138 



NATURE 



\pec. 1 6, 1875 



Red Sea, from which it is separated by a low range of hills. 

 The gentle rains at one part of the year supply sufficient 

 moisture to heat the iron pyrites scattered over the surface of 

 Arrho to a mild glow ; the substance is decomposed, lixiviated, 

 and brought into combination with several combustible matters, 

 and eruptions like those of volcanoes take place ; slime cones are 

 formed from four^to twelve feet in height, from which, as from 

 pipes, issue steam and flame. These generally ephemeral for- 

 mations consist of mud which is mixed with sulphur and salt, 

 and in the dried cones may be found almost pure sublimated 

 sulphur. The general appearance of the place is like that of 

 pulp boiling in a huge, angular, zig-zag cauldron. The pheno- 

 menon continues till increased rains lay the ground under water, 

 after disappearance of which at the end of the rain periods, 

 a hard salt crust, several inches thick, covers the ground. — The 

 trachytes of the island of Cos are described by Dr. Dolter ; 

 Dr. Stache gives an account of the eruptive rocks in the Ortler 

 region and the mountain group of the Zwolfer-Spitz in Upper 

 Vintschgau ; and Dr. Homes, of observations in the district 

 where the Rienz takes its rise. 



SOCIETIES AND ACADEMIES 



London 



Royal Society, Nov. 25. — On the production of Glycosuria 

 by the effect of oxygenated blood on the liver, by F. W. Pavy, 

 M.D.,F.R.S. 



The conclusions arrived at are that the amyloid substance 

 found in the liver is a body which tends to accumulate in certain 

 animal structures under the existence of a limited supply Of 

 oxygen, and that it is through the liver exceptionally receiving 

 the supply of venous blood it does, that the special condition 

 belonging to it is attributable. It is also shown that the undue 

 transmission of oxygenated blood to that organ at once induces 

 an altered state, which is rendered evident by the production of 

 glycosuria. 



On the Structure and Relations of the Alcyonarian Heliopora 

 cccrtdea, with some account of the anatomy of a species of Sarco- 

 phyton ; notes on the structure of species of the genera Mille- 

 pora, Pocillopora, and Stylaster , and remarks on the affinities of 

 certain Paleozoic Corals, by H. M. Mozeley, Naturalist on the 

 Challenger Expedition. 



The title of this paper indicates the nature of its contents. 

 The author has not been able to decide whether Millepora 

 is one of the actinozoa or belongs to the hydrozoa as stated by 

 Prof. Agassiz. Helioperi is undoubtedly alcyonarian. 



Mathematical Society, Dec. 9. — Prof. H. J. S. Smith, 

 F.R.S., president, in the chair. — Major J. R. Campbell and 

 Prof. G. M. Minchin were elected members. — Prof. Clifford 

 read a paper on the transformation of elliptic functions, in which 

 he attempted to apply Jacobi's geometrical representation of the 

 addition-theorem in elliptic functions to the theory of their 

 transformation. — Prof. Cay ley spoke on a system of algebraical 

 equations connected with Malfatti's problem. The communica- 

 tion was an extension of a paper by the same gentleman in the 

 Cambridge and Dublin Mathematical jfournal, torn, iv., 1849, 

 pp. 270-275.— The chairman next communicated three short 

 notes ; i. Oa a problem of Eisenstein's. If p is an uneven 



^p I 



prime, the function 4 = Z can always be expressed in 



the form F" - (-i)^^-^" ''V-^S where ^ and Fare rational 

 and integral functions of x, having integral coefficients. This is 

 a theorem of Gauss. Eisenstein's problem (Crelle's yournal, 

 vol. xxvii. p. 83) is "to determine the cases in which the equa- 

 tion Z = F3 - (- i)*^^~'^/^-2 admits of a multipHcity of 

 solutions, and to ascertain the law connecting the various solu- 

 tions, when there is more than one." The solution of this 

 problem is as follows : — If \T, t/] is any solution whatever 



in integral numbers of the equation T'^—{—\)^^~ ^^p U^^^ 

 and \^X, F] is any one given solution of Gauss's equation, then 

 all the solutions of Gauss's equation are comprised in the formula — 



[k{TX+(- i)^pUV),i{UX+ TY)\. 



