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ELEMENTS OF OONSTEUOTION 
 
 ELECTEO-MAGNETS. 
 
ELEMENTS OF CONSTRUCTION 
 
 ELECTEO-MAG-NETS. 
 
 By count TH. DU MONCEL, 
 
 UEUBRE SE L'INSTITUT DE FRANCE, 
 
 TRANSLATED FROM TEE FRENCH 
 
 BY 
 
 C. J. WHAETON. 
 
 LONDON: 
 E. & F. N. SPON, 16, CHAEING CEOSS, 
 
 NEW YOEK: 
 44, MUERAT STEEET. 
 
 1883. 
 3 
 
PEEFACE. 
 
 In presenting the accompanying translation of the 
 'Determination des Elements de Construction des 
 Electro-Aimants ' by Count Th. du Moncel I need 
 offer no apology, as the researches of this eminent 
 French scientist are well known both in England 
 and America, but to facilitate the studies of those 
 not understanding French I have undertaken the 
 translation which I now put forward as a faithful 
 copy of the original, but in an English dress. The 
 electrical units are almost throughout based on the 
 French system of measurements of a Daniell as the 
 unit of E.M.F. and the metre of telegraph wire as 
 the unit of resistance, but to calculate these values 
 according to the mode now usually obtaining here, 
 viz. the volt and the ohm, the results must be 
 divided by 1 • 079 and 100 respectively. 
 
 With respect to the tables of values, I should 
 mention I have omitted some given by the author 
 containing information as to the weight, price, &e., 
 of copper wires supplied by a Paris maker. 
 
vi "Preface. 
 
 In conclusion, I am requested by Count Th. du 
 Moncel to explain that the experiments undertaken 
 by him in order to determine the ratios of magnetic 
 moment for the standard electro-magnet have not 
 been numerous and varied enough for them to be 
 taken as presenting great exactitude. Further ex- 
 periments in this direction would therefore be of 
 advantage, and it must also be borne in mind that 
 owing to the very varying nature of the iron em- 
 ployed in electro-magnets we can never be certain 
 that the real results correspond with the theoretical. 
 What, however, may be taken as certain is, that 
 electro-magnets constructed and used in accordance 
 with his formulas are employed in the most advan- 
 tageous manner possible. 
 
 c. J. whaeto:n^. 
 
 8 AND 9, HOLBOBN VlADUCT, 
 
 January 1883, 
 
CONTENTS. 
 
 PAGE 
 
 Introduction 1 
 
 CHAPTEE I. 
 
 FoBMULAS roE Eleoteo-Magnets i 
 
 CHAPTEE II. 
 
 Conditions of Maximum: fob Electko-Magnbts on a Simplb 
 
 ClECUIT. 
 
 1. Conditions of maximum in reference to the resistance 
 
 of tlie coils 9 
 
 2. Conditions of maximum relating to the proportion 
 
 which should exist between the depth of the mag- 
 netic coil and the diameter of the iron cores .. . . 12 
 Consequences of the preceding laws 13 
 
 3. Conditions of maximum as regards the length of the 
 
 iron core 15 
 
 CHAPTEE III. 
 
 Conditions of Maximum on Compound Ciecuits .. .. 19 
 
 CHAPTEE IV. 
 
 Application op the Laws of Maximum to the Consteuc- 
 
 TiON OF Electbo-Magnets 25 
 
viii Contents. 
 
 CHAPTER V. 
 
 FAGE 
 
 NtJMEBioAL Examples op the Application op the Pbb- 
 
 OEUINa FOBMCLAS • 37 
 
 CHAPTEE YI. 
 
 Expbbimental Vbeipioation op the Laws op Eleotbo- 
 
 Haonets .. ., ^ 47 
 
 CHAPTER VII. 
 
 Eppeots op a Mobb oe Less CSomplete Maghetio Sattiba- 
 
 TION 56 
 
 CHAPTEE VIII. 
 
 CONDmONS FOE THE GoOD OONSTBCCTIOir OP ELBOTBO- 
 
 Maohbts. 
 
 L Conditions in reference to exterior actions 67 
 
 II. Conditions in reference to the form and application 
 
 of the armature 69 
 
 III. Conditions in reference to the magnetic mass .. .. 72 
 
 CHAPTER IX. 
 
 On the Best GBoupnra op the Cells op a Battebt .. 75 
 
 TABLES 84 
 
ELEMENTS OF CONSTEUCTION 
 
 ELECTEO -MAGNETS. 
 
 INTRODUCTION. 
 
 It is generally complained, and not without reason, 
 that the manner in which the question of electro- 
 magnets is ordinarily treated by scientists is so un- 
 decided, and so little practical in the deductions 
 which are giyen, that it is impossible for a constructor 
 or an inventor to profit by them. It is certain that 
 mathematicians look upon the questions involved 
 from too great a height to occupy themselves with 
 the details of application ; and it must also be ad- 
 mitted that magnetic theories are so vague that it is 
 difficult to translate into mathematical symbols many 
 of the laws which have been pointed out, and of 
 which some are still contested by punctilious spirits. 
 We, who have made many experiments with electro- 
 magnets, are less sceptical, for although we may not 
 have been able to verify with extreme exactness the 
 laws propounded by Lenz, Jacobi, Dub, and MuUer, 
 we have found results approaching near enough to 
 
 B 
 
2 Elements of Cmstruction for Electro-Magnets, 
 
 the truth to be accepted as data for the construction 
 of electro-magnets. The laws of Ohm for electric 
 currents are in the same predicament, for it is diffi- 
 cult to bring into formulas which represent them a 
 crowd of secondary influences which derange more 
 or less the effects enunciated. But these laws are 
 useful guides, and serve as premises for correct 
 deductions, and this is the essential point. 
 
 For scientific data to be of any real use in their 
 application they must be cleared of all the hypo- 
 theses of high science, and of terms which many 
 electricians cannot understand; and further, we 
 must start from experiences obtained under ordinary 
 conditions of application. It is certain that if, to 
 appreciate a magnetic force, we are obliged to base 
 our calculations, through complicated formulas, on 
 the oscillations of a magnetised needle or on the 
 currents induced by this force, constructors would 
 say that this means nothing to their mind, and that, 
 in fact, they know nothing of magnetic force but 
 that which corresponds with some weight lifted or 
 supported ; in short, their desire being to have the 
 greatest possible power under the given conditions, 
 the rest is of no consequence to them. It is, then, 
 under these conditions that we must state the ques- 
 tion to obtain deductions applicable in practice. 
 Now, I have always admitted that the known laws of 
 electro-magnets were sufficient to satisfy constructors 
 in this respect, and it is this which has led me to 
 publish my various pamphlets on the best methods 
 of constructing electro-magnets. Wishing to be 
 
Irvtrodueticm. 3 
 
 certain of my deductions, I have been obliged to 
 make numerous experiments to verify my formulas, 
 and it is only after the most minute experimental 
 researches that I have laid down those given in the 
 present work. 
 
 s 2 
 
4 Elements of Construciion for Electro-Magnets. 
 
 CHAPTEE I. 
 
 FORMULAS FOB ELECTKO-MAGNETS. 
 
 To establish my formulas I have started from the 
 different elements entering into the construction of 
 an electro-magnet — viz. the dimensions of the mag- 
 netic core, the size of the wire in the coils, its length, 
 the number of turns, and its depth, as follows : — 
 
 a. The depth of the magnetising coUs ; 
 
 h. The total length of the bobbins or the two arms 
 of the electro-magnets ; 
 
 e. The bore of the bobbins, or what we may 
 suppose to be identical, the diameter of the magnetic 
 core; 
 
 g. The diameter of the wire of the coils, including 
 the insulating covering ; 
 
 A. The attractive force of the magnetic system ; 
 
 E. The electro-motive force of the battery em- 
 ployed ; 
 
 F. The magnetic moment of the electro-magnet ; 
 H. The total length of the wire in the coUs ; 
 
 I. The intensity of the current in the total 
 circuit ; 
 
 t. The number of turns of the wire ; 
 
 E. The resistance of the exterior circuit, incliiding 
 that of the battery. 
 
Formulas for Electro-Magnets. 5 
 
 It is easy to understand that we can at once re- 
 present the number of turns in each layer bv - , and 
 
 / 9 
 as there are as many layers as g is contained in the 
 depth a, we shall have to represent the total number 
 of turns t, — 
 
 /,, , h a ah 
 (1) t = — x — =—^- 
 9 9 9 
 
 If we want to arrive at the length of each of these 
 turns in the first or in the last layer, we shall find 
 
 that for the first we have 2 tt —9—, and for the last 
 
 2 TT , . — -, and consequently the total length of 
 these two layers will be 
 
 l2^^±^andl2 7r£±l^. 
 9^9 2 
 
 The intermediate layers constitute, with these two, 
 the terms of an arithmetical progression, of which 
 the foregoing expressions are the extremes, and of 
 
 which the number of terms is represented by — ; the 
 
 total length of the wire, or the sum of the lengths of 
 these different layers, will be represented by the 
 formula — 
 
 (2) H = - 27r(c + g' + c + 2a-^) a _ 
 .^ 4 g 
 
 ■7rha(a-\- c) 
 
 —9" 
 
6 Elements of Construction for Electro-Magnets. 
 
 We thus obtain, then, the values of t and H in 
 terms of the different elements entering into the 
 construction of an electro-magnet. These formulas 
 are also useful in themselyes, for they enable us to 
 calculate the length and number of turns in the 
 bobbins of any electro-magnet when we have measured 
 the thickness of the wire and the depth of the coils, 
 which are easy to obtain, since it is only necessary to 
 measure the exact length of the bobbin from one 
 edge to the other, to count the number of wires in 
 this space, divide this length by the number, and 
 take the difference in diameter of the bobbin covered 
 with wire and the iron core. We can also deduce 
 from these formulas the values of a and g, and have 
 different expressions for the values of t and H, 
 according to any particular case. 
 
 To obtain the expression of the electro-magnetic 
 force, I start from the laws of Jacobi, Dub, and 
 Muller, who give for the moment F of an electro- 
 magnet, the product of the intensity I of the current 
 traversing it by the number of turns, and as the 
 value of the attractive force A, the square of this 
 product, which leads to the formulas — 
 
 F = .^andA= ^' ^ 
 
 E-f-H """"- (E_i_H)''* 
 
 And applying to t and H the values already 
 arrived at we obtain the equations — 
 
 F = ^ , , ^,^^ . , and 
 
 A = 
 
 ^g^ + ""ba (a -f c) 
 W a? V 
 
 {'& g'' -\- itI a {a ^ c)Y 
 
Formvlasfor Eledro-Magnets. 7 
 
 which allow us to deduce different conditions of 
 maximum according as we vary the quantities a, b, c, 
 and ff, and which relate : Firstly, to the resistance 
 which we give to the magnetising coils ; Secondly, 
 to the proportion which exists between these coils 
 and the diameter of the magnetic core ; Thirdly, to 
 the dimensions themselves of the electro-magnet. 
 
 It is, however, important to remember that, to 
 obtain the reduced length of the different parts 
 which compose the circuit, and which are repre- 
 sented by very different conductors, it is necessary to 
 transform the resistance E into terms of that of the 
 coils; and for this we shall begin by considering 
 that, as p represents the diameter of the wire, in- 
 cluding the insulating covering, we must divide g by 
 a coefficient / to obtain the diameter of the wire 
 itself, which is all that ought to be taken into con- 
 sideration. This coefficient may be represented 
 practically by 1"6 for the very fine wires, or by 1'4 
 for thicker ones. The diameter of the conducting 
 
 wire wiU then be ^ and if we take q to represent the 
 
 relative condiictibility of K and H, including the 
 constant for unit section, which is "016 mm., we 
 shall have for the reduced value of R the quantity — 
 
 r 
 
 Since, by varying the size of the wire ^ for a con- 
 stant depth a of the coils, we vary not only its 
 resistance but also its length — two properties which 
 
8 Elements of Oonstrueiion for Electro-Magnets. 
 
 vary in the same proportion — the quantity H taken 
 as representing the resistance of the coil instead of 
 being in inverse proportion to g^, will be in inverse 
 
 proportion to ^* ; but the quantity 
 
 r 
 
 which is 
 
 not in the same case as H, because it is not obliged 
 to fill a fixed space, will remain in inverse proportion 
 to p'^ ; so that the denominators of the expressions 
 F and A becoming 
 
 and 
 
 qU ff^ + 'Trig (a + c)f' 
 /V 
 
 hJR^ 
 
 + 7r'ba(a + c)f^ 
 the expressions themselves will be 
 
 (3) 
 
 qR^^+r-Trbaia + c)' 
 
( 9 ) 
 
 CHAPTEE II. 
 
 CONDITIONS OF MAXIMUM FOR ELECTEO-MAGNETS 
 ON A SIMPLE CIECUIT. 
 
 1st. Conditions of maximum in reference to the resis- 
 tance of the coils. — In the preceding formulas the 
 values of A and F may be disputed from several 
 points. We may ask what would be the conditions 
 of maximum, admitting that, having an electro- 
 magnet already constructed with fixed dimensions, 
 we wish to employ it most advantageously with an 
 exterior circuit whose resistance is E; or we may 
 ask — not being limited in respect to the depth of the 
 coils — to what extent it would be advantageous to 
 wind on the bobbins a wire of a given size, to produce, 
 with the given resistance E, the best results ? In 
 the first instance, the variation is in the diameter 
 gf of the wire, in the second in that a of the 
 bobbin. 
 
 If we consider that in the preceding formula the 
 diameter of the conducting wire of the coil is not g 
 
 but jj in which /is a constant, we find that the cal- 
 culation is not so simple as would be imagined at 
 first sight ; and therefore those who first went into 
 this question considered the value of / could be 
 
10 Elements of Construction for Electro-Magneis. 
 
 neglected, and reasoned on the hypothesis that g re- 
 presented the thickness of the wire itself. Even in 
 the preceding formulas, taking into account the value 
 of f, this value cannot be considered constant if we 
 take g as variable, and it will at once be seen that 
 the conclusions will be different. But in considering 
 the question under its most simple conditions, we 
 are able to show with ease, by bringing in the results 
 of the two preceding formulas, that the conditions 
 of maximum will be found in the equation — 
 
 jR^^ 7ria(a +e) 
 
 that is to say, R = H, which means that, for electro- 
 magnets of similar dimensions having bobbins of the 
 same diameter, the most advantageous size of wire 
 for the coils is that which will render their resistance 
 equal to that of the exterior circuit R. 
 
 If we take into account the thickness of the insu- 
 lating material, tlie calculation will show that the 
 best coil will be that of which the resistance will be 
 to the resistance of the exterior circuit, as the 
 diameter of the wire itself is to that of the same 
 wire, incluiling its insulating covering. 
 
 If we vary the depth a of the coils, supposing the 
 action of the turns to be about equal, which we may 
 admit under the ordinary conditions of electro- 
 magnets, taking into account the differences of resis- 
 tance wrought by their greater or less distance from 
 the magnetic core, the conditions of maximum corre- 
 sponding with the cancelment of the results of the 
 
Besisianee of Coils. 11 
 
 preceding expressions as far as concerns a, will show 
 
 that R should be equal to — ^, that is to say, to 
 
 the length H of the wire of the coils divided by 
 
 , or, what comes to the same thing, that H 
 
 a 
 
 should be equal to R (l + - ). Translated into 
 
 ordinary language, this deduction means that: — 
 Between several bobbins wound with the same wire 
 but having a different number of layers, that which 
 will give the best results with a given circuit of 
 resistance will be that of which the resistance is to 
 the exterior resistance, as the depth of the coils, 
 increased by the diameter of the magnetic core, is to 
 the depth of the coils alone. 
 
 As usually in electric applications we start with a 
 given diameter for the magnetic core, and as for 
 other conditions of maximum of which we will treat 
 presently, the depth of layers of wire should be 
 equal to the diameter of this core ; as, on the other 
 hand, the depth of the insulating layers varies, and 
 is undetermined in the researches which we have to 
 make, it is the first of these conditions which ought 
 to be considered in the construction of electro- 
 magnets. But for one who wishes to ascertain what 
 is the resistance of the circuit on which he can most 
 usefully employ a given electro-magnet, it is the 
 second conditions of maximum with which he is con- 
 cerned, and these conditions show that this exterior 
 resistance should be less than half that of the 
 
12 Elements of Constnidion for Electro-Magnets. 
 
 electro-magnet, if the depth of the coils is equal to 
 the diameter of the magnetic core, which ought to be 
 the case. 
 
 2nd. Conditions of maximum relcding to the pro- 
 portion which should exist letween the depth of the 
 magnetic coil, and the diameter of the iron cores. — 
 A second very important point in the construction of 
 an electro-magnet is to know what ought to be the 
 depth of the magnetic bobbins, to make the best use 
 of them. It will be understood that the force of 
 electro-magnets increasing with the diameter of the 
 magnetic cores, and the resistance of the turns of the 
 coil becoming greater in consequence of this increase 
 of diameter, there must be a limit where the advan- 
 tages obtained from the increasing of the diameter 
 are counterbalanced by the increase of resistance of 
 the coil; and the question is to determine this 
 limit. Calculation furnishes the means for resolving 
 this question. 
 
 In the equations (3) expressing the values F and A, 
 the magnetic moment of the electro-magnet and its 
 attractive force, let us vary the quantity c which 
 represents the diameter of the magnetic core, and 
 establish between this quantity and the depth a of 
 the coil an algebraic relation, which is easy, for, sup- 
 posing the coil to be wound on the core itself, its 
 interior diameter is represented by c. We could 
 then, by fulfillmg the conditions of maximum with 
 respect to the resistance of the exterior circuit, obtain 
 an expression susceptible of maximum such that the 
 relation of E to H shall be either of those given in 
 
Proporiion hdween Depth of Coils and Core. 13 
 
 tte first two deductions we have drawn. In repre- 
 senting by \ the coefficient by which the length of 
 the coil must be multiplied to place the total circuit 
 in either of these conditions of maximum, and sup- 
 posing the depth a of the coils, and therefore the num- 
 ber t of the turns, to be invariable, the attractive 
 force A and the magnetic moment F of the electro- 
 magnet are expressed, according to the law of Muller 
 relating to the increase of force with the diameter of 
 the magnetic cores, by 
 
 ^^ ~X-irba{a + c)' and 
 
 [X TT 6 a (a -f- ^)] 
 
 expressions which are susceptible of maximum in 
 reference to c ; but then the quantities R and H are 
 supposed to vary at the same time, in proportion as 
 the coil is lengthened by reason of the increase of 
 the magnetic core. If we take the results of the 
 preceding expressions as regards e considered as 
 variable and reduce them to zero, we find that the 
 maximum conditions answer to a = e, that is to say, 
 to the equality of the depth of the coils and of the 
 diameter of the iron core. This is what I have 
 demonstrated by experiment, as we shall see at the 
 end of this work. 
 
 Consequences of the preceding Laws. — The advan- 
 tages of the laws which we have propounded above 
 are easy of comprehension, for they assist in simple 
 calculations for the construction of electro-magnets. 
 
14 Elements of Construction for Electro-Magnets. 
 
 By this means, in fact, the expression giving the 
 length of the coils becomes for the two conditions of 
 
 maximum which we have considered 5 — , and if 
 
 we take the length h of the electro-magnet in terms 
 of the diameter c, multiplying this by the coefficient 
 m, which practice and the considerations we are 
 about to discuss fix at 12 for the two arms of the 
 electro-magnet, this expression becomes — 
 
 ,_. 2 TTC^ m 75-4 X e' 
 (5) __,_or — ^, 
 
 formulas in which we have only to consider the two 
 quantities e and g, which can be determined accord- 
 ing to the different conditions in which we may find 
 ourselves, by means of the proportion which we will 
 consider presently. On the other hand, we have for 
 the number of turns t, the equation — 
 
 (6) * = i^. 
 
 Where the electro-magnet fulfils the conditions 
 of maximum, E = H, a = c, Z» = c m, as gr is not 
 determined, E must be reduced to terms of g, and 
 
 consequently the eqtiation ^ — > which repre- 
 sents H, and which must be equal to E, leads to the 
 gE^" _ 2 7re ^ m 
 
 r ~ / 
 
 . ., c^ 2irm 
 9 =f 
 
 equation ^g = - — -^ — whence we have 
 
 P2 
 
 E 
 
Length of Iron Gore. 15 
 
 Dince IS a constant composed of known 
 
 quantities and equal to 0-00020106 we have as 
 final*— 
 
 p) ^W5 
 
 •00020106. 
 
 3rd. Conditions of maximum as regards the length 
 of the iron core. — From the deductions which we 
 have just considered, it will easily be understood 
 how important it is to calculate the lengths of the 
 magnetic cores in terms of their diameters ; but can 
 a definite length be fixed by calculation ? This is 
 the question which remains to us to elucidate. It 
 is certain that if we consider purely and simply the 
 length of the magnetic core 6, no maximum con- 
 ditions can be deduced from the formulas giving the 
 values of F and A ; for although, according to the 
 laws of MuUer, the forces increase as the square 
 root of the length of the magnetic cores, these 
 formulas are not susceptible of a maximum when 
 we vary h. But if we render this quantity 6 in 
 terms of the diameter e the attractive force becomes 
 
 then proportional to e x V c m or to c|, and we then 
 
 * In this expression q represents the relative oonduetibility of E 
 
 and H, K being expressed in metres of telegraph wire. As this 
 
 wire is of iron, and that of the coils of copper, this ratio is about 
 
 6. And as the diameter of this telegraph wire is 4 millimetres, 
 
 the square of its section representing unity is •000016 [metre; 
 
 6 2 5r m 
 
 thus q = „„„^,„ = 375000. Therefore the constant = 
 
 ■000016 J 
 
 ^x^-^^^«^^^ = .00020106. 
 375000 
 
16 Elements of Construction for Electro-Magnets. 
 
 obtain for the attractive force A, all the other 
 conditions being taken into consideration, the ex- 
 pression — 
 
 (8) A_ [-R^2_(.2,rc^mP' 
 
 and here E is represented by a given length of 
 the wire of the coil. 
 
