Jan, 26, 1888J 



NATURE 



309 



tare chosen for the observations were Irom - 45° C. to 

 3CX)° C. The experiments were made on rolled bars of the 

 various steels and also on large hammered forgings 5 inches 

 diameter. Details are given in the paper of the general method 

 \ of experimentation, and also of the methods adopted for reducing 

 the metals to the very low temperature employed. The results 

 of an extensive series of experiments are recorded in tabular 

 form in the paper. The coefficients of dilatation were found 

 generally to decrease with the reduced temperature. The author 

 also found such to be the case in his recent observations on the 

 heat dilatation of pure ice from low temperatures. There 

 seemed to be a slightly greater dilatation in the direction of 

 the length of the forged metallic cylinders than when measured 

 across the diameter. It was also noticed that the coefficients of 

 dilatation were greater in the case of steels having a lower per- 

 centage of combined carbon than in those containing a higher 

 percentage. 



January 12. — "Invariants, Covariants, and Quotient Deriva- 

 tives associated with Linear Differential Equations." By A. 

 R. Forsyth, F.R.S. 



The memoir deals with the covariantive forms associated with 

 the general ordinary linear differential equation. The most 

 general transformation to which such an equation can be sub- 

 jected without changing its character is one whereby the 

 dependent variable j is changed to « by a relation/ — ttf[x), 

 and at the same time the independent variable is changed, say, 

 from X to z. When these transformations are effected there are 

 n relations between the coefficients P and Q of the equation in 

 its two forms, and it is shown that from these others can be 

 deduced which are of the form 



.^(P)=( 



dx) 



^{Q)- 



Such a function \p is called an invariant of index p. 



Irreducible invariants are proved to be divisible into two 

 classes, fundamental and derived. Each of the former, which 

 are « - 2 in number, consists of two parts ; one of these is 

 linear in the quantities P and their derivatives, the other is not 

 linear, but has in every term as a factor either Pg or some deri- 

 vative of Pg. It is shown that the differential equation can be 



d" ~ ^ u 

 reduced to a canonical form without any term in — or 



dz"--" 

 ; and hence each of the prior class of invariants is 



d" - 



dz"- 



linear in the coefficients of the canonical form and their deri- 

 vatives. These fundamental invariants are called priminvariants. 

 The derived invariants are obtained from the priminvariants by 

 two processes, which are called the quadriderivative and the 

 Jacobian ; they are most conveniently arranged in classes 

 according to their degrees in the coefficients of the equation. 

 The number of quadrinvariants is 2« — 5 ; the number of in- 

 variants of every degree higher than the second is n - 2. 



The relation between the independent variables of a semi- 

 canonical form and of the canonical form shows that the 

 dependent variable may be considered as a covariant. It is 

 proved that there are other n - 2 associate dependent variables, 

 each satisfying a linear equation and possessing for the canonical 

 form the invariantive property. 



From this aggregate of dependent variables, a set of irre- 

 ducible identical covariants is derived by the two processes 

 formerly used for the invariants ; when the equation is taken in 

 its canonical form, all these covariants up to a certain order 

 involve the dependent variables alone. There is also a set of 

 irreducible mixed covariants which are the Jacobians of each of 

 the dependent variables in turn, and one of the invariants. 



Illustrations of the results are given for the equations of the 

 second, the third, and the fourth orders ; and in this connection, 

 functions, called quotient derivatives, are obtained. Some of 

 their properties are given, one of the most important being that 

 they are covariantive for homographic transformation of both 

 the dependent and the independent variables. 



Finally, the characteristic differential equations satisfied by all 

 concomitants are obtained ; and among other inferences it is 

 proved that the aggregate of concomitants constituted by the 

 invariants and covariants obtained in the earlier part of the 

 memoir is complete, i.e. that any concomitant can be algebraic- 

 ally expressed in terms of the members of that aggregate. 



" Preliminary Note on the Nephridia of Perichaeta." By 

 Frank E. Beddard, M.A. 



The following observations are the result of a study of a 

 species of Perichseta, which is probably identical with Perrier's 

 P. aspergillmn. I owe a number of excellently preserved ex- 

 amples to the kindness of Mr. Shipley, Fellow of Christ's 

 College, Cambridge. 



