May 2 1, 1896] 



NA TURE 



69 



SOCIETIES AND ACADEMIES. 

 London. 

 Physical Society, May 8. — Captain Abney, President, in 

 the chair. — Messrs. Krith and Rogers read a paper on the true 

 resistance of the electric arc. It was pointed out by Prof. Ayrton, 

 at the British -Association meeting at Ipswich, that if the " true 

 resistance " of an arc is defined as the ratio of a small increase 

 of the P.I), between the carbons to the corresponding change in 

 the current, it follows that this "true resistance" must be 

 a negative quantity. In order to measure the " true resistance " 

 without appreciably altering the form of the carbons, i:c., the 

 authors superpose a .small alternating current on the main con- 

 tinuous current. The arc lamp employed was adjusted by hand, 

 and the arc length was measured by projecting an image of the 

 arc by means of a lens. The main (continuous) current and P.D. 

 were measured by a Weston ammeter and voltmeter, while the 

 auxiliary alternating current was measured by means of an air 

 transformer and an electrostatic voltmeter. The authors find 

 that between the limits employed the magnitude of the alter- 

 nating current did not influence the results obtained for the 

 resistance of the arc. The frequency, so long as it lies between 

 the limits 250 - 7 complete alternations per second, and the wave 

 form, do not influence the resistance, since the same results 

 were obtained with a Pyke and Harris alternator, a Ferranti 

 alternator, a Gramme alternator, and a Mordey transformer. 

 For each make of carbon e.\amined, four combinations were 



used : V cored, — cored: + cored, — solid; -)- solid, -cored ; 



+ solid, - solid. The general characteristicof the curves obtained 

 is that for the + solid, — solid combination the " true resist- 

 ance" is ahvays negative; while for + cored, - cored it is 

 always positive ; the other curves lying between these two 

 extremes, those which have the + carbon solid always being 

 more negative than those which have the + carbon cored. In 

 the case of the curves .showing, for solid carbons, the relation 

 between the resistance of the arc and the P.D. between the 

 carbons, the current being constant (10 amperes), a minimum 

 (maximum negative) value for the resistance occurs at about 

 55 volts. With combinations having a cored positive this 

 minimum becomes more strongly marked, and occurs at a lower 

 voltage. The authors find that for cored carbons the position of 

 this minimum is closely connected with the presence or absence 

 of the dark space in the arc. For points on the curve to the 

 right of the minimum point, the dark space is absent ; while for 

 points to the left of the minimum, the dark space is always 

 present. It was found that the effect of using as the -t- carbon 

 a Carre carbon in which the core had been bored out, was to 

 obtain a curve closely resembling that obtained when both car- 

 lions were solid. On filling this hollow carbon with plaster of 

 Paris or kaolin, the resistance of the arc bec.ime positive. The 

 above experiments were made with the 4- carbon uppermost ; 

 other experiments, made with the arc inverted, showed that with 

 .solid carbons the resistance is not appreciably altered by invert- 

 ing the arc. With cored carbons, however, the resistance, as 

 well as the physical character of the arc, is altered ; since, on 

 inversion, the dark space disappears, and the resistance con- 

 siderably diminishes. If, however, the conditions under which 

 the arc is burning are such that the dark space is absent, then 

 inverting the arc does not alter the resistance. Attempts were 

 made to measure the " true resistance " of a direct current hissing 

 arc, but it was found that, even with the alternator at rest, there 

 was a large deflection of the electrometer, showing that the cur- 

 rent through a hissing arc was oscillatory. In order to elucidate 

 the marked ditTerence between their results for cored carbons 

 and those deduced from Mrs. Ayrton's curves, the authors have 

 made a series of measurements at low frequencies. They find 

 that there is a critical frequency above which the resistance has 

 a positive value which is independent of the frequency, and 

 below which it has a negative value, this critical frequency lying 

 between 7 "5 and o. In order to investigate the sign of the resist- 

 ance at low frequencies, the vibrations of the needles of the 

 ammeter and voltmeter were made use of. liy an arrangement 

 of mirrors, the needles and scales of both instruments could be 

 observed simultaneously. In this way it could be seen whether 

 the two needles were, at any instant, vibrating in the same or in 

 opposite directions. If the needles vibrate in the saine phase, 

 i.e. if an increase of P.D. is accompanied by an increase of 

 current, then the resistance must be positive ; while if they are 

 vibrating out of phase, i.e. if an increase of P. I), is accompanied 

 by a decrease in current, then the resistance is negative. An 



NO. 1386, VOL. 54] 



attempt to run the arc oft' a continuous-current dynamo failed, 

 since even with the alternator at rest the electrometer showed a 

 large deflection, evidently due to the oscillation of the current, 

 owing to the commutator of the dynamo having a finite number 

 of segments. Prof. .\. Gray doubted whether it was right to 

 give the name "true resistance" of the arc to the slope of the 

 curve connecting the potential difterence (V) and the current (A). 

