period 7, coT = 2w. 
 Substituting this value of co in the preceding equation 
 and solving for T we have 
 
 T - 27T 
 
12 THE SIMPLE PENDULUM 128 
 
 Method of Coincidences. 
 
 In the present experiment, T is to be measured by 
 comparing, by the "method of coincidences," the period 
 of the simple pendulum with that of a clock pendulum of 
 known period. An electric circuit is completed through 
 an electric bell, the clock pendulum, the simple pendulum, 
 and the contacts at the bottom of each pendulum. As- 
 sume that the period of the clock pendulum is two seconds, 
 that is, that the time of a single swing or half-vibration 
 is one second. If the period of the simple pendulum were 
 the same and the two pendulums be started together, 
 they would vibrate in coincidence and the bell would 
 ring with every passage. If, however, the time of a single 
 swing of the simple pendulum were less than one second, 
 say by l/n th of a second, it would gain on the clock 
 pendulum and thus fall out of coincidence with it, so that 
 the bell would cease to ring until n seconds later, when the 
 two pendulums would be in coincidence again. Let us 
 suppose that the time between these successive coinci- 
 dences is 100 seconds, then we know that in this time the 
 clock pendulum has made one hundred half-vibrations 
 and the simple pendulum one more, or 101 half-vibrations. 
 In other words, the simple pendulum has made 101 half- 
 vibrations in 100 seconds, hence the value of its half- 
 period is 100/101 seconds. If, on the other hand, the 
 simple pendulum had been observed to lag behind the 
 clock pendulum, and the time between successive coinci- 
 dences remained the same, we would know that its half- 
 period is 100/99 seconds. 
 
 (a) The simple pendulum used consists of a brass 
 sphere suspended from a knife-edge by a wire so that the 
 length is adjustable. Adjust the pendulum so that its 
 length is either greater or less, by 2 or 3 cm., than that of a 
 simple pendulum beating seconds. Two different lengths 
 
28] THE SIMPLE PENDULUM 13 
 
 (in successive determinations) should be used such that 
 one is greater and the other less than that of a pendulum 
 beating seconds. In getting the length it is well to measure 
 with a meter rod and square to the top of the ball, and 
 then to determine the diameter of the ball with the cali- 
 pers. The length of the pendulum is the distance from 
 the knife-edge to the center of the ball. Start the ball 
 swinging in an arc of about 10 cm., taking care not to 
 give it a twisting vibratory motion. During the vibra- 
 tions watch the hands of the laboratory clock and record 
 the hour, minute and second of each successive coinci- 
 dence between the simple pendulum and the clock pendu- 
 lum, up to ten or more. If the bell rings for more than 
 one swing during each coincidence, take the mean of the 
 times of the first and last rings as the time of the coin- 
 cidence. 
 
 (6) To obtain from the data a more precise average 
 value of the time between successive coincidences, proceed 
 as follows: Find the difference in time between the first 
 and sixth coincidences, the second and seventh, and so 
 on, and take the mean. From this the average time be- 
 tween successive coincidences may be found and the 
 period calculated. Be sure to note whether the pendulum 
 was gaining or losing on the clock. 
 
 (c) Calculate the value of g for the two cases and take 
 the mean. 
 
 (d) What effect would be produced upon the vibration 
 of a pendulum by carrying it, (1) to a mountain top, (2) 
 from the equator to the pole of the earth? In what way 
 does the pendulum used in this experiment fall short of 
 the requirements for a simple pendulum? What is the 
 object of taking a small amplitude of vibration? 
 
14 MOMENTUM [29 
 
 29. MOMENTUM. 
 
 The object of this experiment is to make a direct test 
 of the truth of the second law of motion in the form in 
 which the law was stated by Newton, viz. : that the rate 
 of change of momentum is proportional to the force 
 producing it. Incidentally it also serves to determine 
 the acceleration of gravity. 
 
 The apparatus consists of a brass plate about fifteen 
 inches long and two and a half inches wide to which 
 is soldered, near the bottom, a funnel of sheet brass 
 with a small re-entrant funnel at its mouth and with a 
 side outlet. At the upper end of the vane are two project- 
 ing knife edges. 
 
 (a) Weigh the vane. Locate its center of gravity 
 and measure the distance from the line of the knife 
 edges to the center of gravity, the distance from the 
 line of the knife edges to the center of the funnel. 
 
 (b) Measure the diameter of the nozzle with the 
 comparator. 
 
 (c) Adjust the vane so that it swings freely about 
 a horizontal axis and direct a small swift stream of water 
 from the nozzle horizontally into the funnel. Arrange 
 a scale parallel to the stream and near one side of the 
 vane. Measure its distance from the knife edge. 
 
 When conditions are steady, make five determinations 
 of the weight of water which is discharged in measured 
 time intervals, noting in each case the deflection of the 
 vane. 
 
 (d) From the diameter of the nozzle and the quantity 
 of water discharged in a second calculate the initial 
 
28-29] MOMENTUM 15 
 
 speed of the water. Assume that the final speed is zero 
 and calculate the change of momentum per second. 
 
 The deflecting force moment is that due to the impact 
 of the water jet, the restoring force moment is that due 
 to the weight of the vane. When the vane is in equilibrium 
 these two moments are equal. 
 
 Let the mass of the vane be M, the angle of deflection 
 a, the force of the water F, the distance from the line 
 of the knife edges to the center of gravity /, the distance 
 from the line of the knife edges to the center of the 
 funnel L, then 
 
 F L cos a = M sin a, and F = M I tan a/ L 
 Using this expression calculate the force F in grams. 
 
 Find the ratio of the rate of change of momentum to 
 the force in each case and determine its mean value. 
 
 The constancy of this ratio affords a verification of 
 the law: and the mean value measures the acceleration 
 of gravity. 
 
 30. MOMENT OF INERTIA. 
 
 A heavy disk is mounted so that it may be set in motion 
 by a mass, m, suspended by a cord wound on its cir- 
 cumference. An electrically driven tuning fork presses 
 a stylus against one side which has been smoked in a 
 gum camphor flame. The opposite side has a circular 
 scale graduated in degrees and two microscopes are 
 mounted, one on either side, to facilitate angular 
 measurements. 
 
 The moment of inertia of a body is defined as 7 = w 1 r 1 2 . 
 For a solid disk this may be shown by elementary calculus 
 
16 MOMENT OF INERTIA [30 
 
 to be M R 2 / 2, when M is the mass and R the radius of 
 the disk. 
 
 When torque L = M g R acts on the disk, an angular 
 acceleration a. is produced such that 
 
 L-Ia (1) 
 
 This may be written in the form I = L / a (constant) (2) 
 If the torque L acts for time t, we may write 
 
 Lt = I at = lu (3) 
 
 where co = a. Ms the final angular velocity of the disk, 
 supposing it to have started from rest. 
 
 This may be written / = L t / co (constant) (4) 
 
 If 6 is the angle turned through in time / starting from 
 rest, 6 = t co/2. Multiply each side of equation (3) by 
 this value and we obtain, on cancelling t, 
 
 L 6 = / coV2 (5) 
 
 The product L 6 represents the work done by the torque 
 in turning through the angle 6, expressed in radians. 
 The work may also be represented by m g s where 5 is 
 the distance the mass, m, falls, in giving the disk its 
 velocity co. Equation (5) may be written in the form 
 
 / = 2L 6 I co 2 (constant) (6) 
 
 If the weight falls for exactly one turn, 6 = 2 TT. 
 
 (a) Set the disk so that one edge of it projects past 
 the edge of the table. Smoke one side of the disk. Tie 
 a thread to the 50 gm. wt. of just sufficient length that 
 it does not quite slip off the peg as the weight touches 
 the floor. Wind the thread up exactly one full turn 
 (one circumference of the disk) as shown by the micro- 
 scope and set the brake. Connect the dry cells to the 
 
30] MOMENT OF INERTIA 17 
 
 two binding posts on the tuning fork and, after starting 
 it, adjust the nut on the end of the axis till the stylus 
 presses tightly against the disk. Release the brake and 
 turn the handle on the rack only slowly so that the 
 trace on the smoked surface will be a spiral with the 
 successive turns pretty close together. Allow the disk 
 to turn at least two full turns after the weight strikes 
 the floor before applying the brake. The negative 
 acceleration due to friction is to be obtained from this 
 part of the trace. 
 
 (6) Take the disk from its bearings and mark with 
 the pin the first distinguishable wave crest and number 
 it 0. Mark the 10th, 20th, 40th, 80th, 120, etc., crests. 
 The falling weight should have touched the floor at the 
 end of exactly one turn. Slip ten or more waves on 
 each side of this point and begin again marking the Oth, 
 
 40th, 80th, etc., wave crest. 
 
 
 
 (c] Replace the disk in its bearings and set the two 
 microscopes. Read the angular positions of the start 
 and of the interval marks and record. If n is the frequency 
 of the tuning fork, 40 / n sec. is the time interval between 
 two successive marks. The difference of successive 
 readings gives the angular distance in degrees passed 
 over in 40 / n sec. The difference of these successive 
 angular distances gives the angular acceleration in 
 degrees per 40 / n sec. per 40 / n sec. In like manner 
 determine the negative acceleration due to friction and 
 correct the previous value. Express the acceleration in 
 radians / sec. / sec. 
 
 (d) Repeat (a) using a 100 gm. suspended weight. 
 
 (e) Plot a curve for each trial using vibrations (0, 
 40, 80, etc.) as ordinates and the corresponding angular 
 position readings as abscissae. Prolong the curve to 
 
18 MECHANICAL EQUIVALENT OF HEAT [30-31 
 
 show the vibration numbers corresponding to the start- 
 ing point of the trace, and 360 later. Read from the 
 curve the . total number of vibrations made while the 
 disk was turning through the first 360. This gives the 
 time interval t. Determine the angular velocity o> from 
 the angular distance passed over in several 40 wave 
 intervals after the weight had struck the floor. 
 
 (/) Substitute the known values for each trial in 
 formulae (2), (4) and (6) and compare the value of / 
 found with the value of MR 2 / 2. Compare m g s with 
 I o> / 2. 
 
 Question: In falling, all the potential energy of the 
 falling mass does not go into kinetic energy of rotation 
 of the disk but some remains as kinetic energy of the 
 falling mass (mv 2 /2). This was ignored in the ex- 
 periment. Are the consequent experimental values of / 
 too large or too small? 
 
 31. MECHANICAL EQUIVALENT OF HEAT 
 
 To determine the mechanical equivalent of heat by 
 Callendar's method. 
 
 The number of units of mechanical work which is equiv- 
 alent to the calorie of heat is called the mechanical equiva- 
 lent of heat. Most of the methods employed in determin- 
 ing it produce the heat by means of mechanical work 
 done against friction. In Callendar's method a measur- 
 able amount of work done against the friction between a 
 stationary silk belt and a revolving vessel is converted 
 into heat in a known mass of water contained in the 
 vessel. The apparatus consists of a brass cylindrical 
 vessel which contains a known mass of water and whose 
 axis is horizontal. This cylinder can be rotated at a 
 moderate speed by hand or by motor. Over the surface 
 
31] MECHANICAL EQUIVALENT OF HEAT 19 
 
 of the cylinder a silk belt is wound so as to make one and a 
 half complete turns. From the ends of this belt are sus- 
 pended known masses, adjusted so as to provide a force- 
 moment which will oppose the rotation of the vessel. An 
 automatic adjustment for equilibrium is secured by the 
 use of a light spring balance which acts in direct opposition 
 to the weight at the lighter end of the belt. This spring 
 balance contributes only a small part to the effective 
 difference of load between the two ends of the belt, hence 
 small errors in its reading are relatively unimportant. 
 The masses suspended from the belt are approximately 
 adjusted by trial to suit the friction of the belt, the final 
 adjustment being automatically effected by the spring 
 balance. A counter registers the number of revolutions; 
 and a bent thermometer, inserted through a central open- 
 ing in the front end of the cylinder, measures the tem- 
 perature. 
 
 If M is the mass at the heavier end of the belt, m the 
 mass at the lighter end, and F the reading of the spring 
 balance, then the force acting to oppose the rotation of 
 the cylinder is (M - m + F) g, where g is the acceleration 
 due to gravity. The work done in overcoming this force 
 during one revolution of the cylinder is 2wr (M - m + F) g. 
 If, in n revolutions, the water of mass W is raised from 
 TV* C. to r 2 C., we have, by equating the work done and 
 the heat generated: 
 
 (1) 2irrn (M - m + F) g = (W + w) (T* - 7\ + R) J, 
 
 where w is the water-equivalent of the cylinder and the 
 thermometer, .R is a temperature-correction to compensate 
 for radiation, conduction, and the viscosity of the water, 
 and J is the mechanical equivalent of heat. 
 
 (a) Fill the cylinder half full of water at room tem- 
 perature. Do not use the bent thermometer to determine 
 
20 MECHANICAL EQUIVALENT OF HEAT [31 
 
 the room temperature, as bending may have shifted its 
 true scale above the bend. Suspend masses from the 
 ends of the belt, so that, when the cylinder is rotated 
 at moderate speed, the masses hang free. 
 
 Read the temperature t\ of the water, loosen the belt 
 so as to eliminate the friction between it and the cylinder, 
 and give 100 turns of the cylinder, at the rate just deter- 
 mined. Record the final temperature 2 . This is done to 
 determine the rate at which the temperature of the water 
 is changing, due to radiation, conduction, and the vis- 
 cosity of the water, just before the run in (b) is made. 
 
 (b) Adjust the belt and masses as at first, read the 
 temperature Ti of the water and rotate the cylinder at 
 uniform speed until the temperature has risen about 3 or 
 4 degrees. Again record the temperature T 2 . Record the 
 number of revolutions n, the masses M and m, and the 
 force F. 
 
 (c) Loosen the belt and give 100 turns at about the 
 the same rate as used in (b), recording the initial and 
 final temperatures t 3 and i*. This is done to determine 
 the rate at which the temperature is changing, due to 
 radiation, conduction, and the viscosity of the water, 
 just after the run in (b) is made. 
 
 (d) For the n revolutions of (b), the loss in temper- 
 ature due to radiation, conduction, and the viscosity 
 of the water is 
 
 (2) R=^L (*.-*. + *.-*) 
 
 100 2 
 
 From the data and equations (1) and (2) determine the 
 value of /. 
 
 (e) Point out the principal sources of error. 
 
32] EFFLUX OF GASES 21 
 
 32. EFFLUX OF GASES. RELATIVE DENSITIES* 
 
 The object of this experiment is to find the relative densities 
 of certain gases from the observation of the relative times of 
 efflux of equal volumes of these gases through a small aperture. 
 
 The ratio of the densities of two gases, under the same 
 conditions as to pressure, is equal, very approximately, 
 to the inverse ratio of the squares of the speeds with 
 which the gases escape through a fine openijig in a dia- 
 phragm. Since the time of escape of a given volume will 
 be inversely as the speed of efflux, it follows that the ratio 
 of the densities of two gases is equal to the direct ratio of 
 the squares of the time of efflux of equal volumes under the 
 same conditions. This relation was experimentally dis- 
 covered by Bunsen. For a proof of it, from the energy 
 relations, see Duff, Sect. 230. 
 
 The gas-holder consists of a glass bulb, at the top of 
 which is a three-way stop-cock and a diaphragm with a 
 fine opening. The bulb is connected with a reservoir of 
 oil of low vapor pressure. The three-way cock allows 
 communication to be made with the outside for filling or 
 with the diaphragm. 
 
 (a) First fill the bulb with dry air. To do this, turn 
 the stop-cock so as to put the bulb in communication 
 with the air, and allow the oil to fill the bulb. This 
 drives out most of the contained gas. Connect the bulb 
 with the calcium chloride drying tube and raise the 
 plunger in the reservoir. This operation will fill the bulb, 
 and by repeatedly emptying and filling the bulb, it will 
 become practically freed of the moist air or other gas 
 previously contained in it. 
 
22 EFFLUX OF GASES [32 
 
 Turning the stop-cock so that the gas in the bulb is 
 in communication with the diaphragm, note the time 
 when the surface of the oil is on a level with the lower 
 mark on the bulb. Again note the time when the oil 
 passes the upper mark on the bulb and quickly close the 
 stop-cock. 
 
 Repeat, making two or three determinations of 
 the time of efflux for the given volume of air, and take 
 the mean. 
 
 (6) Repeat (a), filling the bulb with illuminating gas, 
 following the directions there given for filling the bulb, 
 the bulb being connected directly to the source of the 
 gas used. Note the time of efflux between the same 
 two positions used in (a). This insures the same con- 
 ditions as to pressure in the two cases. 
 
 (c) Repeat the experiment, using dry carbon dioxide. 
 
 (d) Calculate the relative densities, referred to air, of 
 the gas used in (6). Taking the density of dry air under 
 standard conditions to be 0.001293 gm. per cc., find the 
 density, under standard conditions, of the gases used. 
 What laws have been used, or assumptions made, in 
 answering the requirement of the preceding sentence? 
 
33] CALIBRATION OF A SET OF WEIGHTS 23 
 
 33. CALIBRATION OF A SET OF WEIGHTS 
 
 By the calibration of a set of weights is meant the 
 determination of the amount by which each one of them 
 is in error. The simplest way of doing this is by com- 
 paring each weight with a standard of the same nominal 
 (marked) value. When no set of standards is available, 
 it is sufficient to know the variation within a given set 
 of weights from their marked values. This method of 
 calibration* is based on the assumption that one of the 
 weights of the set is correct. The errors in the others 
 are then determined by the following proceedure: 
 
 Assuming provisionally that one of the smaller weights 
 of the set is correct, we determine the errors in all the 
 other weights using this as our standard. When the 
 error in the largest weight is known in terms of this 
 standard, we reverse matters and adopt the largest 
 weight as a permanent standard, and recalculate the 
 errors on this basis. The largest weight is adopted as 
 the permanent standard because it is least likely to be 
 in error. 
 
 (a) The comparisons are best carried out by the 
 method of substitution. Place the 1' gram piece on the 
 left-hand pan and balance it approximately by means 
 of the rough unadjusted 1 gram weight. Take two sets 
 of observations for the determination of the rest point 
 by the method of vibrations, and determine the sensitive- 
 ness of the balance for this load, using the rider, if necess- 
 sary, to secure a rest point which is on the scale. 
 
 Now replace the 1' gram piece by the 1" gram piece to 
 be compared with it. The two are distinguished by 
 
 *Richards, The Jour, of the Am. Chem. Soc., vol. 22, 1900. 
 
24 CALIBRATION OF A SET OF WEIGHTS [33 
 
 small dots punched above and to the right of figure 1. 
 Do not change the load in the right-hand pan (tare). 
 
 Take readings for a new determination of the rest point. 
 Take similar readings for the 1'" gram piece. 
 
 (6) Place the 2 gram piece on the left-hand pan and 
 balance it approximately by means of the unadjusted 
 2 gram weight (tare). Make two determinations of the 
 rest point, and then, replacing the 2 gram weight with 
 the 1' and I" gram weights, make two further deter- 
 minations. 
 
 (c) Proceeding in this way, compare the 5 gram piece 
 with the sum of the 2 ' + I' + I" + I'", and the 10' gram 
 piece with the sum of the 5+2 + 1' + 1" + 1'", deter- 
 mining the sensitiveness with each load. 
 
 (d) From the rest point and sensibility observations 
 in (a), determine the values of the 1" gram and V" gram 
 pieces in terms of the V as a standard, remembering 
 that an increase in weight shifts the rest point towards 
 the larger numbers. 
 
 Using the corrected value of the 1" gram piece, deter- 
 mine from your data taken in (b) the value of the 2 gram 
 piece in terms of your provisional standard. Using these 
 corrected values, determine in a similar fashion the value 
 of the 5 and 10' gram pieces. 
 
 Assuming that the 10' gram piece is to be your standard, 
 find the proportional part of its value which should be 
 assigned to the 5, 2, and 1 gram pieces. Tabulate your 
 data and .results as follows : 
 
 Place in the first column the nominal (marked) values 
 of the weights; in the second column, the preliminary 
 
33] 
 
 CALIBRATION OF A SET OF WEIGHTS 
 
 25 
 
 values in terms of the 1' gram standard. Place in the 
 third column the proportional part of the value of the 
 10' gram weight which should be assigned to each weight 
 on the assumption that the 10' gram weight is your 
 new standard. By subtracting the values in the third 
 column from those in the second we obtain the corrections 
 sought, which should be listed in column four. The 
 values in grams with the 10' gram weight as the standard 
 should be listed in column five. A sample tabulation is 
 given below. 
 
 Calibration of Weights, Set C-7, December, 1915, G. W. R. 
 
 1 1 
 
 63 
 
 O efl 
 
 z> 
 
 rt en *c w 
 
 Z % 6 
 
 PH 
 
 PU 00 f) 
 
 .- 
 
 a ! 
 
 (10 
 
 Standard 
 
 .99984 
 
 + .00016 
 
 1.00016 
 
 d'O 
 
 1.00036 
 
 .99984 
 
 + .00052 
 
 1.00052 
 
 (!'") 
 
 1.00054 
 
 .99984 
 
 + .00070 
 
 1.00070 
 
 (2) 
 
 2.00035 
 
 1.99968 
 
 + .00067 
 
 2.00067 
 
 (5) 
 
 4.99914 
 
 4.99920 
 
 - .00006 
 
 4.99994 
 
 (ioo 
 
 9.99839 
 
 9.99839 
 
 Standard 
 
 10.00000 
 
26 CALIBRATION OF THE TUBE [34 
 
 34. ABSOLUTE CALIBRATION OF A THERMO- 
 METER 
 
 To plot a curve from which the true temperature may be 
 obtained corresponding to each scale-reading of a given 
 mercurial thermometer. 
 
 Such a curve is called the calibration curve of the ther- 
 mometer. The process of obtaining it is absolute since 
 it does not involve comparison with a standard ther- 
 mometer. This process consists, first, in determining 
 the absolute corrections for two separate scale-readings 
 on the thermometer, preferably near its fixed points, 
 that is, the points on its scale corresponding to the tem- 
 perature of melting ice and the temperature of water 
 boiling under standard pressure. The thermometer tube 
 between the two points is then calibrated because of the 
 possibility that its bore may not be uniform, and the 
 relative corrections thus determined are superimposed 
 graphically upon the correction curve resulting from 
 the absolute corrections at or near the two fixed points. 
 
 (a) Calibration of the Tube. Break off a portion 
 of the thread of mercury about ten degrees in length. 
 For this purpose, first invert the thermometer and let 
 enough mercury flow into the small cistern to fill it about 
 half full; then, holding the thermometer in a horizontal 
 position, tap or jar it lightly lengthways to break the 
 mercury in the small cistern loose from the rest. The 
 mercury in the stem will now flow back into the bulb 
 and leave the stem free for the thread of mercury which 
 must be jarred loose from the mercury in the small cis- 
 tern. Ask for assistance, if necessary. 
 
 Jar the lower end of the thread to the zero-point of 
 the scale and read the position of the upper end (which 
 
34] CORRECTION FOR THE FIXED POINTS 27 
 
 will be near the 10 mark) to tenths of a degree. Then 
 jar the lower end of the thread to the 10 mark and read 
 the upper end. Repeat with the lower end at the succes- 
 sive points 20, 30, 40, etc., up to 90. Then take the 
 readings in the reverse order, setting the upper end of 
 the thread successively at 100, 90, etc., down to 10, 
 and reading the position of the lower end each time. 
 The object of these readings is to find the length of the 
 thread in each of the ten intervals between and 100 ; 
 by means of the two series it is possible to find the 
 average value of the length of the thread for each interval. 
 
 (6) Correction at the Lower Fixed Point. Put 
 
 the thermometer through the cork in a test-tube, having 
 filled the latter about half full of distilled water. Place 
 the tube in a freezing mixture of shaved ice and salt, 
 and stir the water around the thermometer until it begins 
 to freeze. Read the thermometer. By warming the tube 
 in the hand and repeating the freezing process, obtain 
 several readings. 
 
 (c) Correction Near the Upper Fixed Point. 
 
 Place the thermometer through the cork in the tube at 
 the top of the boiler, with the bulb well above the sur- 
 face of the water. Boil the water so that the steam issues 
 freely, but not with any perceptible pressure, from the 
 vent. Read the thermometer when it becomes steady. 
 Allow the boiler to cool slightly, and repeat, making 
 three readings in all. If the instrument be provided 
 with a water-manometer, take the manometer-reading 
 simultaneously with the temperature-reading. Read the 
 barometer. 
 
 (d) Let the thermometer cool slowly to about the tem- 
 perature of the room, and repeat (6). If the freezing 
 point observed now is different from that observed in 
 
28 
 
 CORRECTION FOR THE FIXED POINTS 
 
 [34 
 
 (b), use the mean of the two values in the calibration 
 that follows. 
 
 (e) Assume that the temperature of freezing water 
 is 0C. From the Tables take the true boiling-point 
 temperature for the pressure observed in (c), find the 
 corrections of the thermometer for the scale-readings 
 observed in (b) and (c). Record these two corrections 
 by points on coordinate paper, having as abscissae the 
 
 -04 
 
 Fig- 2. This plot shows the calibration curve corresponding to the 
 sample data given on page 18, with an observed freezing point of 
 + 0.2C and boiling point of 99.4C when the true temperature was 
 99.8C. 
 
 scale-readings of the given thermometer in degrees, and 
 as ordinates the corresponding corrections in tenths of 
 a degree but on a magnified scale. (See Fig. 2). Correc- 
 tions should be plus (+) if they are to be added to the 
 observed to give the true temperatures, minus (. ) if 
 they are to be subtracted. 
 
34] THERMOMETER CALIBRATION 29 
 
 Connect these two points by a straight line. The 
 ordinate of this straight line at any point gives the cor- 
 rection of the thermometer at that scale-reading on the 
 assumption that the bore of the thermometer is uniform 
 throughout the whole range. In general this assump- 
 tion is not justified, and there must be added to this 
 correction at each point another correction due to the 
 inequalities of the diameter of the bore. 
 
 In order to find the bore correction, proceed as follows. 
 Determine the mean length of the thread in all of its 
 different positions. If the bore were uniform, each of 
 the observed lengths would equal this mean value. From 
 the deviation between the observed and mean thread 
 lengths, calculate the length which a thread whose mean 
 length was exactly ten degrees would have when placed 
 in the first interval. This would be in the same ratio to 
 the observed thread length as ten degrees is to the mean 
 thread length. We will call this the reduced thread length. 
 An actual rise of exactly ten degrees in temperature will 
 produce an apparent rise equal to this amount. The 
 difference between this reduced thread length and ten 
 degrees when the sign is reversed, gives the bore correction 
 for a temperature of approximately ten degrees. 
 
 Proceed similarly with each interval. The total cor- 
 rection which must be applied at any point of the scale 
 is found, by adding together the corrections for all the 
 intervals below that point. 
 
 Since, by definition, the mean thread length is one 
 tenth the sum of all the observed thread lengths, the 
 total bore correction at 100 will necessarily be zero. 
 
 The sample set of data given below will illustrate the 
 method. / is the observed thread length in a given interval, 
 
so THERMOMETER CALIBRATION [34 
 
 which may be an average of several determinations. L is 
 its mean value for all intervals; r is the reduced thread 
 length in a given interval. Its mean value is ten degrees. 
 10 - r gives the correction for each interval, and the sum 
 of all such corrections gives the total at a certain point 
 of the scale. 
 
 Intervals Average/ r Dif.(10 r) Bore- Corrections 
 
 1st 14.16 10.01 -.01 At 0= .00 
 
 2nd 14.25 10.07 -.07 " 10 =-.01 
 
 " 20 =-.08 
 
 " 30 =-.17 
 
 " 40 =-.36 
 
 11 50 =-.45 
 
 " 60 =-.44 
 
 " 70 =-.43 
 
 " 80 =-.32 
 
 " 90 =-.21 
 
 " 100= .00 
 
 The curve showing the resultant corrections for all 
 scale-readings from to 100 can therefore be obtained by 
 super-imposing the bore-corrections just found upon the 
 line drawn to show the corrections due to the errors in 
 the fixed points. For this purpose plot points whose 
 abscissae are 10, 20, 30, etc., and whose corresponding 
 ordinates are found by measuring from the slanting line, 
 already drawn, distances equal to the corresponding bore- 
 corrections measuring up or down from this line accord- 
 ing as the corrections are plus or minus. The smooth 
 curve, which should now be drawn through these plotted 
 points, is the calibration curve of the thermometer. 
 
 What are the true temperatures corresponding to the 
 scale-readings 0, 25, 50, 75 and 100 on the given 
 thermometer? 
 
 3rd 
 
 14.27 
 
 10.09 
 
 -.09 
 
 4th 
 
 14.42 
 
 10.19 
 
 -.19 
 
 5th 
 
 14.27 
 
 10.09 
 
 -.09 
 
 6th 
 
 14.13 
 
 9.99 
 
 +.01 
 
 7th 
 
 14.13 
 
 9.99 
 
 +.01 
 
 8th . 
 
 14.00 
 
 9.89 
 
 +.11 
 
 9th 
 
 14.00 
 
 9.89 
 
 +.11 
 
 10th 
 
 13.85 
 
 9.79 
 
 +.21 
 
 Mean 
 
 = 14.15 =L 
 
 10.00 
 
 
35] HEAT OF SOLUTION 31 
 
 35. HEAT OF SOLUTION. 
 
 To find the quantity of heat liberated by the solution of 
 one gram of a given salt in water. 
 
 When a salt is dissolved, the temperature may rise 
 or fall, due to the liberation or absorption of heat energy. 
 In the latter case, the heat of solution is said to be nega- 
 tive. The amount of heat evolved depends on the final 
 concentration of the solution, being greatest when the 
 solution is carried to infinite dilution. 
 
 The quantity of heat evolved is measured in terms of 
 the change in temperature of the solution itself. Now 
 the specific heat of a solution may differ very appreciably 
 from the specific heat of water, and can only be accurately 
 determined by experiment. Approximately, however, 
 the heat capacity of a solution is the same as though 
 it consisted of a simple mixture of water and the dissolved 
 material, in the solid form. We will base our calculations 
 on the assumption that error introduced by this approx- 
 imation is negligible. 
 
 The following thermal changes occur during solution: 
 (1) Potential energy is transformed into heat (positive 
 heat of solution) or heat into potential energy (negative 
 heat of solution). (2) Heat is absorbed (or given up) 
 by the water, calorimeter, etc., in changing from the 
 initial to the final temperature. (3) Heat is absorbed 
 (or given up) by the salt in changing from the room 
 temperature to the final temperature reached by the 
 solution. 
 
 Calling M the mass of water used, W the water- 
 equivalent of the calorimeter and accessories, w the 
 water-equivalent of the solid salt, to the room temperature, 
 
32 HEAT OF SOLUTION [35 
 
 ti and t z the initial and final temperatures respectively 
 of the water, and 5 C the heat evolved per gram of solute 
 on forming a solution of concentration c, write the 
 equation which will enable you to calculate 5 C . 
 
 (a) Fill the calorimeter about two-thirds full of water 
 and, by weighing, determine the amount of water used. 
 Weigh out a portion of the salt furnished, sufficient to 
 give a 20 per cent solution (weight of salt 20 per cent 
 of the total.) 
 
 (b) The stopper and thermometer should fit tightly 
 enough so that the whole calorimeter may be shaken to 
 bring about uniform temperature in the water or solution. 
 Shake until the water has reached a constant temperature. 
 Throw in the salt, and after further shaking read the 
 temperature when it has again become constant. Record 
 the room temperature. 
 
 (c) Repeat (a) and (b), using portions of salt giving 
 concentrations of 15, 10, and 5 per cent, approximately. 
 
 (d) Determine the water-eqiuvalent of the calorimeter 
 and accessories by the method explained on page 44, 
 Part 1 (Calorimetry). 
 
 (e) Calculate the heat evolved per gram of solute 
 in each of the three cases. Use the tabular value of 
 specific heat for the solid salt, and assume it to be at 
 room temperature before solution. 
 
 Draw a curve showing the relation between heat 
 evolved and concentration. The curve may be assumed 
 to be a straight line, and when produced to the axis of 
 zero concentration, gives the heat of solution. 
 
35-36] HEAT OH NEUTRALIZATION 33 
 
 (/) To estimate the error of your result, read off the 
 deviation of each plotted point from the curve. The 
 mean deviation, expressed in per cent, gives the approx- 
 imate probable error. 
 
 36. HEAT OF NEUTRALIZATION. 
 
 When an aqueous solution of a strong acid is poured 
 into an aqueous solution of a strong alkali until a neutral 
 mixture is formed, the essential chemical reaction which 
 occurs is the formation of water. For instance, if aqueous 
 solutions of hydrochloric acid and sodium hydroxide are 
 made to form a neutral mixture, although the mixture is 
 a solution of sodium chloride (table salt), the only chemical 
 reaction occurring is the formation of water. The heat 
 generated is called the heat of neutralization. The object 
 of the present experiment is to determine the heat of neutral- 
 ization corresponding to the formation of a gram-molecular 
 weight of water. In the case just mentioned, this will 
 occur when 1000 gm. each of normal solutions of the acid 
 and the alkali are mixed. 
 
 A 0.5 normal solution of each of the above compounds 
 is furnished. By a normal solution is meant one which, 
 in 1000 cubic centimeters of the solution, contains a 
 mass of the compound (which is to enter into the new 
 combination) equal in grams to its molecular weight. 
 Thus the normal solution of sodium hydroxide is a solution 
 which contains, in 1000 cc. of the solution, 40 gm. (23 + 
 16 + 1) of sodium hydroxide, or 23 gm. of sodium, 16 gm. 
 of oxygen, and 1 gm. of hydrogen. The 0.5 normal 
 solution contains one-half as much in -the same volume of 
 solution. 
 
 It is evident that if equal volumes of these solutions 
 be mixed, the reaction will be just completed, and the 
 
34 HEAT OF SOLUTION [36 
 
 result will be a neutral solution of sodium chloride. The 
 solutions are to be mixed in the calorimeter cup at as 
 nearly as possible the same temperature, and the resulting 
 rise of temperature noted. The alkali should be placed 
 in the cup, and the acid added to it. The acid, being 
 immediately neutralized, will then have no action on 
 the metal of the cup. 
 
 (a) Measure out 100 cc. of the sodium hydroxide 
 solution in the cup, and the same volume of the hydro- 
 chloric acid solution in the beaker. Wet the inside of 
 the beaker with the acid solution before pouring the 
 measured amount into it. This is to compensate for the 
 liquid which remains in the beaker when later it is emptied. 
 A small error is introduced by taking the second ther- 
 mometer out of the beaker after reading its temperature, 
 but this may be neglected. 
 
 If care has been taken not to handle the cup and beaker 
 any more than is necessary, the two temperatures should 
 be very nearly the same when ready for use. It may safely 
 be assumed that the resulting solution of sodium chloride 
 has risen to the final temperature from the mean of the 
 two initial temperatures. Make two trials. 
 
 (b) Repeat the work with solutions of potassium 
 hydroxide and sulphuric acid. 
 
 Calculate for each set of data the quantity of heat 
 which would have been evolved if 1000 cc. of the normal 
 solution had been used in each case. Will this cause the 
 formation of one gram-molecular weight of water? 
 
 The specific heat of the sodium chloride solution is 
 0.987 calories, of the potassium sulphate solution 0.985 
 calories per gram. 
 
37] VARIATION OF BOILING POINT 35 
 
 37. VARIATION OF BOILING POINT WITH 
 PRESSURE. 
 
 References. Duff, pp. 227, 228; Kimball, p. 298. 
 
 There are two methods employed in studying the 
 variation in the boiling point of a liquid with the pressure 
 upon its free surface. By the dynamic method the pressure 
 above the boiling liquid is varied by means of an air-pump 
 and the corresponding temperature observed. By the 
 static method the temperature of the liquid, suitably 
 enclosed, is varied by means of baths and the correspond- 
 ing pressure observed. 
 
 The object of the present experiment is to study the variation 
 in the boiling point of water, employing the dynamic method. 
 
 The apparatus consists of an air-tight boiler to hold the 
 liquid, a steam-condenser, around which cold water cir- 
 culates, an air-tight chamber large enough to equalize 
 sudden changes in the pressure, an air-pump for reducing 
 the pressure and a manometer for measuring the same. 
 These are connected up in the order named and made 
 air-tight so far as the air outside is concerned. 
 
 (a) The circulation of water should first be started 
 through the steam-condenser, which is a glass or metal 
 tube to jacket the tube leading from the boiling-flask, 
 thus condensing the steam as it comes from the flask. The 
 thermometer should be passed through the stopper of the 
 flask and so regulated that its bulb will be in the rising 
 steam, and not in the water. The connection with the 
 large glass bottle serves to equalize sudden changes in 
 
-.36 VARIATION OF BOILING POINT [37 
 
 pressure due to irregularities in the boiling. In heating 
 the water do not play the flame on the flask directly under 
 the glass beads, but rather to one side and below the 
 water-line. 
 
 First boil the water at atmospheric pressure, reading 
 the manometer and noting the temperature. Then take 
 a series of readings at intervals of about 5 cm. pressure, 
 until the "bumping" becomes so violent as to render 
 further readings impracticable. Before each reading, 
 after pumping to the pressure desired, close the stop- 
 cock over the jet-pump, wait a short time for the pressure 
 to reach equilibrium, and then make the reading of boiler 
 temperature and corresponding pressure. Put the pump 
 again in connection, obtain a new pressure, and repeat 
 the readings. Before turning off the water at the jet, be 
 sure each time to let air into the apparatus by opening 
 the pinch-cock nearest the pump, otherwise water will 
 flow back into the tubing. 
 
 (b) Take a series of readings with increasing pressures 
 up to atmospheric pressure, choosing values different 
 from the previous ones. 
 
 (c) Plot the observations on coordinate paper, using 
 pressures as ordinates and temperatures as abscissae. 
 From the curve find the boiling point of water at a pressure 
 of i atmosphere. 
 
 (d) Discuss the phenomena of this experiment in 
 connection with the difficulties experienced in cooking 
 food at high altitudes. Could determinations of the 
 boiling point of water be used to measure altitude, and 
 how? 
 
38] CONSTANT VOLUME AIR THERMOMETER 37 
 
 38. CONSTANT-VOLUME AIR- THERMOMETER. 
 
 References. Duff, p. 202. 
 
 The object of this experiment is to study the law of varia- 
 tion of the pressure of a given mass of enclosed air whose 
 volume is kept constant while its temperature is changed. 
 
 The air is enclosed in a glass bulb mounted on a frame. 
 The frame is placed near a table so that the bulb may be 
 surrounded by a water-bath, by shaved ice, or by a steam- 
 bath, the table and an iron stand being made use of to 
 support each bath in turn. A thermometer is placed in 
 the bath to give its temperature. The pressure on the 
 enclosed gas is regulated by raising or lowering the open 
 tube. The value of this pressure may be determined 
 from the barometer-reading and the difference in the 
 levels of the mercury on the two sides of the frame. 
 Each^time before taking the readings, the volume of the 
 air in the bulb is made the same by bringing the mercury 
 meniscus to the level of the wire point inside the glass 
 tube attached to the bulb. 
 
 Caution: The mercury oh the bulb side should always 
 be lowered some distance before changing to a lower 
 temperature. Be especially careful to do this before re- 
 moving the steam-bath when you have taken a reading at 
 the boiling point; otherwise, on cooling, the mercury will 
 run into the bulb. Do not hurry in taking the readings 
 after changing the temperature, but wait until the meniscus 
 set at the wire-point remains stationary. 
 
 (a) Without any bath in the reservoir, while all is at 
 the room-temperature, bring the mercury to the wire 
 point and determine the difference in level of the mercury 
 columns. Record the room-temperature, and the barom- 
 eter-reading. 
 
38 CONSTANT VOLUME AIR THERMOMETER [38 
 
 (b) After having lowered the mercury on the bulb 
 side, surround the bulb with shaved ice, and then deter- 
 mine the pressure with the menicus at the wire point. 
 The temperature may be taken as 0C. 
 
 Melt the ice with warm water, and then make a series 
 of determinations of the pressure when the water in the 
 vessel is successively at a temperature of (approximately) 
 10, 20, 30, 45, 60, and 80C. 
 
 Remove the water-bath, substitute a steam-bath in its 
 place, and make another determination. The temperature 
 of the steam-bath may be found by determining the boiling 
 point of water from the known atmospheric pressure (see 
 Tables). 
 
 Arrange all observations in tabular form. 
 
 (c) Plot on coordinate paper the results of (6), using 
 temperatures as abscissae and the corresponding pressures 
 as ordinates. Draw a smooth curve which will best repre- 
 sent the average position of the points of the plot, and 
 extend the curve until it intersects the line of zero 
 pressure. 
 
 Calculate from the curve the mean increase of pressure 
 per degree increase in temperature from 0C. to 100C., 
 and divide the result by the pressure at 0C., using values 
 taken from the plot. This is the temperature coefficient 
 (ft) of pressure of a gas. Write it as a decimal and find its 
 reciprocal. The negative of this represents what point 
 on the absolute scale of temperatures? 
 
 (d) Write an equation connecting P Q , the pressure at 
 
 0; P t , the pressure at t\ t\ and ft. 
 
 t 
 
 Using this equation and the pressure obtained in (a), 
 calculate the temperature of the room, thus using the 
 
38-39] VAPOR PRESSURE AND TEMPERATURE 39 
 
 apparatus as a thermometer. Compare the result with 
 the room temperature as read from a mercury thermometer. 
 
 Show from your results how the pressure of the gas 
 varies with the absolute temperature, the volume remain- 
 ing constant. 
 
 39. VAPOR-PRESSURE AND TEMPERATURE. 
 
 References. Duff, p. 222; Kimball, p. 296. 
 
 The object of this experiment is to study the relation be- 
 tween the temperature and the pressure of saturated water- 
 vapor. 
 
 The method employed is that referred to in Exp. 37 as 
 the "static" method of determining the boiling point of a 
 liquid at different pressures. Two barometer tubes, 
 filled with mercury, are inverted and mounted side by 
 side in a vessel of mercury. One of the tubes contains, 
 above the mercury, water-vapor with an excess of water 
 present, while the other tube is left to be used as a barom- 
 eter. By means of a water-bath surrounding the upper 
 half of the tubes, the temperature of the water-vapor can 
 be brought to any desired point. The bath is connected 
 to a heater and the change in temperature is brought 
 about by circulation. The pressure of the saturated 
 water-vapor at any temperature will be the difference 
 between the heights of the mercury columns on the two 
 tubes. 
 
 At each temperature the pressure of a saturated vapor 
 of a given liquid has a definite value which depends on the 
 temperature and the nature of the liquid, but is independ- 
 ent of the volume of the vapor. When the temperature is 
 raised, not only is the vapor heated and the pressure 
 raised, but more liquid is vaporized, so there are two 
 
40 VAPOR PRESSURE AND TEMPERATURE [39 
 
 influences tending to increase the pressure of the vapor. 
 The purpose of the present experiment is to plot the curve 
 which shows how rapidly the vapor-pressure increases as 
 the temperature is raised, in the case of water-vapor. 
 
 (a) Read the heights of the mercury columns in the 
 two tubes for ten different temperatures between room- 
 temperature and 80C. Record the height of each 
 column separately. To raise the temperature about 
 5 or 10 at a time, heat the boiler only for two or three 
 minutes, then remove the burner, and stir the water- 
 bath until a uniform temperature prevails. By this 
 time the water-vapor inside the tube will have reached 
 the temperature of the bath. In taking the temper- 
 ature-readings hold the bulb of the thermometer 
 slightly below the center of the space filled with water- 
 vapor. 
 
 Always wait until conditions have become steady before 
 taking readings at a new temperature. 
 
 (b) By replacing the hot water in the bath with cold 
 water, a little at a time, take a second series of readings 
 down to about room temperature. 
 
 (c) Plot a curve from the results of (a) and (b), with 
 the pressures of the saturated water-vapor as ordinates 
 and the temperatures as abscissae. Draw the curve so 
 that it will represent the average positions of all the 
 plotted points. 
 
 Determine from the curve the boiling point of water 
 at a pressure of 30 cm. 
 
 Does the pressure of saturated water-vapor increase 
 with the temperature more or less rapidly than does the 
 pressure of a gas kept at constant volume? 
 
 Would the results be different if the volume of the 
 saturated vapor were kept constant? 
 
40] HYGROMETRY 41 
 
 40. HYGROMETRY. 
 
 References. Duff, p. 279; Kimball, p. 304; Edser, p. 240. 
 
 In this experiment the dew-point and the relative and 
 absolute humidity of the air are to be determined. 
 
 The absolute humidity, d, is the density of the water- 
 vapor present in the air, and is usually expressed in grams 
 per cubic meter. The relative humidity is the ratio of the 
 amount of water-vapor actually present in the air to the 
 amount required to saturate it at the same temperature, 
 the latter quantity being the maximum amount of water- 
 vapor that can be held in suspension at that temperature. 
 The relative humidity is therefore equal to d/D, where D 
 is the maximum density of the water-vapor at the given 
 temperature. The dew-point is the temperature at which 
 the amount of water actually present in the air would sat- 
 urate it, that is, the temperature to which the air must be 
 lowered before the condensation of water will begin. The 
 pressure of water-vapor is the pressure which it would 
 exert by itself if there were no air present in the space 
 considered. By Dalton's law this is the pressure it actually 
 does exert when mixed with air. 
 
 If the unsaturated vapor -be assumed to behave like 
 a perfect gas, the ratio of actual to saturated vapor density 
 equals the ratio of actual to saturated vapor pressure. 
 Since cooling vapor to the dew-point does not change its 
 pressure, the saturated pressure at the dew-point equals 
 the actual pressure at atmospheric temperature, and 
 this may be read from the table without correction. In 
 this way we justify the ordinary proceedure of determin- 
 ing relative humidity as the ratio of the pressure p of 
 the water-vapor in the air to the pressure P of saturated 
 
42 HYGROMBTRY [40 
 
 water- vapor at that temperature; that is, relative humidity 
 is equal to p/P. 
 
 (I) Alluard's Hygrometer. 
 
 (a) Partially fill the hygrometer with ether, and 
 place a thermometer in the liquid. Cool the liquid by 
 pumping air slowly through it. When the tube and the 
 air immediately above it are cooled to the dew-point, 
 moisture appears on the tube, this being detected more 
 easily by comparison with the plate on either side. Note 
 the temperature at which the dew begins to form. Allow 
 the tube to become warm and record the temperature at 
 which the dew disappears. Take the mean of these two 
 as the dew-point. Make three such determinations of 
 the dew-point. 
 
 (6) From the Tables find the pressure of saturated 
 water-vapor at the dew-point and also at the temperature 
 of the room, and calculate the relative humidity. The 
 absolute humidity may be found by multiplying the rela- 
 tive humidity by D, the number of grams of saturated 
 water-vapor in a cubic meter of air at the room-tem- 
 perature (see the Tables). 
 
 II. Wet- and Dry-bulb Hygrometer, or Augustus Psy- 
 chrometer. 
 
 (a) In the wet- and dry-bulb hygrometer, one bulb is cov- 
 ered with wicking which dips into water, so that the bulb is 
 cooled by evaporation. Swing the hygrometer back and 
 forth in the air so as to increase the circulation of air about 
 the wet bulb. After the two thermometers come to con- 
 stant temperatures, record the temperature t of the dry 
 
40] HYGROMKTRY 43 
 
 bulb, and the temperature ti of the wet bulb. Read the 
 barometer. 
 
 (b) The following empirical formula may then be used : 
 p = ^-0.00086 (t-ti), 
 
 where p is the pressure of water-vapor present in the 
 atmosphere and the value of which is to be found; p\ the 
 pressure of saturated vapor at the temperature of the 
 wet-bulb (obtained from the Tables) ; and b is the baro- 
 metric pressure, all being expressed in millimeters of mer- 
 cury. Find the pressure P of saturated water-vapor at 
 the room-temperature from the Tables, and calculate the 
 relative humidity. Find then the absolute humidity as 
 in (b). From the Tables and the readings of the wet- and 
 dry-bulb hygrometer, find the dew-point. 
 
 Compare the values obtained in I and II for the humid- 
 ity and the dew-point. 
 
44 RATIO OF THE TWO SPECIFIC HEATS [41 
 
 41. RATIO OF THE TWO SPECIFIC HEATS OF 
 
 AIR. 
 
 References. Duff, p. 283; Edser, p. 321. 
 
 The object of this experiment is to obtain the value of the 
 ratio y of the specific heat of air at constant pressure to its 
 specific heat at constant volume. 
 
 The method employed is a modification of that used 
 first by Clement and Desormes. A quantity of the gas, 
 compressed in a large flask, is momentarily put in com- 
 munication with the atmosphere to allow its pressure to 
 fall adiabatically to atmospheric pressure, its temperature 
 simultaneously falling a little. The gas, when shut off 
 again from the atmosphere, gradually warms up to its 
 initial temperature, causing an appreciable rise in its 
 pressure. Let pi be the pressure in the compressed gas at 
 the start, Vi the volume of unit mass of the gas and ti its 
 temperature (the same as that of the room). Let p , v t , 
 and t z be the corresponding values of these quantities 
 immediately after communication between the com- 
 pressed gas and the atmosphere is established. Then 
 p 2 , v 2 , and ti will be the values of these same quantities 
 at the end, if p 2 is the final pressure. The gas has now been 
 in three conditions, as follows: 
 
 Condition Pressure Vol. of \gm. Temperature 
 I. pi Vi ti 
 
 II. po V, tt 
 
 III. p t V* h 
 
 The change from I to II was adiabatic, since no time was 
 allowed for heat to pass in or out of the gas by conduction 
 or radiation; hence, by the law for adiabatic changes in a 
 perfect gas, 
 
 (1) v^ pi = v^Y po, or = pi / po. 
 
41] RATIO OF THE TWO SPECIFIC HEATS 45 
 
 The change, from I to III was isothermal; hence, by 
 Boyle's law, 
 
 ( v 2 } 7 ( p l ) 7 
 (2) Vi P i=v>p 2 ,or j^J - j-j 
 
 Hence (pi / p^ = (pi / p ) ; or, taking the logarithm 
 and solving for 7 
 
 log pi - log pz 
 
 The desired ratio may be obtained, experimentally, 
 therefore, by observing the values of the three pressures. 
 
 The apparatus consists of a large carboy with the open- 
 ing ground flat, and covered with a piece of ground glass. 
 The enclosed space may at pleasure be opened to, or shut 
 off from, the atmosphere. The pressure of the enclosed 
 air is measured by an oil manometer, whilst air can be 
 forced into or withdrawn through another inlet. To 
 thoroughly dry the enclosed air, some strong sulphuric 
 acid is poured into the bottom of the carboy. 
 
 (a) Close the carboy, and with a bicycle pump intro- 
 duce enough air in the carboy to give a reasonably large 
 difference of pressure, as indicated by the manometer. 
 Shut off connection between the carboy and the pump, 
 and wait a few minutes until the temperature of the en- 
 closed air is the same as that of the room, which will be 
 when the manometer shows a steady, constant pressure. 
 Read the manometer and the barometer. To get the 
 value of the pressure-difference recorded by the manom- 
 eter, it will be necessary to know the density of the oil 
 used. This is posted on the apparatus. 
 
 (b) Open the carboy for a second, by removing the 
 glass plate, then close again. Wait some time until the 
 
46 THE HEAT OF FUSION OF TIN [41-42 
 
 temperature of the enclosed air has risen again to that 
 of the room, as indicated by a steady, constant difference 
 in pressure; then read the manometer. 
 
 (c) Repeat (a) and (b) twice. 
 
 (d) Using the data in (a) and (6), determine from 
 equation (3) the value of y for air, and take the mean of 
 the results. 
 
 Obtain from the Tables the value of 7 and compare 
 with the result just found by experiment. 
 
 Explain why the specific heat of a gas at constant 
 pressure should be greater than its specific heat at con- 
 stant volume. 
 
 42. THE HEAT OF FUSION OF TIN. 
 
 The object of this exercise is the determination of the heat 
 of fusion of tin from the average of the rate of cooling just 
 before and just after solidification. 
 
 The heat of fusion of any substance is defined as the 
 number of calories required to convert one gram of the 
 solid at the melting point into liquid at the same tem- 
 perature. 
 
 A single nickel-iron thermocouple is connected with a 
 sensitive galvanometer to determine the temperature, 
 the galvanometer deflections being converted into tem- 
 peratures by means of a calibration curve based upon the 
 deflection of the galvanometer for four standard temper- 
 atures: the melting point of zinc 419C., the melting 
 point of tin 232C., the boiling point of water 100C., and 
 0C. 
 
42] THE HEAT OF FUSION OF TIN 47 
 
 (a) Connect the thermocouple in series with the gal- 
 vanometer. Put one of the junctures in a beaker of 
 shaved ice and the other in a small graphite crucible 
 containing a small quantity of pure zinc. After the zinc 
 has been melted allow it to cool off by radiation. Arrange 
 a suitable shunt for the galvanometer, or insert sufficient 
 resistance in series so that when the deflection becomes 
 constant, as it will during the time of solidification of 
 the zinc, this scale deflection is as large as can conven- 
 iently be obtained on the galvanometer scale. After this 
 adjustment, the resistance of the circuit should remain 
 unchanged throughout the rest of the exercise. 
 
 (6) Determine the mass of the tin from the weight 
 of the crucible and the tin, and the given value for the 
 weight of the crucible. Lift the juncture from the zinc, 
 taking care to remove whatever particles of zinc cling to 
 it. Place this juncture in a small crucible containing at 
 least 100 grams of pure tin, heating it up until the gal- 
 vanometer deflection is about as large as that obtained 
 with the zinc. Put out the burner and allow the crucible 
 to cool by radiation when placed in a holder with a three- 
 point asbestos contact, reading the scale deflection every 
 30 seconds. Take five or six readings after the time of 
 constant deflection, which indicated the temperature of 
 solidification of the tin. 
 
 (c) Lift the juncture from the tin, place it in a small 
 beaker of boiling water and note the deflection. Take 
 a final reading with both junctures placed in shaved ice. 
 
 (d) Make a plot showing the relation between tem- 
 perature and the galvanometer deflections, using the 
 constant deflections obtained in (a), (6), and (c), which 
 
48 THE HEAT OF FUSION OF TIN [42 
 
 correspond to the temperatures 419, 232, 100, re- 
 spectively. Draw a smooth curve through these points. 
 
 With the aid of the above plot find the temperatures 
 corresponding to each scale deflection contained in (b). 
 Make a plot with temperatures as ordinates and times as 
 abscissae, using the data in (b) for the cooling of the tin. 
 