 Now, according to a simple reasoning, we may 
 believe that there is, in that case, a limit to the 
 value of m, for the magnetising coil having a given 
 resistance in relation to that of the exterior circuit, 
 and that coil having to possess a thickness equal 
 to the diameter of the magnetic core, this resistance 
 may be more or less completely utilised, according 
 to the relation existing between the diameter and 
 the length of the core upon which the coil is 
 wound. As the electro-magnetic force increases 
 with the diameter of that core, it is best, up to 
 a certain point, to take it as great as possible ; but 
 on the other hand, as the number of turns, for a 
 given length of coil, diminishes with that greatness, 
 it may be preferable not to further increase the 
 diameter, but to lengthen the core, and this ad- 
 vantage, added to the greater number of turns thus 
 obtained, may counterbalance advantageously, in 
 certain conditions, the loss of force resulting from 
 the lessening of the diameter. However, theoretical 
 formulas do not precisely define that limit, because 
 the maximum conditions which may be deduced from 
 
Length of Iron Core. 17 
 
 the preceding formula, taking e as variable, answer to 
 the equation — 
 
 2 TT e'm 
 
 9' 
 
 = HE; 
 
 have m = 11 „ , , and we see that the value of m 
 
 that is to say, that according to the hypothesis which 
 has been admitted, we can increase the dimensions of 
 the magnetic core until the resistance of the mag- 
 netising coil represents eleven times the resistance 
 of the exterior circuit. Under these conditions we 
 Bff' 
 
 then becomes equal to eleven times the proportion 
 
 between B and the resistance of a coil having its 
 
 magnetic core equal in length to the diameter of the 
 
 electro-magnet ; for in this case this coil is expressed 
 
 2 Tt <? 
 by — J— ; now, as the proportion of the resistance E 
 
 to that of the coil ought to be always 1 — for these 
 two resistances must always be equal to meet the 
 conditions of maximum previously laid down — we 
 get the preceding formula m = 11. Numerous 
 calculations that I have given in my ' Eecherches 
 exp&imentales sur les maxima electro-magnetiques ' 
 show the exactitude of these deductions.* 
 
 * Suppose, for example, introduced in a circuit of 64 metres of 
 telegraph wire, three bar electro-magnets with diameters of 8, 7, 
 and 6 millimetres, with lengths of eleven times these diameters, 
 namely, 8'8, 7'7, and 6'6 centimetres. If we compare the results 
 obtained, using wire of • 6 millimetre, we shall find the first mag- 
 net A will have a eoil of 99 metres in length and 1056 metres in re- 
 sistance ; the second magnet B will have a coil of 66 metres in length, 
 
 
 
18 Elements of Construction for Electro-Magnets. 
 
 As the ends of the bobbins must have a tangible 
 thickness, and therefore the poles protrude a little 
 beyond the coils, and there must be a space between 
 the bottom of the ends and the breech of the 
 magnetic core to allow of the egress of the two ends 
 of the wire of the coil, the coefficient 11 must be a 
 little increased, and we have found by experiments 
 that it should be made 12. 
 
 and 704 in resistance ; and the third magnet C a Coil of 41 • 36 metres, 
 and 441 metres in resistance. The number of turns deduced from 
 
 11 c* 
 the formula —r- will be 1955 for A, 1497 for B, 1100 for 0, and the 
 
 g' 
 
 a/ cube of the diameters being -000715, -000585, and -000464, we 
 have for the forces of the three electro-magnets, - 247868, - 252891 
 and -250307. It will be seen from this that the electro-magnet B 
 has the advantage, and has a resistance of 704 metres, or eleven 
 times that of the exterior circuit. But this is no longer the case if 
 we suppose E = 704 metres: the relative forces are then -10037, 
 -07524, and -04871, and it is the large magnet which has the advan- 
 tage because it has a resistance great enough, compared with E, to 
 prevent the latter exercising sufficient influence to greatly modify 
 
 the value of (E 4- H)' which divides the product f c^' 
 
( 19 ) 
 
 CHAPTER III. 
 
 CONDITIONS OF MAXIMUM ON COMPOUND CIECUITS. 
 
 The deductions that we have drawn in the preceding 
 chapter suppose the electric current to be constant 
 and uniform, the reactions of extra currents not to 
 exist, the iron of the electro-magnet to be in that 
 state of magnetic saturation necessary for the laws 
 of Dub and MuUer to be applicable, and finally, the 
 exterior circuit to be perfectly insulated. Except 
 under these conditions the formulas will not exactly 
 apply, and calculations prove that the resistance of 
 the coil must be considerably reduced, as is indis- 
 putably shown by the experiments of Hughes, and 
 confirmed by those of Lenoir made between Paris 
 and Bordeaux ; in which the electro-magnet experi- 
 mented on underwent very rapid magnetisation and 
 demagnetisation . 
 
 With such a diversity of elements it is impossible 
 to fix, for telegraphic electro-magnets, a formula 
 which will give exactly the conditions of maximum 
 for the resistance of magnetic coils ; but on the 
 basis of the preceding formulas, and admitting, as 
 experience has proved, that the electro-magnetic 
 attractions increase in a much more rapid proportion 
 than that of the square of the intensity of the current, 
 
 c 2 
 
20 Elements of Condruction for Electro-Magnets. 
 
 when the state of magnetic saturation is not attained, 
 we may at once assume, as we shall prove further on, 
 that to very rapidly alternately magnetise and de- 
 magnetise, the resistance in the coils must be con- 
 siderably diminished; but this diminution is again 
 considerably augmented in a great measure by the 
 division of the current along the lines, so that it may 
 become necessary to reduce to 40 kilometres the 
 resistance of an electro-magnet for telegraphic pur- 
 poses in use between Paris and Bordeaux, which is 
 about 500 kilometres. All the deductions that we 
 have given above are, however, at fault for this appli- 
 cation of electro-magnets; but it is not the same 
 with a mechanical application where the circuit is 
 weU protected and where the magnetism is free to 
 develop itself in the iron core, and where we may 
 choose the iron to correspond with the point of mag- 
 netic saturation, such as is defined by Muller. We 
 shall have occasion presently to inquire into these 
 properties of the magnetic core ; but to dispose of 
 the laws we have already propounded, we have yet 
 to examine the conditions of maximum in connection 
 with an electro-magnet introduced in one part of a 
 divided circuit. 
 
 Taking the most simple case, that of a single 
 branch u with a resistance I and a common resistance 
 E on leaving the battery, we shall find that the 
 attractive force A of the electro-magnet on I wiU be 
 
 [R (M + Z + H) + M (Z -f- H) ]^ 
 
Compound Oircuits. 21 
 
 and if we substitute for i and H their true values 
 obtained from the equations we have previously- 
 given, we get an expression whose result, in refer- 
 ence to g taken as variable, becomes zero when we 
 have 
 
 rQ\ ggV? I ^'^ \_ '7rha(a+ c) 
 
 W -pV + R + ^J- f~ ' 
 
 an equation which consequently answers to the 
 conditions of maximum. 
 
 In varying the depth a of the coil, the quantity g 
 remaining constant, these conditions of maximum 
 are represented by 
 
 qg^fj, Rm \_''[}^ 
 
 ,2 
 
 /^'V ' R + M/ g 
 
 Now, in the first of these equations, the second 
 side represents the resistance of the wire of the coil 
 and the first the total resistance of the exterior circuit 
 expressed in similar units to those of which we have 
 made use to estimate the length of the wire of the 
 coil. But this total resistance is taken in an inverse 
 sense, for that which we are considering is represented, 
 in fact, by 
 
 R + 
 
 l + u' 
 
 In this case the total resistance must be supposed 
 as if the part common to the two branch currents 
 were represented by I, and as if the part R really 
 common were but a branch. 
 
22 EUmmis of Construction for Electro-Magnets. 
 
 In the second equation the first side represents as 
 before the total resistance of the circuit taken in- 
 versely ; but this total resistance, like the resistance 
 K of a single line, must be considered as before to 
 be less than that of the electro-magnetic coil in 
 
 the proportion of 1 to 1 -^ , to agree with the 
 
 conditions of maximum as regards the variable 
 quantity a. 
 
 In short, we may conclude that the laws of electro- 
 magnetic maximum on divided circuits are the same 
 as those relating to simple circuits, always supposing 
 that the resistance E, on which they are based, is 
 represented by the total resistance of the exterior 
 circuit with its branches, and admitting that this 
 total resistance is considered as if the battery were 
 substituted for the electro-magnet in the circuit. 
 Now, as the total resistance of a divided circuit is 
 less than its proper resistance, the coil ought to have 
 a less resistance than the latter. All these formulas 
 have been verified by experiment, as we shall see 
 further on. 
 
 When the branch circuits have a low resistance 
 and it is possible to combine the elements of the 
 battery as we wish, we arrive by calculation at a de- 
 duction analogous to the preceding, but in an inverse 
 sense. It is then that voltaic combination which 
 makes the resistance of the battery equal to the 
 total resistance of the circuits, which furnishes the 
 most advantageous result. But if, after having 
 
Compoimd Cireuits. 23 
 
 established this voltaic combination we connect the 
 branch circuits direct to the poles of the battery, 
 and try to obtain on each circuit the greatest effect 
 possible, as frequently happens in electric applica- 
 tions, the conditions of resistance of electro-magnets 
 thereon will be quite different from those we have 
 previously considered. This time the electro-magnets, 
 instead of having a resistance less than that of the 
 battery, will have a resistance greater in proportion 
 to the number of branches ; and this is easily under- 
 stood if we consider that, as we said above, the 
 resistance of the battery equals the total re- 
 sistance of these circuits, and this total resistance 
 diminishing as they become more numerous, their 
 Individual resistance must be increased in compensa- 
 tion. If these circuits, with the electro-magnets that 
 they contain, were all equal, this increase would be 
 proportional to their number, for the formula giving 
 the intensity of the current on each would be for an 
 
 ordinary battery rt j. Tt> * representing the 
 
 number of the circuits ; but if they are of unequal 
 resistance, and if we represent by h the number of 
 elements joined up for quantity in each group, and 
 by a the number of groups joined up for tension, the 
 preceding formula wiU be for two branches u and H 
 
 aE 
 
 ^-r{i + -) + r 
 
24 Elements of Construction for Electro-Magnets. 
 
 and the conditions of maximum of this formula in 
 reference to a will answer to 
 
 -^E(1 + — ) = H,orH = -T ^> 
 
 \ li' uo — aa 
 
 an equation which becomes H = 2 -r-Rwhenw = H. 
 
 We shall see later an application of these 
 principles. 
 
( 25 ) 
 
 CHAPTER IV. 
 
 APPLICATION OF THE LAWS OF MAXIMUM TO THE 
 CONSTRUCTION OF ELECTKO-MAGNETS. 
 
 The different laws and formulas that we have just 
 enunciated allow of an easy solution of the problems 
 relating to electro-magnetic attractions ; but for this 
 we must call in MuUer's law relating to magnetic 
 saturation, which law establishes that, to develop in 
 two electro-magnets the same percentage of their 
 magnetic maximum, the intensities of their exciting 
 currents, multiplied by the number of turns of wire, 
 must bear the proportion one to the other of the 
 \/cube of the diameter of each magnet. This law 
 may be thus expressed : — 
 
 It _\/^ 
 
 With this law we can easily understand that it is 
 possible to calculate the conditions required to put 
 an electrd-magnet of a given or calculated diameter 
 into a suitable state of saturation, not only to furnish 
 all the force of which it is capable, but also that the 
 laws of Jacobi, Dub, and Muller may be applicable to 
 
26 Elements of Construction for Electro-Magnets. 
 
 it. For this it is sufficient that two of the terms of 
 the preceding proportion should be supplied by ex- 
 periment, and the given quantities can easily be 
 obtaiued by means of a standard electro-magnet, of 
 which we may augment the magnetic power by 
 increasing the electric intensity till we find that the 
 force produced is as the square of the intensity. 
 From experiments I made with an electro-magnet, 
 the diameter of the iron core of which was one centi- 
 metre, and the resistance of the coil equal to 200 
 kilometres in a circuit of 118,600 metres, I found 
 that the proportion iu question could be obtained 
 when the exciting battery was composed of twenty 
 DanieU cells. As the dimensions of this electro- 
 magnet were known, it was easy, by means of the 
 preceding formulas, to establish constants to be 
 made use of in our calculations. Thus we shall 
 see how the preceding equation joined to the 
 following — 
 
 1ir<?tn -r _ E 
 g^ '""~^' 2E' 
 
 ^ = -:r^^ = — :;!— > I = jrD»^ = H» 
 
 has solved the problem of which we have spoken 
 by bringing back the value of the diameters 
 c to a simple combination of the quantities E 
 and B. 
 
 K, in the first equation given, we suppose the 
 values with an accent known as bearing a proportion 
 to the pattern electro-magnet, and if we substitute 
 for the quantities I and t their value drawn from 
 
Application of Laws. 27 
 
 the conditions of maximum we have explained, we 
 have 
 
 YT 2l{y" '''' 
 
 and as the diameter g is undetermined and must 
 satisfy on one side K = H and on the other a = e, it 
 should be calculated in terms of these two quantities 
 
 by means of the equation E = ^ — > only as ^ is 
 
 undetermined and consequently variable, the quan- 
 tity E must be reduced to terms of g, which gives ^ 
 
 a value represented by — . In substitutiag 
 
 v jE 
 
 this last value for g^ m the preceding equation, we 
 
 have 
 
 e = 
 
 -/ve V2r^71^j' 
 
 a formula in which the quantity in parentheses is a 
 constant which varies according to the system of 
 measures employed, but which refers either to known 
 quantities — as m, which should equal 12, q, which 
 should equal 375,000, nr, which is equal to 3 " 1416 
 — or to quantities considered as given, as they can be 
 obtained from the pattern electro-magnet of which 
 we know the conditions. 
 From another point of view, as iu working out 
 
 for the value ^p-j- the same calculations which were 
 
28 Elements of Construction for Electro-Magnets. 
 
 made for , we get the equation — 
 ,E \/B? /' 
 
 f 
 and as the proportion^ is clearly equal to 1, we can 
 
 rid the constant of the factor /' and consequently 
 the simple formula giving the value of e becomes 
 
 (10) = = ^K, 
 
 K being a constant whose value varies with the units 
 adopted. 
 
 If E represents the electro-motive force with a 
 Daniell cell as unit, and if E is estimated in metres 
 of telegraph wire, K = • 172175, and the %ure 
 obtained represents fractions of a metre. 
 
 In comparing the values of K and E with the 
 system of electric measures established by the 
 British Association, that is to say, the volt and ohm, 
 K= -015957. 
 
 The diameter c being known, all the other 
 elements for the construction of an electro-magnet 
 may be easily determined in conditions of maximum 
 by means of the formulas we have given 
 (5, 6, and 7), which represent the values of g, Z>, t, 
 and H. 
 
 To know the value of the attractive force A, it is 
 
 sufficient to work out the formula P f e^> admitting 
 
ApjpUcation of Laws. 29 
 
 tliat I = — , and that the values of t and e are de- 
 2R' 
 
 termined by the formulas 6 and 10, and to find this 
 
 force in weight we must compare these values with 
 
 those of the pattern electro-magnet, which gives as 
 
 constant '0000855,* and leads to the formula — 
 
 (11) ^ = 1SS55 • • • ^'■™''- 
 
 From the preceding formula important conse- 
 quences result, which are as follows : — 
 
 1. For equal circuit resistances the diameter of 
 an electro - magnet under maximum conditions 
 must be proportional to the electro-motive force 
 employed. 
 
 2. For equal electro-motive forces these diameters 
 must be in inverse ratio to the square root of 
 the resistance of the circuit, including that of the 
 battery. 
 
 3. For equal diameters the electro-motive forces 
 must be proportional to the square roots of the 
 resistance of the circuit. 
 
 4. For a given magnetic force, and with electro- 
 magnets under maximum conditions, the electro- 
 motive forces of the exciting batteries should be 
 proportional to the square roots of the resistance of 
 the circuit (see 12). 
 
 * This coefficient represents the ratio of force calculated from 
 the standard electro-magnet, represented by P P d with the 
 weight in grammes corresponding to this force, which is 26*85 
 grammes at a distance of 1 millimetre. We will refer to this 
 more fully hereafter. 
 
30 Elements of Construction for Eledro-Magnets. 
 
 The preceding formulas also allow of the easy 
 solution of many problems which frequently arise in 
 electric applications, particulariy to calculate the 
 force of the battery and the dimensions of an electro- 
 magnet required to furnish a given attractive force 
 on a given circuit of resistance. It is true that the 
 results obtained often do not correspond exactly with 
 the calculation, by reason of the different nature of 
 the iron, which may be more or less adapted to 
 develop magnetic action, and of the more or less 
 complete magnetic saturation of the iron, but we may 
 always get approximate figures. 
 
 To resolve the problem in question, let us first see 
 how the fourth postulate that we have just formu- 
 lated may be demonstrated, and how, in this way, 
 we can obtain a value representing the attractive 
 magnetic force in terms of the electro-motive force 
 of the battery and of the resistance of the circuit. 
 
 If we consider that the expression 1^ t^ d^, which 
 represents this force, can be converted by successive 
 substitutions * in the values of I, t, and c into the 
 
 * As foUowa : — 
 
 Value of P i» = _f^. 
 
 Value of «» i' = ■^^ 123903. 
 
 E' 
 
 Valueof P i» .. .. P f = r 30976. 
 
 3. 
 
 Value of c2 c==— -OTIDS 
 
 E* 
 
 E* 
 Value of A .. .. J? f c^ = 2228. 
 
AppUeaiion of Laws. 31 
 
 formula — — -4 , 
 
 Q is a constant equal to 2228.* For a like attrac- 
 tive force we get the proportions — 
 
 , ^^ E* K" E Vr 
 
 (12) — . = T^TT or — = 
 
 ^ ^ E" E^ E' v^B/ 
 
 As in the values of E and E, n, n' appear as the 
 number of cells employed, we may easily calculate 
 these numbers, knowing the values of the constants 
 e and p of the element employed, for, starting from 
 the preceding equation, we have — 
 
 n e 
 
 _ \/ n p + r 
 
 and if the accentuated quantities are in proportion 
 to those of the standard electro-magnet which are 
 known, it is easy to deduce from the preceding 
 equation the value of %; it is only an unknown 
 quantity of the second degree. 
 
 If, in the problem now under consideration, we 
 admit that the force of electro-magnetic attraction 
 is expressed in weight, it will be understood that the 
 
 formula -^ —^ could only represent this force, which 
 
 * The unit of electro-motive force is here taken as that of the 
 Daniell cell, and the resistances are calculated in metres of telegraph 
 wire. 
 
32 Elements of Construetion for Electro-Moffneis. 
 
 we will call P, by applying to it a coefficient K, 
 representing r^ deduced from the data of the 
 standard electro-magnet, whose value for an attrac- 
 tion at the distance of 1 millimetre is -^r^r-Tr^- or 
 
 26-85 
 
 ■ 00008555. It will be observed that the quantity 
 F' here represents for the standard electro-magnet, 
 
 s 
 
 the formula P i' <? or its equivalent, and the result 
 will be that by stating the following — 
 
 {n p + rf f 
 
 we shall readily deduce the value of n e, which will 
 be 
 
 . 3/ V/'*P2K2 
 
 ne =V n p + r\/ V -^ ^ . , 
 
 and if we join in one the two constants Q and K, 
 and then apply it, we have — 
 
 n p 
 
 — = (■0225^/^^7^^p, 
 
 In representing by A the quantity in parenthesis, 
 which may easUy be calculated, we have 
 
 ^/m 
 
 A' p ^ ^ / fK' p\^ A?r 
 
 The values of m e and k p being thus determined, 
 it is easy to calculate the dimensions of the electro- 
 
Application of Laws. 33 
 
 magnet by means of the formulas which have already 
 been given. 
 
 In the preceding calculations we suppose the re- 
 sistance of the exterior circuit to be great enough to 
 warrant the battery being joined up for tension ; but 
 when this resistance is small enough to lead us to 
 suppose it would be advantageous to group the 
 elements in series, the calculation will be somewhat 
 modified. There will then be two alternatives ; either 
 the resistance E will be nil, or it will only be less 
 than n r. In the first case, it will be impossible to 
 obtain the individual values of the quantities a and 
 b, representing respectively the number of series, 
 and the number of elements in each series ; for, the 
 interior resistance of the battery (which is repre- 
 sented by T p) and the resistance H of the coils being 
 
 two undetermined quantities which should be equal, 
 we can give them what value we like, within the 
 limits of course that the number of elements in 
 tension includes the whole of the elements in the 
 battery, and that the number of elements in quantity 
 is similarly situated ; * but we can obtain the value 
 
 * This deduction may readily he proved by the following calcu- 
 lations. (See also Chapter IX.) From what we have seen, the 
 attractive force of an electro-magnet may be represented by F H, 
 and for batteries arranged in series, this expression becomes 
 
 a' E^ H a^Wn 
 
 il'*'')' 
 
 4 - P 
 
 Suppose we have 24 cells, and the resistance of each equals 
 
 D 
 
34 Elements of Construction for Electro-Magnets. 
 
 of a h, or the total number n of the elements by 
 means of the preceding formulas, which give under 
 these conditions — 
 
 a6 = 4 -000506 v' ^ pys 
 
 or 
 
 al = -J (-0225 v^ v?/*)'" 
 
 Under these conditions the value of c is determined 
 and shown by the formula — 
 
 (13) c = .-—= -173. 
 
