UNIVERSITY OF CALIFORNIA 
 LOWER DIVISIOH 
 
 LIBRARY 
 
 OF THE 
 
 Received ..... . 
 
 Accessions No<L./.. &. .......... Book No ..... L.. 
 
SOPHOMORE COURSE 
 
 PHYSICAL MEASUREMENTS 
 
 BY 
 
 ELMER R. DREW, B. S. 
 
 INSTRUCTOR IN PHYSICS IN THE UNIVERSITY OF CALIFORNIA 
 
 BERKELEY, CALIFORNIA 
 1808 
 
7 
 
 n 
 
 Entered According to Act of Congress in the Year 1898, by 
 
 ELMER R. DREW 
 
 In the Office of the Librarian of Congress at Washington. 
 
PREFACE. 
 
 THIS Course has been given for the last three years to Sophomore 
 students in the Colleges of Engineering, Chemistry, Agriculture and 
 Natural Sciences, and accompanies a Lecture Course in General 
 Physics. It occupies two 3-hour periods per week throughout the 
 year, being preceded by a Freshman Course of two laboratory periods 
 and one lecture per week for one year, and by a Matriculation Ex- 
 amination in Physics. The students' preparation, as thus indicated, 
 will account for the fact that the present work is confined mainly to 
 directions for handling the apparatus used in the experiments, the 
 theory being either briefly formulated or entirely omitted. 
 
 It is intended to so write the exercises that they can not be carried 
 out without much careful thought on the part of the student. In 
 giving them this form, two ideas have been kept in mind: First, 
 not to require of the student anything which his preparation does 
 not warrant; but within this limit, to compel him to build his own 
 steps, with no more guidance than is really necessary. Second, to 
 take nothing on authority; but to have worked out, in some form, 
 the reasoning underlying every method and formula used. 
 
 These ideas are further carried out, and results tested, in four 
 written examinations which are held at convenient intervals during 
 the year. When the results show it to be necessary, these examina- 
 tions are supplemented by recitations and lectures. Rather than 
 say too much in the directions, it is thought best to say too little, 
 encouraging the student to build up his working knowledge of the 
 subject for himself, according to his ability, and then to assist him 
 in properly completing the structure. 
 
 Little originality can be claimed for the choice of subject-matter 
 673151 (iii) 
 
IV PREFACE. 
 
 for the experiments; but it is hoped that in the sequence in which 
 the exercises are arranged, and in the methods of presenting the de- 
 tails of each experiment, some improvement over the usual texts 
 may be noted. The attempt has been made to have each exercise 
 appear not as an isolated piece of work, but as part of a larger whole, 
 each part depending more or less upon what has preceded it. 
 
 In pursuance of this idea, no measurement has been prescribed 
 merely for practice, but always to serve a definite purpose in obtain- 
 ing a desired result. 
 
 The first two experiments, A and B, do not form an integral part 
 of the course, and are assigned only in special cases, to students who 
 are unable to begin the regular sequence of experiments, or are 
 obliged to interrupt it for any reason. 
 
 After completing the exercises here laid out, about six periods 
 are spent in what is called "Special Problem Work." Each student 
 is required to select some problem, within his capacity and the 
 facilities of the laboratory, which will occupy the required time. 
 The experimental details and adaptation of apparatus are to be ar- 
 ranged by the student, so far as possible. One of the chief aims 
 of the laboratory course has been to give the power to do independ- 
 ent work of a simple character, and this exercise is a test of the 
 student's ability in that direction. 
 
 The author is under obligations to Professor Slate, and all the 
 members of the Physics Department, for advice and constant en- 
 couragement during the building up of the Course. He is also espe- 
 cially indebted to Mr. Arthur Incell for skilful assistance in design- 
 ing and constructing the necessary apparatus, and to Dr. W. P. 
 Boynton for numerous suggestions, the result of his experience in 
 directing the Course during the past year. 
 
 E R DREW. 
 Berkeley, July, 1898. 
 
COURSE IN 
 
 Physical Measurements 
 
 GENERAL DIRECTIONS. 
 
 Each of the following exercises is intended to occupy the 
 time of one laboratory period, unless otherwise stated. There 
 will usually be time, after completing the observations, to 
 calculate and write out the results; but this is not required. 
 
 Carbon copies of completed notes and results must be handed 
 in within two weeks of the date on which the experiment was 
 performed. These papers will be marked in the following four 
 grades: I, excellent; 2, satisfactory; 3, deficient in some part, 
 which will usually be indicated; 4, unsatisfactory as a whole. 
 Papers marked 3 or 4 will be raised to grade 2 if put in satis- 
 factory condition within one week from date of marking; 
 unleso this involves repetition of the experiment, when more 
 time will be allowed. Carbon copies only of such corrections 
 will be received. 
 
 Papers not handed in within the time limits set above will 
 not be marked higher than grade 3, except in special cases. 
 
 All observations should be recorded directly in the note- 
 book. If for any reason it is desirable to take them on loose 
 
 (5) 
 
6 GENERAL DIRECTIONS. 
 
 paper, they should be copied at once into the book. If any 
 observations are rejected, the reason should be stated. 
 
 Neatness and orderly arrangement of notes will be insisted 
 upon. Tabulate numerical data when possible. Always indi- 
 cate the denominations of numbers. Numerical computations 
 may be merely indicated. Make the explanatory matter full 
 enough so that, when taken in connection with the directions, 
 it will make each step intelligible. 
 
 Much time will be saved in numerical work by selecting the 
 method of computation best suited to the particular case, 
 whether logarithms, slide-rule or ordinary arithmetic. A 
 result obtained by one method should always be checked, at 
 ItasJ rou^l^iiy, .try -some other method. 
 
 .. Nevej* f^e ^satisfied -\vjth a single reading or determination of 
 a quaalky. When circumstances will permit, take at least 
 three readings, varying the conditions and the method of 
 reading as widely as possible. 
 
 When two students are working together on an experiment, 
 the work should be so arranged that each observation is taken 
 at least once, usually several times, by each observer. This 
 not only gives each student the full benefit of the experiment, 
 but also tends to eliminate the effect of the so-called "personal 
 equation." The two series of observations should be distin- 
 guished by appropriate initials. 
 
 In using any piece of apparatus, note the number which 
 will usually be found stamped on some part of it. This is 
 necessary, because the same piece may be needed in a later 
 experiment. 
 
 The careful and thoughtful work which this course requires 
 can not be well done in a noisy room. The large number 
 working in the room at once, makes it especially important 
 that each one should carefully avoid making any unnecessary 
 noise, by loud talking or otherwise. 
 
 It is of course expected that the tables and apparatus used 
 will be left in good order at the close of the exercise. 
 
B] MASS OF GIVEN VOLUME OF AIR. 
 
 A. MAGNIFYING POWER OF TELESCOPE. 
 
 Set up a 2-mm. scale in a vertical position, with its divisions 
 in a plane perpendicular to the axis of the telescope, and at a 
 distance of 200 cm. from its object-glass. Focus the telescope 
 on the divisions, by sliding the draw-tube in which the eye- 
 piece is set. It is possible now to see the image in the tel- 
 escope with one eye, at the same time that the scale itself is 
 seen with the other; and by slightly turning the telescope, if 
 necessary, to see the divisions of the image apparently super- 
 posed on those of the scale. 
 
 (a.) When this can be easily seen, count the number of 
 large divisions and fractions which are covered by one large 
 division of the image. This is the magnifying power of the 
 telescope when used on objects at this distance. 
 
 (b.) Repeat for distances of 150, IOO, 75, 60, 50, 40, and 30 
 cm. For the shorter distances, use the i-mm. divisions at one 
 end of the scale. 
 
 (<:.) Plot the results on cross-section paper, using distances 
 as abscissae and corresponding magnifying powers as ordinates. 
 Draw a curve through the points thus obtained, which will 
 give the magnifying power at any distance within its range. 
 
 B. MASS OF GIVEN VOLUME OF AIR. 
 
 To determine the mass of a volume of air inclosed in a 
 flask, over water. 
 
 (a.) Mark the water-level in the neck of the flask by means 
 of a mark on a gummed label, and measure its height (h) 
 above the water-level in the beaker. 
 
 (7>.) Read the barometer (see Exp. 2, a\ From its height 
 (H) and (h), find the pressure on the inclosed air in cm. of 
 mercury. 
 
8 MASS OF GIVEN VOLUME OF AIR. [B 
 
 (r.) The inclosed air may be assumed to be saturated with 
 aqueous vapor. Find from No. 15 of Whiting's tables the 
 maximum pressure of aqueous vapor for the working temper- 
 ature. Subtracting this from (b) gives the pressure due to 
 the dry air in the flask. 
 
 (d.) Empty the flask, wipe off any moisture on the outside, 
 and weigh on trip scales. Fill with water to the mark on the 
 neck, and weigh again. Find from table 22 the volume of I 
 gram of water at the working temperature, and calculate the 
 volume of the water, which is that of the air previously 
 inclosed. 
 
 Why should the interior of the flask not be dried before 
 weighing it empty? 
 
 (e.) Using the formula for (W d ) in (c), Exp. 2, with the 
 proper pressure substituted for H-f, calculate the mass of the 
 inclosed volume of air. 
 
l] USE OF BALANCE. 9 
 
 i. USE OF BALANCE. 
 
 Determination of the weight in air of a billiard-ball (cellu- 
 loid or ivory). 
 
 Put the ring in which the ball is to be set to keep it from 
 rolling about, in the left pan, and its counterpoise in the right 
 pan. 
 
 (a.) Raise the beam of the balance by means of the lever 
 in front. Move the hand near one of the pans if necessary, 
 until the pointer swings over a range of something like ten 
 scale divisions. Then close the case, read and record 5, 7, or 
 any odd number of turning-points of the pointer, in terms of 
 the divisions as numbered on the scale, estimating tenths of 
 the small divisions as closely as possible. Unless successive 
 turning-points differ by a fairly constant amount, showing that 
 the vibration is regularly decreasing, the series should be 
 repeated. 
 
 The second observer may read turning-points at the same 
 time, from the rear of the case, by setting up a plane mirror 
 parallel to and facing the scale. Use both eyes, and look on 
 both sides of the central post at once, in order to avoid 
 parallax. 
 
 Average the turning-points to the right, and those to the 
 left, separately. The average of these two results is the 
 resting-point of the pointier. 
 
 (&) Put the ball in the ring on the left pan, and add weights 
 to thfc right pan until, on raising the beam, the pointer swings 
 about a point near the middle of the scale. Determine the 
 resting-point. 
 
 Always handle the weights with pincers, taking care not to 
 bruise or scratch them. Never change the weights on the 
 pan without first lowering the beam. 
 
 (c^) Increase or decrease the weights used by two or three 
 
to CALIBRATION OF BAKODEIK. [2 
 
 centigrams, so as to move the resting-point toward the center 
 of the scale, then determine this new resting-point. Calculate 
 the change in weight, in milligrams, required to change the 
 resting-point by one scale-division. This is called the sensi- 
 tiveness of the balance for the load used. 
 
 (d.) Calculate the true weight of the ball namely the 
 weights which would give the resting-point found in (a). 
 Carry out the result to milligrams, and correct it by applying 
 to the indicated values of the weights used, the corrections 
 given* on the box. 
 
 (e.) The result given above is the weight required, if the 
 balance-arms are of equal length. To test this, weigh the ball 
 in the right pan. Then (G. and S., p. 100) calling the balance 
 arms R and L and the weights of the ball in the corresponding 
 pans W v and W^ 
 
 = V'W 1 /W, 
 
 and the true weight of the ball, W = W a (R/L). 
 
 In future weighings with this balance, the ratio of the arms 
 as found above should be used if it differs from unity by an 
 amount great enough to affect the last figure desired in the 
 result. 
 
 2. CALIBRATION OF BARODEIK. PART i. 
 
 The Barodeik is an ordinary balance, having a hermetically- 
 sealed flask suspended from one scale-pan, and from the other, 
 as a counterpoise, a glass plate so chosen as to have a surface 
 about equal to the exterior surface of the flask. The reading 
 of the balance pointer on a properly graduated scale gives the 
 density of the surrounding air. 
 
 To find the difference between the barodeik reading and 
 the true density of the air. 
 
 (a.) Set and read the barometer (G. and S., p. 155). The 
 complete setting and reading should be repeated, by each 
 
2] CALIBRATION OF BARODEIK. 1 I 
 
 observer independently, until the readings agree reasonably 
 well, and their average taken. 
 
 (b.) Read the wet and dry bulb thermometers. Apply cor- 
 rections to the readings as indicated, and calculate dew-point 
 by the formula given on the instrument. Then from No. 15 
 of Whiting's tables find the pressure of aqueous vapor in the 
 air, remembering that "dew-point" means the temperature at 
 which the aqueous vapor now in the air would saturate it; or 
 the existing pressure of aqueous vapor in the air would be the 
 maximum pressure if the temperature were lowered to the 
 dew-point. 
 
 (c.) From (a) and (&} calculate the density of the air. The 
 mass of one cu. cm. of dry air, at o (C) and 76 cm. pressure, 
 is 0.001293 grams. The mass of the same volume of aqueous 
 vapor, under the same conditions, is f as much. Then if h be 
 the barometer height,/ the pressure of aqueous vapor, and t 
 the temperature, the mass of dry air in I cu. cm. of moist air 
 is: 
 
 TT_r 
 
 W, = 0.001293 - 
 3 I +at 76 
 
 (a is the coefficient of expansion of a gas.) 
 And the mass of aqueous vapor in the same volume is: 
 
 W v = (I)o.oo, 293 -L 
 
 The sum of these two is the required density. (Deschanel, 
 p. 400-401.) 
 
 (df) Read the barodeik. 
 
 Do not touch the instrument, but by moving the hand near 
 one of the scale-pans, set up a small vibration. Then close 
 the case, and determine the resting-point of the pointer, which 
 is the density of the air as indicated by the instrument. 
 
 (/.) Record the difference between the reading thus ob- 
 tained and the true density from (c), with the proper sign, so 
 
12 CALIBRATION OF BARODEIK. [3 
 
 that when added algebraically to the observed reading it will 
 give the true density of the air. 
 
 This is the absolute correction for the scale-division to which 
 it applies. 
 
 3. CALIBRATION OF BARODEIK. PART 2. 
 
 Relative calibration of barodeik scale. 
 
 (a.) Read the instrument as in Part I (d). Repeat with a 
 2 eg. weight on the right-hand scale-pan, then with the same 
 weight on the left scale-pan. 
 
 (b.) Repeat with a 5 eg. weight. 
 
 Finish the observations before beginning the following cal- 
 culations, so as to make room for those working on Part I. 
 
 (c.) Using the exterior volumes of flask and plate, as given 
 on the instrument, calculate the changes in the density of the 
 air which would produce the same effects on the instrument as 
 the putting of the separate weights on the right pan, and on 
 the left pan. 
 
 From these results construct a table of corrections, with the 
 proper signs, for the different resting-points observed. 
 
