UC-NRLF 
 
THE GALVANOMETER 
 
 A SERIES OF LECTURES 
 
 BY 
 
 EDWARD L. NICHOLS 
 
 PROFESSOR OF PHYSICS IN CORNELL UNIVERSITY 
 
 As printed in ELECTRIC POWER. 
 
 NEW YORK 
 McILROY & EMMET 
 
 1894 
 
UNIVERSITY 
 
GALVANOMETER AT CORNELL UNIVERSITY. 
 
4 MfV. 
 
 THE GALVANOMETER 
 
 A SERIES OF LECTURES 
 
 BY 
 
 EDWARD L NICHOLS 
 
 V 
 
 PROFESSOR OF PHYSICS IN CORNELL UNIVERSITY 
 
 OF THE 
 
 NEW YORK 
 McILROY & EMMET 
 
 1894 
 
Entered according to Act of Congress in the year 1894 by 
 
 McTLROY & EMMET, 
 in the office of the Librarian of Congress at Washington. 
 
X X 
 
 f OF THE 
 
 (UNIVERSITY 
 
 '"PHIS series of lectures was written, in the first place, 
 for the benefit of a class of students of electrical engi- 
 neering- in Cornell University. In it I have endeavored 
 to bring together, in compact form, for the benefit of 
 readers of limited mathematical attainments, the most 
 important features of the theory of the galvanometer, 
 together with some suggestions concerning the methods 
 of using that instrument. ' To this end, free use has 
 been made of the work of many writers. The treatises 
 of Maxwell, and of Mascart and Joubert, in particular, 
 have been repeatedly drawn upon. From various fel- 
 low physicists, also, I have received hints. The services 
 of one of these, Professor W. S. Franklin, I desire es- 
 pecially to acknowledge, since several important features 
 in the theoretical treatment of my subject are due to 
 suggestions made by him. Detailed studies of the per- 
 formance of sensitive galvanometers by Professor 
 Ernest Merritt and Mr. F. J. Rogers, and a variety of 
 details culled from the records, published and unpub- 
 lished, of other workers in the laboratories of the de- 
 partment of physics have been made use of. 
 
 Nearly all that has been written about galvanometers, 
 aside from the theory and details of construction, applies 
 to the types of instrument which were called into ex- 
 istence by the requirements of the testing of cables and 
 telegraph lines and of submarine signalling. Concern- 
 ing the use of such instruments on the one hand and 
 upon the subject of what may be termed voltmeter and 
 ammeter work, from its relation to the practice and meth- 
 
ods of the dynamo laboratory, abundant material is 
 already accessible in the manuals of Kempe, Ayrton, 
 Kittler and many other writers. To this part of the 
 subject I have paid little attention. 
 
 The extraordinary demands upon the sensitiveness of 
 the galvanometer made by the researches of Langley, 
 Aegstrom, Julius, Rubens, Snow, Paschen and others 
 in the domain of radiation, has, however, resulted in the 
 development of a new class of instruments, the delicacy 
 of which has greatly modified the art of using the gal- 
 vanometer. Of these matters I have endeavored to give 
 some account. 
 
 EDWARD L. NICHOLS. 
 January, 1894. 
 
TCTNIVERSITT 
 
 LECTURE I. 
 
 Galvanometers for absolute measurement. Three effects 
 of the voltaic current may be made use of for measure- 
 ment ; the thermal, utilized in electro-calorimetry, the 
 chemical, which is used in voltametry, and the magnetic, 
 upon which the action of the galvanometer depends. 
 The essential parts of this instrument are a magnet 
 needle, suspended in general with freedom of vibration 
 
 FIG. I. 
 
 about a vertical axis, and an electric circuit, consisting 
 of one or more coils of wire, within the field of which 
 the needle swings. 
 
 For purposes of absolute measurement the coils of 
 the galvanometer must be of known dimensions, they 
 must be at a known distance from the needle, and the 
 latter must be situated in a magnetic field of known 
 intensity. The simplest form, which is also that most 
 frequently met with in practice, consists of one or more 
 circular coils mounted vertically in the magnetic meri- 
 dian. Where a single coil is used the needle is in its 
 axis. When there is more than one coil, these have a 
 common axis in which the needle hangs. 
 
The law of action of the galvanometer may be de- 
 rived from the following familiar and well-established 
 principles : 
 
 1. Influence of a current upon a magnetic particle. Given 
 a conductor //, carrying current at right angles to the 
 plane of the paper (Fig. i) in the direction indicated by 
 the lines of force. A particle // situated in the field 
 surrounding L will tend to move along a line of force. 
 
 2. Influence of a circular current. If i is a circular con- 
 ductor and p. be situated in the axis of the ring (Fig. 2) 
 at a distance x from the plane of the latter, and at a 
 distance d from the conductor, the action of each ele- 
 ment (d L) of the ring will be to tend to drive ft along 
 
 d L 
 
 FIG. 2. 
 
 FIG. 3. 
 
 A B, tangent to the line of force at that point with a 
 force f (d L ), such that 
 
 Const, i d L _ Const, i d L 
 
 J(d L) - ff ^ _|_ ^o (1) 
 
 For the entire circle the force fa) is 
 
 /() = Const, ^-qj 
 
 If jj. be a magnet-pole of strength m with freedom of 
 motion only around a vertical axis, the case with which 
 we have to do in considering the galvanometer, the 
 effective component of fa) along the axis is 
 
 ft Const. 
 
 fynriva 
 
 = Const. 
 
 2 TT r 2 i m 
 
 .(3) 
 
When r and x are taken in centimeters the constant is 
 unity and i is expressed in absolute measure. 
 
 The actual case to be considered is that of a needle, 
 swinging in a magnetic field. This field is made up of 
 two active components, the horizontal force (fe) of the 
 earth's magnetism, or of an artificial field which takes 
 its place, and the horizontal component of the force due 
 to the current (/*), see Fig. 3. 
 
 When the couple 2 I f e sin $, due to the action of the 
 earth's field is balanced by the couple 2 I // cos $, the 
 needle is in equilibrium, a condition which is expressed 
 by the equation. 
 
 / /// 
 
 X 
 
 FIG. 4. 
 
 from which, since f e = Jim, where ^Tis the horizontal 
 component of the earth's magnetism and ra is the 
 strength of the magnet pole, we derive the equation of 
 the tangent galvanometer of a single turn. 
 
 (5) 
 
 If the galvanometer consists of any number of sepa- 
 rate coils the radii of which are r lf r 2 , r s , 
 
 etc., at dis- 
 
tances x^x 2) a* 3 , x 4) etc., from the needle, containing re- 
 spectively Wj, ??2, ft 3 , ?? 4 , etc., turns of wire, each coil will 
 have its independent action upon the needle, and the 
 following general equation of the galvanometer may 
 be written. This equation is applicable in every case 
 in which the coils are in the magnetic meridian, and 
 possess a common axis, in which axis the needle, the 
 length of which must be small compared with the radii 
 of the coils, is suspended. This equation is, 
 
 2 7T 
 
 "271 
 
 (rf 
 
 2 it r 
 
 + etc. 
 
 
 FIG. 5. 
 
 The denominator of the right-hand member of this 
 equation is termed the constant of the coils, and it is desig- 
 
 TT 
 
 nated by the letter G^;* the ratio -^ is the constant of the 
 
 galvanometer. Equation (6), then, may be written either 
 in the form 
 
 * The reciprocal of this expression is sometimes taken as the constant of the coils, 
 in which case equation (7) is written 
 
 TT 
 
 i = G H tan $, instead of i = -Q tan $ 
 
or in the form 
 
 i = tan *. (7) 
 
 i = JTtan #. (8) 
 
 As a matter of construction, tangent galvanometers 
 usually have either one or two coils. The large gal- 
 vanometer of Cornell University with six coils is really 
 a combination of several instruments, designed for heavy 
 currents and one for currents of small intensity. 
 
 Of galvanometers with a single coil, the common 
 form has the needle in the plane of the ring. For such 
 
 FIG. 6. 
 
 instruments the quantity x in equation (6) is zero. 
 The equation then takes the following simpler form : 
 
 i = ^- n H tan &. (9) 
 
 The objection to galvanometers with the needle in 
 the plane of the ring is in the nature of the field of 
 force due to the current in a single ring. This field of 
 a circular current was discussed by Lord Kelvin in 
 1869 ;* from whose results, as embodied in Plate XVIII. 
 of the second volume of Maxwell's Treatise on Electric- 
 ity, Fig. 4 is taken. 
 
 * W. Thomson, Transactions of the Royal Society of Edinburgh, vol. 25, p. 217. 
 
In such a field, by the use of a long needle or the dis- 
 placement of the needle from its position in the centre 
 of the coil, considerable errors are introduced. It is to 
 reduce such errors to a minimum that the Helmholtz 
 form of the tangent galvanometer is used. In this in- 
 strument two coils of equal area are placed at a dis- 
 tance apart equal to their radius, and the needle is. 
 
 FIG. 7. 
 
 situated midway between them in their common axis. 
 The field produced by current in coils thus located is 
 very nearly uniform for a considerable region surround- 
 ing the centre of the system, and displacements of the 
 needle have of but slight influence upon the perform- 
 ance of the galvanometer. Figure 5, also from Lord 
 
 10 
 
Kelvin's paper just cited, affords a comparison of the 
 field of two parallel circular currents, with that pro- 
 duced by a single coil. 
 
 The large tangent galvanometer of Cornell Univer- 
 sity, to which brief reference has just been made, affords 
 an interesting example of the application of the Helm- 
 holtz construction. A diagram of this instrument, 
 showing the proportions and dimensions of the various 
 coils, is given in Fig. 6. 
 
 It consists, essentially, of four distinct instruments ar- 
 ranged so as to be used separately or in combination. 
 There are : 
 
 a. A Helmholtz galvanometer (i, 4, i, 4) with coils 
 about two meters in diameter, each consisting of a single 
 turn of heavy wire of about 2.00 cm. diameter. 
 
 b. A similar galvanometer symmetrically placed with 
 reference to the first and acting upon the same needle, 
 with coils of the same heavy wire, the diameter of each 
 coil being about 160 cm. 
 
 FIG. 8. 
 
 c. A Helmholtz galvanometer, for small currents, 
 with 36 turns, divided into two sections of eighteen 
 turns apiece, in each coil, the mean diameter of the 
 coils about 152 cm. The centre of the system coin- 
 cides with that of a and b, and the same needle serves. 
 
 d. A modification of the Kohlrausch instrument for 
 the determination of H. This consists of a coil 100 cm. 
 in diameter with 100 turns of wire. It is suspended in 
 the magnetic meridian by means of a phosphor bronze 
 wire about 200 cm. in length. The axis of the coil is 
 coincident with those of the galvanometer coils already 
 described. The method of using it will be given in the 
 lecture on the determination of H. 
 
 The four coils i, 4 and 2, 3 are connected with a 
 switchboard of massive bronze, shown in diagram in 
 Fig. 7, by means of which any coil can be used by itself 
 or any two or three, or all four in series (directly or dif- 
 ferentially) or in multiple. Thus a considerable range 
 
 ii 
 
 /^^ *4*> 
 
 t 
 
 (UNIVERSITY 
 
 V ^ A ,* 
 
of sensitiveness may be obtained. The time required 
 to change from one combination to another is that 
 necessary to insert and withdraw certain plugs and to 
 throw the switches. 
 
 The following are the dimensions of the six coils and 
 the constants of the galvanometer with various arrange- 
 ments of the coils when the strength of the field is 
 H= 0.1710. 
 
 DIMENSIONS. 
 
 cm 
 
 J 
 
 Coil 
 
 Measurements uy jxyau aim n 
 
 i radius = 
 
 aiiiiiiuii, loot) 
 
 100.1047 
 
 tt 
 
 4 
 
 100 1275 
 
 n 
 
 2 "..,.== 
 
 80.1037 
 
 a 
 
 3 
 
 80.1025 
 
 " 
 
 W mean radius = 
 
 76.0765 
 
 u 
 
 TF a " " = 
 
 76.0919 
 
 Distance 
 
 apart (2 x) i to 4 ; 
 
 99.9770 
 
 
 
 " (2 x) 2 to 3 ; 
 
 80.0274 
 
 (2 x)W 1 toW 2 76.0310 
 
 FIG. Q. 
 
 CONSTANTS (C. G. S.) 
 
 ARRANGEMENT. 
 
 G 
 
 logGH 
 
 *-f 
 
 log AT 
 
 Coil i 
 
 
 
 
 80 06 
 
 
 .044942 
 
 2.652656 
 
 3.8049 
 
 0.580340 
 
 Coils i-f~4 
 
 .089888 
 
 2.053703 
 
 I.9O24 
 
 0.279293 
 
 Coil 2 
 
 
 
 
 
 " 3 
 
 .056159 
 
 2.749421 
 
 3.0449 
 
 0.483575 
 
 Coils 2 -f 3 
 
 .112319 
 
 1.050450 
 
 
 
 :: ^tfe: 
 
 Wound coils in series 
 
 .202205 
 .022430 
 
 .729752 
 
 1.305792 
 2.350831 
 i 863175 
 
 0.84568 
 7.6237 
 0.23433 
 
 1.927204 
 0.882165 
 1.369821 
 
 12 
 
The method of reading the deflections of this gal- 
 vanometer is as follows : There are two circular scales, 
 graduated decimally upon metal strips. These strips 
 form two opposite quadrants of the inner face of a 
 cylinder, the radius of which is 50 cm. (see Fig. 8). At 
 the centre of this cylinder is the mirror, circular in form 
 but cut in two diametrically, the halves hinged at the 
 median line and dropped down to an angle of 45 with 
 the horizontal plane, to which position they are adjusted 
 by means of screws (see Fig. 9). 
 
 The scale is viewed through a telescope (Fig. 10) be- 
 fore the objective of which is placed a large right- 
 
 FIG. 10. 
 
 angled total-reflection prism. This catches the vertical 
 rays proceeding from the two scales to the mirror and 
 reflected upwards from the oblique faces of the latter. 
 Images of those portions of both scales which are 
 directly opposite the halves of the mirror are brought 
 into the eye-piece in superposition. These may be read 
 separately by cutting out one or the other by means of 
 shutters. 
 
 This galvanometer has been briefly described by 
 Professor W. A. Anthony,* under whose direction it 
 was constructed in 1885. 
 
 * The Electrical Engineer (N. Y.), Vol. iv., October, 1885. 
 
 13 
 
LECTURE II. 
 
 The sine galvanometer . In the case of the sine galvano- 
 meter , the coils of which are free to revolve upon a 
 vertical axis, and are made to follow the needle in its 
 deflection, the formulae of the tangent galvanometer 
 are applicable with the following slight modification. 
 
 Since the deflected needle is always finally in the 
 plane of the coils the couple due to the current acting 
 upon it is 2 I fi instead of 2 I f f cos $, and equations 4 
 and 5 become 
 
 /'=/, sin 0. (10) 
 
 TT 
 
 i = -@ sin &. (11) 
 
 In the sine galvanometer the angular movement of 
 the coils may be indicated by verniers upon an astrono- 
 mical circle, a method capable of greater accuracy than 
 the usual methods of reading the deflection of a tan- 
 gent galvanometer. The limit of accuracy is deter- 
 mined, however, not by the fineness of the circle in 
 question, but upon the precision with which the coinci- 
 dence of the needle with the plane of the coils, in the 
 final adjustment of the latter can be ascertained. The 
 most refined device for this purpose is due I believe, to 
 Professor H. A. Rowland. It consists of a small read- 
 ing telescope carried upon an arm which turns with the 
 coils of the galvanometer. This telescope bears a short 
 horizontal scale, the central division of which is in the 
 same vertical plane as the axis of the telescope. A 
 mirror attached to the galvanometer needle gives an 
 image of the scale in the eye-piece of the telescope, the 
 zero falling upon the cross hair when the plane of the 
 coils is parallel to the axis of the needle. 
 
 Standard galvanometers with variable constants. It is 
 often desirable to be able to vary the constant of a 
 standard galvanometer in a determinate manner. To 
 vary /Tin such a manner is not easily practicable, but G 
 may be subjected to perfectly definite changes. The 
 galvanometer of Obach* affords an illustration (Fig. 1 1 
 is from Obach 's original plate) of one method of accom- 
 plishing this end. It depends upon the fact that the 
 
 * Obach : Carl's Repertorium 14, p. 507, 1878. 
 
 14 
 
constant G of a galvanometer, with needle in the plane 
 of the coil, is inversely proportional to the projection 
 of the radius of the coil upon the vertical north and 
 south plane. By mounting the coil of a tangent gal- 
 vanometer upon a horizontal axis and causing it to 
 make any angle 6 with the vertical, the effective con- 
 stant of the coil can be given any value G = G cos 0. 
 The equation of the galvanometer then becomes 
 
 H II 
 
 = -T tan = 
 
 GcosO 
 
 tan #. 
 
 FIG. II. 
 
 Another well-known form of standard instrument 
 with variable constant, is the Thomson graded galvano- 
 meter, in which the needle is moved along the axis of 
 the coil. The sensitiveness of instrument is thus within 
 definite control through a wide range. For the double 
 purpose of further increasing the range of usefulness 
 and of protecting the needle from magnetic disturbance 
 the galvanometer is provided with an artificial field. A 
 further discussion of this feature will be given in the 
 lecture on galvanometers with artificial fields. This 
 
type of instrument like the " swinging-arm " galvano- 
 meter of Moler and other forms does not, however, be- 
 long to the class of galvanometers now under considera- 
 tion, since the constant is determined by calibration in- 
 stead of being computed directly from the dimensions 
 of the coil and the position of the needle. 
 
 Corrections to be applied in the case of galvanometers in 
 which the conditions already described are not fulfilled. In 
 the development of the law of the galvanometer: given 
 in lecture i , the following conditions were assumed : 
 
 1. The needle is in the axis of the coils ; 
 
 2. The coils are vertical and in the magnetic meri- 
 dian ; 
 
 3. The length of the needle is small as compared with 
 the radius of the coils. It is further assumed that the 
 number of turns is so small that they can all be wound 
 within a space such that the cross-section of the bundle 
 of wires is small as compared with the radius of the 
 coil. 
 
 The theory of the action of a circular current upon a 
 needle situated at a distance from the axis is given by 
 Maxwellf also by Mascart and Joubert,^ and has been 
 reproduced in many other treatises. 
 
 It will be possible here to indicate only the outlines 
 of the analysis, and the theoretical considerations upon 
 which the selection of certain forms, notably of the 
 Helmholtz type of galvanometer is based. 
 
 It is usual to base this discussion upon the following 
 considerations : 
 
 1. For any point situated at a distance y from the 
 axis of circular magnetic layer of unit density the po- 
 tential p is 
 
 / > = 2r(/ +/ I y ! +/ 2 y 4 +....etc.) (12) 
 
 in which f J\ /* 8 , etc., are all functions of the distance 
 of the point from the plane of the sheet. 
 
 2. The potential at the same point due to a magnetic 
 shell of unit strength, the boundary of which is the 
 same as that in the previous case is V, where 
 
 3. The x component of the magnetic force due to a 
 circular current traversing the boundary of the mag- 
 netic shell is 
 
 + Maxwell : Electricity and Magnetism, Vol. II., Chap. 14, also in Art. 711, p. 355 
 (Ed., 1892). 
 
 % Mascart et Joubert : Lecons sur 1'electricite et sur le magnetisme, T. 2, pp. 101, 
 etc. 
 
 16 
 
^rN 
 
 UNIVERSITY) 
 
 d V 
 
 The quantities 
 as follows : 
 
 ~ " 2 2 ' d a? 
 
 (14) 
 o, /i, / 2 , etc., are related to each other 
 
 1 d 2 
 
 /_, i 
 
 dafo 
 
 Jn 2 ri* ' d 
 etc. 
 
 '3? (2 . 4 . . 
 
 . nf d a? n 
 
 FIG. 12. 
 
 The value of / , however, from which all the higher 
 coefficients are readily derived, is determined by the 
 value of the potential upon the axis. At the distance 
 x from the plane of the layer for example the potential 
 
 is 
 
 2 TT ( 
 
 a?) ; 
 
 (16) 
 
 where r is the radius of the circular layer, a form which 
 leads by successive differentiation to the same expres- 
 sion for the magnetic action of the current as that given 
 in lecture I. (equation 3). Thus : 
 
 d x ~ 
 
 dx 
 
 r 2 )! 
 
 (18) 
 
 
which is identical with equation 3 when the current is 
 unity. 
 
 Since, however, p is a particular value of p (equation 
 12), in which y = we have 
 
 / = AJM^ , - - * y (19) 
 
 where -w 2 = r 2 -|- ar 2 . 
 
