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 Loud on- 
 
 
 STATL 
 
 , -^ Southern Branch 
 
 of the 
 
 University of California 
 
 Los Angeles 
 
 Form Ll 
 
 JONES' 
 
 BOOK STORE,
 
 This book is DUE on the last date stamped below 
 
 1 1 1924 
 
 V 
 
 1 192, 

 
 A LABORATORY COURSE 
 
 EXPERIMENTAL PHYSICS
 
 A LABORATORY COURSE 
 
 EXPERIMENTAL PHYSICS 
 
 r,\ 
 
 W. J. LOUDON, B.A. 
 
 DEMONSTRATOR IN PHYSICS IN THE UNIVERSITY OF TORONTO 
 AND 
 
 J. C. McLENNAN, B.A. 
 
 ASSISTANT DEMONSTRATOR IN PHYSICS IN THE UNIVERSITY OF TORONTO 
 
 MACMILLAN AND CO. 
 
 AND LONDON 
 
 I8 95 
 All rights reserved
 
 COPYRIGHT, 1895, 
 MACMILLAN AND CO. 
 
 NortoooH $rrss : 
 
 J. S. Gushing & Co. Berwick & Smith. 
 Norwood, Mass., U.S.A.
 
 PREFACE. 
 
 AT the present day, when students are required to gain 
 knowledge of natural phenomena by performing experiments 
 for themselves in laboratories, every teacher finds that as his 
 classes increase in number, some difficulty is experienced in 
 providing, during a limited time, ample instruction in the 
 matter of details and methods. 
 
 During the past few years we ourselves have had such diffi- 
 culties with large classes ; and that is our reason for the 
 appearance of the present work, which is the natural out- 
 come of our experience. We know that it will be of service 
 to our own students, and hope that it will be appreciated by 
 those engaged in teaching Experimental Physics elsewhere. 
 
 The book contains a series of elementary experiments speci- 
 ally adapted for students who have had but little acquaintance 
 with higher mathematical methods : these are arranged as far 
 as possible in order of difficulty. There is also an advanced 
 course of experimental work in Acoustics, Heat, and Electricity 
 and Magnetism, which is intended for those who have taken 
 the elementary course. 
 
 The experiments in Acoustics are simple, and of such a 
 nature that the most of them can be performed by beginners 
 in the study of Physics ; those in Heat, although not requiring 
 more than an ordinary acquaintance with Arithmetic, are more 
 tedious and apt to test the patience of the experimenter ; while
 
 vi PREFACE. 
 
 the course in Electricity and Magnetism has been arranged to 
 illustrate the fundamental laws of the mathematical theory, and 
 involves a good working knowledge of the Calculus. 
 
 The important subject of Physical Optics has been omitted 
 because we have been unable to map out a course of experi- 
 ments which would be in accordance with the ordinarily 
 accepted theory of Fresnel. At an early date, however, we 
 hope to place before the public a connected series of experi- 
 ments (based upon true fundamental laws) in that most 
 interesting of all physical studies. 
 
 A short appendix is given on the methods of determining 
 the value of Gravity and on the use of the Torsion Pendulum. 
 
 Finally, we may say that throughout the work simplifica- 
 tion of method has been our greatest, if not our only, aim ; 
 for therein we believe lies the only true sign of Progress. 
 
 UNIVERSITY OF TORONTO, April, 1895.
 
 LIST OF EXPERIMENTS IN THE ELE- 
 MENTARY COURSE. 
 
 PAGE 
 
 I. The Vernier 3 
 
 II. The Calipers . . > 6 
 
 III. The Cathetometer 8 
 
 IV. The Spherometer 10 
 
 V. Micrometer Screw Gauge ... ... 12 
 
 VI. Dividing Engine .... ... 13 
 
 VII. Specific Gravity Bottle .... . . 15 
 
 VIII. Hydrostatic Balance 16 
 
 IX. Nicholson's Hydrometer ....... 19 
 
 X. Fahrenheit's Hydrometer ....... 20 
 
 XI. Mohr's Balance 22 
 
 XII. Boyle's Law 24 
 
 (A) For pressures greater than an atmosphere. 
 (.#) For pressures less than an atmosphere. 
 
 XIII. The Volumenometer ....... 26 
 
 XIV. Determination of Capillary Constants .... 28 
 XV. The Sextant 33 
 
 XVI. The Goniometer. Two Methods ..... 36 
 
 XVII. Radius of Curvature of a Concave Mirror ... 39 
 
 XVIII. Radius of Curvature of a Convex Mirror ... 41 
 XIX. Radius of Curvature of a Spherical Reflecting Surface. 
 
 General Method , - 44 
 
 XX. Focal Length of a Biconvex Lens. Two Methods . 47 
 vii
 
 viii LIST OF EXPERIMENTS. 
 
 PAGE 
 
 XXI. Focal Length of a Biconcave Lens. Two Methods . 51 
 
 XXII. Indices of Refraction . . . . . 54 
 
 XXIII. Examples of Magnification with Lenses .... 58 
 
 XXIV. Exercises with Photographic Lenses .... 64 
 XXV. Bunsen's Photometer 68 
 
 XXVI. Ayrton's Photometer. Modified from Bunsen's . . 70 
 
 XXVII. Rotating Disc Photometer ...... 73 
 
 XXVIII. Specific Heat of Solids ....... 76 
 
 XXIX. Specific Heat of Liquids ........ 80 
 
 XXX. Latent Heat of Fusion . 83 
 
 XXXI. Level Testing . . . ' . .... 85
 
 PART I. 
 ELEMENTARY COURSE
 
 A LABORATORY COURSE IN EXPERI- 
 MENTAL PHYSICS. 
 
 EXPERIMENTS. 
 
 I. THE VERNIER. 
 
 To measuring instruments provided with a scale there is 
 frequently attached a more finely divided one, by means of which 
 readings can be made with greater precision. This latter scale 
 is called a vernier, and is arranged so as to slide along the scale 
 proper, either edge to edge, or else one overlapping the other. 
 In this latter case the edge of the upper scale is beveled. 
 Generally a length equal to n I divisions on the scale is taken 
 on the vernier, and this length divided into n equal parts, so 
 that one of the vernier divisions is equal to i/n of a scale 
 division. In this way, if the zero line on the vernier coincides 
 with a line on the scale, then the space between line I on the 
 vernier, and the line next to it on the scale is i/n of a scale 
 division, that between line 2 on the vernier, and the line next 
 to it on the scale is equal to 2/n of a scale division, while that 
 between the rth line on the vernier, and the line next to it on 
 the scale is equal to r/n of a scale division, and the ;/th line on 
 the vernier will, just as the zero line, coincide with a line on the 
 
 3
 
 EXPERIMENTAL PHYSICS. 
 
 scale. By following out a process, the converse of that just 
 described, it will be readily seen that if the rth line on the 
 vernier coincides with a line on the scale, then the space 
 between the vernier zero, and the line next behind it on the 
 scale is rjn of a scale division. 
 
 
 
 
 
 
 I 
 
 1 1 1 i 1 1 1 
 
 1 1 1 
 
 
 1 
 
 1 1 
 
 II 1 1 1 1 1 \ 
 
 
 1 1 1 
 
 
 1 
 
 1 1 
 
 
 
 
 
 
 
 
 Fig. 1. 
 
 Different values are given to , according to the purpose for 
 which the vernier is to be used. Figure i is an example of a 
 vernier in which n is equal to 10. In this case, line 5 on the 
 vernier coincides with a line on the scale, so that the space 
 between the vernier zero, and the line on 
 the scale next behind it is T 5 ^ of a scale 
 division. The reading indicated is there- 
 fore 6.5. 
 
 In the vernier on the right-hand side 
 of Fig. 2, n is taken equal to 25. Line 
 18 on the vernier coincides with a line on 
 the scale, and the vernier zero is therefore 
 |-| of a scale division above the line next 
 behind it on the scale. Here one of the 
 small scale divisions is equal to .05 of a 
 large one, and therefore the space men- 
 tioned above will be equal to .036 of a 
 large scale division, and the reading is 
 
 FIP* 2 
 
 30.05 + .036, or 30.086. 
 
 In Fig. 3, A, J3, C, and D are examples of this class of ver- 
 nier, and it will be an instructive exercise for the student to 
 verify the readings there given. In order to avoid the calcula- 
 tion just indicated, verniers are often so numbered that readings
 
 THE VERNIER. 
 
 can be made directly. One of this class is exhibited on the left- 
 hand side of Fig. 2. There n is equal to 20, and since line 19 
 on the vernier coincides with a line on the scale there is there- 
 
 fore $, or .95, of a scale division between the vernier zero and 
 the line next behind it on the scale. From the manner in which 
 the spaces are numbered, it can be readily seen that line 19 is 
 
 Fig. 4. 
 
 represented by 95, and the reading 76.395 may thus be made 
 directly. It frequently happens, when a small value is given 
 to n, that no line on the vernier coincides with one on the scale.
 
 6 EXPERIMENTAL PHYSICS. 
 
 An example of this is shewn in E of Fig. 3, where an approxi- 
 mation has been made. 
 
 Figure 4 is an example of a circular vernier. In this case, n 
 is equal to 30, and since the small scale divisions are each equal 
 to one-half a degree readings can be taken to minutes. The 
 reading indicated is 13 37'. Besides those already described, 
 verniers are sometimes constructed in which n \ of their 
 divisions are equal to n scale divisions. However, in this 
 case no special difficulty will be met with, as the same prin- 
 ciples apply to all classes of verniers. 
 
 II. THE CALIPERS. 
 
 This instrument, shewn in Fig. 5, is essentially a graduated 
 metal scale, to one end of which (and perpendicular to it) is 
 attached a bar EP. Another bar, FQ, of the same form, is capa- 
 
 Fig. 5. 
 
 ble of being moved along AB, and carries with it a slider CD, 
 which can be fixed by the screw V at any point on AB. There 
 is a small rectangular opening in CD, and on one of its bevelled 
 edges a vernier is ruled, which is so placed that when the bars 
 EP and FQ are together, the zero on the vernier corresponds
 
 THE CALIPERS. 7 
 
 with the zero on the scale. In measuring the length of an 
 object, place it between the branches PA and QC, and gently 
 press the slider against it. Then note the reading on the 
 vernier. 
 
 The calipers may also be used to measure the width of an 
 opening in a tube, or other object, provided it is greater than 
 the total thickness of EP and FQ when they are together. In 
 this case, insert AE and CF in the opening, and separate them 
 as far as its walls will permit. To the vernier reading then 
 must be added the thickness of AE and CF when they are 
 
 Fig. 6. 
 
 together, which may be found by means of another instrument. 
 It will be a useful exercise for the student to test, at various 
 times, the accuracy of the calipers, by means of a standard 
 instrument, one of which is shewn in Fig. 6. 
 
 In this standard the object to be measured is placed between 
 the two heads E, E of the branches D, D. These are free 
 to slide in the sockets F, F, so that measurements may be 
 taken between points not accessible by instruments with fixed 
 branches. If it is desired to find the width of an opening, the 
 two heads E, E are inserted in it, and then separated as far as
 
 8 
 
 EXPERIMENTAL PHYSICS. 
 
 possible. By means of a special scale and vernier on the arm 
 AB, readings for such measurements can be taken directly. 
 
 III. THE CATHETOMETER. 
 
 The cathetometer is used to measure the vertical distance 
 between two horizontal planes passing through two points not in 
 general in the same azimuth. As exhibited in Fig. 7, it consists 
 
 Figf. 7-
 
 THE CATHETOMETER. 9 
 
 of a vertical metal cylinder AB with a scale CD inserted in it, 
 is movable about a vertical axis, and is supported on a platform 
 furnished with three adjustment screws. On this platform are 
 two spirit levels, by means of which the perpendicularity of AB 
 may be tested. Attached to this vertical column, or cylinder, 
 is a carriage EF, upon which is placed a telescope with its opti- 
 cal axis horizontal. This carriage is capable of sliding up and 
 down the column, and can be fixed at any point by means of a 
 compression screw. It also has a vernier attached to it, and 
 carries a spirit level to adjust the telescope. 
 
 To make a measurement, sight the telescope on one of the 
 points so that its image coincides with the intersection of 
 the crosswires. Note the reading on the vernier, and then 
 move the carriage, and the telescope up or down, as the case 
 requires, until the image of the other point coincides with the 
 intersection of the crosswires. By means of the reading then 
 made, and the previous one, the vertical distance between the 
 points may be found. The instrument is chiefly used in finding 
 the heights of columns of mercury, examples of which are found 
 in the experiment on the capillary constant, and in that on the 
 absolute expansion of mercury. 
 
 If the two points are at a considerable distance from the 
 cathetometer, it will be found that better results may be 
 obtained by placing a scale, in a vertical position, alongside 
 of these points, and then, by means of the telescope on the 
 instrument, noting what division on the scale corresponds 
 with each of the points respectively. In the construction of 
 the instrument, it is very difficult to make the column AB 
 perfectly rigid, and to keep it so ; and from this cause the 
 readings are often not as accurate as is desirable. The instru- 
 ment may be tested, from time to time, by measuring the 
 spaces between lines, drawn with a dividing engine.
 
 I0 EXPERIMENTAL PHYSICS, 
 
 IV. THE SPHEROMETER. 
 
 The essential part of the spherometer (Fig. 8) is a microm- 
 eter screw DP, which passes through a metal tripod whose feet 
 are of equal length. This screw carries with it, at its upper 
 end, a graduated metallic circular disc 
 LL', whose plane is perpendicular to the 
 screw, and parallel to the plane passing 
 through the three terminals A, B, C, of 
 the supports. To one of these supports 
 there is attached a scale RR' graduated 
 in such a manner that on causing the 
 disc to make one complete revolution, it 
 
 Fig. 8. 
 
 will rise or fall through one of these divi- 
 sions. Usually the scale is divided into spaces one-half a milli- 
 meter in width, and the disc into 500 equal parts so that readings 
 can be taken to y^Vo of a millimeter when any particular meas- 
 urement is about to be made. It is first of all necessary to note 
 the reading when the point P of the micrometer screw is in 
 the same plane as the three terminals A, B, C, and the instru- 
 ment is resting equally on the four supports. Usually to 
 the beginner this presents some difficulty ; but with a little 
 practice, and by repeatedly grasping one of the supports A, 
 B, and C, and giving the instrument a slight rotatory move- 
 ment, one can easily perceive when contact has been made with 
 the screw. If the instrument is standing on a glass, or any 
 reflecting surface, it can easily be ascertained when contact 
 has been made by noting the instant when the point P coincides 
 with its image as seen in the surface. The instrument can be 
 used, (i) to measure small thicknesses, (2) to determine the 
 radius of curvature of a sphere, and (3) to test whether a surface 
 is perfectly plane, or perfectly spherical. If it is required to 
 measure the thickness of a plate with parallel faces, first place 
 the instrument on a perfectly plane surface (usually made of 
 ground glass), and turn the screw until its point P comes in
 
 THE SPHEROMETER. II 
 
 contact with it ; note the reading, and then raise the screw and 
 place under the point P the plate whose thickness is to be deter- 
 mined. Bring the point of the screw in contact with the upper 
 surface of this plate, and again note the reading. The difference 
 between these two readings will give its thickness. It is evident 
 that if the instrument be equally supported on the four points 
 A, B, C, and P in one position, and then caused to glide over a 
 given surface, it can be readily seen whether the surface is per- 
 fectly plane, or not. 
 
 To find the radius of curvature of a spherical surface, place 
 the spherometer on it, and ad- 
 just the micrometer screw until 
 the instrument rests equally on 
 the four supports. Note the 
 reading, and then place the 
 spherometer on a plane surface, 
 and again establish contact at 
 four points, and note the read- 
 ing. The difference between 
 these two will give the height h 
 of the spherical segment which 
 has for its base a plane passing 
 through the terminals A, B, and C, when the first reading was 
 taken. 
 
 If then r be the radius of the base of this segment, h its 
 height, and a the radius of the sphere, we have from Fig. 9, 
 
 Fig. 9. 
 
 2/1 
 
 To find r, place the instrument on a sheet of cardboard rest- 
 ing on the three supports, and the screw, and press it lightly 
 down so as to make small indentations. Since the three points 
 A, B, and C form, an equilateral triangle, r may be determined
 
 12 
 
 EXPERIMENTAL PHYSICS. 
 
 by taking the mean of the distances from each of the holes made 
 by A, B, and C to that made by the point of the screw. 
 
 Whether a surface is perfectly spherical, or not, may be de- 
 termined by adjusting the instrument to four-point contact in 
 one position, and then sliding it over the surface. 
 
 In some instruments the screw is hollow, and throughout its 
 length a slender metallic rod runs. This rod is connected to a 
 lever, and a pointer, which indicates at once the instant when 
 contact is made. 
 
 V. MICROMETER SCREW GAUGE. 
 
 This gauge (Fig. 10) is generally used to measure the diam- 
 eters of fine wires. It is constructed on the same principle as 
 the spherometer, and consists of a bracket or shoulder with two 
 arms FE and FG t in the former of which is screwed a small 
 metal cylinder C, with its protruding face perfectly plane, and 
 
 7/7 /~V 
 
 rj Lr 
 
 Fig. 10. 
 
 at right angles to the axis of the micrometer screw AB, which 
 passes through a threaded tube attached to the arm FG. The 
 end of the screw B is made plane, and is parallel to the face of 
 the cylinder C. To the other end there is attached a cap A, 
 which, when the screw is turned, comes down over the tube on 
 FG. This latter has a scale ruled on it, and the divisions are 
 such that when the screw is made to turn through one complete 
 revolution, the cap A moves over one of the divisions. Just as
 
 DIVIDING ENGINE. 13 
 
 in the case of the disc of the spherometer, the circumference of 
 the cap is divided into a number of equal parts, and readings 
 may thus be made to the fraction of a division. In some gauges 
 the scale is divided into millimeters, and the circumference of 
 the cap into twenty equal parts, so that readings may be taken 
 to 2 a o- of a millimeter; while in others the scale is divided into 
 spaces of ^ of an inch, and the circumference of the disc 
 into 25 equal parts. In this latter case, readings may be taken 
 to joVo f an mcn . For still finer readings a vernier is some- 
 times ruled longitudinally on the back of the tube, in the man- 
 ner shown in the figure. 
 
 In making a measurement, first see that, when the screw is 
 in contact with C, the zero on the cap corresponds with that on 
 the scale ; if not, it may be made to do so by screwing the cyl- 
 inder C either in or out, as the case requires. The wire to be 
 measured is then placed between the face of C and that of the 
 screw, and the latter turned gently until it is felt that contact is 
 made. The reading then indicated will give the diameter of 
 the wire. Care should be taken to insure the same manner of 
 making contact while measuring the wire as in testing for the 
 zero reading:. 
 
 VI. DIVIDING ENGINE. 
 
 The dividing engine, of which the essential parts are shewn in 
 Fig. 1 1, is employed in ruling scales, and gratings, and in measur- 
 ing small horizontal distances. It embodies the same principle 
 as the spherometer, and consists of a screw B, whose pitch, gen- 
 erally one, or one-half a millimeter, is very regular ; a divided 
 circle C attached to one end of it, which can be so adjusted that 
 a fractional part of a rotation may be given to the screw ; and a 
 carriage EE, which slides along the smooth rail / attached to 
 the solid base A. This carriage EE, on which are supported a 
 tracing point .Fwith a handle G, and a microscope H, by means 
 of which the rulings are examined, may be thrown in or out of
 
 EXPERIMENTAL PHYSICS. 
 
 gear with the screw B by turning the rod K, which opens the 
 clutch shown in the figure. 
 
 The object which is to be ruled, or measured, is placed on
 
 SPECIFIC GRAVITY BOTTLE. 15 
 
 the platform J directly beneath the tracing point, and the micro- 
 scope. 
 
 Scales on metal may be ruled either directly by using a 
 hardened steel tracing point, or by ruling on a wax-covered 
 surface, and then etching in with acid. This latter method may 
 also be adopted in the case of glass scales, but these are gener- 
 ally made by using a diamond tracing point. For such fine 
 work as ruling gratings, dividing engines are specially' con- 
 structed with screws whose pitch depends on the degree of 
 delicacy required in the work. 
 
 VII. SPECIFIC GRAVITY BOTTLE. 
 
 Figure 12 shows the specific gravity bottle, with ground glass 
 stopper, generally used in laboratory experiments. Throughout 
 the length of the stopper, a very fine hole runs, which is ter- 
 minated in a small cup-shaped opening at the top. 
 A mark is made on the narrow part of the stopper, 
 and enough liquid is put in the bottle to make it 
 rise to this point when the stopper is in tight. In 
 what follows, the bottle will always be considered 
 full when the surface of the liquid is in this 
 position. 
 
 Great care should be taken to have the bottle 
 perfectly clean. This is done by first washing it 
 out with a solution of caustic potash, and then 
 with hydrochloric acid. After this, wash it several times with 
 ordinary water, and then rinse out with distilled water, and place 
 it in a hot air bath to dry. In filling the bottle with a liquid, 
 the student should be careful to have all the air bubbles 
 removed. 
 
 EXERCISE I. To find the specific gravity of a liquid. 
 
 Let w be the weight of the bottle in air, w' its weight when 
 filled with distilled water, and w" when filled with the given 
 liquid. Equal volumes of the liquid, and of the water weigh
 
 l6 EXPERIMENTAL PHYSICS. 
 
 then w" ' w, and w r w, respectively, and the specific gravity 
 of the former is therefore given by w f ~ w . 
 
 EXERCISE II. To find the specific gravity of a solid broken 
 up into small pieces. 
 
 Let w be the weight of the pieces of the solid, w ! the weight 
 of the bottle rilled with distilled water, and w" the weight of the 
 bottle partly filled with the pieces, and the balance filled with 
 water. Then w + w' w" will be the weight of the water dis- 
 placed by the pieces, and the specific gravity of the solid is 
 therefore 
 
 w + w' w" 
 
 If the solid is soluble in water, the same method may be used 
 to find its specific gravity relative to some liquid in which it is 
 not soluble, and then, the specific gravity of this liquid having 
 been determined, that of the solid may be calculated. 
 
 A suitable exercise on this method is to find the specific 
 gravity of wires used in the experiments on the sonometer. In 
 this case the wire should be cut up into small pieces unless it is 
 very fine. 
 
 If very great accuracy is required in these experiments, cor- 
 rections will have to be made for temperature, and for the 
 buoyancy of the air. 
 
 VIII. HYDROSTATIC BALANCE. 
 
 Experiment I. In Fig. 13 a balance is shewn whose scale 
 pans are at a considerable height above the platform on which 
 it stands. To one of these pans, as shewn in the figure, attach 
 a cylindrical cup A, and also the cylindrical solid B, which just 
 fills this cup. Then balance the two by weights placed in the 
 opposite pan. Having done this, place a beaker of water under 
 A and B, so that B is completely submerged, and shew that on 
 filling A with water equilibrium is restored. This experiment
 
 HYDROSTATIC BALANCE. 17 
 
 shews that the resultant vertical pressure on the cylinder B is 
 equal to the weight of the water it displaces, and this is the 
 principle upon which the following methods are based. 
 
 EXERCISE I. To find the specific gravity of a solid insoluble 
 in water, and heavier than water. 
 
 To one of the arms of the balance suspend the given solid by 
 a very fine thread. Let w be its weight in air, and w' its weight 
 
 Fig. 13. 
 
 when submerged in water. The weight of the water displaced 
 
 by the solid will then be w w', and therefore ^. is its 
 
 w w 
 
 specific gravity. 
 
 EXERCISE II. To find the specific gravity of a liquid. 
 
 Suspend, as in Exercise I., from one of the scale pans a solid 
 heavier than water, or the given liquid, and insoluble in both. 
 Let the weight of the solid when suspended in air be w, in 
 water w', and in the given liquid w". 
 
 The weight of the water displaced by the solid will therefore
 
 18 
 
 EXPERIMENTAL PHYSICS. 
 
 be w iv', and that of the liquid displaced by it w w" . The 
 
 w w 
 specific gravity of the liquid is then , 
 
 it 
 
 w' 
 
 EXERCISE III. To find the specific gravity of a solid lighter 
 than water, and insoluble in it. 
 
 In this exercise it is necessary to use some such substance as 
 a piece of iron in order to cause the given solid to sink in the 
 water. 
 
 Let w and w' be the weights of the given solid, and the sinker 
 respectively in air, w" that of both together in water, and w'" 
 that of the sinker alone in water. 
 
 Then w' w"' is the weight of the water displaced by the 
 sinker, and w + w' w " that of the water displaced by both 
 the solid and the sinker. The weight of the water displaced by 
 the solid is therefore w w" + w'" t and its specific gravity is 
 
 then given by r . 777- 
 
 w w -f- w 
 
 Experiment II. Place a beaker of water on one of the 
 scale pans of an ordinary balance as in Fig. 14, and on the 
 
 Fig. 14. 
 
 other place the cup A, and sufficient weights to produce equi- 
 librium. Then suspend in the water, in the manner indicated, 
 the cylindrical solid B, and shew that on filling the cup A with
 
 NICHOLSON'S HYDROMETER. 19 
 
 water equilibrium is restored. This experiment shews that 
 the reaction of the cylinder B on the water, which is equal to 
 the weight of the water it displaces, is equal to the action of the 
 water on the cylinder. By adopting this method, the weight of 
 the water displaced by a given solid may be found directly, and 
 it will be very instructive for the student to use it in finding the 
 specific gravities indicated in Exercises I., II., and III. 
 
 IX. NICHOLSON'S HYDROMETER. 
 
 The apparatus (Fig. 15) consists of a hollow metallic vessel 
 B, to one end of which is attached a slender stem terminated in 
 a pan D, capable of holding weights, or the solid whose specific 
 gravity is to be determined. To the 
 other end of the instrument there is 
 attached a second platform, or pan C, 
 made very heavy, so that when the 
 instrument is at rest in water it will 
 assume the position indicated in the 
 figure. 
 
 When placed in water, the instru- 
 ment sinks until it is only partially 
 submerged, and by placing weights on 
 the pan D, it may be made to sink to 
 any required depth. 
 
 To find the specific gravity of a solid 
 insoluble in water. 
 
 Let the weight />, when put on the 
 pan D, sink the instrument to the mark 
 A on the stem, in pure distilled water. 
 
 Remove this weight, and place the given solid on the pan. 
 Let / be the weight that must now be added to cause the 
 hydrometer to sink to the mark A ; P p will therefore be 
 the weight of the solid. Leave the weight p on the pan D,
 
 20 EXPERIMENTAL PHYSICS. 
 
 and place the solid then on the pan C, and since the solid 
 will now lose a part of its weight equal to that of the water 
 it displaces, more weights must be placed on the pan D to sink 
 the instrument to the mark A. Let P' be the amount of these 
 weights. Then P' will be the weight of the water displaced by 
 the solid ; and, since its own weight is P /, its specific gravity 
 
 is therefore ~ 
 
 If the solid used is lighter than water, it will tend to rise 
 when placed on the pan C. In this case it is prevented from 
 rising by being placed in a wire cage attached to C, and the 
 experiment is conducted exactly in the manner indicated above. 
 If the solid to be tested is soluble in water, the experiment may 
 be conducted by using some liquid of known specific gravity in 
 which it is not soluble. 
 
 If the stem of the instrument be made very slender, the 
 hydrometer will be sensitive to small variations in the weights 
 placed on the pan ; but owing to the surface tension of the 
 water on this stem, and on the walls of the instrument, the 
 results obtained will not be as accurate as those obtained when 
 either of the two previous methods is adopted. The instru- 
 ment, however, enables us to find rapidly the specific gravity of 
 a solid approximately. Corrections may also be made in this 
 experiment for the temperature of the water. 
 
 X. FAHRENHEIT'S HYDROMETER. 
 
 Fahrenheit's hydrometer (Fig. 16) embodies the same prin- 
 ciple as that of Nicholson, and resembles it in form, except that 
 it is made of glass so that it may be used in all liquids. To its 
 lower extremity there is attached a small glass bulb generally 
 loaded with mercury or small shot. This causes the instrument 
 to float in an erect position. 
 
 To find the specific gravity of a liquid. 
 
 Let P be the weight of the instrument when weighed in air, 
 and p the weight which, when placed on the pan, causes it to
 
 FAHRENHEIT'S HYDROMETER. 
 
 21 
 
 sink in pure distilled water to the mark A on the stem. Let /' 
 be the weight which must be placed in the pan to cause it to 
 sink to this mark when placed in the given liquid. Then, since 
 P+P, and P+ f' are the weights of equal volumes of water, and 
 of the given liquid respectively, the specific gravity of the latter 
 
 is therefore ** 
 
 Hydrometers of constant immersion, such as those of Nichol- 
 son, and Fahrenheit, are but little used outside of laboratories. In 
 actual practice instruments similar to those exhibited in Fig. 17 
 
 Fig. 16. 
 
 Fig. 17. 
 
 are used. They sink to different levels when placed in differ- 
 ent liquids, and from the graduations of the scales attached to 
 their stems the specific gravity of the liquid tested may be 
 determined. Instead of indicating specific gravities, these scales 
 are sometimes graduated to indicate percentages of some sub- 
 stance mixed with, or dissolved in a liquid, such as alcohol, 
 sugar, or salt, in water, or butter fat in milk. When so 
 used the instruments are called alcoholometers, densimeters, 
 salimeters, or lactometers, according to the use to which they 
 are to be put.
 
 22 
 
 EXPERIMENTAL PHYSICS. 
 
 They are sometimes still further distinguished by means of 
 the name of the manufacturer, those generally used being 
 Beaume's, and Tweedel's. Small, hollow glass balls are some- 
 times used to find the specific gravity of a liquid. They are 
 made in a graded series, and numbers are cut in the glass, 
 giving their specific gravities, which have been determined by 
 using them in liquids whose specific gravities are known. To 
 find the specific gravity of a liquid, it is only necessary to find 
 which one of the balls will just float in it. 
 
 By experimenting with different solutions, and mixtures, the 
 student should make himself thoroughly familiar with the use 
 of these practical instruments. 
 
 XI. MOHR'S BALANCE. 
 
 Many instruments have been constructed for the rapid deter- 
 mination of the specific gravity of liquids, and of these the 
 Westphal modification of Mohr's balance shewn in Fig. 18 is 
 
 Fig. 18. 
 
 probably the most convenient. It consists of a weighing beam 
 AB with knife edges resting on the stand C, a glass thermom- 
 eter plummet D suspended from it by a platinum wire, and a set
 
 MOHR'S BALANCE. 23 
 
 of four riders E, E, E, E, whose weights are 5, .5, .05, and .005 
 grams respectively. The beam AB is graduated in ten equal 
 divisions, and the glass plummet, whose weight is 15 grams, is 
 of such a volume that it displaces 5 grams of distilled water at 
 16.7 C. 
 
 If, then, equilibrium exists with the plummet attached to the 
 beam in air, it can be readily seen that it will again be estab- 
 lished, if the plummet be immersed in distilled water at 16.7 C, 
 by hanging the largest, or unity rider to the hook on the end of 
 the beam. If the plummet be immersed in a liquid lighter than 
 distilled water, the unity rider is removed from the hook, and 
 placed upon that point on the beam which will bring it into 
 equilibrium. If this point lies between two division marks, the 
 unity rider is placed upon the lower of these, and the point of 
 equilibrium is sought by a similar application of the next smaller 
 rider, and also, if necessary, of the two following ones. If it 
 should happen that two riders come to find their places on the 
 same division mark, the smaller weight is hung upon the larger. 
 If when the plummet is immersed in such a liquid the riders 
 in order of size are on the points 9, 2, 7, and 6 respectively, 
 the student can easily deduce from elementary principles of 
 mechanics that the weight of the liquid displaced by the plum- 
 met is .9276 that of the distilled water displaced by it ; the 
 specific gravity of the liquid examined is then .9276. 
 
 In determining the specific gravity of a liquid heaver than 
 water, the unity rider is allowed to remain on the hook with the 
 plummet, while a second unity rider, and the three other smaller 
 riders are used on the beam as described above. 
 
 As the instrument illustrates splendidly the principles of 
 mechanics and hydrostatics, the student should make himself 
 thoroughly familiar with its workings.
 
 EXPERIMENTAL PHYSICS. 
 
 XII. BOYLE'S LAW. 
 A. For pressures greater than one atmosphere. 
 
 In Fig. 19, CD and BF are two glass tubes attached to a 
 stand, and connected at the ends D, and B by a metal tube into 
 which both are cemented. Between them a scale is so placed 
 as to indicate equal volumes in the tube BF. The upper end 
 of this tube is closed by two taps F, and G; but if the ex- 
 periment is conducted with a dry gas, only one of these is used, 
 the upper one, G, being then removed. 
 
 Experiment. Open the tap F so as to 
 allow the air in the tube BF to communi- 
 cate with that outside, and pour mercury 
 into the open end C of the tube CD until 
 it rises in both tubes to the level of some 
 chosen mark on the scale. By then closing 
 the tap F a mass of air will be inclosed 
 in the tube BF at atmospheric pressure. 
 On pouring more mercury in at C, the 
 volume occupied by the air will be gradu- 
 ally decreased, and the mercury will rise 
 to different heights in the two tubes. Let 
 //be the height of the barometric column, 
 and Fthe volume occupied by the air at at- 
 H| mospheric pressure. When h is the differ- 
 ence between the heights of the two col- 
 umns of mercury, let v be the volume of 
 the air/ The pressures, therefore, to which 
 
 Fig. 19- 
 
 the air is subjected in the two cases are proportional to //, and 
 H+h respectively. If the experiment is conducted carefully, it 
 will be found that HV=(H+h)v, which proves the law that 
 the volume of a mass of dry air at constant temperature varies 
 inversely as the pressure to which it is subjected. The tube
 
 BOYLE'S LAW. 
 
 BF may be filled with dry air before commencing the experi- 
 ment, by first filling the two tubes with mercury, and then 
 letting it run out through the tap at the bottom of the metal 
 tube, the air entering BF at the tap F after passing through a 
 drying tube filled with calcium chloride. 
 
 The same experiment may be conducted with other gases, 
 and it will be found that they follow the same law. By attach- 
 ing the tap G (shown in section at O, and O') to the tube BF, 
 we may, on account of its peculiar construc- 
 tion, admit a liquid to the tube drop by 
 drop, and the laws of superheated, and 
 saturated vapours may thus be investigated. 
 
 B. For pressures less than one atmosphere. 
 In Fig. 20, PM, a long, narrow vessel, sup- 
 ported on a stand, is nearly filled with mer- 
 cury, and AB, a glass tube closed at one end, 
 is graduated to indicate equal volumes. 
 
 Experiment. Partly fill the tube AB with 
 mercury, invert it in the vessel PM, and 
 then lower it until the surfaces of the mer- 
 cury inside, and outside the tube, are in the 
 same plane. The air in the tube is then 
 at atmospheric pressure. Let V be the 
 volume it then occupies, and H the height 
 of the barometric column. Raise the tube 
 little by little, and note each time the 
 volume the air occupies and the pressure 
 to which it is subjected. In the figure 
 AC is the volume of the air, and CD the height of the mer- 
 cury in the tube at one stage of the experiment. Since the 
 pressure is the same inside, and outside the tube at the level D, 
 it is readily seen that the pressure to which the air is subjected 
 in this case is proportional to the difference between the height
 
 26 
 
 EXPERIMENTAL PHYSICS. 
 
 CD, and that of the barometer. If CD is denoted by 7i, this 
 difference is then equal to Hh. Here again, denoting the 
 volume AC by v, we will have H V equal to (Hk}v, and will 
 thus find that the law holds for 
 pressures less than one atmos- 
 phere. 
 
 Usually there is also a baro- 
 metric tube inverted in the mer- 
 cury in PM, and attached to a 
 support. The quantity (H ' Ji) 
 can then be measured directly, and 
 accurately by means of a cathe- 
 tometer. 
 
 XIII. THE VOLUMENOMETER. 
 
 This instrument is used to find 
 the volume of a substance, such 
 as gunpowder, sugar, or salt, which 
 cannot be placed in contact with 
 water. In Fig. 21, AF, and BG 
 are two glass tubes containing 
 mercury, whose ends F and G are 
 cemented into a metal tube which 
 connects them. In this tube there 
 is a three-way tap which serves 
 different purposes, according to 
 the position in which it is placed. 
 These are indicated in Figs, (i), 
 (2), (3), and (4). 
 In (i) communication is established between the two tubes, 
 
 in (2) the mercury can run out of both tubes, in (3) it can run 
 
 out of ^.Fonly, and in (4) out of BG only. 
 
 On the tube BG are two marks, B and K, and between them 
 
 the tube expands into a little bulb ; above B the tube narrows 
 
 Fig. 21.
 
 THE VOLUMENOMETER. 27 
 
 down, and after being bent is connected to a three-way socket, at 
 one end of which is a tap E, and to the other there is attached 
 a glass globe which can be readily removed, or put on by means 
 of the collar D. 
 
 Let v be the volume of the tube between the marks B and K, 
 Fthat of the tube above B including the globe attached at D, 
 and x the required volume of the given substance. To calculate 
 v, fill the tube BG with mercury up to the mark B, and then 
 let it run out through the tap H until it sinks in the tube to the 
 mark K. By weighing the mercury that runs out v may be 
 determined. 
 
 Experiment. Attach the globe at D, and after opening the 
 tap E fill the two tubes with mercury up to the mark B. Close 
 the tap E, the air thus shut in being at atmospheric pressure, 
 and then allow the mercury to run out of both tubes until it is 
 at the mark K in the tube BG. Let h be the height of the 
 mercury in this tube above that in the tube AF, and applying 
 Boyle's Law, we have therefore, if H is the barometric height, 
 
 HV=(H-h)(V+v). (i) 
 
 Now remove the globe, put the given substance whose volume 
 is to be determined into it, and after again attaching it by the 
 collar D, repeat the operation exactly as described above. If h' 
 be now the height of the mercury in one tube above that in the 
 other, and H' that of the barometric column, we have again, 
 by applying Boyle's Law, 
 
 x + v}. (2) 
 
 From (i) we have y = - (3) 
 
 (4)
 
 28 EXPERIMENTAL PHYSICS. 
 
 .'. from (3) and (4) 
 
 x=v\- 
 
 Ji k 
 
 vH(h'- 
 
 If Wis the weight of the substance in grams, and x its vol- 
 ume in cubic centimeters, its specific gravity is given by 
 
 c r W 
 Sp. Gr. = 
 
 X 
 
 The two tubes AF, and BG are usually graduated in centi- 
 meters, and in this case h and // can be read off directly. 
 Otherwise a cathetometer must be used. 
 
 XIV. DETERMINATION OF CAPILLARY CONSTANTS. 
 
 Various phenomena in nature show that the molecules of a 
 body, whether in a solid or in a liquid state, are subject to the 
 action of two contrary forces, one of which tends to bring them 
 together, and the other to separate them from each other. The 
 
 Fig. 22. 
 
 former of these is a cohesive or attractive force (quite distinct 
 from gravitation), which acts only at very short distances be- 
 tween particles of the same, or of different matter ; while the
 
 DETERMINATION OF CAPILLARY CONSTANTS. 29 
 
 latter is due to a vibratory motion which the molecules are sup- 
 posed to receive on the application of heat. It is with these 
 attractive forces, whose presence is the condition for the exist- 
 ence of matter in the solid state, that we wish especially to deal. 
 
 That they act only at very short distances is evidenced by the 
 fact, that, if we grind up a solid into a fine powder, the particles, 
 when the matter is in this state, are so far apart that these 
 forces exert no influence ; while if we take two pieces of lead 
 or glass with very smooth plane surfaces, and place these in 
 contact, we find that they at once adhere to each other, and it 
 is only on the application of considerable force that we are able 
 to separate them. Although the presence of these forces is 
 chiefly manifested in matter in the solid state, they exert, how- 
 ever, an action between the particles of a liquid, and the general 
 term capillarity is used to denote all the phenomena produced 
 by them. 
 
 If we dip a clean glass rod into water, we find on withdrawing 
 it that a drop clings to its lower extremity. As gravity tends 
 to detach the particles of the drop from each other, and these 
 again from the glass, we learn that there is an attraction between 
 the particles of water forming the drop, and also one between 
 the particles of glass, and of water, which is greater than that 
 exerted by gravity. Again, if we suspend a glass disc horizon- 
 tally from one of the arms of a balance, and allow it to come in 
 contact with the surface of water, we find that in order to detach 
 it a considerable number of weights must be placed in the pan 
 attached to the opposite arm. If instead of water we use 
 mercury, exactly the same action takes place, but a smaller 
 weight is necessary to make the separation. If we replace the 
 glass disc by one of copper of exactly the same form, and dimen- 
 sions, we will find that the same weight as before must be used 
 to detach it from the water, but one different from that used 
 before to make the separation from mercury. The explanation 
 is simple. In the one case a layer of water adheres to the disc, 
 and we separate, not the disc from the water, but a layer of water
 
 30 EXPERIMENTAL PHYSICS. 
 
 from the main body ; while with mercury we separate the disc 
 from the liquid. This shews that the attractions between par- 
 ticles of water are less than those between particles of glass, 
 and of water, and it affords an explanation of the term wetting. 
 A liquid is said to wet a solid when the particles of the solid 
 attract those of the liquid with a force greater than that exerted 
 between the particles of liquid themselves. 
 
 Another class of phenomena caused by these molecular actions 
 is exhibited in the following experiments. If we make a small 
 cup out of a sheet of wire gauze, and, after dipping it in melted 
 paraffine, shake it so that its meshes do not become stopped up, 
 we will find that we can pour into it a large quantity of water 
 before any begins to run out. Or, again, if we rub a common 
 sewing needle with grease so that the water will not wet it, and 
 then place it gently on the surface of clean water, it will float 
 there if not disturbed. These, and many other such experi- 
 ments, lead us to conclude that at the surface of a liquid there 
 is a thin layer of particles which act in the same manner as a 
 flexible elastic membrane spread over the surface. Although 
 it is difficult to explain just why this outside layer acts in this 
 manner, we can, however, readily understand that the particles 
 near the surface of a liquid are subject to actions different in 
 degree from those on the particles in the interior. 
 
 In Fig. 23 let ABC represent a mass of liquid, and P a par- 
 ticle of it situated at the center of a small sphere DE of such a 
 radius that none of the liquid outside of it has any action on 
 the particle at P. If then P is at a point in the liquid whose 
 distance from the surface is greater than the radius of this 
 sphere, it is evident, from symmetry, that the resultant action on 
 the particle is zero. If, however, it is situated at P v it is clear 
 that there is no liquid to compensate the action of the part FGH 
 of the sphere, and there will be, in this case, a resultant 
 action on P 1 depending on the mass FGH, and tending to draw 
 it into the body of the liquid. This will be still more evi- 
 dent if the particle is situated in the surface, at P 2 ; for in
 
 DETERMINATION OF CAPILLARY CONSTANTS. 31 
 
 this case there is the action of a hemisphere of liquid not 
 compensated. 
 
 We thus see that the particles at, or near, the surface of a 
 liquid are subjected to attractions which tend to draw them into 
 the main body, and the effect on the whole mass is precisely 
 the same as if it were surrounded by a thin elastic membrane. 
 
 In the case of liquid films, and soap bubbles we have two 
 sets of surface particles with a thin layer of liquid between them. 
 Delicate experiments have been devised with these to ascertain 
 the force required to separate the particles in the surface layer 
 
 Fig. 23. 
 
 from each other by measuring the radii of bubbles just before 
 they break, and by noting the pressures to which they are sub- 
 jected ; but probably it can best be found in the case of liquids 
 which wet glass by the use of capillary tubes. Figure 22 
 shows a glass vessel nearly filled with such a liquid in which 
 are suspended several capillary tubes A, B, C, and D, a sharp- 
 pointed screw E which can be raised or lowered, and a cathe- 
 tometer L, If the liquid be drawn up through one of these 
 tubes, and then allowed to fall back, it will leave a layer of its 
 particles adhering to the glass on the inside, and will not sink 
 to the level of the outside liquid, but will remain stationary at 
 a height depending on the radius of the tube. 
 
 Since the hydrostatic pressure inside the tube is the same as 
 that outside for points in the same horizontal plane, then the 
 pressure inside just below the surface layer of particles is less
 
 32 EXPERIMENTAL PHYSICS. 
 
 than that at the surface of the main body of the liquid, and we 
 may therefore look upon the column as being held up by the 
 attractions between the particles along the periphery of the sur- 
 face layer, and those adhering to the tube. If, then, T denotes 
 the value of this surface tension per unit length, h the height 
 of the column, and r its radius, we have, since the angle at 
 which the surface of the liquid meets the tube is zero, 
 
 or T=gP.rh. 
 
 As this tension T is exerted between particles of the same 
 kind of matter, it should have a constant value whatever may be 
 the radius of the tube. The product, rh, therefore should be a 
 constant, and this may be illustrated by using tubes of different 
 sizes. 
 
 In order to ascertain the height h at which the liquid stands, 
 lower the screw E until it just grazes the liquid, and then with 
 the cathetometer find the vertical distance between the surface 
 of the liquid in the tube, and the upper extremity of the screw. 
 The difference between this length, and that of the screw, which 
 has been previously determined, will give h. 
 
 The radius of the tube r is found by running a thread of 
 mercury into it. If / be the length of this thread in centimeters, 
 w its weight in grams, and d the density of mercury, then 
 
 = w, 
 
 or 
 
 irld 
 
 This method of finding the surface tension may be adopted 
 in the case of such liquids as water, alcohol, or sulphuric acid, 
 and in order to obtain uniform results, the greatest care must be 
 taken to have tubes, and liquid perfectly clean. 
 
 There are many instructive experiments in capillarity, and 
 students should especially make themselves familiar with those
 
 THE SEXTANT. 33 
 
 devised by Plateau on liquid films, and on the forms that a 
 mass of liquid will take when freed from the action of gravity. 
 
 A full account of the various elementary phenomena of capil- 
 larity is to be found in a little "book entitled " Soap Bubbles " 
 by C. V. Boys. 
 
 XV. THE SEXTANT. 
 
 This instrument, shewn in Fig. 24, consists of a framework 
 in the shape of a circular sector, of which the arc GH is gradu- 
 ated. About the center of this sector turns an index arm EF 
 provided with a vernier and tangent screw. At the center 
 a mirror M is fixed normally to this arm, and moves with 
 
 M 
 
 F 
 
 Fig. 24. 
 
 it ; while to the frame on the left-hand side is attached a 
 second mirror M', which has only the lower half silvered, and 
 may by means of screws be rotated about an axis either parallel, 
 or perpendicular to the frame. Opposite to this mirror, in most 
 sextants, there is attached permanently to the frame a tele- 
 scope, while in others a small hole in a screen at L is substi- 
 tuted for it.
 
 34 
 
 EXPERIMENTAL PHYSICS. 
 
 Experiment. The sextant is used to measure angles sub- 
 tended at the eye by two distant objects. Before taking 
 a measurement it is necessary to know that the two mirrors are 
 parallel, and this is arrived at by first fixing the sliding arm at 
 
 Fig 25. 
 
 zero, and then looking through the telescope and the unsilvered 
 glass M' at some very distant object. 
 
 If the two mirrors are parallel, the observer will notice that 
 
 immediately below the portion of the object viewed directly he 
 
 2 will see an image of the 
 
 other part by means of 
 rays of light coming from 
 it after having been twice 
 reflected at the mirrors M 
 and M'. This will be evi- 
 dent from Fig. 25, in which 
 AB and DC are rays coming 
 from a very distant object. 
 If this is not the case, then 
 the mirrors are not parallel, 
 V\ v but are made so by turn- 
 
 F . 26 ^ ing the mirror M' by the 
 
 screws mentioned previ- 
 ously. This done, hold the sextant so that its plane may pass 
 through both the objects whose angular distance is to be
 
 THE SEXTANT. 
 
 35 
 
 measured ; then look through the telescope and the unsilvered 
 glass at the object to the left, and turn the arm EF until the 
 eye sees the image of the second object, after the rays forming 
 it have been twice reflected, coincide with the object seen 
 directly. The angular distance between the objects is then 
 twice the angle between the mirrors, and therefore twice the 
 angle through which the movable arm has been turned, as is 
 evident from the following proof. 
 
 In Fig. 26 D and F are two distant objects, BA and //"are 
 the two mirrors M and M', and DC and FH are rays of light 
 coming from the objects D and F respectively. 
 
 The angle 
 
 = the angle DCff-the angle NHC 
 = 2 (angle ECH- angle GHC] 
 = 2 (angle ECH+ 90 -angle GtfC-go ) 
 = 2 (angle ACH- angle CHM) 
 = twice angle HMC, which is the angle be- 
 tween the mirrors. 
 
 Besides measuring the altitude of the sun, the angular dis- 
 tance between two stars, or that between two very distant 
 objects, a very instructive a 
 
 exercise with the sextant 
 is to measure the angle 
 subtended at the eye by 
 the sun's or the moon's 
 disc in various positions 
 in the heavens. In meas- 
 uring the sun's diameter, 
 dark glasses may be placed 
 in front of the mirrors M 
 and M'. The sextant may 
 
 also be used to find the height of a tower whose base is 
 supposed inaccessible.
 
 36 EXPERIMENTAL PHYSICS. 
 
 In Fig. 27 let AB represent the tower, and C a point on it in 
 the same horizontal line as the observer's eye. Then since 
 
 cotAEC= , and cot ADC=^, 
 cot ABC- cot ADC= ~ DC 
 
 .: AC= 
 
 AC 
 ED 
 
 cot AEC- cot ADC 
 
 So that if the observer, while in the position EF, measures 
 the angle ABC, subtended at the eye by AC, and then moves 
 up a known distance ED, and observes the angle ADC, he 
 can, by applying the above formula, find the length AC, and 
 adding to this the height of the observer, that of the tower may 
 be obtained. The above formula may be easily adapted to 
 logarithmic calculation. 
 
 XVI. THE GONIOMETER. 
 
 Figure 28 shews this instrument, which is adapted to measure 
 the angle of a crystal. It consists of a divided circle, which 
 has at its center a small glass plat- 
 form concentric with it, and capable 
 of an independent rotation. 
 
 The instrument is provided with a 
 collimator, at one end of which is an 
 adjustable slit which can be illumi- 
 nated, and at the other there is inserted 
 a convex lens to cause the rays coming 
 from the slit to issue parallel to each 
 other. There is also a telescope, at- 
 tached to a sliding arm, that can be 
 rotated about the axis of the divided circle. To the arms of 
 the small glass platform, and the telescope are attached verniers 
 which generally permit readings to be taken to minutes.
 
 THE GONIOMETER. 37 
 
 METHOD I. 
 
 Theory. In Fig. 29 AD is a beam of parallel light, which 
 on striking the angle EDF of the crystal at right angles to the 
 edge is broken up into two beams which are reflected along DB 
 and DG. 
 
 By the laws of reflection, the beam BD has been deviated 
 from its original path AD through an angle equal to twice the 
 angle between AD and ED produced, and also the beam DG 
 through twice the angle between AD and FD produced. 
 Therefore the angle between BD and GD is equal to twice the 
 sum of the angles made by ED and FD produced, with AD, 
 which is twice the angle of the crystal. 
 
 Experiment. Place the crystal upon the central glass plat- 
 form so that its edge is in the axis of the graduated circle, 
 and so turned that the light from the illuminated slit, after 
 passing through the collimator, strikes upon this edge, and 
 being split up into two beams is reflected from each of the two 
 faces forming the angle. This may 
 be accomplished by trial with the 
 telescope. Turn the telescope so 
 as to intercept one of these rays, 
 and note the reading on the 
 divided circle ; then turn it until 
 it receives the other, reflected 
 beam, and again note the reading. 
 The difference between these two 
 readings is twice the angle of the 
 crystal. From the theory the 
 
 student will readily see that great care must be taken, if reliable 
 work is desired, to have the edge of the crystal placed accurately 
 over the axis of the circle, and perpendicular to it. This condi- 
 tion of perpendicularity may be obtained by noting when the 
 edge of the crystal is in line with its image seen reflected in the 
 glass platform.
 
 38 EXPERIMENTAL PHYSICS. 
 
 It is also desirable to choose a part of the edge for the light 
 to fall on which has the smallest number of flaws in it, or else 
 it will be difficult to take the telescope readings with accuracy. 
 
 METHOD II. 
 
 Theory. In Fig. 30 AB is a beam of parallel light incident 
 upon the edge of the crystal DBE at the angle B, BC is the 
 path taken by this beam when reflected by the face BE, and 
 D'BE' is the same crystal in its second position, with its face 
 D'B now in the plane formerly occupied by BE'. From this 
 it is evident that the ray on striking the face D'B will be 
 again reflected along BC. 
 
 Now since BE is in the same straight line as D'B, it is 
 
 clear that, as DB in the first 
 position corresponds with 
 D'B in the second, the crys- 
 tal has been turned through 
 an angle equal to 180, less 
 ^ ===<7 the angle DBE, so that 
 if we measure the angle 
 through which the crystal 
 is turned, and subtract this 
 from two right angles, we 
 get the angle of the crystal. 
 
 Fig. 30. 
 
 Experiment. Place the crystal with its edge as indicated 
 in the previous experiment ; then turn the telescope until it 
 receives the ray BC, and clamp it there. Then note the read- 
 ing on the divided circle as indicated by the vernier on the arm 
 attached to the platform on which the crystal rests. After this 
 turn this arm until, on looking through the telescope, the ob- 
 server again sees the light reflected along BC. This indicates 
 that the crystal is in the second position D'BE', and by again 
 noting the reading on the divided circle the angle through
 
 RADIUS OF CURVATURE. 39 
 
 which the crystal has been turned may be found. To secure 
 accurate results the same precautions must be taken as in 
 finding the angle by the previous method. 
 
 XVII. RADIUS OF CURVATURE OF A CONCAVE SPHERICAL 
 MIRROR. 
 
 Theory. In Fig. 31 DHE is a section of a concave spheri- 
 cal mirror whose center is C. P is the illuminated point, P' 
 
 Fig. 31. 
 
 its image, and PA, AG are the incident and reflected rays. 
 By the law of reflection, the angle PAC is equal to the angle 
 CAG, and 
 
 sinPAC^smP'AC . PC = P'C 
 ' smACP~sinACP'' ' AP AP 1 ' 
 
 Now limit the ray PA by supposing it incident close to H, 
 the point where PC produced meets the mirror. Then AP and 
 AP' will become equal to HP and HP' respectively, 
 
 . PC P'C 
 
 ' ' PH P'H 
 PH-HC HC-P'H 
 
 PH P'H 
 
 I I 2
 
 40 EXPERIMENTAL PHYSICS. 
 
 or, denoting PH t P'H, and HC by /, /', and r respectively, we 
 
 obtain the formula - -\ ; = -, connecting the distances of the 
 p p 1 r 
 
 image and the object from the mirror with the radius of cur- 
 vature. 
 
 Experiment. Place an illuminated object before the mirror, 
 and receive its image on a screen situated so as not to cut off 
 all the rays issuing from P. Then measure / and p' directly, 
 
 and substituting their values in the formula -H - = -, obtain 
 
 p p r 
 
 the value of r. The experiment may be varied by giving p 
 values ranging from infinity to -, noting particularly the case 
 where p = r, in which case p' also equals r, and image and 
 object coincide. For values of / less than -, it is evident from 
 
 the formula that p' is negative. The physical signification of 
 this is that the image is then virtual, and so can no longer be 
 received on a screen. 
 
 For an illuminated object, a candle or small gas flame may be 
 used, or, better still, a fine platinum wire suspended in a Bun- 
 sen flame, or else a dark thread stretched across a small illu- 
 minated opening in a screen. 
 
 For ordinary purposes it will be sufficient to make the 
 measurements with a millimeter scale, but, for greater accu- 
 racy, place the illuminated object and the mirror on stands 
 which slide freely on a bench with scale attached. By 
 using verniers on these stands very close results can be 
 obtained. A very useful exercise in connection with this 
 experiment is the determination of the radii of curvature of 
 small mirrors, such as are used in reflecting telescopes, or 
 galvanometers.
 
 RADIUS OF CURVATURE. 
 