 The authors' method of deducing SX'/S-V was only true if the 

 curve was a straight line ; while in the case of the arc, E and a 

 may both vary with the current. Mrs. Ayrton said, that with 

 reference to the question of the existence of a back E.M.F. the 

 evidence tended to show that it did not exist. By using an 

 exploring carbon, no constant back E.M.F would be found. 

 Prof. .Ayrton said, that considering the arc as consisting of a 

 back E. M. F. and a resistance, it was necessary to separate these 

 two. Simply obtaining one value of the P.D. and the current 

 was of no assistance in solving this question, but a series of 

 values had to be taken. By taking the change in P.D. and 

 current sufticiently small, the curve over the range considered 

 was practically straight. It was curious to note that as long as 

 observers obtained a positive value for the resistance of the arc, 

 no fault was found with the method ; but that now a negative 

 value was found, the accuracy of this method was questioned. 

 If a back E.M.F. does really e.xist, then it follows that the arc 

 must have a negative resistance. Mr. Frith has shown why 

 some people have got positive and some negative values for the 

 resistance of the arc, and also that with an alternating current 

 you may get either one or the other. Mr. Tremlett Carter asked if 

 the fact that the arc had a negative resistance did not imply a back 

 E.M.F. in order that the arc might be stable. If so, was a 

 negative resistance such an absurdity? Mr. Campbell said he 

 was very pleased to see that the authors had applied a method 

 which he (Mr. Campbell) had suggested for measuring pulsating 

 currents. If a pulsating current, such as could be obtained by 

 means of a make and break, were passed through a thermopile, you 

 would get a back E.M.F. ; while if an alternating current were 

 employed, you would not. Mr. Frith, in his reply, said that 

 he had defined the "true resistance " as dV/dA. ^Irs. Ayrton 

 has shown that an arc will not run unless a certain resistance is 

 placed in series with it ; this resistance must be numerically equal 

 to the negative resistance of the arc itself. Prof. Ayrton said 

 Mr. Frith's remarks as to the cause of the want of stability of an 

 arc without outside resistance, were most suggestive. The Chair- 

 man (Captain Abney) said he did not like the expression P. D. 

 He suggested the employment of photography to facilitate the 

 accurate registration of the instrument readings. The further 

 discussion on the paper was adjourned to the next meeting on 

 May 22. 



Mathematical Society, April23. — Major MacMahon, R. A. , 

 F.R.S., President, in the chair. — The President communicated 

 a portion of the following abstract of a paper by Prof. W. Burn- 

 side, F. R. S., on the isomorphism of a group with itself. A one- 

 to-one correspondence between the operations of a group, which 

 leaves the multiplication table of the group unaltered, is spoken 

 of as an isomorphism of the group with itself. Such a corre- 

 spondence may clearly be represented as a substitution performetl 

 on the symbols of the operations of the group, i.e. the isomor- 

 phism may itself be regarded as an operation, and the totality 

 of the isomorphisms of a given group will themselves form a 

 group. This group is known as the "group of isomorphisms" 

 of the given group. The only general theorems connected with 

 the isomorphism of a group with itself hitherto published are 

 due to Herr O. Holder' and Herr G. Frobenius.- In the first 

 part of the present paper I have reproduced such of the definitions 

 due to Herren Holder and Frobenius as are necessary to render 

 it self-contained, and also one fundamental theorem. An isomor- 

 phism is defined to be cogredient or contragredient according as 

 it can or cannot be obtained by transforming all the operations 

 of the group by one of themselves. The theorem is that the 

 cogredient isomorphisms form a self-conjugate sub-group of the 

 complete group of isomorphisms. A definition, due to Herr 

 Frobenius, involving an important new conception, is that of a 

 characteristic sub-group. It is as follows. If a sub-group of a 

 given group is transformed into itself by every isomorphism of 

 which the given group is capable, the sub-group is called a 

 characteristic sub-group. In the second part I have first 



1 Cy. The first ten pages of a memoir with the title " Bildung Zusammen- 

 gesetzter Gruppen." Math, .tm., xlvi. 



- Cf. P.-trts of memoirs with titles " Ueber Endliche Gruppen" and 

 " Ueber auflOsbare Gruppen II.' Berliner Siiziiiigsberichte, 1895. 



