Januabt 22, 1909] 



SCIENCE 



157 



then the sum, difference, product and quotient of 

 the two functions lOi and Wi are functions of the 

 form 



w = p -{■ q- s. 



Let a take all real and complex values and 

 consider the collectivity of all rational functions 

 of z with arbitrary constant real or complex coeffi- 

 cients. These functions form a closed realm, the 

 individual functions of which repeat themselves 

 through the processes of addition, subtraction, 

 multiplication and division, since clearly the sum, 

 the difference, the product and the quotient of 

 two or more rational functions is a rational func- 

 tion and consequently an individual of the realm. 

 This realm is denoted by (»). 



It is evident that if we add (or adjoin) the 

 algebraic quantity s to this realm we will have 

 another realm, the individual functions or ele- 

 ments of which repeat themselves through the 

 processes of addition, subtraction, multiplication 

 and division. This realm includes the former 

 realm. We shall call it the elliptic realm and 

 denote it by (s, n) . 



Owing to a theorem due to Liouville, the most 

 general one-valued doubly periodic function is a 

 rational function of z and s. It is consequently 

 a one-valued function of position in the Riemann 

 surface and belongs to the elliptic realm of ra- 

 tionality (2, s). 



The elliptic or doubly periodic realm of ration- 

 ality («, s), where 



s=± V^(« — Oi) (« — <h) (z — (h) (» — a*) 

 degenerates into the simply periodic realm when 

 any pair of branch-points are equal, say 01 = 02; 

 and into the realm of rational functions when two 

 pairs of branch points are equal, say Oj = O2 and 

 03 = 04. 



Thus the elliptic realm includes the three 

 classes of one- valued functions: (1) the rational 

 functions, (2) the simply periodic functions, (3) 

 the doubly periodic functions. All these one- 

 valued functions, and only these, have algebraic 

 addition-theorems. 



In other words, all functions of the realm {z, s) 

 home algebraio addition-theorems, and no one- 

 valued function that does not belong to this realm 

 has an algebraic addition-theorem. 



We have thus the theorem: The one-valued 

 functions of position on the Riemann surface 



s' = A(z — Oj) (z — Oj) (z — 03) (z — Oi) 

 belong to the closed realm (z, a) a/nd all elements 

 of this realm and no others have algebraio addi- 

 tion-theorems. 



Professor Hancock's paper will be offered to the 

 American Journal of Mathematics for publication. 



12. The paper by Artemas Martin is devoted to 

 an algebraic determination of the point within a 

 triangle at which the sides subtend given angles. 

 The paper is to appear in the Mathematical Maga- 

 zine, which is edited by the author of this paper. 



13. The paper by J. Burkitt Webb is devoted to 

 exhibiting the advantages which would result 

 from the adoption of a system of notation with 

 16 as its base. The success which has attended 

 the movements towards a universal language has 

 inspired the author with hope in the success of a 

 movement towards the selection of a more useful 

 system of notation, and he pointed out the many 

 advantages which the base 16 would offer. 



14. This paper is devoted to a discussion of a 

 series of photographs of comet Morehouse, made 

 at Swarthmore College by J. A. Miller and W. K. 

 Marriott from October 2 to December 3, 1908. 

 The comet was photographed from one to three 

 times every clear night within that period. These 

 photographs show remarkable and, in some in- 

 stances, rapid changes in the form of the comet's 

 tail and in the arrangement of the streamers. 

 The most striking changes occurred on October 4; 

 on October 15, 16, 17, 30, 31 and November 1 

 the changes were sufficiently rapid to enable one 

 to measure an increase of the distance of a con- 

 densation in the tail upon photographs taken less 

 than two hours apart. 



15. The rather prevalent custom of resolving or 

 expressing every natural phenomenon — ^be it 

 periodic or otherwise — by a Bessel or a Fourier 

 series or by spherical harmonic functions, has 

 brought about at times, especially in geophysical 

 and cosmical phenomena, if not direct misapplica- 

 tions, at least misinterpretations of the meaning 

 and value of the derived coefficients. Instead of 

 clarifying the situation our calculations may have 

 actually contributed to befog it. Instead of re- 

 jecting, one must learn to consider the outstand- 

 ing residuals as the true facts of nature and not 

 treat them as though they were " abnormal " or 

 contrary to nature's law. 



Dr. Bauer exemplified these statements in a 

 brief discussion of two cases that are typical in 

 geophysical investigations — ^the one involving an 

 application of spherical harmonic functions to the 

 representation of the distribution of the earth's 

 magnetism over the earth, while the other in- 

 volved the use of Fourier series in the representa- 

 tion of certain diurnal geophysical phenomena. 



The chief purpose of the paper was to recall 