 XVIII. .RADIUS OF CURVATURE OF A CONVEX SPHERICAL 
 MIRROR. 
 
 Theory. In Fig. 32 ABA' is a section of a convex spherical 
 mirror whose center is C. P is the position of a lighted candle, 
 
 Fig. 32. 
 
 or gas jet, P' its image, and PA, AN are the incident and 
 reflected rays. 
 
 By the law of reflection, the angle MAL is equal to the angle 
 MAN, and therefore also equal to the angle P 1 AC. 
 
 smPAC = smP'AC 
 sin ACP~ sin 
 
 PC = P'C 
 PA P'A 
 
 Now limit the ray PA by supposing it incident close to B, 
 the point where PC cuts the mirror. Then PA and P'A will 
 become PB and P'B respectively, and 
 
 PC = P'C 
 ' PB P'B 
 
 PB+BC ^BC-P'B 
 PB P'B
 
 42 EXPERhMENTAL PHYSICS. 
 
 or, denoting PB, P' B, and BC by /, />', and r respectively, we 
 have for a formula connecting the distances of the object and 
 the image from the mirror, with the length of the radius, 
 
 Experiment. Place the mirror in an upright position. Then 
 cover the face of it with a sheet of paper having two holes in it, 
 denoted by A and A', at equal distances from the center of the 
 surface of the mirror. It will probably also be found most con- 
 venient to have them in the same horizontal plane as the center. 
 By this arrangement there will be two rays of light, AN and 
 A'N', reflected from the mirror. Intercept these rays by a 
 screen NN' so made as to permit this, and yet not cut off all 
 the light coming to the mirror. The measurements then to be 
 taken are PB, the distance between the candle and the mirror, 
 BT that of the screen from the mirror, AA' that between the 
 holes in the paper, and NN 1 that between the spots of light 
 formed by the intercepted rays on the screen. 
 
 Then, since the triangles P'AG and P'NT are similar, we 
 
 have = PB ^ BT j or if A A 1 is very small, 
 
 P'B = P'B + BT 
 
 AB~ TN 
 
 p , B= AB-BT = AA'-BT 
 
 TN-AB NN'-AA' 
 
 Therefore P'B or /' can be found, and having measured 
 BP, or p directly, we can, by substituting the values of these 
 
 two quantities in the formula --- = -, obtain ;'. 
 
 / / r 
 
 In order to obtain accurate results in this experiment, the 
 two holes in the paper must be taken as close -together as cir- 
 cumstances will permit ; and, as it is somewhat difficult to meas- 
 ure accurately the distance between the spots of light, it will be
 
 RADIUS OF CURVATURE. 
 
 43 
 
 well to have the holes in the paper very small, and to take 
 great care to have their edges neatly cut. 
 
 Note on the construction of images. By an extension of the 
 theory given in the two preceding experiments, we may find the 
 position of the image 
 of an object which is 
 not a point, as we there 
 supposed it to be. In 
 Fig. 31 let PQ be the 
 object. Join QC, and 
 produce it to meet the 
 section DHE in B. 
 Then just as we found 
 P', the image of P, 
 to lie on PH, and to be 
 given by the formula 
 
 Fig. 33. 
 
 must lie on QB and be given by the relation 1 ?_=_?_. 
 
 QB QB BC 
 
 By joining Q' and P 1 as thus found, Q'P' will be the image of 
 QP. From the above explanation it will be readily seen that 
 Q'P', will not be quite perpendicular to either PH, or QB, and 
 it is therefore for this reason that objects as seen in a spherical 
 
 mirror appear distorted. From the formula -H = -, it will 
 
 p p' r 
 
 be seen that if the object be placed off at an infinite distance 
 the image will be at a distance - from the mirror. This point 
 
 is called the focus, and as all rays coming to the mirror from a 
 point at an infinite distance are practically parallel, we may 
 define it to be the point through which pass all the reflected 
 rays which are incident on the mirror parallel to the axis, PH. 
 Thus we see the ray QA, which is parallel to PH, will, 
 when reflected, approximately pass through the focus F which 
 bisects CH. Again, a ray QC which passes through the center 
 will, when reflected, come back on itself. The point where this
 
 44 
 
 EXPERIMENTAL PHYSICS. 
 
 ray intersects the ray AF will be the image of Q, and by drop- 
 ping a perpendicular from this point on the axis PH, we will 
 
 approximately have the 
 position of P'Q' the 
 image of PQ. 
 
 This method of con- 
 structing images is ex- 
 hibited in Figs. 31, 33, 
 and 34, and it will be 
 an instructive exercise 
 for the student to apply 
 the same method to the 
 
 Fig. 34. 
 
 case of a convex mirror. 
 
 XIX. RADIUS OF CURVATURE OF SPHERICAL MIRRORS. 
 GENERAL METHOD. 
 
 Theory. In Fig. 35 HDG is a section of a convex mirror 
 whose center is C; A and B are two illuminated objects; A 'and 
 B' are their images; EF is a finely divided scale placed before 
 
 G' 
 
 Fig. 35. 
 
 the mirror in the same horizontal plane as A and B; and PG', a 
 telescope placed in the line joining C to the point midway be- 
 tween the objects A and B, has its objective in the line AB. 
 L and L' are the points where the lines A'P and B'P cut EF.
 
 RADIUS OF CURVATURE. 45 
 
 The formula connecting image and object for convex mirrors 
 
 is -=- -; and since A', B', and P' are the images of A, B, 
 p p' r 
 
 and P respectively, we have denoting LL' by y, AB by x, A'B' 
 by x', PD by/, P'D by/', and PP' by/, 
 
 rH ! (I> 
 
 and by comparing the sizes of the image and of the object, we 
 have 
 
 j-TJp- & 
 
 From equation (i) we have 
 
 / r / r 
 
 r ~T = ~p r ' 
 
 P' = r -P' = CP> =*', 
 ' p r+p CP x 
 
 Again, from (3), /' = z > (4) 
 
 r+2/ 
 
 r+2p 
 
 (5) 
 
 From the figure = y . 
 
 y P 
 
 V = P- xr , from (5) 
 
 / r+2p 
 
 ^- - from (4) 
 
 2(r+p) 
 
 :-?L (6) 
 
 x2y
 
 46 EXPERIMENTAL PHYSICS. 
 
 By a process of reasoning similar to this, the student may 
 easily find for himself that in the case of concave mirrors the 
 formula becomes 
 
 - 2 ^ (7) 
 
 Experiment. If this experiment is performed in the dark, 
 two lighted candles, or gas jets may be used for the illuminated 
 points A and B ; and in case the scale FE is not sufficiently 
 illuminated to be easily read, it may be made so by projecting 
 a beam of light on it by means of a convex lens. However, 
 it is better to perform the experiment in a well-lighted room, 
 and to use two porcelain, or ivory scales with divisions made 
 in black, the one to take the position FE, and two marks on 
 the other to replace the illuminated points A and B. Having 
 then placed the mirror, telescope, illuminated objects, and scales 
 in the positions indicated, adjust the telescope so as to be able 
 to read the scale FE. If PD is taken of considerable length, 
 A 1 and B' will be approximately in focus at the same time as 
 FE, and the length LV on the scale, cut off by the space 
 between A' and B', may thus be directly read off by means of 
 the telescope. The lengths p and x are measured with ordinary 
 scales, and for accuracy it would be well to check the work by 
 giving these quantities different values. 
 
 This method may also be used to find the radius of curvature 
 of the surface of a lens. In this case each face of the lens 
 will give a pair of images. By noting that the images produced 
 by a convex reflecting surface are always erect, the student will 
 find no difficulty in distinguishing the two sets.
 
 FOCAL LENGTH OF A BICONVEX LENS. 47 
 
 XX. FOCAL LENGTH OF A BICONVEX LENS. 
 METHOD I. 
 
 Theory. In Fig. 36, AC is a biconvex lens whose focus is 
 at F, PP' is the principal axis, PQ is an illuminated object, and 
 P'Q' its image. If we assume that a ray of light striking the 
 lens at its center passes through without deviation, and that 
 
 Fig. 36. 
 
 a ray parallel to the principal axis on striking it ultimately 
 passes through the focus, we are able to obtain a relation 
 connecting the distance of the object, and of its image from 
 the center of the lens C, with the focal length. 
 
 If QC and QA are these two rays, the point Q' where they 
 intersect will be the image of Q, and dropping a perpendicular 
 Q'P' on PP', P'Q' will be the image of PQ. 
 
 Denoting CP by p, CP' by /, and CF by /, we have, from 
 similar triangles, 
 
 P'Q'~ S' 
 
 Again, by erecting at C a line perpendicular to PP', and ter- 
 minated at the point where it cuts the ray QA, we have 
 
 (2) 
 
 .-. from (i) and (2) 
 
 P'-f _ PL . -I + 1--L 
 
 ~r~ p - p' P /
 
 4 8 
 
 EXPERIMENTAL PHYSICS. 
 
 Experiment. Having placed the object and the lens in the 
 same horizontal line, move the screen until the image is dis- 
 tinctly focussed on it. Then measure directly the distance of 
 the image and of the object from the center of the lens, i.e. 
 p and /', and substituting the values of these quantities in the 
 
 formula -\ = , obtain f, the focal length. As in the case 
 of concave mirrors, although a lighted candle, or gas jet may 
 
 Fig. 37. 
 
 be taken for the illuminated object, it will be found that more 
 accurate results are obtained when an incandescent platinum 
 wire, or a dark thread stretched over an illuminated opening 
 in a screen is used. If this experiment is conducted in a dark 
 room, it will be found that a screen painted white or one 
 made of ground glass will be the most suitable. Great care 
 should be taken to have the image well denned. 
 
 If the object be at a great distance, the formula indicates 
 that the image will then be at the focus, and the focal length 
 may then be measured directly. If/ be taken equal to /', each 
 is then equal to 2/, and in that case, the distance between 
 object and screen need only be measured. If / be taken less 
 than /, the formula indicates that /' becomes negative, which 
 signifies that the image is then virtual. This is the case 
 when a convex lens is used as a simple microscope, and Fig. 
 37 then indicates the relative positions of the image and the 
 object.
 
 FOCAL LENGTH OF A BICONVEX LENS. 49 
 
 METHOD II. 
 
 In Fig. 38 P is an illuminated object, L a biconvex lens, and 
 P' the position of a screen with the image of P clearly defined 
 on it. From the symmetry of this arrangement, as well as from 
 
 L L' 
 
 Fig. 38. 
 
 an inspection of the formula -+- = -, it can be readily seen 
 
 P P f 
 
 that if the lens be moved from L to L 1 so that P'L' is equal to 
 PL, there will again be an image of P on the screen at P'. 
 Denoting then the lengths PP' and LL' by / and a respectively, 
 we have 
 
 V ~~A * ~ 
 
 p - and # = -- 
 2 2 
 
 Substituting these values of / and /' in the formula 
 
 / 4 7 = 7 
 
 we have 
 
 ' /-^ ' < 
 
 Experiment. Place the illuminated object and the lens in 
 position at some chosen distance apart, which, as is evident 
 from Fig. 37, must be greater than the focal length of the lens. 
 Then adjust the screen so that the image of P is quite distinct 
 on it, and measure PP', or / the distance between the object 
 and the image. Measure also the distance a through which 
 the lens must be moved in order to have an image of P focussed 
 on the screen a second time, and substituting these values of 
 / and a, in formula (i), find / the focal length of the lens.
 
 50 EXPERIMENTAL PHYSICS. 
 
 This method is especially applicable in finding the focal 
 lengths of combinations of lenses such as the objectives and 
 eyepieces of optical instruments, or those used in photographic 
 cameras. 
 
 NOTE. If non-achromatized lenses are used in these experi- 
 ments, it is generally difficult for a teginner to decide where 
 the screen should be really placed. 
 
 In Fig. 39 AB represents a simple non-achromatized lens, 
 P an illuminated point, and PA and PB the limiting rays 
 of white light which strike the lens. These, when refracted 
 
 Fig. 39. 
 
 through the lens, become dispersed, and two pencils of coloured 
 rays, DAC&nd FBE, are obtained. AD and BE will therefore 
 be red rays, and AC and BF violet, since the latter are more 
 refracted than the former. It will be readily seen from the 
 figure that any point between G and H may be said to be the 
 image of P. If the screen be placed at G, the image on it 
 will be fringed with a reddish tint, while if placed at H it will 
 have a violet border. Between these two points then there 
 will be a point K where the red and the violet rays overlap, and 
 the image will be of one colour. The screen should therefore 
 be placed at this point.
 
 FOCAL LENGTH OF A BICONCAVE LENS. 51 
 
 XXI. FOCAL LENGTH OF A BICONCAVE LENS. 
 METHOD I. 
 
 Theory In Fig. 4.0 AC is a biconcave lens whose focus is 
 
 at F, PP' is the principal axis, PQ is an illuminated object, and 
 P'Q' is its image. If we assume, as in the case of a convex 
 lens, that a ray, on striking the lens at the center, passes through 
 without deviation, and that one parallel to the principal axis 
 appears ultimately to diverge from the focus, we may also in 
 
 Fig. 40 
 
 this case obtain a formula connecting the distance of the image 
 and of the object from the center of the lens C, with its focal 
 length. QA and QC are two such rays, and as on emerging 
 they appear to come from Q', therefore Q'f, the perpendicular 
 from Q' on PC, will be approximately the image of QP. Denot- 
 ing PC by /, P'C by p\ and FC by /, we have from similar 
 triangles, 
 
 If a line perpendicular to PP' be erected at C to meet the ray 
 QD, we will have pn f 
 
 From (i) and (2) 7 = ' (3) 
 
 TT? 
 
 which is the formula required.
 
 EXPERIMENTAL PHYSICS. 
 
 It will be seen, in this case, that the image is always virtual, 
 and so cannot be measured directly. The following method of 
 finding the focal length is generally adopted. 
 
 In Fig. 41 AB is the lens, one of whose surfaces is covered 
 with paper pierced by two small holes, or slits at E and F, which 
 
 Fig. 41. 
 
 permit two pencils of light PE and PFto pass through the lens, 
 and to form two bright spots, G and ff, on a screen placed 
 behind it. As these pencils on issuing from the lens appear 
 to come from the point P', it is therefore the image of P. 
 
 From similar triangles, GHP' and EFP 1 , we have approxi- 
 mately, 
 
 GH _EF 
 P'C 1 
 
 EF-CS 
 
 or 
 
 P'C+CS 
 P'C 
 
 GH-EF 
 
 (i) 
 
 If we therefore measure the distance between the slits E and 
 F, and that between the spots of light on the screen G and //", 
 and also measure CS, the distance between the screen and the 
 lens, we may, by substituting in equation (i), find P'C, i.e. p' . 
 Having found PC, or p directly, we may then, by means of the 
 
 formula = -, find /"the focal length of the lens. 
 
 P P f
 
 FOCAL LENGTH OF A BICONCAVE LENS. 
 
 53 
 
 Experiment. The results in this experiment may be checked 
 by varying the distance between the holes, or slits in the paper, 
 that between the screen and the lens, or that between the illu- 
 minated point and the lens. A lighted candle, or gas jet, will 
 be sufficient for an illuminated object, and best results are 
 obtained when the slits E and F are made very 'small and 
 taken close together. 
 
 METHOD II. 
 
 Theory. The focal length of a biconcave lens may also be 
 found by using it in combination with a biconvex sufficiently 
 powerful to make the two together act as a convex lens. In 
 
 Fig. 42. 
 
 Fig. 42 AB is a biconvex lens with center C and focal length 
 FC, A'B' is a biconcave lens with center C and focal length 
 F'C, PQ is an illuminated object, P'Q' the image that would 
 be formed, were the concave lens removed, and P"Q" is the 
 image of PQ that is formed by the combination ; while the 
 heavy line joining Q and Q" is the actual path of a ray through 
 both lenses. From the figure it will be evident that Q" is the 
 image of Q' with respect to the lens A'B', because the ray DE, 
 which takes the direction EQ" with the lens A'B' interposed, 
 would follow the path EQ', were it removed. 
 
 By the principle of reversion applicable in the study of optics, 
 if Q" were an illuminated point, and Q"E were a ray of light
 
 54 EXPERIMENTAL PHYSICS 
 
 coming from it, then this ray, after passing through the lens 
 A'B', would, take the path ED, and Q would be the image of Q". 
 
 Since -- = is the formula for a concave lens, the following 
 equation will hold : 
 
 If then CQ' and CQ" can be found, /, the focal length, may be 
 determined. 
 
 Experiment. Take for an illuminated object an incandescent 
 platinum wire ; then place the convex lens at a certain distance 
 from it, which is to remain the same throughout the experiment. 
 On a screen placed at Q' receive P'Q ', the image of PQ formed 
 by the single lens. Measure CQ', and after removing the screen 
 interpose in the path of the rays the concave lens A'B', and 
 again receive on the screen P"Q'', the image of PQ formed by 
 the combination. Measure C'Q" and the distance between the 
 centers of the lenses CO. Then since CQ'=CQ'-CC, we can 
 obtain CQ', and having measured C'Q" directly, we may, by sub- 
 
 stituting these values in the equation 777^ 7^ ^ = , obtain/, 
 
 C'Q C Q f 
 the focal length of the lens A'B'. 
 
 The results obtained may be checked by varying the distance 
 between the lenses, or that of the incandescent wire from the 
 convex lens. In this experiment the student may meet with 
 some difficulty in deciding where the screen should be placed, 
 owing to the lenses not being achromatic; but by noting the 
 precautions given in Experiment XX. this may be easily over- 
 come. 
 
 XXII. INDICES OF REFRACTION. 
 
 Theory -- When a small pencil of white light strikes a prism, 
 as SE in Fig. 43, it is, owing to the unequal refrangibility of 
 the rays composing it, broken up into a series of them, each
 
 INDICES OF REFRACTION. 55 
 
 corresponding to a different colour. In this experiment, as we 
 wish to deal only with a single ray, suppose S to be a simple 
 ray obtained from a source of 
 monochromatic light. 
 
 When the ray S strikes the 
 prism at E, it is refracted, and 
 passing through the prism 
 along EF is again refracted 
 at F, and emerging takes the 
 
 direction FR. EO and FO are normals to the faces of the 
 prism at E and F respectively. Designating the angles SEG 
 and RFH by i and /', and the angles FEO and EFO by r 
 and r 1 , we have, from the law of refraction, 
 
 sin z'=/i sinr, (i) 
 
 sin i'= /j. sin r 1 . (2) 
 
 Also, since the angle EOF is the supplement of the angles 
 r and /, we have 
 
 r+r'=A. (3) 
 
 Again, denoting the angle of deviation by D, we have 
 D=DEF+DFE 
 
 from (3) =i+i'-A. (4) 
 
 From (i), we have 
 
 sinz=/A sin r\ 
 .'. from (3) we have 
 
 sin z = /4sin (A r')=/x sin A cos r 1 p cos A sin ^ 
 = jj. sin A cos r' cos A sin i\ 
 .-. {sinz + cos^ sinz'| 2 = /* 2 sin 2 ^ cos 2 r', 
 .-. {sinz + cos^ sin i '| 2 + sin 2 A sin 2 *' 
 
 = /i 2 sin 2 A cos 2 / + /* 2 sin 2 ^ sin 2 /
 
 56 EXPERIMENTAL PHYSICS. 
 
 from (2) ; 
 
 .;. sin 2 2 + 2 cos A sin i sin r + sin 2 z'=/n 2 sin 2 A 
 
 . 9 . , . o , I COS 21 , I COS 2 t' 
 
 Now snr z -J- sm^ z = 1 
 
 2 2 
 
 I ^ {COS 2 Z + COS 2 Z'} 
 = I COS (z : z') COS (z + Z*), 
 
 and since cos (z z'') cos (i+ z') = 2 sin z sin *', 
 . . /i 2 sin 2 ^ = i - cos (z - z ') cos (z + z ') 
 
 + cos A [cos (z z') cos (z + z')] ; 
 . . since sin 2 A + cos 2 A = i , 
 (fj?- i) sin 2 ^4 = [cos ^4 +cos (z - z -f ) } {cos A -cos (z + z') } ; 
 
 i.e. the product of the two quantities on the right-hand side 
 is a constant quantity. Now from (4) we see that D is least 
 when (z'+z') is least, i.e. when cos(z'+z') is greatest, i.e. when 
 cos^4 cos(z'+z' f ) is least; and since this quantity multiplied by 
 cos^+cos(z z') equals a constant quantity, therefore when 
 D is least, cos^+cos(z z') is greatest, i.e. D is least when 
 cos(z z') is greatest, i.e. when i z'=o. When D, therefore, 
 is a minimum, z is equal to i'. The deviation is a minimum 
 when the ray SE strikes the prism so that the angle of inci- 
 dence equals the angle of emergence. Now when z = z', we 
 have r=f j , and D=2t A, and A2r\ 
 
 .'. since sin Z=/A sin r, 
 
 . D + A 
 
 If, therefore, we are able to get the angle of minimum devia- 
 tion D, and have found the angle of the prism A, the index 
 of refraction p may readily be obtained from this formula.
 
 INDICES OF REFRACTION. 
 
 57 
 
 Experiment. Place the prism on the glass platform attached 
 to the divided circle described in Experiment XVI. in such 
 a way as to allow the light from the illuminated slit to fall on 
 one of its faces EA near the edge A, and move the telescope 
 until it intercepts the emerging ray. Then it will be noticed, 
 if the prism be slightly turned in one direction, that the angle 
 of deviation is increased, while if it is turned in the opposite 
 direction, this angle will be diminished. Continue then to 
 turn it in this latter direction and following the displaced ray 
 with the telescope, it will be noticed that at a certain instant 
 the emerging ray stands still, and on the prism being turned 
 still further, it begins to turn back, and therefore the angle 
 of deviation begins to increase. The deviation, when the 
 emerging ray becomes stationary, is then a minimum, and the 
 angle the incident ray makes with one face of the mirror is 
 equal to the angle the emerging ray makes with the other. 
 When the emerging ray is in this position, note the reading 
 on the divided circle indicated by the vernier attached to the 
 telescope, then remove the prism, and turn the telescope so 
 as to receive the light directly from the slit, and again note 
 the reading. The difference between these two readings will 
 be the angle D of minimum deviation. 
 
 Substituting its value in the formula 
 
 the angle A having been previously measured, p., the index of 
 refraction, may be found. A simple device which will enable 
 the slit to be illuminated with monochromatic light, is to fill 
 with a sodium salt a glass tube drawn to a fine point through 
 which passes a fine platinum wire. If the tube be then 
 placed so that the platinum wire is in the flame with which 
 the slit is to be illuminated, the heated wire melts the salt,
 
 58 EXPERIMENTAL PHYSICS. 
 
 which then runs into the flame and produces the yellow sodium 
 light. In finding the refractive index of a liquid, place it in 
 a glass bottle prism, and proceed just as indicated for a solid 
 prism. 
 
 XXIII. EXAMPLES OF MAGNIFICATION WITH LENSES. 
 
 In estimating size we are guided almost wholly by the angle 
 subtended at the eye by the object viewed. As this angle 
 depends on the distance of the object from the observer, it is 
 necessary to say where it, and its image formed by a lens or 
 system of lenses, should be placed in order that we may 
 proceed to a fit determination of the magnifying power. Com- 
 mon experience teaches us that the eye can accommodate itself 
 to different distances. Persons with normal sight can see 
 distinctly the contour and general appearance of far-removed 
 objects, such as the sun, the moon, mountains, or distant 
 buildings, and can also see distinctly objects placed as close to 
 the eye as fifteen centimeters. Just as, in regard to the sense 
 of touch, we are not able to distinguish between two points of 
 contact, when the distance between them is less than a certain 
 limiting value, so in the case of sight the eye is not able to 
 distinguish points on the body viewed, when the distance 
 between them subtends an angle at the eye less than a certain 
 definite and determinate limit. The angle subtended by two 
 points of course increases as they are brought closer to the 
 observer, and hence it is that the nearer the object, the more 
 detail there is visible. In comparing, therefore, the sizes of 
 objects they should be placed where there is the most detail 
 visible, that is, at the nearest point of distinct vision, and 
 in considering the magnitude of an image the object should be 
 so placed with regard to the lens that the image of it will appear 
 to be at this distance. The smallest distance of distinct vision 
 is, however, very different for different persons, ranging all the 
 way from fifteen to thirty centimeters, and as the magnifying
 
 EXAMPLES OF MAGNIFICATION WITH LENSES. 59 
 
 power of an instrument should give an idea of enlargement for 
 an eye in general, it has been agreed to take the distance of 
 distinct vision as twenty-five centimeters, and in what follows 
 we will always suppose the image, as formed by the lenses, 
 seen there. The magnifying power of an optical instrument is 
 generally expressed as so many diameters, which signifies that 
 the linear dimensions of the image and of the object, and not 
 their areas, are to be compared. In the case of magnifying 
 glasses and microscopes the enlargement is taken to be the 
 ratio of the apparent diameters of the image and of the object, 
 both being seen at the distance of distinct vision. 
 
 The Magnifying Glass. The simplest case of magnification 
 is that produced by means of a single biconvex lens. In Fig. 37 
 let AB be such a lens, O the position of the eye, PQ the object 
 viewed, and P'Q' the image of PQ seen at the distance of 
 distinct vision, OP'. Denoting then, OP' by A, OC by d, CP 
 by p, CP' by/', and the magnification by G, we have 
 
 G _P'Q 
 ' 
 
 Now 
 
 p' p 1 
 
 a 11 J since p' = kd, 
 
 we have G=\-\ = ; (i) 
 
 or, if the eye is close to the lens, G = i + -p 
 
 In order to test this result, take a finely divided scale for the 
 object, and adjust the lens so that the divisions are distinctly 
 visible. If a similar scale be placed at the distance of distinct
 
 6o 
 
 EXPERIMENTAL PHYSICS. 
 
 vision, and be viewed with one eye while the image of the 
 second one is viewed with the other, the observer will be able, 
 after a little practice, to tell how many divisions on the scale 
 seen directly correspond with one division on the image. This 
 number may be taken as a measure of the magnifying power of 
 the lens. A number of trials should be made by placing the 
 eye at different distances from the lens, and in order to correct 
 any error arising from a difference in the eyes of the observer, 
 the one should be used as often as the other in viewing each of 
 the scales. 
 
 Doublets. From formula (i) it is evident that the smaller 
 we take f, the greater does the magnification become. When, 
 however, we increase the curvature of the lens, spherical aberra- 
 
 .#' 
 
 Fig. 44. 
 
 tion increases also, and in order to avoid this, and yet have 
 considerable enlargement, it is customary to use a combination of 
 two lenses separated by an interval, for the eyepiece of a micro- 
 scope or telescope. Such a combination is called a doublet, 
 and its magnifying power may be calculated in the same manner 
 as that of a single lens. 
 
 In Fig. 44 let AB and A'B' be the two lenses composing the
 
 EXAMPLES OF MAGNIFICATION WITH LENSES. 6l 
 
 doublet. Let f and/' be their focal lengths respectively, and 
 let PQ be the object viewed, P i Q 1 its image formed by the 
 lens AJ3, and P Z Q 2 the final image formed by the combination, 
 and seen at the distance of distinct vision A, by an eye placed 
 close to the lens A'B 1 . Denoting then CQ by/, CQ l by /, and 
 CO by D, we have 
 
 H> 
 
 and 
 
 1 ' ' 
 
 A / + > /' 
 The magnification, which is denoted by G, is given by 
 
 G ~~PQ~P& l '~PQ' 
 i.e. 
 
 G-^n' f -- 
 p +> p 
 
 Now from (3) = I H f , and from (2) and (3) 
 
 / = A/' D 
 
 p h /(A + /) /' 
 
 (3) 
 
 
 When the two lenses have the same focal length, and the 
 interval between them is equal to two-thirds of this length, the 
 doublet is called a Ramsden eyepiece. Its magnifying power 
 
 is given by G = - + ^- In the Wollaston Doublet, /' = 3/, D 
 
 3 3 / 
 is made equal to -|/, and its magnifying power is therefore 
 
 5 A _I 
 6/ 2 
 
 The combination most frequently used as the eyepiece for 
 microscopes is that of Huyghens. It is sometimes called a
 
 62 EXPERIMENTAL PHYSICS. 
 
 negative eyepiece, to distinguish it from a Ramsden or positive 
 eyepiece. In it/' = J/, D is taken equal to f /, and its magni- 
 fying power is given by G=- + 2 . When a doublet is used as 
 
 an eyepiece, the lens next the objective is called the field glass, 
 and that next the eye the eyeglass. 
 
 In testing the magnification of a doublet experimentally, 
 exactly the same method is adopted as in the case of a single 
 lens. If the eye is not placed close to the lens in doing this, an 
 allowance must be made in formula (4) for its distance from it, 
 similar to that for enlargement by a magnifying glass. 
 
 The Compound Microscope. In its simplest form the com- 
 pound microscope consists of two condensing lenses, one of 
 which is called an object glass, or objective, and the other 
 an eyeglass, or eyepiece. The objective, however, is usually 
 made up of a system of lenses so constructed as to reduce 
 chromatic and spherical aberration to a minimum, and the 
 eyepiece most generally used is that of Huyghens. Just as 
 in the case of doubtlets, an expression for magnification, by a 
 microscope can be calculated in terms of the focal lengths of 
 the lenses, the distances these are apart, and the distance of 
 distinct vision. Experimentally the same method as that just 
 described for a magnifying glass may be adopted ; but it is 
 usual in the case of the microscope to modify it by placing a 
 camera lucida over the eyepiece, and by placing a scale a little 
 to one side of the instrument, so that, with the same eye the 
 image of the scale under the microscope and the reflected 
 image of the scale at the side are both visible at once. By 
 noting how many divisions on the image formed by reflection 
 correspond with one on that formed by refraction the magni- 
 fying power can be ascertained. 
 
 The microscope is frequently used for measuring small 
 distances. A reference to Fig. 45 will explain how this is 
 accomplished. 
 
 Here AB represents the objective, A l ^ l and A^B^ the field
 
 EXAMPLES OF MAGNIFICATION WITH LENSES. 63 
 
 glass and eyeglass respectively of an Huyghens eyepiece ; PQ 
 the object, P\Q\ the image the rays go to form after passing 
 through the objective, P 2 Q 2 the real image formed by this lens, 
 and the field glass, and P S Q 3 the final image seen at the dis- 
 tance of distinct vision. A micrometer EF (consisting of a 
 thin glass plate on which there is generally ruled a half-centi- 
 meter divided into fiftieths) is placed in the eyepiece in such 
 a position that its image seen through the eye lens is at the 
 distance of distinct vision. When, then, an object PQ is 
 viewed, and its final image P S Q 3 is seen distinctly, the real 
 
 Fig. 45. 
 
 image P Z Q Z of PQ must be in the same plane as the microm- 
 eter. If, therefore, a finely divided scale (usually called a 
 stage micrometer) is taken for the object PQ, the observer 
 can at once see how many divisions on this image correspond 
 with one division on the micrometer, and so ascertain how 
 long an object must be in order that its real image may be 
 equal to the distance between two lines on the micrometer. 
 The micrometer has sometimes a number of small equal 
 squares ruled on it in place of a scale, and by means of these 
 the number of small objects, such as colonies of bacteria, 
 occupying a given area can be ascertained. The magnifying
 
 6 4 
 
 EXPERIMENTAL PHYSICS. 
 
 power of the objective, together with the field glass, depends 
 on the distance between them, and it should be noted that 
 in finding the length of a small object by the method just 
 described, this distance should be the same as in finding the 
 number of divisions on the stage micrometer corresponding 
 to one on that in the eyepiece. 
 
 The Telescope. Expressions can also be calculated for the 
 magnifying power of telescopes. In the case of the astronomi- 
 cal telescope it can be shewn to be the ratio F:f, where F 
 is the focal length of the objective, and / that of the eyepiece. 
 Practically the magnifying power of a telescope is determined 
 by looking through it with both eyes open, at a distant scale, 
 or some object marked with equal divisions, such as a picket 
 fence. The number of divisions on the scale corresponding 
 with one on the image is taken as a measure of the enlarge- 
 ment. 
 
 XXIV. EXERCISES WITH PHOTOGRAPHIC LENSES. 
 
 If the light from objects situated outside, such as trees and 
 buildings, be allowed to enter a darkened room through a small 
 hole in a shutter, there will be formed on a white screen placed 
 before the hole, or on the opposite wall of a room, an inverted 
 ./7 image in true colours of the 
 
 outside objects. This phenom- 
 enon is due to the rectilinear 
 propagation of light, and can 
 be easily explained. 
 
 In Fig. 46 AB is an object 
 situated outside, A'B' is its 
 image formed on a screen, and 
 P is a small aperture in the 
 shutter. A pencil of light coming from the point A will, on pass- 
 ing through the opening, form a spot of light at A' which cannot 
 be distinguished from a single point if P is extremely small. 
 
 Fig. 46.
 
 EXERCISES WITH PHOTOGRAPHIC LENSES. 65 
 
 As no other point on AB can send light to A', its colour and 
 brightness will therefore depend on that of A, and it may thus 
 be said to be the image of A. Similarly, B' will be the image 
 of B, and all points between A and B will have their images 
 between A' and B'. Hence, when light is admitted to a dark 
 room through a small opening, we are able to form an inverted 
 image of the outside objects. As the image of each external 
 point is in reality a spot and not a point, it is evident that if 
 the opening of P is large, those formed by the light coming 
 from different points on the outside objects will overlap, and 
 hence no image will be produced. It is for this reason, there- 
 fore, that no image is obtained when the light is allowed to 
 enter a room through a large opening, such as a window. 
 
 The phenomenon just described affords the simplest method 
 of taking a photograph. If the lens be removed from a camera, 
 and a sheet of tin-foil pierced with a very fine hole be placed 
 over the opening, an image can be formed on a sensitized plate, 
 and a negative thus obtained. Although no "focussing" is 
 required by this method, yet on account of the small amount 
 of light forming the image, the length of time required for a 
 proper exposure makes it generally impracticable. 
 
 The Use of the Lens. When a lens is used, it is because it 
 produces an image of greater intensity, and the times of expos- 
 ure are therefore correspondingly diminished. Intensity, how- 
 ever, is then obtained at the expense of distinctness, and the 
 images formed, owing to chromatic and spherical aberration, are 
 far from being perfect. To remove the defects so produced, or 
 at least reduce them to a minimum, opticians have had recourse 
 to the use of compound lenses and of stops. 
 
 Chromatic Aberration. When white light passes from one 
 medium into another, the violet rays composing it undergo, 
 owing to their greater refrangibility, a greater deviation than 
 the red rays, and the defect caused by this in images formed 
 with lenses is called chromatic aberration. This dispersion, 
 or separation of the rays of different colours, varies with the two
 
 66 EXPERIMENTAL PHYSICS. 
 
 media, and it can be readily shewn by experiments with prisms 
 of different substances, and of different angles that, when the 
 dispersive effects of two prisms are equal, their refractive 
 effects are, in general, not so, and that therefore by suitably 
 combining two prisms of different refracting angles, but hav- 
 ing equal dispersive effects, we may produce refraction of white 
 light without causing dispersion. It therefore follows, since by 
 taking radial sections of a lens we may shew its general effect 
 to be the same as that of a prism, that by suitably combining 
 two lenses of different substances we can bring rays of light 
 to a focus without the presence of colour effects. Hence the 
 
 Fig. 47. 
 
 method of constructing achromatic combinations by uniting 
 lenses made from different kinds of glass. 
 
 A simple and instructive exercise in this connection is to test 
 the objectives of microscopes and telescopes for achromatism 
 by allowing the light from a strongly illuminated pin-hole to 
 pass through the combination. The emerging rays may then 
 be examined by receiving them through a simple eyepiece, or 
 by allowing them to form an image on a white screen. 
 
 Spherical Aberration. When rays of light issuing from an 
 illuminated point fall upon a convex lens, they do not, on being 
 refracted, all intersect in a single point, but, as Fig. 47 indicates, 
 in many of them, situated at different distances from the lens ; 
 and each of these may be taken to be the image of the point 
 from which the rays emanate. This is well exhibited by cover-
 
 EXERCISES WITH PHOTOGRAPHIC LENSES. 67 
 
 ing the lens with a screen in which are cut a number of concen- 
 tric rings. If the light be allowed to pass through only one 
 ring at a time, it will be found that in order to focus an image 
 the screen must be placed nearer to the lens, according as the 
 radius of the ring through which the light passes increases. 
 
 From this it will be evident that when a convex lens is used 
 to form an image of an object, there is in reality a multiplicity 
 of images formed of each point; these images being situated 
 close to each other, but yet in different planes perpendicular to 
 the axis of the lens. When, then, an image of an object is 
 focussed on a screen, it is in fact only so for the set of images 
 which happens to lie in the plane of the screen. Some of the 
 rays are focussed in front of it, and some behind it, and these 
 on striking the screen produce a blurred appearance, and the 
 image is thus lacking in what is technically called definition. 
 This defect in images is said to be due to spherical aberration, 
 and, in order to minimize its effects, recourse, as previously 
 indicated, is had to the use of stops. It will be quite clear that 
 whether these are placed in front of the lens, or behind it, their 
 effect will be to cut off some of the rays which do not focus on 
 the screen, or sensitized plate, and in this way produce better 
 definition. If it were desired to produce an image free entirely 
 from this blurred appearance, it would be necessary to use a 
 stop with a very small opening; and although the image then 
 formed would be sharp and distinct, the illumination would be 
 weak, and there would be little, if any, advantage over the ordi- 
 nary pin-hole method. In reducing the effects of spherical 
 aberration stops are used which, while producing images not 
 perfectly distinct, give moderate intensity, and so permit 
 short exposures. For practical purposes the intensities of the 
 image may be taken proportional to the areas of the open- 
 ings in the stops, and it is an instructive exercise to cal- 
 culate the relative times of exposure for a set of stops, and to 
 test these by taking photographs of an object under constant 
 illumination.
 
 68 EXPERIMENTAL PHYSICS. 
 
 By placing the screen with concentric circular openings on 
 different lenses, it may be shewn that spherical aberration 
 depends on the curvature of the lens, and that it may be 
 diminished by using two lenses of small, instead of one of 
 large, curvature. These lenses can be so chosen that if they 
 are placed a short distance apart they will have, in combina- 
 tion, a focal length equivalent to that of a single lens ; and 
 since by this arrangement there is a gain in definition without 
 a counterbalancing loss in intensity, it is generally adopted in 
 the construction of photographic objectives. 
 
 XXV. BUNSEN'S PHOTOMETER. 
 
 Although we are not able to measure absolutely the intensity 
 of the light coming from a given source, we can, however, 
 compare it with that from another. The eye, which cannot 
 estimate directly, with any degree of accuracy, the relative 
 intensity of two sources of light, is a good judge of the illumi- 
 nations which they can produce ; and it is on the principle of 
 equality of illuminations that the photometers ordinarily em- 
 ployed are constructed. 
 
 The intensity of illumination is taken proportional to light 
 intensity, and is defined to be the quantity of light received on 
 a unit area of surface. The fundamental law in photometry, 
 which is that of the inverse square, is readily demonstrated by 
 considering a small cone of rays coming from an illuminated 
 point. Since the -same amount of light passes through any 
 right section of this cone, the intensity at any point will vary 
 inversely as the area of the right section containing the point ; 
 and again, as any such section varies directly as the square of 
 its distance from the vertex of the cone, the law is evident. 
 
 In Bunsen's photometer the light from some standard, 
 and that from the source to be tested are allowed to fall 
 perpendicularly upon the opposite sides of a sheet of paper 
 with an oiled spot on it. When light falls upon such a
 
 BUNSEN'S PHOTOMETER. 69 
 
 paper, more of it is reflected, and less transmitted by the 
 unoiled portion than by the oiled, and hence the latter, when 
 viewed from the side next the source of light, appears darker 
 than the remainder ; while if viewed from the other side, it 
 appears brighter. If then light be allowed to fall on the two 
 sides of the paper so that both sides are equally illuminated, 
 that from one source, which was lost by transmission, is restored 
 by the light transmitted from the other. 
 
 When the lights have been so adjusted that this condition 
 obtains, the oiled part can no longer be distinguished from 
 the rest of the paper. Let, then, d be the distance of the 
 standard from the paper, and d 1 that of the light to be tested ; 
 A and A ' the intensities of these two lights respectively at 
 a unit distance, and B and B' the intensities at the paper 
 disc. 
 
 By the law of the inverse square : 
 
 B \ B' i 
 
 If, therefore, the intensities are the same at the paper, B=B', 
 and .-.A'= 2 A. 
 
 The intensity of the tested light is then times that of 
 the standard. 
 
 In making a test, different values should be given to d, and 
 corresponding ones found for d', the mean of the results so 
 obtained being taken as the intensity of the light examined. 
 As it is essential that no light should fall upon the paper 
 disc except that coming from the two sources mentioned, the 
 room in which the experiment is conducted should be thor- 
 oughly darkened, and its walls painted black. A simple device 
 to permit both sides of the paper disc to be seen simultaneously
 
 70 EXPERIMENTAL PHYSICS. 
 
 is to place it in a vertical frame, to the back of which are 
 attached two mirrors inclined at an angle to the paper such that 
 the observer can by looking into the mirrors, see images of both 
 sides of the paper at the same time. The oiled spot is gener- 
 ally made circular ; but it is well to check results by making 
 it of some irregular form. A modification of this is obtained 
 by placing together three sheets of paper, in the center one 
 of which is a hole of any desired shape. Owing to the pres- 
 ence of light reflected from the walls of the room, and to 
 unequal absorption by the two parts of the disc, it is rarely 
 possible to arrange the two lights so that the oiled spot entirely 
 disappears. Indeed, the results obtained by the method are 
 only approximate at best. It is assumed in the theory that 
 the sources of light are points, and that the lights are of the 
 same quality, conditions which do not hold in practice. 
 
 Various forms of standard lights have been devised ; but 
 what seems to give the steadiest light is to illuminate a small 
 hole in a screen by means of the flame of a coal-oil lamp burn- 
 ing a broad wick. As, in testing gas flames and incandescent 
 lamps, it is found that the intensity at a given distance varies 
 with the position of the flame in the one case, and with that 
 of the carbon filament in the other, the student should make a 
 table of his results, or plot a curve shewing the intensity of the 
 light in different positions. Bunsen's photometer may also be 
 used to compare the absorptive powers of different transparent 
 media, such as glass, or water, by finding the intensities of the 
 light transmitted by them from a given source. The results 
 obtained, however, will not be exact, as light is always reflected 
 from the surface of the medium employed. 
 
 XXVI. AYRTON'S PHOTOMETER MODIFIED FROM BUNSEN'S. 
 
 When the light to be examined is very great compared with 
 that of the standard, the experiment takes up so much room, if 
 the previous method is adopted, that it is quite impracticable.
 
 AYRTON'S PHOTOMETER. 71 
 
 This difficulty has been overcome by Ayrton, who introduces 
 a biconcave lens between the light tested and the photometer. 
 
 Theory. In Fig. 48 P is the source of the light to be 
 tested, AB the lens, ST the paper disc of the photometer, 
 and CPD a cone of light striking the lens. Let A and A' be 
 the intensities of the standard and of the given light at a unit 
 distance ; B and B' the intensities of the light from the latter 
 at the disc, with the lens interposed, and with it removed 
 respectively, and B" that of the standard at the disc. Also let 
 C be the area of the circle of light on the disc formed by the 
 cone of rays DPC with the lens inserted, and C' with it removed; 
 
 s 
 
 Fig. 48. 
 
 and let a be the distance from the paper disc to the light tested, 
 c that to the standard, and / that to the lens. 
 
 Since the intensity of illumination is the amount of light on 
 a unit area, and since the total amount of light on C and C', 
 under the conditions mentioned above, may be taken to be the 
 same, we have 
 
 BC=B'C t or B'=~B. 
 
 (^ 
 
 Again, from the fundamental law we have 
 
 =0 2 , or A'=a 2 B'; 
 B'
 
 72 EXPERIMENTAL PHYSICS. 
 
 Again, we have = -L O r B" = 
 
 Therefore when the two sides of the disc are equally illu- 
 minated, we have B' =B, 
 
 A^_A . A^_a z C 
 a*C~ c* "' A~*C 
 
 Now denote the circular area of the lens through which the 
 light passes by C", and let the light after refraction appear to 
 come from a point P' at a distance x behind the lens. 
 
 Then = 71^' (0 
 
 Also, since P' is the image of P, 
 
 (3) 
 j-t-a t 
 
 From (i), (2), and (3), 
 
 
 i.e. the intensity of the given light will be 
 
 times the standard. 
 
 Experiment. Place the biconcave lens in a screen, and 
 insert it between the light to be tested and the paper disc 
 of the photometer. Then, as in the previous method, adjust 
 the standard so that the oiled spot cannot be distinguished 
 from the rest of the paper, and measure the lengths a, c, and /.
 
 ROTATING DISC PHOTOMETER. 
 
 73 
 
 By substituting in the formula these values together with 
 that of f, the focal length of the lens, which may be found 
 by either of the methods given in Experiment XXL, the in- 
 tensity of the light may be determined. In making a test, 
 the student should obtain a number of results by giving dif- 
 ferent values to a and /, and therefore to c. 
 
 Ayrton's method is especially applicable in finding the in- 
 tensity of the light from an incandescent lamp, or that from an 
 electric arc. The intensity of the latter should be examined 
 with the carbons making different angles with the vertical, 
 because, just as in the case of gas flames and incandescent 
 lamps, the intensity at a given distance varies with the position 
 of the arc. 
 
 XXVII. ROTATING DISC PHOTOMETER. 
 
 When a number of objects are passed in rapid succession 
 before the eye of an observer, the effect, owing to the persist- 
 
 Fig. 49. 
 
 ence of luminous sensations, is the same as if the retina received 
 the impressions from the different objects simultaneously; and 
 it is for this reason that we are able to produce a white appear-
 
 74 EXPERIMENTAL PHYSICS. 
 
 ance by the rapid rotation of a disc of coloured sectors arranged 
 in proper proportions, or by rapidly oscillating the spectrum 
 .produced by allowing a pencil of the sun's rays to fall on a 
 prism. The principle here involved has been applied with con- 
 siderable success to photometric investigations. In Fig. 49, A, 
 a black circular disc made of thin sheet metal, or of cardboard, 
 has two apertures in it, whose sides are circular arcs concentric 
 with the disc ; B, a double sector of the same material as the 
 disc, is used to reduce, or enlarge the size of these apertures ; 
 and C, D, two double sectors of some gray-coloured material, are 
 of such a size that when they are placed concentric with the disc 
 A the lighter one C extends just beyond the outer edges of the 
 apertures, while the darker one D extends half-way across them. 
 The fundamental hypothesis in the work with rotating discs 
 is that the amount of light coming from a sector is proportional 
 to the angle of this sector. The edges of the plates used play 
 a considerable part in these investigations, and experiment 
 seems to show that, if the angles are not small, and the edges 
 of the plates are made thin by bevelling, the assumption is 
 warrantable. 
 
 EXERCISE I. Comparison of gray tints. 
 
 It is often desirable to have an accurate notion of the relative 
 quantities of light reflected from different papers commonly 
 called white. In order to investigate this, cut two sectors such 
 as C and D from the two sheets of paper to be examined, and 
 place them on the same axis of rotation as the black disc A, and 
 behind it, the darker sector being next to the disc. If, then, 
 this arrangement be rotated in front of a black screen of velvet, 
 or other such material, there will appear two concentric rings 
 of a drab colour produced by the combination of the white of the 
 sector with the black of the disc. As their colour depends only 
 on the amount of the white in combination with the black, the 
 two rings may be made of the same tint by taking suitable angles 
 for the sectors of paper. If great accuracy is desired, the aux-
 
 ROTATING DISC PHOTOMETER. 75 
 
 iliary sector B may be used. If, for example, it is found that 
 the black of the disc combined with 81 of the white of one sec- 
 tor gives nearly the same tint as when combined with 80 of 
 that of the other, then, by adding i to each of the sectors, or 
 by taking i off, we may get the ratios 82 : 81 or 80 : 79, one of 
 which will be a closer approximation to the exact result. If a 
 and b are the angles of the two white sectors, when the rings 
 are of the same tint, and Q, Q' are the quantities of light re- 
 flected from a unit angle of each of these sectors, then aQ = bQ', 
 or the ratio b : a is a measure of the relative whiteness of the 
 two papers. If the edges of the apertures are divided into 
 degrees and parts of a degree, the angles a and b may be read 
 directly. This same method may be applied to a study of the 
 light reflected from walls of different gray tints. It is done by 
 spreading thin layers of the plasters used on different sectors, 
 and then, when it is dry, proceeding just as in the paper tests. 
 In conducting these experiments it can be seen that the colour 
 of the two rings alters slightly with the direction of the inci- 
 dent light, and with the position of the observer. It will be 
 found that the best results are obtained when the disc is 
 illuminated with rays of light incident parallel to the axis of 
 rotation, and when a small part only of the rings is viewed by 
 means of a telescope. 
 
 EXERCISE II. To compare the intensities of light coming 
 from two independent sources. 
 
 To perform this experiment the disc should have two concen- 
 tric sets of apertures a short distance apart, and the light should 
 be allowed to fall on the back of the disc in such a manner that 
 each of the sets is constantly illuminated by the light from one 
 of the two sources. By altering the size of these apertures 
 until the colours of the rings, formed by the rotation of the disc, 
 are the same, a measure of the intensities of the light from the 
 two sources may be obtained. If a and b are the angular open- 
 ings, the intensities are then in the ratio b : a.
 
 76 EXPERIMENTAL PHYSICS. 
 
 As considerable difficulty is likely to be met with in arrang- 
 ing suitable mechanism for this method, a better one is to use 
 two small discs with a single set of apertures in each, and to 
 rotate them in the same plane. In this way the illuminations 
 may be made more simply, and more nearly under the same 
 conditions. It is evident that these methods only apply when 
 the lights examined are of the same quality. Whenever the 
 question of colour comes in, the investigation becomes compli- 
 cated and uncertain. 
 
 EXERCISE III. To compare the absorptive powers of differ- 
 ent media. 
 
 If a screen be made partly of one translucent substance, and 
 partly of another, and light from a single source be allowed to 
 fall on the back of this screen, then that which comes through 
 may be considered as coming from two independent sources. 
 The intensity of the light coming from each part of the screen 
 may then be investigated by either of the methods outlined in 
 Exercise II., and the absorptive powers of the different sub- 
 stances composing the screen compared. 
 
 In Exercises II. and III. every precaution should be taken to 
 insure that no light falls on the apertures of the discs except 
 that which is being examined. 
 
 XXVIII. SPECIFIC HEAT OF SOLIDS. 
 
 The Method of Mixture. In Fig. 50 B is a boiler, in which 
 water may be heated, and AC is a. steam jacket connected to it 
 by a tube indicated in the figure. To the upper portion of the 
 tube DB, leading from the boiler to the outside air, there is 
 fitted a water jacket through which cold water is kept constantly 
 running. The calorimeter G consists of two metallic vessels, 
 generally made of sheet brass, the one resting on wooden sup- 
 ports inside the other. The solid E, whose specific heat is 
 required, is suspended by a string in the inclosure F within the
 
 SPECIFIC HEAT OF SOLIDS. 
 
 77 
 
 steam jacket. The steam after leaving the boiler passes into 
 this jacket, and then out into the tube DB, where, on coming 
 into contact with the cold air, it condenses and falls back into 
 the boiler. In this way a steady supply of steam is led into the 
 jacket, whose temperature is thus kept constant, and there is, 
 therefore, no occasion for renewing the water in the boiler. The 
 steam jacket is generally provided with attachments by means 
 
 Fig. 50. 
 
 of which both ends of the inclosure F may be stopped, and 
 radiation thus prevented. 
 
 The unit of heat generally adopted is the quantity required 
 to raise the temperature of one gram of water one degree centi- 
 grade, and is called a calorie. It is found that, if equal masses of 
 two different substances are subjected to the same heat, under 
 the same circumstances, for a given time, their temperatures 
 will vary considerably.
 
 78 EXPERIMENTAL PHYSICS. 
 
 This shews that different quantities of heat must be imparted 
 to equal masses of different substances to make the same altera- 
 tion in their temperatures, and the specific heat of a substance 
 is defined to be the ratio of the quantity of heat required to raise 
 the temperature of a given mass of this substance one degree, 
 to that required to raise the temperature of an equal mass of 
 water one degree. 
 
 Theory Let M be the mass of the solid in grams, T its 
 
 temperature before it is dropped into the water; m and m* 
 the masses of the water in the calorimeter, and of the calori- 
 meter respectively; f and 6 their initial and final tem- 
 peratures, and c, and c' the specific heats, respectively, of 
 the given solid, and of the substance of which the calorimeter 
 is made. 
 
 The heat given up by the solid will then be equal to 
 cM(T6] calories, while that gained by the water will be equal 
 to m(6 t} calories, and that by the calorimeter m'c'(6t}. 
 Since the heat lost by the solid is gained by the calorimeter 
 and the water it contains, we have 
 
 cM( T- (9) =m(0-t)+ m 'c' (0 - /), 
 
 or 
 
 M(T-6) 
 
 Water Equivalent of the Calorimeter. The expression m'c' 
 in this formula is called the water equivalent of the calorimeter, 
 since from its position it is equivalent to an addition to the 
 mass of water in the calorimeter. Its value is best obtained by 
 finding its mass m' by weighing, and by taking the specific heat 
 c' of the substance of which it is made, from the tables, but it 
 may also be found experimentally in the following manner : 
 Into a mass M^ of water contained in the calorimeter (the 
 temperature of both being t^} pour a mass of water M% of 
 temperature / a , higher than t, and let 6 be the result-
 
 SPECIFIC HEAT OF SOLIDS. 79 
 
 ing temperature. By the same process of reasoning as above 
 we will have 
 
 MM* - 
 m ' c '= 2V 2 
 
 i.e. m'c' is expressed in terms of quantities which can be 
 directly found. 
 
 Experiment. Weigh the solid E whose specific heat is to be 
 determined, and then suspend it together with a thermometer 
 in the inclosure within the steam jacket, as indicated in the 
 figure. Care should be taken to keep a constant stream of cold 
 water running through the vessel attached to the upper part of 
 the tube DB, and not to fill the boiler with water higher than 
 the opening in the tube leading from it to the steam jacket. 
 While the solid is being heated to the temperature of the 
 steam, the student may find the water equivalent of the 
 calorimeter experimentally. In conducting this preliminary 
 part of the experiment care should be taken to have the tem- 
 peratures, and the quantities of water involved as nearly as 
 possible the same as those in the experiment proper. Weigh 
 the calorimeter, including thermometer and stirrer. Partly fill 
 it with water of temperature tf, below that of the room, and 
 again weigh it. The difference between these two weights will 
 give the mass of water M l in it. Then pour into the water M l 
 a sufficient quantity of water of temperature / 2 to bring the 
 resultant temperature as much above that of the room as t was 
 below it. In this way the loss of heat due to radiation may be 
 partly overcome. Again weigh the calorimeter, and the differ-: 
 ence between this weight and the last will give the mass M^ of 
 the water poured in. The quantities t^, / 2 , lt M lt M^ thus 
 being known, the water equivalent may be determined as pre- 
 viously indicated. The vessel from which the water M z is 
 poured should contain more than is actually necessary, and t^
 
 80 EXPERIMENTAL PHYSICS. 
 
 should be the mean of the temperatures of this water before 
 and after M z is poured out. Especial care should be taken to 
 pour the water into the calorimeter in such a manner that the 
 least possible amount of heat is lost in the pouring. 
 
 Having determined the water equivalent, again take the 
 calorimeter, and partly fill it with water of temperature f and 
 weigh it. The difference between this weight and that of the 
 calorimeter empty will give the mass m of the water poured 
 into it. Having made certain that the solid is now at the tem- 
 perature of the steam jacket T, which is noted by reading the 
 thermometer suspended within the inclosure F, or by finding 
 from tables the boiling-point of water corresponding to the 
 reading of a barometer suspended in the room, slide the calo- 
 rimeter under the steam jacket. Then drop the solid into it, 
 and stir the water gently. The temperature of the water will 
 rise gradually, then remain stationary for a time, and finally 
 begin to lower. The stationary temperature will then be that 
 resulting from plunging the heated solid into the water. The 
 mass M of the solid having been previously found, and m, m'c', 
 6, t, and T thus being known, we can, by applying the formula, 
 obtain the specific heat of the solid. 
 