 From the curve find the rate of cooling in seconds per 
 degree from the plot, using a line drawn tangent to the 
 curve just before solidification, and a second value from 
 a line tangent to the curve just after solidification. From 
 the heat capacity of the crucible and the fluid tin, find 
 the rate of cooling in calories per second just before 
 solidification. And using the heat capacity of the crucible 
 and that of solid tin, find a similar value for the rate of 
 cooling just after solidification. From the average rate 
 of cooling and the time as taken from the plot, find the 
 total number of calories given out during solidification, 
 and calculate the heat of fusion of the tin. Specific heat 
 of graphite is 0.200, of porcelain 0.225, of fireclay 0.152, 
 of quartz 0.200, of fluid tin 0.0637, of solid tin 0.0588, 
 calories per gram respectively. 
 
PHYSICAL TABLES 
 
 49 
 
 USEFUL NUMERICAL RELATIONS. 
 Mensuration. 
 
 Circle: circumference = lirr; area = TTr 2 . 
 Sphere: area = 47Tr 2 ; volume = ^TTr 3 . 
 Cylinder: volume = TTr 2 /. 
 
 Angle. 
 
 1 radian = 57.2958 = 3437'.75. 
 1 degree = 0.017453 radian. 
 
 Length. 
 
 1 centimeter (cm.) = 0.3937 in. 1 inch (in.) = 2.540 cm. 
 
 1 meter (m.) = 3.281 ft. 1 foot (ft.) = 0.3048 m. 
 
 1 kilometer (km.) = 0.6214 mi. 1 mile (mi.) = 1.609 km. 
 
 1 micron (M) = 0.001 mm. 1 mil = 0.001 in. 
 
 1 sq. cm. 
 1 sq. m 
 
 0.1550 sq. in. 
 10.674 sq. ft. 
 
 Area. 
 
 Volume. 
 
 1 cc. = 0.06103 cu. in. 
 
 1 cu. in. = 35.317 cu. ft. 
 
 1 liter (1000 cc.) = 1.7608 pints. 
 
 Mass. 
 
 1 gram (gm.) 
 1 kilogram (kg.) 
 
 = 15.43 gr. 
 = 2.2046 Ib. 
 
 1 sq. in. = 6.451 sq. cm. 
 1 sq. ft. = 0.09290 sq. m. 
 
 1 cu. in = 16.386 cc. 
 
 1 cu. ft. = 0.02832 cu. m. 
 
 1 quart = 1.1359 liters. 
 
 1 grain (gr.) = 0.06480 gm. 
 1 pound (Ib.) = 0.45359 kg. 
 
 Density* 
 
 1 gm. per cc. = 62.425 Ib. per cu. ft. 
 1 Ib. per cu. ft. = 0.01602 gm. per cc. 
 
 Thermometric Scales. 
 
 C=5(F 32)/9 F = (9C/5) 4 32 
 
 (C= centigrade temperature; F Fahrenheit temperature) 
 
50 PHYSICAL TABLES 
 
 USEFUL NUMERICAL RELATIONS. 
 
 Force. 
 
 1 gram's weight (gm. wt.) = 980.6 dynes (g = 980.6 cm./sec. 1 .) 
 1 pound's weight (Ib. wt.) = 0.4448 megadynes (g = 980.6.) 
 
 (The "gm. wt.' is here defined as the force of gravity acting on 
 a gram of matter at sea-level and 45 north latitude. The "Ib. wt." is 
 similarly defined.) 
 
 Pressure and Stress 
 
 1 cm. of mercury at 0C. 1 in. of mercury at 0C. 
 
 = 13.596 gm. wt. per sq. cm. = 34.533 gm. wt. per sq. cm. 
 
 = 0.19338 Ib. wt. per sq. in. = 0.49118 Ib. wt. per sq. in. 
 
 Work and Energy. 
 
 1 kilogram-meter (kg. m.) = 7.233 ft. Ib. 
 
 1 foot-pound (ft. Ib.) = 0.13826 kg. m. 
 
 1 joule = 10 7 ergs. 
 
 1 foot-pound = 1.3557 X 10 7 ergs. (&= 980.6 cm./sec. 1 .) 
 
 1 foot-pound = 1.3557 joules (g = 980.6.) 
 
 1 joule = 0.7376 ft. Ib. (g = 980.6.) 
 
 Power (or Activity). 
 
 1 horse-power (H. P.) = 33000 ft. Ib. per min. 
 
 1 watt = 1 joule per sec. = 10 7 ergs per sec. 
 
 1 horse-power = 745.64 watts (g = 980.6 cm./sec. 1 ) 
 
 1 watt = 44.28 ft. Ib. per min (g = 980.6) 
 
 Mechanical Equivalent. 
 
 1 gm.-calorie = 4.187 X 10 7 ergs. 
 
 = 0.4269 kg. m. (g = 980.6 cm./sec. 2 .) 
 = 3.088 ft. Ib. (g = 980.6.) 
 
PHYSICAL TABLES 
 
 51 
 
 DENSITY OF DRY AIR. 
 
 (Values are given in gms. per cc.) 
 
 Temp. 
 
 Barometric Pressure 
 
 (Centimeters of Mercury) 
 
 C. 
 
 72 
 
 73 
 
 74 
 
 75 
 
 76 
 
 77 
 
 
 
 .001225! .001242 
 
 .001259 
 
 .001276 
 
 .001293 
 
 .001310 
 
 1 
 
 220 
 
 237 
 
 254 
 
 271 
 
 288 
 
 305 
 
 2 
 
 . 216 
 
 
 233 
 
 250 
 
 267 
 
 283 
 
 300 
 
 3 
 
 212 
 
 
 228 
 
 245 
 
 262 
 
 279 
 
 296 
 
 4 
 
 207 
 
 
 224 
 
 
 241 
 
 257 
 
 274 
 
 290 
 
 5 
 
 .001203 
 
 .001219 
 
 .001236 
 
 .001253 
 
 .001270 
 
 .001286 
 
 6 
 
 198 
 
 
 215 
 
 232 
 
 248 
 
 265 
 
 282 
 
 7 
 
 194 
 
 
 211 
 
 227 
 
 244 
 
 260 
 
 277 
 
 8 
 
 190 
 
 
 206 
 
 223 
 
 239 
 
 256 
 
 272 
 
 9 
 
 186 
 
 
 202 
 
 219 
 
 235 
 
 251 
 
 268 
 
 10 
 
 .001181 
 
 .001198 
 
 .001214 
 
 .001231 
 
 .001247 
 
 .001263 
 
 11 
 
 177 
 
 
 194 
 
 210 
 
 226 
 
 243 
 
 259 
 
 12 
 
 173 
 
 
 189 
 
 206 
 
 222 
 
 238 
 
 255 
 
 13 
 
 169 
 
 
 185 
 
 202 
 
 218 
 
 234 
 
 250 
 
 14 
 
 165 
 
 
 181 
 
 
 197 
 
 214 
 
 230 
 
 246 
 
 15 
 
 .001161 
 
 .001177 
 
 .001193 
 
 .001209 
 
 .001225 
 
 .001242 
 
 16 
 
 157 
 
 
 173 
 
 
 189 
 
 205 
 
 221 
 
 237 
 
 17 
 
 153 
 
 
 169 
 
 
 185 
 
 201 
 
 217 
 
 233 
 
 18 
 
 149 
 
 
 165 
 
 
 181 
 
 197 
 
 213 
 
 229 
 
 19 
 
 145 
 
 
 161 
 
 
 177 
 
 193 
 
 209 
 
 224 
 
 20 
 
 .001141 
 
 .001157 
 
 .001173 
 
 .001189 
 
 .001204 
 
 .001220 
 
 21 
 
 137 
 
 
 153 
 
 
 169 
 
 185 
 
 200 
 
 216 
 
 22 
 
 133 
 
 
 149 
 
 
 165 
 
 181 
 
 196 
 
 212 
 
 23 
 
 130 
 
 
 145 
 
 
 161 
 
 177 
 
 192 
 
 208 
 
 24 
 
 126 
 
 
 141 
 
 
 157 
 
 - 173 
 
 188 
 
 204 
 
 25 
 
 .001122 
 
 .001138 
 
 .001153 
 
 .001169 
 
 .001184 
 
 .001200 
 
 26 
 
 118 
 
 
 134 
 
 
 149 
 
 165 
 
 180 
 
 196 
 
 27 
 
 114 
 
 
 130 
 
 145 
 
 161 
 
 176 
 
 192 
 
 28 
 
 110 
 
 
 126 
 
 
 142 
 
 157 
 
 172 
 
 188 
 
 29 
 
 107 
 
 
 122 
 
 
 138 153 
 
 169 
 
 184 
 
 30 
 
 .001103 
 
 .001119 
 
 .0011341 .001149 
 
 . 01165 
 
 .001180 
 
 Corrections 
 
 for Moisture in the Atmosphere 
 
 Dew-point Subtract! Dew-point 
 
 Subtract Dew-point 
 
 Subtract 
 
 10 1 .000001 
 
 + 2 
 
 .000003 -i- 14 
 
 .000007 
 
 8 2 
 
 + 4 
 
 4 ' +16 
 
 8 
 
 6 2 
 
 + 6 
 
 4 +18 
 
 9 
 
 4 2 
 
 + 8 
 
 ' 5 +20 
 
 .000010 
 
 2 3 
 
 + 10 
 
 6 +24 
 
 13 
 
 3 
 
 + 12 
 
 6 +28 
 
 16 
 
52 
 
 PHYSICAL TABLES 
 
 DENSITIES AND THERMAL PROPERTIES OF GASES. 
 
 (The densities are given at 0C. and 76 cm. pressure, and the 
 specific heats at ordinary temperatures. The coefficients of cubical 
 expansion (at constant pressure) of the gases listed below are not 
 given in this Table; they are about the same for all the permanent 
 gases, being approximately 1/273 or 0.003663, if referred in each 
 case to the volume of the gas at 0C. The specific heats at constant 
 pressure and at constant volume are represented by the symbols 
 Sp. and Sv) . 
 
 Gas or Vapor 
 
 Formula 
 
 Density 
 (gms. per cc.) 
 
 Molecular 
 Weight 
 
 Sp ~j~ Sv 
 
 S P 
 
 (cals. 
 per gm.) 
 
 A it- 
 
 
 f) f)()1 OQQ 
 
 
 4.1 
 
 0907 
 
 Ammonia 
 Carbon dioxide 
 Carbon monoxide 
 Chlorine 
 Hydrochloric acid 
 Hydrogen 
 Hydrogen sulphide 
 Nitrogen, pure 
 
 NH 3 
 C0 2 
 CO 
 C1 2 
 HC1 
 H 2 
 H 2 S 
 N 2 
 
 0.000770 
 0.001974 
 0.001234 
 0.003133 
 0.001616 
 0.0000896 
 0.001476 
 0.001254 
 
 ft OO1 9^7 
 
 17.06 
 44.00 
 28.00 
 70.90 
 36.46 
 2.016 
 34.08 
 28.08 
 
 .33 
 .29 
 .40 
 .32 
 .40 
 .41 
 1.34 
 1.41 
 
 .530 
 .203 
 .243 
 .124 
 .194 
 3.410 
 .245 
 .244 
 
 Oxygen 
 Steam (100C.) 
 Sulphur dioxide 
 
 2 
 H 2 O 
 S0 2 
 
 0.001430 
 0.000581 
 0.002785 
 
 32.00 
 18.02 
 64.06 
 
 1.41 
 1.28 
 1.26 
 
 .218 
 .421 
 .154 
 
 DENSITY AND SPECIFIC VOLUME OF WATER. 
 
 Temp. 
 C. 
 
 Density 
 (gms. per cc.) 
 
 Specific 
 Volume 
 
 (cc. per gm.) 
 
 Temp. 
 C. 
 
 Density 
 (gms. per cc.) 
 
 Specific 
 Volume 
 
 (cc. per gm.) 
 
 
 
 0.999868 
 
 1.000132 
 
 20 
 
 0.99823 
 
 1.00177 
 
 1 
 
 927 
 
 073 
 
 25 
 
 777 
 
 294 
 
 2 
 
 968 
 
 032 
 
 30 
 
 567 
 
 435 
 
 3 
 
 992 
 
 008 
 
 35 
 
 406 
 
 598 
 
 3.98 
 
 1.000000 
 
 000 
 
 40 
 
 224 
 
 782 
 
 5 
 
 .999992 
 
 008 
 
 50 
 
 .98807 
 
 1.01207 
 
 6 
 
 968 
 
 032 
 
 60 
 
 324 
 
 705 
 
 7 
 
 929 I 071 
 
 70 
 
 .97781 
 
 1.02270 
 
 8 
 
 876 ! -124 
 
 80 
 
 183 
 
 902 
 
 9 
 
 808 
 
 192 
 
 90 
 
 .96534 
 
 1.03590 
 
 10 
 
 727 
 
 273 
 
 100 .95838 
 
 1.04343 
 
 15 
 
 126 
 
 874 
 
 102 
 
 693 
 
 501 
 
PHYSICAL TABLES 
 
 53 
 
 DENSITIES AND THERMAL PROPERTIES OF LIQUIDS. 
 
 (The values given in this Table are mostly for pure specimens 
 of the liquids listed. The student should not expect the properties 
 of the average laboratory specimen to correspond exactly in value 
 with them. With a few exceptions the densities are given for ordi- 
 nary atmospheric temperature and pressure. The specific heats and 
 coefficients of expansion are in most cases the average values be- 
 tween and 100C. The boiling points are given for atmospheric 
 pressure, and the heats of vaporization are given at these boiling 
 points.) 
 
 
 +-* 
 
 
 
 a *3 9 
 
 <u o .2 
 
 bo 
 
 n 
 
 
 
 "ej rt 
 
 *o 'J3 2 
 
 3 o 
 
 O rt 
 
 Liquid 
 
 a 
 
 a> 
 Q 
 
 C& 
 
 1^ | 
 
 !'! 
 
 o o 
 
 
 
 (calories 
 
 
 
 > 
 
 
 
 per gm. 
 
 (per degree 
 
 (degrees 
 
 ( calories 
 
 
 (gms. per cc.) 
 
 per deg.) 
 
 C.) 
 
 C.) 
 
 per gm.) 
 
 Alcohol (ethyl) 
 
 0.794 
 
 .58 
 
 .00111 
 
 78 
 
 205* 
 
 Alcohol (methyl) 
 
 .796 
 
 .60 
 
 .00143 
 
 66 
 
 262t 
 
 Benzene 
 
 .880 
 
 .42 
 
 .00123 
 
 80 
 
 93.2 
 
 Carbon bisulphide 
 
 1.29 
 
 .24 
 
 .00120 
 
 46.6 
 
 84 
 
 Cotton seed oil 
 
 .925 
 
 .47 
 
 .00077 
 
 
 
 Ether 
 
 .74 (0C) 
 
 .55 
 
 .00162 
 
 35 
 
 90 
 
 Glycerine 
 
 1.26 
 
 .576 
 
 .000534 
 
 
 
 Hydrochloric acid 
 
 1.27 
 
 .75 
 
 .000455 
 
 110 
 
 
 Mercury 
 Olive oil 
 
 13.596 (0) 
 .918 
 
 .033 
 .47 
 
 .0001815 
 .000721 
 
 357 
 
 67 
 
 Citric acid 
 
 1.56 
 
 .66 
 
 .00125 
 
 86 
 
 115 
 
 Sea-water 
 
 1.025 
 
 .938 
 
 
 
 
 Sulphuric acid 
 
 1.85 
 
 .33 
 
 .00056 
 
 338 
 
 122 
 
 Turpentine 
 
 .873 
 
 .47 
 
 .00105 
 
 159 
 
 70 
 
 * The heat of vaporization of ethyl alcohol at 0C. is 236.5. 
 t The heat of vaporization of methyl alcohol at 0C. is 289.2. 
 
54 
 
 PHYSICAL TABLES 
 
 DENSITIES AND THERMAL PROPERTIES OF SOLIDS. 
 
 (The values given in this Table are mostly for pure specimens 
 of the substances listed. The student should not expect the prop- 
 erties of the average laboratory specimen to correspond exactly 
 in value with them. As a rule the densities are given for ordi- 
 nary atmospheric temperature. The specific heats and coefficients 
 of expansion are in most cases the average values between and 
 100C. The melting points and heats of fusion are given for 
 atmospheric pressure.) 
 
 The coefficient of cubical expansion of solids is approximately 
 three times the linear coefficient. 
 
 
 * 
 
 o 
 
 111 
 
 <L> bo 
 W) g -*- 
 
 * 
 
 
 1 
 
 1? 
 
 o.S S 
 
 g|| 
 
 rt *^ 
 
 Solid. 
 
 1 
 
 aW 
 
 CO 
 
 1*1 
 
 |3* 
 
 5 3 
 
 
 
 cals. per 
 
 
 degrees 
 
 cals. per 
 
 
 gms. per cc. 
 
 gm. 
 
 per degree C. 
 
 C. 
 
 gm. 
 
 Acetamide 
 
 1.56 
 
 
 
 82 
 
 
 Aluminum 
 
 2.70 
 
 0.219 
 
 .0000231 
 
 658 
 
 
 Brass, cast 
 
 8.44 
 
 .092 1. 0000188 
 
 
 
 " drawn 
 
 8.70 
 
 .092 .0000193 
 
 
 
 Copper 
 
 8.92 
 
 .094 
 
 .0000172 
 
 1090 
 
 43.0 
 
 German-silver 
 
 8.62 
 
 .0946 L000018 
 
 860 
 
 
 Glass, common tube 
 
 2.46 
 
 .186 i. 0000086 
 
 
 
 " flint 
 
 3.9 
 
 .117 .0000079 
 
 
 
 Gold 
 
 19.3 
 
 .0316 .0000144 
 
 1065 
 
 
 Hyposul. of soda 
 
 1.73 
 
 .445 
 
 
 48 
 
 
 Ice 
 
 .918 
 
 .502 
 
 .000051 
 
 
 
 80. 
 
 Iron, cast 
 
 7.4 
 
 .113 
 
 .0000106 
 
 1100 
 
 23-33 
 
 " wrought 
 
 7.8 
 
 .115 
 
 .000012 
 
 1600 
 
 
 Lead 
 
 11.3 
 
 .0315 .000029 
 
 326 
 
 5.4 
 
 Mercury 
 
 13.596 
 
 .0319 
 
 39 
 
 2.8 
 
 Nickel 
 
 8.90 
 
 .109 
 
 .0000128 
 
 1480 
 
 4.6 
 
 Paraffin, wax 
 
 .90 
 
 .560 
 
 .000008-23 
 
 52 
 
 35.1 
 
 liquid 
 
 
 .710 
 
 
 
 Platinum 
 
 21.50 
 
 .0324 .0000090 
 
 1760 
 
 27.2 
 
 Rubber, hard 
 
 1.22 
 
 .331 
 
 .000064 
 
 
 
 Silver 
 
 10.53 
 
 .056 
 
 .0000193 
 
 960 
 
 21.1 
 
 Sodium chloride 
 
 2.17 
 
 .214 
 
 .000040 
 
 800 
 
 
 Steel 
 
 7.8 
 
 .118 
 
 .000011 
 
 1375 
 
 
 Tin 
 
 7.29 
 
 .0588 
 
 .0000214 
 
 232 
 
 14. 
 
 Wood's alloy, solid 
 
 9.78 
 
 .0352 
 
 
 75.5 
 
 8.40 
 
 " " , liquid 
 
 
 .0426 
 
 
 
 
PHYSICAL TABLES 
 
 55 
 
 SURFACE TENSION OF PURE WATER IN CONTACT 
 WITH AIR. 
 
 (The value of the surface tension of a liquid is dependent only 
 upon the character and temperature of the liquid and upon the 
 nature of the gas above the surface of the liquid. It is independent of 
 the curvature of the surface film and of the material of the con- 
 taining vessel.) 
 
 Temp. 
 C. 
 
 Tension 
 (dynes pr. cm) 
 
 Temp. 
 C. 
 
 Tension 
 (dynes pr. cm) 
 
 Temp. 
 C. 
 
 Tension 
 (dynes pr.cm) 
 
 
 5 
 10 
 15 
 20 
 25 
 
 75.5 
 74.8 
 74.0 
 73.3 
 72.5 
 71.8 
 
 30 
 35 
 40 
 45 
 50 
 55 
 
 71.0 
 70.3 
 69.5 
 68.6 
 67.8 
 66.9 
 
 60 
 65 
 70 
 80 
 100 
 
 crit. temp 
 
 66.0 
 65.1 
 64.2 
 62.3 
 56.0 
 0.0 
 
 SURFACE TENSIONS OF SOME LIQUIDS IN CONTACT 
 WITH AIR. 
 
 (The angle of contact between pure water and clean glass vessels 
 of all sizes is 0; the angle of contact between pure water and clean 
 steel or silver is about 90; the angle of contact between mercury 
 and glass is about 132. See the note to Table VIII.) 
 
 
 Dynes 
 per cm. 
 
 
 Dynes 
 per cm. 
 
 Alcohol (ethyl) at 20 
 Alcohol (methyl) at 20 
 Benzene at 15 
 Glycerine at 18 
 
 22-24 
 22-24 
 28-30 
 63-65 
 
 Mercury at 20 
 Olive oil at 20 
 Petroleum at 20 
 Water (pure) at 20 
 
 470-500 
 32-36 
 24- 26 
 
 72- 74 
 
 VISCOSITY OF WATER. 
 
 Temp. Coeff . of Vise. 
 
 C. (C.G.S.Units) 
 
 ; 
 
 Temp. 
 C. 
 
 Coeff. of Vise. 
 
 (C.G.S.Units) 
 
 Temp. 
 C. 
 
 Coeff. of Vise. 
 (C.G.S.Units) 
 
 0.0178 
 5 .0151 
 10 .0131 
 15 .0113 
 20 .0100 
 
 25 
 30 
 35 
 40 
 50 
 
 0.0089 
 .0080 
 .0072 
 .0066 
 .005- 
 
 60 
 70 
 80 
 90 
 100 
 
 0.0047 
 .0041 
 .0036 
 .0032 
 .0028 
 
56 
 
 PHYSICAL TABLES 
 
 (a) BOILING POINT OF WATER AT DIFFERENT BARO- 
 
 METRIC PRESSURES. 
 
 (b) VAPOR-PRESSURE OF SATURATED WATER-VAPOR. 
 
 (This table may be used either (a) to find the boiling point / of 
 water under the barometric pressure P, or (b) to find the vapor- 
 pressure P of water-vapor saturated at the temperature t, the dew- 
 point.) 
 
 t 
 c. 
 
 P 
 cm. 
 
 D 
 gm./cc 
 
 t 
 C. 
 
 P 
 cm. 
 
 D 
 gm./cc. 
 
 t 
 C. 
 
 P 
 cm. 
 
 D 
 
 gm./cc. 
 
 -10j .22 
 
 2.3xlO- 6 
 
 30 : 3.15 
 
 30.1x10- 
 
 88.5 49.62 
 
 
 - 9 .23 
 
 2.5x " 
 
 35 4.18 39.3x' 
 
 89 
 
 50.58 
 
 
 - 8 
 
 .25 
 
 2.7x" 
 
 40 5.49 
 
 50.9x ' 
 
 89.5 51.55 
 
 
 - 7 
 
 .27 
 
 2.9x " 
 
 45 7.14 65.3x ' 
 
 90 
 
 52.54 
 
 428.4xlO- 6 
 
 - 6 
 
 .29 
 
 3.2x " 
 
 50 9.20 
 
 83.0x ' 
 
 90.5 53.55 
 
 
 - 5 
 
 .32 
 
 3.4x " 
 
 55 111.75 
 
 104.6x ' 
 
 91 
 
 54.57 
 
 
 - 4 
 
 .34 
 
 3.7x " 
 
 60 
 
 14.88 
 
 130.7x 
 
 91.5 
 
 55.61 
 
 
 - 3 
 
 .37 
 
 4.0x " 
 
 65 
 
 18.70 
 
 162.1x' 
 
 92 
 
 56.67 
 
 
 - 2 
 
 .39 
 
 4.2x " 
 
 70 
 
 23.31 
 
 199.5x ' 
 
 92.5 
 
 1 57.74 
 
 
 - 1 
 
 .42 
 
 4.5x " 
 
 71 
 
 24.36 
 
 
 93 
 
 i 58.83 
 
 
 
 
 .46 
 
 4.9x " 
 
 72 
 
 25.43 
 
 
 93.5 
 
 59.96 
 
 
 1 
 
 .49 
 
 5.2x " 
 
 73 
 
 26.54 
 
 
 94 
 
 61.06 
 
 
 2 
 
 .53 
 
 5.6x " 
 
 74 
 
 27.69 i 
 
 94.5 
 
 62.20 
 
 
 3 
 
 .57 
 
 6.0x ' 
 
 75 
 
 28.88 
 
 243.7" 
 
 95 
 
 63.36 
 
 511.1" 
 
 4 
 
 .61 
 
 6.4x ' 
 
 75.5 
 
 29.49 
 
 
 95.5 
 
 64.54 
 
 
 5 
 
 .65 
 
 6.8x ' 
 
 76 
 
 30.11 
 
 
 96 
 
 65.74 
 
 
 6 
 
 .70 
 
 7.3x ' 
 
 76.5 
 
 30.74 
 
 
 96.5 
 
 66.95 
 
 
 7 
 
 .75 
 
 7.7x ' 
 
 77 
 
 31.38 
 
 
 97 
 
 68.18 
 
 
 8 
 
 .80 
 
 8.2x ' 
 
 77.5 
 
 32.04 
 
 
 97.5 
 
 69.42 
 
 
 9 
 
 .85 
 
 8.7x ' 
 
 78 
 
 32.71 
 
 
 98 
 
 70.71 
 
 
 10 .91 
 
 9.3x " 
 
 78.5 
 
 33.38 
 
 
 98.2! 71.23 
 
 
 11 
 
 .98 
 
 lO.Ox " 
 
 79 
 
 34.07 
 
 
 98.4 
 
 71.74 
 
 
 12 
 
 1.04 
 
 10.6x " 
 
 79.5 
 
 34.77 
 
 
 98.6 
 
 72.26 
 
 
 13 
 
 1.11 
 
 11.2x" 
 
 80 
 
 35.49 
 
 295.9 " 
 
 98.8 
 
 72.79 
 
 
 14 
 
 1.19 
 
 12.0x " 
 
 80.5 
 
 36.21 
 
 
 99 
 
 73.32 
 
 
 15 
 
 1.27 
 
 12.8x " 
 
 81 
 
 36.95 
 
 
 99.2 
 
 73.85 
 
 
 16 
 
 1.35 
 
 13.5x" 
 
 81.5 
 
 37.70 
 
 
 99.4 
 
 74.38 
 
 
 17 
 
 1.44 
 
 14.4x " 
 
 82 
 
 38.46 1 
 
 99.6 
 
 74.92 
 
 
 18 
 
 1.53 
 
 15.2x " 
 
 82.5 
 
 39.24 ! 
 
 99.8 
 
 75.47 
 
 
 19 
 
 1.63 
 
 16.2x " 
 
 83 
 
 40.03 
 
 100 
 
 76.00 
 
 606.2 " 
 
 20 
 
 1.74 
 
 17.2x " 
 
 83.5 
 
 40.83 
 
 100.2 
 
 76.55 
 
 
 21 
 
 1.85 
 
 18.2x " 
 
 84 
 
 41.65 
 
 100.4 
 
 77.10 
 
 
 22 
 
 1 5o 
 
 19.3x " 
 
 84.5 
 
 42.47 
 
 
 00.6 
 
 77.65 
 
 
 23 
 
 2.09 20.4x " 
 
 85 
 
 43.32 
 
 357.1 " 
 
 00.8 
 
 78.21 
 
 
 24 2.2221.6x" 
 
 85.5 
 
 44.17 
 
 01 
 
 78.77 
 
 
 25 2.35 22.9x " 
 
 86 
 
 45.05 
 
 
 02 
 
 81.60 
 
 
 26 2.50 24.2x " 
 
 86.5 
 
 45.93 
 
 
 03 
 
 84.53 
 
 
 27 ! 2.65 
 
 25.6x " 
 
 87 
 
 46.83 
 
 
 05 
 
 90.64 
 
 715.4 " 
 
 28 2.81 27.0x " 
 
 87.5 47.74 
 
 
 07 
 
 97.11 
 
 
 29 2.97 28.5x " 
 
 88 148.68 
 
 
 10 
 
 107.54 
 
 840.1 " 
 
PHYSICAL TABLES 
 
 57 
 
 THE WET- AND DRY- BULB HYGROMETER. DEW-POINT. 
 
 (This Table gives the vapor-pressure, in mercurial centimeters, of 
 the water-vapor in the atmosphere corresponding to the dry-bulb 
 reading /C. (first column) and the difference (first row) between 
 the dry-bulb and wet-bulb readings of the hygrometer. Having 
 obtained from this Table the value of the vapor-pressure for a given 
 case, the dew-point can be found by consulting the table. The data 
 given below are calculated for a barometric pressure equal to 76 cm.) 
 
 *c 
 
 Difference between Dry-bulb and Wet-bulb Readings. 
 
 1 2 3 4 5 6 7 8 9 10 
 
 
 cm. 
 
 cm. cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 cm. 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 10 
 
 .92 
 
 .81 
 
 .70 
 
 .60 
 
 .50 
 
 .40 .31 
 
 .22 
 
 .13 
 
 
 
 11 
 
 .98 
 
 .87 
 
 .76 
 
 .65 
 
 .55 
 
 .45 .35 
 
 .26 
 
 .17 
 
 
 
 12 
 
 .105 
 
 .93 
 
 .82 
 
 .71 
 
 .60 
 
 .50 .40 
 
 .30 
 
 .21 
 
 .12 
 
 .03 
 
 13 
 
 1.12 
 
 .00 
 
 .89 
 
 .76 
 
 .65 
 
 .55 
 
 .45 
 
 .35 
 
 .25 
 
 .16 
 
 .07 
 
 14 
 
 1.19 
 
 .07 
 
 .94 
 
 .83 
 
 .71 
 
 .61 
 
 .50 
 
 .40 
 
 .30 
 
 .20 
 
 .11 
 
 15 
 
 1.27 
 
 .14 
 
 1.01 
 
 .90 
 
 .78 
 
 .66 
 
 -.55 
 
 .45 
 
 .34 
 
 .25 
 
 .15 
 
 16 
 
 1.35 
 
 .22 
 
 1.09 
 
 .97 
 
 .84 
 
 .73 
 
 .60 
 
 .50 
 
 .40 
 
 .30 
 
 .19 
 
 17 
 
 1.44 
 
 .30 
 
 1.17 
 
 1.04 
 
 .91 
 
 .80 .67 
 
 .56 
 
 .45 
 
 .35 
 
 .24 
 
 18 
 
 1.54 
 
 1.39 
 
 1.25 
 
 1.12 
 
 .99 
 
 .86 .74 
 
 .63 
 
 .51 
 
 .40 
 
 .30 
 
 19 
 
 1.63 
 
 1.49 
 
 1.34 
 
 1.20 
 
 1.07 
 
 .94 
 
 .81 
 
 .69 
 
 .57 
 
 .46 
 
 .35 
 
 20 
 
 1.74 
 
 1.59 
 
 .43 1.29 
 
 1.15 
 
 1.02 
 
 .88 
 
 .76 
 
 .64 
 
 .52 
 
 .41 
 
 21 
 
 1.85 
 
 1.69 
 
 .53 1.38 
 
 1.24 
 
 1.10 .96 
 
 .84 
 
 .71 
 
 .59 
 
 .47 
 
 22 
 
 1.97 
 
 1.80 .64 
 
 1.48 
 
 1.33 
 
 1.19 
 
 .05 
 
 .91 
 
 .78 
 
 .66 
 
 .54 
 
 23 
 
 2.09 
 
 1.92 
 
 .75 
 
 1.59 
 
 1.43 
 
 1.28 
 
 .13 
 
 .00 
 
 .86 
 
 .73 
 
 .61 
 
 24 
 
 2.22 
 
 2.04 
 
 .86 
 
 1.70 
 
 1.53 
 
 1.38 
 
 .231 .09 
 
 .94 
 
 .81 
 
 .68 
 
 25 
 
 2.35 
 
 2.17 
 
 .99 
 
 1.81 
 
 1.64 
 
 1.48 
 
 .33| .18 
 
 1.03 
 
 .90 
 
 .76 
 
 26 
 
 2.50 
 
 2.31 
 
 2.1l{ 1.94 
 
 1.76 
 
 1.59 
 
 .43 
 
 .28 
 
 1.13 
 
 .98 .84 
 
 27 
 
 2.65 
 
 2.45 
 
 2.25 2.07 
 
 1.88 
 
 1.71 
 
 1.54 
 
 .38 
 
 1.23 
 
 1.08 
 
 .93 
 
 28 
 
 2.81 
 
 2.60 
 
 2.40; 2.20 2.01 
 
 1.83 
 
 1.66 
 
 .49 
 
 1.33 
 
 1.18 
 
 1.02 
 
 29 
 
 2.98 
 
 2.76 
 
 2.55 2.35 
 
 2.15 
 
 1.96 
 
 1.78 
 
 .61 
 
 1.44 
 
 1.28 
 
 1.12 
 
 30 
 
 3.15 
 
 2.931 2.71 2.50 2.29 : 2.10 L9l] 1.73 
 
 1.55 1.39 
 
 1.23 
 
 Miscellaneous. 
 
 (1.) Heat of Neutralization. 
 
 Any strong acid with any strong alkali evolves (+) about 
 
 761- calories for every gm. of water formed. 
 (2.) Heat of Solution in water. 
 
 For Calcium oxide (CaO), -f- 327 cals. per gm. 
 
 " Sodium chloride (NaCl), - 21 " " " 
 
 hydroxide (NaOH), + 248 " " " 
 
 hyposulphite (Na 2 S 2 O 3 -f 5H 2 O), - 44.8 " " " 
 
 (3.) Fuel value of illuminating gas is 5500 to 6500 calories per liter, 
 its density is .00058 gm. per cc. at 0C and 76 cm. pressure. 
 
 Fuel value of ethyl alcohol is 7400, of methyl alcohol 5700, 
 calories per gram. 
 
58 
 
 NATURAL SINES. 
 
 
 0' 
 
 6 
 
 12' 
 
 18 
 
 24' 
 
 30' 
 
 36 
 
 42' 
 
 48' 
 
 54' 
 
 123 
 
 4 5 
 
 
 
 ~T~ 
 2 
 3 
 
 oooo 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 oi57 
 
 369 
 
 12 15 
 
 0175 
 0349 
 0523 
 
 0192 
 0366 
 0541 
 
 0209 
 0384 
 0558 
 
 0227 
 0401 
 0576 
 
 0244 
 0419 
 0593 
 
 0262 
 0436 
 0610 
 
 0279 
 
 0454 
 0628 
 
 0297 
 0471 
 0645 
 
 0314 
 0488 
 0663 
 
 0332 
 0506 
 0680 
 
 369 
 369 
 369 
 
 12 15 
 12 15 
 
 12 15 
 
 4 
 5 
 6 
 
 0698 
 0872 
 1045 
 
 0715 
 0889 
 1063 
 
 0732 
 0906 
 1080 
 
 0750 
 0924 
 1097 
 
 0767 
 0941 
 i"5 
 
 0785 
 0958 
 1132 
 
 0802 
 0976 
 "49 
 
 0819 
 
 0993 
 II6.7 
 
 0837 
 ion 
 1184 
 
 0854 
 1028 
 
 1 201 
 
 369 
 369 
 369 
 
 12 15 
 12 14 
 12 14 
 
 7 
 8 
 9 
 
 1219 
 
 1392 
 1564 
 
 1236 
 1409 
 1582 
 
 1253 
 1426 
 
 '599 
 
 1271 
 1444 
 1616 
 
 1288 
 1461 
 1633 
 
 1305 
 1478 
 1650 
 
 *323 
 1495 
 1668 
 
 1340 
 1513 
 1685 
 
 1357 
 1530 
 1702 
 
 1374 
 1547 
 1719 
 
 369 
 369 
 369 
 
 12 14 
 12 14 
 12 14 
 
 10 
 
 1736 
 
 1754 
 
 1771 
 
 1788 
 
 1805 
 
 1822 
 
 1840 
 
 1857 
 
 1874 
 
 1891 
 
 369 
 
 12 14 
 
 11 
 12 
 13 
 
 1908 
 2079 
 
 2250 
 
 1925 
 2096 
 2267 
 
 1942 
 2113 
 
 2284 
 
 1959 
 2130 
 2300 
 
 1977 
 2147 
 2317 
 
 1994 
 2164 
 2334 
 
 201 1 
 
 2181 
 
 2351 
 
 2028 
 2198 
 2368 
 
 2045 
 2215 
 2385 
 
 2062 
 2232 
 2402 
 
 369 
 369 
 368 
 
 II 14 
 II 14 
 II 14 
 
 14 
 15 
 16 
 
 24*19 
 
 2588 
 2756 
 
 2436 
 2605 
 2773 
 
 2453 
 2622 
 2790 
 
 2470 
 2639 
 2807 
 
 2487 
 2656 
 2823 
 
 2504 
 2672 
 2840 
 
 2521 
 2689 
 2857 
 
 2538 
 2706 
 2874 
 
 2554 
 2723 
 2890 
 
 2571 
 2740 
 2907 
 
 368 
 368 
 368 
 
 II 14 
 II 14 
 II 14 
 
 17 
 18 
 19 
 
 2924 
 3090 
 3256 
 
 2940 
 3107 
 3272 
 
 2957 
 3123 
 3289 
 
 2974 
 3140 
 
 3305 
 
 2990 
 3156 
 3322 
 
 3007 
 3173 
 3338 
 
 3024 
 3190 
 
 3355 
 
 3040 
 3206 
 3371 
 
 3057 
 3223 
 
 3387 
 
 3074 
 3239 
 3404 
 
 368 
 368 
 3 5 8 
 
 II 14 
 II 14 
 II 14 
 
 20 
 
 3420 
 
 3437 
 
 3453 
 
 3469 
 
 3486 
 
 3502 
 
 35i8 
 
 3535 
 
 3551 
 
 3567 
 
 3 5 8 
 
 II 14 
 
 21 
 22 
 23 
 
 3584 
 3746 
 
 397 
 
 3600 
 3762 
 3923 
 4083 
 4242 
 4399 
 
 3616 
 3778 
 3939 
 
 3633 
 3795 
 3955 
 
 3649 
 3811 
 397i 
 
 3665 
 3827 
 3987 
 
 3681 
 
 3843 
 4003 
 
 3697 
 3859 
 4019 
 
 3714 
 3875 
 4035 
 
 3730 
 3891 
 4051 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 II 14 
 II 14 
 II 14 
 
 24 
 25 
 26 
 
 4067 
 4226 
 4384 
 
 4099 
 4258 
 4415 
 
 4H5 
 4274 
 
 4431 
 
 *i3i 
 4289 
 
 H4& 
 
 4M7 
 4305 
 4462 
 
 4163 
 4321 
 4478 
 
 4179 
 4337 
 4493 
 
 4195 
 4352 
 4509 
 
 4210 
 4368 
 4524 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 II 13 
 II 13 
 
 10 13 
 
 27 
 28 
 
 29 
 
 4540 
 4695 
 
 4848 
 
 4555 
 4710 
 4863 
 
 4571 
 4726 
 4879 
 
 4586 
 4741 
 4894 
 
 4602 
 4756 
 t909 
 
 4617 
 4772 
 4924 
 
 4633 
 4787 
 4939 
 
 4648 
 4802 
 4955 
 
 4664 
 4818 
 4970 
 
 4679 
 4833 
 4985 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 10 13 
 10 13 
 10 13 
 
 30 
 
 5000 
 5150 
 5299 
 5446 
 
 5015 
 
 5030 
 
 5045 
 
 5060 
 
 5075 
 
 5090 
 
 5105 
 
 5120 
 
 5135 
 
 3 5 8 
 
 10 13 
 
 31 
 32 
 33 
 
 5165 
 53M 
 5461 
 
 5180 
 5329 
 5476 
 
 5195 
 5344 
 5490 
 
 5210 
 
 5358 
 5505 
 
 5225 
 5373 
 5519 
 
 5240 
 
 5388 
 5534 
 
 5255 
 5402 
 
 5548 
 
 5270 
 5417 
 5563 
 
 5284 
 5432 
 
 5577 
 
 2 5 7 
 257 
 257 
 
 10 12 
 10 12 
 10 12 
 
 34 
 35 
 36 
 
 5592 
 5736 
 
 5878 
 
 5606 
 5750 
 5892 
 
 5621 
 5764 
 5906 
 
 5635 
 5779 
 5920 
 
 5650 
 5793 
 5934 
 
 5664 
 5807 
 5948 
 
 5678 
 5821 
 5962 
 
 5693 
 5835 
 5976 
 
 5707 
 5850 
 599 
 
 5721 
 5864 
 6004 
 
 2 5. 7 
 257 
 2 5 7 
 
 10 12 
 10 12 
 
 9 12 
 
 37 
 38 
 39 
 
 6018 
 
 6i57 
 6293 
 
 6032 
 6170 
 6307 
 
 6046 
 6184 
 6320 
 
 6060 
 6198 
 6334 
 
 6074 
 6211 
 
 6347 
 
 6088 
 6225 
 6361 
 
 6101 
 6239 
 6374 
 
 6115 
 6252 
 6388 
 
 6129 6143 
 6266 6280 
 6401 6414 
 
 257 
 257 
 247 
 
 9 12 
 
 9 II 
 
 9 'I 
 
 40 
 
 6428 
 
 6441 
 
 6455 
 
 6468 
 
 6481 
 
 6494 
 
 6508 
 
 6521 
 
 6534 
 
 6547 
 
 247 
 
 9 ii 
 
 41 
 
 42 
 43 
 
 6561 
 6691 
 6820 
 
 6574 
 6704 
 
 6833 
 
 6587 
 6717 
 6845 
 
 6600 
 6730 
 
 6858 
 
 6613 
 6743 
 6871 
 
 6626 
 6756 
 6884 
 
 6639 
 6769 
 6896 
 
 6652 
 6782 
 6909 
 
 6665 
 6794 
 6921 
 
 6678 
 6807 
 6934 
 
 247 
 246 
 246 
 
 9 II 
 9 ii 
 8 ii 
 
 44 
 
 6947 
 
 6959 
 
 6972 
 
 6984 
 
 6997 
 
 7009 
 
 7022 
 
 7034 
 
 7046 
 
 7059 
 
 246 
 
 8 10 
 
NATURAL SINES. 
 
 59 
 
 
 0' 
 
 6' 
 
 12 
 
 18 
 
 24 
 
 30 
 
 36 
 
 42 
 
 48 
 
 54 
 
 123 
 
 4 5 
 
 45 
 
 ~46~ 
 47 
 48 
 
 7071 
 
 7083 
 
 7096 
 
 7108 
 
 7120 
 
 7133 
 
 7M5 
 
 7157 
 
 7169 
 
 7181 
 
 2 4 6 
 
 8 10 
 
 7193 
 7314 
 743i 
 
 7206 
 7325 
 7443 
 
 7218 
 7337 
 
 7455 
 
 7230 
 7349 
 7466 
 
 7242 
 7361 
 
 7478 
 
 7254 
 7373 
 7490 
 
 7266 
 
 7385 
 750i 
 
 7278 
 7396 
 7513 
 
 7290 
 7408 
 7524 
 
 7302 
 7420 
 7536 
 
 246 
 246 
 246 
 
 8 10 
 8 10 
 8 10 
 
 49 
 50 
 51 
 ~52~ 
 53 
 54 
 
 7547 
 7660 
 
 7771 
 
 7558 
 7672 
 7782 
 
 7570 
 7683 
 7793 
 7902 
 8007 
 8m 
 
 758i 
 7694 
 7804 
 
 7593 
 7705 
 T^IS 
 
 7923 
 8028 
 8131 
 
 7604 
 7716 
 7826 
 
 7615 
 
 7727 
 7837 
 
 7627 
 7738 
 7848 
 
 7638 
 7749 
 759 
 
 7649 
 7760 
 7869 
 
 246 
 
 246 
 245 
 
 8 9 
 7 9 
 7 9 
 
 7880 
 7986 
 8090 
 
 7891 
 
 7997 
 8100 
 
 7912 
 8018 
 8121 
 
 7934 
 8039 
 8141 
 
 7944 
 8049 
 8151 
 
 7955 
 8059 
 8161 
 
 7965 
 8070 
 8171 
 
 7976 
 8080 
 8181 
 
 245 
 2 3 5 
 2 3 5 
 
 7 9 
 7 9 
 7 8 
 
 55 
 
 8192 
 
 8202 
 
 8211 
 
 8221 
 
 8231 
 
 8241 
 
 8251 
 
 8261 
 
 8271 
 
 8281 
 
 2 3 5 
 
 7 8 
 
 56 
 57 
 58 
 
 829Q 
 
 8387 
 8480 
 
 8572 
 8660 
 
 8746 
 8829 
 
 8910 
 
 8988 
 
 8300 
 8396 
 8490 
 
 8581 
 8669 
 8755 
 8838 
 8918 
 8996 
 
 8310 
 8406 
 8499 
 8590 
 8678 
 8763 
 
 8320 
 8415 
 8508 
 
 8329 
 8425 
 
 8517 
 
 8339 
 8434 
 8526 
 
 8348 
 8443 
 8536 
 
 8358 
 8453 
 
 8545 
 
 8368 
 8462 
 8554 
 
 8377 
 8471 
 
 8563 
 
 235 
 2 3 5 
 2 3 5 
 
 6 8 
 6 8 
 6 8 
 
 59 
 60 
 61 
 
 8599 
 8686 
 8771 
 
 8607 
 8695 
 8780 
 
 8616 
 8704 
 
 8788 
 
 8625 
 8712 
 8796 
 
 8634 
 8721 
 8805 
 
 8643 
 8729 
 8813 
 
 8652 
 8738 
 8821 
 
 i 3 4 
 i 3 4 
 i 3 4 
 
 6 7 
 6 7 
 6 7 
 
 62 
 
 63 
 64 
 
 8846 
 8926 
 9QQ3 
 9078 
 
 8854 
 8934 
 9011 
 
 8862 
 8942 
 9018 
 
 8870 
 
 8949 
 9026 
 
 8878 
 8957 
 9033 
 
 8886 
 8965 
 9041 
 
 8894 
 8973 
 9048 
 
 8902 
 8980 
 9056 
 
 i 3 4 
 
 i 3 4 
 i 3 4 
 
 5 7 
 5 6 
 5 6 
 
 65 
 
 9063 
 
 9070 
 
 9143 
 9212 
 9278 
 
 9085 
 
 9092 
 9164 
 9232 
 9298 
 
 9100 
 
 9i7i 
 9239 
 9304 
 
 9107 
 
 9114 
 
 9121 
 
 9128 
 
 I 2 4 
 
 5 6 
 
 66 
 67 
 68 
 
 9135 
 
 9205 
 
 9272 
 
 9'50 
 9219 
 9285 
 
 9157 
 9225 
 9291 
 
 9178 
 9245 
 93U 
 
 9184 
 9252 
 9317 
 
 9191 
 9259 
 9323 
 
 919^ 
 9265 
 9330 
 
 I 2 3 
 I 2 3 
 I 2 3 
 
 5 6 
 4 6 
 4 5 
 
 69 
 70 
 71 
 72 
 73 
 74 
 
 9336 
 
 9397 
 9455 
 
 9342 
 9403 
 9461 
 
 9348 
 9409 
 9466 
 
 9354 
 9415 
 9472 
 
 9361 
 9421 
 
 9478 
 
 9367 
 9426 
 
 9483 
 
 9373 
 9432 
 9489 
 
 9379 
 9438 
 9494 
 
 9385 
 9444 
 
 939 i 
 9449 
 9505 
 
 I 2 3 
 
 I 2 3 
 
 i 2 3 
 
 4 5 
 4 5 
 4 5 
 
 9500 
 
 95U 
 9563 
 9613 
 
 >5i6 
 
 9568 
 9617 
 
 95 2 i 
 9573 
 9622 
 
 9527 
 9578 
 9627 
 
 9532 
 9583 
 9632 
 
 9537 
 9588 
 9636 
 
 9542 
 9593 
 9641 
 
 9548 
 9598 
 9646 
 
 9553 
 9603 
 9650 
 
 9558 
 9608 
 9655 
 
 I 2 3 
 
 I 2 2 
 122 
 
 4 4 
 3 4 
 3 4 
 
 75 
 76 
 77 
 78 
 
 9659 
 
 0664 
 
 9668 
 
 9673 
 
 9677 
 
 9681 
 
 9686 
 
 9690 
 
 9694 
 
 9699 
 
 112 
 
 3 4 
 
 9703 
 9744 
 9781 
 
 9707 
 9748 
 9785 
 
 9711 
 
 9751 
 9789 
 
 9715 
 9755 
 9792 
 
 9720 
 9759 
 9796 
 
 9724 
 9763 
 9799 
 
 9728 
 9767 
 9803 
 
 9732 
 9770 
 9806 
 
 9736 
 9774 
 9810 
 
 9740 
 9778 
 9813 
 
 112 
 112 
 112 
 
 3 3 
 3 3 
 
 2 3 
 
 79 
 80 
 81 
 
 9816 
 9848 
 9877 
 
 9820 
 
 9851 
 9880 
 
 9823 
 
 9854 
 9882 
 
 9826 
 9857 
 9885 
 
 9829 
 )86o 
 )888 
 
 )833 
 9863 
 9890 
 9914 
 9936 
 9954 
 
 9836 
 9866 
 9893 
 
 9839 
 9869 
 
 9895 
 
 9842 
 9871 
 9898 
 
 9845 
 9874 
 9900 
 
 I 2 
 O I 
 
 o 
 
 2 3 
 2 2 
 2 2 
 
 82 
 83 
 84 
 
 9903 
 99 2 5 
 9945 
 
 9905 
 9928 
 
 9947 
 
 9907 
 9930 
 9949 
 
 9910 
 9932 
 995i 
 
 9912 
 9934 
 9952 
 
 9917 
 9938 
 9956 
 
 9919 
 9940 
 9957 
 
 9921 
 9942 
 9959 
 
 9923 
 9943 
 9960 
 
 O 
 O 
 O 
 
 2 2 
 I 2 
 I I 
 
 85 
 
 9962 
 
 9963 
 
 9965 
 
 9966 
 
 9968 
 
 9969 
 
 997i 
 
 9972 
 
 9973 
 
 9974 
 
 
 
 I I 
 
 86 
 87 
 88 
 
 9976 
 9986 
 9994 
 
 9977 
 9987 
 9995 
 
 9978 
 9988 
 9995 
 
 9979 
 9989 
 9996 
 
 9980 
 9990 
 9996 
 
 9981 
 9990 
 9997 
 
 9982 
 9991 
 9997 
 
 9983 
 999 2 
 9997 
 
 9984 
 
 9993 
 9998 
 
 9985 
 9993 
 9998 
 
 
 O O O 
 O O O 
 
 I I 
 I I 
 
 
 89 
 
 9998 
 
 9999 
 
 9999 
 
 9999 
 
 9999 
 
 1.000 
 nearly 
 
 1. 000 
 nearly 
 
 1.000 
 
 nearlv 
 
 1. 000 
 nearly 
 
 1. 000 
 nearly 
 
 o o o 
 
 O O 
 
60 
 
 NATURAL TANGENTS. 
 