 The coefficients belonging to the series for X may 
 readily be obtained by differentiation. 
 
 Thus we have 
 
 (20) 
 
 /o = u x, 
 
 _r \. CL 
 
 /O 
 
 = -<? / 8 ,etc.; 
 
 c 'W a? 
 aJ a? ~~ ^ 
 
 d a? ~ u 5 
 
 ft u 3 y 2 (4 a? r 2 ) 
 
 <# a? 4 ~ 'w 7 
 
 By means of these values we may write the series for 
 X to the fourth power of x. 
 
 3 4^ ^ 
 
 3 2 .. 5 y* 12 r 2 a? 2 -f 8 a? 4 
 
 The series, the lower members of which are given in 
 22 leads to a result only when y < u, which is always 
 the case in dealing with galvanometers of ordinary 
 form. 
 
 When the needle lies in the axis, y becomes zero, and 
 equation 22 becomes 
 
 T-_ f 27rr* 2*7* 
 
 -/*- ^3 -pqr^; 
 
 which is identical, of course, with 18 and with 3. 
 
 Professor W. S. Franklin has suggested to the writer 
 that an expression for X corresponding to 22 may be 
 obtained without recourse to the artificial conception of 
 the potential of a magnetic shell, 
 
 18 
 
His method is as follows : 
 
 To find a development of the component in the direction 
 of the axis of the magnetic field due to a circular coil. 
 
 The value of this component at points in the axis (at 
 these points the component being the total field) is ob- 
 tained as follows : 
 
 Let a magnetic pole of strength m be placed in the 
 axis of the coil at a distance x from its plane. The 
 
 m m 
 
 magnetic field / at the wire is / = ^ = y2 . ^. An ele- 
 ment d I of the coil will be acted upon by a force d F 
 at right angles to / and to d I, such that d F = f i d I. 
 i being the strength of the current in the coil. The 
 component of d Fin the axial direction is / i d I Sin 0, 
 and this same force acting upon every element d I of 
 the coil gives for the total force acting on the coil 
 (which force is in the axial direction from symmetry) 
 > = / Hsin 0. 
 
 T 
 
 where I = 2 n T and sin 6 
 
 FIG. 13. 
 
 Substituting in this value for F the values of 
 f =. a.*, I, and cos 6 we have 
 
 (23) 
 
 = 2 TT m i a 
 
 + flty 
 
 Owing to the equality of action and reaction this 
 same force must act upon the pole m, but a force acting 
 on a magnetic pole is always the product of the strength 
 of the pole into the strength of the magnetic field at the pole 
 so that the factor by which m is multiplied in (i) is the 
 required strength X of the magnetic field due to the 
 
circular coil at a distance from x the plane of the coil 
 and in its axis so that 
 
 2 7i i 
 r 
 
 in which expression , _* may easily be expanded 
 
 in powers of ?!_, giving a series for -5T with definite 
 r 
 
 coefficients. 
 
 The component in any given direction of a magnetic 
 field satisfies La Place's equation so that we may ex- 
 pand such a component when symmetrical to an axis in 
 a series of zonal harmonics, i.e., 
 
 X=A Q Pv (cos ^)+A 1 dP z (cos0+A4A(cos^). . . (25) 
 
 where the ^.'s are undetermined coefficients X is the 
 required component at distance d from centre of circular 
 coil d making an angle <p with the axis of the coil (see 
 Fig. 13). When <p = then (25) must reduce to the 
 value (24) for JTat points in the axis. Substituting in 
 
 1 
 
 (24) the expansion of / cc 2 \f and placing this series 
 
 equal to (25) with <p = we have (since P n (cos <p) 1 
 when <p and d becomes x when <p = 0), by placing 
 the various coefficients of x, x*, x 3 , etc., equal each to 
 each a means for determining the values of the A's so 
 that (25) is then the completely determined. 
 The series for JTis made up thus : 
 
 PQ (COS <p) 1 
 
 P l (cos (p) cos (p 
 
 P 2 (cos f) = i (3 cos 2 <p 1) (26) 
 
 P 3 (cos tp) = J (5 cos 3 ^ 3 cos ^ ), etc. 
 
 The development thus formed holds only when x < r. 
 For values of a; > r must be broken up so as to intro- 
 
 duce the development of /r 3 \| when we may pro- 
 
 V?+ 
 
 ceed as before. 
 
 20 
 
The other rectangular components of the magnetic 
 field due to a circular coil cannot be developed in a 
 series of zonal spherical harmonics. 
 
 The value of the series (22) lies in the fact that by 
 means of it we may discover the influence of deviations 
 of the galvanometer needle from the axis when the 
 needle is sitiiated at various distances from the plane 
 of the coils. This has been done by a graphical method 
 in Mascart and Joubert's treatise (Vol. II., p. 103). 
 Curves showing the sign and value of the two members 
 containing y* and y* respectively, show that the former 
 which is the more important becomes zero when x = % r 
 while the member containing y* is very small for that 
 value of x. It is in accordance with the results of this 
 analysis, therefore, that in the Gaugain galvanometer 
 and in Helmholtz's pattern also the distance from the 
 needle to the plane of the coils is always one half the 
 radius of the latter. 
 
 It is not always allowable in computing the constant 
 of a galvanometer, to take a mean radius r as applicable 
 to all the turns which the coil contains. Particularly in 
 the case of galvanometers with many turns of the wire 
 it becomes necessary to consider the cross-section of 
 the coil. For the general discussion of this case the 
 reader is referred to Maxwell's treatise.* 
 
 Correction for the. length of the needle. The influence of 
 the length of the needle is a subject which scarcely 
 needs attention when dealing with modern galvano- 
 meters of the type now under consideration. The cor- 
 rection is a small one in all ordinary cases. It is a mini- 
 mum in the Helmholtz galvanometer in which the dis- 
 tance from the plane of the coil to the needle is - r. 
 In instruments of this type the correction is as follows, 
 where length of needle is 2 I. 
 
 2 I = . 2 r ; correction . 001 
 
 2 = .1667-; " .0005, etc. 
 
 These lengths are much greater than any in use in 
 modern tangent galvanometers, in the case of which 
 instruments the correction becomes entirely neglible. 
 
 Correction for torsion. In tangent galvanometers the 
 correction for the torsion of the suspension fibre is so 
 small that the following simple approximate method of 
 determining it is entirely adequate. 
 
 To estimate the correction for torsion we twist the 
 
 * Maxwell : A Treatise on Electricity and Magnetism, Vol. II., Chap,. 15 p. 354 
 (edition, 1892). 
 
 21 
 
upper end of the suspension fibre through an angle 8. 
 and note the movement of the needle resulting there- 
 from. Let the angle be ft ; then 
 
 (27) 
 u 
 
 is the approximate factor. 
 
 Case in which the coils of the galvanometers are not in the 
 magnetic meridian. Let the coils make an angle a. with 
 the meridian. Then upon reversing the direction of the 
 current through the instrument we get equilibrium for 
 the following positions of the needle. 
 
 TT 
 
 i cos (# + a) = 77 sin #. (28) 
 
 TT 
 
 i cos (0 1 a) = -Q sin tf 1 . (29) 
 
 Adding these equations we have 
 
 i [cos & cos a sin $ sina -f- cos $' cos a -|- sin #' sin = 
 
 ^(sin # + sin #') (30) 
 from which by the use of the usual conversion formulas 
 
 f # #' i - sin # #'-] 
 
 * [cos a cos g- + sin a g- (31) 
 
 H & + #' # #' 
 0.tan2 -cos -3- 
 
 When a is very small, the usual case, the member 
 
 cit"| 7jr ^-XTI i 7/ 1 
 
 sin a - ^ - which is the product of two small 
 
 quantities disappears and cos a is nearly unit. We may 
 use then as an approximate form 
 
 When a is not small the complete expression must be 
 obtained as follows : 
 
 From 28 and 29 we have, 
 
 H sin# 
 cos # cos a sm # sm a -/> j and 
 
 77 sin #' . jj 
 
 cos # cos a-j- sin # sm a = 75 : > ais 
 
 22 
 
2 H sin & sin 
 
 cos a = 
 
 a s 
 
 * sn 
 
 , 2 jy sin # sin #' 
 COS 
 
 , { 2 jy sin # sin #' ) 
 <M a sin (#+#')) "" 
 
 . n A/ 4r^T 8 sin ^sin 2 ^ ^sin & 
 "^^sin 2 (^ + ^) r ^^ 
 
 (33) 
 When solved for $ this becomes 
 
 ^ _ # 2 jsin ^ cos ^ sin $ cos #7 + sin 2 & sin 2 # x j 
 (? 2 (sin # cos ^ + cos # sin ^) 2 
 
 which may be written in the following more convenient 
 form 
 
 , _ jH* (tan &' tan #) 2 + 4 tan 2 # tan 2 &' ,.. 
 G* (tan & + tan #') 8 
 
 Since in nearly all cases the galvanometer is placed 
 approximately with its coils in the magnetic meridian, 
 the simpler expression (32) is of sufficient accuracy. 
 
LECTURE III. 
 
 Ballistic Methods, It is frequently desirable to deter- 
 mine current values by means of a single " throw " or 
 " kick " of the galvanometer needle, instead of by the 
 method of permanent deflections. In the measurement 
 of induced currents, or of transient or rapidly fluctuat- 
 ing current of any kind this method is almost the only 
 one. 
 
 A galvanometer thus used, an instrument, that is to 
 say, which is arranged for measuring the time-integral 
 Q or total charge transmitted during the flow of an 
 electric current of short duration, is called a ballistic 
 galvanometer, but the term should be applied to the 
 method rather than to the instrument since any galva- 
 nometer may be used ballistically. It is true, however, 
 that the method demands certain qualities which are 
 not essential nor desirable in instruments intended for 
 use in the method of permanent deflections. Thus 
 there has arisen a type of galvanometer especially de- 
 signed for the method of the first throw. 
 
 An expression for the performance of a galvanometer 
 when used ballistically may be derived in the following 
 manner. The force (//) (see Lecture I.) due to a cur- 
 rent i flowing through the galvanometer coils, the con- 
 stant of which is 6r, is 
 
 /, = G i m, (35) 
 
 and the moment of the couple at the needle pole is 
 GiZlm = GiM. 
 
 This couple acting during the time , upon a needle 
 the moment of inertia of which is 7", will produce an an- 
 gular velocity 
 
 (36) 
 
In the case of galvanometers used ballistically, it is 
 not with the current itself but with the time-integral 
 
 ( Q = / i d t) that we are concerned, and equation (36) 
 becomes 
 
 , . & -M- * (^T\ 
 
 From this equation, G, M and I being known, Q 
 could be obtained ; provided that the angular velocity 
 could be directly observed. 
 
 A slightly different statement of this point is the 
 following : 
 
 A current i in the coils of the galvanometer, which is 
 supposed to be placed in the earth's field or in an arti- 
 ficial field the strength of which is jET, the plane of the 
 coils being in the magnetic meridian, produces a field 
 perpendicular to H. The strength of this field at the 
 centre of the coil is 
 
 F= Gi (38) 
 
 where 6r, as in previous equations, is the constant of 
 the coils. Let to be the angular velocity of the sus- 
 pended parts of the galvanometer at a given instant of 
 time. If any torque $ act upon the moving system, to 
 will change in such a manner that 
 
 (39) 
 
 Now the field F, being perpendicular to the axis of 
 the needle exerts upon it a torque , which is equal to 
 
 FM, or to G i M (38). Since, however, i = -=-^, 
 
 a t 
 
 we have 
 
 /^L-f? G M ~z (40) 
 
 dt dt 
 
 If to is zero at the instant of closing the galvano- 
 meter circuit, equation (40) becomes 
 
 I to = GMQ, (41) 
 
 which is the same as (37), the fundamental equation of 
 the ballistic galvanometer. 
 
 In order to render the ballistic method feasible, it is 
 necessary to avoid the determination of the moment of 
 inertia (I) and the magnetic moment ( M ) of the needle, 
 also the impracticable operation of observing the angu- 
 
 25 
 
lar velocity (to) ; and to substitute the easily measured 
 time of vibration T and the observation of the ampli- 
 tude of the first throw ($.) That it is possible to do so 
 will appear at once from the consideration of the work 
 done upon the needle, to deflect it through the angle & 
 of the first throw. To move the north -pointing pole N 
 (Fig. 14) the strength of which is m, from a^ to b, and 
 the south-pointing pole S, of like strength, through a 
 similar path, requires work against the force Jim. 
 This amounts to 
 
 2 H m X di<h 2 H M (1 cos #) 
 
 where M is the magnetic moment of the needle. 
 
 Expressed in terms of the moment of inertia (7) and 
 the angular velocity (co) of the returning needle, at the 
 instant when the angle # becomes zero, the equivalent 
 
 FIG. 14. 
 of this work is 1 1 to 2 , whence 
 
 to = 2 . sin ! # (42) 
 
 / 2 
 
 By combining equations 37 and 42 we obtain 
 
 - w^W 7 \ = '-r^m* > (43) 
 
 and by use of the expression for the time of vibration 
 
 of the needle, T = n V -J (44) 
 
 H. M. 
 
 we may reduce the equation of the ballistic galvano- 
 meter to the final form, 
 
 L#. (45) 
 
 26 
 
Absolute measurements with the ballistic galvanometer 
 involve : 
 
 1. A knowledge of the constant (r, to be determined 
 by measurement. 
 
 2. A knowledge of Hio be obtained by one of the 
 methods to be described in Lecture IV. 
 
 3. A knowledge of ^in seconds. 
 
 4. The observation of the throw $. 
 
 The value of 7 7 is that for infinitesimal amplitudes, 
 which is to be obtained by the application of a correc- 
 tion similar to that used in determining the rate of 
 vibration of a magnetometer needle. (See Lecture IV.) 
 
 The separate determination of H and G, which latter 
 quantity can not be satisfactorily ascertained from 
 measurements of the coils of the forms of galvano- 
 meter usually employed in ballistic work, may be 
 avoided by calibration of the instrument. If a known 
 current i p be sent through the galvanometer and the 
 permanent deflection <p be noted, we have 
 
 (46) 
 
 . 
 
 a tan <p 
 
 and the expression for Q becomes 
 
 Q= **' . rsinlfl. (47) 
 
 TT tan <p . 2 
 
 Equation (47) affords an expression for $in which all 
 the factors are either numerical or capable of computa- 
 tion from direct observation. If the value of Q is to 
 be free from errors other than those involved in the 
 determination of ip y T, <p, and # however, the condi- 
 tions implied in the equations upon which the expres- 
 sion (47) is based must be complied with. These condi- 
 tions may be stated as follows : 
 
 Conditions implied by equation 41 : 
 
 1. The needle must be at rest when the discharge 
 through the coils begins. 
 
 2. The coils must be in the plane of the directing 
 field (H.) 
 
 3. The field (F) due to the current in the coils must 
 remain sensibly perpendicular to the axis of the needle 
 during the entire discharge, so that the torque may be 
 equal to G- i M through the whole time of its action. 
 This condition compels a slow vibration period of the 
 needle excepting in cases in which the duration of the 
 discharge is very brief. 
 
 27 
 
Conditions implied by equation 42 : 
 
 i. The whole of the kinetic energy ( I aF) must be 
 employed in turning the needle against the directing 
 field. This means that there must be no damping, a 
 condition which can not be fulfilled in practice, and for 
 the failure to fulfil which proper corrections must be 
 applied. 
 
 Conditions implied by equation 44 : 
 
 i. The suspending fibre must be free from torsion. 
 
 FIG. 15. 
 
 2. The damping must be so slight as to produce no 
 sensible influence upon the time of vibration of the 
 needle. 
 
 Condition implied by equation 46 : 
 
 i. The diameter of the coils must be large compared 
 
 28 
 
with the length of the needle ; in other words, the law 
 of the tangent galvanometer must hold true for such 
 deflections as are produced by the current used in cali- 
 brating the instrument. 
 
 Many of these conditions coincide with those desir- 
 able in the magnetometer and accordingly the earlier 
 forms of ballistic galvanometer, of which Fig. 15 affords 
 an illustration, were simply magnetometer bars with 
 one or more elongated coils of wire surrounding them. 
 The requirements of modern practice and the extension 
 of the ballistic method to operations of the highest 
 delicacy have essentially modified the type. 
 
 The equation of the ballistic galvanometer already 
 given (Equation 37) involves the throw of an undamp- 
 ed needle. In the case of a needle moving through 
 a resisting medium (damped by friction) or in the neigh- 
 borhood of metallic bodies within which induced cur- 
 
 FIG. 16. 
 
 rents are generated by the moving field of the needle 
 (damped by induction) the deflection # will always be 
 less than that which would be observed in the case of 
 an undamped needle, by an amount which may be 
 estimated as follows : 
 
 Determination of the effect of damping : Damping affects 
 the values of T and of $. To determine its influence 
 on these quantities we take advantage of the following 
 law : 
 
 Law of logarithmic decrements: The amplitudes of succes- 
 sive oscillations form a decreasing geometric series and the 
 natural logarithms of successive swings have a constant 
 difference. This difference is known as the logarithmic 
 decrement. 
 
 29 
 
The relation of the damped to undamped vibra- 
 tions of the same period may be shown graphically by 
 deriving the curve for the former by a method analogous 
 to that in which the curve of sines is derived. 
 
 Tait* has shown that the logarithmic spiral (Fig 16) 
 gives the law of damped vibrations in the same way 
 that the circle leads to the curve of sines. We derive 
 
 FIG. 17. 
 
 the latter curve by considering the motion of a radius 
 of the circle traveling with a uniform angular velocity 
 and plotting a curve (Fig. 17) with times as abscissae 
 and the corresponding vertical distances of the moving 
 end of the radius as ordinates. A similar procedure 
 applied to the case of the radius vector of the spiral 
 (Fig. 1 8) gives us the curve of damped vibrations. 
 
 FIG. 
 
 In order that the decrement shall be strictly logarith- 
 mic the resistance offered to the needle must be pro- 
 portional to the velocity. Atmospheric resistance is 
 thus proportional for low velocities and for high ones it 
 deviates from that law and becomes more and more 
 nearly proportional to the square of the velocity. For 
 
 *Tait ; Proceedings of the Royal Society of Edinburgh, 1867 ; also Maxwell's 
 Treatise, vol. ii., p. 375. 
 
 30 
 
all velocities reached by galvanometer needles, however, 
 the former law may be assumed. Even in the case of 
 instruments of the D' Arson val type the divergence is 
 not marked. Fig. 19 is from the photographic trace of 
 a very small light mirror and needle mounted in a 
 strong field for the purpose of following the changes in 
 rapidly fluctuating currents. [Messrs. Hotchkiss and 
 
 FIG. 19. 
 
 Millis, 1893.] It shows the sudden throw a v b upon 
 making circuit and the decrement of the oscillations as 
 the needle comes gradually to rest in the new position 
 at c. The period was about IODO vibrations per second. 
 The record was obtained by shooting a sensitive photo- 
 graphic plate rapidly across the field of a camera and 
 
 t \ N 
 
 FIG. 2O. 
 
 developing the image of the spot of light thrown upon 
 the same from the vibrating mirror. 
 
 Working formula and approximate correction for damping: 
 For work requiring no very great accuracy equation 
 (47) may be greatly simplified. Let a (Fig. 20) be the 
 observed throw, in scale divisions, produced by the dis- 
 
charge. Let b be the permanent deflection, in scale 
 divisions produced by the current i p and let D be the 
 distance of the scale from the mirror. Then we have 
 
 approximately Sin \ & = _^ and tan <p = _ whence 
 
 4 _Ls 2i dJ 
 
 Equation (47) becomes. 
 
 - - (48) 
 
 When a vibrating body has reached its extreme posi- 
 tion on either side of its position of equilibrium it is 
 said to in elongation. 
 
 The zero point (using telescope and scale) is the scale 
 reading when the body is in its position of equilibrium. 
 The zero point is most easily determined by observing 
 an odd number of successive elongations, an even num- 
 ber to the right and an odd number to the left. The 
 mean of the right hand readings and the mean of the 
 left hand readings are taken. The mean of these two 
 gives the zero point. When the damping is great this 
 gives a distinctly erroneous value for the zero point. 
 In such a case it is better to allow the vibrations to 
 cease so that the zero point may be read off directly. 
 
 The distance (or angle) from the position of equili- 
 brium to an elongation is called also an elongation. No 
 confusion need arise from this double use of the word 
 elongation. 
 
 The successive elongations of a damped vibrating 
 body as has already been shown, form ordinarily a de- 
 creasing geometric series, in which if k be the ratio of 
 an elongation to the next following, 
 
 a m 
 
 (49) 
 
 Wherein is an elongation and a^ is the m th follow- 
 ing elongation, loth of which are easily observed. The 
 symbol k is the ratio of damping. 
 
