Decemkhk 14, 18S3.] 



SCIENCE, 



769 



work, which certainly places the village-com- 

 munity theory upon the defensive, and over- 

 throws a considerable part of its assumptions ; 



and, apart from its controversial character, as 

 a ' history of land-holding ' it possesses the 

 highest value. 



WEEKLY SUMMARY OF TEE PROGRESS OF SCIENCE. 



MATHEMATICS. 



Hyperelliptic integrals. — The full title of this 

 p.ipor by >I. St.imle is " Goometiische deutung iler ad- 

 ditiou«theoreme der liypcrelliplischeu iiitegiale und 

 fnnciioneii t'rster oidiuing im systenie der confocalen 

 fliicUen zweiteu grades." Only a brief notice of M. 

 Staiule".s paper is possible in this place, although its 

 importance makes it worthy of a much more ex- 

 tended one. The paper is divided into five chapters. 

 In the first chapter the author considers the gemuetric 

 sigiUticaiicc of the symmetric algebraic functions of 

 two independent variables, and the differenti.als 

 of the integral functions of an hyperelliptic form 

 {ijehilde) of deficiency (iieschleclil). The second chap- 

 ter treats of the repre-^entation of the (/ehiUle in 

 systems of cmifocal surfaces by aid of hyperelliptic 

 functions, and opens by the introduction of certain 

 transcendental parameters in place of the usual 

 elliptic co-ordinates. An expression is also given 

 of the homogeneous point co-ordinates in space in 

 terras of products of the double thet.a-functions, and 

 also of homogeneous plane co-ordinates in space 

 by aid of products of two doub!n theta-functions. 

 The third chapter is of particular interest from a 

 purely geometrical point of view. In this the author 

 considei-s the relations of the addition theorem for 

 hyperelliptic integrals to systems of eonfocal sur- 

 faces, treating particularly the reduction of given 

 suras of three integrals tosum-^of two integrals of the 

 same kind. The fourth and fifth chapters have not 

 yet appeared, but the author mentions their con- 

 tents. C'liapter four is to treat of the ray-systems of 

 common tangents to two eonfocal surfaces ; and chap- 

 ter five is to be devoted to a geometrical interpreta- 

 tion of -Vbel's addition theorem, by aid of which the 

 reduction of the sum of any four of the integrals in 

 question to the sum of two integrals of the same 

 kind is arrived at by a purely geometrical process. — 

 (il/«(/i. rtJiH.. xxii.) T. c. [471 



Discontinuous groups of linear substitutions. 

 — The complete title of M. Picard's ijaper is "Sur 

 une classe de groupcs discontinus de substitutions 

 lin^aires ct snr les fonctions de deux variables ind6- 

 pendantes rest.ant invariable p.ar ces substitutions." 

 The theory of the elliptic functions has given the first 

 .example of a uniform function of a variable which 

 does not change for a group of an infinite number 

 of linear non-permutable substitutions effected upon 

 the variable. The modular functions, i.e., the func- 

 tions arising from considering the modulus as given 

 by the ratio of the two periods, was for the first con- 

 sidered by M. nermite. M. Poincar^ h.as treated in 

 his theory of the Fuchsian functions, in all its gen- 

 er.ality, the subject of functions of one variable which 



are reproduced by a group of an infinite number of 

 linear substitutions. 51. Picard, in the present me- 

 moir, proposes to consider functions of two independ- 

 ent variables which may be considered as analogous 

 to the elliptic modular functions. He shows, first, that 

 the Abelian functions do not conduct to functions 

 entirely analogous to the modular functions, and 

 illustrates this by the Abelian functions of the first 

 order. But by taking the case of the Abelian func- 

 tions of the second order, i.e., of three variables, he 

 has found an indication of the desired extension, and 

 hopes in a future paper to enter more fully into the 

 subject of functions of two variables which .are anal- 

 ogous to the modular functions. The present paper 

 is interesting as pointing out the difficulties, and indi- 

 cating the manner of overcoming tliem, in an entirely 

 new department of the theory of functions. — {Acta 

 math., i.) t. c. [472 



PHYSICS. 

 Target-shooting. — From Liagre's theory that 

 errors in target-shooting are compounded of errors 

 in sighting and in levelling, each of which follow in- 

 dependently the law of error, it was .shown by Mr. 

 C. H. Knmmell that shots of equal probability are 

 arranged in ellipses, which can be reduced to circles 

 of shots uniformly distributed, the integration being 

 much simplified by using the reduced distances and 

 directions. Sir J. Ilerschers 'even-chance circle' 

 (ellipse, more generally), the one hit or missed with 

 equal probability, can be deduced from the shots 

 actually found in any given circle (ellipse), the most 

 reliable result being given by the one containing the 

 greatest number of shots, whose radius (mean semi- 

 diameter) is the most probable shot. The number 

 of shots falling within this ellipse shouM be about 

 thirty-nine and one-half per cent. The equations be- 

 tween the even-chance shot (p), the most probable 

 shot (e), and the average shot (ro), are — 



rv/2«2. r„ = t\|.> 



In determining those from the sums of squares of 

 the vertical and horizontal co-ordinates of the sepa- 

 rate shots, the number that miss the target .should be 

 considered. The probable position of centre and 

 axes should not be calculated from the observations, 

 unless the true positions are unknown. A target of 

 ninety shots at eight hundred yards' range, by the 

 Irish team at Creedmoor in 1874, gave discrepancies 

 of less than five per cent between observation and 

 theory, in the number of shots within successive 

 rings. One of fifty pistol-shots, at fifty yards' range, 

 showed a similar agreement. — {Phil. sue. Wash., 

 mcUU. sect. ; meeting 'Sov. 21.) [473 