 XXIX. SPECIFIC HEAT OF LIQUIDS. 
 
 In Fig. 51, AB is a boiler containing water, and CD is a 
 hollow, metallic reservoir almost wholly immersed in it. The 
 calorimeter consists of three parts, EF resting on a stand, GH 
 resting on wooden supports inside of EF, and K immersed in 
 water contained in GH. The different parts of the calorimeter 
 are generally made of thin brass, the part K being made of such 
 a form as to expose as much surface as possible to the water. 
 As the figure indicates, it is connected to the reservoir by a 
 tube in which there is a tap. 
 
 Theory. In this case, also, the method is one of mixture. 
 The liquid whose specific heat is to be determined, is first heated
 
 SPECIFIC HEAT OF LIQUIDS. 
 
 81 
 
 in the reservoir CD, and then allowed to run into the calo- 
 rimeter, where it gives up its heat to the water, and the vessel 
 containing it. Let M v M 2 , and M be the masses, in grams, of 
 the calorimeter, the water in it, and the given liquid respec- 
 tively, /j and 6 the initial and final temperatures of the water 
 in the calorimeter, and t the temperature of the liquid before 
 it is run into the calorimeter. Also let c and c l be the specific 
 
 Fig. 51. 
 
 heats of the given liquid and of the metal of which the calo- 
 rimeter is made, respectively. 
 
 The heat given out by the liquid, on being run into the 
 calorimeter, is then equal to cM(t 6} calories, while that 
 gained by the calorimeter is c l M 1 (0 ^ l ), and that by the 
 water in it M0 f. 
 
 or 
 
 M 
 
 (t-6}
 
 82 EXPERIMENTAL PHYSICS. 
 
 Water Equivalent of the Calorimeter. In this experiment, 
 as in the last, c l M l is the water equivalent of the calorimeter, 
 and may be found by either of the methods there given. If 
 the experimental method is adopted, the error in this case aris- 
 ing from the water cooling when being poured into the calo- 
 rimeter, may be avoided by heating it in the reservoir CD, 
 and then allowing it to run through the tube connecting the 
 two vessels. 
 
 Experiment. First weigh the two parts of the calorimeter 
 GH and K, together with the stirrer and a thermometer. 
 Then pour in water sufficient to cover the part K, as indicated 
 in the figure, and again weigh them. The difference between 
 the two weights will give the mass, M v of the water poured in. 
 In the mean time, suspend a thermometer in the boiler, and 
 place in the reservoir CD enough of the liquid whose specific 
 heat is to be determined to nearly fill the vessel K, and allow 
 it to become heated by the surrounding water. 
 
 In order to prevent evaporation, care should be taken to keep 
 the top of the reservoir closed during the heating process, and 
 not to let its temperature rise above the boiling-point of the 
 given liquid corresponding to atmospheric pressure. When it 
 has been sufficiently heated, the temperature t of this liquid 
 may be noted by reading the thermometer suspended in the 
 water. Also note the temperature tf of the water in the calo- 
 rimeter, and then turn the tap, and allow the liquid to run into 
 it through the tube. 
 
 Gently stir the water in the calorimeter, and when its tem- 
 perature has become stationary, note its value 6. Then weigh 
 the calorimeter and contents, and the difference between this 
 weight and the last will give the mass M of the liquid that ran 
 in. This result may be checked by first weighing the liquid 
 before it is poured into the reservoir. The student will thus 
 see that the only essential difference between this experiment, 
 and the last is that the substance whose specific heat is desired,
 
 LATENT HEAT OF FUSION OF ICE. 83 
 
 is not in this case allowed to come in contact with the water in 
 the calorimeter. 
 
 Especial care should be taken to choose conditions favourable 
 to overcoming the errors due to evaporation, as in this experi- 
 ment considerable inaccuracies of this kind are almost certain 
 to be met with by the student. 
 
 The specific heat of a given liquid may also be calculated by 
 finding the specific heat of a solid relative to it, and also relative 
 to water. If C l and 7 2 are these two specific heats respectively, 
 and C the specific heat of the liquid, we have 
 
 or 
 
 XXX. LATENT HEAT OF FUSION OF ICE. 
 
 If a certain mass of water at o C. is placed in one beaker, and 
 an equal mass of ice at o C. in another, and if the two are then 
 placed in a hot-water bath, it is found that at the instant the ice 
 is all melted the temperature of the water produced by it is 
 still o C., while that of the water in the other beaker is about 
 80 C. From the similarity of the conditions to which the two 
 vessels are subjected, it is evident that the same quantity of 
 heat passes into each during the time they are in the hot water. 
 In the one case the heat is used up in raising the temperature 
 of a substance, while in the other it is used up in changing the 
 physical structure of the substance without altering its tempera- 
 ture. From this fact the heat absorbed in the latter case has 
 been called latent heat, and technically the latent heat of fusion 
 of a substance is defined to be the number of units of heat 
 required to convert a unit mass of it from the solid to the 
 liquid state without raising its temperature. Adopting the 
 units of heat indicated in Experiment XXVIII., the latent 
 heat of fusion of ice is the number of calories required to con-
 
 84 EXPERIMENTAL PHYSICS. 
 
 vert one gram of ice at o C. into water without altering its 
 temperature. 
 
 Theory. Into a quantity of water M contained in a calo- 
 rimeter of mass M lt drop a mass of ice M 2 of temperature o C. 
 Let /j and 6 be the initial and final temperatures of the water 
 respectively, L the latent heat of fusion of ice, and ^ the specific 
 heat of the substance of which the calorimeter is made. LM Z 
 calories is then the quantity of heat absorbed in melting the 
 ice, and M Z 6 calories that in raising the temperature of the 
 water produced by it from o C. to 6 C. Since the quantity 
 of heat given up by the calorimeter and the water in it is 
 - (9), we have 
 
 -0)=LM 2 
 L= (M 
 
 Experiment. The masses M, M ly and M% are found by weigh- 
 ing, and just as in the two previous experiments, the water 
 equivalent of the calorimeter c-^M^ may be determined either 
 by a reference to a table of specific heats, or by the experi- 
 mental method. The initial temperature t^, and the masses M 
 and M 2 of the water and ice respectively, should be so selected 
 that the final temperature 6 is as much below the tempera- 
 ture of the room as t- is above it. These quantities may 
 be ascertained approximately by a calculation, but the student 
 will find it exceedingly instructive to adopt a tentative proc- 
 ess, and in this way arrive at the proper proportions that 
 should obtain in an exact determination of L. As it is essen- 
 tial that the ice should melt as rapidly as possible, it is best to 
 break it up into small pieces, and to dry each piece thoroughly 
 with blotting paper before it is placed in the calorimeter. Care 
 should also be taken to have the temperature of the ice at 
 o C., by placing it in a warm room a short time before the 
 experiment is commenced.
 
 LEVEL TESTING. 85 
 
 XXXI. LEVEL TESTING. 
 
 The spirit level is made of a glass tube slightly curved, so 
 that its axis forms part of a circle of large radius. Usually 
 when tubes come from the manufacturer they have a slight 
 curvature at each point, and it is only necessary to perfect this 
 in the piece chosen by grinding the inside with emery powder. 
 The tube is filled with alcohol, or ether, excepting a small space 
 containing a bubble of air which tends to occupy the highest 
 part of the tube. It is sealed as shewn in Fig. 52, and then 
 firmly cemented in a brass tube, which is itself attached to a 
 brass plate (Fig. 53) in such a manner that when the plate 
 rests on a level surface the air bubble remains stationary at the 
 center of the opening in the upper side of the brass tube. In 
 
 Fig. 52. Fig. 53. 
 
 some levels a thin brass strip across the opening marks this 
 position of the bubble ; while in others its position is marked 
 by the central division of a scale ruled on the glass. 
 
 For fine work, a level should be very sensitive, and should be 
 ground to a true curvature, which is indicated by a uniform 
 run of the bubble when it is given a slight inclination to the 
 horizon. As the sensitiveness of a level should be in strict 
 keeping with the instrument to which it is attached, it is neces- 
 sary that each one should be thoroughly tested before being 
 used for any particular work. 
 
 Buff and Berger s Level Trier, an instrument for this pur- 
 pose, is shewn in Fig. 54. It consists of an iron plate upon 
 which is mounted an iron bar having at one end two pivotal 
 centers resting in receptacles provided for them in the base 
 plate, and at the other a micrometer screw carrying a graduated 
 disc. The bar is provided with fixed wyes in which levels to
 
 86 
 
 EXPERIMENTAL PHYSICS. 
 
 be tested may be placed, and also with a scale, to be used in 
 case none is ruled on the level. The points of the pivots and 
 
 of the micrometer screw are 
 made very hard, and the latter 
 bears on a plate with a hard 
 and polished surface. This 
 plate is often made to move 
 eccentrically with regard to the 
 screw, so that the point of rest 
 can be changed in case of wear. 
 
 Theory. Let x be the height 
 through which the left-hand side 
 of the instrument is raised on 
 turning the micrometer screw, 
 and let y be the distance be- 
 tween the screw point and a 
 point midway between the two 
 pivots. The circular measure 
 of the angle through which the 
 instrument has been turned is 
 
 then , and if this ratio is multi- 
 
 y 
 
 plied by 3437.748 or 206264.9, 
 the angle is given in minutes or 
 seconds respectively. 
 
 EXERCISE I. To investigate 
 the run of the bubble. 
 
 Place the level to be tested on 
 the wyes on the bar, and then 
 raise, or lower the latter until 
 the bubble is at one end of the 
 scale, or, if the level has no 
 scale, at the point which is the
 
 LEVEL TESTING. 
 
 limit of the run of the bubble in practice. The micrometer 
 disc is then turned over equal spaces, and the run of the bubble 
 carefully noted. When the bubble has been moved over its 
 course, it should be moved in the opposite direction in the same 
 manner, and the whole operation repeated several times. The 
 mean value of all the observations may then be determined, and 
 the value of one division on the level expressed in minutes or 
 seconds of an arc, as the case may be. If the level to be tested 
 cannot be easily removed, the entire instrument to which it is 
 attached may be placed on the trier, resting on points provided 
 for it directly over the pivots. 
 
 In the following table, exhibiting the manner in which a test 
 was made, A and B are the right and left hand sides of the air 
 bubble respectively : 
 
 Number of 
 Trials. 
 
 Microm- 
 eter 
 Readings. 
 
 Level Scale Readings. 
 
 Differences. 
 
 Length of 
 Bubble. 
 
 A End. 
 
 BEnd. 
 
 A End. 
 
 BEnd. 
 
 I 
 
 7 
 
 9 .8 
 
 51.8 
 
 4-4 
 
 4.6 
 
 6l.6 
 
 2 
 
 17 
 
 14.2 
 
 47.2 
 
 4-3 
 
 4-5 
 
 61.4 
 
 3 
 
 27 
 
 18.5 
 
 42.7 
 
 4- S 
 
 4.2 
 
 6l.2 
 
 4 
 
 37 
 
 23.0 
 
 .38.5 
 
 4.0 
 
 4.1 
 
 6l. S 
 
 5 
 6 
 
 7 
 
 47 
 57 
 67 
 
 27.0 
 31-5 
 
 35-8 
 
 34-4 
 30.0 
 
 25-7 
 
 4-5 
 4-3 
 4-4 
 
 4.4 
 4-3 
 4-7 
 
 61.4 
 6l. 5 
 6l. 5 
 
 8 
 
 77 
 
 40.2 
 
 21.0 
 
 
 
 6l.2 
 
 8 
 
 77 
 
 40.0 
 
 21.2 
 
 
 
 61.2 
 
 7 
 
 67 
 
 35-5 
 
 25-5 
 
 4-5 
 
 4-3 
 
 61.0 
 
 6 
 
 57 
 
 
 30.2 
 
 4.4 
 
 4-7 
 
 61.3 
 
 5 
 
 47 
 
 26.7 
 
 34-6 
 
 4-4 
 
 4-4 
 
 61.3 
 
 4 
 
 37 
 
 22.3 
 
 38.8 
 
 4-4 
 
 4.2 
 
 61.1 
 
 3 
 
 27 
 
 18.2 
 
 42.8 
 
 4.1 
 
 4.0 
 
 61.0 
 
 2 
 
 *7 
 
 14.0 
 
 47.1 
 
 4.2 
 
 4-3 
 
 61.1 
 
 I 
 
 7 
 
 9-5 
 
 S l -S 
 
 4-5 
 
 4-4 
 
 61.0 
 
 
 
 
 
 60.9 
 
 61.1 
 
 
 Mean value of differences, 4.36 twentieths of an inch. 
 
 The pitch of the micrometer screw in the instrument used 
 was one-sixtieth of an inch, its disc divided into one hundred
 
 88 EXPERIMENTAL PHYSICS. 
 
 parts, the level scale graduated to twentieths of an inch, reading 
 right and left from zero at the center, and the distance between 
 the micrometer-screw point and the line joining the two pivots 
 was 17.9 inches. By a simple calculation it may be seen that 
 the angle corresponding to a rotation of 10 divisions on the 
 micrometer disc is 19.2 seconds, and that therefore an inclina- 
 tion of 4.4 seconds caused the bubble in the level tested to move 
 over one-twentieth of an inch. This then may be taken as an 
 indication of the sensitiveness of the level. The uniformity 
 exhibited in the displacements of the bubble shews that the 
 curvature of the level was very regular. 
 
 EXERCISE II. To find the radius of curvature of a level. 
 
 This is found by rotating the level through some known 
 angle, and noting the displacement of the bubble produced. If 
 
 z is this displacement, and - the circular measure of the angle 
 of rotation, the radius of curvature of the level is given by 
 
 A great source of error in spirit levels, increasing with their 
 sensitiveness, is an unequal heating of the level tube. The 
 bubble will always move towards the warmer spot, or end, on 
 account of a changed condition in the adhesiveness of the 
 fluid, and so a spirit level, while being tested, should never be 
 touched by the fingers, nor breathed upon, and it should always 
 be protected from the heat of the sun, or of artificial lights. 
 Care should also be taken to allow sufficient time for the bubble 
 to settle before taking a reading.
 
 PART II. 
 
 ADVANCED COURSE. 
 
 ACOUSTICS, HEAT, ELECTRICITY AND MAGNETISM, WITH 
 
 AN APPENDIX ON THE DETERMINATION OF 
 
 GRAVITY, AND ON THE TORSION 
 
 PENDULUM.
 
 ACOUSTICS. 
 
 LIST OF EXPERIMENTS. 
 
 PAGE 
 
 I. THE SONOMETER 93 
 
 Method of plucking strings ; tuning in unison ; exercises on 
 scale formation. 
 
 II. THE SCALE OF EQUAL TEMPERAMENT 95 
 
 Tuning a fork to any desired pitch; counting beats with 
 clock or stop watch ; manner of bowing forks. 
 
 III. LAWS OF THE TRANSVERSE VIBRATIONS OF STRINGS . . 98 
 
 Tuning a string to a fork. 
 
 IV. DETERMINATION OF PITCH 101 
 
 (a) Graphically, as in the case of a tuning fork. 
 
 (b) With the siren, as for an organ pipe. 
 
 (c) By a calculation, as in a string. 
 
 V. LISSAJOUS' OPTICAL METHOD OF TUNING .... 104 
 
 (1) By mirrors. 
 
 (2) With the vibration microscope. 
 
 Mode of constructing tuning forks ; and how they may be 
 sharpened or flattened. 
 
 VI. HARMONIC MOTION 108 
 
 (1) Composition of parallel or rectangular vibrations. 
 
 (2) Blackburn's pendulum. 
 
 (3) Wheatstone's wave apparatus. 
 
 VII. OVERTONES 114 
 
 (1) In strings. 
 
 (2) In forks. 
 
 (3) In pipes.
 
 92 EXPERIMENTAL PHYSICS. 
 
 PAGE 
 
 VIII. THE CHRONOGRAPH 117 
 
 IX. THE CLOCK FORK 118 
 
 X. MELDE'S EXPERIMENTS WITH STRINGS 120 
 
 XI. HELMHOLTZ'S APPARATUS FOR COMBINING SIMPLE TONES . 122 
 
 XII. KONIG'S ANALYZER 124 
 
 The principle of the manometric flame 
 
 XIII. THE MANOMETRIC FLAME 125 
 
 General application. 
 
 XIV. VELOCITY OF SOUND 129 
 
 (1) By measurement of a wave length. 
 
 (2) By resonance. 
 
 (3) Bosscha's method. 
 
 (4) Kundt's method. 
 
 (5) Regnault's method. 
 
 XV. DOPPLER'S PRINCIPLE . . . . . . . . 135
 
 EXPERIMENTS. 
 
 I. THE SONOMETER. 
 
 THIS is a large sounding box, a little more than a meter in 
 length, provided with a bridge at each end, over which eight 
 strings are stretched, either by means of weights, or by fixed 
 pins and thumbscrews, to which the strings are attached. The 
 distance between the bridges is usually one meter, and a wooden 
 scale is provided alongside, divided into millimeters. Clamps 
 with weights are also provided, or else sliding bridges, to enable 
 one to set any desired portion of a string in vibration. The 
 method of using the instrument is to obtain steel strings about 
 half a millimeter in diameter, and, stretching them over the 
 bridges, to tune them all in unison with any chosen pitch. 
 This may be done very accurately (even by what is termed an 
 unmusical ear) after a little practice, by listening for the beats 
 which are produced when two sets of waves, differing slightly 
 in their periodic times, are sent from two separate sources 
 simultaneously to the ear. The tension of any particular string 
 is increased or diminished until it begins to beat with the one 
 chosen as a standard ; the beats will be at first very rapid, and 
 cause a peculiar rolling effect in the ear, which must be heard 
 to be appreciated ; as the tension is properly altered they be- 
 come slower and slower, and finally disappear. The two strings 
 are then in unison, and the same process is repeated with the 
 others, until all give the same note. In order to vibrate a string 
 properly, it should be plucked with the thumb and forefinger at 
 its center ; or, better still, the ball of the thumb is placed on the 
 
 93
 
 94 EXPERIMENTAL PHYSICS. 
 
 middle point of the string, the direction of the thumb being 
 nearly at right angles to the string, and then, with a light pres- 
 sure, it is allowed to slip off, thus producing the fundamental 
 note, comparatively free from the higher harmonics. Steel 
 strings are preferable to others, as they can be obtained almost 
 perfectly uniform throughout, give a loud tone, and last indefi- 
 nitely; but they must be plucked properly, or else they give most 
 disagreeable harmonics, which may interfere with tuning. 
 When all the strings are in unison, the major diatonic scale may 
 be formed by choosing the first open string as the fundamental 
 or starting note, and then adjusting the clamps with weights on 
 the other strings, so that the lengths put in vibration are to the 
 length of the fundamental in the ratios : 
 
 This simply assumes the definition of the major diatonic scale 
 to be one whose notes are as the ratios : 
 
 and also that the pitch of a string varies inversely as its length. 
 The scale of equal temperament may be obtained for any open 
 string by taking the other lengths in the ratios : 
 
 The ordinary Minor and Harmonic Minor scales may be had 
 by choosing lengths similar to the major diatonic ; except that 
 for the former the ratio -| is made x ff = f> a minor third ; and 
 for the latter | is made f, a minor third, while f is made 
 f xff = f, a minor sixth. 
 
 The harmonic scale is formed by taking lengths on a stretched 
 string inversely proportional to the natural numbers i, 2, 3, 4, ... 
 and so on. 
 
 By listening carefully a great many of the notes of this scale 
 can be heard in the case of any freely vibrating string, their
 
 THE SCALE OF EQUAL TEMPERAMENT. 95 
 
 presence there being theoretically established by what is known 
 as Fourier's theorem. 
 
 The notes of this scale, when heard as auxiliaries to the 
 fundamental note of a string, are usually called harmonics, and 
 are of the greatest importance in the theory of Music. 
 
 The formation of the scales is based upon the assumption 
 that the strings are all of the same material, and uniform 
 throughout ; and, the tension of all being the same, the pitch 
 of any portion of a string depends inversely on its length. 
 See Exp. 3. 
 
 II. THE SCALE OF EQUAL TEMPERAMENT. 
 
 The accurate scale of physicists, determined by the series of 
 vibration, ratios, 
 
 i> |, t, t> I. f> > 2, 
 
 is seldom used for musical purposes, on account of the difficul- 
 ties which have been met with in arranging mechanism for 
 instruments to play in different keys. A musical instrument 
 like the piano, which would play accurately, according to the 
 true scale of the physicist, in all keys, would require more than 
 twenty-five notes to the octave. This can easily be seen by 
 writing down all the notes necessary for playing in the various 
 keys, and then striking out those which are repeated. Thus, 
 if we commence at middle C on the piano, we can obtain the 
 true major scale by tuning eight strings, according to the ratios 
 expressing pitch. If we did this throughout the keyboard, we 
 should have a complete major scale (all white notes) in perfect 
 order. But suppose now we wish to play perfect intervals in 
 the key of D : it is evident we must introduce new notes which 
 may not correspond at all with those in the key of C; and as 
 we proceed from one key to another we find additional notes 
 necessary for each change of key. And then, if we take 
 account of flats and sharps, we are introducing fresh difficulties.
 
 96 EXPERIMENTAL PHYSICS. 
 
 Many attempts were made to overcome this mechanical diffi- 
 culty by reducing the number of notes on the keyboard ; but the 
 only method which allowed playing in all the keys was the one 
 finally adopted. In the scale of equal temperament the octave 
 was defined to be twelve semitones ; and, as the pitch of the 
 octave is twice that of the fundamental, a scale of thirteen notes 
 in the octave was chosen, with twelve intervals, the pitch of each 
 being obtained from the next lower by multiplying by the twelfth 
 root of two, which corresponds to the interval called a semitone. 
 
 In most stringed instruments, the scales are approximately 
 scales of equal temperament, although in the case of the piano 
 more latitude is allowed the tuner, who generally makes some 
 intervals a little greater than others, being guided more by his 
 individual taste than by the factor which is represented mathe- 
 matically by ty~2. 
 
 One may easily construct a scale of equal temperament for 
 any fundamental C note by using a tonometer or series of 
 forks, differing from one another by 8 or 16 vibrations per 
 second. The set used for experimental purposes generally 
 commences with 256 or 512 v.s.,* and runs up to 1024. For 
 example, to construct the tempered scale commencing with 512, 
 one would calculate the vibration numbers corresponding to 
 the other twelve notes by using the factor ^5. The complete 
 scale would then be given by the numbers, 512, 542.4, 574.7, 
 608.9,645.1, 683.4, 724.1, 767.1, 812.7, 861.1, 912.3,966.5, 1024. 
 
 To obtain the vibration number 542.4, two forks of the tonom- 
 eter are chosen, 536 and 544, and these, if put in vibration to- 
 gether, give four beats per second ; if now the pitch of 544 be 
 
 * Note on Pitch Notation, Confusion often arises in the calculation of pitch from 
 the fact that the English use the term -vibration in the same sense as the complete 
 vibration of a pendulum, to and fro, while the French System takes it to be a simple 
 vibration ; so that a French fork numbered 512 would be the same as an English 
 256. The difficulty is overcome by marking v.s., or v.d., after the pitch, as is done 
 by Konig on his standard forks, to indicate vibrations simples (simple vibrations), 
 and vibrations doubles (complete vibrations). We have adhered to this in the 
 present course.
 
 THE SCALE OF EQUAL TEMPERAMENT. 97 
 
 altered by placing small pellets of wax on the prongs so that it 
 moves more slowly, it is evident that its pitch can be reduced to 
 the required number, 542.4. In other words, when the loaded 
 fork gives, with 536, 3.2 beats per second, or 32 beats in 10 
 seconds, then the former is vibrating 542.4 times per second. 
 The process then becomes one of counting beats with a clock 
 or stop watch, and loading the forks with wax until the necessary 
 numbers of beats are obtained in each case. 
 The following precautions should be taken : 
 
 1. When vibrating, the prongs of a tuning fork move out and 
 in, together: so, if one sets a fork in vibration with a bow, it 
 should be drawn across the upper end of one prong only, and in 
 such a way that the plane of the horse hair is nearly in the 
 plane containing the two ends of the prongs. If inclined at an 
 angle, a pure note is not obtained, and the horse hair is very 
 liable to get torn. The fork may be put in vibration by bowing 
 at the side of the prongs ; but in that case the fundamental tone 
 of the fork is often accompanied by a disagreeable high note. 
 SeeExp. 7(3). 
 
 2. In counting the beats, the beginner is very apt to count 
 the maximum from which he starts as one : a little practice 
 soon enables him to avoid this. 
 
 As large a number as possible should be counted in every 
 case to minimize the error ; usually the counting can be carried 
 on for thirty or forty seconds, except in the case of high forks 
 which have small amplitudes and soon stop vibrating. 
 
 3. In loading the forks, when very slight differences are de- 
 sired, it will be found more convenient to move the pellets of 
 wax up or down the prongs, instead of increasing or diminishing 
 the loads themselves. 
 
 4. Tuning forks are always made of steel and change rapidly 
 if rusted. Care must then be always taken of them by the 
 experimenter ; and when an experiment is finished they should 
 be carefully cleaned with a soft rag moistened with alcohol, 
 dried, and put away in a dry place.
 
 98 EXPERIMENTAL PHYSICS. 
 
 III. LAWS OF THE TRANSVERSE VIBRATIONS OF STRINGS. 
 
 If a string of any material be stretched beyond its natural 
 length between two fixed points and then be disturbed from 
 its position of equilibrium, it vibrates to and fro, and emits 
 certain musical notes, of which one is much more prominent than 
 any other : this is the lowest or fundamental note of the open 
 string, and its pitch or vibration number is usually spoken of as 
 the pitch of the string. This pitch will evidently depend on the 
 length of the string, its diameter, tension, and the material of 
 which it is made ; and a relation connecting these elements can 
 be found theoretically in the following manner : 
 
 Let the string be stretched between two fixed points O and 
 A, Fig. i, with a tension T. And suppose that it is uniform 
 throughout, of radius r, density p, and length /. 
 
 Then, if disturbed slightly, and the motion be referred to 
 three rectangular axes at O, of which OA is the axis of x, it 
 c 
 
 o^-- 
 
 Fig. 1. 
 
 is obvious that any element such as PQ whose mass is pds will 
 have motions parallel to OB and to OC. The equations of 
 motion of this element, on the hypothesis that the tension 
 remains constant and that the length ds is equal to its pro- 
 jection dx, will be
 
 THE TRANSVERSE VIBRATIONS OF STRINGS. 99 
 
 the coordinates of the point P being x, y, z. For, since the ten- 
 sion at P along the tangent is T, the tensions along lines 
 
 through P, parallel to OB and OC, will be T-j- y T-j- ; and at 
 Q, in the same directions, the tensions will be 
 
 ^dy d( rr dy\. , ^dz d ( ~dz\ . 
 
 T^ + -r( T^r)ds, and T-J- + -J-I T- T )ds. 
 
 ds ds\ dsj ds ds\ dsj 
 
 These become 
 
 d*y_T d?y_ 
 dt*~ p' dx* 
 
 since ds dx ; and they express analytically the fact that the 
 string may move in the plane of xz, or in the plane of xy, or it 
 may have both motions combined. 
 The solution of one gives 
 
 y =f( x - at) + F ( x + ') 
 
 where a 2 = 
 P 
 
 This represents the transmission of arbitrary forms along the 
 string with velocity a, where ar = 2 /, T being the periodic time 
 of vibration of the fundamental note. 
 
 Hence T =- = 2rl\l, 
 
 a ^Tg 
 
 where d is the specific gravity of the string referred to water as 
 
 unity, and p therefore equal to ^ , the cross-section of the 
 
 g 
 string being circular. Therefore the pitch of the fundamental 
 
 note of the string, which is equal to -, becomes
 
 100 EXPERIMENTAL PHYSICS. 
 
 In this formula are included all the laws of the transverse 
 vibrations of strings. It shews us that the pitch varies inversely 
 as the length, and radius of the string, directly as the square 
 root of the stretching force, and inversely as the square root 
 of the density. All these facts may be verified experimentally 
 by varying the elements of the string; and different experi- 
 ments can easily be arranged to prove any particular law. The 
 most useful and instructive exercise for the student, however, 
 is to stretch a string over such a sonometer, as is shewn in Fig. 2. 
 
 Fig. 2. 
 
 The string is fixed at one end, passes over two bridges which 
 determine the two fixed points of the previous theory, and then, 
 running over a pulley, is kept stretched by means of known 
 weights. Its mean radius is determined with the wire gauge, 
 its length is measured, or observed directly on the scale of the 
 sonometer, and its specific gravity may be either taken from 
 tables or found in the usual way. Then the pitch is calculated 
 by the formula and verified by means of a standard fork. 
 
 This latter determination is sometimes difficult on account of 
 the peculiar monotonous tone of the fork, which leads one to 
 imagine that it is an octave lower than its real pitch, as well as 
 the presence of loud overtones in the string (especially if made 
 of steel) which an unmusical ear often chooses instead of the 
 fundamental. 
 
 The easiest method of using the fork is to move the sliding 
 bridge, usually provided with the sonometer, along until it cuts 
 off a portion of the string which will vibrate in unison with 
 the fork ; then the pitch of the whole string will be to that of the
 
 DETERMINATION OF PITCH. IOI 
 
 fork in the inverse ratio of the length of the open string to the 
 part cut off by the bridge. The fork should be chosen of a 
 suitable pitch so that the portion of the string cut off by the 
 bridge may not be too small. 
 
 If the sonometer is provided with two sets of pins and pulleys, 
 the experiment may be modified by stretching two strings of 
 the same material, length, and diameter with different weights ; 
 or by keeping the material, length, tension constant, and vary- 
 ing the diameters. 
 
 The most useful strings for experimental purposes are the 
 fine steel E strings used for the mandolin and guitar. Copper 
 or brass ones may also be used, as they can be obtained in all 
 sizes, are comparatively uniform throughout, and require little 
 tension to give a distinct note ; they are, however, if of copper, 
 apt to stretch and break. Gut strings can be used, but they 
 are hygroscopic, and it is difficult to get them uniform through- 
 out their length. 
 
 IV. DETERMINATION OF PITCH. 
 
 (a) Forks. 
 
 The method employed for determining the pitch of a tuning 
 fork is graphical. A slight style or writer of inappreciable 
 weight is attached to the fork with wax; the fork being so 
 arranged that the style just rests against a cylinder turning 
 at a known rate, and covered with a smoked paper. As the 
 cylinder is turned the writer, owing to the vibrations of the 
 fork, makes a wavy line on the paper, and the number of 
 these vibrations made in a given time can then be counted, and 
 the pitch of the fork found. 
 
 Generally, a standard fork is placed alongside of the fork 
 whose pitch is to be found, and a comparison made between the 
 two tracings on smoked paper ; thus rendering the determina- 
 tion independent of the rate at which the cylinder is turning if 
 the points of the two styles are side by side. The style may be
 
 102 
 
 EXPERIMENTAL PHYSICS. 
 
 made of a bristle or stiff hair, or of thin sheet brass ; but in the 
 last case a small correction will have to be made unless the fork 
 is very large. 
 
 (b) Organ Pipes. 
 
 The pitch of an organ pipe may be found most readily and 
 simply by comparison with a fork, which may be chosen so as 
 to give a small number of beats per second. Care must be 
 taken, however, that the fundamental note of the pipe and not 
 
 Fig. 3. 
 
 one of the overtones is chosen. Forks with sliding weights 
 to alter their pitch are best adapted for this purpose. 
 
 One may also find the pitch of a pipe by means of Helm- 
 holtzs siren, which is shewn in Fig. 3. 
 
 The siren is attached to a bellows and driven at constant 
 speed. Counters, shewn in the figure, are attached to the ver- 
 tical axis of the instrument and enable one to observe the 
 number of vibrations in any given time. To determine the
 
 DETERMINATION OF PITCH. 103 
 
 pitch of a pipe, the discs are driven until the note emitted by 
 the siren is in unison with it, and if the note is maintained for 
 some seconds without variation, the time and corresponding 
 number of vibrations can be found. The method is tedious, 
 and, unless great care is taken to maintain a constant pres- 
 sure in the air chest of the bellows, no accurate result can be 
 obtained. If the axis carrying the two discs is driven by a 
 small electromotor, thus making the driving power independ- 
 ent of the note, better results may be obtained ; the instru- 
 ment is, however, much better suited for illustrative purposes 
 on the lecture table, and is not strictly a measuring instrument. 
 It may be mentioned here that the simplest form of siren, and 
 one that may be used to some advantage for determining pitch, 
 is a circular plate of brass with small openings symmetrically 
 arranged. When driven uniformly by any rotation apparatus, 
 and air, blown through a small orifice, is directed against the 
 openings perpendicular to the face of the plate, a note is pro- 
 duced whose pitch can easily be found from the number of 
 holes and the time of rotation. Such a plate provided with 
 concentric rings of holes properly arranged may be used for 
 lecture purposes to shew the formation of scales and combina- 
 tions of notes, and has the advantage over all other forms of 
 sirens on account of its perfect simplicity. 
 
 (c) Strings. 
 
 The pitch of a stretched string is found in the manner already 
 explained, by comparison with a standard fork directly, or by 
 calculation from the elements of the string where they are 
 known. 
 
 When the fork chosen is not near the string in pitch, or 
 where it is not possible to obtain a fork close enough, then the 
 sliding bridge may be used on the string, and a portion cut off 
 which vibrates in unison with the standard : the pitch of the 
 the open string can then be inferred. 
 
 For very low or very high sounds, some difficulty is always 
 experienced in determining pitch.
 
 104 EXPERIMENTAL PHYSICS. 
 
 In the case of low sounds a siren may be used, but the diffi- 
 culty then is to maintain a constant pressure in the air chest. 
 The method devised by Savart may be used both for very low 
 and very high notes. A wheel with a large number of teeth is 
 driven uniformly at any desired speed, and a small strip of stiff 
 cardboard or metal is held with an edge just touching the teeth ; 
 the noise, which is heard at first when the wheel rotates slowly, 
 gradually develops into a decided musical note that becomes 
 higher as the speed increases, the upper limit depending only on 
 the rate at which it is possible to drive the wheel. When used 
 to determine pitch, its note is made identical with the one to be 
 observed, and the pitch calculated from the speed and the 
 number of teeth. Conversely, the speed of a rapidly rotating 
 wheel or disc may be best inferred from the note produced by 
 some attachment that makes periodic sounds which change to a 
 musical note under high speed. This method was used by 
 Wheatstone in his measurement of the velocity of electricity, 
 and is of some importance. 
 
 For very high notes, the sonometer with a sliding bridge, and 
 a very tightly strung fine steel wire may also be used to advan- 
 tage, and will give fairly accurate measurements of pitch as high 
 as 10,000 vibrations per second. 
 
 Steel cylinders are also made, to furnish standards of pitch 
 for very high notes, and to test the upper limit of audibility. 
 They range from 8192 vibrations to 40,000, and are usually 
 made to give the notes of the major diatonic scale. By means of 
 them one may locate any particular high note within a few hun- 
 dred vibrations, by the unassisted ear. 
 
 V. LISSAJOUS' OPTICAL METHOD OF TUNING. 
 
 (i) With mirrors attached to the forks. 
 
 Instead of tuning by means of the ear, another method, 
 devised by Lissajous, is used for the accurate comparison of 
 forks which may be either close to one another in pitch, or very
 
 LISSAJOUS 1 OPTICAL METHOD OF TUNING. 105 
 
 far apart. Mirrors are fixed rigidly to the prongs of the forks, 
 which are so arranged that they can vibrate in two planes at 
 right angles to one another ; and a beam of light from a small 
 luminous source is allowed to fall in succession on the two 
 mirrors, and then received on a screen, viewed directly by the 
 eye, or allowed to pass into a small telescope focussed so that 
 the image of the luminous source is clear and sharp. On 
 vibrating the forks, the image in the telescope is seen to pass 
 through various forms owing to the compound motion ; and, by 
 loading the forks until certain known forms are obtained, the 
 ratio of the two vibration numbers can be inferred. 
 
 Theory. When the two forks vibrate, their motions are 
 periodic, and the displacement of the spot of light seen after 
 reflection from the two mirrors will therefore be equivalent to 
 two displacements obtained by solving the equations of motion : 
 
 x=a COS27T 
 
 These give 
 
 Where a and b are the amplitudes of vibration, Tand T the 
 periodic times, and d a constant, called the phase, which depends 
 on the initial circumstances of motion. On eliminating t, we 
 shall get an equation to a curve, which is the form seen in the 
 telescope. 
 
 i. If T= T', the forks will be in unison, and the curve seen 
 will have the form 
 
 r- 
 tr ao
 
 io6 
 
 EXPERIMENTAL PHYSICS. 
 
 which represents an ellipse or straight line, as shewn in (a) Fig. 4, 
 according to the values of S, determined by the initial conditions 
 of vibration. 
 
 If T differs slightly from T', then, whatever be the initial 
 form, the curve begins to change, and runs through all the 
 forms shewn ; on loading the forks properly with small pellets 
 of wax the change takes place more slowly, until when T 
 becomes equal to T' the figure initially obtained preserves its 
 form throughout the motion. 
 
 2. If T' = 2 T, then, on eliminating / as before, a complicated 
 equation is obtained, which represents the curves shewn in (b). 
 
 (Os) 
 
 Fig. 4. 
 
 In a particular case, where S = J, the equation reduces to a 
 parabola. 
 
 3. The figures in (c} and (d) shew the forms obtained when 
 r' = 3 T, and 2 r = 3 T. 
 
 There are many ways of drawing these curves geometrically ; 
 and an excellent exercise for the student is to take a rectangle 
 whose sides represent the amplitudes and map out a series of 
 points from the relations for the displacements, for any chosen
 
 LISSAJOUS' OPTICAL METHOD OF TUNING. IO/ 
 
 ratio, assuming some particular value for 8. The curve for that 
 particular phase is then easily drawn. 
 
 (2) TJie vibration microscope. 
 
 For the more practical purpose of comparing forks with 
 standards, an instrument known as the vibration microscope is 
 used. It is represented in Fig. 5. A standard fork of known 
 pitch, which can usually be altered by sliding weights, carries 
 
 C: 
 
 
 Fig. 5. 
 
 on one prong a small lens, whose axis is perpendicular to the 
 plane of vibration of the fork ; and a fixed microscope is 
 arranged independently behind, so that its axis can be placed 
 in the prolongation of the axis of the lens. The lens on the 
 fork forms the objective of the microscope. The fork to be 
 examined is then placed so that its plane of vibration is per- 
 pendicular to that of the standard ; and any small mark on it 
 is viewed through the microscope, which thus shews an image
 
 108 EXPERIMENTAL PHYSICS. 
 
 of the mark going through a compound motion, when the 
 two forks are vibrating, similar to that obtained with two 
 mirrors and a spot of light. The pitch of the fork is inferred 
 as in the former experiment. Practically, this furnishes an 
 exact way of reproducing a standard ; by filing slightly the ends 
 of the prongs the pitch can be raised as much as one pleases, 
 and if it becomes too high a slight filing on the inside of 
 the prongs lowers it ; and by operating in this way until 
 the stationary ellipse or straight line is obtained, a standard 
 may be made in a short time. The accuracy of the method 
 is such that one may get two forks so close in pitch that 
 they will not differ from one another by more than one beat 
 in a minute, a thing which no system of tuning by ear could 
 ever accomplish. 
 
 VI. HARMONIC MOTION. 
 
 (i) Composition of parallel and rectangular vibrations. 
 A motion represented by the equation 
 
 is said to be periodic or harmonic. The prong of a tuning fork 
 gives such a motion, as may be seen graphically by arranging a 
 small writer on the end of one prong of the fork ; when put in 
 vibration and drawn so that the writer moves uniformly along a 
 piece of smoked glass, the tracing is an accurate curve of sines. 
 
 Two parallel or rectangular harmonic motions may be com- 
 pounded, by aid of two tuning forks, with the apparatus shewn 
 in Fig. 6. 
 
 One fork is fixed in position, and holds on one prong a plate 
 of smoked glass which vibrates with it ; on the other prong is a 
 small sliding weight which is used as a counterpoise to the glass 
 plate. The other fork, provided with a small writer, is arranged 
 independently so that it can be drawn backwards while the
 
 HARMONIC MOTION. 
 
 109 
 
 writer rests gently on the smoked glass. To compound two 
 parallel motions, the forks are so placed that their prongs and 
 the plate of glass are all parallel ; the position for angular com- 
 
 position is shewn in the figure. Both forks are set vibrating, 
 and then the one carrying the writer is drawn along as uni- 
 
 
 
 
 .. .. ^ ^ ^ ^ 
 
 J \J \J \j \J \J \J V 
 
 v/ 
 
 _ 
 
 \r\r 
 
 Fig. 7. 
 
 formly as possible, and the resulting curve traced on the glass 
 shews the composition. 
 
 These curves may also be drawn geometrically, and, as in
 
 110 EXPERIMENTAL PHYSICS. 
 
 the case of Lissajous' curves, the drawing is a good training for 
 the proper appreciation of wave motion. 
 
 Figure 7 shews two compositions of parallel motions. The 
 lower represents two curves of sines of which one wave length 
 is twice the other, and their composition is also shewn. The 
 upper one shews the composition of two curves whose wave 
 lengths are as I to 3. 
 
 Figure 8 shews the composition of two curves described by 
 forks whose vibration numbers are in the ratio of 4 to 5. 
 
 \ /\ /\ /\ /\ /\ /\ /\ /^ 
 
 \7 V7 \^7 \y \^7 \y \s \y 
 /\ /N /\ /\ /\ /\ /\ /~\ /\ /~\ 
 7 \s \s VT \I7 \y \37 \37 \7 \7 \s 
 
 - ^ ~ 
 
 Fig. 8. 
 
 Figures 9, 10, 11 shew compositions arising from forks of 1:2, 
 i : 3, i : 4 very nearly. One is exactly in the ratio i : 4. The 
 curious chainlike appearance presented by them is caused by 
 the slight difference of phase. 
 
 Figure 12 shews the composition of two vibrations nearly 
 rectangular. 
 
 Figure 13 represents to the eye the effect produced on the 
 ear by two sets of waves which have very nearly the same 
 periodic time. It is, in fact, a graphical representation of a beat. 
 
 (2) Blackburn '.r pendulum. 
 
 The composition of two motions at right angles to one another 
 may also be exhibited by means of a very simple contrivance 
 known as Blackburn's pendulum. It is shewn in Fig. 14. 
 
 Three wires are joined together at D, the two upper ones 
 being fastened to a horizontal beam, while the lower carries a 
 heavy sphere to which a small writer is attached. If the sphere 
 be moved perpendicularly to the plane of the paper, it oscillates 
 about the point E, its time of oscillation varying as the square
 
 HARMONIC MOTION. 
 
 1 II 
 
 Fig 9. 
 
 Fig. 10. 
 
 Fig. 11. 
 
 Fig. 12. 
 
 Fig. 13.
 
 EXPERIMENTAL PHYSICS. 
 
 root of EC, C being the center of the sphere. If disturbed in 
 the plane of the paper, it oscillates about D, with a different 
 periodic time. The lengths, EC, 
 DC, can be adjusted so that the 
 times of oscillation are in any 
 desired ratio. If then the sphere 
 be moved indifferently in any direc- 
 tion, and a piece of smoked glass 
 be placed underneath so that the 
 writer is always in contact with it, 
 a closed curve will be traced out, 
 arising from the compositions of the 
 two rectangular motions. 
 
 Figure 16 is a photograph of an 
 actual tracing made with a rough 
 pendulum consisting of an iron ball 
 The writer was a piece of thin sheet 
 
 Fig. 14. 
 
 suspended by steel wires, 
 brass. 
 
 Fig. 15. 
 
 (3) Wheatstones wave apparatus. 
 
 This shewn in Fig. 15 is an admirable way of studying the
 
 114 EXPERIMENTAL PHYSICS. 
 
 composition of parallel and rectangular vibrations ; also for 
 illustrating the general phenomena of wave motion. 
 
 The machine has a number of stiff iron rods, each made in 
 the form of a cross and carrying beads at the ends of the upper 
 and the two side pieces. These rods are constrained to move 
 in one plane. Boards are supplied with the apparatus, which 
 have curves of sines cut along their edges, and are made in 
 pairs so that the curves fit one another. The iron rods are so 
 arranged that when the boards are placed in position and 
 pushed through the machine, the motion of the beads shews 
 the propagation of the wave form, which may be either simple 
 or complex. If the horizontal boards are used alone, the beads 
 have rectilinear motions, which indicate the propagation of a 
 plane horizontal wave ; if the vertical boards are used, plane 
 vertical waves, resembling somewhat water waves, are obtained ; 
 and, by combining vertical with horizontal, compound waves 
 like those of light may be formed. 
 
 It will be noticed in all cases that, whatever boards or com- 
 binations be used, the curve on top is the resultant of the two 
 curves shewn at either side by the sets of fixed and movable 
 beads. 
 
 VII. OVERTONES. 
 
 (i)' In strings vibrating freely, the complex nature of the 
 vibration, as determined analytically from the equations of 
 motion by the aid of Fourier's theorem, may be seen directly 
 by the eye, or observed by the ear with the assistance of what 
 are called resonators ; these are hollow globes or cylinders 
 of metal or glass, constructed of various sizes so that the air 
 inside vibrates in sympathy with certain corresponding notes, 
 and thereby reinforcing them renders it possible for a person 
 to hear tones which, without the resonators, would not be 
 audible. They are usually made in a series corresponding to 
 the notes of the harmonic scale, and may be either fixed or 
 variable. Figure 17 represents two kinds, one fixed and spherical
 
 OVERTONES. 1 1 5 
 
 in shape, the other cylindrical and capable of adjustment so 
 as to be used for several tones. 
 
 To use them properly, the smaller end is either placed 
 directly in the aperture of the ear or connected with it by 
 means of a short rubber tube ; the other end being then 
 brought into the vicinity of the complex note to be examined. 
 Success in the experiment is best attained by closing, at the 
 same time, the aperture of the other ear. To obtain good 
 results from a string, choose a steel one about six feet long 
 and stretch it between two fixed supports until it gives a pitch 
 of 256 v.s. With the series of resonators corresponding to 
 this pitch as fundamental, it is possible to recognize eighteen 
 or twenty pure overtones of the harmonic scale. In fact, after 
 
 Fig. 17- 
 
 a little practice with the resonators, they may be discarded 
 and most of the overtones still heard by the unassisted ear. 
 With steel and copper strings, the third harmonic is often 
 louder than the fundamental ; and it is, no doubt, owing to the 
 presence of these overtones in the string that great difficulty 
 is always experienced in tuning a string to a fork, as the ear 
 is apt to choose, instead of the fundamental note, one of the 
 more prominent overtones. 
 
 (2) In the case of organ pipes, the complex nature of the note 
 obtained may also be shewn by using the resonators ; but the 
 best method for the examination of the notes produced by both 
 closed and open pipes is that of the manometric flame, in con- 
 nection with the resonator. See Exps. 12 and 13.
 
 Il6 EXPERIMENTAL PHYSICS. 
 
 (3) The timing fork is, in reality, a combination of two steel 
 rods which vibrate in unison with one another ; and its pitch 
 varies directly as the thickness of the prongs (measured in the 
 plane of vibration), inversely as the square of the length, and is 
 practically independent of the thickness of the prongs (measured 
 perpendicularly to the plane of vibration). The prongs in vibrat- 
 ing, may divide into segments, just like a string; and there- 
 fore may be made to give tones superior to what is commonly 
 called the pitch of the fork. These tones, however, are not true 
 harmonics : the first one being nearly six times, and the next 
 
 Fig. 18. 
 
 higher, seventeen times, the pitch of the fork. To attain 
 the first, the fork is bowed at the side, about half way down ; the 
 second may be obtained by bowing at a point one third of the 
 way from the bottom. Or they may be obtained along with 
 the fundamental by bowing the fork first in the usual way on 
 top and then along the sides. Figure 18 shews tracings of a 
 fork accompanied by its superior tones. 
 
 The lower right-hand tracing is made by the fundamental 
 with its first superior tone ; the lower left hand one is the funda- 
 mental with the second superior tone ; and the upper one is a 
 combination of the fundamental with the first two superior 
 tones.
 
 THE CHRONOGRAPH. 
 
 117 
 
 VIII. THE CHRONOGRAPH. 
 
 This is an instrument which is used for the measurement of 
 small intervals of time by means of an electric interrupter and 
 tuning fork. The arrangement is shewn in Fig. 19. 
 
 A tuning fork, held upright in a rigid support, can be put in 
 vibration by a current of electricity, which passes along one 
 prong of the fork, through a small contact piece that acts as an 
 
 Fig. 19. 
 
 interrupter, and then through two electro magnets situated one 
 on each side of the fork, as shewn in the figure. To the end of 
 one of the prongs of this fork is attached a writer of brass which 
 rests against a piece of smoked paper, so arranged in a roll that 
 it can be drawn along continuously by the handle shewn at the 
 upper part of the apparatus. Two small writers, one on each 
 side of the tuning-fork writer, act as interrupters, when the 
 currents passing through the electro-magnets which carry the 
 writers are in any way broken. When the fork is running elec-
 
 Il8 EXPERIMENTAL PHYSICS. 
 
 trically, it makes a wavy line on the smoked paper as it is un- 
 rolled, each distance from crest to crest corresponding to - 
 
 n 
 seconds, where n is the pitch of the fork in double vibrations. 
 
 The two interrupters at the side make continuous straight 
 lines. If now the interrupters are arranged in a circuit which 
 can be broken so as to indicate the beginning and the end of any 
 particular phenomenon, then, if the interruptions are properly 
 made while the paper is moving and the fork vibrating, marks 
 will be visible on the smoked paper where the tracers of the 
 interrupters have departed from a straight line. By counting 
 the number of waves between these marks the time of the event 
 is found. Instead of using two interrupters, one only may be 
 used ; although in using both, one serves as a check upon the 
 other. The two interrupters can also be arranged in separate 
 circuits independent of one another, while the tuning-fork cir- 
 cuit is, of course, independent of both. 
 
 With such an instrument as this, and with due care, one may 
 measure with great accuracy the time of vibration of a body, or 
 the time of a falling body, or any such interval where it is possi- 
 ble to form electrical circuits with the interrupters. 
 
 The auxiliary figure represents an apparatus for smoking the 
 roll of paper in a proper manner. A hollow brass cylinder is 
 fixed in position between two wooden supports, and the paper 
 is unrolled from one grooved wheel to another, passing around 
 the cylinder as shewn in the figure and being blackened by a coal 
 oil lamp which is placed underneath the cylinder. To avoid 
 burning the paper the cylinder is partially filled with water. 
 
 IX. THE CLOCK FORK. 
 
 This (Fig. 20) consists of a fork of 100 v.d., held upright in a 
 firm support and connected to a clockwork movement in such 
 a manner that it regulates it through the escapement just 
 as a pendulum would ; but it receives at each oscillation a small 
 impulse which enables it to keep up a steady vibration.
 
 THE CLOCK FORK. 1 19 
 
 The two branches of the fork carry sliding weights attached 
 to micrometer screws so that the period of vibration can be 
 regulated with great precision. In addition, one of the prongs 
 carries the objective of a microscope whose ocular is placed on 
 an independent support ; thus, it may be used in the same man- 
 
 Fig. 20. 
 
 ner as the optical comparator of Lissajous. A thermometer 
 placed between the prongs of the fork gives the temperature. 
 
 The instrument gives perfect results when properly regulated, 
 and enables one to observe the effects of temperature on the 
 period of vibration of a tuning fork by using an auxiliary fork 
 whose temperature is gradually increased, and noticing the
 
 120 EXPERIMENTAL PHYSICS. 
 
 figures obtained with the aid of the vibration microscope. It 
 serves also to study in a similar way the effect of reinforcement 
 on the pitch of a fork by observing the vibration of a fork 
 placed on supports of different materials and forms. It may be 
 stated here that the ordinary steel forks between 128 v.s. and 
 1024 v.s. have their pitch diminished by about -^ for each 
 increase of temperature of i C. This coefficient increases 
 slightly as the pitch of the fork increases, and also changes 
 considerably with the material of the fork. 
 
 X. MELDE'S EXPERIMENTS. 
 
 A tuning fork has a string attached to one prong and stretched 
 over a pulley by a small weight, or else depending vertically 
 from the fork. When put in vibration, the fork sends waves 
 along the string which will be transversal or longitudinal accord- 
 ing to the position of the string relatively to the plane of vibra- 
 tion of the fork. The string then divides up into loops and 
 nodal points according to the well-known laws of stretched 
 strings explained in Exp. 3. A simple experiment for trans- 
 verse vibrations is to alter the length and tension, and prove 
 that the number of loops obtained varies inversely as the square 
 root of the tension. The fork may be kept vibrating electri- 
 cally by using an ordinary dry or mercurial interrupter. Strings 
 of various kinds may be used to prove the law of density : silk 
 cord, fine steel, copper, or platinum wires give good results. 
 
 The complex vibration of a string attached to two forks may 
 be shewn by means of the apparatus indicated in Fig. 21, where 
 the forks are vibrated electrically and the string is stretched 
 between them. 
 
 Curious and instructive complex motions are obtained by 
 using a silk cord about a millimeter in diameter and a meter in 
 length stretched from a fork (vibrating electrically) of 128 v.s. 
 On placing the string so that it makes an angle of 45 with the 
 fork in its plane of vibration, and stretching it with the proper
 
 MELDE'S EXPERIMENTS. 121 
 
 weight (which can easily be found by trial or by calculation), 
 one sees it divide up into two portions which resemble solid 
 figures whose cross-sections are usually parabolas or lemnis- 
 cates, and in the center is a plane. 
 
 This seems to arise from the two sets of waves which are 
 travelling along the string with velocities in the ratio i : 2. 
 
 Tyndalfs experiment with the luminous wire may also be 
 shewn by using an electrically vibrating fork and a platinum 
 wire a millimeter in diameter and a meter long, which runs over 
 a brass pulley and is stretched by the proper weight. An 
 independent current is then passed along the wire by one prong 
 
 Fig. 21. 
 
 and the brass pulley which gives good contact ; the wire being 
 then made red hot by a current of 10 or 15 amperes, the fork is 
 set vibrating; when immediately one sees the loops become 
 dark, owing to cooling by their rapid motion, and the nodes 
 grow brighter. More current is then turned on until the loops 
 again become bright, and the whole outline is seen ; the wire is 
 then very apt to break at the pulley, or at a nodal point, owing 
 to the unequal distribution of heat. A heavy steel wire may 
 also be used for this purpose, but in that case the weights must 
 be adjusted after it becomes red hot, on account of its losing its 
 elasticity. No doubt the best results would be obtained by 
 using a carbon filament, such as is seen in an incandescent 
 lamp, since it possesses the advantage of having its resistance 
 diminished as the temperature is increased.
 
 122 
 
 EXPERIMENTAL PHYSICS. 
 
 XI. HELMHOLTZ'S APPARATUS FOR SYNTHESIS. 
 
 This is shewn in the adjoining figure. A series of forks 
 running from ut^ to mt s are arranged in the same electrical cir- 
 cuit with an interrupting fork. Resonators are placed behind 
 each fork, and are capable of small adjustments so that they 
 may be moved backwards and forwards. 
 
 Fig. 22. 
 
 Stoppers are provided for the resonators and connected by 
 means of strings with a keyboard in front, in such a way that 
 on depressing any key the corresponding fork is heard vibrating 
 loudly. When the keys are all down and the stoppers closed, 
 all that is heard is a gentle humming sound.
 
 HELMHOLTZ'S APPARATUS. 
 