 The observed throw, a, in equation (48) is smaller in 
 the ratio, nearly, of i : V k than it would be were there 
 no damping so that we may write \' k- a for a in that 
 equation, whence : 
 
 (50) 
 
 TC.l 
 
 If k is very nearly unity this equation leaves only 
 very small outstanding reduction errors. 
 
 32 
 
There are three distinct kinds of error which affect 
 the results of a set of observations, (a) Observational 
 errors, (b) Instrumental errors, (c) Reduction errors. 
 Errors of observation arise from incorrect determina- 
 tions of the immediately observed quantities ; Instru- 
 mental errors arise from incomplete realization of the 
 conditions assumed in the derivation of the formulae 
 for reduction ; Reduction errors arise from the use of 
 approximate reduction formulae, often indeed from 
 the use of approximate formulae in the allowance for 
 instrumental error. 
 
 Example : Errors in the observed values of <p, & 
 and Tin equation (47) lead to observational errors in Q. 
 Any failure in the realization of the conditions assumed 
 in the derivation of (47), if ignored, leads to an instru- 
 mental error in Q. The use of approximate formulae, 
 such as (48), and (50) in the calculation of Q leads to 
 reduction errors in Q. 
 
 When great accuracy is desired allowance should be 
 made, if possible, for all instrumental error, and approxi- 
 mate formulae should not be used unless the reduction 
 errors they introduce are decidedly smaller than the 
 observational error and the outstanding instrumental 
 error. Instrumental errors are so called from the fact 
 that the conditions assumed in the derivation of reduc- 
 tion formulae refer to the arrangement and construction 
 of the apparatus used in taking the observations. Cal- 
 culations, which are made in the allowance for instru- 
 mental error, are always based upon observed values of 
 such quantities as characterize the incompleteness in 
 the realization of the assumed instrumental conditions. 
 
 Rigorous correction for damping. Every actual case of 
 vibration is damped. This damping is due to resisting 
 forces which oppose the motion of the vibrating body. 
 It is assumed in the following discussion that the resist- 
 ing forces are strictly proportional to the velocity of 
 the vibrating body. The errors left outstanding by 
 this assumption can always be made inconsiderable by 
 arranging that the vibrating body be massive ; that it 
 vibrate slowly ; and, if it vibrate about an axis, that it 
 be circularly symmetrical about that axis and smooth 
 so as to stir the air as little as possible. With such 
 limitations the following discussion leads to rigorous 
 correction for damping in the use of the ballistic galva- 
 nometer. 
 
 Let (p be the angle measured from the position of 
 equilibrium of a vibrating body to its instantaneous 
 
 position. Then, ordinarily, a torque 1& = c (f> f ~ 
 
 (> t 
 
 33 
 
acts upon the body and since this torque must be equal 
 
 d 2 d> 
 to 1 - I. , / being the moment of inertia of the body 
 
 Ct 11 
 
 and c and / being constant, the only condition which 
 must, in general, be satisfied by the angle (j> is ex- 
 pressed by the differential equation 
 
 in which 2 /9 is written for ! and f for . 
 
 Equation (51) leads, for our present purpose, to the 
 following expression for the instantaneous value of the 
 angle <p : 
 
 <p = A e~^ t Sin to t (52) 
 
 in which A is an undetermined constant, 
 
 o) = y r __0a (53) 
 
 and t is the elapsed time reckoned from the instant of 
 one of the passages of the body through its position of 
 equilibrium 
 
 The time interval, T, which elapses between succes- 
 sive null values of <p is called the period of the vibra- 
 
 tion and is evidently equal to or 
 
 CO 
 
 <= ^T (54) 
 
 Let T' be the undamped time of vibration. From 
 (53) and (54) with the . condition /3 o we have 
 
 T' = -2u , whence 
 
 V r ' 
 
 The successive elongations are maxima and minima 
 values for (p and the values of t at the instants of these 
 
 elongations are to be found from ( V o . Applying 
 
 a t 
 this method to equation (52) we easily find 
 
 t = Arc tan -^- + T n , (56) 
 
 co p 
 
 in which n is any whole number. [At the first elonga- 
 
 34 
 
tion n o , at the second n = i, etc.] Substituting the 
 value of t from (56) in equation (52), we have for the 
 first elongation <p ' : 
 
 *>:=.. -g_ ; ^~" Arctan t (57) 
 
 For the second elongation <p ", we have similarly 
 
 also <!>"' = e~ <!>" 
 
 &c. &c. 
 
 The ratio of damping, k, however, is by definition 
 equal to ^ whence 
 
 k = e ft T , (58) 
 
 or 
 
 ;, = Log. nat. k = ft T. (59) 
 
 This quantity ft T , which for convenience will here- 
 
 after be represented by A is the logarithmic decrement of 
 
 the vibrations. Expressed in terms of the common 
 
 system of logarithms it is, 
 
 / = 2.306 Log 10 k . 
 Noting the relations between ^ ft, 71 and w ; viz : 
 
 we may evidently write equation (55) in the form 
 
 T ' = T - J- (60) 
 
 The value of (p ' given in equation (57) is the observed 
 throw of the ballistic galvanometer. If there had been 
 no damping the throw would have been A, since ^/ re- 
 duces to A when ft = o. Solving equation (57) for A 
 and eliminating ft and to by use of (54) and (59) we 
 have 
 
 ?. 7T 
 
 TO" - arc tan -r- 
 + ^.e" { -f (61) 
 
 7T 
 
 or 
 
 35 
 
1 7T 
 
 / ;-* arc tan -^ 
 
 A = V 1 + ^ . k * <!>' (62) 
 
 Equation (61) or (62) enables us to calculate the un- 
 damped throw (A) from the observed throw (^'). 
 
 The calculation of Q, having observed T, k, y>, #, i p 
 is as follows : 
 
 From the observed value of k, equation (49), X is cal- 
 culated. From the observed period T the undamped 
 period T' [Same as t in equation (47)] is calculated by 
 means equation (60). From the observed throw y ' the 
 undamped throw A is calculated by equation (62). 
 Then T' is substituted for t and A for # in equation (47) 
 from which Q is calculated. The steps in this calcula- 
 tion are tedious and cannot be greatly simplified unless 
 approximate formulae are used. 
 
 A formula somewhat more exact than (50) is obtained 
 as follows : 
 
 T 
 
 Write -7= == for T in (48) 
 
 V 1 H 
 [See equation (60)]. 
 
 1 7T 
 
 / J-ST arc tan 
 Write V 1 + . k n aior am (48) 
 
 [See equation (62)]. 
 We thus have 
 
 1 7T 
 
 </ T n Tf ~ arc tan ^~ 
 Q _ ^ 2 - a ' fc * l 9 (63) 
 
 in "which Tis the observed period, a is the observed 
 throw, etc. The exponent of k in (63) becomes ^ when 
 k is very nearly unity or A very small ; so that for this 
 case (63) approximates to equations (50). 
 
LECTURE IV. 
 
 Methods of Measuring the Magnetic Field. Since, in 
 the use of the galvanometer, the sensitiveness depends 
 upon the value of H y that is to say, upon the strength 
 of the earth's field, or of the artifically created field 
 within which the needle swings, it is necessary to be 
 able to determine the strength of 'this field with a high 
 degree of accuracy. 
 
 Two of the earliest methods for the absolute deter- 
 mination of H were those developed by Gauss and by 
 his co-worker in Gb'ttingen, Wilhelm Weber. Gauss' 
 method comprises two operations: 
 
 (i.) The determination of the rate of vibration of a 
 suspended bar magnet. 
 
 (2.) The observation of the deflection which this 
 magnet is capable of producing when acting upon an- 
 other suspended magnet at a known distance. 
 
 From the time of vibration we get 
 
 Where Tis the time of vibration, JTthe moment of 
 inertia of the magnet and m the strength of the mag- 
 net pole. 
 
 From the determination of the deflection we get 
 
 772. 
 
 - = d s sin 
 
 or -- tan 
 
 (65) 
 
 The latter expression applies when the deflecting 
 magnet is stationary, in a position due east or west of 
 the magnetometer, the former is used when the deflect- 
 ing magnet is mounted on a swinging arm as in the 
 Kew magnetometer. 
 
 Equations (64) and (65) each contain m and H. The 
 quantity m, therefore, may be eliminated, and H may 
 be obtained in terms which involve only the funda- 
 mental quantities length, mass and time. 
 
 37 
 
There are few operations in experimental physics 
 which can be carried out with a higher degree of pre- 
 cision than this first operation of Gauss, which consists 
 in the determination of the time of vibration. When a 
 chronograph is accessible, it is convenient to use it in 
 obtaining a record of the times of passage. Fig. 2 1 is 
 from a portion of such a sheet, upon which have been 
 recorded the successive transits of a needle possessing 
 a period of five seconds. The equidistant notches are 
 the clock records. The records of transit are marked 
 a, b, c, etc. 
 
 Otherwise the period may be determined with all 
 sufficient accuracy by the eye and ear method. In the 
 latter case the observer counts the beats of the chrono- 
 meter or clock while watching the vibrations of the mag- 
 netometer needle, and estimates to tenths of a second 
 the successive times of transit. This estimation of tenths 
 is a matter requiring rather more practice in the case 
 
 FIG. 21. 
 
 of time intervals than where one has to do with linear 
 measurements ; but it is an art readily acquired by 
 practice. From the observed time of vibration the true 
 period must be computed by making corrections. 
 
 (a) for torsion. 
 
 (b) for induction. 
 
 (c) for temperature. 
 
 (d) for the arc of vibration. 
 
 (e) for the rate of the chronometer. 
 
 Under all ordinary circumstances these corrections 
 are individually small, but they are none of them en- 
 tirely negligible. 
 
 The method of correcting for torsion is similar to that 
 already described in the case of the galvanometer, viz. : 
 
 The ratio of the torsion of the fibre to the directive 
 
 38 
 
force of the earth's magnetic field is determined by 
 twisting the head to which the suspension fibre is at- 
 tached through 9o Q , and noting the deflection (ti) of the 
 magnet. 
 
 This ratio* is 
 
 1 11 IQP\ 
 
 7 = 90<nrv 
 
 The coefficient of induction (//), by means of which the 
 earth's inductive action in changing the magnetic con- 
 dition of the needle or bar is expressed, is the increase 
 in magnetic moment due to the action of a unit in- 
 ducing force. 
 
 Lament's method of determining p. consists in deter- 
 mining the deflection produced by the magnet to be 
 tested is placed vertically, first, with the north seeking 
 pole upwards, and then downwards at a constant dis- 
 tance from a suspended needle. Under these condi- 
 tions equation (65) becomes respectively 
 
 m ~ff F// = i d* sin #, (67) 
 
 and m+Vfj. = ^ ^ 
 
 where Vis the vertical component of the directive force 
 of the earth. 
 
 By the combination of (67) and (68) we obtain 
 
 V ft _ sin #' sin & tan j (#' #) , ~ 
 
 ' ' 
 
 m ~ sin &' + sin & ~ tan J (#' + 0) 
 
 By making use of the relation V = H tan , in which 
 i is the magnetic inclination, and of the equation 
 
 = i- sin u which expresses the action of the magnet 
 H 
 
 when used as a deflecting bar at unit distance, we may 
 write 
 
 sin u tan ^ (#' d) 
 f 2 '" 2 tan i ' tan i (&' + #)' 
 
 The induction coefficient is a small quantity. It is 
 given by Figg and Whipple, in the case of Kew mag- 
 netometer No. 47, for example as 0.0000050. It is 
 usually determined once for all by the method just de- 
 scribed.f 
 
 * In this case and in the discussion of the subsequent corrections the notation is in 
 the main that given by the Kew Observatory in the official sheets. 
 t See Stewart and Gee : Practical Physics, 11., p. 488. 
 
 39 
 
The temperature correction is more important. The 
 bar to be tested is placed in a bath, its axis east and 
 west. Its deflecting power upon a magnetometer 
 needle suspended at a suitable distance is noted at 
 various temperatures throughout a range of about 
 40 Q C. The influence of a rise of temperature upon 
 the strength of magnetization is always to diminish it, 
 and the change can be expressed by an equation of the 
 form : 
 
 m t = m (1 a t I f). (71) 
 
 where m t is the magnetic moment at the temperature 
 
 i ooo 
 
 .998 
 
 .996 
 
 .994 
 
 .992 
 
 .990 
 
 TEMP, COEFF, or A MAGNET 
 
 m 
 
 FIG 22. 
 
 of observation, ra that at o p (7, and a and b are coeffi- 
 cients. The values of a and b vary somewhat with 
 individual magnets. The following are average co- 
 efficients: 
 
 a = 0.000289. 
 
 b = 0.00500077. 
 
 This effect of temperature is not wholly temporary. 
 If, for example, measurements be made upon a mag- 
 net, giving for rising temperatures the curve 7", Fig. 22, 
 the succeeding curve for falling temperatures (77) will 
 
 40 
 
not coincide with it, but will lead to a lower final value 
 ra '. The amount of this divergence depends upon the 
 age, temper and previous history of the magnet. 
 
 The corrections for tr^e arc of vibration are made by 
 means of the well-known formula 
 
 7 _ T II 
 
 -*- *- nha I J- r>/ 4 r\f\ ~~~ 
 
 a a 
 
 In this expression s is the number of seconds gained 
 in a day by the chronometer or clock, and a, a' are the 
 initial and final values of the semi-arc of vibration. 
 
 The second operation of Gauss may be carried out in 
 two ways: the first of which is called the method of 
 sines ; and the second, the method of tangents. The 
 method of tangents is the more convenient in cases in 
 which the determinations are made repeatedly in a 
 
 FIG. 23. 
 
 single locality ; the method of sines, on the other hand, 
 is better adapted for portable instruments. Of such 
 instruments, the Kew magnetometer is the best known. 
 Its essential features are a central box or house, con- 
 taining the suspended magnet C (Fig. 23), an arm ex- 
 tending to the north of the axis of the instrument upon 
 which is mounted a small reading telescope (t) bearing 
 a short circular scale (s). By means of this, observa- 
 tions are made upon the position of the magnetometer 
 needle. A second bar accurately graduated, as to length, 
 extends to the east and west at right angles to the 
 arm carrying the telescope. Upon this, at fixed points, 
 is placed, for the purpose of the second operation, the 
 deflecting magnet (Z>). The entire instrument, includ- 
 
 
ing the two arms just mentioned and the magnetometer 
 box itself, are capable of rotation upon the vertical 
 axis, which axis corresponds with that of the suspended 
 magnet. The angles, through which this instrument is 
 turned, are measured upon a horizontal circular scale 
 similar to that with which sine galvanometers are pro- 
 vided ; and the method of making the readings is the 
 same. 
 
 The expression made use of in the second operation 
 of Gauss, see equation 65, is derived as follows : 
 
 Consider a magnet, D (Fig. 23), of strength of pole, 
 
 (* - if 
 
 FIG. 24. 
 
 /, acting to deflect the suspended magnet, <7, the 
 strength of pole of which is /'. 
 
 The action between pole S and pole N' is 
 
 in which d is the distance between the centres of /> and 
 C, while I is half the distance between the poles of the 
 former. 
 
The action between JVand N' is, however, 
 
 and the total force upon N' (Fig. 24), is 
 
 ' d (73) 
 
 The total force upon S 1 is, in the same way, 
 
 and the couple due to the deflecting magnet is 
 
 (d* _ ) 
 when Z> remains fixed in the east and west direction, or 
 
 Zff'lld 
 (d* - ^ 
 
 for the case of the Kew magnetometer. 
 
 When the suspended magnet has come to rest at the 
 deflection #, we have 
 
 f:=f'lH,m. (74) 
 
 
 Now the magnetic moment of the deflecting mag 
 net is 
 
 and when I is small compared to d (a condition which 
 should always be fulfilled in performing this operation) 
 we may use the approximation (d z i* = d*, and reduce 
 (75) to the same form as (65), viz.: 
 
 m d 3 . 
 Tt = "2 sm * 
 
 The following are the corrections to be applied in the 
 computation of the results of the second operation : 
 
 (a) for temperature of the bar holding the deflecting 
 magnet. 
 
 (b) for the distribution of magnetism in the same. 
 
 (c) for the temperature of the magnet. 
 
 (d) for induction. 
 
 43 
 
For the first of these it is sufficient to assume an 
 average coefficient for brass and to correct the observed 
 value (d') to the proper value (d Q ) by noting the tem- 
 perature at which the observation is made, and the in- 
 terval to the temperature at which the scale upon the 
 bar is right. 
 
 The form is 
 
 d' d Q (1 + 0.000018 (f 4). (76) 
 
 The correction for distribution is obtained by making 
 two sets of readings of deflection with the magnet D 
 at very different distances from C. The most favor- 
 
 d l 3 
 
 able relation to be that in which -7 = -; 
 
 a z 4: 
 
 The correction for distribution (/>) is applied by means 
 of the formula 
 
 m /my / p \ 
 ~H ~- : \H) \ l "d 2 )' 
 
 Where ^ is the corrected and ( ^i the observed 
 H \H) 
 
 value. 
 
 p is determined thus: 
 
 A 1 = i d? sin $! 
 
 A 2 = i d% sin #>. (78) 
 
 in which A l and A* are the two observed values of jr 
 
 at distances d^ and d 2 . 
 
 From (77) it follows, however, that 
 
 /- \ 
 
 (79) 
 
 or 
 
 .From equation 80, which contains only d l7 d 2 , ^,and/>, 
 the correction value can be computed numerically. 
 The expression (77) is only an approximation, it is true, 
 but since the value of /> is very small, it is not neces- 
 sary to use the higher terms which appear in a more 
 accurate formula. 
 
 The methods of obtaining the temperature coefficient 
 (q) and the correction for induction (//), have already 
 been indicated. It should be noted, however, that // 
 depends upon the position of the magnet D in the field, 
 
 44 
 
as well as upon II. In the tangent method of Gauss it 
 disappears altogether. In the sine method, where the 
 axis of D makes a final angle ft with the east and west 
 direction, the inductive effect on D is p. II sin ft. 
 
 From this expression we may obtain an approximate 
 
 form by use of the equation sin $ = 8 . 
 The form commonly used is 
 
 m w ( 1 -f- ~jjf ) (81) 
 
 in which m is the corrected moment and m Q the observed 
 value. 
 
 Kohlrausch 's Method for H. Next in importance to 
 Gauss' method comes that of Kohlrausch, where abso- 
 lute determinations of H are desired. Kohlrausch's 
 method, indeed, in those cases in which the knowledge 
 of If is to be used as a factor in the constant of the 
 galvanometer, is much to be preferred to any other, 
 because it permits of the determination of H in the 
 precise region to which the value is to be applied ; 
 whereas in the case of the Kew magnetometer it is 
 oftentimes necessary to make determinations which are 
 strictly applicable only to regions distant several feet 
 from the precise locality from which we desire to know 
 the intensity of the field. The Kohlrausch apparatus 
 consists of the tangent galvanometer, which may be, 
 and should be, the galvanometer for the calibration of 
 which the value of H is desired, and a swinging coil. 
 The coil is held vertically in the magnetic meridian by 
 means of a bi-filar suspension. The suspension con- 
 sists of two wires which serve to carry current into and 
 out of the suspended coil, and at the same time give it 
 the necessary directive force. This coil should be sus- 
 pended as nearly as possible in the region containing 
 the needle of the galvanometer which we wish to cali- 
 brate. In the case of tangent galvanometers of the 
 Helmholtz pattern, it is oftentimes entirely practicable 
 to mount the swinging coil midway between the fixed 
 coils of the galvanometer, so that the needle will be 
 within the plane of the former and in its axis. Such 
 an arrangement exists in the large tangent galvano- 
 meter described in the first lecture. The method of 
 procedure is as follows : A current i is sent through the 
 coils of a galvanometer of known dimensions, and in 
 the case in question this should be the galvanometer to 
 be calibrated. We have then 
 
 i = -^ tan #. (82) 
 
 45 
 
The same current is then sent through the suspended 
 coil. If the galvanometer coils or the suspended coils 
 are of appreciable resistance, as compared with the re- 
 mainder of the circuit, it is necessary to have elsewhere 
 resistances which can be inserted and removed, the 
 values of which have been previously adjusted so as to 
 correspond precisely to the resistance of the galvano- 
 meter and of the swinging coil respectively. A con- 
 venient arrangement for doing this is represented in 
 Fig. 25, in which fi g and K a are resistances, non-in- 
 ductively wound, which are to be substituted in turn 
 for the galvanometer and for the swinging coil by 
 means of the switch S. When the current i is sent 
 through the swinging coil it will deflect the latter. This 
 deflection is to be read by means of a suitably adjusted 
 telescope and scale. We have then 
 
 ^ = : 
 
 , 
 tan a 
 
 (83) 
 
 FIG. 25. 
 
 in which M b is the moment of the bi-filar suspension, A 
 is the effective area of the suspended coil, the dimen- 
 sions of which must have been determined by previous 
 measurement, and is the deflection of the coil from 
 the magnetic meridian. By combining equations 82 
 and 83, we obtain the following expression for II Z in 
 terms of the constants of the galvanometer and of the 
 swinging coil, the moment of the latter and the ratio of 
 the tangents of the deflections, viz.: 
 
 In the hands of Kohlrausch this method has afforded 
 some of the most precise determinations of the abso- 
 
 46 
 
lute value of the horizontal component of the earth's 
 magnetic field which have ever been made. He availed 
 himself of it, for example, in connection with his re-de- 
 terminations of the electro- chemical equivalents of 
 silver and copper. Such operations demand the most 
 precise knowledge of all the factors which enter into 
 the operation. 
 
