April ^, 1888] 



NA TURE 



551 



tion as before is formed with regard to iher^e new variables, the 

 equation 



* 



U(X,V)/ 



is satisfied. 



The transformations for which any detailed results are given 

 are of the general homographic type. The characteristic 

 properties of such invariants are : — 



(i.) Every invariant is explicitly free from the variables, but 

 necessarily contains both the differential coefficients / and q of 

 the first order. 



(ii. ) It is homogeneous in the differential coefficients, and is 

 of uniform and the same grade in differentiations with regard 

 to each of the independent variables. 



(iii.) It is symmetric or skew symmetric with regard to these 

 differentiations. 



(iv. ) It satisfies four differential form-equations and two 

 index-equations, all linear and partial of the first order. 



An invariant is said to be proper to the rank «, when the 

 highest differential coefficient of 2 which occurs in it is of order 

 n. By means of the solutions of the form-equations, the follow- 

 ing propositions relating to irreducible invariants in a single 

 dependent variable, z, are established : — 



Invariants can be ranged in sets, each set being proper to a 

 particular rank. 



There is no invariant proper to the rank i; there is one 

 proper to the rank 2 ; there are three invariants proper to the 

 rank 3. 



For every value of n greater than 3 there are « + i invariants 

 proper to the rank «, which can be chosen so as to be linear in 

 the partial differential coefficients of order n. 



Every invariant can be expressed in terms of this aggregate of 

 irreducible invariants ; and the expression involves invariants 

 proper to rank no higher than the order of the highest 

 differential coefficient which occurs in that invariant. 



Some of the properties of the irreducible invariants involving 

 differential coefficients of two dependent variables are obtained, 

 and, in particular, it is shown that there is a single irreducible 

 simultaneous invariant proper to the rank i, and that there are 

 four such invariants proper to the rank 2. 



The theory of eduction is next considered, with some 

 examples. Finally, it is shown that the theory of binary forms 

 can be partly connected with functional invariants. 



March 22. — "Second Preliminary Note on the Development 

 of Apteryx." By T. Jeffery Parker, B.Sc, C.M.Z.S., Pro- 

 fessor of Biology in the University of Otago. Communicated 

 by W. K. Parker, F.R.S. 



Chemical Society, March 15. — Mr. W. Crookes, F.R.S. , 

 in the chair. —The following papers were read : — The nature of 

 solutions as elucidated by the heat evolved on their dilution ; 

 Part I, calcium chloride, by Mr. S. U. Pickering. To deter- 

 mine the nature of the action which takes place on diluting 

 aqueous solutions, the author has examined calcium chloride, 

 and, in a series of elaborate experiments, has obtained results 

 which form a curve of great regularity. This regularity, how- 

 ever, is only apparent, since on differentiation a number of 

 independent curves are obtained, each of which on further 

 differentiation gives a straight line. The points at which these 

 lines meet, when produced, indicate percentages of water cor- 

 responding to distinct hydrates of the salt, and moreover coin- 

 cide in every case, within the limits of experimental error, with 

 the points obtained by treating in a similar manner the curve 

 expressing the densities of the various solutions. The author 

 contends that these results, taken in conjunction with the fact that 

 the variation in the electrical conductivity and the density of 

 sulphuric acid on diluting with water also point to the existence 

 of certain hydrates in solution, make it no longer reasonable to 

 doubt that solutions do in reality consist of such hydrates, and is 

 of opinion that any theory of the nature of solutions which 

 ignores their existence must be rejected absolutely and for ever. 

 A new form of mixing calorimeter, devised for these experi- 

 ments, was exhibited. — The action of thiocyanates on aldehyde- 

 ammonias, by Dr. A. E. Dixon. — Carboxy-derivatives of quinone, 

 by Dr. J. U. Nef. Ethylic paradiketohexamethylenecarboxylate, 

 obtained by the reduction of ethylic quinonetetracarboxylate 

 with zinc dust, exists apparently in three distinct modifications, 

 only two of which, however, have been studied — the one modi- 

 fication IS green and crystallizes in needles, the other is yellow 



and crystallizes in plates ; after fusion, the former appears dark 

 yellow and the latter bright yellow. If either modification be 

 separately dissolved in carbon bisulphide, a solution is ob- 

 tained from which the two substances crystallize out together ; 

 the solution also has the same colour and the same absorption 

 spectrum whichever modification be dissolved. The author 

 calls attention in the paper to a number of similar cases of 

 dimorphism. — The action of acetone on ammonium salts of 

 fatty acids in the presence of dehydrating agents, by Dr. S. 

