^26 



NATURE 



\Sept. 29, 1 88 1 



papers essentially economic : witness tlie numerous interesting 

 papers read to Sections A and G, on the applications of electric 

 energy, and, best of all, the ]'residential addresses to those sec- 

 tions, of- which both were devoted to economic physics, or rather, 

 as the subject should be called, physical economics. 



Finally, if the preceding axioms and the soundness of the 

 above classification of social facts be not disproved, it follows 

 that three out of the four great reforms demanded in Mr. 

 Ingram's presidential address,' and repeated and enforced in Mr. 

 Grant Duff's, are here introduced into the conduct of economic 

 research, namely, "(I) that the study of the economic pheno- 

 mena of society ou^iht to be systematically combined w ith that 

 of the other aspects of social existence ; (2) that the exce-sive 

 tenriency to abstraction and to unreal simplifications should be 

 checked ; (3) that the h priori deductive method should be 

 changed for the historical ; while the fourth, that economic laws 

 and the practical prescriptions founded on these should be con- 

 ceived and expressed in a less absolute form," would readily also 

 be exemplified if the limits of the present paper permitted refer- 

 ence to generalisation and to practice. Again, it is sufficient to 

 quote Mr. Ingram's concluding proposals to show that these have 

 been substantially adopted. The field of the section should be 

 enlarged so as to comprehend the whole of sociology, " since 

 the economic facts of society . . . cannot be scientifically con- 

 sidered apart, and there is no reason why the researches of Sir 

 Henry Maine or Mr. Spencer should not be as much at home 

 here as those of Mr. Fawcett or Prof. Price. Many of the 

 subjects, too, at present included in the artificial assemblage of 

 heterogeneous inquiries known by the name of anthropology 

 really connect themselves with the laws of social development, 

 and if our section bore the title of the sociological, the studies of 

 Mr. Tylor and Sir John Lubbock would find in it their most 

 appropriate place. I prefer the name sociology to that of social 

 science." - 



F 



THE PROPER PROPORTIONS OF RESISTANCE 

 IN THE WORKING COILS, THE ELECTRO- 

 MAGNETS, AND THE EXTERNAL CIRCUITS 

 OF DYNAMOS^ 



OR the electro-magnet ; 

 Let L be the length of the wire, 



B ,, bulk of the whole space occupied by wire and 



insulation, 

 H ,, ratio of this whole space to the bulk of the 



copper alone (that is, let - 5 the bulk of the 



copper), 



A ,, the sectional area of wire and insulator, 



R ,, the resistance of the wire. 

 ■ the working coil, let the corresponding quantities be L', B', 

 R'. Lastiv, let s be the specific resistance of the copper. 



Hence, 



and similarly 



A 



B = AL 



r I 



R — ns — = *, 



A 

 _^{nsB) __ 



A- 

 K 



(I) 



(2) 



■sJK s'R 



^ ^(h'i'B' ) ^ JC^ 

 s/R -JR ' 



where A" and A" denote constants. 



Now, let c be the current through the magnet coil, and c' that 

 through the working coil, and let v be the velocity of any chosen 

 point of the working coil. Denoting by / the average electro- 

 motive force between the two ends of the working coil, we have — 



, = /£-!.. . . . (3) 



where /is a quantity depending on the forms, magnitudes, and 

 relative positions of B and B', and on the magnetic susceptibihty 

 of iron ; diminishing as the susceptibility diminishes with in- 

 creased strength of current, or with any change of R and R' 

 which gives increase of magnetising force. 



In the single-circuit dynamo (that is, the ordinary dynamo) c' 

 is equal to c, but not so in the shunt-dynamo. In each, the 



' " On the Present Position and Prospects of Political Economy " (British 

 Association. Dublin. 18781. 



= Quoted by Mr. Grant Duff. Presidential Address to Section F, 1881. 



3 Paper read at the British Association, York, 1881, by Sir Wllham 

 Thomson, F.R.S. 



whole electric activity (that is, the rate of doing work) is /> c' ; 

 or. by (3)— 



or, by (i) and (2) — 



I^(RK')cc 



(4) 



(5) 



(9) 



RTR- 



Of this whole work, the proportions ^^hich go to waste in heating 

 the coils and to work in the external circuit are — 



Rc^ + R'/- . . . waste . (6) 

 ^^^A-A-'^^^ ~ (^ f= -I- ^ c'') . useful work (7) 



By making v sufficiently great, the ratio of (6) to (7) (waste to 

 useful work) may be made as small as we please. Our question 

 is, how ought A' and R' to be proportioned to make the ratio of 

 waste to work a minimum, with any given speed ? or, w hich 

 comes to the same thing, to make the speed required for a given 

 ratio of work to waste a minimum ? To answer it, let r be the 

 ratio of the v\hole work to the waste. We have, by (5) and (6) — 

 _ / s/(RR')cc' V 



'^ " Rc^ + Rc'"- TT-' ■ ■ '*'■' 

 For the single-circuit dynamo we have c = c', and (8) becomes — 



^_ I^{RR') V 

 R + R' K'K' 



where S ^ R ^ R' (11) 



Suppose now S io be given, and suppo-e for a moment I to 

 be constant. The prof'lem of making r a maximum with v 

 given, ox V 2l minimum with ;■ given, requires simply that R 

 {S- R] be a maximum ; which it is \Ahen A' = J 5', that is, when 

 the resistances in the working coil and the electro-magnet are 

 equal. But in reality / is not constant ; it diminishes with 

 increase of the magnetising force. As it generally depends 

 chiefly on the soft iron of the electro-magnet, and comparatively 

 but little on the soft iron of the moving armature, or on iron 

 magnetised by the current through the moving coils, it will gene- 

 rally be the case xh-A I wVA, catcris paribus, h^ diminished by 

 increasing R and diminishing R'. Hence the maximum oi rjv is 

 shown by (10) to require A' to be somewhat greater than ^S : 

 how much greater »e cannot find from the formula, without 

 knowing the law of the variation of /. 



Experience and natural selection seem to have led in most of 

 the ordinary dynamos, as now made, to the resistance in the 

 electro-magnet being somewhat less than the resistance in the 

 working coil, which is in accordance Avith the preceding theory. 

 Whether the useful work of the dynamo be light-giving, or 

 power, or beating, or electro-metallurgy, we may, for_simplicity, 

 reckon it in any possible case by referring t** the convenient 

 standard case of a current through a conductor of given resistance 

 E connecting the working terminals of the dynamo. This con- 

 ductor, in accordance with general usage, I call the "external 

 circuit," which is an abbreviation for the part of the whole 

 circuit which is external to the dynamo. In the case of the 

 single-circuit dynamo the current in the external circle is equal to 

 that through the working coil and electro-magnet, or c of our 

 notation. Hence, by Ohm's law — 



,- P 



or, by (3), (l), and (2), 



E + R + R • 

 I^{R R')v 



R') 



Hence either 



The case of c ■ 



KK' (E + R 

 c= O . 



/= A" A" [E + R + R) 

 \/iRR')z' 

 O is that in which 



A' A" (E + R + R' 



v< -' 



(12) 



(13) 

 (14) 

 (15) 



(16) 



/„ ^ (R R) 



where /„ denotes the value of /for c = 0. To understand it, 

 remember we are supposing no residual magnetism. For any 

 speed subject to (16), the dynamo produces no current. When 

 this limit is exceeded the electric equilibrium in the circuit 

 becomes unstable ; an infinitesimal current started in either 

 direction rises rapidly in strength, till it is limited by equation 

 (15), through the diuiinution of /, which it produces. Thus, 