 The most troublesome features of this method are 
 those which dealwith the temperature changes of J/ h , 
 the secular changes of the same and variation in the 
 resistance of the suspension wires. These wires to give 
 sufficient delicacy to the method must be of small size, 
 and they must consequently be Subjected to high 
 current densities. 
 
 In the case of the large tangent galvanometer at 
 Cornell University, reference to which was made in the 
 first lecture, an arrangement was perfected for the de- 
 termination of H by a modification of the method of 
 Kohlrausch. A large swinging coil, the diameter of 
 which was one meter, was suspended by means of a 
 phosphor bronze wire two meters long, the upper end 
 of which was attached to a torsion head carrying a circle 
 and vernier. The method consists in bringing the 
 swinging coil back to its zero position by twisting the 
 suspension wire. The position of the coil was read by 
 means of a telescope and scale at a distance of three 
 meters to the south of the instrument, for which pur- 
 pose the torsion head could be given a slow motion of 
 rotation by means of a tangent screw operated by the 
 observer at the reading telescope. The suspension 
 wire served also to introduce current to the suspended 
 coil, which consisted of 100 turns of No. 18 copper 
 wire. The other terminal of the coil consisted of a 
 wire situated in the axis of rotation, the end of which 
 was dipped in a mercury cup at the base of the instru- 
 ment. This form of suspension gave greater delicacy 
 than could be obtained by means of any bi-filar suspen- 
 sion, which would be capable of carrying the currents 
 which it was necessary to introduce into the coil. The 
 method possessed also the advantages common to what 
 are known as zero methods. The equations, by means of 
 which H is determined with this instrument, differ from 
 those which apply to the Kohlrausch method only in 
 two particulars. In the first place, the force of torsion 
 used in returning the wire to its original position is pro- 
 portional to the angle through which the suspension wire 
 is twisted. In the second place, we have to substitute 
 for the moment of the bi-filar suspension the moment of 
 torsion of the wire. Making these changes in equation 
 83, we have the following : 
 
 47 
 
t 
 
 ^ =: jf-jfO, (85) 
 
 IP __ GM* . _JL_ (86) 
 
 A tan # 
 
 The moment of torsion is determined by substituting 
 for the swinging coil, which is so adjusted as to be 
 readily unmounted and removed from its position, a 
 cylindrical brass weight the moment of inertia of which 
 can be determined directly from its mass and from its 
 dimensions. This cylinder, in the instrument under 
 consideration, weighs 4954.22 grammes, and its diameter 
 is 7.8747 cm. The'weight is hung in place of the coil 
 and its period of oscillation is accurately determined 
 by the aid of the chronograph. The expression for the 
 moment of torsion takes the usual form of the equation 
 for the torsion pendulum, viz.: 
 
 " ; ; (87) 
 
 in which, when we know, the moment of inertia K and 
 the period of oscillation T, we have all the factors 
 necessary to the computation of H in absolute measure. 
 Substituting in equation (86) the above value of the 
 moment of torsion, we have 
 
 or 
 
 (88) 
 
 The quantity C in equation (88) is a constant such 
 that : 
 
 When this .method was first put into operation in 
 1886, two serious sources of error, one of which was en- 
 tirely unexpected, arose. The first of these was an 
 error due to the influence of temperature upon the 
 moment of torsion of the suspension wire. This error 
 has its basis in a property of matter which is perfectly 
 well known. It is not a difficult matter to determine once 
 for all the temperature coefficient of torsion for the 
 material used, and to apply the correction. The diffi- 
 culty in maintaining a vertical wire, two meters long, 
 at anything approximating a constant temperature 
 throughout its entire length, however, was found to be 
 unsurmountable under the conditions which existed in 
 the observatory where the galvanometer was situated, 
 
 48 
 
and no satisfactory correction for the temperature of 
 the wire was reached until after many expedients had 
 been tried and abandoned ; the following method of 
 ascertaining the average temperature of the wire at the 
 precise time when each observation was made came to 
 be adopted. 
 
 This method of integrating the temperature for 
 the entire length of the wire consisted in placing a 
 No. 40 copper wire parallel to the suspension and as 
 close to the same as could be without actual contact. 
 This copper wire was drawn back and forth several 
 times. It was placed in series with a compensated re- 
 sistance, the value of which was approximately the 
 
 SECONDS 
 
 8.704 8.708 8.712 8- 7 l 
 
 FIG. 26. 
 
 same as its own and a suitable current sent through the 
 circuit containing the two. A sensitive galvanometer 
 was mounted in another part of the observatory, by 
 means of which the flow of potential through the com- 
 pensated resistance and through the temperature wire 
 just described could be compared. The ratio of these 
 deflections gave the average temperature of the wire 
 with a high degree of accuracy. 
 
 Before mounting the fine copper for this purpose, its 
 temperature coefficient had been determined, and the 
 ratio of the deflections when the galvanometer was 
 
 49 
 
shunted across its terminal, to that obtained when the 
 galvanometer was shunted across the terminals of the 
 compensated resistance, had been ascertained for a 
 sufficient range of temperatures. By the use of this 
 simple device, the difficulties arising from difference of 
 temperatures in the suspension wire were eliminated. 
 
 The other source of error was of a more serious 
 character. It was found that the torsional elasticity of 
 the suspension wire varied continually with age. The 
 change, which was very marked, indeed, at first, dimin- 
 ished slowly as time passed ; but it never became a 
 negligible quantity. The time curve of this wire has 
 been taken with great care, and it now covers an inter- 
 val of nearly ten years. By means of this curve the mo- 
 ment of torsion of the wire for any desired date can be 
 ascertained with a sufficient degree of accuracy; but with- 
 
 FIG. 27. 
 
 out this correction the values for H determined by this 
 method would be seriously at fault. The character of 
 the first of these two variations, that due to tempera- 
 ture and is shown graphically in Fig. 26, which gives 
 the relation of the rate of vibration of the wire when 
 attached to the calibration weight already described. 
 This curve was made on February 26, 1887, and is from 
 measurements by Professor H. J. Ryan. Determina- 
 tions at later dates would afford data showing a slower 
 period. The rate at 10, on October 28, 1889; for ex- 
 ample, according to measurements by Mr. N. H. 
 Genung was 8.689 + seconds. 
 
 To control these factors, upon which accuracy de- 
 pends, is a matter of considerable difficulty, and the 
 method of Kohlrausch for the determination of H is 
 rendered a laborious one because of them. 
 
 50 
 
The Determination of H by Means of the Copper Volta- 
 meter. For all ordinary operations with the tangent 
 galvanometer, a sufficiently accurate determination of 
 H can be obtained by a method which is much more 
 convenient than those of Gauss or Kohlrausch. This 
 method consists in sending through the galvanometer 
 a current, the intensity of which is measured by means 
 of copper voltameters placed in the circuit, and noting 
 the deflection produced. The requisites are a steady 
 source of current, such as a storage battery of consider- 
 able capacity, a fine balance, a fairly accurate time- 
 piece and a copper voltameter of proper construction. 
 The form of voltameter which has shown itself best 
 adapted to accurate work is one which is at the same 
 time the most easily constructed. I refer to the spiral 
 coil voltameter of Professor Ryan.* This consists of 
 two suitable jars containing a slightly acidulated solu- 
 tion of the sulphate of copper, two coils of pure copper 
 wire about five centimeters in diameter which are to 
 form the losing electrodes, two coils of smaller diameter 
 
 FIG. 28. 
 
 constructed from the same wire, which are to constitute 
 the gaining electrodes, and any simple device for hold- 
 ing these coils pair-wise in the cells with a common 
 vertical axis corresponding to the axis of the jar (see 
 Fig. 27). To construct these coils, it is only necessary 
 to take a few feet of copper, size No. 10 or No. 12, to 
 strip the same of its isolation, to clean the wire thor- 
 oughly by clamping one end of it and drawing sand 
 paper, grasped in the hand, briskly, over its entire length 
 several times. The wire is then wound upon cylinders 
 of suitable diameter so as to form two large and small 
 coils such as have been already described. When com- 
 pleted, these coils will have a length along the axis of 
 about three inches, and their diameters will be respec- 
 tively for the losing coils, five, and for the gaining coils, 
 two centimeters. 
 
 The smaller coils are carefully weighed upon a bal- 
 ance of high precision, are then mounted within the 
 
large coils in the two jars, Fig. 28, and concentric with 
 the same ; the jars are filled with the electrolytic solu- 
 tion, electric connections are completed, and the time 
 of making circuit is carefully noted. In the course of 
 a half hour or thereabouts, during which the current is 
 flowing through the cells and through the galvano- 
 meter, a number of readings of the deflection are taken. 
 Th e current is then broken at a time accurately noted, 
 and the inner or gaining coils are removed from the 
 voltameter. They are rinsed with water and then with 
 alcohol, after which they are dried without friction with 
 filter paper, or by holding them at a safe distance over 
 the flame of a bunsen burner. 
 
 If the operation has been a successful one, the sur- 
 face will possess a uniform and beautifully tinted sur- 
 face characteristic of freshly deposited electrolytic 
 copper. Any marked granulation of the surface would 
 indicate too great a current density, and would subject 
 the results of the measurement to suspicion. Under such 
 circumstances the calibration should be repeated. The 
 amount of copper deposited upon two plates, as shown 
 by the comparison of the weighings before and after 
 should agree to within two-tenths of one per cent. Pro 
 perly carried out, therefore, this method will give the 
 value of H to a like degree of precision. 
 
LECTURE V. 
 GALVANOMETERS WITH ARTIFICIAL FIELDS. 
 
 i. Instruments with Strong Fields . The magnetic field 
 of the galvanometer is frequently strengthened artifici- 
 ally for one or more of the following reasons: 
 
 (a) To increase the constant , thereby securing an 
 
 instrument suitable to the measurement of heavy cur- 
 rents. 
 
 (b) To diminish the period of vibration of the needle. 
 
 (c) To obtain immunity from magnetic disturbances, 
 such as the daily fluctuations which take place in the 
 value of JET, and the accidental variations brought about 
 by the proximity of masses of iron or by the inductive 
 influence of the dynamo motors, and of line wires 
 carrying current. 
 
 The most important instruments of the kind under 
 consideration are the galvanometers of the D' Arson val 
 type. In these well-known galvanometers a strong field 
 is obtained by means of a nearly closed magnetic circuit. 
 Within the air space of this magnetic circuit is placed a 
 coil of wire, through which the current to be measured 
 is allowed to pass. This coil has freedom of rotation 
 upon an axis at right angles to the lines of force, and 
 also at right angles to the axis the coil itself. In order 
 to hold such a coil in place in the very strong fields 
 which are made use of in these instruments the suspen- 
 sion is by means of wires vertically fastened above and 
 below. 
 
 The original type described by Deprez and D'Arson- 
 val,* is shown in Fig. 29. A coil thus held between 
 tense suspension wires vibrates rapidly, and a short 
 period of oscillation is, therefore, one of the character- 
 istic features of such instruments. 
 
 The D'Arsonval galvanometer has been subjected to 
 a great variety of modifications. We have, for example, 
 the moving coil without an iron core, a moving coil 
 
 * Deprez and D'Arsonval: Comptes Rendus 94, p. 1347, 1882. 
 
 53 
 
with an iron core, a moving coil the interior of which 
 is rilled with a stationary piece of soft iron. The 
 field in which the coil swings is sometimes that of a 
 permanent magnet of the horseshoe type, sometimes 
 that of an electromagnet, and sometimes that of a sole- 
 noid without iron. The air gap also is of various sizes, 
 from the very large air gap of the Thomson graded 
 galvanometer to the exceedingly small one employed 
 in instruments of the Breguet form. The advantages 
 
 FIG. 29. 
 
 of all these galvanometers may be summed up in the 
 statement that they are exceedingly quick of action 
 and remarkably free from outside influences. As to the 
 permanency of their indications, it is evident that any 
 instrument, the constant of which depends upon the 
 maintenance of an unchanged field due to permanent 
 magnets, must be subject to a certain amount of secular 
 change. Whether the time change in the field of such 
 instruments can be reduced to an inappreciable quan- 
 
 54 
 
tity is a subject about which there has been consider- 
 able discussion. Dr. Koepsel,* for example, in a paper 
 read before the Electro-technical Congress at Frank- 
 fort, in 1891, took the ground that the use of steel mag- 
 nets in instruments for the measurement of electric 
 current should be altogether abandoned on account of 
 the lacK of permanence. This view has been combatted, 
 however, on the part of those who have had much ex- 
 perience in making instruments in which permanent 
 magnets are used. The permanence of such magnets, 
 undoubtedly, depends in part on the size of the air gap, 
 and increases as the latter is reduced. 
 
 Instruments such as the Thomson graded galvanome- 
 ters, on the one hand, in which the magnetic circuit is 
 nearly half through the air, exhibit much more rapid de- 
 cadence than instruments of the Deprez type in which 
 the air gap is reduced to a minimum. A graded galva- 
 nometer with a home-made magnet which had been con- 
 structed to take the place of the original magnet be- 
 longing to the instrument showed, for example, a change 
 of constant in one year from 7.00 to 6.6 1. This marked 
 falling off in the strength of the field may, with justice, 
 be ascribed in part to the inadequate treatment of the 
 permanent magnet in preparing it for use in such an 
 instrument. The original magnet belonging to a simi- 
 lar galvanometer showed, however, a scarcely better re- 
 cord. The constant fell off in this second case from 
 4.52 to 4.44 in one year. An ammeter of the D' Arson - 
 val- Deprez type showed somewhat greater permanence. 
 A current, which, on the 3oth of November, 1892, pro- 
 duced a deflection of 
 
 36.2 scale divisions to the right, 
 
 35.8 scale divisions to the left, 
 
 was found in October, 1893, to produce, respectively 
 35.1 scale divisions to the right, 
 
 34.9 scale divisions to the left. 
 
 As originally constructed, the calibration curves of 
 the D'Arsonval galvanometer were by no means 
 straight. Figure 30 shows the curve for right and left 
 deflections in the case of the ammeter just referred to. 
 It is obvious, however, that the law of deflections in all 
 such instruments is. under control by modifying the 
 shape and disposition of the pole pieces. Professors 
 Ayrton and Perry have shown that by this device the 
 curve can be readily straightened. 
 
 An interesting example of the application of the prin- 
 ciples upon which galvanometers with strong fields de- 
 
 * Koepsel : Verhandlungen des internationalen Elektrotechniker-Consrresses z u 
 Frankfurt, 1892, Zweite Halfte, p. 3. 
 
 55 
 
pend is found in the Moler curve writing voltmeter.* 
 This is essentially a galvanometer of the D'Arsonval 
 type in which a needle of soft iron is mounted in the 
 strong field between the poles of a powerful permanent 
 magnet. 
 
 The needle carries a short aluminium pointer which 
 records its oscillations upon the smoked drum, Fig. 3 1 . 
 The vibrations of the needle of this instrument are so 
 rapid that by means of it one can follow the fluctuations 
 of current during a single revolution of a dynamo or 
 motor. The amplitude of vibration is necessarily quite 
 small, but the indications of the instrument are never- 
 theless exact, and when measured and duly magnified 
 they are found to correspond excellently with results 
 obtained by other methods. 
 
 It is possible, by means of an instrument of this kind, 
 to make interesting studies of a great variety of pheno- 
 
 FIG. 30. 
 
 mena in which rapid fluctuations of current occur. The 
 instrument was originally devised for the purpose of 
 exploring the field of dynamos and motors, a purpose 
 to which it is well adapted. It can, however, be used in 
 many other ways. 
 
 Fig. 32 shows the tracing obtained by means of this 
 instrument in the study of the performance of arc 
 lamps. The voltmeter was placed across the terminals 
 of a direct current lamp the construction of which is 
 such that the carbons are held apart by a spring until 
 they are brought together by the action of the current 
 through the shunt coil, after which they are separated 
 again and the arc is formed. The point marked w in the 
 figure is that at which the circuit was closed. The ver- 
 tical distance between the upper and lower tracings at 
 
 * American Institute of Electrical Engineers, vol. 9, p. 223. 
 
NI v 
 
 
 b is 50 volts which is the normal potential difference of 
 the lamp. It will be seen that immediately after clos- 
 ing- the circuit atw there is a rise of potential to a much 
 higher value during the interval before the shunt coil 
 comes into operation ; furthermore, that the carbons do 
 not come into contact for a considerable time, viz., that 
 which elapses between the point marked w and that 
 
 f* D 
 
 FIG. 31. 
 
 marked x. There is also a further interval of time from 
 x to y, during which the contact between the carbons, 
 which had been poor at first, improved as the tips grew 
 hot. This appears from the fact that the potential dif- 
 ference continues to fall off until y is reached, at which 
 time it is virtually reduced to zero. Finally, the adjust- 
 ing mechanism of the lamp begins to act, and the car- 
 bons are drawn apart to their normal condition. 
 
 FIG. 32. 
 
 2. Instruments with Weak Fields. A more important 
 modification of the field of the galvanometer than that 
 which we have just considered, is in the direction of 
 reducing its intensity. The object sought in such cases 
 is increase of sensitiveness. The limit to which such 
 increase can be carried is reached only when the natural 
 or artificial changes of the field thus produced become 
 
 57 
 
so great as to cause troublesome drifting of the gal- 
 vanometer needle. A magnetized needle in the weak- 
 ened field shows motions corresponding to those of the 
 declination needle, but the amplitude of fluctuation is 
 increased. The sensitized galvanometer, therefore, 
 drifts more and more as its sensitiveness increases. 
 
 The following simple demonstration of the relation 
 between sensitiveness and the fluctuation of the direc- 
 tion of the resultant force of the weakened field is due 
 to Mr. F. J. Rogers.* In Fig. 33, let o E represent the 
 direction and the directive force of the earth's magnetic 
 field which has been partly counteracted by the intro- 
 duction into the neighborhood of a controlling magnet 
 which produces a field represented by the line o M. 
 The resultant field is o R. If now from any cause the 
 
 original field be subjected to a change of direction and 
 intensity that it must be represented by o E', which 
 makes an angle a with o E, the new resultant formed 
 by the combination of o M with o E' will take a new 
 direction o R', which makes a much larger angle a! with 
 the first resultant o R. To the same writer is due a 
 very interesting experimental study of the behavior of 
 sensitive galvanometers. 
 
 For this purpose two galvanometers were taken, one 
 of which was highly sensitized by the action of a con- 
 trolling magnet, while the other possessed a field due to 
 the uncounteracted intensity of the earth's magnetism 
 at the point at which the instrument was set up. The 
 
 * F. J. Rogers, The Crank^ vol. 6, 1892, p. 270. 
 
 58 
 
first of these was studied on three successive days for 
 the purpose of observing the range of fluctuation of 
 the zero point under the ordinary changes in the direc- 
 tion of the earth's lines. 
 
 The course followed by the needle is shown in Fig. 
 34. It was the same in all essential particulars during 
 the three days in question, reaching a maximum of 
 elongation from its mean position daily just before 
 2 p. M. These curves correspond very closely with 
 those which would be obtained by the observation of 
 the movement of a declination needle at times when 
 there was no marked magnetic storm. The amplitudes 
 of oscillation, however, are very much greater owing 
 to the weakness of the field. On each of these days 
 the other galvanometer, the needle of which was sus- 
 pended in the earth's field, went through the same 
 range with about one-twentieth of the amplitude shown 
 in Fig. 34. 
 
 74 s d. 
 
 66 s, d. 
 
 X 
 
 58 s. 
 
 6 P.M. 
 
 FIG. 34. 
 
