April 26, 1894] 



NA TURE 



619 



morphous with the icosahedral group, or a group of order less 

 , than 60, which is necessarily solvable. In the first case the 

 solutinn of the given equation cannot be made to depend on an 

 equation of lower degree than the 6th ; in the second case the 

 roots of the equation are rationally expressible in terms of the 

 roots of an equation of the 5th degree ; and in the last case 

 the equation can be solved by radicals. For a given equa- 

 • tion with numerical coefficients the cases are distinguished by 

 forming the resolvents of the 6th and loth degrees and deter- 

 mining whether either of these have a rational root. If the re- 

 solvent of the loth degree has a rational root the equation can 

 be solved by radicals, and if that of the 6th degree has a rational 

 root the solution depends on a quintic. It is this latter 

 resolvent which is calculated in the present paper, on the sup- 

 position that the sextic is reduced to the standard form 



x^ -\- ttx- + vx -{■ w = o, 



which is always possible by solving a cubic equation. Repre- 

 senting the roots of the equation by o, I, 2, 3, 4, 5, a transitive 

 icosahedral group is generated (see Sarret, " Cours d'Alg. S.ip. " 

 vol. ii.) by the two even permutations 



(i234S)>nd (01) (25). 



There is no function of the roots of the and degree that is 

 invariable for this group, but it is easily verified that 



012 + 023 + 034 + 04s -f 051 -f 124 + 235 + 341 -1- 452 + 513 



is such a function ; and therefore that this function takes 6 values 

 for all even permutations of the 6 roots. If 



y^ + P\y^ + p-iy^ + /sJ* + P^y^ +p5y + A = o 



is the equation whose roots are these 6 values, p^ &c., must be 

 rational in u, v, tv, and ^/A, where A is the discriminant of the 

 sextic. By comparing the degrees of these functions it is seen 

 that 



/^ = o, p.2 = tnfcv, /g = ni^uv, p^ = 7n-^u^ -I- 7n^dj" 



\ where the w's are numbers ; and it^ is further easily shown that 



: Finally, by choosing suitable special cases, the values of the 

 other 7;/'s are completely determined. The final result is that 

 this sextic resolvent has the form 



y^ + 30<cy* -)- {id^zu"- - 4u^)}'- + 2^11-v''' - 8o7«/^ + 64?^^^ 

 = j^/A. 

 It is obvious that the twelve values that the function 



012 -t- 023 -f +513 



takes for a// permutations of the roots are, for the standard 

 sextic equal and opposite in pairs, so that y- is a 6-valued 

 function for the symmetric group. If then the above equation 

 be squared, while z is written for y^ and its value in terms of 

 71, V, zu substituted for A, 



(s^ + ^QWZ^ + ( I65W- - ^11^)% -\- 2SU-V- - 80OT^ + 64u^zv)- - Az = o 



is the resolvent, a rational root of which will indicate that the 

 ' solution of the sextic depends on that of a quintic. — Mr. Perigal 

 : exhibited some diagrams illustrating circle-squaring by dissec- 

 tion. 



Entomological Society, April 11. — Henry John Elwes, 

 President, in the chair. — The Hon. Walter Rothschild exhibited 

 male and female spacimens of Oniitkoplera pxradisea, Stdgr., 

 from Fmislerre Mountains, New Gamea ; O. trojana, Stdgr., 

 : from Palawan ; O. androinache, Std^r., from Kim Balu, Borneo ; 

 (Enetiis mirabilis, R -thsch. , from Cedar Bay, Qaeensland, and 

 a few other splendid species from the Upper Amazons. The 

 President, Mr. J. J. Walker, R.N., Mr. Osbert Silvin, F.R.S., 

 , Lord Walsingham,F. R S., Colonel Lang, R. E., Mr. Champion, 

 I and Mr. Hampson made remarks on the geographical distribu- 

 tion of some of the species and the elevation at which they were 

 taken. — Mr. H. Gjss exhibited, for Mr. G. A. J. Rothney, 

 ; several speciinens of a species of Hemiptera ( Serinetka augur, 

 I Fab. ), and of a species of Lepidoptera (/"/iawa^tz fianimans, 

 ' Walk.), the latter of which closely resembled and mimicked 

 I the former. He said that Mr. Ro;hney had found both species 

 abundantly on the roots and trunks of trees in Mysore, in 

 ; November last, in company with ants (several species of Cam- 

 \ ponotus and Cremasto^aster). The Hemiptera appeared to be 



I distasteful to the ants, as they were never molested by them, 

 NO. 1278, VOL. 49] 



and he thought that the species of Lepidoptera was undoubtedly 

 protected from attack by its close imitation of the Hemipteron. 