 Note that this is a relative calibration. That is, it gives the 
 corrections to be applied to certain readings, as compared 
 with one reading (namely, that when no weights were used), 
 which is assumed correct. 
 
 (d.) In Part I, the absolute correction for a certain reading 
 was found. That reading was not far from the one assumed 
 correct above, so the same absolute correction may be applied 
 to the latter. By means of this, convert the table of relative 
 corrections (c) into a table of absolute corrections. This com- 
 pletes the absolute calibration of the instrument. 
 
 (e.) Plot on cross-section paper the readings on the barodeik 
 scale as abscissae, and the relative corrections of (c) as ordinates, 
 but on a much larger scale. Draw a curve through the 
 
5] HYDROSTATIC BALANCE. 13 
 
 plotted points, which will give the correction for any reading 
 on the scale. 
 
 Show how the curve can be made to indicate absolute cor- 
 rections, instead of relative, by moving the horizontal axis of 
 reference up or down by a proper amount. This converts it 
 into an absolute calibration curve for the instrument. 
 
 4. WRITTEN EXERCISE. 
 
 In place of the regular laboratory exercise, write out the 
 following requirements in the note-book, and hand in copy 
 as usual. 
 
 Make the statements clear and complete, and illustrate by 
 diagrams when possible. Ask for assistance on any points 
 which are not understood after a reasonable amount of study. 
 
 1. Why is an odd number of turning-points used in deter- 
 mining the resting-point of a balance pointer? 
 
 2. Suggest any consideration which would cause the sen- 
 sitiveness of a balance to change for a change of load. 
 
 3. Show that the calculation of the weight in i(d) is 
 simply a process of interpolation. 
 
 4. Show that the method used for finding the resting- 
 point of a balance pointer gives a better result than could 
 be obtained by reading the position of the pointer after 
 allowing it to come to rest. 
 
 5. Give a full explanation of the action of the barodeik. 
 Why should the flask and plate have equal surfaces exposed 
 to the air? 
 
 5. HYDROSTATIC BALANCE. 
 
 Determination of mass, volume, and density of billiard 
 ball. Use the same balance, weights, and ball as in (i). 
 The resting-point with pans unloaded should be determined 
 
14 THERMOMETER CALIBRATION. [6 
 
 anew each time the balance is used. The sensitiveness need 
 not be re-determined unless the load is quite different. 
 
 Put the arch over the left pan, and on it a beaker of water. 
 Suspend the ball in its wire cage from the hook above the 
 pan so that the cage is entirely immersed, and only the fine 
 suspending wire cuts the surface of the water. 
 
 Take the temperature of the water before and after the 
 weighing, and consider the mean of the two to be the tem- 
 perature of both water and ball during the experiment. 
 
 (a.) Weigh as in (i), and read the barodeik. 
 
 (b.) Remove the ball, fill up the beaker until the suspend- 
 ing wire of the cage is immersed to the same point as before, 
 and weigh the cage. 
 
 (c.) Calculate the volume of the ball, assuming volume of 
 I gram water = one cu. cm. as an approximate value to be 
 used in the subsequent work. 
 
 (d.} Calculate the following: 
 
 Weight of air displaced by the ball when weighed in (i). 
 
 Weight of air displaced by the weights used, assuming 
 their density to be 8.4. 
 
 Weight of the ball in vacuo. 
 
 Mass of the ball. 
 
 (e.~) Calculate the weight of air displaced by the weights 
 representing the weight of the ball in water in (a) and (^). 
 From this find what the ball would have weighed in water, 
 if the weighing had been done in vacuo. 
 
 (f.) Find the weight in vacuo of the water displaced by 
 the ball, and from table 23, calculate the exact volume of the 
 ball. From this volume and the mass, calculate the density 
 of the ball at the temperature of the determination. 
 
 6. THERMOMETER CALIBRATION ABSOLUTE. 
 
 Calibration of a 100 thermometer, reading to i, which 
 is to be qsed in subsequent work. 
 
6] THERMOMETER CALIBRATION. 1 5 
 
 (a.) Set a wire-gauze cone into the mouth of a beaker^ 
 and fill it with crushed ice. Put the thermometer bulb into 
 the middle of the ice, and let it remain for a few minutes 
 after the reading has become apparently constant. 
 
 Record the reading (to o. i) and the difference between 
 it and o with the proper sign as the correction to the 
 reading at o. 
 
 To avoid parallax in the readings, always turn the stem 
 so that the reflected images of the divisions in the mercury- 
 thread can be seen, apparently as prolongations of the divi- 
 sions themselves. 
 
 (b.) See that the steam-heater contains a sufficient supply 
 of water. Hang the thermometer in the central vertical tube, 
 with the IOO mark just above the cork, arranging the tubes 
 so that the steam will pass up the central tube, down the 
 space between the tubes, and out at the bottom of the outer 
 tube. Trjus the thermometer stem is surrounded by steam, 
 and this again by a jacket of steam. 
 
 After steam has issued for a few minutes, adjust the lamp 
 so as to give a very feeble current Why is this necessary? 
 
 Record the reading as before, when it has become steady. 
 Read the barometer, and find corresponding temperature of 
 steam from table. Record the difference between this and 
 the observed reading, with the proper sign, as the correction 
 at 100. 
 
 (c.) Let the thermometer cool slowly to about the tem- 
 perature of the room, then repeat (a). If the result differs 
 from that previously obtained, ask for advice. 
 
 (</.) Ask for assistance in breaking off" a thread of mercury 
 about 50 in length. 
 
 1. Set the lower end at the o mark and read the upper end. 
 Set the upper end at the 100 mark and read the lower end. 
 
 2. Repeat with a thread slightly longer or shorter, accord- 
 ing a.s the first was too short or too long, 
 
1 6 THERMOMETER CALIBRATION. 
 
 3. Calculate for each case what the readings would have 
 been if the ends of the thread had been set at the true o 
 and 1 00 marks. 
 
 4. The mean of the four readings in (3) gives the middle 
 point of the thermometer, or the true 50 point. From this 
 find the correction at 50. 
 
 (*?.) Break off a thread 25 in length, and using the o and 
 50 marks as starting-points, find as in (d) the first quarter- 
 point, or true 25 point, and the corresponding correction. 
 
 (/) With the 50 and 100 marks as starting-points, find 
 the third quarter-point, or true 75 point, and the correction. 
 
 (g.} Plot on cross-section paper the divisions on the ther- 
 mometer as abscissae, and the corrections on a much larger 
 scale as ordinates. Draw as smooth a curve as possible 
 through the plotted points. This gives a calibration-curve 
 for the instrument. 
 
 7. THERMOMETER CALIBRATION COMPARATIVE. 
 
 Calibration of a 50 thermometer reading to o.i, by com- 
 parison with a standard. This standard is a thermometer 
 like the one to be calibrated, but which has been carefully 
 compared with a reliable standard It should always be 
 returned to the desk after using. 
 
 Tie your thermometer to the standard with soft cotton 
 twine, winding it between the stems so as to separate them 
 slightly. Put the bulbs nearly opposite each other; and see 
 that corresponding divisions are as nearly opposite as is con- 
 sistent with this condition. Suspend the two securely with 
 the bulbs in the middle of a kettle of water, and steady the 
 stems by catching them loosely, without pressure, in a clamp. 
 The thermometers are to be read by a paper-tube telescope, 
 which slides easily on the vertical rod of an iron stand. This 
 should be set with its object-glass at a distance of 40-50 cm. 
 
7] CALORIMETRY. 17 
 
 from the thermometers, which should be perpendicular to its 
 axis. When taking a reading, always set the telescope so 
 that the top of the mercury column appears in the middle 
 of the field of view, not near its upper or lower edge, in 
 order to avoid parallax. 
 
 (a.} Take a careful series of readings to o.oi, at intervals 
 of 2 or 3, from about 12 to 45. Keep the water well 
 stirred, and keep the temperature fairly constant for a few 
 minutes before each reading. A good plan is to take a 
 preliminary reading of each thermometer, in order to see 
 about where the reading is going to come. The two exact 
 readings can then be made so quickly as to be practically 
 simultaneous. 
 
 Read again in a few seconds, taking the thermometers in 
 reverse order. Repeat if necessary until the differences 
 obtained from two such readings agree fairly well. 
 
 ($.) Let the observers change places, and take a similar 
 descending series, cooling the water by dipping out hot and 
 adding cold. 
 
 (c.} Ask to see the calibration curve of the standard used, 
 and from it construct a table of corrections to your thermome- 
 ter at the temperatures used. 
 
 (d.} Plot a calibration curve as in (5). 
 
 CALORIMETRY GENERAL. 
 
 The calorimeter apparatus used consists of a light copper 
 cup, the calorimeter cup proper, and a brass water-jacket, 
 which is wrapped with felt and closed with a wooden cover. 
 The cup rests on three corks in the central space of the 
 jacket, and is thus surrounded by a jacket of air, except for 
 the cork supports and the wooden cover, both poor con- 
 ductors. A central opening in the cover, above the cup, is 
 closed by a cork through which the thermometer passes. 
 
1 8 WATER EQUIVALENT OF CALORIMETER. L 8 
 
 Two stirrers, one for the cup and one for the jacket, are 
 attached to a common rod which is moved vertically by means 
 of a lever attached to a wooden stand on which the apparatus 
 is placed. 
 
 Use the thermometer which was calibrated in (7), and read 
 it with a telescope to o.oi, as was done there. This will be 
 called the "first thermometer." A second thermometer will 
 be furnished when necessary, to take the temperature of a 
 substance which is to be put into the calorimeter. When this 
 is done, the second thermometer should be compared with the 
 first in a water-bath at nearly the same temperature, in 
 order to find the correction to be applied to its reading. 
 Substances to be poured into the calorimeter should be within 
 i or 2 of the temperature o'f the room, in order to guard 
 against change of temperature in pouring. 
 
 Before putting either thermometer into a hot liquid, always 
 test with a 100 thermometer to make sure that its tempera- 
 ture is under 50. 
 
 The difference in temperature between cup and jacket is 
 usually made as small as possible by adjusting the temperature 
 of the water in the jacket to that necessary for the cup. The 
 rate of change of temperature of the cup, due to radiation 
 across the air-space which is caused by this difference, is thus 
 made as small as possible, and being fairly constant, can be 
 observed and allowed for. 
 
 All weighings should be made accurate to about 0.2 per 
 cent, which can be done on trip scales except where otherwise 
 specified. 
 
 8. WATER EQUIVALENT OF CALORIMETER. 
 
 Determination of the thermal capacity of the cup, with its 
 stirrer and the portion of the thermometer which is immersed 
 in water when the cup is nearly filled, as in ordinary use. 
 
8] WATER EQUIVALENT OF CALORIMETER. 19 
 
 This being the number of grams of water to which the appa- 
 ratus is thermally equivalent, is called its "water equivalent." 
 
 (a.) Weigh a beaker containing enough water to half fill 
 the cup, and set it on the stand beside the calorimeter so that 
 the second thermometer can be hung with its bulb in the 
 water. Adjust the thermometers so that either can be read 
 easily by merely turning the telescope, without altering its 
 focus. 
 
 Weigh the cup, fill ^ full of water at about 40, and weigh 
 again. Fill the jacket with water at the same or a somewhat 
 higher temperature, pouring it in through a small funnel. Be 
 careful always to avoid getting water into the air-space in the 
 jacket, and also to avoid wetting the felt wrapping. 
 
 (.) While one observer keeps the stirrers in gentle but 
 continuous motion (one stroke in 4 or 5 seconds is usually 
 sufficient) let the other read the thermometer, at intervals of 
 one minute, for 4 or 5 minutes. By this time the difference 
 between successive readings should be quite constant. At 
 some time during the series, stir the water in the beaker with 
 the thermometer bulb, and read its temperature between two 
 readings of the first thermometer. Now raise the cork from 
 the calorimeter cover, and pour in enough water from the 
 beaker to nearly fill the cup. This should be done by the one 
 who has been handling the stirrer, and so timed that the mixture 
 is made at a half minute, between two readings. Resume the 
 stirring at once, and continue the thermometer readings at 
 each minute for a few minutes as before. 
 
 (c.^ From the average rate of change of temperature of the 
 cup before and after making the mixture, calculate the tem- 
 perature of the cup at the moment before the mixture was 
 made, and at the moment after, assuming that the change in 
 temperature due to the mixture was instantaneous. 
 
 Reweigh the beaker to find weight of cold water used. 
 
 Denoting by x, the thermal capacity of the apparatus, 
 
2O WATER EQUIVALENT OF CALORIMETER. [8 
 
 which is the quantity to be determined, construct an equation 
 involving the following quantities: 
 
 1. Quantity of heat lost by the hot water in the cup. 
 
 2. Quantity of heat lost by cup, stirrer, and thermometer. 
 
 3. Quantity of heat gained by cold water poured in. 
 From this equation find x. Use corrected temperatures in 
 
 calculating. 
 
 (e,) Repeat with initial temperature a few degrees lower, 
 and take the mean of the two results. 
 
 (/.) As a check on the preceding result, calculate the 
 thermal capacity of the cup and stirrer from their weights and 
 the specific heats of copper and brass, which may be found in 
 the tables. To this must be added the thermal capacity of 
 the immersed portion of the thermometer, which may be 
 experimentally determined with sufficient accuracy as follows: 
 
 Set up the calorimeter, leaving out the cork, with water at 
 the room temperature in both cup and jacket, having first 
 weighed the cup and contained water. Record the tempera- 
 ture when it has become steady, then take out the thermom- 
 eter and immerse it, to the same depth as is usual in the cup, 
 in water at about 45. After a few minutes, read the temper- 
 ature to o.i, then as quickly as possible take out the ther- 
 mometer and put it into tke usual position in the calorimeter, 
 shaking off superfluous water on the way. Stir until the 
 temperature becomes steady and record the reading. Then 
 calculate the thermal capacity of the thermometer according 
 to the general method used in (d). 
 
 (.) If this result differs from the final result in (e) by more 
 than one unit, ask for advice. If it does not, use the mean of 
 the two as the water equivalent of the apparatus in future 
 work. 
 
 The different sets of apparatus are nearly enough of the 
 same dimensions so that this result may be used with any 
 of them. 
 
I0 1 HEAT OF SOLUTION. AMMONIUM NITRATE. 21 
 
 9. SPECIFIC HEAT OF LEAD SHOT. 
 
 Set up the apparatus, take observations and calculate 
 results as in Exp. 8, (a), (/;), (c) and (if), with the following 
 modifications: 
 
 (#.) For the beaker of water, substitute a copper cup con- 
 taining about 700 grams of dry shot Fill the calorimeter 
 cup only about % with water, to avoid risk of running it over 
 when the shot is poured in. 
 