 To further establish the relationship between the 
 movements of a sensitive galvanometer and those of 
 the declination needle, in other words, to ascertain 
 whether the movements, familiar to all who use sensi- 
 tive instruments of the class in question, are due to 
 local disturbances, or to those widespread magnetic 
 fluctuations which cause magnetometer needles over 
 the entire continent to move together, observations 
 were made with the two instruments already described 
 upon a day when there was marked magnetic disturb- 
 ance. The result is shown in Fig. 35. 
 
 In this diagram the amplitude of fluctuation of the 
 needle, which was suspended in the earth's field, was 
 multiplied by a constant factor such as to bring it to 
 the same scale as that of the more sensitive galvano- 
 meter. To make sure that the fluctuations which af- 
 fected the two galvanometers simultaneously through 
 
 59 
 
the day, although they were mounted in different rooms 
 of the laboratory, were not due to local causes, but 
 were the result of changes in the magnetic field of the 
 earth itself, records were obtained for the day in ques- 
 tion from the magnetic observatory at Washington. 
 These were multiplied by the proper factor to bring 
 them to the exaggerated scale applicable to the sensi- 
 tized galvanometer, and were plotted upon the same 
 sheet. In Fig. 35, to which reference has just been 
 made, the unbroken line represents the changes in the 
 position of the declination needle as recorded at Wash- 
 ington on the day in question. The two broken 
 curves are those obtained by making readings with the 
 two galvanometers in the laboratory at Ithaca. It will 
 be seen that the three curves agree in a remarkable 
 manner throughout. 
 
 The consideration of these results makes it obvious 
 that in order to use a sensitive galvanometer in opera- 
 tions of precision, two things are necessary: 
 
 FIG. 35. 
 
 1. A knowledge of Hm the locality where the instru- 
 ment is in use at the time when calibration of the latter 
 is made. 
 
 2. Some method of following the fluctuations of H 
 from moment to moment. 
 
 A discussion of the means of meeting this second 
 requirement will be given in a subsequent lecture. 
 
 The methods of determining H, described in the 
 fourth lecture are very laborious, and it is desirable, 
 therefore, to substitute for them some means for com- 
 paring the strength of unknown fields with that of 
 known fields previously determined for this purpose the 
 method of Wilhelm Weber is most convenient. It is 
 described below. 
 
 Weber's Method for the Determination of H. The pro- 
 cedure consists in turning a coil of known dimensions 
 (the earth inductor) suddenly through 180, and noting 
 
 60 
 
the throw of the slow moving needle of a ballistic gal- 
 ganometer placed in circuit with this coil. Fig. 36 shows 
 the instrument in its simplest form, while Fig. 37 gives 
 the arrangement of the electrical circuit. 
 
 The earth inductor is a coil of considerable area, and 
 consisting of many turns of copper wire. The dimen- 
 sions of the coil, the number of turns and the resist- 
 ance of the instrument will depend upon the character 
 of the galvanometer with which it is to be used. 
 
 The coil is mounted in a strong wooden frame with 
 
 FIG. 36. 
 
 freedom to revolve upon a vertical axis. Stops are so 
 placed as to limit this motion to exactly 180. The 
 frame itself is free to turn upon a horizontal axis, with 
 stops to facilitate the adjustment of it in a horizontal or 
 in a vertical plane. 
 
 The base of the earth inductor should be provided 
 with a good level, and with levelling screws. In the 
 operations to be considered here it is desired to com- 
 pare the value of S in a locality where that quantity is 
 known, as, for example, in the magnetic observatory, 
 
 61 
 
with the value of H where the galvanometer is mounted. 
 For this purpose the earth inductor is carefully levelled 
 in the former locality with its coil vertical and against 
 the stops, the axis of the coil being in the magnetic 
 meridian. It is connected with a line leading to the 
 galvanometer. The earth inductor is in series with the 
 latter and with a resistance R. These three portions of 
 the circuit, viz., the coils of the galvanometer of the 
 earth inductor and of the resistance R, should include 
 nearly all the resistance, that of the line being negligible. 
 
 The circuit having been completed, the observer 
 watches the galvanometer while an assistant swings the 
 earth inductor through 180. A series of readings are 
 thus made, using the galvanometer ballistically. 
 
 It is necessary to accuracy, that the period of vibra- 
 tion of the instrument be much longer than the inter- 
 val of time occupied by the semi-revolution of the 
 inductor. Otherwise one of the conditions indicated in 
 Lecture III , viz., that the entire quantity of electricity 
 due to the motion of the coil should traverse the coils 
 of the galvanometer before the needle had moved 
 through a considerable angle, will not be fulfilled. 
 
 FIG. 37. 
 
 After the completion of a series of readings the earth 
 inductor is removed to the spot in which the intensity 
 of the field is to be compared with that of the locality 
 which that instrument had occupied. After proper ad- 
 justment of the coil in its new place, a new series of 
 semi-revolutions is made and the corresponding deflec- 
 tions are noted. 
 
 It is by the comparison of these deflections that the 
 two magnetic fields are brought into relative measure- 
 ment. The changes of the field in the second locality 
 can, moreover, be followed by repetition from time to 
 time of the second series of determinations. 
 
 The expression for quantity of electricity generated 
 by the motion of the earth inductor is 
 
 Q = ^ % sin J # = 2 a tf g jsin \ & (90) 
 
 an equation, the development of which has been given 
 
 62 
 
in the lecture on the ballistic galvanometer The 
 quantity , however, depends upon the area A of the 
 earth inductor coil, and the resistance R of the entire 
 circuit as well as upon the strength of the field within 
 which the rotation of the coil occurs. This relationship 
 is expressed by the formula 
 
 combining equations 90 and 91 we may write 
 
 This equation may be used in several ways : 
 
 1. Given # e , A, R, G and T, 
 
 H K , the strength of the field in which the galvanometer 
 needle swings may be determined. 
 
 2. Given the constant of the galvanometer in the 
 field H g also A and R, any field ff e in which the earth 
 inductor is turned may be computed in absolute meas- 
 ure. 
 
 3. The operations just described yield two equations 
 of the type 90, viz.: 
 
 sin^, (94) 
 
 By combining these we are able to eliminate all the 
 constants, and to obtain a ratio between the fields to be 
 explored, viz.: 
 
 ff. _ sin j # e (95) 
 
 H x ~ sin # x 
 
 The method of the earth inductor is especially useful 
 when a very strong field, such as> that which exists in 
 the air gap of an electromagnet, is to be compared 
 with H e . 
 
 For such measurements a coil should be constructed, 
 the cross-section of which is not too great to admit it 
 entirely within the field to be measured. 
 
 The total area of this coil is to be determined as 
 accurately as possible at the time of winding. 
 
 It is generally better to pull this coil out of the field 
 than to attempt to give it a motion of rotation. Some- 
 times this can be most conveniently accomplished by 
 the aid of gravitation, the coil being attached to a 
 weight, as in Fig. 38, and released by breaking the cir- 
 cuit which animates the small electromagnet (m). 
 
 63 
 
Sometimes it is more convenient to make use of a 
 spring, by means of which the coil may be removed 
 from the field with the desired speed. In either case 
 the interval of time should be comparable with that 
 necessary to turn the earth inductor through 180, 
 otherwise the impedance of the circuit will not be quite 
 the same for the two operations. A more important 
 matter is that of the placing of the coil within the field. 
 The distribution of lines of force in the fields of the 
 character of those for the exploration of which the de- 
 vice under consideration is applied is by no means uni- 
 form. The number of lines which will penetrate the 
 coil before it is taken from the field in successive trials, 
 
 FIG. 38. 
 
 will often be found to vary considerably unless the 
 greatest care is taken to bring it each time to precisely 
 the same position. 
 
 In carrying out this method it will frequently be 
 found necessary to vary the resistance R so as to render 
 the deflections, due to the motion of the earth inductor 
 and of the small coil, comparable. We have then in the 
 computation of the ratio of the fields H^. and Zf e , two 
 areas (those of the respective coils) A x and A vt and two 
 resistances 72 X and E e . 
 
 Equation 86 takes the following form when modified 
 
 6 4 
 
to express the relations existing in this application of 
 the method, viz.: 
 
 Sill 
 
 2 R^ A e sl 
 
 (96) 
 
 G 
 
 
 
 - f 
 
 6 ' 
 
 FIG. 39. 
 
 It is in most cases convenient to place the coils A x and 
 A e in series with one another. The arrangement of 
 connections is that shown in Fig. 39. 
 
 Where the comparison of fields is for the purpose of 
 
 o 
 
 .174' 
 
 o 
 
 .1728 
 
 o 
 .1750 
 
 o 
 
 .1748 
 
 o 
 
 1750 
 
 o 
 
 .18.9 
 
 o 
 
 .1836 
 
 o 
 
 .1809 
 
 o 
 
 ..805 
 
 o 
 
 o 
 
 O 
 
 o 
 
 .-363 
 
 '553 
 
 '349 
 
 .1600 
 
 o 
 
 O 
 
 
 
 
 .1684 
 
 .188. 
 
 .2772. 
 
 
 o 
 
 .'653 
 
 o 
 1717 
 
 O 
 ,1730 
 
 O 
 
 .1770 
 
 FRANKLIN HALL. 
 
 o 
 
 .r68 5 
 
 FIG. 40. 
 
 determining H in the locality where a galvanometer is 
 placed it is important to make the measurement as close- 
 ly as possible for the region occupied by the needle. It 
 will not do to assume that the value of H throughout 
 an ordinary laboratory room is nearly constant. R. W. 
 
 65 
 
Wilson* has pointed out the wide range in the values of 
 H within the limits of the Jefferson Laboratory at 
 Cambridge, and his experience has been abundantly 
 confirmed elsewhere. Fig. 40 shows the results of a 
 similar survey recently made under the direction of the 
 writer by Messrs. C. E. Hewitt and A. W. Smith. The 
 exploration covered the interior and surroundings of 
 the annex to Franklin Hall. The latter building is the 
 physical laboratory of Cornell University, and the 
 annex is a one-storied brick structure, 100 feet X 36 feet, 
 situated a few feet to the north of the main laboratory. 
 It contains but little iron aside from gas and steam 
 pipes, some cast-iron wall brackets and some rods which 
 serve to strengthen the roof. 
 
 In the accompanying map, Fig. 40, A is the annex, 
 and the various small circles are stations at which H 
 was determined. The values in c. G. s. units for each 
 station are indicated. It will be seen that within the 
 building 77 varied between .1363 and .2772, .1728 being 
 the normal value in localities distant from local sources 
 of disturbance. 
 
 It is interesting to note that the stations just north of 
 the annex, also those along the north wall of Franklin 
 Hall show values of H above the normal, while the row 
 of stations just within the north wall all have low 
 values. A similar survey of the neighborhood of the 
 Magnetic Observatory of Cornell University, made by 
 F. J. Rogers, shows the same phenomenon, and it seems 
 probable that walls of masonry always exert a magnetic 
 influence of the kind described. 
 
 * R. W. Wilson: Am.rican Journal of Science , vol. 39. p. 87. 1890. 
 
 66 
 
LECTURE VI. 
 
 THE CONSTRUCTION OF GALVANOMETERS OF EXTREME 
 SENSITIVENESS. 
 
 There is probably no instrument of precision used in 
 physics at the present day which possesses so wide a 
 range of sensitiveness as the galvanometer. The ana- 
 lytical balance which for a long time was the most re- 
 markable of all instruments in this respect has fallen 
 into second place on account of the remarkable develop- 
 ments as regards extreme sensitiveness which have 
 been made in the construction of the galvanometer 
 within a few years. 
 
 The equation of the galvanometer given in the first 
 lecture indicates the lines along which increase in sensi- 
 tiveness is to be attained. The constant of the gal- 
 vanometer is made up of two parts, H the horizontal 
 component of the earth's magnetism, and G a factor 
 which depends upon the dimensions of the coil and its 
 distance from the needle. 
 
 It is evident from equation 7, 
 
 that any method which will increase the value of G or 
 diminish H will increase the sensitiveness of the instru- 
 ment. The latter of these two processes has been dis- 
 cussed at some length in Lectures IV. and V., in the 
 course of which it has been shown that an artifical field 
 may be substituted for the earth's magnetic field, this 
 field being either stronger or weaker than the earth's 
 field according to the purpose for which the instrument 
 is designed. 
 
 The final result of weakening the field around the 
 magnet needle is to produce greater and greater insta- 
 bility of zero until finally the drifting of the needle 
 becomes so rapid as to make it impossible to obtain 
 readings. Thus a limit to the usefulness of the method 
 of weakening the field for the purpose of increasing 
 
 67 
 
the sensitiveness of the instrument is reached. In many 
 operations, also, the lengthening of the period of vibra- 
 tion would in itself bring us to a limit of usefulness in- 
 jlependent of the matter of magnetic drift. 
 
 Not less important than the reduction of H to small 
 values is the increase of the quantity G in the constant 
 of the galvanometer; and since this factor increases as 
 the distance between the needle and the wire diminishes 
 and increases also with the number of turns of wire in 
 the coil, the problem of construction with view to ex- 
 treme sensitiveness consists in part of reducing to a 
 minimum the mean distance of the windings from the 
 needle and of getting the largest number of complete 
 turns for a given electrical resistance in the wire used 
 in the construction of the instrument. 'A third, and 
 very important, factor which enters into the considera- 
 tion of the construction of galvanometers, is the light- 
 ness of the moving parts. 
 
 It is true that the sensitiveness of a galvanometer 
 which is used following the method of permanent de- 
 flections is independent of the mass and moment of 
 inertia of the moving parts, and independent of the 
 magnetic moment of the needle also. In order to ren- 
 der a galvanometer, the suspended parts of which 
 possess a large moment of inertia, as sensitive as one 
 in which the moving parts are light, it is necessary, 
 however, to increase the period of oscillation; and for 
 many purposes this consideration taken by itself would 
 dictate the reduction of the mass of the suspended por- 
 tions to a minimum. 
 
 In nearly all operations of extreme sensitiveness, gal- 
 vanometers are used ballistically, and under these con- 
 ditions, both the moment of inertia and the magnetic 
 moment are involved in the question of sensitiveness. 
 The problem of the maker of such instruments, there- 
 fore, includes the question of securing as large a mag- 
 netic moment, and as small a moment of inertia as 
 possible. 
 
 The sensitive galvanometer owes its origin to the 
 demands of the student of radiant energy, and it was 
 at the hands of Nobili and of Melloni that two of the 
 important steps toward increased sensitiveness were 
 made. The first of these was the introduction of the 
 astatic pair in place of a single needle, a device which 
 in modified and refined form holds its place in nearly 
 all modern instruments. The other step consisted in 
 the use of the telescope and mirror. The galvanometer 
 is a direct descendant of the magnetic compass, and the 
 user of the galvanometer inherited from his forerunner, 
 the mariner who steered by the aid of the compass, the 
 crude device of a metallic pointer moving over a divided 
 
 68 
 
circle. The substitution of the angular movement of a 
 ray of light, noted by the aid of the telescope and scale, 
 was a great advance. 
 
 The next important step resulted from the demands 
 of the needs of sub marine telegraphy, and the require- 
 ments of this branch of applied electricity were com- 
 pletely and beautifully met in the mirror golvanometers 
 of Thomson. In these well-known instruments the 
 mass of the moving parts was for the first time reduced 
 to a small quantity. The needle which, at the time of 
 Melloni, was still really a needle, taken without any 
 modification from the hands of the seamstress, and 
 which in the later galvanometers of Siemens, Wiede- 
 mann, Edelmann and others, had undergone a series of 
 transformations, none of which, however, had been in 
 the direction of diminishing its mass materially, was 
 reduced by Kelvin to a system of short, thin strips of 
 steel, the length of each which was but a few milli- 
 meters, while the aggregate mass was a few milli- 
 grams. 
 
 In the hands of Kelvin, also, the mirror was reduced 
 in weight in like proportion, by the substitution of 
 microscopic cover glass for polished metal, or for the 
 thick sheets of glass which had been used in the gal- 
 vanometers of previous designers. In his instruments 
 we find also, for the first time, the coil brought into 
 really close proximity to the needles. The result of 
 these changes was an instrument, the sensitiveness of 
 which far exceeded that of any instruments which had 
 previously existed, while the quickness of action neces- 
 sary in cable signalling was secured by the reduction 
 of the mass of the moving parts. 
 
 The discussion of the proper form and method of 
 winding galvanometer coils to secure a maximum effect 
 from a given weight or resistance of copper has been 
 given by Maxwell in his treatise * The two most im- 
 portant points to be considered are the winding of the 
 coil with different sizes of wire beginning with the 
 smallest diameter, and the construction of the coil in 
 such a manner as to bring the largest number of turns 
 within a given effective distance from the needle. 
 
 If we consider the action upon a needle at JV(Fig. 41) 
 of a single turn of length I, causing a current 2, we 
 have for the strength of the component of the magnetic 
 field at tf, parallel to the axis of the galvanometer (see 
 equation 3), 
 
 / -- * 7 sin 
 ~ 
 
 where d is the distance between the wire and the 
 needle. 
 
 * Electricity and Magnetism, vol. ii., p. 360. 
 
 69 
 
Since it is upon this field that the action of the gal- 
 vanometer depends, it is clear that the problem consists 
 in placing the winding, the radius of which is d sin 6, 
 where it will make a field with the largest component 
 in the required direction. 
 
 Maxwell has shown, in the paragraph just cited, that 
 if a surface, the polar equation of which is 
 
 d? = x? sin 6. (98) 
 
 be constructed, any circular winding of length / will 
 produce a greater effect when it lies within the surface 
 than when it lies outside it. It follows, therefore, that 
 if a completed coil be of such shape that its surface is 
 not of the above form, we may shift windings from 
 
 FIG. 41. 
 
 without the surface to a position within the same, thus 
 improving its action without changing the amount of 
 wire used. In a word, each layer of an ideal coil will 
 always lie in a surface having an equation of the form 
 of (98), and the value of x in the expression 
 
 a? = 
 
 sn 
 
 (99) 
 
 will be constant for all its turns. 
 
 Fig. 42 is a diagram showing cross-sections of three 
 such surfaces [Maxwell, ii., p. 361]. 
 
 As regards the diameter of wire to be used in wind- 
 
ing, the chief results of the discussion, cited above, are 
 stated by Maxwell, as follows: 
 
 i. "If the method of covering the wire and of wind- 
 ing it is such that the space occupied by the metal 
 bears the same proportion to the space between the 
 wires whether the wire is thick or thin, then 
 
 Y 
 
 y 
 
 dy 
 
 (100) 
 
 (where y is the radius of the wire and Y 9 is the area of 
 the quadrilateral whose angles are the sections of the 
 axes of four neighboring wires of the coil by a plane 
 through the axis of the latter), and we must make both 
 
 FIG. 42. 
 
 FIG. 43. 
 
 y and Y proportional to x (see equation 99); that is to 
 say, the diameter of the wire in any layer must be proportional 
 to the linear dimension of that layer'' 
 
 2. " If the thickness of the insulating covering is 
 constant and equal to b, and if the wires are arranged 
 in square order 
 
 T='2(y + b) (101) 
 
 and the condition is 
 
 (2/+ ft) 
 
 = constant. 
 
 (102) 
 
 V 
 
 In this case the diameter of the wire increases with 
 the diameter of the layer of which it forms a part, but 
 not at so great a rate." 
 
3- " If increase of resistance is not regarded as a de- 
 fect, as when the external resistance is far greater than 
 that of the galvanometer, or when our only object is to 
 produce a field of intense force, we may make y and T 
 constant. In this case the value of G increases uni- 
 formly as the dimensions of the coil are increased so 
 that there is no limit to the value of G except the labor 
 and expense of making the coil/'* 
 
 Aside from the questions of the shape of coils and 
 the grading of the wire, the construction of a delicate 
 galvanometer depends, as has been indicated already, 
 upon the lightness of suspended parts, and the arrange- 
 ment of same with reference to a minimum value of 
 the moment of inertia, upon the strength of the needles, 
 and upon the reduction of the space within which the 
 suspended parts swing. 
 
 In all three of these particulars, modern practice 
 seems to have been carried to a definite limit, beyond 
 which it is difficult to proceed. The independent efforts 
 of three or four of the most recent workers in this field 
 have, indeed, led to the simultaneous development of 
 instruments essentially identical and possessing very 
 nearly the same relative figure of merit. 
 
