Fkbbuaby 23, 1883.] 



SCIENCE. 



71 



of magnesiutn-wire, fed by clock-work, in the focus 

 of a parabolic reflector, gives an excellent light; but 

 this, like the electric light, is too expensive for ordi- 

 nary geodetic uses. The U.S. coast and geodetic sur- 

 vey has used kerosene student-lamps in place of the 

 magnesium wire in connection with parabolic reflect- 

 ors, on lines of twenty-five miles, with satisfactory 

 results. At a station in Virginia, occupied by C. O. 

 Boutelle, angles measured by day were duplicated at 

 night, and the mean error of the night-work was 

 only two-thirds of that done in the daytime. 



In 1881 Mr. AVilson procured a small locomotive 

 head-light with a twelve-inch reflector, and two cast 

 semaphore lenses, one twelve and the other fourteen 

 inches in diameter. Each of these lenses was mounted 

 in the end of a box in which a kerosene-lamp with a 

 'mammoth-leader' burner was placed at the focus 

 of the lens. These three lights, being set near each 

 other, were readily seen through a small telescope at 

 a distance of thirty-five miles, and little if any differ- 

 ence of brilliancy was detected. The magnesium 

 apparatus and the locomotive head-light each cost 

 about thirty-five dollars; but the magnesium wire 

 being expensive, and this light requiring constant 

 attention, the cost of maintaining it is several times 

 greater than that of operating the locomotive head- 

 light. The cost of a semaphore lens mounted in a 

 galvanized-iron box is from ten to fifteen dollars, 

 according to the size. The exj^euse of maintaining it 

 is small, — not more than -fifty cents a night, kerosene 

 being cheap, and no attention being required after 

 the lamp has been properly trimmed, and lighted a 

 short time. These lamps have been seen by the 

 naked eye at a distance of forty miles. 



In order to diminish as little as possible the light 

 in tlie field of the telescope, a series of mirrors was so 

 arranged upon and within tlie tube as to illuminate 

 the wires, and leave the field dark. It is believed 

 that this has not before been done with small tele- 

 scopes, the one used in this instance having an aper- 

 tm-e of only two and a half inches. Kerosene hand- 

 lamps, protected for use in the wind, were devised 

 and successfully used for reading the circle and illu- 

 minating the wires. The night observations thus 

 made at state survey stations in 1882 were apparently 

 fully equal to those taken in the daytime by means 

 of heliotrope signals; and about half of the primary 

 observations were actually made in the time thus 

 saved. 



For readily finding a distant signal light at night, 

 a reference lantern was placed a short distance from 

 the observing-station. By this, rough settings were 

 made for the signal-liglit needed, which could then 

 be brought into view by a slight vertical movement 

 of the telescope. — {Alb. inst.; meeting Jan. 30.) [173 

 MATHEMATICS. 



Conjugate quadrangles. — M. Stephanos, in seek- 

 ing to generalize a kinematical proposition an- 

 nounced by M. Tohekychcf in his memoir Sur les plus 

 simples systemes articulesquifournissentun mouvement 

 rectiligne apjiroximatif au quatri'eme et an cinquieme 

 ordre (St. Petersburgh, 1881), has arrived at a num- 

 ber of properties of conjugate quadrangles. M. Ste- 

 phanos defines conjugate quadrangles as being formed 

 by two systems of four points (Ai, Aj, A:i, A4), (Bi, 

 B2, B3, B4), when, being placed upon a plane in any 

 manner, without altering their respective dimensions, 

 the corresponding points (Ai and Bi) form four pairs 

 of conjugate points with respect to a circle. There 

 is an infinite number of quadrangles B, conjugate to 

 a given quadrangle A; and all of the B-quadrangles 

 are similar one to another. If A and B are two con- 

 jugate quadrangles, the areas of the triangles Aj, A3, 



A4, etc., are proportional to the areas of the trian- 

 gles B,, B3, 64, etc. The respective ratios are de- 

 noted by /li :^-2 : /I3 -.^.^ with 1X, = 0. ^i, ^2, and 

 A3 are given in terms of the cotangents of the angles 

 oif the triangles A.j, A3, A4, and Bj, B3, B4. Con- 

 sidering two conjugate quadrangles A and B situated 

 in the same plane, and denoting by Pi, p.;, P;,, P4, the 

 distances between corresponding summits, it is 

 shown, that, whatever be the relative positions of 

 the two quadrangles in the saine plane, we have 

 always the relation: — 



Ai P]- -\- X, p._,2 -^ A3 Pj- -|- A4 Pi'^ = C; 

 where C is a constant depending only on the dimen- 

 sions of the two quadrangles. — {Compies rendus, 

 Oct. 16, 1882.) T. c. [174 



Conical umbilics. — The following is taken from 

 a report by MM. Bouquet and Jordan upon a me- 

 moir presented by M. de Salvert to the Academy of 

 sciences. M. de Salvert studies the sections of a sur- 

 face F (x,y,z} ^0, in those singular points where 

 the tangent cone is of the second degree by planes 

 passing through the axis of the tangent cone. Each 

 section consists of two branches crossing at the mul- 

 tiple point, and having for tangents in this point the 

 two opposite generatrices of the cone : it is proposed 

 to find the curvature of these two branches. The 

 author finds a foi'mula for this curvature, of which 

 he shows the analogy to the known expression for the 

 determination of the radii of curvature of a normal 

 section at, an ordinary jioint. An application is made 

 to the case of the wave surface, and then the author 

 seeks the necessary conditions that the assumed point 

 shall be a conical umbilic: i.e., a point such, 1°, that 

 the tangent cone shall be one of revolution; 2°, that 

 the branches of the curve which correspond to its 

 different generatrices shall all have the same curva- 

 ture. The first of these conditions leads only to 

 known results ; the second introduces six new equa- 

 tions Involving the third derivatives of F. — {Comptes 

 rendus, Jan. S, 1883. ) T. c. [175 



Subdeterminants of a symmetric system. — 

 In July, 1882, Prof. Kronecker presented to the Berlin 

 academy a memoir in which he established certain 

 linear relations between the subdeterminants (mi- 

 nors) of a symmetric system. M. Kunge deals with 

 the same subject in the present paper, and claims to 

 show that relations found by Kronecker are the only 

 ones existing, inasmuch as all others can be expressed 

 by linear combinations of Kroneeker's relations. He 

 also finds a method for the determination of a system 

 of linearly independent subdeterminants in terms of 

 which all the remaining subdeterminants of the same 

 order are linearly expressible. — {Journ. reine angew. 

 math., xciii. 1882.) T. c. [176 



Ternary quartics. — In continuation of his re- 

 searches on the ternary quartic Xi^,X2 + Xz^ x^ + x^s^Xi, 

 and on systems of conies, Prof. Gordan discusses the 

 typical representation of the system formed by this 

 quartic and a conic. He finds that the coefficients 

 in this representation are entire functions of only 

 twelve simultaneous invariants, five of which are 

 expressible as rational functions of the other seven, 

 which are themselves connected by an algebraic equa- 

 tion of the sixth degree; and all these relations are 

 explicitly given. These relations reduce the number 

 of independent invariants to six, which is evidently 

 the actual number. The last part of the article is 

 devoted to the solution of the converse problem of 

 determining a conic when the invariants above men- 

 tioned are given. — (Math, ann., xxiv. 1882.) r. p. [177 



Equations of the seventh degree. — In this pa- 

 per, Prof. Gordan applies the results obtained by him 