 (&) When the shot is poured in, the temperature will not 
 be well equalized by the ordinary stirring, as the stirrer will 
 not penetrate the mass of shot. It is necessary to unhook 
 the thermometer from its support and stir the shot very care- 
 fully with its bulb for about one minute. Then replace the 
 cork, hang up the thermometer and resume the readings, one 
 of which has thus been omitted. From this point the ordinary 
 stirring is sufficient. 
 
 (c.) In writing the equation, x should now denote the 
 specific heat of the shot, and the thermal capacity of the 
 apparatus is known. 
 
 (d.} Repeat with a fresh portion of shot, and a higher initial 
 temperature of about 46. 
 
 Wash the shot well after use, and spread it in a thin layer 
 on a cloth to dry. 
 
 10. HEAT OF SOLUTION. AMMONIUM NITRATE. 
 
 The quantity of heat absorbed in the solution of one gram 
 of a substance is called its heat of solution. If heat is given 
 out in the solution, the quantity is considered negative. 
 
 If the temperature of the salt after solution be different 
 from that at which it was poured into the water, it will be 
 necessary to consider its specific heat also. According to the 
 following method the heat of solution and the specific heat 
 
22 LATENT HEAT OF VAPORIZATION. [10 
 
 are both determined, although the former is the main object 
 of the experiment. 
 
 On one of the Becker balances weigh out two portions of 
 the salt, each of 10 grams, to o.oi gram. This can be con- 
 veniently done by putting two equal pieces of paper in the 
 two pans, and pouring the salt on one of them. Never put 
 the salt directly into the pan. Put each portion, after weigh- 
 ing, into a smallest-size beaker which is thoroughly dry. 
 
 This amount of salt, when dissolved in about 2OO grams of 
 water, will lower its temperature a little over 3. It is best 
 to have the cup about 3 degrees warmer than the jacket, 
 because the larger part of the salt dissolves in a few seconds, 
 so that the cooling during this time is small; and the tem- 
 perature being then reduced to about that of the jacket, there 
 is no cooling during the longer time required for the complete 
 solution of the salt. 
 
 (a.) With the jacket at the room temperature and the cup 
 about 3 higher, proceed as in (8). In removing the second 
 thermometer from the beaker of salt, after taking its tempera- 
 ture, see that no salt is removed. After pouring in the salt, 
 stir rather vigorously to hasten the solution, and record the 
 temperature when it has become steady. 
 
 (.) Make a similar trial with the second portion of salt, 
 having the jacket at 40 or higher. Make sure that there is 
 the proper difference of temperature between the cup and 
 the jacket before beginning the observations. 
 
 (c.) Calling the specific heat of the salt x and its heat of 
 solution y t write the proper equation for each case, and solve 
 the two equations for x and y. 
 
 CAUTION. Do not leave the solution standing in the copper 
 cup. Wash it out as soon as possible. 
 
 ii. LATENT HEAT OF VAPORIZATION. 
 Examine the construction of the steam-trap, and see that 
 
II] 
 
 LATENT HEAT OF VAPORIZATION. 
 
 when held vertically, with the exit tube down, it will catch 
 and discharge through the side vent all water which is con- 
 densed on the way from the boiler, and permit only dry steam 
 to issue from the exit. Fill the boiler nearly full of water, 
 heat it to boiling, then remove the lamp until ready to use 
 the steam. 
 
 The steam will be condensed in a copper tube, closed at the 
 bottom, which projects downward from the lower surface of 
 the cork into the calorimeter cup. This tube, closed with a dry 
 cork and well dried on the outside, is to be weighed, before and 
 after using, on one of the delicate balances. These balances 
 must be handled with especial care. See that the pans are 
 protected by sheets of filter-paper. Adjust the weights to the 
 nearest centigram, then use the rider to obtain the nearest 
 milligram. Never touch weights or rider without first arrest- 
 ing the pans. 
 
 The cut is a section through the cork, 
 showing the proper arrangement of the 
 trap and tube when in operation. See 
 cut. The cup must be filled until the 
 water rises almost, but not quite, to the 
 lower surface of the cork, so that the 
 copper tube may be immersed as deeply 
 as possible. The proper amount of 
 water should be determined by a prelim- 
 inary trial. 
 
 Lay a small piece of blotting-paper over the hole in the 
 corkj and punch a corresponding hole through the paper, 
 using the exit tube as a punch. When steam is issuing from 
 the tube, an occasional drop of water forms at the mouth, 
 which will be absorbed by the paper. The object of these 
 precautions is to make sure that the water which is collected 
 and weighed in the tube is simply that which, in condensing 
 on the walls of the tube, gives up its heat of vaporization to 
 the water in the cup. 
 
 i 
 
24 HEAT OF CHEMICAL COMBINATION. 12 
 
 Put in the cup the proper amount of water at a temperature 
 about 3 below that of the jacket, which should be at the 
 room temperature. Set the lamp under the boiler, and adjust 
 its height so that when steam is issuing freely the water stands 
 at about 25 cm. in the manometer tube. This gives a vigorous 
 current of steam, without raising the boiling point too much. 
 The stirring'of the jacket may be dispensed with, and the cup 
 stirred directly by hand. After reading temperatures for a 
 few minutes, put the steam-trap in position, with the side 
 vent pointing away from the thermometer, and press down so 
 tightly that no steam escapes around the cork. Stir gently 
 until the temperature has risen about 6, then remove the 
 trap, continue stirring, and record the highest temperature. 
 Corrections for cooling are practically unnecessary, since the 
 range of temperature is distributed equally above and below 
 that of the jacket. 
 
 In calculating results, remember to account for the quantity 
 of heat given out by the condensed steam in cooling from the 
 boiling-point (which may be taken as 100) to the temperature 
 of the cup. 
 
 Make a second trial, which should give a good result, with 
 the practice in handling the apparatus acquired during the 
 first trial. 
 
 12. HEAT OF CHEMICAL COMBINATION. 
 
 Determination of the heat generated by the combination of 
 NaOH with HC1 to form NaCl. 
 
 A 0.5 normal solution of each of the above compounds is 
 furnished. By a normal solution is meant one which, in 1,000 
 grams of the solution, contains a weight of the element which 
 is to enter into the new combination equal in grams to its atomic 
 weight. Thus the normal solution of NaOH is a solution 
 which contains, in 1,000 grams, 40 grams (23+16+1) of 
 NaOH, or 23 grams of Na. The 0.5 normal solutions used 
 contain one-half of these proportions. 
 
J 3] COEFFICIENT OF EXPANSION OF A LIQUID. 25 
 
 It is evident that if equal weights of these solutions be 
 mixed, the reaction will be just completed, and the result will 
 be a neutral solution of NaCl. 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. 
 
 Weigh out 100 grams of the NaOH solution in the cup, 
 and the same amount of the HC1 solution in the beaker. The 
 latter should be weighed out roughly at first, poured back 
 into the bottle, then the wet beaker counterpoised and the 
 amount weighed accurately. This will then be quite accu- 
 rately the weight which is afterward poured into the calori- 
 meter. A small error is introduced by taking the second 
 thermometer 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 necessary, the two temperatures should be 
 very nearly the same when ready for use. Since the amounts 
 used are equal, it may be safely assumed that the resulting 
 solution of NaCl has risen to the final temperature from the 
 mean of the two initial temperatures. 
 
 A direct determination of the specific heat of the NaCl 
 solution is impracticable. The value 0.987, which has been 
 -calculated by interpolation from tabulated results, may be 
 used for this case. 
 
 Mfcke two trials, and calculate for each the number of heat 
 units which would have been evolved had normal solutions 
 been used. 
 
 13. COEFFICIENT OF EXPANSION OF A LIQUID. 
 
 The method consists in determining the weights of alcohol 
 filling a specific gravity bottle at several different temperatures, 
 
26 COEFFICIENT OF EXPANSION OF A UQUID. [*3 
 
 and from these data calculating the coefficient of expansion. 
 Four determinations should be made, at intervals of about 8, 
 beginning with the room temperature. 
 
 (a.) Fill the bottle with alcohol and set it on a platform in 
 a kettle of water, so that the water comes as high as the 
 alcohol in the bottle. See that the stopper is set in very lightly. 
 Hang your 50 calibrated thermometer in the bath beside 
 the bottle, and keep the bath well stirred for about 5 minutes. 
 The temperature of the bath should remain constant within 
 o.i during this time, and at the end of it the alcohol will 
 have the same temperature within o.i. 
 
 Set the stopper in tightly, using no more pressure than 
 necessary, take the bottle out of the bath, wipe dry and weigh 
 to o.ooi gram on one of the delicate balances. 
 
 (b.) Repeat with the bath at about each of the higher tem- 
 peratures selected, holding the temperature steady for 10 
 minutes by holding the lamp under the kettle for a few 
 seconds occasionally. Before putting the bottle in the bath, 
 loosen the stopper to allow the excess of alcohol to escape, 
 and try it occasionally to see that it remains loose during the 
 warming. Careful trial has shown that, after this treatment, 
 the temperature of the alcohol at the center of the bottle, as 
 shown by the thermometer in the stopper, is about o. I lower 
 than that of the bath, and therefore the average temperature 
 of the alcohol is the same as that of the bath to within o.i. 
 
 (c.) Empty the alcohol into the bottle from which it was 
 taken, dry the bottle with the jet-pump, and weigh. (Ask for 
 directions as to the use of the jet-pump.) 
 
 (*/.) Find the weight of alcohol filling the bottle at each 
 temperature. Plot the results, with the corrected tempera- 
 tures, starting from o, as abscissae, and changes in weights 
 as ordinates. Assuming that the expansion is uniform, draw 
 the straight line which best represents the plotted points, and 
 from it find the weight filling the bottle at o. 
 
14] VOLUMENOMETER. 2/ 
 
 (e.) From the weights filling the bottle at o and at 40, 
 calculate the relative volume of a given weight of alcohol at 
 40, referred to its volume at o. From this calculate the 
 coefficient of cubical expansion of the alcohol for the given 
 range of temperature. (Coefficient of expansion is expansion 
 of I cu. cm. from o to i.) 
 
 (/.) Note that what you have obtained is not the absolute 
 coefficient for alcohol, since the bottle also expands. Find 
 from the tables the coefficient of cubical expansion of glass, 
 and by applying it to the above result, find the absolute 
 coefficient for the alcohol. Remember that the expansion of 
 the interior of a glass bottle is equal to the expansion of the 
 same volume of solid glass. 
 
 14. VOLUMENOMETER. 
 
 Density of NaCI Crystals. 
 
 The apparatus is designed to measure the volume Fof'air 
 inclosed above a certain mark on the glass tube. The given 
 volume is inclosed over mercury, and its pressure p meas- 
 ured by the manometer. By raising or lowering the manom- 
 eter tube, the volume of the inclosed air is changed by an 
 amount v lt which can be measured, and the corresponding 
 pressure p l is read. By Boyle's law, pv const. Hence 
 pV=p^ V lt where V l = F-H'i, from which V, the original 
 volume, is determined. The temperature is assumed to re- 
 mam constant. 
 
 CAUTION. Put no dirty mercury into the instrument, and 
 see that there is a mercury-tight tray on the floor under- 
 neath it. 
 
 (a.) Having the brass tube unscrewed from the upper end 
 of the fixed glass tube, so that the latter is open to the air, 
 set the movable tube, pouring in more mercury if necessary, 
 
28 VOLUMENOMETER. [14 
 
 so that the top of the column in the fixed tube is at the 
 middle mark. See that the brass tube is empty except for a 
 plug of cotton. Clean off the old grease from its end, and 
 from the shoulder on the brass cap, smear a little fresh grease 
 evenly on the screw-threads of the cap, and screw on the 
 tube, making sure that it fits tightly against the shoulder. 
 
 Readjust the open tube so that the edge of the meniscus 
 in the closed tube is again at the mark. The volume now 
 inclosed above the mercury is the volume Fto be determined. 
 
 Read the position of the top of each column on the meter 
 rod by means of a straight edge laid across its face, and from 
 the difference in level and the barometer height, find the 
 pressure/ on the inclosed air. 
 
 (b) Raise the open tube, pouring in more mercury if neces- 
 sary, until the mercury rises to the upper mark in the closed 
 tube. Let stand a few minutes to allow the inclosed air to 
 return to the temperature of the surrounding air, then read 
 both columns. Let stand five minutes, and read again. Any 
 difference is probably due to leakage in the screw-joint, which 
 must be made tight. 
 
 Repeat the setting three or four times, bringing the column 
 in the closed tube to the mark and reading the open tube each 
 time. 
 
 From the heights of the two columns, find the pressure p v 
 
 (c.) Lower the open tube, catching the overflow in a beaker, 
 until the column in the closed tube sinks to the lower mark. 
 Read both columns as before, and find the pressure p, 2 for 
 this case. 
 
 (d) Bring the inclosed air back to nearly atmospheric 
 pressure, and unscrew the brass tube. Take out the' cotton 
 plug from the brass tube, fill it nearly full of clean NaCl 
 crystals, and replace the plug so that they will not fall out. 
 Repeat (a), (b), and (c), the volume V now being less than 
 before by the volume of the crystals. 
 
15] CALIBRATION OF THE TUBE. 2Q 
 
 (e.) A separate tube is furnished which contains the volumes 
 z\ and 2A 2 between marks corresponding to those on the vol- 
 umenometer. To find these volumes, fill the tube to the 
 marks with mercury, from a separate supply, pour into a 
 weighed beaker and weigh, using Becker balance. The vol- 
 ume of one gram of mercury at the room temperature may 
 be found in the tables. 
 
 (/) Calculate the value of Ffor each case, and so find the 
 volume of the crystals used. 
 
 Weigh the crystals, and calculate their density. 
 
 15, 16, 17, 18. MEASUREMENTS OF CHANGE OF 
 VOLUME WITH TEMPERATURE. 
 
 The apparatus used is practically a thermometer with a 
 large bulb, open at the end. This opening can be closed by 
 a stopper consisting of a flat surface of rubber which is pressed 
 against it by springs. The constants of the instrument are 
 determined in the first two experiments, for use in the two 
 succeeding. 
 
 The instrument furnished to each pair of students will be 
 used by them only, until the four experiments are completed. 
 
 15. CALIBRATION OF THE TUBE. 
 
 Always handle mercury over a mercury-tight tray. 
 
 (a.) The cm. divisions nearest the ends of the scale will be 
 considered as the ends of the scale to be used, corresponding 
 to the o and 100 marks on the ordinary thermometer scale. 
 Pour a little mercury into the bulb and run it into the tube, 
 so as to secure a continuous thread as long as the scale. 
 Adjust by pouring out a drop at a time from the end of the 
 tube. Read the length of the column, find its weight, to I 
 eg., and from this find the volume of the tube between the 
 two ends of the scale. 
 
3O EXPANSION OF THE THERMOMETER. [16 
 
 (b.) Find the middle and quarter points by the method used 
 in 6, and also find the eighth points, midway between the 
 quarter points. The bore of the tube is apt to vary consider- 
 ably in different parts, so that the quarter points will not give 
 a sufficiently good calibration. 
 
