176 



SCIENCE. 



[N. S. Vol. XXI. Xo. 527. 



to 5 

 5 to 10 

 10 to 15 

 15 to 20 

 20 to 25 

 25 to 30 

 30 to 35 



24.56 days. 

 24.79 " 



25.03 • " 

 25.20 " 

 25.45 " 

 25.99 " 

 25.31 " 



1 r(m-lVf/-l) 

 Wi L <r = l 



P(0,1, 2, 



Determination of all Non-divisible Groups 

 of Order p"' ■ q which Contain an Ahelia7i 

 Subgroup of Order and Type 



[1, 1,1 ■■■ to m units]: Mr. O. E. 

 Glenn, University of Pennsylvania, 

 Philadelphia. 



Bnrnside remarks, at the beginning of 

 Chapter XV. of his 'Theory of Groups,' 

 that the most general problem of finite 

 group theory is the determination and an- 

 alysis of all distinct types of groups whose 

 order is a given integer. The author sug- 

 gests as a more comprehensive problem the 

 generalization of all types belonging to a 

 given integer. In the paper is given a de- 

 termination of the sets of defining relations 

 which include as special cases all groups of 

 order pq, p^q, and a family of the known 

 groups of order p'-'q. 



When q is a proper divisor of 2^"' — ^, G 

 is defined by the relations 



PiP =Q'' = \, PiPj = PjP,, Q-'P,Q r= p,+ , 



(1 = 1, 2, . . ., m; j=l, 2, . . ., m ; k=l, 2, . . ., m-l.) 



Q-^PmQ ^ PiC-i)"-' p/-iy--2.(Mi' . . . . . . ^ 



A being a mark of the G-F{p"') and a 

 primitive root in that field of the con- 

 gi'uence 



>.'/ = l(mo(l p.). 



In case j) = \ (mod. q) the group is de- 

 fined by 



P'" r^Qi \, PiPi= PjPi , Q-i n, Q =- /'.'-■'(r, 1 ), 

 ai = 1( mod. p). 



T\n; first .set of rcliilions represents a single 

 type. The number of types in the secotid 

 set is <ri ven bv 



P being Cayley's form of the partititjn 

 symbol and ij/ a determinate function of 

 m and q. 



A Note on Groups of Order 2'" which Con- 

 tain Self-conjugate Groups of Order 

 2">-2. Dj, q Hallett, Univei'sity of 

 Pennsylvania, Philadelphia. 

 In the list of groups of the character in- 

 dicated above which is given in Buruside's 

 'Theory of Groups,' there are six types. 

 There appears to be a simple type of group 

 which is non-isomorphic to any one of these 

 six groups. The object of the paper is to 

 set up the defining relations of this type, 

 viz., 



P^' = 1 , Q^^ P^"'--\ Q-'PQ^P-K 



Biology and Mathematics: Professor G. B. 



Halsted, Kenyon College. Gambler, 0. 



In Professor Halsted 's paper attention 

 was called to certain analogies which have 

 been assumed to exist between the mathe- 

 matical doctrine of continuity and the evo- 

 lution of new species through natural selec- 

 tion. He then proceeded to show that the 

 analogy between mathematics and biology 

 is much closer if we emphasize, on the one 

 hand, the idea of discontinuity as it ap- 

 pears in modern mathematics and, on the 

 other, those phases of the process of evo- 

 lution supposed to be more readily ex- 

 ])lained by the theory of mutations. 



The Path of the Shadoiv of a Plummet 



Bead: Profes.sor Ellen Hayes, AVellesley 



College, Wellesley, Mass. 



The equation to the path of the shadow 

 of a plummet bead was derived, and dis- 

 cussed for various latitudes and for dif- 

 ferent seasons of the year. 



The interest and value which this 

 gnomon conic possesses as an observation 

 exercise for beginners in elementary prac- 

 tical astronomy wei'c made apparent. 