 The use of miscroscope cover glass for the galvano- 
 meter mirror necessitates the careful study of the 
 materials used, since glass in these thin layers is in- 
 variably badly warped. One plan has been to silver a 
 very large number of covers, using Draper's solutions 
 or the rather more convenient ones recommended by 
 Kohlrausch. It is a matter of great difficulty to find 
 among a lot of mirrors thus silvered, even one of any 
 considerable size which presents a plane surface; but 
 fortunately the reduction of the face of the mirror to a 
 minimum is a desirable thing where we are seeking to 
 construct an instrument with very small moment of 
 inertia. One may rest content, therefore, with a few 
 square millimeters of surface, provided by means of 
 these the scale can be read. 
 
 Undoubtedly the best procedure is that recommended 
 by Snow, Franklin and others, which consists in using 
 a glass plate with a plane surface as a test plate and of 
 laying down upon the same, one after another, the 
 various pieces of cover glass from which mirrors are to 
 be selected. If these be properly cleaned, interference 
 bands will show themselves and from the shape of these 
 it will be possible to determine whether any portion 
 of the surface of each is approximately plane. Those 
 which show the best surfaces are to be laid aside and 
 silvered; the others are useless for the making of mir- 
 
 * Maxwell : Treatise ii., pp. 363-364. 
 
 72 
 
U N I v i: 
 
 rors. These selected glasses having been silvered in 
 the usual manner, should then be cut into small rectan- 
 gular pieces of the sizes desired. 
 
 The best size of mirror for galvanometers of the 
 highest sensitiveness is the smallest size which will 
 admit of readings being made with the telescope and 
 scale. It is found that when such a mirror is cut to a 
 width of less than two millimeters, diffraction fringes 
 begin to disturb the image seriously. This, therefore, 
 may be taken as the limiting size of such a mirror. 
 We may gain something in surface without increasing 
 the moment of inertia, appreciably by making the 
 mirror oblong in shape and mounting it with its longer 
 diameter parallel to the suspension rod. The best size 
 for many purposes would seem to be a length of four 
 to five, with a width from two to two and one-half 
 millimeters. 
 
 Experience shows that one of the best materials for 
 the suspension bar or rod upon which the elements of 
 the astatic pair, together with the mirror, are to be 
 mounted, is a slender fibre of glass. This must be as 
 nearly straight as possible, since it forms the axis of 
 rotation of the system. If a considerable number of 
 glass fibers are made by drawing in the flame and are 
 cut to the proper length, the straight ones may be 
 selected by laying all upon a flat surface and rolling 
 them both back and forth under the finger. 
 
 The question of the best size and shape for galvan- 
 ometer needles, where the object is delicacy, is one upon 
 which some further investigations should be made. At 
 present it is generally conceded that a system of three 
 or five small needles arranged side by side, instead of 
 one heavier needle, gives a better result. Snow, in his 
 galvanometer constructed in Berlin for the exploration 
 of the bright line spectra of the metals, used six strips 
 in each element of his astatic pair. These were arrang- 
 ed pair wise, back to back, one long between two shorter 
 pairs. (Fig. 43.) Professor W. S. Franklin and the 
 writer, in the course of a recent investigation requiring 
 the very highest attainable sensitiveness in the galvan- 
 ometer, made use of an instrument in which the needles 
 were prepared as described below.* 
 
 " The elements of the astatic pair contained four mag- 
 nets each. They were very nearly equal in strength, 
 and were only two inches apart, the mirror being 
 placed below the coils instead of between them. By 
 this arrangement the galvanometer was rendered com- 
 paratively insensible to magnetic disturbances. The 
 galvanometer was provided with freshly made magnets 
 
 * See Physical Review, Vol. i. p. 437. 
 
 73 
 
just before being used, a precaution which should 
 always be taken in preparing for any important work 
 requiring the last degree of sensitiveness, and care was 
 taken to send no currents of any ordinary strength 
 through the instrument. In the preparation of the 
 magnets the following precautions were taken. Piano 
 wire ^ mm. in diameter was straightened by subjecting 
 it to slight tension at a low red heat, and was cut into 
 two-inch lengths. These were placed, two or three at 
 a time, in an acute V-shaped iron trough, and atter 
 being heated uniformly to a cherry-red heat (800 C.) 
 in a Bunsen flame, were quickly dropped into cool water. 
 Two small pieces of exactly the same length were then 
 cut from the central portion of each, and magnetized 
 under similar conditions. The cutting was done by 
 placing the hardened wire upon a smooth block of hard 
 wood, and pressing an edged tool against it. If this 
 procedure be carefully followed a highly astatic pair 
 may always be obtained. 
 
 The two small magnets thus made from each piece 
 were used, one in each of the elements of the astatic 
 system. The galvanometer had a resistance of 15 ohms 
 with its coils in series. When so arranged, and with a 
 half -period of seven seconds, and scale distance of 120 
 cm., a deflection of i mm. corresponded to 6 x io~ 10 
 amperes." 
 
 This refinement of the moving parts of the galvano- 
 meter would have been of little use but for the discov- 
 ery of the remarkable qualities of quartz fibres made 
 some years ago by Professor C. V. Boys. It has been 
 abundantly shown by that physicist, and his statements 
 have been verified by many others, that quartz pos- 
 sesses a strength be} r ond that of any other known ma- 
 terial when drawn into fine threads or fibres, and that 
 it is free from the structural defects of cocoon silk, 
 which had been previously the best of known materials 
 for the suspension of galvanometer needles. 
 
 The method of obtaining quartz fibres described by 
 Boys is one the execution of which demands a con- 
 siderable amount of manipulative skill and no little ex- 
 perience. The following is a more simple procedure. 
 
 The apparatus needed is a oxyhydrogen blowpipe, 
 two pairs of crucible tongs and some bits of white 
 quartz. The common variety of quartz crystal, known 
 as milky quartz, serves well for this purpose. The ma- 
 terials should be crushed into small granules about 
 three or four millimeters in diameter. These show a 
 tendency to disintegration when first heated ; when 
 fused, however, the material goes over into a condition 
 such that it may be placed in the flame over and over 
 
 74 
 
again after becoming cold without further rupture. 
 The first step consists in making from these bits of 
 pulverized quartz a number of short rods of the molten 
 silica. These should be long enough so that they can 
 be held in the flame, each end being within the jaws of 
 a pair of tongs. When thus exposed to the hottest part 
 of the gas jet, the middle softens readily and the rod 
 can be drawn out into a fibre. These fibres are still 
 much too heavy for use in the 
 suspension of a delicate galvano- 
 meter, but they can be reduced to 
 the desired fineness by the very 
 simple process of holding them in 
 the flame until they soften, when 
 the draught of heated gas from the 
 nozzle of the burner will be found 
 sufficient to carry the softened 
 fibre with it, drawing it out to a 
 thinness which renders it suitable 
 for the purpose now under con- 
 sideration. With a little care these 
 attenuated fibres, which are too 
 small to be readily seen, can be 
 secured, since they are attached 
 at one end to the larger filament 
 held in the hand. It is not al- 
 ways possible to secure the fibres 
 in this way, many of them being 
 torn loose and swept away in the 
 currents of air. By placing at a 
 safe distance above the flame a 
 piece of canton flannel, however, 
 these stray fibres will be driven 
 against the rough surface of the 
 cloth and will become entangled. 
 In a very short time hundreds of 
 them may be collected in this man- 
 ner over the oxyhydrogen flame. 
 Many of these will be of consider- 
 able length, and nearly all of them 
 will be of sufficient fineness for 
 use in galvanometers of the high- 
 est delicacy. 
 
 The advantage of the astatic pair in the construction 
 of galvanometers having been once recognized, it was 
 a very natural extension of the principle to introduce a 
 second set of coils, so that each element of the pair 
 might be brought into a stronger field due to the cur 
 rent. Kelvin made use of such an arrangement in one 
 of his types of galvanometer, bringing the mirror to a 
 
 FIG. 44. 
 
 75 
 
position midway between the two groups of needles, a 
 procedure which has been widely followed by others. 
 Fig. 44 shows an instrument in which four coils are used 
 with the mirror placed between them as above de- 
 scribed. Fig. 45 shows the same instrument on a 
 larger scale with the short coils swung away so as to 
 show the interior. Fig. 46 shows the arrangement of 
 
 FIG. 45. 
 
 the needles and mirror in this instrument about life 
 size. 
 
 The suspended parts of such galvanometers having 
 been reduced to a minimum as regards mass and moment 
 of inertia, it was a natural mistake to suppose that even 
 the quartz fibre must be of great length in order that 
 its moment of torsion should remain inappreciable. 
 Thus in the galvanometer just depicted a fibre half a 
 
meter long was used, and of many other galvanometers 
 made at that time the same thing is true. While it is 
 easy to obtain quartz fibres of the requisite length and 
 fineness, it is a much more serious matter to mount a 
 long fibre than a short one, and after the instrument 
 has been successfully set up, the question of keeping it 
 adjusted, as to level, so that the suspended parts shall 
 be free within the very narrow space allotted to them, 
 becomes a difficult one. Subsequent experience has 
 shown that a fibre five to ten centimeters long is 
 sufficient, even in the case of the lightest galvanometers. 
 There are certain advantages in bringing the elements 
 
 II 
 
 FIG. 46. 
 
 a b 
 
 FIG. 47. 
 
 of an astatic pair near to one another, and to attain this 
 end, the mirror is sometimes placed at the bottom of the 
 suspension rods as shown in Fig. 47 (a) which gives 
 .the arrangement of mirror and needles in such a gal- 
 vanometer. Fig. 48 shows an instmment of this 
 type, constructed as were also the galvanometers shown 
 in Fig. 44 and 49, by F. C. Fowler, instrument maker 
 to the Department of Physics in Cornell University. 
 This instrument has four coils placed pair- wise one 
 above another and adjustable as to the distance between 
 them. A somewhat similar instrument with eight coils 
 and four sets of needles mounted in four equally distant 
 
 77 
 
groups with a mirror at the bottom is shown in Fig. 
 49. A diagram of the suspended parts is given in Fig. 
 47 (b) Figures 50, 51 and 52 give some details of the 
 galvanometer shown in Fig. 49. , 
 
 The only factor in the construction of a sensitive gal- 
 vanometer, which we have still to consider, is that of 
 the distance between the opposite coils. With mirrors 
 two millimeters across and needles three or four milli- 
 meters long, it would seem that the clearance necessary 
 
 FIG. 48. 
 
 to give freedom of action might be extremely small. 
 In this regard, however, a limit is soon reached on 
 account of the difficulties which arise through the mag- 
 netic properties of the materials of which the coils 
 themselves are constructed. If the layers of wire lying 
 next the needle be insulated with the usual green silk, 
 it will be found that these when brought near the sus- 
 pended needles will attract the same strongly and will 
 
soon interfere altogether with the requisite freedom of 
 motion. The substitution of white for the green cover- 
 ed wire seems to mitigate this trouble to some degree, 
 but it is a matter of great difficulty to find thoroughly 
 non-magnetic insulated copper wire. Then, again, in 
 the process of handling the instrument for mounting, 
 the silk covering, and the shellaced surfaces, tend to 
 become electrified and much annoyance arises from this 
 source. It is, indeed, sometimes necessary to cover the 
 
 FIG. 49. 
 
 entire inner face of the coils with gold leaf and to 
 ground the same before the needles can be made to 
 swing freely in close proximity to the coils. The writer 
 has never found it practicable to work with galvano- 
 meters in which the average clearance space was re- 
 duced to less than one millimeter. Fig. 53, which shows 
 the position of the coils of Snow's galvanometer, in- 
 dicates that in his instrument, the clearance was approx- 
 imately that just mentioned. 
 
 79 
 
As Regards the sensitiveness of galvanometers con- 
 structed in accordance with the principles laid down in 
 this lecture, the following data may be of interest: 
 
 B. W. Snow, 1892, constructed the galvanometer, the 
 coils of which are shown in Fig. 53, and the suspended 
 parts in Fig. 43. He obtained a figure of merit 1.5 X io~ n 
 amperes, with a vibration period of 20 seconds and a 
 scale 300 cm. from the instrument. The above applies 
 to a deflection of one millimeter. The resistance of the 
 
 FIG. 50. 
 
 FIG. 51. 
 
 FIG. 52. 
 
 instrument was 140 ohms. In the same year E. F. 
 Nichols and the writer used the galvanometer with long 
 suspension fibre, shown in Fig. 44. The aggregate 
 weight of the moving parts of this instrument was 
 48 mg. With the coils in multiple the resistance was 
 9.3 ohms. The sensitiveness reached i X icr 10 amperes 
 with the coils thus connected. The scale was about 150 
 cm. from the instrument, and the period about 10 
 seconds. W. S. Franklin and the writer found for the 
 
 80 
 
galvanometer used in their study of the condition of 
 the ether surrounding a moving body, the method of 
 constructing the magnets of which instrument has just 
 been described, a figure of merit of 6 x io~ 10 amperes 
 with a resistance of 150 ohms, a period of 7 seconds and 
 a scale of 1 20 cm. distant. In this case also the figure 
 of merit refers to a millimeter of deflection.* 
 
 Paschenf also, who constructed a special galvano- 
 
 
 
 o 
 
 FIG. 53 
 
 meter for the exploration of the very weak spectra 
 afforded by the diffraction grating, obtained i 6 X io~ n 
 amperes for one millimeter with 20 ohms, a period of 
 30 seconds and a scale 270 cm. distant. 
 
 * For these three cases see Physical Review, vol. i. 
 t Paschen : Wiedemanrfs Annalen, 48, p. 284. 
 
LECTURE VII. 
 
 SPECIAL APPLICATIONS OF THE GALVANOMETER TO THE 
 MEASUREMENT OF CURRENT AND RESISTANCE. 
 
 i. Measurement of feeble currents. One of the import- 
 ant uses of the galvanometer is for the detection and 
 measurement of currents of exceedingly small intensity, 
 for which purpose the instrument must be especially 
 adapted by constructing it with reference to the reduc- 
 tion of the constant to the smallest possible value. 
 This reduction, as has already been pointed out (Lecture 
 VI.), is attained by bringing the wire as near as pos- 
 sible to the needle, by reducing the moment of inertia 
 of the moving parts to a minimum, by making use of 
 the astatic system and by the artificial reduction of the 
 magnetic field within which the needle swings to a very 
 small intensity. 
 
 Since the constant of such instruments cannot be 
 determined by computation from the dimensions of the 
 coils the galvanometer must, be calibrated. In ad- 
 dition to the absolute calibration, means must be de- 
 vised for repeated re-determinations of the fluctuations 
 to which the figure ot merit of galvanometers of extreme 
 delicacy are subject. 
 
 The calibration may be made: 
 
 (i.) By the use of the Clark's cell, the greatest care 
 being taken to fulfill the conditions under which this 
 form of cell affords reliable results. A detailed de- 
 scription of this cell and of the method of using it will 
 be given in the Lecture VIII. 
 
 (2.) The galvanometer may be calibrated also by 
 placing it in shunt around a known resistance (see Fig. 
 54). This method involves the use of a second galvano- 
 meter, G, of known constant, by means of which the 
 current flowing through the resistance in question can 
 be measured. The size of this resistance, R S , will de- 
 
 82 
 
pend upon the figure of merit of the instrument. It 
 must be of such size that when a measurable current 
 traverses it, the difference of potential between its ter- 
 minals, shall give a suitable deflection to the galvano- 
 meter to be tested. In case of instruments of the 
 highest delicacy this involves the % reduction of the cur- 
 rent to values too small to be measured upon a stand- 
 
 FIG. 54. 
 
 ard instrument, or the increase of the resistance to such 
 an extent as to render accurate knowledge of its value, 
 a matter of difficulty. In such cases it is better to use 
 a multiple shunt, the arrangement of which is shown in 
 
 Fig- 55- 
 
 This device makes it possible to standardize instru- 
 ments of extreme sensitiveness with a fair degree of 
 accuracy. It is true that increased error is introduced 
 by the use of the second shunt, but it is also true that 
 
 FIG. 
 
 very precise determinations of the figure of merit of 
 galvanometers of the highest delicacy are rendered use- 
 less from the fact that such galvanometers are subject 
 to continual changes of constant. Almost in proportion 
 as the sensitiveness of the instrument rises beyond a 
 certain value, the possibility of precise determination 
 of its sensitiveness diminishes. 
 
Arrangements for the calibration of the* instrument 
 having been completed, it is necessary to provide some 
 means of keeping pace with the fluctuations of constant 
 already referred to. Probably the very best means for 
 this purpose is a subsidiary coil placed at a suitable dis- 
 tance behind the galvanometer. This coil will act upon 
 the needle, but the current through it necessary to pro- 
 duce a deflection will be very much larger than that 
 which would give the same result when sent through 
 the coils of the galvanometer itself. The number of 
 turns of this subsidiary coil may be few, but its dis- 
 tance from the needle is necessarily very considerable. 
 
 The subsidiary coil should be placed in shunt circuit 
 
 FIG. 56. 
 
 with a compensated resistance, r s , Fig. 56; through 
 which a known current flows. Upon closing this sub- 
 sidiary circuit a deflection will be produced by the 
 proper adjustment of the resistance in the circuit. The 
 deflection may be brought to size approximately equal 
 to that of the deflections which the galvanometer will 
 give in the operations to which it is to be subjected by 
 further adjustment of the resistance r ? . Immediately 
 after the completion of the absolute calibration, the de- 
 flection due to the subsidiary coil with known resistance 
 flowing through the compensated resistance should be 
 noted, and this deflection be made to serve as a refer- 
 
ence factor in all subsequent operations. In order to 
 keep track of changes in the constant of the galvano- 
 meter occurring from time to time thereafter, it will 
 only be necessary to send the same current through the 
 subsidiary coil as that which was sent through it at 
 the time of the calibration. The range in the deflec- 
 tions thus produced will then represent the range of the 
 figure of merit of the galvanometer.- The arrangement 
 of the subsidiary coil and its circuit is shown in Fig. 
 56, to which reference has just been made. 
 
 Another device for obtaining the same end, which has 
 the advantage of not requiring the use of a standard 
 instrument or ammeter, is as follows : 
 
 A thermo-element consisting of an iron or German 
 silver wire about one meter long, the ends of which are 
 
 FIG. 57. 
 
 soldered to copper wires, is constructed. One junction 
 of this thermo-element is packed in ice, the other is 
 placed in a steam bath, the pressure of which can be 
 regulated so as to make a delicate thermometer situated 
 in the same bath read constantly 100 degrees. Such a 
 thermo-element will give a small but perfectly constant 
 electromotive force so long as a constant difference of 
 temperature between its terminals is maintained. This 
 thermo-element may be used two ways: (i) to produce 
 deflections by placing it in circuit with the galvano- 
 meter coils themselves from time to time (see Fig. 58); 
 (2) in the case of galvanometers of extreme sensitive- 
 ness, by placing the thermo-element in circuit with the 
 subsidiary coil. 
 
 In all cases in which it is necessary to reduce the gal- 
 vanometer to its condition of maximum sensitiveness, a 
 
 85 
 
point is reached at which the drift of the zero due to 
 magnetic disturbances introduces serious error. It has 
 already been shown in the lecture on galvanometers 
 with artificial fields, that in the case of instruments 
 with weak fields every fluctuation in strength of the 
 earth's magnetic forces has an exaggerated effect upon 
 the needle. It is, indeed, oftentimes impossible to 
 maintain a sensitive galvanometer with approximately 
 fixed zero long enough to obtain a permanent deflec- 
 tion. 
 
 In all such cases the only remedy is to make use of 
 the ballistic method in which the zero of the galvano- 
 meter at the instant before closing circuit is noted, and 
 the circuit is closed during a single swing of the instru- 
 ment. It is then opened, the reading at the end of this 
 
 FIG. 58. 
 
 swing is noted, and finally the point which the needle 
 reaches in its first return swing is observed. If the 
 average between the reading of the return swing and 
 the original zero be taken as a corrected zero, the drift 
 of the galvanometer in the short intervening interval 
 will be almost entirely eliminated. Deflections com- 
 puted in this way should be interpreted by means of a 
 calibration performed in a corresponding manner in- 
 stead of a calibration by permanent deflections. 
 
 In some extreme cases, such as occur in work with 
 the thermo-pile or bolometer, it is found to be impos- 
 sible to open and close a switch in the galvanometer 
 circuit at all without producing disturbances of great 
 magnitude. In these cases fortunately, it will be found 
 possible to use the galvanometer in a permanently 
 
 86 
 
closed circuit, the deflection being produced by the ex- 
 posure of the thermo-pile or bolometer, to the source 
 of radiation under observation, by the sudden removal 
 of an intervening screen. 
 