Thus if / = 4« -H 3 the equation admits of but one solution (the 

 four solutions [± X, ± F] being regarded as but one) except in 

 the case / = 3, when it admits of three ; if /> = 4 « + i, the 

 equation admits of an infinite number of solutions. That the 



functions [h{TX + pUY), ii{UX+ T Y)\ are all of them 

 solutions of Gauss's equation is evident ; the proof that this formula 

 comprises all the solutions of the equations is less elementary, 

 because it depends on the irreducibility of the function Z. There 

 exists a general theory of the representation of raiional and 

 integral functions of ;r by quadratic forms ; such representation 

 being, of course, only possible when the given function of x is 

 capable of resolution into two factors by the adjunction of a 

 quadratic surd. — 2. On the joint invariants of two conies, or two 

 quadrics. Let P and Q be two conies, and let i 23 be any 

 triangle self-conjugate with regard to P. Let also P-^, P^, P-^, 

 be the rectangles of the points I, 2, 3, with regard to the conic 

 P, these rectangles being taken upon transversals measured in 

 any fixed direction ; and let Q^, Q^, Q.^, have similar meanings 

 with regard to the conic Q, the direction of the transversals being 



also fixed. Then the expression "5 + ^ + jr has the same 



^\ -^1 -^3 

 value for all self-conjugate triangles of P ; and is, in fact, that 

 invariant of P, Q, which is linear with regard to Q, and quad- 

 ratic with regard to P, and the evanescence of which expresses 

 that Q harmonically circumscribes P. The corresponding theorem 

 in the geometry of the straight line is : " If Q^, Q^, Pj, P^, are 

 two pairs of fixed points on a line, and if ^j, A^, is any pair of 

 harmonic conjugates of /"^ P^, the value of the expression 



^2 Qi ■ ^2 Qi 

 A^P^ ■ A,P,.A,P, 

 is independent of the particular pair A^ A^ considered." From 

 this theorem, the result given above for two conies follows imme- 

 diately ; from it the corresponding property for two quadrics 

 may be inferred, viz. : 



Qi + Qi+^ + Qa = constant ; 

 -^1 •'a -^3 -^4 

 and so on for quadratic functions containing any number of in- 

 determinates. 3. On the equation P x D — constant, of the 

 geodesic lines of an ellipsoid. From this equation (in which P 

 is the perpendicular from the centre upon the tangent plane at 

 any point of the geodesic, and D is the semi-diameter parallel to 

 the tangent line of the geodesic) it is convenient to be able to 

 infer directly the principal properties of the geodesic line, without 

 having first to transform the equation into M . Liouville's form, 

 f^"^ cos'^ i + p^ sin^ i = a^. In Dr. Salmon's "Geometry of 

 Three Dimensions," the theorem of the constancy of the 

 sum or difference of the geodesic radii vectores, drawn from 

 any point of a line of curvature to two umbilics, is thus 

 demonstrated. And it is worth while to add (though it is 

 very improbable that the point has not been noticed before) that 

 a proof of the theorem that two geodesic tangents of a Ime of 

 curvature which intersect at right angles, intersect on a sphero- 

 conic, may similarly be obtained without transforming the equa- 

 tion. Let Q be the point where the two gcDdesic tangents inter- 

 sect at right angles, O the centre of the ellipsoid ; let c = O Q, 

 and let a, b be the semi-axes of the central section parallel to the 

 tangent plane at Q. The two geodesies make angles of 45° with the 

 lines of curvature at Q : hence, for either of these geodesic lines, 



2a* ^2 

 Z?^ = 2 . , a. Let (/ be a second point where two geodesic 



tangents to the same line of curvature intersect at right angles ; 

 ,, 'zP'^a'^b'^ _ iP'^a'-ib'-" 



tnen — „ ■ /., , »a — ' because P x D has the same 



a'- -ir 0- a^ ■\- o* 



value for all geodesic lines touching the same line of curvature. 

 But/*^a^3* = P'^ a'^ b'^, because parallelepipeds circumscribing 

 an ellipsoid with their faces parallel to conjugate diametral planes 

 are equal. Hence a^ + b^ = a''^ + b'^. But also a^ + b" + c^ — 

 ^'2 ^ ^'2 ^ ^2 ._ ^ _ ^'^ and Q and Q^ lie in the same sphero-conic. 

 Mr. Tucker (in the absence of the author) brought before the 

 Society a paper by Mr. H. W. Lloyd Tarmer on the solution of 

 certain partial differential equations of the second order, having 

 more than two independent variables. The equations con^ 

 sidered afe included in the form — 



« = « 



2 



1 = 1 y =; I dXiflx-^ 



where V-i^V^ are functions of 



+ Vo = o 



(I) 



•^n* 



'^^(-^)---^°(-^> 



and it is proposed to investigate the conditions that ( i ) should 

 be soluble in ternis of arbitrary functions, the arguments of 