 I2 3 
 
 The object of the apparatus is to combine the fundamental 
 note nt z with any of the harmonics in the series, and observe 
 the quality or timbre of the corresponding tone. It does not 
 admit of accurate comparison, but yet affords an instructive 
 exercise for the student. The noise of the interrupting fork 
 may be overcome to a large extent by dividing the current, or 
 
 Fig. 23. 
 
 introducing a great resistance into the circuit ; or the interrupt- 
 ing apparatus, which is quite independent of the other part of the 
 instrument, may be carried to another room, and the circuit 
 through the forks completed by wires leading from one room to 
 the other.
 
 124 EXPERIMENTAL PHYSICS. 
 
 XII. KONIG'S ANALYZER. 
 
 This is the reverse of the apparatus described in the preced- 
 ing experiment : it is used to analyze the constituents of a 
 compound note. The general appearance of the instrument is 
 shewn in Fig. 23. 
 
 The principle used is that of the manometric flame, devised 
 by Konig, combined with the rotating mirror of Wheatstone. In 
 Fig. 24 a section of the manometric capsule is given. A small 
 wooden box encloses a membrane M, which is stretched between 
 two points B and C and separates an air chamber from a gas 
 chamber. The gas is supplied at G, and burns at F with a 
 small narrow flame about half an inch in height. The air from 
 any source enters at A. 
 
 In the analyzing apparatus a series of these manometric 
 capsules have their air chambers connected with the resonators 
 
 by tubes at the back (not 
 shewn in the figure). The 
 f tubes seen in front lead from 
 
 * 1 1 the gas chambers to the large 
 
 tube at the bottom which sup- 
 plies gas to them all. Stop- 
 cocks are provided for each gas 
 jet, so that any one may be 
 
 Fig. 24. 
 
 shut off if necessary. 
 
 In order to examine any compound note, such as that pro- 
 duced by an open pipe, its harmonics are calculated from the 
 pitch of its fundamental, the resonators are adjusted for these 
 harmonics, and when the pipe is blown and held near the front 
 of the apparatus, while the mirror (shewn at the right) is 
 turning, certain of the flames will be seen to give wavy appear- 
 ances in the mirror, shewing that they are vibrating, by the 
 intermediary of the elastic membranes, in sympathy with the 
 air in certain of the resonators. 
 
 The instrument furnishes a useful means of observing over- 
 
 K
 
 MANOMETRIC FLAMES. 
 
 125 
 
 tones from any source of sound ; but it must be adjusted with 
 considerable care in order to perform its work with accuracy. 
 
 XIII. MANOMETRIC FLAMES. 
 
 The principle of the manometric flame may also be used for 
 many other purposes : notably for observing the complexity of 
 the vowel sounds or of notes sung by the human voice, for 
 
 Fig. 25. 
 
 shewing interference of waves of sound, and for the general 
 composition of wave motions. 
 
 Figure 25 shews how it may be used for the examination of 
 the vowels sung according to any chosen pitch. 
 
 The manometric capsule, a section of which is at A, is 
 attached by means of a tube to a funnel-shaped mouthpiece for 
 collecting the waves of sound. The rotating mirror is shewn 
 at M. 
 
 Figure 26 explains the manner of comparing the vibrations of 
 two columns of air in organ pipes ; and, by connecting the two
 
 126 
 
 EXPERIMENTAL PHYSICS. 
 
 gas tubes leading from the capsules in the organ pipes with the 
 two ends of a T-shaped tube, a single gas jet can be obtained 
 which is actuated by the two sets of vibrations simultaneously. 
 A number of these combinational vibrations are represented 
 
 Fig. 26. 
 
 below: they have been obtained in the manner indicated, and 
 correspond to the intervals of the major diatonic scale. 
 
 The manometric flames may also be employed in conjunction 
 with the resonators of tuning forks. In most of the resonators 
 supplied by Konig with his forks, there is a small brass tube at 
 the back provided with a stopper, the use of which is to con-
 
 MANOMETRIC FLAMES. 
 
 127 
 
 nect the resonator with a manometric flame by means of a long 
 rubber tube and thus observe the vibrations of the air within. 
 Any number of forks may thus be used and the composition 
 
 Fig. 27. 
 
 of their vibrations studied by connecting their resonators all 
 to a common gas jet. 
 
 If forks forming a harmonic scale be chosen, one may com-
 
 128 EXPERIMENTAL PHYSICS. 
 
 bine them optically and see the effect which is obtained by the 
 ear with the apparatus of Helmholtz. 
 
 Perhaps the most striking experiment is that of Konig for 
 
 Fig. 28. 
 
 shewing interference. The apparatus shewn above is constructed 
 in accordance with the principle introduced by Herschel
 
 VELOCITY OF SOUND IN AIR. 
 
 I2 9 
 
 A tuning fork sends vibrations, reinforced by a resonator, to a 
 tube which, between its extremities, is divided into two branches, 
 the length of one of which can, by being drawn out, be altered 
 at will. The vibrations are sent to two independent gas jets, 
 after having travelled different dis- 
 tances, and a third gas jet enables 
 one to see the effect of compound- 
 ing the two sets of waves. The 
 manometric capsules are inserted 
 between the gas jets and the tubes. 
 
 The instrument admits of perfect 
 adjustment, and when complete 
 interference of the two wave mo- Fi g. 29. 
 
 tions takes place the appearance 
 
 of the three flames, as seen in a rotating mirror, is as shewn 
 in Fig. 29. 
 
 XIV. VELOCITY OF SOUND IN AIR. 
 
 The velocity with which sound is propagated through any 
 medium will evidently depend on the elasticity and density of 
 this medium, and theoretically it may be calculated from the 
 formula 
 
 where E is the coefficient of elasticity and D the density. 
 
 In solids E and D can be found experimentally, and the 
 determination of the velocity presents no difficulty. 
 
 For liquids the above formula becomes 
 
 where p is the density of mercury, H the height of the normal 
 barometer, K the coefficient of compressibility, and D the den- 
 sity of the liquid at the observed temperature. But in the case
 
 130 EXPERIMENTAL PHYSICS. 
 
 of air or any gas an experimental method of finding V directly 
 is preferable on account of the difficulty of determining accu- 
 rately the heat constants which are involved in E. 
 
 A method that at first sight appears simple is the direct one 
 of firing a gun at a known distance from an observer, who esti- 
 mates in some way the time which elapses between the percep- 
 tion of light and that of the sound ; but the personal errors in 
 the appreciation of this interval seriously interfere with the 
 accuracy of the result, and the method at best is but approxi- 
 mate. 
 
 Other methods are given here to illustrate the various ways 
 in which the velocity of sound in air may be measured ; modi- 
 fications in some of them would, no doubt, insure greater accu- 
 racy. 
 
 (i) By measurement of a wave length. 
 
 We know that when a periodic disturbance, for example that 
 from a vibrating fork, is propagated through the air, any parti- 
 cle of this air is in the same state of vibration at successive 
 times separated by an interval T which is therefore called the 
 periodic time and is equal to , where N is the pitch in double 
 
 vibrations ; and the wave length is defined to be the distance 
 which the wave form travels in this time T. 
 Hence we have 
 
 X = VT, 
 
 if V be the velocity of propagation and X the wave length. If 
 then we take a tuning fork and measure X in any way, we can 
 calculate V. 
 
 With forks of very low pitch and consequently of great wave 
 length this can be done with some degree of exactness by aid 
 of the ear, either unassisted or with a resonator. The fork is 
 put in vibration in front of a reflecting surface, and the observer 
 chooses points in the surrounding medium where maximum or 
 minimum effects are produced on the ear. A measurement of 
 the distance between successive maxima or minima gives half
 
 VELOCITY OF SOUND IN AIR. 131 
 
 the wave length ; and the pitch being generally given accurately 
 on the fork, we have 
 
 V= \N. 
 
 An improvement on this method might be made by using reso- 
 nators in connection with manometric flames. 
 
 The most accurate way perhaps to estimate the wave length 
 of a fork is to use the apparatus shewn in Fig. 28, and, after 
 obtaining complete interference, to measure the wave length by 
 means of the scale attached to the apparatus. 
 
 Assuming the pitch of the fork, the velocity of sound can 
 then be inferred. The same apparatus may also be used for 
 comparing the velocities of sound in different gases. 
 
 (2) By resonance. 
 
 If a hollow cylinder, open at one end, be fitted at the other 
 with a tight plug and piston rod which can be moved backwards 
 or forwards, and a tuning fork be put in vibration and held with 
 the prongs over the open end, it will be found that, by adjusting 
 the movable plug, a point is reached where the air column 
 within the cylinder vibrates in sympathy with the fork, and acts 
 as a resonator. In this case, since X = VT, 
 
 where / is the length of the vibrating air column and N the 
 pitch of the fork in double vibrations. 
 
 The method is not exact, a considerable error arising from the 
 size of the open end of the cylinder. Rayleigk's correction for 
 this gives 
 
 F= 4 (/+r) N 
 
 where r is the radius of the tube supposed cylindrical. If the 
 tube be of sufficient length, it is evident that a series of maxi- 
 mum reinforcements may be found, and then 
 
 
 2 + I 
 
 where n is zero or any integer.
 
 132 
 
 EXPERIMENTAL PHYSICS. 
 
 The adjoining figure shews how the experiment may be tried. 
 A tube AB a few feet in length is attached by means of a clamp 
 at B to another tube with a reservoir R, and is provided with 
 two stopcocks /, /, for the regulation of the supply of water 
 from the reservoir. Adjustment is then made until the distance 
 AL gives perfect resonance. 
 (3) Basse/la's method. 
 
 This method, although not more exact than the preceding, 
 involves a new idea, that of noting by ear the coincidence of 
 
 two sounds. The apparatus, de- 
 vised by Konig, consists of two 
 boxes provided with sounders 
 and electro-magnets, which are 
 inserted in an electrical circuit 
 with a mercurial interrupter, 
 whose period can be regulated 
 by means of adjustable masses. 
 The axis of the interrupter car- 
 ries a mirror, and a fixed tuning 
 fork of 40 v.cl. also carrying a 
 mirror is so situated that the 
 image of a silvered bead is seen 
 by reflection from the two mir- 
 rors. The fork and interrupter 
 vibrate in perpendicular planes ; 
 and the time of interruption is 
 found from the figure given by the image of the bead. It is 
 usually made one tenth, or one eighth of a second, the corre- 
 sponding figures having then four or five loops. When the 
 adjustment to get the correct time of interruption has been 
 made the sounders are set in motion, and of course give 
 simultaneous raps when heard by an ear near them ; one of 
 them is then carried away by the observer, who then hears 
 the raps coming alternately and then together again. He 
 notes the distance at which they appear coincident : and this 
 
 v 
 
 Fig. 30.
 
 VELOCITY OF SOUND IN AIR. 133 
 
 distance, divided by the period of interruption, must give the 
 velocity of sound. 
 
 If then V= velocity of sound in air at zero, 
 t mean temperature of the air, 
 T= period of interruption, 
 d = distance of coincidence, 
 d 
 
 then 
 
 where = . 003665, and / is the mean temperature of the air. 
 By taking two or three coincidences and varying the period of 
 interruption fairly good results may be obtained ; but the error 
 in estimating a single coincidence by the ear, however skilled it 
 may be, may amount to as. much as one foot : so that the velocity 
 is liable to be in error eight or ten feet per second. 
 
 (4) Kundi 's method. 
 
 This enables one to compare velocities in gases with one 
 another ; and also the velocity in air or any gas with that in a 
 solid. It is susceptible of great accuracy and the apparatus is 
 simple and easily arranged. 
 
 A glass tube AB rests freely on two supports, and at one end 
 is a movable plunger C, and at the other end a cork B t through 
 
 Fig. 31. 
 
 which passes a metal rod DF, fixed at its center E, and having 
 at D a small circular disc which fits the tube loosely. If DFbc 
 put in longitudinal vibration and the length CD properly ad- 
 justed by the plunger, it will be found that nodes and loops 
 are formed in the air inside ; and if lycopodium be strewn lightly
 
 134 
 
 EXPERIMENTAL PHYSICS. 
 
 within the tube it will collect and form little heaps where the 
 vibrating air is at rest ; and it is evident that the rate of propa- 
 gation of sound in air is to that in the metal DFas the distance 
 between two loops (or which is the same thing, two nodes) is to 
 
 the length of the rod. If 
 then v is the velocity of 
 sound in air, V that in the 
 metal, / the distance be- 
 tween the little loops of 
 powder, L the length of the 
 rod, 
 
 By placing a similar tube 
 in connection with the other 
 end F, and using another 
 gas in it, the velocities of 
 sound in air and in the other 
 gas will evidently be to one 
 another as the distances be- 
 tween the respective nodal 
 points. 
 
 (5) Regnaulfs method. 
 By far the most accurate 
 method is that devised by 
 Regnault, who determined 
 by direct measurement of 
 distance and time the ve- 
 locity of propagation of 
 
 sound in tubes of various diameters ; and the measurement of 
 time was made independent of the observer, thus eliminating 
 the personal error. A modified form of the apparatus used by 
 him, which enables one to perform the experiment in a limited 
 space, is shewn in Fig. 32. 
 
 Fig. 32.
 
 DOPPLER'S PRINCIPLE. 135 
 
 A series of tubes is arranged, forming one long tube, and an 
 electro-magnetic arrangement at one end is attached to a pistol 
 which, when discharged, produces a disturbance that travels to 
 the other end and there causes a mark to be made on some re- 
 cording apparatus, or can, if necessary, be reflected, return and 
 again travel to the other end, and so on. The pistol is also 
 connected with the recording apparatus, which may be either a 
 cylinder with tuning-fork attachment or the chronograph of 
 Exp. 8. 
 
 The distance along the axis of the tube can easily be 
 measured : in the apparatus constructed by Konig it is about 
 30 meters. 
 
 Besides being used for measuring the velocity of sound, this 
 instrument enables one to investigate all the phenomena of 
 reflected sound waves by attaching at the reflecting end tubes 
 of various lengths, which are put in communication with mano- 
 metric flames. 
 
 XV. DOPPLER'S PRINCIPLE. 
 
 When a source of sound has a motion of translation an 
 observer who listens will hear the sound gradually changing in 
 pitch. This may very often be noticed in the case of a railway 
 locomotive, which whistles as it is coming into a station : to an 
 observer at the station the pitch of the note seems gradually to 
 rise. 
 
 If V be the velocity of propagation, v the velocity of trans- 
 lation, and \, \' the wave lengths of the two notes, then 
 
 X'_ Vv 
 \~ V 
 
 according as v is in the same or opposite direction. One of the 
 simplest ways to study this is by means of two tuning forks of 
 rather high pitch (512 and 520 v.s.), which give four beats per 
 second when sounded together ; if an observer stations himself
 
 136 
 
 EXPERIMENTAL PHYSICS. 
 
 at a distance of thirty or forty feet, and an assistant carries one 
 fork towards him, it will be found that the number of beats 
 alters ; if the higher fork be carried at the rate of about two 
 feet per second, then nearly five beats a second will be heard ; 
 if the lower fork, then about three. 
 
 This method might be made accurate by 
 having the tuning fork slide along, at a 
 uniform rate, a board twenty or thirty feet 
 long, by means of a pulley and weights, 
 which could be adjusted to give the proper 
 speed. 
 
 The experiment can also be easily tried 
 by rotating a tuning fork, or by placing it 
 on the slide of a slowly moving horizontal 
 engine. 
 
 The apparatus of Mac/i, to illustrate this 
 principle, consists of a hollow brass tube 
 with reed attachment at one end or at both 
 ends. This can be rotated while the reed is 
 vibrated through a side tube, as shewn in 
 Fig- 33> ar) d an observer stationed in front 
 hears only one note, while in the plane of 
 Fl? ' 33 ' rotation he hears the alternate increase and 
 
 diminution in pitch. The instrument in this state, however, 
 hardly admits of giving measurements.
 
 HEAT. 
 
 LIST OF EXPERIMENTS. 
 
 I. Determination of the Zero and Boiling-point of a Ther- 
 mometer ......... 139 
 
 II. Calibration of a Tube ........ 140 
 
 III. Coefficient of Expansion of Mercury ..... 142 
 
 IV. Weight Thermometer. Cubical Expansion of a Glass En- 
 
 velope . . . . . . . . . . 143 
 
 V. Coefficient of Expansion of Dry Air at Constant Volume . 146 
 
 VI. Coefficient of Expansion of Dry Air at Constant Pressure . 149 
 
 VII. Coefficient of Expansion of Dry Air. Third Method . 151 
 
 VIII. The Air Thermometer 153 
 
 IX. Favre and Silbermann's Calorimeter . . . . . 155 
 
 X. Latent Heat of Steam. Regnault's Method . . . 156 
 
 XI. Latent Heat of Steam. Berthelot's Method . .159 
 
 XII. Latent Heat of Vaporization of Volatile Liquids . . 160 
 
 XIII. Weight of a Liter of Dry Air at Normal Pressure and Tem- 
 
 perature ......... 161 
 
 XIV. Determination of the Hygrometric State of the Air . . 164 
 
 (0) The Chemical Hygrometer. 
 () Condensation Hygrometers. 
 
 (1) Regnault's. 
 
 (2) Alluard's. 
 
 (c} Regnault's Experiment on Saturated Vapour of Water. 
 137
 
 138 EXPERIMENTAL PHYSICS. 
 
 XV. Density of a Vapour. Dumas' Method . . . . 1 70 
 
 XVI. Coefficient of Expansion of Metals 173 
 
 XVII. Specific Heat of Dry Air at Constant Pressure . . . 174 
 
 XVIII. Pressure of Vapours for Low Temperatures . . . 176 
 
 (1) Between Zero and 100 C. 
 
 (2) Below Zero. 
 
 XIX. Regnault's Apparatus for the Determination of the Pres- 
 sure of Vapours at High Temperatures . . . 178
 
 EXPERIMENTS. 
 
 I. DETERMINATION OF THE ZERO AND BOILING-POINT OF 
 A THERMOMETER. 
 
 To find the zero point of a thermometer, any small cylindri- 
 cal vessel is filled with pieces of clean ice, and the thermometer 
 inserted so that the zero marked on the stem is just visible 
 over the ice. Holes made in the bottom of the vessel allow 
 the water to drain off, and the thermometer is left in until the 
 column of mercury becomes stationary. This takes about half 
 an hour. The position of the column is then noted and gives 
 the error of the zero marked on the stem. 
 
 Figure i shews the arrangement for determining the boiling- 
 point at a given pressure. The thermometer is placed in a 
 vessel in which water is boiling, but so 
 that the bulb does not touch the water. 
 The steam circulates around it, and 
 passes, as shewn by the arrows, through 
 an outer jacket into the air, thus insur- 
 ing that the inner temperature will be, 
 that of the steam. A small water mano- 
 meter may be placed at M to shew any 
 small difference in pressure between the 
 steam inside and the air outside. When 
 the mercurial column has become station- 
 
 Figr. 1. 
 
 ary, its position is noted and at the same time the barometer 
 is read and reduced to zero. The true temperature of the 
 steam corresponding to the reduced height is obtained from 
 the Tables, and the difference between this and the observed 
 
 139
 
 140 EXPERIMENTAL PHYSICS. 
 
 reading on the stem gives the error of the boiling-point at the 
 pressure noted. 
 
 Thus the true temperature, corresponding to any reading of 
 the thermometer, can easily be found. 
 
 In most thermometers sent out by reliable makers, the zero 
 point is generally correct within a tenth of a degree, though 
 changing very slightly from time to time; while the boiling- 
 point may be in error as much as half a degree, and changes 
 every time the thermometer is heated up to or above 100 C. 
 
 In connection with the reading of temperatures it may be 
 noticed that, in a great many experiments where the thermome- 
 ter is used, it is a difference of temperature that is required : 
 in this case, an ordinary thermometer may be used throughout 
 the experiment, as its errors to a large extent disappear. This 
 is notably the case in finding specific and latent heats. And 
 when two or more thermometers are used for different measure- 
 ments in the same experiment, it is evident they should be com- 
 pared with one another before and after the experiment. 
 
 II. CALIBRATION OF A TUBE. 
 
 To examine the bore of a thermometer, the apparatus shewn 
 in the adjoining figure may be used. 
 
 A thread of mercury of any desired length is detached from 
 
 Fig. 2. 
 
 the column, and the spaces it occupies at various points in the 
 tube noted. Usually, however, before the thermometer is made,
 
 CALIBRATION OF A TUBE. 141 
 
 a number of capillary tubes are examined by means of this 
 instrument, and only a tube of uniform bore chosen for the 
 manufacture of the thermometer. 
 
 To examine the bore of a capillary tube, open at both ends, 
 a small thread of mercury is introduced by attaching a piece of 
 rubber tubing K to one end, and either allowing the mercury to 
 run in, or else to be drawn up by suction. 
 
 The tube is then placed under the microscopes M, N, which 
 are supported on two movable stands C, D, alongside of a scale 
 graduated into millimeters ; and the lengths occupied by the 
 thread in different positions are then measured, by means of 
 this scale, and the micrometer glasses in M and N. 
 
 A curve may be constructed, of which the ordinates represent 
 the lengths occupied by the same thread, and the abscissae are 
 the distances of one end of the thread from some fixed mark 
 on the tube, which may be taken as the origin of coordinates. 
 
 Most fine tubes will be found slightly conical in bore. 
 
 A tube of rather large bore is calibrated by simply pouring 
 into it equal volumes of mercury determined by weighing, and 
 making corresponding marks on the tube. 
 
 In many cases, one may wish to calibrate a large tube which 
 has already divisions on it indicating equal distances. The 
 readiest and most accurate way to do this is to fill the tube with 
 mercury to a known mark ; and then, allowing the mercury to 
 flow out for a small number of divisions, weigh this quantity ; 
 again allow the mercury to flow out for a small number of divis- 
 ions (not necessarily the same as before), weigh, and repeat the 
 operation as far as is desired. Then a curve can be constructed 
 which will enable one to read off by interpolation the volume 
 corresponding to any division ; the fixed mark being the origin 
 of co-ordinates, the numbers of divisions the abscissae, and the 
 volumes the ordinates. If necessary, the fixed mark may be 
 chosen as one end of the tube, which will then be completely 
 filled with mercury at the first operation.
 
 I 4 2 
 
 EXPERIMENTAL PHYSICS. 
 
 III. COEFFICIENT OF EXPANSION OF MERCURY. 
 
 The most important constant, and one upon which nearly all 
 others are ultimately based, in experiments with heat, is the 
 coefficient of expansion of mercury. By this is meant the 
 fraction which represents the volumetric increase in a quantity 
 of mercury for each increase of temperature of i C. 
 
 The method used is that based upon a principle suggested by 
 Boyle, and devised by Dulong and Petit. If a bent tube, such 
 
 as is shewn in Fig. 3, con- 
 
 _ a z? 
 
 tain in its two arms fluids 
 of different densities, the 
 weight of a cylindrical col- 
 umn of fluid in one arm 
 standing upon any base 
 is equal to the weight of 
 a cylindrical column in the 
 other arm standing on an 
 equal base in the same 
 horizontal plane. 
 c J) By filling the tube with 
 
 Fi - 3 - mercury, and surrounding 
 
 A C and BD with envelopes 
 
 so that AC may be cooled to zero and BD heated to the temper- 
 ature of steam, we obviously can calculate the coefficient of 
 expansion by observing the heights of H Q and H t from the com- 
 mon level below. For, conceive two small cylindrical columns 
 standing on equal bases, whose heights from CD are measured. 
 The mass of these two columns being the same, since their 
 weights are the same, their volumes are simply to one another 
 as their heights, since they stand upon equal bases. 
 Let HQ= height of cold column. 
 H t = height of hot column. 
 V Q = volume of small cylindrical column in AC. 
 V t = volume of small cylindrical column in BD. 
 f= difference in temperature.
 
 WEIGHT THERMOMETER. 143 
 
 Then 
 
 H Q _H t 
 
 V VI 
 H Q _ V n 
 
 Z! rtfi+JB)' 
 
 since the relation F= V Q (i+Kt] determines the coefficient of 
 expansion K. 
 
 H, -ff a 
 
 The apparatus used commonly for the determination of this 
 constant is made by surrounding AC and BD with two brass 
 jackets, one of which holds ice, and the other has a tube leading 
 into it which supplies steam from an adjoining boiler. The 
 measurements must not be taken until the temperature of the 
 hot column, as indicated by a very accurate thermometer, is 
 steady. The ice must be frequently renewed, as it melts 
 rapidly. 
 
 The greatest difficulty in the ordinary form of the apparatus 
 is to determine accurately the mean temperature of the heated 
 column. Measurements of heights must be made as exact as 
 possible ; a small error means a great deal when the experiment 
 is performed with ice and steam, the difference in level being 
 in that case not quite a centimeter. Hot air or boiling oil may 
 be used instead of steam ; but the latter is exceedingly dirty 
 and gives off disagreeable odours : it has the advantage, how- 
 ever, of giving a greater range of temperature. 
 
 IV. WEIGHT THERMOMETER: CUBICAL EXPANSION OF A 
 GLASS ENVELOPE. 
 
 The absolute expansion of mercury being assumed, the volu- 
 metric expansion of a glass envelope can easily be determined 
 by filling it with mercury at zero, and then raising it to some 
 known temperature, when a certain quantity of mercury es-
 
 144 EXPERIMENTAL PHYSICS. 
 
 capes, representing the expansion of the enclosed mercury less 
 the expansion of the envelope. 
 
 Theory. 
 
 Let W Q = weight of mercury filling the envelope at zero. 
 W t = weight of mercury filling the envelope at temperature t. 
 
 k= coefficient of expansion of mercury. 
 
 S= coefficient of expansion of glass. 
 Then, weights being proportional to volumes, we have 
 
 For, the left-hand side of this relation represents the internal 
 capacity of the envelope at temperature /; and W t represents 
 a quantity of mercury such that, if allowed to expand from zero 
 to /, it would just fill the envelope at that temperature. 
 
 From this & is found, all the other quantities being known. 
 
 Experiment. To perform the experiment, any arrangement 
 will suffice whereby a glass vessel can be completely filled with 
 
 mercury at zero, and then 
 heated to some known 
 temperature ; but the fol- 
 lowing apparatus will be 
 found very convenient. 
 
 A glass envelope, with 
 a capillary tube attached 
 and bent twice, is placed 
 in a receptacle as shewn in 
 the figure below. 
 
 The end of the capillary 
 tube dips into a small iron 
 vessel containing mercury, 
 and the bulb is prevented 
 from breakage by being 
 Fi g. 4 . surrounded with a small
 
 WEIGHT THERMOMETER. 145 
 
 iron basket ; the whole being supported in a circular brass plate 
 with a handle attached. The outer vessel is of thick copper, 
 and, when heated from below with a gas-burner, the air inside 
 the bulb expands and bubbles up through the mercury in the 
 iron cup. Then it is cooled by lifting out bodily the plate by 
 the handle, and exposing to the cool air. This causes the air 
 in the bulb to contract, and some mercury is then drawn in ; 
 and, by alternately heating and cooling, the bulb is finally filled 
 with mercury. If one wishes to perform the experiment more 
 rapidly, the outer copper vessel may be removed, and the basket 
 holding the bulb heated directly with a spirit lamp : this, how- 
 ever, must be done with care, as there is danger of breakage. 
 
 Then it is surrounded with ice and cooled to zero ; and, 
 finally, water being put in the copper vessel, it is heated to the 
 temperature of steam. 
 
 Three weighings are necessary : 
 
 1. Weight of the envelope when empty and perfectly clean 
 and dry. 
 
 2. Weight of the envelope after heating to the temperature 
 of steam and allowing to cool properly. 
 
 3. Weight of mercury which escapes between zero and tem- 
 perature /. 
 
 These weighings give W Q and W t . 
 
 Precautions. Heat slowly, and boil the mercury in the bulb 
 to remove air and moisture. Cool slowly to avoid breakage. 
 The experiment throughout must be performed slowly and 
 carefully to insure success. 
 
 It is evident that the envelope, when once filled with mer- 
 cury, and its elements determined, may serve as a weight ther- 
 mometer. For, to find the temperature at any point where it 
 may be placed, it is sufficient to fill it at zero and then bring 
 it to the place in question, and notice how much mercury 
 escapes. Then, as before, 
 
 and the quantities W Q , IV,, S, k being known, t is found.
 
 146 EXPERIMENTAL PHYSICS. 
 
 The weight thermometer is theoretically a perfect instru- 
 ment ; for, by its use, we determine temperatures in terms of 
 masses, which can always be measured with great accuracy ; the 
 great objection to its general use is the length and tediousness 
 of the process. 
 
 V. COEFFICIENT OF EXPANSION OF DRY AIR AT CON- 
 STANT VOLUME. 
 
 In this experiment dry air is taken as the type of a perfect 
 gas, and its expansion determined from the increase of pressure 
 necessary to keep its volume constant when its temperature is 
 raised. 
 
 The apparatus used is that devised by Regnault, and is repre- 
 sented in Fig. 5. 
 
 The air to be operated upon is inclosed in a glass bulb A, 
 which is placed in a boiler clamped to a stand at K. The water 
 can be run off from this boiler by a small stopcock below, and 
 replaced, when necessary, by ice, without disturbing the bulb. 
 A special perforated jacket of tin surrounds the bulb inside to 
 insure a constant temperature when steam is generated. A 
 capillary tube of copper, attached to the bulb, is connected 
 at m, n by means of a three-way stopcock, with a drying 
 apparatus H, and an exhausting pump P, and also with a 
 manometer of mercury, which is itself provided with a three- 
 way tap at the bottom of the tube BC. These three-way taps 
 have openings in the form of a T, and thereby enable one to 
 make communication between any two of three sources, or to shut 
 any two off, or all three ; the directions in which the openings 
 run are shewn on the outside by small lines. The drying tubes 
 contain pumicestone moistened with sulphuric acid. 
 
 The connections are all made so that no leakage takes place, 
 and the manometer tubes are filled with mercury until they both 
 stand at the same level a, everything being in communication 
 with the air. Then the three-way tap at the bottom of C is
 
 COEFFICIENT OF EXPANSION OF DRY AIR. 147 
 
 closed, so that the tube BC does not communicate with DE. 
 The process of drying the air then commences. Water is 
 placed in the boiler and steam generated, and then exhaustion 
 is made by P, through the drying tubes ; finally air is allowed 
 to enter by means of a central tap placed at the bottom of P 
 which makes direct connection with the air. This operation is 
 
 Fig. 5. 
 
 repeated a number of times, the water still boiling, and, at 
 length, when the air is dry, the tap at C is opened, and the 
 apparatus throughout brought to atmospheric pressure. Then 
 the hot water in the boiler is run off, replaced by ice, and the 
 air in A cooled to zero ; and when the ice has been in about 
 twenty minutes, the stopcock at n is turned so that A com-
 
 148 EXPERIMENTAL PHYSICS. 
 
 municates only with the manometer. The barometer is then 
 read, and the levels of the two columns of mercury in the 
 manometer (which should be the same) noted. 
 
 There is now a volume of dry air inclosed at zero, and at a 
 known pressure. The next operation is to heat this air to the 
 temperature of steam, and as it expands, driving before it the 
 column aC, to pour mercury in ED so that it keeps the air 
 always at constant volume, and the mercury in BC therefore 
 always at the mark a. This second operation will last about 
 fifteen minutes ; and when the column in DE remains station- 
 ary the difference in level in the two tubes is noted and the 
 barometer again read. 
 
 Theory. 
 
 Let V = volume of air inclosed in A at zero. 
 7' = volume between the marks ;/ and a. 
 H Q = barometric reading when air is at zero. 
 H barometric reading when air is heated. 
 // = difference in level in the two tubes at the end of the 
 
 experiment. 
 
 T= temperature of steam when h is read. 
 /= temperature of room. 
 
 8 = coefficient of expansion of the glass envelope. 
 = coefficient of expansion of dry air. 
 
 Then, by Boyle's law, 
 
 i+a/ l+aT l+ 
 
 where V Q (i + BT) represents the internal capacity of the glass 
 envelope at the temperature 7! The coefficient of expansion of 
 glass may be obtained from the tables ; but this is only approxi- 
 mate, as its value depends more on the form of the envelope than 
 on the quality of the glass, and for accurate purposes should be
 
 COEFFICIENT OF EXPANSION OF DRY AIR. 149 
 
 determined in each case. The capacity of the bulb at zero may 
 be found in the usual way and marked on the outside. 
 
 In the preceding formula a first approximation to the value of 
 a is found by neglecting v. This value is then used to reduce the 
 
 expression - in the second approximation. The experi- 
 i + at 
 
 ment may also be performed in the reverse order, by filling the 
 bulb with hot air at atmospheric pressure and then cooling it to 
 zero : in this case represents a contraction, and the formula 
 will be altered correspondingly. 
 
 Precautions. In surrounding the bulb with ice, after it has 
 been raised to the temperature of steam, care must be taken 
 not to place ice on the metal socket of the glass too suddenly. 
 It is better to run off the hot water, then cool the metal boiler 
 with cold water, and finally the bulb. The stopcock at C must 
 be closed when drying the air, otherwise the mercury will run 
 over into the drying tubes and bulb. When finished, the appa- 
 ratus should be opened to the air and the mercury in the tubes 
 let down to the same level. The use of the three-way tap at C 
 and at n must be fully understood. 
 
 VI. COEFFICIENT OF EXPANSION OF DRY AIR AT CON- 
 STANT PRESSURE. 
 
 The mode of operation in this experiment is similar to that in 
 the previous one, the only change being that when the dry air 
 is heated from zero to the temperature of steam the mercury is 
 allowed to flow out until the two columns are at the same level. 
 The preceding formula then applies, h being zero, and v on the 
 left-hand side becoming v' on the right-hand side. 
 
 The apparatus used is shown in Fig. 6. 
 
 The tubes BC, ED are inclosed in a water jacket to insure 
 constancy of temperature, the bulb A is placed as before in a 
 boiler (not shewn), and the connections at m and n lead by a 
 three-way tap to drying tubes and an air-pump.
 
 150 EXPERIMENTAL PHYSICS. 
 
 FJ 
 
 Theory. Fi e- 6 - 
 
 Let VQ = volume of bulb at zero. 
 
 v = small volume to a chosen mark. 
 
 v' volume when BC is lowered. 
 H Q = barometric pressure at zero. 
 H ' = barometric pressure at temperature of steam. 
 
 T= temperature of steam. 
 
 / = temperature of water. 
 
 g = coefficient of expansion of glass. 
 
 a = coefficient of expansion of air at constant pressure.
 
 COEFFICIENT OF EXPANSION OF DRY AIR. 
 
 Then, by Boyle's law, 
 . v 
 
 ~^H. 
 
 I + at 
 
 In this v' will be much greater than v\ both are found by filling 
 the tube BC with mercury to the tap n, and then allowing it to 
 run out to the marks which define v and v' ; the volumes are 
 then inferred from the weights of the quantities of mercury 
 which escape. First and second approximations are made as 
 before, and the general precautions are the same as in the 
 former experiment. 
 
 VII. COEFFICIENT OF EXPANSION OF DRY AIR. THIRD 
 METHOD. 
 
 In this experiment dry air is cooled from the temperature of 
 steam to zero, and both volume and pressure change. A small 
 cylindrical bulb of glass with 
 a bent capillary stem is used, 
 and filled with dry air by being 
 placed in the apparatus for 
 determining the boiling-point 
 of a thermometer (see Fig. i), 
 in connection with a drying 
 tube and an exhausting air- 
 pump. When filled with dry 
 air by continual exhaustion and 
 heating, the end is sealed up, 
 after detaching the drying- 
 tube a few seconds to allow it 
 to come to atmospheric pres- 
 sure. It is then placed, when 
 cool, in the apparatus below 
 (Fig. 7). 
 
 AB is the glass envelope inverted in a trough of mercury and 
 surrounded with melting ice placed in a vessel shewn in section
 
 152 EXPERIMENTAL PHYSICS. 
 
 by EE. D is a small outlet for the melted ice. At C is an 
 arm which can be moved up and down, and to this is attached a 
 small pipe-shaped piece of metal whose bowl is filled with wax ; 
 this can be pushed in against the end of the capillary tube, and 
 enables one to close it under the mercury. H is a screw which 
 can be adjusted to touch the surface of the mercury. 
 
 Order of the Experiment. i. Determine the coefficient of 
 expansion of the glass bulb in the same manner as given in 
 Exp. 4. 
 
 2. Place it in the apparatus shewn in Fig. i, and connect 
 with drying-tubes and an air-pump ; and when the air in it 
 is completely dry, allow it to come to atmospheric pressure by 
 separating it from the drying-tubes (while the steam is still 
 circulating around it) for a few seconds. Then seal the end 
 with a blow-pipe, at the same time noting the barometric 
 pressure. 
 
 3. Arrange as in Fig. 7, and break off the point underneath 
 the mercury. This is done most readily by making a small file 
 mark near the end before inverting it, and then with a pair of 
 pliers breaking it off under the mercury, at the same time hold- 
 ing firmly the stem at B. 
 
 The mercury then rises into the capillary tube, and on filling 
 EE with ice, it finally takes up a permanent position. Then 
 the screw H is moved down until its lower point touches the 
 surface of the mercury, and the bowl of soft wax is pressed in 
 against the end of the tube, sealing it up, and at the same instant 
 the barometer is read. The ice being next carefully removed, 
 and everything allowed to come to the temperature of the room, 
 the height from the top of //"to the top of the column of mer- 
 cury is measured with a cathetometer, and then, finally, the 
 length of the screw H. The bulb, is then removed, and the 
 mercury taken from it and weighed. 
 
 Theory. 
 
 Let V Q = volume of mercury filling the bulb at zero. 
 8= coefficient of expansion of the glass bulb.
 
 THE AIR THERMOMETER. 153 
 
 HQ = barometric reading when the end is sealed up under 
 
 the mercury. 
 H= barometric reading when sealed at the temperature 
 
 of steam. 
 
 T= temperature of steam. 
 h = height of column. 
 
 v= volume of mercury which runs into bulb when cooled. 
 a = coefficient of expansion of dry air. 
 
 Then, by Boyle's law, 
 
 ( r -v)(*f -/i)(i +aT)= F (i +&T)H, 
 
 from which a is determined. 
 
 Precautions. Care must be taken in determining V Q and 8. 
 Some special apparatus should be used for filling the bulb with 
 mercury, either by the application of heat or by means of a 
 good air-pump and a three-way tap. The air must be well 
 dried. In finding v by weighing the mercury, some difficulty 
 will be found in getting it out of the bulb ; it is well to weigh 
 the bulb previous to the experiment, and then weigh bulb and 
 mercury at the end of the experiment, checking, if possible, by 
 weighing separately the mercury. A small error is introduced 
 by the wax and the piece broken off ; but these can be allowed 
 for without any difficulty. The values of // , H, h are sup- 
 posed to be taken for some standard temperature, usually zero. 
 
 VIII. THE AIR THERMOMETER. 
 
 The principle of the air thermometer is that a perfect gas, 
 such as dry air, if kept at constant volume, will have its pressure 
 increased as its temperature increases : the constant of increase 
 being 0.003665 for each degree centigrade. Any of the preced- 
 ing instruments (Figs. 5, 6, 7) may be used as an air thermome- 
 ter, assuming the value of . The most convenient apparatus 
 is one similiar to that of Exp. 5. The air thermometer itself 
 is a glass bulb, or (for very high temperatures) a cylindrical
 
 154 EXPERIMENTAL PHYSICS. 
 
 or spherical reservoir of Bayeux porcelain, with a long capillary 
 tube of copper, so that it may be carried, if necessary, some 
 distance from the apparatus. This thermometer is connected 
 to a manometer, drying apparatus, and air-pump, just as in 
 the former experiments. Then, having obtained dry air, it is 
 cooled to zero, and the barometer read : the two columns of the 
 manometer being at the same level. Then the stopcock lead- 
 ing to the air-pump is closed, and the bulb is placed where the 
 temperature is desired, and as the air in it expands, the volume 
 is kept constant and the increase of pressure and barometric 
 reading taken. Then the formula of Exp. 5 applies : 
 
 } 
 
 V 
 
 I +aT I +a/ 
 
 where a is now assumed, and T, the unknown temperature, is 
 to be found. This process can be used to graduate a mercurial 
 thermometer : the reservoir of air is raised from zero to tempera- 
 tures gradually increasing, and the temperatures calculated 
 from the formula are compared with the indications of the 
 mercurial thermometer. 
 
 For rough calculations v may be neglected, when it is very 
 small, and also 8 ; then it is not necessary to know F" , and we 
 get the temperature from the simple relation 
 
 and 
 
 0.003665 
 
 The essential precaution in the use of the air thermometer 
 is to have the air perfectly dry: this is insured by the use 
 of a good Sprengel or Geissler pump in connection with a 
 reservoir of sulphuric acid. 
 
 The porcelain reservoir is used for the determination of 
 temperatures beyond that at which glass begins to melt.
 
 FAVRE AND SILBERMANN'S CALORIMETER. 
 
 155 
 
 IX. FAVRE AND SILBERMANN'S CALORIMETER. 
 
 This is in reality a large thermometer, consisting of a reservoir 
 A (Fig. 8), which contains about fifty pounds of mercury and 
 which has a graduated stem CD with a fine bore into which the 
 mercury of the reservoir when heated can expand. 
 
 A microscope L provided with a micrometer glass enables one 
 to estimate the divisions on CD to twentieths of a millimeter. 
 A special contrivance at B, with a handle, is used to put pressure 
 
 Fig. 8. 
 
 on the interior of the reservoir and move the thread of mercury 
 to any desired point ; and there are two or three openings in A 
 in which heated bodies may be placed and thereby give out 
 their heat to the mercury in the reservoir. 
 
 The instrument is first graduated by taking a known weight 
 of water at a known temperature, and inserting it, as shewn at 
 E, and noticing the number of divisions over which the thread 
 D moves, as the heat from this water is imparted to A.
 
 156 EXPERIMENTAL PHYSICS. 
 
 The theory of the instrument is simple. 
 Let M=mass of water introduced. 
 
 T= temperature when introduced. 
 
 0=;its temperature when the thread is stationary. 
 
 = number of divisions over which the thread has moved. 
 
 Then ' = -^- = value of one division in units of heat 
 
 To measure, then, a specific heat of a liquid, for example, 
 Let P=mass of liquid introduced. 
 
 T= its initial temperature. 
 
 0' = its final temperature. 
 
 x=its specific heat. 
 
 m number of divisions over which the thread moves. 
 
 Then Px(T-6') = mc, 
 
 from which x is found. 
 
 The instrument can be used for measuring specific heats of 
 liquids, heats of combustion, and even specific heats of solids. 
 Care only must be taken in determining the constant c, which 
 should be found from the mean of a number of observations. 
 
 X. LATENT HEAT OF STEAM. REGNAULT'S METHOD. 
 
 This is essentially a method of mixture, the apparatus for 
 performing the experiment being shewn in Fig. 9. 
 
 B is a boiler in which steam is generated either at atmos- 
 pheric or other pressures. The steam is led from this boiler 
 into a brass globe G placed in a calorimeter C containing a 
 known weight of water. The handle H when turned in a cer- 
 tain direction allows the steam to pass, by a peculiar arrange- 
 ment of two hollow tubes working within one another, into G ; 
 when turned in all other directions the steam escapes through
 
 LATENT HEAT OF STEAM. 
 
 157 
 
 an exhaust pipe E and condenses in a vessel below. The pipes 
 P, P are used when pressures greater than atmospheric are 
 needed ; they lead to a brass box, which can be connected with 
 a mercurial manometer, or gauge of any kind to indicate the 
 pressure at which the steam is generated. The steam passes 
 then into G, condenses, and gives out heat to the surrounding 
 
 .f 
 
 Fig. 9. 
 
 water in the calorimeter, which rises correspondingly in tem- 
 perature. After a certain time has elapsed, the steam is shut 
 off by turning H, and then, waiting a few minutes to allow the 
 condensed steam in G to come to its lowest temperature, the 
 tap T is opened and the water of condensation collected and 
 weighed or estimated by volume. The temperature of the 
 water in the calorimeter, at the beginning and end of the 
 experiment, is taken with a thermometer reading directly to 
 fifths or tenths. 
 
 Theory. 
 
 Let W= weight of water, together with the water equivalent 
 of the calorimeter. 
 
 /=its temperature. 
 
 w= weight of condensed steam. 
 
 T= temperature of the steam.
 
 158 EXPERIMENTAL PHYSICS. 
 
 T'= temperature of condensed steam when run out from 
 
 the calorimeter. 
 
 = final temperature of the water in the calorimeter. 
 x= latent heat of the steam. 
 
 Then, by the equivalence of heat lost and gained, 
 
 For the left-hand side of this relation represents the heat 
 absorbed by the water and calorimeter; and the right-hand side 
 is the heat given out by the steam in condensing, together with 
 the heat given out by the condensed steam in falling from the 
 boiling-point to its final temperature when run off. 
 
 Precautions. i. The greatest difficulty, in such an experi- 
 ment as this, is to determine accurately the water equivalent of 
 the calorimeter, which may either be found by an independent 
 experiment, using the method of mixture, or determined in 
 conjunction with the latent heat of steam by the method of 
 successive approximation. 
 
 2. The water in the calorimeter should be in the first place 
 as cold as possible. 
 
 3. The steam, in passing into the calorimeter, is made to heat 
 the water as much above the temperature of the room as it 
 previously was below. This is to equalize radiation and absorp- 
 tion. If the temperature of the room be above 18 C., then the 
 initial temperature of the water should not be more than 8 C. ; 
 otherwise the final temperature of the water will be so high that 
 rapid evaporation takes place from its surface. 
 
 4. A series of temperatures should be taken as the condensed 
 steam is running out, and the mean in some way accurately 
 found. 
 
 5. The water should be well stirred just before the steam is 
 let into the calorimeter, and its temperature carefully read. 
 Stirring should be kept up throughout the experiment, espe- 
 cially if the surface of the water is near the pipe which enters
 
 LATENT HEAT OF STEAM. 
 
 159 
 
 into G, for in that case the surface water becomes very hot and 
 gives off steam. 
 
 6. Care should be taken not to let the boiler run dry, and 
 when H is closed, to open the exhaust pipe E, or the outlets P, P. 
 
 XI. LATENT HEAT OF STEAM. BERTHELOT'S METHOD. 
 
 Although the preceding method can be applied to calculate 
 the latent heat of steam and other vapours at different pressures, 
 yet it necessitates large quantities of liquid, and the errors of 
 experiment are not easily corrected. Berthelot's arrangement 
 is more simple, avoiding thereby several sources of error; it 
 can be used for any volatile substance, requires but a small 
 quantity of liquid with which to operate, and gives accurate and 
 rapid results. 
 
 It consists of a calorimeter C (Fig. 10) of special construc- 
 tion, filled with water or other non-conducting material, and 
 covered with felt, and an inner 
 vessel c in which a known 
 quantity of water is placed. 
 
 The boiler of the preceding 
 experiment is replaced by a 
 bottle-shaped vessel G, in 
 which the liquid to be exam- 
 ined is placed ; and this, being 
 heated from below, sends va- 
 pour, as shewn by the arrows, 
 through a tube AD into a 
 receptacle B, where it con- 
 denses and gives out its heat ; 
 
 Fig. 10. 
 
 owing to the compact form of the apparatus, no heat is lost in 
 the passage from G to B. At D is a detachable clamp by 
 means of which one can separate the boiler from B ; and a 
 spiral tube generally runs from D to B, so that condensation is 
 more easily effected. The exit at E insures that vaporization
 
 i6o 
 
 EXPERIMENTAL PHYSICS. 
 
 takes place at atmospheric pressure. The boiler and receptacle, 
 with attached tubes, may be made of glass or metal ; if of glass, 
 then the instrument can be used for almost any volatile liquid, 
 and has the advantage of being transparent. 
 
 The water equivalent of the calorimeter in this case is easily 
 found by detaching B and weighing it ; and the heat from the 
 burner which is placed under G can be prevented from heating 
 the water in c by placing over the lid of the apparatus a layer 
 of asbestos. 
 
 The thermometer / should be graduated in tenths of degrees. 
 
 To perform the experiment, unfasten the clamp D, weigh B 
 and connecting tubes, and determine its water equivalent ; then 
 invert G, pour in the liquid to be heated, and arrange as in the 
 figure. The method being one of mixture, the calculation is 
 the same as in the preceding experiment. 
 
 XII. LATENT HEAT OF VAPORIZATION OF VOLATILE 
 LIQUIDS. 
 
 For liquids which evaporate rapidly at ordinary temperatures 
 
 a reverse process devised by Regnault may be used. 
 
 It consists in evaporating a quantity of the liquid at a low 
 
 pressure in a vessel surrounded with water, and from the 
 
 diminution in tempera- 
 ture of this water cal- 
 culating the heat nec- 
 essary for vaporization. 
 The apparatus used is 
 similar to that shewn in 
 Fig. ii. 
 
 The liquid to be evap- 
 orated is poured into a 
 calorimeter C through a 
 
 tube at O, which is then closed with a tight-fitting screw cap. 
 
 The calorimeter portion consists of three vessels : an inner
 
 WEIGHT OF A LITER OF DRY AIR. 161 
 
 closed one in which the liquid is placed, a second to hold 
 the water to be cooled, and an outer one to prevent direct 
 radiation or absorption of heat. A tube leads from the inner- 
 most closed vessel to another small calorimeter c, and then to 
 an air pump connected at A. The tube M is connected with 
 a pressure gauge or manometer. 
 
 A freezing mixture is generally placed in c, so that most of 
 the vapour produced by evaporation in C condenses, and can 
 be collected again by opening the tap t. A stirrer and ther- 
 mometer are also necessary. 
 
 Theory. 
 
 Let W= weight of water in the calorimeter. 
 w = water equivalent of the calorimeter. 
 T= initial temperature of the water. 
 
 t= final temperature of the water. 
 P = weight of liquid. 
 
 c = its specific heat. 
 
 x= latent heat of vaporization. 
 
 6 = temperature of vaporization. 
 
 Then 
 
 (W+w)(T-t)+Pc(T-0)=Px, 
 
 from which x is found. 
 
 In addition to the term Px there will be another term (due to 
 the vapour leaving the calorimeter becoming heated), which would 
 involve the specific heat of the vapour ; but it will generally be 
 very small. There is also a slight uncertainty in the determina- 
 tion of the temperature of ebullition (0). 
 
 XIII. WEIGHT OF A LITER OF DRY AIR AT NORMAL 
 PRESSURE AND TEMPERATURE. 
 
 The method employed is that of Regnault, in which two 
 hollow glass globes are blown as nearly as possible identical in 
 every respect, the capacity of each being about seven liters. One
 
 162 
 
 EXPERIMENTAL PHYSICS. 
 
 of these is sealed up permanently, and the other is provided 
 with a stopcock. They are suspended by hooks from the scale 
 pans of a large balance and are brought into a state of equi- 
 librium by the addition, if necessary, of small weights ; if a very 
 accurate experiment is required, they are next suspended in 
 water, and due allowance made for any difference in their exter- 
 nal volume shewn by the unequal upward pressures. 
 
 In this way one obtains two bodies which displace equal 
 amounts of air, whatever be its pressure, temperature, and 
 hygrometric state. 
 
 The globe provided with a stopcock is then removed and 
 placed in connection with an air pump and mercurial gauge, 
 and drying apparatus, as shewn in the adjoining figure. 
 
 G is the globe, which may be surrounded either with water 
 of a known temperature or, better, with ice. T' is the stop- 
 
 \M 
 
 Fig. 12. 
 
 cock, which when open makes communication with a three-way 
 tap /'attached to a stand 5. At A is an outlet leading to a 
 drying apparatus, gauge, and air pump. 
 
 When T' and T are so adjusted that the globe G communi- 
 cates with A, it may be filled with perfectly dry air at a known 
 temperature, usually zero : it is then left in communication with 
 the air for a few seconds and the barometric reading taken. 
 
 After assuming the temperature of the room, it is again put
 
 WEIGHT OF A LITER OF DRY AIR. 163 
 
 in the scale pan, and equilibrium established with the compen- 
 sating globe and small weights. Once more it is removed, 
 placed in connection with the air pump and gauge, and exhausted 
 as completely as possible ; and the indication of the gauge along 
 with the barometric reading taken. Finally, after closing T' t 
 it is placed on the scale pan, and equilibrium restored by the 
 addition of certain new weights, which must represent the 
 weight of the dry air removed from the globe. 
 
 Then it only remains to determine the internal capacity of G 
 at zero, which is done by taking the exhausted globe and plac- 
 ing it, by means of T, in connection with the outlet M, without 
 opening T 1 . A tube being now led from M into a vessel of 
 water, T' is opened, and the water rushes from M into G and 
 almost completely fills it. The opening in the top of the globe 
 is usually so small that it would take some time to fill it in any 
 other way. 
 
 Theory. 
 
 Let V Q = internal capacity of the globe at zero. 
 
 // = mean of the two barometric readings taken. 
 h = reading of the gauge when the globe is exhausted. 
 P = weight necessary in the final weighing to produce 
 equilibrium. 
 
 Then, by Boyle's law, the volume of air removed is 
 v H-h 
 
 V ~IT' 
 
 and therefore the weight of a liter of air at o C. and 760 mm. is 
 
 F H-h 
 
 Precaution. Care must be taken, when determining the 
 capacity of the glass globe, not to allow the water, by any mis- 
 take in turning T, to get into the air pump through A. 
 
 It is evident that the above method can be readily applied to 
 the determination of the densities of gases.
 
 1 64 EXPERIMENTAL PHYSICS. 
 
 XIV. DETERMINATION OF THE HYGROMETRIC STATE 
 OF THE AIR. 
 
 The hygrometric state of the air will be known when we 
 know the quantity of water vapour in a given volume of air, 
 and also the pressure which this vapour exerts ; but the deter- 
 mination of either of these, although simple in theory, is 
 attended with great practical difficulties. 
 
 The hygrometric coefficients may be found by 
 
 (a) The chemical hygrometer. 
 
 This consists simply in passing a known volume of air 
 through drying tubes which are weighed before and after the 
 experiment. 
 
 The weight of vapour in a given quantity of air is thus found, 
 and if we wish the pressure which this vapour exerts, we may 
 find it in the following way. 
 
 Let w = weight of water in grains. 
 ?/=its volume in liters. 
 /=its temperature. 
 ;r=its pressure. 
 
 Then, since i liter of air at o C. and 760 mm. pressure 
 weighs 1.293 grams, and since water vapour is .622 times 
 denser than air, we have 
 
 _vxx 1.293 x .622 
 (1+^)760 
 
 where 
 And 
 
 Conversely, if we know x, we can find w. 
 
 If we require to find the weight of a given volume of moist 
 air, 
 
 Let V= volume in cc. 
 
 H = barometric pressure.
 
 HYGROMETRIC STATE OF THE AIR. 
 
 I6 5 
 
 f pressure of vapour of water in it, found from the dew- 
 point and the tables. 
 t temperature of air. 
 = . 003665. 
 
 Then the weight of dry air in V 
 
 Fx.ooi2Q3 x 
 
 And the weight of vapour 
 = .622 Fx. 001293 x 
 
 Therefore the weight required 
 _Fx.ooi293 tZ7 
 
 760 
 
 Fx " 1293 (#-f/). 
 760(1+ /) V 
 
 To perform the experiment (a), a set of drying tubes is 
 attached to an aspirator (shewn in section in Fig. 13), consist- 
 ing of two reservoirs R, R, 
 capable of holding water, 
 and at the same time of 
 rotating about a fixed 
 horizontal axis AB. By 
 means of the tubes C, D, 
 the water falling from the 
 upper reservoir into the 
 lower drags air through the 
 opening at A and forces it 
 through the exit E, as in 
 the figure. It will also be 
 seen that when the lower 
 reservoir is full, the instru- 
 ment can be inverted, and 
 the same process repeated 
 without disturbing any con- 
 nections made at A or E. 
 
 Any required volume of air 
 
 Fig. 13.
 
 166 EXPERIMENTAL PHYSICS. 
 
 can then be operated upon in a short time. The drying tubes 
 are, of course, connected at A. 
 
