Apbil 27, 1883.] 



SCIENCE. 



339 



the whole phenomena of the continuous and bright 

 line spectrum in the neighborhood of tlie nucleus 

 may be accounted for." He also discusses briefly 

 some of the polarization phejiomena of comets, and 

 the envelopes which appear near the nucleus. The 

 article is a very interesting and suggestive one; but 

 in view of the fact that comets' tails sometimes grow, 

 not a hundred thousand, but more than a million 

 miles a day, it is doubtful whether the proposed hy- 

 potliesis can be regarded as s^tfficient. — (Astr. recj., 

 March.) c. A. r. [689 



GEODESY. 

 Altitude of Lake Constance. — Part of the work 

 laid out by the European geodetic commission con- 

 sists in carrying an accurate series of levels across 

 the country, and a share of this has recently been 

 completed by the royal Prussian geodetic institute. 

 It is published as the Gradmessungn-nwellement zwi- 

 schen Swinemiinde und' Konstanz, by W. Seibt (Ber- 

 lin, 1882), and records the altitudes of a large number 

 of points from the Baltic, where the datum plane is 

 the mean water-level from fifty-four years' observa- 

 tions, to Lake Constance, where connection is made 

 with the Swiss triangulation. The railway station in 

 Constance is 399.990 met. above the Baltic. — [Verh. 

 ges.f. erdk., Berlin, 1882, .514, 538.) w. u. D. [690 



MATHEMATICS. 



Symmetric functions. — Previous mention has 

 been made of Mr. Durfee's tables for the twelfthic. 

 By a curious coincidence, M. Kehorovsky of Prague 

 has, almost simultaneously with Mr. Durfee, com- 

 puted the same tables. M. Eehorovsky's tables differ 

 from those of Mr. Durfee only in arrangement. The 

 tables as arranged by the former are identical in form 

 with those given by Prof. Cayley for the first ten 

 orders in the PJiil. trans., vol. 147; while those of 

 Mr. Durfee are arranged symmetrically, and cannot 

 be included in a half-square, as M. Eehorovsliy's are. 

 — ISitzungsb. alcad. wissensch. Wien, 1882.) t. c. 



[691 



Maximum value of a determinant. — The el'e- 

 ments of a determinant being restricted ^to lie be- 

 tween ( — a) and {-\-a), Mr. Davis finds, tliat, for all 

 determinants whose order is greater than 2, a numer- 

 ical maximum is found by making all the elements of 

 the principal diagonal = — a, and all the remaining 

 elements of the determinant = -f- a. In the maxi- 

 umm cubic determinant D*' a", all of the strata are 

 made identical, and equal to DJ^"'. The value of this 

 determinant is ± n! D^*"' a". Formulae are also 

 given for hyperspace determinants. — (Johns Hopk. 

 univ. circ, No. 20.) T. c. [692 



Functions of several variables. — M. Combes- 

 cure seeks to develop completely tlie immediate 

 conditions to be satisiied by an analytic function of 

 several imaginary variables. Assuming z,, Zo . . . Zn 

 as the variables, these are defined by the equations 

 Zj = ocj + iyj, where J = 1, 2 . . . «. Then the func- 

 tion to be considered is F{zi, ?.... . . . Zn) = <p + if. 



The differential co-efficients of ^ and i> of the first 

 order are connected by relations precisely similar 

 to those connecting these quantities when there is 

 only one variable, z : so, when one of the functions 

 (j> or iji is given, the other may be found by simple 

 quadratures. It is shown that the group of con- 

 ditions for the determination of <p reduces itself to 



n(n + l) 

 the ^ partial differential equations of the 



second order, As,t ^ ^ 0, where 



A;,,t = 



d^ 



+ 



d- 



dxh dxk dyh dijic' 

 for 7i, fc = 1, 2 . . . 71, and, of course, including the 

 cAses where h = k. These are the necessary and 

 sufficient conditions to be satisfied by (j). A means is 

 given of representing <p analytically by an exponen- 

 tial series, the eo-efiicients of which depend upon 

 the sines and cosines of (a, x, -|- . . . + a^Xn) and 

 (/8iZ/i + • • • + Pnijn); a /3, as well as the constant 

 co-efficients of these sines and cosines, being indeter- 

 minate real quantities, to which we can give any 

 values we please. — ( Comptes rendus, Jan. 22. ) T. c. 



[693 



-Homologies and conies. — If L and M are two 

 fixed points on a conic, K, and P a variable point, then 

 P H, perpendicular to L M, cuts again the circle L M P 

 in a point, H, which describes a conic, K'. If the cir- 

 cle on L M as diameter cuts K again in E F, then 

 L M and E F are the axes, and the point at infinity in 

 the direction PH is the common centre of two of the 

 twelve homologies which two conies in general de- 

 termine. The ratio of corresponding areas of K and 

 K' is constant, — a function of the eccentricity of K 

 and of the inclination of L M to the focal axis of K. 

 Given, on the other hand, the centre and axes of the 

 homology, two triply infinite systems of conies, K and 

 K', can "be determined ; the conies of each system be- 

 ing similar and similarly placed, and the common 

 points at infinity of one system being orthogonal to 

 those of the other. All the conies of the plane are 

 thus distributed into a doubly infinite number of 

 triply infinite systems. The net of conies determined 

 by three arbitrary points in a plane will give a doubly 

 infinite number of conies, one out of each system, 

 and hence will produce all the homologies of the plane, 

 and each once only. There is therefore a (2,1) corre- 

 spondence between the doubly pointed plane and the 

 plane of the homologies. The discussion of these 

 points by Luigi Certo is followed by an investigation 

 of the variation of the ratio of corresponding areas, 

 first, with the variation of the eccentricity, and, sec- 

 ond, with the variation of the direction of the line 

 L M. He also considers the distribution in the plane 

 of the pairs of similar conies of which the system of 

 conies through four points on a circle is composed. — 

 (Giorn. mat, xx.) c. L. F. [694 



PHYSICS. 



Optics. 

 Color of -water. — W. Spring reviews the several 

 explanations suggested to account for blue and green- 

 ish colors of water in lakes and seas, — Bunsen's idea 

 of inherent coloi-, Tyndall's theory of reflection, and 

 others, — and concludes that some further' study of 

 the question is needed. Blue from reflection would 

 imply red by transmission, but this is not observed 

 from diving-bells. The author concludes provision- 

 ally that the color depends on the presence of certain 

 salts, especially calcic carbonate in solution. The 

 more complete the solution, the bluer the water. — 

 [Rev. sclent., 1883, 161.) W. m. d. [695 



{Plioiomeiry.) 

 Spectrum photometry. — MM. J. M. de L^pinay 

 and W. Nicati have recently completed an inves- 

 tigation of the relative brilliancy of white surfaces 

 when illuminated by different colored lights and by 

 different portions of the same spectrum. In the pre- 

 liminary experiments, two lights were employed, — a 

 yellow and a blue one, — the blue light being the 

 fainter. Their intensity was compared by means of 

 a Rumford photometer, casting very small shadows. 