 (<:.) On a sheet of co-ordinate paper, lay off the divisions 
 of the scale as abscissae, and at each of the points on the 
 scale found above, including the upper end of the scale, erect 
 an ordinate which will represent the volume of the tube from 
 the zero up to that point. A curve through these points will 
 then give the volume of the tube between any two points on 
 the scale. Such a curve can be best drawn with a flexible 
 ruler, such as a strip of spring brass on edge, held at the ends 
 so as to pass through all the points. 
 
 16. EXPANSION OF THE THERMOMETER. 
 
 (a.) Fill the thermometer with clean mercury, getting the 
 right quantity by trial so that the top of the column will rise 
 in the tube to the lower part of the scale when the stopper is 
 set in place so as to inclose no air-bubbles. 
 
 Always steady the bulb with one hand when handling the 
 stopper, as a slight twist of the bulb may break the tube. 
 
 (b.) Immerse the bulb in a water-bath so that the tube will 
 be vertical, and suspend your 100 calibrated thermometer 
 beside it. Read the height of the column in the tube at 
 intervals of about 10 up to 60, holding the temperature of 
 the bath steady at each chosen point for a few minutes after 
 the reading of the expansion thermometer has become steady, 
 and reading the temperature to o.i. Then cool the bath, and 
 take a similar set of readings at temperatures intermediate 
 between those previously used. 
 
 If there is time, heat the bath again, taking a second series 
 of readings. 
 
17] EXPANSION OF COTTON-SEED OIL. 3! 
 
 Find the weight of mercury used, by carefully pouring it 
 out into a weighed beaker, and from this find its volume. 
 Use trip scales. 
 
 (c.) Plot the results, using corrected temperatures starting 
 from o as abscissae, and corresponding volumes of the tube, 
 from the volume-curve, as ordinates. Draw the straight line 
 which best represents the plotted points. Through the point 
 where this line intersects the ordinate at o, draw an axis 
 parallel to the axis of abscissae. The apparent expansion of 
 the mercury in the thermometer, from o to any given temper- 
 ature, will then be represented by the corresponding ordinate 
 measured from this axis to the plotted line. 
 
 (d.) At the highest temperature used, erect an ordinate 
 from this axis which will represent the absolute expansion of 
 the mercury used, from o to this temperature, and diaw a 
 line from this point to the o point. The difference between 
 corresponding ordinates to the two lines gives the expansion 
 of the apparatus. Construct a third plot which will represent 
 this expansion measured from a horizontal axis. 
 
 (>.) Assuming that the apparatus is made entirely of glass, 
 calculate the coefficient of expansion of the glass. 
 
 17. EXPANSION OF COTTON-SEED OIL. 
 
 Fill the instrument with oil, in the same way that it was 
 filled with mercury in the last experiment. There will be less 
 trouble with air-bubbles if the oil is poured into the bulb 
 slowly, in a steady stream. 
 
 (a.) Set the thermometer in a water-bath as before, and take 
 a similar set of ascending and descending readings. The oil is 
 such a poor conductor that it will be necessary to keep the 
 temperature of the bath constant at each reading for a much 
 longer time than with the mercury. 
 
 ($.) Plot the observations as in 16, (<r). To correct for the 
 
32 EXPANSION OF Nad CRYSTALS. . L 1 ^ 
 
 expansion of the apparatus, measure from this line at some 
 convenient temperature an ordinate, in the proper direction, to 
 represent the expansion of the thermometer from o to that 
 temperature, taken from the final plot in 16. Then a line 
 drawn through this point and the point where the first line 
 intersects the ordinate of o will represent the true expansion 
 of the oil. 
 
 (c.) Calculate the coefficient of expansion of the oil. The 
 volume at o of the oil used can easily be calculated, but if 
 the result be found in terms of the volume at the lowest tem- 
 perature used, it will be nearly enough the same. 
 
 Leave the oil in the thermometer for the next experiment. 
 
 ' '18, EXPANSION OF NaCl CRYSTALS. 
 
 The bulb is now to be filled with a weighed amount of 
 NaCl crystals, leaving the spaces between them filled with the 
 oil used in the last experiment. 
 
 Set the thermometer, still containing the oil, in the bath 
 with the bulb partly immersed, its axis being vertical, and 
 heat to 70. 
 
 Pick out and weigh enough clean NaCl crystals to more 
 than fill the tube. Remove the stopper from the bulb and 
 drop in the crystals one at a time. The heat, and gentle 
 stirring around with a bit of wire, will set free the air-bubbles 
 which at first are caught by the rough surfaces. After getting 
 in the proper amount of crystals, fill up with oil arid take a 
 set of readings as in 17. Weigh the remaining crystals, and 
 calculate the volume used from the density found in 14. 
 
 Plot the observations, and correct the line obtained for the 
 expansion of the thermometer, and of the volume of oil used. 
 
 From the resulting line representing the true expansion of 
 the crystals, calculate their coefficient of expansion. 
 
i8] ELECTRICITY. 33 
 
 ELECTRICITYGENERAL DIRECTIONS. 
 
 To set up a Daniell cell : The zinc plate is left standing in 
 water in the outer cup, and the inner porous cup, filled with 
 CuSO 4 solution and containing the copper plate, is kept soak- 
 ing in a jar of the solution. Clean off any scale from the 
 surface of the zinc by rubbing it with a rag or the fingers, 
 under the tap, and put it in place. Rinse off the outside of a 
 porous cup, which is filled to within 3 or 4 cm. of the top with 
 CuSO 4 solution, and put it inside the zinc. Fill the space 
 around it with dilute H 2 SO 4 from one of the labeled bottles. 
 
 When you have finished using the cell, put it away as it 
 was found, returning the acid to the bottle. 
 
 If gas is given off when the acid is first poured in, the zinc 
 needs amalgamating. Ask for directions. 
 
 When these precautions are observed, the electromotive 
 force and resistance of the cell will be very nearly the same 
 at one time as at another. The resistance may change slightly 
 for a few minutes on first setting up. 
 
 In setting up a circuit of any kind, clean all the connections, 
 including those at the battery terminals, with knife, sandpaper, 
 or file. Never make connections by twisting the ends of wires 
 together, as the resistance of such a connection is very variable. 
 Always use binding screws, and make sure that the ends of 
 the wire are firmly held by the screws. 
 
 If at any time the current seems to be variable, or too 
 weak, examine the connections first, as they are very likely to 
 be the seat of the trouble. 
 
 Do not move tangent galvanometers without first asking 
 permission; and handle them carefully, to avoid breaking the 
 silk fiber suspensions. If the pointer does not swing freely 
 after adjusting the leveling screws, ask for assistance. 
 
 In using galvanometers with pivot suspension, tap the 
 
34 TANGENT GALVANOMETER. [19-21 
 
 instrument with some soft object before taking a reading. 
 Raise the needle from the pivot before leaving the instrument. 
 
 Binding screws and connecting wires will be found at the 
 northwest corner of the room; return them to their places 
 after use. 
 
 In making connections, do not use wires which are longer 
 than necessary, and make a practice of arranging wires so that 
 currents in opposite directions shall be as near each other as 
 possible, in order to minimize effect on galvanometer needle. 
 Use wires of small diameter, except in special cases where it 
 is desirable to make the resistance of connections as small as 
 possible. 
 
 In using resistance boxes, take care to protect the plugs, 
 and the tapered holes into which they fit, from bruises and 
 dirt. Never interchange plugs between boxes without per- 
 mission. Keep boxes covered when not in use. 
 
 19, 20, 21. TANGENT GALVANOMETER. 
 
 GENERAL THEORY. A unit magnetic pole is defined as one 
 which, when placed in air at a distance of I cm. from a pole 
 of equal strength, will exert upon it a force of I dyne. 
 
 A unit current is defined as one which, flowing in a wire 
 I cm. long, bent into an arc of I cm. radius, will act on a 
 unit magnetic pole at the center of the arc with a force of 
 I dyne. 
 
 We shall speak of the field at the center of the coil, due 
 to the current flowing in it, as the force in dynes exerted by 
 the current on a unit pole at the center of the coil; let this 
 be denoted by F. 
 
 For a unit current, flowing in a circular coil of I turn, 
 radius I cm., F=2n; for n turns, F=2*n. If the radius of 
 the coil be r, F=2xn/r; since the field due to a given length 
 of wire varies as i/r 2 , but the length of wire in the coil is r 
 
19] TEST OF TANGENT LAW. 35 
 
 times as great This value of F t namely the field produced 
 at the center of the coil by a unit current flowing in the coil, 
 is called the constant K of the galvanometer. 
 
 //denotes 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 in directions at right 
 angles to each other, upon each pole of the needle. Assum- 
 ing 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 c cause 
 the needle to deflect through an angle a from its zero posi- 
 tion, show that in general tan a = FIH=cKjH. From this 
 c = (HjK) tan a. 
 
 c is here expressed in c. g. s. units. The commercial 
 unit, the ampere, is o.i of the c. g. s. unit. Hence if c be 
 expressed in amperes, c = (\oHjK) tan a. \H\K is called 
 the reduction factor R of the galvanometer. 
 
 19. TEST OF TANGENT LAW. 
 
 (a.) Set the galvanometer with its plane in the magnetic 
 meridian. With the instruments used, this is done by bring- 
 ing the ends of the pointer to o and 180. Set up a Daniell 
 cell, and connect it in series with the galvanometer and 
 resistance box. On the galvanometer set the plug of the 
 small switch-board in the hole marked 50; the current then 
 flows through a coil of 50 turns. Take out plugs from the 
 box until the deflection is about 50. 
 
 Determine the deflection by reading both ends of the 
 pointer, to o.i, then reversing the terminals to get a deflec- 
 tion 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 or 6, down to about 10. 
 
36 REDUCTION FACTOR. [20 
 
 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. 
 
 (.) Show the truth of the relation expressed on the pre- 
 ceding page, tan a F]H. By combining this with Ohm's 
 law, Ecr, where r is the total resistance in the circuit, show 
 that EjR = r tan # = r/cotan a. 
 
 If then you make a plot with resistances as abscissae and 
 cotangents of corresponding angles of deflection as ordinates, 
 the points obtained should lie on a straight line if the gal- 
 vanometer obeys the tangent law. 
 
 (<r.) 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? 
 
 The same galvanometer must be used for the next two 
 experiments. 
 
 20. REDUCTION FACTOR. 
 
 (#.) Set up your galvanometer with its needle as nearly as 
 possible over one of the numbered brass nails on the tables. 
 Straighten out two of the copper wire spirals from the 
 voltameter cells, clean and smooth them well with fine 
 emery paper, and re- wind on a brass rod 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 rod, and having the other end secured, 
 
20] REDUCTION FACTOR. 37 
 
 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 cells with fresh electrolyte 
 solution from the bottle. 
 
 Connect the two voltameters in series with a Daniell 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 German silver wire if necessary to reduce the deflec- 
 tion to about 25. 
 
 (b.) When the deflection remains practically constant for 
 several minutes, lift the spirals out of the solution and see 
 if they are uniformly covered with a bright and clean deposit. 
 If so, plunge them at once in 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 I mg. 'on the delicate 
 balance. 
 
 The battery should be allowed to send a current through 
 some convenient resistance while not in use. 
 
 (c.) Replace the spirals in the voltameters, put them in 
 circuit as before, and let the current run for 50 minutes 50 
 seconds. Read the deflection at each minute, reversing its 
 direction at about the middle of the 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. 
 
 - (d.) Remembering that the weight of copper in grams 
 deposited by the current in the given time is numerically 
 equal to the current strength in amperes, calculate the reduc- 
 tion factor for the coil used. If the deflection has not 
 remained constant, the mean of the tangents of the observed 
 angles should be used in the calculation. However, if the 
 variation has not been large, the tangent of the mean angle 
 may be used instead. % 
 
38 CONSTANT. [2! 
 
 Do not replace the used electrolyte solution in the bottle, 
 but pour it in the jar containing porous cups. 
 
 21. CONSTANT. 
 
 (a.) To calculate the constants of the i-turn, 5-turn and 
 5O-turn coils, from their measured dimensions. 
 
 Measure with a beam-compass the diameter of the ledge on 
 the face of the ring, which is 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 i-turn coil. The next layer, of No. 16 wire, con- 
 sists 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, which 
 may be connected in series with the five turns, to make the 
 5O-turn coil. 
 
 The coils of galvanometers I and 2 are not wound accord- 
 ing to the plan here described. If they are used, ask for 
 the plan. 
 
 Samples of No. 14 and No. 16 wire are furnished. From 
 the measured diameters of these, and the inner and outer 
 diameters of the whole coil, find the mean diameter of each 
 coil. Calculate the constant for each coil. 
 
 From the definition of the constant it is evident that the 
 constant for two coils in series is the sum of the two con- 
 stants taken separately. Find thus the required constants for 
 the i-turn, 5-turn and 5O-turn coils. 
 
22] VARIOMETER. 39 
 
 (b.) From the constants, and the reduction factor for the 
 5-turn coil, calculate the reduction factors for the l-turn and 
 5O-turn coils. From the plot of Exp. 19, take a pair of values 
 of r and cotan a, and find the electromotive force of the cell 
 used from these and the proper reduction factor. 
 
 (c.) Find the value of H at the point where the needle of 
 the galvanometer was situated in Exp. 20. Locate this point 
 by measuring, by means of two meter rods placed end to end 
 in succession, the distance (x) from the west wall of the 
 laboratory, (y) from the south wall, and (z) from the floor. 
 This gives the three co-ordinates of the point, referred to 
 rectangular axes with origin at the southwest corner of the 
 floor. 
 
 22. VARIOMETER. 
 
 From the value of H found in the last experiment, to deter- 
 mine its value at another point in the room. The point to 
 be used is No. 10, in the northeast part of the room. The 
 instrument is a tangent galvanometer, adapted to the purpose 
 by attaching to the top of the coil a carriage rotating in a 
 horizontal plane for the control magnet. As the strength of 
 this magnet is supposed to remain constant throughout the 
 experiment, it must not be jarred, nor brought into contact 
 with iron. 
 
 (a.) Release the galvanometer needle by raising the pin at 
 one side of the scale, and, having the control magnet removed 
 a 'short distance and stood up vertically to neutralize its 
 magnetic effect, turn the coil and scale until the pointer 
 stands nearly at o. Read both ends accurately. See that 
 there are no masses of iron on the table, or on the persons of 
 the observers, which might affect the readings. Now place 
 the control magnet in its carriage, which should be placed 
 approximately at right angles to the plane of the coil, and 
 
4O VARIOMETER. [22 
 
 clamped just firmly enough so that it will rotate with some 
 difficulty. 
 
 (b.) Read both ends of the pointer. Lift the control mag- 
 net from the carriage, and after reversing its direction, replace 
 it without moving the carriage. Read the pointer as before. 
 Calculate the true deflection in each direction from the zero 
 found in (a). If the two deflections agree, show that the 
 axis of the control magnet must be at right angles to the 
 earth's lines of force. If they do not agree, see which way 
 to rotate the carnage in order to make the difference less, and 
 by repeated trials reduce it to a small quantity. 
 