 Ruhemann and Mr. D. J. Carnegie. — A method of estimating 

 nitrites either alone or in pre sence of nitrates and chlorides, by 

 Mr. T. C. Day. 



Paris. 



Academy of Sciences, March 26. — M. Janssen in the 

 chair. — New theory of the equatorial coud^ 2ind of equatorials in 

 general (continued), by MM. Loewy and P. Puiseux. Here are 

 given the general formulas promised in the previous communica- 

 tion, together with the terms depending on the position of the 

 outer glass. — On the relations of atmospheric nitrogen with 

 vegetable humus, by M. Th. Schloesing. A detailed account is 

 given of the experiments carried out according to the already 

 described method for the purpose of ascertaining whether gaseous 

 nitrogen is fixed by vegetable soil. The disappearance of the 

 oxygen shows in six different cases that the combustion of the 

 organic substances takes place in various degrees depending on the 

 quantity and nature of such substances. During this combustion 

 nitric acid is formed with disappearance of the ammonia. The 

 volume of gaeeous nitrogen contained in the soil does not per- 

 ceptibly vary. — On the absorption of saline substances by plants 

 (continued), by MM. Berthelot and G. Andre. The experiments 

 here described deal with the acetate of potassa, an organic salt 

 analogous to those present or produced in the plants ; also with 

 the nitrate of potassa, the formation or accumulation of which is 

 characteristic of certain species, especially of the Amaranthus 

 group. This accumulation is shown to depend rather on the 

 period of vegetation than on the proportion of the salt in the 

 ground. — New nebulse of a remarkable character discovered in 

 the Pleiades, by means of photography, by MM. Henry, and 

 described by M. Mouchez. Besides a new nebula round. 

 Maia in the Pleiades, the more recent researches of MM. 

 Henry have revealed a great mass of cosmic matter 

 covering a large part of this constellation. But the most 

 remarkable discovery, and one of an absolutely unique charac- 

 ter, is a rectilinear thread of nebular matter projected from 

 the central mass nearly in the direction from east to west 

 for a distance of 35' to 40' of arc, but with a thickness of no 

 more than 3" to 4". This thread crosses on its path seven star.", 

 which it seems to string together like the beads on a rosary, 

 and slightly changes its direction at the point where it meets the 

 largest of these stars. A second streak, somewhat similar, but 

 shorter, is perceptible in the middle of the nebular mass. — Pre- 

 liminary work for the execution of the photographic chart of the 

 firmament, by M. Mouchez. Reference is made to the publica- 

 tion of a Bulletin specially devoted to this object. Two more 

 Observatories, those of Potsdam and Oxford, are announced as 

 intending to take part in this great work, making thirteen sta- 

 tions altogether. These, it is stated, are already sufficient to 

 secure the completion of the undertaking in the course of four 

 or five years. — Treatment of auriferous sands by amalgamation, 

 in ancient times, by M. Berthelot. The second part, just pub- 

 lished, of the already noticed "Collection des Alchimistes 

 grecs," contains the works of Zosimus, a writer of the third 

 century of the new era, dealing with the extraction of gold by 

 means of its natural ores treated with mercury. This process 

 appears to have been substituted for a still more ancient method, 

 in which the ore was fused with lead, salt, a little tin, and barley 

 bran, and submitted to a genuine process of refining. — Obser- 

 vations of the Comet 1888a, made at the Paris Observatory with 

 the equatorial of the West Tower, by M. G. Bigourdan. The 

 observation here recorded was taken on March 25, when the 

 comet, discovered at the Cape, on February 18, by M. Sawer- 

 thal, was approaching the northern hemisphere.— On^ a new 

 mercury-bath for the obsers'ation of the nadir, by M. Perigaud. 

 This valuable appliance at last gives the long-sought solution of 

 the problem, how to employ the mercury-bath for deterniining 

 the vertical, and for taking observations by reflection in all 

 states of the weather, and on ground subject to the constant 

 vibrations produced by carriage traffic, as in large towns. 