 
 0' 
 
 6 
 
 12' 
 
 18 
 
 24' 
 
 30' 
 
 36 
 
 42 
 
 48' 
 
 54' 
 
 123 
 
 4 5 
 
 
 
 .0000 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 oi57 
 
 369 
 
 12 14- 
 
 1 
 
 2 
 3 
 
 .0175 
 
 .0349 
 .0524 
 
 0192 
 0367 
 0542 
 
 0209 
 0384 
 0559 
 
 0227 
 0402 
 0577 
 
 0244 
 0419 
 0594 
 
 0262 
 
 0437 
 0612 
 
 0279 
 0454 
 0629 
 
 0297 
 0472 
 0647 
 
 0314 
 0489 
 0664 
 
 0332 
 0507 
 0682 
 
 369 
 369 
 369 
 
 12 15 
 
 12 15 
 
 12 I 5 
 
 4 
 5 
 6 
 
 .0699 
 
 .0875 
 .1051 
 
 0717 
 0892 
 1069 
 
 0734 
 0910 
 1086 
 
 0752 
 0928 
 1104 
 
 0769 
 0945 
 
 1122 
 
 0787 
 0963 
 
 "39 
 
 0805 
 0981 
 "57 
 
 0822 
 0998 
 H75 
 
 0840 
 1016 
 1192 
 
 0857 
 1033 
 
 1210 
 
 369 
 369 
 369 
 
 12 15 
 12 15 
 12 I 5 
 
 7 
 8 
 9 
 
 .1228 
 .1405 
 .1584 
 
 1246 
 1423 
 1602 
 
 1263 
 1441 
 1620 
 
 1281 
 
 1459 
 1638 
 
 1299 
 1477 
 1655 
 
 1317 
 1495 
 1673 
 
 1334 
 1512 
 1691 
 
 f352 
 1530 
 1709 
 
 1370 
 1548 
 1727 
 
 1388 
 1566 
 1745 
 
 369 
 
 3 6 9 
 369 
 
 12 15 
 12 15 
 12 I 5 
 
 10 
 
 1763 
 
 1781 
 
 1799 
 
 1817 
 
 1835 
 
 1853 
 
 1871 
 
 1890 
 
 1908 
 
 1926 
 
 369 
 
 12 15 
 
 11 
 12 
 13 
 
 .1944 
 .2126 
 
 .2309 
 
 1962 
 2144 
 2327 
 
 1980 
 2162 
 2345 
 
 1998 
 2180 
 2364 
 
 2016 
 2199 
 
 2382 
 
 2035 
 2217 
 2401 
 
 2053 
 
 2235 
 2419 
 
 2071 
 2254 
 2438 
 
 2089 
 2272 
 2456 
 
 2107 
 2290 
 2475 
 266l 
 2849 
 3038 
 
 369 
 3 6 9 
 3 6 9 
 
 12 15 
 12 15 
 12 15 
 
 14 
 15 
 16 
 
 .2493 
 .2679 
 .2867 
 
 2512 
 2698 
 2886 
 
 2530 
 2717 
 2905 
 
 2549 
 2736 
 2924 
 
 2568 
 2754 
 2943 
 
 2586 
 
 2773 
 2962 
 
 2605 
 2792 
 2981 
 3172 
 3365 
 356i 
 
 2623 
 28ll 
 3000 
 
 2642 
 2830 
 3019 
 
 369 
 369 
 369 
 
 12 l6 
 
 13 '<< 
 
 13 1 6 
 
 17 
 18 
 19 
 
 .3057 
 3249 
 
 3443 
 
 3076 
 3269 
 3463 
 
 3096 
 
 3288 
 3482 
 
 3"5 
 3307 
 3502 
 
 3134 
 3327 
 3522 
 
 3153 
 3346 
 3541 
 
 3I9 1 
 
 3385 
 3581 
 
 3211 
 3404 
 3600 
 
 3230 
 3424 
 3620 
 
 3 6 10 
 3 6 10 
 3 6 10 
 
 13 16 
 13 6 
 13 '7 
 
 20 
 
 3640 
 
 3659 
 
 3679 
 
 3699 
 
 3719 
 
 3739 
 
 3759 
 
 3779 
 
 3799 
 
 3819 
 
 3 7 '0 
 
 13 '7 
 
 21 
 22 
 23 
 
 .3839 
 .4040 
 
 4245 
 
 3859 
 4061 
 4265 
 
 3879 
 4081 
 4286 
 
 3899 
 4101 
 
 4307 
 
 3919 
 4122 
 
 4327 
 
 3939 
 4142 
 
 4348 
 
 3959 
 4163 
 4369 
 
 3978 
 4183 
 4390 
 
 4000 
 4204 
 4411 
 
 4O2O 
 4224 
 4431 
 
 3 7 >o 
 3 7 10 
 3' 7 Jo 
 
 '3 7 
 4 i7 
 14 17 
 
 24 
 25 
 26 
 
 4452 
 .4663 
 .4877 
 
 4473 
 4684 
 4899 
 
 4494 
 4706 
 4921 
 
 4515 
 4727 
 4942 
 
 4536 
 4748 
 4964 
 
 4557 
 4770 
 4986 
 
 4578 
 4791 
 5008 
 
 4599 
 4813 
 5029 
 
 4621 
 4834 
 5051 
 
 4642 
 4856 
 5073 
 
 4 7 ii 
 4 7 ii 
 
 ,4 18 
 
 15 18 
 
 27 
 28 
 29 
 
 5095 
 5317 
 
 5543 
 
 5U7 
 5340 
 5566 
 
 5139 
 5362 
 5589 
 
 5161 
 
 5384 
 5612 
 
 5184 
 5407 
 5635 
 
 5206 
 5430 
 5658 
 
 5228 
 5452 
 5681 
 
 5250 
 5475 
 5704 
 
 5272 
 5498 
 5727 
 
 5295 
 5520 
 5750 
 
 4 7 I' 
 
 4 8 ii 
 4 8 12 
 
 15 18 
 15 '9 
 '5 19 
 
 30 
 
 5774 
 
 5797 
 
 5820 
 
 5844 
 
 5867 
 
 5890 
 
 5914 
 
 5938 
 
 596i 
 
 5985 
 
 4 8 12 
 
 l6 20 
 
 31 
 32 
 33 
 
 .6009 
 .6249 
 .6494 
 
 6032 
 6273 
 6519 
 
 6056 
 6297 
 6544 
 
 6080 
 6322 
 6569 
 
 6lO4 
 6346 
 6594 
 
 6128 
 6371 
 6619 
 
 6152 
 6395 
 6644 
 6899 
 7159 
 7427 
 
 6176 
 6420 
 6669 
 6^24 
 7186 
 7454 
 
 6200 
 6445 
 6694 
 
 6224 
 6469 
 6720 
 
 4 8 12 
 4 8 12 
 4 13 
 
 l6 2O 
 
 16 20 
 
 17 21 
 
 34 
 35 
 36 
 
 6745 
 .7002 
 .7265 
 
 6771 
 7028 
 7292 
 
 6796 
 7054 
 7319 
 
 6822 
 7080 
 7346 
 
 6847 
 7107 
 
 7373 
 
 6873 
 7133 
 7400 
 
 6950 
 7212 
 7481 
 
 6976 
 
 7239 
 7508 
 
 4 9 13 
 4 9 '3 
 5 9 '4 
 
 17 21 
 
 l8 22 
 l8 23 
 
 37 
 38 
 39 
 
 7536 
 7813 
 .8098 
 
 7563 
 7841 
 8127 
 
 7590 
 7869 
 8156 
 
 7618 
 7898 
 8185 
 
 7646 
 7926 
 
 8214 
 
 7673 
 7954 
 8243 
 
 7701 
 7983 
 8273 
 
 7729 
 8012 
 8302 
 
 7757 
 8040 
 8332 
 
 7785 
 8069 
 836! 
 
 5 9 '4 
 5 M 
 5 o 15 
 
 1 8 23 
 19 24 
 
 20 =4 
 
 40 
 
 .8391 
 
 8421 
 
 8451 
 
 8481 
 
 8511 
 
 8541 
 
 8571 
 
 8601 
 
 8632 
 
 8662 
 
 5 o 15 
 
 20 25 
 
 41 
 42 
 43 
 
 .8693 
 .9004 
 9325 
 
 8724 
 9036 
 9358 
 
 8754 
 9067 
 
 9391 
 
 8785 
 9099 
 9424 
 
 8816 
 9131 
 9457 
 
 8847 
 9163 
 9490 
 
 8878 
 9195 
 9523 
 
 8910 
 9228 
 9556 
 
 8941 
 9260 
 9590 
 
 8972 
 9293 
 9623 
 
 99 6 5 
 
 5016 
 5 i 16 
 6 . 17 
 
 21 26 
 
 zi 27 
 
 22 28 
 
 44 
 
 9657 
 
 9691 
 
 9725 
 
 9759 
 
 9793 
 
 9827 
 
 9861 
 
 9896 
 
 9930 
 
 6 ii 17 
 
 23 2 9 
 
NATURAL TANGENTS. 
 
 61 
 
 
 
 
 6' 
 
 12' 
 
 18 
 
 24' 
 
 30 
 
 36 
 
 42' 
 
 48 
 
 54' 
 
 123 
 
 4 5 
 
 45 
 
 1. 0000 
 
 0035 
 
 0070 
 
 0105 
 
 0141 
 
 0176 
 
 0212 
 
 0247 
 
 0283 
 
 0319 
 
 6 12 18 
 
 24 30 
 
 46 
 47 
 48 
 
 1-0355 
 1.0724 
 
 1.1106 
 
 0392 
 0761 
 H45 
 
 0428 
 0799 
 1184 
 
 0464 
 0837 
 1224 
 
 0501 
 
 0875 
 1263 
 
 0538 
 0913 
 1303 
 
 0575 
 095 1 
 1343 
 
 0612 
 0990 
 1383 
 
 0649 
 1028 
 1423 
 
 0686 
 1067 
 1463 
 
 6 12 18 
 6 J 3 19 
 7 13 20 
 
 25 3' 
 25 32 
 26 33 
 
 49 
 50 
 51 
 
 1.1504 
 1.1918 
 1-2349 
 
 1544 
 1960 
 2393 
 
 1585 
 
 2OO2 
 
 2437 
 
 1626 
 2045 
 2482 
 
 1667 
 2088 
 2527 
 
 1708 
 2131 
 2572 
 
 I75C 
 2174 
 2617 
 
 1792 
 2218 
 2662 
 
 1833 
 2261 
 2708 
 
 1875 
 2305 
 2753 
 
 7 4 21 
 7 H *> 
 8 15 23 
 
 28 34 
 29 36 
 30 38 
 
 52 
 53 
 54 
 55 
 
 1.2799 
 1.3270 
 1-3764 
 
 2846 
 3319 
 3814 
 
 2892 
 
 3367 
 3865 
 
 2938 
 34i6 
 3916 
 
 2985 
 3465 
 3968 
 
 3032 
 3514 
 4019 
 
 3079 
 3564 
 4071 
 
 3127 
 3613 
 4124 
 
 3175 
 3663 
 4176 
 
 3222 
 
 3713 
 4229 
 
 8 16 23 
 8 16 25 
 9 17 26 
 
 3 39 
 33 4i 
 
 34 43 
 
 1.4281 
 
 4335 
 
 4388 
 
 4442 
 
 4496 
 
 4550 
 
 4605 
 
 4659 
 
 4715 
 
 4770 
 
 9 18 27 
 
 36 45 
 
 56 
 57 
 58 
 
 1.4826 
 
 1-5399 
 1.6003 
 
 4882 
 5458 
 6066 
 
 4938 
 5517 
 6128 
 
 4994 
 5577 
 6191 
 
 5051 
 5637 
 6255 
 6909 
 7603 
 8341 
 
 5108 
 
 5697 
 6319 
 
 5166 
 5757 
 6383 
 
 5224 
 5818 
 6447 
 
 5282 
 5880 
 6512 
 
 5340 
 5941 
 6577 
 
 10 19 29 
 
 10 20 30 
 II 21 32 
 
 38 48 
 40 50 
 43 53 
 
 59 
 60 
 61 
 
 1.6643 
 1.7321 
 1.8040 
 
 6709 
 
 739 1 
 8115 
 
 6775 
 
 7461 
 8190 
 
 6842 
 7532 
 8265 
 
 6977 
 7675 
 8418 
 
 7045 
 7747 
 8495 
 
 7"3 
 7820 
 8572 
 
 7182 
 
 7893 
 8650 
 
 725J 
 7966 
 8728 
 
 ii 23 34 
 
 12 24 36 
 13 26 38 
 
 45 56 
 48 60 
 51 64 
 
 62 
 63 
 64 
 
 1.8807 
 1.9626 
 2.0503 
 
 8887 
 97" 
 0594 
 
 8967 
 
 9797 
 0586 
 
 9047 
 9883 
 0778 
 
 9128 
 9970 
 0872 
 
 9210 
 0057 
 0965 
 
 3292 
 0145 
 1060 
 
 9375 
 0233 
 
 H55 
 
 9458 
 6323 
 1251 
 
 9542 
 0413 
 1348 
 
 14 27 41 
 
 '5 29 44 
 16 31 47 
 
 55 68 
 5 73 
 63 78 
 
 65 
 
 2-1445 
 
 1543 
 
 1642 
 
 1742 
 
 1842 
 
 1943 
 
 2045 
 
 2148 
 
 2251 
 
 2355 
 
 '7 34 5' 
 
 68 85 
 
 66 
 67 
 
 68 
 
 2.2460 
 2-3559 
 2.4751 
 
 2566 
 3673 
 4876 
 
 2673 
 
 3789 
 5002 
 
 2781 
 3906 
 5129 
 
 2889 
 4023 
 5257 
 
 2998 
 4142 
 5386 
 
 3109 
 4262 
 5517 
 
 3220 
 4383 
 5649 
 
 3332 
 4504 
 5782 
 
 3445 
 4627 
 59'6 
 
 18 37 55 
 20 40 60 
 22 43 65 
 
 74 92 
 79 99 
 87 108 
 
 69 
 70 
 71 
 
 2.6051 
 
 2-7475 
 2.9042 
 
 6187 
 7625 
 9208 
 
 6325 
 7776 
 
 9375 
 
 6464 
 7929 
 9544 
 
 6605 
 8083 
 9714 
 
 6746 
 8239 
 
 9887 
 
 6889 
 8397 
 6061 
 
 7034 
 8556 
 0237 
 
 7179 
 8716 
 0415 
 
 7326 
 8878 
 0595 
 
 24 47 7' 
 
 26 52 78 
 29 58 87 
 
 95 "8 
 104 130 
 "5 '44 
 
 72 
 73 
 74 
 75 
 
 3-0777 
 32709 
 3.4874 
 
 0961 
 2914 
 5105 
 
 1146 
 3122 
 5339 
 
 1334 
 3332 
 
 5576 
 
 1524 
 3544 
 5816 
 
 1716 
 
 3759 
 6059 
 
 1910 
 
 3977 
 6305 
 
 2106- 
 4197 
 6554 
 
 2305 
 4420 
 6806 
 
 2506 
 4646 
 7062 
 
 32 64 96 
 36 72 108 
 
 41 82 122 
 
 129 161 
 144 180 
 162 203 
 
 3-7321 
 
 7583 
 
 7848 
 
 8118 
 
 8391 
 
 8667 
 
 8947 
 
 9232 
 
 9520 
 
 9812 
 
 46 94 139 
 
 i 86 232 
 
 76 
 77 
 78 
 79 
 80 
 81 
 ~82 
 83 
 84 
 
 4.0108 
 
 4-3315 
 4.7046 
 
 0408 
 3662 
 7453 
 
 0713 
 4015 
 7867 
 
 IO22 
 
 4374 
 8288 
 
 1335 
 4737 
 8716 
 
 1653 
 5107 
 9152 
 
 1976 
 
 5483 
 9594 
 
 2303 
 5864 
 0045 
 
 2635 
 6252 
 0504 
 
 2972 
 6646 
 0970 
 
 53 107 160 
 62 124 1 86 
 73 146 219 
 
 214 267 
 248 310 
 292 365 
 
 5.1446 
 5.6713 
 
 1929 
 7297 
 3859 
 
 2422 
 7894 
 4596 
 
 2924 
 8502 
 
 3435 
 9124 
 6122 
 
 3955 
 9758 
 6912 
 
 4486 
 0405 
 
 5026 
 1066 
 
 8548 
 
 5578 
 1742 
 9395 
 
 6140 
 ?1?? 
 
 87 175 262 
 
 35 437 
 
 6.313^ 
 
 5350 
 
 7920 
 
 6264 
 
 
 
 7-H54 
 8.144.3 
 9-5I44 
 
 2066 
 2636 
 9.677 
 
 3002 
 3863 
 9-845 
 
 3962 
 5126 
 
 IO.O2 
 
 4947 
 6427 
 
 IO.2O 
 
 5958 
 7769 
 10.39 
 
 6996 
 9 ! 52 
 10.58 
 
 8062 
 
 0579 
 10.78 
 
 9158 
 2052 
 10.99 
 
 0285 
 3572 
 
 11.20 
 
 Difference - col- 
 umns cease to be 
 useful, owing to 
 the rapidity with 
 which the value 
 of the tangent 
 changes. 
 
 85 
 86 
 87 
 88 
 
 89 
 
 n-43 
 
 11.66 
 
 11.91 
 
 12.16 
 
 1243 
 
 12.71 
 
 13.00 
 
 13-30 
 
 13-62 
 
 13-95 
 
 14.30 
 19.08 
 28.64 
 
 14.67 
 19.74 
 30.14 
 
 15-06 
 20.45 
 31.82 
 
 15.46 
 21.20 
 33-6Q 
 
 15.89 
 22.02 
 
 35-8o 
 
 16.35 
 22.90 
 38.19 
 
 16.83 
 23.86 
 40.92 
 
 17-34 
 24.90 
 44.07 
 
 17.89 
 26.03 
 47-74 
 
 18.46 
 27.27 
 52.08 
 
 57-29 
 
 63.66 
 
 71.62 
 
 81.8 
 
 95-49 
 
 114.6 
 
 143-2 
 
 191.0 
 
 286.5 
 
 573-0 
 
62 
 
 ANTI-LOGARITHMS 
 
 Hants, 
 
 01234 56789 
 
 PROPORTIONAL PARTS. 
 
 123 
 
 456 
 
 789 
 
 .00 
 .0 
 .02 
 .03 
 .04 
 
 1000 1002 1005 1007 1009 
 1023 1026 1028 1030 1033 
 1047 1050 1052 1054 1057 
 1072 1074 1076 1079 1081 
 1096 1099 1102 1104 1107 
 
 1012 1014 1016 1019 1021 
 1035 1033 1040 1042 1045 
 1059 1062 1064 1067 1069 
 1084 1086 1089 1091 1094 
 1109 1112 1114 1117 1119 
 
 1 
 001 
 1 
 
 1 1 
 1 1 
 1 1 
 
 222 
 222 
 222 
 
 1 1 
 
 1 1 
 
 222 
 
 .05 
 .06 
 .07 
 .08 
 .09 
 
 1122 1126 1127 1130 1132 
 1148 1151 1153 1156 1159 
 1175 1178 1180 1183 1186 
 1202 1205 1208 1211 1213 
 1230 1233 1236 1239 1242 
 
 1135 1138 1140 1143 1146 
 1161 1164 1167 1169 1172 
 1189 1191 1194 1197 1199 
 1216 1219 1222 1225 1227 
 1245 1247 1250 1253 1256 
 
 Oil 
 1 1 
 Oil 
 1 1 
 1 1 
 
 1 1 
 1 1 
 1 1 
 112 
 112 
 
 222 
 222 
 222 
 223 
 223 
 
 .10 
 
 .11 
 .12 
 
 1 
 
 1259 1262 1265 1268 1271 
 1288 1291 1294 1297 1300 
 1318 1321 1324 1327 1330 
 1349 1352 1355 1358 1361 
 1380 1384 1387 1390 1393 
 
 1274 1276 1279 1282 1285 
 1303 1306 1309 1312 1315 
 1334 1337 1340 1343 1346 
 1365 1368 1371 1374 1377 
 1396 1400 1403 1406 1409 
 
 1 1 
 Oil 
 1 1 
 Oil 
 1 1 
 
 113 
 122 
 122 
 122 
 122 
 
 228 
 223 
 223 
 233 
 233 
 
 '.li 
 .17 
 .18 
 .19 
 
 1413 1416 1419 1422 1426 
 1445 1449 1452 1455 1459 
 1479 1483 1486 1489 1493 
 1514 1517 1521 1524 1528 
 1549 1552 1556 1560 1563 
 
 1429 1432 1435 1439 1442 
 1462 1466 1469 1472 1476 
 1496 1500 1503 1507 1510 
 1531 1535 1538 1542 1545 
 1567 1570 1574 1578 1581 
 
 Oil 
 1 1 
 Oil 
 1 1 
 Oil 
 
 122 
 122 
 1 2 2 
 122 
 
 233 
 233 
 233 
 233 
 
 
 
 20 
 .21 
 .22 
 .23 
 .24 
 
 1585 1589 1592 1596 1600 
 1622 1626 1629 1633 1637 
 1660, 1663 1667 1671 1675 
 legs' 1702 1706 1710 1714 
 1738 1742 1746 1750 1754 
 
 1603 1607 1611 1614 1618 
 1641 1644 1648 1652 1656 
 1679 1683 1687 1690 1694 
 1718 1722 1726 1730 1734 
 1758 1762 1766 1770 1774 
 
 1 1 
 Oil 
 1 1 
 Oil 
 1 1 
 
 
 
 222 
 222 
 
 333 
 333 
 
 222 
 
 334 
 
 .25 
 
 .26 
 .27 
 .28 
 .29 
 
 1778 1782 1786 1791 1795 
 1820 1824 1828 1832 1837 
 1862 1866 1871 1875 1879 
 1905 1910 1914 1919 1923 
 1950 1954 1959 1963 1968 
 
 1799 1803 1807 1811 1816 
 1841 1845 1849 1854 1858 
 1884 1888 1892 1897 1901 
 1928 1932 1936 1941 1945 
 1972 1977 1982 1986 1991 
 
 Oil 
 1 1 
 
 
 
 Oil 
 1 1 
 
 
 
 .30 
 .31 
 .32 
 .63 
 .84 
 
 1995 2000 2004 2009 2014 
 2042 2046 2051 2056 2061 
 2089 2094 2099 2104 2109 
 2138 2143 2148 2153 2158 
 2188 2193 2198 2203 2208 
 
 2018 2023 2028 2032 2037 
 2065 2070 2075 2080 2084 
 2113 2118 2123 2128 2133 
 2163 2168 2173 2178 2183 
 2213 2218 2223 2228 2234 
 
 Oil 
 
 ] 1 
 1 1 
 
 223 
 
 344 
 
 223 
 
 344 
 
 1 2 
 
 
 
 .35 
 .36 
 .87 
 .38 
 .39 
 
 2239 2244 2249 2254 2259 
 2291 2296 2301 2307 2312 
 2344 2350 2355 2360 2366 
 2399 2404 2410 2415 2421 
 2455 2460 2466 2472 2477 
 
 2265 2270 2275 2280 2286 
 2317 2323 2328 2333 2339 
 2371 2377 2382 2388 2393 
 2427 2432 2438 2443 2449 
 2483 2489 2495 2500 2506 
 
 1 2 
 
 
 
 1 2 
 1 2 
 
 
 
 .40 
 .41 
 .42 
 .43 
 .44 
 
 2512 2518 2523 2529 2535 
 2570 2576 2582 2588 2594 
 2630 2636 2642 2649 2655 
 2692 2698 2704 2710 2716 
 2754 2761 2767 2773 2780 
 
 2541 2547 2553 2559 2564 
 2600 2606 2612 2618 2624 
 2661 2667 2673 2679 2685 
 2723 2729 2735 2742 2748 
 2786 2793 2799 2805 2812 
 
 1 2 
 1 2 
 1 2 
 1 2 
 1 2 
 
 
 
 234 
 
 455 
 
 334 
 834 
 
 456 
 456 
 
 45 
 
 .46 
 .47 
 .48 
 .49 
 
 2818 2825 2831 2838 2844 
 2884 2891 2897 2904 2911 
 2951 2958 2965 2972 2979 
 020 3027 3034 3041 3048 
 090 3097 3105 3112 3119 
 
 2851 2858 2864 2871 2877 
 2917 2924 2931 2938 2944 
 2985 2992 2999 3006 3013 
 3055 3062 3069 3076 3083 
 3126 3133 3141 3148 3155 
 
 1 2 
 1 2 
 1 2 
 1 2 
 1 2 
 
 
 
 334 
 334 
 344 
 344 
 
 556 
 556 
 666 
 566 
 
ANTI-LOGARITHMS 
 
 63 
 
 Hants. 
 
 01234 56789 
 
 PROPORTIONAL PARTS. 
 
 J. 2 3 
 
 456 
 
 789 
 
 .50 
 .51 
 .52 
 .53 
 .54 
 
 3162 3170 3177 3184 3192 
 3236 3243 3251 3258 3266 
 3311 3319 3327 3334 3342 
 3388 3396 3404 3412 3420 
 3467 3475 3483 3491 3499 
 
 3199 3206 3214 3221 3228 
 3273 3281 3289 3296 3304 
 3350 3357 3365 3373 3381 
 3428 3436 3443 3451 3459 
 3508 3516 3524 3532 3540 
 
 112 
 1 2 2 
 122 
 122 
 122 
 
 
 
 
 
 .55 
 .56 
 .57 
 .58 
 .59 
 
 3548 3556 3565 3573 3581 
 3631 3639 3648 3656 3664 
 3715 3724 3733 3741 3750 
 3802 3811 3819 3828 3837 
 3890 3899 3908 3917 3926 
 
 8589 3597 3606 3614 3622 
 3673 3681 3690 3698 3707 
 3758 3767 3776 3784 3793 
 3846 3855 3864 3873 3882 
 3936 3945 3954 3963 3972 
 
 
 
 677 
 
 123 
 
 
 
 .60 
 
 .61 
 .62 
 .63 
 .64 
 
 3981 3990 3999 4009 4018 
 4074 4083 4093 4102 4111 
 4169 4178 4188 4198 4207 
 4266 4276 4285 4295 4305 
 4365 4375 4385 4395 4406 
 
 4027 4036 4046 4055 4064 
 4121 4130 4140 4150 4159 
 4217 4227 4236 4246 4256 
 4315 4325 4335 4345 4355 
 4416 4426 4436 4446 4457 
 
 128 
 
 
 
 128 
 
 
 
 .65 
 
 .66 
 .67 
 .68 
 .69 
 
 4467 4477 4487 4498 4508 
 4571 4581 4592 4603 4613 
 4677 4688 4699 4710 4721 
 4786 4797 4808 4819 4831 
 4898 4909 4920 4932 4943 
 
 4519 4529 4539 4550 4560 
 4624 4634 4645 4656 4667 
 4732 4742 4753 4764 4775 
 4842 4853 4864 4875 4887 
 4955 4966 4977 4989 5000 
 
 
 
 
 128 
 
 467 
 
 8 9 10 
 8 9 10 
 
 
 
 
 .70 
 
 .71 
 .72 
 .73 
 .74 
 
 6012 5023 5035 5047 5058 
 5129 5140 5152 5164 5176 
 5248 5260 5272 5284 5297 
 5370 5383 5395 5408 5420 
 5495 5508 5521 5534 5546 
 
 6070 5082 5093 6105 5117 
 5188 5200 5212 5224 5236 
 6309 5321 5333 5346 5358 
 5433 5445 5458 5470 6483 
 6659 5572 6585 6598 6610 
 
 
 
 8 10 11 
 
 
 
 9 10 11 
 9 10 12 
 
 .75 
 .76 
 .77 
 .78 
 .79 
 
 5623 5636 5649 5662 5675 
 5754 5768 5781 5794 5808 
 5888 5902 5916 5929 5943 
 6026 6039 6053 6067 6081 
 6166 6180 6194 6209 6223 
 
 6689 5702 5715 5728 6741 
 5821 6834 6848 5861 5875 
 6957 5970 5984 5998 6012 
 6095 6109 6124 6138 6152 
 6237 6252 6266 6281 6295 
 
 
 
 9 10 12 
 9 11 12 
 10 11 12 
 10 11 18 
 10 11 18 
 
 
 
 .80 
 .81 
 .82 
 .83 
 .84 
 
 6310 6324 6339 6353 6368 
 6457 6471 6486 6501 6516 
 6607 6622 6637 6653 6668 
 6761 6776 6792 6808 6823 
 6918 6934 6950 6966 6982 
 
 6383 6397 6412 8427 6442 
 6531 6546 6561 6577 6592 
 6683 6699 6714 6730 6745 
 6839 6855 6871 6887 6902 
 6998 7016 7031 7047 7063 
 
 
 
 10 12 18 
 11 12 14 
 11 12 14 
 11 13 14 
 11 18 10 
 
 
 
 .85 
 .86 
 .87 
 .88 
 .89 
 
 7079 7096 7112 7129 7145 
 7244 7261 7278 7295 7811 
 7413 7430 7447 7464 7482 
 7586 7603 7621 7638 7656 
 7762 7780 7798 7816 7834 
 
 7161 7178 7194 7211 7228 
 7328 7345 7362 7379 7396 
 7499 7516 7534 7551 7568 
 7674 *91 7709 7727 7745 
 7852 7870 7889 7907 7925 
 
 8 
 3 
 8 
 
 4 
 4 
 
 7 8 10 
 7 8 10 
 7 9 10 
 7 9 11 
 7 911 
 
 12 18 15 
 12 13 16 
 12 14 16 
 12 14 16 
 18 14 16 
 
 .90 
 
 .91 
 .92 
 .93 
 .94 
 
 7943 7962 7980 7998 8017 
 8128 8147 8166 8185 8204 
 8318 8337 8356 8375 8395 
 8511 8531 8651 8570 8590 
 8710 8730 8750 8770 8790 
 
 8035 8054 8072 8091 8110 
 8222 8241 8260 8279 8299 
 8414 8433 8453 8472 8492 
 8610 8630 8650 8670 8690 
 8810 8831 8851 8872 8892 
 
 4 
 
 4 
 4 
 
 4 
 4 
 
 7 9 11 
 9 11 
 10 12 
 10 12 
 10 12 
 
 18 10 17 
 13 16 17 
 14 15 17 
 14 16 18 
 14 16 18 
 
 .95 
 
 .96 
 .97 
 .98 
 .99 
 
 8913 8933 8954 8974 8995 
 9120 9141 9162 9183 9204 
 9333 9354 9376 9397 9419 
 9550 9572 9594 9616 9638 
 9772 9795 9817 9840 9863 
 
 9016 9036 9057 9078 9099 
 9226 9247 9268 9290 9311 
 9441 9462 9484 9506 9528 
 9661 9683 9705 9727 9750 
 9886 9908 9931 9954 9977 
 
 4 
 4 
 2 4 
 2 4 
 2 5 
 
 10 12 
 
 11 18 
 11 13 
 11 13 
 11 14 
 
 10 17 10 
 15 17 19 
 15 17 20 
 16*18 20 
 16 1820 
 
64 
 
 LOGARITHMS. 
 
 10 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 .5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 123 
 
 456 
 
 789 
 
 0000 
 
 0043 
 
 0086 
 
 0128 
 
 0170 
 
 0212 
 
 0253 
 
 0294 
 
 0334 
 
 0374 
 
 Use Table second 
 page following 
 
 11 
 
 12 
 13 
 14 
 15 
 16. 
 17 
 18 
 19 
 20 
 
 0414 
 0792 
 1139 
 
 0453 
 
 0828 
 
 "73 
 
 0492 
 0864 
 1206 
 
 0531 
 0899 
 1239 
 
 0569 
 0934 
 1271 
 
 0607 
 0969 
 1303 
 
 0645 
 1004 
 1335 
 
 0682 
 1038 
 1367 
 
 6719 
 1072 
 1399 
 
 0755 
 1106 
 
 T430 
 
 4811 
 3 7 10 
 3 6 10 
 
 15 19 2 3 
 
 14 17 21 
 
 13 16 19 
 
 26 30 34 
 24 28 31 
 23 26 29 
 
 1461 
 1761 
 2041 
 
 1492 
 1790 
 2068 
 
 1523 
 1818 
 2095 
 
 1553 
 1847 
 
 2122 
 
 1584 
 1875 
 2148 
 
 1614 
 1903 
 2175 
 
 1644 
 !93i 
 
 22OI 
 
 1673 
 
 1959 
 2227 
 
 17^3 
 1967 
 
 2253 
 
 1732 
 2014 
 2279 
 
 3 6 9 
 368 
 35 8 
 
 12 15 18 
 n 14 17 
 ir 13 16 
 
 21 2 4 2 7 
 2O 22 25 
 I 8 21 24 
 
 2304 
 
 2553 
 2788 
 
 2330 
 
 2577 
 2810 
 
 2355 
 2601 
 
 2833 
 
 2380 
 2625 
 2856 
 
 2405 
 2648 
 2878 
 
 2430 
 2672 
 2900 
 
 2455 
 2695 
 2923 
 
 3139 
 
 2480 
 2718 
 2945 
 
 2504 
 2742 
 2967 
 
 2529 
 2765 
 2989 
 
 5 7 
 5 7 
 4 7 
 
 IO 12 15 
 
 9 12 14 
 9 ii 13 
 
 17 20 22 
 l6 19 21 
 
 16 18 20 
 
 3010 
 
 3032 
 
 3054 
 
 3075 
 
 3096 
 
 3Il8 
 
 3160 
 
 3181 
 
 3201 
 
 4 6 
 
 8 ii 13 
 
 15 17 19 
 
 21 
 22 
 23 
 
 3222 
 3424 
 
 3617 
 
 3243 
 3444 
 3636 
 
 3263 
 3464 
 3655 
 
 3284 
 3483 
 3674 
 
 3304 
 3502 
 3692 
 
 3324 
 3522 
 37H 
 
 3345 
 354i 
 3729 
 
 3365 
 3560 
 3747 
 
 3385 
 3579 
 3766 
 
 3404 
 3598 
 3784 
 
 4 6 
 4 6 
 4 6 
 
 8 IO 12 
 8 10 12 
 
 7 9 ii 
 
 14 16 18 
 14 IS -7 
 i3 '5 *7 
 12 14 16 
 
 12 14 IS 
 II 13 15 
 
 24 
 26 
 26 
 
 3802 
 
 3979 
 41^ 
 
 3820 
 
 3997 
 4166 
 
 3838 
 4014 
 4183 
 
 3856 
 4031 
 42OO 
 
 3874 
 4048 
 4216 
 
 3892 
 4065 
 4232 
 
 3909 
 4082 
 4249 
 
 3927 
 4099 
 4265 
 
 3945 
 4116 
 4281 
 
 3962 
 4133 
 4298 
 
 4 5 
 3 5 
 3 5 
 
 7 9 ii 
 7 9 10 
 7 8 10 
 
 27 
 28 
 
 29 
 
 30 
 
 43M 
 4472 
 4624 
 
 4330 
 W87 
 4639 
 
 4346 
 4502 
 4654 
 
 4362 
 4518 
 4669 
 
 4378 
 4533 
 4683 
 
 4393 
 
 4548 
 4698 
 
 4409 
 4564 
 4713 
 
 4425 
 4579 
 4728 
 
 4440 
 4594 
 4742 
 
 4456 
 4609 
 4757 
 
 3 5 
 3 5 
 3 4 
 
 689 
 689 
 679 
 
 II 13 14 
 II 12 14 
 10 12 13 
 
 477i 
 
 1786 
 
 4800 
 
 4814 
 
 4829 
 
 4843 
 
 4857 
 
 4871 
 
 4886 
 
 4900 
 
 3 4 
 
 679 
 
 IO II 13 
 
 31 
 32 
 33 
 
 4914 
 5051 
 
 5185 
 
 1928 
 5065 
 5198 
 
 4942 
 
 5079 
 5211 
 
 4955 
 5092 
 5224 
 
 4969 
 5105 
 
 5237 
 
 4983 
 5"9 
 5250 
 
 4997 
 5132 
 5263 
 
 5011 
 5145 
 5276 
 
 5024 
 
 5159 
 5289 
 
 5038 
 5172 
 5302 
 
 3 4 
 3 4 
 3 4 
 
 678 
 
 5 7 8 
 568 
 
 IO II 12 
 9 11 12 
 
 9 10 12 
 
 34 
 35 
 36 
 37 
 38 
 39 
 
 5315 
 544i 
 5563 
 
 5328 
 5453 
 5575 
 
 5340 
 5465 
 5587 
 
 5353 
 5478 
 5599 
 
 5366 
 5490 
 5611 
 
 5378 
 5502 
 5623 
 
 5391 
 5514 
 5635 
 
 5403 
 
 5527 
 5647 
 
 54i6 
 
 5539 
 5658 
 
 5428 
 
 5551 
 5670 
 
 3 4 
 4 
 
 4 
 
 568 
 5 6 7 
 5 6 7 
 
 9 10 ii 
 9_io u 
 8 10 ii 
 
 5682 
 5798 
 59" 
 
 5694 
 SSog 
 5922 
 
 5705 
 5821 
 
 5933 
 
 5717 
 5832 
 5944 
 
 5729 
 5843 
 5955 
 
 5740 
 5855 
 5966 
 
 5752 
 5866 
 5977 
 
 5763 
 5877 
 5988 
 
 5775 
 5888 
 
 5999 
 
 5786 
 
 5899 
 6010 
 
 3 
 3 
 3 
 
 5 6 7 
 5 6 7 
 457 
 
 8 9 10 
 8 9 10 
 8 9 10 
 
 40 
 
 6021 
 
 6031 
 
 6042 
 
 605.3 
 
 6064 
 
 6075 
 
 6085 
 
 6096 
 
 6107 
 
 6117 
 
 3 
 
 4 5 6 
 
 8 9 10 
 
 41 
 
 42 
 43 
 
 6128 
 6232 
 6335 
 
 6138 
 6243 
 6345 
 
 6149 
 6253 
 6355 
 
 6160 
 6263 
 6365 
 
 6170 
 6274 
 6375 
 
 6180 
 6284 
 6385 
 
 6191 
 6294 
 6395 
 
 6201 
 6304 
 6405 
 
 6212 
 6314 
 6415 
 
 6222 
 
 6325 
 6425 
 
 3 
 3 
 r 3 
 
 456 
 4 5 6 
 4 5 6 
 
 789 
 7 8 9 
 7 8 Q 
 
 44 
 45 
 46 
 
 6435 
 6532 
 6628 
 
 6444 
 6542 
 6637 
 6730 
 6821 
 6911 
 
 6454 
 6551 
 6646 
 
 6464 
 6561 
 6656 
 
 6474 
 
 6571 
 6665 
 
 6484 
 6580 
 6675 
 
 6493 
 6590 
 6684 
 
 6503 
 65-99 
 6693 
 
 6513 
 6609 
 6702 
 
 6522 
 6618 
 6712 
 
 3 
 3 
 3 
 
 456 
 4 5 6 
 4 5 6 
 
 7 8 9 
 
 7 8 9 
 7 7 8 
 
 47 
 48 
 49 
 
 6721 
 6812 
 6902 
 
 6739 
 6830 
 6920 
 
 6749 
 6839 
 6928 
 
 6758 
 6848 
 6937 
 
 6767 
 
 6857 
 6946 
 
 6776 
 6866 
 6955 
 
 6785 
 6875 
 6964 
 
 6794 
 68& 
 972 
 
 6803 
 
 6893 
 6981 
 
 3 
 
 455 
 
 6 7 8 
 
 3 
 
 445 
 
 6 7 8 
 
 50 
 51 
 52 
 63 
 
 6990 
 
 >99S 
 
 7007 
 
 7016 
 
 7024 
 
 7033 
 
 7042 
 
 7050 
 
 7059 
 
 7067 
 
 3 
 
 3^ 4 5 
 
 6 7 * 
 
 7076 
 7160 
 7243 
 
 7084 
 7168 
 7251 
 
 7093 
 7177 
 7259 
 
 7101 
 
 7185 
 7267 
 
 7110 
 7193 
 
 7275 
 
 7356 
 
 7118 
 7202 
 7284 
 
 7126 
 7210 
 7292 
 
 7135 
 7218 
 7300 
 
 7M3 
 7226 
 7308 
 
 7152 
 7235 
 73i6 
 
 ' 2 3 
 "V*i 2 
 
 122 
 
 345 
 345 
 345 
 
 6 7 8 
 6 7 7 
 667 
 
 54 
 
 7324 
 
 7332 
 
 7340 
 
 7348 
 
 7364 
 
 7372 
 
 738o 
 
 7388 
 
 7396 
 
 122 
 
 345 
 
 667 
 
LOGARITHMS 
 
 65 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 123 
 
 456 
 
 789 
 
 if) IVO l> CO IO\ O H 
 if) | if) if) if) 1 if) VO VO 
 
 7404 
 
 7412 
 
 7419 
 
 7427 
 
 7435 
 
 7443 
 
 745 1 
 
 7459 
 
 7466 
 
 7474 
 
 I 2 2 
 
 345 
 
 5 6 7 
 
 7482 
 7559 
 7634 
 
 7490 
 7566 
 7642 
 
 7497 
 7574 
 7649 
 
 7505 
 7582 
 7657 
 
 7513 
 7589 
 7664 
 
 7520 
 
 7597 
 7672 
 
 7528 
 7604 
 7679 
 
 753<> 
 7612 
 7686 
 
 7543 
 7619 
 7694 
 
 755J 
 7627 
 7701 
 
 
 
 I I 2 
 
 3 
 
 5 6 7 
 
 7709 
 
 7782 
 7853 
 
 77i6 
 7789 
 7860 
 
 7723 
 7796 
 7868 
 
 773i 
 7803 
 7875 
 
 7738, 
 7810 
 7882 
 
 7745 
 7818 
 7889 
 
 7752 
 7825 
 7896 
 
 7760 
 7832 
 7903 
 
 7767 
 
 7839 
 7910 
 
 7774 
 7846 
 
 79^7 
 
 I I 2 
 I I 2 
 112 
 
 3 
 3 
 3 
 
 5 6 7 
 5 6 t 
 5 6 fc 
 
 62 
 63 
 64 
 
 7924 
 
 7993 
 8062 
 
 7931 
 Sooo 
 8069 
 
 7938 
 8007 
 8075 
 
 7945 
 8014 
 
 8082 
 
 7952 
 8021 
 8089 
 
 7959 
 8028 
 8096 
 
 7966 
 8035 
 8102 
 
 7973 
 8041 
 8109 
 
 7980 
 8048 
 8116 
 
 7987 
 805 q 
 812? 
 
 I 1 2 
 112 
 I I 2 
 
 112 
 
 3 3 
 3 3 
 3 3 
 
 5 6 6 
 
 5 S 6 
 
 s s e 
 
 65 
 
 8129 
 
 8136 
 
 8142 
 
 8149 
 
 8156 
 
 8162 
 
 8169 
 
 8176 
 
 8182 
 
 8189 
 
 3 3 
 
 5 S <- 
 
 66 
 67 
 68 
 
 8195 
 8261 
 8325 
 
 8202 
 8267 
 8331 
 
 8209 
 8274 
 8338 
 
 8215 
 8280 
 8344 
 
 8222 
 8287 
 8351 
 
 8228 
 8293 
 8357 
 
 8235 
 8299 
 
 8363 
 
 8241 
 8306 
 8370 
 
 8248 
 8312 
 8376 
 
 8254 
 8319 
 8382 
 
 8445 
 8506 
 8567 
 
 I I 2 
 I I 2 
 112 
 
 3 3 
 3 3 
 3 3 
 
 5 5 6 
 5 5 6 
 4 5 6 
 
 69 
 70 
 71 
 
 8388 
 845J 
 8^13 
 
 8395 
 8457 
 8519 
 
 8401 
 8463 
 8S2* 
 
 8407 
 8470 
 
 853i 
 
 8414 
 8476 
 
 8*37 
 
 8420 
 8482 
 8543 
 
 8426 
 8488 
 8549 
 
 8432 
 8494 
 
 8555 
 
 8439 
 8500 
 8561 
 
 112 
 
 *\3 
 
 2 ^ 
 
 4 5 ( 
 
 72 
 73 
 74 
 
 S573 
 8633 
 8692 
 
 8579 
 8639 
 8698 
 
 8585 
 8645 
 8704 
 
 859 1 
 8651 
 8710 
 
 8597 
 8657 
 8716 
 
 8603 
 8663 
 8722 
 
 8609 
 8669 
 8727 
 
 8615 
 8675 
 8733 
 
 8621 
 8681 
 8739 
 
 8627 
 8686 
 874S 
 
 112 
 I I 2 
 
 2 3 
 2 3 
 
 455 
 455 
 
 75 
 
 875J 
 
 8756 
 
 8762 
 
 8768 
 
 8774 
 
 8779 
 
 8785 
 
 8791 
 
 8797 
 
 8802 
 
 112 
 
 233 
 
 4 5 5 
 
 76 
 77 
 78 
 
 8808 
 8865 
 8921 
 
 8814 
 8871 
 8927 
 
 8876 
 8932 
 
 8882 
 8938 
 
 8831 
 8887 
 8943 
 
 8837 
 8893 
 8949 
 
 8842 
 8899 
 8954 
 
 8848 
 8904 
 8960 
 
 8^4 
 8910 
 8965 
 
 885-9 
 8915 
 8971 
 
 112 
 112 
 I I 2 
 
 233 
 233 
 
 4 5 
 
 4 5 
 
 79 
 80 
 81 
 
 8976 
 9031 
 9085 
 
 8982 
 9036 
 9090 
 
 8987 
 9042 
 9096 
 
 8993 
 ^047 
 9101 
 
 8998 
 
 9053 
 9106 
 
 9004 
 9058 
 9112 
 
 9009 
 9063 
 9117 
 
 9015 
 9069 
 9122 
 
 9020 
 9074 
 9128 
 
 9025 
 9079 
 9*33 
 
 I I 2 
 112 
 112 
 
 233 
 233 
 233 
 
 4 5 
 4 5 
 4 5 
 
 82 
 83 
 84 
 
 9138 
 9191 
 9243 
 
 9*43 
 9196 
 9248 
 
 9149 
 9201 
 9253 
 
 9154 
 9206 
 9258 
 
 9159 
 9212 
 9263 
 
 9 l6 5 
 9217 
 9269 
 
 9170 
 9222 
 9274 
 
 9175 
 9227 
 
 9279 
 
 9180 
 9232 
 9284 
 
 9186 
 9238 
 9289 
 
 I I 2 
 
 2 3 3j 4 4 5 
 
 
 
 
 
 
 
 85 
 
 9294 
 
 9299 
 
 9304 
 
 9309 
 
 9315 
 
 9320 
 
 9325 
 
 9330 
 
 9335 
 
 9340 
 
 
 
 
 86 
 87 
 88 
 
 9345 
 9395 
 944S 
 
 9350 
 9400 
 9450 
 
 9355 
 9405 
 9455 
 
 9360 
 9410 
 9460 
 
 9365 
 9415 
 9465 
 
 9370 
 9420 
 9469 
 
 9375 
 9425 
 9474 
 
 9380 
 9430 
 9479 
 
 9385 
 9435 
 9484 
 
 9390 
 9440 
 9489 
 
 
 
 
 I I 
 
 2 2 3 
 
 3 4 4 
 
 89 
 90 
 91 
 
 9494 
 9542 
 959 
 
 9499 
 9547 
 9595 
 
 9504 
 9552 
 9600 
 
 9509 
 9557 
 9605 
 
 9513 
 9562 
 9609 
 
 95i8 
 9566 
 9614 
 
 9523 
 957' 
 9619 
 
 9528 
 9576 
 9624 
 
 9533 
 958i 
 9628 
 
 9538 
 9580 
 9^33 
 
 Oil 
 Oil 
 Oil 
 
 2 2 3 
 
 223 
 223 
 
 3 4 A 
 3 4 A 
 344 
 
 92 
 93 
 94 
 
 9638 
 9685 
 973' 
 9777 
 
 9643 
 9689 
 9736 
 
 9647 
 9694 
 9741 
 
 9652 
 9699 
 9745 
 9791 
 
 9657 
 9703 
 9/50 
 
 9661 
 9708 
 9754 
 
 9666 
 97*3 
 9759 
 
 9671 
 9717 
 9763 
 
 9675 
 9722 
 9768 
 
 9680 
 9727 
 9773 
 
 I 1 
 Oil 
 I I 
 
 223 
 223 
 
 344 
 344 
 
 
 
 95 
 
 9782 
 
 9786 
 
 9795 
 
 9800 
 
 9805 
 
 9809 
 
 9814 
 
 9818 
 
 Oil 
 
 223 
 
 3 4 4 
 
 96 
 97 
 98 
 
 9823 
 9868 
 9912 
 
 9827 
 9872 
 9917 
 
 9832 
 9877 
 9921 
 
 9836 
 9881 
 9926 
 
 9841 
 9886 
 9930 
 
 9845 
 9890 
 
 9934 
 
 9850 
 9894 
 9939 
 
 9854 
 9899 
 
 9943 
 
 9859 
 9903 
 9948 
 
 9863 
 9908 
 9952 
 
 O I I 
 I I 
 I I 
 
 223 
 223 
 223 
 
 344 
 344 
 344 
 
 99 
 
 9956 
 
 9961 
 
 9965 
 
 9969 
 
 9974 
 
 9978 
 
 9983 
 
 9987 
 
 999 1 
 
 9996 
 
 I I 
 
 223 
 
 334 
 
66 
 
 LOGARITHMS. 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 100 
 
 00000 
 
 043 
 
 087 
 
 130 
 
 173 
 
 217 
 
 s6o 
 
 303 
 
 346 
 
 389 
 
 101 
 102 
 103 
 
 432 
 860 
 
 01 284 
 
 475 
 93 
 326 
 
 5i8 
 945 
 368 
 
 561 
 988 
 410 
 
 604 
 030 
 452 
 
 647 
 072 
 494 
 
 689 
 US 
 536 
 
 732 
 157 
 578 
 
 775 
 199 
 620 
 
 817 
 242 
 662 
 
 104 
 105 
 106 
 
 703 
 
 02 IIQ 
 
 531 
 
 745 
 1 60 
 572 
 
 787 
 202 
 612 
 
 828 
 243 
 653 
 
 870 
 284 
 694 
 
 912 
 325 
 
 735 
 
 953 
 366 
 776 
 
 995 
 407 
 816 
 
 036 
 449 
 
 857 
 
 078 
 490 
 898 
 
 107 
 108 
 109 
 
 938 
 03342 
 
 743 
 
 979 
 
 383 
 
 782 
 
 019 
 
 423 
 
 822 
 
 060 
 
 463 
 862 
 
 100 
 
 503 
 902 
 
 141 
 
 543 
 941 
 
 181 
 
 583 
 981 
 
 222 
 623 
 O2 1 
 
 262 
 663 
 060 
 
 302 
 703 
 
 100 
 
 To find the logarithm of a number: First, locate in the 
 table the mantissa which lies in line with the first two figures of the 
 number and underneath the third figure, then increase this mantissa 
 by an amount depending upon the fourth figure of the number and 
 found by means of the interpolation columns at the right; secondly, 
 determine the characteristic, or the exponent of that integer power 
 of 10 which lies next in value below the number; for example, 
 log 600= 0.7782 -h 2. ; log 73.46= 0.8661 + 1. ; 
 
 log .006-0.7782-3.; log .7346=0.8661-1.; 
 
 log 6.003= 0.7784 + 0. ; log 7349= 0.8662 + 3. 
 
 The logarithm of a product of two or more numbers is the sum of 
 the logarithms of its factors; for example, 
 
 log. (.0821 X 463.2) = (0.9143 - 2.) + (0.6658 + 2.) = 0.5801 + 1. 
 The logarithm of a quotient is the difference between the logar- 
 ithms of the dividend and divisor; for example, 
 
 log. (.5321 -f- 916) = (0.7260 - 1 .) - (0.9619 + 2.) = 0.7641 - 4. 
 The logarithm of a power or root of a number is the exponent 
 times the logarithm of the number; for example, 
 
 log V^63) 3 =3/2 X (0.9360 1.) = 0.9040 - 1. 
 
 To find the number from its logarithm: Locate in the table 
 the mantissa next less than the given mantissa, then join the figure 
 standing above it at the top of the table to the two figures at the 
 extreme left on the same line as the mantissa, and finally to these 
 three join the figure at the top of the interpolation column which 
 contains the difference between the two mantissae. In the four- 
 figure number thus found, so place the decimal point that the 
 number shall be the product of some number, that lies between 
 1 and 10, by a power of 10 whose exponent is the characteristic 
 of the logarithm. For example, 
 
 antilog (0.6440 + 3) = 4405; 
 antilog (0.3069 - 2) = .02027. 
 
 Caution. In adding and subtracting logarithms it is well to 
 remember that the mantissa is always essentially positive and may 
 or may not therefore, have the same sign as its characteristic. 
 
INDEX 
 
 Absolute, humidity - 41 
 
 Acceleration, uniform - 5 
 
 due to gravity - - 10 
 
 apparatus, described - 6 
 
 normal - 8 
 
 Adiabatic changes, law for 41 
 
 Air thermometer, constant 
 
 volume - 37 
 
 Auguste's Psychrometer 42 
 
 Boiling Point, correction - 27 
 Boiling Point, variation 
 
 with pressure - 35 
 
 Boyle's Law, used - - 45 
 Calibration 
 
 absolute, of thermometer 25 
 
 plot of - 28 
 
 sample set of data - 30 
 
 of thermometer tube - 26 
 
 of set of weights - 23 
 
 Centripetal force - - 8 
 
 Charles' Law, tested - 38 
 
 Clement & Desormes, 
 
 method of - - 44 
 Coincidences, method of 12 
 Coefficient of expansion of 
 air, with air thermome- 
 ter - - 37 
 Dew point, defined - 41 
 Efflux of gases 21 
 Force, centripetal - - 8 
 Freezing Point, correction 27 
 Fusion, heat of, of tin - 46 
 "G" determination of, 
 
 with fall- machine - 5 
 with pendulum - - 10 
 Heat, mechanical equiva- 
 lent of ... 18 
 
 Heat of Neutralization - 33 
 
 Heat of Solution 31 
 
 Humidity, absolute, de- 
 fined - - 41 
 relative, defined - - 41 
 
 Hygrometer, formula for 
 
 use with - - 43 
 
 Alluard's 42 
 
 wet-and-dry-bulb - 42 
 
 Mechanical Equivalent of 
 
 Heat, defined - 18 
 
 by Calendar's method 18 
 
 Moment of inertia of a 
 
 disk 15 
 
 Melting points, zinc, tin - 46 
 
 Momentum - 14 
 
 Motion, study of uni- 
 formly accelerated - 5 
 
 Normal Solution - - 34 
 
 Pendulum, simple - 12 
 
 Period, of vibration of 
 
 pendulum - - 10 
 
 Ratio of specific heats - 44 
 
 Specific heat, of graphite, 
 
 porcelain, fire clay - 48 
 
 Thermometer, constant 
 
 volume air - 37 
 
 Uniform, accelerated mo- 
 tion . - - - - 5 
 circular motion - - 8 
 
 Vapor pressure and tem- 
 perature - - 39 
 
 Velocity, average 7 
 
 Wet- and dry-bulb, hy- 
 grometer * - - 42 
 

 
PHYSICAL MEASUREMENTS 
 
 MINOR 
 
 PART III. MAGNETISM AND ELECTRICITY 
 1916-1917 
 
PHYSICAL MEASUREMENTS 
 
WETZEL BROS. PRINTING CO. 
 
 2110 ADDISON STREET 
 
 BERKELEY, CAL. 
 
PHYSICAL MEASUREMENTS 
 
 A Laboratory Manual in General Physics 
 For Colleges 
 
 by 
 RALPH S. MINOR, PH. D. 
 
 Associate Professor of Physics, University of California 
 
 IN FOUR PARTS 
 
 PART III. 
 MAGNETISM AND ELECTRICITY 
 
 Edited in Collaboration With 
 RAYMOND B. ABBOTT, M. S. 
 
 Instructor in Physics, University of California 
 
 Berkeley, California 
 1916 
 
Copyrighted in the year 1916 
 
 by 
 Ralph S. Minor 
 
LIST OF EXPERIMENTS 
 
 Magnetism Page 
 
 General Statements and Directions - - 7 
 
 51. Plotting of Magnetic and Electric Fields 9 
 
 52. Laws of Magnetic Force - - 12 
 
 53. Relative Determination of H 14 
 
 54. Absolute Determination of H - - 15 
 
 55. Magnetic Induction 
 
 B and H Curve - 19 
 
 Electricity 
 
 General Directions - - 23 
 
 Theory of the Tangent Galvanometer - 23 
 
 61. Test of the Tangent Law 24 
 
 62. Reduction Factor of Galvanometer 
 
 By Deposition of Copper - 26 
 
 63. Measurement of Resistance 
 
 Wheatstone Bridge - 29 
 
 64. Comparison of Resistances 
 
 Carey Foster's Method - 30 
 
 65. Absolute Determination of Resistance 
 
 Calorimeter Method - 32 
 
 66. Electromotive Force 
 
 PoggendorfTs Method. The Potentiometer - 34 
 
 67. Thermo-Electromotive Force 
 
 Calibration of a Galvanometer as a Voltmeter 37 
 
 68. The Earth-Inductor 
 
 Relative Calibration of a Galvanometer - 40 
 
 69. Resistance of an Electrolyte 
 
 Kohlrausch's Method. Alternating Current- 
 Telephone - - 42 
 
 70. Study of Polarization Effects in a Leclanche 
 
 Cell 
 
 Galvanometer used as a Voltmeter - 45 
 
 71. Comparison of Capacities - 46 
 
 72. Efficiency of Electrical Heating Devices - - 47 
 
REFERENCES 
 
 Brooks and Poyser: Electricity and Magnetism. 
 Duff: Textbook of Physics (Fourth Edition.) 
 Ganot: Textbook of Physics (18th Edition). 
 Kimball: College Physics. 
 Thompson: Electricity and Magnetism. 
 
 Kaye and Laby: Physical and Chemical Constants. 
 I/andolt and Bernstein: Physical and Chemical Tables. 
 Smithsonian Institute: Physical Tables. 
 
MAGNETISM 
 
 General Statements and Directions. 
 