 V p 
 
 In the second case, since to satisfy the conditions 
 of maximum as regards the electro-magnet, its re- 
 sistance should equal r- p -t- r, and that, to obtain 
 the minimum value of the resistance of the simple 
 circuit {-=p-{-r\ corresponding to the maximum of 
 
 12 metres, and the electio-motive force equals 1 ; when all the 
 cells are joined for tension, we shall have 
 
 when for quantity 
 
 1 X 288 _ 
 
 , 24= X 1 X -5 
 
 ^= 4X12' =-^- 
 
 when they are in series of cells in quantity with o = 6, we shall 
 have 
 
 . 24 X 1 X 12 
 
 all identical values. 
 
Hon of Laws. 35 
 
 the value I, on whieli resistance is based that of the 
 
 electro-magnet, ^ p must equal r, we shall then be 
 
 able to determine the quantities a and b as soon as 
 we have ascertained the value of a 6 or n, and this 
 quantity is determined by the preceding equation 
 
 giving the value of a b, because -^ p + »• then equals 
 2 T P- From this we have — 
 
 
 
 a =\/ — and 6 = \/ — or _. 
 ^ p V r a 
 
 If, instead of a simple circuit, we consider a com- 
 pound circuit with a number x of branches issuing 
 directly from the poles of the battery, as frequently 
 occurs in electric applications, and if on these 
 circuits be introduced electro-magnets of similar 
 resistance and dimensions, the value of the intensity 
 
 of the current on each branch will be 
 
 a 
 
 b 
 6E 
 
 i^i^ p + R 
 
 or rr — , and the value of a 6 or w will be the same 
 2xp 
 
 as in the most simple case, only multiplied by x. 
 
 Let us now see how, the dimensions and construc- 
 tion of an electro-magnet being given, we can deter- 
 mine the number of elements and the arrangement 
 of the battery that will best produce in this magnet 
 the given force P on each of the x branches issuing 
 from the battery. 
 
 D 2 
 
36 Elements of Constriidion for Electro-Magnds. 
 
 According to the results then obtained, the quantity 
 
 H in the equation I = — is ascertained, 
 
 x-p + B. 
 
 and we know that for the conditions of maximum 
 
 X rp should equal H. On the other hand, we 
 
 know that we should have P t^, e^ = P K. Now as 
 the quantities t^ and c^ may be easily calculated, as 
 c is given, the value of I may be deduced from the 
 preceding equation in which — 
 
 From this we havi 
 
 I '^^ 
 
 0* 
 
 2IH a X p 
 
 a = — :p — and o = — o"— 
 
( 37 ) 
 
 CHAPTER V. 
 
 NUMEBIOAL EXAMPLES OF THE APPLICATION OF 
 THE PRECEDING POEMULAS. 
 
 To facilitate putting into practice the formulas 
 already given and the manner of calculating them, 
 we will give several numerical examples. 
 
 First, we will suppose that we wish to find the 
 best dimensions to give an electro-magnet before 
 working it with a single ordinary Bunsen cell, on a 
 circuit having scarcely any resistance beyond that of 
 the coil. We will suppose that the electro-motive 
 force of this element, taking the Daniell as unity, is 
 1 • 86 and its resistance 57 metres of telegraph wire, 
 or -57 ohm. These quantities show the necessity of 
 knovdng the values of the voltaic constants, and at 
 the end of this work will be found a table of the 
 principal batteries employed. 
 
 According to formula No. 10 we shall have for 
 the value of the diameter of the iron core of the 
 electro-magnet 
 
 1-86 
 e = -y= -172175 = -0424 metre. 
 
 Each of the arms will thus have a diameter of 
 4*25 centimetres and a length of 25 '5 centimetres. 
 
38 Elements of Construction for Electro-Magnets. 
 
 The diameter g of the wire of the coil will be 
 according to formula No. 7 
 
 / / ■ 0424* 
 
 ^ = V 1 -4 V —57- • 00020106 = 4-865 millim., 
 
 including its insulating covering, or 3 • 36 millimetres 
 without this covering, supposing g to be divided by 
 / or 1 • 4. The length of the wire will be, by formula 
 No. 5, 243 metres, and the attractive force in weight at 
 
 1 millimetre, according to the formula, A = 
 
 UUUUoOO 
 
 wiU be 23 • 112 kilogrammes. 
 
 It is needless to say that if the quantity E is given 
 in ohms instead of metres of telegraph wire, the con- 
 stant of formula No. 7, instead, of being • 00020106, 
 should be considered as -0000020106, so that the 
 formula giving the value of g will then be 
 
 '=va7 
 
 g = 'W /V ^ -0000020106. 
 
 The number of turns is calculated by means of 
 formula No. 6, 
 
 Let us now suppose that this electro-magnet is to 
 
 be introduced on a line of 100 kilometres, which 
 
 with the resistance of the battery consisting of 20 
 
 DanieU cells, wiU constitute a circuit of 118620 
 
 metres of resistance in units of telegraph wire ; we 
 
 shall have 
 
 20 
 : »j^gg2Q "172175... = 1 centimetre; 
 
 c = 
 
Numerical Examples. 39 
 
 or, using the English units of volt and ohm, 
 
 21-58 volts niKnRn 1 i- i 
 
 e = —======-= •015957... = 1 centimetre. 
 
 Vll86 ohms 
 
 Each of the arms will then be 1 centimetre in 
 diameter and six centimetres in length ; the diameter 
 of the wire will be • 2597 millimetre with its covering, 
 or •1583 millimetre without this covering, and its 
 length 1116 metres. The attractive force will be 
 26 "85 grammes. These figures show that electro- 
 magnets on long circuits should be of small dimen- 
 sions and covered with fine wire, and that, on the 
 contrary, they should be very large when the circuit 
 is short and the battery supplies electricity of 
 quantity. 
 
 These dimensions for an electro-magnet excited by 
 a Bunsen cell may appear surprising to many ; but 
 this would be still more the case if the iron had a 
 less magnetic saturation. Thus a tubular electro- 
 magnet whose diameter was 10 centimetres, the 
 length of each arm 30 centimetres, and the thickness 
 of the tube 1 centimetre, has been made to lift even 
 without the addition of iron mouthpieces at the polar 
 extremities, a weight of 160 kilogrammes, under the 
 influence of a single Bunsen cell and of a coil of 
 copper wire 4 millimetres in diameter (not including 
 the insulating covering), only 182 metres in length 
 and taking only 482 turns. Now, with an ordinary 
 electro-magnet 2 centimetres in diameter, wound 
 with a copper wire of 1 millimetre, I have never 
 
40 Elements of Construction for Electro-Magnets. 
 
 been able to obtain with the same Bunsen more than 
 12 kilogrammes of attractive force on contact. 
 
 Let us say that we wish to obtain a given force on 
 a given circuit of resistance, and we wish to have an 
 attractive force of 273 grammes (the armature being 
 at a distance of 1 millimetre) on a circuit of 50 kilo- 
 metres of resistance (or 500 ohms) with a battery of 
 gravitation bichromate cells. In this battery the 
 electro-motive force of each element is 2 (that of 
 Daniell being 1), and the internal resistance p is 
 about 1000 metres of telegraph wire (or 10 ohms). 
 According to the formulas on p. 32, we shall have 
 
 A = • 0225 /(/\/nfS<^73^ = • 09 and A^ = • 0081, 
 and 
 
 8-1, //S-IV , -0081x50000 h.tok 
 '* = ^ + V V~8"^ + 4 = 11 125, 
 
 whence 
 
 11-125x2 ,„„ , ,,„ 
 c = — /r.-,-,^^ X '173 = 1'553 centimetre. 
 Voll25 
 
 This gives, for the length of each bobbin, 9 '32 
 centimetres. 
 
 For the diameter of the wire, with its covering, 
 •3894 millimetre. 
 
 For the diameter of the same wire, without its 
 covering, • 2842 millimetre. 
 
 For the length of the said wire, 1861 metres. 
 
 For the number of turns, 19078. 
 
 For the intensity of the current, ' 0001859. 
 
! Numerical Exam^ples. 41 
 
 For the value of e*, -001935. 
 
 By squaring the values of I and t, and multiplying 
 by c^, we get •0243778517, which represents the 
 electro-magnetic force ; and this value, compared 
 with that of the standard electro-magnet which is 
 •002297, gives the proportion 10 • 6, which is very 
 near that of the two weights 273 and 26 • 85 repre- 
 senting in grammes the attractive forces, particularly 
 when we consider that some figures have been forced 
 and decimals neglected to simplify the calculations. 
 
 We will now suppose that we wish to obtain 
 separately, from six electro-magnets put in direct 
 communication with the battery, a force of 200 
 grammes, employing a battery similar to the pre- 
 ceding. The total number of elements a h will be 
 given by the formula 
 
 ah=-^ — X • 000506/^ v'200* x r4«=10 • 778, 
 
 or eleven elements ; but, as we cannot group eleven 
 elements into multiple series, we will suppose the 
 battery to be composed of 12 elements. With 
 this number the battery may be grouped in 3 sets of 
 4 elements, or two sets of six ; but in any case we 
 shall have 
 
 ■\/T2 
 c = . X 2 X '173 = 1^53 centimetre, 
 
 \/l000x6 
 
 and consequently the length of each arm of the 
 electro-magnet will be "0918, or 10 centimetres. 
 
42 Elements of Construction for Electro-Magnets. 
 
 And here are the values of ff, t, I, &c., which we 
 shall obtain in each case. 
 
 1st. Where a = 3, and 6 = 4: — 
 
 g' = -000000560075; g= -0007484; -^ = 
 
 •0005346. 
 H = 482 metres. 
 t = 5015 ; e = 25150225. 
 I = - 000666 ; P = -000000443556. 
 c?= -001892. 
 
 r ^ = 11 • 155533 ■,Ff(^ = - 021084. 
 A = 247 grammes. 
 
 2nd. Where a = 2, and 6 = 6:— 
 
 ^2= •00000083986;^=- 000916 ;-| = -0006543. 
 
 H = 321 metres. 
 
 t = 3345 ; f = 11189025. 
 
 1= -001; P= -000001. 
 
 c^= -001892. 
 
 P f = 11-189025; T' P ^ = -0211686353. 
 
 A = 247 grammes. 
 We need not, however, assume that to preserve 
 perfect magnetic saturation, and to maintain a con- 
 stant force in an electro-magnet, we are obliged to 
 vary its dimensions according to the resistance of the 
 exterior circuit; we can, in certain applications, 
 maintain the same type of electro-m?ignet very well 
 by varying the force of the supplying battery and 
 the size of the wire of its magnetising coil. In fact, 
 as the electro-motive force of the battery employed, 
 for a like diameter of electro-magnet and a like 
 
Numerical Examples. 43 
 
 magnetic force, is in proportion to the square root of 
 the exterior resistance of the circuit, we may conclude 
 that if we increase the E. M. F. in proportion as the 
 resistance of the circuit increases, we may maintain 
 the same diameter for the electro-magnet, and the 
 same force. Consequently, if, instead of introducing 
 our standard electro-magnet of 1 centimetre in 
 diameter, on a circuit of 100 kilometres, we introduce 
 it on a circuit of 400 kilometres with a double 
 number of elements, nearly the same effect will be 
 produced. But the size of the wire must be changed, 
 for this varies for a like diameter of electro-magnet 
 in the inverse ratio of the ^ of the resistance of the 
 exterior circuit. This diameter, then, the wire of 
 the standard electro-magnet being • 1583 millimetre, 
 must be, in the present example, 
 
 as -C/IOOOOO : \/400000, or as 17-8 : 25-1, 
 
 which makes it -112 millimetre. We see by this 
 that we may without inconvenience maintain for 
 telegraphic relays the allotted dimensions. 
 
 We have said above that by doubling the number 
 of elements in the battery we should obtain nearly 
 the same force on a circuit of 400 kilometres as on 
 that of 100 kilometres. This word "nearly" must 
 not be overlooked, for the resistance of the circuit, in 
 the example we have cited, is not 400 kilometres, 
 but 437240 metres, and this number is not exactly 
 four times 118620, which represents the resistance 
 in metres of the first circuit ; it is less, and conse- 
 quently, to obtain exactly the same effect in both 
 
44 Elements of Construction fw Eleetro-Magnds. 
 
 cases, the battery of 20 elements must be increased 
 to 38-4 elements, or 39 instead of 40. 
 
 We will now employ two electro-magnets of small 
 dimensions, each, having a resistance of 2200 metres, 
 wound with No. 16 wire ('44 millimetre), and con- 
 nected direct to a battery of eight Daniell cells 
 arranged for tension. Experiment shows that the 
 force obtained from each of them will be 70 
 grammes at a distance of 1 millimetre. Conse- 
 quently the total force developed by both will be 
 140 grammes. Now if, instead of employing two 
 electro-magnets we use but one, this force will be 
 200 grammes. Thus we lose 60 grammes of power 
 by employing two electro-magnets instead of one, 
 although the two are put in direct communication 
 with the poles of the battery. We might thint, at 
 first sight, that there was some fault in the electric 
 communications, but our calculations prove it to be 
 otherwise. 
 
 In applying the formulas of Ohm to the two cases 
 we find : — 
 
 1st. That in the case of a single electro-magnet, the 
 value of the intensity of the current, the E. M. F. 
 of the Daniell battery being taken as 5973, 
 is 4-95. 
 
 2nd. That in the case in which the two electro- 
 magnets are on two circuits, this intensity is repre- 
 sented by 2-79. 
 
 The combined forces of these electro-magnets 
 then, arranged in these two ways, will be as the 
 squares of the two quantities 4*95 and 2*79, and if 
 
Numerical Examples. 45 
 
 we admit that the force of the electro-magnet in the 
 simple circuit would be 200 grammes, the following 
 formula will giye the force of the two others : — 
 
 2-79 X 200 „. 
 ^ — or o4 grammes ; 
 
 4-95 
 
 so the joint force of the two electro-magnets wiU be 
 128 grammes instead of 200, as produced by a single 
 one. This is a still more marked diflference than 
 that shown by experiment. This effect is in conse- 
 quence of the arrangement of the battery, which is 
 already out of proportion to the resistance of the 
 exterior circuit, as its resistance is nearly four times 
 as great as the latter, being still less in proportion to 
 the circuit composed of two branches. Indeed, the 
 
 formula r- ==, which represents in this case 
 
 2np + H 
 
 the intensity of the current on each branch, 
 
 and which, in the case of a battery arranged in 
 
 series, gives for conditions of maximum 2 ^ p = H 
 
 or =- p = -7r> and this shows that iu this case the re- 
 ft 2 
 
 sistance of the battery should be half that of each 
 circuit. Consequently, the number a of the ele- 
 ments in tension should be obtained by the formula 
 
 a = \/ "ij— , which in the present case gives 
 
 = 3-074. 
 
 2p 
 
 ^8 X 2200 
 2 X 931 
 
 V' 
 
46 Elements of Construction for Electro-Magnets. 
 
 As we cannot divide a cell into fractions, and as the 
 couplings require numbers which are even divisors 
 of n, the voltaic combination which will best answer 
 to these conditions of maximum will be that in 
 which the elements, separate and combined, will 
 come the nearest to a, h, and n, and we shall see that, 
 in the present case, it is that which contains three 
 elements in tension, each composed of three elements 
 in quantity, which will be the most advantageous. 
 Now, under these conditions the attractive force of 
 the single electro-magnet would be 267 grammes, 
 and that of the two combined 316 grammes. Thus 
 we gain by employing two electro-magnets. 
 
 We see by this how important it is that the con- 
 struction of electro-magnets and the arrangement of 
 the battery in electric applications should be 
 preceded by calculation, and how much would be 
 gained if the theory we are now propounding were 
 well understood by constructors. 
 
( 47 ) 
 
 CHAPTER VI. 
 
 EXPEBIMENTAL VEEIPICATION OF THE LAWS OP 
 ELECTED -MAGNETS. 
 
 As I have already said, the formulas I have given 
 have all been verified by experiment, except that 
 generally adopted, which presents great difficulties 
 in its proof, as we shall presently see. This experi- 
 mental verification has seemed to me the more 
 important, as originally many scientists did not 
 accept these formulas without discussion, and besides, 
 mathematical deductions are not always sufficient to 
 convince. I have, therefore, undertaken several series 
 of experiments to prove the truth of my deductions, 
 and the following is the result : — 
 
 1st. The experimental demonstration of the prin- 
 ciple, establishing the fact that the resistance of the 
 electro-magnet should equal that of the exterior 
 circuit, is. rather delicate, by reason of the difficulty 
 we meet with in obtaining in all the wires of com- 
 merce of different diameters conductibility exactly 
 proportional to the squares of those diameters; The 
 experiments I have made have been rather contra- 
 dictory, and I may say contrary to theoretic deduc- 
 tions, which made me hesitate, in my early investi- 
 gations, in admitting them, although they were 
 
48 Elements of Construction for Electro-Magnets. 
 
 generally accepted ; however, as the other deductions 
 have been confirmed by experience, and as the non- 
 success of those I undertook on this matter might 
 arise from accidental causes, I have been obliged to 
 admit the principle, at least in cases where it is 
 really applicable ; that is to say, when, the depth and 
 length of the bobbins being first fixed, we wish to 
 know what diameter of wire will correspond best 
 with a circuit of a given resistance. This is 
 evidently the case in which we find ourselves when 
 we have determined the elements for the construction 
 of an electro-magnet, for we must, above all, propor- 
 tion its core to the electric intensity which excites it, 
 the depth of the coils being determined by a second 
 principle which must not be lost sight of. 
 
 It is not the same, however, if we wish to ascertain 
 what should be the resistance of the exterior circuit 
 on which we could most usefully employ a given 
 electro-magnet or galvanometer, without taking into 
 account its dimensions. In this case the conditions 
 of maximum that I have established show that the 
 resistance of the electro-magnet should be to that of 
 the exterior circuit as the depth of the magnetising 
 coil, including that of the magnetic core, is to the 
 depth of the coil alone. 
 
 This deduction may easily be demonstrated, and 
 this is how I have drawn up my experiences. 
 
 I first wound with extreme care on the same 
 bobbin, 6 • 1 centimetres in length between the ends 
 and with an exterior diameter of 1 • 1 centimetre, two 
 lengths of 60 metres of No. 16 wire, and another of 
 
Verification of Laws. 
 
 49 
 
 57 "25 metres, which formed a third coil. These 
 three coils had their ends out and distinct one from 
 another, so that they could be worked separately or 
 all together. The first gave a resistance of 1080 
 metres of telegraph wire, the second nearly the 
 same, which constituted for both together a resistance 
 of 2160 metres, and the third added to these made a 
 total resistance of 3200 metres. 
 
 When I tried this magnet in my electro-magnetic 
 balance with one pole only (that covered with the 
 bobbin), I obtained, under the influence of a 
 Leclanche battery of three elements, the total resis- 
 tance of which was about 1200 metres, the following 
 results : — 
 
 Eesistance of Exterior 
 
 Coil A of 
 
 Coil B of 
 
 Coil C of 
 
 Circuit ill metres. 
 
 1080 metres. 
 
 2160 metres. 
 
 3200 metres. 
 
 
 gr- 
 
 gr- 
 
 gr. 
 
 1200 + = 1200 
 
 P = 112 
 
 P' = 122 
 
 P" = 112 
 
 1200 + 400 = 1600 
 
 P = 73 
 
 P' = 92 
 
 P"= 95 
 
 1200 + 1000 = 2200 
 
 P= 47 
 
 P' = 66 
 
 P"= 70 
 
 1200 + 2000 = 3200 
 
 P= 27 
 
 . P' = 43 
 
 P" = 50 
 
 1200 + 3000 = 4200 
 
 P= 17 
 
 F'= 29 
 
 F" = 36 
 
 1200 + 4000 = 5200 
 
 P= 12 
 
 P' = 22 
 
 P"= 28 
 
 The attractive forces F, F', F" were measured at a 
 distance of 1 millimetre. This table shows that it 
 is the coil B which most nearly approaches the 
 theoretical requirements ; that is to say, with a 
 
 resistance of , or 982 metres, which gives the 
 
 maximum effects; and it is only when the resis- 
 tance of the exterior circuit reaches 1600 metres 
 
 E 
 
50 Elements of Construction for Electro-Magnets. 
 
 that is, 5^. that the coil C, the most resisting, 
 begins to show its preponderance. 
 
 With a battery of two elements and an exterior 
 resistance of 800 metres represented by that of the 
 battery, the force of the coil B still preponderates, 
 being 60 grammes, while those of both the other 
 coils, A and C, were but 57 grammes ; but with a 
 single cell, and consequently an exterior resistance 
 of 400 metres, the coil A had the advantage, and the 
 forces were : for A 21 grammes, for B 19 grammes, 
 and for C 17 grammes. 
 
 These experiments, repeated with a galvanometer, 
 were still more decisive, as may be seen from the 
 experiments, the results of which I published in the 
 'Comptes rendus des seances de I'Academie des 
 Sciences ' of August 13th, 1877, page 379. 
 
 2nd. To demonstrate that on a divided circuit the 
 resistance which should serve as basis for that of a 
 given electro-magnet or galvanometer, is represented 
 by the total resistance of the exterior circuit, taken 
 inversely ; that is to say, substituting the generator 
 for the electro-magnet, I introduced in a very feeble 
 circuit * a sensitive Euhmkorff galvanometer, with 
 two coils, the resistance of one of which was repre- 
 sented by 783 kilometres, and of the other by 237 
 kilometres. I added a rheostat to the circuit, and 
 placed a second in a branch circuit, joining the two 
 ends of the wire of my galvanometer. Under these 
 conditions, taking K to represent the battery circuit 
 
 * The generator was composed of an iron wire and one of copper 
 joined together ; it had an electro-motive force of ^ of a DanieU 
 and a resistance of 272 kilometres of telegraph wire. 
 