 The operation consists in reading the instrument with 
 the circuit closed, and with the screen intervening be- 
 tween the source of light and the thermo-pile or bolo- 
 meter. This gives the zero point of the deflection. 
 The screen is then removed during one swing of the 
 galvanometer and is restored to its place. The extreme 
 reading of the swing is read, also the return swing, and 
 the deflection is computed from these three observations 
 as already indicated. Theory would lead us to expect 
 that the deflection thus obtained would be proportional 
 
 3* 
 
 
 FIG. 5 9 . 
 
 to the permanent deflection of a galvanometer, the 
 zero of which was fixed. Professor Ernest Merritt* 
 has shown experimentally that these conditions are ful- 
 filled in practice. Fig. 59 gives a curve obtained by 
 him from observation of a galvanometer used in this 
 manner on -closed circuit, the readings being carried on 
 not only during the first swing, but for a much longer 
 period, up to the time, indeed, when the needle reached 
 its final position. 
 
 In this figure curve I gives the observed movement 
 of the needle, the galvanometer being in closed circuit 
 with a thermo pile which was suddenly exposed to heat 
 at the time, marked O seconds. The analysis of this 
 
 * Merritt ; American Journal of Science, vol. xli., p. 417. 
 
curve shows it to be the resultant of curves II and III, 
 the latter, of which is a trace of logarithmic decrements 
 the period of which agrees with the free swing of the 
 needle. 
 
 Merritt found that in all cases the first throw of the gal- 
 ganometer needle was proportional to its permanent 
 deflection. 
 
 2. Measurement of heavy currents. Another important 
 problem in the measurement of currents is that where 
 the strength of the current is too great for direct deter- 
 minations. Under these circumstances the method by 
 fall of potential is to be preferred. The only difficulty 
 of carrying out this method successfully lies in the 
 maintenance of a constant temperature in the resist- 
 
 0.0030 
 
 
 
 t 
 
 
 1 
 
 
 1 
 
 0.0025 
 
 * 
 
 
 
 
 
 1 
 
 0.0020 
 
 t 
 
 
 \ 
 
 
 V 
 
 
 1 
 
 0.0015 
 
 1 
 
 
 \ 
 
 
 X 
 
 
 \ 
 
 OOU10 
 
 1 
 
 \ 
 
 
 V 
 
 
 1 
 
 
 \ 
 
 0.0005 
 
 
 
 \ 
 
 
 X 
 
 
 V 
 
 
 
 24 
 
 6 
 
 FIG. 60. 
 
 ance around which the galvanometer is shunted, but 
 this difficulty has been found so great as to lead to the 
 condemnation of the method on the part of many in- 
 vestigators. Two remedies have been proposed: (i) 
 the use of a material for the shunt resistance which 
 does not vary with the temperature. Certain alloys of 
 manganese and copper, or manganese, nicKel and copper, 
 are known to possess this property, but their other pro- 
 perties have not been studied as yet with sufficient 
 thoroughness to warrant unreserved confidence in their 
 stability. In a word, we do not know whether shunts 
 constructed of such material will remain of constant 
 resistance when subjected to the action of heavy cur- 
 rents. 
 
 88 
 
From the measurements made upon this class of 
 alloys, however, it appears that unless very particular 
 attention is paid to the matter of annealing, etc., 
 changes of specific resistances are sure to occur as the 
 result of every subsequent fluctuation of temperature. 
 B. H. Blood* in the course of an examination of an 
 alloy containing .8082 copper and .1912 ferro-manganese, 
 found that eight successive heatings to ico C. with 
 alternate coolings to 20 C., gave the results indicated 
 in the following table. 
 
 EFFECT OF REPEATED HEATING AND COOLING UPON THE 
 RESISTANCE OF AN ALLOY OF FERRO-MANGANESE AND 
 COPPER (HARD DRAWN). 
 
 Observation. 
 
 Temperature. 
 
 Specific Resistance. 
 
 Relative Resistance. 
 
 
 Degrees. 
 
 
 
 
 20 
 
 30.380 
 
 1. 0000 
 
 
 100 
 
 30.186 
 
 99331. 
 
 
 20 
 
 30.163 
 
 .99287 
 
 
 100 
 
 30.151 
 
 99255 
 
 
 20 
 
 30.138 
 
 .99202 
 
 6 
 
 100 
 
 30.121 
 
 .99180 
 
 7 
 
 20 
 
 30.118 
 
 99134 
 
 8 
 
 100 
 
 30.118 
 
 99 I 34 
 
 9 
 
 20 
 
 30.105 
 
 .99093 
 
 10 
 
 100 
 
 30.099 
 
 .99072 
 
 n 
 
 2O 
 
 30.092 
 
 .99051 
 
 12 
 
 100 
 
 30.104 
 
 .99092 
 
 13 
 
 20 
 
 30.079 
 
 .99007 
 
 *4 
 
 100 
 
 30.104 
 
 .99092 
 
 15 
 
 20 
 
 30.072 
 
 .98985 
 
 As regards temperature coefficients of resistance, 
 however, provided a method of treatment can be found 
 to check the behavior just referred to, this class of al- 
 loys leaves little to be desired. The coefficient, indeed, 
 seems to depend directly upon of the percentage of ferro- 
 manganese present, and to pass from positive to negative 
 values when 18 per cent, of that material is combined 
 with the copper. This is shown in the curve (Fig. 60) 
 platted from results obtained by the same observer. In 
 this diagram ordinates are coefficients and abscissae are 
 percentages of ferro-manganese. 
 
 Since the currents, the measurement of which we are 
 now considering, are too heavy to allow the use of the 
 compensated resistance of copper and carbon, which is 
 to be described in Lecture VIII., and since the only 
 known available material which is without a coefficient 
 is questioned, recourse must be had to a metallic shunt, 
 and to some device by means of which its temperature 
 can be controlled. This, indeed, is the plan prescribed 
 
 * American Journal of Science^ vol. xxxix., p, 473. 
 
 8Q 
 
in what is known as the Vienna method. The use of a 
 bath has been found to introduce such errors, however, 
 as to lead to the very general abandonment of the 
 method. 
 
 The principal source of error is that due to the as- 
 sumption of a false value for the resistance of the 
 metallic shunt. The resistance is, of course, a function 
 of the temperature of the metal, and since a conductor 
 carrying a current beyond its normal capacity and de- 
 veloping a large amount of heat energy must necessar- 
 ily possess a temperature considerably above that of its 
 surroundings, it is not allowable to assume that a 
 thermometer placed in a bath will indicate the true 
 temperature of the shunt. 
 
 FIG. 6l. 
 
 To make the method a reliable one, it is before all 
 necessary to determine the precise resistance of the 
 shunt for all the conditions under which it is to be used. 
 This may be accomplished in one of the folio wing ways: 
 
 (i.) The shunt resistance may be constructed in such 
 a form that it can be readily surrounded by a coil of 
 fine insulated copper wire fitting snugly to its surface 
 (see Fig. 61.) The temperature coefficient of the resist- 
 ance of this copper coil may be determined once for all. 
 Its resistance will indicate much more closely than can 
 be done by means of a mercury thermometer inserted 
 in a bath, the temperature of the conductor around 
 which it is wound. This method of measuring tem- 
 peratures is of sufficient delicacy; it gives an integration 
 of the temperature for the entire surface of the shunt 
 resistance, and it is, indeed, open only to a single 
 
 90 
 
objection; that the temperature measured is that of the 
 region lying just without the surface of the metal, which, 
 in the cases under consideration, will always be slightly 
 lower than the average temperature within the mass. 
 This error, which is a small one, may be further reduc- 
 ed by using thin strips of metal for the shunt resistance 
 so that the temperature differences between the surface 
 and the interior shall be extremely small. Another, 
 and still more effectual, method of reducing the error 
 of difference between the temperature indicated by the 
 calibrating coil and that which it is desired to know, 
 consists in making the shunt resistance in the form 
 of a tube, or series of tubes, around which the cali- 
 brating coil is wound longitudinally so that one-half its 
 length will lie within the core of the tube, in which 
 position it will reach a temperature very nearly as high 
 as that of the mass of metal itself. 
 
 (2.) The other method of determining the true tem- 
 perature of the shunt-resistance is free from the errors 
 which have just been pointed out. This method consists 
 in studying the temperature of the shunt as a function 
 of the time, a time curve being taken immediately after 
 the close of each measurement of current. The best 
 apparatus for getting this curve is a calibrating coil of 
 the kind described under method (i.) The procedure 
 is as follows: 
 
 The current to be measured having been sent through 
 the shunt resistance for a sufficient amount of time to 
 allow the latter to reach its final temperature, and 
 the readings by fall of potential having been made, 
 the circuit is broken at an instant of time accurately 
 noted, and a succession of measurements of the resist- 
 ance of the calibrating coil are made at times likewise 
 carefully observed. From the moment of breaking the 
 circuit, the shunt resistance will begin to cool, approach- 
 ing gradually to the temperature of its surroundings. 
 The measurements of resistance and time just described 
 will permit of the platting of a curve in which abscissas 
 are times and ordinates are temperatures. By extend- 
 ing this curve backwards to the origin of time, it is 
 possible, through extrapolation, to obtain a very accu- 
 rate determination of the temperature of the shunt at 
 the instant when the current ceased to flow. 
 
 Carried out in this way, the Vienna method affords 
 an entirely reliable means of measuring heavy currents. 
 Its accuracy depends only upon the precision with 
 which the absolute resistance of the shunt can be deter- 
 mined. By means of the second method just outlined, 
 the change of resistance, which the shunt undergoes on 
 account of the heating effect of the current which 
 traverses it, can be accurately ascertained. 
 
 91 
 
LECTURE VIII. 
 
 THE MEASUREMENT OF ELECTROMOTIVE FORCE BY MEANS 
 OF THE GALVANOMETER. 
 
 The Determination of the Electromotive Force in Absolute 
 Measure. The best instrument for this practice is a tan- 
 gent galvanometer of considerable sensitiveness, the con- 
 stant of which has been determined by measurement of 
 the coils. The resistance of the galvanometer itself should 
 be as low as possible, and it should be used in series 
 with a variable resistance, the absolute values of all the 
 coils of which are accurately known. Fig. 62 shows the 
 arrangement of this apparatus for the determination of 
 the difference of potential between points a and b in any 
 circuit. 
 
 The method is capable of very wide range. The up- 
 per limit is reached when the electromotive force to be 
 measured is so great that the resistance R P cannot be 
 made large enough to reduce the deflection to a read- 
 able size. The lower limit is reached when R P becomes 
 zero and the deflections are too small for accuracy. 
 The method may be extended in the direction of high 
 electro-motive forces by the use of a galvanometer with 
 two coils capable of being placed in series, or in mul- 
 tiple, or of being used differentially. 
 
 Fig. 63 is from a photograph of a small tangent gal- 
 vanometer designed especially for this purpose by Prof. 
 W. A. Anthony. The windings are arranged in four 
 parts which may be thrown together in either of the 
 three ways indicated above, by the use of a small divided 
 block with plugs placed upon the base of the instrument. 
 The range of usefulness is very wide, extending from 
 a thousandth of a volt per centimeter deflection with no 
 resistance in circuit, to over twenty volts per centimeter 
 deflection, when used in series with a megohm box. 
 Used differentially, the instrument extends to quite as 
 high potentials as one could expect to measure by such 
 a method. 
 
 92 
 
In a case of very low electromotive forces, the stand- 
 ard galvanometer must be replaced by one of high 
 sensitiveness, which it is necessary to calibrate before 
 using. Probably the best method of calibrating such 
 a galvanometer consists in looping the same with suit- 
 able resistance in circuit around a compensated resist- 
 ance through which a known current flows. (See Fig. 
 64.) With a suitable standard instrument, A, for the 
 measurement of current flowing through this compen- 
 sated resistance and with adjustable resistance R S and R g 
 in the main circuit and in the circuit of the galvano- 
 meter to be tested, the latter can be accurately calibrated 
 throughout its entire range. 
 
 The compensated resistance coil suitable for this 
 purpose, is made of copper, which, as is well known, 
 possesses a positive temperature coefficient of about 
 0.004 placed in multiple with a rod of graphitic carbon, 
 the temperature coefficient of which is always negative 
 with a value varying considerably, but always much 
 smaller than that of copper. (See Fig. 65.) Some data 
 concerning the variations of the temperature coefficient 
 for resistance to be met with in testing various kinds of 
 carbon are given in the following table. 
 
 INFLUENCE OF TEMPERATURE UPON THE RESISTANCE OF 
 VARIOUS VARIETIES OF CARBON. 
 
 Qbserver. 
 
 Variety of Carbon. 
 
 Tempera- 
 ture 
 Interval 
 
 Mean co- 
 efficient 
 per deg. 
 
 Werner Siemens ( Wiede- ( 
 ntanns Annalen, 10, p. < 
 5 6o) 1 
 
 Borgmatm (Journal der f 
 
 Gas carbon 
 Pressed gas carbon 
 
 Charcoal 
 
 tO 200 
 
 o to 200 
 26 to 260" 
 
 .000345 
 .000301 
 
 .00370 
 
 
 Anthracite 
 
 20 to 250 
 
 .00265 
 
 lischen Gesellschaft, 9, p. 1 
 167 ) 
 
 Graphite 
 Coke 
 
 25 to 250 
 26 to 245 
 
 .00082 
 .00026 
 
 
 Gas carbon (coarse) .... 
 
 18 to 200 
 
 .000285 
 
 Kemlein (Wtedemanns A n- j 
 nalen, 12, p. 73.) "l 
 
 Gas carbon (fine) 
 Carry's carbons . .... 
 
 1 8 tO 200 
 
 18 to 100 
 
 .000287 
 .000321 
 
 f 
 
 Paris retort carbons 
 
 
 .000300 
 
 
 
 
 
 
 
 
 .000425 
 
 Muroaka ( Wiedemann s 
 
 Gaudoin's carbons . . 
 
 
 .000415 
 .000370 
 
 Annalen, 13, p. 307.) 
 
 
 
 
 
 
 
 .0001 s6 
 
 I 
 
 Siberian graphite 
 Faber's lead pencil graphite. . 
 
 
 .000739 
 .000588 
 
 For the range of temperatures through which it is 
 desired that this resistance shall be constant, we may 
 assume that the coefficients are both constant quantities. 
 
 The problem to be solved in the construction of a 
 
 93 
 
compensated coil for the range of temperatures in ques- 
 tion consists in determining the proper amount of 
 carbon and of copper to be placed in multiple circuit so 
 that the total resistance of the combination shall not 
 vary. This condition will be met, provided the rise in 
 resistance on the part of the copper is exact.ly counter- 
 balanced by the increased conductivity of the carbon. 
 These conditions will be approximately fulfilled when 
 
 ) = S t ' (,--?-.), (103) 
 
 < c &/ \1 -f- a m t) 
 
 in which equation R m ' is the resistance of copper, R ' 
 that of the carbon at a given temperature, (say 2oC), 
 while a m and a c are the respective coefficients and t is a 
 temperature (say iooC), up to which compensation is 
 desired. 
 
 FIG. 62. 
 
 Assuming that the combination would be subject to 
 the law of parallel circuits, we have for the total resist- 
 ance R: 
 
 where R m is the resistance of the metal and R c the re- 
 sistance of the carbon. 
 
 The equation of condition, for complete compensation 
 will be 
 
 R B - ^ 
 
 where R m ' R c ' are the resistances of the components and 
 
 94 
 
fi m " -^c*. the corresponding resistances at any other 
 temperature within the limits of temperature for which 
 compensation exists. 
 
 Now the variation of the resistance of a metal with 
 the temperature may be expressed by an equation of 
 the form : 
 
 - m *=-m' (1 + at W\ (106) 
 
 where a and b are coefficients to be determined by ex- 
 periment. 
 
 FIG. 63. 
 
 In the case of carbon, the coefficient a will have a 
 negative value, and the equation will take the following 
 form : 
 
 7? c " = R c ' (i at If). . (107) 
 
 Between o and 100 the value of b in the case both 
 of copper and carbon is very small. A determination 
 of the coefficients for copper, made in the Physical 
 
 95 
 
Laboratory of Cornell University, for example, yielded 
 the equation : t 
 
 R m " = E m ' (1 + .00380 t + .00000047 t 2 ). (108) 
 
 The experiments, which covered a range of 100, are 
 in close agreement with the results published by 
 Matthiessen. 
 
 In the above equation, for t 100, we have bt 2 = 
 .0047. If we neglect the coefficient b and adopt for a 
 the mean coefficient between o and io~, we may write 
 the equation in the simpler form, 
 
 fi = ^ m ' (1 + .003847 t\ (109) 
 
 which will give results agreeing with the complete form 
 at o and ioc and will have a maximum error, at 50, 
 of .0008. 
 
 In the case of carbon the coefficient b may also be 
 neglected without appreciable error. 
 
 R. 
 
 FIG. 64. 
 
 A Carre pencil, measured in the same laboratory, 
 gave as mean coefficient between o and 100, the value 
 .000235. 
 
 We may, therefore, write the equation for this variety 
 of carbon as follows : 
 
 R^ = E c ' (1 .000235 t). (110) 
 
 In combining copper and carbon in such proportions 
 that the resulting resistance shall be independent of the 
 temperature, the equation of condition must be satisfied. 
 This equation will be satisfied only for a range of 
 temperatures throughout, which b 2 t is negligible, in the 
 case of both substances. 
 
 For such a range of temperatures we have, however, 
 as already indicated, 
 
 J3 m = B m ' (1 + a m t) 
 R; =.' (I -a, t); (111) 
 
 when a c and a m are the coefficients -for carbon and 
 copper respectively. 
 
 96 
 
Within such limits 
 
 t) 
 
 -- a c t) 
 
 m c m a m 
 
 which is readily reducible with a sufficient degree of 
 approximation to the form given in (103). 
 
 A convenient form for such a resistance is shown in 
 Fig. 66. It consists of a rod of carbon about twenty 
 centimeters in length, the ends of which have been 
 copper plated and then soldered to massive copper ter- 
 minals. These are bent at right angles and amalga- 
 mated for convenience in making connections by means 
 of mercury cups. (See Fig. 65.) The copper compen- 
 
 FIG. 65. 
 
 sation may be obtained by means of a insulated wire of 
 proper resistance, coiled snugly in spiral form around 
 the rod from end to end. A glass tube protects the 
 apparatus from damage. 
 
 Compensated resistances can be made, in a variety of 
 other forms, according to the materials in hand and the 
 requirements of the case. A very simple and excellent 
 form consists of an incandescent lamp with german 
 silver wire placed in series or in multiple with the fila- 
 ment. This form is of much higher resistance than 
 that depicted in Fig. 66, and will carry less current. It 
 is available, indeed, only in cases where the heating 
 effect of the current would be inappreciable. 
 
 97 
 
The use of a compensated resistance, of the character 
 first described, is very convenient, since it does away 
 with the necessity of considering- fluctuations to which 
 an uncompensated conductor will be subject as the 
 result of the heating effect of the current flowing in it, 
 and of temperature disturbances from without. In 
 Lecture VII, it has been shown, however, that it is 
 entirely practicable to determine accurately the resis- 
 tance of a metallic conductor when carrying current. 
 By the application of the methods therein described, 
 especially of the second method of the time curve, en- 
 tirely satisfactory calibrations of a sensitive galvanom- 
 eter for the measurement of low electromotive forces 
 may be obtained. 
 
 An important example of the use of sensitive galvan- 
 ometers in the determination of electromotive force is 
 found in the comparison of standard cells. The gal- 
 vanometer for such purposes should be sensitive to 
 one hundred-thousandth of a volt, and its figure of 
 merit should be known with a fair degree of accuracy. 
 The conditions of such determinations involve the ready 
 measurement of small differences of potential with the 
 
 FIG. 66. 
 
 expenditure of as little current as possible. The usual 
 method of proceedure is to place the two cells which 
 are to be compared in opposition to each other, and to 
 use the galvanometer ballistically, a sufficient amount 
 of resistance being placed in circuit to place its deflec- 
 tion to a suitable quantity.* One of these two cells will 
 usually be the standard with which others are to be 
 compared. The standard cell must be maintained at a 
 constant temperature, viz., that for which its electro- 
 motive force has been previously determined. By sub- 
 stituting vsuccessively the various cells which are to be 
 compared, these being placed in such a direction as to 
 oppose the standard, very accurate determinations of 
 the ratios of their electromotive forces to that of the 
 standard may be obtained. In the case of the Clark 
 cell the temperature coefficient has been made a matter 
 of careful study by Clark, Rayleigh, Wright, Helm- 
 holtz, Kittler, and more recently by Carhart. The value 
 of this coefficient, according to the different observers, 
 
 * For details of methods of testing, see Carhart ; Primary Batteries. 
 
 9 8 
 
varies through a considerable range, as will appear from 
 the following table : 
 
 Loss per degree 
 Observer. Centigrade. 
 