 How could this method be used to set the plane of the coil 
 of a tangent galvanometer in the magnetic meridian, by means 
 of the current in its coil ? 
 
 (<:.) Let each observer read several sets of deflections in 
 both directions. Take the mean of the series as the true 
 deflection. 
 
 (//.) Leaving the carriage clamped in position, raise the 
 needle from the pivot, and set the instrument with the needle 
 as nearly as possible at the point where the value of H is 
 known. Adjust the pointer to o, and take a series of read- 
 ings of deflections as before. If a galvanometer is in use near 
 by, determine whether it will affect your work by taking a 
 careful reading when a current is flowing in the coil, and 
 again when it is not. If the effect is appreciable, try to obtain 
 your readings when no current is flowing. 
 
 (e.) Again set the instrument in its first position, and repeat 
 the first set of readings. If the result does not agree with 
 that previously obtained, it is evidence that the strength of 
 the control magnet has probably changed. The mean of the 
 two results should be taken as corresponding most nearly to 
 the condition of things at the time when the readings in (c) 
 were taken. 
 
 (/.) Noting the similarity between the field produced at the 
 
23] VOLTMETER CALIBRATION. 41 
 
 center of the coil by the control magnet and that which would 
 be produced the-re by a current in the coil, show how to cal- 
 culate the ratio of the values of H at the two points, and from 
 this its absolute value at the new point. Find the co-ordinates 
 of this point as before. 
 
 23. VOLTMETER CALIBRATION. 
 
 Any galvanometer may be used as a voltmeter if its resist- 
 ance be sufficiently high. A mirror galvanometer is here 
 used, its deflections being measured by means of a telescope 
 and scale. 
 
 The galvanometer used has not the necessary high resist- 
 ance. Its coils are to be connected in series with a resistance 
 coil, and the galvanometer and resistance coil taken together 
 then constitute the voltmeter. 
 
 The method used for the calibration depends upon the fact 
 that if a constant current be maintained through a circuit of 
 given resistance, the E. M. F. between any two points on that 
 circuit varies as the resistance included between them. The 
 voltmeter is so arranged as to read the E. M. F. between 
 various pairs of points on such a circuit, which consists of a 
 wire of uniform resistance. 
 
 (a.) On the rheostat provided, each of the six loops and 
 the stretched wire are of the same length, and may for the 
 present be assumed to have the same resistance. This equality 
 will be tested afterward. Set up a Daniell cell at once, and 
 connect it to the terminals A and B of the rheostat, in order 
 that the current may become steady by the time the readings 
 are to be made. 
 
 Notice that if all three of the plugs provided are in place, 
 in any three holes, the current from the cell will flow through 
 four meters of wire. By changing the plugs from one end of 
 the board to the other, the straight meter on which the sliding 
 
42 VOLTMETER CALIBRATION. [23 
 
 contact works may be made the first, second, third or fourth 
 of the four meters; so that the sliding contact may be made 
 to move from one end to the other of the four meters through 
 which the current is flowing. The circuit as now set up will 
 be called the main circuit. 
 
 (b.) If the galvanometer does not work properly at any 
 time, do not attempt to adjust it, but ask for assistance. 
 
 Set the telescope about on the normal to the mirror, which 
 may be easily located by seeing the reflection of your eye in 
 it. Make the distance of the scale from the mirror (measured 
 to the axis of the tube containing it) 150 cm. Point the 
 telescope tube by sighting along it, directly at the mirror. 
 Raise or lower the scale until its image is seen in the mirror 
 on looking along the tube. If this is carefully done, the 
 image of the scale should be distinctly seen in the telescope 
 when it is properly focused. Move the telescope sideways, 
 making the last small adjustment with the slow-motion screw, 
 until the image of the center division of the scale, marked o 
 or 20, coincides with the vertical cross-wire in the eye-piece. 
 
 For each setting the true deflection is to be obtained by 
 taking the mean of the deflections to right and left on revers- 
 ing the current. 
 
 (c.) If the current as it is now flowing in the main circuit 
 causes an appreciable deflection of the galvanometer needle 
 (tested by making and breaking the circuit), try to reduce it 
 by proper arrangement of connecting wires. 
 
 See that the inner edges of the slides to which the galvano- 
 meter coils are attached are at division 10 on the scale, if the 
 Edelmann galvanometer is used; if the other instrument is 
 used, set the outer edges of the slides at 12.5. 
 
 (d.} It will be best now to make a sketch on paper of the 
 whole scheme of connections, and ask to have this verified 
 before setting up. For this purpose the four meters of wire 
 in the main circuit may be represented as a single straight 
 
23] VOLTMETER CALIBRATION. 43 
 
 piece, and the galvanometer circuit added according to the 
 following description. 
 
 Such a sketch of connections should accompany the notes 
 of each of the following experiments on electricity. 
 
 Connect the resistance coil (500 ohms for the Edelmann, 
 1,000 for the other) in series with the galvanometer coils, to 
 make up the necessary high resistance for the galvanometer 
 circuit Connect one terminal of this circuit to one, terminal 
 of the rheostat, the other to the binding-screw c, at the end of 
 the brass rod on which the sliding contact moves. The gal- 
 vanometer circuit may thus be put in parallel with any desired 
 portion of the main circuit by proper arrangement of plugs. 
 
 Include in the galvanometer circuit a mercury-cup com- 
 mutator, in such a way that tipping it from one side to the 
 other will reverse the direction of the current in the galvano- 
 meter circuit, without affecting that in the main circuit. It 
 must evidently be put between the rheostat and the galvano- 
 meter. Include in the galvanometer circuit also a key near 
 the telescope, enabling the observer to make and break that 
 circuit at pleasure. 
 
 It is evident that in setting up these connections, small 
 wires may be used throughout. 
 
 (e.) Reckoning from that rheostat terminal to which the 
 galvanometer terminal is attached, arrange the plugs so that 
 the stretched meter shall be the first of the four. Let one 
 observer set the sliding contact at the first marked point, which 
 is 20 cm. from the end, thus putting the galvanometer circuit 
 in parallel with 20 cm. of the main circuit, and let the other 
 read the deflection; then reverse the commutator and read 
 the opposite deflection. Repeat for the other marked points, 
 at 50 and 80 cm. Learn to control the vibrations of the 
 needle, so as to bring it to rest quickly, by proper use of the 
 key. 
 
 Change the plugs so as to make the stretched meter sue- 
 
44 WHEATSTONE BRIDGE. [24 
 
 cessively the second, third and fourth of the series, and for 
 each case read the deflections for each of the marked points. 
 Record for each case the total length of wire on the main 
 circuit with which the galvanometer circuit is in parallel, and 
 also the numbers of the loops used in making up each com- 
 bination for the main circuit. 
 
 The observers should change places at about the middle of 
 the series. 
 
 (/) If there is time, repeat the series in reverse order, 
 moving the galvanometer terminal which was at one terminal 
 of the rheostat to the other one. The lengths of wire now 
 included between the galvanometer terminals will be measured 
 from the opposite end of the four meters. Use the same 
 combinations as before in making up the main circuit, for 
 convenience. 
 
 If the resistance in the galvanometer circuit remain constant, 
 show that the currents indicated by the deflections will be 
 proportional to the E. M. F.'s between its terminals; and that, 
 if the resistance be sufficiently large compared with that of 
 the main circuit, these E. M. F.'s will be proportional to the 
 resistances of those portions of the main circuit included be- 
 tween the terminals. 
 
 Finally connect the terminals of the galvanometer circuit 
 directly to those of the cell, leaving out the rheostat entirely, 
 and read the deflection. Show that the current here indicated 
 should be proportional to the E. M. F, of the cell, as deter- 
 mined in Exp. 21. 
 
 Calculations may be deferred until Exp. 25, and the papers 
 for this exercise and that handed in together. 
 
 24. WHEATSTONE BRIDGE. 
 
 If a current be sent through two wires in parallel, there 
 will be a certain E. M. F. between the two ends a and b of this 
 
24] WHEATSTONE BRIDGE. 45 
 
 portion of the circuit. If a point c be chosen on one of the 
 wires, there will be a certain E. M. F. between a and c. A 
 point d can then be found on the other wire, such that the 
 c same E. M. F. exists 
 
 between a and d. It 
 is then evident that 
 if the points c and d 
 d be connected by a 
 
 wire, there will be no E. M. F. between them along this wire, 
 and hence no current will flow in the wire. The point d 
 may then be found by inserting a delicate galvanoscope as 
 part of the circuit cd, and moving d along adb until no current 
 is indicated. 
 
 By applying Ohm's law to each of the four portions ac % cb 
 ad> db, show that when this adjustment has been made rjx=. 
 pjq, where r denotes the resistance of ac, x that of cb, etc. 
 
 In the apparatus used, adb is a uniform stretched wire of 
 German silver I meter long, and d is a sliding contact on this 
 wire, x is an unknown resistance to be determined, and ris a 
 known resistance (for this experiment a coil whose resistance 
 in ohms is marked upon it). 
 
 It is evident that a constant cell is not required, since the 
 relations obtained above are independent of the current- 
 strength in the main circuit (They will not be true, how- 
 ever, for rapid variations in that current, since Ohm's law 
 does not hold for such cases.) It is customary to use a 
 Leclanche cell, dry cell, or other inconstant element, and to 
 include in the battery circuit a key which is only depressed 
 while taking an observation, since such cells can not send a 
 strong current for any length of time without injury. 
 
 Be careful never to connect such a cell so that it will send 
 a continuous current. 
 
 To determine the resistance per cm, of No. 19 German 
 silver wire: 
 
46 WHEATSTONE BRIDGE. [24 
 
 (a.) Set up the apparatus according to the plan outlined 
 above, with about two meters of the wire as the unknown 
 resistance. Connect the ends of the wire to the proper points 
 on the circuit through binding-posts, measuring the length of 
 wire included between them. In connecting the wire and 
 box to the bridge, use pieces of large copper wire as short as 
 possible, and well cleaned at the points of contact. It is also 
 well to have the resistances of the connecting wires used on 
 the two sides of c as nearly equal as possible if the position of 
 d is near the middle of the wire, as it usually is. Small wire 
 should of course be used for the galvanometer and battery 
 circuits. 
 
 Ask to have your scheme of connections verified before 
 beginning work. 
 
 See that the galvanometer is not affected by a current in 
 any branch of the circuit except its own. 
 
 (b.) Find by trial the position of the contact on the bridge- 
 wire which gives zero deflection of the galvanometer. In 
 making a trial, hold the sliding contact down with a gentle 
 pressure, at a point near one end of the bridge, then depress 
 the battery key just long enough to note the direction of the 
 deflection produced. Make a similar trial near the other end 
 of the bridge, and if the deflection is in the opposite direction, 
 the position sought must lie between these limits. 
 
 (c.) Leaving the connecting wires in position, interchange 
 the wire and the coil, and make a new determination. The 
 sliding contact will now come on the other side of the center 
 of the bridge, so that its position can readily be found; and 
 the mean of the two results from these observations will 
 be nearly free from small errors due to lack of uniformity in 
 the stretched wire and scale, and to the resistance of the 
 connections. 
 
 (d) Make a similar pair of determinations for a new length 
 of wire, shorter by a few cm. than the first 
 
25] TESTING EQUALITY OF RESISTANCES. 47 
 
 Calculate from each result the resistance of the wire in 
 ohms per cm. Wire cut from the same piece will be used 
 as a known resistance in later experiments, for the determina. 
 tion of small resistances. 
 
 (e.) Measure the diameter of the wire with the micrometer 
 caliper at several points, and calculate the specific resistance 
 of this sample of German silver that is, the resistance of a 
 piece I sq. cm. in cross-section and I cm. long. This value 
 must not be taken as applying to German silver in general, 
 since different samples vary greatly in specific resistance. 
 
 25. TESTING EQUALITY OF RESISTANCES. 
 
 (a.) Secure the rheostat used in Exp. 23, and with a Wheat- 
 stone bridge apparatus similar to that used in the last experi- 
 ment, compare the resistance of each of the six loops with 
 that of the stretched meter. The stretched meter will then con- 
 stitute the r branch for each case, and the loop used, the x 
 branch. Connections may be readily made by means of the 
 binding-screw plugs which fit the blocks of the rheostat. See 
 that the connecting wires are properly equalized, and use 
 the method of interchange described in Exp. 24 (c). 
 
 (b) Assuming that the wire on the rheostat is of uniform 
 resistance throughout, calculate the length of each loop, that 
 of the stretched meter being 100 cm. 
 
 (c.) For each of the combinations of loops used with the 
 voltmeter, calculate the true length of wire, in terms of the 
 above results, and the true position of the slider on this length 
 for each reading. Then reduce the position of the slider to 
 what it would have been had the true length been 400 cm., by 
 proportion. A correction of less than I cm. on any one loop 
 may be disregarded. 
 
 Plot the results, using as abscissae the corrected lengths of 
 wire in the main circuit with which the voltmeter was in 
 
48 CALIBRATION OF RHEOSTAT. [26 
 
 parallel, and as ordinates the corresponding deflections of the 
 voltmeter. If the galvanometer obeys the tangent law, show 
 that the plot should be very nearly a straight line, so that any 
 ordinate will be proportional to the corresponding E. M. F. 
 
 Show how to convert this relative calibration curve into 
 an absolute one, which will read E. M. F. in volts, by means 
 of the deflection corresponding to the known E. M. F. of the 
 Daniell cell. 
 
 26. CALIBRATION OF RHEOSTAT. 
 
 The bridge apparatus here used is a more compact form, 
 in which only 50 cm. of the wire is stretched along a scale, 
 the remainder of the meter being made up of two portions of 
 25 cm. each which may be connected to the first portion by 
 plugs. This connection may be made by means of the two 
 plugs in three different ways, so that the 50 cm. along the 
 scale may be either end or the middle of the meter. The 
 latter arrangement will be most used, and the scale is num- 
 bered to correspond to it. For any combination, the terminals 
 of the meter are evidently the blocks which are not connected 
 by a plug, and the binding-screw plugs should be inserted in 
 these blocks, to replace the binding-screws at the ends of the 
 wire in the ordinary form. 
 
 A rheostat is furnished, the coils of which have nominal 
 resistances of I, 2, 2 and 5 ohms. Determine as accurately 
 as possible the resistance of each of these coils, using as known 
 resistances the corresponding coils on a standard rheostat of 
 similar pattern, which has been carefully adjusted. 
 
 After finding the resistance of each coil separately, deter- 
 mine that of the four in series as a check. 
 
 The rheostat which has thus been calibrated will be used 
 for known resistances in the next experiments. 
 
27] RESISTANCE OF GALVANOMETER COILS. 49 
 
 27. RESISTANCE OF GALVANOMETER COILS. 
 
 Spools are provided whose resistances are equal respectively 
 to those of the 5o-turn and the 5-turn coils on tangent gal- 
 vanometer No. 6. Determine accurately the resistance of 
 these spools. 
 