 The phenomena of magnetism and current electricity 
 are almost inseparably connected. A magnetic field of 
 force always exists in the medium surrounding a conduc- 
 tor carrying a current of electricity, and if a conductor 
 be moved across a magnetic field an electromotive force is 
 always induced in the conductor. Whether a magnetic 
 field be due to the magnetism of the earth, to a system of 
 magnets, or to electric currents in conductors, the field at 
 any point is completely determined when the direction 
 and intensity of the force at the point are known. 
 
 1. The direction of the magnetic force is arbitrarily de- 
 fined as the direction in which an isolated north-seeking 
 magnet pole would be urged by the magnetic force if placed 
 in a magnetic field. Imaginary lines showing at all points 
 the direction in which the magnetic force acts are called 
 lines of force. Magnetic lines of force are closed curves, 
 i. e., they have no free ends. 
 
 2. The strength or intensity of the magnetic force at 
 any point in a magnetic field is numerically equal to the 
 force in dynes which would be exerted by the field upon a 
 unit magnetic pole placed at that point, a unit magnetic 
 pole being defined as a magnetic pole of such strength 
 that when placed in air at a distance of one centimeter 
 from a similar pole it will repel it with a force of one dyne. 
 Dimensionally the intensity of the magnetic field is force 
 divided by pole strength. The unit of magnetic field 
 intensity is called the gauss. 
 
 The intensity of the earth's magnetic field is usually 
 represented by the letter H, field intensity in general will 
 be designated by H. This might be expressed by saying 
 
8 MAGNETISM 
 
 that the magnetic force would produce H lines per square 
 centimeter in air. 
 
 3. If a piece of iron or other magnetizable substance 
 be placed in the magnetic field some of the magnetic 
 lines pass through the substance and magnetize it; in 
 fact, more lines pass through iron than through the air 
 which it displaces. The actual number of magnetic lines 
 that run through unit area of cross-section in the iron 
 or other material is denoted by the letter B and is a 
 measure of the magnetization. 
 
 The ratio of B to H is called the permeability, /x. 
 
 4. The poles of a magnet are sometimes thought of 
 as points, usually near the ends of a magnet, at which 
 the magnetic charge may be regarded as concentrated. 
 In any given magnet, however, the magnetization is not 
 concentrated in two points, but is distributed over the 
 magnet. A more extended conception of a magnetic pole 
 may be obtained by considering a magnet placed in a 
 uniform magnetic field. Then the forces acting on the 
 elementary positive magnetic charges will be a system of 
 parallel forces all acting in the same direction. By com- 
 position of forces these parallel forces may be replaced by 
 a single force acting at a point TV which is the center for 
 this system of parallel forces. This point, N, is the posi- 
 tive or north-seeking pole of the magnet. In the same 
 way the forces acting on the elementary negative charges 
 may be replaced by a single resultant force acting at 5, 
 the negative pole of the magnet. 
 
 If in the above case the strength of the uniform field is 
 of unit intensity then the magnitude of the force acting 
 at N or 5 is equal numerically to the pole strength of 
 the magnet. 
 
51] MAPPING MAGNETIC LINES OF FORCE 9 
 
 If the field is not uniform the resultant of all the forces 
 acting will not pass through the points N or S, so that as 
 far as the magnet is concerned its poles are a convenient 
 but highly idealized conception. 
 
 5. The magnetic moment of a magnet, however, is an 
 important quantity with a perfectly definite physical 
 meaning. It is found that the tendency of a magnet to 
 turn or to be turned by another magnet depends not only 
 upon the strength of its poles, m, but also upon their 
 distance apart, 21. The product of these two quantities 
 mx2l is called the magnetic moment, M, of the magnet. 
 If the magnet be placed in a field of intensity, H, and the 
 turning moment (G) be found, when the axis of the 
 magnet is at right angles to the lines of force, then 
 
 M H = G. 
 
 6. In handling magnets care must be taken not to 
 bring the magnets into strong magnetic fields, or to jar 
 them. Rough treatment may change the strength of the 
 magnet and this change in the middle of an exercise will 
 necessitate the repetition of all previous measurements 
 involving the magnetic moment of the magnet. Always 
 protect the jewel of the compass needle, never move nor 
 leave the compass without first releasing the needle from 
 the pivot. 
 
 51. PLOTTING OF MAGNETIC AND ELECTRIC FIELDS. 
 
 Equipotential Lines. 
 A. Magnetic Field. 
 
 The object of this part of the exercise is to plot the magnetic 
 field due to the combination of the fields of two magnets 
 with the earth's magnetic field. 
 
 (a) Fasten two Robison spherical-ended magnets with 
 their axes vertical to the underside of the laboratory 
 
10 PLOTTING AN ELECTRIC FIELD [51 
 
 table. The magnets should be about 20 centimeters 
 apart and the line through them should coincide with 
 the magnetic meridian. Upon the top of the table, over 
 the magnets, fasten a large sheet of paper. To trace a 
 force line, place the compass needle, which should not 
 be over 2 cm. long, on the paper and make a dot on the 
 paper under each end of the needle. Now move the 
 needle until the S-pole is over the dot just made under 
 the N-pole, and make a dot under the new position of 
 the N-pole. Continue this process until you reach a 
 place where the compass suddenly reverses its direction, 
 or the line runs off the paper; and then, working in the 
 opposite direction from the starting point, proceed until 
 a second such point is found. In this way plot four or 
 five lines symmetrically located, enough to show clearly 
 the complete outline of the field. 
 
 Plot with especial care the ends of the lines near the 
 singular points, i. e., points in which the needle will 
 assume no definite direction. 
 
 (6) Using red ink draw a second series of lines which 
 are everywhere approximately perpendicular to the first 
 set. These then are the lines of equipotential. 
 
 B, Electric Field. 
 
 The object of this part of the exercise is to plot the equipo- 
 tential lines in an electric field and to draw the electric force 
 lines. 
 
 (c) Fill the glass bottomed tray to a depth of three 
 or four millimeters with a solution of ammonium chloride. 
 Connect one cell of storage battery to the primary of the 
 small induction coil and then connect the secondary 
 terminals of the coil with two electrodes A and B which 
 
51] PLOTTING AN ELECTRIC FIELD 11 
 
 have their pointed ends turned downward so as to dip 
 into the solution. These electrodes should be about 20 
 centimeters apart. 
 
 When the coil is in operation the points A and B will 
 be maintained at different potentials, and although this 
 is a constantly varying potential difference the equipoten- 
 tial lines will preserve the same configuration, although, 
 of course, assuming different absolute values. This fact 
 enables us to use a telephone receiver in exploring the 
 equipotential lines. For if both of its terminals lie on the 
 same equipotential surface, no current will pass through 
 the receiver. If, however, the two terminals lie on sur- 
 faces of different potentials, a charge is urged through 
 the receiver from the point of high potential to that of 
 low potential and the receiver responds by buzzing. 
 
 Connect the telephone receiver to one fixed electrode 
 and one exploring electrode furnished with a handle. 
 Place the fixed electrode about three centimeters from 
 the electrode A on the line A B and with the exploring 
 electrode find enough points for which there is no buzzing 
 in the receiver so that the equipotential line through the 
 position of the fixed electrode can be plotted. 
 
 Move the fixed electrode about three centimeters 
 toward B and locate a second equipotential line, and 
 continue in this way until four or five lines covering the 
 entire field have been explored. 
 
 (d) Plot on co-ordinate paper to one half scale the 
 lines so found. The lines of force will form a system 
 everywhere at right angles to these equipotential lines. 
 Draw this system in red ink. 
 
12 LAWS OF MAGNETIC FORCE [52 
 
 52. LAWS OF MAGNETIC FORGE. 
 
 In every real magnet there is always an equal quantity 
 of both kinds of magnetism. It is impossible to isolate a 
 region of positive magnetism ; if the magnet be broken 
 each piece is found still possessed of equal quantities 
 of both kinds of magnetism. If we have a bar magnet 
 the free magnetism is strongest in a region near the end. 
 In short, thick magnets this region is comparatively 
 large; as the length increases and the diameter decreases 
 the free magnetism is found nearer the end. In a very 
 long thin filament the free magnetism will be at the end. 
 
 The object of this experiment is to study the variation in 
 field intensity with distance in the magnetic field near one end 
 of a long magnet and in a plane perpendicular to its axis. 
 
 The magnet is a piece of steel ribbon about 150 centi- 
 meters long. To avoid the formation of consequent poles 
 it should be magnetized by placing it inside a long coil of 
 wire carrying a heavy current, the magnetic circuit being 
 completed outside the coil, from one end of the magnet to 
 the other, by means of pieces of soft iron. 
 
 The compass needle is a magnetized steel disk, 20 mil- 
 limeters in diameter, with a long aluminum pointer set 
 at right angles to the magnetic axis of the disk. 
 
 The magnet should be mounted with its axis vertical 
 and one end on a level with the compass needle. 
 
 (a) With the magnet removed to a considerable dis- 
 tance the compass should first be set so that the pointer 
 reads zero and then leveled so that the pointer clears the 
 scale. 
 
52] LAWS OF MAGNETIC FORCE 13 
 
 Place he magnet on the east-and-west line, west of the 
 compassjlneedle, and read the deflections produced for 
 the series of distances 20, 21, 22, 28, 29, 30, centimeters 
 between the compass needle and the magnet. Readings 
 should be taken from both ends of the pointer. 
 
 (b) Repeat the readings for the same distances to 
 the east of the compass needle. 
 
 (c) Show that if the compass needle is deflected by 
 a horizontal force acting in an east-and-west direction, 
 the magnitude of the force is proportional to the tangent 
 of the angle of deflection. 
 
 State the law for the mutual action of magnet poles. 
 
 If the magnetic force varies as some power, n, of the 
 distance the product d n tan a should be a constant. That 
 is, d 22 n tan a 22 = k = d 2s n tan a 28 etc. From which we get 
 d 28 n / d 22 a = tan a 22 / tan a 2S , and n = (log tan a^ - log tan 
 a 2s) I (log d 2s - log dv). 
 
 Average the readings for each distance. By pairing 
 each of the first three values of d n tan a with each of the 
 last three in turn, calculate nine values for n and take 
 their average. Find the average difference in percent 
 between these values and your result. 
 
 What error is eliminated by reading both ends of the 
 pointer? by taking readings both east and west of the 
 needle? 
 
 X 
 
 (d) Two assumptions have been made whose effect 
 should be considered. The force due to one pole has been 
 neglected when considering that of the other. Calculate 
 the force discarded, assuming the inverse square law, 
 (n = 2) and express it in per cent of the force used, for 
 
14 RELATIVE DETERMINATION OF H [52-53 
 
 the largest and smallest values of the distance, d. Second- 
 ly, the force lines from the magnet pole to compass needle 
 have been assumed parallel. Calculate the error this 
 produces, in per cent of the force used, for the same 
 distances, the distance between the poles of the needle 
 being taken as 18 millimeters and its direction being 
 assumed perpendicular to the east-and-west line. 
 
 53. RELATIVE DETERMINATION OF H. 
 A. Method of Oscillations. 
 
 If a magnet vibrating as a torsion pendulum, at a point 
 where the horizontal intensity has the value Hi, has a 
 period Ti, and at a second point the period is found to be 
 T 2 , then the horizontal intensity, H 2 , at the second point 
 is given by the relation Hi : H 2 = TV : TV. 
 
 (a) Determine the value of H, using the method of 
 oscillations. The amplitude of the vibration should be 
 small and care taken to prevent the magnet from swinging. 
 
 B. Using a Tangent Galvanometer, an Ammeter and an 
 adjustable Resistance.* 
 
 Set up the tangent galvanometer at the point where 
 the value of H is to be found and connect in series with it 
 an ammeter, a single storage or Daniell cell, and an ad- 
 justable resistance. The ammeter should be placed at 
 some distance from the galvanometer. 
 
 Adjust the resistance until the galvanometer reads the 
 angle whose tangent is equal to one tenth of the galvanometer 
 constant, the ammeter reading will then give H directly. 
 
 *See R, B. Abbott, School Science and Mathematics, Vol. XII, 
 page 533, 1913. 
 
53-54] ABSOLUTE DETERMINATION OF H 15 
 
 10 TJ 
 
 Since in general I = - tan a, where n is the number of 
 2-rrn/r 
 
 turns and r the mean radius of the coils (see Theory of Tangent 
 Galvanometer, p. 24). If then, we so adjust the current that 
 
 tan a = - X - it follows by substitution 
 10 r 
 
 I = - -X--X lirn/r = H, or I = H numerically. 
 lirn/r 10 
 
 54. ABSOLUTE DETERMINATION OF H. 
 
 The following absolute determination of the horizontal 
 intensity of H of the earth's magnetic field at any point 
 requires the measurement of the values of the quotient 
 M/ H and the product M H, where M is the moment of 
 the magnet. 
 
 If a magnet be placed with its axis at right angles 
 to the meridian, from the observed deflection of a com- 
 pass needle or magnetometer placed on the axis of the 
 magnet the quotient M / H may be calculated. 
 
 If this same magnet be put in a bifilar suspension with 
 the plane of the wires at right angles to the meridian, 
 from the deflection produced by the earth's field in turn- 
 ing the magnet toward the magnetic meridian the prod- 
 uct M H may be found. From these two quantities either 
 H or M may be found. Either method may be used alone 
 to compare the moments of two magnets. 
 
 The magnet used must be kept away from other mag- 
 nets or magnetic bodies and be carefully handled until 
 both experiments have been completed so as not to change 
 the amount or distribution of the magnetism. 
 
 A. Finding M/H by Deflection in End-on Position. 
 
 (a) As deflecting magnet, one of the Robison type will 
 be used since these have practically true poles, located at 
 
16 ABSOLUTE DETERMINATION OF H [54 
 
 the centers of the spherical ends. Measure the distance, 
 21, between the centers of the spheres. 
 
 (b) Place a magnetometer at the point at which H is 
 to be measured and place the magnet with its axis on the 
 east-and-west line, east or west of the compass at a dis- 
 tance which is relatively large compared with length of 
 the magnet in the magnetometer. Measure the distance 
 from the center of the magnetometer to the nearest end 
 of the magnet, and read the deflection of the magneto- 
 meter using a telescope and scale placed one meter away. 
 Reverse the magnet and again note the deflection. Repeat 
 the measurements with the magnet on the other side of 
 the magnetometer and the same distance away. With the 
 aid of the data taken in (a) find L, the mean distance 
 from the center of the magnetometer to the point midway 
 between the poles of the magnet. 
 
 (c) Assuming the inverse square law derive an ex- 
 pression for the strength of field due to a magnet at any 
 point, s, on its axis. Call the distance between the poles 
 of the magnet 21, its pole strength m, and L the distance 
 from the point s to a point midway between the poles. 
 
 Show that, if this force acting in an east-and-west 
 direction deflects a compass needle, the magnitude of 
 the force is proportional to the tangent of the angle of 
 deflection, a, and if the length of the compass needle be 
 negligible in comparison to the distance L show that 
 
 M (L 2 -/ 2 ) 2 
 -H' ~^L tana 
 
 where 2 / m = M = the moment of the magnet. 
 
 Using this relation and the above data calculate the 
 numerical value of M / H. 
 
54] ABSOLUTE DETERMINATION OF H 17 
 
 B. Finding M H by the bifilar method. 
 
 (a) Suspend the carriage for the magnet by two 
 lengths of fine braided fish-line, adjustable from above 
 so that they are in a plane perpendicular to the magnetic 
 meridian. The vane extending from the carriage should 
 be placed in a vessel of water to "damp" the vibrations. 
 
 To find the zero position place a brass rod of about the 
 same size as the magnet in the carriage and adjust the 
 telescope and scale, placed about a meter from the car- 
 riage, until the zero of the scale seen in the telescope 
 after reflection in the mirror coincides with the cross-hairs. 
 Measure the distance from the mirror to the scale. 
 
 Remove the brass rod, place the magnet in the carriage, 
 and read the deflection. It is not necessary to wait until 
 the system comes to rest before taking a reading. Where 
 the oscillations are small determine the resting point by 
 reading the turning points as in weighing. Reverse the 
 magnet and again read the deflection. Make several read- 
 ings, reversing the magnet each time, and take the mean. 
 
 (b) Weigh the magnet and carriage on the trip scales 
 to one decigram. Measure the distance d between the two 
 wires of the bifilar suspension and also the length of the 
 wires from the point of suspension to the top of the car- 
 riage. 
 
 (c) Since the angle of deflection a is small, A A' is 
 approximately perpendicular to AB and may be ex- 
 pressed in terms of quantities measured in (a) A A' = 
 AO sin a. = (d/2) sin a. 
 
 Assume that the weight of the magnet and carriage 
 (mg dynes) is equally divided between the two wires of 
 
18 
 
 ABSOLUTE DETERMINATION OF H 
 
 [54 
 
 the suspension. In the force parallelogram, figure (c), 
 we see that the nearly horizontal restoring force F which 
 
 D 
 
 Fig 1. Illustrates Experiment 54. (c.) 
 
 acts along the line A A' is the resultant of the upward 
 tension of the string T and half the weight of the magnet 
 and carriage mg/2. An equal force acts along BB'. The 
 restoring couple is thus = 
 
 d*mg sin a 
 
 4CA 
 
 Show with the aid of a diagram that the moment of 
 the magnetic deflecting couple is = H m 21 cos a = 
 H M cos a. 
 
 Calculate the numerical value of the product M H. 
 
 Combine the results for M/ H and MH and find the 
 value of H. 
 
55] MAGNETIC INDUCTION 19 
 
 55. MAGNETIC INDUCTION. 
 B and H Curve. 
 
 The magnetic field intensity at the center of a long 
 solenoid is, for air, 
 
 (1) H = 4 TT n I/Wl 
 
 where n is the number of turns, I is the current in amperes, 
 and / is the length of the solenoid in centimeters. The 
 total number of lines of force threading the solenoid is 
 H A, where A is the mean area of its cross section, to be 
 found by taking the average of the cross-sections of each 
 separate layer. 
 
 A secondary winding placed around the solenoid will 
 have the same number of lines threading it as pass through 
 the solenoid. 
 
 A ballistic galvanometer connected to the secondary 
 winding may be calibrated to indicate the change in the 
 number of lines of force which thread the coil, by the 
 amount of its deflection when a current / is suddenly sent 
 through the primary winding. 
 
 Calibration of a Ballistic Galvanometer. 
 
 (a) The solenoid is mounted on a base provided with 
 a commutator, a key K, and four knife switches, num- 
 bered 1, 2, 3 and 4 for changing the resistance in the cir- 
 cuit. Connect the primary of the solenoid in series with 
 six volts of storage battery. The secondary winding 
 should be connected to the galvanometer. 
 
 After the connections have been made, with the switches 
 1, 2, 3 and 4 open note the deflection of the galvanometer 
 when the circuit is closed with the key K. Read the amme- 
 
20 MAGNETIC INDUCTION [55 
 
 ter. After the galvanometer has returned to zero, again 
 note the deflection when the circuit is broken. 
 
 Now close switch number 1 and note the deflection 
 when the circuit is closed. Repeat the ammeter reading 
 and take a second reading of the deflection when the 
 circuit is suddenly opened as before. Then close switch 
 number 2, leaving number 1 closed, and repeat the scale 
 and ammeter readings. Continuing in this way secure 
 scale readings for five different values of the current. 
 
 (6) Measure the inside and outside diameter of the 
 primary winding of the solenoid with the calipers. Note 
 the number of turns n, the number of layers, and measure 
 the length /. 
 
 (c) Using your data and formula (1) compute H for 
 each value of the current. Plot a curve using the values 
 of the galvanometer deflections as abscissae and the cor- 
 responding changes in the number of lines of force (values 
 of HA) as ordinates. 
 
 B and H Curves. 
 
 If a rod of some magnetic substance longer than the 
 solenoid be placed within the coil, the magnetic field 
 intensity B within the rod is B = JJL H where JJL is the 
 permeability of the substance. The total number of lines 
 <t> threading the coil is 
 
 (2) Ba + H (A - a) = 
 where a is the cross sectional area of the rod. 
 
 The value of A$ and 0, ($ = S A <), can be found from 
 the plot previously made and the scale deflections ob- 
 served with the magnetic substance in the solenoid. 
 
55] 
 
 MAGNETIZATION OF IRON 
 
 21 
 
 H may be calculated from equation (1) as before and 
 thus B may be calculated from equation (2). 
 
 To get a series of values of B and H and at the same 
 time not allow them to be reduced to zero values at each 
 step it will be necessary to proceed by increments; i. e. 
 to measure the deflections produced by successive changes 
 in current. In this case the value of <j> is to be found 
 by summing up the separate increments, A 0, due to 
 successive changes in current. 
 
 The following table, containing partial data and cal- 
 culations, indicates a convenient form of tabulating the 
 results. 
 
 Reading Deflection 
 
 C hange 
 in flux 
 
 Total flux 
 
 I 
 
 
 I 
 
 7? 
 
 
 Number 
 
 (cms.) 
 
 A , </> = 2A< ^P^ S 
 
 K, closed 
 
 +6.7 
 
 +902.0 
 
 + 902.0 
 
 +0.15 
 
 + 12.48 
 
 +7950 
 
 1 
 
 +6.5 
 
 +875.0 
 
 + 1777.0 
 
 +0.50 
 
 +41.52 
 
 + 15030 
 
 2 
 
 +2.6 
 
 +350.0 
 
 +2127.0 
 
 +0.95 
 
 +79.00 
 
 + 17100 
 
 3 
 
 + 1.6 
 
 +212.6 
 
 +2339.6 
 
 + 1.42 
 
 + 118.00 
 
 + 17860 
 
 4 
 
 + 1.3 
 
 + 175.0 
 
 +2514.6 
 
 + 1.85 
 
 +154.00 
 
 + 18380 
 
 4, opened 
 
 -1.4 
 
 - 198.4 
 
 +2316.2 
 
 + 1.42 
 
 + 118.00 
 
 + 17650 
 
 3 
 
 -1.5 
 
 - 202.0 
 
 +2104.2 
 
 +0.95 
 
 +79.00 
 
 + 16900 
 
 2 
 
 -2.1 
 
 - 282.6 
 
 + 1821.6 
 
 +0.50 
 
 +41.52 
 
 + 15450 
 
 1 
 
 -3.7 
 
 - 498.0 
 
 +1323.6 
 
 +0.15 
 
 + 12.48 
 
 + 11760 
 
 K, " 
 
 -4.0 
 
 - 538.5 
 
 +795.1 
 
 0.00 
 
 +00.00 
 
 +7300 
 
 Reverse 
 
 
 
 
 
 
 
 K, closed 
 
 -12.9 
 
 - 1738.0 
 
 - 942.9 
 
 -0.15 
 
 - 12.48 
 
 -8260 
 
 1 
 
 -7.2 
 
 - 970.0 
 
 - 1912.9 
 
 -0.50 
 
 -41.52 
 
 - 16280 
 
 (a) Measure the diameter of the specimen under test 
 with the calipers. Demagnetize it by slowly withdrawing 
 it from a solenoid operated by an alternating current, and 
 place it in the solenoid. 
 
 (b) Note the deflection of the galvanometer when 
 the circuit is closed, and read the ammeter. Next, leav- 
 ing K closed, close the switch number 1, noting the deflec- 
 
22 MAGNETIZATION OF IRON [55 
 
 tion as before. Read the ammeter. Then by closing 
 the switches in the order, K, 1, 2, 3, 4, and opening 
 them in the reverse order, 4, 3, 2, 1, K, reversing the 
 current with the commutator and continuing the cycle 
 take a series of readings of the deflections of the galvan- 
 ometer and corresponding changes of the current such 
 that the values of H pass from zero to a positive maxi- 
 mum back to zero then to a negative maximum and back 
 to zero then again to a positive maximum. Never open 
 any switches just closed, nor close any just opened, in 
 order to repeat a reading. If repetition is necessary repeat 
 the entire cycle, including the demagnetization. 
 
 (b) From the calibration curve and from equation (2) 
 tabulate and plot the values of B and H for the specimen 
 furnished. Tabulate and plot the values of JJL and H. 
 
 Values of H are to be the abscissae in all the curves. 
 
ELECTRICITY 
 
 General Directions. 
 
 In setting up any electric circuit see that all connections 
 are clean and well made. Never make connections by 
 twisting ends of wire together, as the resistance of such a 
 connection is variable. Always use the binding screws. 
 Make a practice of arranging wires so that currents op- 
 posite directions are as near together as possible in order 
 to minimize the magnetic effect. 
 
 When using galvanometers with pivot suspension never 
 move or leave them without first releasing the needle from 
 the pivot. 
 
 In using resistance boxes, take care to protect the plugs 
 and the tapered holes into which they fit, from bruises 
 and dirt. 
 
 The brass tapers of the plugs should on no account be 
 touched with the fingers or held in the palm of the hand; 
 but should be at once put into the idle sockets, or, if these 
 are wanting, the plugs may be stood on end or laid on any 
 clean surface. 
 
 THEORY OF THE TANGENT GALVANOMETER. 
 
 1. A unit magnet pole is defined as one which, when 
 placed in air at a distance of one centimeter from a pole 
 of equal strength, will exert upon it a force of one dyne. 
 
 2. A unit current is defined (in electro-magnetic units) 
 as one which, flowing in a conductor one centimeter long, 
 bent into an arc of one centimeter radius, will act on a unit 
 magnet pole at the center of the arc with a force of one 
 dyne. 
 
24 TEST OF THE TANGENT LAW [61 
 
 3. We shall speak of the field at the center of the coil, 
 due to current flowing in it, as the force in dynes exerted 
 by the current on a unit magnet pole at the center of the 
 coil : let this be denoted by F. 
 
 4. For a unit current, flowing in a circular coil of 
 1 turn, radius 1 centimeter, F= 27r; for n turns, F= 2-jrn. 
 
 If the radius of the coil be r, F = 27rn/r; since the field 
 due to a given length of wire varies as 1/r 2 , but the 
 length of wire in the coil is r times as great. This value 
 of F, namely the field produced at the center of the coil 
 by a unit current flowing in the coil, is called the constant, 
 G, of the galvanometer. 
 
 5. Let H denote the horizontal component of the 
 earth's magnetic field at the center of the coil. Then if 
 the coil be set with its plane vertical and in the magnetic 
 meridian, the two fields of force denoted by F and H will 
 act at right angles to each other, upon each pole of the 
 needle. Assuming the needle to be so short that the field 
 at each pole is the same as at the center of the coil, if the 
 current I cause the needle to deflect through an angle a 
 from its zero position, it may be shown that in general 
 tan a = F/H =IG/H. From this I = (H / G) tan a. 
 
 6. I is here expressed in c. g. s. units. The commercial 
 unit, the ampere, is 0.1 of the c. g. s. unit. Hence if I be 
 expressed in amperes I = (10 H / G ) tan a. 
 
 10 H I G is called the reduction factor, K, of the gal- 
 vanometer. I = K tana. 
 
 61. TEST OF THE TANGENT LAW. 
 
 The tangent galvanometer used by each student is to 
 be set with its needle directly over a particular numbered 
 brass nail in the table-top. Record the number of the 
 galvanometer and the number of the spot assigned. 
 Always refer to the galvanometer by number. 
 
61] TEST OF THE TANGENT LAW 25 
 
 (a) Set the galvanometer with its plane in the magnetic 
 meridian. With the instrument used this is done by bring- 
 ing the ends of the pointer to and 180. 
 
 Connect a portable storage cell in series with the gal- 
 vanometer and a resistance box, and insert a commutator 
 in the circuit for reversing the direction of the current in 
 the galvanometer. On the galvanometer switch-board 
 make contact with the 50-turn coil, by turning in the screw 
 marked 50, the other screws being out about one turn; 
 the current then flows through a coil of 50 turns. 
 
 Take out plugs from the resistance box until the deflec- 
 tion is about 50. Determine the deflection by reading 
 both ends of the pointer to 0.1, then reversing the cur- 
 rent to get a deflection in the opposite direction and 
 reading both ends again. The mean of the four readings 
 is the deflection required. Increase the resistance in the 
 box so as to obtain deflections at intervals of 5 to 6 
 down to about 10. 
 
 In what follows, the electromotive force and resistance 
 of the cell are supposed to remain constant throughout 
 the experiment. To test this, repeat the first reading. 
 If it does not agree with the former value, repeat the series 
 in the same order as before to a point where the new 
 values do agree with the old. 
 
 (6) Show the truth of the relation expressed on the 
 preceding page, tan a = F/ H. By combining this with 
 Ohm's law, E = IR where R is the total resistance in the 
 circuit, show that 
 
 E/ K = R tan a = R/ cotan a 
 
 If then a plot be made with resistances as abscissae and 
 cotangents of corresponding angles of deflection as ordin- 
 
 5 
 
26 REDUCTION FACTOR OF GALVANOMETER [61-62 
 
 ates, the points obtained should lie on a straight line if 
 the galvanometer obeys the tangent law. 
 
 (c) Make such a plot from your observations, using 
 box resistances as abscissae, and show how to obtain from 
 it the constant resistance in the circuit outside the box. 
 Do this first by reading it directly from the plot, and then 
 more accurately, by constructing the equation of the line 
 and calculating from it. 
 
 What does this resistance include? 
 
 62. REDUCTION FACTOR OF GALVANOMETER. 
 By Deposition of Copper. 
 
 (a) Set up the galvanometer in the same place as 
 before. Record the number of the galvanometer and the 
 number of the spot assigned. Straighten out two of the 
 copper wire spirals from the voltameter cells, clean and 
 smooth them well with fine sand paper, and rewind on a 
 brass tube provided for the purpose. Do not touch the 
 wire with the fingers except at the ends, after cleaning; 
 catch one end through the small hole in the brass tube, 
 and, having the other end secured, stretch the wire while 
 winding, so as to get a smooth spiral. If the surface of 
 the wire flakes off in cleaning, ask for new wire. Clean 
 the copper plates of the voltameters, replace them and the 
 spirals, and fill the cell with fresh electrolyte solution. 
 
 Connect the two voltameters in series with a portable 
 storage cell and the 5-turn coil of the galvanometer, in 
 such a way that copper will be deposited by the current 
 on the spirals. Insert an adjustable rheostat or a piece 
 of German silver wire in the circuit if necessary to reduce 
 the deflection to about 25. A current-reversing switch 
 should also be placed in the circuit, so that reversing the 
 
62] REDUCTION FACTOR OF GALVANOMETER 27 
 
 direction of the current through the galvanometer will leave 
 unchanged the direction of the current in the voltameters. * 
 
 (b) After the circuit has been closed for several minutes 
 lift the spirals out of the solution and see if they are uni- 
 formly covered with a bright and clean deposit. If so 
 plunge them at once into a beaker of clean water, then 
 wash thoroughly under the tap and dry by gentle heat, no 
 greater than may be easily borne by the hand. 
 
 Weigh the spirals separately to 1 mg. on a delicate 
 balance. Replace the spirals in the voltameters, put them 
 in circuit as before and let the current run for a measured 
 time interval not less than twenty minutes. Keep the 
 deflection constant by adjusting the resistance in the 
 circuit, reversing the direction of the current through the 
 galvanometer at about the middle of this period. Repeat 
 the washing, drying and weighing. The gain in weight 
 should be the same for the two spirals. If it is nearly the 
 same, take the mean. 
 
 (c) The international ampere, the unit of current, is 
 represented sufficiently well for practical use by the un- 
 varying current which, when passed through a solution 
 of copper sulphate will deposit under conditions similar 
 to those used in this exercise, 0.000328 grams of copper 
 per second. Using this fact, calculate the reduction factor 
 for the coil used. 
 
 To calculate the Constants of the 1-Turn, 5-Turn and 
 5Q-Turn Coils, From Their Measured Dimensions. 
 
 (d) Measure with a beam-compass or outside calipers, 
 the diameter of the ledge on the face of the ring, which is 
 
 *If available a direct current ammeter may be placed in series 
 in this circuit and its calibration tested by comparison of its read- 
 ing with the weight of copper deposited. 
 
28 GALVANOMETER CONSTANT? [62 
 
 the same as the diameter of the bottom of the groove, or 
 the inner diameter of the coil. Also measure the outer - 
 diameter of the wire coil. In each case take several 
 measurements along different diameters. 
 
 The first layer, in the bottom of the groove, consists of six turns 
 of No. 14 wire, so connected that the current flows through all 
 six in parallel, thus making one effective turn. This is the 1-turn 
 coil. 
 
 The next layer, of No. 16 wire, consists of two wires in parallel 
 wound four times around, thus making four turns. These eight 
 wires just fill the width of the groove, as do the six wires of the 
 first layer. The four turns and the one may be connected in series, 
 making the 5-turn coil. 
 
 Outside of this are ^45 turns of No. 22 wire, wound in three layers 
 of 15 turns each, which may be connected in series with the five 
 turns, to make the 50-turn coil. 
 
 Samples of Nos. 14, 16 and 22 wire are furnished. From 
 the measured diameters of these (insulated), and the inner 
 and outer diameters of the whole coil, find the mean 
 diameter of each layer of wire. 
 
 Calculate the constants 2w n / r for the 1-, 4-, and each 
 of the -three layers of 15-turns. Giving n and r their 
 appropriate value in each case. 
 
 From the definitions of the constant it is evident that 
 the constant for two coils in series is the sum of the two 
 constants taken separately. Find thus the required con- 
 stants for the 5-turn, and 50-turn coils. Also calculate 
 the reduction factors of the 1-turn and the 50-turn coils. 
 
 (e) From your data in this experiment find the value 
 of H at the point where the needle of the galvanometer 
 is located. 
 
63] MEASUREMENT OF RESISTANCE 29 
 
 63. MEASUREMENT OF RESISTANCE. 
 Wheatstone Bridge. 
 
 The object of this exercise is to calibrate a resistance box, 
 using a slide-wire Wheatstone bridge and a standard resis- 
 tance box ; and to test the laws for resistances connected in 
 series and in parallel. 
 
 (a) Connect a I/eclanche cell to the bridge-wire with 
 a key, k, in the circuit, so that the wire forms the arms, 
 p and q, of the bridge. To eliminate the effect of induction 
 and capacity in any parts of the 'apparatus or the un- 
 known resistance the key, k, in the battery branch should 
 
 be closed before the key, K, 
 ^XTV^ (usually a sliding contact) is 
 
 pressed and should be kept 
 closed until K is opened. 
 
 To avoid polarization, the 
 battery circuit should be 
 closed only while taking an 
 observation. 
 
 . 2.-The Wheatstone Bridge. Af ter finding the resistance 
 
 of the coils separately, deter- 
 mine the resistance of all of them in series as a check. 
 
 Interchange the known and unknown resistances, 
 leaving the connections as they were, and repeat the 
 observations. The average of the two determinations of 
 the length p and the length q (q being always the length 
 opposite the unknown resistance x) will eliminate, in 
 part, errors due to inequalities of the bridge wire, and 
 the resistance of the connecting wires. 
 
 (6) Determine the value of the unknown resistance 
 furnished. (Doorbell, sounder, relay, or incandescent 
 lamp.) Make a determination of the resistance of two 
 resistances connected in parallel, and compare your 
 result with the calculated value. 
 
30 
 
 COMPARISON OF RESISTANCE 
 
 164 
 
 64. COMPARISON OF RESISTANCE 
 Carey Foster's Method. 
 
 The following extremely accurate method of comparing 
 two nearly equal resistances was devised by Foster. 
 
 The two branches p and q of the Wheatstone bridge 
 are extended by means of resistances A and B, using 
 connecting wires of resistance a and b respectively. The 
 resistances r and x should be of about the same magnitude 
 as A or B, but their actual value need not be known. 
 
 Then for balance 
 ^_ (A+a+pk) 
 x "" (B+b+qk). 
 
 where k is the resistance 
 per centimeter of the 
 wire. If A and B are 
 interchanged together 
 with their connecting 
 
 Fig. 3. Carey Foster's Method. . . 
 
 wires, and a new posi- 
 tion of balance found, then, r/x = (B+b+p f k)/(A+a+q f k). 
 
 Adding unity to each member of the equation gives: 
 
 r+x A+a+pk + B+b+qk B+b+p'k+A+a+q'k 
 x B+b+qk A + a + q'k 
 
 Remembering that p+q = p'+q', we see that 
 
 k(q - q') = (A + a) - (B + b) = A -B+ (a-b). Or in words 
 the resistance of the part of the bridge wire whose length 
 is q - q' is equal to the difference between the resistances 
 with which the bridge was extended, plus the difference 
 in the resistances of the connecting wires. 
 
 If we determine two corresponding positions of balance 
 #o and q' , with no resistances in the extended arms 
 
64] ADDITIONAL EXERCISES 31 
 
 except that of the connecting wires later to be used, 
 k(q q'o) = ab, and by substitution we have finally, 
 
 k[q-q' ~ fao-g'o)] = A-B. 
 
 This is the principle of Carey Foster's method of 
 calibrating a bridge wire, and with a calibrated wire is 
 the most accurate method available for the calibration 
 of resistances by comparison with standards. 
 
 The apparatus is provided with a double-throw com- 
 mutator for interchanging A and B together with their 
 connecting wires. 
 
 (a) To determine the resistance of the bridge wire by 
 the above method connect a standard box at A and a box 
 which is later to be calibrated at .5, using heavy connecting 
 wires of the same length. Before inserting any resistance 
 from either box in the circuit (by removing plugs or ro- 
 tating the dial) take readings both direct and reversed; 
 the difference in the two settings gives a length of wire 
 having a resistance equal to the difference in resistance 
 of the connections as they now stand. This length should 
 be added, algebraically, to the following readings with A. 
 
 Make A = 0.1 ohm, leaving B unchanged. Find a rest 
 point for both the direct and reversed positions. Increase 
 the value of A by 0.1 ohm steps within the range per- 
 mitted by the resistance of the bridge-wire. 
 
 (b) Compare in this way the coils in the box B with 
 similar coils in the standard box A. 
 
 ADDITIONAL EXERCISES WITH THE CAREY FOSTER 
 
 BRIDGE. 
 
 1. Determine the specific resistance of some pure metal. 
 
 After checking the calibration of the bridge-wire in (a) 
 above, calculate the specific resistance of a sample of wire 
 of measured dimensions from the readings taken with 
 
32 ABSOLUTE DETERMINATION OF RESISTANCE [64-65 
 
 two heavy copper staples at A and B, and the readings 
 with a measured length of the sample inserted in series 
 with the staple A. 
 
 2. Determine the temperature coefficient of resistance 
 of some pure metal. 
 
 The metal is furnished in the form of a coil of wire. 
 Connect the coil as resistance A in the bridge and balance 
 its resistance at 0C., by suitable resistance from a resis- 
 tance box as B. 
 
 Measure the resistance at 100C., or some measured 
 higher temperature and from your observations compute 
 the value of the temperature coefficient of resistance of 
 the metal. 
 
 65. ABSOLUTE DETERMINATION OF RESISTANCE. 
 Calorimeter Method. 
 
 Resistance may be denned as that property of a con- 
 ductor by virtue of which energy is expended when elec- 
 tricity is transferred from one point to another along 
 the conductor. If therefore the amount of energy ex- 
 pended can be measured, the resistance may be deter- 
 mined. Electrical energy may be transformed into energy 
 of motion, chemical separation, or heat. If the conditions 
 be so chosen that the first two transformations cannot 
 occur then the energy will all be transformed into heat and 
 can readily be measured. 
 
 A unit quantity of electricity may be defined as the quan- 
 tity of electricity passing a given point in a conductor in 
 one second when the current in the conductor is one 
 unit. Where I denotes the current (assumed to be con- 
 stant) in a conductor, Q, the quantity passing any given 
 point, and T the time, then we have the equation, 
 Q = I T. 
 
65] ABSOLUTE DETERMINATION OE RESISTANCE 3S 
 
 The difference of potential between two points on a 
 conductor is defined as the energy which would be ex^ 
 pended in transferring a unit quantity of electricity from 
 one point to the other. Therefore W = Q (Vi - V,), 
 where W is the energy expended and Vi - V* is the, dif- 
 ference in potential. If / is measured in amperes, and 
 Vi, F 2 in volts, then W is measured in a unit called the 
 joule. (1 joule = 10 7 ergs). We have then 
 
 W = (Vi - V*) I T x 10 7 ergs . 
 
 From Ohm's law I R = Vi - V 2 , R being the resistance 
 of the conductor between the points whose difference in 
 potentials is Vi - V 2 . 
 
 .'. W = I 2 R T x 10 7 ergs. 
 
 Since this energy is all expended in heating the con- 
 ductor, W = H J where H is the amount of heat pro- 
 duced in calories and / is the mechanical equivalent of 
 
 heat - /. H J = I* RT x 10 7 . 
 
 Inserting the value of /, (J = 4.19 x 10 7 ), and solving 
 for R * 4.19 H 
 
 (a) To determine the water equivalent of the thermos 
 bottle (which is to be used as a calorimeter), first weigh 
 it with thermometer and heating coil inserted. Then 
 add about 25 gms. of hot water, replacing the coil and 
 thermometer, and determining the exact amount by 
 weighing the bottle after the water has been added. 
 Move the bottle about so that the water will come in 
 contact with all parts of the inside, hold it inverted 
 the greater part of the time but do not shake it. Note 
 the temperature after it becomes constant. 
 
 Next add about 40-50 gms. more of water at a known 
 temperature near that of the room. Determine as 
 
34 ELECTROMOTIVE FORCE [65-66 
 
 before the exact amount by weighing and after moving 
 it about and inverting it as before note the final steady 
 temperature. Equating the heat lost by the bottle, 
 thermometer, coil, and water first put in to that gained 
 by the water added later will give the desired water- 
 equivalent. 
 
 (6) Connect the heating coil, whose resistance, R, 
 is to be determined in series with an ammeter, leaving 
 the circuit open. Put some water in the thermos bottle, 
 determining the exact amount by weighing the bottle 
 after the water is poured in, immerse the heating coil 
 and thermometer and after moving the bottle about 
 and inverting it record the initial temperature. 
 
 Close the circuit and let the current run for a time 
 interval, T, measured in seconds, then move the bottle 
 about to secure a uniform temperature and note its 
 final constant value. , 
 
 Make a second trial, using your results as a guide in 
 varying the time and amount of water used. 
 
 (c) Measure the resistance of the heating coil using 
 a postoffice bridge. 
 
 (d) From your data in (a) and (b) calculate the resis- 
 tance of the coil used, and compare your results with 
 the value found in (c). 
 
 66. ELECTROMOTIVE FORGE. 
 Poggendorff's Method. The Potentiometer. 
 
 The B. M. F. of a cell is equal to the maximum differ- 
 ence of potential which it is capable of producing at its 
 terminals. This maximum occurs when the external cir- 
 cuit is broken, i. e., on open circuit. 
 
66] ELECTROMOTIVE FORCE 35 
 
 Poggendorff's method of measuring the E. M. F. of a 
 cell, b 2 , consists in finding two points A and C in an 
 independent circuit (61, G\ t A B) between which the dif- 
 ference in potential exactly equals that of the terminals of 
 the cell, 6 2 , on open circuit. If these points be connected 
 so as to oppose the cell no current will flow through it 
 and the difference in potential between the two points, 
 calculated from the current and resistance between A 
 ^-^ j ^ and C, equals the B. M. F. 
 
 ~\/ ^T ~^\ of the cell. 
 
 B A wire or other resistance 
 
 I 
 
 like the wire A B is 
 called a potentiometer, since 
 for constant current the dif- 
 ference in potential between 
 
 Fig. 4.-Poggendorff's method of twQ pomts j s proportional tO 
 
 measuring electromotive-force. v 
 
 the resistance between them. 
 
 The simplest form of potentiometer is a uniform stretched 
 wire. If the points of balance obtained in two cases 
 are C x and C s , while the corresponding values of the 
 E. M. F. are E x and E s , then 
 
 _*_ AC X 
 E s AC S ' 
 
 (a) Connect a storage cell b l in series with a tangent 
 galvanometer, G lt and a high resistance wire, AB, whose 
 resistance per centimeter is known.* 
 
 Carefully adjust the tangent galvanometer. Its reduc- 
 tion factor, accurately determined for its present position, 
 is given. 
 
 Form a parallel circuit to the wire, consisting of a Le- 
 clanche cell, 6 2 , a sensitive galvanometer G 2 , and a key, 
 at C. The + terminal of the Leclanche should be con- 
 
 *If the resistance of the wire is not known it should be deter- 
 mined, using a Postoffice bridge. 
 
36 ELECTROMOTIVE FORCE [66 
 
 nected to the same end of the wire as the + terminal of 
 the storage battery. 
 
 In making a test, set the slider at the desired point on 
 the wire and press the key just long enough to determine 
 the direction of the deflection produced, as a long con- 
 tact will cause polarization of the Leclanche. 
 
 Determine first whether the Leclanche is properly con- 
 nected, which will be the case if it is possible to find two 
 points of contact for which the deflections are in opposite 
 directions. Then adjust the slider until the galvanometer 
 indicates zero current, and read its position. Read the 
 tangent galvanometer, reversing its deflection as usual. 
 
 As the error in your result will be directly proportional 
 to the error in the determination of current endeavor to 
 reduce the error in the tangent galvanometer reading to 
 one tenth of one percent or less. 
 
 From Ohm's law calculate the difference in potential 
 between the end of the wire, A, and the sliding contact, C, 
 which is equal to the E. M. F. of the Leclanche cell. 
 
 Replace the Leclanche cell with a dry battery and 
 determine its E. M. F. in the same way. 
 
 (6) Check your determinations of the E- M. F. of the 
 cells just used by comparison with a standard cell of 
 known E. M. F. 
 
 Such a cell is used as a standard of E. M. F., its con- 
 stancy depends upon careful handling, and upon having 
 only minute currents sent through it. A standard cell is 
 intended only for comparison purposes by compensation 
 or open circuit methods. Do not test its E. M. F. on a 
 voltmeter. 
 
 Knowing that the E. M. F. is 1.0183 volts, it will be 
 possible by a little calculation to set the slider very 
 near to the true point at the start. 
 
66-67] THERMO-ELECTROMOTIVE FORCE 37 
 
 Never use a standard cell without such a preliminary 
 test with an ordinary cell. If it is impossible to approxi- 
 mate closely to the true position of the slider, a resistance 
 of several thousand ohms should be included in series with 
 the cell to protect it against currents of dangerous strength. 
 
 67. THERMO-ELEGTROMOTIVE FORCE. 
 Calibration of a Galvanometer as a Direct-Reading Voltmeter 
 
 Whenever contact is established between two dissimilar 
 metals an electric potential difference is set up between 
 their "free" ends. If now these free ends be joined this 
 juncture will be the seat of a new difference in potential 
 which will oppose that of the first juncture. The magni- 
 tude of this potential difference depends not only upon the 
 metals but also upon the temperature of the juncture. If 
 therefore the two junctures be maintained at different 
 temperatures, and the circuit be closed, the algebraic 
 sum of the electromotive forces in the circuit will not be 
 zero, and a current of electricity will be obtained. The 
 object of this experiment is to find the relation between 
 the difference in temperature of the junctures and the di- 
 rection and magnitude of the resulting B. M. F. for a 
 limited range of temperature. 
 
 (a) Set the telescope so that the center of the scale 
 is seen in the galvanometer mirror. Connect the thermo- 
 electric couples in series with the galvanometer and an 
 adjustable resistance. Keep one of the cups at the tem- 
 perature of melting ice and heat the other until the water 
 boils. Adjust the resistance until the galvanometer 
 reading is on the scale and the deflection as large as 
 possible. 
 
 Then remove the source of heat and take the galvano- 
 meter readings as the water cools for about every 10 
 
38 
 
 THERMO-ELECTROMOTIVE FORCE 
 
 [67 
 
 change in temperature, down to a temperature of 10. 
 Keep the water in the cups well stirred and record as nearly 
 simultaneously as possible the temperature of the two 
 junctures and the corresponding galvanometer reading. 
 Record the resistance in series with the galvanometer and 
 also the distance from the scale to the galvanometer. 
 
 (6) In order to find the potential difference in volts 
 between the terminals of the thermo-electric couples, for 
 any given temperatures, it will be necessary to calibrate 
 the galvanometer as a voltmeter. This may be done by 
 finding the current necessary to produce deflections over 
 the same range as those in (a), then if the resistance of 
 the circuit is known, the B. M. F. may be found from 
 Ohm's law. 
 
 Pass a current from a storage battery, B, through a 
 resistance box containing 0.1, 0.2, 0.3, 0.4 ohm coils and 
 an ammeter, A, connected in series. 
 
 Remove the thermo-couple from the d' Arson val 
 galvanometer circuit and attach a traveling plug to 
 each end of this circuit. Insert these traveling plugs on 
 either side of the 0.1 ohm coil whose resistance should 
 be in the circuit. Leave the resistance box, R, in the 
 d'Arsonval galvanometer circuit with the resistance un- 
 
 Fig. 5. Showing the connection when calibrating the d'Arsonval 
 galvanometer used with resistance R as a voltmeter. 
 
67] THERMO-ELECTROMOTIVE FORCE 39 
 
 changed. The conditions as to resistance in this circuit 
 must be the same as in (a). The resistance of the thermo- 
 couple is small as compared with the resistance of the 
 remainder of the circuit and may be neglected. 
 
 The resistance of the d'Arsonval galvanometer circuit 
 is now one branch of a parallel circuit, the 0.1 ohm 
 resistance of the box between the galvanometer terminals 
 being the other branch. Since the resistance in the 
 galvanometer circuit is very large compared to this 0.1 
 ohm, the difference in potential between the terminals 
 of the galvanometer circuit can be considered as the 
 product of the total current times this 0.1 ohm resistance. 
 The direction of the current should be such as to give a 
 deflection in the same direction as the deflection in (a). 
 
 Adjust the resistance in the box connected in series 
 with the storage cell to give a suitable deflection of the 
 d'Arsonval galvanometer. Obtain four such values 
 covering the range of deflections observed in (a), reading 
 the ammeter in each case. 
 
 (c) Knowing the direction of the current producing 
 the deflection, find which way the current was flowing 
 in (a) across the heated juncture. 
 
 (d) Make a plot of the results in (a) plotting deflections 
 as ordinates and differences in temperature as abscissae. 
 From the data, obtained in (b) make another curve on 
 the same sheet in which the abscissae give the electro- 
 motive force in volts, the ordinates being scale deflections 
 as before. From these curves find the E. M. F. in micro- 
 volts for a single element when the difference in temperature 
 between the junctures is 1C. 
 
40 THE EARTH-INDUCTOR [68 
 
 68. THE EARTH-INDUCTOR. 
 Relative Calibration of a Galvanometer. 
 
 (a) A sensitive galvanometer is necessary in this ex- 
 periment. Since the galvanometer may not obey any sim- 
 ple law, such as the tangent law, a relative calibration 
 curve must be obtained. To do this connect a portable 
 storage cell in series with a stretched wire and a suitable 
 resistance coil; set the galvanometer in the position in 
 which it is to be used in the experiment. If the galvano- 
 meter terminals be now connected to two points on this 
 stretched wire, separated by a suitable distance, a de- 
 flection will be obtained. Since the galvanometer re- 
 sistance is high compared to the resistance of the stretched 
 wire between the galvanometer terminals, it follows that 
 the current through the galvanometer is approximately 
 proportional to the distance between the galvanometer 
 terminals. Obtain about ten galvanometer deflections, 
 reversing the current through the galvanometer in each 
 case, note also the distance between the galvanometer 
 terminals along the stretched wire corresponding to each 
 deflection. Plot your results, taking relative electromo- 
 tive forces as abscissae and galvanometer deflections as 
 ordinates. 
 
 (6) Connect the galvanometer with the earth-inductor, 
 placing the latter as far as possible from the galvano- 
 meter. Place the earth-inductor so that the two sta- 
 tionary, upright supports are in an east-and-west line and 
 set the circle so that its axis of rotation is horizontal. 
 Turn the circle slowly into a horizontal position, let the 
 galvanometer needle come to rest, and then turn the cir- 
 cle suddenly through 180, noting the effect on the gal- 
 vanometer. Explain the cause of the current produced. 
 
68] THE EARTH INDUCTOR 41 
 
 Turn the circle in the same direction through another 
 180, and compare the direction of the induced current 
 with that just obtained. Account for the directions being 
 as you find them in the two cases. 
 
 (c) Rotate the coil continuously and uniformly, re- 
 cording the number of turns per minute and the deflection 
 of the galvanometer. Now rotate the coil continuously 
 and uniformly at a rate either one-half or twice as great 
 as the rate just used, recording again the rate and the 
 deflection. From the calibration curve of the galvano- 
 meter determine the relative electromotive forces in the 
 two cases. Assuming the resistance of the circuit constant, 
 what effect do you find upon the induced electromotive 
 force when the rate of rotation is doubled? 
 
 When the coil is being turned at a constant rate are 
 the tubes or lines of magnetic force being cut at a constant 
 rate? Is the electromotive force induced under these 
 conditions constant or variable? If the current is variable 
 why is the galvanometer reading fairly constant when 
 the turning is uniform? 
 
 (d) Set the coil with the axis of rotation vertical, and 
 rotate the coil at the same rate as in one of the cases in 
 (c). To what component of the earth's magnetic field is 
 the induced electromotive force proportional in this case? 
 To what component was it proportional in (c) ? From the 
 relative currents obtained in the two cases calculate the 
 angle of dip. 
 
 (e) By varying the angle of inclination of the coil, find 
 a position for which there will be no electromotive force 
 induced when the coil is rotated. Read the angle of in- 
 clination of the axis of rotation on the graduated circle, 
 and compare the angle thus found with the angle of dip 
 calculated in (d). 
 
42 RESISTANCE OF AN ELECTROLYTE [68-69 
 
 (/) Turn the base of the earth-inductor through 90 
 and rotate the coil continuously about a vertical axis at 
 the same rate as in (d). Compare the deflection here 
 obtained with that obtained in (d), and explain the dif- 
 ference if there be any difference. 
 
 (g) From the absolute value of H find the total 
 strength of the earth's field. 
 
 69. RESISTANCE OF AN ELECTROLYTE. 
 Kohlrausch's Method. Alternating Current-Telephone. 
 
 The specific resistance of a material is the resistance 
 between two opposite faces of a cube of the material each 
 edge of which is 1 centimeter in length. If this material 
 is a substance in solution, the specific resistance always 
 refers to a particular concentration which should be 
 stated. The reciprocal of the specific resistance is called 
 the specific conductivity. 
 
 Metallic conductors, as far as we know, suffer no per- 
 manent change by the passage of electricity through them; 
 in electrolytes, however, the transference of electricity 
 from the anode to the kathode is accompanied by the 
 transport of matter. 
 
 If we assume that the electricity is carried wholly by 
 the positively and negatively charged ions into which 
 some of the molecules are split up when in solution it 
 follows as a matter of course that the ease with which 
 the current will flow (as measured by the conductivity) 
 in a solution containing a definite number of molecules, 
 will depend on the number which are dissociated. 
 
