442 



SCIENCE. 



[N. S. Vol. IV. No. 91. 



14. On the TiypotTiesis of the successive transmission of 

 gravity and the possible perturbative effect on the eartWs 

 orbit. Peof. J. McMahon. 



15. The continuity of chance. Pkof. E. W. Davis. 



16. A method of finding icithout a table the number cor- 

 responding to a given logarithm. Dk. Aetemas 

 Martin. 



17. Table giving the first forty roots of the Bessel equa- 

 tion Jo {x)^ 0, and the corresponding values of Ji {x) 

 Prof. B. O. Peiece and Mr. E. W. Wilson. 



18. On the projective group. Prof. H. Tabee. 



Dr. Blake's first paper gave a classifica- 

 tion of the methods which have been used 

 for defining monogenic functions, with ex- 

 ample, references and historical notes. His 

 second paper contained a demonstration of 

 the existence of monogenic functions satis- 

 fying a relation which had presented itself 

 as a greatly generalized form of the addi- 

 tion theorem of elliptic functions. 



Prof. Osgood's first paper gave a geomet- 

 ric method, consisting in the study of the 

 approximation of curves y=S^^(x), for dis- 

 cussing the manner in which S^^(x) con- 

 verges (uniformly or non-uniformly) toward 

 its limit f (x) when w=oo . Applications 

 were made to the allied problems in double- 

 limits of integrating and differentiating a 

 series term by term. Numerous examples 

 served as illustrations. His second paper 

 studied the most general manner of the 

 convergence of S^X^) toward its limit /(a;), 

 where /S^(a;), / (x) are continuous functions, 

 and treated the problem of determining 

 when 



j:/{^)dx=j^^^fi^s^{x)d.. 



Broader sufficient conditions than those 

 generally known for the integration of a 

 series of continuous functions term by term 

 were obtained. The former of these pa- 

 pers will appear in the Bulletin of the Amer- 

 ican Mathematical Society and the latter in the 

 American Journal of Mathematics. 



Jordon's decomposition of the general 

 linear group on m indices, and of the 

 group of Abelian substitutions on 2 m indi- 

 ces, leads to two doubly-infinite systems of 



simple groups. By generalizing to the 

 Galois field of order j)", p being prime, Dr. 

 Dickson reaches two systems of simple 

 groups whose orders depend on three inde- 

 pendent parameters m, n and p. His paper 

 is intended for the Annals of Mathematics. 



Prof. Shaw considered, in his paper, all 

 the algebras in which three independent 

 units occur. Dr. Hill used the general 

 cubo-cubic transformation in three cases 

 where the principal systems of the two 

 spaces degenerate, to pass from a general 

 cubic surface to a septic surface. In the 

 first case the surface is distiilguished by 

 three triple and three double lines; in the 

 second by a triple line and a double quin- 

 tic, and in the third by a triple conic and 

 a double quartic of the second kind. The 

 quartic surface discussed in Dr. Hutchin- 

 son's paper is the locus of the vertex of a. 

 cone of the second order passing through 

 six given points. If these six points be in 

 involution on the twisted cubic through 

 them, the coordinates of a point of the sur- 

 face can be expressed in terms of elliptic 

 functions. The object of Prof. Eoe's paper 

 was to obtain general formulae for the sum- 

 mation of the integral and integro-geomet- 

 ric series. 



All the cross-ratios of n quantities are 

 expressible rationally in terms of any n — 3 

 independent ratios. Starting with a certain 

 such system of w — 3 rations, and permuting 

 the quantities one obtains in all n ! systems 

 of the same type. The expression of these 

 n ! systems in terms of any one leads to the 

 group studied in Prof. Moore's paper. For 

 n=4 it is the well-known group of six linear 

 practical substitutions generated by X'^=l/ky 

 X'=l—X. For ?i^5 it is holoedrically iso- 

 morphic to the symmetric group in n let- 

 ters. It contains a sub-group of (n-1)! 

 collineations permuting amongst them selves- 

 certain n-1 fundamental points, and first 

 given by Klein. Prof. Moore's group results 

 from extending that of Prof. Klein by an. 