 The volume of air dragged through the aspirator at one 
 operation may be taken approximately to be the capacity of 
 the reservoir, which is supposed to be initially full of water. 
 If a very accurate experiment is required, account will have 
 to be taken of the change in pressure owing to the final air 
 in the aspirator being saturated with water vapour and at a 
 different temperature from that which came to the drying 
 tubes. 
 
 It is preferable to have three drying tubes, two for the 
 absorption of the moisture from the air upon which one 
 operates ; and the third, placed next the aspirator, to absorb 
 any moisture which may chance to get back from the aspira- 
 tor, which is filled with water : the two outside ones are then 
 weighed before and after the experiment. 
 
 Tubes filled with calcium chloride or pumice stone and 
 sulphuric acid may be used, and instead of making them air- 
 tight with paraffin or wax, it is better that they should be 
 provided with perforated glass stoppers : they can then be 
 opened or closed at will. 
 
 (b) Condensation hygrometers. 
 
 The determination of the hygrometric state of the air by 
 chemical means is slow, and does not give accurate informa- 
 tion of the variations in humidity during short intervals of time: 
 it is really a mean of the states which the air had during the 
 time of the experiment. More minute and rapid information 
 is obtained by the hygrometers of condensation, which are 
 arrangements whereby the temperature inside a vessel is so 
 lowered that dew appears on the outside owing to condensation 
 of moisture from air in its vicinity. A glass of ice water on 
 a hot day is the simplest form of hygrometer ; and the forma- 
 tion of dew on the earth when it loses its heat by radiation 
 into space shews how the earth itself, under favourable cir- 
 cumstances, becomes a huge hygrometer.
 
 HYGROMETRIC STATE OF THE AIR. 
 
 I6 7 
 
 I. Regnault's hygrometer. 
 
 In this instrument, which is shewn in section in Fig. 13 (a), 
 there are two small tubes A and B, the upper portions being 
 of glass, the lower, caps of silver or silvered copper : these are 
 attached to a stand so as to be near one another. In A is 
 placed a cork with a thermometer T' ; in B a cork with a ther- 
 mometer T and two glass 
 tubes C, D. When in use B 
 is partially filled with alcohol 
 or ether, and the tube C con- 
 nected with an aspirator ; or 
 else air is forced through the 
 tube D. The evaporation of 
 the alcohol or of the ether thus 
 produced lowers the tempera- 
 ture of the metal cap, and dew 
 is seen to form on the out- 
 side. 
 
 point. At the same instant T' is read, and the two tempera- 
 tures T, T' determine the humidity. 
 
 Fig. 13 (). 
 
 The temperature then shewn by T is called the Dew- 
 
 Let / = weight of moisture in a given volume of the surround- 
 
 ing air. 
 P = weight of moisture in the same volume of the surround- 
 
 ing air, supposed to be saturated. 
 f= pressure of moisture in/. 
 F= pressure of moisture in P. 
 
 Then by what has already been shewn, 
 
 P F 
 
 Now from Regnault's tables of vapour tension, / and F are 
 known when we know T and T'. Hence the relative humidity is 
 at once given by these two temperatures. Usually, for practical
 
 1 68 
 
 EXPERIMENTAL PHYSICS. 
 
 purposes, this ratio - is expressed in terms of a unit of satu- 
 ration taken to be i or 100. Thus, if T=6 C. and T' = 15 C., 
 the humidity would be .55, or 55. 
 
 Precautions. In finding the dew-point, the aspirator is 
 adjusted so that the dew appears and disappears within small 
 limits of temperature, and a mean taken. The two caps should 
 present the same appearance and be placed in a suitable light. 
 The thermometers may be read by a telescope placed at a 
 distance. In warm weather alcohol may be used in B ; but in 
 winter, when the dew-point sometimes 
 falls below freezing point, ether must be 
 used. 
 
 2. AlluarcTs hygrometer. 
 This is an improved form of Regnault's, 
 in which the two metal caps are replaced 
 by a brass box A plated with gold, and 
 a piece of the same material B (Fig. 14) 
 placed near it, but not in contact, so that 
 a comparison can be readily made between 
 the two surfaces, and the presence of dew 
 on A at once detected. Inserted in the 
 box A is a thermometer t, which can be 
 read if necessary through the glass win- 
 dow in A. Another thermometer t' gives 
 the temperature of the surrounding air. 
 The liquid to be evaporated is poured 
 in at E, which is provided with a cork 
 to prevent evaporation, and two metal 
 tubes D, C lead into the box, the former just entering it, and 
 the latter running to the bottom. Two stopcocks are provided 
 at F, G. 
 
 To use the instrument, ether is poured in at E and the cork 
 replaced ; the stopcocks F, G are opened, and a tube leading 
 from an aspirator attached to /. The dew is made to appear 
 
 Fig. 14.
 
 HYGROMETRIC STATE OF THE AIR. 169 
 
 and disappear a number of times, and a mean temperature 
 taken, /' being read at the same time. When finished, one 
 may remove the ether by blowing into /, and having another 
 tube leading from H into a small bottle. 
 
 The instrument is compact and can easily be carried about in 
 a box ; and instead of using an aspirator one may blow through 
 the tube attached to H in order to induce evaporation of the 
 ether. 
 
 (c) Regnaulfs experiment. 
 
 This is to shew that in a given space occupied by a gas a 
 liquid will produce a quantity of vapour whose weight and ten- 
 
 Fig. 15. 
 
 sion are the same as if the space were a vacuum ; or, in other 
 words, vapour from any liquid forms in presence of a gas just as 
 in a vacuum, and its pressure is added to that of the gas. That 
 the latter is true may be readily shewn by the apparatus used 
 for proving Boyle's law (Exp. 12, Elementary Course); and the 
 following experiment enables us to shew that the weight of 
 vapour formed in presence of a gas is equal to that produced in 
 vacuo. 
 
 Fig. 15 shews the apparatus. A glass bulb A, open at one 
 end, is fil]ed with pieces of sponge saturated with water ; the
 
 1 70 EXPERIMENTAL PHYSICS. 
 
 other end fits into a closed brass vessel BC, with a sieve inside 
 covered with wet pieces of cotton waste ; another tube leads 
 from the inside of the sieve, through two drying tubes D, E, to 
 an aspirator, such as is used in the previous experiments. When 
 the aspirator is started, air is drawn through A, filters through 
 BC, and thereby becomes thoroughly saturated with water 
 vapour ; it then deposits its moisture in the drying tubes D, E, 
 and enters the aspirator, where it is again saturated. A ther- 
 mometer gives the temperature of the saturated air. 
 
 If now V= volume of aspirator in cubic centimeters, 
 
 F= tension of vapour, found from t and the tables, 
 / = temperature of vapour, 
 
 then, if the space V were a vacuum, the weight of vapour pro- 
 duced in it in presence of water would be 
 
 ^ x . 001 293 X. 622 
 760(1 + at} 
 
 since a liter of dry air at o C. and 760 weighs 1.293 grams, 
 and the density of water vapour is .622. 
 
 This expression should give the same result as would be 
 obtained by getting the difference in weight of the drying 
 tubes before and after the experiment. 
 
 The weight of vapour is then calculated from the foregoing 
 expression, and also observed directly by weighing the drying 
 tubes : the two results should be the same. 
 
 XV. DENSITY OF A VAPOUR (DUMAS' METHOD). 
 
 This method is one of direct weighing, a glass envelope 
 being first weighed when empty, and then when filled with the 
 vapour whose density is required ; and finally its internal capacity 
 found by filling it with water and again weighing. 
 
 The experiment, though theoretically simple, is attended with
 
 DENSITY OF A VAPOUR. 
 
 I/I 
 
 Fig. 16. 
 
 many practical difficulties, and if performed with due care is a 
 most instructive exercise for the student. 
 
 The apparatus is simple : a glass bulb with a bent stem, after 
 being carefully dried and weighed, has a small quantity of the 
 liquid whose vapour density 
 is required introduced into 
 it, and it is then placed in a 
 vessel of water, as shewn in 
 Fig. 16. 
 
 On heating this to the tem- 
 perature at which the liquid 
 boils, the air in the bulb is 
 driven out gradually along 
 with the vapour ; and finally, 
 when all the liquid has dis- 
 appeared, the end of the stem 
 is sealed up, and the tempera- 
 ture of the bath and barometric reading noted. 
 
 It is then removed, dried, and again weighed, and then, the 
 point being broken off under water, it is filled by water rushing 
 into the vacuum. It is again weighed in order to determine its 
 volume. 
 
 For a very exact determination, it will be necessary to read 
 the barometer and hygrometer when weighing, and also to take 
 the temperature in the balance case. 
 
 Theory. 
 
 Let / = weight of bulb (open) in grams. 
 
 p' = weight of bulb filled with vapour and sealed. 
 H= barometric reading when sealed. 
 //' = barometric reading when weighing bulb filled with 
 
 vapour. 
 
 tj f= corresponding temperature of room, and pressure of 
 vapour.
 
 172 EXPERIMENTAL PHYSICS. 
 
 VQ= capacity of bulb at o C. in cubic centimeters. 
 T== temperature of bath at which bulb is sealed. 
 k= coefficient of expansion of glass. 
 = coefficient of expansion of air. 
 P=. weight of bulb when filled with water. 
 ff" t /',/' = corresponding barometric reading, temperature 
 
 of room, and pressure of vapour. 
 x= density of vapour (compared with dry air at o C. 
 
 and 760 m.m. 
 d= weight in grams of I c.c. of water at temper- 
 
 ature /'. 
 
 In the above, usually H=H' = H", and /=/', and/=/'. 
 Then/' / = weight of vapour less weight of displaced air of 
 room, 
 
 and Pp = weight of water less weight of displaced air of 
 room, 
 
 and on dividing one relation by another, we get x by eliminat- 
 
 ing ^o- 
 
 These are rigorous relations from which x may be found. For 
 ordinary purposes the following approximate formula may be 
 used. 
 
 Let / = weight open. 
 
 p' = weight filled with vapour and sealed. 
 P = weight filled with water. 
 
 /, H temperature of bath, and barometric reading when 
 sealed.
 
 COEFFICIENT OF EXPANSION OF METALS. 173 
 
 \ H'= temperature in balance case and barometric 
 
 reading when weighing. 
 d= density of air referred to water for f, H'. 
 x= density of vapour referred to air at o and 760 m.m. 
 
 ft-p i }H'(l+at} 
 
 -^ .-f- 1 ? -^-= > 
 
 = . 003665. 
 
 Precautions. i . The bulb must be dry, clean, and carefully 
 weighed to the tenth of a milligram. 
 
 2. Sufficient of the liquid whose vapour density is required 
 must be put in the bulb, so that the air may be all driven out. 
 
 3. When the liquid has all disappeared, the end must be 
 sealed and the temperature of the bath and barometric reading 
 noted. Some difficulty will be found in doing this ; with vapours 
 such as chloroform or ether the escaping vapour may be burned, 
 and in that way watched. On no account must the bulb be 
 raised out of the water when nearing the end of the experiment, 
 as the cooling causes air to be drawn in, and this is rarely all 
 driven out again. 
 
 4. Distilled water must be used for finding the volume. If 
 the bulb has been properly sealed, then, when the point is 
 broken off under water, it should be almost completely filled, 
 leaving only a small air bubble about a centimeter in diameter. 
 For ordinary volatile vapours a water bath is used ; for iodine 
 vapour, sulphuric acid or olive oil ; and if a higher temperature 
 is required, some kind of metallic alloy may be used. 
 
 XVI. COEFFICIENT OF EXPANSION OF METALS. 
 
 The most accurate method for determining a coefficient of 
 expansion of a metal is that devised by Ramsden, in which two 
 microscopes, fixed in position, are provided with micrometer 
 eyepieces and movable cross-wires, the micrometer head being 
 graduated so that each division corresponds to a very small
 
 1/4 EXPERIMENTAL PHYSICS. 
 
 fraction of an inch. The metal to be examined, made into a 
 suitable bar, is placed under the microscopes, which are so 
 adjusted laterally that the cross-wires coincide with certain 
 fixed marks on the bar at a certain temperature. 
 
 As this temperature is increased or diminished, the fixed 
 marks are seen to change position when viewed through the 
 microscopes ; and the corresponding increase or diminution in 
 length can easily be calculated from the number of turns of the 
 micrometers necessary to bring the cross-wires again in coin- 
 cidence with the fixed marks. The method is simple, satisfac- 
 tory, and, with due precaution, very exact. 
 
 Usually, for the purposes of comparing standards of length 
 and finding their coefficients, a machine known as a compara- 
 tor is used, which consists essentially of two microscopes as 
 described above, and special arrangements with rollers to level 
 the bars or to raise or lower them : some way must also be 
 found for maintaining them at constant temperatures. 
 
 The bars may be made of any form, and fine lines ruled upon 
 them with a dividing engine ; but for standards of length they 
 are made of square cross-section, and near each end are drilled 
 two circular openings to the mid-depth of the bar. On the 
 bottoms of these surfaces lines are engraved, which thus lie in 
 the neutral axis of the bar, and are not so liable to change their 
 position owing to flexure or strain. 
 
 XVII. SPECIFIC HEAT OF DRY AIR AT CONSTANT PRESSURE. 
 
 Regnault's experimental determination of this is the most 
 reliable. His method, which is one of mixture, consists in 
 heating a given mass of air to a high temperature, and then 
 allowing it to pass at a constant pressure (nearly atmospheric) 
 into a calorimeter containing a known quantity of water. The 
 equivalence of heat lost and gained determines the specific heat 
 of the air. 
 
 An apparatus similar to that devised by Regnault is shewn in 
 Fig. 17.
 
 SPECIFIC HEAT OF DRY AIR. 
 
 175 
 
 A large vessel R filled with water contains an inner reservoir 
 of about 35 liters' capacity, in which dry air is compressed by 
 means of an air pump attached at A. A manometer at M, 
 which may be of any kind (although for exact results a mercurial 
 one is preferable), gives the pressure. Leading from this reser- 
 voir is a tube which passes to a heating apparatus C, runs 
 through it in a spiral form, and emerges at the bottom, and 
 finally leads into a small calorimeter c. A conical stopcock V 
 regulates the flow of air into C, and by watching a small water 
 
 Fig. 17. 
 
 gauge G the operator can maintain a pressure which is never 
 very much greater than atmospheric, and is perfectly steady. 
 A stirrer 5 and thermometer are placed in C, and a stirrer s 
 and thermometer / in c. The air emerges eventually at e. 
 
 Theory. 
 
 . Let H= total pressure of air in reservoir. 
 V capacity of reservoir in liters. 
 // = pressure of barometer. 
 T= temperature in C.
 
 1/6 EXPERIMENTAL PHYSICS. 
 
 / = initial temperature in c. 
 f' = final temperature in c. 
 W = weight of water in c. 
 w = water equivalent of the calorimeter. 
 x specific heat of air at the mean pressure observed. 
 
 Then, after the experiment is finished, there will be a quan- 
 tity of air left in the reservoir equal to V, the pressure of 
 which is h ; and therefore by Boyle's law the volume which 
 has escaped is 
 
 h 
 
 And, since one liter weighs 1.293 grams, weight of air 
 = j - V x 1.293 grams, 
 
 and, by the equivalence of heat lost and gained, 
 H h jT- . . 
 
 Precautions. There are many corrections to be made in this 
 formula if an accurate result is required. The air in the reser- 
 voir should be dry ; if not, then a complicated correction is nec- 
 essary, both for temperature and pressure of vapour contained 
 in it. Evidently h will be corrected by the mean reading of G; 
 and the usual precautions with regard to maintaining a constant 
 temperature in C, c, by use of the stirrers, and also to insure 
 equal radiation and absorption in c, must be taken. 
 
 XVIII. PRESSURE OF VAPOURS FOR LOW TEMPERATURES. 
 
 (i) Between zero and 100 C. 
 
 The apparatus for this is that devised by Dalton and modified 
 by Regnault : it is shewn in the following figures. 
 
 A barometer A is placed alongside of another tube B which
 
 PRESSURE OF VAPOURS FOR LOW TEMPERATURES. 177 
 
 Fig. 19. 
 
 Fig. 18. 
 
 has an attachment at its upper end shewn in Fig. 19 : this is 
 simply a small glass globe which is detachable at D and provided 
 with a stopcock 7! A vessel C con- 
 taining water, a stirrer, and a ther- 
 mometer, surrounds the upper part of 
 A and B. 
 
 When in use, the liquid whose 
 vapour tension is to be examined is 
 
 placed in G, and an air 
 
 pump being attached, 
 
 exhaustion is carried 
 
 on until only vapour is 
 
 left, and T is then 
 
 closed. The differ- 
 ences in the readings 
 
 of A and B at different 
 temperatures give the required pressures. 
 (2) For the temperatures below zero. 
 
 For temperatures below the freezing point of water the appara- 
 tus is slightly altered as in Fig. 20. 
 A and B are two barometer 
 tubes : at the upper end of B is a 
 glass envelope G in which is the 
 liquid, so that the space above 
 the mercury in B is filled with 
 vapour. Surrounding G is a freez- 
 ing mixture (liquid) contained in a 
 vessel C. The tension of the 
 vapour can thus be found for tem- 
 peratures below freezing point, 
 and, since the two mercurial por- 
 tions are exactly in the same 
 conditions, the tension is given 
 directly by the difference in level 
 between A and B. 
 
 Fig. 20.
 
 178 EXPERIMENTAL PHYSICS. 
 
 The method is exact, if one takes care to insure the absence 
 of air in the tube B, which might be done more readily by using 
 an arrangement provided with a stopcock, as in Fig 19. 
 
 XIX. REGNAULT'S APPARATUS FOR THE DETERMINATION 
 OF THE PRESSURE OF VAPOURS AT HIGH TEMPERATURES. 
 
 This is represented in Fig. 21. A closed boiler B contains 
 water (or other liquid), the temperature of which is given by the 
 thermometer t. A large reservoir R is connected to the boiler, 
 and a compression pump at A, with a manometer at M, com- 
 pletes the arrangement. A water jacket CD surrounds the 
 
 Fig. 21. 
 
 tube leading from B to R, so that the steam condenses and runs 
 back into the boiler. 
 
 The mode of operation is to compress the air in R, deter- 
 mine the pressure by means of the manometer and barometer, 
 and finally take the temperature indicated by /. Tables are then 
 constructed, which may be graphically arranged to exhibit the 
 relation between temperature and pressure.
 
 ELECTRICITY AND MAGNETISM. 
 
 LIST OF EXPERIMENTS. 
 
 PACK 
 
 I. MAGNETIC LINES OF FORCE 181 
 
 II. DETERMINATION OF MAGNETIC MOMENTS . . . 184 
 
 III. THE DECLINATION COMPASS 187 
 
 IV. THE INCLINATION COMPASS 190 
 
 V. DETERMINATION OF THE INTENSITY OF THE EARTH'S 
 
 MAGNETIC FIELD . . . . . . . 195 
 
 VI . MAGNETIC FIELD OF A CURRENT BIOT AND SAVART'S 
 
 LAW 205 
 
 VII. MEASUREMENT OF CURRENT INTENSITY .... 210 
 
 (1) The Tangent Galvanometer 
 
 (2) The Sine Galvanometer ..... 
 
 VIII. HYDROGEN VOLTAMETER 216 
 
 IX. COPPER VOLTAMETER 219 
 
 X. CALIBRATION OF GALVANOMETERS 222 
 
 XL GALVANIC BATTERIES 227 
 
 XII. DETERMINATION OF RESISTANCE 239 
 
 XIII. TEMPERATURE COEFFICIENT OF RESISTANCE . . . 249 
 
 XIV. GALVANOMETER RESISTANCE ...... 255 
 
 XV. RESISTANCE OF BATTERIES OR CELLS .... 256 
 
 XVI. DETERMINATION OF ELECTROMOTIVE FORCES . . . 261 
 
 XVII. ABSOLUTE DETERMINATION OF RESISTANCE BY USE OF 
 
 CALORIMETER _. ' 271 
 
 179
 
 ELECTRICITY AND MAGNETISM. 
 
 I. MAGNETIC LINES OF FORCE. 
 
 When a magnetized steel bar is plunged into a vessel con- 
 taining iron filings, it is found on withdrawing it that these 
 arrange themselves in tufts or bunches, not along the whole 
 length of the bar, but round its ends, leaving the central por- 
 tion quite free. 
 
 From this it would appear that the centers of attraction, or, 
 as they are commonly called, the magnetic masses, are situated 
 in the regions near the ends of a magnet, and it is customary 
 to call the points in these regions to which the attractions are 
 the greatest the poles of the magnet, and the straight line 
 joining them its axis. 
 
 If a bar magnet be suspended so that it can move freely in a 
 horizontal plane, it will take up a position in space pointing 
 nearly north and south, and if it is displaced from this posi- 
 tion, it will come back to it and will always present the same 
 end towards the north. For this reason the end of a magnet 
 which points to the north is called the north or north-seeking 
 pole, and the opposite end the south or south-seeking pole. 
 The terms positive and negative are also frequently used to 
 designate the same poles. 
 
 Further, if the north pole of a second magnet is presented 
 to each of the poles of the suspended one, the north pole of 
 the latter will be repelled and its south attracted. Hence, 
 the law of magnetic action : Two magnetic poles of the same 
 kind repel, and two poles of the opposite kind attract each 
 other. 
 
 181
 
 I 82 
 
 EXPERIMENTAL PHYSICS. 
 
 If a bar magnet be placed on a piece of cork or wood, and 
 the whole be then floated on the surface of water or mercury, 
 it will be found that under the action of the earth the magnet 
 is subjected to a pure rotation, and not to any motion of trans- 
 lation. As the action of the earth may be taken to be con- 
 stant in magnitude and direction for points near the floating 
 magnet, and since all the forces acting on the latter are thus 
 reduced to a couple, we conclude that positive and negative 
 magnetic masses are present in equal quantities in every 
 magnet. 
 
 Again, if we attempt to separate the two poles of a magnet 
 by breaking it into two pieces, we find that instead of accom- 
 plishing this, we only produce two magnets in place of one, 
 and no matter into how many pieces a magnet is divided, each 
 
 of the fragments will pos- 
 sess polarity just as the mag- 
 net did as a whole. We 
 therefore conclude that it is 
 impossible to obtain a quan- 
 tity of positive magnetism 
 which is not associated with 
 an equal quantity of the 
 opposite kind. Although 
 we cannot obtain a magnet 
 with only one pole, it is 
 often useful in facilitating 
 calculation to consider a 
 single pole as existing by 
 itself, always remembering, 
 of course, that it is a purely 
 mental conception. 
 
 The magnetic mass of a 
 unit pole is defined to be such that if concentrated in a point 
 and placed at a unit distance from that of a similar pole, it 
 will repel the latter with unit force.
 
 MAGNETIC LINES OF FORCE. ^3 
 
 The field of a magnet is the region in which it exerts any 
 force, and the intensity of a magnetic field at any point is the 
 force exerted by the magnet on a unit pole, supposed placed 
 at this point. If we were to imagine a free unit pole placed 
 in a magnetic field, and moving under the forces exerted by 
 the magnet, the path which it would describe is called a line 
 of force, and it is such that the tangent to it at each point 
 is the direction of the resultant force at that point. The lines 
 of force for a given magnet may be plotted by placing a very 
 short compass needle at various points in its field, and noting 
 in what direction the needle points while in each position. 
 
 The actions of the magnet on the positive and negative poles, 
 respectively, of such a needle will be equal in magnitude and 
 opposite in direction, and therefore the direction which the 
 needle takes up is that of the resultant force at the point 
 where it is placed. 
 
 Probably one of the best ways to observe these lines of force 
 is to place a plate of glass over a magnet in contact with it, 
 and then to sprinkle a quantity of very fine iron filings over 
 the plate. On gently tapping the glass, the small particles, 
 which act like so many compass needles under these circum- 
 stances, will arrange themselves under the influence of the 
 magnetic forces into curves which coincide with the lines of 
 force. If it is desired to preserve copies of such curves, the 
 glass plate may be replaced by a sheet of white paper coated 
 over with a very thin layer of paraffine. If this paper be then 
 heated without disturbing the filings, these will sink into the 
 melted paraffine, and on its cooling will remain embedded in 
 it. Copies of this may then be taken by the ordinary photo- 
 graphic processes. 
 
 A better method is to use blue printing paper in place of 
 that coated with paraffine, care being taken while exposing it 
 to the sunlight not to disturb the filings. A still better method 
 is to use an ordinary sensitized photographic plate, and, as 
 with the blue paper, to perform the experiment in a darkened
 
 !84 EXPERIMENTAL PHYSICS. 
 
 room. The curves formed by the particles may be impressed 
 on the plate by illuminating it with an electric spark, or by 
 quickly turning on and off the light from an incandescent 
 lamp. From a plate prepared in this way any number of 
 prints can be readily taken. 
 
 A diagram of the lines of force taken by this method is 
 exhibited in Fig. i, in which case they are due to the com- 
 bination of two dissimilar poles. The student will find it 
 exceedingly interesting and instructive to make for himself a 
 set of plates showing the lines of force produced by (i) an 
 ordinary bar magnet, (2) a horseshoe magnet, (3) various 
 irregular combinations of magnetic poles, (4) the pieces of a 
 broken bar magnet placed close to each other, and (5) the pres- 
 ence of a piece of soft iron near the poles of a magnet. 
 
 From an inspection of Fig. i it can readily be seen that the 
 lines of force are closer together near the poles of the magnet 
 than they are at a distance from it. As the force exerted by 
 the magnet on a unit pole decreases with the distance of the 
 pole from the magnet, we may consider the number of lines of 
 force perpendicularly crossing a given small area, taken at any 
 point in the field, to be proportional to the force intensity at 
 that point. In a field of uniform intensity, therefore, the lines 
 of force will be so distributed that the same number will cut 
 any such area placed as indicated. 
 
 II. DETERMINATION OF MAGNETIC MOMENTS. 
 
 From an inspection of its lines of force it is evident that a 
 bar magnet cannot be considered as having a single pair of 
 poles or centers of attraction. In fact, such a magnet possesses 
 an indefinite number of pairs of poles, those of each pair being 
 of equal strength and of opposite sign. 
 
 If, then, a bar magnet be suspended in a uniform horizontal 
 magnetic field, similar to that represented in Fig. 2, where 
 P denotes the direction towards which a north pole tends to
 
 DETERMINATION OF MAGNETIC MOMENTS. 
 
 I8 5 
 
 move, it will be acted on by a number of pairs of forces, 
 whose resultant effect is that of a couple, and under the action 
 
 Fig. 2. 
 
 of this couple the magnet will tend to take up a position such 
 that its axis coincides with the direction of the lines of force 
 of the field. 
 
 If we represent the mass- 
 es of the various pairs of 
 poles by ^, -^; /4 2 , -yu 2 ; 
 etc., and the distances be- 
 tween those forming each 
 pair by l v / 2 , etc., and if H 
 is the intensity of the mag- 
 netic field, the turning cou- 
 ple or moment of the forces 
 about the point of suspension 
 is equal to /f 2/*/ sin 0, 
 the angle Q being that which 
 the magnet makes in any 
 position with the direction 
 of the lines of force. 
 
 The expression 2/^/, which 
 is generally written M, is 
 a constant for each magnet, 
 irrespective of how it may 
 be situated, and is called its 
 
 Fig. 3. 
 
 magnetic moment. 
 
 One of the best methods for comparing the magnetic moments
 
 lS6 EXPERIMENTAL PHYSICS. 
 
 of different magnets is that of the torsion balance, and in case 
 absolute determinations are required these can be made with 
 a fair degree of accuracy by means of the same instrument. 
 
 Such a balance as ordinarily used (Fig. 3) consists of a 
 glass case provided with a scale placed round its sides, and 
 graduated so as to indicate degrees. Situated in the same 
 plane as the scale is the magnet to be tested. It is suspended 
 by a fine wire which, after passing through a glass tube, C, 
 attached perpendicularly to the cover of the case, has its 
 upper end fastened to a brass torsion head, A. This piece 
 rests on a graduated brass collar, B, surrounding the upper 
 portion of C, and by means of these graduations, and a mark 
 on the torsion head, any angle through which the latter may 
 be turned can be directly ascertained. 
 
 Having placed the balance in some suitable position, it is 
 first necessary to find what two graduations on the scale lie in 
 the earth's magnetic meridian. This is best done by placing a 
 compass needle, resting easily on a pivot, at the center of the 
 case, or by suspending it from A by a torsionless fiber, such as 
 a single strand of silk. The needle will, under these conditions, 
 lie in the meridian, and the graduations on the scale towards 
 which it points may then be easily noted. 
 
 A bar of brass or copper of about the same dimensions as 
 the magnet to be examined, should then be suspended by the 
 wire, and the torsion head rotated until this bar rests in the 
 meridian. If the bar of brass or copper be then detached 
 without disturbing the wire, and the magnet put in its place, 
 the latter will, after oscillating for some time, finally come to 
 rest in the meridian without being subjected to any torsional 
 strain. If then A be turned through some angle , the magnet 
 will be displaced from the meridian by an angle 9, and (a 0), 
 which is generally denoted by <, will be the amount of torsion 
 in the wire. 
 
 When the magnet is in this position the torsion couple is 
 balanced by the action of the forces due to the earth's magnetic
 
 THE DECLINATION COMPASS. 
 
 I8 7 
 
 field, and since this may be taken to be uniform, we have, by 
 taking moments about the point of suspension, the relation 
 
 HMs'm0=C<f>, 
 
 " _ "=ra : 
 
 H being the horizontal intensity of the earth's field, M the mag- 
 netic moment, and C the constant of torsion for the wire. (See 
 Appendix B for method of determining C.) 
 
 A method of finding H is given in Experiment V., and as 
 both this quantity and C can be ascertained by preliminary 
 investigations, it only remains to determine an accurate value 
 
 for-*-- 
 sin 
 
 This is done simply by repeating the experiment just de- 
 scribed, a number of times, care being taken at each trial to 
 give to A a different amount of rotation. 
 
 If it is desired to make a comparison of two magnets, it is 
 
 only necessary to determine a value for for each of them, 
 
 sintf 
 
 provided the same torsion wire is used and both experiments 
 are conducted in the same place. The ratio of these two quan- 
 tities will be equal to that of the two magnetic moments, and if 
 one of them is known, the other can then be found in terms of it. 
 
 The method is especially suitable for magnets which are long 
 and slender. In case they are very short the angular deflec- 
 tions can be measured with much greater accuracy by adopting 
 the method exhibited in Fig. 13. 
 
 In this experiment the best results are obtained by using fine 
 silver wire or quartz fibers for the suspensions. 
 
 III. THE DECLINATION COMPASS. 
 
 It has already been pointed out that if a bar magnet be sus- 
 pended so as to rotate freely in a horizontal plane, it will take 
 up a position with its axis pointing in a northerly direction.
 
 1 88 
 
 EXPERIMENTAL PHYSICS. 
 
 In Experiment I., it has also been shewn that the field of a 
 magnet can be explored by a small compass needle being placed 
 at different points in it, since the needle will come to rest in 
 each position with its axis always coinciding with the direction 
 of the lines of force. 
 
 /Such considerations as these lead us to the conclusion that 
 the earth is an immense magnet surrounded with a field of 
 
 force, and that therefore 
 the bar magnet suspended 
 as above comes to rest 
 pointing towards the north 
 simply because it performs 
 the same function with 
 regard to the earth's field 
 that the small compass 
 needle does with regard 
 to that of a given magnet. 
 In order to obtain exact 
 knowledge of the earth's 
 magnetic action at a point, 
 it is necessary to deter- 
 mine (i) the direction of 
 the lines of force and (2) 
 the intensity of the mag- 
 netic action at that point. 
 The directions of ter- 
 restrial lines of force are 
 defined with respect to the geographical meridian, and to the 
 horizontal plane through any point. The vertical plane passing 
 through the axis of a freely-suspended horizontal magnet at rest 
 is called the magnetic meridian, and the angle between it and 
 the geographical meridian, the declination. 
 
 Since the magnetic poles of the earth lie in the magnetic 
 meridian, the lines of force at any place will be parallel to this 
 plane, and, therefore, the first step to take in ascertaining their 
 
 Fig. 4.
 
 THE DECLINATION COMPASS. 189 
 
 direction, is to make an accurate determination of the declina- 
 tion. 
 
 This is found by means of an instrument called the declina- 
 tion compass. It consists, as represented in Fig. 4, of a circular 
 brass or copper box AB, attached at right angles to, and freely 
 rotating about, a foot P, which is itself supported by a tripod. 
 On the bottom of this box there is placed a divided circle M, 
 at whose center a lozenge-shaped compass needle is delicately 
 pivoted. To the box there are fastened two uprights supporting 
 a horizontal axis X, to which is fixed an astronomical telescope, 
 movable in a vertical plane. The foot P also carries a fixed 
 azimuthal circle QR, and by means of it and the vernier Kany 
 rotation given to the box may be determined. The graduated 
 arc x, attached to the upright and the vernier K, which moves 
 with the telescope, afford a means of measuring any angle 
 through which the latter may be turned. 
 
 In determining the declination-, the divided scale in the bottom 
 of the box is first made horizontal by using the level suspended 
 from the telescope's support as an indicator, and the screws in 
 the tripod for making the adjustment. If the geographical 
 meridian through the point of observation has been previously 
 determined, the telescope is then sighted upon some distant 
 mark that also lies in it. 
 
 Since the divided scale is so placed that its zero diameter is 
 in the same vertical plane as the axis of the telescope, it will 
 also lie in the geographical meridian when this is done, and the 
 angle between it and the axis of the compass needle in its posi- 
 tion of rest will be the declination. 
 
 In case the geographical meridian has not been located, it will 
 be necessary to sight the telescope upon the sun at noon, or on 
 some star whose time of transit is given in the Nautical Almanac. 
 When the telescope is so adjusted, the diameter of the azimuthal 
 circle M lying in the meridian can then be ascertained, and the 
 declination will as before be the angle between the axis of the 
 compass needle and this diameter. As it frequently happens
 
 190 
 
 EXPERIMENTAL PHYSICS. 
 
 that owing to imperfect magnetization the magnetic axis does 
 not coincide with the geometrical axis of the needle, it should 
 be inverted so that what was its lower face in the first case 
 will be the upper in the second. The mean of the two readings 
 obtained should be taken for the declination, and the whole 
 operation repeated several times to insure accuracy. 
 
 The declination is said to be east or west according as the 
 north pole of the needle is to the east or west of the geograph- 
 ical meridian. The results obtained from a series of observa- 
 tions, extending over a long term of years, indicate a gradual 
 change at each point, the declination being at one time east or 
 west, and after increasing up to a certain limit, then diminishing 
 and passing to the other side of the meridian. Besides this 
 change, which is very slow, there are yearly and daily variations, 
 and, in addition to these, abrupt changes frequently occur owing 
 to the existence of so-called magnetic storms. 
 
 Maps are issued each year from various magnetic observa- 
 tories, shewing the declination at various places, and on these 
 are drawn lines, called isogonic, which pass through those points 
 where the declination is the same. 
 
 IV. THE INCLINATION COiMPASS. 
 
 Having determined the magnetic meridian, or plane, parallel 
 to the lines of force, it then remains to find the angle their 
 direction makes with the horizontal. This angle is called the 
 inclination or dip, and is determined by the inclination compass 
 or dipping needle. 
 
 A common form of the instrument is exhibited in Fig. 5, 
 where / is an azimuthal circle supported on three feet supplied 
 with leveling screws, and A is a plate which carries a spirit 
 level, and is free to rotate about a vertical axis. A frame r, 
 resting on two uprights, supports the graduated circle M in a 
 vertical plane, and ab, a magnetic compass needle, rotates 
 about a horizontal axis through its center of gravity, the resting
 
 THE INCLINATION COMPASS. 
 
 191 
 
 points being pieces of agate embedded in the frame at the 
 center of the vertical circle. The instrument is so constructed 
 that when the plate A is 
 level, the zero diameter of 
 the circle M is horizontal. 
 
 In determining the incli- 
 nation, the instrument must 
 first be leveled and then any 
 of the following methods 
 may be applied : 
 
 METHOD I. Turn the 
 frame A very slbwly about 
 its vertical axis of rotation, 
 and observe the values of 
 the inclination during the 
 motion. The greatest dip 
 noted will be the true incli- 
 nation at the point of obser- 
 vation, as it is obtained when 
 the vertical circle is in the magnetic meridian, and the needle is 
 then acted upon by the total resultant of the earth's magnetic 
 action. 
 
 METHOD II. Rotate the plate A until the needle assumes 
 a vertical position and points towards the ninety-marks on the 
 circle M. Since the needle is suspended at its center of 
 gravity, and in this position is acted upon only by the vertical 
 component of the terrestrial field, the circle Mmust then be at 
 right angles to the magnetic meridian. By next turning it 
 through a right angle, as indicated by the azimuthal circle ?#, 
 the needle will be in the meridian, and the inclination may be 
 noted. 
 
 METHOD III. By this method the vertical circle is placed 
 successively in two planes which are at right angles to each 
 other, and are situated one on each side of the meridian. The 
 angle between the needle and the horizontal diameter is noted 
 
 Fig. 5.
 
 192 
 
 EXPERIMENTAL PHYSICS. 
 
 in each case, and from these angles the inclination at the point 
 of observation is determined. 
 
 The disposition is that shewn in Fig. 6. AHB represents 
 the magnetic meridian, AEB a plane making an angle 6 with 
 it, and ACB a plane at right angles to AEB. D 1 and D 2 
 
 denote the angles made by the needle with the zero diameter 
 of the vertical circle when it is at rest in each of these positions. 
 
 If D is the true value of the inclination at points in the 
 immediate neighbourhood of the observing station, and F is the 
 resultant intensity of the earth's magnetic action on the needle 
 there, the latter can be resolved into two components in the 
 magnetic meridian : one horizontal FcosD, and the other verti- 
 cal FsinD. The component FcosD can again be resolved in 
 an infinite number of ways into two others in the horizontal 
 plane. In a direction making an angle 6 with the meridian, it 
 has a component Fcos D cos 6, and in one perpendicular to this, 
 a component FcosD sin 0. These forces are represented in 
 the figure as acting on the needle when it is at rest in the 
 plane AEB. 
 
 As the force FcosD sm 6 acts perpendicularly to this plane, 
 the position occupied by the needle is not affected by it, and
 
 THE INCLINATION COMPASS. ! 9 3 
 
 the angle D l is therefore completely determined by the relation 
 
 *'- ^sin^ ' 
 or cot D l = cot D cos 6. (i) 
 
 From similar considerations it can be readily seen that when 
 the vertical circle is turned into the plane ACB the component 
 FcosDcosQ does not affect the displacement of the needle, 
 and the angle D z is therefore given by 
 
 cot Z> 2 = cot D sin 6. (2) 
 
 Eliminating 6 from (i) and (2), it follows that 
 cot 2 D = cot 2 >! + cot 2 Z> 2 . 
 
 If, therefore, the apparent inclinations D and Z> 2 are found 
 in any two planes at right angles to each other, the true inclina- 
 tion can be calculated by applying this relation. 
 
 METHOD IV. If when the divided circle M is in the mag- 
 netic meridian the needle be given a slight displacement from 
 its position of rest, it will oscillate harmonically, and if F is 
 the intensity of the earth's field, M 1 the magnetic moment of 
 the needle, and M 2 its moment of inertia about the axis of rota- 
 tion, we have the time of a small oscillation given by 
 
 / 1 = 2 
 since the equation of motion is 
 
 If the circle be now turned so that its plane is at right 
 angles to the magnetic meridian, and the needle be again 
 made to perform small oscillations, its periodic time is given 
 
 by / 2 = 2?r\ ? , since in this case the vertical compo- 
 
 nent of the earth's attraction alone is acting.
 
 i 9 4 
 
 EXPERIMENTAL PHYSICS. 
 
 t 2 
 We have, therefore, sin D=^ and from this equation the 
 
 inclination may be calculated. The times t and f 2 should be 
 found by allowing the needle to oscillate for a considerable 
 time. 
 
 Method III. will probably give the most satisfactory results, 
 but the sources of error are so numerous in any case that the 
 greatest care must be taken in making the adjustments if accu- 
 rate determinations are required. 
 
 Precautions. I. It frequently happens that from various 
 causes the vertical circle becomes displaced so that the zero 
 diameter is not horizontal when the plate A is level. It then 
 becomes necessary to ascertain the true horizontal diameter. 
 
 Since the needle will be vertical when its plane of rotation is 
 at right angles to the magnetic meridian, it will also be vertical 
 when the circle M is turned through an angle of 180 from 
 this position. Two points on this circle must therefore be 
 found such that the needle points towards them when it is in 
 one position, and also when the same circle is given a rotation 
 of 1 80. This will determine at the same time both the mag- 
 netic meridian and the two points in the vertical diameter of 
 the graduated circle. The horizontal diameter can then be 
 inferred. 
 
 II. Readings should be taken at both ends of the needle in 
 order to correct any error arising from the axis of rotation not 
 passing exactly through the center of the vertical circle. 
 
 III. The magnetic axis of the needle may not coincide with 
 its geometrical axis, and to correct errors from this source, 
 the method of reversion indicated in Experiment III. should 
 be adopted. Instead of removing the needle, this can be ac- 
 complished by simply rotating the vertical circle through 180 
 when any reading has been taken, and again observing the 
 inclination. 
 
 IV. The center of inertia may not lie in the axis of suspen- 
 sion. In this case, the inclination will be affected by gravity, and
 
 HORIZONTAL INTENSITY. 
 
 195 
 
 should be corrected by repeating the operation after the needle 
 has been remagnetized so that its poles are reversed. As in 
 the case of declination, maps are used shewing the inclination 
 at various places. The lines which pass through those points 
 where the inclination is the same are termed isoclinic. 
 
 V. DETERMINATION OF THE ABSOLUTE INTENSITY 
 OF THE EARTH'S MAGNETIC FIELD. 
 
 In Experiment IV. it has been shewn that if the dipping needle 
 be slightly displaced from its position of equilibrium in the mag- 
 netic meridian, it will perform small oscillations whose periodic 
 
 time is 2 TT'Y L, M-^ being the moment of inertia of the needle 
 
 about the axis of rotation, M its magnetic moment, and F the 
 resultant intensity of the earth's magnetic action. It follows 
 
 from this that MF= ^ -, and since M l and / can be easily 
 
 ascertained, a relation can thus be established between M 
 and F. Owing to friction, however, an accurate result cannot 
 be determined in this way, and it is customary to determine 
 instead the product MH, H being the horizontal component 
 
 J[/T 
 
 of F, and then by means of a second relation , to evaluate 
 
 H 
 both the quantities H and M. 
 
 Fig. 7. Fig. 8. 
 
 After H has been found in this manner, and the inclination 
 determined at the point of observation by a separate investiga- 
 tion, the resultant intensity F can then be calculated from the 
 relation F=H-secD.
 
 EXPERIMENTAL PHYSICS. 
 
 The apparatus employed in this experiment consists of a 
 telescope (Fig. 7) mounted so as to be capable of rotation in 
 a horizontal and a vertical plane, a finely divided scale VV 
 (Fig. 8), and a magnetometer. 
 
 This instrument (Fig. 9) is composed essentially of a cylin- 
 drical brass or copper box B resting on a tripod provided with 
 leveling screws. It is ca- 
 pable of rotation about a 
 vertical axis, carries with it 
 a graduated circle C at- 
 tached to its base, and has 
 its anterior face pierced by 
 a circular opening in which 
 is inserted a convergent 
 lens O whose focal length 
 is about one meter. A 
 metallic column V has its 
 base inserted in a beveled 
 plate resting on the top of 
 B, and carries at its upper 
 end a reel on which is 
 wound the silk strand which 
 supports the stirrup and 
 the bar magnet A. This 
 stirrup carries a vertical 
 mirror M mounted so as 
 to be at right angles to 
 
 the magnet when the latter is in position. A second mirror 
 M' is fitted into a frame fastened to the bottom of the box, 
 and can be adjusted either horizontally or vertically by means 
 of the screws E, E. A divided circle C indicates any rotation 
 given to the column V. 
 
 Theory. Determination of MH. When a bar magnet is sus- 
 pended so that it is free to move in a horizontal plane, it will, 
 
 Fig. 9.
 
 HORIZONTAL INTENSITY. 
 
 197 
 
 when displaced, oscillate about its mean position. If H is the 
 horizontal intensity of the terrestrial field, and M the magnetic 
 moment of the bar, the equation of motion is 
 
 The time of a small oscillation is therefore given by 
 
 and denoting the moment of inertia of the magnet by M v this 
 equation gives the relation 
 
 (2) 
 
 The quantity t can easily be ascertained by counting the 
 oscillations performed in some given time, and M-^ can either 
 be calculated from the dimensions of the magnet or found 
 experimentally. 
 
 In the case of very exact determinations allowance will have 
 to be made for the torsion couple of the suspending fiber. 
 Adopting this correction, the relation becomes 
 
 (3) 
 
 the constant C being the coefficient of torsion, which must be 
 ascertained by a preliminary investigation. (See Appendix B.) 
 
 Determination of -- The relation- is determined from 
 H H 
 
 the reciprocal actions exerted by two magnets situated at a 
 distance from each other, which is great compared with their 
 lengths. After having accurately determined the time of vibra- 
 tion of the magnet A, it is next used to deflect a second one 
 which is inserted in the stirrup in its place. 
 
 Either of two methods may be followed. In one the disposi- 
 tion is that shewn in Fig. 10, where JVS represents the magnet
 
 198 EXPERIMENTAL PHYSICS. 
 
 in the stirrup at rest in the meridian, and AB is the magnet A 
 resting on a support in the same horizontal plane as NS, but 
 pointing east and west. In the second, the arrangement is 
 
 # 
 
 1 
 
 Fig. 10. 
 
 shewn in Fig. 12. The magnet NS is suspended in the 
 meridian as before, but the deflecting magnet AB is now placed 
 with its center in this plane and its axis at right angles to it. 
 
 METHOD I. Denoting the lengths of AB and NS (Fig. 10) 
 by / and / v respectively, and the strengths of their poles by 
 
 Fig. 11. 
 
 fi and /*!, it follows if r, the distance between their centers, is 
 considerable, that the action exerted by AB on each of the 
 poles of NS is approximately equal to
 
 HORIZONTAL INTENSITY. 
 
 199 
 
 and may be taken as acting parallel to the line joining the 
 centers of the magnets. Under these actions and those exerted 
 by the earth's field, the magnet NS will take up the position 
 indicated in Fig. u. Taking moments about the point of 
 suspension, it follows that 
 
 
 -- r 
 
 2 
 
 from which r 3 tan 6 = 2 fi +-^A (4) 
 
 If now this experiment be repeated with the centers of the 
 magnets at a distance r lt apart, we will have a second relation, 
 
 and is therefore given by the equation, 
 H 
 
 M _ . 
 
 ~ 
 
 METHOD II. Denoting again the distance between the cen- 
 ters of the magnets by r, and their lengths by / and / 1} the 
 
 S 
 
 Fig. 12. 
 
 actions exerted by AB on 5 and N respectively (Fig. 12) are 
 perpendicular to the meridian, and are approximately equal to
 
 2QO EXPERIMENTAL PHYSICS. 
 
 Under these forces, and those exerted by the earth's field, the 
 suspended magnet will again be deflected from the meridian 
 (neglecting any small motion of translation) through an angle 
 6, the equation of moments being, in this case, 
 
 (7) 
 
 From this it follows that 
 
 and repeating the experiment, taking the distance between the 
 centers to be r lt we have also 
 
 Combining (8) and (9), a second equation for determining 
 is given by Jf 
 
 H r*r^ 
 
 Experiment. When the apparatus is arranged for taking 
 observations, the several parts are situated as in Fig. 13, where 
 AB represents the meridian, CA the telescope, VV the scale, 
 and GH and KL the mirrors attached to the base of the 
 magnetometer and to the stirrup respectively. The first step 
 to take in making this disposition is to adjust the mirror GH, 
 so that it is perpendicular to the meridian. This is arrived at by 
 placing a copper bar in the stirrup, and, after taking all the 
 torsion out of the silk strand, by turning the column V until 
 the bar lies approximately in the meridian, as indicated by a 
 delicately poised compass needle. On replacing the copper bar 
 by the magnet, the latter will then come to rest in the meridian 
 without being subjected to any torsion, and the mirror KL will 
 therefore be perpendicular to it. The magnetometer is next 
 turned until the mirror GH is parallel to KL. This can be
 
 HORIZONTAL INTENSITY. 
 
 2O I 
 
 performed very accurately by observing with the telescope the 
 images of the scale (GH being slightly inclined to the vertical) 
 formed by the two mirrors, the latter being parallel when the 
 divisions on one scale image are seen directly above the corre- 
 sponding ones on the other. 
 
 In order to adjust the telescope so that its axis will be in 
 the meridian, place the scale with its stand on the same sup- 
 port as the former, and then, after noting the scale division, 
 
 generally zero, which is situated in the same plane as the 
 axis of the telescope and the vertical diameter of its objective, 
 move the support until the images of this division, formed 
 by the mirrors, are seen on the vertical cross-hair in the eye- 
 piece. The axis of the telescope will then be in the meridian, 
 and the scale at right angles to it. In determining the quantity 
 MH, it is first necessary to find accurately the time of a single 
 oscillation. This is done by giving the magnet A initially a 
 small displacement, by bringing near to it a piece of iron or 
 steel, and when the motion has become steady, by ascertaining 
 with a stop watch the time occupied in performing a large
 
 202 EXPERIMENTAL PHYSICS. 
 
 number (one hundred, for example) of small oscillations. The 
 time of a single one can then be calculated, and by taking the 
 mean of the results obtained by repeating this operation a 
 number of times a very close approximation can be found. In 
 order to make certain that the stop watch marks correct time 
 it should be compared, both before and after the experiment, 
 with a standard chronometer. 
 
 As already indicated, the moment of inertia of the magnet 
 A and the stirrup upon which it rests may be determined 
 experimentally by the method indicated in Appendix B ; but 
 for a rough approximation that of the stirrup may be neglected, 
 and that of the magnet calculated from its dimensions. 
 
 In determining , a second magnet is placed in the stirrup, 
 H 
 
 and the magnet A rests on a support attached to a copper 
 bar (not shewn in the figure), which together with a counter- 
 poise rotates about the cylindrical column connecting the box 
 of the magnetometer to the tripod. Divisions are marked on 
 this bar, shewing the distance of the center of the support 
 from the axis of rotation of the instrument, and therefore that 
 between the centers of the magnets. The mirror GH is gen- 
 erally attached to the base of the magnetometer so that its 
 lower edge lies in the ninety-diameter of the divided circle C, 
 and therefore when the bar carrying A is turned so that its 
 index points to this division it is perpendicular to the meridian, 
 
 and may then be found by Method I. 
 H 
 
 If it is desired to adopt Method II., the bar should be turned 
 through a right angle from this position. 
 
 The method of Poggendorff is followed in measuring the 
 angle through which the auxiliary magnet is deflected. When 
 this magnet is in the meridian, one sees the image of the 
 zero division (Fig. 13) formed on the cross-hair of the telescope, 
 but when the mirror KL is turned through an angle 6, one 
 sees formed there the image of some other division, N.
 
 HORIZONTAL INTENSITY. 
 
 203 
 
 It follows then that 
 
 tan 26 = 
 
 AN 
 AS 
 
 1 AN 
 
 or 0=-. approximately. 
 
 2 *LJJ 
 
 In ascertaining the value of 6 corresponding to a given value 
 of r the magnet A should first be placed with its positive pole 
 
 Fig. 14. 
 
 pointing east, for example, and the deflection noted, and it 
 should then be turned with its negative pole in this direction, 
 and the deflection again observed. After this the bar should
 
 204 
 
 EXPERIMENTAL PHYSICS. 
 
 be rotated through 180 and the same process repeated, the 
 mean of the four values so obtained being the one inserted in 
 the formula with the corresponding value of r. 
 
 The apparatus is designed specially for the determination of 
 the quantity H, and is not suited for finding the magnetic 
 moments of magnets of different sizes. 
 
 When, however, H has been determined accurately for the 
 point of observation, these can readily be obtained with a 
 slightly modified form of the instrument, by simply performing 
 the first operation in this experiment. 
 
 Fig. 15. 
 
 In the British magnetic observatories, and also, in fact, in 
 many laboratories, the Kew portable magnetometer is used for 
 finding the horizontal intensity. 
 
 In Fig. 14 it is shewn adjusted for determining the quantity 
 MH. The vibration magnet M is suspended at the center of 
 a wooden box by a silk thread attached to the torsion head A. 
 The magnet is a hollow steel cylinder having a glass scale at
 
 HORIZONTAL INTENSITY. 205 
 
 one end and a lens at the other, the former being in the focal 
 plane of the lens. This scale is illuminated by a small mirror, 
 C, and the oscillations of the magnet are observed by means 
 of the telescope T v 
 
 The moment of inertia of the magnet is found experiment- 
 ally, by inserting a cylindrical brass or copper bar whose 
 moment of inertia can be calculated, in the brass collar which 
 is shewn immediately above the magnet. 
 
 The arrangement adopted in determining is exhibited in 
 
 H 
 
 Fig. 15. The wooden box is removed and a metallic one put 
 in its place. The vibration magnet M l is placed on the support 
 C, which is movable along the bar SS' previously referred to, 
 and the telescope T, with its accompanying scale s, is attached 
 to the lower part of the end of the large hollow cylinder. 
 The auxiliary magnet, M z , and the mirror for reflecting the 
 scale divisions are shewn in position in the figure. 
 
 In determining H by this instrument the same theory and 
 the same general method of manipulation as that just described 
 apply. 
 
 VI. MAGNETIC FIELD OF A CURRENT. BIOT AND 
 SAVART'S LAW. 
 
 Oerstedt discovered in 1820 that a magnetic needle, if sus- 
 pended in the neighbourhood of a rectilinear current, tended to 
 place itself at right angles to the conducting wire. This dis- 
 covery at once suggested that surrounding a rectilinear current 
 there is a magnetic field and that its lines of force are a series 
 of concentric circles having the conductor as a common axis. 
 The truth of this conclusion has been amply verified by experi- 
 ment and can be clearly exhibited by the use of iron filings. 
 
 If a straight wire in which a current is established be passed 
 up through a piece of cardboard which is held horizontally, and 
 fine iron filings be then sprinkled upon it, these will at once 
 arrange themselves into the series of circles described ; while, on
 
 206 
 
 EXPERIMENTAL PHYSICS. 
 
 the other hand, if the wire be laid flat on the paper, the filings 
 will arrange themselves into a series of straight lines cutting 
 the line of the current at right angles. 
 
 Since then the lines of force are circles, it only remains to 
 state the sense in which a north magnetic pole tends to move, 
 in order to define completely the action exerted by a given cur- 
 rent on a magnet. The following rule has been proposed by 
 Ampere for guidance in this connection : The positive pole of 
 a magnet tends to turn towards the left hand of an observer look- 
 ing at it, and supposed swimming in the conductor in the direc- 
 tion in which it has been agreed to consider the current as passing. 
 
 A. The magnetic field of an infinitely long rectilinear current. 
 As in the case of other magnetic fields the intensity of that 
 produced by a long rectilinear current may be investigated by 
 the method of oscillations. The current is passed through a 
 vertical conductor as in Fig. 16, and 
 a small magnet is suspended by a 
 torsionless fiber at different points 
 along a horizontal line passing 
 through the wire perpendicularly to 
 the direction of the magnetic meri- 
 dian at that point. 
 
 If the needle is taken very small 
 and is placed not too close to the 
 wire, the action of the current on it 
 will reduce to that of a couple which 
 acts in conjunction with, or in opposition to, the horizontal 
 component of the earth's magnetic field according to the direc- 
 tion in which the current passes. 
 
 By applying this method Biot and Savart found that when a 
 
 constant current was maintained in the wire the intensity of the 
 
 field was inversely proportional to the distance from the conductor. 
 
 Denoting the horizontal component of the earth's action near 
 
 the current by H, and the intensity of the field due to the cur- 
 
 Fig. 16.
 
 MAGNETIC FIELD OF A CURRENT. 
 
