February i6, 1893] 



NATURE 



383 



This is not an invariant of a single quantic, but of the 

 three 



a^x- + Txi^xy + a.^v- 

 a„-r + a^y. 



It bears, however, a definite relation to the first of these 

 three quantics, viz. : it is a seminvciriartt of that quantic, being 

 in fact the source of its cubic-covariant J. The paper points 

 out that all seminvariants are thus invariants of two or more 

 quantics, and can therefore be represented by graphs ; the 

 difference between a graph representing an invariant of a 

 quantic and one representing a seminvariant of the same quantic 

 consisting merely in this, that the simple forms, i.e. the Small 

 circles or nuclei of the graphs in the former case are all of the 

 same " valence," i.e. have the same number of bonds, while in 

 the latter, though of like marks, they differ in valence. The 

 classification of seminvariants, according to the valences of the 

 simple forms composing them, or, in other words, according to 

 the orders of the quantics of the systems of which they are re- 

 spectively invariants, obviously throws considerable light upon 

 their structure. 



The paper also deals with the breaking up of graphs into 

 simpler ones ; and gives a theorem upon the subject which leads 

 to some interesting results. It points out, moreover, how the 

 graphs representing the sources of covariants can be instan- 

 taneously derived from those representing the covariants them- 

 selves. 



On the evaluation of a certain surface-integral and its applica- 

 tion to the expansion of the potential of ellipsoids in series, Dr. 

 Hobson. 



On the vibrations of an elastic circular ring, by Mr. A. E. H. 

 Love. — The ring is supposed to be of small circular section of 

 radius c, and the elastic central-line a circle of radius a. There 

 are four ways of displacing the ring. A point on the central-line 

 may move along the radius of the circle which is its primitive 

 form, or perpendicular to the plane of this circle, or along the 

 tangent to this circle ; and the circular sections may be dis- 

 placed by rotation about the central-line. The modes of vibra- 

 tion fall into four classes, of which two are physically import- 

 ant : — Class I. Flexural vibrations in plane of ring. — These 

 were investigated by Hoppe in 1871 (Cr^&, bd. Ixxiii.). The 

 motion of a point on the elastic central-line is compounded of a 

 displacement in and out along the radius and a displacement along 

 the tangent to the circle, so proportioned that the central-line 

 remains unstretched, and the nodes of the former displace- 

 ment are the antinodes of the latter. There must be at least 

 two wave-lengths to the circumference, and the frequency ( //27r) 

 of the mode in which there are n wave-lengths to the circumfer- 

 ence is given by the equation 



^2 = ^ «M«^ 



I? 



in which E is the Young's modulus, and p^ the density of the 

 material. Except for the numerical coefficient this is precisely 

 similar to the formula for the lateral vibrations of a straight bar 

 ofthe same material and section and of length tra (for which the 

 fundamental tone has the same wave-lengths). The sequence 

 of component tones when n is very great is ultimately identical 

 vith that ofthe tones of a free-free bar of length wa, but the 



■quence for the low tones is quite different to that for a bar. 



lass II. Flexural vibrations perpendicular to the plane of the 

 ring. — It is found to be impossible to make the ring vibrate 

 freely so that each particle of the elastic central-line moves per- 

 pendicular to the plane of the ring, unless at the same time the 

 sections turn about the central-line through a certain angle. 

 The flexure perpendicular to the plane of the ring is always 

 accompanied by torsion. As in Class I. there must be at least two 

 wave-lengths to the circumference, and the frequency of the 

 niode in which there are n wave-lengths to the circumference is 

 eiven by the equation 



1 + <r +n^ p^ a* 

 where (t is the Poissons ratio for the material and the other 

 constants have the same meaning as before. (For most hard 

 solids <T is about \.) Since « must be at least 2 the sequence of 

 tones is very nearly the same as in the vibrations of Class I., 

 but the pitch is slightly lower, the ratio of the frequencies for 

 the gravest tones being ^ / \\ , which is ver}' little less 



than a comma. 