 Mr. Goss said he was indebted to Mr. C. J. Gahan for deter- 

 mining the species. A discussion followed on the mimicking 

 species, in which the President, Mr. Waterhouse, Mr. J. 1. 

 Walker, Colonel Swinhoe, and Mr. Hampson took part. — Mr. 

 J. W. Tutt exhibited a typical specimen of Lyctiua co>ydon, 

 captured in July 1893 ; a hybrid male (Z. coiydon and L. 

 adonis), taken in copicld with a typical female L- adonis, May 

 20, i8q3; a typical male L. adonis. May 20, 1893 ; a female 

 L. adonis, the pigment failing in one hind wing ; a pale van of 

 L. corydon, probably to be referred to var. apennina ofZeller, 

 usually taken in Italian mountains, or var. albicans, H. S., taken 

 in Andalusia. Mr. Tutt remarked that, of the first, Staudinger 

 (Cat. p. 12) says "pallidior," of the latter "albicans." He 

 also remarked that the hybrid retains the external features of 

 the species corydon, but has taken on to a great extent the 

 coloration of L. adonis. It was captured in copula with a 

 female L. adonis, at a time when Z. adonis was very abundant, 

 and some weeks before L. corydon occurred. — The question 

 having been raised by the President as to the number of meet- 

 ings of the Society which it was desirable to hold during the 

 year, and the most convenient dates for such meetings, a long 

 discussion on the subject ensued, in which Mr. Waterhouse, 

 Mr. Salvin, the Hon. Walter Rothschild, the Rev. T. Wood, 

 Mr. S. Stevens, the Rev. Seymour St. John, and others took 

 part. 



Royal Meteorological Society, April 18. — Mr. Richard 

 Inwards, the President, delivered an address on some phe- 

 nomena of the upper air. He said that there are three principal 

 ways in which the higher atmosphere may be studied : (l) by 

 living in it on some of the great mountain chains which pierce 

 many miles into the air in various parts of the globe ; (2) by 

 ascending into it by means of balloons ; and (3) by the study of 

 the upper currents as shown to our sight by the movements of 

 the clouds. After describing the effects of rarified air on animal 

 life and natural phenomena, Mr. Inwards proceeded to give an 

 account of various balloon ascents which had been undertaken 

 with the object of making meteorological observations. In 1850 

 Messrs. Barral and Bixio, when they had ascended to 20,000 

 feet, found the temperature had sunk to 15° F. ; but this was in 

 a cloud, and on emerging from this 3000 feet higher, the tem- 

 perature fell as low as — 38°, or 70° below freezing point. 

 In 1862, Mr. Glaisher and Mr. Coxwell made their famous 

 ascent when they reached an altitude of about seven miles from 

 the earth. A short time ago a balloon without an aeronaut, but 

 having a set of self-recording instruments attached, was sent up 

 in France, and from the records obtained it is shown that a 

 height of about ten miles was attained, and that the temperature 

 fell to - 104° F. Clouds are simply a form of water made 

 visible by the cooling of the air which previously held the water 

 in the form of invisible vapour. Every cloud maybe regarded as 

 the top of an invisible warm column or current thrusting its way 

 into a colder body of air. After referring to the various classifica- 

 tions and nomenclatures of clouds, of which that proposed by 

 Luke Howard in 1803 is still in general use, Mr. Inwards said that 

 whatever system of naming and classifying clouds be adopted, it 

 should depend on the heights of the various clouds in the air, 

 and he gave a few rough rules by which the comparative alti- 

 tudes of the clouds may be judged when there is no time or 

 opportunity to make exact measurements. Among the indica- 

 tions by which a cloud's height in the air may be gathered are 

 its form and outline, its shade or shadow, its apparent size and 

 movement, its perspective effect, and the lengih of time it 

 remains directly illuminated after sunset. By the last method 

 some clouds have been estimated to have been at least ten miles 

 above the surface of the earth. The cloud velocities at high 

 altitudes have been carefully noted at the Blue Hill Obssrva- 

 tory, Mass., U.S., and show, practically, that at about five 

 miles' height, the movement is three times as fast in summer, 

 and six times in winter, as compared with the currents on the 

 earth's surface. After showing a number of lantern slides illus- 

 trating the various types and forms of clouds, the aurora borealis, 

 rainbows, &c. , Mr. Inwards concluded his address by urging 

 the desirabdity of establishing a good cloud observatory some- 

 where in the British Isles. At the close of the maeiing, the 

 Fellows and their friends inspected the exhibition of instru- 

 ments, photographs, and drawings relating to the representation 

 and measurement of clouds, which had been arranged in the 