 207 
 
 rent at the distances a and b by F^ and F 2 respectively, the 
 equations of motion for the magnet when acted on by (i) H 
 alone, (2) F l and H together, and (3) F z and H together, are 
 
 (i) 
 , (2) 
 
 and Sair 8 ^ = - (ff+FJMsm 0. (3) 
 
 If , lf and # 2 are the number of oscillations performed in 
 each of these cases respectively in a given time, we have, if the 
 amplitudes of the vibrations are small, 
 
 (4) 
 (5) 
 (6) 
 
 where K is a constant depending on the moment of inertia, and 
 the magnetic moment of the magnet. 
 From (4), (5), and (6) it follows that 
 
 7^~ 2 2 - 2 
 
 If then the law of the inverse distance holds, we should have 
 
 FI b r b n 2 7z 2 
 
 =-, and therefore -=-L -, i.e. we should have b(ns 2 ) 
 
 F 2 a a n z 2 -n z 
 
 =a(n l z n z )= a constant for all positions of the magnet along a 
 line through the conductor perpendicular to the meridian. 
 
 The apparatus for performing this experiment is similar to 
 that shewn in Fig. 9. The various parts are arranged just as 
 indicated there excepting the wire conductor, which is movable 
 about the vertical through the point of suspension of the 
 magnet, and is so placed that the perpendicular on it from 
 the magnet is at right angles to the meridian.
 
 208 
 
 EXPERIMENTAL PHYSICS. 
 
 In order to make this adjustment accurately the wire is first 
 placed approximately in position, and the magnet is brought to 
 rest in the meridian. If the current is then turned on and the 
 magnet remains still at rest the adjustment is perfect ; if, how- 
 ever, there is a displacement of the magnet the wire must be 
 rotated through a small angle in a sense opposite to that of such 
 displacement and the same test again applied. 
 
 As the chief difficulty in this experiment arises from the 
 inconstancy of the current, a sensitive galvanometer should be 
 inserted in the circuit along with an adjustable resistance, so 
 that any variation in the current strength may be at once 
 detected, and corrected by altering the resistance. A battery 
 of Grove cells will be found to produce a very steady current 
 for a considerable time. 
 
 In order to correct still further any variations in the current, 
 the oscillations of the magnet should be counted a number of 
 times for each position. 
 
 B. Magnetic field produced by an angular current. 
 
 Biot and Savart modified the experiment just described by 
 
 observing the action of a cur- 
 rent traversing two very long 
 rectilinear conductors BA and 
 AC, so connected as to make 
 an angle at A, Fig. 17, upon 
 a short magnet placed with its 
 center P on the line bisecting 
 
 the angle BAG. 
 
 As can be seen from the 
 figure, the circular lines of 
 force due to the current pass- 
 ing along BA and AC have a 
 common tangent at P, and if 
 these wires are so placed that 
 their plane is vertical and per-
 
 MAGNETIC FIELD OF A CURRENT. 
 
 209 
 
 pendicular to the magnetic meridian through A, the action 
 exerted on a short magnet suspended horizontally at P will 
 again be reduced to two equal and opposite forces whose direc- 
 tion is in the meridian at that point. 
 
 The results of experiment indicate that the action of such 
 a disposition of conductors is given by 
 
 where A is the angle BAP, and r is the length AP, K being a 
 constant. 
 
 The same method is adopted in verifying this law as in that 
 of the inverse distance for an infinite rectilinear current, and 
 similar precautions are necessary regarding the constancy of 
 the current, and the plane BAG being perpendicular to the 
 meridian. 
 
 Law of Laplace. 
 
 It follows from the law just investigated that the action of 
 an infinite current, supposed commencing at A (Fig. 18), upon 
 a magnetic pole placed at P can be represented by 
 
 2 r 
 where BAP is the angle A, and AP is the length r.
 
 2io EXPERIMENTAL PHYSICS. 
 
 It follows, therefore, that 
 
 = K- 
 
 P 
 
 where p=PN, and since the action of AB on P may be con- 
 sidered as the sum of the elementary actions of its parts, that 
 of an element A A' is given by 
 
 sin d cos - 
 dF=K - - - -dA 
 
 dA 
 i.e. since 
 
 AA 
 
 and p=r sin A, 
 
 or, denoting the length AA' by ds, we have Laplace's law for 
 the action of an element of current, 
 
 VII. MEASUREMENT OF CURRENT INTENSITY. 
 
 Of the many effects that can be produced by an electric cur- 
 rent, two have been selected as being most suitable for the 
 determination of its intensity, the action exerted by the 
 current on a magnetic pole, and the decomposition of chemical 
 compounds. 
 
 If a circuit be so arranged that a constant current in pass- 
 ing can produce (i) magnetic, (2) heating, (3) chemical, and 
 (4) magnetizing effects, it will be found that if a measure of 
 each of these be taken, only those of the magnetic and the 
 chemical effects will bear the same ratio to the new values 
 obtained when the current is by any means altered in intensity.
 
 MEASUREMENT OF CURRENT INTENSITY. 
 
 211 
 
 It has therefore been agreed to consider current strength as 
 being proportional to the action in each of these cases ; and 
 vice versa, the action exerted on a magnetic pole, and the 
 amount of chemical separation produced, will therefore be pro- 
 portional to the intensity of the current causing such action. 
 
 If, then, a magnetic pole of strength p be placed near a 
 conductor bearing a current whose intensity is C, Laplace's 
 law, when combined with the above convention, gives 
 
 . r _ Cds sin A 
 
 as the force exerted on the pole by an element of current ds, 
 A being the angle between the line r and the element ds. 
 
 The absolute unit of current strength in the electro-magnetic 
 system is defined to be such that if one centimeter length of 
 the circuit be bent into an arc of one centimeter radius, the 
 current in it exerts a force of unit intensity upon a magnetic 
 pole of unit strength placed at the center of the arc. On the 
 
 basis of this definition, therefore, /* s -^- will represent the 
 
 force in absolute units exerted on the magnetic pole /j, placed 
 as previously indicated. 
 
 Fig. 19. 
 
 Action of a circular current on a magnetic pole. 
 Let AC represent the circular conductor bearing the current 
 whose intensity is C (Fig. 19), AB or ds an element of this
 
 212 EXPERIMENTAL PHYSICS. 
 
 current, and P the position of a magnetic pole of strength p., 
 supposed placed on CD, the axis of the circuit. 
 
 Since in this case the angle A is a right angle, the force 
 
 exerted by the element ds on the pole p is equal to ^ 2 S . It also 
 
 acts in a direction at right angles to APB. If a is the radius of 
 the circle AC, and PD be denoted by x, the component of this 
 
 force parallel to CD is equal to **' ' s ' a Since a similar 
 
 (0 2 +* 2 )* 
 expression may be obtained for each element of the circuit, the 
 
 action of the whole conductor resolved along CD is given by 
 
 and from the symmetry of the arrangement it is evident, more- 
 over, that this is the resultant action of the current. 
 
 If instead of a single coil there are n of them, the action will 
 be given by 
 
 *-*:&$ (2> 
 
 and if the pole is situated at the center of the coil, this further 
 reduces to 
 
 F=**!f*. (3) 
 
 The tangent galvanometer. 
 
 A form of the tangent galvanometer commonly used is that 
 shewn in Fig. 20. It consists of a coil of insulated wire wound 
 on the vertical circle AB, which is capable of rotation about an 
 axis coinciding with its vertical diameter. At the center of this 
 circle a short magnet is suspended from C by a torsionless 
 thread of silk, and immediately below it there is placed a hori- 
 zontal divided circle DE, which can rotate about the same 
 axis as AB, but independently of it. The divided circle FG, 
 which is provided with a vernier, indicates the amount of any
 
 MEASUREMENT OF CURRENT INTENSITY. 
 
 213 
 
 rotation given to the coil bearing the current, and an alumin- 
 ium pointer attached to the magnet serves to shew the angle 
 through which the latter may be deflected. 
 
 In adjusting the instrument 
 for a measurement, the circles 
 DE and FG are first made 
 horizontal by means of the 
 leveling screws, and the coil 
 AB is then turned until the 
 magnet when at rest lies ap- 
 parently in its plane, which 
 will then coincide approxi- 
 mately with the magnetic 
 meridian. 
 
 Since the magnet is short, 
 the actions exerted on its poles 
 by the current will form a 
 couple which will tend to 
 make it take up a position at 
 right angles to the meridian. 
 Owing, however, to the pres- 
 ence of the terrestrial field, it 
 will come to rest in a posi- 
 tion such as that indicated in 
 
 Fig. 21, where AB is the meridian, CD the magnet, and 6 the 
 angle of deviation. 
 
 When, therefore, the current is passing and the magnet has 
 come to rest, this angle should be noted ; and if, when the 
 current is reversed, the magnet is deflected through the same 
 angle but on the other side of the meridian, the plane of the 
 coil coincides with the latter, and the instrument is ready for a 
 measurement. If, however, this condition does not obtain, the 
 coil AB must be given a slight displacement, and the test 
 again applied.
 
 214 
 
 EXPERIMENTAL PHYSICS. 
 
 When the magnet is displaced, as in Fig. 21, it follows, by 
 taking moments about the point of suspension that, 
 
 (4) 
 
 .Ha 
 2 HIT 
 
 tan0, 
 
 and since the right-hand member of this equation involves quan- 
 tities which can be determined with precision, this method 
 
 affords a means of accurately measuring the intensity of a cur- 
 rent in absolute units. 
 
 The ampere, or practical unit of current strength, is equal to 
 one-tenth of the absolute unit. The relation for determining 
 the intensity of a given current in amperes is, therefore, 
 
 . ZL7"_ 
 
 (5) 
 
 To obtain satisfactory results, H must be found by means 
 of the magnetometer at the point where the galvanometer is 
 placed, and extreme care must be taken to have the coil AB 
 accurately in the plane of the meridian. In order to overcome 
 any imperfection in this adjustment, the current should always 
 be reversed and the mean of the two readings taken.
 
 MEASUREMENT OF CURRENT INTENSITY. 
 
 215 
 
 For very close determinations, it is best to have only one turn 
 of wire in the circuit, as in that case the radius a can then be 
 measured very accurately. 
 
 In order to dampen the oscillations of the magnet, a plate of 
 copper is frequently placed immediately below it within the 
 circle DE; and, as it generally takes a considerable time for 
 the magnet to cease vibrating, even when this precaution is 
 taken, its position of equilibrium is generally deduced by noting 
 three of its successive positions of rest. 
 
 In many cases the angle of deflection is determined by means 
 of a small mirror attached to the magnet. A scale and telescope 
 may then be used just as in Experiment V., and the angle 
 through which the needle is deflected calculated by the method 
 there explained. 
 
 Besides being used for measuring given currents, the tangent 
 galvanometer is frequently employed to test the accuracy of 
 direct reading ammeters. These again, when they have been 
 accurately calibrated, may be used to calculate the constant K 
 in the formula C=K tan 0, for a tangent galvanometer whose 
 construction is of such a character that it is difficult to measure 
 with accuracy the dimensions of its coil. 
 
 Sine galvanometer. 
 
 The instrument just described may also be used as a sine 
 galvanometer. _ The same precautions are to be taken in regard 
 to placing the coil AB initially in the meridian, and the same 
 adjustments made, except that when the current is turned on, 
 and the magnet is deflected, the coil AB is then slowly and 
 carefully rotated until it overtakes the magnet, which when 
 it comes to rest does so with its axis in the plane of the coil. 
 
 Figure 22 exhibits the forces which then maintain the magnet 
 in equilibrium, AB denoting the meridian. By again taking mo- 
 ments about the point of suspension, we obtain the relation, 
 
 C = ^Lsm0, (6) 
 
 2;/?r
 
 216 
 
 EXPERIMENTAL PHYSICS. 
 
 and from it the intensity of any current can be calculated. 
 The angle 6, which is that through which the coil is rotated, 
 can be ascertained by means of the graduations on FG. Es- 
 pecial care should be taken to have the pointer attached to the 
 magnet directed in its initial and final positions to the same 
 division on the graduated circle DE, which rotates with the coil. 
 
 Fig. 22. 
 
 VIII. HYDROGEN VOLTAMETER. 
 
 When a current of electricity is made to decompose an elec- 
 trolyte it is found that the amount of a gas liberated, or the 
 quantity of a metal deposited, is directly proportional to the 
 intensity of the current, and, within very wide limits, is not 
 affected by the size or shape of the vessel containing the 
 electrolyte. 
 
 The mass of a substance liberated in one second by one 
 ampere of current is termed its electro-chemical equivalent. 
 When, therefore, a constant current is passed through a voltam- 
 eter and a known weight of gas is liberated in a second, the 
 strength of this current is perfectly defined, and can readily be 
 calculated when the electro-chemical equivalent of the gas is 
 known. That of hydrogen is .00x301038 grams.
 
 HYDROGEN VOLTAMETER. 
 
 217 
 
 A simple form of hydrogen voltameter is shewn in Fig. 23. 
 The acidulated water is placed in the vase AB, whose base is 
 pierced by two holes in which are cemented two small strips of 
 platinum. These are connected to the two binding poles C 
 and D, at which the current is made to enter and leave the 
 
 Fig. 23. 
 
 voltameter. The hydrogen is collected in the graduated glass 
 tube E, and the current is made to pass through the mercury 
 in the small cup F, so that the circuit can be rapidly made or 
 broken. 
 
 When the current is established, the oxygen which appears 
 at the positive platinum partly recombines with the water, and 
 since the quantity of it which might be collected would not 
 then be an exact measure of the decomposition which has taken 
 place, it is customary to allow it to escape and to collect only 
 the hydrogen, by placing the tube E over the negative platinum. 
 
 When an experiment is about to be performed the circuit is 
 first broken at F, and the tube E is then carefully filled with 
 the acidulated water and inverted over the negative electrode. 
 A thermometer is also placed in the liquid in the vase. 
 
 The circuit is then closed by inserting the leading wire in 
 the mercury, and a stop watch, reading to one-fifth of a second,
 
 2I g EXPERIMENTAL PHYSICS. 
 
 is started at the same instant. When the tube has been filled 
 with hydrogen until the liquid in it stands at the same level 
 as that in the vase outside, the circuit is broken and the watch 
 stopped. The time, /, during which the current was passing 
 can thus be accurately found, and the volume, V, in cubic centi- 
 meters, occupied by the hydrogen read directly from the gradu- 
 ations on the tube. The temperature of the gas, T. C, may be 
 taken to be the same as that of the liquid in the vase. The 
 pressure, f, exerted by the water vapour corresponding to this 
 temperature can be found from the tables, and if H is the 
 reduced height of the barometer, the pressure to which the 
 hydrogen is subjected is proportional to (//"/). 
 
 Since the weight of I c.c. of dry air at 760 mm. pressure and 
 o C. is .001293 grams, and the density of hydrogen is .06926, 
 the weight of the gas liberated is given by 
 
 W= Vx -OP 1 293 x .06926 x (H-f) ( . 
 
 760(1 +.003665 T) 
 
 The average intensity of the current in amperes is therefore 
 found from the relation 
 
 c= .- , 
 
 /x 760(1 +.003665 T) 
 
 For very fine determinations, a correction will have to be 
 applied to / owing to the water being acidulated ; but for 
 ordinary purposes, the tension may be taken to be the same 
 as that exerted by the vapour of pure water. 
 
 If it is desired to produce the decomposition rapidly, the 
 electrolyte should be so prepared as to be of maximum con- 
 ductivity, its density being then about 1.25 at 15 C. It is 
 not necessary that the liquid should stand at the same level 
 both inside and outside the tube E when the circuit is broken. 
 By adopting this procedure, however, the necessity of finding 
 the density of the electrolyte after each measurement is 
 removed.
 
 COPPER VOLTAMETER. 
 
 2I 9 
 
 An improved form of hydrogen voltameter is shewn in 
 Fig. 24. The graduated tube 
 E narrows down at its upper 
 extremity until the opening is 
 of capillary dimensions. It 
 then expands into a little bulb, 
 and afterwards bends over so 
 as to rest on a support. It 
 is filled by aspiration through 
 the rubber tube T, and the 
 capillary opening possesses the 
 property of permitting this, 
 and yet at the same time pre- 
 venting the escape of the gas, 
 provided that a little of the 
 liquid is left in the bulb. A 
 glass collar M attached to the 
 vase G, when filled with water, 
 keeps the gas in the tube at / 
 a uniform temperature, and 
 enables it to be determined 
 exactly. Fig . 24 . 
 
 IX. COPPER VOLTAMETER. 
 
 In finding the intensity of an electric current by means of 
 the hydrogen voltameter, errors of temperature, pressure, and 
 volume are almost certain to be met with, and for this reason, 
 the results obtained are often not as accurate as is desirable. 
 When, however, current strength is measured by the amount 
 of a metal deposited, the determinations are more accurate, 
 since the method is one of direct weighing. It is, besides, 
 more convenient and more cleanly in operation, and is therefore 
 generally ado'pted. The metals commonly deposited are copper 
 and silver, and the solutions employed are copper sulphate and 
 nitrate or chlorate of silver.
 
 220 
 
 EXPERIMENTAL PHYSICS. 
 
 When a copper voltameter is used, care must be taken to 
 have it of such form that the density of the current, i.e. the 
 quotient of the intensity by the surface of the electrodes, is 
 very small. Otherwise, the deposit is made in little globules, 
 which either do not adhere at all, or if they do they fall off on 
 the slightest disturbance. 
 
 One of the best forms of the instrument is exhibited in Fig. 
 25, where the copper plates A and B, which are partly immersed 
 in a solution of copper sulphate, are the positive and negative 
 electrodes, respectively. The solution should be in the propor- 
 tion of five parts of water to one 
 of copper sulphate, and should 
 contain a slight trace of sulphuric 
 acid. 
 
 In setting up the voltameter, 
 both electrodes must be thor- 
 oughly cleaned by first rubbing 
 them with fine emery paper, or 
 dipping them in a solution of 
 nitric acid, and afterwards rinsing 
 them for some time in running 
 water. 
 
 The curved plate A is then 
 placed in position as indicated in 
 the figure. B is well dried by 
 rolling it on blotting paper and then placing it in a hot-air bath. 
 In case the time in which the experiment may be performed 
 is limited, the plate may be rapidly dried by dipping it in 
 strong alcohol before rolling it in the blotting paper. Any 
 moisture which may adhere to it after this will evaporate very 
 rapidly. 
 
 After weighing it accurately to the tenth of a milligram, this 
 plate is attached to its support and lowered into the solution 
 until each point on its surface is at about the same distance 
 from some portion of the positive electrode. 
 
 Fig. 25.
 
 COPPER VOLTAMETER. 221 
 
 The circuit should also, in this case, contain a mercurial con- 
 tact breaker, and the connections be so made that the current 
 will enter the voltameter by A and leave by B. In making 
 a test, the current should be allowed to pass for at least fifteen 
 minutes, and especial care should be taken to note this time 
 accurately. 
 
 On removing the plate B from the voltameter after the de- 
 posit has been made, the various operations already given in 
 regard to thoroughly cleaning and drying it are to be repeated. 
 Care must also be taken while doing this to handle the plate 
 very carefully, else portions of the copper may be detached. 
 
 The amount of metal deposited is found by again weighing 
 the plate. If W is this amount in grams, the average intensity 
 of the current passing during the test is given in amperes by 
 
 TJT 
 
 C= where / is the number of seconds the circuit was 
 
 tx. 000328 
 
 complete, and .000328 grams is the electro-chemical equivalent 
 of copper. 
 
 Theoretically, it should make no difference in which direction 
 the current is passed through the voltameter, as the plate B in 
 the one case should lose the same amount that it gains in the 
 other. It is found, however, that secondary chemical actions 
 take place in the neighbourhood of the positive plate, which 
 cause a loss in it without a corresponding deposit on the nega- 
 tive electrode. ' For this reason the amount of metal deposited 
 rather than that dissolved is taken as a measure of the current 
 passing. 
 
 A modification of the voltameter just described is obtained 
 by using spiral coils instead of plates for electrodes. The two 
 coils being of different sizes are placed one within the other in 
 the solution, the negative generally being placed in the center. 
 This form of the instrument possesses the advantage of present- 
 ing a very large surface to the action of the liquid. 
 
 In finding the strength of a current by the amount of silver 
 deposited, Poggendorff's voltameter is generally used. The nega-
 
 222 EXPERIMENTAL PHYSICS. 
 
 tive electrode is a platinum bowl resting on a metallic plate to 
 which the leading wire is attached. 
 
 The solution is placed in this vessel, and the positive elec- 
 trode, which is a disc of silver, is suspended in it so that its 
 edge is equidistant from the sides and the bottom of the bowl. 
 
 It is customary to surround the silver disc by a small bag 
 made of fine gauze or filter paper, so that the particles of metal 
 which are separated by other than electrolytic action may be 
 prevented from being deposited on the platinum. 
 
 The solution used, in this voltameter contains from fifteen to 
 twenty per cent of silver nitrate. 
 
 X. CALIBRATION OF GALVANOMETERS. 
 
 In Experiment VII. it has been shewn that when an electric 
 current is measured by passing it through a galvanometer whose 
 dimensions can be readily determined, its intensity is given 
 by a relation either of the form C=Ktan0, or of CK sin#, 
 according to the disposition adopted. 
 
 Many galvanometers, however, are constructed in such a 
 manner that although their deflections follow the tangent, or 
 the sine law, yet it is difficult and inconvenient to ascertain 
 the number of windings in the coil, and quite impossible to 
 obtain even an approximation to its mean radius. 
 
 The constant K, which depends on these quantities and on 
 the intensity of the terrestrial field, cannot then be calculated, 
 and must be determined experimentally. 
 
 METHOD I. A standard tangent galvanometer whose con- 
 stant KI is known is joined in series in a circuit with the one 
 examined. 
 
 If for a given current the deflections of the two instruments 
 are respectively l and 6, then we have
 
 CALIBRATION OF GALVANOMETERS. 
 
 223 
 
 as a relation for determining the constant. The value of K 
 should be found by taking the mean of a number of readings 
 obtained by varying the resistance in the circuit, and by revers- 
 ing the direction of the current through the galvanometers. 
 When this method is followed, care must be taken to place the 
 instruments at a considerable distance apart, else the needle 
 of the one may be affected by the magnetic field of the other. 
 
 METHOD II. In this method the standard galvanometer of 
 the last is replaced by a copper or a silver voltameter of one of 
 the forms described in Experiment IX. The current passed 
 through is kept as steady as possible by means of a variable 
 resistance, and any variations which do occur are corrected by 
 taking the mean of the deflections observed at stated intervals 
 during the test. 
 
 If 6 is the mean deflection, the constant is given by 
 
 K= , where M is the amount of the metal deposited in 
 
 zt tan 6 
 
 grams, z its electro-chemical equivalent, and / the number of 
 seconds the current was passing. For very exact determina- 
 tions the test should be continued for at least two hours, and 
 the circuit so arranged that the deflection on the galvanometer 
 is about 45, this being the deflection which gives the most 
 accurate results. 
 
 METHOD III. This, which is a very simple method, and 
 probably the most rapid, is a direct application of Ohm's law. 
 A battery or cell of constant electromotive force is joined up 
 in circuit with the galvanometer, and the current is varied by 
 the insertion of one or more of a set of graduated resistances. 
 If the electromotive force of the cell is denoted by E, and the 
 deflection of the galvanometer corresponding to a given current 
 
 intensity is 0, the constant is given by K= -, where R is 
 
 J\. tan 6 
 
 the total resistance in the circuit. Usually the inserted resist- 
 ance is very high, and in that case the resistances of the bat- 
 tery and the galvanometer may be neglected ; otherwise these 
 must be determined by some one of the methods given later.
 
 22 4 
 
 EXPERIMENTAL PHYSICS. 
 
 It frequently happens that the current cannot be readily 
 reduced to the proper intensity by the insertion of resistances 
 directly in the circuit. The galvanometer should then be pro- 
 vided with a variable shunt, since by suitably adjusting it any 
 desired modification of the intensity can be easily obtained. 
 The current in the main circuit being known, that passing 
 
 through the galvanometer can 
 then be calculated by the law of 
 divided circuits. 
 
 In many experiments the in- 
 vestigation does not consist in 
 ascertaining the intensity, but 
 rather in detecting the presence 
 of weak currents, and in deter- 
 mining their direction. 
 
 Galvanometers which are used 
 for this purpose are of special 
 construction, and are so designed 
 as to be extremely sensitive. 
 
 One of the most useful of this 
 class is that known as Thomson's 
 (Fig. 26). 
 
 The magnets which form an 
 astatic couple are very small, and 
 are connected by a fine strip of 
 
 aluminium, which is itself suspended from the support V by a 
 strand of unspun silk. This strip carries a thin plate of mica for 
 damping the oscillations of the magnets, and a small concave 
 mirror attached to it serves to indicate the sense of the deflections. 
 The coils, which are so arranged that a magnet is at the 
 center of each, are divided into two halves, and these are sup- 
 ported on two brass plates, one of which, LL', is shewn in the 
 figure. The binding poles BB' serve to connect the two parts 
 of each coil. 
 
 A large steel bar feebly magnetized is supported above the 
 
 Fig. 26.
 
 CALIBRATION OF GALVANOMETERS. 
 
 instrument, and as it can rotate about the vertical, or slide up 
 and down, the earth's magnetic field and the sensibility of the 
 instrument can be modified as desired. 
 
 On account of the great sensitiveness of this instrument it 
 is generally used in experiments of precision. It however 
 requires very delicate han- 
 dling, and must be far re- 
 moved from the presence 
 of moving masses of iron. 
 
 Another form of galva- 
 nometer which is very sen- 
 sitive, but which is more 
 stable, is that of Depretz 
 and D'Arsonval. It is 
 shewn in Fig. 27. 
 
 It is the coil which 
 moves in this instrument 
 and not the magnet. The 
 former is made up of a 
 number of turns of very 
 fine wire, carries a small 
 
 concave mirror, and is suspended by two silver wires between 
 the poles of a horseshoe magnet, as shewn in the figure. These 
 wires also serve to lead the current to and from the coil. A 
 tube of soft iron, which is held within the coil, and moves with 
 it, serves to increase the intensity of the magnetic field. 
 
 Owing to induction the damping in this instrument is so 
 rapid that when a current is passed through the coil it almost 
 immediately assumes its position of rest. 
 
 When observations are made with these galvanometers two 
 different arrangements can be made. For very exact work, that 
 shewn in Fig. 13 is adopted. There a scale is placed in front 
 of the mirror, and the images of its graduations are observed 
 by means of a telescope whose axis lies in the perpendicular 
 from the center of the mirror on the scale. 
 
 Fig. 27.
 
 226 
 
 EXPERIMENTAL PHYSICS. 
 
 Ordinarily, when rapid determinations are necessary the scale 
 is ruled on some translucent substance such as celluloid, and is 
 supported on a screen. In this screen there is a small opening 
 immediately below the scale, with a dark thread stretched verti- 
 cally across it. When this opening is illuminated with parallel 
 rays of light from some source an image is formed by the 
 mirror on the scale, and any displacement of this image will at 
 once indicate the deflections of the instrument. In order to 
 have a distinct image when the latter method is adopted, it is 
 best to place the scale at a distance from the mirror equal to 
 the length of its radius of curvature. 
 
 Fig. 28. 
 
 Galvanometers of this class may also be calibrated to indi- 
 cate absolute intensity. Different currents are passed through 
 the instrument, and their strengths are determined by one of the 
 methods given above. Curves are then plotted by taking the 
 deflections as indicated by the scale graduations for abscissae, 
 and the corresponding currents for ordinates. By a reference 
 to these curves, the intensity of a current can readily be deter- 
 mined when the deflection it produces is known. 
 
 In Fig. 28 there is shewn one of a class of direct reading 
 galvanometers, usually called ammeters, since they are graduated 
 to give the value of the current strength in amperes. The 
 principle of this instrument can be easily understood by a refer- 
 ence to Fig. 29.
 
 GALVANIC BATTERIES. 
 
 227 
 
 A powerful magnetic field is formed by two semicircular per- 
 manent magnets, whose like poles are A A' and BB\ A double 
 coil of wire CO is fixed obliquely between the two pairs of 
 poles, and at its center there is 
 pivoted a small bar of soft iron, 
 which carries the pointer indicated 
 in Fig. 28. This bar always takes 
 up a position with its axis coin- 
 ciding with the direction of the 
 lines of force, and when a current 
 is passing through the coil, the mag- 
 netic field which it produces modi- 
 fies that of the magnets, and the 
 bar is therefore deflected. 
 
 In these instruments the inten- 
 sities corresponding to different 
 deflections are recorded on the dial ; and as the accuracy 
 of the graduations depends on the magnets retaining their 
 initial strength, they should be tested from time to time, and, 
 if necessary, recalibrated. The range of the instrument is 
 extended by attaching to it, by the brass strips m and n, a shunt 
 CD (Fig. 28), whose resistance bears a simple ratio to that of 
 the instrument proper. 
 
 Fig. 29. 
 
 XI. GALVANIC BATTERIES. 
 
 The following article is devoted to short descriptions of a 
 few of the cells commonly used in electrical laboratories. No 
 attempt is made to trace the development of this part of the 
 subject, and complete details are omitted, as these can be found 
 in any one of the many treatises dealing at length with this 
 department of electrical investigation. 
 
 A few exercises are appended, however, for the purpose of 
 illustrating fundamental notions in regard to the action of cells 
 and the flow of currents. Table XX. contains the electromotive
 
 228 
 
 EXPERIMENTAL PHYSICS. 
 
 forces of a number of elements, and Table XIX., their resistances. 
 As the latter depend very much on the dimensions of the cell 
 and on the strengths of the solutions used, the results are, 
 of course, only approximate. 
 
 When possible, the laboratory should be provided with a 
 dynamo, as currents generated in this way, especially if of 
 strong intensity, are produced at a much smaller cost than by 
 the use of batteries. 
 
 I. Daniell's Cell. Meidinger type. In this cell there is an 
 outer glass vessel made of two cylinders connected by a shoul- 
 der, a glass tumbler resting on 
 the bottom of this vessel, and an 
 inverted flask whose volume is 
 about two liters. 
 
 In setting up the cell, the posi- 
 tive pole, consisting of a copper 
 coil or a thin plate of that metal 
 rolled into a cylinder, is placed in 
 the tumbler and then covered with 
 a saturated solution of copper sul- 
 phate. The flask, after being filled 
 as far as possible with crystals of 
 this substance, and the balance of 
 the space with the saturated solu- 
 tion, is inverted, and placed with 
 its mouth under the surface of the liquid in the tumbler. The 
 negative pole, a cylinder of zinc, is next placed in position as 
 shewn in the figure, and after both tumbler and flask are lowered 
 into the outer vessel a ten per cent solution of zinc sulphate is 
 gently poured into the latter by means of a rubber tube until it 
 covers the zinc plate. Owing to the density of this liquid being 
 less than that of the copper sulphate, it will, when not agitated, 
 float on the top of the latter in the tumbler without mixing. 
 Wires to which clamps may be attached lead from the two 
 
 Fig. 30.
 
 GALVANIC BATTERIES. 
 
 22 9 
 
 electrodes, that from the positive being insulated by gutta- 
 percha in order to prevent its being eaten away by chemical 
 action at the surface of separation of the liquids. 
 
 In this cell, the sulphuric acid acting on the negative elec- 
 trode forms zinc sulphate and liberates hydrogen. The latter 
 unites with the copper sulphate to form sulphuric acid, which 
 is again ready to react upon more zinc, while the copper, which 
 is set free, is deposited on the positive plate. 
 
 The reactions are expressed in the following equations : 
 
 Zn + H 2 SO 4 = ZnSO 4 + 2 H, 
 2H + CuSO 4 =Cu + H 2 SO 4 . 
 
 In mounting this cell, and, in fact, all cells, care should be 
 taken to have all the chemicals pure, and to use only distilled 
 water in making the solutions. Ordinary commercial zinc con- 
 tains many impurities, such as iron and lead, so that when a 
 plate of such metal is used in a cell these particles form with 
 the adjacent zinc a series of small voltaic elements, and local 
 actions are set up which waste away the zinc without doing any 
 useful work. This waste may be avoided, and the plate made 
 to act practically as if it were pure by covering it with an 
 amalgam. This is done by plunging it for a few seconds in 
 dilute sulphuric acid, and then rubbing it over with mercury. 
 After doing this it should be thoroughly washed before being 
 used. Instead of having the positive plate of this cell all of 
 copper, it is sometimes replaced by one of lead which has been 
 copper-plated. This modification possesses the advantage* of 
 readily permitting the removal of the copper deposit. With 
 ordinary usage a cell set up as described will last without 
 renewal for about a year, and as it has a very constant electro- 
 motive force, and produces a very steady current, it is admira- 
 bly adapted for laboratory work. 
 
 II. Grove Cell. This element in its common form consists 
 of a rectangular outer vessel AE, made of porcelain, or ebonite,
 
 230 
 
 EXPERIMENTAL PHYSICS. 
 
 ;i U-shaped plate B of amalgamated zinc, a porous cup C, and a 
 
 thin strip of platinum D. 
 
 Sulphuric acid diluted in the ratio of one to ten is placed in 
 
 the outer vessel, and concentrated nitric acid in the porous cup. 
 The platinum and the zinc form the posi- 
 tive and the negative poles of the cell 
 respectively. The following equations ex- 
 press the reactions for the element : 
 
 4 + 2H, 
 
 
 When the cell has been allowed to act 
 for some time the nitric acid becomes 
 used up by the chemical action, and the 
 sulphuric acid mixed with zinc sulphate. 
 
 There is a consequent increase in the internal resistance, but 
 since this cell has a high electromotive force, it is capable of 
 producing a strong current for a considerable time. 
 
 The U form of the zinc is adopted in order to thereby reduce 
 the distance between the electrodes, but it is not very economical 
 as the zinc is almost invariably eaten away at its lowest point 
 first, and a large quantity of the metal is thereby rendered 
 useless. 
 
 The chief objections to an extensive use of Grove cells arise 
 from the high price of platinum, the injurious effects produced 
 by the nitric peroxide gas given off, and from the nitric acid 
 diffusing through the porous cup and acting directly on the 
 zinc. The element is, however, an extremely convenient one, 
 and is generally used when a strong current is required. 
 
 III. Grenet Cell. This cell as ordinarily constructed con- 
 sists of a large glass bottle, whose volume is about two liters, 
 and whose shape is similar to that shewn in Fig. 32. A cover 
 
 * The oxidizing action may go much beyond this phase, and can be represented 
 by the equation 2 H% HNO 3 = H 2 O + HNO,.
 
 GALVANIC BATTERIES. 
 
 231 
 
 of ebonite rests on the brass collar surrounding the upper 
 portion of the neck, and carries two parallel slabs C and C of 
 carbon which form the positive plates of the cell. The negative 
 plate Z is made of amalgamated zinc, and is attached to a brass 
 rod a which can slide freely through a socket 
 in the cover, and so permits this electrode to be 
 raised out of the liquid when the cell is not in 
 use. 
 
 A piece of ebonite attached to the inside 
 of each of the carbon plates serves to guide 
 the zinc plate when it is being lowered, and 
 at the same time prevents contact. 
 
 The solution for the cell is composed of 
 bichromate of potash, sulphuric acid, and water. 
 Various proportions have been suggested, but 
 the following composition is considered one of Fig 32 . 
 
 the best : 
 
 Water, 
 
 80 parts by weight. 
 
 Pulverized bichromate of potash, 12 
 Sulphuric acid, 36 
 
 On mixing these together chromic acid and potassium sul- 
 phate are formed, and when the zinc is lowered and the current 
 established the chemical action consists in the chromic acid 
 combining with the sulphuric to form chromium sulphate, and 
 the oxygen, which is then set free, uniting with the hydrogen 
 to form water. 
 
 The reactions are as follows : 
 
 H 2 O + H 2 S0 4 + K 2 Cr 2 O 7 = K 2 SO 4 + 2 H 2 CrO 4 , 
 3 Zn + 3 H 2 S0 4 = 3 ZnSO 4 + 6 H, 
 
 6H + 3 H 2 S0 4 + 2 H 2 Cr0 4 = Cr 2 (SO 4 ) 3 + 8 H 2 O. 
 
 When fresh the solution is of an orange colour, but as the 
 potash bichromate becomes used up it changes to a green
 
 232 
 
 EXPERIMENTAL PHYSICS. 
 
 or blue owing to the formation of chromic sulphate which 
 combines with the potassium sulphate to form chrome alum.* 
 As a consequence the current becomes rapidly weaker and 
 more bichromate should then be added. If, however, the 
 action becomes weak without this change of colour, either the 
 supply of sulphuric acid has become exhausted, and should 
 be renewed, or the carbons have become coated with hydrogen 
 bubbles. These may be removed either by the chemical action 
 of the solution if the circuit be broken for a short time, or 
 by violently shaking up the liquid. 
 
 This element when it is first set up has a high electromotive 
 force. It is well suited for the excitation of induction coils, but 
 polarizes very rapidly. In some forms of the bichromate ele- 
 ment the zinc plate is not movable, and local action is pre- 
 vented by separating it from the chromic acid by means of a 
 porous cup. 
 
 IV. Leclancht Cell. The Leclanche" cell in its original form 
 is represented in Fig. 33, where A is a square glass vessel, B, 
 the negative electrode, is a rod of amalgamated zinc, C, a porous 
 
 Fig. 33. 
 
 Fig. 34. 
 
 cup, contains equal quantities of peroxide of manganese and 
 small pieces of carbon, and D, the positive plate, is a strip of 
 carbon inserted in the mixture. 
 
 The exciting liquid is a saturated solution of sal ammoniac, 
 *KCr(SO 4 ) 2 + I2H 2 O.
 
 GALVANIC BATTERIES. 
 
 233 
 
 which about half fills the vessel A. When the current is pass- 
 ing it acts on the zinc, and forms hydrogen, ammonia, and zinc 
 chloride. The peroxide of manganese is slowly reduced to a 
 sesquioxide, and the oxygen given off unites with the hydrogen. 
 The following are the reactions : 
 
 2 HN 4 Cl + Zn = ZnCl 2 + 2 NH 3 + 2 H, (i) 
 
 2H + 2MnO 2 =Mn 2 O 3 -t-H 2 O. (2) 
 
 The current from this cell falls off very rapidly on closing the 
 circuit, but as the element has the property of rapidly building 
 itself up, it is exceedingly well adapted for open circuit work, 
 such as the ringing of electric bells. 
 
 The strip of carbon is generally surmounted by a lead cap, to 
 which the leading wire may be attached, and in order to prevent 
 the liquid rising by capillary action, and increasing the resist- 
 ance of the cell by its action on the lead, the upper part of the 
 strip is soaked in melted paraffine. This on cooling solidifies in 
 the pores, but on its being carefully removed from the outside 
 of the carbon, a good conducting surface is obtained, and, at the 
 same time, capillary action is prevented. 
 
 This cell possesses many important advantages over other 
 elements. It has a high electromotive force and a compara- 
 tively low resistance ; the materials used in its construction can 
 be obtained at a very low price ; it emits no noxious gases and 
 contains no poisonous liquids. The zinc is acted on by the sal 
 ammoniac only when the current is passing, and, as the liquid 
 freezes at a very low temperature, the element is much used in 
 very cold climates. 
 
 The cell as constructed now is shewn in Fig. 34. There the 
 porous cup is dispensed with, and the carbon electrode is held 
 between two plates made of carbon, manganese dioxide, and 
 shellac, by subjecting the mixture to very high pressures. The 
 zinc is also placed close to these plates, and the whole is held 
 together by rubber bands.
 
 234 
 
 EXPERIMENTAL PHYSICS. 
 
 This modification has a somewhat lower internal resistance, 
 and a slightly higher electromotive force than the cell as 
 originally devised. 
 
 V. Latimer-Clark 's Cell. Many efforts have been put forth 
 to devise some form of cell which would possess a constant 
 electromotive force, and could therefore be adopted as a stan- 
 dard for comparative purposes. That proposed by Latimer- 
 Clark more nearly satisfies the requirements than any other as 
 yet constructed. It is prepared by dissolving sulphate of zinc 
 in boiling distilled water. After cooling, this is decanted and 
 the saturated solution thus obtained is used to make a thick 
 paste with mercurous sulphate. This is heated to 100 C. in 
 order to expel all the air, and is then poured upon the surface of 
 warm mercury contained in some vessel of approved form. A 
 zinc plate is then suspended in this paste, and the whole sealed 
 in by pouring over it melted paraffine. The mercury and the 
 zinc are the positive and the negative electrodes respectively, 
 and each has a conducting wire leading from it, that from the 
 mercury being of platinum, and passing up through a zinc glass 
 tube embedded in. the mixture. The reaction for the Clark 
 cell is given by the relation 2 H + Hg 2 SO 4 = 2 Hg + H 2 SO 4 . 
 
 The electromotive forces of cells prepared in this way are 
 found to be exceedingly uniform, their variations not exceeding 
 .0005 volts. Owing to the possession of this property they are 
 admirably suited for comparative work, and are greatly superior 
 to standard Daniell cells, whose electromotive forces are found 
 to depend largely on the concentration of the solutions used. 
 
 Careful determinations have been recently made by Messrs. 
 Glazebrook and Skinner on this form of element, their experi- 
 ments involving the examination of hundreds of cells. The 
 results of their work shew that its electromotive force at 15 C. 
 is 1.4342 volts. In constructing cells of this class now the paste 
 is baked, so that on cooling it becomes very hard. This has the 
 effect of increasing the internal resistance to about 15,700 ohms.
 
 GALVANIC BATTERIES. 
 
 235 
 
 The cell can then be used in a closed circuit without any appre- 
 ciable diminution in the electromotive force. 
 
 EXERCISE I. Static charge. 
 
 Insert a plate of copper and one of amalgamated zinc in a 
 dilute solution of sulphuric acid, and attach a wire to each of 
 them ; connect to the lower disc of a condensing electroscope 
 the wire leading from the zinc, and to the upper disc that from 
 the copper plate. 
 
 Then remove the wires, and lift the upper disc by means 
 of the insulated handle. If the electroscope be very delicate, 
 the gold leaves will diverge, shewing that they are electrified. 
 The same phenomenon can be exhibited by interchanging the 
 wires and repeating the operations described. 
 
 The experiment, which may be conducted much more suc- 
 cessfully with a battery of three or four Grove cells, or a 
 dynamo, indicates that when the circuit of a current generator 
 is open, the two terminals are statically charged, and are at a 
 difference of potential. 
 
 If the laboratory is provided with a quadrant electrometer, 
 much better results can be obtained by using this instrument 
 instead of the electroscope. 
 
 EXERCISE II. Polarization. 
 
 Place, as in Exercise I., a plate of amalgamated zinc and one 
 of copper in a vessel containing a dilute solution of sulphuric 
 acid, and as soon as the circuit is made observe the current 
 strength as indicated by a sensitive galvanometer. It will begin 
 to diminish the moment the circuit is closed. If, after the cell 
 has been allowed to act for about five minutes the circuit be 
 opened for another five, it will be found on again closing it that 
 it has regained almost entirely its initial intensity. 
 
 In seeking the cause of this, one has only to watch closely 
 the action in the element. As soon as the current begins to 
 pass, hydrogen is deposited in little bubbles on the copper elec-
 
 236 EXPERIMENTAL PHYSICS. 
 
 trode, and, after some time, these form a gaseous layer between 
 the sulphuric acid and the plate. If the solution be shaken up, 
 or the plates agitated separately or together in the liquid, or the 
 surface of the copper plate be rubbed over with a brush, it may 
 be shewn that the diminution of the current strength depends 
 only on the presence of this gaseous layer, and that the dis- 
 appearance of the hydrogen, however brought about, from the 
 positive electrode, is always accompanied by an increase in 
 the current. 
 
 As the strength of the current depends only on the electro- 
 motive force of the cell and on its resistance, the presence of 
 the gas must affect one or both of these. That it does affect 
 the resistance is evident, since the active surface of the cop- 
 per is at once diminished when it is present. It will be seen 
 later that a diminution in the electromotive force also takes 
 place. 
 
 In order to prevent this polarization, as it is called, cells 
 generally contain some liquid or substance which will act 
 chemically on the hydrogen. Examples of this are found in 
 the use of nitric acid in the Grove, bichromate of potash in the 
 Grenet, and peroxide of manganese in the Leclanch6 cell. 
 
 EXERC i SE III. Resistance. 
 
 In this exercise a long rectangular wooden box, a plate of 
 copper and one of zinc, and a set of wires of homogeneous 
 structure and of constant cross-section, are required. A solu- 
 tion of sulphuric acid is put in the box, and the two plates are 
 also inserted in it at a known distance apart. Two of the wires 
 are then selected, and these, together with a sensitive galva- 
 nometer, whose coil is also made of this wire, are used to com- 
 plete the circuit. The initial intensity of the current produced 
 is noted and then modifications of this arrangement are made by 
 altering the distance between the plates, or the amount of their 
 surface immersed, and also by using different lengths of wires 
 and wires of different diameters.
 
 GALVANIC BATTERIES. 
 
 237 
 
 If the initial intensity of the current corresponding to each 
 disposition be carefully noted, it will be found that the results 
 obtained go to establish the law that the current produced by a 
 cell varies inversely as the resistance of the circuit including 
 that of the cell, and that the resistance of a uniform conductor 
 of any substance varies directly as its length and inversely as 
 its cross-section. 
 
 EXERCISE IV. Electromotive Force. 
 
 Many experiments establish the fact that when two electri- 
 fied bodies which are at a difference of potential are brought 
 close together, or are connected by a wire or other conductor, 
 a discharge or current passes between them, and they are in 
 consequence reduced to the same electrical state. 
 
 In Exercise I., it has been shewn that a difference of poten- 
 tial always exists between the terminals of a battery or cell in 
 open circuit. In this case, however, the current produced when 
 the terminals are joined does not consist of a single discharge, 
 and is not followed by the establishment of electrical equilib- 
 rium between the two electrodes. The current is continuous 
 and lasts as long as chemical action is kept up in the cell ; 
 while the difference of potential which this action produces 
 between the terminals of the battery when the current is estab- 
 lished is, owing to the existence of this current, intermediary 
 between the potential difference in open circuit and the zero 
 difference which obtains when the element ceases to act. 
 
 Owing to a battery or cell possessing this property of main- 
 taining its electrodes at a difference of potential when they are 
 connected by a conductor, it is said to be the seat of an electro- 
 motive force. 
 
 The electromotive force of an element which may thus be 
 considered to be the cause of the current is measured by the 
 greatest difference of potential which exists between its termi- 
 nals when the circuit is open or when the circuit contains a 
 resistance so great that practically no current passes.
 
 238 EXPERIMENTAL PHYSICS. 
 
 It is independent of the dimensions of the cell, and depends 
 only on the materials which enter into its composition. In 
 order to make clear that this is so, it will suffice to perform a 
 few experiments with cells of different types and of different 
 dimensions. 
 
 If two zinc-copper elements, of the type described in Exercise 
 I., be constructed exactly similar in every way, the currents 
 produced by them will be practically of the same intensity. If, 
 then, they be joined in opposition, i.e. so that they tend to send 
 currents in opposite directions in the same circuit, it will be 
 found that no current is produced. If, now, instead of having 
 the two cells of exactly the same dimensions, the plates of one 
 of them are made much larger than those of the other, the larger 
 cell will (the external circuit being the same) produce much the 
 stronger current. When the two cells are opposed to each 
 other, however, there will again be no current produced, owing 
 to the fact that, although the cells are of different dimensions, 
 their electromotive forces are the same. 
 
 Further, if, instead of performing this experiment with two 
 zinc-copper elements, one of these be used and an iron-copper 
 element of exactly the same dimensions be taken for the other, 
 it will be found that when these are joined in opposition there 
 is a current produced which takes the same direction as it would 
 if the zinc-copper element only were acting. 
 
 From this it follows that the electromotive force of the latter 
 is greater than that of the iron-copper element, and further that 
 the materials of which a cell is composed determines its electro- 
 motive force. In order to make this still clearer, the last experi- 
 ment can be modified by using a small zinc-copper element and 
 a very large iron-copper one. On joining these in circuit in 
 opposition, a current is again obtained whose direction is the 
 same as that of the current obtained in the preceding experi- 
 ment. Many similar exercises can be readily devised by using 
 different exciting liquids and plates of various other materials. 
 These, however, will be sufficient to substantiate the statement
 
 DETERMINATION OF RESISTANCE. 
 
 239 
 
 given above, that it is the materials used in t/ie construction of 
 an element, and not its dimensions, which determine its electro- 
 motive force. 
 
 The method adopted in these experiments can be applied 
 with advantage to shew that when polarization takes place in 
 an element, there is a consequent diminution in its electro- 
 motive force as well as an increase in its resistance. To do this, 
 it is only necessary to oppose a freshly prepared cell to one 
 exactly similar in dimensions and composition which has been 
 allowed to generate a current for some time. The resulting 
 current at once indicates that the former has the greater elec- 
 tromotive force. 
 
 XII. DETERMINATION OF RESISTANCE. 
 
 Numerous experiments shew that when two points, which are 
 kept at a constant difference of potential, are connected by a 
 conductor, the intensity of the current which passes depends 
 on the dimensions of the conductor, and on the substance of 
 which it is composed. 
 
 If the conductor is prismatic in form, and of uniform cross- 
 section, the strength of the current varies inversely as the 
 length and directly as the cross-section. Further, if different 
 currents are made to pass through the same conductor by vary- 
 ing the potential .difference of its terminals, the ratio of the 
 current strength to the corresponding potential difference is 
 within wide limits a constant quantity. This constant is termed 
 the resistance of the conductor. 
 
 These statements which represent the laws governing the 
 flow of electric currents are summarized in the relation called 
 
 Ohm's Law, C = > where C is the current intensity, E the 
 R 
 
 potential difference of the terminals of the conductor, and R its 
 resistance, which when the conductor is homogeneous is equal 
 
 to s > / being the length of the conductor, A its cross-section, 
 
 A 
 
 and s a constant called the specific resistance of the substance 
 composing it.
 
 240 
 
 EXPERIMENTAL PHYSICS. 
 
 A conductor therefore possesses an absolute unit of resist- 
 ance when a unit difference of potential between its terminals 
 causes a current of unit strength to pass through it. This unit 
 being inconveniently small for practical purposes, that chosen is 
 the ohm, which is equal to io 9 absolute units in the C. G. S. sys- 
 tem, and is the resistance of a conductor when a difference of 
 potential of one volt between its ends causes a current of one 
 ampere to flow through it. 
 
 The ohm is realized practically in the resistance offered to a 
 steady electric current by a uniform column of pure mercury 
 106.3 centimeters long and one square millimeter in cross- 
 section. 
 
 Standard ohms are usually constructed of mercury, or of 
 some metal whose resistance varies but little with slight changes 
 in temperature. 
 
 One form of a mercurial standard is shewn in Fig. 35. The 
 mercury is contained in a fine capillary tube, bent into a num- 
 ber of turns, and connected to large open 
 tubes, also partly filled with mercury. This 
 capillary tube is calibrated with extreme 
 care, and in determining its length allowance 
 is made for the resistance of the mercury 
 in the large tubes. When in use, it is sus- 
 pended as shewn in the figure, in a glass 
 vessel filled with pieces of melted ice. 
 
 These standards are extremely fragile, and 
 it is customary in laboratory practice to 
 replace them by those made of metal wire. 
 The wire, Fig. 36, is coiled in a double spiral, 
 and its ends being attached to two bent cop- 
 per rods A, B, it is then placed between two 
 brass cylinders E and C, and the intervening 
 Fig . 35- space is filled with paraffine. When the 
 
 standard is being used, the free ends of the 
 rods A, B are placed in mercury cups, which form part of
 
 DETERiMINATION OF RESISTANCE. 
 
 241 
 
 the circuit, the cylinder C is partly immersed in a water bath, 
 and the temperature of the wire is ascertained by means of a 
 thermometer inserted in the cylindrical opening D. The tem- 
 
 Fig. 36. 
 
 perature at which the standard is accurate is usually inscribed 
 on it, and having been thoroughly tested, the coil is accom- 
 panied by a statement shewing its variations, with changes of 
 
 Fig. 37. 
 
 temperature. In order that these may be small the wire is 
 usually made from an alloy of platinum and silver. 
 
 The resistance of a conductor is determined by comparison
 
 242 
 
 EXPERIMENTAL PHYSICS. 
 
 with others whose resistances are known. These which are ac- 
 curate copies of standards are generally made in coils, and are 
 arranged conveniently in boxes provided with some device for 
 rapidly throwing one or more of them in or out of circuit. Such 
 an arrangement is exhibited in Fig. 37, and by reference to Fig. 
 38, the manner in which the coils are inserted may be readily 
 seen. The wire corresponding to any particular resistance is 
 bent double at its middle point, and is wound as a double spiral 
 (to prevent induction) about either one or two bobbins, such as 
 A and B. These are attached to a plate of ebonite, which forms 
 the cover of the box, by means of two large brass or copper 
 screws, which pass completely through the ebonite, and have 
 
 their ends firmly embedded in two brass plates, C and D. These 
 plates, which form part of the circuit, are connected electrically 
 by soldering an end of the spiral to each of the screws. Plugs 
 of brass, such as E, are made to fit snugly into grooves provided 
 for them between the brass plates. When these are inserted 
 the current 'passes directly from one plate to the other without 
 going through the coil, since the plugs offer practically no resist- 
 ance to its passage. In order, therefore, to throw any particu- 
 lar resistance into circuit it is only necessary to remove the 
 corresponding plug. In the manufacture of these coils German 
 silver is extensively used, but for very high resistances they are 
 generally made from an alloy of platinum and silver.
 
 DETERMINATION OF RESISTANCE. 
 
 243 
 
 Resistances of Solids. METHOD I. By substitution. 
 
 In order to determine the resistance in question, it is placed 
 in the same circuit as a sensitive galvanometer, and a cell 
 or battery of constant electromotive force. When the current 
 has become steady, the deflection obtained is noted. The un- 
 known resistance is then replaced by a box of coils, and by with- 
 drawing the plugs resistances are thrown in circuit sufficient 
 to reestablish the original intensity of the current, the sum 
 of those so inserted being equal to that of the unknown. 
 
 The method is simple but primitive, and the results obtained 
 are only approximate. It is difficult to obtain a cell which will 
 remain constant during the experiment, and the resistance 
 boxes ordinarily used are not graded with sufficient delicacy 
 to be used in this way. 
 
 METHOD II. By ^lse of a differential galvanometer. 
 
 In a differential galvanometer there are two distinct coils 
 of equal resistance, which are so wound that when currents of 
 equal intensity are passed through them in opposite directions 
 no effect is produced on the needle. In determining the re- 
 sistance of a conductor, it is placed in circuit .with one of these 
 coils, and a rheostat or resistance box with the other. The 
 current from a battery is made to divide so that part goes 
 through one coil in one direction, and the remainder in the 
 opposite direction through the other. 
 
 The needle is then deflected, and suitable resistances are 
 thrown in circuit by means of the resistance box to bring the 
 needle back to its initial position. The sum of the resistances 
 so inserted is in this case also equal to that of the conductor 
 tested. This method possesses the advantage of being sensi- 
 tive and also of being independent of all variations in the inten- 
 sity of the current, but it is difficult to construct a galvanometer 
 which will have the properties assumed. 
 
 The accuracy of the instrument can be tested by noting 
 whether there is any deflection of the needle when the same
 
 244 
 
 EXPERIMENTAL PHYSICS. 
 
 current is passed through the galvanometer by one coil and 
 back by the other, and also when any given current is allowed 
 to divide freely between the two coils when their corresponding 
 terminals are connected. 
 
 METHOD III. By use of a tangent galvanometer. 
 
 A circuit is made so as to include a constant battery, a finely 
 graduated tangent galvanometer, a rheostat, and a known resist- 
 ance r, approximately equal to that of the unknown. 
 