Verificalion of Laws. 
 
 51 
 
 u the galvanometer branch, and I the branch in 
 which the galvanometer was introduced, the equation 
 giving the maximum conditions was 
 
 , R « irla^ 
 
 It is true I could not then figure in the calculations, 
 for the points of bifurcation of the two branches cor- 
 responded with the two ends of the galvanometer 
 wire; but, by means of the two rheostats, I could 
 vary the resistances u and R so as to form a total 
 resistance, starting from the galvanometer, which 
 was equal to, less or greater than, the resistance of 
 the less resisting galvanometer coils ; and for this it 
 was sufficient, R being given, to calculate u by 
 
 means of the formula u = 5 =• , H representing 
 
 It — Jd 
 
 the resistance of the coil. Below are the results I 
 
 obtained, taking all precautions not to vary the 
 
 conditions during the experiments. 
 
 Circuits. 
 
 R = 512 
 
 M= 86 
 E = 512 
 
 u = 128 
 E = 512 
 
 « = 200 
 K = 512 
 
 « = 256 
 E = 512 
 
 « = 512 
 E = 256 
 
 M = 200 
 E = 256 
 
 « = 512 
 
 + 2721 
 + 272| 
 + 272> 
 + 272| 
 + 2721 
 + 272| 
 + 272) 
 
 Total 
 BeBistance, 
 
 kilometres 
 78 
 
 110 
 
 160 
 
 193 
 
 309 
 
 145 
 
 260 
 
 OoUof 
 kilometres. 
 
 :27i° 
 
 :32| 
 
 :36| 
 
 :40 
 
 :46 
 
 = 36 
 
 = 47 
 
 Coil of 
 ?33 kilometres. 
 
 r = 24i° 
 r = 30 
 r = 36i 
 r = 4i 
 r = 5i 
 
 V = 33i 
 
 r = 52 
 £"2 
 
52 Elements of Constntction for Electro-Magnets. 
 
 We see by tliis table that, as I had stated, the 
 high-resisting coil preserves the advantage, even 
 when the total resistance of the circuit, outside the 
 galvanometer, is nearly equal to that of the less- 
 resisting coil, and it is only when this total resistance 
 falls below 160 kilometres, or 145 kilometres (accord- 
 ing to the value of R), the point where the two coils 
 have the same sensitiveness, that the superiority of 
 the less-resisting coil begins to show itself. As with 
 the galvanometer experimented on, the proportions 
 of the two resistances (those of the exterior circuit 
 and of the galvanometer) should be, according to the 
 conditions of maximum deduced from the proportion 
 as 1 • 9 : 2 • 425 ; the resistances of the exterior circuit 
 on which we could most usefully employ the two 
 coils would be 98 kilometres for the low resistance 
 and 386 kilometres for the high resistance one. And 
 the point where they become equal is the mean pro- 
 portion between these two numbers ; this being 193, 
 and that shown by experience 160, we see that the 
 formulas enunciated may be regarded as sufficiently 
 demonstrated. We can also prove it in another 
 way by calculating what resistance in u would 
 equalise the two coils. This resistance is shown by 
 the formula 
 
 B, {t'K-t H') 
 t (K -i- H') - «' (E + H) ' 
 
 which gives, under similar conditions, 224 kilometres 
 of telegraph wire, and the experiments give 200 
 kilometres. 
 
Verifieation of Laws. 53 
 
 3rd. The deduction whicli shows that the depth of 
 the magnetising coils of an electro-magnet should be 
 equal to the diameter of the cores which they cover, 
 may easily be verified. In order to demonstrate 
 this experimentally, I took three electro-magnets 
 with bobbins of the same length but very different 
 diameters. 
 
 One of these magnets had a diameter of 2 centi- 
 metres, another of 1 centimetre, and the third of 
 •65 centimetre. I applied one pole only to the 
 balance, and each bobbin, covered with No. 16 wire, 
 had 23 layers of 111 turns in each, or 2553 turns in 
 all. Nothing of the thickness of paper even was 
 introduced between the layers, and all the turns 
 were pressed closely one against another, which gave 
 these coils a uniform depth of 1 centimetre. The 
 result was that the electro-magnet of which the core 
 was 1 centimetre in diameter was the only one which 
 answered to the maximum conditions previously 
 given. This magnet had a resistance of 3200 metres, 
 the large one of 5200 metres, and the small one of 
 2800 metres. The following (p. 54) are the results 
 I obtained in passing through these different electro- 
 magnets the current from a Leclanche battery con- 
 taining from 1 to 3 cells, each haviag an internal 
 resistance of about 400 metres, estimating the 
 attractive force at a distance of 1 millimetre. 
 
 We see by this table, that, for a like resistance of 
 exterior circuit, and with a sufficient electric inten- 
 sity, it is the electro-magnet of which the depth of 
 the coil equals the diameter of the core, which has 
 
54 Elements of Construction for Eleetro-Magnets. 
 
 
 Electro-Magnets. 
 
 Reaistances of Exterior circuit. 
 
 
 
 Small 
 
 in metres. 
 
 Large, 
 2 centimetres 
 
 Mean, 
 1 centimetre. 
 
 •65 centi- 
 metre. 
 
 
 gr. 
 
 gr. 
 
 gr- 
 
 ^(^1200+ 0= 1,200 
 
 76 
 
 112 
 
 86 
 
 g 1200 + 1,600 = 2,800 
 
 48 
 
 57 
 
 (44) 
 
 a {1200+ 2,000= 3,200 
 
 43 
 
 (50) 
 
 39 
 
 ^ 1200+ 4,000= 5,200 
 
 (28) 
 
 29 
 
 22 
 
 « U200 + 10,000 = 11,200 
 
 13 
 
 10 
 
 8 
 
 4S /- 800 + = 800 
 
 38 
 
 57 
 
 46 
 
 g 800+ 2,000= 2,800 
 
 22 
 
 26 
 
 (20) 
 
 S 800+ 2,400= 3,200 
 
 20 
 
 (22) 
 
 16 
 
 ■S 800+ 4,400= 5,200 
 
 (14) 
 
 13 
 
 10 
 
 eq I 800 + 10,000 = 10,800 
 
 7 
 
 5 
 
 4 
 
 ■S MOO + = 400 
 
 12 
 
 18 
 
 15 
 
 § 400+ 2,400= 2,800 
 
 7 
 
 7 
 
 (6) 
 
 i 400+ 2,800= 3,200 
 
 6 
 
 (6) 
 
 5 
 
 ■3 400+ 4,800= 5,200 
 
 (5) 
 
 4 
 
 4 
 
 '-' I 400 + 10,000 = 10,400 
 
 3 
 
 2 
 
 2 
 
 the advantage; this advantage is found to be in- 
 variable when we compare the forces produced on 
 exterior circuits of different resistance, adapted to 
 the resistances of the magnets, the only case which 
 has been discussed in the formulas. It is only when 
 the electric force is so weak that the increase of the 
 magnetic action with the diameter is not very 
 palpable, that the maximum of the electro-magnet 
 of 1 centimetre loses ground a little ; and this is as 
 it should be, for MuUer's law, which supposes the 
 attractive forces to be proportional to the diameter 
 of the magnetic cores, is only true when these cores 
 are magnetised to a pitch approaching that of their 
 magnetic saturation; and by this word saturation 
 must be here understood that magnetic state which 
 
Verification of Laws. 
 
 55 
 
 the electro-magnet would preserve, if, instead of 
 being iron, it were tempered steel magnetised. When 
 the magnetic force developed is much below this 
 point, it is on the electric intensity that the attractive 
 force produced chiefly depends, and this is naturally 
 greatest with the least resisting circuit. The figures 
 in parentheses in the preceding table point out, in 
 each of the three series of experiments made with 
 different electric intensities, the forces corresponding 
 to the maximum conditions with reference to the 
 circuit, and these conditions were naturally esta- 
 blished, supposing the resistance of the exterior 
 circuit to be equal to that of the coil ; for I started 
 in the construction of my electro-magnets from a 
 given depth of bobbin. But if I based my calcula- 
 tions on the maximum conditions relative to a given 
 electro-magnet without considering its dimensions, 
 the preceding law is still further verified, for the 
 maximum resistances of the exterior circuit then 
 become 1400 metres for the small magnet, 2600 
 metres for the large one, and 1600 metres for the 
 other. Thus the attractive forces of these three 
 electro-magnets are as follows : — 
 
 Electro-magnet of 1 centimetre 
 „ „ „ '65 centimetre 
 
 „ „ „ 2 centimetres 
 
 With 3 
 ElemeDts. 
 
 94 
 79 
 50 
 
 With 2 
 Eieiuents. 
 
 gr- 
 46 
 41 
 25 
 
 Withl 
 Element. 
 
 14 
 13 
 
56 Elements of Construction for Electro-Magnets. 
 
 CHAPTER VIL 
 
 EFFECTS OF A MOEE OB LESS COMPLETE 
 MAGNETIC SATURATION. 
 
 As I have said, the laws of electro-magnets are 
 not as definite as might be desired, by reason of the 
 various effects resulting from the state of saturation 
 of their magnetic core ; but this cause of perturba- 
 tion is far from being as great as certain scientists 
 would have it thought, and it interferes in a less 
 proportion with the definite results deduced, than do 
 the effects of polarisation with the calculated results 
 of the laws of electric currents. At the time of my 
 last investigations concerning electro-magnets, I 
 wished to assure myself of the importance of this 
 disturbing cause, and I undertook a great number 
 of experiments which seem to me interesting to 
 report here, for in consequence of the development 
 that takes place every day in the application of 
 electro-magnetism, it is, above all, important to be 
 well grounded in the conditions for the good con- 
 struction of magnets on which depends the success 
 of these applications. 
 
 In the first part of this work I gave several 
 general deductions which I had drawn from experi- 
 
On Magnetic Saturation. 57 
 
 ence, but I did not insist on my experiments them- 
 selves, for they led to no law besides those given 
 by Dub and Muller; but it is important that I 
 should explain myself more explicitly in this re- 
 spect ; I will, therefore, begin by demonstrating 
 that if we take 11 as. the value of the coefScient m, 
 by which the diameter of the magnetic core must be 
 multiplied to find its length, so as to satisfy the 
 different conditions of maximum relating to it, its 
 attractive force always increases in proportion to its 
 length. In fact, if we start with a given length of 
 wire representing the resistance of the exterior 
 circuit, and wind it on electro-magnets of different 
 diameters so as to get a depth of coil equal to their 
 diameters, their lengths must be different and calcu- 
 lated to satisfy the equations 
 
 27r6e2 27r&'e'' , „ ,, ,, 
 
 r- = 5 — or bc^ = b e^ 
 
 9 9 
 
 and then these lengths will be inversely proportional 
 to the squares of their diameters. In this .case the 
 factor m is no longer constant and becomes propor- 
 tional to the cube of the diameters ; but then the 
 law which supposes the E. M. F. proportional to the 
 squares of the number of turns multiplied by the 
 \/cube of the diameter, is no longer applicable, and 
 we must then, to compare the forces, have recourse 
 to the law which takes the latter to be proportional 
 to the number of turns multiplied by the diameters 
 of the Cores, and the square roots of the lengths. 
 
58 Elements of CondrvMion for Eleetro-Magmets. 
 We have then for a like intensity of current — 
 
 A 
 
 c'Vc\/b 
 
 <? ¥ (? e'5 e'^ h 
 
 A' 
 
 c'^Vc'Vb' 
 
 d^v^~d^<f ~ g" ~ y 
 
 which shows that the forces are then in inverse 
 proportion to the squares of the diameters, or pro- 
 portional to the lengths, admitting, however, that 
 the magnetic cores satisfy the desired conditions of 
 saturation, so that the laws of Dub and Muller may 
 apply. 
 
 To ascertain the modifications in the laws we have 
 been considering, caused by the more or less com- 
 plete saturation of a magnet in conjunction with its 
 length and diameter, I wound on three different bar 
 magnets of 8, 7, and 6 millimetres in diameter re- 
 spectively, a similar length of wire (No. 12 of -059 
 millimetre including the insulating covering). The 
 length of this wire was 71 "47 metres, equalling in 
 resistance 722 metres of telegraph wire, or 7-22 
 ohms ; and I calculated the length of my bobbins so 
 that the depth of the coils was always equal to the 
 diameter of the magnetic core. These bobbins had 
 consequently lengths of 5-9, 7-7, and 9-8 centi- 
 metres respectively, and the numbers of turns 
 were 1470, 1677, and 1842. The following are 
 the results obtained with a battery of from 1 to 
 4 elements (latest type, Leclanche), the resistance 
 of each cell being 113 metre, or 1-13 ohm. The 
 attractive force was measured at a distance of 1 
 millimetre. 
 
On Magnetic Saturation. 
 
 59 
 
 Resistances of Exterior 
 
 li^ 
 
 
 
 ■sf 
 
 li- 
 
 Circuit, ia metres. 
 
 ^ » ^ 
 
 o a 03 
 
 ill 
 
 
 Hi 
 
 
 @a a 
 
 MSlil 
 
 H;-a 
 
 Ss& 
 
 HiS 
 
 
 gr- 
 
 
 gr- 
 
 
 gr. 
 
 
 ^452+ 0= 452 
 
 225 
 
 1-018 
 
 221 
 
 •9210 
 
 240 
 
 ■§ 
 
 452+ 300= 752 
 
 151 
 
 1-049 
 
 144 
 
 -9660 
 
 149 
 
 § 
 
 452+ 400= 852 
 
 134 
 
 1-0635 
 
 126 
 
 •9770 
 
 129 
 
 8 
 
 452 + 1000 = 1452 
 
 73 
 
 1-0900 
 
 67 
 
 1-0307 
 
 65 
 
 % 
 
 452 + 2000 = 2452 
 
 36 
 
 1-161 
 
 31 
 
 1-0333 
 
 30 
 
 •* 
 
 452 + 3000 = 3452 
 
 2] 
 
 1-235 
 
 17 
 
 1-0630 
 
 16 
 
 V452 + 4000 = 4452 
 
 14 
 
 1-272 
 
 11 
 
 1-1000 
 
 10 
 
 
 ' 339 + = 339 
 
 169 
 
 1-0432 
 
 162 
 
 •953 
 
 170 
 
 S 
 
 339 + 300 = 639 
 
 109 
 
 1-0792 
 
 101 
 
 1-010 
 
 100 
 
 § 
 
 339+ 400= 739 
 
 95 
 
 1-092 
 
 87 
 
 1-0117 
 
 86 
 
 1 
 
 339 + 1000 = 1339 
 
 50 
 
 1-168 
 
 48 
 
 1-0238 
 
 42 
 
 »2 
 
 9 
 
 339 + 2000 = 2339 
 
 23 
 
 1-2105 
 
 19, 
 
 1-0555 
 
 18 
 
 CO 
 
 389 + 3000 = 3339 
 
 13 
 
 1-300 
 
 10 
 
 1-000 
 
 10 
 
 
 .339 + 4000 = 4339 
 
 8 
 
 1-333 
 
 6 
 
 1-000 
 
 6 
 
 
 226 + = 226 
 
 100 
 
 1-111 
 
 90 
 
 1-000 
 
 90 
 
 S 
 
 226+ 300= 526 
 
 60 
 
 1-154 
 
 52 
 
 1-0196 
 
 51 
 
 § 
 
 226 + 400 = 626 
 
 62 
 
 1-1818 
 
 44 
 
 1-0232 
 
 43 
 
 "o 
 
 226 + 1000 = 1226 
 
 26 
 
 1-8000 
 
 20 
 
 1-000 
 
 20 
 
 226 + 2000 = 2226 
 
 11 
 
 1-375 
 
 8 
 
 1-000 
 
 8 
 
 (M 
 
 226 + 3000 = 8226 
 
 6 
 
 1-500 
 
 4 
 
 1-000 
 
 4 
 
 
 226 + 4000 = 4226 
 
 4 
 
 2-000 
 
 2 
 
 1-000 
 
 2 
 
 
 fll8+ 0= 113 
 
 35 
 
 1-207 
 
 29 
 
 1-036 
 
 28 
 
 "S 
 
 113 + 300 = 413 
 
 19 
 
 1-2666 
 
 15 
 
 1-071 
 
 14 
 
 s 
 
 113 + 400 = 518 
 
 16 
 
 1-3338 
 
 12 
 
 1-0909 
 
 11 
 
 i 
 
 113 + 1000 = 1113 
 
 7 
 
 1-1666 
 
 6 
 
 1-200 
 
 5 
 
 "3 
 
 113 + 2000 = 2113 
 
 2 
 
 1-0000 
 
 2 
 
 2-000 
 
 1 
 
 T-l 
 
 113 + 3000 = 3113 
 
 1 
 
 1-0000 
 
 1 
 
 2-000 
 
 
 
 
 113 + 4000 = 4113 
 
 
 
 1-0000 
 
 
 
 2-000 
 
 
 
 These results show that in conformity with theory, 
 the longest electro-magnet is usually the strongest, 
 but in a ratio which is only in proportion to the 
 lengths when the diameters are approximately equal, 
 and for a certain intensity of current which probably 
 corresponds with the point of magnetic saturation. 
 
60 Elements of Construction for Electro-Magnets. 
 
 This intensity for the magnets of 9 • 8 and 7 • 7 centi- 
 metres, varied between -001082 and -001030, and 
 between -00143 and -00113 in the four series of 
 experiments ; but in comparing the electro-magnets 
 of 9-8 and 5-9 centimetres, we do not find this 
 proportion for any intensity of current, and the 
 ratio of force is always less than the length. Further 
 eyen, it will be observed that the strength of these 
 electro-magnets is so subject to the state of magnetic 
 saturation of the cores, that for high intensities or 
 low resistance circuits, it is the short, thick magnet 
 which has the advantage. On the contrary, the 
 preponderance of the long thin magnet becomes 
 more and more marked as the electric intensity 
 diminishes; whether this diminution arises from 
 fewer cells being employed, or from higher resistances 
 being introduced into the circuit, and we may judge 
 of the importance of these variations from the pro- 
 portions in the second and fourth columns in the 
 preceding table. It will be understood that this 
 should be the case, as for high electric intensities 
 the diameter of the long electro-magnet is not pro- 
 portional to these intensities, and its point of mag- 
 netic saturation is passed when it is hardly reached 
 by the short, thick magnet. On the other hand 
 the advantage of the small diameter is explained, 
 when we consider that the magnetic mass of the bar 
 being great enough to correspond with the electric 
 force developed, it receives the full benefit of the 
 greater number of turns in its magnetising coil. It 
 may then be concluded that the dimensions to be 
 
On Magnetic Saturation, 61 
 
 given to an electro-magnet, must essentially depend 
 upon the electric force which has to act upon it, and 
 on the resistance of the circuit in which it is intro- 
 duced. When the circuit is long and the electricity 
 of low intensity, they should be long, and small in 
 diameter ; when, on the other hand, the circuit is 
 short and the electricity of high intensity, the core 
 should certainly be of large diameter. We also arrive 
 
 E 
 at this deduction from the formula c = —i= • 173 
 
 vE 
 
 which I have given for calculating the diameter to 
 give to an electro-magnet according to the conditions 
 in which it may be employed. 
 
 As to the influence of the magnetic saturation of 
 the cores, it is somewhat difficult to formulate pre- 
 cisely. Joule, de Haldat, MuUer and Pobinson 
 long ago discovered that, at the commencement of 
 the action of the current, and when the magnetic 
 state of the iron is far from the point of saturation, 
 the attractive force, instead of increasing as the 
 square of the intensity of the current, increases in a 
 much more rapid proportion, which may be beyond 
 the third or even the fourth power of this intensity ; 
 but they also showed that as the electro-motive 
 force develops, this proportion diminishes rapidly, 
 till the point of saturation is reached, when it 
 remains some instants stationary, and then begins 
 to decrease beyond the point of saturation until it 
 becomes simply proportional to the intensity of the 
 current. 
 
 If this be the case as the magnetic force is de- 
 
62 Elements of Consitruction for Eledro-Magnels. 
 
 veloped, it must be the same when this force, being 
 capable of complete development, is produced by 
 cores of which the dimensions for a given electric 
 intensity receive a different magnetic saturation; 
 and we may judge by the results of the experiments 
 cited above of the truth of this deduction. In fact, 
 if we take the ratios of force of each of the three 
 electro-magnets in question when they are excited 
 by different electric intensities, we find that for each 
 of them there is an intensity with which the force 
 increases as the square of the intensity, and above or 
 below which it increases in a greater or less propor- 
 tion. It will also be noticed that this limit of 
 intensity varies according to the dimensions of the 
 electro-magnet. Thus, on compiling the following 
 table from these proportions, we see, firstly, that the 
 electro-magnet B best satisfies the law of the squares 
 of electric intensities ; secondly, that the large 
 magnet C gives a more rapid proportion; thirdly, 
 that the magnet A of small diameter gives a slower 
 proportion ; fourthly, that the ratio of forces for 
 these three electro-magnets is so much more rapid 
 compared with what it should be, according to the 
 law of the squares of the intensities, as these in- 
 tensities become weakened, whether this weakening 
 results from lessening the number of cells in the 
 battery or from an increase in the resistance of 
 the circuit. It is true that these ratios, notwith- 
 standing their great value, more nearly approach the 
 squares of the intensities than the cubes, but it will 
 be noticed that the three electro-magnets have 
 
On Magnetic Saturation. 
 
