■Septebibek 25, 1896.] 



SCIENCE. 



443 



'■ inversion ' having n-2 of the fundamental 

 points as critical points and the remaining 

 one as a fixed point. This paper is in- 

 tended for the 3fathematische Annalen. 



In Dr. Snyder's paper Lie's haxaspherical 

 coordinates were emploj^ed to define, with- 

 out the use of line-geometry, a Dupin's 

 cyclide.' By associating the three simul- 

 taneous linear equations of definition with 

 the point-complex and the plane-complex, 

 one obtains determinants the signs of which 

 indicate the reality of spheres common to 

 the four complexes and thus show the pres- 

 ence of nodes. Dr. Snyder's paper will ap- 

 pear in the Annals of Mathematics. 



On a convex surface of deficiency zero 

 Euler's equation, together with the require- 

 ment of numerical regularity, gives three 

 sets of integers for vertices, faces and 

 edges of a polyedron. These by duality 

 become five, corresponding to the five reg- 

 ular polyedra. On a surface of deficiency 

 greater than unity the modified equation 

 of Euler, together with similar limitations, 

 gives again a finite number of sets of in- 

 tegers for vertices, faces and edges. These 

 sets are of two sorts : ' derivative,' ob- 

 tained from sets belonging to lower defi- 

 ciencies ; and ' special,' not so obtainable, 

 but peculiar to the deficiency in question. 

 These sets of characteristic numbers can be 

 realized on concrete models. Prof. White 

 discussed the subject in detail and exhib- 

 ited models, constructed by Mr. O. H. Bas- 

 quin, for deficiency 2 and for the ' special ' 

 sets of deficiency 3. For deficiency 2 

 there were 13 card models and 6 of plaster ; 

 for deficiency 3 there were 7 card models. 

 Prof. "White's paper will appear in the Bul- 

 letin of the American Mathematical Society. 



In Prof. Hyde's paper, which is intended 

 for the Annals of Mathematics, j) is a variable 

 point, e is a fixed point, x and y are scalars 

 varying from — go to + go, and f and 4' are 

 linear points, functions of the form : 



(j)q=I,(A!cejc.ek \ q) , i'q = 'S {Buek. eic | q). 



The curve and surface represented by the 

 equations given in the title of the paper are 

 studied, and many interesting properties 

 which thej possess are described. 



In Prof. McMahon's paper, also intended 

 for the Annals of Mathematics, the sun is as- 

 sumed to move in a straight line with con- 

 stant velocity, which is shared by the whole 

 solar system ; and the gravitational influence 

 is supposed to issue from the sun in waves 

 that move outward with constant velocity 

 (equals, perhaps, to that of light). When 

 any wave reaches the earth the latter is 

 attracted towards the wave center, or point 

 of space from which the wave issued. This 

 effective center of acceleration is at a dis- 

 tance from the sun which varies between 

 the limits ha (1— e) and ka (l + e), where 

 h is the ratio of the velocity of the sun to 

 that of gravitation, a is the semi-axis major 

 of the earth's orbit, and e the eccentricity. 

 Thus the orbit of the earth relatively to the 

 sun is that which would be due to a center 

 of force that performs small oscillations 

 about its mean position. The law of this 

 oscillatory motion is first determined, and 

 then the equations of acceleration of the 

 earth in its orbit, along and perpendicular 

 to the radius-vector, are corrected for this 

 small disturbance ; and appropriate solu- 

 tions of these differential equations are given, 

 correct as far as terms in ke'. The most 

 important perturbative terms are examined, 

 and their effect on the orbit determined. 



Prof. Davis maintained that there existed 

 a necessary and intimate association be- 

 tween the notions of continuity and chance. 

 His paper will be published in the Neh-aska 

 University Studies. Dr. Martin's paper con- 

 tained several series suitable for calculating 

 a number when its logarithm is given. He 

 intends to publish the paper in the Mathe- 

 matical Magazine. 



The first ten values of x, for which Bes- 

 sel's function of the zeroth order Jq{x) van- 

 ishes, were given to ten places of decimals 



