174 



SCIENCE. 



[Vol. I., No. 6. 



WEEKLY SUMMARY OF THE PROGRESS OF SCIENCE. 



GEODESY. 

 The lake survey. — There has just been issued by 

 the chief of engineers, in a quarto of 920 pp. with 

 thirty plates, a detailed report of the operations in 

 the prosecution of the survey of the Great Laljes. 

 This imijortant work is now finislied, and tlie report 

 presents in a comx^reliensiye manner the methods 

 used and results obtained. While omitting the vast 

 amount of uninteresting detail with which such works 

 are usually encumbered, all important features are 

 given mention; and the whole volume is indexed with 

 such care that any particular subject may be instantly 

 found. The report starts with a historical account 

 of the survey, from its inception in 1S41, to its com- 

 pletion; gives a synopsis of the work accomplished 

 under the various officers who from time to time have 

 had charge of the survey; gives an account of the 

 standards of length upon which the surveys depend, of 

 the measuring-bars used and methods of using them, 

 and of the results obtained iioth in the measurement 

 of the base lines and in tlie results of their connection 

 by triangulation, and of the geodetic and astronomi- 

 cal work. The part devoted to the discussion of the 

 base apparatus will be found of special interest to 

 geodeticians. Full account is given of the determina- 

 tion of the constants of the apparatus used, and of 

 the co-efficients of expansion. Also there is a discus- 

 sion of the ' set ' of a zinc bar when heated. A por- 

 tion of the book is devoted to the consideration of the 

 mean levels of the Great Lakes, and the methods by 

 which the results were obtained. The question of 

 tides in the lakes had been previously considered (Re- 

 port of chief of enr/ineers, 1S7$). The tides are x^cr- 

 ceptible, but of scientific rather than practical impor- 

 tance, the maximum being less than two inches. — 

 {Professional papers, corps of engineers, no. 24.) 

 H. W. B. [346 



MATHEMATICS. 



Elliptic-function formulas. — Integral forms are 

 given for certain products and quotients of the ellip- 

 tic functions en?i, enn, and dnn. The author, Mr. 

 Craig, starts out from a formula of Mr. Glaisher's for 

 the second derivative of the f unctioii cnfii. — (Amer^ 

 journ. >7iath., v., 18S2.) T. c. [347' 



Intersections of circles and spheres. — Gen. 

 Alvord gives geometrical solutions of the problems, 

 — to draw a circle cutting three given circles at the 

 same given angle, to draw a circle cutting four given 

 circles at the same unknown angle, and the analo- 

 gous problems for spheres.' — [Amer. journ. math., v. 

 1882. ) T. c. [348 



Symmetric functions. — Mr. Durfee has given 

 tabulated values of the functions (of weight twelve) of 

 the co-efficients of the twelfthic in terms of the sym- 

 metric functions of its roots, also the values of these 

 symmetric functions in terms of the co-efficients. — 

 (Amer. journ. math., Y. 1S82.) t. c. [349 



Elliptic functions. — This is the first part of a 

 paper by Otto Rausenberger, in which he introduces 

 a new idea into the theory of elliptic functions. In- 

 stead of, as usual, considering doubly periodic elliptic 

 functions, he considers that an advantage is gained 

 by considering what may be called transcendants, 

 with simply multiplicate periods (einfachcr mullipli- 

 catorischer periode) ; that is, functions satisfying the 

 equation/ (px) =/ (,r). The notation which he has 

 adopted is made to conform as nearly as possible 

 with that employed by KiJnigsberger in his ' Vorle- 

 suncjen iXber die theorie der elliptischen functionen.' 



He defines certain functions, ri„, VnVz, Vi, which are 

 analogous to the ordinary theta-functions, and gives 

 the values of functions S (p,x), C (p,x), D (p,x), 

 which correspond in the ordinary notation to sn x, 

 en X, dn x, in terms of the.se )?-functions. The 

 equations are identical in form with those giving sn x, 

 etc., in terms of the 9-functions. In conclusion a 

 discussion of some of the properties of multiplicate- 

 periodic functions is given. — [Jown. reine angew. 

 math., xciii.) T. c. [350 



Binary quintics. — An extensive discussion of tlie 

 Hessian of the binary quintic is given by Mr. F. Lin- 

 demann. The expressions for the invariants and 

 quadratic covariants of this sextic covariant, in terms 

 of the invariants and covariants of the quintic to 

 which it belongs, are obtained, and a relation found 

 to exist between them, which is the necessary and 

 sufficient condition that a given sextic may be the 

 Hessian of a quintic. The typical expression of the 

 Hessian by means of its quadratic covariants is next 

 found. In the course of obtaining this, it is observed, 

 that, when a certain invariantive condition is fulfilled, 

 the quintic is reducible to a known soluble form. 

 The remainder of the article contains the investiga- 

 tion of the peculiarities which attach to the Hessian 

 on the supposition of any peculiarity in the quintic, 

 and vice versa ; the determination of a quintic whose 

 Hessian is given; and, finally, a geometrical interpre- 

 tation of the condition satisfied by any sextic which 

 is thfe Hessian of a quintic. — (Math, ann., xxi. 1, 

 188.3.) p. r. [351 



Theory of numbers. — In an article on power- 

 residues (potenzreste) F. Hofmann employs the de- 

 vice of representing the residues of the successive 

 powers of a number with respect to a prime-number 

 modulus as the successive vertices of a regular poly- 

 gon inscribed in a circle, to prove Gauss's theorems 

 concerning the sums of the primitive roots of the 



J)- 1 _ 

 binomial congruence, x =1 (mod. p). He makes 

 some remarks on binomial equations, and their con- 

 nection with binomial congruences. — (Math, ann., 

 XX. 4, 1882.) F. F. [352 



PHYSICS. 



Aconstica. 

 Range of sounds in air. — Allard has deduced a 

 formula for the intensity of a sound in terms of the 

 work done in producing it (T), the rate of vibration 

 in), and the extreme range (x). The table given by 

 him shows that the intensity of the sound in air 

 decreases more rapidly than is indicated by the law 

 of inverse squares. At the extreme range, all the 

 sounds are reduced to tlie same intensity; while the 



T 



values of ~j vary, for the six instruments used, from 



X 



0.10 to 13.46. 



A cause of this enfeeblement of .sound is the re- 

 flecting action of the successive layers of air of differ- 

 ent density when the atmosphere is not homogeneous. 

 A formula is deduced which takes this action into 

 account, which, with its constants determined from 

 the e.xperiments described, gives for a moderate acous- 

 tic transparency of the air, — 



T (0.473 )-^- = 0.0000277 n x-. 



The work necessary to cause a given increase of 

 range, and the range of sounds of different pitch 

 produced by the same expenditure of energy, can also 

 be determined from the formula. The difference of 