 63 
 
 somewliat similar diameters, and are very nearly in 
 the same condition of saturation, and that the forces 
 have been measured at their maximum magnetisation. 
 It is evident that it would not have been the same 
 with electro-magnets of diameters differing more 
 widely, or if the forces could have been measured 
 instantaneously at the moment of magnetisation, and 
 during a very short closing of the circuit. In this 
 case we might have found a ratio even higher than 
 85 ■ 600, the cube. It is besides on this principle 
 
 Eatio of the Squares 
 of the Intensities. 
 
 Eatio of the Magnetic Force of 
 
 Magnet A of 
 9-8 centimetres. 
 
 Magnet B of 
 1'1 centimetres. 
 
 Magnet C of 
 5-9 centimetres. 
 
 
 f 1-576 
 
 1-49 
 
 1 54 
 
 1-61 
 
 1 
 
 1-797 
 
 1-68 
 
 1-75 
 
 1-86 
 
 S 
 i 
 
 3-429 
 
 3-082 
 
 3-30 
 
 3-70 
 
 7-309 
 
 6-25 
 
 7-13 
 
 8- 
 
 -ffl 
 
 12-640 
 
 10-71 
 
 13- 
 
 15- 
 
 "H 
 
 1 19-423 
 
 16-07 
 
 .20-09 
 
 24- 
 
 ™ { 1'645 
 
 1-55 
 
 1-60 
 
 1-70 
 
 1 1-896 
 
 1-78 
 
 1-86 
 
 1-97 
 
 2 \ 3-773 
 
 3 1 8-323 
 
 3-38 
 
 3-77 
 
 4-05 
 
 7-35 
 
 8-52 
 
 9-44 
 
 "3 14-650 
 
 13- 
 
 16-20 
 
 17- 
 
 " V22-753 
 
 21-12 
 
 27- 
 
 28-33 
 
 „ f 1-733 
 
 1-66 
 
 1-73 
 
 1 76 
 
 •g 2-022 
 
 1-92 
 
 2-04 
 
 209 
 
 2 4-222 
 S 9-670 
 
 3-84 
 
 4-50 
 
 4-50 
 
 9-09 
 
 11-25 
 
 11-25 
 
 ■3 17-343 
 
 16-66 
 
 22-5 
 
 22-5 
 
 =*■ 1,27-242 
 
 25- 
 
 45- 
 
 45- 
 
 / 1'847 
 
 1-84 
 
 1-93 
 
 2- 
 
 "S 2-187 
 
 2-19 
 
 2-42 
 
 2-55 
 
 i , 4-829 
 
 5- 
 
 4-83 
 
 5-60 
 
 » \ 11-527 
 
 17-50 
 
 14-50 
 
 28- 
 
 Z 21-094 
 
 35- 
 
 29- 
 
 
 V33-529 
 
 
 
 
64 Elements of Construction for Electro-Magnets. 
 
 that the action of slow-working magnets in some 
 printing telegraphs is based, and which are only slow- 
 working because their mass being relatively greats 
 they take a certain time to magnetise. However, the 
 foregoing is the table to which I have referred. 
 
 We may deduce from the preceding results, that 
 the law of the proportion of attractive force to the 
 square of the intensity of the current is only true 
 within certain limits, and under certain conditions 
 and that electro-magnets, on a circuit which is rapidly 
 made and broken, are more or less exempt there- 
 from. And as this is the case in which it is most 
 important to ascertain what are the best conditions, 
 it has appeared to me that the matter should be 
 restudied from this point of view, not to fix new 
 arbitrary conditions of maximum, but to ascertain in 
 what way those already deduced must be modified to 
 suit any particular case. 
 
 The question to resolve is this : When the electro- 
 magnetic force increases in a greater ratio than that 
 of the square of the intensity of the current, as, for 
 instance, as the cube of this intensity, should the 
 resistance of the magnetising coils be greater or less 
 than that of .the exterior circuit ? 
 
 To solve this question we have only to change in 
 the formula giving the value of the electro-motive 
 force, the indices of the quantities in proportion with 
 this quantity ; in a word, to transform I^ into P. We 
 thus obtain a reduced expression 
 
 8/2 7r&a(a-fc) = 42E/; 
 
On Magnetic Saturation. 65 
 
 when we vary g, which gives for conditions of 
 maximum 
 
 ■irha{a + c) qRg^ u -^ 
 
 g^ - 2/2 . or, M _ - 
 
 and when we vary a these conditions will answer to 
 the equation 
 
 T, „ 7rha(2a + — ] 
 qBg^ \ ^ 2) 
 
 ,2 
 
 r ~ g 
 
 These two equations show that with the force pro- 
 portional to the cube of the intensity, the coils must 
 always have half the resistance of the exterior 
 circuit, when we take g as variable, and in the pro- 
 
 ft 
 
 portion of2a-f-^toa4-e when a is variable. We 
 
 may thus conclude that on circuits where the inter- 
 ruptions of current are rapid, the resistance of the 
 electro-magnet coils must be as much less as the 
 closing of the circuit is of short duration ; and it is 
 for this reason, as well as on account of the defective 
 insulation of telegraph lines and the development of 
 extra currents, that Mr. Hughes first, and the elec- 
 tricians of the telegraph department afterwards, 
 have considerably reduced the resistance of electro- 
 magnets introduced on long circuits. 
 
 It is also of interest to know how the resistance of 
 an electro-magnet should be modified, when, the 
 point of saturation being passed, the forces increase 
 less rapidly than the squares of the intensities. If 
 
66 Elements of Construction for Electro-Magnets. 
 
 we suppose that this increase is in a ratio simply 
 proportional to the intensity of the current it will 
 be seen that there will be no maximum possible, 
 and therefore we may advantageously increase the 
 resistance beyond the limits which have been assigned. 
 If we now endeavour to ascertain how the condi- 
 tions of maximum, as regards the depth of the mag- 
 netising coil, may be modified in consequence of the 
 complete saturation of the core, we shall see that we 
 may then increase this depth, which might with 
 advantage be double the diameter of this core, if the 
 force increased as the cube of the intensity of the 
 current. The maximum conditions of the formula 
 
 r\ TT & a ( a + c) 
 
 9 
 
 3 
 
 give, in fact, for conditions of maximum, taking c as 
 variable, a = 2 c. No maximum is, however, possible, 
 if we consider that the forces are simply proportional 
 to the intensities of the current. 
 
( 67 ) 
 
 CHAPTER VIII. 
 
 CONDITIONS FOE THE GOOD CONSTEUOTION OF 
 ELEOTEO-MAGNETS. 
 
 It is not only necessary to calculate accurately the 
 dimensions tQ give to an electro-magnet, but also to 
 employ it in such a way that it may most efficiently 
 develop its proper force, taking into account any 
 secondary action, upon which its useful application 
 may greatly depend. It is this which we will now 
 consider on the basis of the different results I have 
 obtained from very numerous experiments made by 
 me during the last twenty years and more. 
 
 I. Conditions in reference to exterior actions. — 
 Descartes, and several other scientists after him, 
 showed that if a mass of iron be applied to one pole 
 of a bar magnet, the attractive force of the other is 
 increased in proportion to the size of the iron. Dub, 
 in applying this phenomenon to the laws of electro- 
 magnets, stated that this increase of force arises 
 from the fact that the magnetic core is thereby 
 lengthened, and that the forces ought to increase in 
 consequence : but this increase, according to him, 
 would only be as the square roots of the lengths, 
 and according to my experiments this increase is 
 infinitely greater, since I have often found the forces 
 
 P 2 
 
68 Elements of Construction for Electro-Magnets. 
 
 tripled. I have also found that this increase may 
 be produced with masses of iron of all shapes, which 
 do not materially increase the length of the core, 
 and even when the iron is not in contact with the 
 magnetic core. I found also that the development 
 of the magnetised surface has great influence on 
 this phenomenon, and I have in consequence come 
 to the conclusion that this increase of energy results 
 principally from magnetic condensation towards the 
 iron, having the effect of stimulating the opposite 
 free pole of the magnet, since the opposite poles of a 
 magnet must correspond in force. 
 
 I also directly demonstrated this deduction by 
 placing on a long rod of soft iron, magnetising coils 
 of different lengths but composed of equal lengths of 
 wire, and I found that it was the coil that entirely 
 covered this rod that produced the least attractive 
 force although it had the greatest number of turns. 
 The coil which gave the best result was the one 
 which only covered one-third of the rod. When, 
 however, these same coils covered iron cores of the 
 same lengths as themselves, the result was quite 
 different ; the force then increased with the length 
 of the coils in a proportion less rapid than that of 
 the lengths, but which, from my experiments, seemed 
 to me to be in that of an arithmetical progression, 
 when the lengths increased as a geometrical progres- 
 sion; and the terms of this progression seemed to 
 increase in proportion to the number of cells em- 
 ployed. I only give these results, however, as 
 applying to a particular case. 
 
Form and Afplication of Armature. 69 
 
 The effects which I have described above, explain 
 the cause of the relatively great force developed by 
 horse-shoe magnets, with a bobbin on only one of 
 the arms ; electro-magnets which I was the first to 
 employ, and to which I gave the name of one- 
 legged (boiteiix) electro-magnets. 
 
 II. Conditions in reference to the form and appli- 
 cation of the armature. — The results of my experi- 
 ments in this respect may be numbered as follows : — 
 
 1st. The attractive force of any electro-magnet is 
 greater in proportion as the surface of its armature, 
 which directly receives the magnetic influence, is 
 extended, as it is thus brought into better relation 
 with the magnetic energy of the electro-magnet. 
 
 2nd. It follows that the attractive force of a horse- 
 shoe electro-magnet at a distance, is greater with a 
 prism-shaped armature placed lengthwise before the 
 poles, than with the end presented, although the re- 
 verse is the case when the attractive force is exerted 
 on contact. For an attraction at a distance the 
 effects produced may be in the proportion of 59 to 92. 
 
 3rd. Armatures moving at an angle with the line of 
 the poles of an electro-magnet, i.e. working on a 
 joint near one of the poles, are much more effective 
 than armatures moving parallel to this line or fitted 
 cross-wise to a rocking lever. This advantage is 
 especially noticeable with the one-legged electro- 
 magnets, and the force varies in the proportion of 
 125 to 64. 
 
 4th. Prismatic armatures are attracted in propor- 
 tion as the surface is extended, but the form of this 
 
70 Elements of Construction for Electro-Magnets. 
 
 surface has an immense influence on the attraction, 
 on account of the mean distance of all the points 
 which are under the influence of the magnet, which 
 distance may vary very much with this form. Thus 
 a cylindrical armature of the same surface as a 
 square one is attracted with much less force that the 
 latter, in the proportion of perhaps 85 to 44. 
 
 5th. In like manner the lateral attraction of an 
 electro-magnet whose ends protrude a little beyond 
 the bobbins, is infinitely less than the normal 
 attraction, i.e. that in line with the axis of the 
 poles, in the proportion of 33 to 18. 
 
 6th. Armatures formed of permanent magnets do 
 not increase the attraction except at a distance, and 
 when moving parallel to the line of the poles. In 
 other cases, the reverse obtains, as the magnetic 
 action on the steel is much less than on iron. 
 
 7th. The attractive force obtained by a momentary 
 closing of the circuit for a similar distance, is greater 
 than that obtained by the same current continuously 
 applied in the hope of overcoming the resistance. 
 This is on account of the vis viva, and of the effects 
 of polarisation of the battery. The proportion of 
 these forces is as 136 is to 95. 
 
 8th. When the attractive force of an electro- 
 magnet is divided over several armatures, the total 
 attractive force is increased, but the individual force 
 of each is diminished in proportion to their number. 
 This increase, however, is only shown up to a certain 
 limit, which is attained when the mass of the arma- 
 ture equals that of the electro-magnet. 
 
Form and Apjplicaiion of Armature. 71 
 
 9th. The attractive force of an electro-magnet 
 and armature which have never been used, is greater 
 for a given electric force, than that of the same 
 magnet and armature after being strongly magnetised, 
 and to obtain from the same magnet and armature 
 a nearly equal force to that first obtained, the current 
 must be reversed ; still this greater strength is then 
 only to be found on first closing the circuit. 
 
 10th. The attraction at a distance is less, when 
 from some cause the first closing of the circuit has not 
 been followed by a complete attraction of the arma- 
 ture ; this is explained, as also the preceding result, 
 by the effect of the residual magnetism. 
 
 11th. The repulsive force developed by electro- 
 magnets on a magnetised armature is very far from 
 corresponding with the attractive force which may 
 be obtained by reversing the poles of the magnet. 
 This fact, recognised in the earliest researches on 
 magnetism, and fully investigated by Mussembroeck 
 and the Abbot NoUet, is explained by the fact of the 
 magnet acting inductively, thus tending to develop 
 in the armature a polarity opposite to itself. In 
 attraction, this influence increases the result, whereas 
 in repulsion it has the contrary effect. 
 
 12th. When the iron cores protrude beyond the 
 magnetising coils their strength is diminished, but 
 if iron plates are fitted round the cores, the attractive 
 force is increased, and the maximum effect is obtained 
 when the distance between the two rings is about a 
 quarter of the distance between the poles. This is 
 occasioned by the greater attractive surface pre- 
 
72 Elements of Construction for Electro-Magnets. 
 
 sented, which then nearly corresponds with that of 
 the armature. 
 
 13th. According to the experiments of Dub, the 
 best results are obtained when the different parts 
 of the electro-magnet (arms of the magnet, breech 
 and armature) are equal in mass. 
 
 14th. This conclusion is of course subordinate to 
 the conditions of application, for it is certain that if 
 we wish to obtain a quick movement of the armature 
 we must have a light one ; but we may make up for 
 it by providing the poles with iron rings. 
 
 III. Conditions in reference to the magnetic mass. — 
 The conditions of force of electro-magnets with 
 reference to their diameter and their degree of 
 magnetic saturation, may, in certain circumstances, 
 be contradictory one to the other ; for, if we gain in 
 force by increasing the diameter of the magnetic 
 cores, we lose when, their diameter being too large, 
 the cores are not saturated, and this is in consequence 
 of secondary reactions exercised on the outside of 
 the cores by the central part, which is inert and 
 plays the part of a second armature. It has been 
 tried to obviate this inconvenience by employing 
 tubular cores, which at the same time facilitated 
 rapid magnetisation and demagnetisation ; but with 
 this system a considerable lessening of the attractive 
 force was the result, and it was abandoned for the 
 moment. In consequence, I undertook the study of 
 the conditions of force of this sort of electro- 
 magnet, and after numerous experiments recorded 
 in a memorandum presented to the Academie 
 
Conditions in Beference to Magnetia Mass. 73 
 
 des Sciences in 1862, I arrived at the following 
 conclusions : 
 
 1st. The greater force of electro-magnets with 
 solid cores is not in consequence of their greater 
 mass of metal, but chiefly depends on the disposition 
 of the polar surfaces with regard to the armature. 
 
 2nd. If the polar extremity of a tubular core is 
 provided, inside the tube, with an iron stopper, 
 which may be very thin, the force of the electro- 
 magnet is nearly the same as with a solid core ; but 
 this is not the case if we increase the polar surface 
 by surrounding the end of the tube with an iron 
 ring, which shows that it is not so much the increase 
 of the polar surface as its disposition which acts 
 upon the attractive force. 
 
 3rd. If we consider that the ring acts as an 
 armature to the detriment of the external attraction, 
 as when an electro-magnet acts at the same time on 
 several armatures, while with the iron stopper we 
 have a concentration of the magnetic effects pro- 
 duced by the different parts of the internal walls of 
 the tube, as much below the stopper as laterally, 
 we can understand that there must be a considerable 
 difference of force in the two cases ; and we may 
 easily see the effect of the magnetic concentration 
 produced in the second instance, by the projection 
 of the iron stopper out of the tube at the moment of 
 magnetisation, provided that the stopper is not fitted 
 in too firmly. 
 
 4th. It results from these effects, that, to usefully 
 employ tubular electro-magnets, they should be 
 
74 Elements of Oonsiruction for Electro-Magnets. 
 
 provided at the polar ends with iron stoppers of a 
 thickness equal at least to that of the tube. By 
 this arrangement an increase of force is gained, 
 which in telegraphic electro-magnets might be in 
 the proportion of 25 to 38. 
 
 As to the thickness of the tubular cores, it should 
 be in proportion to the magnetising current. Hughes 
 found by experience that it should be a quarter 
 of the diameter of the tube; but I have found, 
 by experiment, that for electro-magnets of large 
 diameter, this thickness may be reduced in still 
 greater proportion. In a memorandum inserted in 
 my 'Eecherches sur les meilleurs conditions de 
 construction des electro-aimants ' (page 112), I 
 treated this question at length, and showed that the 
 diameter c' of the tube should be expressed by 
 
 "' = « \/. 
 
 4(0!- 1)' 
 
 c being the diameter of a solid iron core, susceptible 
 of being magnetised to saturation, and x the divisor 
 of o' to represent the thickness. This value of x 
 may in some cases be as great as 7. 
 
 5th. The disadvantages of solid electro-magnets 
 result chiefly from the effects of residual magnetism, 
 and from magnetic condensation, but they may be 
 considerably reduced by magnetically isolating the 
 arms of the electro-magnet from the breech by means 
 of rings of copper, or by cutting in two the breech 
 itself by a thickness of copper. 
 
( 75 ) 
 
 CHAPTER IX. 
 
 ON THE BEST GROUPING OF THE CELLS OP A 
 BATTEKT. 
 
 As it is important to arrange an electro-magnet in 
 such a manner as to place it in its maximum con- 
 ditions with regard to a given circuit of resistance, 
 so we should endeavour to combine the cells of a 
 given battery so as to make it of most use on a 
 given circuit with a given electro-magnet. Ohm 
 has established for this particular case a formula, 
 generally given in electrical treatises, but not very 
 applicable in practice, and this has induced me, 
 since 1860, to endeavour, by another method, to 
 ascertain the best way of grouping the elements of a 
 battery. What is most important indeed to know, 
 is what is the most advantageous mode of grouping 
 the elements of a given battery to correspond with 
 a known exterior circuit of resistance E, and what 
 number of groups there should be, also the number 
 of elements in each. Thus, for a battery with a 
 number n of elements, the point in question is, whether 
 for a circuit of resistance R, it would be better to 
 form the elements of this battery into four groups in 
 tension, each consisting of three elements in quantity, 
 or six groups of double elements, or two groups of 
 
76 Elements of Construction for Electro-Magnets. 
 
 six elements, &c. Calculation will give us the desired 
 answer. 
 
 Let us suppose tliat we have a battery composed 
 of n elements ; we will take a for the number of 
 groups which should be joined for tension — i.e. by 
 opposite poles — and i for the number of elements 
 joined for quantity in each group — by similar poles. 
 If we want an expression for the intensity of the 
 current of each of these groups, by representing 
 by r the resistance of one element, by E the resist- 
 ance of the circuit, and E the electro-motive force 
 of a single element, we have, according to Ohm's 
 formula with respect to the grouping for quantity 
 
 1= ^ 
 
 
 and if we join for tension all these groups, of which 
 the number is represented by a, we have 
 
 a E ^, ab'E 
 
 T = or i = 
 
 ar ar-^oR 
 
 -J+ a 
 
 This formula, as well as that of Ohm, is susceptible 
 
 of maximum, ior al = n and I = - ; so that the 
 
 a 
 
 equation becomes 
 
 nE E 
 
 r = 
 
 ar + - R 
 a 
 
 ar 
 
 
 R 
 
 
 + 
 
 
 n 
 
 
 a 
 
 and the denominator being, with respect to a, con- 
 
which is a/ -^- or simply 
 
 Arrangement of the Battery. 77 
 
 r E 
 
 sidered as variahle, ? ; we obtain the maximum 
 
 n a 
 
 conditions by making ^ = 0, ov a = \/ » 
 
 and as a = -=- , we can also obtain the value of b, 
 
 
 
 n 
 R "' "'"^^•' a 
 
 We could, however, perfectly demonstrate this 
 deduction by simple reasoning, without the inter- 
 vention of differential calculation, as I have shown 
 in the first volume of my ' Expose des applications 
 de I'electricite ' (page 158). 
 
 There result from these conditions of maximum 
 some rather important consequences : — 
 
 1st. That when a battery is placed in its maximum 
 conditions with respect to a given exterior circuit 
 of resistance R, this resistance is equal to that of 
 
 the battery ; and from the equation a = » / — 
 
 get E = - r, and the second side of this equation 
 
 represents the resistance of the battery. 
 
 2nd. That we can know the most advantageous 
 method of arranging the elements of a battery, 
 starting from a given intensity I, provided this in- 
 tensity does not answer to a resistance E greater 
 than n r. And as, in its maximum conditions, we have 
 
 wE wE wE 
 
 we 
 
 ar + ili 2ar 2b B 
 
78 Elements of Construction for Electro-Magnets. 
 
 finally we haye from this 
 
 2IE ,^ 2Ir 
 
 a = -^ and 6 = -^ , 
 
 very conTenient formulas and extremely simple. 
 