 Let us take as the /number of molecules, the number 
 found in 1 c.cm. of normal solution of the substance. In 
 a 0.5-normal solution this number would be scattered 
 
69] RESISTANCE OF AN ELECTROLYTE 43 
 
 through 2 c.cm. of the solution, if the degree of dissocia- 
 tion were the same, the specific conductivity would be 
 only one-half as great, since only half as many ions are 
 present. Dividing the specific conductivity by the con- 
 centration will give us the conductivity due to the num- 
 ber of molecules in 1 c.cm. normal solution the molecu- 
 lar conductivity. 
 
 The limit to the molecular conductivity will evidently 
 be reached when all the molecules are dissociated. 
 
 The resistance of an electrolyte cannot be found by the 
 ordinary methods since polarization enters, producing a 
 back E. M. F. There are several methods for measuring 
 the resistance of an electrolyte, one of these methods, due 
 to Kohlrausch, almost wholly avoids the effects due to 
 polarization by using a rapidly alternating current. To 
 secure such a current the secondary current from a small 
 induction coil will be used. 
 
 (a) Connect a storage cell to the primary of the in- 
 duction coil and connect the secondary of the induction 
 coil to the bridge. Place the electrolyte in one branch 
 and a resistance box in the other branch. Since a gal- 
 vanometer will not detect a rapidly alternating current 
 a telephone will be used in place of a galvanometer. Avoid 
 any loops of wire; since an alternating current is used, 
 self-induction is present and should be made as small as 
 possible. 
 
 If k be the specific conductivity of a solution of known 
 concentration and W the resistance of this electrolyte be- 
 tween the fixed electrodes measured with the bridge, then 
 the constant, h, for this pair of electrodes is h = W k. 
 
 Measure W, using a 1/50 normal solution of KCL The 
 position of balance will be the position in which there is 
 
44 RESISTANCE OF AN ELECTROLYTE [69 
 
 a minimum of sound in the telephone. From the value of 
 k given in the tables calculate h. 
 
 The determination of k for the various concentrations 
 of the substance whose conductivity is desired involves 
 only measurement of W for each concentration. 
 
 (6) Determine the resistance of the substance fur- 
 nished for the following concentrations: 3-, 2-, 1-, 0.1-, 
 0.01-, 0.001-, and 0.0001- normal, keeping the temper- 
 ature constant. 
 
 Determine the resistance of the water used in preparing 
 the solutions. 
 
 Care should be taken not to disturb the electrodes 
 while changing the concentration of the solution. 
 
 (c) Calculate from the observed values of the resist- 
 ance, the specific conductivity and the molecular conduc- 
 tivity, presenting the results in tabular form. The 
 conductivity of the water is to be subtracted from that of 
 the solutions. 
 
70] POLARIZATION EFFECTS IN A CELL 45 
 
 70. STUDY OF POLARIZATION EFFECTS IN A 
 
 LECLANCHE CELL. 
 Galvanometer Used as Voltmeter 
 
 Use a d'Arsonval galvanometer, which has the proper 
 resistance connected in series with it, as a voltmeter. In 
 this experiment the closing of a key produces a deflection 
 of the galvanometer corresponding to an E. M. F., which 
 is changing rather rapidly, so that a reading must be 
 obtained as soon as possible. 
 
 (a) Connect the terminals of the Leclanche cell directly 
 with the terminals of the galvanometer, and so get a 
 reading which is proportional to the total E. M. F., E, 
 of the cell. 
 
 Throughout the succeeding series of observations the volt- 
 meter should remain connected to the terminals of the cell, 
 except that it should be disconnected at convenient intervals 
 to check the zero reading on the scale. 
 
 (6) Connect the terminals of the cell through a spool 
 of 4-ohms resistance and a break-circuit key (one which 
 breaks connection instead of making it on depressing the 
 lever). Complete this parallel circuit at a noted time and 
 immediately read the galvanometer deflection, e. Be 
 careful not to connect the cell in such a way as to send a 
 current through it until ready to begin operations. 
 
 (c) At the end of five minutes read e again, disconnect 
 the resistance, by means of the break-circuit key, and as 
 quickly as possible read E. This reading must be taken 
 quickly because E, which has been reduced by polarization 
 caused by the passage of the current, immediately begins 
 to return to its former value. 
 
 (d) Re-connect the resistance, and note the time at 
 which e again has the same value as that read in (c). 
 Count from this another 5-minute interval, then repeat 
 
46 COMPARISON OF CAPACITIES [70-71 
 
 the readings as in (c). By this means the readings ob- 
 tained are very nearly the same as they would have 
 been if no time had been lost in taking them, so that the 
 current could flow continuously. 
 
 Continue the readings in this way until four periods of 
 five minutes each have been covered. 
 
 (e) Leaving the resistance disconnected, after taking 
 the last reading under (d), take readings of E at 5-minute 
 intervals for about 45 minutes. 
 
 At the end of this interval secure a cell whose E. M. F 
 has been previously determined, or is known, and by 
 getting the galvanometer deflection corresponding to its 
 E. M. F., obtain a factor for reducing your readings to 
 volts. 
 
 Why could not a standard cell, whose resistance is 
 about 1,000 ohms, be used for this purpose? 
 
 (/) Plot a curve with time as abscissae and corres- 
 ponding values of E as ordinates, for the whole time of 
 the experiment. Remember that during the first part, the 
 5-minute intervals only are to be counted. 
 
 Derive from Ohm's law an expression for the resistance 
 of the cell in terms of E, e, and the known external re- 
 sistance. Using this expression calculate the resistance 
 of the cell for each pair of values of E and e, tabulate the 
 results, and show that for this purpose the scale readings 
 corresponding to E and e may be used without knowing 
 their values in volts. 
 
 71. COMPARISON OF CAPACITIES. 
 
 A comparison of capacities may be made by a method 
 which is analogous to the Wheatstone Bridge method of 
 determining the resistance of electrolytes, using an alter- 
 nating current and telephone (as in Experiment 69), 
 
71-72] EFFICIENCY OF HEATING DEVICES 47 
 
 where a condition of balance is indicated by no buzzing 
 in the telephone. 
 
 If two condensers, whose capacities may be represented 
 by Ci and C 2 be placed in the r and x branches of the 
 bridge (see fig. 2, p. 29) and the parts of the slide wire be 
 replaced by two resistances p and q when the condition 
 of balance is obtained it may be shown that d/C 2 = q/p. 
 
 (a) Determine the capacity of the condenser fur- 
 nished by comparing it with the standard condenser. 
 The resistance box p should have a range of from 1-10000 
 ohms. Give to p a fixed value and vary q until a balance 
 is obtained. Approach the position of balance from the 
 side of too large resistances as well as from the side of 
 too small resistances, using the mean in your calculations. 
 
 (6) Change the dielectric of the condenser and make a 
 second determination of its capacity. 
 
 (c) From the two values of the capacity of the con- 
 denser determine the specific inductive capacity of the 
 substance used in (b). 
 
 (d) Establish the truth of the relation used in calcula- 
 ting the capacity of the condenser. It may be assumed, 
 in the proof, that neither absorption nor leakage occur. 
 
 72. EFFICIENCY OF ELECTRICAL HEATING DEVICES. 
 
 The object of this exercise is to determine the efficiency of 
 a hot plate when used to heat water. 
 
 (a) Connect the hot plate in series with an ammeter 
 having a range of 1 to 10 amperes and shunt a voltmeter 
 having a range of 1 to 120 volts across its terminals. 
 
 Weigh the kettle, dry and empty, and again when 
 
48 EFFICIENCY OF HEATING DEVICES [72 
 
 about half full of water. Close the circuit through the 
 stove and let the current run for one minute before placing 
 the kettle upon it. During this time stir the water in the 
 kettle and take its temperature. 
 
 Start the stop clock as the kettle is placed upon the 
 stove. Read the voltmeter and the ammeter every 
 minute, arranging the readings in tabular form so that 
 the average may be conveniently found for use in your 
 computations. Stir the water occasionally. 
 
 At the end of 9i minutes begin to stir the water and 
 minute later read the temperature of the water in the 
 kettle just as you take it from the stove. 
 
 Immediately open the circuit. 
 
 (b) Make a second determination allowing the water 
 to come to the boiling point. 
 
 (c) Repeat the experiment with the kettle full of 
 water. 
 
 (d) From your average ammeter and voltmeter read- 
 ings calculate the resistance of the stove and using this 
 value find the number of calories consumed from the 
 relation, Heat (in calories) = 0.24 C 2 R t where C repre- 
 sents the current in amperes and R the resistance in 
 ohms and t the time in seconds. Find the heat received 
 from the weight of water and the temperature change, 
 taking into account the water equivalent of the kettle. 
 
 Express the efficiency of the stove in per cent in each 
 case. At local rates for electricity find the cost of one 
 liter of boiling hot water. 
 
PHYSICAL TABLES 
 
 49 
 
 DENSITIES AND THERMAL PROPERTIES OF SOLIDS. 
 
 (The values given in this Table are mostly for pure specimens 
 of the substances listed. The student should not expect the prop- 
 erties of the average laboratory specimen to correspond exactly 
 in value with them. As a rule the densities are given for ordi- 
 nary atmospheric temperature. The specific heats and coefficients 
 of expansion are in most cases the average values between and 
 100C. The melting points and heats of fusion are given for 
 atmospheric pressure.) 
 
 The coefficient of cubical expansion of solids is approximately 
 three times the linear coefficient. 
 
 
 
 CJ 
 
 +J . 
 
 OJ bJO 
 
 +-4 ^ 
 
 
 a 
 
 rt 
 
 "3 S 
 
 'O.g 8 
 
 rt-S d 
 
 o " 
 -> -2 
 
 Cj w 
 
 Solid. 
 
 Q 
 
 o,W 
 
 CO 
 
 o w 
 
 I!* 2 
 
 o> 3 
 
 
 
 cals. per 
 
 
 degrees 
 
 cals. per 
 
 
 gms. per cc. 
 
 gm. 
 
 per degree C. 
 
 C. 
 
 gm. 
 
 Acetamide 
 
 1.56 
 
 
 
 82 
 
 
 Aluminum 
 
 2.70 
 
 0.219 
 
 .0000231 
 
 658 
 
 
 Brass, cast 
 
 8.44 
 
 .092 
 
 .0000188 
 
 
 
 " drawn 
 
 8.70 
 
 .092 
 
 .0000193 
 
 
 
 Copper 
 
 8.92 
 
 .094 
 
 .0000172 
 
 1090 
 
 43.0 
 
 German-silver 
 
 8.62 
 
 .0946 
 
 .000018 
 
 860 
 
 
 Glass, common tube 
 
 2.46 
 
 .186 
 
 .0000086 
 
 
 
 flint 
 
 3.9 
 
 .117 
 
 .0000079 
 
 
 
 Gold 
 
 19.3 
 
 .0316 
 
 .0000144 
 
 1065 
 
 
 Hyposul. of soda 
 
 1.73 
 
 .445 
 
 
 48 
 
 
 Ice 
 
 .918 
 
 .502 
 
 .000051 
 
 
 
 80. 
 
 Iron, cast 
 
 7.4 
 
 .113 
 
 .0000106 
 
 1100 
 
 23-33 
 
 " wrought 
 
 7.8 
 
 .115 
 
 .000012 
 
 1600 
 
 
 Lead 
 
 11.3 
 
 .0315 
 
 .000029 
 
 326 
 
 5.4 
 
 Mercury 
 
 13.596 
 
 .0319 
 
 
 39 
 
 2.8 
 
 Nickel 
 
 8.90 
 
 .109 
 
 .0000128 
 
 1480 
 
 4.6 
 
 Paraffin, wax 
 
 .90 
 
 .560 
 
 .000008-23 
 
 52 
 
 35.1 
 
 liquid 
 
 
 .710 
 
 
 
 
 Platinum 
 
 21.50 
 
 .0324 
 
 .0000090 
 
 1760 
 
 27.2 
 
 Rubber, hard 
 
 1.22 
 
 .331 
 
 .000064 
 
 
 
 Silver 
 
 10.53 
 
 .056 
 
 .0000193 
 
 960 
 
 21.1 
 
 Sodium chloride 
 
 2.17 
 
 .214 
 
 .000040 
 
 800 
 
 
 Steel 
 
 7.8 
 
 .118 
 
 .000011 
 
 1375 
 
 
 Wood's alloy, solid 
 
 9.78 
 
 .0352 
 
 
 75.5 
 
 8.40 
 
 " " , liquid 
 
 
 .0426 
 
 
 
 
50 
 
 PHYSICAL TABLES 
 
 SPECIFIC RESISTANCE OF VARIOUS SUBSTANCES. 
 
 R=r(l+aO 
 
 R= resistance at tC. 
 
 r= resistance at 0C. 
 
 Ohms per 
 mil foot 
 at 0C. 
 
 Specific 
 
 Resistance 
 
 Microhms per 
 
 centimeter 
 
 cube 
 
 Aluminum 17.38 2.889 
 
 j 2400 
 
 Carbon, graphite I 42000 
 
 Carbon, arc light 4000. 
 
 Constantine _ 51. 
 
 Copper, annealed. 9.44 1.570 
 
 Copper, hard 9.64 1.603 
 
 German Silver 125.0 20.76 
 
 Gold 2.07 
 
 la. la., hard 50.2 
 
 la. la., soft 47.1 
 
 Iron 58.0 9.64 
 
 Iron, wire 58.0 9.50 
 
 Iron, telegraph wire 90.0 15.0 
 
 Lead 19.6 
 
 Manganin 47.5 
 
 Mercury 94.34 
 
 Nickeline, No. I, hard. 43.6 
 
 Nickeline, No. I, soft... 40.7 
 
 Nickeline, No. II, hard. 33.9 
 
 Nickeline, No. II, soft... 32.3 
 
 Platinum 54.03 8.98 
 
 Silver, annealed 1.49 
 
 Silver, hard 1.62 
 
 Steel 82. 13.3 
 
 Liquids at 18C. 
 
 Dilute H N O 3> 30%._ 129. XlO 4 
 
 H 2 S0 4 , 5% 486. XlO 4 
 
 H 2 S 4 , 30%._ 137. XlO 4 
 
 H 2 S 4 , 80% 918. XlO 4 
 
 Zn S 4 , 24 % 214. XlO 5 
 
 Water 26.5X10 8 
 
 Megohms 
 
 Benzine 14. X 10 6 
 
 Ebonite 28. XlO 9 
 
 Glass, 20C._ 91. XlO 6 
 
 Glass, 200C 22.7 
 
 Gutta-percha, 24C 4.5 XlO 8 
 
 Mica..._ 84. XlO 6 
 
 Paraffine 34. XlO 9 
 
 Paraffine, oil._.. 8. XlO 6 
 
 Shellac 9. XlO 9 
 
 Wood Tar.... 167. XlO 7 
 
 Temperature 
 
 Coefficient a 
 
 Divide by 10 5 
 
 309. 
 
 388. 
 
 28 to 44 
 365. 
 -1.1 
 0.5 
 453. 
 
 387. 
 1. 
 88. 
 7.6 
 7.7 
 16.8 
 18.1 
 247. 
 377. 
 
PHYSICAL TABLES 
 
 51 
 
 (IN Ohm- 1 cm" 1 ) OF 
 
 THIS SPECIFIC CONDUCTIVITY k 
 
 STANDARD-SOLUTIONS FOR DETERMINING 
 RESISTANCE-CAPACITY OF VESSELS. 
 (Kohlrausch and Holborn, p. 204.) 
 
 15 
 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24 
 25 
 26 
 27 
 28 
 29 
 30 
 
 NaCl 
 
 Saturated 
 k 
 
 0.2014 
 0.2062 
 0.2111 
 0.2160 
 0.2209 
 0.2259 
 0.2309 
 0.2360 
 0.2411 
 0.2462 
 0.2513 
 0.2565 
 0.2616 
 0.2669 
 0.2721 
 0.2774 
 
 KCl 
 
 Normal 
 k 
 
 0.09252 
 0.09441 
 0.09631 
 0.09822 
 0.10014 
 0.10207 
 0.10400 
 0.10594 
 0.10789 
 0.10984 
 0.11180 
 0.11377 
 0.11574 
 
 KCl 
 
 KCl 
 
 KCl 
 
 0.1-Normal 
 
 0.02-Normal 
 
 0.01-Normal 
 
 k 
 
 k 
 
 k 
 
 0.01048 
 
 0.002243 
 
 0.001147 
 
 0.01072 
 
 0.002294 
 
 0.001173 
 
 0.01095 
 
 0.002345 
 
 0.001199 
 
 0.01119 
 
 0.002397 
 
 0.001225 
 
 0.01143 
 
 0.002449 
 
 0.001251 
 
 0.01167 
 
 0.002501 
 
 0.001278 
 
 0.01191 
 
 0.002553 
 
 0.001305 
 
 0.01215 
 
 0.002606 
 
 0.001332 
 
 0.01239 
 
 0.002659 
 
 0.001359 
 
 0.01264 
 
 0.002712 
 
 0.001386 
 
 0.01288 
 
 0.002765 
 
 0.001413 
 
 0.01313 
 
 0.002819 
 
 0.001441 
 
 0.01337 
 
 0.002873 
 
 0.001468 
 
 0.01362 
 
 0.002927 
 
 0.001496 
 
 0.01387 
 
 0.002981 
 
 0.001524 
 
 0.01412 
 
 0.003036 
 
 0.001552 
 
 Rate of Electrolytic Deposition. 
 
 Mg. per 
 Element. Coulomb 
 
 Aluminum 0935 
 
 Copper 3282 
 
 Gold.. 0679 
 
 Hydrogen 0104 
 
 Lead .1072 
 
 Mercury.. 1 .038 
 
 Mg. per 
 Element. Coulomb 
 
 Nickel 3043 
 
 Oxygen 0831 
 
 Platinum 1.0095 
 
 Silver 1.1186 
 
 Tin 6097 
 
 Zinc.... .3365 
 
 Specific Inductive Capacities. 
 
 Medium 
 
 Specific 
 Inductive 
 Capacity. 
 
 Air, 760 mm .'. 1.0 
 
 Alcohol 26.0 
 
 Beeswax 1.8 
 
 Ebonite 2.2-3.2 
 
 Glass, light flint 6.72 
 
 Glass, dense flint 7.38 
 
 Glass, hard crown ' 6.96 
 
 Glass, plate 5.8-8.5 
 
 Gutta-percha 2.5 
 
 Kerosene.... .. 2.-2.S 
 
 Specific 
 
 Medium Inductive 
 
 Capacity. 
 
 Mica.. 6.64 
 
 Paraffine, solid 1.96-2.3 
 
 Paraffine, oil 1.92 
 
 Petroleum 2.05 
 
 Shellac. 2.7-3.7 
 
 Sulphur 2.8-3.9 
 
 Turpentine 2.2 
 
 Vacuum 999 
 
 Water.... 76.0 
 
52 
 
 NATURAL SINES. 
 
 
 0' 
 
 6' 
 
 12 
 
 18 
 
 24 
 
 30' 
 
 36 
 
 42' 
 
 48' 
 
 54' 
 
 123 
 
 4 5 
 
 
 
 0000 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 oi57 
 
 369 
 
 12 15 
 
 1 
 
 2 
 3 
 
 0175 
 
 0349 
 0523 
 
 0192 
 0366 
 0541 
 
 0209 
 0384 
 0558 
 
 0227 
 0401 
 0576 
 
 0244 
 0419 
 0593 
 
 0262 
 0436 
 0610 
 
 0279 
 
 0454 
 0628 
 
 0297 
 0471 
 0645 
 
 0314 
 0488 
 0663 
 
 0332 
 0506 
 0680 
 
 369 
 369 
 369 
 
 12 15 
 12 15 
 12 15 
 
 4 
 5 
 6 
 
 0698 
 0872 
 1045 
 
 0715 
 0889 
 1063 
 
 0732 
 0906 
 1080 
 
 0750 
 0924 
 1097 
 
 0767 
 0941 
 i"5 
 
 0785 
 0958 
 1132 
 
 0802 
 0976 
 "49 
 
 O8ig 
 
 0993 
 1 1(>7 
 
 08-37 
 ion 
 1184 
 
 0854 
 1028 
 
 I2OI 
 
 369 
 369 
 369 
 
 12 15 
 12 14 
 12 14 
 
 7 
 8 
 9 
 
 1219 
 
 1392 
 1564 
 
 1236 
 1409 
 1582 
 
 1754 
 
 1253 
 1426 
 
 '599 
 
 1271 
 1444 
 1616 
 
 1288 
 1461 
 1633 
 
 1305 
 1478 
 1650 
 
 *323 
 1495 
 1668 
 
 1340 
 1513 
 I68 5 
 
 1357 
 1530 
 1702 
 
 1374 
 1547 
 1719 
 
 369 
 369 
 369 
 
 12 14 
 12 14 
 12 14 
 
 10 
 
 1736 
 
 1771 
 
 1788 
 
 1805 
 
 1822 
 
 1840 
 
 1857 
 
 1874 
 
 1891 
 
 369 
 
 12 14 
 
 11 
 12 
 13 
 
 1908 
 
 2079 
 2250 
 
 1925 
 2096 
 2267 
 
 1942 
 2113 
 2284 
 
 T 959 
 2130 
 2300 
 
 1977 
 2147 
 2317 
 
 1994 
 2164 
 2334 
 
 2OII 
 
 2181 
 
 2351 
 
 2028 
 2198 
 2368 
 
 2045 
 2215 
 2385 
 
 2062 
 2232 
 24O2 
 
 369 
 369 
 368 
 
 II 14 
 II 14 
 II 14 
 
 14 
 15 
 16 
 
 24*19 
 2588 
 2756 
 
 2436 
 2605 
 2773 
 
 2453 
 2622 
 2790 
 
 2470 
 2639 
 2807 
 
 2487 
 2656 
 2823 
 
 2504 
 2672 
 2840 
 
 2521 
 2689 
 2857 
 
 2538 
 2706 
 2874 
 
 2554 
 2723 
 2890 
 
 2571 
 2740 
 2907 
 
 368 
 
 368 
 368 
 
 II 14 
 II 14 
 II 14 
 
 17 
 18 
 19 
 
 2924 
 3090 
 3256 
 
 2940 
 3107 
 3272 
 
 2957 
 3123 
 3289 
 
 2974 
 3140 
 3305 
 
 2990 
 
 3156 
 3322 
 
 3007 
 3173 
 3338 
 
 3024 
 3T90 
 
 3355 
 
 3040 
 3206 
 3371 
 
 3057 
 3223 
 
 3387 
 
 3074 
 3239 
 3404 
 
 368 
 
 368 
 
 3 5 8 
 
 II 14 
 II 14 
 II 14 
 
 20 
 
 3420 
 
 3437 
 
 3453 
 
 3469 
 
 3486 
 
 3502 
 
 35i8 
 
 3535 
 
 355i 
 
 3567 
 
 3 5 8 
 
 II 14 
 
 21 
 22 
 23 
 
 35B4 
 3746 
 397 
 
 3600 
 3762 
 39 2 3 
 
 3616 
 3778 
 3939 
 
 3633 
 3795 
 3955 
 
 3649 
 3811 
 
 3971 
 
 3665 
 3827 
 3987 
 
 3681 
 
 3843 
 4003 
 
 3697 
 3859 
 4019 
 
 3714 
 3875 
 4035 
 
 3730 
 3891 
 4051 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 II 14 
 II 14 
 II 14 
 
 24 
 25 
 26 
 
 4067 
 4226 
 4384 
 
 4083 
 4242 
 4399 
 
 4099 
 4258 
 4415 
 
 4"5 
 4274 
 4431 
 
 4i3i 
 4289 
 4446 
 
 4M7 
 4305 
 4462 
 
 4163 
 4321 
 
 4478 
 
 4179 
 4337 
 4493 
 
 4195 
 4352 
 4509 
 
 42IO 
 
 4368 
 4524 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 II 13 
 II 13 
 
 10 13 
 
 27 
 28 
 
 29 
 
 4540 
 4695 
 
 4848 
 
 4555 
 4710 
 4863 
 
 4571 
 4726 
 4879 
 
 4586 
 4741 
 4894 
 
 4602 
 4756 
 4909 
 
 4617 
 4772 
 4924 
 
 4633 
 4787 
 4939 
 
 4648 
 4802 
 4955 
 
 4664 
 4818 
 4970 
 
 4679 
 4833 
 4985 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 10 13 
 10 13 
 10 13 
 
 30 
 ~31~ 
 32 
 33 
 
 5000 
 
 5015 
 
 5030 
 
 5045 
 
 5060 
 
 5075 
 
 5090 
 
 5105 
 
 5120 
 
 5135 
 
 3 5 8 
 
 10 13 
 
 5150 
 5299 
 5446 
 
 5592 
 5736 
 5878 
 
 5165 
 53M 
 546i 
 
 5180 
 5329 
 5476 
 
 5195 
 5344 
 5490 
 
 5210 
 
 5358 
 5505 
 
 5225 
 5373 
 5519 
 
 5240 
 
 5388 
 5534 
 
 5255 
 5402 
 
 5548 
 
 5270 
 5417 
 5563 
 
 5707 
 5850 
 5990 
 
 5284 
 5432 
 
 5577 
 
 257 
 2 5 7 
 
 2 5 7 
 
 IO 12 
 10 12 
 10 12 
 
 34 
 35 
 36 
 
 5606 
 5750 
 5892 
 
 5621 
 
 5764 
 5906 
 
 5635 
 5779 
 5920 
 
 5650 
 5793 
 5934 
 
 5664 
 5807 
 5948 
 
 5678 
 5821 
 5962 
 
 5693 
 5835 
 5976 
 
 572i 
 5864 
 6004 
 
 257 
 
 2 5 7 
 2 5 7 
 
 10 12 
 IO 12 
 
 9 12 
 
 37 
 38 
 39 
 
 6018 
 
 6i57 
 6293 
 
 6032 
 6170 
 6307 
 
 6046 
 6184 
 6320 
 
 6060 
 6198 
 6334 
 
 6074 
 6211 
 6347 
 
 6088 
 6225 
 6361 
 
 6101 
 6239 
 6374 
 
 6115 
 6252 
 
 6388 
 
 6129 
 6266 
 6401 
 
 6143 
 6280 
 6414 
 
 2 5 7 
 257 
 247 
 
 9 12 
 9 ii 
 9 ii 
 
 40 
 
 428 
 
 6441 
 
 6455 
 
 6468 
 
 6481 
 
 6494 
 
 6508 
 
 6521 
 
 6534 
 
 6547 
 
 247 
 
 9 ii 
 
 41 
 
 42 
 43 
 
 6561 
 6691 
 6820 
 
 6574 
 6704 
 
 6833 
 
 6587 
 6717 
 
 6845 
 
 6600 
 6730 
 6858 
 
 6613 
 6743 
 6871 
 
 6626 
 6756 
 6884 
 
 6639 
 6769 
 6896 
 
 6652 
 6782 
 6909 
 
 6665 
 6794 
 6921 
 
 6678 
 6807 
 6934 
 
 247 
 2 4 6 
 2 4 6 
 
 9 ii 
 9 ii 
 8 ii 
 
 44 
 
 6947 
 
 6959 
 
 6972 
 
 6984 
 
 6997 
 
 7009 
 
 7022 
 
 7034 
 
 7046 
 
 7059 
 
 2 4 6 
 
 8 10 
 
NATURAL SINES. 
 
 53 
 
 
 0' 
 
 6' 
 
 12' 
 
 18 
 
 24 
 
 30 
 
 36 
 
 42' 
 
 48 
 
 54 
 
 123 
 
 4 5 
 
 45 
 
 7071 
 
 7083 
 
 7096 
 
 7108 
 
 7120 
 
 7133 
 
 7M5 
 
 7157 
 
 7169 
 
 7181 
 
 2 4 6 
 
 8 10 
 
 46 
 47 
 48 
 
 7193 
 73M 
 743i 
 
 7206 
 
 7325 
 7443 
 
 7218 
 7337 
 7455 
 
 7230 
 7349 
 7466 
 
 7242 
 7361 
 
 7478 
 
 7254 
 7373 
 7490 
 
 7266 
 7385 
 750i 
 
 7278 
 7396 
 7513 
 
 7290 
 7408 
 7524 
 
 7302 
 7420 
 7536 
 
 2 4 6 
 246 
 246 
 
 8 10 
 8 10 
 8 10 
 
 49 
 50 
 51 
 52 
 53 
 54 
 
 7547 
 7660 
 777i 
 
 7558 
 7672 
 7782 
 
 7570 
 7683 
 1793 
 7902 
 8007 
 8in 
 
 758i 
 7694 
 7804 
 
 7593 
 7705 
 7815 
 
 7923 
 8028 
 8131 
 
 7604 
 7716 
 7826 
 
 7615 
 
 7727 
 7837 
 
 7627 
 7738 
 7848 
 
 7638 
 7749 
 7859 
 
 7649 
 7760 
 7869 
 
 246 
 246 
 245 
 
 8 9 
 7 9 
 7 9 
 
 7880 
 7986 
 8090 
 
 7891 
 7997 
 8100 
 
 7912 
 8018 
 8121 
 
 7934 
 8039 
 8141 
 
 7944 
 8049 
 8151 
 
 7955 
 8059 
 8161 
 
 7965 
 8070 
 8171 
 
 7976 
 8080 
 8181 
 
 245 
 235 
 235 
 
 7 9 
 7 9 
 
 7 8 
 
 55 
 
 8192 
 
 8202 
 
 8211 
 
 8221 
 
 8231 
 
 8241 
 
 8251 
 
 8261 
 
 8271 
 
 8281 
 
 235 
 
 7 8 
 
 56 
 57 
 58 
 
 8290 
 
 838? 
 8480 
 
 8572 
 8660 
 
 8746 
 
 5829 
 8910 
 
 8988 
 
 8300 
 8396 
 8490 
 
 8310 
 8406 
 8499 
 
 8320 
 8415 
 8508 
 
 8329 
 8425 
 8517 
 8607 
 8695 
 8780 
 
 8862 
 8942 
 9018 
 
 8339 
 8434 
 8526 
 
 8616 
 8704 
 8788 
 8870 
 8949 
 9026 
 
 834 
 8443 
 8536 
 
 8358 
 8453 
 8545 
 
 8368 
 8462 
 8554 
 
 8377 
 8471 
 8563 
 
 2 3 5 
 235 
 235 
 
 6 8 
 6 8 
 6 8 
 
 59 
 60 
 61 
 
 S 5 8i 
 8669 
 8755 
 8838 
 8918 
 8996 
 
 8590 
 8678 
 8763 
 
 8599 
 8686 
 
 8771 
 
 8854 
 
 8934 
 9011 
 
 8625 
 8712 
 8796 
 
 8634 
 8721 
 8805 
 
 8643 
 8729 
 8813 
 
 8652 
 8738 
 8821 
 
 i 3 4 
 i 3 4 
 
 i 3 4 
 
 6 7 
 6 7 
 6 7 
 
 62 
 
 63 
 64 
 
 8846 
 8926 
 9<>Q3 
 9078 
 
 8878 
 8957 
 9033 
 
 8886 
 8965 
 9041 
 
 8894 
 8973 
 9048 
 
 8902 
 8980 
 9056 
 
 i 3 4 
 i 3 4 
 i 3 4 
 
 5 7 
 5 6 
 5 6 
 
 65 
 
 9063 
 
 9070 
 
 9085 
 
 9092 
 
 9100 
 
 9107 
 
 9114 
 
 9121 
 
 9128 
 
 I 2 4 
 
 5 6 
 
 66 
 67 
 68 
 
 9135 
 
 9205 
 
 9272 
 
 9M3 
 9212 
 9278 
 
 9150 
 9219 
 9285 
 
 9157 
 9225 
 9291 
 
 9164 
 9232 
 9298 
 
 9171 
 9239 
 9304 
 
 9178 
 9245 
 93ii 
 
 9184 
 9252 
 9317 
 
 9191 
 9259 
 9323 
 
 919^ 
 9265 
 9330 
 
 i 2 3 
 i 2 3 
 i 2 3 
 
 5 6 
 4 6 
 4 5 
 
 69 
 70 
 71 
 
 9336 
 
 9397 
 9455 
 
 9342 
 9403 
 9461 
 
 9348 
 9409 
 9466 
 
 9354 
 9415 
 9472 
 
 9361 
 9421 
 
 9478 
 
 9367 
 9426 
 
 9483 
 
 9373 
 9432 
 9489 
 
 9379 
 9438 
 9494 
 
 9385 
 9444 
 
 939 1 
 9449 
 9505 
 
 i 2 3 
 i 2 3 
 i 2 3 
 
 4 5 
 4 5 
 4 5 
 
 9500 
 
 72 
 73 
 74 
 
 95ii 
 9563 
 9613 
 
 9516 
 9568 
 9617 
 
 952i 
 9573 
 9622 
 
 9527 
 9578 
 9627 
 
 9532 
 9583 
 9632 
 
 9537 
 9588 
 9636 
 
 9542 
 9593 
 9641 
 
 9548 
 9598 
 9646 
 
 9553 
 9603 
 9650 
 
 9558 
 9608 
 9655 
 
 i 2 3 
 
 122 
 122 
 
 4 4 
 3 4 
 3 4 
 
 75 
 
 9659 
 
 9664 
 
 9668 
 
 9673 
 
 9677 
 
 9681 
 
 9686 
 
 9690 
 
 9694 
 
 9699 
 
 112 
 
 3 4 
 
 76 
 77 
 78 
 
 9703 
 9744 
 9781 
 
 9707 
 9748 
 9785 
 
 9711 
 9751 
 9789 
 
 9715 
 9755 
 9792 
 
 9720 
 
 9759 
 9796 
 
 9724 
 9763 
 9799 
 
 9728 
 9767 
 9803 
 
 9732 
 9770 
 9806 
 
 9736 
 9774 
 9810 
 
 9740 
 9778 
 9813 
 
 I 2 
 I 2 
 I 2 
 
 3 3 
 3 3 
 
 2 3 
 
 79 
 80 
 81 
 
 9816 
 9848 
 9877 
 
 9820 
 
 9851 
 9880 
 
 9823 
 
 9854 
 9882 
 
 9826 
 
 9857 
 9885 
 
 9829 
 ;S6o 
 9888 
 9912 
 9934 
 9952 
 
 9833 
 9863 
 9890 
 
 9836 
 9866 
 9893 
 
 9839 
 9869 
 
 9895 
 
 9842 
 9871 
 9898 
 
 9845 
 9874 
 9900 
 
 I 2 
 O I 
 
 o i 
 
 2 3 
 2 2 
 2 2 
 
 82 
 83 
 84 
 
 9903 
 9925 
 9945 
 
 9905 
 9928 
 
 9947 
 
 9907 
 9930 
 9949 
 9965 
 
 9910 
 9932 
 9951 
 9966 
 
 9914 
 9936 
 9954 
 
 9917 
 9938 
 9956 
 
 9919 
 9940 
 9957 
 
 9921 
 9942 
 9959 
 
 9923 
 9943 
 9960 
 
 I 
 I 
 P I 
 
 2 2 
 
 2 
 
 I 
 
 85 
 
 9962 
 
 9963 
 
 9968 
 
 9969 
 
 9971 
 
 9972 
 
 9973 
 
 9974 
 
 O O I 
 
 I 
 
 86 
 87 
 88 
 
 89 
 
 9976 
 9986 
 9994 
 
 9977 
 9987 
 9995 
 
 9978 
 9988 
 9995 
 
 9999 
 
 9979 
 9989 
 9996 
 
 9980 
 9990 
 9996 
 
 9981 
 9990 
 9997 
 
 1.000 
 nearly 
 
 9982 
 9991 
 9997 
 
 9983 
 9992 
 
 9997 
 
 9984 
 
 9993 
 9998 
 
 9985 
 9993 
 9998 
 
 I 
 
 o o o 
 o o o 
 
 I 
 I 
 O O 
 
 9998 
 
 9999 
 
 9999 
 
 9999 
 
 1. 000 
 
 nearly 
 
 1.000 
 
 nearl" 
 
 I.OOO 
 nearly 
 
 I.OOO 
 nearly 
 
 000 
 
 
 
54 
 
 NATURAL TANGENTS. 
 
 
 0' 
 
 6' 
 
 12 
 
 18 
 
 24' 
 
 30' 
 
 36 
 
 42' 
 
 48' 
 
 54' 
 
 123 
 
 4 5 
 
 
 
 .0000 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 oi57 
 
 369 
 
 12 I 4 
 
 1 
 2 
 3 
 
 .0175 
 
 0349 
 .0524 
 
 0192 
 0367 
 0542 
 
 0209 
 0384 
 0559 
 
 0227 
 0402 
 0577 
 
 0244 
 0419 
 0594 
 
 0262 
 
 0437 
 0612 
 
 0279 
 
 0454 
 0629 
 
 0297 
 0472 
 0647 
 
 0314 
 0489 
 0664 
 
 0332 
 0507 
 0682 
 
 369 
 369 
 369 
 
 12 I 5 
 
 1 "5 
 i 15 
 
 4 
 5 
 6 
 
 .0699 
 .0875 
 .1051 
 
 0717 
 0892 
 1069 
 
 0734 
 0910 
 1086 
 
 0752 
 0928 
 1104 
 
 0769 
 0945 
 
 1122 
 
 0787 
 0963 
 "39 
 
 0805 
 0981 
 "57 
 
 0822 
 0998 
 "75 
 
 0840 
 1016 
 1192 
 
 0857 
 1033 
 
 1210 
 
 369 
 369 
 369 
 
 i 15 
 i IS 
 
 i i5 
 
 7 
 8 
 9 
 
 .1228 
 .1405 
 .1584 
 
 1246 
 
 1423 
 1602 
 
 1263 
 1441 
 1620 
 
 1281 
 
 1459 
 1638 
 
 1299 
 1477 
 1655 
 
 1317 
 1495 
 1673 
 
 1334 
 1512 
 1691 
 
 '352 
 1530 
 1709 
 
 1370 
 1548 
 1727 
 
 1388 
 1566 
 1745 
 
 369 
 369 
 369 
 
 i 15 
 i '5 
 i 15 
 
 10 
 
 1763 
 
 1781 
 
 1799 
 
 1817 
 
 1835 
 
 1853 
 
 1871 
 
 1890 
 
 1908 
 
 1926 
 
 369 
 
 12 15 
 
 11 
 12 
 13 
 
 .1944 
 
 .2126 
 .2309 
 
 1962 
 2144 
 2327 
 
 1980 
 2162 
 2345 
 
 1998 
 2180 
 2364 
 
 2016 
 2199 
 
 2382 
 
 2035 
 
 2217 
 
 2401 
 
 2053 
 2235 
 2419 
 
 2071 
 2254 
 2438 
 
 2089 
 2272 
 2456 
 
 2107 
 2290 
 
 247_5 
 2661 
 2849 
 3038 
 3230 
 3424 
 3620 
 
 369 
 369 
 369 
 
 12 1 5 
 12 15 
 12 15 
 
 14 
 15 
 16 
 
 2493 
 .2679 
 .2867 
 
 2512 
 2698 
 2886 
 
 2530 
 2717 
 2905 
 
 2549 
 2736 
 2924 
 
 2568 
 2754 
 2943 
 
 2586 
 2773 
 
 2962 
 
 2605 
 2792 
 2981 
 
 2623 
 28ll 
 3000 
 
 2642 
 2830 
 3019 
 3211 
 
 3404 
 3600 
 
 3 6 9 
 369 
 369 
 
 12 l6 
 
 13 16 
 13 16 
 
 17 
 18 
 19 
 
 3057 
 3249 
 
 3443 
 
 3076 
 3269 
 3463 
 
 3096 
 3288 
 3482 
 
 3"5 
 3307 
 3502 
 
 3134 
 3327 
 3522 
 
 3153 
 3346 
 3541 
 
 3172 
 3365 
 356i 
 
 3191 
 
 3385 
 3581 
 
 3 6 10 
 
 3 6 10 
 3 6 10 
 
 13 16 
 13 16 
 13 17 
 
 20 
 
 3640 
 
 3659 
 
 3679 
 
 3699 
 
 3719 
 
 3739 
 
 3759 
 
 3779 
 
 3799 
 
 3819 
 
 3 7 1 
 
 >3 7 
 
 21 
 22 
 23 
 
 3839 
 .4040 
 
 4245 
 
 3859 
 4061 
 4265 
 
 3879 
 4081 
 
 4286 
 
 3899 
 4101 
 4307 
 
 39'9 
 4122 
 
 4327 
 
 3939 
 4142 
 
 4348 
 
 3959 
 4163 
 4369 
 
 3978 
 4183 
 4390 
 
 4000 
 4204 
 44" 
 
 4020 
 4224 
 4431 
 
 3 7 10 
 3 7 10 
 3 7 10 
 
 13 '7 
 >4 '7 
 14 17 
 
 24 
 25 
 26 
 
 4452 
 .4663 
 .4877 
 
 4473 
 4684 
 4899 
 
 4494 
 4706 
 4921 
 
 4515 
 4727 
 4942 
 
 4536 
 4748 
 4964 
 
 4557 
 4770 
 4986 
 
 4578 
 4791 
 5008 
 
 4599 
 4813 
 5029 
 
 4621 
 
 4834 
 5051 
 
 4642 
 4856 
 5073 
 
 4 7 10 
 4 7 ii 
 4 7 ii 
 
 14 18 
 14 18 
 15 18 
 
 27 
 28 
 29 
 
 5095 
 5317 
 
 5543 
 
 5H7 
 3340 
 5566 
 
 5139 
 5362 
 5589 
 
 5161 
 
 5384 
 5612 
 
 5184 
 5407 
 5635 
 
 5206 
 5430 
 5658 
 
 5228 
 5452 
 5681 
 
 5250 
 5475 
 5704 
 
 5272 
 5498 
 
 5727 
 
 5295 
 5520 
 5750 
 
 4 7 H 
 8 ii 
 
 8 12 
 
 15 18 
 15 >9 
 '5 '9 
 
 30 
 
 5774 
 
 5797 
 
 5820 
 
 5844 
 
 5867 
 
 5890 
 
 59M 
 
 5938 
 
 596i 
 
 5985 
 
 8 12 
 
 16 20 
 
 31 
 32 
 33 
 
 .6009 
 .6249 
 .6494 
 
 6032 
 6273 
 6519 
 
 6056 
 6297 
 6544 
 
 6080 
 6322 
 6569 
 
 6104 
 6346 
 6594 
 
 6128 
 
 6371 
 6619 
 
 6152 
 
 6395 
 6644 
 
 6176 
 6420 
 6669 
 6924 
 7186 
 7454 
 
 6200 
 
 6445 
 6694 
 
 6224 
 6469 
 6720 
 6976 
 
 7239 
 7508 
 
 8 12 
 8 12 
 
 S 13 
 
 16 20 
 10 20 
 17 21 
 
 34 
 35 
 36 
 
 6745 
 .7002 
 .7265 
 
 6771 
 7028 
 7292 
 
 6796 
 7054 
 7319 
 
 6822 
 7080 
 7346 
 
 6847 
 7107 
 7373 
 
 6873 
 7133 
 7400 
 
 6899 
 7159 
 7427 
 
 6950 
 7212 
 7481 
 
 9 '3 
 
 4 9 3 
 5 9 4 
 
 17 21 
 
 l8 22 
 l8 23 
 
 37 
 38 
 39 
 
 7536 
 7813 
 .8098 
 
 7563 
 7841 
 8127 
 
 7590 
 7869 
 8156 
 
 7618 
 7898 
 8185 
 
 7646 
 7926 
 
 8214 
 
 7673 
 7954 
 
 8243 
 
 7701 
 7983 
 8273 
 
 7729 
 8012 
 8302 
 
 7757 
 8040 
 8332 
 
 7785 
 8069 
 8361 
 
 5 9 J 4 
 5 '4 
 5 o 15 
 
 18 23 
 19 24 
 
 20 =4 
 
 40 
 
 .8391 
 
 8421 
 
 8451 
 
 8481 
 
 8511 
 
 8541 
 
 8571 
 
 8601 
 
 8632 
 
 8662 
 
 5 o 15 
 
 20 25 
 
 41 
 42 
 43 
 
 .8693 
 .9004 
 9325 
 
 8724 
 9036 
 9358 
 
 8754 
 9067 
 9391 
 
 8785 
 9099 
 9424 
 
 8816 
 9131 
 
 9457 
 
 8847 
 9163 
 9490 
 
 8878 
 9195 
 9523 
 
 8910 
 9228 
 9556 
 
 8941 
 9260 
 9590 
 
 8972 
 9293 
 9623 
 
 99 6 5 
 
 5 o 16 
 5 i 16 
 6 i 17 
 
 21 26 
 ZI 2 7 
 22 28 
 
 44 
 
 9657 
 
 9691 
 
 9725 
 
 9759 
 
 9793 
 
 9827 
 
 9861 
 
 9896 
 
 9930 
 
 6 n 17 
 
 23 2 9 
 
NATURAL TANGENTS. 
 
 55 
 
 45 
 
 46 
 47 
 48 
 
 
 
 6' 
 
 12 
 
 18 
 
 24' 
 
 30 
 
 36 
 
 42' 
 
 48 
 
 54' 
 
 123 
 
 4 5 
 
 1. 0000 
 
 0035 
 
 0070 
 
 0103 
 
 0141 
 
 0176 
 
 0212 
 
 0247 
 
 0283 
 
 0319 
 
 6 12 18 
 
 .24 3 
 
 1-0355 
 1.0724 
 1.1106 
 
 0392 
 0761 
 H45 
 
 0428 
 0799 
 1184 
 
 0464 
 
 0837 
 1224 
 
 0501 
 
 0875 
 1263 
 
 0538 
 0913 
 1303 
 
 0575 
 0951 
 1343 
 
 0612 
 0990 
 1383 
 
 0649 
 1028 
 1423 
 
 0686 
 1067 
 1463 
 
 6 12 18 
 6 43 19 
 7 13 20 
 
 25 3' 
 25 3 
 26 33 
 
 49 
 50 
 51 
 52 
 53 
 54 
 ~55 
 
 1.1504 
 
 1.1918 
 
 1-2349 
 
 1544 
 1960 
 2393 
 
 1585 
 
 2OO2 
 
 2437 
 
 1626 
 2045 
 
 2482 
 
 1667 
 2088 
 2527 
 
 1708 
 2131 
 2572 
 
 I75C 
 2174 
 2617 
 
 1792 
 2218 
 2662 
 
 1833 
 2261 
 2708 
 
 1875 
 2305 
 2753 
 
 7 '4 21 
 7 14 2* 
 8 15 23 
 
 28 34 
 29 36 
 30 38 
 
 1.2799 
 1.3270 
 1-3764 
 
 2846 
 3319 
 3814 
 
 2892 
 3367 
 3865 
 
 2938 
 34i6 
 3916 
 
 2985 
 3465 
 3968 
 
 3032 
 3514 
 4019 
 
 3079 
 3564 
 4071 
 
 3127 
 36i3 
 4124 
 
 3175 
 3663 
 4176 
 
 3222 
 
 3713 
 4229 
 
 8 16 23 
 8 16 25 
 9 17 26 
 
 3 39 
 33 4> 
 
 34 43 
 
 1.4281 
 
 4335 
 
 4388 
 4938 
 5517 
 6128 
 
 4442 
 
 4496 
 
 4550 
 
 4605 
 
 4659 
 
 4715 
 
 4770 
 
 9 18 27 
 
 36 45 
 
 56 
 57 
 58 
 
 1.4826 
 
 1-5399 
 1.6003 
 
 4882 
 5458 
 6066 
 
 4994 
 5577 
 6191 
 
 5051 
 5637 
 6255 
 
 5108 
 
 5697 
 6319 
 
 5166 
 
 5757 
 6383 
 
 5224 
 5818 
 6447 
 
 5282 
 5880 
 6512 
 
 5340 
 594i 
 6577 
 
 ip 19 29 
 
 10 20 30 
 II 21 32 
 
 38 48 
 40 50 
 43 53 
 
 59 
 60 
 61 
 62 
 63 
 64 
 65 
 
 1.6643 
 1.7321 
 1.8040 
 
 6709 
 
 739 1 
 8115 
 
 6775 
 7461 
 8190 
 
 6842 
 7532 
 8265 
 
 6909 
 7603 
 8341 
 
 6977 
 
 7675 
 8418 
 
 7045 
 7747 
 8495 
 
 7"3 
 7820 
 
 8572 
 
 7182 
 
 7893 
 8650 
 
 725' 
 7966 
 8728 
 
 11 23 34 
 
 12 24 36 
 
 13 26 38 
 
 45 56 
 48 60 
 Si 64 
 
 1.8807 
 1.9626 
 2.0503 
 
 8887 
 9711 
 0594 
 
 8967 
 
 9797 
 0686 
 
 9047 
 9883 
 0778 
 
 9128 
 9970 
 0872 
 
 9210 
 0057 
 0965 
 
 5292 
 0145 
 1060 
 
 9375 
 6233 
 
 "55 
 
 2148 
 
 9458 
 0323 
 1251 
 
 9542 
 0413 
 1348 
 
 14 27 41 
 
 5 29 44 
 
 16 31 47 
 
 55 68 
 58 73 
 63 7 8 
 
 2.1445 
 
 1543 
 
 1642 
 
 1742 
 
 1842 
 
 1943 
 
 2045 
 
 2251 
 
 2355 
 
 i? 34 5 
 
 68 85 
 
 66 
 67 
 68 
 
 2.2460 
 2-3559 
 2-4751 
 
 2566 
 
 3673 
 4876 
 
 2673 
 
 3789 
 5002 
 
 2781 
 3906 
 5129 
 
 2889 
 4023 
 5257 
 
 2998 
 4142 
 5386 
 
 3IP9 
 4262 
 
 5517 
 
 3220 
 4383 
 5649 
 
 3332 
 4504 
 5782 
 
 3445 
 4627 
 59 l6 
 
 18 37 55 
 20 40 60 
 22 43 65 
 
 74 92 
 79 99 
 
 87 108 
 
 69 
 70 
 71 
 72 
 73 
 74 
 75 
 
 2.6051 
 
 2-7475 
 2.9042 
 
 6187 
 7625 
 9208 
 
 6325 
 7776 
 9375 
 
 6464 
 7929 
 9544 
 
 6605 
 8083 
 9714 
 
 6746 
 8239 
 9887 
 
 6889 
 8397 
 6061 
 
 7034 
 8556 
 0237 
 
 7179 
 8716 
 0415 
 
 7326 
 8878 
 0595 
 
 24 47 7' 
 26 52 78 
 
 29 58 87 
 
 95 "8 
 104 130 
 "5 M4 
 
 3-0777 
 32709 
 34874 
 
 0961 
 2914 
 5105 
 
 1146 
 3122 
 5339 
 
 1334 
 3332 
 5576 
 
 1524 
 
 3544 
 5816 
 
 1716 
 3759 
 6059 
 
 1910 
 3977 
 6305 
 
 2106 
 4197 
 6554 
 
 2305 
 4420 
 6806 
 
 2506 
 4646 
 7062 
 
 32 64 96 
 36 72 108 
 
 41 82 122 
 
 129 1161 
 144 180 
 162 203 
 
 3-7321 
 
 7583 
 
 7848 
 
 8118 
 
 8391 
 
 8667 
 
 8947 
 
 9232 
 
 9520 
 
 9812 
 
 46 94 139 
 
 i 86 232 
 
 76 
 77 
 78 
 
 4.0108 
 
 4-33I5 
 4.7046 
 
 0408 
 3662 
 7453 
 
 0713 
 4015 
 7867 
 
 J022 
 
 4374 
 8288 
 
 1335 
 4737 
 8716 
 
 1653 
 5107 
 9152 
 
 1976 
 5483 
 9594 
 
 2303 
 5864 
 0045 
 
 2635 
 6252 
 0504 
 
 2972 
 6646 
 0970 
 
 53 107 160 
 62 124 186 
 73 146 219 
 
 214 267 
 248 310 
 292 365 
 
 79 
 80 
 81 
 
 5.1446 
 5-6713 
 6. 3 i3fc 
 
 1929 
 7297 
 3859 
 
 2422 
 7894 
 4596 
 
 2924 
 8502 
 5350 
 
 3435 
 9124 
 6122 
 
 3955 
 9758 
 6912 
 
 4486 
 0405 
 7920 
 
 5026 
 1066 
 
 8548 
 
 557 
 1742 
 9395 
 
 6140 
 ?13?, 
 
 87 175 262 
 
 350 437 
 
 6264 
 
 Difference - col- 
 umns cease to be 
 useful, owing to 
 the rapidity with 
 which the value 
 of the tangent 
 changes. 
 