 The resistance of the 5-turn coil is so small that the ordinary 
 method can not be relied upon to give good results, since the 
 resistance of the connections is here relatively important. 
 In such cases a substitution method may be employed to 
 advantage. Set up the bridge with the spool as the r branch, 
 and a piece of German silver wire for x, of a length which will 
 bring the slider to a position near the middle of the bridge. 
 Adjust the slider accurately for balance. Then remove the 
 spool and substitute for it a piece of the No. 19 wire, whose 
 resistance per cm. is known, stretching it between the same 
 binding-posts which held the spool, so that the connections 
 are exactly as before. With the slider in the same place, 
 adjust the length of wire between the binding-posts until 
 balance is again obtained. It is then evident that the resist- 
 ance of the length of wire used is exactly equal to that of the 
 spool for which it was substituted. 
 
 This method should be used always for low resistances, as 
 well as for any work where great accuracy is required, since 
 it entirely elimimites errors due to connections, as well as to 
 faults of any kind in the bridge itself. 
 
 FQI- the 5<D-turn coil, use the ordinary bridge method with 
 the calibrated rheostat as known resistance. 
 
 From the data of Exp. 19, find the resistance of the battery 
 there used, neglecting the resistance of the connecting wires. 
 Unless the plot was very accurately made, the probable error 
 of this result may be a large fraction of the battery resistance. 
 This method will, however, give accurate results when the bat- 
 
5O RESISTANCE OF GALVANOMETER AND BATTERY. O 8 
 
 tery resistance is of about the same magnitude as the average 
 resistance which was used in the box in taking measurements 
 for the plot. 
 
 28. RESISTANCE OF GALVANOMETER AND 
 BATTERY. 
 
 Resistance of Galvanometer Thompson's Method. 
 
 Put the galvanometer in the branch x of the bridge, and let 
 the ordinary galvanometer branch be replaced by a wire. 
 Then it is evident, first, that there will always be a current 
 through the galvanometer when the battery circuit is com- 
 pleted; second, that if the slide be so placed that r]x-=plq t 
 where x is the resistance of the galvanometer, no current will 
 flow in the wire cd on making contact to the bridge-wire by 
 means of the slide. If the slide be placed in any other posi- 
 tion, there will be a current in cd on making contact, which 
 means a redistribution of currents throughout the system, and 
 hence a change in the current through the galvanometer. 
 When no current flows through cd on making contact, there 
 is no such redistribution, and hence no change in the current 
 through the galvanometer. 
 
 The method therefore consists in adjusting the slide until, 
 on making contact with the slide on the bridge-wire, there is 
 no change in the deflection of the galvanometer. The current 
 from the battery must be allowed to run steadily through the 
 galvanometer, so that the deflection may become steady before 
 making a test; therefore a Daniell cell should be used. The 
 current used will produce so large a deflection that small 
 changes can not be noticed; it can be reduced to the proper 
 amount by connecting the two poles of the cell directly by a 
 wire, to serve as a shunt 
 
29] RESISTANCE OF BRIDGE- WIRE. 5! 
 
 Resistance of Battery Mance's flethod. 
 
 The galvanometer and battery b being in their usual posi- 
 tions, let a battery B be put in the branch x. Then for any 
 position of the slider, the current through the galvanometer 
 branch will be the algebraic sum of the currents due to 
 the batteries separately. That is, the effect of one is super- 
 posed on that of the other. If the slide be so placed that 
 r\xp\q^ where x is the resistance of the battery B, then the 
 current through the galvanometer due to the battery b is zero; 
 so that removing from the system the E. M. F. due to b would 
 not change the galvanometer deflection. It can be shown, 
 further, that for this adjustment, the current through the 
 galvanometer is independent of both the E. M. F. and the 
 resistance in the battery branch. (See Cumming's Theory of 
 Electricity, p. 182.) The battery b may then be replaced by 
 a key, and when the slide is in the proper position, making 
 and breaking the contact with this key will not change the 
 galvanometer deflection. 
 
 Thus the only battery used is the one whose resistance is 
 to be measured. Use the Daniell cell as before, and use a 
 shunt in the appropriate place to reduce the galvanometer 
 deflection to a proper amount. 
 
 29. RESISTANCE OF BRIDGE-WIRE. 
 
 The two branches 
 p and q of the bridge 
 are extended by 
 means of the two un- 
 equal known resist- 
 ances A and B; the 
 
 battery terminals must then be connected as shown; rand 
 x are unknown resistances, of about the same magnitude as 
 A or B. Then for balance, 
 
52 RESISTANCE OF BRIDGE-WIRE. 1.29 
 
 r/x = (A+p)/(B + q) 
 
 Now interchange A and B, leaving other connections as before, 
 and find the new positions for balance, for which 
 
 r/x = (B + p')/(A + q') 
 
 Add unity to each member of the equations. This gives 
 r + xA + p + B + q _B + p / +A + q / 
 
 _ 
 B + q A + q' 
 
 Remembering that p + q = p'H-q', show that q q' = A B. 
 
 This is the principle of Carey Foster's method for calibrat- 
 ing a bridge- wire. 
 
 (#.) For A, use the i-ohm spool of your rheostat; for B, a 
 short piece of heavy copper wire. A B will then be the true 
 value of the i-ohm spool simply. For r and x use a meter of 
 No. 25 G. S. wire with copper terminals, the point c being a 
 binding-post slipped on the wire, so that the ratio r>x may be 
 varied at will. The wire on the bridge used consists of two 
 equal meters of No. 30 G. S. wire, one of which is coiled at 
 one end of the board and the other stretched along the scale. 
 The measurements are of course to be taken on the stretched 
 wire simply. 
 
 Make several determinations on different parts of the wire, 
 by varying r\x. 
 
 CAUTIONS. Handle the wire with great care. On account 
 of its small diameter it is easily stretched or broken. 
 
 Do not press down on the contact. The weight of the 
 slide is sufficient to secure good contact, and any additional 
 pressure may injure the wire. 
 
 Do not slide the contact along the wire. Always raise the 
 slide before moving it. 
 
 (b.) Make another series of determinations, using the 2-ohm 
 spool in place of the i-ohm. 
 
 (c.) From all of the results, calculate the average resistance 
 per cm. of the wire.' 
 
 The same wire is to be used in the next experiment. 
 
30] ELECTROMOTIVE FORCE. 53 
 
 30. ELECTROMOTIVE FORCE. 
 
 Uae the same wire as in the preceding experiment. The 
 two wires, used by the two sets of observers, and the 5o-turn 
 coil of the tangent galvanometer, should be connected in 
 series with two cells of the storage battery, giving an electro- 
 motive force of about four volts. 
 
 Ask for storage battery connection. 
 
 Do not disturb the galvanometer in any way. If it is out 
 of order ask for assistance. Its reduction factor, accurately 
 determined for its present position, is given. 
 
 Form a parallel circuit to the wire, consisting of one of the 
 Leclanche cells bearing the number of this experiment, and a 
 Wheatstone galvanometer. Connect one terminal of this 
 circuit to the sliding contact, the other to the key at one end 
 of the board, so that depressing the key will connect the 
 parallel circuit to the coiled end of the bridge-wire. 
 
 There is an E. M. F. of about 2 volts distributed along the 
 wire. Therefore an E. M. F. equal to that of the Leclanche, 
 which is less than two volts, exists between the end of the 
 wire and some point along it. If the sliding contact be placed 
 at this point, and if the + terminal of the Leclanche be con- 
 nected to the same end of the wire as the + terminal of the 
 storage battery, then there will be no current through the 
 parallel circuit. If the Leclanche be connected in the opposite 
 direction, however, there will always be a current through the 
 parallel circuit, for any position of the slider; and this current 
 wjU obviously be always in the same direction. 
 
 (a.) In making a test, set the slider at the desired point on 
 the wire, then depress the key at the end of the board just 
 long enough to determine the direction of the deflection 
 produced. A long contact will cause polarization of the 
 Leclanche, and also disturb the current in the main circuit 
 which is being 'used by the other observers. 
 
54 ELECTROMOTIVE FORCE. [30 
 
 Determine first whether the Leclanche is properly con- 
 nected. Then adjust the slider until the galvanometer indi- 
 cates zero current, and read its position. Read the tangent 
 galvanometer, reversing its deflection as usual. This reading, 
 as well as the final careful adjustment of the slider, should be 
 made when the other pair of observers are not making trial 
 adjustments which will disturb the main current. 
 
 ($.) From Ohm's law, calculate the E. M. F. between the 
 end of the wire and the sliding contact, which is equal to that 
 of the Leclanche. 
 
 Make sure that this method is well understood before going 
 on, as mistakes with the standard cell might be dangerous. 
 
 (c.) Repeat with the standard cell furnished. This is a cell 
 used as a standard of electromotive force. Its constancy 
 depends upon careful handling, and upon having only minute 
 currents sent through it. 
 
 The 1 terminal coming from the upper end of the cell is the 
 zinc or negative, corresponding to the zinc of the Leclanche. 
 Knowing also that the E. M. F. is very nearly one volt, it 
 will be possible by a little calculation to set the slider very 
 near to the true point at the start. 
 
 Never use a standard ceil without such a preliminary test 
 with an ordinary cell. If it is impossible to approximate 
 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. 
 
 (*/.) How could the E. M. F. of any cell be determined in 
 terms of the known E. M. F. of the standard by this method, 
 without using the tangent galvanometer? 
 
 Determine in this way the E. M. F. of a bichromate (Grenet) 
 cell, if one is to be had. In using this cell, the zinc should 
 be lowered into the acid only while a current is wanted. 
 
 (e.) If there is time, let each set of observers determine, as 
 in (c), the E. M. F. of the standard previously used by the 
 other. 
 
3 1 ] BATTERY RESISTANCE. 55 
 
 A wire or other resistance used as in this experiment is 
 called a potentiometer. 
 
 31. BATTERY RESISTANCE. 
 
 Let Ebe the E. M. F. of a cell measured when it is sending 
 no current, as in the last experiment. Let e be the E. M. F. 
 measured between its terminals when it is sending a current c 
 through an external resistance r. Then in this latter case, E 
 is evidently the E. M. F. which sends the current c through 
 the whole resistance in circuit, including that of the battery; 
 while e is the E. M. F. which sends the same current through 
 the external resistance r. 
 
 Then calling the resistance of the battery b, E=c(r+fy, 
 e=cr. From these equations b can be found if E and e are 
 known, and either c or r. In this experiment we use known 
 resistances. Use a Daniell cell and resistance box, and 
 measure the E. M. F.'s on a potentiometer supplied with 
 current from storage battery. The potentiometer is thus 
 used as a voltmeter. Ask for storage battery connections. 
 
 (a.) Determine the E. M. F. of the Daniell by comparison 
 with one of the standards whose value was determined in 
 the last experiment. Use the Daniell first, remembering 
 that its E. M. F. is about i.i volts, in order to get the con- 
 nections right and the slide at about the right point, for the 
 standard. 
 
 The result here obtained for the Daniell is evidently the 
 value of E required. 
 
 .(.) Having the Daniell connected to the potentiometer 
 as at first, connect its terminals also to those of the resist- 
 ance box, using connecting wires of low resistance. The 
 potentiometer may now be used as a voltmeter, to determine 
 the E. M. F. existing between the terminals of the cell while 
 it is sending a current through the known resistance in the 
 
56 CALIBRATION OF A COPPER-WIRE THERMOMETER. 1.32 
 
 box. In this way find the values of e for a number of resist- 
 ances, from 10 units down to 0.5. After each determination, 
 disconnect the resistance box and test the value of E. The 
 reading with the standard cell should be repeated at about 
 the middle of the series, and again at the end, in order to 
 make sure that the storage battery current has remained 
 constant throughout. 
 
 (c.) Calculate the resistance of the battery, and also the 
 current flowing through it, for each case. 
 
 (d.) Plot the results obtained in (c\ so as to show the 
 relation between the resistance of the cell and the current 
 passing through it. 
 
 32. CALIBRATION OF A COPPER-WIRE 
 THERMOMETER. 
 
 The electrical resistance of all conductors changes with 
 change of temperature, and for pure metals the change is 
 found to be very nearly proportional to the change of tem- 
 perature. In this experiment the rate of change of resistance 
 with temperature is to be determined for a copper wire, 
 which may then be used as a thermometer if necessary. 
 If the calibration curve be a straight line, it is evident that 
 the thermometer can safely be used for temperatures beyond 
 the limits of the mercury thermometer. Another advantage 
 of this form of thermometer is that the average temperature 
 of any body may be determined very accurately, when it is 
 not at a uniform temperature throughout, by distributing 
 the copper wire uniformly around or through it. 
 
 The wire used is No. 32 insulated copper, of about 4 ohms 
 resistance, wound on a sheet of mica, its terminals soldered 
 to No. 22 lead-wires. 
 
 A spool is furnished on which are wound two equal 
 resistances. If these be made the branches p and q of a 
 
32] ' CALIBRATION OF A COPPER-WIRE THERMOMETER. 57 
 
 bridge arrangement in which the thermometer is the branch 
 x and the bridge-wire, measured from one end to the sliding 
 contact, is the branch r, then it is evident that if r can be 
 adjusted, by means of the sliding contact, so as to give no 
 current in the galvanometer circuit, the resistance in r is 
 exactly equal to that of the thermometer wire, and its lead- 
 wires. In order to eliminate the effect of these leads, a pair 
 of " compensating leads," consisting of a simple loop of wire 
 of the same size and length as the thermometer leads, is 
 wound with them, so that the resistance of the two pairs of 
 leads will be the same under all circumstances. If these 
 compensating leads be put in r, in series with the bridge- 
 wire, the resistance of the bridge-wire in r will then be 
 equal to that of the No. 32 wire of the thermometer simply. 
 
 The two 25 cm. lengths 
 on the bridge need not 
 be used. The 50 cm. 
 along the scale is to be 
 extended by a spool k 
 equal in resistance to 250 
 cm. of the wire, so that 
 the actual length of wire 
 up to the sliding contact is the length on the scale added to 
 250. 
 
 In setting up the arrangement, notice that the point a is 
 the junction of one thermometer lead with one end of the 
 spool carrying/ and q, b is the binding-screw on the sliding 
 contact, to which the other end of the spool is attached, c is 
 the junction of the second thermometer lead with one of the 
 compensating leads, the other being connected to the coil k 
 at the end of the bridge, and d is the junction of p and q. 
 
 The thermometer leads are tied with a piece of twine near 
 the ends, to distinguish them from the compensating leads. 
 
 (a.) Use a kettle for a water-bath in which to immerse the 
 
5# STUDY OF POLARIZATION EFFECTS. [33 
 
 copper thermometer and the attached mercury thermometer. 
 Record the position of the slider for temperatures at intervals 
 of about 5 through the range from 20 to 40. For each 
 reading keep the temperature steady, within o.i, and the 
 bath well stirred until the position of the slider for balance 
 remains constant, showing that the wire has attained the 
 same temperature as that indicated by the thermometer. 
 Take care to make the contacts in the battery circuit as 
 short as possible, in order to avoid heating the thermometer 
 wire by the testing current. 
 
