50 



SCIENCE. 



[Vol. II., No. 23. 



tion has been in use at the^Harvard college obser- 

 vatory for the past two years. Prof. Pickering, 

 however, very wisely prefers to compare the eclipsing 

 satellite with one of the other satellites, or with an 

 image of the planet, rather than with an artificial 

 star; and he uses polarization apparatus instead of a 

 cat's eye to equalize the brightness of the objects 

 compared. — {Comptes rendus, June 4.) c. A. T. 



[40 

 MATHEMATICS. 

 Theory of functions. — In a series of three me- 

 moirs, M. Appell has reproduced in a more extended 

 form a number of investigations which he has 

 recently communicated to the French academy of 

 sciences. The first of the three memoirs treats of 

 uniform functions of an analytical point (x, y); the 

 term ' analytical point' meaning simply the system of 

 values of {x, y) formed by any arbitrary value of x 

 and the, say, m corresponding values of y. The first 

 section of the memoir contains three theorems con- 

 cerning the development in rational fractions of such 

 functions. In the second section a uniform function 

 is defined, and also the poles and essential singular 

 points (points singuliers essentiels, Weierstrass' wesent- 

 Uche singulcire stelle). Functions with a finite num- 

 ber of singular points are then taken up, and a 

 generalization of a known theorem concerning the 

 coefiieients in the development of a uniform func- 

 tion is given: viz., if F{x, y) is a uniform function 

 of the analytical point (x, y), having a finite number 

 of singular points (a,, 6,), and if Rt are the residues 

 relatively to these points; if, further, in a certain 



region of the analytical point (» = <», Jim — = Cj ), 

 we haveii'(x, ?/) = 2 ^^*'-,— then we have 



v = —'x " X" 



the relation 



A['' + A'P + ... + ^'i"" = R,+Ito + ... + R„. 

 In this, i has all values from 1 up to n, and k has all 

 values from 1 up to m ; m denoting the number of 

 values of y corresponding to a given value of x. 

 After a brief review of some of the properties of the 

 Abelian integrals, the author gives a generalization 

 of a holomorphic function of x in the interior of a 

 circle whose centre is a in terms of ascending powers 

 of (X — a). The subject of functions with an in- 

 finite number of singular points is then taken up, and 

 a generalization is first given of Mittag-Zeffler's 

 theorem concerning these functions; viz., if a series 

 of distinct analytical points (a,, 6i) . . . (a^, bj . . . 

 are such that lim. (a^, 6^) = (a b) for v = co, and if 

 ■Fi(^) y), F2(x, y) . . . F^(x, y) is a series of rational 

 functions of x and y which become infinite only in 

 the two points (a,„ b^) and (a, b) respectively, then 

 there exists a uniform function ^(x, y) having only 

 the point (a, b) as an essential singular point, and 

 admitting as poles the points (a^, 6,,) in such a man- 

 ner that the difference *(x, y) — F^ (x, y) Is regular 

 in the point (o,„ b^). 



The second memoir by M. Appell is a continuation 

 of the first. In it he considers the decomposition into 

 prime factors of a uniform function of an analyti- 



cal point (x, y) having only one essential singular 

 point, and also gives a theory of doubly periodic 

 functions with essential singular points. The author 

 examines, first, functions having in a parallelogram of 

 periods a finite number of singular points, and gives 

 an interesting theorem; viz., the sura of the residues 

 of F (u) relative to the singular points situated in a 

 given parallelogram of periods is equal to zero. A 

 general expression is then obtained for a doubly 

 periodic uniform function F {«) having in a given 

 parallelogram of periods only one singular point. 



In the third memoir, M. Appell considers the de- 

 velopment of functions in series inside an area 

 bounded by arcs of circles. These three memoirs 

 by M. Appell, taken with a memoir by M. Poincar^, 

 which precedes them, and which has already been 

 I'eferred to in these pages, constitute a very valuable 

 series of papers on the modern theory of functions. 

 — (Acta math., i. no. 2.) T. c. [41 



PHYSICS. 



Acoustics, 



Upper limit of audibility. — Pauchon and Ber- 

 trand have investigated the question of the effect of 

 the intensity of the sound upon this limit. A siren 

 blown by steam with pressures varying from 0.5 to 

 1.5 atmospheres gave from 24,000 to 30,000 double 

 vibrations as a limit; but, with certain modifications 

 and a higher pressure (2^ atmospheres), the most acute 

 sound that could be produced by the instrument, due 

 to 36,000 vibrations, was still heard. Metallic rods 

 of different lengths, set into longitudinal vibration 

 in the usual manner, gave the following results: 1. 

 The length of the rod giving the highest perceptible 

 sound is independent of its diameter ; 2. For steel, 

 copper, and silver, the lengths are proportional to 

 the velocity of sound in those media. These results 

 disagree with those reached with the siren. The 

 authors find, however, that, if the ear is aided by a 

 resonating trumpet, the limit is slightly raised ; 

 that the limit is raised witli substances like rosin, 

 producing the most energetic friction ; and that the 

 sound, even when too high to affect the ear, still acts 

 on a sensitive flame. 



These results of Pauchon with the siren agree with 

 the fact observed several years since by Dr. H. P. , 

 Bowditch of Boston, that, with a Konig's bar of 

 exceedingly large diameter, the limit of audibility is 

 higher than with one of the ordinary size. — (Comp- 

 tes rendus, April 9.) c. r. c. [42 



Production of •whispered voTvels. — Lefort 

 calls attention to the wide range of whispered vowels 

 that can he artificially produced by blowing across 

 resonant tubes or spheres: ou, o (closed), o (open), 

 u, eu, e, i, e (closed), e (open), — all being produced as 

 the capacity of the resonator is diminished. By 

 diminishing the length of an open tube the vowels 

 &, h, e, eu, u, e, e, i, are successively heard, while 

 ou, 6, 0, are obtained by closing the upper end of the 

 tube more or less. — (Comptes rendus, April 23.) 

 C. B. c. [43 



Transmission of sounds by gases. — Ney- 

 reneuf has studied the relative transmission of sound 