 NO. ] 



y 



For the higher tones, as we should expect, there 

 2X6. VOL. 47] 



is no sensible difference. These two classes include all that 

 have much physical importance. The remaining types can be 

 classified as : — Class III. Extensional vibrations. — The motion 

 may be purely radial or partly radial and partly tangential. In 

 the second case there will be an integral number of wave- 

 lengths, and when this number is ;/ we have the formula for the 

 frequency 



/2 = (I -f 71-^) ^ 4 



Putting n = zero we find the frequency of the purely radial 

 vibrations. The pitch of any mode of extensional vibration of 

 the ring is of the same order of magnitude as the pitch of the 

 corresponding longitudinal vibration of a bar of length equal to 

 half the circumference, the formula for the latter being in fact 

 derived by writing n~ for i + ir. Class IV. Torsional vibra- 

 tions. — The motion consists of an angular displacement of the 

 sections about the elastic central-line accompanied by a rela- 

 tively very small displacement of the points on this line per- 

 pendicular to the plane of the ring. When there are n wave- 

 lengths to the circumference the frequency is given by the 

 formula 



p-i = {i + „ + „•-') f^ a'-, 

 Po 

 in which ju is the rigidity of the material. There is one sym- 

 metrical mode for which n is zero, and since 2ju (1 + 0-)=: E, 

 the frequency of this mode is ^ V 2 of that of the radial vibra- 

 tions. The pitch of the torsional vibrations is comparable with 

 that for a straight rod of length equal to half the circumference, 

 the formula for the latter being in fact derived by writing «'- in 

 place of I H- <r -t- «-. Formulae equivalent to those given in 

 connection with Classes II. and IV. have been obtained by Mr. 

 Basset (Proc. Dec. 1891), but he has not interpreted his results. 

 Entomological Society, February 8. — Mr. Henry John 

 Elwes, president, in the chair. — The President announced that 

 he had nominated Mr. F. DuCane Godman, F.R.S., Mr. 

 Frederic Merrifield, and Mr. George H. Verrall as Vice- 

 Presidents during the Session 1893-1894. — Mr. S. Stevens ex- 

 hibited a specimen of Charocampa celerio, in very fine condi- 

 tion, captured at light, in Hastings, on September 26 last, by 

 Mr. Johnson. — Mr. A. J. Chitty exhibited specimens of Gib- 

 Hum scotias and Pentarthrum huttoni, taken by Mr. Rye in a 

 cellar in Shoe Lane. He stated that the Giblmim scotias lived 

 in a mixture of beer and sawdust in the cellar, and that when 

 I this was cleaned out the beetles disappeared. The Pentarthrum 

 huttoni lived in wood in the cellar. — Mr. McLachlan exhibited 

 a large Noctuid moth, which had been placed in his hands by 

 Mr. R. H. Scott, F.R.S., of the Meteorological OfSce. It 

 was stated to have been taken at sea in the South Atlantic, in 

 about lat. 28° S., long. 26° W. Colonel Swinhoe and the 

 President made some remarks on the species, and on the migra- 

 tion of many species of Lepidoptera. — Mr. W. F. H. Blandford 

 exhibited larvae and pupae of Khynchophorus palmarum, L., 

 the Gru-gru Worm of the West Indian Islands, which is eaten 

 as a delicacy by the Negroes and by the French Creoles of 

 Martinique. He stated that the existence of post-thoracic stig- 

 mata in the larva of a species of Rhynchophorus had been 

 mentioned by Candeze, but denied by Leconte and Horn. 

 They were certainly present in the larva of R. palmarum, but 

 were very minute. — Mr. G. T. Porritt exhibited two varieties of 

 Arctia lubricipeda from York ; an olive-banded specimen of 

 Bombyx quercus from Huddersfield ; and a small melanic 

 specimen of Melanippe hastata from Wharncliffe Wood, York- 

 shire. — Mr. H. Goss exhibited species of Lepidoptera, Coleop- 

 tera, and Neuroptera, sent to him by Major G. H. Leathern, 

 who had collected them, last June and July, whilst on a shoot- 

 ing expedition in Kashmi territory, Bengal. Some of the 

 specimens were taken by Major Leathern at an elevation of 

 from 10,000 to 11,000 feet, but the majority were stated to have 

 been collected in the Krishnye Valley, which drains the glaciers 

 on the western slopes of the Nun Kun range. Mr. Elwes re- 

 marked that some of the butterflies were of great interest. — 

 Mr. G. F. Hampson exhibited a curious form of Parnassitis, 

 taken by Sir Henry Jenkyns, K.C.B., on June 29 last, in the 

 Gasternthal, Kandersteg. — Mr. J. M. Adye exhibited a long 

 series of remarkable varieties of Boarmia repandata, taken last 

 July in the New Forest.— Mr. C. O. Waterhouse exhibited a 

 photograph of the middle of the eye of a male Tabanus, show- 

 ing square and other forma of facets, multiplied twenty-five 

 times. — Mr. R. Trimen, F.R.S., communicated a paper entitled 

 " On some new, or imperfectly known, species of South African 



