December io, 1891] 



NATURE 



141 



Bidder has himself been working at Naples for the last five 

 years. His appreciatory remarks on Dendy's researches prove 

 how much of interesting and new matter lies in manuscript 

 in the laboratory at Naples, and leads us to express the hope 

 that Mr. Bidder will soon follow the example of his senior, and 

 give us a monograph of the Calcarea Homocoela of the Gulf of 

 Naples ; with more details on the glandular ectoderm. 



Travaux tie la SocUte des Naturalisles de St. Petersbotirg, 

 Section de Zoologie et de Physiologic, tome xxi., livr. i (Russian). — 

 On the influence of temperature, and the distance from the section 

 of a nerve, on its electrical irritability, by B. F. Verigo. — Ob- 

 servations on the Araneina, by V. Wagner. — The minutes of 

 proceedings contain several interesting notes : namely, a list of 

 the Bryozoa of the Murman coast of Russia, by M. Khvoro- 

 stansky, containing eighty-one species ; on the blood of some 

 invertebrates, by V. Wagner, from which it appears that it 

 always contains two different kinds of cells -the granulous and 

 coloured ones, and the amoeboid ones or leucocytes, besides some 

 other cells which, however, must be considered as derived 

 from the above two kinds. — M. Shimkevitch's remarks on the 

 artificial incubation of ostriches in the ostrich park at Helio- 

 polis are also worth mentioning. 



Bulletin de la Societe des Naturalisles de Moscou, 1 89 1, 

 No. I.— On the group of the s-illimanite and the part played by 

 aluminium in the silicates, by W. Wernadsky (in Russian, 

 summed up in French). The paper contains, besides the 

 description of the experiments already published in the Comptes 

 rendus, a discussion of the facts, which brings the author to 

 the following conclusions : the compounds of silicon with 

 aluminium have an acid reaction ; they may be embodied in 

 one group, that of the sillimanite. Some of them are hydrates, 

 and some others are salts of these, or of other possible an- 

 hydrides. Polymorphic varieties arise in this group with the 

 change of physical conditions, without any perceptible change 

 in the chemical composition.— On the morphology and classi- 

 fication of the Chlamydomonads, by Prof. Goroshankin (in 

 German, with three coloured plates) ; being a full monograph on 

 the family, in which the following new species are described : 

 Chlamydomonas De-Baryana, C. Perty, C. Steinii, C. Kuteini- 

 kowi, C. reticulata, and C. ^/^r^w^^r^u.— On some peculiarities 

 in the development and the structure of the skull of Pelobatcs 

 fuscus, by A. N. Sewertzow. — Note on the Hipparion crassum, 

 by Marie Pavloff (French).— On the fossil plant-bearing depo- 

 sits of East Russia and Siberia, by C. Kosmovsky (in French). 

 The close similarity between the supposed Jurassic fresh-water 

 deposits of East Russia and Siberia and the "Artinsk" series 

 is briefly indicated. 



SOCIETIES AND ACADEMIES. 

 London. 

 Royal Society, November 19.—" On the Loci of Singular 

 Points and Lines which occur in connection with the Theory 

 of the Locus of Ultimate Intersections of a System of Surfaces." 

 By M. J. M. Hill, M.A., Sc.D., Professor of Mathematics at 

 University College, London. Communicated by Prof. Henrici, 

 F.R.S. 



Introduction. 



In a paper "On the c- and /-Discriminants of Ordinary 

 Integrable Differential Equations of the First Order," published 

 in vol. xix. of the Proceedings of the London Mathematical 

 Society, the factors which occur in the ^-discriminant of an 

 equation of the form/(x, y, c) = o, where /(x, y, c) is a rational 

 integral function of x, y, c, are determined analytically. 



It is shown ^ that if E = o be the equation of the envelope 

 locus of the curves/(;«r, y,c)^o; if N = o be the equation of 

 their node locus ; if C = o be the equation of their cusp locus, 

 then the factors of the discriminant are E, N^, C^. 



The object of this paper is to extend these results to surfaces. 



Part I.— The Equation of the System of Surfaces is a Rational 

 Integral Function of the Co-ordinates and one Arbitrary 

 Parameter. 