 By means of the rheostat the intensity of the current is 
 modified so that the needle comes to rest, making an angle 
 of about 45 with the meridian. The unknown resistance x 
 is then added to the circuit, and when the needle again comes 
 to rest its deflection is noted. Finally, a reading is taken when 
 both r and x are removed from the circuit. If a, /3, and 7 are 
 the deflections in these cases respectively, we have, 
 E 
 
 G + r 
 E 
 
 (i) 
 (2) 
 
 G + r+x 
 
 | = ^tan 7 , (3) 
 
 where E is the electromotive force of the battery, and G the 
 resistance of the whole circuit exclusive of r and x. 
 From (i), (2), and (3) the relation 
 
 r+x _tan tan 7 tan /3 
 r tan/3 tan 7 tana 
 
 is obtained, and from it x can be calculated. 
 
 METHOD IV. The Wheatstone bridge. 
 
 While the methods previously given are of importance in 
 special cases, that generally adopted in laboratory practice 
 is the method of the Wheatstone bridge. Its sensitiveness is 
 limited only by that of the galvanometer, and it excels in admit- 
 ting of rapid and extremely accurate determinations.
 
 DETERMINATION OF RESISTANCE. 
 
 245 
 
 The principle of the method is illustrated in Fig. 39. 
 
 The main circuit of a battery E divides at A into two 
 branches, ABC and ADC, which contain the resistances a, x 
 and b, c, respectively. Between the terminals of these resist- 
 ances B and D a galvanometer G is inserted, and as a current 
 of short duration only is required, both this circuit and the 
 main one contain contact keys to permit the current to be 
 established or broken at will. 
 
 In making a test the unknown resistance x is inserted between 
 B and C, and the other resistances a, b, and c are so adjusted 
 
 Fig. 39. 
 
 that on depressing the key L, no current passes through the 
 galvanometer. The points B and D are then at the same 
 potential. Denoting it by V v and the potentials of the points 
 A and C by V and V z respectively, it follows from Ohm's law, 
 since the intensity of the current in each of the circuits ABC 
 and ADC is the same throughout its length, that 
 
 and 
 
 (i) 
 (2)
 
 246 
 
 EXPERIMENTAL PHYSICS. 
 
 Dividing each member of (i) by the corresponding one of (2), 
 we obtain the relation 
 
 ax 
 
 -=- or 
 be 
 
 If, therefore, the resistance of a or c, and either the resist- 
 ances of the other two, or their ratio, be known, that of x can 
 be calculated. It is best to take b equal to a at first, and then 
 to vary c until by trial it is approximately equal to x. By 
 
 Fig. 40. 
 
 then giving a a certain value, and gradually increasing b, that 
 of x can be determined with a degree of accuracy which is 
 limited only by the resistances available, and as already indi- 
 cated by the sensitiveness of the galvanometer. 
 
 A practical form of the bridge consists of a resistance box 
 containing three sets of coils. One of the best arrangements 
 is that shewn in Fig. 40, where a and b of the theoretical 
 diagram are represented by two sets of ten, one hundred, and 
 one thousand ohm coils. These are inserted between the brass
 
 DETERMINATION OF RESISTANCE. 
 
 247 
 
 plates bearing these numbers, and the two brass strips shewn 
 in the figure connected to A and H. In order to place any 
 one of them in circuit it is only necessary to insert a plug in 
 the corresponding aperture. The third resistance c is given a 
 wider range, and consists of thirty-six coils placed between B 
 and G, and graded in powers of ten so as to give any resistance 
 from i to 9999 ohms. 
 
 The unknown resistance is placed between A and B, the 
 galvanometer between F and E, and the battery between C 
 and D. The two keys, K' and K, correspond to those in 
 Fig. 39 indicated by K and L respectively, and the dotted 
 lines exhibit the paths of the current through the instrument. 
 
 Besides being exceedingly compact, this form of bridge 
 possesses the advantages of having a minimum number of 
 plugs, and of permitting readings to be made directly. In 
 determining a resistance care should be taken to depress 
 the battery key K' before that of the galvanometer K, and to 
 release them in the reverse order, else the disturbing effects 
 of induction currents will be experienced and unnecessary delay 
 be caused. 
 
 When a preliminary test is being made the galvanometer key 
 should not be pressed down firmly, otherwise the contact will 
 be good, and the galvanometer needle may be so violently 
 deflected as to break the suspending fiber. On account of its 
 constant electromotive force, a Daniell cell is peculiarly suited 
 for work on the Wheatstone bridge. If, however, such a cell 
 is not available, others may be used, and in case the current is 
 too strong part of it may be short-circuited by the insertion 
 of a shunt. 
 
 In comparing the conducting powers of different materials it 
 is necessary to know their specific resistances. In the case of 
 metals, or other solids, these can be readily determined for any 
 
 temperature by finding the resistances of long uniform wires or 
 
 jt 
 rods of the substances, and then applying the relation s = R
 
 248 EXPERIMENTAL PHYSICS. 
 
 Resistances of Liquids. In determining the specific resist- 
 ance of a liquid such as mercury a capillary tube is attached by 
 rubber tips to two small steel or wooden cups which have short 
 tubes protruding from their sides. 
 
 The liquid in one of these cups is forced through the capil- 
 lary tube, expelling the air, and is made to gradually fill up the 
 cup to which the other end is attached. 
 
 Terminal wires are then inserted in the mercury in the cups, 
 and the resistance, which will be practically that of the liquid in 
 the capillary tube, is determined by one of the methods given 
 above. It only remains to calculate the dimensions of this tube 
 and to apply the relation already given. 
 
 For very accurate results the tube must be carefully cali- 
 brated by the method adopted in the construction of ther- 
 mometers, but for ordinary purposes that explained on page 32 
 may be followed. 
 
 In order to insure a uniform temperature for making the test, 
 the cups and the capillary tube should be placed in a water or 
 oil bath. 
 
 In case the liquid is an electrolyte, a serious difficulty is met 
 with in the polarization of the electrodes. On passing a current 
 through such a liquid, its resistance is altered owing to changes 
 in its condition near the electrodes, and to the presence of the 
 chemical products deposited there. 
 
 The existence of an electromotive force opposed to that pro- 
 ducing the current still further complicates the problem, and 
 renders an accurate determination of the resistance difficult. 
 
 It has been found when the liquid examined is a solution of a 
 metallic salt, that if the electrodes are composed of the metal 
 forming the base of this salt the polarizing effects are small, 
 and the resistance of the liquid may be found approximately by 
 the ordinary methods. 
 
 One of the early devices adopted was to place the liquid in a 
 rectangular trough, and to have movable electrodes inserted in 
 it. The resistance of a certain length of the liquid was first
 
 TEMPERATURE COEFFICIENT OF RESISTANCE. 
 
 249 
 
 determined, and then the distance between the electrodes being 
 shortened by a known length, the current was restored to its 
 original intensity by means of a rheostat placed in the circuit. 
 The amount of the wire resistance thus added was taken as a 
 measure of the resistance of the liquid through which the 
 electrode was moved. 
 
 Alternating currents have been applied with considerable 
 success to finding the resistances of electrolytes. A Wheat- 
 stone bridge of special construction is used, but the theoretical 
 arrangements are the same as have already been described. 
 
 An electro-dynamometer, however, takes the place of the 
 galvanometer, and this instrument again is sometimes replaced 
 with advantage by a telephone. When the latter is used, and 
 the alternating currents are produced by means of a rapidly 
 vibrating induction coil, a humming sound is heard, which, 
 when the proper proportions obtain among the resistances, 
 weakens to a minimum and nearly disappears. 
 
 In this method polarization is avoided since the current is 
 passed through the liquid alternately in opposite directions. 
 
 XIII. TEMPERATURE COEFFICIENT OF RESISTANCE. 
 SLIDE WIRE BRIDGE. 
 
 The resistance of a conductor depends not only on its dimen- 
 sions, and on the substance of which it is composed, but also 
 on its temperature. Owing to the extensive use of resistance 
 coils for comparative purposes, it becomes therefore a matter of 
 great importance to select materials for their construction which 
 vary but little with temperature, and to know exactly the law 
 governing such variations. 
 
 The resistances of alloys such as platinum silver, German 
 silver, gold silver, and platinoid, increase much less rapidly with 
 a rise in temperature than do pure metals, while carbon forms a 
 notable exception to the general law for elemental substances 
 in that its resistance decreases as its temperature rises.
 
 250 EXPERIMENTAL PHYSICS. 
 
 In determining a law for this variation in the resistance of 
 metals a wire of suitable length of the substance in question 
 is selected, and a series of experiments are performed to ascer- 
 tain exactly its resistances at different temperatures. 
 
 By means of the results obtained the law is then exhibited 
 graphically by plotting a curve, taking temperatures for abscissae, 
 and resistances for ordinates ; but for the purposes of calcu- 
 lation it is better to follow the method of Least Squares, and to 
 establish an empirical formula connecting the resistances of the 
 conductor with its corresponding temperatures. 
 
 This relation is usually obtained in the form 
 
 in which R is the resistance of the wire at zero, and the 
 coefficients a and are calculated from the observed results. 
 
 As these coefficients depend only on the substance of the 
 wire, this relation can therefore be applied to ascertain the resist- 
 ance of any conductor of the same material at any given tem- 
 perature when its value at o C. is known. In order to arrive at 
 results which will be reliable, every precaution must be taken 
 to measure the resistances of the wire accurately, and to note 
 its temperatures with the greatest precision. 
 
 The wire, which should be well insulated, is wound on a thin 
 hollow bobbin made of wood, and is inserted in a thin glass 
 tube closed at one end. A delicate thermometer is also placed 
 in this tube, and the whole is suspended in a liquid bath which 
 should be provided with a stirrer, and a second thermometer to 
 check the readings of the first. 
 
 The resistance of the wire is generally determined by means 
 of a Wheatstone bridge, the connections being made by solder- 
 ing two stout wires to its terminals. 
 
 When a test is about to be made a Bunsen flame is applied 
 to the bath until it reaches a suitable temperature. It is then 
 withdrawn, and the liquid is stirred until the two thermometers 
 shew that its temperature has become steady. The resistance
 
 TEMPERATURE COEFFICIENT OF RESISTANCE. 
 
 251 
 
 is then determined, and the corresponding temperature of the 
 wire is found by taking the mean of the readings of the two 
 thermometers. More heat is then applied, and the same opera- 
 tions are repeated until the whole range of temperatures chosen 
 has been covered. 
 
 If it is desired to make this somewhat extensive, an oil bath 
 should be used, but as it emits an exceedingly offensive odour 
 it is better for ordinary purposes to use water instead of oil. 
 To increase the accuracy of the work the test should be con- 
 ducted with falling as well as rising temperatures. 
 
 The errors which necessarily accompany the results obtained 
 by using the ordinary form of the Wheatstone bridge cannot be 
 
 Fig. 41. 
 
 neglected in a delicate experiment such as this, and a simplified 
 form of the instrument known as the slide wire bridge is gen- 
 erally used. 
 
 In connection with Fig. 39, it has been shewn that when the 
 points B and D are at the same potential, the resistances are 
 
 connected by the relation x = ^c. In the slide wire bridge, a 
 
 b 
 
 single wire of homogeneous structure and of constant cross- 
 section takes the place of the resistances b and c, and in 
 determining that of x, the galvanometer terminal is moved 
 along this wire until a point D is reached, which is at the 
 same potential as that of B.
 
 252 
 
 EXPERIMENTAL PHYSICS. 
 
 Since the resistance of any portion of this wire is propor- 
 tional to its length, that of the unknown is given by x = -j<z, 
 
 *i 
 where /j and / 2 are respectively the lengths AD and DC of 
 
 the wire AC. 
 
 In its practical form, the bridge is shewn in Fig. 41. The 
 wire, which may be of brass or an alloy of platinum, is one 
 meter long, and is tightly stretched between the points C and 
 K. A heavy brass bar, on which is ruled a meter scale with 
 centimeter divisions, is so placed in front of this that the ter- 
 minals of the wire are opposite the limiting divisions of the 
 scale. 
 
 A slider A provided with a vernier, and with a key B for 
 making contact with the wire, can move freely along this bar, 
 and a series of brass strips, shewn in the figure, supply the 
 remaining connections. 
 
 The unknown resistance and a standard are introduced into 
 the circuit by means of four mercury cups, two of which are 
 of brass and two of ebonite, and the relative positions of these 
 can be reversed by means of the commutator D, constructed 
 of two U-shaped brass or copper rods, whose terminals also 
 rest in the mercury cups. 
 
 The battery terminals are attached to the bridge at E and F, 
 and those of the galvanometer to the binding poles G and //", 
 which are connected to the graduated brass bar and to the 
 central metallic strip supporting the mercury cups respectively. 
 The slider A can be clamped in any position, and the key is 
 so arranged that by depressing it a spring is acted on which 
 causes a rounded metallic knife-edge to press gently but uni- 
 formly against the stretched wire. 
 
 Best results are obtained when / x and / 2 are nearly equal, 
 and the unknown resistance, therefore, should be always so 
 selected that it is at o C. approximately equal to that of the 
 standard. 
 
 Care must be taken not to touch the wire or the other metal-
 
 GALVANOMETER RESISTANCE. 
 
 253 
 
 lie parts of the bridge by the hands during a test, otherwise 
 thermo-electric currents are at once set up, which greatly dis- 
 turb the readings. 
 
 The slide wire bridge is especially useful in the construction 
 of resistance coils which are intended to be very accurate 
 copies of standards. 
 
 XIV. GALVANOMETER RESISTANCE. 
 
 If the laboratory is provided with a second galvanometer 
 sufficiently sensitive, the coils of the one to be examined may 
 be treated as an ordinary conductor, and their resistance deter- 
 mined by one of the methods outlined in the last experiment. 
 
 If, however, a second instrument is not at hand or is not 
 conveniently adjusted, either the method of equal deflections 
 or that known as Thomson's may be applied, and in these 
 tests, the galvanometer whose resistance is required is itself 
 used as the current indicator. 
 
 (i) Method of equal deflections. 
 
 The arrangement is that shewn in Fig. 42. The current 
 which is supplied by a constant battery E divides at C, part 
 
 Fig. 42. 
 
 going through the galvanometer G, and part through a shunt 
 5. The main circuit also contains a variable resistance AB.
 
 254 
 
 EXPERIMENTAL PHYSICS. 
 
 Denoting the resistance of the galvanometer by g, that of the 
 shunt by s, and that of the battery and leading wires by r, 
 we have, when the resistance of AB is R lt the intensity of 
 the main current given by 
 
 c E 
 
 L' 9 / x 
 
 and that of the part passing through the galvanometers by 
 
 w 
 
 If now the shunt be removed from the circuit, and the re- 
 sistance of AB altered until the galvanometer indicates that 
 the original intensity of the current passing through it is 
 restored, the strength of the current is given by 
 
 where R z is the new value of AB. 
 
 Equating the right-hand members of (2) and (3), it follows 
 that the resistance of the galvanometer may be found from the 
 relation, 
 
 As the theory indicates, it is absolutely necessary for a suc- 
 cessful application of this method to use only a battery which 
 will remain constant for a considerable time. The method is 
 somewhat tedious, owing to the necessity of previously deter- 
 mining the internal resistance of the battery, and best results 
 are obtained when a battery is selected whose resistance is so 
 small that it can be neglected.
 
 GALVANOMETER RESISTANCE. 
 
 255 
 
 In that case, if the resistance of the leading wires is also 
 neglected, that of the galvanometer is given by 
 
 (5) 
 
 (2) Thomson s method. 
 
 This is an application of the principle of the Wheatstone 
 bridge, similar to that described in Experiment XII. The gal- 
 vanometer is disposed, as shewn in Fig. 43, between the two 
 points C and D, and a contact key is placed in the circuit 
 between C and B. 
 
 If, when the key is raised and a current is passing through 
 the galvanometer, the points C and B are not at the same 
 
 potential, then a current will pass between these points when 
 the key is depressed, and a consequent alteration in the deflec- 
 tion of the galvanometer will occur. 
 
 If, however, the resistances a, b, and c are so adjusted that 
 C and B are at the same potential, it is quite immaterial 
 whether the key is raised or depressed, as no current will pass 
 between these points, and there will therefore be no change
 
 256 EXPERIMENTAL PHYSICS. 
 
 in the intensity of the current passing through the galva- 
 nometer. 
 
 When this condition obtains the ordinary relation for the 
 equilibrium of the Wheatstone bridge applies, and the resistance 
 of the galvanometer is given by 
 
 The method is exceedingly simple, and is preferable to the for- 
 mer, in that it is quite independent of the resistance of the battery. 
 
 In case the battery used does not remain constant during 
 the experiment, the deflection of the galvanometer needle will 
 gradually change. This, however, will not affect the application 
 of the method, as the test consists entirely in noting whether the 
 intensity of the current passing through the galvanometer is 
 affected by raising or depressing the contact key. 
 
 In case the current through the galvanometer produces too 
 great a deflection, a shunt should be inserted between the points 
 A and D. By giving different values to this, the current pass- 
 ing through the bridge may then be suitably modified. 
 
 XV. RESISTANCE OF BATTERIES OR CELLS. 
 
 From the exercises on batteries in Experiment XL, it is 
 evident that a cell or battery acts in the same manner as an 
 ordinary conductor, and offers a resistance to the passage of the 
 current which it produces. 
 
 The value of this resistance depends to some extent on the 
 electrodes of the cell, on the amount of their surfaces submerged, 
 and on the distance they are apart. It also varies within very 
 wide limits with the composition of the battery solutions, and 
 with their concentration. 
 
 It, however, cannot be considered a definite quantity, since 
 polarization occurs in most cells as soon as the current is estab- 
 lished, and the resistance gradually increases, its value at any 
 instant depending both on the strength of the current then
 
 RESISTANCE OF BATTERIES OR CELLS. 257 
 
 passing and on the length of time the circuit has been closed. 
 In determining the value of this quantity, therefore, for a cur- 
 rent of given intensity, the result obtained can be taken only as 
 an approximation to its correct value under other circumstances ; 
 but such determinations, however, are sufficient for ordinary 
 purposes, and may be made by one or other of the following 
 methods : 
 
 (i) Ohms method. 
 
 This method is a direct application of Ohm's law to a com- 
 plete circuit, including the battery to be tested, a sensitive 
 galvanometer, and a variable resistance R. 
 
 Denoting the electromotive force of the cell by E, the resist- 
 ance of the galvanometer and leading wires by g, and that of 
 the cell by r, the relation 
 
 will represent the intensity of the current produced when R l is 
 the resistance given to R. On changing the variable resistance 
 
 to R v a second relation 
 
 is obtained ; and, assuming that E remains constant during the 
 test, it follows from (i) and (2) that 
 
 If the readings of the galvanometer follow the tangent law, 
 this relation becomes 
 
 _ pr 
 
 _ 
 
 tan #!- tan 2 
 
 in which d l and 2 are the deflections corresponding to the cur- 
 rents C^ and C z .
 
 2 5 8 
 
 EXPERIMENTAL PHYSICS. 
 
 In adopting this method a number of readings should be 
 taken by varying the resistance R, and in each case the results 
 should be checked by reversing the current through the galva- 
 nometer. 
 
 (2) Thomsons method. 
 
 As shewn in Fig. 44, the battery E, whose resistance is 
 required, is joined in series with a galvanometer G and a 
 variable resistance AB, while a shunt 5 is inserted between 
 the points F and H. 
 
 In applying the method a suitable deflection is first given to 
 the galvanometer needle by properly adjusting the resistances 
 AB and 5. The shunt is then removed, and the rheostat 
 
 IHitHi 
 
 Fig. 44. 
 
 altered, until the original intensity of the current passing 
 through the galvanometer is reestablished. 
 
 If in the first operation s and R l are the values given to vS 
 and to AB, and in the second R 2 that given to the rheostat, 
 the expressions for the current passing through G in the two 
 cases are, 
 
 and 
 
 (2)
 
 RESISTANCE OF BATTERIES OR CELLS. 
 
 259 
 
 By equating the right-hand members of these equations, the 
 relation r=^-f. ^ may be deduced, and from it the internal 
 
 resistance of the battery may be calculated. 
 
 (3) Mances method. 
 
 In applying the previous methods considerable time is re- 
 quired to make the two adjustments, and unless the element 
 examined polarizes very slowly, the results obtained cannot be 
 taken *as accurate. 
 
 In Mance's test, however, the time which elapses between 
 the two observations is exceedingly short, being merely that 
 spent in closing a circuit, and it is therefore not sufficient for 
 the battery resistance to undergo any perceptible change. For 
 
 this reason this method is generally followed in determining 
 the resistance of a cell, or battery, which polarizes very rapidly. 
 The arrangement is that shewn in Fig. 45. There the cell 
 is inserted in one of the arms of the bridge, the galvanometer 
 in one of its diagonals, and a coarse wire of very small resist- 
 ance, containing a contact key in the other.
 
 2 6o EXPERIMENTAL PHYSICS. 
 
 The test consists simply in so adjusting the resistances a, b, 
 and c that no difference is observed in the strength of the 
 current passing through the galvanometer when the key is 
 raised or depressed. 
 
 When the resistances are so disposed that the proper con- 
 
 ditions obtain, they are connected by the relation r =7 c, where 
 
 b 
 
 r is the resistance of the cell. 
 
 This may be shewn by considering the currents passing 
 through the galvanometer under the two arrangements. .In the 
 first the current divides at B, and takes the paths BAG and 
 BC to reach the point C. The strength of the current in 
 this case is given by 
 
 E 
 
 In the second, depressing the key amounts to the same thing 
 as bringing the two points D and A together. Denoting this 
 hypothetical point of union by DA, the current may be con- 
 sidered as dividing at B, and then passing partly by BA to 
 DA, and partly by BC, and its subdivisions CD and CA to the 
 same point, and thence back to the battery. 
 
 Under these circumstances the current intensity is given by 
 
 be 
 
 
 If, then, Ci is equal to C v it will be found on equating the 
 right-hand members of (i) and (2) that the resistances satisfy 
 the relation 
 
 ~f<- 
 
 This can also be readily seen by considering the intensities 
 of the currents' in the various branches of the bridge when the
 
 DETERMINATION OF ELECTROMOTIVE FORCES. 26 1 
 
 key is depressed, as being the resultant of two steady systems, 
 the first being that which exists when the key is raised, and the 
 second being some system in which the points B and C are at 
 the same potential. As is evident from Method IV., Experi- 
 ment XII., this can be considered to be an arrangement in 
 which an electromotive force is inserted in the circuit AKD, 
 and in which consequently the resistances of the arms of the 
 bridge are connected by the formula 
 
 rrf* 
 
 In applying this method it will be found that in most cases 
 the galvanometer deflection will be inconveniently large. This 
 defect may be remedied by shunting the galvanometer, or if 
 the instrument is provided with a directing magnet, by so dis- 
 posing it that it will act in an opposite direction to that of 
 the current, and so reduce the deflection. 
 
 If the laboratory is provided with a potential galvanometer, 
 or finely graduated voltmeter, the resistance of a cell may be 
 determined by observing the differences of potential between 
 the terminals of the element first in open circuit, and then 
 when the circuit is completed by a small known resistance R. 
 Denoting these differences of potential by E and V respectively, 
 it follows from Ohm's law that 
 
 E = V 
 r+R R 
 
 or that 
 
 DETERMINATION OF ELECTROMOTIVE FORCES. 
 
 In Experiment XL a few exercises are indicated as being 
 suitable for giving clear and accurate notions regarding the 
 electromotive force of a cell or battery. The following article 
 is devoted to a description of some of the best methods that may 
 be followed in determining this constant for a given element.
 
 262 EXPERIMENTAL PHYSICS. 
 
 As already stated, the electromotive force of a cell is meas- 
 ured by the greatest difference of potential between its terminals 
 in open circuit. The unit potential difference in the C. G. S. 
 system is inconveniently small for practical purposes, and that 
 generally adopted is the volt. It is equal to io 8 units in the 
 C. G. S. system, and is the difference of potential that must be 
 maintained between the terminals of a conductor of one ohm 
 resistance in order that the intensity of the current produced 
 in it may be one ampere. 
 
 While the first three methods are for comparative determina- 
 tions by reference to some standard cell whose constant is 
 known, the following ones indicate how an absolute measure of 
 the electromotive force of an element may be obtained in terms 
 of resistance and current intensity. 
 
 I. Comparative determination of electromotive forces. 
 
 (i) W/teatstones method. 
 
 A standard cell, such as that of Daniell or Clark, is joined in 
 simple circuit with a galvanometer, and a rheostat, or resistance 
 box. The latter is adjusted to produce a convenient deflection 
 of 0, and then a known resistance r is added to the circuit, and 
 a new deflection of 6^ is obtained. 
 
 The standard is next replaced in the circuit by the cell, whose 
 electromotive force is required, and by means of the rheostat 
 the current is modified until the initial deflection of 6 is repro- 
 duced. It only remains then to determine what resistance 1\ 
 must now be added to reestablish the second deflection of 0^, 
 and from these known quantities the unknown electromotive 
 force -*! can be calculated in terms of that of the standard E. 
 
 Denoting the resistances of the whole circuit in the two cases, 
 corresponding to the deflection of 6 by R and R I} it follows 
 from Ohm's law, that 
 
 <> 
 andthat
 
 DETERMINATION OF ELECTROMOTIVE FORCES. 263 
 By combining these we have, 
 
 and therefore the electromotive force E 1 is - 1 times that of the 
 standard. 
 
 The method is simple, and is very accurate if the cells used 
 are constant. If not, however, the method cannot be success- 
 fully applied. 
 
 (2) Lnmsden or Lecoine's method. 
 
 The arrangement for this method is shewn in Fig. 46. The 
 standard A and the cell to be tested, B, are mounted in series 
 
 i 
 
 1 
 
 Fig. 46. 
 
 with their opposite terminals connected in a circuit, which also 
 contains the variable resistances E and F. 
 
 A shunt, which includes a sensitive galvanometer, connects 
 the points C and D. 
 
 In applying this method a certain value is given to one of the 
 adjustable resistances, and then the other is varied until no 
 current passes through the galvanometer. If E and E denote 
 the electromotive forces of the cells A and B respectively, and 
 r and ^ their internal resistances, these quantities are connected 
 by the relation 
 
 =^7- W
 
 264 EXPERIMENTAL PHYSICS. 
 
 R and R 1 being the resistances of E and F respectively (in- 
 cluding that of their connecting wires). This will be readily 
 seen from a consideration of the following theory : In all such 
 cases as this where the circuit has many branches, and contains 
 a number of electromotive forces, the principle of superposition 
 is to be applied. Each electromotive force in combination with 
 the others has precisely the same effect as it would have were 
 it the only one in the circuit, and hence the problem of deter- 
 mining the current passing in any one of the branches can be 
 greatly simplified by considering the action of each of the cells 
 separately. 
 
 In the above arrangement the two cells tend to send a cur- 
 rent through the galvanometer circuit in opposite directions, and 
 since no deflection is produced in this instrument, the currents 
 which they would send through, were they acting separately, 
 must then be equal. 
 
 That which would pass through it, were the standard the only 
 one acting, is given by 
 
 E r + R 
 
 r\ R | 
 
 while if the cell B were the only one producing a current, the 
 amount which it would send through the galvanometer is given 
 by 
 
 1 _ r+R 
 
 l+ ^+*) 
 
 1 g+r+R 
 If, then, C is equal to C v it follows that 
 
 or that 
 
 Z7 ., i T? 
 
 (3) 
 
 If, now, r and ^ are known, E can be at once found in terms 
 of E, but if not, a second observation is made by increasing R
 
 DETERMINATION OF ELECTROMOTIVE FORCES. 
 
 26 5 
 
 by some value p, and then determining what resistance must be 
 added to R to again reduce the current in CD to zero. Denot- 
 ing this by p lt we have, 
 
 E r+R+p ' 
 and combining (3) and (4), it follows that 
 
 This method, just as the preceding one, can only be ap- 
 plied successfully to constant elements. If, however, one of 
 the cells is constant, it affords a means of studying the proc- 
 ess of polarization in the other. The special advantage of 
 the method is that both cells are working under exactly the 
 same conditions. 
 
 (3) Poggendorjfs method. 
 
 This, which is probably the most exact of all the methods 
 devised, involves a new idea, that of determining the electro- 
 motive force of an element by balancing it against a potential 
 difference maintained in a circuit by means of a standard cell. 
 
 -. - Hi - 
 
 K 
 
 Fig. 47. 
 
 As Fig. 47 indicates, the standard A is joined in series with 
 a rheostat F, and two carefully constructed sliding resistances 
 L and M, B the cell to be tested together with a galvanometer
 
 2 66 EXPERIMENTAL PHYSICS. 
 
 G, being inserted as a shunt between the points C and D in such 
 a manner that its current tends to oppose that produced by the 
 standard cell. Contact keys are provided at K and H, and the 
 various connections of the circuit are made of stout brass or 
 copper strips, in order that their resistance may be negligible. 
 
 When the arrangements described have been made the rheo- 
 stat F is varied until on depressing the contact keys the galva- 
 nometer shews that no current is passing along CD. 
 
 This condition implies that a difference of potential exists 
 between the points C and D exactly equal to the electromotive 
 force of the cell B. 
 
 The accuracy of this adjustment can be tested by first giving 
 the rheostat F a value slightly above that which it then has, 
 and afterwards one slightly below it. This should result in the 
 galvanometer indicating a current passing first in one direction, 
 and then in the other along CD. 
 
 After this, a known resistance R is thrown in circuit by 
 means of the sliding resistance M, and then a corresponding 
 amount R l is added at L to reproduce the condition of zero 
 current in CD. 
 
 The electromotive force of the element B is then given in 
 terms of that of the standard E by the relation 
 
 EI=E R 
 
 This follows from first applying Ohm's law to the whole 
 circuit ACKD, and then to the part CKD. Denoting the 
 potential difference between C and D by V^ F^, the resist- 
 ance of F by p, and that of the battery A by r, we have, for 
 the first adjustment, 
 
 E V V,, 
 
 and for the second, 
 
 r+p + R + R~ p+R
 
 DETERMINATION OF ELECTROMOTIVE FORCES. 267 
 Combining (i) and (2), we have, 
 
 ''R + Ri 
 
 and since the potential difference V-^ V^ is equal to the re- 
 quired electromotive force E l the relation given previously is 
 evident. 
 
 Owing to the determination being made with the cell practi- 
 cally in open circuit, errors due to polarization cannot arise, and 
 the method can therefore be readily applied to all classes of 
 cells, including those which polarize even very rapidly. The 
 standard selected should be a constant element, but even should 
 it vary, errors due to this cause may be avoided by closing the 
 circuit only for short intervals at each test. In case the galva- 
 nometer used is a reflecting one, it will suffice to merely depress 
 the keys K and H for an instant. If the cell tested is not 
 properly balanced, the momentary contact will be sufficient to 
 indicate this. 
 
 Like the preceding, this method is well adapted for studying 
 the effects of polarization in a cell by measuring its electro- 
 motive force after it has been allowed to produce currents of 
 different intensities for varying intervals of time. 
 
 As the theory indicates, the electromotive force of the stand- 
 ard must be higher than that of the element tested ; if, how- 
 ever, it is lower, the method can still be applied by joining a 
 number of standard cells in series until an electromotive force 
 sufficiently high is obtained. 
 
 (4) Clark's method. 
 
 This method is based on the same principle as the last, but it 
 is so modified that the standard and the unknown element are 
 compared under precisely similar circumstances. 
 
 The disposition is that outlined in Fig. 48. The current is 
 produced by a constant battery C, of resistance r, whose circuit 
 CDEF contains a rheostat M, and a sliding resistance EF.
 
 268 
 
 EXPERIMENTAL PHYSICS. 
 
 The standard A, together with a galvanometer, is inserted as 
 a shunt between the points D and P, and B the cell to be 
 examined and another galvanometer are inserted in a second 
 derived circuit EKN. 
 
 The standard cell is first balanced by properly adjusting the 
 resistance of the rheostat, and then the contact key K is 
 
 Fig. 48. 
 
 depressed and the terminal .V is moved along EF, until a 
 point is reached when no deflection is observed in the gal- 
 vanometer G. 
 
 The cells A and B are then both balanced, and the current is 
 produced by the battery C alone. 
 
 Denoting the electromotive forces of A, B, and C by E, E lt 
 and E z respectively, the resistances of the two parts of EF by 
 R and R I} and that of the rheostat by p, we have, 
 
 (I) 
 and _A._,_ = ^. ( 2 ) 
 
 The unknown electromotive force is then given by the 
 
 relation, 
 
 R 
 
 . E. 
 
 (3)
 
 DETERMINATION OF ELECTROMOTIVE FORCES 269 
 
 Here, again, the electromotive force of C must be greater 
 than of either, of the elements A or B, and in case E 1 is 
 greater than E, the dispositions of these two elements must be 
 interchanged. The sliding resistance EF is generally made 
 of platinum-iridium wire of constant cross-section, and it is so 
 arranged that it can be minutely subdivided. That employed 
 by Latimer-Clark in his investigations was of 40 ohms resist- 
 ance, and was so wound on an insulated cylinder that it could 
 be read to one twenty-thousandth of its length. 
 
 II. Absolute determination of electromotive forces. 
 
 From the relation C= > it is evident that if C and R be 
 J\. 
 
 expressed in absolute units, a value for E can at once be 
 deduced in the same system. 
 
 So far as C is concerned, this can be readily accomplished by 
 the use of a tangent galvanometer, but in the case of R, since 
 it includes the internal resistance of the cell, difficulties arise in 
 connection with direct determinations which make the results 
 so obtained practically worthless. Hence, methods are adopted 
 in which a determination of the internal resistance is avoided. 
 
 For constant cells Ohm's method may be followed, but it is 
 better both with constant and variable elements to apply that 
 outlined by Latimer-Clark, which is a method of compensation, 
 the cell being balanced against a difference of potential main- 
 tained by an auxiliary generator. 
 
 (i) OJims method. 
 
 The circuit consists of a rheostat, or resistance box, a tangent 
 galvanometer, and the element whose electromotive force is 
 to be determined. 
 
 A convenient deflection B 1 is given to the galvanometer by 
 suitably adjusting the rheostat, and then a known resistance 
 R is added to the circuit, and a second deflection # 2 is 
 obtained.
 
 270 
 
 EXPERIMENTAL PHYSICS. 
 
 Denoting the resistance of the circuit corresponding to the 
 first deflection by p, it follows that 
 
 and 
 
 P+R 
 K being the constant of the galvanometer. 
 
 Eliminating p from these equations, the relation 
 
 (i) 
 
 (2) 
 
 tan 0j tan 2 
 is obtained, and from it E can be found in absolute units. 
 
 (2) Clark's method. 
 
 The disposition adopted by Latimer-Clark is similar to that 
 devised by Poggendorff for comparative measurements. A tan- 
 gent galvanometer G (Fig. 49), whose constant K is known, 
 
 Fig. 49. 
 
 is inserted along with a rheostat H in the circuit of a constant 
 battery B, and the cell A, whose electromotive force is to be 
 found, together with an auxiliary galvanometer, form the derived 
 circuit CD.
 
 ABSOLUTE DETERMINATION OF RESISTANCE. 271 
 
 The corresponding terminals of the two cells are connected, 
 so that they both tend to send a current through the galva- 
 nometer G in the same direction. 
 
 The rheostat is varied until the galvanometer F indicates 
 that the electromotive force of A is balanced by the potential 
 difference between the points C and D. It only remains then 
 to note the reading on the tangent galvanometer, and to measure 
 the resistance of the partial circuit CLD. Denoting this resist- 
 ance by R, the galvanometer deflection by 0, and the electro- 
 motive force of A by E, these quantities are connected by the 
 relation 
 
 E=KR tan0. 
 
 As K, R, and 6 can be readily determined with precision, 
 the method affords a means of making very accurate absolute 
 determinations. 
 
 The method is exceedingly simple, and should any errors arise 
 from a falling off in the intensity of the current from B, these 
 may be corrected by closing the circuit only for short intervals 
 at a time during the test. 
 
 Since the constant K is given by 
 
 K, 
 
 2 mr 
 
 it is evident that the method may be applied to finding H, the 
 horizontal component of the earth's magnetic action, when the 
 laboratory is provided with a cell whose electromotive force is 
 accurately known. 
 
 XVII. ABSOLUTE DETERMINATION OF RESISTANCE 
 BY USE OF A CALORIMETER. 
 
 When an electrical current is established in a conductor it 
 is produced and maintained by an expenditure of energy, me- 
 chanical, thermal, or chemical, and consequently it may itself
 
 272 
 
 EXPERIMENTAL PHYSICS. 
 
 be considered to be a form of energy which is distributed along 
 the conductor, and which reappears in the form of motion, 
 chemical separation, or heat. 
 
 If the conditions to which the conductor is subjected are 
 such that the transformation cannot take place into the first 
 two of these forms, the electrical energy will be expended in heat- 
 ing the conductor, and the amount 
 of heat so produced will be an 
 exact equivalent of the work done 
 in producing the current. 
 
 The conductor may then be 
 viewed as a series of points be- 
 tween which, while a current is 
 passing, a certain amount of en- 
 ergy is being expended, and the 
 potential difference between any 
 two points will represent the work 
 done between them in a unit of 
 time when the electric current is 
 of unit intensity. 
 If, then, YI and V^ represent in volts the potentials of the 
 two terminals of a conductor forming part of a circuit, and C 
 denotes in amperes the intensity of the current passing between 
 them, C(V^ V^T watts will represent the energy transformed 
 into heat in the conductor in T seconds. 
 
 Denoting by W the amount of this energy in ergs, we have, 
 W=C(V l V^)T io 7 ergs, and combining this again with the 
 
 
 
 F 
 
 
 
 1 
 
 K 
 
 > ~ 
 
 El 
 ~ 
 
 S 
 
 & 
 
 # 
 
 
 
 y 
 
 _-3 
 
 1? 
 
 
 
 
 
 Fig. 50. 
 
 relation C- 
 
 ', where R is the resistance of the conductor 
 
 in ohms, we have, 
 
 W=C*RT> io 7 ergs. 
 
 The mechanical equivalent of heat which is the number of 
 units of work required to be expended to heat one gram of 
 water through one degree centigrade has been found, by experi- 
 ment, to be about 4.16 x io 7 ergs.
 
 ABSOLUTE DETERMINATION OF RESISTANCE. 
 
 273 
 
 Adopting this value, the number of calories or units of heat 
 developed in the conductor is given, therefore, by 
 
 calories, 
 
 4.16- 10' 
 or H=C 2 RTx 24 calories, 
 
 and the resistance of the conductor may be found in terms of 
 //, T, and C from the relation 
 
 The amount of heat developed in the conductor is ascertained 
 by immersing it in a known mass of water, and by noting the 
 gradual rise in the temperature of the latter when the current 
 is passed through it. The apparatus is similar to that shewn in 
 Fig. 50. The water is contained in the vessel AB, which is 
 made of sheet brass, and in order to reduce the errors due 
 to radiation and absorption, this is itself placed on wooden 
 supports in an outer vessel CD, made of the same material. 
 The conductor, which is coiled into a spiral, is attached to two 
 insulated binding poles inserted in the cover of the calorimeter, 
 and is so arranged that when the cover is in position it is com- 
 pletely immersed in the water, and yet does not come in contact 
 with the sides or bottom of AB. The apparatus is also pro- 
 vided with a delicate thermometer E and a stirrer F. 
 
 In conducting the experiment the weight of the stirrer and 
 the vessel AB is first determined in grams, and then a known 
 mass of water is placed in it, and the whole lowered into the 
 outer vessel CD. The cover, with coil attached, is then placed 
 in position, and the thermometer is inserted in the opening 
 prepared for it. 
 
 The current which is passed through the conductor may be 
 generated by a powerful battery, or, better still, by a dynamo, 
 and the circuit should contain either a sensitive galvanometer,
 
 274 EXPERIMENTAL PHYSICS. 
 
 or a voltameter and a galvanometer, for determining its inten- 
 sity, as well as a rheostat for modifying it if desired. 
 
 When the initial temperature / 1 C. of the water has been 
 accurately ascertained, the current is then turned on, and at 
 the same instant the indication on a stop watch, or other deli- 
 cate time recorder, is noted. 
 
 By means of the rheostat the current is kept constant, and at 
 intervals, as the experiment goes on, both the time and the 
 corresponding temperature of the water is noted, care being 
 taken to keep the latter well stirred. 
 
 The temperature of the water should always be initially below 
 that of the room, and the current should be allowed to pass 
 until its temperature has risen as much above that of the room 
 as it was at first below it. 
 
 Denoting the final temperature by / 2 C. the time occupied by 
 the experiment by T, and the specific heats of the vessel AB, 
 and the conductor by s l and s%, we have, if m 2 grams is the mass 
 of the conductor, and m 1 and m those of the vessel, AB and 
 the water in it respectively, 
 
 Jf=(M + m l s l + MySi)(( z -tj, (i) 
 
 and if the intensity of the current is denoted by C, the resist- 
 ance of the conductor may therefore be found from the relation 
 
 K 4.10 
 
 The uncertainty which prevails as to the exact value of the 
 mechanical equivalent of heat rather unfits the method for very 
 accurate determinations, and as besides the resistance of a con- 
 ductor varies with its temperature, the values obtained can only 
 be considered approximations. The method is, however, instruc- 
 tive, and involves an excellent training in the use of the funda- 
 mental units of electrical measurements. 
 
 A slight modification of the experiment is to determine the 
 resistance offered by the carbon filament of an incandescent
 
 ABSOLUTE DETERMINATION OF RESISTANCE. 
 
 275 
 
 lamp to the passage of a current. This may be accomplished 
 by covering the socket of the lamp with paraffine wax, and 
 noting the heat evolved when it is immersed in the water in 
 the calorimeter. The results obtained illustrate very clearly 
 that the resistance of a carbon conductor rapidly decreases as 
 its temperature increases.
 
 APPENDIX A. 
 
 DETERMINATION OF GRAVITY BY THE PENDULUM. 
 
 1. If a pendulum of any form be allowed to make small oscillations 
 under the action of gravity, we have the time of a complete oscillation 
 
 given by the relation t = 2 TTA/ , where / is the length of the equiva- 
 * S 
 
 T g i t2 
 
 lent simple pendulum and equal to If, now, / be observed by 
 
 means of a clock, and h and k found, we have the value of g. This 
 method is one of the most accurate known for finding the intensity of 
 the earth's attraction at different points on its surface. Various forms 
 have been given to these pendulums from time to time, in order to 
 insure accuracy of measurement, and the most important of those 
 which have been used for the scientific determination of gravity are 
 described below : 
 
 (a) Borders pendulum. 
 
 Borda (1792) constructed his pendulum so as to realize as nearly as 
 possible the simple pendulum. It was made of a sphere of known 
 radius (a). To render it very heavy it was composed of platinum, and 
 was suspended by a very fine wire about one meter in length. 
 
 The knife edge which carried the wire and sphere was so arranged 
 by means of a movable screw as to oscillate in the same time as the 
 complete pendulum. The time was determined by the method of 
 coincidences, and g was found from the relation 
 
 g 
 
 where / is the length from the knife edge to the center of the sphere, 
 a the radius of the sphere, and a half the angle of a single oscillation. 
 
 277
 
 278 EXPERIMENTAL PHYSICS. 
 
 (b) Rater's pendulum. 
 
 In 1818, Captain Kater determined the value of gravity at London 
 by applying to the pendulum the principle discovered by Huyghens, 
 that the centers of suspension and of oscillation are reversible. 
 
 He made a pendulum of a bar of brass about an inch and a half 
 wide and an eighth of an inch in thickness. This bar was pierced in 
 two places, and triangular knife edges of hard steel were inserted, so 
 that the distance between them was nearly 39 inches. 
 
 A large mass in the form of a cylinder was placed near one of the 
 knife edges, being slid on by means of a rectangular opening cut in it ; a 
 smaller mass was also attached to the pendulum in such a way as to 
 admit of small motions either way. The pendulum was then swung 
 about the two axes, and adjustment of the masses made until the 
 time of small oscillations was the same. This time being noted, 
 and the distance between the knife edges being accurately measured, 
 g was readily calculated. A small difference being generally found in 
 the two times, it can be shewn that the length of the seconds pendu- 
 lum is found from the expression, 
 
 (h, 
 
 where h lt h t are the distances of the center of gravity from the two 
 knife edges, and / t the corresponding times of oscillation. 
 
 (c) Repsold's pendulum. 
 
 It was noticed in experimenting with pendulums made like Kater's, 
 that the vibration is differently affected by the surrounding air according 
 as the large mass is above or below. This led to the form known as 
 Repsold's, in which the two ends are exactly similar externally, but 
 the pendulum (which is cylindrical) is hollow at one end. The center 
 of gravity of the figure is equidistant from the knife edges, but the 
 true center of gravity of the whole mass is at a different point. 
 
 2. Many observers have, during the present century, conducted 
 observations at different points on the earth's surface in order to deter- 
 mine not only the length of the seconds pendulum but also the 
 excentricity of the earth considered as a spheroid.
 
 APPENDIX A. 
 
 279 
 
 Helmert in his Geodesy has collected the results of nearly all the 
 more important expeditions, and the following table gives some of 
 the principal stations, with the corresponding lengths of the seconds 
 pendulum and the name of the observer. To find g from this table 
 for any place, the relation 
 
 log g = 2 log TT + log / 
 
 may be used, where / is the length of the seconds pendulum in centi- 
 meters. The places are arranged geographically in order of their 
 latitudes, and shew thereby the gradual increase in the length of the 
 seconds pendulum as we go from the Equator to the Pole. 
 
 PLACE. 
 
 LATITUDE. 
 
 /. 
 
 OBSERVER. 
 
 Rawak 
 St. Thomas 
 
 o I'S. 
 o 24 N. 
 o 32 N. 
 i 27 S. 
 7 55 S. 
 8 29 N. 
 10 38 N. 
 12 46 N. 
 13 4 N. 
 15 56 S. 
 17 56 N. 
 
 22 3 2 N. 
 
 22 55 S. 
 33 2 S. 
 34 54 S. 
 38 28 N. 
 40 44 N. 
 41 41 N. 
 43 7 N. 
 44 50 N. 
 45 24 N. 
 48 40 N. 
 5 37 N. 
 51 28 N. 
 51 28 N. 
 51 31 N. 
 52 30 N. 
 54 46 S. 
 55 5i S. 
 55 58 N. 
 57 3N. 
 59 46 N. 
 59 56 N. 
 60 45 N. 
 
 99.0966 
 99.1134 
 99.1019 
 99.0948 
 99.1217 
 99.1104 
 99.1091 
 99.1227 
 99.1168 
 99.1581 
 99.1497 
 99.1712 
 99.1712 
 99.2500 
 99.2641 
 
 99-397 
 99.3191 
 99.3190 
 99.3402 
 
 99-347 
 99.3623 
 99.3858 
 99.4042 
 99.4169 
 
 99-4I43 
 99.4140 
 
 994235 
 99.4501 
 
 99-45 6 5 
 99-455 
 99.4621 
 99.4854 
 99.4876 
 99-4959 
 
 Freycinet. 
 Sahine. 
 Hall. 
 Foster. 
 
 Sabine. 
 
 Basevi and Heaviside. 
 Basevi and Heaviside. 
 
 Sabine. 
 Basevi and Heaviside. 
 
 Liitke. 
 Foster. 
 Biot. 
 
 Duperrey. 
 Biot. 
 Biot. 
 
 Kater. 
 
 Foster. 
 Foster. 
 
 Liitke. 
 Sawitsch. 
 
 Galapagos 
 Para 
 
 
 Trinidad 
 Aden 
 Madras 
 St. Helena 
 Jamaica 
 Calcutta 
 Rio Janeiro 
 
 Valparaiso 
 
 Lipari 
 Hohoken N. J 
 
 Tiflis 
 Toulon 
 Bordeaux 
 Padua 
 
 Paris 
 Shanklin Farm (Isle of Wight) . 
 Kew ... 
 
 Greenwich 
 London 
 Berlin 
 Staten Island 
 Cape Horn 
 Leith 
 
 Sitka . ... 
 
 Pulkovva 
 Petersburg 
 Unst
 
 2 8o EXPERIMENTAL PHYSICS. 
 
 Those places in the preceding table for which the lengths of the 
 seconds pendulum have been calculated from a number of observations 
 made by different observers are indicated by a dash. 
 
 3. During the past few years several observers have made observa- 
 tions on the value of g at different points in North America. Professor 
 Mendenhall, of the U. S. Coast Survey, during the summer of 1891, 
 visited a number of places on the Pacific coast between San Francisco 
 and Alaska, and in his report of the expedition gives a table of the 
 values determined, with the places and corresponding latitudes. He 
 made use of a half-seconds pendulum, inclosed in an air-tight chamber, 
 which could be exhausted with an air-pump. A special method was 
 used for noting the coincidences. (See U. S. Coast and Geodetic Survey, 
 Report for 1891, part 2.) 
 
 4. Defforges, one of the greatest living authorities on methods of 
 gravity determination, crossed from Washington to San Francisco during 
 the summer of 1893, and made a number of observations, which are 
 given in the following table. The value of g alone is given : 
 
 Washington 980.169 
 
 Montreal 980.747 
 
 Chicago 980.375 
 
 Denver 980.983 
 
 Salt Lake City 980.050 
 
 Mt. Hamilton 979. 916 
 
 San Francisco 980.037 
 
 These are all reduced to sea-level. 
 
 NOTE. An excellent book of reference on the subject of gravity determination 
 is Memoir es rtlatifs h la Physique, Tome IV, Paris, 1889.
 
 APPENDIX B. 
 
 THE TORSION PENDULUM. 
 
 IN many instruments of precision designed for investigations in Elec- 
 tricity, Magnetism, and Gravitation, certain portions of the apparatus, 
 which are movable about a vertical axis, are suspended either by 
 metallic wires or elastic fibers. 
 
 The suspension may be either unifilar or bifilar. If the former method 
 is adopted and suspension is all that is required, fibers are selected, such 
 as silk cocoon threads, which offer practically no resistance to torsion, 
 and are capable of supporting a considerable mass. If, however, a couple, 
 or a number of couples, is applied to the movable part of the instrument, 
 and this system is equilibrated and measured by the torsion in the sus- 
 pending fibers or wires, the latter are generally made from some metal, 
 such as silver, or from fine glass or quartz threads. 
 
 In the bifilar suspension the movable apparatus is attached to two 
 threads or fibers of the same length, which have their upper ends fas- 
 tened to two fixed points. When the suspended system is displaced 
 from its position of rest, it is its weight which gives rise to the direc- 
 tive force, and the restorative couple is but slightly affected by any 
 torsion which may exist in the suspending threads. 
 
 The elastic properties of substances used for suspension purposes may 
 be investigated by means of the Torsion Pendulum. In such an appara- 
 tus a wire or fiber of the material in question is suspended from some 
 point to which it is firmly attached, and a heavy body called a vibrator 
 is securely fastened to its lower extremity. 
 
 Various forms have been suggested for this vibrator. It is generally 
 made of some non-magnetic substance such as brass or copper, and it 
 must be of such dimensions that its moment of inertia about the axis of 
 the suspending wire can be readily calculated. Professor Andrew Gray 
 recommends a hollow circular cylinder of brass or copper, as it is best 
 suited for a correct determination of the moment of inertia, and as 
 besides its oscillations are but slightly affected by the presence of the air. 
 
 281
 
 282 EXPERIMENTAL PHYSICS. 
 
 Professor Threlfall, in his article* on " The Elastic Constants of Quartz 
 Threads," states that the vibrators used by him in his investigations 
 "were (i) a pure silver anchor ring supported by a disc of aluminium 
 foil through the center of which passed a short bit of aluminium wire 
 to which the thread was attached, (2) a cylindrical vibrator of brass, 
 vibration being about the axis of the cylinder, and (3) a disc of brass, 
 vibration taking place about its axis of revolution." 
 
 When the vibrator is given a rotation about the axis of the suspending 
 wire or thread, and is then left to itself, it will perform oscillations about 
 its mean position which within the limits of perfect elasticity are isochro- 
 nous. This at once indicates that the motion is harmonic, and that 
 therefore the moment of the restorative couple is proportional to the 
 angle through which the suspended body is rotated. This moment is 
 also proportional to the fourth power of the diameter of the wire, and 
 varies inversely as its length. As formulated originally by Coulomb 
 these laws are exhibited in the relation 
 
 where d is the diameter of the wire, / its length, and the angular dis- 
 placement. 
 
 The time of an oscillation is therefore given by 
 
 -5V? 
 
 K being the moment of inertia of the vibrator about the axis of rotation. 
 
 In order to verify these laws, it is only necessary to select a number 
 of wires of different dimensions, but of the same material. If the vibra- 
 tor be suspended by each of them in turn, and the time of an oscillation 
 observed with each, it will be found that if these values together with 
 the corresponding ones for d, /, and K be substituted in equation (2), a 
 constant value will be obtained for p.. 
 
 This quantity, which is termed the constant, or coefficient of torsion, 
 depends only on the nature of the suspending wire, and on its tempera- 
 ture. In the C.G.S. system of units it is the numerical expression for 
 the couple which, if applied to one base of a circular cylinder of wire, 
 whose length and diameter are each one centimeter, would twist this 
 base through a unit angle, the other base being fixed. 
 
 * Phil. Mag., July, 1890.
 
 APPENDIX B. 
 
 283 
 
 The following table gives the values of this coefficient for a number of 
 metals, and for silk and quartz fibers : 
 
 SUBSTANCE. 
 
 COEFFICIENT OF TORSION. 
 
 SUBSTANCE. 
 
 COEFFICIENT OF TORSION. 
 
 Aluminium 
 Silver 
 
 2.5516 x io 10 
 
 2.6144 
 
 Copper 
 German Silver 
 
 4.2458 x io 10 
 
 4.7412 
 
 Gold 
 
 2.6958 
 
 Platinum 
 
 6.6659 
 
 Zinc 
 
 3-2559 
 
 Iron 
 
 7.4448 
 
 Brass 
 
 3-3727 
 
 Quartz (fibers) 
 
 2.8289 
 
 Platinum-Silver 
 
 (lPt + 2Ag) 
 
 3.5581 
 
 * Silk (fibers) 
 
 
 * Professor T. Gray in 1886 made a thorough investigation of the elastic properties 
 of silk threads, and found that the torsion factor for fibers of Japanese silk one centi- 
 meter in length, and of diameters ranging from .0008 to .0015 cms. varied from .00096 
 to .0025 of a dyne. 
 
 Metallic wires have been found quite unsuitable for suspension pur- 
 poses when delicate manipulation is demanded. Their coefficients of 
 torsion vary considerably with changes in temperature, and are slightly 
 diminished when the wires have been allowed to oscillate for a long 
 period. Permanent deformations also soon appear in them unless the 
 amplitudes of the oscillations are very small, and consequently the 
 angular displacements cannot be relied on in making accurate measure- 
 ments of applied couples. 
 
 The latter difficulty was experienced by Professor C. V. Boys* in his 
 recent investigation on the Newtonian Constant of Gravitation. After 
 experimenting with many substances he found fibers of quartz to possess 
 elastic properties more nearly perfect than those of any other material. 
 Such fibers can be drawn to any required length, and so exceedingly fine 
 that even a powerful microscope fails to reveal their presence. They 
 are, however, very regular, are as strong as steel, and remain constant for 
 a long period. With them the suspended body can be permitted to 
 oscillate through large angles without any deformation or displacement 
 of the zero position appearing. On account of these many excellent 
 properties, they are, therefore, most suitable for suspensions in investiga- 
 tions demanding accuracy and delicacy of manipulation. 
 
 When a magnet whose magnetic moment is known is suspended in a 
 field of determined intensity, the coefficient of torsion can also be found 
 
 * Phil. Mag., June,
 
 284 EXPERIMENTAL PHYSICS. 
 
 by observing the deflections of the magnet from its position of rest when 
 the suspending fiber or wire is subjected to different amounts of torsion. 
 If denote the deflection of the magnet corresponding to the torsion 
 in the wire <, and H and M the intensity of the field and the magnetic 
 moment respectively, these quantities are connected by the relation 
 
 (3) 
 
 and the constant of torsion is therefore given by 
 HML sin 
 
 When this method is adopted, care must be taken to have the magnet 
 initially at rest, without the suspending wire being subjected to any 
 torsional strain. Further details of the method have already been given 
 on page 187. 
 