 82 
 83 
 84 
 
 7-II54 
 8.1443 
 95M4 
 
 2066 
 2636 
 9.677 
 
 3002 
 3863 
 9-845 
 
 3962 
 5126 
 IO.O2 
 
 4947 
 6427 
 
 10.20 
 
 5958 
 7769 
 10.39 
 
 6996 
 9152 
 10.58 
 
 8062 
 
 0579 
 10.78 
 
 9158 
 2052 
 10.99 
 
 0285 
 3572 
 
 11.20 
 
 85 
 86 
 87 
 88 
 
 89 
 
 U-43 
 
 ri.66 
 
 11.91 
 
 I2.I6 
 
 1243 
 
 12.71 
 
 13.00 
 
 13-30 
 
 13.62 
 
 13-95 
 
 14.30 
 19.08 
 28.64 
 
 14.67 
 19.74 
 30.14 
 
 15-06 
 20.45 
 31.82 
 
 15.46 
 21.20 
 33-69 
 
 15.89 
 
 22.02 
 35-80 
 
 16.35 
 22.90 
 38.19 
 
 16.83 
 23.86 
 40.92 
 
 17-34 
 24.90 
 44.07 
 
 17.89 
 26.03 
 47-74 
 
 18.46 
 27.27 
 52.08 
 
 57-29 
 
 63.66 
 
 71.62 
 
 81.85 
 
 95-49 
 
 114.6 
 
 143.2 
 
 [91.0 
 
 286,5 
 
 573-c 
 
56 
 
 LOGARITHMS. 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 123 
 
 456 
 
 789 
 
 10 
 
 oooc 
 
 0043 
 
 oose 
 
 OI2 
 
 0170 
 
 0212 
 
 025 
 
 029^ 
 
 0334 
 
 0374 
 
 Use Table on p. 58 
 
 11 
 
 12 
 13 
 14 
 15 
 
 16 
 
 041 
 079 
 
 "3 
 
 0453 
 0828 
 
 "73 
 
 0492 
 0864 
 1206 
 
 053 
 089 
 1239 
 
 0569 
 0934 
 1271 
 
 0607 
 0969 
 1303 
 
 064 
 1004 
 1335 
 
 0682 
 1038 
 1367 
 
 0719 
 1072 
 1399 
 
 0755 
 1106 
 7430 
 
 4811 
 3710 
 3610 
 
 15 19 23 
 
 14 17 21 
 
 13 16 19 
 
 26 30 34 
 24 28 31 
 23 26 29 
 
 146 
 176 
 204 
 
 1492 
 1790 
 2068 
 
 1523 
 
 1818 
 209^ 
 
 155 
 184 
 2122 
 
 i54 
 '875 
 2148 
 
 1614 
 1903 
 2175 
 
 1644 
 193 
 
 220 
 
 1673 
 
 1959 
 2227 
 
 17^3 
 1967 
 
 2253 
 
 1732 
 2014 
 2279 
 
 36 9 
 36 8 
 3 5 8 
 
 12 15 18 
 II 14 17 
 
 21 24 27 
 2O 22 25 
 
 18 21 24 
 
 17 
 18 
 19 
 
 2304 
 
 2553 
 2788 
 
 2330 
 
 2577 
 2810 
 
 2355 
 2601 
 
 2833 
 
 2380 
 2625 
 2856 
 
 2405 
 2648 
 
 2878 
 
 2430 
 26 7 2 
 2900 
 
 2455 
 2695 
 2923 
 
 2480 
 2718 
 2945 
 
 2504 
 2742 
 2967 
 
 2529 
 2765 
 2989 
 
 257 
 257 
 247 
 
 IO 12 15 
 
 9 12 14 
 9 " 13 
 
 17 20 22 
 
 16 19 21 
 16 18 20 
 
 20 
 
 3 OI 9 
 
 3032 
 
 3054 
 
 3075 
 
 3096 
 
 3H8 
 
 3139 
 
 3160 
 
 3181 
 
 320 
 
 246 
 
 8 ii 13 
 
 15 17 19 
 
 21 
 22 
 23 
 
 3222 
 
 3424 
 3617 
 
 3243 
 3444 
 3636 
 
 3263 
 3464 
 
 3655 
 
 3284 
 3483 
 3674 
 
 3304 
 3502 
 3692 
 
 3324 
 3522 
 3711 
 
 3345 
 3541 
 3729 
 
 3365 
 356o 
 3747 
 
 3385 
 3579 
 3766 
 
 3404 
 3598 
 3784 
 
 2 4 6 
 246 
 246 
 
 8 10 12 
 8 10 12 
 
 7 9 n 
 
 14 16 18 
 14 15 i-7 
 13 '5 17 
 
 24 
 25 
 26 
 
 3802 
 3979 
 4i$6 
 
 3820 
 
 3997 
 4166 
 
 3838 
 4014 
 4183 
 
 3856 
 4031 
 4200 
 
 3874 
 4048 
 4216 
 
 3892 
 4065 
 4232 
 
 39<>9 
 4082 
 
 4249 
 
 3927 
 4099 
 4265 
 
 3945 
 4116 
 4281 
 
 3962 
 4133 
 4298 
 
 2 45 
 235 
 235 
 
 7 9 ii 
 7 9 10 
 7 8 10 
 
 12 14 16 
 12 14 15 
 ii 13 15 
 
 27 
 28 
 29 
 
 4314 
 4472 
 4624 
 
 4330 
 4487 
 4639 
 
 4346 
 4502 
 4654 
 
 4362 
 4518 
 4669 
 
 4378 
 4533 
 4683 
 
 4393 
 4548 
 4698 
 
 4409 
 4564 
 4713 
 
 4425 
 4579 
 4728 
 
 4440 
 4594 
 4742 
 
 4456 
 4609 
 4757 
 
 235 
 235 
 * 3 4 
 
 689 
 689 
 679 
 
 ii 13 14 
 
 II 12 I. 
 10 12 13 
 
 30 
 31 
 
 33 
 
 477i 
 
 4786 
 
 4800 
 
 4814 
 
 4829 
 
 4843 
 
 4857 
 
 4871 
 
 4886 
 
 4900 
 
 1 3 4 
 
 6 7 9 
 
 10 ii 13 
 
 4914 
 051 
 
 185 
 
 J928 
 5065 
 198 
 
 4942 
 
 5079 
 5211 
 
 -f955 
 5092 
 5224 
 
 4969 
 105 
 237 
 
 4983 
 5"9 
 5250 
 
 4997 
 5132 
 5263 
 
 5011 
 5145 
 5276 
 
 5024 
 
 5159 
 5289 
 
 5038 
 5172 
 5302 
 
 1 3 4 
 134 
 * 3 4 
 
 6 7 8 
 
 5 7 8 
 5 6 8 
 
 10 II 12 
 
 9 ii 12 
 9 10 12 
 
 34 
 35 
 36 
 37 
 38 
 39 
 
 315 
 441 
 
 563 
 
 328 
 453 
 575 
 
 5340 
 5465 
 5587 
 
 5353 
 5478 
 5599 
 
 366 
 490 
 611 
 
 5378 
 5502 
 5623 
 
 5391 
 
 5514 
 5635 
 
 5403 
 5527 
 5647 
 
 54i6 
 
 5539 
 5658 
 
 5428 
 555i 
 5670 
 
 i 3 4 
 i 4 
 i 4 
 
 5 6 8 
 5 6 7 
 5 6 7 
 
 9 10 ii 
 9 jo ii 
 8 10 ii 
 
 682 
 798 
 911 
 
 694 
 SSog 
 922 
 
 5705 
 5821 
 
 5933 
 
 5717 
 5832 
 5944 
 
 729 
 843 
 955 
 
 5740 
 5855 
 5966 
 
 5752 
 5866 
 5977 
 
 5763 
 5877 
 5988 
 
 5775 
 5888 
 
 5999 
 
 5786 
 
 5899 
 6010 
 
 i 3 
 1 3 
 
 i 3 
 
 567 
 5 6 7 
 457 
 
 8 9 10 
 8 910 
 8 910 
 
 40 
 
 021 
 
 o3i 
 
 6042 
 
 605.3 
 
 6064 
 
 6075 
 
 6085 
 
 6096 
 
 6107 
 
 6117 
 
 1 3 
 
 4 5 6 
 
 8 9 10 
 
 41 
 
 42 
 43 
 
 128 
 
 232 
 
 335 
 
 138 
 243 
 345 
 
 6149 
 6253 
 6355 
 
 6160 
 6263 
 6365 
 
 6170 
 274 
 375 
 
 6180 
 6284 
 6385 
 
 6191 
 6294 
 6395 
 
 6201 
 6304 
 6405 
 
 6212 
 
 6314 
 6415 
 
 6222 
 
 6325 
 6425 
 
 i 3 
 i 3 
 r 3 
 
 4 5 6 
 4 5 6 
 4 5 6 
 
 789 
 
 7 8 9 
 7 8 g 
 
 44 
 45 
 46 
 47 
 48 
 49 
 
 435 
 532 
 628 
 
 444 
 542 
 637 
 
 6454 
 6551 
 6646 
 
 6464 
 6561 
 6656 
 
 474 
 57i 
 665 
 
 6484 
 6580 
 6675 
 
 6493 
 6590 
 
 6684 
 
 6503 
 
 6599 
 6693 
 
 6513 
 6609 
 6702 
 
 6522 
 6618 
 6712 
 
 1 3 
 i 3 
 1 3 
 
 4 5 6 
 4 5 6 
 4 5 6 
 
 7 8 9 
 7 8 9 
 7 7 8 
 
 721 
 812 
 902 
 
 730 
 821 
 911 
 
 6739 
 6830 
 6920 
 
 6749 
 6839 
 6928 
 
 758 
 848 
 937 
 
 6767 
 6857 
 6946 
 
 6776 
 6866 
 6955 
 
 6785 
 6875 
 6964 
 
 6794 
 6884 
 6972 
 
 6803 
 
 6893 
 6981 
 
 1 3 
 
 i 3 
 1 3 
 
 455 
 
 6 7 8 
 
 445 
 
 6 7 8 
 
 50 
 
 990 
 
 998 
 
 7007 
 
 7016 
 
 024 
 
 7033 
 
 7042 
 
 050 
 
 7059 
 
 7067 
 
 * 3 
 
 345 
 
 6 7 8 
 
 51 
 
 52 
 53 
 
 076 
 1 60 
 243 
 
 084 
 168 
 251 
 
 7093 
 7177 
 7259 
 
 7101 
 
 7185 
 7267 
 
 no 
 193 
 
 275 
 
 7118 
 7202 
 7284 
 
 7126 
 7210 
 7292 
 
 135 
 218 
 300 
 
 7M3 
 7226 
 7308 
 
 7152 
 7235 
 7316 
 
 i 3 
 
 I 2 
 I 2 
 
 345 
 345 
 345 
 
 678 
 6 7 7 
 667 
 
 54 
 
 324 
 
 332 
 
 7340 
 
 348 
 
 356 
 
 7364 
 
 7372 
 
 380 
 
 7388 
 
 7396 
 
 122 
 
 3 4 5 
 
 6 6 7 
 
LOGARITHMS. 
 
 57 
 
 55 
 
 
 
 74CM 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 123 
 
 456 
 
 789 
 
 7412 
 
 7419 
 
 7427 
 
 7435 
 
 7443 
 
 745i 
 
 7459 
 
 7466 
 
 7474 
 
 I 2 2 
 
 345 
 
 5 6 7 
 
 66 
 57 
 58 
 
 742 
 7559 
 7634 
 7709 
 7782 
 7853 
 
 7490 
 7566 
 7642 
 
 7497 
 7574 
 7649 
 
 7505 
 7582 
 7657 
 
 7513 
 7589 
 7664 
 
 7520 
 
 7597 
 7672 
 
 7528 
 7604 
 7679 
 
 753t> 
 7612 
 7686 
 
 7543 
 7619 
 7694 
 
 755' 
 7627 
 7701 
 
 I 2 2 
 
 
 5 6 7 
 5 6 7 
 
 I I 2 
 
 344 
 
 59 
 60 
 61 
 
 7716 
 7789 
 7860 
 
 7723 
 7796 
 
 7868 
 
 773i 
 7803 
 
 7875 
 
 7738, 
 7810 
 
 7882 
 
 7745 
 7818 
 7889 
 
 7752 
 7825 
 7896 
 
 7760 
 7832 
 7903 
 
 7767 
 
 7839 
 7910 
 
 7774 
 7846 
 
 79 T 7 
 
 112 
 I I 2 
 112 
 
 344 
 344 
 344 
 
 5 6 7 
 5 6 ( 
 5 6 6 
 
 62 
 63 
 64 
 
 7924 
 
 7993 
 8062 
 
 7931 
 Sooo 
 8069 
 
 7938 
 8007 
 8075 
 
 7945 
 8014 
 8082 
 
 7952 
 8021 
 8089 
 
 7959 
 8028 
 8096 
 
 7966 
 8035 
 8102 
 
 7973 
 8041 
 8109 
 
 7980 
 8048 
 8116 
 
 7987 
 8055 
 812? 
 
 I 1 2 
 112 
 
 334 
 334 
 
 5 6 6 
 5 5 6 
 5 5 * 
 
 65 
 
 8129 
 
 8136 
 
 8142 
 
 8149 
 
 8156 
 
 8162 
 
 8169 
 
 8176 
 
 8182 
 
 8189 
 
 112 
 
 334 
 
 5 S C 
 
 66 
 67 
 68 
 
 8195 
 826) 
 8325 
 
 8202 
 8267 
 8331 
 
 8209 
 8274 
 8338 
 
 8215 
 8280 
 8344 
 
 8222 
 8287 
 8351 
 
 8228 
 8293 
 8357 
 
 8235 
 8299 
 8363 
 
 8241 
 8306 
 8370 
 
 8248 
 8312 
 8376 
 
 8254 
 8319 
 8382 
 
 112 
 ( I 2 
 I I 2 
 
 334 
 3 3 4 
 334 
 
 5 5 6 
 5 5 6 
 4 5 6 
 
 69 
 70 
 71 
 
 8388 
 8451 
 S$I3 
 
 8395 
 8457 
 8519 
 
 8401 
 8463 
 
 8525 
 
 8407 
 8470 
 853i 
 
 8414 
 8476 
 8537 
 
 8420 
 8482 
 8543 
 
 8426 
 8488 
 8549 
 
 8432 
 8494 
 
 8555 
 
 8439 
 8500 
 8561 
 
 8445 
 8506 
 
 8567 
 
 I I 2 
 
 2 3 
 
 456 
 
 
 
 
 72 
 73 
 74 
 
 8573 
 8633 
 8692 
 
 8579 
 8639 
 8698 
 
 8585 
 8645 
 8704 
 
 859 1 
 8651 
 8710 
 
 8597 
 8657 
 8716 
 
 8603 
 8663 
 8722 
 
 8609 
 8669 
 8727 
 
 8615 
 8675 
 8733 
 
 8621 
 8681 
 8739 
 
 8627 
 8686 
 8745 
 
 I I 2 
 112 
 112 
 
 2 3 
 
 2 3 
 
 2 3 
 
 435 
 455 
 455 
 
 75 
 "76 
 77 
 78 
 
 8751 
 
 8756 
 
 8762 
 
 8768 
 
 8774 
 
 8779 
 
 8785 
 
 8791 
 
 8797 
 
 8802 
 
 I I 2 
 
 
 
 8808 
 8865 
 8921 
 
 8814 
 8871 
 8927 
 
 8820 
 8876 
 8932 
 
 8825 
 8882 
 8938 
 
 8831 
 8887 
 8943 
 
 8837 
 8893 
 8949 
 
 8842 
 8899 
 8954 
 
 8848 
 8904 
 8960 
 
 8854 
 8910 
 8965 
 
 885-9 
 8915 
 897) 
 
 112 
 112 
 
 233 
 233 
 
 455 
 
 445 
 
 79 
 80 
 81 
 
 8976 
 9031 
 9085 
 
 8982 
 9036 
 9090 
 
 8987 
 9042 
 9096 
 
 8993 
 9047 
 9101 
 
 8998 
 
 9053 
 9106 
 
 9004 
 9058 
 9112 
 
 9009 
 9063 
 9117 
 
 9015 
 9069 
 9122 
 
 9020 
 9074 
 9128 
 9180 
 9232 
 9284 
 
 9025 
 9079 
 9133 
 
 I I 2 
 112 
 I I 2 
 
 233 
 233 
 
 445 
 445 
 
 82 
 83 
 84 
 
 9138 
 9191 
 9 2 43 
 
 9143 
 9196 
 9248 
 
 9149 
 9201 
 9253 
 
 9154 
 9206 
 9258 
 
 9*59 
 9212 
 9263 
 
 9165 
 9217 
 9269 
 
 9170 
 9222 
 9274 
 
 9175 
 9227 
 
 9279 
 
 9186 
 9238 
 9289 
 
 112 
 I I 2 
 112 
 
 233 
 233 
 233 
 
 445 
 
 445 
 445 
 
 85 
 
 9294 
 
 9299 
 
 9304 
 
 9309 
 
 9315 
 
 9320 
 
 9325 
 
 9330 
 
 9335 
 
 9340 
 
 I 1 2 
 
 233 
 
 445 
 
 86 
 87 
 88 
 
 9345 
 9395 
 9445 
 
 9350 
 9400 
 9450 
 
 9355 
 9405 
 9455 
 
 936o 
 9410 
 9460 
 
 9365 
 9415 
 9465 
 
 9370 
 9420 
 9469 
 
 9375 
 9425 
 9474 
 
 9380 
 9430 
 9479 
 
 9385 
 9435 
 9484 
 
 9390 
 9440 
 9489 
 
 1 I 2 
 
 1 I 
 Oil 
 
 233 
 
 2 3 
 
 2 3 
 
 4 4 5 
 344 
 344 
 
 89 
 90 
 91 
 
 9494 
 9542 
 959 
 
 9499 
 9547 
 9595 
 
 9504 
 9552 
 9600 
 
 9509 
 9557 
 9605 
 
 9513 
 9562 
 9609 
 
 95i8 
 9566 
 9614 
 
 9523 
 957" 
 9619 
 
 9528 
 9576 
 9624 
 
 9533 
 958i 
 9628 
 
 953& 
 9586 
 
 9633 
 
 Oil 
 Oil 
 Oil 
 
 2 3 
 
 2 3 
 
 2 '3 
 
 3 4 4 
 344 
 3 4 4 
 
 92 
 93 
 94 
 
 9638 
 9685 
 9731 
 
 9 6 43 
 9689 
 9736 
 
 9647 
 9694 
 974i 
 
 9652 
 9699 
 9745 
 
 9657 
 9703 
 9750 
 
 9661 
 9708 
 9754 
 
 9666 
 97*3 
 9759 
 
 9671 
 9717 
 9763 
 
 9675 
 9722 
 9768 
 
 9680 
 9727 
 9773 
 
 Oil 
 Oil 
 Oil 
 
 2 3 
 2 3 
 
 2 - 3 
 
 344 
 344 
 3 4 4 
 
 95 
 
 9777 
 
 9782 
 
 9786 
 
 9791 
 9836 
 9881 
 9926 
 
 9795 
 
 9800 
 
 9805 
 
 9809 
 
 9814 
 
 9818 
 
 1. I 
 
 2 3 
 
 3 4 4 
 
 96 
 97 
 98 
 
 9823 
 9868 
 9912 
 
 9827 
 9872 
 9917 
 
 9832 
 9877 
 9921 
 
 9841 
 9886 
 9930 
 
 9845 
 9890 
 9934 
 
 9850 
 9894 
 9939 
 
 9854 
 9899 
 
 9943 
 
 9859 
 9903 
 9948 
 
 9863 
 9908 
 9952 
 
 I I 
 Oil 
 Oil 
 
 2 3 
 2 3 
 2 3 
 
 344 
 
 344 
 344 
 
 99 
 
 9956 
 
 9961 
 
 99 6 5 
 
 9969 
 
 9974 
 
 9978 
 
 9983 
 
 9987 
 
 999 1 
 
 9996 
 
 I 1 
 
 223 
 
 334 
 
58 
 
 LOGARITHMS. 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 100 
 
 ooooo 
 
 043 
 
 087 
 
 130 
 
 173 
 
 217 
 
 260 
 
 303 
 
 346 
 
 389 
 
 101 
 102 
 103 
 
 432 
 860 
 oi 284 
 
 475 
 903 
 326 
 
 5i8 
 945 
 368 
 
 56i 
 ciSS 
 410 
 
 604 
 030 
 452 
 
 647- 
 072 
 
 494 
 
 689 
 "5 
 536 
 
 732 
 157 
 578 
 
 775 
 199 
 620 
 
 817 
 242 
 662 
 
 104 
 105 
 106 
 
 703 
 
 02 119 
 
 531 
 
 745 
 1 60 
 572 
 
 787 
 202 
 612 
 
 828 
 -43 
 653 
 
 870 
 284 
 694 
 
 912 
 325 
 
 735 
 
 953 
 366 
 
 77* 
 
 995 
 407 
 816 
 
 036 
 
 449 
 857 
 
 078 
 490 
 898 
 
 107 
 108 
 109 
 
 938 
 03342 
 
 743 
 
 979 
 
 383 
 782 
 
 019 
 
 423 
 822 
 
 060 
 463 
 
 862 
 
 100 
 
 503 
 902 
 
 .41 
 
 543 
 94i 
 
 181 
 
 583 
 981 
 
 222 
 623 
 021 
 
 262 
 663 
 
 060 
 
 302 
 703 
 
 100 
 
 To find the logarithm of a number: First, locate in the 
 table the mantissa which lies in line with the first two figures of the 
 number and underneath the third figure, then increase this mantissa 
 by an amount depending upon the fourth figure of the number and 
 found by means of the interpolation columns at the right; secondly, 
 determine the characteristic, or the exponent of that integer power 
 of 10 which lies next in value below the number; for example, 
 log 600- 0.7782 t 2. ; -log 73.46= 0.8661 + 1. ; 
 
 log .006= 0.7782-3. ; log .7346= 0.8661 - 1. ; 
 
 log 6.003= 0.7784 + 0. ; log 7349= 0.8662 + 3. 
 
 The logarithm of a product of two or more numbers is the sum of 
 the logarithms of its factors; for example, 
 
 log. (.0821 X 463.2) = (0.9143 - 2.) + (0.6658 + 2.) = 0.5801 + 1. 
 The logarithm of a quotient is the difference between the logar- 
 ithms of the dividend and divisor; for example, 
 
 log. (.5321-*- 916)= (0.7260-1.) - (0.9619 + 2.) = 0.7641 -4. 
 The logarithm of a power or root of a number is the exponent 
 times the logarithm of the number; for example, 
 
 log V / :863) T =3/2X(0.9360-1.) = 0.9040-1. 
 
 To find the number from its logarithm: Locate in the table 
 the mantissa next less than the given mantissa, then join the figure 
 standing above it at the top of the table to the two figures at the 
 extreme left on the same line as the mantissa, and finally to these 
 three join the figure at the top of the interpolation column which 
 contains the difference between the two mantissae. In the four- 
 figure number thus found, so place the decimal point that the 
 number shall be the product of some number, that lies between 
 1 and 10, by a power of 10 whose exponent is the characteristic 
 of the logarithm. For example, 
 
 antilog (0.6440 + 3) = 4405; 
 antilog (0.3069 - 2) = .02027. 
 
 Caution. In adding and subtracting logarithms it is well to 
 remember that the mantissa is always essentially positive and may 
 or may not therefore, have the same sign as its characteristic. 
 
INDEX 
 
 Ampere, defined - - 24 
 
 Ammeter, tested - - 27 
 
 Bifilar suspension - 17 
 
 Bridge, Wheatstone, used 29 
 Calibration, relative of 
 
 Galvanometer - - 37 
 Capacity, specific induc- 
 tive, determination of - 47 
 Capacities, comparison of 47 
 Cell, standard, use of - 36 
 Coefficient of resistance, 
 temperature, deter- 
 mined - 31 
 
 Commutator, use of - 25 
 
 Conductivity, specific - 42 
 
 Conductivity, molecular - 43 
 
 Connections, electrical - 23 
 
 Current, unit, defined - 23 
 Demagnetization, method 
 
 of - 20 
 Earth inductor - 40 
 Electrolyte, resistance of - 42 
 Electromotive forces, com- 
 parison of - - 34 
 End-on position - - 15 
 Experiments, list of 5 
 
 Field, electric, plotting - 10 
 
 Magnetic, location of - 7 
 
 Plotting - - 9 
 Galvanometer, ballistic, 
 
 calibration of - 19 
 
 Tangent, constants of - 27 
 
 " reduction factor of 26 
 
 theory of - 23 
 
 used as voltmeter - 45 
 
 "H" defined 7 
 relative determination 
 
 of - f - 14 
 absolute determination 
 
 of - 15 
 
 Heating devices, electrical 47 
 
 Hot-plate, efficiency of - 47 
 
 Induction, magnetic - 19 
 Inclination, or dip, angle 
 
 of - 41 
 
 Law, tangent, test of - 24 
 
 Lines of force, direction 
 
 of magnetic 7 
 
 intensity of magnetic - 7 
 
 tracing of electric - 10 
 
 tracing of magnetic - 9 
 
 Magnetic force, direction 
 
 of, defined - - 7 
 
 intensity of, denned - 7 
 
 laws of - 13 
 
 Magnetometer, mirror, 
 
 used - - 17 
 
 Magnets, care of 9 
 
 Method, Carey Foster 30 
 
 of oscillations - - 14 
 
 Kohlrausch's - 43 
 
 Poggendorff's - - 34 
 
 Moment of magnet - 9 
 
 Permeability, defined - 8 
 
 Poles, of magnet, defined 8, 23 
 
 Polarization, in Leclanche 
 
 cell - - 45 
 Postoffice bridge, used - 34 
 Reduction factor, of gal- 
 vanometer, defined - 24 
 Reference books 6 
 
 Resistance, absolute deter- 
 mination of - 32 
 boxes, care of - 23 
 comparison of - - 30 
 defined - 32 
 specific, determined - 31 
 
 Robison magnet - - 15 
 
 Solenoid, field intensity 
 
 for - - 19 
 
 Suspension, bifilar, used - 17 
 
 Tables, of logarithms - 56 
 
 physical - - 49 
 
 of sines and tangents - 52 
 
 Thermo electromotive 
 
 force. - - - - 37 
 
 Tangent galvanometer, 
 
 constants of - 27 
 
 reduction factor of - 26 
 
 Voltameter, copper, used - 26 
 
 Water, equivalent of heat- 
 ing coil - - 33 
 Wheatstone bridge, used - 29 
 
PHYSICAL MEASUREMENTS 
 
 MINOR 
 
 PART IV. SOUND AND LIGHT 
 1917 
 
PHYSICAL MEASUREMENTS 
 
 A LABORATORY MANUAL IN GENERAL PHYSICS 
 FOR COLLEGES 
 
 In Four Parts 
 
 BY 
 RALPH S. MINOR, Ph. D. 
 
 Associate Professor of Physics, University of California 
 
 PART IV 
 SOUND AND LIGHT 
 
 In collaboration with 
 Raymond B. Abbott, M. S. 
 
 Instructor in Physics, University of California 
 
 BERKELEY, CALIFORNIA 
 1917 
 
Copyrighted in the year 1917 by 
 Ralph S. Minor 
 
 Wetzel Bros. Printing Co. 
 
 21 10 Addiwn Street 
 
 Berkeley. Calif. 
 
LIST OF EXPERIMENTS 
 
 Sound Page 
 
 81. Velocity of Sound in Solids 5 
 
 82. The Natural Scale. Harmony - 8 
 
 83. Longitudinal Vibration of Wires - 9 
 
 Light 
 
 The Convention of Signs for Mirrors, Single 
 
 Refracting Surfaces and Lenses 11 
 
 Path of Any Oblique Ray 12 
 
 Spherical Mirrors - 12 
 
 The Spherometer 13 
 
 91. Radius of Curvature. By Reflection - 15 
 
 92. Refraction at a Single Plane Surface - 17 
 
 93. The Refraction Trough 
 
 Refraction at a Single Spherical Surface 20 
 
 A Study of the Model Eye 21 
 
 94. The Spectrometer. Measuring, the Angles of a 
 
 Prism 24 
 
 95. The Spectrometer. Index of Refraction of a 
 
 Prism 28 
 
 96. Refractive Index by Total Reflection 
 
 Pulfrich's Refractometer 30 
 Thin Lenses - - 33 
 Direct Measurement of Focal Length of Con- 
 vex Lenses - 33 
 
 97. Refraction Through a Single Thin Lens 33 
 
 98. Thick Lenses 36 
 
 99. Astronomical Telescope - - 40 
 
 100. The Spectroscope 44 
 
 101. Spark Spectra of Metals - 47 
 
 102. Wave-Length of Sodium Light 
 
 Newton's Rings - 48 
 
 103. The Diffraction Grating 
 
 The Grating Constant 50 
 
 104. Measurement of Wave Length 
 
 With Spectrometer and Grating - 52 
 
 105. Reflection of Light on. Transparent Substances - 53 
 
REFERENCE BOOKS 
 
 Clay : Treatise on Practical Light. 
 
 Duff: Text Book of Physics. (Fourth Edition). 
 
 Kdser: Light for Students. 
 
 Ganot: Text Book of Physics. (18th Edition.) 
 
 Kimball: College Physics. 
 
 Kaye and Laby: Physical and Chemical Constants. 
 Landolt and Bernstein: Physical and Chemical Tables. 
 Smithsonian Institute: Physical Tables. 
 
SOUND 
 
 81. VELOCITY OF SOUNDS IN SOLIDS. 
 Kundt's Method. 
 
 Kundt's method of determining the velocity of sound in 
 solids and gases consists in setting up stationary waves 
 in a horizontal glass tube by means of the longitudinal 
 vibrations of a rod, the form of the stationary wave being 
 made evident by the recurring patterns into which lyco- 
 podium powder, or other dust, is thrown when the 
 resonance is sufficiently vigorous. A great variety of 
 forms will be observed in these dust figures, but the 
 exact conditions which result in any pattern are difficult 
 to define as much depnends upo the manner of rubbing 
 the vibrating rod, the size of the dust particles, the dry- 
 ness of the walls of the tube, the amount of dust present, 
 the length of the glass tube, the length of the vibrating 
 column of air, and the space between the disk on the 
 end of the rod and the glass tube. 
 
 In every pattern, however, the distance between corres- 
 ponding points of adjacent waves will be exactly one-half a 
 wave length of the sound in air. 
 
 It should be particularly noted that irregularities may 
 occur, in some patterns, both near the closed end of the 
 tube and near the end of the vibrating rod, so that in 
 general, measurements should not include the first half 
 wave length at either end. 
 
 Figure t is the pattern usually obtained. Figure 2 is 
 often obtained when the length of the vibrating column is 
 
8 THE NATURAL SCALE. HARMONY [82 
 
 Additional Exercise. 
 
 Determine the velocity of sound in tubes of various 
 diameters presenting your results in the form of a plot 
 showing the relation between the velocity of sound in 
 air and the diameter of the tube. 
 
 82. THE NATURAL SCALE. HARMONY. 
 
 The sonometer should be provided with two strings of 
 the same material and diameter. The strings should be 
 tuned in unison. 
 
 (a) Place a bridge under one of the strings and grad- 
 ually change the length of the vibrating portion until 
 you have secured a tone which in your judgment sounds 
 well when heard with the string whose length has not 
 been changed. 
 
 Shorten the vibrating part still further until you have 
 a second tone which goes well with the fundamental 
 tone. Strike these three notes in rapid succession. They 
 form a major triad. Together with the octave of the first 
 note they form a major chord. 
 
 From your measurements determine the ratio of the 
 vibration rates for these four notes. 
 
 Place a bridge under the first string so that it gives 
 the second of the new tortes found and see if with this 
 tone as your starting point you can find two other notes 
 which sound well with it. That is, find another major 
 triad. Find the ratio of their vibration rates. 
 
 Starting with the octave of your first fundamental 
 tone find another triad by increasing the length of the 
 string until a pleasing combination similar to the two first 
 found is the result. 
 
 Find the ratios of the vibration rates of the eight tones 
 in terms of the fundamental, i. e. the tone given out by 
 the entire string. 
 
82-83] THE NATURAL SCALE 9 
 
 Strike these tones in succession, in order of increasing 
 vibration rate and you have the natural or diatonic scale. 
 
 Why is this scale not used on the piano? 
 
 (b) In order to determine the physical basis for har- 
 mony, test the triads separately for sympathetic vibra- 
 tions of the second string for any of the positions found 
 when the two sound well together. In what way must 
 the first string be vibrating in order that such sympa- 
 thetic vibrations may be set up? Is the intensity of the 
 sympathetic vibration greater in the case of two notes 
 of a triad than for two notes not so related. If possible 
 repeat this test on a piano. 
 
 After considering the overtones possible on a string in 
 transverse vibration together with your experimental 
 work make a statement as to the secret of harmony. 
 
 83. LONGITUDINAL VIBRATION OF WIRES. 
 
 The object of this experiment is to determine Young's 
 modulus for the material of a wire from measurements of 
 the wave-length of its longitudinal vibrations and its density. 
 
 A wire about ten meters long is clamped at each end 
 by means of two vises. The attachment at one end being 
 to the movable jaw of the vise so that the tension may 
 be changed. The length of the vibrating part is varied 
 by means of a movable bridge. 
 
 The points at which the wire meets the vise and the 
 bridge are nodes, and the distance / between two consecu- 
 tive nodes is one half of a wave-length (for the funda- 
 mental). Therefore the velocity v = 2ln = V E / d, from 
 which Young's modulus E may be found. 
 
 (a) With the wire under slight tension set up longi- 
 tudinal vibrations in it by rubbing it with the fingers, 
 
10 LONGITUDINAL VIBRATION OF WIRES [83 
 
 which have been touched with powdered rosin. At the 
 same time sound a tuning fork of known frequency, and 
 adjust the vibrating length of the wire until it gives the 
 same note. Approach the position of unison from the 
 side of too great a length and again from the side of too 
 small a length and take the mean as the value of /. 
 
 The formula given above states that the velocity of a 
 longitudinal vibration is independent of the tension of 
 the wire. If this is the case the pitch of the note should 
 not be altered when the tension of the wire is varied with- 
 in wide limits (while the length is kept constant). Try 
 this experiment, taking care not to stretch the wire beyond 
 the elastic limit of its material. 
 
 (6) Determine the density of the wire from the sample 
 furnished by any method you select. 
 
 (c) Calculate Young's modulus for the material of 
 the wire. 
 
LIGHT 
 
 The Convention of Signs for Mirrors, Single Refracting 
 Surfaces and Lenses. 
 
 In writing the formulae which show the relation be- 
 tween the object distance u, the image distance v, and 
 the constants of mirrors, single refracting surfaces and 
 lenses the following convention of signs will be used in 
 the manual and also in the lecture course given in con- 
 nection with the laboratory work: 
 
 1. Distances measured over a path actually traveled by 
 the light are called positive, +; distances measured over a 
 projected path of the ray are called negative, . The ray 
 being the normal to the wave-front. 
 
 This applies to the object distance u, the image dis- 
 tance v, and the focal length /. In accordance with this 
 convention the distance from the mirror or lens to a real 
 object, a real image, or a real focus, is always positive; 
 negative values always mean that the object, image or 
 focus, is virtual. 
 
 2. The radius of curvature of a surface which tends to 
 produce a real focus is called positive; one which tends to 
 produce a virtual focus is negative. Accordingly the 
 radius of a concave mirror is positive, since it brings light 
 to a real focus, while the radius of curvature of a concave 
 refracting surface is negative, as it renders a parallel 
 beam divergent. Both radii of curvature of a double 
 convex lens are positive; both radii of curvature of a 
 double concave lens are negative. 
 
 This convention of signs places the real physical quantity and 
 the positive sign together. It has also the practical advantage of 
 being the one uniformly used by opticians, manufacturers and 
 dealers. Any convergent system has a positive focal length. A 
 divergent system has a negative focal length. 
 
12 
 
 THE CONVENTION OF SIGNS 
 
 1 1 
 
 For a mirror ~^~ ~ ~ 
 
 (i) 
 
 For a single refracting surface + = - 
 
 (2) 
 (3) 
 
 For a lens 4~ + ~T = ^ n ~ 1 ) ^~7T + ^ = ~J~ 
 
 For a combination of two thin lenses in contact, whose 
 focal lengths are /j, and f,, respectively, the focal length 
 of the combination F, is given by the formula 
 
 111 , . 
 
 Tr+TT^-Tr (4) 
 
 Path of Any Oblique Ray Through a Single Thin Lens. 
 
 Given the focal points FI and F 2 and the optical center 
 0. For any oblique ray AB draw a parallel to AB from 
 the optical center to its intersection with the second 
 
 
 
 Fig. 5. 
 
 Path of any oblique ray. 
 
 focal plane at C, then the refracted ray is BC, since 
 parallel rays intersect in the focal plane. 
 
 Spherical Mirrors. 
 
 The following simple optical methods are useful in 
 determining the radius of curvature of spherical mirrors 
 or lens surfaces: 
 
THE SPHEROMETER 13 
 
 Concave Mirrors. 
 
 1. Place the concave surface in the path of a beam of parallel 
 light sunlight, light rendered parallel by means of a lens, or light 
 from a distant object tilt it slightly and find the reflected image 
 with a card. When sharply in focus the distance from the reflect- 
 ing surface to the image will be the focal length, or one-half the 
 radius of curvature. 
 
 For an object not very distant, the radius of curvature could 
 similarly be determined by measuring both object and image 
 distances. 
 
 2. Turn the concave surface to be tested toward an illuminated 
 aperture in a white screen. A slit with a single cross-hair will be 
 found convenient. Adjust until the image is seen on the screen 
 beside the slit. The distance from the reflecting surface to the 
 screen will then be equal to the radius of curvature. 
 
 Convex Mirrors. 
 
 1. Use a convergent lens to form a real image of a narrow slit 
 in a white screen. Adjust a needle, using the parallax test, so that 
 it coincides with the image. Now place the convex surface to be 
 tested between the image and the lens (convex side toward the 
 lens) and adjust until an image of the slit is formed on the white 
 screen, beside the slit. In this case the distance between the needle 
 and the convex surface is equal to its radius of curvature. 
 
 The Spherometer. 
 
 The spherometer furnishes an easy mechanical method 
 for determining the radius of curvature of spherical 
 surfaces. It is used to best advantage with surfaces of 
 medium curvature. 
 
 A. The Ring Spherometer. 
 
 A rotating arm is provided which slips over the point 
 of the spherometer screw. By means of this the upper 
 brass ring should be adjusted so that its axis coincides 
 with the axis of rotation of the screw and then clamped in 
 position by means of the small thumb-screws underneath. 
 The internal radius of this ring, which is the perpendi- 
 cular distance from the axis of the screw to the inner edge 
 of the ring, is called the "Span," 5, of the spherometer. 
 
14 
 
 THE SPHEROMETER 
 
 Fig. 6. A section through the lens 
 and spherometer ring. 
 
 If the difference in the 
 reading of the spherometer 
 screw when the ring is 
 covered with a piece of 
 plate glass and when a 
 lens of radius of curvature 
 R is placed upon it be d, 
 see figure, then it may be 
 shown that the radius of 
 curvature is determined 
 from these measurements 
 by means of the formula 
 
 d) 
 
 2d 
 
 If the pitch of the spherometer screw is not standard it should 
 be determined by measuring with the micrometer calipers the 
 thickness of a piece of glass, whose thickness has been determined 
 in terms of the pitch of the spherometer screw. 
 
 B. The Peg-Spherometer. 
 
 The peg-spherometer consists' of a tripod, the legs of 
 which form an aquilateral triangle, with the screw at its 
 center. If / represents the average distance between the 
 legs of the tripod and d the mean distance from the point 
 of the screw when resting against the surface of a lens, 
 to the plane through the ends of the spherometer legs, it 
 may be shown that the radius of curvature, R, is given 
 
 by the expression 
 
 (2) 
 
 C. The Aldis Peg- Spherometer. 
 
 With the Aldis peg-spherometer, in which the points 
 on the ordinary spherometer are replaced by small 
 spheres, of average radius r, the relations are very similar, 
 R + r taking the place of R in equation (2). 
 
91] RADIUS OF CURVATURE 15 
 
 91. RADIUS OF CURVATURE 
 By Reflection. 
 
 The following method of determining the radius of 
 curvature of spherical reflecting surfaces is particularly 
 useful in measuring surfaces of small radius, as well as 
 surfaces inaccessible to measurement by other methods. 
 
 In the figure DCE is a section of the reflecting surface 
 whose center is 0. A and B are two objects and A' and 
 B' are their images. 
 
 Fig. 7. Illustrates the method of finding the radius of curvature by reflection. 
 Let P C =d, P 1 C = -x, O C = -r, A B = L, A 1 B> = /. 
 
 112 r d 
 
 Then - -- -- .-. *= - - n\ 
 
 d x r r + 2d 
 
 The first equation, however may be written 
 
 r- + - --- From which follows 
 d r x r 
 
 r + d 
 
 From the figure -=- =< , .-. - - = JL 
 
 L r + d L d 
 
16 RADIUS OF CURVATURE [91 
 
 The radius of curvature, r, of a mirror (or lens) may 
 thus be determined from measurements of the distance 
 apart, L, of two objects and the distance, /, between their 
 images formed by the mirror when d centimeters away, 
 To measure the distance / between the images, a traveling 
 telescope should be placed midway between the objects 
 A and B with its axis parallel to the axis, OP, of the 
 mirror. The objects, A and B, as well as the telescope 
 and the mirror should be in a common horizontal plane. 
 
 In finding the radius of curvature of a lens surface it 
 must be remembered that each surface of the lens will 
 form a pair of images. These may be distinguished by 
 remembering that the images produced by a convex re- 
 flecting surface are always erect. The telescope, however, 
 inverts them. It should be noted that formula (2) does 
 not hold for the images formed by the second surface of a 
 lens, as the light which strikes it has suffered refraction 
 at the first surface. 
 
 To find the radius of curvature of a lens. 
 
 (a) Measure with a meter rod the distance, d, meas- 
 uring from the line joining the two objects to the reflect- 
 ing surface; and the distance, L, between the centers of 
 the objects A and B, (slits in a screen, illuminated in 
 turn by a frosted bulb incandescent lamp). 
 
 (6) To find / set the cross-hairs of the eyepiece on 
 one of the images and read the scale and head of the 
 micrometer screw. Make several settings, always turn- 
 ing the screw in the same direction in coming to the 
 final position, and take the average. Then turn the screw 
 until the cross-hairs coincide with the other image, take 
 several readings, averaging them as before, being careful 
 in making any setting to turn the screw in the same 
 direction as in the first case in coming to the position of 
 coincidence between image and cross-hairs. 
 
91-92] REFRACTION AT A SINGLE PLANE SURFACE 17 
 
 (c) Calculate the value of r from equation (2). 
 What modification in equations (1) and (2) would be 
 introduced if the reflecting surface were concave? 
 
 92. REFRACTION AT A SINGLE PLANE SURFACE. 
 
 The object of this experiment is to observe the visual effects 
 due to the astigmatism which results when spherical waves 
 are refracted at a plane surface, and to plot the caustic curve 
 of the refracted wave. 
 
 AIR 
 
 AIR 
 
 WATER 
 
 Fig. 8. 
 
 In the figure let the point be a source in water from 
 which spherical waves emanate passing into air across the 
 plane surface A B. The dotted line aa represents the 
 position an advancing spherical wave would have reached 
 at a given instant if it had not passed out of the water 
 into the air, and the line bbb represents the actual wave in 
 air. This wave bbb is not spherical as we see by compari- 
 son with the arc C V; that is, refraction at a plane surface 
 is subject to spherical aberration. Furthermore, a small 
 portion of the wave at dd has two radii of curvature; or 
 in other words, the pencil of rays near dd is an astigmatic 
 pencil, that is, oblique refraction at a plane surface is 
 subject to astigmatism. The intersection of the wave bb 
 with the plane of the paper forms at dd the arc of a circle 
 
18 REFRACTION AT A SINGLE PLANE SURFACE [92 
 
 whose center is at K. Suppose a plane to be passed 
 through the line L K d perpendicular to the plane of the 
 paper. The intersection of the wave bb with this plane 
 at dd is the arc of a circle whose center is at L. 
 
 Suppose a person to look at the point O, allowing the 
 pencil of rays dd to enter the two eyes. If the head is held 
 erect so that the eyes lie in a horizontal line (perpendicular 
 to the plane of the paper) then the curvature of the por- 
 tion dd of the wave front about the vertical axis deter- 
 mines the location of as perceived by the two eyes, and 
 the point appears at L. If, on the other hand, the head 
 is held on one side so that the two eyes lie in a vertical 
 line (in the plane of the paper) then the curvature of the 
 portion dd of the wave front about a horizontal axis 
 determines the location of 0, as perceived by the two 
 eyes, and the point O appears at K. A short horizontal 
 line through K and a short vertical line at L are the two 
 focal lines of the astigmatic pencil dd. 
 
 The caustic curve 0' K is the locus of the center 
 
 of curvature of the 
 wave front bbb, or the 
 caustic curve may be 
 defined as follows: 
 
 Imagine lines to be 
 . T r drawn normal to the 
 
 Fig. 9. Showing the focal lines KK andLL 
 of the astigmatic pencil dddd. Curve bb Sit each 
 
 point; the envelope of these lines will be the caustic curve. 
 
 The apparatus consists of a metal strip carrying a ver- 
 tical scale VH and a horizontal scale A B in a jar of 
 water, the scale A B lying on a level with the surface of 
 the water. The lower end of the extension H is turned 
 upward and has a narrow straight edge of celluloid in- 
 serted in the end. The rest of this extension as well as 
 the inside of the jar is painted black so as to make the 
 
92] REFRACTION AT A SINGLE PLANE SURFACE 19 
 
 edge clearly visible and to enable one to see clearly the 
 reflected image of the vertical scale in the water surface. 
 (a) Note that the polished edge O appears to lie against 
 the face of the reflected image of the vertical scale when 
 the line joining the observer's eyes 
 is horizontal, whereas it seems very 
 much nearer to the eyes than the 
 reflected image of the vertical scale 
 when the line joining the eyes is 
 vertical. 
 
 To determine the caustic curve, 
 look at the polished edge O with one 
 eye placed at E, in the plane of the 
 scales V H - AB, and record the 
 reading on the reflected image of the vertical scale of 
 the point L where the polished edge appears to stand, 
 and record the corresponding reading of the point K where 
 the line of sight crosses the horizontal scale. 
 
 Observe the readings L and K for a series of positions 
 of the eye E covering as wide a range as possible. 
 
 Measure the distance A from the top of the scale 
 A B to the edge 0. 
 
 (b) Make a plot showing the scales O V and A B as 
 axes, lay off each pair of observed values of H K andH L 
 and draw each straight line LK. Add to the plot the 
 point O' calculated from the measured distance A O and 
 the refractive index, and draw the totally reflected ray, 
 then draw the caustic curve. 
 
 State Huygens' principle. Taking the refractive index 
 of water with reference to air as 4/3 find by graphical 
 application of this principle a wave front in air emanating 
 from a point source below a plane boundary between air 
 and water. Show by comparison that this wave front is 
 not spherical. 
 
20 REFRACTION AT A SINGLE SPHERICAL SURFACE [93 
 93. THE REFRACTION TROUGH. 
 
 A. Refraction at a Single Spherical Surface. 
 
 The exercise is a study of the conjugate relation be- 
 tween object and image for a single refracting surface, 
 together with a determination of the constants (radius of 
 curvature, refractive index, and focal length) of such a 
 surface. The apparatus consists of a long trough with a 
 thin spherical watch glass set in one end, so that the 
 liquid filling the trough will have a convex surface out- 
 wards. The refraction of the watch glass is to be neglected. 
 
 (a) Determine the radius of curvature of the surface 
 or surfaces to be used by any of the methods, described 
 in a separate exercise. 
 
 (b) Place the object screen on the air side of the sur- 
 face at distances from the surface which gives a real 
 image in the liquid. Locate this real image by means of 
 a movable screen placed in the liquid. 
 
 Immerse an object screen in the liquid, illuminating 
 it, if necessary, through the plate glass end of the trough, 
 place the screen at such a distance from the surface that 
 a real image is formed in air. Locate the image with a 
 screen and eyepiece. See if image and object distances 
 are reversible by taking an object distance equal to one 
 previously used and find the image. 
 
 (c) Find the refractive index of the liquid from each 
 of the observed values of u and v. 
 
 (d) Show that for a single refracting surface 
 
 n J_ n - l 
 
 V u r 
 
93] THE MODEL EYE 21 
 
 where n is the refractive index of the medium into which 
 the light passes relative to the medium containing the 
 object. The radius of curvature of a convex refracting 
 surface is considered positive, the radius of curvature of 
 a concave surface is considered negative. 
 
 Define conjugate points. 
 
 The first principal focal distance is defined as the value 
 of u when l/v = 0, i. e. t when the light is parallel after 
 leaving the surface. The second principal focal distance 
 is defined as the value of v for 1 / u = 0, i. e., parallel 
 light is incident on the surface. Calculate the first and 
 second principal focal distances for the surfaces or the 
 surface used, assuming the object in air. 
 
 Additional Exercise with the Refraction Trough. 
 
 Having determined the constants, (radii of curvature 
 and index) of a convex lens, place it in the refraction 
 trough in place of the watch glass and determine its focal 
 length on the water side from measurements of conjugate 
 foci. Compare this value with the calculated value. 
 
 B. Study of a Model Eye. Myopia, Hypermetropia, Astig- 
 matism. 
 
 The refraction trough with the meniscus is to represent 
 the optical system of the eye. The white screen corres- 
 ponding to the retina, and the refraction of the meniscus 
 corresponding to the total refraction of the normal eye. 
 Abnormal vision in this model eye will be produced by 
 placing a single lens close to the meniscus, while the 
 correcting glasses will be placed in the two groves 
 slightly in advance of the first one, corresponding to the 
 distance at which eye glasses are ordinarily worn. 
 
22 THE MODEL EYE [93 
 
 As a test object an incandescent lamp with a clear 
 globe or a small arc light is to be used. It should be 
 placed behind a small circular opening at a distance of 
 about two feet, with space on the table to move the 
 lamp about a foot further away from the trough. 
 
 (a) Focus the image sharply on the white screen 
 with the lamp about two feet away and consider the 
 position of the white screen thus found as the proper 
 position of the retina for the normal eye. 
 
 (6) The Myopic Eye. In the myopic eye the image 
 of a distant object seen without accommodation falls in 
 front of the retina. Either because the eyeball is too 
 long, the usual case, or because the lens is too strong. 
 Bringing the object nearer will throw the image further 
 back and hence improve vision. A person with myopic 
 eyes is called "near-sighted." 
 
 Represent the myopic eye by placing a lens marked 
 M just in front of the meniscus. Note the change in the 
 appearance of the image. Test for improvement of the 
 image by moving the object nearer to the "eye." Place 
 the object in its original position and find a correcting 
 lens which will bring the image to its former sharpness. 
 Note the power and form of this lens. 
 
 (c) The Hypermetropic Eye. In the hypermetropic 
 eye the image of a distant object seen without accommo- 
 dation falls behind the retina. Hither because the 
 eyeball is too short or the refracting system is too weak. 
 Vision will be best for objects furthest away, hence the 
 term "far-sightedness." Represent the hypermetropic 
 eye by placing a lens marked H in front of the meniscus. 
 Test for improvement of the image by moving the 
 object away from the eye. Place the object in the 
 
93] THE MODEL EYE 23 
 
 original position and find the lens which brings the 
 image to its proper focus on the "retina." Note the 
 power and form of this correcting lens. 
 
 (d) The Astigmatic Eye. In the astigmatic eye 
 the images of points are not points but lines. The 
 effect of the refracting system is that of a combination 
 of a spherical lens with a cylinder. The power of the 
 eye is greater in one plane than in a plane at right angles 
 to it. 
 
 Represent the astigmatic eye by placing the lens 
 marked A in front of the meniscus. Try to secure im- 
 provement of vision by changing the position of the 
 object; and, with the object in its initial position, by 
 the use of various correcting glasses. 
 
 What is the relation between the axis of the correcting 
 lens and the plane of the image seen without this lens? 
 
 With a +4 cylinder in front of the eye study the 
 form of the cone of light when the wave surface is 
 astigmatic. 
 
 (e) Draw diagrams showing the initial and final 
 positions of the object, eye, correcting lenses, and retina, 
 in the cases studied above. 
 
24 THE SPECTROMETER [94 
 
 94. THE SPECTROMETER. 
 
 Measuring the Angles of a Prism. 
 
 The spectrometer consists of a graduated circle, 
 generally fixed in a horizontal position, to which is at- 
 tached a collimator and a telescope. The collimator 
 consists of a tube containing an achromatic lens at one 
 end and a vertical slit at the principal focus of the lens. 
 The movable arm carries an astronomical telescope which 
 is always directed towards the center of the graduated 
 circle. The position of the telescope with reference to 
 the fixed circle may be read by means of a vernier. 
 
 Above the center of the graduated circle is a horizontal 
 table, called the table of the spectrometer, which is capa- 
 ble of rotation about the vertical axis of the circle. The 
 position of the circle table can be determined by means 
 of the vernier. 
 
 Adjustment of the Spectrometer, Using a Gauss Eyepiece. 
 
 1. The Eyepiece. The eyepiece should be moved in or out 
 until the cross-hairs are distinctly seen. 
 
 2. The Telescope. As the adjustment secured by focusing the 
 telescope on a distant object is only approximate the following 
 method of focusing the telescope for parallel light should be used. 
 Cover the objective of the telescope with a plane mirror. Place a 
 light at right angles to the axis of the telescope opposite the open- 
 ing in the Gauss eyepiece so that the cross-hairs will be illuminated 
 and the glass plate, inclined at 45 to the axis of the tube, sends 
 some light down the barrel of the telescope. Move the tube carry- 
 ing the eye-piece and cross-hairs until there is no parallax between 
 the cross-hairs and their image, formed by the light reflected from 
 the plane mirror. The cross-hairs are then in the focal plane of 
 the telescope. 
 
 Set the cross-hairs at 45 with the vertical. 
 
 3. 1*he Collimator. Illuminate the slit, turn the adjusted 
 telescope into line with the collimator and then, while looking 
 through the telescope, move the slit in or out till there is no parallax 
 between its image and the cross-hairs. The slit is then in the focal 
 plane of the collimator lens. 
 
94] ADJUSTMENT OF THE SPECTROMETER 25 
 
 This is an important adjustment since when the slit is in focus 
 the light coming from the collimator has a plane wave front, and 
 the curvature of the wave-front will not be altered by the intro- 
 duction of the prism or by any lateral displacement of the prism. 
 
 After the above adjustments have been made, if there is any 
 difficulty in seeing the cross-hairs, the eyepiece may be moved, 
 but not the cross-hairs themselves. 
 
 For work of extreme accuracy the axes of the telescope 
 and collimator must lie in one plane, and always be per- 
 pendicular to the axis about which the telescope rotates, 
 and the faces of a prism or the plane of a grating placed 
 on the spectrometer table should be parallel to this axis. 
 
 These adjustments require patient and careful manipu- 
 lation, and in the following work with the spectrometer 
 the student may assume that they have been made. 
 The methods of adjustment which have been used are 
 given below. 
 
 To adjust the Telescope perpendicular to the Axis of the 
 Spectrometer: Use a Gauss eyepiece, illuminating the cross-hairs 
 with some convenient source of light. Place a plane-parallel plate 
 of glass upon the table of the spectrometer with one edge parallel to 
 any two leveling screws (G F. See figure illustrating first method). 
 Turn the table so that the glass reflects light back down the tele- 
 scope and adjust the table until the reflected image of the cross- 
 hairs is seen in the field of view. If the cross-hairs do not coincide 
 vertically with their image correct half the distance by adjusting 
 the table, by means of leveling screw H, the other half by moving 
 the telescope. Rotate the table 180 until the reflected image is 
 again in the field. If the two images coincide for both positions of 
 the table, the telescope is perpendicular to the axis of the instru- 
 ment and the plane of the glass plate is parallel to this axis. 
 
 To adjust the Collimator perpendicular to the Axis of the 
 Spectrometer: Illuminate the slit and rotate the telescope until 
 the image of the slit is seen in the telescope, adjust the collimator 
 until the image of the cross-hairs on the collimator slit coincide with 
 the intersection of the cross-hairs in the field of the telescope. The 
 collimator is then perpendicular to the axis of the instrument. 
 
26 
 
 MEASURING THE ANGLES OF A PRISM 
 
 [94 
 
 To adjust the Faces of the Prism parallel to the Axis of the 
 Spectrometer: Set the prism on the spectrometer table with the 
 face AC at right angles to a line through F G. The inclination of 
 A C may be changed by turning either F or G. Using the Gauss eye- 
 piece as directed above adjust A C until the cross-hairs coincide 
 with their image in the field of view. 
 