 In the different works I have undertaken on this 
 
 subject, which were successively presented to the 
 
 Academie des Sciences in June and August 1860, 
 
 and in September 1869, I have pointed out the 
 
 limits of the resistance of the exterior circuit, between 
 
 which it is most advantageous to employ such and 
 
 such a mode of grouping ; and these limits, for the 
 
 n Y 
 coupling of 6 elements joined for quantity are t — r 
 
 n T 
 and ■ , , that is to say, for double elements ; 
 
 n Y nT 
 these resistance limits of E are -„- and -^ for triple 
 
 A O 
 
 n r n r n r 
 
 elements ; -w- and -^ for quadruple elements ; r^ 
 
 and -^, &c. ; and I have shown also, as Ohm had 
 
 established, that the resistances K corresponding to 
 the maximum of the different sorts of coupling are 
 
 represented by p-, that is to say by — when the 
 
 battery is arranged in double elements, by — when 
 
 y 
 
 it is arranged in triple elements, by — when in 
 
 quadruple elements, &c., &c. I thus show that the 
 limits of resistance of the exterior circuit, to give 
 
Arrangement of the Battery. 79 
 
 the same intensity with double, triple, quadruple 
 cells, &c^ compared with cells coupled up for tension 
 or for quantity, correspond on the one hand with the 
 half, third, or fourth of the total resistance of the 
 feattery, and on the other, with the half, third, or 
 fourth of the resistance of a single cell.* 
 
 Thus we shall know by means of the preceding 
 calculations, that, if we have a battery of 20 Daniell 
 cells, the resistance of the exterior circuit which 
 would necessitate its arrangement in double ele- 
 ments, will be between 10 kilometres and 3333 metres 
 
 * These different deductions are obtained by siiccesBiYely equal- 
 ising the different expressions representing the intensity of the 
 current furnished by the different groupings of the battery, and 
 extracting the value of E. We may even employ this means to 
 demonstrate without the assistance of the differential calculus, the 
 conditions of maximum of the formula, for, supposing h and 6' to 
 represent the number of cells in quantity in two nearly similar 
 groupings, we find on equalling the two expressions, 
 
 a r 
 whence 
 
 + 5E = a'r' + 5'E, or, E(6 - 5') = r^|, - ^V 
 
 
 which may be transformed into 
 
 according to whether we compare the two groupings in one way or 
 the other. But if we so approximate one to the other that b' = b 
 supposing that the battery corresponds with the maximum arrange- 
 ment, the preceding expression becomes E = -^ , and thence we 
 
 arrive at the equations expressing the maximum values of o and 5 
 given above. 
 
80 Elements of Construction for Electro-Magnets. 
 
 of telegraph wire. When the resistance is between 
 3333 metres and 1666 metres, we shall gain by- 
 arranging the cells three in series, and between 1666 
 and 1000 metres we ought to group them in fours. 
 
 From what we have said above, it will be seen 
 that nothing is easier than to calculate directly what 
 arrangement it will be best to make of a given 
 number of cells, to correspond with a given circuit 
 of resistance ; since if we divide successively by 4, 9, 
 16, &c., the total resistance of n elements, we find 
 which is the nearest to the given external resistance. 
 The figure of the divisor gives the square of the 
 number of elements in each group, and the number 
 of groups is obtained by dividing the total number 
 of elements by the number in quantity in each 
 group. 
 
 These calculations, as may be seen, are exceedingly 
 easy and within the reach of any intelligence, and I 
 have always been astonished that inventors have not 
 given them more attention, and thereby avoided 
 loss of time and money in useless researches, when 
 they might at once have ascertained the maximum 
 conditions obtainable. 
 
 Some people have objected that these calculations 
 cannot be precisely correct, unless we can get whole 
 numbers for a final result, as the elements of a 
 battery cannot be divided into fractions ; but in 
 electricity we must be content with approximate 
 data. Certainly the elements of a battery cannot be 
 divided up, and owing to that we are rarely able 
 exactly to satisfy the conditions of theoretical 
 
Comparison of Batteries. 81 
 
 maximum ; but we have a guide, and it the nearest 
 whole number which must be made use of. Thus, 
 for example, in the case mentioned in Chapter V. 
 we find that the value of a is 3 •074; and we must 
 evidently take 3 groups in tension as the nearest we 
 can get ; and as 8 divided by 8 gives 2-7, we must 
 put 3 elements in quantity as the best arrangement 
 we can make.* 
 
 The formulas which we have given above enable 
 us also to compare the forces of the different electric 
 generators. There are, however, in these calculations', 
 certain considerations which not having been looked 
 at from the same point of view by different scientists, 
 have led to regrettable disagreements. Thus, ac- 
 cording to Jacobi, two different batteries cannot be 
 compared unless they are both placed in their maxi- 
 mum conditions ; but for this it would be necessary 
 for the external resistance of their circuits to be 
 equal to their internal resistance,, and therefore their 
 
 intensity would be represented by „ — or nr-p- 
 
 And if, for the battery we are taking as the basis of 
 comparison, we take w = 1, reducing the preceding 
 
 * When the results given by calculation do not correspond with 
 whole numbers, the conditions of maximum cannot evidently be 
 exactly fulfilled by taking the whole numbers nearest to them. 
 Gangain mathematically examined this phase of the question, but 
 only arrived at undecided and complicated solutions, for he found 
 that sometimes a problem had two and even more solutions, while 
 others had only one. (See Annales Telegraphiques for January 
 1861.) At any rate the method described above is sufiScient to 
 obtain a practical maximum arrangement of the cells of a batterjf 
 in any given case. 
 
 G 
 
82 Elements of Construction for Electro-Magnets. 
 
 formula to „— , we have, supposing the intensities 
 
 equal, 
 
 w E _ E' n^ ^ E^ 
 
 2rr~2^'2b^~2r" 
 
 which may also be written, 
 
 bE _ E' ^ g E _ E' 
 2"r~2V ^"""^ 2R~ 2r" 
 
 and from which we may obtain the values of a and 6, 
 if we suppose the resistance R equal to the resis- 
 tance of the test cell. We have then, 
 
 E' ^ , E' r 
 
 and the total number of elements is then given by 
 multiplying a by h. 
 
 In this manner we find that 5 Bunsen cells equal 
 9 Daniell cells of 11 times the surface, or in other 
 words that 20 Daniells of ordinary size are equal to 
 one Bunsen. We also find that 3 Bunsen cells equal 
 in intensity 4 sulphate of mercury cells with 3^ the 
 surface of the Bunsen, or that to equal one Bunsen 
 we require 5 sulphate of mercury elements. 
 
 If the idea of Jacobi in giving the preceding 
 system of comparison has been well understood, it 
 will be seen that he endeavoured to avoid giving to 
 the exterior resistance E a definite value; but in 
 fact this is represented by that of the battery serving 
 as basis for the comparison, and in the preceding 
 
Comparison of Batteries. 83 
 
 examples it is equal to 153 metres of telegraph wire. 
 
 And it will be seen that as this resistance is increased 
 
 or diminished, the figures given above may be 
 
 singularly modified; for the value of R may then 
 
 become less and less important, or may become more 
 
 and more so, in the value of the denominator of the 
 
 formula giving the intensity of the current. This, 
 
 then, has caused many scientists to criticise this 
 
 system. We might take, as a basis for the resistance 
 
 R, the mean of the resistance of the batteries to be 
 
 compared, but this would not advance matters much, 
 
 and it could only be applied to short circuits. In 
 
 this case we might start with the intensity I of the 
 
 test battery and obtain the values of a and h of the 
 
 2 T R 
 battery composed according to the formulas a = -^ 
 
 21r . . 
 
 and h = , which in the case of batteries of 
 
 Daniells and Bunsens, for a resistance R of 158 metres, 
 
 would give 
 
 I = 36-55, 
 
 2 X 36-55 X 153 , ._ 
 
 ^ = 5973 =l-»7, 
 
 and 
 
 , 2 X 36-55 X 931 
 
 5973 
 
 = 11-4, 
 
 which bring us back to the conclusions previously 
 
 arrived at, for then R is equal to the resistance of the 
 
 test cell ; but if R equals 1000, I becomes equal to 
 
 9 • 64 ; a = 3 - 2 and 6 = 3, and we see that instead 
 
 of employing 20 Daniell cells, we have only to employ 
 
 9 to be equal to the Bunsen cell. 
 
 G 2 
 
84 Elements of Construetion for Electro-Magnets. 
 
 TABLE I. 
 
 Voltaic Constants. 
 
 
 Electro- 
 motive 
 force. 
 £. 
 
 Internal reeistanoe 
 in metres. 
 
 Specific 
 
 Batteries in most 
 general use. 
 
 With high 
 
 external 
 
 resistance. 
 
 With low 
 
 external 
 
 resistance. 
 
 oJE. 
 
 Daniell cell (sulphate of 
 copper solution) French 
 telegraph pattern. Small 
 size 
 
 Bunsen, with amalgamated 
 zinc, shortly after charging. 
 Same size as foregoing . . 
 
 Delanrier cell, with chromic 
 acid and solution of com- 
 mon salt. Same size as 
 foregoing 
 
 Bichromate of potash, one 
 liquid (Ohutaux). Largesize 
 
 Duchemin cell, perchloride 
 of iron and solution of 
 salt. Large size 
 
 Sulphuric acid and water 
 cell, the two liquids sepa- 
 rated. Small size . . 
 
 Marie-Davy cell, sulphate of 
 mercury and water, the 
 two liquids separated. 
 Small size 
 
 Leclanch^ cell, peroxide of 
 manganese and solution of 
 sal-ammoniac. Large size 
 
 De la Eue's fused chloride of 
 silver and zinc. Diminu- 
 tive size 
 
 Sulphate of lead cell (Prud- 
 homme). Same size as 
 ordinary Bunsen 
 
 5973 
 11123 
 
 12413 
 11400 
 
 9640 
 
 8547 
 
 8192 
 7529 
 5596 
 3301 
 
 931 
 153 
 
 366 
 600 
 
 942 
 
 880 
 
 550 
 400 
 
 748 
 880 
 
 180 
 57 
 
 » 
 
 » 
 It 
 
 225 
 
 1-00 
 1-86 
 
 2-08 
 1-91 
 
 1-61 
 
 1-43 
 
 1-37 
 
 1-26 
 
 •94 
 
 •55 
 
( 85 ) 
 
 TABLE II. 
 
 CALcniiATBD Values for the Wires of Bleoteo-Magnets.* 
 
 
 Wires 
 
 Approxi- 
 
 a 
 
 
 
 
 fiO 
 
 
 
 (French 
 
 mate 
 
 fors 
 
 g 
 
 / 
 
 f 
 
 
 g' 
 
 
 gauge). 
 
 B. W. G. 
 
 / 
 
 
 
 
 S'" 
 
 
 
 No. 
 
 No. 
 
 metres 
 
 metres 
 
 
 
 
 
 
 32 
 
 34 
 
 •00014 
 
 •00023 
 
 
 6428 
 
 2-6988 
 
 816-3 
 
 •0000000529 
 
 "S 
 
 28 
 
 32 
 
 •00022 
 
 ■00033 
 
 
 5000 
 
 2^2500 
 
 330-6 
 
 •0000001089 
 
 fc 
 
 24 
 
 31 full 
 
 ■00027 
 
 •00040 
 
 
 4814 
 
 2-1934 
 
 219-5 
 
 ■0000001600 
 
 M 
 
 n 
 
 20 
 
 28 
 
 •00035 
 
 •00048 
 
 
 3714 
 
 1^8796 
 
 130-6 
 
 •0000002304 
 
 16 
 
 27 
 
 •00040 
 
 •00055 
 
 
 3750 
 
 1-8906 
 
 100-0 
 
 •0000003025 
 
 12 
 
 25 bare 
 
 ■00049 
 
 •00065 
 
 
 3265 
 
 1-7582 
 
 66-6 
 
 ■ 0000004225 
 
 w 
 
 P 
 
 24 Ml 
 
 •00058 
 
 •00077 
 
 
 3275 
 
 1-7609 
 
 47-5 
 
 0000005929 
 
 
 1 
 
 23 bare 
 
 •00060 
 
 •00122 
 
 
 0333 
 
 4-1831 
 
 44-4 
 
 ■000001488 
 
 
 2 
 
 22 „ 
 
 •00070 
 
 ■00134 
 
 
 9143 
 
 3-6634 
 
 32^6 
 
 •000001795 
 
 
 3 
 
 21 „ 
 
 •00080 
 
 •00146 
 
 
 8250 
 
 3-3306 
 
 25^0 
 
 •000002132 
 
 
 4 
 
 20 full 
 
 •00090 
 
 •00158 
 
 
 7555 
 
 3^0800 
 
 19-8 
 
 •000002396 
 
 
 5 
 
 19 bare 
 
 ■00100 
 
 •00172 
 
 
 7200 
 
 2 •9584 
 
 16^0 
 
 000002958 
 
 "S 
 
 6 
 
 19 full 
 
 •00110 
 
 •00184 
 
 
 6727 
 
 2^7989 
 
 13^2 
 
 ■000003385 
 
 ^ 
 
 7 
 
 18 bare 
 
 •00120 
 
 •00196 
 
 
 6333 
 
 2,-6687 
 
 11-1 
 
 ■000003842 
 
 1 
 a 
 
 8 
 
 18 full 
 
 •00130 
 
 •00208 
 
 
 6000 
 
 2 •5600 
 
 9^4 
 
 ■000004326 
 
 9 
 
 17 bare 
 
 •00140 
 
 •00220 
 
 
 5714 
 
 2-4670 
 
 8^2 
 
 ■000004840 
 
 10 
 
 17 full 
 
 •00150 
 
 •00232 
 
 
 5466 
 
 2^3901 
 
 7^1 
 
 •000005382 
 
 ■g 
 
 11 
 
 16 full 
 
 •00170 
 
 ■00254 
 
 
 4941 
 
 2^2320 
 
 5^5 
 
 ■000006451 
 
 O 
 
 12 
 
 15 bare 
 
 •00180 
 
 ■00266 
 
 
 4777 
 
 2-1844 
 
 4^9 
 
 ■000007075 
 
 
 13 
 
 14 full 
 
 •00200 
 
 ■00288 
 
 
 4444 
 
 2-0851 
 
 4-0 
 
 ■000008294 
 
 
 14 
 
 14 
 
 •00210 
 
 •00300 
 
 
 4285 
 
 2-0892 
 
 3^6 
 
 •000009000 
 
 
 15 
 
 13 bare 
 
 ■00235 
 
 •00327 
 
 
 3915 
 
 1-9349 
 
 2^9 
 
 •000010690 
 
 
 16 
 
 13 full 
 
 •00260 
 
 •00354 
 
 1-3615 
 
 1-8523 
 
 2-4 
 
 •000012532 
 
 * These diameters of the wire, wrth or without its covering, are not 
 exactly the results that would be obtained by actually measuring the 
 wire ; but these dimensions have been calculated from electro-magnets 
 already made, whereby empty spaces in the coils and irregularities in 
 the winding, which cannot be avoided, are brought into the values, 
 and although the figures are in consequence not mathematically exact, 
 the result is that we thus have much more correct data for practical 
 purposes. 
 
86 Elements of Construction for Electro-Magnets. 
 
 TABLE ni. 
 
 Table of the Resistaitobs op Copper Wiee Used in 
 Blbotkio Applications. 
 
 Diameters in 
 milUmetres. 
 
 Lengths per 
 
 kilogramme in 
 
 metres. 
 
 Eesistances per 
 
 kilogramme 
 
 in olims. 
 
 Eesistances per 
 
 kilometre in 
 
 ohms. 
 
 •02 
 
 355584 
 
 1803084 
 
 52317 
 
 •10 
 
 14369 
 
 30377 
 
 2113 
 
 •20 
 
 3614 
 
 1922 
 
 531 
 
 •30 
 
 1607 
 
 378 
 
 235 
 
 •40 
 
 902 
 
 119 
 
 133 
 
 •50 
 
 576 
 
 49 
 
 85 
 
 •60 
 
 401 
 
 24 
 
 59 
 
 •70 
 
 294 
 
 13 
 
 43 
 
 •80 
 
 226 
 
 7 
 
 5 
 
 33 
 
 •90 
 
 178 
 
 4 
 
 6 
 
 26 
 
 1^00 
 
 144 
 
 3 
 
 
 
 21 
 
 1^50 
 
 64 
 
 
 59 
 
 9-2 
 
 2-00 
 
 36 
 
 
 19 
 
 5^3 
 
 2^50 
 
 23 
 
 
 078 
 
 3-39 
 
 3'00 
 
 16 
 
 
 037 
 
 2-36 
 
 3-50 
 
 12 
 
 
 020 
 
 1-72 
 
 4-00 
 
 9 
 
 
 Oil 
 
 1-32 
 
 4-60 
 
 7 
 
 
 0074 
 
 1-04 
 
 5-00 
 
 6^76 
 
 
 0049 
 
 •84 
 
 5-50 
 
 4-71 
 
 
 0033 
 
 •70 
 
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 Steam Rolling— Road Metal and Breaking — Pitched Pavements— Asphalte — Wood Pavements 
 — Footpaths — Kerbs and Gutters — Street Naming and Numbering— Street Lighting — Sewer- 
 age — Ventilation of Sewers — Disposal .of Sewage — House Drainage — Disinfection— Gas and 
 Water Companies, &c., Breaking up Streets — Improvement of Private Streets — Borrowing 
 Powers — Artizans' and Labourers' Dwellings — Public Conveniences — Scavenging, including 
 Street Cleansing — Watering and the Removing of Snow — Planting Street Trees — Deposit of 
 Plans — Dangerous Buildings — Hoardings — Obstructions — Improving Street Lines — Cellar 
 Openings — Public Pleasure Grounds — Cemeteries — Mortuaries — Cattle and Ordinary Markets - 
 — Public Slaughter-houses, etc. — Giving numerous Forms of Notices, Specifications, and 
 General Information upon these and other subjects of great importance to Municipal Engi- 
 neers and others engaged in Sanitary Work. 
 
6 CATALOGUE OF SCIENTIFIC BOOKS 
 
 Tables of the Principal Speeds occurring in Mechanical 
 
 Engineering, expressed in metres in a second. By P. Keerayeff, Chief 
 Mechanic of the Obouchoff Steel Works, St. Petersburg ; translated by 
 Sergius Kern, M.E. Fcap. 8vo, sewed, td. 
 
 A Treatise on the Origin, Progress, Prevention, and 
 
 Cure of Dry Rot in Timber; with Remarks on the Means of Preserving 
 Wood from Destruction by Sea- Worms, Beetles, Ants, etc. By Thomas 
 Allen Britton, late Surveyor to the Metropolitan Board of Works, 
 etc., etc. With lo plates, crown 8vo, cloth, yj. ()d. 
 
 Metrical Tables. By G. L. Molesworth, M.I.C.E. 
 
 32mo, cloth, IS, 6d. 
 
 Contents. 
 
 Genera! — Linear Measures — Square Measures — Cubic Measures — Measures of Capacity — 
 Weights — Combinations — ^Thermometers. 
 
 Elements of Construction for Electro-Magnets. By 
 
 Count Th. Du Moncel, Mem. de I'lnstitut de France. Translated from 
 the French by C. J. Wharton. Crown 8vo, cloth, 4J. bd. 
 
 Electro -Telegraphy. By Frederick S. Beechey, 
 
 Telegraph Engineer. A Book for Beginners. Illustrated. Fcap. 8vo, 
 sewed, dd. 
 
 Handrailing : by the Square Ctit. By John Jones, 
 
 Staircase Builder. Fourth edition, with seven plates, 8vo, cloth, 3J. dd. 
 
 Handrailing : by the Square Cut. By John Jones, 
 
 Staircase Builder. Part Second, with eight plates, 8vo, cloth, 3J-. dd. 
 
 Practical Electrical Units Popularly Explained, with 
 
 numerous illustrations and Remarks. By James Swinburne, late of 
 J. W. Swan and Co., Paris, late of Brush-Swan Electric Light Company, 
 U.S.A. i8mo, cloth, is. dd. 
 
 Philipp Reis, Inventor of the Telephone: A Biographical 
 
 Sketch. With Documentary Testimony, Translations of the Original 
 Papers of the Inventor, &c. By SiLVANUS P. Thompson, B.A., Dr. Sc, 
 Professor of Experimental Physics in University College, Bristol. With 
 illustrations, 8vo, cloth, 7j. dd. 
 
 A Treatise on the Use of Belting for the Transmis- 
 sion of Power. By J. H. CooPER. Second edition, illustrated, 8vo, 
 cloth, 1 5 J. 
 
PUBLISHED BY E. & F. N. SPOjST. 
 
 A Pocket-Book of Useful Formulcs and Memoranda 
 
 for Civil and Mechanical Engineers, By Guilford L. Molesworth, 
 Mem. Inst. C.E., Consulting Engineer to the Government of India for 
 State Railways. With mitnerous illustrations, 744 pp.^ Twenty-first 
 edition, revised and enlarged, 32mo, roan, 6j. 
 
 Synopsis of Contents: 
 
 Surveying, Levelling, etc.— Strength and Weight of Materials— Earthwork, Brickwork, 
 Masonry, Arches, etc.— Struts, Columns, Beams, and Trusses— Flooring, Roofing, and Roof 
 Trasses— Girders, Bridges, etc.— Railways and Roads— Hydraulic Formulae— Canals. Sewers, 
 Waterworks, Docks— Irrigation and Breakwaters— Gas, Ventilation, and Warming — Heat, 
 Light, Colour, and Sound— Gravity : Centires, Forces, and Powers— Millwork, Teeth of 
 Wheels, Shafting, etc.— Workshop Recipes— Sundry Machinery— Animal Power— Steam and 
 the Steam Engine— Water-power, Water-wheels, Turbines, etc.— Wind and Windmills- 
 Steam Navigation, Ship Building, Tonnage, etc. — Gunnery, Projectiles, etc.r-Weights, 
 Measures, and Money— Trigonometry, Conic Sections, and Curves — Telegraphy— Mensura- 
 tion — Tables of Areas and Circumference, and Arcs of Circles — Logarithms, Square and 
 Cube Roots, Powers — Reciprocals, etc. — Useful Numbers — Differential and Integral Calcu- 
 lus—Algebraic Signs— Telegraphic Construction and Formulae. 
 
 Spans Tables and Memoranda for Engineers; 
 
 selected and arranged by J. T. HURST, C.E., Author of 'Architectural 
 Surveyors' Handbook,' 'Hurst's Tredgold's Carpentry,' etc. Fifth edition, 
 64mo, roan, gilt edges, is. ; or in cloth case, \s. dd. 
 
