Apbil 19, 1907] 



SCIENCE 



607 



The compasses may be superseded by the 

 simpler 'transferrer of line-segments,' for 

 which the name 'sect-carrier ' has been 

 adopted. Thus without the circle or com- 

 passes all the problems of elementary geom- 

 etry are solved in the first edition of Hal- 

 sted's 'Rational Geometry.' But a remark- 

 able additional simplification has now been 

 achieved, and this paper makes public for 

 the first time the simple demonstration 

 which makes it available for the elements 

 of geometry. This advance is the substi- 

 tution of the set-sect for the sect-carrier. 

 The transference of only a single sect need 

 be assumed for the solution of all the prob- 

 lems of elementary geometry. Consequently 

 the power to take a centimeter on a given 

 straight line is found to be assumption 

 enough to supersede the circle, the com- 

 passes, and even the sect-carrier. Nothing 

 now is needed but a ruler and a set-sect. 



On a Fundamental Theorem of W eierstrass 

 hy Means of which the Theory of Elliptic 

 Functions may he Established: Professor 

 Hakeis Hancock, University of Cincin- 

 nati, Cincinnati, 0. 



The theorem in question is stated by 

 Weierstrass in the ' Theorie der Abelschen 

 Functionen ' {Crelle's Journ., bd. 52, § 7; 

 and ' Math. Werke,' bd. I., p. 349). 



By means of his theorem it may be 

 shown that the p-function may be ex- 

 pressed as the quotient of two series which 

 are both convergent for all values of the 

 variable; the same is true of the functions 



Vp" — f A (A = l, 2, 3). 



It follows directly from Weierstrass 's 

 theorem that the a-function may be ex- 

 pressed as a convergent series for all 

 values of the variable. 



The different series are calculated and it 

 is interesting to compare the results usually 

 obtained from the well-known theorem 

 also due to "Weierstrass, that every one- 

 vdlAied function that has not an essential 



singularity in the fiwite portion of the 

 plane, may he expressed through the quo- 

 tient of two power-series, which a/re con- 

 vergent for all values of the variahle. 



Weierstrass 's theorem is also generalized 

 and applied to differential equations of a 

 higher order. 



Dynamical Trajectories: Dr. Edward 



Kasner, Columbia University, New York 



City. 



Professor Kasner discusses two general 

 questions, of interest in connection with 

 celestial mechanics, relating to the geometry 

 of dynamical trajectories. The first is sug- 

 gested by the problem of binary stars and 

 Bertrand's discussion of the interdepend- 

 ence of Keplei's laws. It is shown that 

 two distinct fields of force can have only a 

 certain multiplicity of trajectories in com- 

 mon. It is then possible to determine a 

 field from a minimum number of trajec- 

 tories. In particular, the Newtonian law 

 may be deduced without assuming, as Ber- 

 trand does, that all the orbits are conies. 



The second part of the paper relates to 

 the problem of n bodies, and extends some 

 of the results which hold for a single par- 

 ticle (see Trans. Amer. Math. Soc, 1906, 

 1907). For example, the locus of the 

 centers of the osculating spheres, under 

 prescribed initial conditions, is a cubic 

 curve; in the case of a single particle, on 

 the other hand, it is a straight line. The 

 results obtained are true for all interacting 

 particles. 



The Stream Function for a Straight Chan- 

 nel with a Circular Island: James Mc- 

 Mahon, Cornell University, Ithaca, N. Y. 

 This is one of the standing problems in 

 two-dimensional fluid motion. A solution 

 is here obtained by imagining a doublet 

 placed mid-stream in a uniform current so 

 that the line from the source to the adjoin- 

 ing sink points in the direction of the un- 

 disturbed current. The appropriate stream 