 Raising or lowering H will simply move the face A C in its own 
 plane so that if F and G are left adjusted the face A C may be 
 adjusted by turning H until the reflected image from this face also 
 coincides with the cross-hairs. Adjust the face A B in this manner. 
 All three faces of the prism are then vertical, that is, parallel to the 
 axis about which the telescope (or the spectrometer table) rotates. 
 
 (a) Adjust the eyepiece, telescope and collimator of 
 the spectrometer using the methods outlined above. 
 
 Measure by either one of the following methods the 
 three angles of the prism, recording the number of the 
 prism and distinguishing the angles by their numbers or 
 letters. Make several settings for each angle. 
 
 First Method. Keeping the Prism fixed. The prism 
 is placed on the table of the spectrometer, (see figure) with 
 
 its faces A B, A C vertical 
 while the parallel beam from 
 the collimator falls partly on 
 the face A C, and partly on 
 the face A B. From each of 
 these faces a parallel beam is 
 reflected, and if either of 
 these beams falls on the ob- 
 jective of the telescope, it 
 will be brought to a focus on 
 the cross-hairs of the latter. 
 Turn the telescope so that 
 the image of the slit reflected 
 from one face coincides with 
 
 {fa^ intersection of the tWO 
 
 Fig. 11. Illustrates the adjustment 
 of the prism and the first method of 
 
 measuring the angles of a prism. 
 
 . ., , 
 
 cross-hairs and read the ver- 
 
94] 
 
 MEASURING THE ANGLES OF A PRISM 
 
 27 
 
 niers. Move the telescope until coincidence is again ob- 
 tained between the cross-hairs and the image reflected 
 from the other face and again read the verniers. 
 
 A quick method of finding the reflected image of the 
 slit is to locate the reflected beam with the eye first, 
 then on swinging the telescope in front of the eye, the slit 
 will be found in the field of view on looking through the 
 telescope. The arm should then be clamped fast, the 
 final adjustment always being made by means of the 
 slow motion screw. 
 
 (6) Show with the aid of a diagram that the difference 
 of the two readings (the angle swept out by rotation of 
 the telescope) is twice the angle of the prism formed 
 by the intersection of the two faces, provided the incident 
 light is parallel. Tabulate the values of the angles of 
 the prism given by your data. 
 
 Second Method. Keeping the Telescope fixed. Clamp 
 the telescope so that its axis makes any convenient angle 
 with the axis of the collimator. Turn the table carrying 
 the prism until the image reflected from one face of the 
 prism coincides with the intersection of the cross-hairs, 
 
 and read the vernier at- 
 tached to the spectrom- 
 eter table. Turn the table 
 until coincidence between 
 the cross-hairs and the im- 
 age of the slit as reflected 
 from another face of the 
 prism is obtained and 
 again read the vernier. 
 With this method, the 
 
 Fig. 12. Illustrates the second method error due tQ non _ paral l e l_ 
 of measuring the angles of a prism. 
 
 ism of the light is avoided. 
 
28 INDEX OF REFRACTION OF PRISM [94-95 
 
 To avoid the refracted images which are sometimes 
 present in measuring the angles of a 90 prism it may be 
 found convenient to shut off all light except that striking 
 the reflecting surface by means of a small block of wood, 
 or piece of cardboard. 
 
 (b) Show that the difference in the two readings sub- 
 tracted from 180 is the angle between the two reflecting 
 surfaces and hence the angle of the prism. 
 
 Present in tabular form the values of the angles of 
 the prism calculated in this way from your data. 
 
 95. THE SPECTROMETER. 
 Index of Refraction of a Prism. Dispersion. 
 
 Use sodium light as a source and adjust the spectrom- 
 eter as in the preceding experiment. 
 
 (a) Measure one angle of the prism using the second 
 method described above. 
 
 (6) With the prism removed, turn the telescope to 
 view the light coming directly from the collimator, and 
 adjust until the intersection of the cross-hairs coincides 
 with the image of the center of the slit. Displace the 
 telescope and make another setting and read the vernier. 
 Repeat several times and use the average of the results. 
 
 Replace the prism on the spectrometer table, in the path 
 of the light from the collimator, so that the light is re- 
 fracted through that angle of the prism which has just been 
 measured. Turn the prism so that the refracted image 
 appears to move towards the direction of the incident light, 
 continuing the motion until the image appears to stop. 
 Further turning in the same direction will now cause the 
 image to move away from the direction of the incident 
 light. Having thus roughly found the position of mini- 
 mum deviation, turn the telescope to view the image of 
 
95] DISPERSION OF GLASS PRISM 29 
 
 the slit. Again turn the prism slightly, first one way and 
 then the other, seeking to find the position of the prism 
 for which turning in either direction causes the image 
 to move away from the direction of the incident light. 
 Having apparently found this position, carefully set the 
 intersection of the cross-hairs on the center of the slit 
 image, and again slightly turn the prism first one way 
 and then the other. Any small motion of the image 
 toward less deviation can now readily be detected, the 
 cross-hairs serving as reference. Having the prism finally 
 set at the position of minimum deviation and the inter- 
 section of the cross-hairs coincident with the center of 
 the image of the slit, read the telescope vernier. 
 
 Displace the telescope and prism, set them again and 
 take a second reading. Repeat several times and take 
 the average. The difference between the minimum devi- 
 ation reading and the direct reading gives the minimum 
 deviation D. 
 
 (c) Using a Bunsen burner and a salt of Lithium, 
 determine the position of minimum deviation for another 
 wave-length. 
 
 (d) At the position of minimum deviation of a prism 
 the angle of incidence and the angle of emergence are 
 equal. From this relation and Snell's law show that 
 _ sin%(D + A) 
 
 sin ^ A where A is the angle of the prism. 
 
 Insert the values A and D previously found and cal- 
 culate the indices of refraction of the prism. 
 
 Put your results in tabular form giving the number 
 of the prism, the angle used, and for each wave-length 
 the angle of minimum deviation and the index. 
 
 The index of refraction of any transparent liquid can 
 similarly be found by using a hollow glass prism. 
 
30 PULFRICH'S REFRACTOMETER [96 
 
 96. REFRACTIVE INDEX BY TOTAL REFLECTION. 
 
 Pulfrich's Refractometer. 
 
 In Pulfrich's refractometer the light is usually received 
 at grazing incidence on the boundary surface examined*. 
 A lens throws a beam of light on the lower edge of the 
 glass cylinder containing the liquid, whose refractive index 
 is to be determined. The greater the angle of incidence 
 a ray makes with the boundary line between the liquid 
 
 Fig. 13. Pulfrich's 
 Refractometer. 
 
 and the glass the greater will be the angle of refraction. 
 For a ray just parallel to the boundary (grazing incidence) 
 the angle of refraction reaches a maximum value called 
 the "critical angle." 
 
 If now the angle i (see figure) be the angle which the 
 ray refracted at the critical angle makes with the normal 
 to the vertical face of the prism on emergence, then, since 
 
 *The more elaborate forms of Pulfrich's refractometer, which 
 are fitted with a cube or cylinder of glass in place of the 90 degree 
 prism here shown, also permit of observations by the method of 
 total internal reflection. 
 
96] PULFRICH'S REFRACTOMETER 31 
 
 none of the emerging rays can make a smaller angle, the 
 field of the observing telescope must be dark when its 
 axis makes a smaller angle than i. The angle i thus deter- 
 mined by the boundary between the bright and dark 
 regions evidently depends upon the angle, A, and refrac- 
 tive index of the glass as well as upon the refractive index of 
 the liquid. If the first two are known the refractive index 
 of any liquid (provided it is less than that of the glass 
 prism) may be found from a measurement of the angle i. 
 
 (a) Place a sodium flame about 10-15 centimeters 
 from the lens of the refractometer so that it is seen in line 
 with the cylinder on looking through the lens. Fill the 
 cylinder with water. With the slow-motion screw released 
 and starting with the axis of the telescope as nearly 
 vertical as possible raise it slowly while looking in the 
 field for a yellow band of light. When this has been 
 found tighten the slow motion screw and set the inter- 
 section of the cross-hairs on the boundary line between 
 the yellow and the black, remembering that it is the 
 upper edge of the boundary that is determined by the 
 critical angle of the liquid used. 
 
 Take several readings, moving the telescope so as to 
 approach the boundary line first from one side and then 
 from the other. 
 
 Note the temperature of the water in the cylinder. 
 
 To find the angle i which is measured from the normal 
 to the prism face a second reading should be taken before 
 the prism has been disturbed in any way. Set the tele- 
 scope near the zero of the scale, illuminate the Gauss eye- 
 piece using an incandescent lamp set at right angles to 
 the axis of the eyepiece and by means of the slow-motion 
 screw set the image of the cross-hairs (formed by light 
 reflected on the vertical face of the prism) to coincide 
 with the cross-hairs themselves, or, if this is not possible, 
 
32 PULFRICH'S REFRACTOMETER [96 
 
 place the centers of the two crosses in the same horizon- 
 tal line. The scale reading in this position subtracted 
 from the reading first taken gives the true value of the 
 angle i, since in the second case the incident and reflected 
 beams coincide and the axis of the telescope is thus 
 normal to the vertical face of the prism. 
 
 (6) Determine by the method outlined above the 
 angle i for the other liquids furnished, noting the tem- 
 perature in each case. 
 
 (c) If A be the angle of the prism, n the index of 
 refraction of the glass, and N the index of refraction 
 of the liquid above the glass and the angles i, r 2 , n, be, 
 as indicated in the figure. 
 
 From the geometry of the figure A = TI + r 2 . (1) 
 
 For refraction at the critical angle on the boundary 
 
 liquid-glass we have ~ in n ~jj~ (2) And from the 
 
 refraction on emergence a third relation sin rj =n (3) 
 
 Show from these relations that 
 
 N = sin A \/ n * - sin 1 i cos A sin i (4) 
 
 If A does not differ from 90 more than 20', a condition 
 fulfilled in this case, then sin A 1, and cos A = very 
 approximately, and the formula then becomes 
 N = \/n 2 - sin*i 
 
 Using this relation and the tabular values for the 
 refractive index of water at the temperature of your 
 observation calculate the refractive index of the glass 
 prism. 
 
 From this value of the refractive index of the prism, 
 and the values of the angle observed above calculate the 
 refractive indices of the other liquids. 
 
 Carry five significant figures throughout all calculations. 
 
97] REFRACTION THROUGH A THIN LENS 33 
 
 Thin Lenses. 
 
 A thin lens is a lens whose thickness may be neglected 
 in comparison with its focal length. In measuring to 
 the lens, the student should measure to the center of a 
 symmetrical lens and to the convex surface of a plano- 
 convex lens. 
 
 Direct Measurement of the Focal Length of Convex Lenses 
 With Parallel Light. 
 
 The following simple methods of measuring the focal 
 length of convergent lenses directly with the aid of 
 parallel light will be found convenient in checking results 
 obtained by other methods. 
 
 1. Form an image of the sun upon a screen with the lens; the 
 distance between the lens and the screen when the image is most 
 sharply defined is the focal length of the lens. 
 
 2. Mount the lens on the optical bench with a plane mirror 
 behind it. Adjust its distance from the object screen (a narrow 
 slit with a cross hair) until an image of the slit is formed on the 
 screen beside the slit. The light between the lens and mirror is 
 then parallel and the distance from the lens to the screen is the 
 focal length of the lens. 
 
 3. Focus a telescope accurately for parallel light. Place the 
 lens in front of the objective of the telescope and view through the 
 telescope some fixed mark a narrow slit or a fine point. Adjust 
 the distance between the lens and object until there is no parallax 
 in the field of the telescope. The distance from the lens to the 
 object is the focal length of the lens. 
 
 97. REFRACTION THROUGH A SINGLE THIN LENS. 
 Chromatic and Spherical Aberration and Astigmatism. 
 
 The object of this exercise is to determine the constants 
 (radii of curvature and refractive index) of a single thin 
 lens and to study the phenomena of chromatic and 
 spherical aberration and astigmatism. 
 
34 CHROMATIC ABERRATION [97 
 
 The optical bench should be provided with three slides, 
 one to support the object screen, one for the lens holder, 
 and a third for the image screen. One or more cross-hairs 
 across an aperture in the large screen or a piece of wire 
 gauze may serve as object. 
 
 Constants of the Lens. 
 
 (a) Place the object screen at one end of the bench, 
 slide the lens somewhat past the middle of the bench and, 
 leaving these two fixed, move the image screen away 
 from the lens until the image seems to be in the same place 
 as the cross-hairs in the eyepiece, tested by seeing that 
 there is no relative motion (absence of parallax) between 
 the two on moving the eye from side to side. Measure the 
 distance of the lens from each screen. Next move the 
 image screen past the position of best definition and ap- 
 proach it again, moving the screen tQward the lens. Meas- 
 ure the distance from the image screen to the lens. The 
 average of the two values obtained for the image distance 
 v with the value of the object distance u, constitute one 
 pair of values for the determination of the focal length. 
 
 Determine a second pair of values in which u and v 
 are about equal. 
 
 Take the necessary measurements with the spherom- 
 eter for the calculation of the radii of curvature of the 
 lens surfaces. 
 
 Chromatic Aberration. 
 
 (6) Since the focal length of a lens depends upon its 
 index of refraction, which varies for different wave-lengths, 
 it is evident that the value of / determined in (a) will be 
 an average value for the brighter portion of the spectrum 
 of the light used. In order to determine / for two fairly 
 definite regions of the spectrum, first put in front of the 
 
97] SPHERICAL ABERRATION 35 
 
 object a piece of red glass, and determine a pair of con- 
 jugate foci. Then replacing the red glass with a piece 
 of blue glass, determine a second pair of conjugate foci 
 for blue light. As neither of the colors are absolutely 
 pure it will not be possible to eliminate parallax entirely. 
 
 Spherical Aberration. 
 
 (c) In order to avoid confusing chromatic abberation 
 with spherical aberration, observations must be taken 
 with light which is fairly monochromatic. This is best 
 secured by using the red glass as it transmits a fairly 
 narrow region of the spectrum. 
 
 Cover the lens with a diaphragm having a one centi- 
 meter central .opening and after locating the image 
 measure the conjugate foci. 
 
 Make a second determination of the image after the 
 diaphragm has been replaced by one which covers the 
 center of the lens and lets light pass only through the 
 outer portion. 
 
 Astigmatism. 
 
 (d) With the red glass still in place and using the 
 diaphragm with central opening rotate the lensholder in 
 its slide 45 degrees about a vertical axis. This will make 
 the wave-front emerging from the lens astigmatic and the 
 definite focus previously obtained will disappear. Bach 
 point of the object will now have two line images but 
 no point image as before. Since the object consists of 
 vertical and horizontal lines there will be one position in 
 which the vertical lines are in focus and another in which 
 the horizontal lines are in focus. 
 
3t> THICK LENSES [98 
 
 L/ocate both of these positions for some particular 
 value of u. 
 
 (e) From the data taken in (a) and (b) calculate the 
 radii of curvature of the lens surfaces, its focal length, 
 and refractive index for red, blue and white light. 
 
 Defining the difference between the focal length for 
 red light and the focal length for blue light divided by 
 the mean focal length as a measure of the chromatic 
 aberration, calculate its value from your data. 
 
 From the data taken in (c) determine the difference 
 between the focal length for the center of the lens and 
 for the edge divided by the mean focal length as a measure 
 of the spherical aberration. 
 
 Illustrate with diagrams, chromatic and spherical 
 aberration and astigmatism. (See figure 8.) 
 
 98. THICK LENSES. 
 Principal Points, Nodal Points and Focal Length. 
 
 The conjugate focal relation l/u + l/v = I//, in which 
 u, v, and / are measured .from the center of the lens, 
 holds only so long as the thickness of the lens may be 
 neglected in comparison with these quantities. For a 
 thick lens, it is possible to select two points on the axis 
 of the lens, so situated that if the object distance U be 
 measured from the one of them, and the image distance V 
 be measured from the other, we obtain* a similar conju- 
 gate focal relation, namely I/ U + l/V = l/F. 
 
 These two axial points are called the "principal" 
 points of the thick lens. Planes drawn perpendicular to 
 
 *Proof of this proposition is given in Edser's Light for Students, 
 Chapter VII. 
 
98] 
 
 PRINCIPAL POINTS AND NODAL POINTS 
 
 37 
 
 the axis through the principal points are called the 
 principal planes of the lens. It may be shown that the 
 principal planes are planes of unit magnification. As a 
 consequence any ray directed towards a point n in the 
 first principal plane, at a distance x from the axis, will 
 give rise to a transmitted ray proceeding from a point o 
 in the second principal plane, on the same side of the 
 axis at a distance x from it. 
 
 The graphic determination of images follows readily 
 
 from the properties of the principal planes (See Figure 14). 
 
 O 
 
 Fig. 14. Showing the graphic determination of the image / of an object using 
 the principal planes PI, Pj. 
 
 There are two other points on the axis of a thick lens 
 which possess important properties. A ray of light 
 directed toward the first of these points on the axis will 
 after refraction by the lens proceed from the second 
 point on the axis in a direction parallel to that of the 
 incident ray. These points are termed the first and 
 second nodal points respectively. When image and 
 object are situated in the same media, for instance a 
 glass lens is used in air, the principal points and the 
 nodal points coincide. 
 
 The two principal foci, the two principal points, and the 
 two nodal points are the six cardinal points of a thick lens. 
 
 The characteristic of the nodal points may be made 
 the basis of their determination. If we have a ray ac 
 incident through the first nodal point Ni from a distant 
 
38 
 
 PRINCIPAL POINTS AND NODAL POINTS 
 
 [98 
 
 object, it will emerge parallel through N- 2 and cut the 
 screen at C. C will be the image of the point of the 
 object from which ac came. As the object is supposed 
 to be at a great distance its image will be formed in the 
 focal plane. 
 
 Now suppose the lens system rotated about the second 
 nodal point N 2 , as ac is coming from a great distance, 
 the ray from the same point of the object aci incident 
 through N! will still be practically parallel to ac. Thus 
 the ray that emerges from N 2 has not moved, and the 
 image at C will remain at rest. Any movement of N z , 
 however, will cause C to move also. (See Figure 15a). 
 
 C 
 
 Fig. 150 156 15C 
 
 Finding the nodal points and focal length of a lens system 
 
 Thus to find the second nodal point N 2 we have to 
 find the point about which rotation of the lens system 
 produces no movement of the image of a distant object. 
 This is easily accomplished experimentally because if 
 the second nodal point N 2 be in front of the axis of rota- 
 tion 0, i. e., too near the screen (Figure 156), then a 
 small counter-clockwise rotation will carry N 2 to A/V 
 and therefore C to C'. If, on the contrary, the second 
 nodal point N 2 is behind the axis, i. e., too far from the 
 screen (Figure 15c), a similar counter-clockwise rotation 
 will take it to A/V' and the image C moves to C" . 
 
 (a) Place a piano convex lens in the nodal slide at 
 a distance approximately equal to the focal length of 
 the lens, and adjust the plane mirror back of it so that 
 the light from the slide rendered parallel by the lens will 
 
98] PRINCIPAL POINTS AND NODAL POINTS 39 
 
 be reflected backwards and form an image of the opening 
 on the white screen in the plane of the slit itself. Using 
 the method outlined above, locate the second nodal 
 point of the lens and measure ' the distance from the 
 center of rotation to the screen. This is the focal length 
 F. Note the position of the lens holder on the nodal slide. 
 
 Rotate the slide 180, and find the other nodal point, 
 measuring the focal length and noting again the position 
 of the lens holder on the slide. 
 
 Measure the thickness of the lens with the calipers. 
 
 . (b) Determine the nodal points and focal length of 
 the combination of a convex and a concave lens separated 
 two or three centimeters. Measure the thickness of 
 each lens used and also the distance apart of the inner 
 surface of the two lenses. 
 
 (c) From the data taken in (a) find the distance 
 between the nodal points expressed as a decimal part of 
 the thickness of the lens. Draw an enlarged diagram 
 of the lens, and on the axis of the lens show the position 
 of the lens surfaces and the two nodal points. 
 
 Show in a diagram drawn to scale the results obtained 
 in (b). 
 
40 RESOLVING POWER OF TELESCOPE [99 
 
 99. ASTRONOMICAL TELESCOPE. 
 MAGNIFYING POWER. RESOLVING POWER. 
 
 Magnifying Power of Telescope. 
 Reference. Duff, p. 615; Kimball, p. 653. 
 
 The magnification produced by a telescope, when 
 focussed on a distant object and adjusted to suit the 
 unaccomodated eye, is equal to the focal length of the 
 objective divided by the focal length of the eyepiece. 
 
 (a) Remove the objective and determine its focal 
 length by any method you please. (No. 2, page 33, 
 for instance.) 
 
 (b) To determine the focal length of the eyepiece, 
 first put a micrometer eyepiece on the telescope in place 
 of the objective. Using the eyepiece whose focal length 
 is to be determined as an objective form a magnified 
 image of a standard 0.2 mm. glass scale in the field of the 
 micrometer eyepiece. Measure with the micrometer the 
 dimensions of this image. 
 
 Now extend the draw tube a measured distance and 
 again measure the image formed. 
 
 Resolving Power of Telescope. 
 
 The action of a lens, S, in forming a real image, /, 
 of a point source, O, consists in transforming the spherical 
 waves, such as a b c, diverging from O into spherical 
 waves, such as d e f, converging toward /. From geo- 
 metrical considerations alone we might conclude that the 
 image at I is a point, but the wave theory shows that 
 this cannot be so. 
 
99] 
 
 RESOLVING POWER OF TELESCOPE 
 
 41 
 
 To determine the effect at / of the wave, d e f, let us 
 apply Huygens principle and consider the resultant 
 effect as due to wavelets with their centers in the surface 
 d e f. These wavelets will all reach / in the same phase, 
 since it is equidistant from all of them, hence 7 must 
 be a point of maximum brightness. At points outside 
 
 of I differences of phase will exist, and there must be 
 some point, H, which is on the average a half wave 
 length farther off from the upper half of the surface d e f 
 than it is from the lower half. At that point waves 
 from one half of the surface will interfere with those 
 from the other half and produce complete darkness. 
 But between / and H the interference is only partial 
 and consequently the light intensity must fade off from 
 J to H. Since the light is symmetrical about e I, there 
 must be a little spot of light formed in the image plane 
 having the distance I H as its radius. An optical image 
 is thus essentially a diffraction figure. The height of the 
 curve in Fig. 16(6), is proportional to the intensity of the 
 light in the image plane. It is evident that if we decrease 
 the aperture of the lens the corresponding point H 1 will 
 be farther from / than H, and the intensity curve will 
 be broader, as in Fig. 17 (a). And conversely increasing 
 the aperture will steepen the curve. Consequently if 
 a lens forms an image of two small objects, while the 
 central maximum of each image is fixed by. the .distance 
 
42 
 
 RESOLVING POWER OF TELESCOPE 
 
 [99 
 
 between the objects and their distance from the lens, 
 the amount of overlapping of the intensity curves will 
 then depend upon the aperture only. 
 
 If we use a rectangular aperture with the lens, the 
 conditions for the formation of maxima and minima will 
 correspond to those of a narrow slit. The "limit of 
 resolution" is reached when the first maximum of the 
 image of one object coincides with the first minimum 
 
 Fig. 17(c) 
 Intensity Curve, Narrow Slit 
 
 Fig. 17(6) 
 Limit of Resolution. 
 
 of the other as in Fig. 17(6). From this it may be shown 
 that if d is the distance between two details of an object 
 at a distance D from an aperture whose width parallel 
 to these details is a, they may be distinguished if 
 
 where X represents the wave-length of the light used. 
 This ratio is called the "resolving power" of the aperture. 
 
 (c) Illuminate the metal gauze with sodium light 
 and observe it through the telescope, which should have 
 an adjustable slit placed just before its objective. 
 
 Adjust the width of the slit until the wires parallel with 
 its length may be just distinguished and then measure the 
 
99] RESOLVING POWER OF TELESCOPE 43 
 
 width of the slit a with the comparator. The distance 
 D will be measured with sufficient accuracy with the meter 
 rod. For the wave-length, X, consult the tables. The 
 distance d between the centers of the adjacent wires of 
 the gauze may be conveniently determined by measuring 
 the distance occupied by a known number of wires. 
 
 (d) Letting Wi = Vi/Ui be the magnification of the 
 first image measured in (b), m 2 = v 2 /u 2 the magnification 
 of the second image and d = v 2 -Vi be the extension of 
 the draw tube prove that the focal length of the eyepiece 
 / = d/(m z - Wi). 
 
 Using your data in (a) and (6) calculate the magni- 
 fying power of the telescope. 
 
 Calculate the ratio d/D and compare it with the ratio 
 
 X/a. 
 
 ._- 
 
 Explain, using a diagram, the formula given above for 
 the resolving power. 
 
44 THE SPECTROSCOPE [100 
 
 100. THE SPECTROSCOPE. 
 Drawing Spectra. Calibration of Spectroscope. 
 
 The direct vision spectroscope consist of two tubes, H, 
 r, the first contains a slit 5, adjustable by means of a 
 
 H 
 
 \L 
 
 Fig. 18. Direct vision spectroscope. 
 
 knurled ring K, and protected by means of a glass disk g, 
 placed just in front of it; the second tube movable within 
 the first, carries a short focus lens L, and the direct vision 
 prism P. When adjusted for the unaccomodated eye the 
 lens renders the light from the slit parallel before it enters 
 the prism. 
 
 A small tube, attached at right angles to the axis of the 
 movable tube contains a scale s, a lens system /, which 
 may be moved along the tube, and an adjustable mirror M, 
 for illuminating the scale. The light from the scale, having 
 been rendered parallel by the lens, is reflected on the 
 slanting end of the prism and reaches the eye at E, with 
 the light which has gone through the prism. To adjust 
 the spectroscope, the lens system /, in front of the scale, 
 should be moved by means of the pin d, until the scale is 
 clearly seen and then the tube containing the prism should 
 be moved in or out until the spectrum is sharply defined 
 and is seen, without parallax, partly overlapping the 
 image of the scale. 
 
100] CALIBRATION OF SPECTROSCOPE 45 
 
 The position of lines of a particular wave-length on the 
 spectroscope scale depends not only upon the scale, 
 which is arbitrary, but also upon the dispersion of the 
 glass prism used. Scale readings of a spectroscope must 
 accordingly be put into wave-lengths for purposes of 
 comparison with the values obtained by other methods. 
 Data for a calibration curve consists of a series of scale 
 readings for lines of known wave-length. These should 
 be distributed with some uniformity over the spectrum- 
 Suitable lines may be found in the spectrum of sun- 
 light the "Fraunhofer lines" or in the flame spectra 
 of the light metals. If both sources are available the 
 calibration curve obtained using one may be used to 
 determine the wave-length of the lines in the spectrum 
 of the other. 
 
 To examine the flame spectra of the light metals. 
 
 (a) Color the Bunsen flame with the salts furnished 
 and examine their spectra. 
 
 Note in each case the general color of the flame and 
 the colors of the various bands and lines. 
 
 To observe successfully the potassium spectrum it will 
 be necessary to open the slit somewhat and insert a piece 
 of cobalt glass between the flame and slit. Remove the 
 light illuminating the slit until the line is seen, then 
 replace the light and take the scale reading. 
 
 The sodium spectrum will probably be ever present, 
 but it is readily distinguished from that of the salt under 
 examination. Protect the eye from side light, especially 
 when locating the fainter lines. 
 
 To examine the solar spectrum with the spectroscope. 
 
 (6) Reflect sunlight so that it falls on the slit, nar- 
 rowing the latter to the width of a fine hair. If the 
 
46 ABSORPTION SPECTRA [100 
 
 4aBC D Eb c F d eGg h HK 
 
 Red, yellow, green, blue. indigo, violet. 
 
 Fig. 19. The Fraunhofer lines. 
 
 instrument has been properly focused a continuous 
 spectrum crossed by fine vertical dark lines "Fraun- 
 hofer's lines" will be visible. 
 
 Note the position and color of the principal lines, 
 identify them with the aid of the figure. Some direct 
 vision prisms have a smaller dispersion in the red and 
 much larger dispersion in the extreme violet than here 
 indicated. 
 
 Absorption Spectra. 
 
 (c) Draw the spectrum of a luminous flame seen 
 through red, yellow, green, and blue glass. 
 
 (d) Plot a curve with the scale readings taken in (6) 
 as ordinates, and the corresponding wave-lengths given 
 in the tables as abscissae. From this plot determine the 
 wave-length of the sharpest lines observed in (a). 
 
 Draw the spectra, observed in (c), upon plotting paper, 
 the spectroscope scale being taken as abscissae and each 
 spectrum occupying a separate horizontal strip about a 
 centimeter high. 
 
101] SPARK SPECTRA OF METALS 47 
 
 101. SPARK SPECTRA OF METALS. 
 
 The spectroscope which has previously been adjusted 
 and calibrated should be used for this exercise. 
 
 The spark spectra of various metals are to be observed 
 and the wave-length of the lines to be determined from 
 their position on the scale as calibrated. 
 
 (a) Connect the two electrodes of the metal with 
 the secondary of the induction coil. The storage battery 
 (6-8 volts) should be connected to the primary of the 
 coil. The spark gap should be horizontal and about 
 1 millimeter in width. Start the coil and observe the 
 spark directly. Describe the change in the appearance 
 of the spark when a Ley den jar is connected in parallel 
 with the secondary of the coil. 
 
 (b) Leave the Ley den jar connected as in (a). Set 
 the spark gap in front of the slit of the spectroscope and 
 adjust the height of the spark so that the spectrum is 
 seen in the telescope above the illuminated scale. Locate 
 the lines on the scale, tabulating the readings as follows: 
 metal, color, scale reading and wave-length, as read from 
 the calibration curve. After color place a figure 1, 2 or 
 3, to designate the intensity. The brightest lines being 1. 
 
 (c) Introduce self-induction into the circuit by con- 
 necting the coil furnished in series with the spark gap. 
 Note the change in the spectrum. By comparing the 
 present spectrum with the readings of (b) make a list 
 of the lines which ha,ve entirely disappeared. A con- 
 venient method of making the comparison is to use a 
 commutator irt making the connections, then by reversing 
 the commutator the self-induction may be thrown in 
 or out. 
 
 (d) Repeat (b) and(c) with each of the other metals 
 furnished. To what substance do the lines belong which 
 diasppear when self-induction is put into the circuit? 
 
48 
 
 NEWTON'S RINGS 
 
 [102 
 
 102. WAVE-LENGTH OF SODIUM LIGHT. 
 Newton's Rings. 
 
 If a plano-convex lens be placed upon a plane piece of 
 glass, a thin air film will be formed between them. Near 
 the point of contact the thickness of the film will be small 
 compared with the wave-length of light, consequently 
 there will be a circular black spot around the point of 
 contact when this is viewed by reflected light. The air 
 film increases in thickness from the point of contact 
 outwards in every direction, and, since the lower surface 
 of the lens is spherical, points of equal thickness form 
 concentric circles with the point of contact as the center. 
 
 Using monochromatic light to illuminate the lens and 
 plate, the central black spot will be seen, by reflected 
 light, surrounded by concentric bright circles separated 
 by dark intervals. 
 
 Interference* has taken 
 place between the light re- 
 flected from the lower sur- 
 face of the lens and that 
 reflected from the glass plate, 
 the optical path of the latter 
 being greater, by twice the 
 thickness of the air film, 
 than that of the first. 
 
 (a) Use sodium light as 
 a source, placing the burner 
 at the focus of the lens. The 
 plate glass (G) should be set 
 so as to reflect the light downward. (See fig.) If a broad 
 
 Fig. 18. 
 
 Arrangement of the apparatus 
 for observing Newton's Rings. 
 
 "This is a pure interference phenomenon, the difference of phase 
 necessary for the formation of the rings being introduced without 
 violation of any of the laws of geometrical optics. 
 
102] NEWTON'S RINGS 49 
 
 flame is used as a source, the lens L, may be dispensed 
 with. Remove the microscope from its holder and focus 
 the eyepiece on the cross-hairs. Adjust the plate glass 
 (G) so that the light reflected downwards, returns to the 
 eye after reflection on the lens and plate at (P). Find 
 the rings, move the microscope holder to get them 
 roughly in the center of the opening, and then insert the 
 microscope, focusing carefully with the aid of the cross- 
 hairs. In the proper position there will be no relative 
 motion of the rings with reference to the cross-hairs on 
 moving the source of light. 
 
 Take care to prevent lost motion (backlash) by always 
 turning the screw in the same direction when making a 
 setting. 
 
 Measure the diameter of several rings near the cen- 
 ter, then skip to the tenth, or beyond, and measure sev- 
 eral more, finding the difference in diameter as accurately 
 as possible by repeated measurement of successive rings 
 on one side, then move to the opposite side and repeat 
 the measurements. The lens should not be disturbed 
 during the entire series of observations. 
 
 (b) Show with the aid of a drawing that if r is the 
 radius of curvature of the lens surface, Di, D 2 , D 3 , D 4 , 
 Dn, the diameters of successive bright rings, counting 
 from the center and X the wave-length of the light used 
 
 2r(2n-l] 
 
 For the (n-f AOth ring, X = - (2) 
 
 Solving for the numerators in (1) and (2) and taking 
 their difference we have 
 
50 THE GRATING CONSTANT [102-103 
 
 This final form gives us an expression for the wave- 
 length which is independent of the actual thickness of 
 the air film at the point of contact. It also shows that the 
 results will be mare accurate the farther apart the rings 
 are whose diameters are used. 
 
 In what two ways is the difference of phase between 
 the interfering rays produced? 
 
 (c) Suggest a modification of the experiment by which 
 the center would appear bright instead of dark by reflected 
 light. 
 
 (d) From the known value of r and your data, calcu- 
 late the value of the wave-length of sodium light. 
 
 Find the mean error in per cent in your result. 
 
 Using the average value for the wave-length obtained 
 in (d) calculate the actual thickness of the air film having 
 the same diameter as the third bright ring, assuming the 
 lens and plate in actual contact. 
 
 103. THE DIFFRACTION GRATING. 
 
 The Grating Constant. 
 Reference Edser, Light for Students, p. 452. 
 
 A monochromatic light is required. For this purpose 
 use a Bunsen burner specially arranged to give a strong 
 flat sodium flame. Set in front of this the screen with its 
 plane vertical and the slits horizontal. In use, the light 
 is allowed to shine through one slit on each side of the 
 central vertical line of the screen. 
 
 The grating is mounted in a slide on the optical bench, 
 with the ruled surface toward the screen and the lines 
 parallel to the slits. On looking at the slits through the 
 grating, a series of diffracted images of each slit is seen. 
 The two sets of images move relatively to each other as 
 
103] 
 
 THE GRATING CONSTANT 
 
 51 
 
 the grating is moved back and forth. A position of the 
 grating may be found for which one of the images in one 
 set appears to form a continuous line with one in the 
 other set. If the lines do not quite meet, it is evident that 
 the lines of the grating are not quite parallel to the slits. 
 
 (a) Using the two slits which are closer together, set 
 the grating so that the first image of one slit coincides in 
 the way above indicated, with the first image of the other 
 slit, at the point half-way between the two slits. Make a 
 number of settings and for the mean of these settings 
 determine the distance of the ruled surface of the grating 
 from the screen. 
 
 (b) Repeat (a), using the second diffracted image of 
 each slit, instead of the first. 
 
 (c) Repeat (a) and (6) for the pair of slits which are 
 wider apart. 
 
 (d) Measure the distance between the slits. Deter- 
 mine the mean distance from measurements on the edges 
 taking a number of readings for each distance. 
 
 (e) Explain the formation of both the second and 
 fifth diffracted images in the final position .secured in (b) 
 with the aid of a drawing, lettered like the figure given. 
 Are the images real or virtual? Derive an expression for 
 
 Fig. 19. Determination of the Grating Constant using two slits. 
 
52 MEASUREMENT OF WAVE-LENGTH [103-104 
 
 the distance apart of the lines of the grating the grat- 
 ing constant, in terms of the wave-length of the light 
 used, the distance from the screen to the grating and the 
 distance between the slits. 
 
 (/) Assuming the wave-length of sodium light to be 
 0.00005893 cm., calculate the grating constant. 
 
 104. MEASUREMENT OF WAVE-LENGTH. 
 With Spectrometer and Grating. 
 
 The telescope and collimator should be focused for 
 parallel light. (See directions Bxp. 94.) 
 
 Mount the grating on the table of the spectrometer, 
 with the lines vertical. 
 
 To set the grating perpendicular to the collimator 
 clamp the telescope at an angle of 90 with the collimator, 
 then, loosening the table rotate it until the slit is seen 
 by light reflected from the face of the grating. Noting 
 this position, move the table through 45 and the grating 
 will be perpendicular to the collimator. 
 
 In case the mounting of the grating does not permit 
 the use of the 'above method, find, by trial, a position of 
 the grating such that the deviation for the spectrum of 
 the first order is the same on both sides, within half a 
 degree. This may be done without reading the vernier, 
 the accurate determinations of the angle being made later. 
 
 (ai) Work this or the following section. Determine 
 the wave-length of the principal lines in the spectrum* 
 of the salts furnished. Take observations both to the 
 right and the left for as many orders as permit of accurate 
 measurements. Arrange the data in tabular form, giving 
 
 *The most satisfactory method of securing a spectrum of great 
 intensity is to use a carbon arc, fed with salts, a separate pair of 
 carbons being used for each salt. 
 
104-105] REFLECTION ON TRANSPARENT SUBSTANCES 53 
 
 for each metal the color of the lines and the average 
 values for the wave-length. 
 
 (a 2 ) Determine the wave-length of the two ")" lines 
 of the sodium flame. 
 
 (b) Show with the aid of a diagram, the relation which 
 exists between the wave-length of the light used, the 
 grating constant, and the angular distance of the spectra 
 of the first, second, and third orders, measured from 
 the image of the slit seen direct. 
 
 The constant of the grating used will be furnished. 
 
 105. REFLECTION OF LIGHT ON TRANSPARENT 
 SUBSTANCES. 
 
 As a source of light use the flame of a candle, or better, 
 a sodium flame, if available, against a dark background. 
 The reflecting surface is glass, the back surface of which 
 has been blackened; an over-exposed photographic plate 
 works well when black glass is not to be had. 
 
 I. To examine the light coming direct from the flame. 
 
 (a) Place the flame between the two reflectors with 
 the flame at the same height as the axis of rotation of 
 the plate. Find the image of the flame in the glass; 
 rotate the glass about the incident light as an axis. Do 
 you note any change in the intensity of the flame? 
 Change the angle of incidence by shifting the flame. If 
 you detect any change in the intensity of the flame as 
 the reflecting surface is rotated to what may it be due? 
 
 Diagram the apparatus as arranged and make a state- 
 ment covering your observation. 
 
 II. To examine the light from the flame after one re- 
 flection on glass. 
 
 (b) Use two reflecting surfaces having the second 
 movable about the incident ray. Place the flame so that 
 
54 REFLECTION ON TRANSPARENT SUBSTANCES [105 
 
 it may be seen in surface 2, the light being first reflected 
 on surface 1. Examine the image as surface 2 is rotated. 
 
 Diagram the apparatus as now arranged. 
 
 In case you observe no appreciable change in either 
 (a) or (6) repeat both with different angles of incidence. 
 
 If you note a change in the intensity of the flame, try 
 to increase this change by varying successively the angle 
 of incidence of the light on the first surface and then by 
 rotating the second surface. In case repeated trials fail 
 to show any difference, rest the eyes for a few minutes 
 and avoid looking at any bright light while making a 
 new trial; do not look at the flame except by reflection. 
 Ask for assistance in case your efforts are not successful. 
 If you find that light once reflected behaves differently 
 from light coming directly from the source, make the 
 two planes of incidence coincide and describe the change 
 in the appearance of the flame as the second surface is 
 rotated from this initial position. 
 
 (c) Diagram the apparatus as arranged to show a 
 maximum of the observed effect, indicating the planes of 
 incidence of the light on the two 'surfaces. Formulate 
 the results of your observation so as to cover the follow- 
 ing points. 
 
 1. This phenomenon has been called the "polarization" 
 of light by reflection. From this experiment alone what 
 meaning would be attached to the word "polarized" in 
 the phrase "the light reflected from the first piece of glass 
 is polarized?" To what conception of the effect of 
 reflection upon the incident light do we owe this word? 
 
 2. How many times does the change observed occur 
 in one complete revolution of the second plate? 
 
 3. State a position of surface 2 with reference to sur- 
 face 1, such that turning an equal number of degrees to 
 
105] 
 
 BREWSTER'S LAW 
 
 55 
 
 the right or to the left produces the same effect. The 
 beam is evidently symmetrical about such a position. 
 
 Planes of symmetry are often taken as reference planes. 
 Find a second plane of symmetry stating the angle which 
 it makes with the first. The plane of symmetry (with 
 reference to the changes in intensity} in which the light is 
 most plentifully reflected is called the plane of polarization. 
 
 4. Would the phenomenon observed be possible if 
 light were wholly a longitudinal disturbance of the ether? 
 
 How much does this experiment 
 tell you about the direction of the 
 displacement in the wave front? 
 
 Suggestions for Further Work. 
 
 Brewster's Law. Set the two 
 
 plates for complete extinction of the 
 light and then with a couple of 
 meter rods determine the angle of 
 incidence for the light striking the 
 first plate of glass. 
 
 State the relation which Brewster 
 showed exists between the angle of 
 incidence and the refractive index 
 of the substance. 
 
 Nicol's Polarizing Prism. Ex- 
 amine the model of a Nicol prism and 
 describe its construction briefly, 
 using a diagram. 
 
 Determine the .plane of polariza- 
 tion of a Nicol prism mounted in a 
 brass tube, stating your result with 
 reference to the angles or diagonals 
 on the end face. 
 
 Scctfon of Nicol's Prism 
 
 Figs. 20. 21. Nicol's Prism 
 
56 PHYSICAL TABLES 
 
 DENSITIES OF SOLIDS. 
 
 Solid Density Solid Density 
 
 gms. per cc. gms. per cc. 
 
 Aluminum 2.70 Iron, cast 7.4 
 
 Brass, cast 8.44 Iron, wrought... 7.8 
 
 Brass, drawn 8.70 Lead 11.3 
 
 Copper 8.92 Nickel._ 8.90 
 
 German-silver 8.62 Platinum 21.50 
 
 Glass, crown 2.6 Rubber, hard... 1.10-1.27 
 
 Glass, flint 3.9 Silver 10.53 
 
 Gold 19.3 Steel .. 7.8 
 
 STANDARDS OF PITCH. 
 
 The French Standard, "Diapason Normal" of 1859 (which adopts 
 a fork having c"=522 at 20 C.) is -coming into general adoption 
 for organs and pianos in England, the Continent, and America, as 
 the result of a makers' conference in 1899. Other scales in vogue 
 are Concert Pitch (c"=546), Society of Arts (c"=528), Tonic 
 Sol-fa (c"=507), Philharmonic (c"=540). (The "middle" c of the 
 piano is c'). 
 
 ELASTIC CONSTANTS OF SOLIDS. 
 (Approximate Values.) 
 
 In C. G. S. Units. 
 
 Substance Bulk- Simple Young's 
 
 Modulus Rigidity Modulus 
 
 Aluminum 5.5 x 10" 2.5 x 10 11 . 6.5 x 10 11 
 
 Brass, drawn 10.8 x " 3.7 x " 10.8 x " 
 
 Copper 16.8 x " 4.5 x " 12.3 x " 
 
 German Silver 4.5 x " 12.8 x " 
 
 Glass 2.4 x " 7.0 x " 
 
 Iron, wrought 14.6 x " 7.7 x " 19.6 x " 
 
 Steel 18.4 x " 8.2 x " 21.4 x " 
 
PHYSICAL TABLES 57 
 
 VELOCITY OF SOUND IN METERS PER SECOND. 
 
 Aluminum 5100 
 
 Brass 3200 to 3600 
 
 Copper.... ....3500 to 3900 
 
 Glass, ...5000 to 5900 
 
 Iron..... .5000 to 5100 
 
 Nickel 4970 
 
 Steel.... ....4900 to 5000 
 
 Gases at C. 
 
 Air 332 
 
 Carbon dioxide.., 261 
 
 Chlorine 206 
 
 Illuminating Gas 490 
 
 Hydrogen.... 1285 
 
 Oxygen.... .. 317 
 
 PRINCIPAL LINES IN SPARK-SPECTRA OF 
 
 Cadmium Zinc Tin Lead Air Copper 
 
 3982 4680 4525 4058 4631 4023 
 
 4413 4722 5563 4247 5003 4063 
 
 4678 4811 5589 4387 5006 4275 
 
 4800 4912 5799 5373 5679 4378 
 
 5059 4925 6453 5608 5933 4481 
 
 5338 6103 6657 5943 4587 
 
 5379 6362 6563 5106 
 
 6438 5153 
 
 5218 
 5700 
 5782 
 
 The ; colors of the spectrum are approximately: violet 3600 to 
 4240, blue 4240 to 4920; green 4920 to 5350; yellow 5350 to 5860; 
 orange 5860 to 6470; red 6470 to 8100. 
 
58 PHYSICAL TABLES 
 
 INDEX OF REFRACTION OF VARIOUS SUBSTANCES. 
 
 For Sodium Light D Line, X = 5893xlO- 8 cm. 
 
 Air, Dry . ....1.0002945 
 
 " 20 1.0002773 
 
 Alcohol 1.3616 
 
 Benzene 1.5005 
 
 Calcite, ordinary ray 1.6585 
 
 " extraordinary ray._ 1.4864 
 
 Canada Balsam, hard 1.54 
 
 Carbon Bisulphide, 20 1.62772 
 
 Cassia Oil, 17.5 1.6053 
 
 " " Light... 1.5153 
 
 Glass, crown 1.465 to 1.6112 
 
 " flint 1.515 to 1.75 
 
 densest flint '. 1.9626 
 
 Glycerine 1.47 
 
 Cornea, aqueous and vitreous humors of eye. 1.3365 
 
 Crystalline lens 1.4371 
 
 a monobromnaphthelene 15 ^ 1.6603 
 
 20 _.. 1.6581 
 
 Methyl Iodide, 15._ 1.7429 
 
 " 20 1.7419 
 
 Quartz, ordinary ray 1.5442 
 
 " extraordinary ray.... 1.5533 
 
 Rock Salt 1.5443 
 
 Olive Oil- 1.47 
 
 Flourspar 1.43085 
 
 Diamond : 2.4173 
 
 Water, 15 1.33339 
 
 " 20 _,. 1.33300 
 
 22 1.33280 
 
 " 25 1.33253 
 
 30 1.33194 
 
 Ice, ordinary ray 1.3087 
 
 " extraordinary ray 1.3119 
 
 Turpentine 1.47 
 
PHYSICAL TABLES 59 
 
 WAVE-LENGTHS OF THE PRINCIPAL FRAUNHOFFER 
 LINES IN AENGSTROEM'S UNITS (ten millionth mm.) 
 (In air at 20C and 760 mm.) 
 
 Line 
 
 Element Line 
 
 K 3934 Ca 
 
 H 3968 Ca 
 
 h 4102 H, In 
 
 g 4227 Ca 
 
 G 4308 Ca, Fe 
 
 e 4384 Fe 
 
 d. 4646 Fe 
 
 F.... ....4862.... ...H 
 
 Element 
 
 .5173. 
 
 -Mg 
 
 b t 5184 Mg 
 
 E. 5270 Fe 
 
 D 2 ._ 5889.965....Na 
 
 D!.. 5895.932....Na 
 
 D. average 5893 Na 
 
 C. 6563 H 
 
 B. 6870 O 
 
 a 7202 Atm 
 
 A._. ....7608 O 
 
 LINES OF THE FLAME-SPECTRA OF SEVERAL METALS. 
 
 In the following table * denotes an average for lines near to- 
 gether, or the center of a band; e the edge of a band nearest the 
 D line; m that the line belongs to the metal itself. 
 
 The lines most useful in identifying the substance are printed 
 in bold face type; those suitable for the calibration of a spectro- 
 scope scale are marked c. 
 
 K 
 
 Na 
 
 4046mc 
 
 5893 *mc 
 
 Faint con- 
 
 
 tinuous 
 
 Li 
 
 spectrum 
 
 6102m 
 
 in the 
 
 6708mc 
 
 blue and 
 
 
 green 
 
 Tl 
 
 
 5351mc 
 
 7680 *mc 
 
 Ca 
 
 4227mc 
 
 5530* 
 
 5728* 
 
 5817 
 
 5933 
 
 6055* 
 
 6191 
 
 6265* 
 
 6441 
 
 Sr 
 
 4607mc 
 6045* 
 
 6233* 
 6350* 
 6464* 
 6597* 
 6694* 
 6827* 
 
 Ba 
 
 4873* 
 
 5089e 
 
 5215e 
 
 5346e 
 
 5492e 
 
 5536m 
 
 5661* 
 
 5719* 
 
 5881* 
 
 6044* 
 
 6297* 
 
 Cs 
 
 4560mc 
 
 4597mc 
 
 6007m 
 
 6219m 
 
 Rb 
 
 4202mc 
 4216mc 
 6207m 
 6297m 
 7811 me 
 
 *Spectres Lumineux, Lecoq de Boisbaudran. 
 
60 
 
 NATURAL SINES. 
 
 
 0' 
 
 6' 
 
 12' 
 
 18 
 
 24 
 
 30' 
 
 36 
 
 42 
 
 48' 
 
 54' 
 
 123 
 
 4 5 
 
 
 
 0000 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 oi57 
 
 369 
 
 12 15 
 
 1 
 
 2 
 3 
 
 0175 
 
 0349 
 
 0523 
 
 0192 
 0366 
 0541 
 
 0209 
 0384 
 0558 
 
 0227 
 0401 
 0576 
 
 0244 
 0419 
 0593 
 
 0262 
 0436 
 0610 
 
 0279 
 
 0454 
 0628 
 
 0297 
 0471 
 0645 
 
 0314 
 0488 
 0663 
 
 0332 
 0506 
 0680 
 
 369 
 369 
 369 
 
 12 15 
 
 12 15 
 12 15 
 
 4 
 5 
 6 
 
 0698 
 
 0872 
 1045 
 
 071507320750 
 08890906 0924 
 1063 1 080,1097 
 
 0767 0785 
 0941 0958 
 ni5|ii32 
 
 0802 
 0976 
 1149 
 
 *323 
 1495 
 1668 
 
 0819 
 
 0993 
 1167 
 
 0837 
 ion 
 1184 
 
 0854 
 1028 
 
 1 201 
 
 369 
 369 
 369 
 
 12 15 
 12 14 
 12 J4 
 
 7 
 8 
 9 
 
 1219 
 
 1392 
 1564 
 
 1236 
 1409 
 1582 
 
 1253 
 1426 
 
 1599 
 
 1271 
 1444 
 1616 
 
 1788 
 
 1288 
 1461 
 1633 
 
 1305 
 1478 
 1650 
 
 [340 
 1513 
 1685 
 
 1357 
 1530 
 1702 
 
 1374 
 1547 
 1719 
 
 369 
 369 
 369 
 
 12 14 
 12 14 
 12 14 
 
 10 
 
 1736 
 
 1754 
 
 1771 
 
 1805 
 
 1822 
 
 1840 
 
 i8 5 7|i8 7 4 
 
 I8 9 I 
 
 369 
 
 12 14 
 
 11 
 12 
 13 
 
 1908 
 2079 
 2250 
 
 1925 
 2096 
 2267 
 
 1942 
 2113 
 
 2284 
 
 r 959 
 2130 
 2300 
 
 1977 
 2147 
 2317 
 
 1994 
 2164 
 2334 
 
 2OII 
 
 2181 
 
 2351 
 
 2028 
 2198 
 2368 
 
 2045 
 2215 
 
 2385 
 
 2062 
 2232 
 24O2 
 
 369 
 369 
 368 
 
 II 14 
 II 14 
 II 14 
 
 14 
 15 
 16 
 
 24*9 
 
 2588 
 
 2756 
 
 2436 
 2605 
 2773 
 
 2453 
 2622 
 2790 
 
 2470 
 2639 
 2807 
 
 2487 
 2656 
 2823 
 
 2504 
 2672 
 2840 
 
 2521 
 2689 
 2857 
 
 2538 
 2706 
 2874 
 
 2554 
 2723 
 2890 
 
 2571 
 2740 
 2907 
 
 368 
 368 
 368 
 
 II 14 
 II 14 
 II 14 
 
 17 
 18 
 19 
 
 2924 
 3090 
 3256 
 
 2940 
 3107 
 3272 
 
 2957 
 3123 
 3289 
 
 2974 
 3140 
 
 3305 
 
 2990 
 
 3156 
 3322 
 
 3007 
 3173 
 3338 
 
 3024 
 3190 
 
 3355 
 
 3040 
 3206 
 3371 
 
 3057 
 3223 
 3387 
 
 3074 
 3239 
 3404 
 
 368 
 
 368 
 
 3 5 8 
 
 II 14 
 II 14 
 II 14 
 
 20 
 
 3420 
 
 3437 
 
 3453 
 
 3469 
 
 3486 
 
 3502 
 
 35i8 
 
 3535 
 
 355i 
 
 3567 
 
 3 5 8 
 
 II 14 
 
 21 
 22 
 23 
 
 3584 
 3746 
 397 
 
 3600 
 3762 
 3923 
 
 3616 
 3778 
 3939 
 
 3633 
 3795 
 3955 
 
 3649 
 3811 
 3971 
 
 3665 
 3827 
 3987 
 
 3681 
 
 3843 
 4003 
 
 3697 
 3859 
 4019 
 
 3714 
 3875 
 4035 
 
 3730 
 3891 
 4051 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 II 14 
 II 14 
 II 14 
 
 24 
 25 
 26 
 
 4067 
 4226 
 
 4384 
 
 4083 
 4242 
 4399 
 
 4099 
 4258 
 4415 
 
 4H5 
 4274 
 
 4431 
 
 4i3i 
 4289 
 4446 
 
 4M7 
 4305 
 4462 
 
 4163 
 4321 
 
 4478 
 
 4179 
 
 4337 
 4493 
 
 4195 
 4352 
 4509 
 
 4210 
 4368 
 4524 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 II 13 
 II 13 
 10 13 
 
 27 
 28 
 
 29 
 
 4540 
 4695 
 
 4848 
 
 4555 
 4710 
 4863 
 
 4571 
 4726 
 4879 
 
 4586 
 4741 
 4894 
 
 4602 
 4756 
 4909 
 
 4617 
 4772 
 4924 
 
 4633 
 4787 
 4939 
 
 4648 
 4802 
 4955 
 
 4664 
 4818 
 4970 
 
 4679 
 4833 
 4985 
 
 3 5 8 
 3 5 8 
 3 5 8 
 
 10 13 
 10 13 
 10 13 
 
 30 
 
 5000 
 
 5015 
 
 5030 
 
 5045 
 
 5060 
 
 5075 
 
 5090 
 
 5105 
 
 5120 
 
 5135 
 
 3 5 8 
 
 10 13 
 
 31 
 32 
 33 
 
 5150 
 5299 
 5446 
 
 5165 
 53M 
 5461 
 
 5180 
 5329 
 5476 
 
 5195 
 5344 
 5490 
 
 5210 
 5358 
 5505 
 
 5225 
 5373 
 5519 
 
 5240 
 
 5388 
 5534 
 
 5255 
 5402 
 
 5548 
 
 5270 
 5417 
 5563 
 
 5284 
 5432 
 5577 
 
 257 
 
 2 5 7 
 257 
 
 10 12 
 10 12 
 
 IO 12 
 
 34 
 35 
 36 
 
 5592 
 5736 
 
 5878 
 
 5606 
 5750 
 5892 
 
 5621 
 5764 
 5906 
 
 5635 
 5779 
 5920 
 
 5650 
 5793 
 5934 
 
 5664 
 5807 
 5948 
 
 5678 
 5821 
 5962 
 
 5693 
 5835 
 5976 
 
 5707 
 5850 
 5990 
 
 5721 
 5864 
 6004 
 
 257 
 257 
 2 5 7 
 
 10 12 
 10 12 
 
 9 12 
 
 37 
 38 
 39 
 
 6018 
 
 6157 
 6293 
 
 6032 
 6170 
 6307 
 
 6046 
 6184 
 6320 
 
 6060 
 6198 
 6334 
 
 6074 
 6211 
 6347 
 
 6088 
 6225 
 6361 
 
 6101 
 6239 
 6374 
 
 6115 
 6252 
 6388 
 
 61296143 
 6266 6280 
 6401 6414 
 
 257 
 2 5 7 
 247 
 
 9 12 
 9 ii 
 9 ii 
 
 40 
 
 6428 
 
 6441 
 
 6455 
 
 6468 
 
 6481 
 
 6494 
 
 6508 
 
 652165346547 
 
 247 
 
 9 " 
 
 41 
 42 
 43 
 
 6561 
 6691 
 6820 
 
 6574 
 6704 
 6833 
 
 6587 
 6717 
 6845 
 
 6600 
 6730 
 
 6858 
 
 6613 
 
 6743 
 6871 
 
 6626 
 6756 
 6884 
 
 6639 
 6769 
 6896 
 
 6652 
 6782 
 6909 
 
 6665 
 6794 
 6921 
 
 6678 
 6807 
 6934 
 
 247 
 246 
 246 
 
 9 ii 
 9 ii 
 8 ii 
 
 44 
 
 6947 
 
 6959 
 
 6972 
 
 6984 
 
 6997 
 
 7009 
 
 7022 
 
 7034 
 
 7046 
 
 7059 
 
 246 
 
 8 10 
 
NATURAL SINES. 
 