 ^This work is printed in ^a pearl type, and is so small, measuring only si in. by li in. by 
 ■J m. thick, that it may be easily carried in the waistcoat pocket. 
 
 " It is certainly an extremely rare thing for a reviewer to be called upon to notice a volume 
 measuring but a^ in. by if in., yet these dimensions faithfully represent the size of the handy 
 little book before us. The volume — which contains 118 printed pages, besides a few blank 
 pages for memoranda— -is, in fact, a true pocket-book, adapted for being carried in the waist- 
 coat pocket, and containing a far greater amount and variety of information than most people 
 
 would imagine could be compressed into so small a space The little volume has been 
 
 compiled with considerable care and judgment, and we can cordially recommend it to our 
 readers as a useful little pocket companion." — Engineering, 
 
 A Practical Treatise on Natural and Artificial 
 
 Concrete, its Varieties and Constructive Adaptations. By Henry Reid, 
 Author of the ' Science and Art of the Manufacture of Portland Cement.' 
 New Edition, with 59 woodcuts and ^plates, Svo, cloth, 15^. 
 
 Hydrodynamics : Treatise relative to the Testing of 
 
 Water- Wheels and Machinery, with various other matters pertaining to 
 Hydrodynamics. By James Emerson. With numerous illustrations, 
 360 pp. Third edition, crown Svo, cloth, 4^. dd. 
 
 Electricity as a Motive Power, By Count Th. Du 
 
 MoNCEL, Membre de I'lnstitut de France, and Frank Geraldy, Inge- 
 nieur des Fonts et Chaussees. Translated and Edited, with Additions, by 
 C. J. Wharton, Assoc. Soc. Tel. Eng. and Elec. With 113 engravings 
 and diagrams, crown Svo, cloth, yj. €d. 
 
 Hints on Architectural Draughtsmanship. By G. W. 
 
 Tuxford Hallatt. Fcap. Svo, cloth, u. dd. ■ 
 
CATALOGUE OF SCIENTIFIC BOOKS 
 
 Treatise on Valve-Gears, with special consideration 
 
 of the Link-Motions of Locomotive Engines. By Dr. GuSTAV Zeuner, 
 Professor of Applied Mechanics at the Confederated Polytechniltum of 
 Zurich. Translated from the Fourth German Edition, by Professor J. F. 
 Klein, Lehigh XJniversity, Bethlehem, Pa. Illustrated, 8vo, cloth, \2s. 6d. 
 
 The French- Polisher s Manual. By a French- 
 
 Polisher; containing Timber Staining, Washing, Matching, Improving, 
 Painting, Imitations, Directions for Staining, Sizing, Embodying, 
 Smoothing, Spirit Varnishing, French-Polishing, Directions for Re- 
 polishing. Third edition, royal 32mo, sewed, dd. 
 
 Hops, their Cultivation, Commerce, and Uses in 
 
 various Countries. By P. L. SiMMONDS. Crown 8vo, cloth, 4J. (>d. 
 
 A Practical Treatise on the Manufacture and Distii- 
 
 bution of Coal Gas. By William Richards. Demy 410, with numerous . 
 ■wood engravings and 29 plates, cloth, 2&f. * 
 
 Synopsis of Contents : 
 
 Introduction — History of Gas Lighting — Chemistry of Gas Mfinufactiire, by Lewis 
 Thompson, Esq., M.R.C.S. — Coal, with Analyses, by J. Paterson, Lewis Thompson, and 
 G. R, Hislop, Esqrs. — Retorts, Iron and Clay — Retort Setting — Hydraulic_ Main — Con- 
 densers — Exhausters — Washers and Scrubbers — Purifiers — Purification — History of Gas 
 Holder — Tanks, Brick and Stone, Composite, Concrete, Cast-iron, Compound Annular 
 Wrought-iron — Specifications — Gas Holders — Station Meter — Governor — Distribution — 
 Mains — Gas Mathematics, or Formula: for the Distribution of Gas, by Lewis Thompson, Esq. — 
 Services — Consumers' Meters — Regulators — Burners — Fittings — Photometer — Carburization 
 of 'Gas — Air Gas and Water Gas — Composition of Coal Gas, by Lewis Thompson, Esq. — 
 Analyses of Gas — Influence of Atmospheric Pressure and Temperature on Gasy— Residual 
 Products — Appendix — Description of Retort Settings, Buildings, etc., etc. 
 
 Practical Geometry, Perspective, and Engineering 
 
 Drawing; a Course of Descriptive Geometry adapted to the Require- 
 ments of the Engineering Draughtsman, including the determination of 
 cast shadows and Isometric Projection, each chapter being followed by 
 numerous examples ; to which are added rules for Shading, Shade-lining, 
 etc., together with practical instructions as to the Lining, Colouring, 
 Printing, and general treatment of Engineering Drawings, with a chapter 
 on drawing Instruments. By George S. Clarke, Capt. R.E. Second 
 edition, with 21 plates. 2 vols., cloth, loj. dd. 
 
 The Elements of Graphic Statics. By Professor 
 
 Karl Von Ott, translated from the German by G. S. Clarke, Capt. 
 R.E., Instructor in Mechanical Drawing, Royal Indian Engineering 
 College. With 93 illustrations, crown 8vo, cloth, 5^. 
 
 The Principles of Graphic Statics. By George 
 
 Sydenham Clarke, Capt. Royal Engineers. With 112 illustrations. 
 4to, cloth, I2J. dd. 
 
 Dynamo-Electric Machinery : A Manual for Students 
 
 of Electro-technics. By Silvanus P. Thompson, B.A., D.Sc, Professor 
 of Experimental Physics in University College, Bristol, etc., etc. Illus- 
 trated, 8vo, cloth, I2S. 6d. 
 
PUBLISHED BY E. & F. N. SPON. 
 
 The Netp Formula for Mean Velocity of Discharge 
 
 of Rivers and Canals. By W. R. KuTTER. Translated from articles in 
 the 'Cultur-Ingenieur,' by Lowis D'A. Jackson, Assoc. Inst. C.E. 
 8vo, cloth, I2s. td. 
 
 Practical Hydraulics ; a Series of Rules and Tables 
 
 for the use of Engineers, etc., etc By THOMAS Box. Fifth edition, 
 numerous plates, post 8vo, cloth, 5^. 
 
 A Practical Treatise on the Construction of Hori- 
 zontal and Vertical. Waterwheels, specially designed for the use of opera- 
 tive mechanics. By William CULLEN, Millwright and Engineer. With 
 II plates. Second edition, revised and enlarged, small 4to, cloth, I2j. td. 
 
 Tin: Describing the Chief Methods of Mining, 
 
 Dressing and Smelting it abroad ; with Notes upon Arsenic, Bismuth and 
 Wolfram. 'By Arthur G. Charleton, Mem. American Inst, of 
 Mining Engineers. With plates, 8vo, cloth, I2s. 6d. 
 
 Perspective, Explained and Illustrated. By G. S. 
 
 Clarke, Capt. R.E. Witk illustrations, 8vo, cloth, 3^. dd. 
 
 The Essential Elements of Practical Mechanics ; 
 
 based on the Principle of Work, designed for Engineering Students. By 
 Oliver Byrne, formerly Professor of Mathematics, College for Civil 
 Engineers. Third edition, with 148 wood engravings, pos't 8vo, cloth, 
 is. td. 
 
 Contents : 
 
 Chap. I. How Work is Measured by a Unit, botli witli and without reference to a Unit 
 of Time — Chap. 2. The Worlc of Living Agents, the Influence of Friction, and introduces' 
 one of the most beautiful Laws of Motion^-rChap. i. The principles expounded in the first and 
 second chapters are applied to the Motion of Bodies — Chap. 4. The Transmission of Work by 
 simple Machines — Chap. 5. Useful Propositions and Rules. ^ 
 
 The Practical Millwright and Engineers Ready 
 
 Reckoner; or Tables for finding the diameter and power of cog-wheels, 
 diameter, weight, and power of shafts, diameter and strength of bolts, etc. 
 By Thomas Dixon. Fourth edition, i2mo, cloth, y. 
 
 Breweries and Maltings : their Arrangement, Con- 
 struction, Machinery, and Plant. By G. Scamell, F.R.I.B.A. Second 
 edition, revised, enlarged, and partly rewritten. By F. Colyer, M.I.C.E., 
 M.I.M.E. With 20 plates, Svo, cloth, i&s. 
 
 A Practical Treatise on the Manufacture of Starch, 
 
 Glucose, Starch-Sugar, and Dextrine, based on the German of L. Von 
 Wagner, Professor in the Royal Technical School, Buda Pesth, and 
 other authorities. By Julius Frankel ; edited by Robert Hutter, 
 proprietor of the Philafielphia Starch Works. VHth 58 illustrations, 
 344 pp., 8vo, cloth, l8f. 
 
 B 3 
 
CATALOGUE OF SCIENTIFIC BOOKS 
 
 A Practical Treatise on Mill-gearing, Wheels, Shafts, 
 
 Riggers, etc.; for the use of Engineers. By Thomas Box. Third 
 edition, with \i plates. Crown 8vo, cloth, Is. 6d. 
 
 Mining Machinery : a Descriptive Treatise on the 
 
 Machinery, Tools, and other Appliances used in Mining. By G. G. 
 Andre, F.G.S., Assoc. Inst. C.E., Mem. of the Society of Engineers. 
 Royal 4to, uniform with the Author's Treatise on Coal Mining, con- 
 taining 182 plates, accurately drawn to scale, with descriptive text, in 
 2 vols., cloth, 3/. 12s. 
 
 Contents : 
 
 Machinery for Prospecting, Excavating, Hauling, and Hoisting — Ventilation — Pumping — 
 Treatment of Mineral Products, including Gold and Silver, Copper, Tin, and Lead, Iron, 
 Coal, Sulphur, China Clay, Brick Earth, etc. 
 
 Tables for Setting out Curves for Railways, Canals, 
 
 Roads, etc., varying from a radius of five chains to three miles. By A. 
 Kennedy and R. W. Hackwood. Illustrated, 32mo, cloth, zs. 6d. 
 
 The Science and Art of the Mamifacture of Portland 
 
 Cement, with observations on some of its constructive applications. With 
 66 illustrations. By Henry Reid, C.E., Author of 'A Practical 
 Treatise on Concrete,' etc., etc. 8vo, cloth, iSj. 
 
 The Draughtsman's Handbook of Plan and Map • 
 
 Drawing; including instructions for the preparation of Engineering, 
 Architectural, and Mechanical Drawings. With numerous illustrations 
 in the text, and 33 plates (15 printed in colours). By G. G. Andre, 
 F.G.S., Assoc. Inst. C.E. 4to, cloth, gj. 
 
 Contents : 
 
 The Drawing' Office and its Furnishings — Geometrical Problems — Lines, Dots, and their 
 Combinations — Colours, Shading, Lettering, Bordering, and North Points — Scales — Plotting 
 — Civil Engineers' and Surveyors' Plans — Map Drawing — Mechanical and Architectural 
 Drawing — Copying and Reducing Trigonometrical Formula, etc., etc. 
 
 The Boiler-maker^ s andiron Ship-builder s Companion, 
 
 comprising a series of original and carefully calculated tables, of the 
 utmost utility to persons interested in the iron trades. By James Foden, 
 author of ' Mechanical Tables,' etc. Second edition revised, with illustra- 
 tions, crown 8vo, cloth, 5^. 
 
 Pock Blasting: a Practical Treatise on the means 
 
 employed in Blasting Rocks for Industrial Purposes. By G. G. Andr£, 
 F.G.S., Assoc. Inst. C.E. With 56 illustrations and 12 plates, 8vo, cloth, 
 los. dd. 
 
 Painting and Painters' Manual: a Book of Facts 
 
 for Painters and those who Use or Deal in Paint Materials. By C. L. 
 Condit and J. Scheller. Illtistrated, 8vo, cloth, 10s. 6d. 
 
PUBLISHED BY E. & F. N. SPON. 
 
 A Treatise on Ropemaking as practised in public and 
 
 private Rofe-yards, with a Description of the Manufacture, Rules, Tables 
 of Weights, etc., adapted to the Trade, Shipping, Mining, Railways, 
 Builders, etc. By R. Chapman, formerly foreman to Messrs. Huddart 
 and Co., Limehouse, and late Master Ropemaker to H.M. Dockyard, 
 Deptford. Second edition, i2mo, cloth, 3^. 
 
 Laxtons Builders' and Contractors Tables ; for the 
 
 use of Engineers, Architects, Surveyors, Builders, Land Agents, and 
 others. Bricklayer, containing 22 tables, with nearly 30,000 calculations. 
 4to, cloth, 5j. 
 
 Laxtons Builders and Contractors' Tables. Ex- 
 cavator, Earth, Land, Water, and Gas, containing 53 tables, vfith nearly 
 24,000 calculations. 4to, cloth, Sj. 
 
 Sanitary Engineering: a Guide to the Construction- 
 
 of Works of Sewerage and House Drainage, with Tables for facilitating ' 
 the calculations of the Engineer. By Baldwin Latham, C.E., M. Inst. 
 C.E., F.G.S., F.M.S., Past-President of the Society of Engineers. Second 
 edition, with numerous plates and woodcuts, 8vo, cloth, i/. loj. 
 
 Screw Cutting Tables for Engineers and Machinists, 
 
 giving the values of the different trains of Wheels required to produce 
 Screws of any pitch, calculated by Lord Lindsay, M.P., F.R.S., F.R.A.S., 
 etc. Cloth, oblong, 2s. 
 
 Screw Cutting Tables, for the use of Mechanical 
 
 Engineers, showing the proper arrangement of Wheels for cutting the 
 Threads of Screws of any required pitch, with a Table for making the 
 Universal Gas-pipe Threads and Taps. By W. A. Martin, Engineer. 
 Second edition, oblong, cloth, \s., or sewed, dd. 
 
 A Treatise on a Practical Method of Designing Slide- 
 
 Valve Gears hy Simple Geometrical Construction, based upon the principles 
 enunciated in Euclid's Elements, and comprising the various forms of 
 Plain Slide- Valve and Expansion Gearing ; together with Stephenson'5, 
 Gooch's, and Allan's Link-Motions, as applied either to reversing or to 
 variable expansion combinations. • By Edward J. Cowling Welch, 
 Memb. Inst. Mechanical Engineers. Crown 8vo, cloth, (is. 
 
 Cleaning and Scouring : a Manual for Dyei's, Laun- 
 dresses, and for Domestic Use. By S. Christopher. i8mo, sewed, (sd. 
 
 A Handbook of House Sanitation ; for the use of all 
 
 persons seeking a Healthy Home. A reprint of those portions of Mr. 
 Bailey-Denton's Lectures on Sanitary Engineering, given before the 
 School of Military Engineering, which related to the "Dwelling," 
 enlarged and revised by his Son, E. F. Eailey-Denton, C.E., B.A. 
 With 140 illuHrations, 8vo, cloth, 8r. dd. 
 
CATALOGUE OF SCIENTIFIC BOOKS 
 
 A Glossary of Terms used in Coal Mining. By 
 
 William Stukeley Gresley, Assoc. Mem. Inst. C.E., F.G.S., Member 
 of the North of England Institute of Mining Engineers. Illustrated with 
 numerous woodcuts and diagrams, crown 8vo, clotli, ^s. 
 
 A Pocket-Book for Boiler Makers and Steam Users, 
 
 comprising a variety of useful information for Employer and Workman, 
 Government Inspectors, Board of Trade Surveyors, Engineers in charge 
 of Works and Slips, Foremen of Manufactories, and the general Steam- 
 using Public. By Maurice John Sexton. Second edition, royal 
 32mo, roan, gilt edges, %s. 
 
 The Strains upon Bridge Girders and Roof Trusses, 
 
 including the Warren, Lattice, Trellis, Bowstring, and other Forms of 
 Girders, the Curved Roof, and Simple and Compound Trusses. By 
 Thos. Cargill, C.E.B.A.T., CD., Assoc. Inst. C.E., Member of the 
 Society of Engineers. With 64 illustrations, drawn and worked out to scale, 
 8vo, cloth, I2J-. (>d. 
 
 A Practical Treatise on the Steam Engine, con- 
 taining Plans and Arrangements of Details for Fixed Steam Engines, 
 with Essays on the Principles involved in Design and Construction. By 
 Arthur Rigg, Engineer, Member of the Society of Engineers and of 
 the Royal Institution of Great Britain. Demy 410, copiously illustrated 
 with woodcuts and 96 plates, in one Volume, half-bound morocco, 2/. 2,s. ; 
 or cheaper edition, cloth, 25^. 
 
 This work is rot, in any sense, an elementary treatise, or history of tlie steam engine, but 
 is intended to describe examples of P'ixed Steam Engines without entering into the wide 
 domain of locomotive or marine practice. To this end illustrations will be given of the most 
 recent arrangements of Horizontal, Vertical, Beam, Pumping, Winding, Portable, Semi- 
 portable, Corliss, Allen, Compound, and other similar Engines, by the most eminent Firms in 
 Great Britain and America. _ The laws relating to the action and precautions to be observed 
 in the construction of the various details, such as Cylinders, Pistons, Piston-rods, Connecting- 
 rods, Cross-heads, Motion-blocks, Eccentrics, Simple, Exp.insion, Balanced, and Equilibrium 
 Slide-valves, and Valve-gearing will be minutely dealt with, -In this connection will be found 
 articles upon the Velocity of Reciprocating Parts and the Mode of Applying the Indicator 
 Heat and Expansion of Steam Governors, and the like. It is the writer's desire to draw 
 illustrations from every possible source, and give only those rules that present practice deems 
 correctj 
 
 Barlow s Tables of Squares, Cubes, Square Roots, 
 
 Cube Roots, Reciprocals of all Integer Numbers up to 10,000. Post 8vo 
 cloth, ds. 
 
 Camus (M.) Treatise on the Teeth of Wheels, demon- 
 strating the best forms which can be given to them for the purposes of 
 Machinery, such as Mill-work and Clock-work, and the art of finding 
 their numbers. Translated from the French, with details of the present 
 practice of Millwrights, Engine Makers, and other Machinists by 
 Isaac Hawkins. Third edition, ic/M l8//afej-, Svo, cloth, Jj. ' 
 
PUBLISHED BY E. & F. N. SPON. 
 
 A Practical Treatise on the Science of Land and 
 
 Engineering Surveying, Levelling, Estimating Quantities, etc., with a 
 general description of the several Instruments required for Surveying, 
 Levelling, Plotting, etc. By H. S. Merrett. Third edition, 41 ;plates 
 with illustrations and tables, royal 8vo, cloth, \2s, dd. 
 
 Principal Contents : 
 
 Part I. Introduction and the Principles of Geometry. Part 2. Land Surveying ; com- 
 prising General Observations — The Chain — OiTsets Surveying by the Chain only — Surveying 
 Hilly Ground— To Survey an Estate or Parish by the Chain only — Surveying with the 
 Theodolite — Mining and Town Surveying — Railroad Surveying — Mapping — Division and 
 Laying out of Land — Observations on Enclosures — Plane Trigonometry. Part 3. Levelling — 
 Simple and Compound Levelling — ^The Level Book — Parliamentary Plan and Section—' 
 Levelling with a Theodolite — Gradients — ^Wooden Curves — To Lay out a Railway Curve- 
 Setting out Widths. Part 4.^ Calculating Quantities generally for Estimates — Cuttings and 
 Embanlcments — Tunnels — Brickwork — Ironwork — ^Timber Measuring. Part 5. Description 
 and Use of Instruments in Surveying and Plotting — The Improved Dumpy Level — Troughton's 
 Level — The Prismatic Compass — Proportional Compass — Box Sextant — Vernier — Panta- 
 graph — Merrett's Improved Quadrant — Improved Computation Scale — ^The Diagonal Scale — 
 Straight Edge and Sector. Part 6. Logarithms of Numbers — Logarithmic Sines and 
 Co-Sines, Tangents and Co-Tangents — Natural Sines and Co-Sines — Tables for Earthwork, 
 for Setting out Curves, and for various Calculations, etc., etc., etc. 
 
 Saws : the History, Development, Action, Classifica- 
 tion, and Comparison of Saws of all kinds. By Robert Grimshaw. 
 PVith 220 illustrations, 410, cloth, 12J. dd. 
 
 A Supplement to the above ; containing additional 
 
 .practical matter, more especially relating to the forms of Saw Teeth for 
 special material and conditions, and to the behaviour of Saws under 
 particular conditions. With 120 illustrations, cloth, gj. 
 
 A Guide for the Electric Testing of Telegraph Cables. 
 
 By Capt. V. Hoskicer, Royal Danish Engineers. With illustrations f- 
 second edition, crown 8vo, cloth, 4J. (>d. 
 
 Laying and Repairing Electric Telegraph Cables. By 
 
 Capt. V. Hoskicer, Royal Danish Engineers. Crown Svo, cloth, 
 y. 6d. 
 
 A Pocket- Book of Practical Rules for the Proportions 
 
 ■ of Modern Engines and Boilers for Land and Marine purposes. By N. P. 
 Burgh. Seventh edition, royal 32mo, roan, 4J. dd. 
 
 The Assayers Manual: an Abridged Treatise on 
 
 the Docimastic Examination of Ores and Furnace and other Artificial 
 Products. By Bruno Kerl. Translated by W. T. Brannt. With 65 
 illustrations, Svo, cloth, 12s. 6d. 
 