In transverse sections of the anterior segments the nephridia 

 are seen to form numerous tufts of glandular tubules closely 

 related to the body-wall and to the septa. This appearance, 

 which is also seen in dissections, is very different from that of 

 most earthworms, and has been commented upon by other 

 observers. 



The remarkable appearance of the nephridia led me to infer 

 that I should find the external apertures in each segment to be 

 numerous, as I showed to be the case in Acanthodrilus. I am 

 now able to state that this is also the case in Perichaeta (in 

 all probability in other species besides P. aspergillum). The 

 external pores lie between the setae, but have no regularity in 

 their arrangement ; frequently there were three or four between 

 two successive setae, as often there seemed to be only one or two. 

 The minute structure of the terminal section of nephridia is 

 slightly different from that of Acanthodrilus. Another point, to 

 which I wish to direct attention in this communication, is that 

 in Perichceta there is a connection between the nephridia of 

 successive segments. 



Quite recently, Ed. Meyer and Cunningham have shown that 

 in Lanice conchilega the nephridia of each side are connected by 

 a continuous longitudinal duct. This discovery is in accord with 

 the presumed origin of the Annelid from the Platyhelminth 

 excretory system, and also with the development of Polygordius 

 (Hatchek) and Lumbricus. In Perichaeta the connection be- 

 tween the nephridial tufts of successive segments is not brought 

 about by a continuous longitudinal duct, one on each side of the 

 body, but by numerous tubules which perforate the interseg- 

 mental septa. Thus it appears that the nephridial system of 

 Perichceta consists of a network of tnbules. In this respect 

 Perichaeta agrees with the leech Pontobdella, but differs in the 

 presence of numerous nephridiopores in each segment. These 

 facts appear to lend further support to the view that it is 

 possible to derive the Annelid from the Platyhelminth excretory 

 system. 



Lang has pointed out that the " secondary " pores by which 

 the excretory organ of the Platyhelminths communicates with 

 the exterior have probably given rise to the nephridial pores in 

 the Annelida ; by a subsequent arrangement of these in a meta- 

 meric fashion, and by the breaking up of the nephridial net- 

 work, the paired nephridia have originated. The longitudinal 

 canal has disappeared, except in the cases that I have already 

 mentioned. In some Platyhelminths the longitudinal canals are, 

 partly at least, broken up into a network ; and it is this con 

 dition which has persisted in Perichaeta and Pontobdella ; more 

 over, in some Platyhelminths, where the " secondary " pores have 

 become metamerically arranged, there are more than one pair 

 to each " segment." For this reason it is perhaps allowable to 

 regard the condition of the nephridia in Perichaeta as more 

 archaic than Pontobdella. The disappearance of the connection 

 between the nephridia of successive segments leads to the con- 

 dition which exists in Acanthodrilus ; the reduction of the ex- 

 ternal pores, already perceptible in the posterior segments of 

 A. mu/tiportts, calm'mntes in the disappearance of all but two in 

 each segment. The irregularity in the position of these, which 

 is best marked in Plutellus, is the last trace of the presence of 

 multiple nephridiopores in each segment. 



Royal Meteorological Society, January 18. — Mr. W. Ellis, 

 President, in the chair. — The paper read was on the non-instru- 

 mental meteorology of England, Wales, and Ireland, by Mr. G. 

 M. Whipple. This is a discussion of the observations of wind, 

 cloud, thunderstorms, hail, snow, &c., made at the stations of 

 the Royal Meteorological Society during the eight years 1878- 

 85, and published in the Meteorological Record. The S.W. 

 wind is the most prevalent, and blows on the average seventy- 

 four days in the year ; the W. wind occurs almost as frequently, 

 blowing sixty-five days. The least dominant winds are tlie 

 S.E. and N., which occur on twenty-seven days, and the N.E. 

 on thirty-two days. Thunderstorms are most frequent in the 

 eastern and midland counties, and least frequent in the north of 

 Wales. — After the reading of this paper, the annual general 

 meeting was held. The report of the Council showed t'^'^" 

 Society to be in a satisfactory condition, the number of Fel' 