 When there is only one arbitrary parameter, each surface of 

 the system intersects the consecutive surface in a curve, whose 

 equations are the equation of the surface and the equation ob- 

 tained by differentiating it with regard to the parameter. (These 



' The theorem was originally given by Prof. Cayley, in the Messenger of 

 Mathematics, vol. ii., 1872, pp. 6-12. 



NO. II 54, VOL. 45] 



equations will be called the fundamental equations in this part.) 

 Hence each surface touches the envelope along a curve. It is 

 known that the equation of the envelope may be obtained by 

 eliminating the parameter from the fundamental equations and 

 equating a factor of the result to zero. But it frequently happens 

 that there are other factors of the result (or discriminant) which, 

 when equated to zero, do not give the equation of the envelope. 



These factors are connected with loci of singular points. 

 If each surface have one singular point, the locus of all the 

 singular points of the surfaces of the system is a curve. Its 

 equations, therefore, cannot be found by equating a factor of 

 the discriminant to zero. But if each surface of the system 

 have upon it a nodal line, then the locus of the nodal lines of 

 all the surfaces is a surface, and its equation may be found by 

 equating to zero a factor of the discriminant. 



The singular points in space, the form of which depends only 

 on the terms of the second order, when the origin of co-ordinates 

 is taken at the singular point, are : — 



(i.) The conic node. 



(ii.) The biplanar node or binode. 



(iii.) The uniplanar node or unode. 



It is shown that a surface cannot have upon it a curve at every 

 point of which there is a conic node. Hence there are two 

 varieties of nodal lines to be considered ; the first, being such 

 that every point is a binode, may be called a binodal line ; and 

 the second, being such that every point on it is a unode, may be 

 called a unodal line. 



It is shown that if E = o be the equation of the envelope 

 locus, B = o the equation of the locus of binodal lines, U = o 

 the equation of the locus of unodal lines, then the factors of the 

 discriminant are, in general, E, B'-, U^, respectively. 



This is the general theorem, but it is assumed in the course of 

 the investigation, when the discriminant is being formed, that 

 the fundamental equations are satisfied by only one value of the 

 parameter at each point on the envelope locus or on a locus of 

 binodal or unodal lines. 



The investigation is accordingly carried a step further, and it 

 is shown that if the fundamental equations are satisfied by two 

 equal values of the parameter at points on an envelope locus, 

 or on a locus of binodal or unodal lines, the factors of the 

 discriminant are E^, B^ U*, respectively. 



The geometrical meaning of the condition that the fundamental 

 equations are satisfied by two equal values of the parameter in 

 the case of the envelope is that the line of contact of the envelope 

 with each surface of the system counts three times over as a 

 curve of intersection, instead of twice as in the ordinary case. 

 The meaning of the condition in the case of the loci of singular 

 lines is that each of these loci is also an envelope. 



Part II. — The Equation of the System of Surfaces is a Rational 

 Integral Function of the Co-ordinates and two Arbitrary 

 Parameters. 



When there are two arbitrary parameters in the equation of 

 the system of surfaces, the equation of the locus of ultimate 

 intersections is found by eliminating the parameters between 

 this equation and the two equations obtained by differentiating 

 it with regard to the parameters. (These equations will in this 

 part of the investigation be called the fundamental equations.) 



In general the locus of ultimate intersections is a surface. 

 The exceptional cases in which it is not a surface are enumerated 

 at the end of the paper. These include the case where the 

 equation of the system of surfaces is of the first degree in the 

 parameters. Hence it will be supposed that the degree of the 

 equation of the system of surfaces in the parameters is above 

 the first. 



In general, the locus of ultimate intersections possesses the 

 envelope property, and the equation of the envelope is deter- 

 mined by equating the discriminant, or a factor of it, to zero. 



If factors of the discriminant exist which, when equated to 

 zero, give surfaces not possessing the envelope property, then 

 these surfaces are connected with loci of singular points. 



Now the locus of singular points of a system of surfaces whose 

 equation contains two arbitrary parameters is in general a curve. 

 Hence its equations cannot be determined by equating to zero a 

 factor of the discriminant. 



But if every surface of the system have a singular point, then 

 in general its co-ordinates may be expressed as functions of the 

 two parameters of the surface to which it belongs. Hence the 

 locus of the singular points is a surface. It will be proved that 

 it is a part of the locus of ultimate intersections. Hence its 