 Clark 1 0.0006 
 
 Helmholtz 2 0.0008 
 
 Kittler 3 0.0008 ' 
 
 Rayleigh* 0.00077 
 
 Wright 5 0.00041 
 
 Von Ettingshausen 6 0.00068 
 
 Carhart 7 0.000387 
 
 The construction of Rayleigh's form of the Clark cell 
 is shown in Fig. 67. Such cells give constant and com- 
 parable values only under very careful treatment, and 
 
 ire and Zinc Rod. 
 
 U 9 
 
 Glue. 
 
 Cork 
 
 Solntion of Zn S 0, 
 
 I/ff, S 0, (pasta.) 
 
 20 80 40 
 
 FIG. 68. 
 
 Pt Wire. 
 
 FIG. 67. 
 
 even when handled in a manner which ensures their 
 integrity, they possess a high temperature coefficient. 
 
 The source of the variation in the temperature coffi- 
 cient of Clark cells, has been clearly pointed out by 
 Carhart, who has succeeded by modifying the cell in 
 such a way as to reduce the coefficient to a minimum 
 value, by eliminating one of its elements, the variation 
 in the concentration of the solution of zinc sulphate. 
 
 This done, by the use of a solution which is saturated 
 at o (or at some temperature below that at which the 
 cell is to be maintained) the variation due to changes in 
 
 1. Latimer Clark; Journal of the Society of Telegraph Engineers, Vol. 7, p. 53. 
 
 2. Von Helmholtz; Sitzungs Berichte der Berliner Akademi, 1882, p. 26. 
 
 3. Kittler; Wiedemann's Annalen, 17, p. 890. 
 
 4. Rayleigh and Sidgwick; Proceedings, Royal Society, 17, 1884. 
 
 5. Wright; Philos Magazine, 5, Vol. 16. p. 25, 1883. 
 
 6. Von Ettingshausen; Zeitschrift fur Electrotechniker, (Wien), 1884, p. i. 
 
 7. Carhart; Primary Batteries, p. 93. 
 
 99 
 
 ERSITT 
 
the density of this electrolyte vanishes and the tempera- 
 ture coefficient falls to its normal value (0.000387). The 
 coefficient of such cells can be expressed graphically, 
 as function of the temperature, as in Fig. 68 or by the 
 equation, 
 
 E t =E^(l 0.00038T(rf 15)+0.0000005( 15) 2 (113) 
 
 The Clark cell was made use of by the Chamber of 
 Delegates of the Chicago International Congress of 
 Electricians in the establishment of a practical unit 
 of electromotive force. Their definition was as follow : 
 
 " The International volt is the electromotive force 
 which steadily applied to a conductor whose resistance 
 is one international ohm, will produce a current of one 
 international ampere, and which is represented suffi- 
 ciently well for practical use by |-| of the electromotive 
 force between the poles of electrodes of the voltaic cell, 
 known as Clark's cell, at a temperature of 15." 
 
 If one is in posession of two Clark cells, the tempera- 
 ture coefficients of which are well known, these may be 
 used in combination as a means of procuring a standard 
 of very small electromotive force. By maintaining the 
 two at slightly different temperatures ; and placing 
 them in circuit, back to back, they may be made to pro- 
 duce a perfectly definite and very small electromotive 
 force by their differential action, and in this combination 
 may be used for the purpose of calibration. A much 
 more convenient source of small electromotive force 
 which serves as an admirable secondary standard, is a 
 thermo-element of copper-iron, copper-german silver or 
 platinum-iridium, according to the range desired. 
 
 It is not possible, in constructing such a thermo-ele- 
 ment out of the materials ordinarily attainable, to secure 
 a standard of electromotive force which is absolute in 
 the sense of giving the same values in a case of different 
 individual elements, since the differences between the 
 various members of a series of such elements, even 
 when these are constructed as nearly as possible from 
 like materials, will be found to be considerable. Once 
 constructed, however, such thermo-elements are not 
 subject to marked fluctuations in their character. It is 
 possible, therefore, to make a thermo-element and to 
 calibrate it once for all. It may then be used as a sec- 
 ondary standard of small electromotive forces ; the 
 only further precautions being those involved in bring- 
 ing the two junctions to a known temperature difference 
 and maintaining them there. The temperatures most 
 easily maintained are, of course, those of melted ice and 
 of steam at normal pressure. The arrangement of such 
 a standard is described in a previous lecture. 
 
 100 
 
LECTURE IX. 
 
 THE USE OF THE GALVANOMETER FOR THE MEASURE- 
 MENT OF TEMPERATURE. 
 
 The types of instrument iiseful for the determination 
 of temperatures are those of maximum sensitiveness, 
 described in Lecture VI. In thermometric work the 
 galvanometer is used : 
 
 (1) In measuring changes in the resistance of a wire 
 by the method of fall of potential. 
 
 (2) In the Wheatstone bridge. 
 
 (3) In circuit with a thermo-pile or thermo-electric 
 couple. 
 
 The first method has a very wide range of useful- 
 ness. If, for example, temperatures are to be deter- 
 mined in a locality in which the mercury thermometer 
 cannot be used, a coil of pure copper wire may be pre- 
 pared, the resistance of which at a known temperature 
 is accurately determined once for all, as also its tem- 
 perature coefficient for the entire range of temperatures 
 under consideration. This coil having been placed in 
 the locality for which the temperatures are to be 
 measured is connected with the galvanometer by line 
 wires of negligible resistance. A comparison coil, the 
 resistance of which should be as nearly as is convenient 
 the same as that of the temperature coil, is placed in a 
 bath of constant temperature. This comparison coil 
 should be constructed of material having as small a 
 coefficient temperature as possible, or it may be com- 
 pensated by the methods described in Lecture VIII. 
 The temperature coil, R t , and the comparison coil, R C , 
 are placed in series with some suitable source of current 
 (B), and the circuit is permanently closed. The gal- 
 vanometer is placed alternately in shunt with the two 
 coils (see Fig. 69) and the ratio of the deflections thus 
 obtained gives the resistance of the temperature coil in 
 terms of that of the other. As has already been pointed 
 out in the lecture just referred to, this method of alter- 
 
 101 
 
nate deflections eliminates the fluctuations in the figure 
 of merit in the galvanometer and affords a means of 
 ready and accurate measurement of the variations of 
 resistance of the temperature coil and so indirectly of 
 the changes of temperature which occur in the locality 
 in which it is placed. When it is desired to integrate 
 the temperatures existing throughout a region of con- 
 siderable extent, the wire instead of being wound into 
 a coil is carried through the entire region to be studied. 
 If, for example, we desire to know the average temper- 
 ature of a standard bar, the length of which is to be 
 measured, the wire may be wound around the bar 
 longitudinally a sufficient number of turns being made 
 to give the desired resistance to the temperature coil. 
 
 FIG. 69. 
 
 This method has been found to be very satisfactory in 
 determining the coeffiicient of the expansion of such 
 bars.* 
 
 Another example of the application of this method is 
 found in the determination of the average temperature 
 of the phosphor-bronze suspension wire of the swing- 
 ing coil described in Lecture IV. This wire was 
 stretched vertically through a distance of two meters, 
 and it formed the suspension of the coil. All attempts 
 to get its temperature by means of mercury thermo 
 meters proved very unsatisfactory, but by means of a No. 
 40 copper wire carried several times the entire length 
 of the suspension tube, very good results were ob- 
 tained.f 
 
 * See the report of Joseph Le Conte upon the coefficient of expansion of a standard 
 meter made by the Societe Genevoise ; reports of the Physical Laboratory of Cornell 
 University, 1891. 
 
 + See N. H. Genung's Thesis on the Electro-Chemical Equivalent of Silver, Cor- 
 nell University Library. 
 
 102 
 
Still another illustration of the application of this 
 method of measuring temperatures is afforded by the 
 experiments of Messrs. Child, Quick and Lanphear* 
 upon the distribution of temperature along a copper 
 bar, one end of which was heated. The object of the 
 experiment was to determine the thermo-conductivity 
 of the bar. For this purpose the distribution of temper- 
 atures after the bar had reached its final condition was 
 necessary. A collar consisting of a single layer of very 
 fine insulated copper wire fitting closely around the 
 bar made it possible to measure the temperature at 
 different points throughout the entire length of the 
 latter with great accuracy. In this case, however, the 
 measurements were made by the method of the Wheat- 
 stone bridge. 
 
 The best form of apparatus for the application of this 
 method is the slide bridge. In Fig. 70, which shows the 
 connections, A B is the slide wire of platinum iridium, p. 
 
 FIG. 70. 
 
 the sliding contact, R t and R C are the temperature coil 
 and compensated resistance respectively, while r v and 
 r z , taken together with the two parts of the slide wire, 
 are the other arms of the bridge. 
 
 With this arrangement of the apparatus the temper- 
 ature may be conveniently expressed in terms of the 
 position of the sliding pointer p. 
 
 The resistance of copper is well adapted for the meas- 
 urement of temperature, since its changes, through a 
 very wide range, are nearly proportional to the temper- 
 ature. It is, indeed, only above 200 and below io<. 
 that the change in the coefficient is such as to introduce 
 grave errors. 
 
 The coefficients of different wires vary somewhat, 
 however, in absolute value, so that the specimen of 
 which the temperature coil is to be made should always 
 
 * Physical Review, Vol. II. 
 
 103 
 
be calibrated. Many samples of modern commercial 
 copper show a coefficient of resistance greatly in excess 
 of Matthiessen's value for the pure metal. Kennelly 
 and Fessenden*, for example, in 1893 found for a copper 
 wire 0.004065, with variations from that value at 27.8 
 of 0.000058 and at 255 of -f- 0.000005. 
 
 Values lying between .0041 and .0042 are frequently 
 observed in modern practice. Thus Dewar and Flem- 
 
 ing found for low temperatures 0.00410 ; Quick and 
 Lanphear, between 38.6 C. and -j- i.5C. obtained 
 0.004147 ; Cailletet and Bouty's value was higher than 
 any of these, viz.: 0.00423. 
 
 The very nearly constant value of the resistance co- 
 efficient is shown graphically in Figs. 71 and 72, 
 
 ^Physical Review, vol. x. 
 
 104 
 
which are plotted from the results of Kennelly and 
 Fessenden and of Quick and Lanphear respectively. 
 
 These curves are plotted to different scales and they 
 apply to different specimens of copper, not of the same 
 quality as to the coefficient of resistance. The meth- 
 ods of calibration also were distinct. Both are straight 
 lines, however, indicating an unvarying coefficient 
 through wide range of temperatures, including 
 the important interval below 40 C., which 
 
 cannot be reached with mercury thermometers. 
 
 The calibration of a temperature coil, should always 
 be made under conditions as nearly as possible, identical 
 with those under which it is to be used. Otherwise the 
 temperature lag, of the coil with reference to the body 
 the temperature of which is to be determined, (or vice 
 versa) will introduce errors which will always be ap- 
 preciable excepting when temperatures have become 
 strictly stationary, and which may sometimes rise to un- 
 suspected size. 
 
 105 
 
The nature of this error in a typical case, that in 
 which the temperature of a copper bar, was to be meas- 
 ured by means of a collar of fine wire surrounding it, is 
 indicated in Fig. 73 (from determinations by the observ- 
 ers just cited). In this diagram abscissas are tempera- 
 tures of the bar, as indicated by a thermometer, the 
 bulb of which was immersed in a mercury capsule with- 
 in the body of the metal, while ordinates are readings 
 on the slide bridge. The bridge gives relative readings 
 
 of the temperature of the collar. The crosses and cir- 
 cles show the temperatures of the bar at which the 
 bridge readings were the same, for rising and falling 
 temperatures respectively. The curves of heating and 
 cooling to which these apply were very nearly identical 
 and the temperature difference at any point, between 
 an observation and the median line, gives the lag (posi- 
 tive or negative). It will be noted that in this case the 
 error of assuming the temperature of the collar to be 
 that of the bar would have been about one degree. 
 
 106 
 
For very high temperatures, copper is not available 
 for electrical thermometry and consequently many 
 attempts have been made to substitute platinum for that 
 metal. The law of resistance for platinum, must how- 
 ever be determined for each specimen and this calibra- 
 tion is a matter of extreme difficulty. 
 
 A comparison of the various formulae proposed by 
 Matthiessen, Siemens, and Benoit, for determining tem- 
 peratures from the resistance of platinum was made 
 some years ago by the writer.* His curves showing the 
 
 2400* 
 
 " ^a" i ' *"* ^*/ /&* ' 
 ^^ ' / <k* ' 7 /'2* 
 
 2200' 
 2000' 
 1800 
 
 : l^^ ? " 
 
 1600 
 
 \kj/ / 
 
 
 i i> / * 
 
 1400 
 
 !!///'/ 
 
 1200* 
 1000 C 
 SOO 3 
 
 I III// 
 ' i .''.'/ 
 
 ' IW 
 
 600' 
 
 /// 
 /"' 
 
 
 
 400 
 
 
 200 e 
 
 
 ] 
 
 \ 2345578 
 
 
 FIG. 74. 
 
 divergent character of the results which would be ob- 
 tained by the application of their formulae, are given 
 in Fig. 74. 
 
 Another method, applicable alike to very high and to 
 very low temperatures, is that of the thermo-element of 
 platinum platinum-iridium. The same caution must 
 be observed, however, with reference to commercial 
 specimens as in the method previously described. 
 
 * Am. Journal of Science, vol. 22, p 363. 
 
 107 
 
Platinum wire purchased from leading dealers in this 
 country, and supposed to be pure, gave in the hands of 
 the writer curves of E. M. F. and temperature of the 
 character shown in Fig. 75, one being concave and the 
 other convex to the base line. These wires were com- 
 bined with the same quality of platinum-iridium in the 
 construction of the thermo elements. 
 
 It will be seen that a deflection which corresponds to 
 600 C. in the one case would be reached, with the other 
 thermo element at 900 C. 
 
 Barus, in his exhaustive research upon the measure- 
 ment of high temperatures, found similar peculiarities 
 
 400 600 
 
 FIG. 75. 
 
 in commercial wires, but when he used an element 
 composed of standard materials he obtained a curve 
 which is very nearly straight through a wide range of 
 temperatures. 
 
 Excellent results have also been reported with this 
 couple at very low temperatures, but it should be noted 
 that some commercial samples give reversal at a tem- 
 perature slightly above o. 
 
 The methods thus far described can be pursued with 
 galvanometers of medium delicacy ; when, however, 
 we come to the exceedingly small temperature differ- 
 
 108 
 

 ences with which the student of radiant heat has to 
 deal, instruments of the highest sensitiveness are es- 
 sential. 
 
 Nearly all measurements in this domain are of a rel- 
 ative character ; Knut Angstrom, however, (1893) has 
 described a bolometric method by means of which ab- 
 solute determinations may be made, and the results 
 expressed in gram-calories per second per c m 9 . 
 
 The principle of Angstrom's method, as described in 
 his paper,* is briefly as follows : " Given two thin strips 
 of metal, A and B (Fig. 76), which are as nearly as pos- 
 sible identical. The sides of these, which are exposed 
 to the source of heat, are blackened, and the strips are 
 arranged in such a way that it is possible to determine 
 accurately when they are of the same temperature. 
 These strips are so placed that a current of any desired 
 strength can be sent through them. If one of them, 
 for example, A, is exposed to the source of heat, while 
 
 FIG. 76. 
 
 B is protected by a screen, w r e may restore the balance 
 of temperature which has been disturbed by the ab- 
 sorption of heat on the part of A, by sending a current 
 of proper intensity through B. When the temperatures 
 are the same, then the amounts of energy which A and 
 B have received are equal to one another. Let / and b 
 be the length and width of the strips, r the resistance 
 of the same, and / the current. Then since the heat 
 absorbed by A is the equivalent of that produced by 
 the electric current in b, we may write 
 
 where q is the radiant energy received by a unit of sur- 
 face. In order to counteract the inequality of the 
 strips, they are interchangeable, B being illuminated 
 and the current sent through A. 
 
 *Angstrora ; Trans. Royal Soc. of Sc. Upsala, 1893 ; also Physic. I Review, vol. i, 
 P- 365- 
 
 I0 9 
 
" In the practical application of this principle of com- 
 pensation, one may follow various methods. The equal- 
 ity of temperature may be determined in a variety of 
 ways. If thermo-elements are used for this purpose, 
 one needs a sensitive galvanoscope, and the measure- 
 ment of the current can be made upon an instrument of 
 ordinary delicacy. It is possible, however, to carry on 
 the investigation without any difficulty, using only one 
 galvanometer, a plan which I pursued in the case of the 
 first apparatus which I constructed. Fig. 77 gives a 
 diagram of the connections. 
 
 "The metallic strips, A and B, cut simultaneously from 
 two thin sheets of platinum laid one upon another, are 
 0.154 cm. wide and 1.80 cm. in length. They are black- 
 ened in the usual way upon the side exposed to radia- 
 tion, and are mounted side by side in a frame of ebonite. 
 
 FIG. 77- 
 
 This frame is inserted in a tube. Upon the back of the 
 strips are laid two exceedingly thin leaves of mica, up- 
 on which very minute junctions of copper and German 
 silver, are inserted. To hold the mica and the therm o- 
 j unctions together, I used marine glue in as thin a layer 
 as possible. 
 
 " One of the strips is subjected to radiation ; the cir- 
 cuit is then closed through the other one ; the thermo- 
 element is brought into circuit with the galvanometer 
 by means of the commutator ; the sliding contact is ad- 
 justed until the galvanometer stands at zero ; the com- 
 mutator is then reversed, and the strength of the cur- 
 rent used in heating the strip is determined. The 
 switch is then reversed, the shutter is placed in front of 
 the other strip, and the setting and current measure- 
 
ment are repeated. There are two other ways in which 
 one can use this apparatus for the measurement of rad- 
 iation, viz : 
 
 First Variation of the Method, One of the strips, for 
 example, A, is exposed to radiation, while the other is 
 screened. One notes the deflection of the galvanom- 
 eter, which becomes constant in about fifteen seconds. 
 A is then also screened, and by means of the current is 
 brought to the same temperature to which radiation 
 had previously brought it. The strength of the current 
 producing this rise of temperature is then measured. 
 The advantage of this arrangement is that the same 
 strip is warmed by means of radiation and then through 
 the agency of the current, so that it is not necessary to 
 be so painstaking as regards the identity of the two 
 strips. 
 
 Second Variation of the Method. One of the strips, for 
 example, A, is exposed to radiation, the other, B, is 
 screened ; the deflection of the galvanometer is observed 
 after the thermo-current has become constant. The 
 quantity of heat which A has received is calculated from 
 Newton's law of cooling. 
 
 where 6 is the difference of temperature indicated by 
 the galvanometer, and k is the constant of cooling of 
 the strip. If we now send a current through A, without 
 making any change of conditions, the difference of tem- 
 perature will become greater still. This will be indic- 
 ated by the resulting deflection 0^ The strength of the 
 current producing this additional heating effect may be 
 measured in the manner already described." 
 
 Galvanometer, suitable for investigations are easily 
 constructed, following the general principles of design 
 given in Lecture VI. 
 
 In undertaking any extended bolometric work, or 
 other research of great delicacy, the galvanometer 
 should be especially constructed for the particular pur- 
 pose which it must serve. The question of the period 
 of oscillation is sometimes an important one ; on account 
 of local magnetic disturbances. The proper adapt- 
 ation of the resistance to the circuit in which "the instru- 
 ment is to be used is of especial significance. 
 
 Very often, the experimenter with the bolom- 
 eter is compelled to use his galvanometer under condi- 
 tions of the highest attainable delicacy. Then, if he is 
 to obtain readings which possess value, his patience and 
 skill will be taxed to the utmost. For the difficulties 
 which arise in such researches no general prescription 
 can be written. The vagaries of the needle under the 
 
 in 
 
combined influences of diurnal magnetic drift, local mag- 
 netic disturbance, thermo-electric differences, and the 
 obscure thermal fluctuations to which the bolometer 
 and its accessories are subject must be studied as they 
 arise, and overcome. These causes can be detected, 
 and, after due experience, ingenuity and tireless pat- 
 ience will bring their effects under control ; but more 
 remote sources of disturbance are perpetually at work 
 against the bolometrist : A moderate gale of wind, an 
 auroral display, almost too faint to be visible, even a 
 storm in the solar atmosphere, will drive him from his 
 seat at the reading telescope in despair. 
 
 These are difficulties to be surmounted only by him 
 who can wait ; he it is, alone, who may enter the realms 
 of research which lie along the very boundary line of 
 human attainment ; in his hands alone do we see the 
 highest performance of that remarkable instrument, the 
 galvanometer. 
 
 FWIS. 
 
 
 ME 
 
 ERSITT 
 
 112 
 
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