 61 
 
 45 
 
 46" 
 47 
 48 
 
 0' 
 
 6' 
 
 12' 
 
 18 
 
 24' 
 
 30 
 
 36 
 
 42' 
 
 48 
 
 54 
 
 123 
 
 4 5 
 
 7071 
 
 7083 
 
 7096 
 
 7108 
 
 7120 
 
 7133 
 
 7M5 
 
 7157 
 
 7169 
 
 7181 
 
 2 4 6 
 
 8 10 
 
 7i93 
 7314 
 743i 
 
 7547 
 7660 
 
 777i 
 
 7206 
 7325- 
 7443 
 
 7218 
 7337 
 7455 
 
 7230 
 
 7349 
 7466 
 
 7242 
 736i 
 7478 
 
 7254 
 7373 
 7490 
 
 7266 
 
 7385 
 75oi 
 
 7278 
 7396 
 7513 
 
 7290 
 7408 
 7524 
 
 7302 
 7420 
 7536 
 
 246 
 246 
 246 
 
 8 10 
 8 10 
 8 10 
 
 49 
 50 
 51 
 
 7558 
 7672 
 7782 
 
 7570 
 7683 
 7793 
 
 758i 
 7694 
 7804 
 7912 
 8018 
 8121 
 
 7593 
 7705 
 78_L5 
 7923 
 8028 
 8131 
 
 7604 
 7716 
 7826 
 
 7615 
 7727 
 7837 
 
 7627 
 7738 
 7848 
 
 7638 
 7749 
 7859 
 
 7649 
 7760 
 7869 
 
 246 
 246 
 245 
 
 8 9 
 7 9 
 7 9 
 
 52 
 53 
 54 
 
 7880 
 
 7986 
 8090 
 
 7891 
 7997 
 8100 
 
 7902 
 8007 
 8111 
 
 7934 
 8039 
 8141 
 
 7944 
 8049 
 8151 
 
 7955 
 8059 
 8161 
 
 7965 
 8070 
 8171 
 
 7976 
 8080 
 8181 
 
 245 
 235 
 2 3 5 
 
 7 9 
 7 9 
 
 7 8 
 
 55 
 
 8192 
 
 8202 
 
 8211 
 
 8221 
 
 8231 
 
 8241 
 
 8251 
 
 8261 
 
 8271 
 
 8281 
 
 235 
 
 7 8 
 
 56 
 57 
 58 
 
 8290 
 
 8387 
 8480 
 
 8300 
 8396 
 8490 
 
 8310 
 8406 
 8499 
 8590 
 8678 
 8763 
 8846 
 8926 
 90Q3 
 9078 
 
 8320 
 8415 
 8508 
 
 8329 
 8425 
 8517 
 
 8339 
 8434 
 8526 
 
 8348 
 8443 
 8536 
 
 8358 
 8453 
 
 8545 
 
 8368 
 8462 
 8554 
 
 8377 
 8471 
 
 8563 
 
 235 
 235 
 235 
 
 6 8 
 6 8 
 6 8 
 
 59 
 60 
 61 
 
 8572 
 8.660 
 8746 
 
 8581 
 8669 
 8755 
 8838 
 8918 
 8996 
 
 8599 
 8686 
 8 77 i 
 
 8607 
 8695 
 8780 
 
 8616 
 8704 
 8788 
 
 8625 
 8712 
 8796 
 
 8634 
 8721 
 8805 
 
 8643 
 8729 
 8813 
 
 8652 
 8738 
 8821 
 
 3 4 
 
 3 4 
 3 4 
 
 6 7 
 
 6 7 
 
 6 7 
 
 62 
 
 63 
 64 
 
 8829 
 8910 
 8988 
 
 8854 
 8934 
 9011 
 
 8862 
 8942 
 9018 
 
 8870 
 8949 
 9026 
 
 8878 
 8957 
 9033 
 
 8886 
 8965 
 9041 
 
 8894 
 8973 
 9048 
 
 8902 
 8980 
 9056 
 
 3 4 
 
 3 4 
 3 4 
 
 5 7 
 5 6 
 5 6 
 
 65 
 "66" 
 67 
 68 
 ~69~ 
 70 
 71 
 72 
 73 
 74 
 75" 
 76 
 77 
 78 
 
 9063 
 
 9070 
 
 9143 
 9212 
 9278 
 
 9085 
 
 9092 
 
 9100 
 
 9107 
 
 9114 
 
 9121 
 
 9128 
 
 2 4 
 
 5 6 
 
 9135 
 9205 
 9272 
 
 -9150 
 9219 
 9285 
 
 9157 
 9225 
 9291 
 
 9164 
 9232 
 9298 
 
 9361 
 9421 
 9478 
 9532 
 9583 
 9632 
 
 9171 
 9239 
 9304 
 
 9367 
 9426 
 
 9483 
 
 9537 
 9588 
 9636 
 
 9178 
 9245 
 93ii 
 
 9184 
 9252 
 9317 
 
 9191 
 9259 
 9323 
 
 9 i 9 h 
 9265 
 9330 
 
 2 3 
 2 3 
 2 3 
 
 5 <> 
 4 6 
 
 4 5 
 
 9336 
 9397 
 9455 
 
 9342 
 9403 
 9461 
 
 9348 
 9409 
 9466 
 
 9354 
 9415 
 9472 
 
 9527 
 9578 
 9627 
 
 9373 
 9432 
 
 9489 
 
 9379 
 9438 
 9494 
 
 9385 
 9444 
 
 939 T 
 9449 
 9505 
 
 2 3 
 2 3 
 2 3 
 
 4 5 
 4 5 
 4 5 
 
 9500 
 
 95ii 
 9563 
 9613 
 
 9659 
 9703 
 9744 
 9781 
 
 95i6 
 9568 
 9617 
 
 9521 
 9573 
 9622 
 
 9542 
 9593 
 9641 
 
 ;68G 
 
 9548 
 9598 
 9646 
 
 9553 
 9603 
 9650 
 
 955 
 9608 
 9655 
 
 2 3 
 2 2 
 2 2 
 
 4 4 
 3 4 
 3 4 
 
 9664 
 
 9668 
 
 9673 
 
 9677 
 
 9681 
 
 9690 
 
 9694 
 
 ,699 
 
 I 2 
 
 3 4 
 
 9707 
 9748 
 9785 
 
 9711 
 
 9751 
 9789 
 
 9715 
 9755 
 9792 
 
 9720 
 
 9759 
 9796 
 
 9724 
 9763 
 9799 
 
 9728 
 9767 
 9803 
 
 9732 
 9770 
 9806 
 
 9736 
 9774 
 9810 
 
 9740 
 9778 
 9813 
 
 2 
 2 
 2 
 
 3 3 
 3 3 
 2 3 
 
 79 
 80 
 81 
 
 9816 
 9848 
 9877 
 
 9820 
 
 9851 
 9880 
 
 9823 
 9854 
 9882 
 
 9907 
 9930 
 9949 
 
 9826 
 
 9857 
 9885 
 
 9829 
 9860 
 9888 
 9912 
 9934 
 9952 
 
 9833 
 9863 
 9890 
 
 9914 
 9936 
 9954 
 
 9836 
 9866 
 9893 
 
 9839 
 9869 
 
 9895 
 
 9842 
 9871 
 9898 
 
 9845 
 9874 
 9900 
 
 I 2 
 I 
 1 
 
 2 3 
 2 2 
 2 2 
 
 82 
 83 
 84 
 
 9903 
 9925 
 9945 
 9962 
 
 9905 
 9928 
 
 9947 
 9963 
 
 9910 
 9932 
 9951 
 9966 
 
 9917 
 9938 
 9956 
 
 9919 
 9940 
 9957 
 
 9921 
 9942 
 9959 
 
 9923 
 9943 
 9960 
 
 O 
 
 
 
 2 2 
 I 2 
 I 1 
 
 85 
 
 9965 
 
 9968 
 
 9969 
 9981 
 9990 
 9997 
 
 9971 
 
 9972 
 
 9973 
 
 9974 
 
 O O 
 
 I I 
 
 86 
 87 
 88 
 
 9976 
 9986 
 9994 
 
 9977 
 9987 
 9995 
 
 9978 
 9988 
 9995 
 
 9979 
 9989 
 9996 
 
 9980 
 9990 
 9996 
 
 9982 
 9991 
 9997 
 
 9983 
 9992 
 9997 
 
 9984 
 9993 
 9998 
 
 9985 
 9993 
 9998 
 
 O 
 O 
 000 
 
 I I 
 I I 
 O O 
 
 89 
 
 9998 
 
 9999 
 
 9999 
 
 9999 
 
 9999 
 
 1.000 
 nearly 
 
 1. 000 
 
 nearly 
 
 I.OOO 
 icarlv 
 
 I.OOO 
 nearly 
 
 i I.OOO 
 nearly 
 
 000 
 
 
 
62 
 
 NATURAL TANGENTS 
 
 
 
 
 6 
 
 12 
 
 18 
 
 24' 
 
 30' 
 
 36 
 
 42 
 
 48' 
 
 54' 
 
 123 
 
 4 5 
 
 
 
 .0000 
 
 0017 
 
 0035 
 
 0209 
 0384 
 0559 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 oi57 
 
 0332 
 0507 
 0682 
 
 369 
 
 12 14- 
 
 1 
 
 2 
 3 
 
 0175 
 
 0349 
 .0524 
 
 0192 
 0367 
 0542 
 
 0227 
 0402 
 0577 
 
 0244 
 0419 
 0594 
 
 0262 
 
 0437 
 0612 
 
 0279 
 
 0454 
 0629 
 
 0297 
 0472 
 0647 
 
 0314 
 0489 
 0664 
 
 369 
 369 
 369 
 
 12 15 
 12 1 5 
 12 I 5 
 
 4 
 5 
 6 
 
 .0699 
 .0875 
 .1051 
 
 0717 
 0892 
 1069 
 
 0734 
 0910 
 1086 
 
 0752 
 0928 
 1104 
 
 0769 
 0945 
 
 1122 
 
 0787 
 0963 
 "39 
 
 0805 
 0981 
 "57 
 
 1334 
 1512 
 1691 
 
 0822 
 0998 
 
 "75 
 
 0840 
 1016 
 1192 
 
 0857 
 1033 
 
 1210 
 
 369' 
 369 
 369 
 
 12 15 
 12 I S 
 12 I 5 
 
 7 
 8 
 9 
 
 .1228 
 .1405 
 .1584 
 
 1246 
 
 1423 
 1602 
 
 1263 
 1441 
 1620 
 
 1281 
 
 1459 
 1638 
 
 1299 
 1477 
 1655 
 
 1317 
 1495 
 1673 
 
 1352 
 1530 
 1709 
 
 1370 
 
 1548 
 1727 
 
 1388 
 1566 
 1745 
 
 369 
 369 
 369 
 
 12 15 
 12 15 
 12 I 5 
 
 10 
 
 1763 
 
 1781 
 
 1799 
 
 1817 
 
 1835 
 
 1853 
 
 1871 
 
 1890 
 
 1908 
 
 1926 
 
 3 6 9^ 
 
 12 15 
 
 11 
 12 
 13 
 
 .1944 
 .2126 
 .2309 
 
 1962 
 2144 
 2327 
 
 1980 
 2162 
 2345 
 
 1998 
 2180 
 2364 
 
 2016 
 2199 
 
 2382 
 
 2035 
 2217 
 2401 
 
 2053 
 2235 
 2419 
 
 2071 
 2254 
 2438 
 
 2089 
 2272 
 2456 
 
 2107 
 2290 
 2475 
 2661 
 2849 
 3038 
 
 369 
 369 
 369 
 
 12 15 
 12 15 
 12 15 
 
 14 
 15 
 16 
 
 2493 
 .2679 
 .2867 
 
 2512 
 2698 
 2886 
 
 2530 
 2717 
 2905 
 
 2549 
 2736 
 2924 
 
 2568 
 2754 
 2943 
 
 2586 
 
 2773 
 2962 
 
 2605 
 2792 
 2981 
 
 2623 
 2811 
 3000 
 
 2642 
 2830 
 3019 
 
 369 
 369 
 369 
 
 12 l6 
 
 13 16 
 J3 J6 
 
 17 
 18 
 19 
 
 3057 
 3249 
 3443 
 
 3076 
 3269 
 3463 
 
 3096 
 3288 
 3482 
 
 3H5 
 3307 
 3502 
 
 3134 
 3327 
 3522 
 
 3153 
 3346 
 3541 
 
 3172 
 3365 
 356i 
 
 3I9 1 
 
 3385 
 358i 
 
 3211 
 3404 
 3600 
 
 3230 
 3424 
 3620 
 
 3 6 10 
 3 6 10 
 3 6 10 
 
 13 16 
 13 16 
 13 17 
 
 20 
 
 3640 
 
 3659 
 
 3679 
 
 3699 
 
 3719 
 
 3739 
 
 3759 
 
 3779 
 
 3799 
 
 3819 
 
 3 7 '0 
 
 ij J7 
 
 21 
 22 
 23 
 
 .3839 
 .4040 
 
 4245 
 
 3859 
 4061 
 4265 
 
 3879 
 4081 
 4286 
 
 3899 
 4101 
 
 4307 
 
 3919 
 4122 
 
 4327 
 
 3939 
 4142 
 
 4348 
 
 3959 
 4163 
 
 4369 
 
 3978 
 4183 
 4390 
 
 4000 
 4204 
 44" 
 
 4020 
 4224 
 4431 
 
 3 7 Jo 
 3 7 I0 
 3' 7 JO 
 
 J3 J7 
 J4 J7 
 14 17 
 
 24 
 25 
 26 
 
 4452 
 .4663 
 .4877 
 
 4473 
 4684 
 4899 
 
 4494 
 4706 
 4921 
 
 45i5 
 4727 
 4942 
 
 4536 
 4748 
 4964 
 
 4557 
 4770 
 4986 
 
 4578 
 
 479' 
 5008 
 
 4599 
 4813 
 5029 
 
 4621 
 4834 
 5051 
 
 4642 
 4856 
 5073 
 
 4 7 J 
 4 7 ii 
 4 7 ii 
 
 14 18 
 14 18 
 15 18 
 
 27 
 28 
 29 
 
 5095 
 5317 
 
 5543 
 
 5"7 
 5340 
 5566 
 
 5139 
 5362 
 
 5589 
 
 5161 
 
 5384 
 5612 
 
 5184 
 5407 
 5635 
 
 5206 
 5430 
 5658 
 
 5228 
 5452 
 5681 
 
 5250 
 5475 
 5704 
 
 5272 
 5498 
 5727 
 
 5295 
 5520 
 5750 
 
 4 7 M 
 4811 
 4 8 12 
 
 15 J 
 J5 '9 
 J5 '9 
 
 30 
 
 5774 
 
 5797 
 
 5820 
 
 5844 
 
 5867 
 
 5890 
 
 5914 
 
 5938 
 
 596i 
 
 5985 
 
 4 8 .2 
 
 16 20 
 
 31 
 32 
 33 
 
 .6009 
 .6249 
 .6494 
 
 6032 
 6273 
 6519 
 6771 
 7028 
 7292 
 
 6056 
 6297 
 6544 
 
 6080 
 6322 
 6569 
 
 6lO4 
 6346 
 6594 
 
 6128 
 6371 
 6619 
 
 6152 
 
 6395 
 6644 
 
 6899 
 7159 
 7427 
 
 6176 
 6420 
 6669 
 6924 
 7186 
 7454 
 
 6200 
 6445 
 6694 
 
 6224 
 6469 
 6720 
 
 4 8 12 
 4812 
 4 S 13 
 
 1 6 20 
 16 20 
 
 I 7 21 
 
 34 
 35 
 36 
 
 6745 
 .7002 
 7265 
 
 6796 
 7054 
 7319 
 
 6822 
 7080 
 7346 
 
 6847 
 7107 
 
 7373 
 
 6873 
 7133 
 7400 
 
 6950 
 7212 
 7481 
 
 6976 
 
 7239 
 7508 
 
 4 9 J3 
 4 9 J3 
 5 9 J4 
 
 17 21 
 
 l8 22 
 
 18 23 
 
 37 
 38 
 39 
 
 7536 
 7813 
 .8098 
 
 7563 
 
 7841 
 8127 
 
 7590 
 7869 
 8156 
 
 7618 
 7898 
 8185 
 
 7646 
 7926 
 8214 
 
 7673 
 7954 
 
 8243 
 
 7701 
 7983 
 
 8273 
 
 7729 
 8012 
 8302 
 
 7757 
 8040 
 8332 
 
 7785 
 806 9 
 8361 
 
 5 9 J4 
 5 J J4 
 5 J J 5 
 
 18 2 3 
 19 24 
 20 =4 
 
 40 
 
 .8391 
 
 8421 
 
 8451 
 
 8481 
 
 8511 
 
 8541 
 
 8571 
 
 8601 
 
 8632 
 
 8662 
 
 5 io 15 
 
 2O 25 
 
 41 
 42 
 43 
 
 .8693 
 .9004 
 9325 
 
 8724 
 9036 
 9358 
 
 8754 
 9067 
 9391 
 
 8785 
 9099 
 
 9424 
 
 8816 
 9131 
 9457 
 
 8847 
 9163 
 9490 
 
 8878 
 9195 
 9523 
 
 8910 
 
 9228 
 9556 
 
 8941 
 9260 
 9590 
 
 8972 
 9 2 93 
 9623 
 
 5 io 16 
 5 ii 16 
 6 ii 17 
 
 21 26 
 SI 2 7 
 22 28 
 
 44 
 
 9657 
 
 9691 
 
 9725 
 
 9759 
 
 9793 
 
 9827 
 
 9861 
 
 9896 
 
 9930 
 
 99&5 
 
 6 ii 17 
 
 23 29 
 
NATURAL TANGENTS. 
 
 63 
 
 
 
 
 6' 
 
 12 
 
 18 
 
 24' 
 
 30 
 
 36 
 
 42' 
 
 48 
 
 54' 
 
 123 
 
 4 5 
 
 45 
 
 46 
 47 
 48 
 
 I.OOOO 
 
 0035 
 
 0070 
 
 0105 
 
 0141 
 
 0176 
 
 O2I2 
 
 0575 
 095 J 
 1343 
 
 0247 
 
 0283 
 
 0319 
 
 6 12 18 
 
 24 30 
 
 1-0355 
 1.0724 
 
 1.1106 
 
 0392 
 0761 
 H45 
 
 0428 
 0799 
 1184 
 
 0464 
 0837 
 1224 
 
 0501 
 
 0875 
 1263 
 
 0538 
 0913 
 1303 
 
 0612 
 0990 
 1383 
 
 0649 
 1028 
 1423 
 
 0686 
 1067 
 1463 
 
 6 12 18 
 
 6 13 19 
 7 13 20 
 
 25 3' 
 25 3 2 
 26 33 
 
 49 
 50 
 51 
 
 1.1504 
 1.1918 
 1-2349 
 
 1544 
 1960 
 
 2393 
 
 1585 
 
 2002 
 
 2437 
 
 1626 
 2045 
 2482 
 
 1667 
 2088 
 2527 
 
 1708 
 2131 
 2572 
 
 I75C 
 2174 
 2617 
 
 1792 
 2218 
 2662 
 
 1833 
 2261 
 2708 
 
 1875 
 2305 
 2753 
 
 7 *4 21 
 7 H 22 
 8 15 23 
 
 28 34 
 29 36 
 30 3 8 
 
 52 
 53 
 54 
 
 1.2799 
 1.3270 
 1.3764 
 
 2846 
 3319 
 3814 
 
 2892 
 3367 
 
 3865 
 
 2938 
 34i6 
 3916 
 
 2985 
 3465 
 3968 
 
 3032 
 3514 
 4019 
 
 3079 
 3564 
 4071 
 
 3127 
 3613 
 4124 
 
 3175 
 3663 
 4176 
 
 3222 
 
 3713 
 4229 
 
 8 16 23 
 8 16 25 
 9 7 * 6 
 
 3' 39 
 33 4' 
 34 43 
 
 10 IVO OCO 
 
 10 |iO 10 10 
 
 1.4281 
 
 4335 
 
 4388 
 
 4442 
 
 4496 
 
 4550 
 
 4605 
 
 4659 
 
 4715 
 
 4770 
 
 9 18 27 
 
 36 45 
 
 1.4826 
 
 J-5399 
 1.6003 
 
 4882 
 5458 
 6066 
 
 4938 
 5517 
 6128 
 
 4994 
 5577 
 6191 
 
 5051 
 5637 
 6255 
 
 5108 
 
 5697 
 6319 
 
 5166 
 
 5757 
 6383 
 
 5224 
 5818 
 6447 
 
 5282 
 5880 
 6512 
 
 5340 
 5941 
 6577 
 
 10 19 29 
 io 20 30 
 
 11 21 32 
 
 38 48 
 40 50 
 43 53 
 
 59 
 60 
 61 
 62 
 63 
 64 
 
 1.6643 
 1.7321 
 1.8040 
 
 6709 
 7391 
 
 8115 
 
 6775 
 7461 
 
 8190 
 
 6842 
 7532 
 8265 
 
 6909 
 7603 
 8341 
 
 6977 
 
 7675 
 8418 
 
 7045 
 7747 
 8495 
 
 7"3 
 
 7820 
 8573 
 
 7182 
 
 7893 
 8650 
 
 7251 
 7966 
 8728 
 
 Ji 23 34 
 
 12 2 4 3 6 
 13 26 3 8 
 
 45 56 
 48 60 
 5i 64 
 
 1.8807 
 1.9626 
 2.0503 
 
 8887 
 9711 
 0594 
 
 8967 
 
 9797 
 0886 
 
 9047 
 
 9883 
 0778 
 
 9128 
 9970 
 0872 
 
 9210 
 
 6057 
 0965 
 
 3292 
 0145 
 1060 
 
 9375 
 0233 
 
 "55 
 
 9458 
 6323 
 1251 
 
 9,542 
 0413 
 1348 
 
 I 4 2 7 41 
 5 29 44 
 16 31 47 
 
 S5 68 
 53 73 
 63 78 
 
 ~68~8s 
 
 65 
 
 2.1445 
 
 1543 
 
 1642 
 
 1742 
 
 1842 
 
 1943 
 
 2045 
 
 2148 
 
 2251 
 
 2355 
 
 '7 34 5 
 
 66 
 67 
 68 
 
 2.2460 
 2-3559 
 24751 
 
 2566 
 
 3673 
 4876 
 
 2673 
 
 3789 
 5002 
 
 2781 
 3906 
 5129 
 
 2889 
 4023 
 5257 
 
 2998 
 4142 
 5386 
 
 3109 
 4262 
 5517 
 
 3220 
 4383 
 5649 
 
 3332 
 4504 
 5782 
 
 3445 
 4627 
 59*6 
 
 '8 37 55 
 
 20 40 60 
 
 43 6 5 
 
 74 9 2 
 79 99 
 87 108 
 
 69 
 70 
 71 
 
 2.6051 
 
 2-7475 
 2.9042 
 
 6187 
 7625 
 9208 
 
 6325 
 7776 
 9375 
 
 6464 
 7929 
 9544 
 
 6605 
 8083 
 9714 
 
 6746 
 8239 
 9887 
 
 6889 
 8397 
 0061 
 
 7034 
 8556 
 0237 
 
 7179 
 8716 
 6415 
 
 7326 
 
 8878 
 0595 
 
 24 47 7' 
 
 26 52 78 
 29 58 87 
 
 95 "8 
 104 130 
 115 M4 
 
 72 
 73 
 74 
 
 3-0777 
 32709 
 
 3.4874 
 
 0961 
 2914 
 5105 
 
 1146 
 3122 
 5339 
 
 1334 
 3332 
 5576 
 
 1524 
 3544 
 5816 
 
 1716 
 
 3759 
 6059 
 
 1910 
 3977 
 6305 
 
 2106 
 4197 
 6554 
 
 2305 
 4420 
 '6806 
 
 2506 
 4646 
 7062 
 
 32 64 96 
 36 72 108 
 
 41 82 122 
 
 129 1161 
 
 144 180 
 162 203 
 
 75 
 
 3.7321 
 
 7583 
 
 7848 
 
 8118 
 
 839 1 
 
 8667 
 
 8947 
 
 9232 
 
 9520 
 
 9812 
 
 46 94 139 
 
 186 232 
 
 76 
 77 
 78 
 
 4.0108 
 
 4-3315 
 4.7046 
 
 0408 
 3662 
 7453 
 
 0713 
 4015 
 7867 
 
 JO22 
 
 4374 
 
 8288 
 
 1335 
 4737 
 8716 
 
 1653 
 5107 
 9152 
 
 1976 
 5483 
 9594 
 
 2303 
 5864 
 0045 
 
 2635 
 6252 
 0504 
 
 2972 
 6646 
 0970 
 
 53 107 160 
 62 124 186 
 73146219 
 
 214 267 
 248 310 
 292 365 
 
 79 
 80 
 81 
 
 5.1446 
 5.6713 
 
 1929 
 7297 
 3859 
 
 2422 
 7894 
 4596 
 
 292^ 
 8502 
 5350 
 
 3435 
 9 I2 4 
 6122 
 
 3955 
 9758 
 6912 
 
 4486 
 0405 
 7920 
 
 5026 
 1066 
 
 8548 
 
 5578 
 1742 
 9395 
 
 6140 
 
 ?<n? 
 
 87 175 262 
 
 350 437 
 
 6.313^ 
 
 0264 
 
 
 
 82 
 83 
 84 
 
 7-H54 
 
 8.1443 
 9-5144 
 
 2066 
 2636 
 9.677 
 
 3002 
 3863 
 9-845 
 
 3962 
 5126 
 IO.O2 
 
 4947 
 6427 
 
 IO.2O 
 
 5958 
 7769 
 10.39 
 
 6996 
 9152 
 10.58 
 
 8062 
 
 0579 
 10.78 
 
 9158 
 2052 
 10.99 
 
 0285 
 3572 
 n. 20 
 
 Difference - col- 
 umns cease to be 
 useful, owing to 
 the rapidity with 
 which the value 
 of the tangent 
 changes. 
 
 85 
 86 
 87 
 88 
 
 n-43 
 
 n.66 
 
 11.91 
 
 12.16 
 
 1243 
 
 12.71 
 
 13.06 
 
 13-30 
 
 13-62 
 
 1395 
 
 14.30 
 19.08 
 28.64 
 
 14.67 
 19.74 
 30.14 
 
 15-06 
 20.45 
 31-82 
 
 15.46 
 21.20 
 33.69 
 
 15.89 
 22.02 
 35-80 
 
 16.35 
 22.90 
 38.19 
 
 16.83 
 23-86 
 40.9 
 
 17-34 
 24.90 
 44.07 
 
 17.89 
 26.0; 
 47-74 
 
 18.46 
 27.27 
 52.08 
 
 89 
 
 57-29 
 
 63-66 
 
 71.62 
 
 81.8 
 
 95-49 
 
 114.6 
 
 143-2 
 
 191.0 
 
 286.5 
 
 573.0 
 
64 
 
 LOGARITHMS. 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 123 
 
 456 
 
 789 
 
 10 
 
 oooo 
 
 0043 
 
 0086 
 
 0128 
 
 0170 
 
 O2I2 
 
 0253 
 
 0294 
 
 0334 
 
 0374 
 
 Use Ta le second 
 page fo ! lowin;4 
 
 11 
 
 12 
 13 
 
 0414 
 0792 
 1139 
 
 0453 
 0828 
 
 "73 
 
 0492 
 0864 
 1206 
 
 0531 
 0899 
 1239 
 
 0569 
 0934 
 1271 
 
 0607 
 0969 
 1303 
 
 0645 
 1004 
 1335 
 
 0682 
 1038 
 1367 
 
 6719 
 1072 
 1399 
 
 0755 
 1106 
 T430 
 
 4811 
 3 7 10 
 3 6 10 
 
 15 19 23 
 
 14 17 21 
 
 13 16 19 
 
 26 30 34 
 24 28 31 
 23 26 29 
 
 14 
 15 
 IS 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 
 1461 
 1761 
 2041 
 
 1492 
 1790 
 2068 
 
 1523 
 1818 
 2095 
 
 1553 
 1847 
 
 2122 
 
 1584 
 1875 
 2148 
 
 1614 
 1903 
 2175 
 
 1644 
 !93i 
 
 2201 
 
 1673 
 1959 
 2227 
 
 i?>3 
 1967 
 2253 
 
 1732 
 2014 
 
 2279 
 
 369 
 368 
 3 5 8 
 
 12 15 18 
 
 II 14 17 
 II 13 16 
 
 21 24 27 
 20 22 25 
 
 18 21 24 
 
 2304 
 
 2553 
 2788 
 
 2330 
 2577 
 2810 
 
 2355 
 2601 
 
 2833 
 
 2380 
 2625 
 2856 
 
 2405 
 2648 
 
 2878 
 
 2430 
 2672 
 29OO 
 
 2455 
 2695 
 2923 
 
 2480 
 2718 
 2945 
 
 2504 
 2742 
 2967 
 
 2529 
 2765 
 2989 
 
 5 7 
 5 7 
 4 7 
 
 10 12 15 
 
 9 12 J 4 
 9 " *3 
 
 17 2O 22 
 
 16 19 21 
 16 18 20 
 
 3010 
 
 3032 
 
 3054 
 
 3075 
 
 3096 
 
 3118 
 
 3139 
 
 3160 
 
 3181 
 
 3201 
 
 4 6 
 
 8 ii 13 
 
 15 17 19 
 
 3222 
 3424 
 
 3617 
 
 3243 
 3444 
 3636 
 
 3263 
 3464 
 
 3655 
 
 3284 
 3483 
 3674 
 
 3304 
 3502 
 3692 
 
 3324 
 3522 
 
 37" 
 
 3345 
 354i 
 3729 
 
 3365 
 3560 
 3747 
 
 3385 
 3579 
 3766 
 
 3404 
 3598 
 3784 
 
 4 6 
 4 6 
 4 6 
 
 8 10 12 
 8 10 12 
 
 7 9 ii 
 
 14 16 18 
 14 15 17 
 13 '5 7 
 
 24 
 25 
 26 
 
 3802 
 
 3979 
 4rto 
 
 3820 
 
 3997 
 4166 
 
 3838 
 4014 
 4183 
 
 3856 
 4031 
 4200 
 
 3874 
 4048 
 4216 
 
 3892 
 4065 
 4232 
 
 3909 
 4082 
 
 4249 
 
 3927 
 4099 
 4265 
 
 3945 
 4116 
 4281 
 
 3962 
 4133 
 4298 
 
 4 5 
 3 5 
 3 5 
 
 7 9 ii 
 7 9 10 
 7 8 10 
 
 12 14 16 
 
 12 14 15 
 II 13 15 
 
 27 
 28 
 
 29 
 
 4314 
 4472 
 4624 
 
 4330 
 H87 
 4639 
 
 4346 
 4502 
 4654 
 
 4362 
 4518 
 4669 
 
 4814 
 
 43.78 
 
 4533 
 4683 
 
 4393 
 4548 
 4698 
 
 4409 
 4564 
 4713 
 
 4425 
 4579 
 
 4728 
 
 4440 
 4594 
 4742 
 
 4456 
 4609 
 4757 
 
 3 5 
 3 5 
 3 4 
 
 6>8 9 
 689 
 679 
 
 II 13 14 
 II 12 14 
 
 10 12 13 
 
 30 
 31 
 32 
 33 
 
 477i 
 
 4786 
 
 4800 
 
 4829 
 
 4843 
 
 4857 
 
 4871 
 
 4886 
 
 4900 
 
 3 4 
 
 679 
 
 10 II 13 
 
 4914 
 5051 
 5185 
 
 1928 
 5065 
 5198 
 
 4942 
 5079 
 5211 
 
 4955 
 5092 
 5224 
 
 4969 
 5105 
 
 5237 
 
 4983 
 5"9 
 5250 
 
 4997 
 5132 
 5263 
 
 5011 
 
 5145 
 5276 
 
 5024 
 
 5159 
 5289 
 
 5038 
 5172 
 5302 
 
 3 4 
 3 4 
 3 4 
 
 678 
 5 7 8 
 5 6 8 
 
 10 II J2 
 9 11 12 
 
 9 10 12 
 
 34 
 35 
 36 
 37 
 38 
 39 
 
 5315 
 544i 
 5563 
 
 5328 
 5453 
 
 5575 
 
 5340 
 5465 
 5587 
 
 5353 
 5478 
 5599 
 5717 
 5832 
 5944 
 
 5366 
 
 549<> 
 5611 
 
 5378 
 5502 
 5623 
 
 5391 
 
 5514 
 5635 
 
 5403 
 
 5527 
 5647 
 
 5416 
 
 5539 
 5658 
 
 5428 
 
 5551 
 5670 
 
 3 4 
 4 
 
 4 
 
 5 6 8 
 5 6 7 
 5 6 7 
 
 9 10 ii 
 9-io ii 
 8 10 ii 
 
 5682 
 5798 
 59 11 
 
 5694 
 S8o 9 
 5922 
 
 5705 
 5821 
 
 5933 
 
 5729 
 5843 
 5955 
 
 5740 
 5855 
 5966 
 
 5752 
 5866 
 5977 
 
 5763 
 5877 
 5988 
 
 5775 
 5888 
 
 5999 
 
 5786 
 
 5899 
 6010 
 
 3 
 3 
 
 3 
 
 5 6 7 
 5 6 7 
 457 
 
 8 9 10 
 8 9 10 
 8 9 10 
 
 40 
 41 
 42 
 43 
 
 6021 
 6128 
 6232 
 6335 
 
 6031 
 
 6042 
 
 605.3 
 
 6064 
 
 6075 
 
 6085 
 
 6096 
 
 6107 
 
 6117 
 
 3 
 
 4 5 6 
 
 8 9 10 
 
 6138 
 6243 
 6345 
 
 6149 
 6253 
 6355 
 
 6160 
 6263 
 6365 
 
 6170 
 6274 
 6375 
 
 6180 
 6284 
 6385 
 
 6191 
 6294 
 6395 
 
 6201 
 6304 
 6405 
 
 6212 
 6314 
 6415 
 
 6222 
 6325 
 6425 
 
 3 
 3 
 r 3 
 
 456 
 4 5 6 
 4 5 6 
 
 789 
 7 8 9 
 7 8 Q 
 
 44 
 45 
 46 
 
 6435 
 6532 
 6628 
 
 6444 
 6542 
 6637 
 
 6454 
 6551 
 6646 
 
 6464 
 6561 
 6656 
 
 6474 
 
 6571 
 6665 
 
 6484 
 6580 
 6675 
 
 6493 
 6590 
 6684 
 
 6503 
 65-99 
 6693 
 
 6513 
 6609 
 6702 
 
 6522 
 6618 
 6712 
 
 3 
 3 
 3 
 
 456 
 4 5 6 
 456 
 
 7 8 9 
 7 8 9 
 7 7 8 
 
 47 
 48 
 49 
 
 6721 
 6812 
 6902 
 
 6730 
 6821 
 6911 
 
 6739 
 6830 
 6920 
 7007 
 
 6749 
 
 6839 
 6928 
 
 6758 
 6848 
 6937 
 
 6767 
 
 6857 
 6946 
 
 6776 
 6866 
 6955 
 
 6785 
 6875 
 6964 
 
 6794 
 6884 
 6972 
 
 6803 
 6893 
 6981 
 
 3 
 3 
 3 
 
 455 
 
 445 
 445 
 
 6 7 8 
 6 7 fe 
 6 7 8 
 
 50 
 51 
 52 
 53 
 
 6990 
 
 6998 
 
 7016 
 
 7024 
 
 7033 
 
 7042 
 
 7050 
 
 7059 
 
 7067 
 
 3 
 
 345 
 
 6 7 8 
 
 7076 
 7160 
 7243 
 
 7084 
 7168 
 7251 
 
 7093 
 
 7177 
 7259 
 
 7101 
 7185 
 7267 
 
 7348 
 
 7110 
 7193 
 
 7275 
 
 7118 
 7202 
 7284 
 
 7126 
 7210 
 7292 
 
 7135 
 7218 
 7300 
 
 7143 
 7226 
 7308 
 
 7152 
 7235 
 7316 
 
 3 
 
 2 
 2 
 
 345 
 345 
 345 
 
 6 7 8 
 6 7 7 
 667 
 
 54 
 
 7324 
 
 7332 
 
 7340 
 
 7356 
 
 7364 
 
 7372 
 
 738o 
 
 7388 
 
 7396 
 
 I 2 2 
 
 345 
 
 6 6 7 
 
LOGARITHMS 
 
 65 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 123 
 
 456 
 
 789 
 
 55 
 
 740-1 
 
 7412 
 
 7419 
 
 7427 
 
 7435 
 
 7443 
 
 745J 
 
 7459 
 
 7466 
 
 7474 
 
 I 2 2 
 
 345 
 
 5 6 7 
 
 56 
 57 
 58 
 
 7482 
 7559 
 7634 
 
 7490 
 7566 
 7642 
 
 7497 
 7574 
 7649 
 
 7505 
 7582 
 7657 
 
 7513 
 7589 
 7664 
 
 7520 
 
 7597 
 7672 
 
 7528 
 7604 
 7679 
 
 7536 
 7612 
 7686 
 
 7543 
 7619 
 7694 
 
 7551 
 7627 
 770) 
 
 I 2 
 I 2 
 I 2 
 
 3 4 5 
 345 
 344 
 
 5 6 7 
 5 6 7 
 5 6 7 
 
 59 
 60 
 61 
 
 7709 
 7732 
 
 78<n 
 
 7716 
 7789 
 7860 
 
 7723 
 7796 
 7868 
 
 773i 
 7803 
 7S75 
 
 7735 
 7810 
 7882 
 
 7745 
 7818 
 7889 
 
 7752 
 7825 
 7896 
 
 7760 
 7832 
 7903 
 
 7767 
 
 7839 
 7910 
 
 7774 
 7846 
 79 T 7 
 
 I 2 
 I 2 
 I 2 
 
 344 
 344 
 344 
 
 5 6 7 
 5 6 t 
 5 6 6 
 
 62 
 63 
 64 
 
 7924 
 
 7993 
 8062 
 
 7931 
 Sooo 
 8069 
 
 7938 
 8007 
 
 8075 
 
 7945 
 8014 
 8082 
 
 7952 
 8021 
 8089 
 
 7959 
 8028 
 8096 
 
 7966 
 8035 
 8102 
 
 7973 
 8041 
 8109 
 
 798o 
 8048 
 8116 
 
 7987 
 8055 
 812? 
 
 I 2 
 
 I 2 
 I 2 
 
 334 
 334 
 334 
 
 5 6 6 
 
 5 5 6 
 5 5 ( 
 
 65 
 
 8129 
 
 8136 
 
 8142 
 
 8149 
 
 8156 
 
 8162 
 
 8169 
 
 8176 
 
 8182 
 
 8189 
 
 1 '2 
 
 334 
 
 5 5 <- 
 
 66 
 67 
 68 
 
 8195 
 8261 
 
 8325 
 
 8202 
 8267 
 8331 
 
 8209 
 8274 
 8338 
 
 8215 
 8280 
 8344 
 
 8222 
 8287 
 8351 
 
 8228 
 8293 
 8357 
 
 8235 
 8299 
 
 8363 
 
 8241 
 8306 
 8370 
 
 8248 
 8312 
 8376 
 
 8254 
 8319 
 8382 
 
 
 
 
 I 2 
 
 334 
 
 4 5 6 
 
 69 
 70 
 71 
 
 8388 
 8451 
 8513 
 
 8395 
 8457 
 8519 
 
 8401 
 8463 
 
 8525 
 
 8407 
 8470 
 853i 
 
 8414 
 8476 
 8537 
 
 8420 
 8482 
 8543 
 
 8426 
 8488 
 8549 
 
 8432 
 8494 
 
 8555 
 
 8439 
 8500 
 8561 
 
 8445 
 8506 
 
 8567 
 
 1 2 
 
 234 
 
 4 5 6 
 
 I ' 2 
 
 234 
 
 455 
 
 72 
 73 
 74 
 
 8573 
 8633 
 8692 
 
 8579 
 8639 
 8698 
 
 8585 
 8645 
 8704 
 8762 
 
 S59 1 
 8651 
 8710 
 
 8597 
 8657 
 8716 
 
 8603 
 8663 
 8722 
 
 8609 
 8669 
 8727 
 
 8615 
 8675 
 8733 
 
 8621 
 8681 
 8739 
 
 8627 
 8686 
 
 8745 
 
 I 2 
 
 I 2 
 I 2 
 
 234 
 2 3 4 
 234 
 
 455 
 455 
 455 
 
 75 
 
 875i 
 
 8756 
 
 8768 
 
 8774 
 
 8779 
 
 8785 
 
 8791 
 
 8797 
 
 8802 
 
 I 2 
 
 233 
 
 4 5 5 
 
 76 
 77 
 78 
 
 8808 
 8865 
 8921 
 
 8814 
 8871 
 8927 
 
 8820 
 8876 
 8932 
 
 8825 
 8882 
 8938 
 
 8831 
 8887 
 8943 
 
 8837 
 8893 
 8949 
 
 8842 
 8899 
 8954 
 
 8848 
 8904 
 8960 
 
 8854 
 8910 
 8965 
 
 885-9 
 8915 
 
 8971 
 
 I 2 
 
 I 2 
 
 233 
 233 
 
 445 
 445 
 
 79 
 80 
 81 
 
 8976 
 9031 
 
 9085 
 
 8982 
 9036 
 9090 
 
 8987 
 9042 
 9096 
 
 8993 
 9047 
 9101 
 
 8998 
 9053 
 9106 
 
 9004 
 9058 
 9112 
 
 9009 
 9063 
 9117 
 
 9015 
 9069 
 9122 
 
 9020 
 9074 
 9128 
 9180 
 9232 
 9284 
 
 9025 
 9079 
 9133 
 
 
 
 4 4 5 
 
 
 
 445 
 
 82 
 83 
 84 
 
 9138 
 9191 
 9 2 43 
 
 9 r 43 
 9196 
 
 9248 
 
 9149 
 9201 
 9253 
 
 9*54 
 9206 
 9258 
 
 9159 
 9212 
 9263 
 
 9165 
 9217 
 9269 
 
 9170 
 9222 
 9274 
 
 9175 
 9227 
 
 9279 
 
 9186 
 9238 
 9289 
 
 I 2 
 I 2 
 
 233 
 233 
 
 445 
 445 
 
 85 
 
 9294 
 
 9299 
 
 9304 
 
 9309 
 
 9315 
 
 9320 
 
 9325 
 
 9330 
 
 9335 
 
 9340 
 
 
 
 
 86 
 87 
 88 
 
 9345 
 9395 
 9445 
 
 9350 
 9400 
 9450 
 
 9355 
 9405 
 9455 
 
 936o 
 9410 
 9460 
 
 9365 
 9415 
 9465 
 
 9370 
 9420 
 9469 
 
 9375 
 9425 
 9474 
 
 9380 
 9430 
 9479 
 
 9385 
 9435 
 9484 
 
 9390 
 9440 
 9489 
 
 1 2 
 
 1 
 O I 
 
 233 
 
 2 3 
 
 2 3 
 
 445 
 3 4 4 
 3 4 4 
 
 89 
 90 
 91 
 
 9494 
 9542 
 959 
 
 9499 
 9547 
 9595 
 
 9504 
 9552 
 9600 
 
 9509 
 9557 
 9605 
 
 9513 
 9562 
 9609 
 
 95i8 
 9566 
 9614 
 9661 
 9708 
 9754 
 
 9523 
 957' 
 9619 
 
 9528 
 9576 
 9624 
 
 9533 
 958i 
 9628 
 
 953S 
 9586 
 9 6 33 
 
 I 
 I 
 I 
 
 2 3 
 
 2 3 
 
 2 3 
 
 ~*~ 3 
 
 2 3 
 2 3 
 
 3 4 4 
 344 
 
 3 4 4 
 
 92 
 93 
 94 
 
 9638 
 9685 
 973' 
 
 9643 
 9689 
 9736 
 
 9647 
 9694 
 9741 
 
 9652 
 9699 
 9745 
 
 9657 
 9703 
 9750 
 
 9666 
 9713 
 9759 
 
 9671 
 9717 
 9763 
 
 9675 
 9722 
 9768 
 
 9680 
 9727 
 9773 
 
 O 1 
 O 1 
 O I 
 
 344 
 344 
 3 4 4 
 
 95 
 
 9777 
 
 9782 
 
 9786 
 
 9-791 
 
 9795 
 
 9800 
 
 9805 
 
 9809 
 
 9814 
 
 9818 
 
 J 
 
 
 
 96 
 97 
 98 
 
 9823 
 9868 
 9912 
 
 9827 
 9872 
 9917 
 
 9832 
 9877 
 
 ) 9 2I 
 
 9836 
 9881 
 9926 
 
 9841 
 9886 
 9930 
 
 945 
 9890 
 9934 
 
 9850 
 994 
 9939 
 
 9854 
 9899 
 
 9943 
 
 9859 
 9903 
 9948 
 
 9863 
 99 oS 
 9952 
 
 I 
 I 
 I 
 
 2 3 
 2 3 
 2 3 
 
 344 
 
 344 
 344 
 
 99 
 
 9956 
 
 9961 
 
 9965 9969 
 
 9974 
 
 9978 
 
 9983 
 
 9987 
 
 999 * 
 
 9996 
 
 O I 1 
 
 223 
 
 334 
 
66 
 
 LOGARITHMS. 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 100 
 
 ooooo 
 
 043 
 
 087 
 
 130 
 
 173 
 
 217 
 
 '^60 
 
 303 
 
 346 
 
 389 
 
 101 
 102 
 103 
 
 432 
 860 
 
 01284 
 
 475 
 903 
 326 
 
 5i8 
 945 
 368 
 
 561 
 c,88 
 410 
 
 604 
 030 
 452 
 
 647 
 072 
 494 
 
 689 
 H5 
 536 
 
 732 
 157 
 
 578 
 
 775 
 199 
 620 
 
 817 
 242 
 662 
 
 104 
 105 
 106 
 
 703 
 
 02 119 
 
 531 
 
 745 
 160 
 572 
 
 787 
 202 
 612 
 
 828 
 243 
 653 
 
 870 
 284 
 694 
 
 912 
 325 
 
 735 
 
 953 
 366 
 776 
 
 995 
 407 
 816 
 
 036 
 
 449 
 
 857 
 
 078 
 490 
 898 
 
 107 
 108 
 109 
 
 938 
 03342 
 
 743 
 
 979 
 
 383 
 782 
 
 019 
 
 423 
 
 822 
 
 060 
 463 
 
 862 
 
 100 
 
 503 
 902 
 
 141 
 
 543 
 941 
 
 181 
 
 583 
 981 
 
 222 
 623 
 021 
 
 262 
 663 
 060 
 
 302 
 703 
 
 IOO 
 
 To find the logarithm of a number: First, locate in the 
 table the mantissa which lies in line with the first two figures of the 
 number and underneath the third figure, then increase this mantissa 
 by an amount depending upon the fourth figure of the number and 
 found by means of the interpolation columns at the right; secondly, 
 determine the characteristic, or the exponent of that integer power 
 of 10 which lies next in value below the number; for example, 
 log 600;= 0.7782 -(- 2. ; log 73-46= 0.8661 + 1. ; 
 
 log ,006=0.7782-3.; log .7346=0.8661-1.; 
 
 log -6.003= Q-.7784 + 0. ; log 734j9= 0.8662 + 3. 
 
 The logarithm of a product of two or more numbers is the sum of 
 the logarithms of its factors; for example, 
 
 log. (.0821 X 463.2)= (0.9143-2.) + (0.6658 + 2.) = 0.5801 + 1. 
 The logarithm of a quotient is the difference between the logar- 
 ithms of the dividend and divisor; for example, 
 
 log. (.5321 -916) =(0.7260-1.) - (0.9619 + 2.) = 0.7641-4. 
 The logarithm of a power or root of a number is the exponent 
 times the logarithm of the number; for example, 
 
 log N/.863) 8 =3/2 X (0.9360 l.) = 0.9040-l. 
 
 To find the number from its logarithm: Locate in the table 
 the mantissa next less than the given mantissa, then join the figure 
 standing above it at the top of the table to the two figures at the 
 extreme left on the same line as the mantissa, and finally to these 
 three join the figure at the top of the interpolation column which 
 contains the difference between the two mantissae. In the four- 
 figure number thus found, so place the decimal point that the 
 number shall be the product of some number, that lies between 
 1 and 10, by a power of 10 whose exponent is the characteristic 
 of the logarithm. For example, 
 
 antilog (0.6440 + 3) = 4405; 
 antilog (0.3069 -2) = .02027. 
 
 Caution. Tn adding and subtracting logarithms it is well to 
 remember that the mantissa is always essentially positive and may 
 or may not therefore, have the same sign as its characteristic. 
 
INDEX 
 
 Aberration, chromatic 
 
 study of - - 34 
 
 spherical study of - 35 
 
 Angles, measuring, with 
 
 spectrometer 24 
 
 Astigmatic pencil 18 
 
 Astigmatism, in lens - 35 
 Brewster's Law 55 
 
 Cardinal Points of Thick 
 
 Lens 37 
 
 Caustic Curve - 18 
 
 Chord, Major - - 8 
 
 Convention of Signs, opti- 
 cal - - 11 
 Conjugate Points - - 21 
 Deviation, Angle of mini- 
 mum - - 28 
 Dust Figures - 5 
 Eyepiece, Gauss, use of - 25 
 focal length of 40 
 Focal Distance, first and 
 second, principal, defined 21 
 direct measurement of 33 
 Formula, for mirror, sin- 
 gle refracting surface, 
 lens - - 12 
 Fraunhofer lines, the - 46 
 Grating Constant, the - 51 
 Harmony - 9 
 Image, formed by convex 
 
 reflecting surface - 15 
 
 Image: graphic determin- 
 ation of; using principal 
 planes 37 
 
 Kundt's methqd for de- 
 termining the velocity 
 of sound in solids - 5 
 Lenses, thin, defined - 33 
 Magnifying power of tele- 
 scope - - 40 
 
 Mirrors, concave, method 
 of determining radius of 
 
 curvature - 13 
 
 convex - 13 
 
 Model Eye, study of 21 
 
 Newton's Rings 48 
 
 Nicol's polarizing prisms - 52 
 Nodal Points of Thick 
 
 Lenses: Defined - 37 
 
 Determined 38 
 
 Path of any oblique ray - 12 
 
 Polarization, plane of - 55 
 
 angle of maximum - 55 
 Principal Points of Thick 
 
 Lenses - 36 
 Radius of curvature, by 
 
 reflection - - 15 
 Refraction trough 20 
 Refraction, at a single 
 plane surface - 17 
 at a single spherical sur- 
 face - 20 
 Refractometer, Pulfrich's 30 
 Scale, the natural 8 
 Spectra - - 45 
 Spectrometer, adjustments 24 
 Spectroscope, direct vision 44 
 Spherometer, the Ring - 13 
 the Peg - 14 
 the Aldis peg - 14 
 Telescope, resolving power 
 of - 40 
 magnifying power of - 40 
 Thick Lenses, defined 36 
 Triad, major - 8 
 Velocity of sound, in tubes 7 
 in rods - 5 
 Vibration of wires 9 
 Young's Modulus,of wires, 
 from longitudinal vi- 
 brations - - 9 
 of rods, from dust figures 7 
 

UNIVERSITY OF CALIFORNIA LIBRARY 
 BERKELEY 
 
 Return to desk from which borrowed. 
 This book is DUE on the last date stamped below. 
 
 NOV 21 1947 
 
 
 !5Apr52LU 
 
 . 26ANov'52FA 
 
 * 8 
 
 JAN 5 195t 
 
 ; 
 
 3lan'58WWXi 
 REC'D LD 
 
 DEC 61957 
 
 16Mar'59Tj 
 
 REC'D LD 
 
 1959 
 
 REC'D 
 
 APR 30 i3o 
 
 REC'D LD 
 SIP 8 1962 
 
 LD 21-100m-9, 5 47(A5702sl6)476 
 
ru I 
 
 7 
 
 ' 
 
 UNIVERSITY OF CALIFORNIA LIBRARY 
 
 . > 
 
 Si? "' ; '%