 The Steam Engine considered as a Heat Engine : a 
 
 Treatise on the Theory of the Steam Engine, illustrated by Diagrams, 
 Tables, and Examples from Practice. By Jas. H. Cotterill, M.A., 
 F.R.S., Professor of Applied Mechanics in the Royal Naval College. 
 Svo, cloth, I2J-. 6d. • 
 
14 CATALOGUE OF SCIENTIFIC BOOKS. 
 
 Electricity: its Theory, Sources, and Applications. 
 
 By J. T. Sprague, M.S.T.E. Second edition, revised and enlarged, -with 
 numerous illustrations, crown Svo, cloth, IS^. 
 
 The Practice of Hand Turning in Wood, Ivory, Shell, 
 
 etc., with Instructions for Turning such Work in Metal as may be required 
 in the Practice of Turning in Wood, Ivory, etc. ; also an Appendix on 
 Ornamental Turning. (A book for beginners.) By Francis Campin. 
 Third edition, with wood engravings, crown Svo, cloth, 6r. 
 Contents : 
 
 On Lathes — Turning Tools — Turning Wood — Drilling — Screw Cutting — Miscellaneous 
 Apparatus and Processes — Turning Particular Forms — Staining — Polishing — Spinning Metals 
 —Materials — Ornamental Turning, etc. 
 
 Health and Comfort in House Building, or Ventila- 
 tion with Warm Air by Self-Acting Suction Power, with Review of the 
 mode of Calculating the Draught in Hot- Air Flues, and with some actual 
 Experiments. By J. Drysdale, M.D., and J. W. Hayward, M.D. 
 Second edition, with Supplement, with plates, demy Svo, cloth, "Js. 6d. 
 
 Treatise on Watchwork, Past and Present. By the 
 
 Rev. H. L. Nelthropp, M.A., F.S.A. With 32 illustrations, crown 
 Svo, cloth, ds. 6d. 
 
 Contents : 
 
 Definitions of Words and. Terms used in Watchwork — Tools — Time — Historical Sum- 
 mary — On Calculations of the Numbers for Wheels and Pinions; their Proportional Sizes, 
 Trains, etc. — Of Dial Wheels, or Motion Work — Length of Time of Going without Winding 
 up — The Verge — The Horizontal — The Duplex — The Lever — The Chronometer — Repeating 
 Watches — Keyless Watches — ^The Pendulum, or Spiral Spring — Compensation — Jewelling of 
 Pivot Holes— Clerkenwell — Fallacies of the Trade — Incapacity of Workmen — How to Choose 
 and Use a Watch, etc. 
 
 Notes in Mechanical Engineering. Compiled prin- 
 cipally for the use of the Students attending the Classes on this subject at 
 the City of London College. By Henry Adams, Mem. Inst. M.E., 
 Mem. Inst. C.E., Mem. Soc. of Engineers. Crown Svo, cloth, 2s. 6d. 
 
 Algebra Self-Taught. By W. P. Hi.ggs, M.A., 
 
 D.Sc, LL.D., Assoc. Inst C.E., Author of ' A Handbook of the Differ- 
 ential Calculus,' etc. Second edition, crown Svo, cloth, 2,s. 6d. 
 ' Contents : 
 
 Symbols and the Signs of Operation — 'Die Equation and the Unknown Quantity — 
 Positive and Negative Quantities— Multiplicafton — Involution — Exponents— Negative Expo- 
 nents — Roots, andithe Use of Exponents as Logarithms — Logarithms — Tables of Logarithms 
 and Proportionate Parts — Transformation of System of Logarithms— Common Uses of 
 Common Logarithins— Compound Multiplication and the Binomial Theorem— Division, 
 Fractions, and Ratio — Continued Proportion — ^The Series and the Summation of the Series — 
 Limit of Series— Square and Cube Roots — Equations — List of Formvlas, etc. 
 
 Spons' Dictionary of Engineering, Civil, Mechanical, 
 
 Military, and Naval; with technical terms in French, German, Italian, 
 and Spanish, 3100 pp., and nearly 8000 engravings, in super-royal Svo, 
 in S divisions, 5/. %s. Complete in 3 vols., cloth, 5/. 5^. Bound in a 
 superior manner, half-morocco, top edge gilt, 3 vols., dl. I2J. 
 
In super-royal 8vo, 1168 pp., with 2400 iUustrations, in 3 Divisions, cloth, price 13*. td, 
 each ; or 1 vol., cloth, 2/. ; or half-morocco, 2/, Zs. 
 
 A SUPPLEMENT 
 
 TO 
 
 SPONS' DICTIONARY OF ENGINEERING. 
 
 Edited by ERNEST SPON, Memb. Soc. Engineers. 
 
 Abacus, Counters, Speed 
 Indicators, and Slide 
 Rule. 
 
 Agricultural Implements 
 and Machinery. 
 
 Air Compressors. 
 
 Animal Charcoal Ma- 
 chinery. 
 
 Antimony, 
 
 Axles and Axle-boxes. 
 
 Barn Machinery. 
 
 Belts and Belting. 
 
 Blasting. Boilers. 
 
 Brakes. 
 
 Brick Machinery. 
 
 Bridges. 
 
 Cages for Mines. 
 
 Calculus, Differential and 
 Integral. 
 
 Canals. 
 
 Carpentry. 
 
 Cast Iron. 
 
 Cement, Concrete, 
 Limes, and Mortar. 
 
 Chimney Shafts. 
 
 Coal Cleansing and 
 Washing. 
 
 Coal Mining. 
 
 Coal Cutting Machines. 
 
 Coke Ovens. Copper. 
 
 Docks. Drainage. 
 
 Dredging Machinery. 
 
 Dynamo - Electric and 
 Magneto-Electric Ma- 
 chines. 
 
 Dynamometers. 
 
 Electrical Engineering, 
 Telegraphy, Electric 
 Lighting and its prac- 
 ticaldetails,Telephones 
 
 Engines, Varieties of. 
 
 Explosives. Fans. 
 
 Founding, Moulding and 
 the practical work of 
 the Foundry. 
 
 Gas, Manufacture of. 
 
 Hammers, Steam and 
 other Power. 
 
 Heat. . Horse Power.i 
 
 Hydraulics. 
 
 Hydro-geology. 
 
 Indicators. Iron. 
 
 Lifts, Hoists, and Eleva- 
 tors. 
 
 Lighthouses, Buoys, and 
 Beacons. 
 
 Machine Tools. 
 
 Materials of Construc- 
 tion. 
 
 Meters. 
 
 Ores, Machinery and 
 Processes employed to 
 Dress. 
 
 Piers. 
 
 Pile Driving. 
 
 Pneumatic Transmis 
 sipn. 
 
 Pumps. 
 
 Pyrometers. 
 
 Road Locomotives. 
 
 Rock Drills. 
 
 Rolling Stock. 
 
 Sanitary Engineering. 
 
 Shafting. 
 
 Steel. 
 
 Steam Navvy. 
 
 Stone Machinery. 
 
 Tramways. 
 
 Well Sinking. 
 
 London : E. & F. N. SPON, 125, Strand. 
 New York : 35, Murray Street. 
 
NOW COMPLETE. 
 
 JViih nearly 1500 illustrations, in super-royal 8vo, in 5 Divisions, cloth. 
 Divisions i to 4, i3j-. td. each ; Division $, lis. 6d. ; or 2 vols., cloth, £2, los. 
 
 SPONS' ENCYCLOPEDIA 
 
 OF THE 
 
 INDUSTRIAL ARTS, MANUFACTURES, AND COMMERCIAL 
 PRODUCTS. 
 
 Edited by C. G. WARNFORD LOCK, F.L.S. 
 
 Among the more important ■ of the subjects treated of, are the 
 following : — 
 
 Acids, 207 pp. 220 figs. 
 Alcohol, 23 pp. 16 figs. 
 Alcoholic Liquors, 13 pp. 
 Alkalies, ^9 pp. 78 figs. 
 Alloys. Alum. 
 
 Asphalt. Assaying. 
 Beverages, 89 pp. 29 figs. 
 Blacks. 
 
 Bleaching Powder, 1 5 PP- 
 Bleaching, 5 1 pp. 48 figs. 
 Candles, 18 pp. 9 figs. 
 Carbon Bisulphide. 
 Celluloid, 9 pp. 
 Cements. Clay. 
 Coal-tar Products, 44 pp. 
 
 14 figs. 
 Cocoa, 8 pp. 
 Coffee, 32 pp. 13 figs. 
 Cork, 8 pp. 17 figs. 
 Cotton Manufactures, 62 
 
 pp. 57 figs- 
 Drugs, 38 pp. 
 Dyeing and Calico 
 
 Printing, 28 pp. 9 figs. 
 Dyestuffs, 16 pp. 
 Electro-Metallurgy, 13 
 
 pp. 
 Explosives, 22 pp. 33 figs. 
 Feathers. 
 Fibrous Substances, 92 
 
 pp. 79 figs. 
 Floor-cloth, 16 pp. 21 
 
 figs- 
 Food Preservation, 8 pp. 
 Fruit, 8 pp. 
 
 Fur, S pp. 
 
 Gas, Coal, 8 pp. 
 
 Gems. 
 
 Glass, 45 pp. 77 'figs. 
 
 Graphite, 7 PP- 
 
 Hair, 7 pp. 
 
 Hair Manufactures. 
 
 Hats, 26 pp. 26, figs. 
 
 Honey. Hops. 
 
 Horn. 
 
 Ice, 10 pp. 14 figs. 
 
 Indiarubber Manufac- 
 tures, 23 pp. 17 figs. 
 
 Ink, 17 pp. 
 
 Ivory. 
 
 Jute Manufactures, 1 1 
 pp., II figs. 
 
 Knitted Fabrics — 
 Hosiery, 15 pp. 13 figs. 
 
 Lace, 13 pp. 9 figs. 
 
 Leather, 28 pp. 3 1 figs. 
 
 Linen Manufactures, 16 
 pp. 6 figs. 
 
 Manures, 21 pp. 30 figs. 
 
 Matches, 17 pp. 38 figs. 
 
 Mordants, 13 pp. 
 
 Narcotics, 47 pp. 
 
 Nuts, 10 pp. 
 
 Oils and Fatty Sub- 
 stances, 125 pp. 
 
 Paint. 
 
 Paper, 26 pp. 23 figs. 
 
 Paraffin, S pp. 6 figs. 
 
 Pearl and Coral, 8 pp. 
 
 Perfumes, 10 pp. 
 
 Photography, 13 pp. 20 
 
 figs. 
 Pigments, 9 pp. 6 figs. 
 Pottery, 46 pp. 57 figs. 
 Printing and Engraving, 
 
 20 pp. 8 figs. 
 Rags. 
 Resinous and Gummy 
 
 Substances, 75 PP- '^ 
 
 figs. 
 Rope, i5pp. 17 figs. 
 Salt, 31 pp. 23 figs. 
 Silk, 8 pp. 
 Silk Manufactures, 9 pp. 
 
 II figs. 
 Skins, 5 pp. 
 Small Wares, 4 pp. 
 Soap and Glycerine, 39 
 
 PP- 45 figs. 
 Spices, 16 pp. 
 Sponge, 5 pp. 
 Starch, 9 pp. 10 figs. 
 Sugar, 15s pp. 134 
 
 figs. 
 Sulphur. 
 Tannin, 18 pp. 
 Tea, 12 pp. 
 Timber, 13 pp. 
 Varnish, 15 pp. 
 Vinegar, S pp. 
 Wax, 5 pp. 
 Wool, 2 pp. 
 Woollen Manufactures, 
 
 £8 pp. 39 figs. 
 
 London: E. & F. N. SPON, 125, Strand, 
 
 New York : 35, Murray Street. 
 
Crown 8vo, cloth, with illustrations, 5j. ^ 
 
 WOEKSHOP EECEIPTS, 
 
 FIRST SERIES. 
 
 By ERNEST SPON. 
 
 Synopsis of Contents. 
 
 Freezing. 
 
 Fulminates. 
 
 Furniture Creams, Oils, 
 Polishes, Lacquers, 
 and Pastes. 
 
 Gilding. 
 
 Glass Cutting, Cleaning, 
 Frosting, Drilling, 
 Darkening, Bending, 
 Staining, and Paint- 
 ing. 
 
 Glass Making. 
 
 Glues. 
 
 Gold. 
 
 Graining. 
 
 Gums. 
 
 Gun Cotton. 
 
 Gunpowder. 
 
 Horn Working. 
 
 Indiarubber. 
 
 Japans, Japanning, and 
 , kindred processes. 
 
 Lacquers. 
 
 Lathing. 
 
 Lubricants. 
 
 Marble Working. 
 
 Matches. 
 
 Mortars. 
 
 Nitro-Glycerine. 
 
 Oils. 
 
 Bookbinding. 
 
 Bronzes and BrSnzlng. 
 
 Candles. 
 
 Cement. 
 
 Cleaning. 
 
 Colourwashing. 
 
 Concretes. 
 
 Dipping Acids. 
 
 Drawing Office Details. 
 
 Drying Oils. 
 
 Dynamite. 
 
 Electro - Metallurgy — 
 
 (Cleaning, Dipping, 
 
 Scratch-brushing, Bat- 
 teries, Baths, and 
 
 Deposits of every 
 
 description). 
 Enamels. 
 Engraving on Wood, 
 
 Copper, Gold, Silver, 
 
 Steel, and Stone. 
 Etching and Aqua Tint. 
 Firework Staking , — 
 
 (Rockets, Stars, Rains, 
 
 Gerbes, Jets, Tour- 
 billons, Candles, Fires, 
 
 Lances,Lights, Wheels, 
 
 Fire-balloons, and 
 
 minor Fireworks). 
 Fluxes. 
 Foundry Mixtures. 
 
 Besides Receipts relating to the lesser Technological matters and processes, 
 such as the manufacture and use of Stencil Plates, Blacking, Crayons, Paste, 
 Putty, Wax, Size, Alloys, Catgut, Tunbridge Ware, Picture Frame and 
 Archi'tectural Mouldings, Compos, Cameos, and others too numerous to 
 mention. 
 
 London : E. & F. N. SPON, 185, Strand. 
 
 New York : 35, Murray Street. 
 
 Paper. 
 
 Paper Hanging. 
 
 Painting in Oils, in Water 
 Colours, as well as 
 Fresco, House, Trans- 
 parency, Sign, and 
 Carriage Painting. 
 
 Photography. 
 
 Plastering. 
 
 Polishes. 
 
 Pottery — (Clays, Bodies, 
 Glazes, Colours, Oils, 
 Stains, Fluxes, Ena- 
 mels, and Lustres). 
 
 Scouring. 
 
 Silvering. 
 
 Soap. 
 
 Solders. 
 
 Tanning. 
 
 Taxidermy. 
 
 Tempering Metals. 
 
 Treating Horii, Mother- 
 o'-Pearl, and like sub- 
 stances. 
 
 Varnishes, Manufacture 
 and Use of. 
 
 Veneering. 
 
 Washing. 
 
 Waterprofing. 
 
 Welding. 
 
Crown 8vo, cloth, 485 pages, with illustrations, Sj. 
 
 WOKKSHOP BECEIPTS, 
 
 SECOND SERIES. 
 
 By ROBERT HALDANE. 
 
 Acidimetry and Alkali- 
 metry. 
 Albumen. 
 Alcohol . 
 Alkaloids. 
 Baking-powders. 
 Bitters. 
 Bleaching. 
 Boiler Incrustations. 
 Cements and Lutes. 
 Cleansing. 
 Confectionery. 
 Copying. 
 
 Synopsis of Contents. 
 
 Disinfectants. 
 
 Dyeing, Staining, and 
 
 Colouring. 
 Essences. 
 Extracts. 
 Fireproofing. 
 Gelatine, Glue, and Size. 
 Glycerine. 
 Gut. 
 
 Hydrogen peroxide. 
 Ink. 
 Iodine. 
 Iodoform. 
 
 Isinglass. 
 
 Ivory substitutes. 
 
 Leather. 
 
 Luminous bodies. 
 
 Magnesia. 
 
 Matches. 
 
 Paper. 
 
 Parchment. 
 
 Perchloric acid. 
 
 Potassium oxalate. 
 
 Preserving. 
 
 Pigments, Paint, and Fainting : embracing the preparation of 
 Figments, including alumina lakes, blacks (animal, bone, Frankfort, ivory, 
 lamp, sight, soot), blues (antimony, Antwerp, cobalt, coeruleum, Egyptian, 
 manganate, Paris, Peligot, Prussian, smalt, ultramarine), browns (bistre, 
 hinau, sepia, sienna, umber, Vandyke), greens (baryta, Brighton, Brunswick, 
 chrome, cobalt, Douglas, emerald, manganese, mitis, mountain, Prussian, 
 sap, Scheele's, SchweinfUrth, titanium, verdigris, zinc), reds (Brazilwood lake, 
 carminated lake, carmine, Cassius purple, cobalt pink, cochineal lake, colco- 
 thar, Indian red, madder lake, red chalk, red lead, vermilion), whites (alum, 
 baryta, Chinese, lead sulphate, white lead — by American, Dutch, French, 
 German, Kremnitz, and Pattinson processes, precautions in making, and 
 composition of commercial samples — whiting, Wilkinson's white, zinc white) , 
 yellows (chrome, gamboge, Naples, orpiment, realgar, yellow lakes) ; Faint 
 (vehicles, testing oils, driers, grinding, storing, applying, priming, drying, 
 filling, coats, brushes, surface, water-colours, removing smell, discoloration ; 
 miscellaneous paints — cement paint for carton-pierre, copper paint, gold paint, 
 iron paint, lime paints, silicated paints, steatite paint, transparent paints, 
 tungsten paints, window paint, zinc paints) ; Fainting (general instructions, 
 proportions of ingredients, measuring paint work ; carriage painting — priming 
 paint, best putty, finishing colour, cause of cracking, mixing the paints, oils, 
 driers, and colours, varnishing, importance of washing vehicles, re-varnishing, 
 how to dry paint ; woodworlc painting). 
 
 London : E. «& F. N. SPON, 125, Strand. 
 New York: 35, Murray Street. 
 
JUST PXJBHSHED. 
 
 Crown 8vo, cloth, 480 pages, with 183 illustrations, S-f- 
 
 WOEKSHOP EECEIPTS, 
 
 THIRD SERIES. 
 By C. G. WARNFORD LOCK. 
 
 TTniform with the First and Second Series. 
 Synopsis of Contents. 
 
 Alloys. 
 
 Indium. 
 
 Rubidium. 
 
 Aluminium. 
 
 Iridium. 
 
 Ruthenium. 
 
 Antimony. 
 
 Iron and Steel. 
 
 Selenium. 
 
 Barium. 
 
 Lacquers and Lacquering. 
 
 Silver. 
 
 Beryllium. 
 
 Lanthanum. 
 
 1 
 
 Slag. 
 
 Bismuth. 
 
 Lead. 
 
 Sodium. 
 
 Cadmium. 
 
 Lithium. 
 
 Strontium. 
 
 Caesium. 
 
 Lubricants. 
 
 Tantalum. 
 
 Calcium. 
 
 Magnesium. 
 
 Terbium. 
 
 Cerium. 
 
 Manganese. 
 
 Thallium. 
 
 Chromium. 
 
 Mercury. 
 
 Thorium. 
 
 Cobalt. 
 
 Mica. 
 
 Tin. 
 
 Copper. 
 
 Molybdenum. 
 
 Titanium. 
 
 Didymium. 
 
 Nickel. 
 
 Tungsten. 
 
 Electrics. 
 
 Niobium. 
 
 Uranium. 
 
 Enamels and Glazes. 
 
 Osmium. 
 
 Vanadium. 
 
 Erbium. 
 
 Palladium. 
 
 Yttrium. 
 
 Gallium. 
 
 Platinum. 
 
 Zinc. 
 
 Glass. 
 
 Potassium. 
 
 Zirconium. 
 
 Gold. 
 
 Rhodium. 1 
 
 Aluminium 
 
 London : E. & F. W. SPON, 1S5, Strand, 
 
 New York : 35, Murray Street. 
 
JUST IPTJBLISHLEO. 
 
 In demy 8vo, cloth, 600 pages, and 1420 Illustrations, 6s. 
 
 SPONS' 
 MECHANIC'S OWN BOOK; 
 
 A MANUAL FOR HANDIGRAFTSIVIEN AND AMATEURS. 
 
 Contents. 
 
 Mechanical Drawing — Casting and Founding in Iron, Brass, Bronze, 
 and other Alloys — Forging and Finishing Iron — Sheetmetal Working 
 — Soldering, Brazing, and Burning — Carpentry and Joinery, embracing 
 descriptions of some 400 Woods, over 200 Illustrations of Tools and 
 their uses. Explanations (with Diagrams) of 1 16 joints and hinges, and 
 Details of Construction of Workshop . appliances, rough furniture, 
 Garden and Yard Erections, and House Building — Cabinet-Making 
 and Veneering — Carving and Fretcutting — Upholstery — Painting, 
 Graining, and Marbling — Staining Furniture, Woods, Floors, and 
 Fittings — Gilding, dead and bright, on various grounds — Polishing 
 Marble, Metals, and Wood — Varnishing — Mechanical movements, 
 illustrating contrivances for transmitting motion — Turning in Wood 
 and Metals — Masonry, embracing Stonework, Brickwork, Terracotta 
 
 and Concrete — Roofing with Thatch, Tiles, Slates, Felt, Zinc, &c. 
 
 Glazing with and without putty, and lead glazing — Plastering and 
 Whitewashing— Paper-hanging— Gas-fitting— Bell-hanging, ordinary 
 and electric Systems — Lighting — Warming — Ventilating — Roads 
 Pavements, and Bridges — Hedges, Ditches, and Drains — Water 
 Supply and Sanitation— Hints on House Construction suited to new 
 •countries. 
 
 London : E. «S: F. N. SPON, 125, Strand. 
 
 New York : 35, Murray Street.-