 Take a similar series of descending readings, at about the 
 middle points of the temperature intervals previously used. 
 
 (&) Plot the observations, using temperatures as abscissae 
 and changes in position of slider as ordinates, each on as 
 large a scale as the paper will permit. The calibration curve 
 for the mercury thermometer used will be furnished on 
 application. Since the length of the bridge-wire is always 
 proportional to the resistance of the thermometer wire, 
 changes in position of the slider will be proportional to 
 corresponding changes in that resistance; and a knowledge 
 of the actual resistance of the wire is unnecessary. 
 
 (c.) Calculate the coefficient of increase of resistance per 
 degree, in terms of the resistance at o. 
 
 (</.) Calculate the temperature at which the resistance of 
 the wire would become zero. What must you assume? 
 
 33. STUDY OF POLARIZATION EFFECTS. 
 
 Use the same voltmeter as in Exp. 23, and see that the 
 proper resistance coil is connected in series with it. Also 
 use a Leclanche cell which bears the number of this experi- 
 ment, and no other. 
 
 In this experiment the closing of a key produces a deflec- 
 tion of the voltmeter corresponding to an E. M. F. which is 
 
33] STUDY OF POLARIZATION EFFECTS. 59 
 
 changing rather rapidly, so that a reading must be obtained 
 as soon as possible. The needle, however, does not come to 
 rest until after several vibrations; so that the best plan is 
 to read three consecutive turning-points (the first three if 
 possible) and calculate from them the resting-point, as was 
 done with the balance. A little practice at this before begin- 
 ning the experiment will be of advantage. 
 
 (#.) Connect the terminals of the cell directly with the 
 terminals of the voltmeter, and so get a reading which is 
 proportional to the total E. M. F. E of the cell. It may be 
 necessary to move the voltmeter coils farther away from the 
 needle in order to get a deflection which can be read on the 
 scale. This deflection should be at least 3 cm. from the 
 end of the scale, for convenience in reading turning-points 
 later on. 
 
 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. 
 
 (^.) Connect the terminals of the cell through a spool of 2 
 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 voltmeter deflection e by means of three turning- 
 points. Be careful not to connect the cell in such a way as 
 to send a current through it until ready to begin observations. 
 
 (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, by means of three turning-points. 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 
 
60 WRITTEN EXERCISE ELECTRICITY. [34 
 
 this another 5 -minute interval, then repeat the readings as 
 in (c). By this means the readings obtained are very nearly 
 the same that 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 $ 
 minutes each have been covered. 
 
 (>.) Leaving the resistance disconnected, take readings of 
 E at 5-minute intervals for about 45 minutes. 
 
 At some time during this interval, secure if possible the 
 Leclanche cell whose E. M. F. was determined in Exp. 30, and 
 by getting the voltmeter 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 times as abscissae and corresponding 
 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. 
 
 Calculate, as in the last experiment, the resistance of the 
 ceil from each pair of values of e and E, tabulate the results, 
 and show that for this purpose the scale-readings correspond- 
 ing to E and e may be used without knowing their values 
 in volts. 
 
 34. WRITTEN EXERCISE ELECTRICITY. 
 
 (i.) Arrange the details of a method for calibrating the 
 sensitive galvanometer whose resistance was determined in Exp. 
 28, using your tangent galvanometer to determine the strengths 
 of the currents used, and dividing these currents so as to send 
 known fractions through the sensitive galvanometer. 
 
 The curve obtained should read amperes directly. Will this 
 probably be a tangent curve? 
 
 (2.) Having given this calibration curve for the sensitive 
 galvanometer, show how you could use it as a voltmeter. 
 
35-36] INDEX OF REFRACTION. 6 1 
 
 (3.) Show that the scale readings in Exp. 23 are only approx- 
 imately proportional to the tangents of the corresponding 
 angles of deflection, using the simple law of reflection at a 
 plane mirror. 
 
 (4.) Show that the Carey Foster method could be used 
 to find the difference between two nearly equal resistances, 
 A and B. 
 
 (5.) Show how to modify the copper- wire thermometer 
 apparatus in 32 so that it will be capable of reading to o.ooi 
 through a range of 5. 
 
 35-36. INDEX OF REFRACTION. (Two Periods.) 
 
 A small piece of plate glass is set vertically, with its upper 
 edge horizontal, at the center of a rotating table. The arm 
 to which the glass is directly attached, by means of soft wax, 
 can be turned independently of the table, but moves with 
 the table when the latter is rotated. A millimeter scale is 
 placed with its divisions vertical, in such a position that when 
 a telescope at the same height as the upper edge of the glass 
 is focused on the scale, its divisions will be seen partly 
 through he glass plate, and partly above its upper edge. 
 The scale should be normal to the line of sight, with its 
 middle division, marked zero, behind the middle of the glass 
 plate. Set the eye- piece of the telescope so that the cross- 
 wires are at an angle of 45 with the vertical, and with their 
 intersection on the zero line above the glass. The image seen 
 in the telescope will then consist of two parts, one formed 
 by light coming through the glass plate, the other by light 
 coming through the air above the plate. These two images 
 will appear to coincide when the plate is normal to the line 
 of sight ; but when the plate is rotated in either direction 
 about a vertical axis, one portion of the image is displaced 
 with reference to the other. For a certain angle of rotation, 
 
62 
 
 INDEX OF REFRACTION. 
 
 [35-36 
 
 the lines of the scale will again appear to be continuous, 
 instead of broken. This evidently means that one portion of 
 the image has been displaced with reference to the other 
 by I mm. Continuing the rotation, the lines of the scale 
 will again appear to be continuous for displacements of 
 2, 3. etc., mm. 
 
 In the figure, PBOT repre- 
 sents the path of the pencil, 
 from the scale to the tel- 
 escope, by which part of 
 one scale division is seen 
 through the glass; QAOT, 
 that by which part of the 
 adjacent division is seen above 
 the glass. The two paths co- 
 incide after leaving the glass, 
 so that the two parts of the 
 two lines appear to coincide 
 and form one line in the image. 
 / is thickness of plate. 
 
 d is distance which one image is displaced with reference to 
 the other. 
 
 i is angle of incidence (or emergence). 
 r is angle of refraction. 
 
 The index of refraction is to be obtained from the formula 
 n = s'm z/sin r. 
 
 All of the above quantities will be determined directly, 
 except r, which can be obtained from the relation tan r=tan i- 
 (d/t cos i). Show the truth of this relation from the figure. 
 (a.) To determine the values of i corresponding to displace- 
 ments of i, 2, 3, and 4 mm. Either the arm to which the 
 glass is attached, or the whole table, can be rotated by the 
 observer at the telescope by means of a wooden rod with a 
 peg at the end, which can be set in one of the holes made for 
 the purpose. 
 
35-36] INDEX OF REFRACTION. 63 
 
 Although i may be observed directly, a better result will be 
 obtained by measuring 21. To do this, set the plate so that a 
 displacement of I mm. in one direction is obtained, then read 
 the angle through which the table must be rotated to cause 
 an equal displacement in the other direction. 
 
 It will soon be seen that the accuracy of a single setting is 
 not very great, so that it is desirable to average a large number 
 of settings. This can be done without reading each setting, 
 as follows: Set the circle so that the index reads zero ; set the 
 glass, by means of the movable arm, to give a displacement 
 of I mm.; turn the table until an equal displacement in the 
 opposite direction is secured; leaving the table in this position, 
 turn the glass back to its original position by means of the 
 movable arm; again turn the glass into its second position by 
 means of the table. It is evident that each time the operation 
 is repeated, the table is turned through an angle 2t; and thus 
 any number of observations may be automatically added. 
 
 Make a series of determinations in this way until the sum 
 of the angles reaches three or four times 360. The final 
 reading should be taken at a point as near to the zero of the 
 circle as possible, which will reduce the effect of errors in the 
 graduation or mounting of the circle. The second observer 
 should note the number of settings made, and the number of 
 times the zero of the circle passes the index. Calculate the 
 mean value of i for the series. 
 
 In this way let each observer make a determination of i for 
 each displacement of i, 2, 3, and 4 millimeters. 
 
 (.) To determine /, by means of the OPTICAL LEVER. 
 
 The optical lever is a short lever of brass resting by three 
 points on a bed-plate of brass. Two of the points, at one end, 
 rest in minute conical depressions in the bed-plate, which keep 
 them in definite position, and serve as the fulcrum of the lever. 
 The perpendicular distance from the third point to the line 
 joining these two points is the effective length of the lever. 
 
64 INDEX OF REFRACTION. [35~36 
 
 The distance through which the third point is raised when 
 any object is placed under it, or the thickness of the object, is 
 to be calculated from the length of the lever and the angle 
 through which it is rotated. 
 
 This angle is to be measured by means of a modification of 
 the telescope and scale method. A scale is set in a vertical 
 position at a distance of 75 to 100 cm. in front of the mirror 
 which is attached to the lever. A slider attached to the scale 
 is pierced with a small round hole, and has on the side toward 
 the mirror a horizontal line whose direction passes through 
 the center of the hole. Adjust the relative positions of scale 
 and lever so that when the slider is at the proper height, the 
 reflected image of the horizontal line can be seen in the mirror, 
 on looking through the hole. Adjust the slider until the 
 image of the line appears to bisect the small round ink-spot 
 near the center of the mirror. The adjustment can be made 
 more closely if the eye be held a short distance from the hole, 
 and so that the spot on the mirror appears to be exactly in 
 the center of the hole. It is now evident that the normal to 
 the mirror, at the center of the spot, passes through the center 
 of the hole. 
 
 The scale should now be vertical, and the normal to the 
 mirror horizontal, that they may form two sides of a right 
 triangle. Adjust the former by sighting on window casings 
 or other convenient vertical lines. Test the latter by measur- 
 ing from the table the height of the line on the slider and of 
 the spot on the mirror, and make any necessary adjustment 
 by means of the screw to which the third point is attached. 
 Having these adjustments made, read the position of the 
 slider by means of the index. 
 
 The thickness of the plate is only needed over the portion 
 actually used in (a). The settings made should therefore be 
 confined to this portion, and should cover it with some 
 attempt at uniformity, so that the mean thickness of the plate 
 as used will be determined. 
 
37-38] SPHEROMETER. 65 
 
 Carefully lift the third point of the lever, and insert the 
 plate, letting it rest on the three projections from the bed- 
 plate. See that the two points at the other end of the lever 
 have not been disturbed from their position, then adjust and 
 read the slider as before. Make several settings of the slider 
 for different points over the desired area of the plate, then 
 remove the latter and check the zero reading. 
 
 Measure the horizontal distance from the line on the slider 
 to the face of the mirror, and calculate the mean angle through 
 which the lever has been rotated. 
 
 Repeat for at least two other distances of the scale from 
 the mirror. Note that if the distance be made just I meter, 
 the difference between the two readings of the slider gives 
 the tangent of the angle directly. 
 
 Calculate the mean angle from the whole series. 
 
 To find the length of the lever, lay it on the table with the 
 points up, and lay on them a half-millimeter scale ruled on 
 glass, ruled side down. Set the two points at one end of the 
 lever in the long zero division, and read the position of the 
 third point as closely as possible with a magnifying glass. 
 This evidently gives the desired distance, since the zero divi- 
 sion is perpendicular to the length of the scale. 
 
 This gives all the data needed for calculating the thickness 
 of the working portion of the glass plate. 
 
 (c.) Calculate the index of refraction of the glass, from each 
 pair of values of i and d obtained in (a\ and find the mean of 
 these results. 
 
 37-38. SPHEROMETER. (Two Periods.) 
 
 * 
 
 Radii of Curvature of Lens Surfaces. 
 
 PRELIMINARY. Testing vernier calipers. 
 Secure the billiard ball whose volume was determined in 
 Exp. 5, and from that volume find the mean diameter of the 
 
66 SPHEROMETER. [37~38 
 
 ball (Whiting's Tables, 3H). This gives a standard of length 
 whose value is independent of any direct measurement on a 
 scale. (See Whiting, p. 72.) By measuring the diameter of 
 the ball with the vernier calipers, the scale of the calipers may 
 be compared with this standard. 
 
 As the ball is not a perfect sphere, it will be necessary to 
 measure a number of diameters in different directions, uni- 
 formly distributed, in order to get a result which may be 
 compared with the diameter calculated from the volume. In 
 order to get a uniform distribution of the measurements, draw 
 lightly in pencil on the ball two great circles at right angles, 
 and a third* perpendicular to both. The intersections of these 
 will give three diameters, and points on each circle midway 
 between the intersections will give six more. Four others 
 may be taken connecting the centers of the triangles into 
 which the surface has been divided. 
 
 Only one pair of calipers is available, so that while one set 
 of observers is using it, the other should be taking the spher- 
 ometer readings in (a) and (b). 
 
 In using the calipers, great care should be taken not to 
 strain the jaws. Ask for special directions as to the necessary 
 precautions. Make a coarse adjustment by taking hold of 
 the movable jaw (not the small slide attached to it) and sliding 
 it along the scale. Then clamp the small slide, and make the 
 final adjustment by means of the slow-motion screw, turning 
 it until the object is touched lightly by both jaws, but not held 
 firmly between them. 
 
 To set on a given diameter of the ball, hold it with one 
 marked end of the diameter against the fixed jaw, just inside 
 the end. Then adjust the movable jaw until, on moving the 
 ball slightly back and forth, it just grazes the corresponding 
 point on this jaw. Never clamp the ball so tightly that its 
 weight can be supported by the calipers. The accuracy of the 
 results will depend largely on the uniformity of the pressure 
 with which these settings are made. 
 
37-38] SPHEROMETER. 6/ 
 
 After making the settings on the ball, close the movable 
 jaw up against the fixed jaw with as nearly as possible the 
 same pressure used on the ball, and determine the zero error. 
 If any is found, allow for it on the measurements taken. 
 
 Find the factor by which the mean diameter in terms of the 
 caliper scale must be reduced to the calculated value, and if 
 this factor differs from unity by an appreciable amount, use it 
 in future work. 
 
 (a.) Pitch of spherometer screw. 
 
 A rotating arm is provided which slips over the point of 
 the spherometer screw. By means of this, adjust the upper 
 brass ring so that its axis coincides with the axis of rotation 
 of the screw, and clamp it in that position by means of the 
 small clamp-screws underneath. 
 
 Fasten a small piece of plate glass to a larger piece with soft 
 wax, making sure that the surfaces are clean and accurately 
 in contact. Set the plate on the ring of the spherometer, with 
 the attached piece downward, and while holding it down with 
 one hand, turn the screw upward by means of the milled head 
 underneath the disk. Never turn the screw by means of the 
 disk itself. The screw turns very easily, so that the moment 
 of contact can be felt, and care should be taken not to strain 
 the screw by turning beyond this point. Find the number of 
 turns of the screw corresponding to the thickness of the piece 
 of glass at two selected points near its edges, by setting the 
 screw first on the selected points, and then on the plate when 
 the piece has been removed. 
 