 Determination of moments of inertia. 
 
 By a reference to equation (2) it can be seen that if the constant of 
 torsion for a given wire /*. is known, the moment of inertia K of the sus- 
 pended body can be at once deduced by finding the time of an oscilla- 
 tion. 
 
 It is better, however, owing to the difficulty in obtaining a value for p 
 sufficiently exact for this purpose, to adopt a comparative method. The 
 body whose moment of inertia is required is first attached to the sus- 
 pending wire, and its time of oscillation found. It is then replaced by 
 some body whose moment of inertia is known or can be readily calcu- 
 lated, and the oscillation period is again determined. 
 
 Since the directive couple in the two cases is the. same, the constant 
 of torsion can then be eliminated. If /, and / are the two periods of 
 oscillation, and K l and K the required moment of inertia and that which 
 is known respectively, it follows from equation (2) that 
 
 K, = K'. 
 
 A modification of this method is to use a bar magnet for the auxiliary 
 body, and to determine the vibration period first with the given body 
 attached, and then with it removed. The restorative couple will in this 
 case depend on the intensity of the magnetic field as well as on the 
 coefficient of torsion of the wire. The vibrations must also be taken of 
 small amplitude.
 
 TABLES. 
 
 TABLE I. 
 
 5, =.3183, 
 77^ = 9.8696, VTT =1.7724. 
 Circle, diameter 2 a : 
 
 circumference = 2 ira, 
 area = 77-0*. 
 Ellipse, axes 2 a, 2 b : 
 
 area = irab. 
 Triangle, base b, altitude a : 
 
 area = \ ab. 
 Sphere, radius a : 
 
 surface = 4 ira 2 , 
 volume 
 
 Right circular cylinder, radius a, length I : 
 
 surface (including ends) = 2 ira(l+ a), 
 volume = 7T0 2 /. 
 
 Right circular cone, radius of base b, altitude a : 
 
 surface = TT( V<z 2 -M 2 + b), 
 volume = -nab*. 
 
 TABLE II. 
 
 i meter = 100 centimeters = 1000 millimeters. 
 
 i liter = 1000 cubic centimeters. 
 
 i kilogram = 1000 grams. 
 
 i gram = 10 decigrams = 100 centigrams = 1000 milligrams. 
 
 285
 
 286 
 
 EXPERIMENTAL PHYSICS. 
 
 i meter (i m.) =39-37* inches, 
 
 i inch = 25.4 millimeters, 
 
 i millimeter (i mm.) =.039371 inches. 
 
 i cubic centimeter of water at 4C. = i gram. 
 
 i cubic inch = 16.386 cubic centimeters (cc.). 
 i pint = 567.93 cubic centimeters, 
 
 i liter =1.76 pints. 
 
 i ounce avoirdupois = 28.35 g rams - 
 i pound avoirdupois = 453.593 grams. 
 i gram = 15.432 grains, 
 
 i kilogram = 2.2046 pounds. 
 
 TABLE III. 
 
 Specific Gravities of Solids. 
 
 Amber 
 Antimony .... 
 Bismuth 
 
 i.i 
 
 . 6.7 
 . o 8 
 
 Iceland spar .... 
 India rubber .... 
 Iron (cast) . -. 
 
 2-7 
 
 99 
 
 7.2 
 
 Bone . ... 
 
 I 
 
 Iron (wrought) 
 
 7-70 
 
 Brass | 
 
 84 
 
 Iron (steel) .... 
 
 7-79 
 
 Bronze j 
 Carbon (gas) 
 
 i 8 
 
 Ivory 
 Lead 
 
 1.92 
 
 114 
 
 Copper . 
 
 . 8.QS 
 
 Lignum vitse .... 
 
 I.* 
 
 Cork 
 Diamond .... 
 Ebony 
 German silver 
 
 .24 
 
 3-5 
 . 1.19 
 8 62 
 
 Nickel 
 Platinum 
 Sand 
 Silver (925 fine) 
 
 8-57 
 21.5 
 1.42 
 10.18 
 
 Glass (reen) 
 
 2 64. 
 
 Silver (pure) 
 
 IO.^ 7 
 
 Glass (crown) 
 
 2 C 
 
 Starch 
 
 I .cr -2 
 
 Glass (flint) . . . 
 Gold (18 caret) 
 
 3-3 
 14 88 
 
 Sugar (cane) .... 
 Salt 
 
 1.6 
 
 .Q2 
 
 Gold (pure) 
 
 
 Tin 
 
 7.3 
 
 Graphite .... 
 Gunpowder 
 
 2.2 
 2.O? 
 
 Wax (bees') .... 
 Zinc (rolled) .... 
 
 . 9 6 
 
 7.2 
 
 Guttapercha 
 
 
 Zinc (cast) 
 
 6.86 
 
 Human body . 
 
 I.O7 
 

 
 TABLES. 
 
 28; 
 
 Specific Gravities of Liquids. 
 
 Alcohol (ethyl) at 15 C 79 
 
 Ammonia at 15 C 761 
 
 Benzine at 15 C 89 
 
 Bisulphide of carbon at 15 C. . . . 1.28 
 
 Chloroform at 15 C 1.5 
 
 Ether at 15 C 72 
 
 Glycerine at 15 C 1.26 
 
 Hydrochloric acid ati5C 1.2 
 
 Mercury at zero C. . . . . . . . 13.596 
 
 Milk (cows') at zero C 1.03 
 
 Nitric acid at 15 C 1.52 
 
 Olive oil at 15 C 915 
 
 Sea water at o C 1.026 
 
 Turpentine at 15 C 872 
 
 Water at 4 C i. 
 
 Specific Gravities of Gases referred to Air at o C. and 760 mm. 
 
 Air i. 
 
 Oxygen 1.10563 
 
 Nitrogen 97 J 37 
 
 Hydrogen 06926 
 
 Carbon dioxide 1.52901 
 
 Vapours. 
 
 Alcohol (ethyl) 1.61 at 78.4 C. 
 
 Chloroform ...... 4.20 at 6o.8 
 
 Water 64 at 100 
 
 Ether ........ 2.59 at 35.5 
 
 Iodine 8.72 at 175 
 
 Mercury . . . . . . 6.98 at 350 
 
 i liter of dry air at o C. and 760 mm. pressure weighs 1.293 grams.
 
 288 
 
 EXPERIMENTAL PHYSICS. 
 
 TABLE IV. 
 
 Specific Heats of Solids and Liquids referred to Water. 
 Aluminium .202 i 
 
 Liquids at 15 C. 
 
 Alcohol. . .. . ... .615 
 
 Chloroform 233 
 
 Ether ...". 517 
 
 Mercury 033 
 
 Gases at Constant Pressure. 
 
 Air . . ... . V . . .2375 
 
 Steam 4805 
 
 Antimony 
 Bismuth 
 Brass 
 
 . .0507 
 -0305 
 
 .OQ4 
 
 Copper 
 
 Glass 
 
 . .095 
 
 2 
 
 Gold 
 
 O"?24 
 
 Graphite 
 
 
 Ice 
 Iron ... ... 
 
 -5 
 
 .112 
 
 Lead . . 
 
 .03 I tJ 
 
 Platinum 
 Quartz 
 
 .035 6 
 
 Silver 
 Sulphur 
 Tin . 
 Zinc. 
 
 -0559 
 . .184 
 -0559 
 
 .GO ^=; 
 
 TABLE V. 
 Latent Heats of Fusion. 
 
 Beeswax 97.22 
 
 Lead 5.37 
 
 Steam 536. 
 
 Sulphur 9.35 
 
 Water 79- 2 5 
 
 Zinc 28.15 
 
 TABLE VI. 
 
 Cubical Expansion of Solids and Liquids for i C. 
 
 Copper 00005 
 
 Glass 00002328 
 
 Iron 0000355 
 
 Platinum 000026 
 
 Silver 0000583 
 
 Tin 000069 
 
 Zinc . .000089
 
 TABLES. 289 
 
 Liquids (Mean Expansion). 
 
 Alcohol 00108 
 
 Chloroform 0014 
 
 Ether 0021 
 
 Mercury . . ... .0001815 
 
 Turpentine 00105 
 
 Water 00012 at 15 C. to 
 
 .00074 at 90 C. 
 
 Gases expand .003665 of their volume for each degree Centigrade. 
 
 TABLE VII. 
 
 Expansion of Water from o to 100. 
 Volume of i Gram of Water in Cubic Centimeters. 
 
 TEMPERATURE. VOLUME. INCREASE PER i". 
 
 O I.OOOI 
 
 4 .... i.oooo 
 10 ... i.ooo-? 
 
 J ..... 00012 
 1C .... I.OOO9 , 
 
 y ..... 00016 
 
 20 I ' OI . .00024 
 
 . . . 00028 
 
 3 I '43 , , . 00032 
 
 35 ' ' ' ' I '59 . 00036 
 
 40 .... 1.0077 
 
 ' ' ..... 00040 
 
 4? .... 1.0097 
 
 , . . .00046 
 
 SO .... I.OI20 
 
 ..... 00048 
 CC . . . . I.OI44 
 
 ..... OOOC2 
 60 .... I.OI70 
 
 ..... OOOS4 
 65 .... I.OI97 
 
 V/ ..... 00060 
 7O .... I.O227 ,. 
 
 ' ..... 00062 
 715 .... I.O2i;8 .. 
 
 J ..... 00064 
 80 .... 1. 0200 
 
 9 ..... 00066 
 
 ..... 00070 
 oo .... 1.0358 
 
 OJ ..... 00074 
 95 .... 1.0^9=5 
 
 J7J ..... 00074 
 100 .... 1.0432 
 
 u
 
 2QO 
 
 EXPERIMENTAL PHYSICS. 
 
 TABLE VIII. 
 
 Boiling Temperature of Water (/) at Barometer Pressure 
 (after Regnaulf). 
 
 b 
 
 t 
 
 b 
 
 / 
 
 b 
 
 / 
 
 b 
 
 / 
 
 b 
 
 t 
 
 680 
 
 9 692 
 
 700 
 
 970.72 
 
 720 
 
 9849 
 
 740 
 
 99 .26 
 
 760 
 
 IOO.00 
 
 68 1 
 
 .96 
 
 701 
 
 75 
 
 721 
 
 53 
 
 741 
 
 .29 
 
 761 
 
 .04 
 
 682 
 
 97 .00 
 
 702 
 
 79 
 
 722 
 
 57 
 
 742 
 
 33 
 
 762 
 
 .07 
 
 683 
 
 .04 
 
 73 
 
 83 
 
 723 
 
 .61 
 
 743 
 
 37 
 
 763 
 
 .11 
 
 684 
 
 .08 
 
 704 
 
 .87 
 
 724 
 
 65 
 
 744 
 
 .41 
 
 764 
 
 15 
 
 685 
 
 .12 
 
 75 
 
 .91 
 
 725 
 
 .69 
 
 745 
 
 44 
 
 765 
 
 .18 
 
 686 
 
 .16 
 
 706 
 
 95 
 
 726 
 
 .72 
 
 746 
 
 .48 
 
 766 
 
 .22 
 
 687 
 
 .20 
 
 707 
 
 99 
 
 727 
 
 .76 
 
 747 
 
 52 
 
 767 
 
 .26 
 
 688 
 
 .24 
 
 708 
 
 98 -03 
 
 728 
 
 .80 
 
 748 
 
 56 
 
 768 
 
 29 
 
 689 
 
 .28 
 
 709 
 
 .07 
 
 729 
 
 .84 
 
 749 
 
 59 
 
 769 
 
 33 
 
 690 
 
 32 
 
 710 
 
 .11 
 
 73 
 
 .88 
 
 75 
 
 63 
 
 770 
 
 36 
 
 691 
 
 .36 
 
 711 
 
 15 
 
 73i 
 
 .92 
 
 75 ! 
 
 .67 
 
 771 
 
 .40 
 
 692 
 
 .40 
 
 712 
 
 .19 
 
 732 
 
 95 
 
 752 
 
 7 
 
 772 
 
 44 
 
 693 
 
 44 
 
 7*3 
 
 .22 
 
 733 
 
 99 
 
 753 
 
 74 
 
 773 
 
 47 
 
 694 
 
 .48 
 
 7H 
 
 .26 
 
 734 
 
 99 -03 
 
 754 
 
 .78 
 
 774 
 
 5 1 
 
 695 
 
 52 
 
 7i5 
 
 3 
 
 735 
 
 .07 
 
 755 
 
 .82 
 
 775 
 
 55 
 
 696 
 
 56 
 
 716 
 
 34 
 
 736 
 
 .11 
 
 756 
 
 85 
 
 776 
 
 58 
 
 697 
 
 .60 
 
 717 
 
 38 
 
 737 
 
 .14 
 
 757 
 
 .89 
 
 777 
 
 .62 
 
 698 
 
 .64 
 
 718 
 
 .42 
 
 738 
 
 .18 
 
 758 
 
 93 
 
 778 
 
 65 
 
 699 
 
 .68 
 
 719 
 
 .46 
 
 739 
 
 .22 
 
 759 
 
 .96 
 
 779 
 
 .69 
 
 700 
 
 97 . 7 2 
 
 720 
 
 9 849 
 
 740 
 
 99 .26 
 
 760 
 
 IOO.OO 
 
 780 
 
 IOO.72
 
 TABLES. 
 
 2 9 I 
 
 TABLE IX. 
 For Hygrometry. 
 
 Pressure of aqueous vapour e and weight of water f contained in I cubic meter of 
 air with dew-point /; or, when at the temperature /, the air would be saturated with 
 aqueous vapour. 
 
 t 
 
 
 
 / 
 
 t 
 
 ' 
 
 / 
 
 t 
 
 
 
 / 
 
 t 
 
 ' 
 
 / 
 
 -10 
 
 mm. 
 2.0 
 
 2.1 
 
 
 
 mm. 
 4.6 
 
 gr- 
 4-9 
 
 10 
 
 9-i 
 
 gr- 
 9-4 
 
 20 
 
 mm. 
 17.4 
 
 gr- 
 I 7 .2 
 
 - 9 
 
 2.2 
 
 2-4 
 
 I 
 
 4-9 
 
 5-2 
 
 ii 
 
 9.8 
 
 IO.O 
 
 21 
 
 I8. 5 
 
 18.2 
 
 - 8 
 
 2.4 
 
 2. 7 
 
 2 
 
 5-3 
 
 5-6 
 
 12 
 
 10.4 
 
 10.6 
 
 22 
 
 19.7 
 
 19-3 
 
 - 7 2.6 
 
 3.O 
 
 3 
 
 5-7 
 
 6.0 
 
 13 
 
 ii. I 
 
 "3 
 
 23 
 
 20.9 
 
 20.4 
 
 - 6 2.8 
 
 3-2 
 
 4 
 
 6.1 
 
 6.4 
 
 14 
 
 11.9 
 
 12.0 
 
 24 
 
 22.2 
 
 21. 5 
 
 -5 3-i 
 
 3-5 
 
 5 
 
 6.5 
 
 6.8 
 
 5 
 
 12.7 
 
 12.8 
 
 25 
 
 2 3 .6 
 
 22.9 
 
 - 4 
 
 3-3 
 
 3-8 
 
 6 
 
 7.0 
 
 7-3 
 
 16 
 
 13-5 
 
 13-6 
 
 26 
 
 25.0 
 
 24.2 
 
 - 3 
 
 3-6 
 
 4.1 
 
 7 
 
 7-5 
 
 7-7 
 
 17 
 
 14.4 
 
 14-5 
 
 27 
 
 26.5 
 
 2 5 .6 
 
 - 2 
 
 3-9 
 
 4-4 
 
 8 
 
 8.0 
 
 8.1 
 
 18 
 
 15.4 
 
 I5-I 
 
 28 
 
 28.1 
 
 27.0 
 
 - i 
 
 4.2 
 
 4.6 
 
 9 
 
 8-5 
 
 8.8 
 
 19 
 
 16.3 
 
 16.2 
 
 29 
 
 29.8 
 
 28.6 
 
 
 
 4.6 
 
 4-9 
 
 10 
 
 9.1 
 
 9-4 
 
 20 
 
 17.4 
 
 17.2 
 
 30 
 
 31-6 
 
 3 O.I 
 
 TABLE X. 
 
 SCALE OF PHYSICISTS. 
 
 ut 
 
 256 
 
 5 12 
 
 ut 
 
 256 
 
 re 
 
 288 
 
 576 
 
 ut* 
 
 271.22 
 
 mi 
 
 320 
 
 640 
 
 re 
 
 287-35 
 
 fa 
 
 341-33 
 
 682.66 
 
 re* 
 
 34-437 
 
 sol 
 
 384 
 
 768 
 
 mi 
 
 3 22 -539 
 
 la 
 
 426.66 
 
 853-33 
 
 fa 
 
 34i-7 J 9 
 
 si 
 
 480 
 
 960 
 
 fa* 
 
 362.038 
 
 
 
 
 sol 
 
 383-566 
 
 
 
 
 sol* 
 
 406.374 
 
 
 
 
 la 
 
 43-539 
 
 
 
 
 la* 
 
 456.140 
 
 
 
 
 si 
 
 483.263 
 
 SCALE OF EQUAL TEMPERAMENT.
 
 292 
 
 EXPERIMENTAL PHYSICS. 
 
 TABLE XI. 
 
 Mean Indices of Refraction, and Dispersions of Several Bodies. 
 
 INDEX OF REFRACTION. 
 1-53 
 
 Crown glass (mean) .... 
 
 Flint glass i .60 . . 
 
 Water 1.336 . . 
 
 Alcohol r -37 2 
 
 Carbon disulphide 1.68 .. 
 
 Canada balsam 1.54 . . 
 
 Air 1.000294 . 
 
 DISPERSION. 
 .022 
 .042 
 .0132 
 
 .0837 
 
 TABLE XII A. 
 
 Elements of Terrestrial Magnetism at Toronto. 
 Latitude, 43 39' 36". Longitude, 5 h. 17 m. 34.65 s. 
 
 DATE. 
 
 DECLINATION. 
 
 INCLINATION. 
 
 HORIZONTAL FORCE 
 IN C.G.S. UNITS. 
 
 
 (westerly) 
 
 
 
 I88 5 
 
 3S9'-8 
 
 74 5 * '-6 
 
 I 65579 
 
 1886 
 
 4 2'.I 
 
 74 48'-9 
 
 .165717 
 
 1887 
 
 4 4'. 
 
 74 47 '-6 
 
 165875 
 
 1888 
 
 4 8'. 3 
 
 74 46 '-5 
 
 165993 
 
 1889 
 
 4 1 2 '.o 
 
 74 44 '-7 
 
 .l66lll 
 
 1890 
 
 4 i8'.2 
 
 7442 ( .2 
 
 .166150 
 
 1891 
 
 4 2 3 '-3 
 
 7437'-5 
 
 .166170 
 
 1892 
 
 4 29'.2 
 
 74 37'- 2 
 
 .166229 
 
 1893 
 
 4 3 6 '-4 
 
 74 36'.2 
 
 .166354 
 
 1894 
 
 4 42 '-3 
 
 74 34 '-7 
 
 .166334
 
 TABLES. 
 
 293 
 
 TABLE XII B. 
 Elements of Terrestrial Magnetism at Other Places. 
 
 DATE. 
 
 PLACE. 
 
 DECLINATION. 
 
 INCLINATION. 
 
 HORIZONTAL FORCE 
 IN C.G.S. UNITS. 
 
 
 
 (westerly) 
 
 
 1887 
 
 Greenwich 
 
 1 7 49 '.i 
 
 67 26' 26" 
 
 .18175 
 
 1888 
 
 
 
 i74o'.4 
 
 6 7 2 5 ' 2 5 " 
 
 .18204 
 
 l88 9 
 
 " 
 
 I 734'-9 
 
 67 24' 9 " 
 
 .18201 
 
 1890 
 
 " 
 
 i728'.6 
 
 67 22' 52" 
 
 .18232 
 
 1891 
 
 " 
 
 i7 a 3'-4 
 
 67 2l' 24" 
 
 .18254 
 
 1889 
 
 Washington 
 
 4i'3i" 
 
 7i 5'59" 
 
 .198693 
 
 1890 
 
 a 
 
 4 5 '45" 
 
 ?i 4'3 lM 
 
 .198604 
 
 1891 
 
 " 
 
 4 9' 43" 
 
 7i 5' 4" 
 
 .198550 
 
 1892 
 
 <i 
 
 4 14' 12" 
 
 7i 3'55" 
 
 .198485 
 
 1886 
 
 Paris 
 
 1 6 oo '.9 
 
 65 i5'. 
 
 J 9439 
 
 1887 
 
 it 
 
 i554'-8 
 
 65 i4'-7 
 
 .19470 
 
 1888 
 
 u 
 
 i549'-7 
 
 65 i4'-5 
 
 .19496 
 
 l88 9 
 
 " 
 
 15 44 '.6 
 
 65 I2'.6 
 
 .19522 
 
 1890 
 
 u 
 
 i538'.7 
 
 65 u'. 
 
 I 9543 
 
 1891 
 
 It 
 
 i532'-8 
 
 65 IQ'.I 
 
 1 9S5 8
 
 294 
 
 EXPERIMENTAL PHYSICS. 
 
 TABLE XIII. 
 
 Cross-Section of Round Wires, with Resistance, Conductivity, and Weight of 
 Pure Copper Wires, according to the Birmingham Wire Gauge. 
 
 TEMPERATURE 15 C. 
 
 B. 
 W.G. 
 
 DIAMETER. 
 
 AREA OF 
 CROSS-SECTION. 
 
 RESISTANCE. 
 
 CONDUC- 
 TIVITY. 
 
 WEIGHT 
 (Density = 8.95). 
 
 
 Ins. 
 
 Cms. 
 
 Sq. Ins. 
 
 Sq. Cms. 
 
 Legal 
 Ohms' 
 per 
 Yard. 
 
 Legal 
 Ohms 
 
 M P e1e, 
 
 Yards 
 per 
 
 te! 
 
 Meters 
 per 
 Legal 
 Ohm. 
 
 Lbs. 
 Y^rd. 
 
 Grams 
 Meter. 
 
 0000 
 
 454 
 
 I-I53 
 
 .162 
 
 1.0444 
 
 .000150 
 
 .000165 
 
 6640 
 
 6072 
 
 1.884 
 
 934-7 
 
 000 
 
 425 
 
 1.079 
 
 .142 
 
 .915 
 
 .000172 
 
 .000188 
 
 5819 
 
 5321 
 
 1.651 
 
 819-1 
 
 oo 
 
 .380 
 
 965 
 
 "3 
 
 732 
 
 .000215 
 
 .000235 
 
 4653 
 
 4254 
 
 1.320 
 
 654.8 
 
 
 
 34 
 
 .864 
 
 .0908 
 
 .586 
 
 .000269 
 
 .000294 
 
 3735 
 
 3415 
 
 1.056 
 
 524.2 
 
 j 
 
 .300 
 
 .762 
 
 .0707 
 
 456 
 
 .000345 
 
 .000378 
 
 2899 
 
 2652 
 
 .822 
 
 408.1 
 
 2 
 
 .284 
 
 .721 
 
 .0633 
 
 .409 
 
 .000385 
 
 .000420 
 
 2599 
 
 2377 
 
 737 
 
 365-8 
 
 3 
 
 .259 
 
 .658 
 
 .0527 
 
 340 
 
 .000463 
 
 .000506 
 
 2162 
 
 1996 
 
 613 
 
 304.2 
 
 4 
 
 5 
 
 .238 
 
 .220 
 
 .605 
 559 
 
 0445 
 .0380 
 
 .287 
 245 
 
 .000642 
 
 .000599 
 .000701 
 
 1825 
 1561 
 
 1669 
 1427 
 
 .518 
 442 
 
 256.9 
 219-5 
 
 6 
 
 203 
 
 .516 
 
 .0324 
 
 .209 
 
 .000754 
 
 .000824 
 
 1328 
 
 1214 
 
 377 
 
 186.9 
 
 7 
 
 !i8o 
 
 457 
 
 .0254 
 
 .164 
 
 .000958 
 
 .00105 
 
 1044 
 
 1004 
 
 .296 
 
 146.9 
 
 8 
 
 .165 
 
 .419 
 
 .0214 
 
 .138 
 
 .00114 
 
 .00125 
 
 877 
 
 802 
 
 .249 
 
 123.5 
 
 9 
 
 .148 
 
 .376 
 
 .0172 
 
 
 
 .00155 
 
 706 
 
 645 
 
 
 99-3 
 
 
 134 
 
 34 
 
 .0141 
 
 .0910 
 
 .00173 
 
 .00189 
 
 578 
 
 529 
 
 SJ 
 
 Eu4 
 
 , 
 
 .120 
 
 35 
 
 .0113 
 
 .0730 
 
 .00216 
 
 .00235 
 
 463 
 
 424 
 
 132 
 
 65-5 
 
 2 
 
 .109 
 
 .277 
 
 00933 
 
 .0602 
 
 .00261 
 
 .00286 
 
 382 
 
 350 
 
 .109 
 
 53-9 
 
 3 
 
 095 
 
 .241 
 
 .00709 
 
 0457 
 
 .00344 
 
 .00376 
 
 291 
 
 266 
 
 .0825 
 
 40.9 
 
 4 
 
 083 
 
 .211 
 
 00541 
 
 349 
 
 .00451 
 
 .00492 
 
 221 
 
 203 
 
 .0630 
 
 31.2 
 
 5 
 
 .072 
 
 .183 
 
 .00407 
 
 .0263 
 
 .00599 
 
 .00655 
 
 l6 7 
 
 J 53 
 
 .0474 
 
 23-5 
 
 6 
 
 065 
 
 .165 
 
 .0033! 
 
 .0214 
 
 00735 
 
 .00804 
 
 I 3 6 
 
 124 
 
 .0386 
 
 19.2 
 
 7 
 
 .058 
 
 147 
 
 .00264 
 
 .0170 
 
 .00923 
 
 .0101 
 
 108 
 
 98.7 
 
 .0307 
 
 I5.3 
 
 8 
 
 .049 
 
 .124 
 
 .00189 
 
 .0122 
 
 .0130 
 
 .0141 
 
 77-3 
 
 70.7 
 
 .0220 
 
 10.9 
 
 9 
 
 .042 
 
 .107 
 
 .00139 
 
 .00894 
 
 .0176 
 
 .0194 
 
 5 -8 
 
 52.O 
 
 .Ol6l 
 
 
 
 035 
 
 .0889 
 
 .000962 
 
 .00621 
 
 .0253 
 
 .0277 
 
 39-4 
 
 36.1 
 
 .0122 
 
 5-Sfi 
 
 t 
 
 .032 
 
 .0813 
 
 .000804 
 
 .00519 
 
 .0304 , .0331 
 
 32.0 
 
 30.1 
 
 .00936 
 
 4-64 
 
 2 
 
 .028 
 
 .0711 
 
 .000616 
 
 .00397 
 
 395 
 
 0433 
 
 25-3 
 
 23-1 
 
 .00716 
 
 3-55 
 
 3 
 
 .025 
 
 .0635 
 
 .000491 
 
 .00317 
 
 .0496 
 
 0543 
 
 20.2 
 
 18.4 
 
 .00571 
 
 2.83 
 
 4 
 
 .022 
 
 0559 
 
 .000380 
 
 .00245 
 
 .0642 
 
 .0701 
 
 15-6 
 
 14-3 
 
 .00442 
 
 2.19 
 
 5 
 
 .020 
 
 .0508 
 
 .000314 
 
 .00203 
 
 .0778 
 
 .0849 
 
 12.8 
 
 11.7 
 
 .00367 
 
 1.82 
 
 26 
 
 .0 8 
 
 .0457 
 
 .000254 
 
 .00164 
 
 959 
 
 .105 
 
 102 
 
 9-53 
 
 .00296 
 
 1-47 
 
 27 
 
 .0 6 
 
 .0406 
 
 .000201 
 
 .00130 
 
 
 133 
 
 8^25 
 
 7-54 
 
 .00234 
 
 1.16 
 
 28 
 
 .0 4 
 
 .0356 
 
 .000154 
 
 .000993 
 
 ilp 
 
 173 
 
 6. 3 I 
 
 5-77 
 
 .00179 
 
 .889 
 
 29 
 
 o 3 
 
 .0330 
 
 .000133 
 
 .000856 
 
 .184 
 
 .201 
 
 541 
 
 4.98 
 
 .00154 
 
 .766 
 
 30 
 
 .0 2 
 
 0305 
 
 
 .000732 
 
 .216 
 
 235 
 
 4.64 
 
 4.24 
 
 .00132 
 
 .653 
 
 3i 
 
 .010 
 
 .0254 
 
 .0000785 
 
 .000507 
 
 3" 
 
 339 
 
 3-23 
 
 2.95 
 
 .000915 
 
 454 
 
 32 
 
 .009 
 
 .0229 
 
 .0000636 
 
 .000410 
 
 384 
 
 .419 
 
 2-5 1 
 
 2.39 
 
 .000746 
 
 367 
 
 33 
 
 .008 
 
 .0203 
 
 .0000503 
 
 .000324 
 
 .486 
 
 530 
 
 2.06 
 
 1.88 
 
 .000585 
 
 .290 
 
 34 
 
 .007 
 
 .0178 
 
 .0000385 
 
 .000248 
 
 634 
 
 .693 
 
 1.58 
 
 i-45 
 
 .000442 
 
 
 1 
 
 .005 
 .004 
 
 .0127 
 
 .0000196 
 
 .000127 
 .OOOoSlI 
 
 1.25 
 1.94 
 
 '35 
 2.13 
 
 .806 
 .516 
 
 736 
 47' 
 
 2 
 
 "3 
 .0726
 
 TABLES. 
 
 295 
 
 TABLE XIV. 
 
 Cross-Section of Round Wires, with Resistance, Conductivity, and Weight of 
 Hard-Drawn Pure Copper Wires, according to the New Standard Wire Gauge 
 (Legalized Aug. 23, 1883, Great Britain and Ireland). 
 
 TEMPERATURE 15 C. 
 
 1 
 
 DIAMETER. 
 
 AREA OF 
 CROSS-SECTION. 
 
 RESISTANCE. 
 
 CONDUC- 
 TIVITY. 
 
 WEIGHT 
 (Density = 8.95). 
 
 f 
 
 Ins. 
 
 Cms. 
 
 Sq. Ins. 
 
 Sq. Cms. 
 
 Legal 
 Ohms 
 
 Y P ard. 
 
 Legal 
 Ohms 
 per 
 Meter. 
 
 Yards 
 per 
 Legal 
 Ohm. 
 
 Meters 
 per 
 Legal 
 Ohm. 
 
 Lbs. 
 
 Grams 
 Meier. 
 
 0000000 
 
 .500 
 
 I270 u- > 
 
 1.267 
 
 .000125 
 
 .000136! 8055 
 
 7365 
 
 .285 
 
 "34 
 
 000000 
 00000 
 
 .464 
 432 
 
 I.'^f ' 15 
 
 1.091 
 
 .946 
 
 .000144 
 .ooo!66 
 
 .0001571 6937 
 .000182 6013 
 
 6343 
 S49 8 
 
 .970 
 .706 
 
 976.3 
 846.3 
 
 oooo 
 
 .400 
 
 i.$5 .1257 
 
 .811 
 
 .000194 
 
 .000213 554 
 
 47H 
 
 463 
 
 725.6 
 
 000 
 
 .372 
 
 945 
 
 .1087 
 
 .701 
 
 .000225 
 
 .000245 4459 
 
 4077 
 
 .265 
 
 627.6 
 
 oo 
 
 348 
 
 .884 
 
 .0951 
 
 .6,4 
 
 .000256 
 
 .000280 3901 
 
 3568 
 
 .107 
 
 549-6 
 
 
 324 
 
 .823 .0824 
 
 S3 2 
 
 .000296 
 
 .000323 3384 
 
 393 
 
 .960 
 
 476.1 
 
 
 
 .762 ;.0707 
 
 456 
 
 .000345 
 
 .000377 2899 
 
 2652 
 
 .823 
 
 408.1 
 
 
 ' 76 
 
 .701 1.0598 
 
 .386 
 
 .000408 
 
 .000446 
 
 2454 
 
 244 
 
 .696 
 
 345-4 
 
 L 52 
 
 
 .0499 
 
 .322 
 
 .000489 
 
 .000536 
 
 2046 
 
 871 
 
 .58 
 
 2880 
 
 32 
 
 .589 
 
 .0423 
 
 273 
 
 .000577 
 
 .00063, 
 
 1734 
 
 586 
 
 49 
 
 244.1 
 
 
 . 12 
 
 538 
 
 353 
 
 .228 
 
 .00069, 
 
 .000756 
 
 1451 
 
 324 
 
 
 203.8 
 
 6 
 
 92 
 
 .488 
 
 .0290 
 
 .187 
 
 .000842 
 
 .000921 
 
 "12 
 
 086 
 
 33 
 
 166.8 
 
 1: 11 
 
 447 
 
 .0243 
 
 *S7 
 
 00122" 
 
 roi 10 
 
 988 
 
 9I 2 
 
 .28 
 
 140.5 
 
 9 44 
 
 !s66 
 
 .0163 
 
 .130 
 
 .00149 
 
 .00164 
 
 
 
 .190 
 
 94.0 
 
 10 . 28 
 
 325 
 
 .0129 
 
 .0830 
 
 .00190 
 
 .00208 
 
 528 
 
 482 
 
 .150 
 
 74-3 
 
 ii . 16 
 
 295 
 
 .0!06 
 
 .068* 
 
 .00230 
 
 .00252 
 
 434 
 
 396 
 
 .123 
 
 61.0 
 
 12 . 04 
 
 .264 
 
 .00849 
 
 .0548 
 
 .00287 
 
 .00314 
 
 348 
 
 
 .0989 
 
 49.0 
 
 13 .092 
 
 234 
 
 .00665 
 
 .0429 
 
 .00367 
 
 .00402 
 
 273 
 
 2 5 
 
 774 
 
 38.4 
 
 14 .080 
 
 .203 
 
 .00503 
 
 .0324 
 
 .00485 
 
 .00530 
 
 206 
 
 188 
 
 .0585 
 
 29.0 
 
 15 .072 
 
 16 *' 
 
 'III 
 
 .00407 
 
 .0263 
 
 .00599 
 
 .00657 
 
 .00839 
 
 167 
 
 153 
 
 .0474 
 
 11 
 
 ID 
 11 
 
 vv t 
 .056 
 .048 
 
 .103 
 .142 
 
 .122 
 
 .00246 
 00181 
 
 .0159 
 .0117 
 
 .00752 
 .0099 
 0135 
 
 .0108 
 .0147 
 
 74-2 
 
 Ci 
 
 .0287 
 
 14.2 
 IO.4 
 
 9 
 
 .O4O 
 
 .IO2 
 
 .00126 
 
 .00811 
 
 .0194 
 
 .0212 
 
 1.6 
 
 47 i 
 
 !oi 4 6 
 
 7.26 
 
 
 .036 
 
 .0914 
 
 00102 
 
 .00657 
 
 .0239 
 
 .0262 
 
 1.8 
 
 
 .on8 
 
 5-88 
 
 21 
 
 .032 
 
 .0813 
 
 .000804 
 
 .00519 
 
 .0304 
 
 0331 
 
 29 
 
 30.1 
 
 .00936 
 
 4.64 
 
 22 
 
 .028 
 
 .0711 
 
 .OOo6l6 
 
 .00397 
 
 .0396 
 
 0433 
 
 53 
 
 23.0 
 
 .00717 
 
 3-56 
 
 23 
 
 .024 
 
 .O6l0 
 
 .000452 
 
 .00292 
 
 539 
 
 .0589 
 
 8.5 
 
 170 
 
 .00526 
 
 .61 
 
 2 4 
 
 .022 
 
 559 
 
 .000380 
 
 .00245 
 
 .0642 
 
 .0701 
 
 5-6 
 
 14.3 
 
 .00443 
 
 .19 
 
 25 
 
 O2O 
 
 .0508 
 
 .000314 
 
 .00203 
 
 .0778 
 
 .0849 
 
 2.8 
 
 
 .00366 
 
 .80 
 
 26 
 
 018 
 
 0457 
 
 .000254 
 
 .00164 
 
 .0958 
 
 .105 
 
 0.4 
 
 9-54 
 
 .00296 
 
 47 
 
 27 
 
 0164 
 
 .0417 
 
 .000211 
 
 .00136 
 
 .116 
 
 123 
 
 8.65 
 
 793 
 
 .00246 
 
 .22 
 
 28 
 
 0148 
 
 .0376 
 
 .000172 
 
 .001 i i 
 
 .141 
 
 
 7.07 
 
 6-45 
 
 .00200 
 
 893 
 
 2 9 
 
 0136 
 
 345 
 
 .000145 
 
 .000937 
 
 .168 
 
 .183 
 
 5-95 
 
 5-45 
 
 .00169 
 
 839 
 
 
 0124 
 
 0315 
 
 .000121 
 
 .000779 
 
 .202 
 
 .221 
 
 4.86 
 
 4-53 
 
 .00141 
 
 .697 
 
 3' 
 
 0116 
 
 .0295 
 
 .000106 
 
 .000682 
 
 .230 
 
 .252 
 
 4-34 
 
 
 .00123 
 
 .610 
 
 32 
 
 0108 
 
 .0274 
 
 .0000916 
 
 .000591 
 
 .266 
 
 .291 
 
 3-75 
 
 3-44 
 
 .00107 
 
 529 
 
 33 
 
 OIOO 
 
 .0254 
 
 .0000785 
 
 .000507 
 
 3 11 
 
 339 
 
 3-22 
 
 94 
 
 .000914 
 
 
 34 
 
 0092 
 
 .0234 
 
 .0000665 
 
 .000429 
 
 367 
 
 .402 
 
 2.73 
 
 50 
 
 .000774 
 
 .384 
 
 35 
 
 0084 
 
 .O2I3 
 
 .0000554 
 
 .000358 
 
 44 
 
 .481 
 
 2.27 
 
 .08 
 
 .000645 
 
 .320 
 
 36 
 37 
 
 .0076 
 
 .0068 
 
 .0193 
 .0173 
 
 .0000454 
 .0000363 
 
 .000293 
 .000234 
 
 .540 
 .672 
 
 $ 
 
 1.86 
 1.49 
 
 s 
 
 .000548 
 .000423 
 
 .262 
 
 .210 
 
 38 
 
 .0060 
 
 .0152 
 
 .0000283 
 
 
 944 
 
 1.16 
 
 .04 
 
 .000329 
 
 .163 
 
 39 
 
 .0052 
 
 .0132 ,. 0000212 
 
 .000137 
 
 T - T 5 
 
 1.26 
 
 .870 
 
 .796 
 
 .000247 
 
 .123 
 
 40 
 
 .0048 
 
 .0122 
 
 .0000181 
 
 .000117 
 
 1.32 
 
 1.47 
 
 759 
 
 679 
 
 .000211 
 
 .104 
 
 41 
 
 .0044 
 
 .0112 
 
 .0000152 
 
 .0000981 
 
 i. 60 
 
 i-75 
 
 .624 
 
 570 
 
 .000177 
 
 .0878 
 
 42 
 
 .0040 
 
 
 .0000811 
 
 1.94 
 
 2.13 
 
 516 
 
 .471 
 
 .000146 
 
 .0726 
 
 43 
 
 .0036 
 
 .00914;. 0000102 
 
 .0000657 
 
 2.39 
 
 2.62 
 
 .418 
 
 
 
 .0588 
 
 44 
 
 .0032 
 
 .00813 
 
 .00000804 
 
 .0000519 
 
 3-4 
 
 3-32 
 
 330 
 
 .301 
 
 .0000936 
 
 .0464 
 
 45 
 
 .0028 
 
 .0071! 
 
 .00000616 
 
 .0000397 
 
 3-96 
 
 4-33 
 
 253 
 
 .230 
 
 .0000717 
 
 .0356 
 
 46 
 
 .0024 
 
 .oo6io| .00000452 
 
 .0000292 
 
 5-39 
 
 5-9 
 
 185 
 
 .170 
 
 .0000527 
 
 .0261 
 
 47 
 
 .0020 
 
 .00508 .00000314 
 
 .0000203 
 
 7.76 
 
 8-49 
 
 .128 
 
 
 .0000366 
 
 .Ol8l 
 
 48 
 
 .0016 
 
 .004061.00000201 
 
 .0000130 
 
 12.20 
 
 13-3 
 
 .0824 
 
 0754 
 
 .0000234 
 
 .OIl6 
 
 49 
 
 .0012 
 
 .00305.00000113 
 
 .00000730 
 
 21.6 
 
 23-5 
 
 .0464 
 
 .0425 
 
 .0000132 
 
 00653 
 
 5 
 
 .0010 
 
 .00254 .000000785 
 
 .00000507 
 
 31-1 
 
 33-9 
 
 .0322 
 
 .0294 
 
 .00000914 
 
 .00453
 
 2 9 6 
 
 EXPERIMENTAL PHYSICS. 
 
 TABLE XV. 
 
 Specific Resistances of Wires of Different Metals and Alloys. 
 
 NAME OF CONDUCTOR. 
 
 SPECIFIC RESIST- 
 ANCE IN OHMS. 
 
 RESISTANCE IN OHMS. 
 
 i MBTER WEIGH- 
 ING x GRAM. 
 
 ioo METERS i MM. 
 IN DIAMETER. 
 
 Silver (annealed) . . . 
 
 I.492XIO" 6 
 
 JS 1 ? 
 
 1.899 
 
 Silver (hard drawn) . . . 
 
 1.62 
 
 .165 
 
 2.062 
 
 Copper (annealed) . 
 
 1.584 
 
 .1415^ 
 
 2.017 
 
 Copper (hard drawn) . . 
 
 1.621 
 
 Mi 
 1443 
 
 2.063 
 
 Gold (annealed) .... 
 
 2.041 
 
 .4007 
 
 2.598 
 
 Gold (hard drawn) . . . 
 
 2.077 
 
 .4076 
 
 2.644 
 
 Aluminium (annealed) . . 
 
 2.889 
 
 0743 
 
 3-^S- 
 
 Zinc (compressed) . 
 
 5.58 
 
 3995 
 
 7- I0 5 
 
 Platinum (annealed) . 
 
 8.981 
 
 1.925 
 
 11 -435 
 
 Iron (annealed) .... 
 
 9.636 
 
 75** 
 
 12.27 
 
 Nickel (annealed) . . . 
 
 12.356 
 
 1.052 
 
 15-73 
 
 Tin (annealed) .... 
 
 13-103 
 
 95 6 4 
 
 16.68 
 
 Lead (compressed) . . . 
 
 19.465 
 
 2.217 
 
 24.78 
 
 Antimony (compressed) 
 
 35- 21 
 
 2 -37 
 
 44-83 
 
 Bismuth (compressed) . 
 
 130.1 
 
 12.8 
 
 165.60 
 
 Mercury (liquid*) . . . 
 
 94-34 
 
 12.826 
 
 120. II 
 
 Alloy (2 Pt + i Ag) . . . 
 
 24.187 
 
 2.907 
 
 30-79 
 
 Alloy (2 Au+ i Ag) . V 
 
 10.776 
 
 1.638 
 
 13.72 
 
 Alloy (9 Pt + i Ir) . . . 
 
 21.633 
 
 4.651 
 
 2 7-54 
 
 German silver 
 
 20.76 
 
 1.817 
 
 26.43 
 
 * According to a very careful determination by Lord Rayleigh and Mrs. Sidgwick 
 (Phil. Trans., Part I., 1883), a column of mercury, one square millimeter in cross- 
 section, which at o C. has a resistance of an ohm, is 106.21 centimeters in length. 
 At an International Conference at Paris in 1884 it was agreed to define the "legal 
 ohm " as " the resistance of a column of mercury 106 centimeters long and one square 
 millimeter in section at the temperature of melting ice" The Chamber of Delegates 
 at the Chicago Electrical Congress in 1 893 adopted " as a unit of resistance the 
 international ohm, which is based upon the ohm equal to IO 9 units of resistance of 
 the C.G.S. system of electromagnetic units, and is represented sufficiently well by the 
 resistance offered to an unvarying electric current by a column of mercury at the 
 temperature of melting ice, 14.4521 grams in mass, of a constant cross-sectional area, 
 and of a length of 106.3 centimeters.
 
 TABLES. 
 
 297 
 
 TABLE XVI. 
 
 Conductivities of Pure Metals at t C* 
 Conductivity at o = i. 
 
 METAL. 
 
 
 CONDUCTIVITY AT /> C. 
 
 Silver 
 Copper 
 Gold 
 Zinc 
 
 
 .0038278 / + .000009848 / 2 
 .0038701 /+ .OOOOOpOOQ t~ 
 .0036745 /4- .000008443 /- 
 .0037047 t -\- .000008274 t~ 
 
 Cadmium 
 
 
 .0036871 t -\- .000007575 t" 
 
 Tin 
 
 
 0036029 /+ .000006 136 t* 
 
 Lead 
 
 
 0038756 /-)- 000009 146 t 2 
 
 \rsenic 
 
 
 0038996 /+ 0000088 79 / 2 
 
 Antimony 
 Bismuth 
 Iron 
 
 
 .0030826 /-f .000010364 / 2 
 .0035216/4- .000005 728 / 2 
 .0051182 t+ .00001 2916 t~ 
 
 * From Matthiessen.
 
 2 9 8 
 
 EXPERIMENTAL PHYSICS. 
 
 TABLE XVII. 
 
 Conductivity and Resistance of Pure Copper at Temperatures from 
 o C. to 40 C. 
 
 TEMPERATURE. 
 
 CONDUCTIVITY. 
 
 RESISTANCE. 
 
 TEMPERATURE. 
 
 CONDUCTIVITY. 
 
 RESISTANCE. 
 
 
 
 1. 0000 
 
 1. 0000 
 
 21 
 
 .9227 
 
 1.0838 
 
 I 
 
 .9961 
 
 1.00388 
 
 22 
 
 .9192 
 
 1.0879 
 
 2 
 
 9923 
 
 1.00776 
 
 2 3 
 
 .9158 
 
 1.0920 
 
 3 
 
 .9885 
 
 1.0116 
 
 24 
 
 .9123 
 
 1.0961 
 
 4 
 
 .9847 
 
 1.0156 
 
 2 5 
 
 .9089 
 
 I.I003 
 
 5 
 
 .9809 
 
 1.0195 
 
 26 
 
 9054 
 
 1.1044 
 
 6 
 
 9771 
 
 1.0234 
 
 2 7 
 
 .9020 
 
 1.1085 
 
 7 
 
 9734 
 
 1.0274 
 
 28 
 
 .8987 
 
 I.II27 
 
 8 
 
 .9696 
 
 I-03I3 
 
 29 
 
 8953 
 
 1.1169 
 
 9 
 
 9659 
 
 "0353 
 
 3 
 
 .8920 
 
 I.I2II 
 
 10 
 
 .9622 
 
 '0393 
 
 3 1 
 
 .8887 
 
 I-I253 
 
 ii 
 
 9585 
 
 1-0433 
 
 3 2 
 
 .8854 
 
 I.I2 9 5 
 
 12 
 
 9549 
 
 1-0473 
 
 33 
 
 .8821 
 
 I-I337 
 
 ^3 
 
 .9512 
 
 l.<>5*3 
 
 34 
 
 .8788 
 
 I-I379 
 
 14 
 
 .9476 
 
 *-553 
 
 35 
 
 .8756 
 
 I.I42I 
 
 '5 
 
 .9440 
 
 I -593 
 
 36 
 
 .8723 
 
 1.1464 
 
 16 
 
 .9404 
 
 1.0634 
 
 37 
 
 .8691 
 
 1.1506 
 
 i? 
 
 .9368 
 
 1.0675 
 
 38 
 
 .8659 
 
 I.I5 4 8 
 
 18 
 
 9333 
 
 1.0715 
 
 39 
 
 .8628 
 
 I-I59I 
 
 T 9 
 
 .9297 
 
 1.0756 
 
 40 
 
 .8596 
 
 1-1633 
 
 20 
 
 .9262 
 
 1.0797 
 
 

 
 TABLES. 
 
 299 
 
 TABLE XVIII A. 
 
 Liquid Resistances. 
 
 SUBSTANCE DISSOLVED. 
 
 COMPOSITION. 
 
 TEMPERATURE. 
 
 SPECIFIC 
 RESISTANCE. 
 
 Sulphuric acid .... 
 
 f SO 3 HO 
 ' SO 3 HO+ 1 4 HO 
 S0 3 HO + 1 3 HO 
 LS0 3 HO + 499HO 
 
 15 c. 
 
 19 C. 
 
 22 C. 
 22 C. 
 
 9.146 ohms 
 1.336 " 
 1.256 
 I743I " 
 
 Sulphate of zinc . 
 
 f ZnOSO 3 + 23 HO 
 ZnOS0 3 + 24 HO 
 ZnOSO 3 + 105 HO 
 
 23 C. 
 23 C. 
 23 C. 
 
 18.31 
 
 1 8.02 
 
 33.04 
 
 Sulphate of copper . 
 
 CuOS0 3 + 45 HO 
 \ CuOSO 3 + 105 HO 
 
 22 C. 
 12 C. 
 
 19.10 " 
 
 31.42 
 
 Sulphate of magnesia 
 
 f MgOSO 3 + 34 HO 
 \ MgOSO 3 + 107 HO 
 
 22 C. 
 22 C. 
 
 18.44 " 
 
 30.06 " 
 
 Hydrochloric acid 
 
 fHCl + 7.sHO 
 I HC1 + 250 HO 
 
 23 C. 
 23 C. 
 
 1.285 " 
 8.177 "
 
 300 
 
 EXPERIMENTAL PHYSICS. 
 
 TABLE XVIII B. 
 
 ecific Resistances of Sulphuric Acid at 22 C. 
 (Kohlrausck and Nippoldt.} 
 
 DENSITY OF THE 
 SOLUTION. 
 
 PROPORTION OF ACID. 
 
 SPECIFIC RESISTANCE. 
 
 RELATIVE INCREASE OF 
 CONDUCTIVITY FOR iC. 
 
 .9985 
 
 .0 
 
 70.41 
 
 .47-IO-- 
 
 1. 0000 
 
 .2 
 
 41.05 
 
 47 
 
 1.0504 
 
 3-3 
 
 3-252 
 
 -653 
 
 1.0989 
 
 14.2 
 
 1.787 
 
 .646 ' 
 
 1.1431 
 
 20.2 
 
 1.414 
 
 799 
 
 1.2045 
 
 28.0 
 
 1.239 
 
 I-3 1 ? 
 
 1.2631 
 
 35- 2 
 
 1.239 
 
 1.259 
 
 1-3163 
 
 4i-5 
 
 1-347 
 
 1.410 
 
 1-3547 
 
 46.0 
 
 1.487 
 
 1.674 
 
 1-3994 
 
 5-4 
 
 1.672 
 
 1.582 
 
 1.4482 
 
 55- 2 
 
 1.962 
 
 1.417 
 
 1.5026 
 
 60.3 
 
 2.412 
 
 1.794 
 
 TABLE XIX. 
 
 Internal Resistances {Approximate} of Batteries. 
 
 ELEMENT. 
 
 TYPE. 
 
 RESISTANCE IN OHMS. 
 
 Daniell 
 
 Callaud 
 
 4.00 to 5.00 
 
 Daniell 
 
 Meidinger 
 
 4.00 to 9.00 
 
 Grove 
 
 Ordinary 
 
 .26 to .45 
 
 Bunsen 
 
 " 
 
 .06 to .24 
 
 Grenet 
 
 " 
 
 .75 to i. oo 
 
 Leclanchd 
 
 " 
 
 5.50 to 6.00 
 
 Latimer-Clark 
 
 Beetz 
 
 I57OO.OO
 
 TABLES. 
 
 3 0i 
 
 TABLE XX. 
 
 Electromotive Forces of Batteries. 
 
 VOLTS. 
 
 Daniell 
 
 Daniell 
 
 Grove 
 
 Bunsen 
 
 Grenet 
 
 Leclanche" 
 
 Volta 
 
 Latimer-Clark 
 
 ( Amalgamated zinc, 
 i sulphuric acid -f 4 water, 
 Saturated solution of copper sulphate, f 
 Copper. 
 
 Amalgamated zinc, 
 i sulphuric acid 4-12 water, 
 Saturated solution of copper sulphate, 
 [ Copper. 
 
 f Amalgamated zinc, 
 
 I i sulphuric acid + 4 water, 
 
 Fuming nitric acid, 
 
 Platinum. 
 
 j Amalgamated zinc, 
 i sulphuric acid + 12 water, 
 Fuming nitric acid, 
 Carbon. 
 
 When freshly set up, 
 
 f Amalgamated zinc, 
 
 -, Solution of sal-ammoniac, 
 
 I Binoxide of manganese and carbon. 
 
 rZinc, 
 
 \ Ordinary water, 
 
 I Copper. 
 
 f Zinc, 
 
 Sulphate of zinc, 
 j Sulphate of mercury in paste, 
 
 Mercury. 
 
 1.07 
 97 
 
 '95 
 
 1.94 
 
 2.03 
 1.46 
 
 .98 
 1-434
 
 3 02 
 
 EXPERIMENTAL PHYSICS. 
 
 TABLE XXI. 
 
 Electro-chemical Equivalents. 
 
 ELEMENTS. 
 
 ATOMIC 
 WEIGHT. 
 
 VALENCY. 
 
 CHEMICAL 
 EQUIVALENTS. 
 
 ELECTRO-CHEMICAL 
 EQUIVALENTS, OR 
 GRAMS PER COULOMB 
 OF ELECTRICITY. 
 
 Electro-positive. 
 
 
 
 
 
 Hydrogen .... 
 
 I 
 
 I 
 
 I 
 
 .00001038 
 
 Potassium .... 
 
 39-3 
 
 I 
 
 39-3 
 
 .000405 I 
 
 Sodium 
 
 2 3- 
 
 I 
 
 2 3- 
 
 .0002387 
 
 Gold 
 
 196.2 
 
 3 
 
 65-4 
 
 .0006789 
 
 Silver 
 
 107.7 
 
 i 
 
 107.7 
 
 .OOIIlS 
 
 Copper (cupric) . . 
 
 63.18 
 
 2 
 
 3 T -59 
 
 .0003279 
 
 Copper (cuprous) 
 
 63.18 
 
 I 
 
 63.18 
 
 .0006558 
 
 Mercury (mercuric) . 
 
 199.8 
 
 2 
 
 99-9 
 
 .001037 
 
 Mercury (mercurous) 
 
 199.8 
 
 I 
 
 199.8 
 
 .002074 
 
 Tin (stannic) . . . 
 
 117.4 
 
 4 
 
 29-35 
 
 .0003046 
 
 Tin (stannous) . . 
 
 117.4 
 
 2 
 
 58.7 
 
 .0006093 
 
 Iron (ferric) . ... 
 
 55-88 
 
 3 
 
 18.63 
 
 .0001934 
 
 Iron (ferrous) . 
 
 55-88 
 
 2 
 
 27.94 
 
 .OOO29OO 
 
 Nickel 
 
 58.6 
 
 2 
 
 20 ? 
 
 .0003042 
 
 Zinc 
 
 64.88 
 
 2 
 
 *y*o 
 
 32-44 
 
 .0003367 
 
 Lead 
 
 206.4 
 
 2 
 
 10^.2 
 
 .OOIO7I 
 
 Aluminium .... 
 
 27.04 
 
 3 
 
 A V O * 
 
 9.01 
 
 .0000935 
 
 Electro-negative. 
 
 
 
 
 
 Oxygen 
 
 15.96 
 
 2 
 
 7.98 
 
 .00008283 
 
 Chlorine . . . -.-. .- 
 
 35-37 
 
 I 
 
 35-37 
 
 .0003671 
 
 Iodine 
 
 126.54 
 
 I 
 
 126.54 
 
 .0013134 
 
 Bromine 
 
 70 76 
 
 I 
 
 70 76 
 
 OOO8270 
 
 Nitrogen . . 
 
 /y- /" 
 
 14.01 
 
 3 
 
 / 7 1 
 4 .6 7 
 
 ,*-/w~j ^ y y 
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