 Measure the thickness of the piece at each of the two points, 
 making several settings of the calipers in each case. Calcu- 
 late the pitch of the screw, or the distance in centimeters 
 which its point advances for one revolution. 
 
 Also measure with the calipers the internal diameter of the 
 upper ring on the spherometer, and check the measurement 
 with the glass scale used in the last experiment. The radius 
 
68 FOCAL LENGTH OF LENS. [39 
 
 of this ring, which is the perpendicular distance from the axis 
 of the screw to the inner edge of the ring, is called the 
 " Span" s of the spherometer. 
 
 When you have finished with the calipers wipe them care- 
 fully with a dry cloth before putting them in the case, to guard 
 against rust. 
 
 (b.y Set the ball in the ring of the spherometer, and bring 
 up the screw as in (a) against its lower surface. In this way 
 make a setting on each of the previously marked points, in 
 order to find the mean distance d from the point of the screw, 
 when thus resting against the surface of the ball, to the plane 
 of the upper surface of the ring. 
 
 To calculate the diameter of the ball from these data, con- 
 sider a section of the ball made by a plane through the 
 common axis of the screw and the ring. In this figure, if D 
 be the vertical diameter of the ball, show that 
 
 s 2 = d(D-d) 
 
 If the value of D, calculated from this formula, agrees closely 
 with the value first obtained from the volume, it is evidence 
 that the constants of the spherometer have been accurately 
 determined in terms of your absolute length standard, and 
 that the method of calculation is well understood. 
 
 (<;.) Find the radius of curvature of each surface of the given 
 lens from a number of readings on different parts of the 
 surface. 
 
 The same lens is to be used in several succeeding experi- 
 ments. When not in use it should be kept in the rack pro- 
 vided for the purpose. Be careful at all times to avoid 
 scratching the surfaces of the lens in using. 
 
 39. FOCAL LENGTH OF LENS. 
 
 The focal length of the lens is first to be determined by the 
 method of conjugate foci. Fix the lens on one of the slides 
 
391 FOCAL LENGTH OF LENS. 69 
 
 of the optical bench, which is fitted to hold it. A large screen., 
 at one end of the bench is fitted with fine threads stretched 
 across a small hole, to serve as objects. The similar threads 
 stretched across the hole in the small screen on the second 
 slide of the bench are used to locate the image, formed by the 
 lens, of the object. 
 
 Move the two slides as near to the large screen as the 
 bench will allow, and adjust the two screens so that the cen- 
 ters of the openings and the center of the lens shall be on the 
 same> straight line parallel to tl.^ bench. The centers will then 
 remain on this line as the slides are moved. This adjustment 
 can be made well enough by eye, or by rough measurement 
 from the bench. 
 
 (a.) For each observation, the distance of the lens from 
 each screen is needed; and it is evident that these distances 
 can not be directly obtained by reading the positions of the 
 slides on the meter scale. To provide the means of getting 
 the true distances from any set of readings, set the slides at 
 any convenient positions, read their indices, and measure with 
 a beam caliper the distance from each set of threads to the 
 corresponding lens surface. Take three sets of measurements 
 for different positions of the slide. 
 
 The lens is assumed to be a thin lens, or one whose thick- 
 ness may be neglected. The approximation arising from this 
 assumption will be less, for the double convex lens used, if 
 the measurements are made from the center instead of from 
 the surfaces. The thickness of the lens, for this purpose, can 
 be measured with sufficient accuracy as follows: Put two 
 small wooden blocks on the two slides, setting them so that 
 the adjacent surfaces are in contact, and read the distance 
 between the indices. Then separate the slides, insert the lens 
 between the blocks, and read the distance again. 
 
 (.) Bring the image screen back nearly to the opposite end 
 of the bench from the object, and bring the lens toward it until, 
 
7O FOCAL LENGTH OF LENS. [39 
 
 on looking toward the lens from about 30 cm. behind the 
 image screen, the image of the object is seen through the 
 hole in the screen. Adjust the lens then until the image 
 seems to be in the same plane as the threads in the screen^ 
 tested by seeing that there is no parallax between the two on 
 moving the eye from side to side. 
 
 To make a closer adjustment, hold a magnifying glass so 
 that the threads are clearly seen. If the image be not in the 
 same plane, it will not be clearly in focus at the same time, 
 and any remaining parallax will be magnified. Find the object, 
 distance u. 
 
 Leaving the lens where it is, so that u will remain the same, 
 move the image screen and then readjust it. Practice in this 
 way until settings can be made which agree within I or 2 mm. r 
 then from the mean of several settings find the image distance v^ 
 
 Move the lens toward the object until, by rough trial, v is 
 found nearly equal to u. Then determine n and v as before. 
 
 (c.) Lay off on co-ordinate paper the first pair of values of 
 u and v along the axes of X and Y respectively, on a scale as 
 large as possible, and connect the two points by a straight line- 
 Do the same for the other pair of values. Draw the lines 
 accurately, with a fine pointed hard pencil. 
 
 Write the equation of either of the lines, and show that if 
 
 i/v+ i/u= i/f, 
 
 the point, each of whose co-ordinates is/, will lie on this line, 
 and hence will be the point of intersection of two such lines. 
 It will of course also lie on the line making an angle of 45 
 with each axis. Draw this 45 line. Then the best value of 
 / will be found by locating the point which represents the 
 average of the points in which the ?/, v lines intersect the 45 
 line. 
 
 Find several additional pairs of values of u and v, choosing 
 them so that the resulting lines will cover as wide a range in 
 direction as possible, and use the average from all. For some 
 
4] FOCAL LENGTH OF LENS. /I 
 
 of them, v should be made greater than u, so as to give a 
 magnified image. 
 
 If this magnified image is indistinct, so that it is difficult to 
 make an accurate setting, cut a hole about I cm. in diameter 
 in a piece of paper, and put it in front of the lens so that only 
 the central portion of the lens will be effective in forming the 
 image. 
 
 (d.~) Move the opening in the piece of paper out near one 
 edge of the lens, so that the image is now formed by a portion 
 of the lens away from the center. Determine the focal length 
 as in (c) for this part of the lens. Then explain the advan- 
 tage of a diaphragm with a small central opening for securing 
 a distinct and definite image ; and see also that in looking at the 
 image with the eye, the image formed by any desired portion 
 of the lens can be seen by holding the eye in the appropriate 
 position. The office of the diaphragm in this case is merely 
 to make it certain that the eye sees the desired image and no 
 other. 
 
 40. FOCAL LENGTH AND INDEX OF REFRACTION 
 FOR RED AND BLUE LIGHT. 
 
 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 the last experiment was an 
 average value for the brighter portion of the spectrum of the 
 light used. In order to determine f for two fairly definite 
 colors of the spectrum, red and blue, put in front of the object 
 threads a piece of blue glass, which, it will be remembered, 
 transmits some red light as well as blue. Illuminate the 
 threads by a kerosene lamp placed behind them. This light, 
 being richer in red as compared with blue than ordinary day- 
 light, will give more nearly equal brightness in the red and 
 blue transmitted by the glass. We will now have two images 
 
/2 INDEX OF REFRACTION OF A LIQUID. [41 
 
 formed by the lens one by red and one by blue light. On 
 setting the image-screen and applying the magnifying glass it 
 will be seen that there are two positions of the screen which 
 give a clear image. At one position the threads, and the 
 margin of the aperture in the screen, appear red and the 
 spaces bluish; at the other the threads and margin appear blue 
 and the spaces reddish. Decide as to which image is formed 
 by blue and which by red light, and explain the color 
 phenomena in the two cases. 
 
 For convenience, the magnifying glass is attached to the 
 image-screen, so that it can be focused on the thread and 
 moved with the screen, or turned one side for a rough 
 setting. 
 
 (.) Determine, by the method of the last experiment, the 
 focal length of the lens for red and for blue light. Locate 
 both images for each position of the lens, and so carry on 
 both series at once. 
 
 ($.) From the radii of curvature of the lens surfaces, and/" 
 as found in the last experiment, calculate its index of refrac- 
 tion for the light there used. In the same way, from the data 
 obtained in this experiment, calculate its index of refraction 
 for red and for blue light. 
 
 41. INDEX OF REFRACTION OF A LIQUID. 
 
 (a.) Inclose a little water between a piece of plate glass and 
 the lens previously used, keeping it in place by a cell made 
 from a piece of sheet rubber with a round hole cut from its 
 center. Mount the whole on the optical bench, and determine 
 the focal length of the combination, for daylight. Use the 
 following method: 
 
 For a given position of the lens, let u + v=/ t the distance 
 between the screens. Then leaving the screens fixed, move 
 the lens to another position, such that u v v and ^ = u. If the 
 
42] INDEX OF REFRACTION OF LENS. 73 
 
 first image was reduced, the second will be enlarged; and the 
 distance through which the lens was moved (read directly on 
 the scale) is v v l = v u=a. Show then that /= (/ 2 a*) / 4!. 
 This does not involve any measurements to the lens surfaces, 
 which might be awkward in this case. 
 
 The plate glass used is of the same piece as that whose 
 index of refraction was determined in Exp. 36. If u above in- 
 clude this plate glass, show how to correct it so as to get the 
 value which u would have in air; and show that this correction 
 should be applied to / as measured above. 
 
 (&) Calculate the focal length and index of refraction of the 
 liquid lens. 
 
 42. INDEX OF REFRACTION OF LENS MICRO- 
 SCOPE METHOD. 
 
 The measurements to be taken are the same as in the 
 measurement of refractive index of a plate namely, the real 
 thickness, and the apparent thickness, of the lens, both taken 
 along its axis. It is evident that the formula for the plate will 
 not apply here, since the wave-front in the air, after refraction 
 at the upper spherical surface of the lens, will differ in form 
 from that due to refraction at the plane upper surface of the 
 plate. By means of the known relations for refraction at a 
 single spherical surface, show that 
 
 AB(rAb\ where AB = real thickness of lens 
 
 n = Ab( -AB\ ^4^ = apparent " 
 
 r = radius of upper surface. 
 
 ^ Set the lens, resting on a piece of plate glass, on the stage 
 of the microscope, with its axis as nearly as possible in the 
 axis of the microscope. A little chalk dust rubbed with the 
 finger from a crayon and smeared very lightly on the upper 
 surfaces of the lens and the plate glass, will provide objects 
 on which to focus the microscope. Focus roughly on the 
 
74 WRITTEN EXERCISE. LENSES. [43 
 
 upper surface of the lens by sliding the tube in its clamp, then 
 clamp the tube and focus more closely by means of the 
 micrometer screw, which moves the tube vertically. See 
 that the tube is parallel to the screw, and that it can be 
 moved downward through a distance fully equal to the thick- 
 ness of the lens. 
 
 In making a setting, always move the screw in the same 
 direction to the desired point; and if that point be accidentally 
 passed, turn the screw backward some distance, in order to 
 approach the point again from the original direction. This 
 precaution is! necessary because there is likely to be some 
 lost motion in changing the direction of rotation of the screw. 
 
 Use the parallax test, as far as possible, to decide whether 
 the image is accurately in the plane of the cross-wires. 
 
 The pitch of the screw is 0.025 cm. 
 
 After practicing until settings can be made which agree 
 fairly well, let each observer make a series of settings on (a) 
 the upper surface of the lens, (&) the upper surface of the plate 
 as seen through the lens, and (c) the upper surface of the plate 
 after removing the lens. 
 
 From these find AB and Ab, and calculate n, the index of 
 refraction. 
 
 Tabulate all the indices of refraction which have been 
 found. 
 
 43. WRITTEN EXERCISE. LENSES. 
 
 1. Show how the optical bench, as used in Exps. 39, 40, 
 corresponds to the type of (a) the astronomical telescope, or 
 (b) the compound microscope, according to the position of 
 the large lens with reference to the two screens. 
 
 2. Taking either of the cases in (i), trace the light from the 
 object which falls on the effective aperture of the large lens, 
 until it finally forms an image of the object on the retina of 
 the eye; indicating the position and kind of image formed 
 by refraction at each lens of the system. 
 
44] DIFFRACTION GRATING. 75 
 
 3. Describe the essential improvements in the working 
 telescope and microscope over the optical bench types. 
 
 4. Certain approximations are made in deriving the funda- 
 mental formulae relating to lenses which you have used in 
 Exps. 3942. State these approximations, and show that 
 under the working conditions, their use is justified. 
 
 5. Explain the theory of the following method of finding 
 the focal length of a diverging (concave) lens: 
 
 Threads stretched across an aperture in a screen, and 
 brightly illuminated, form the object o. This is put at one 
 end of an optical bench, next to it a converging lens, then 
 the diverging lens, and at the other end of the bench a plane 
 mirror, facing the object, and with its plane at right angles 
 to the axis of the bench. 
 
 The lenses are now so adjusted that the light from the 
 object, passing through the lenses to the mirror, and there 
 reflected back so as to again pass through the lenses in the 
 reverse direction, forms an image i of the object on the 
 screen beside the aperture that is, the image is formed in 
 the same plane as the object. 
 
 If now the diverg- 
 ing lens be re- 
 moved, the con- 
 verging lens will 
 form an image i of the object at some point beyond the 
 place where the diverging lens was. This image may be 
 located by any of the usual methods, and its position was 
 the principal focus of the diverging lens when that was in 
 position. Then of course the distance from this point to the 
 point where the lens was, is its focal length. 
 
 44. DIFFRACTION GRATING. 
 
 A monochromatic light is required. For this purpose 
 use a Bunsen burner with an asbestos cap which can be 
 
76 DIFFRACTION GRATING. [44 
 
 soaked in a solution of sodium chloride, so as to give a 
 strong sodium flame. Set in front of this the brass 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 third slit being covered. 
 
 The grating is to be mounted in a slide on a meter rod, 
 with the ruled surface toward the screen and the lines parallel 
 to the slits. The rod is to be supported by a clamp so that 
 it will carry the grating along a line perpendicular to the 
 screen at the lower slit. 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 the 
 grating is moved back and forth; and 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 evidence 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. 
 
 (&) Repeat (a), using the second diffracted image of each 
 slit, instead of the first. 
 
 (<r.) Repeat (a) and (b) for the pair of slits which are wider 
 apart. 
 
 (d.) Measure the distance between the slits with the y 2 mm. 
 glass scale. Determine the mean distance from measure- 
 ments on the edges, taking a number of readings, on different 
 parts of the scale, for each distance. 
 
 (e.) Assuming the wave-length of sodium light to be 
 0.0000589 cm., determine the distance apart of the lines in 
 the grating. Explain fully your method of calculation, and 
 show that it applies to this case. 
 
673151 ^7 
 
 LOWER DIVISION 
 
 UNIVERSITY OF CALIFORNIA LIBRARY