462 



SCIENCE. 



[N. S. Vol. XXIII. No. 58G. 



This paper will be printed in the 'Astro- 

 nomical Papers of the American Ephem- 

 eris,' Vol. VIII., Pt. 3. 



A Class of Central Forces: Dr. Edwaed 

 Kasnee, Columbia University, New 

 York City. 



There exists no field of force in Avhich a 

 particle started from an arbitrary position 

 with arbitrary velocity will describe a 

 circular path. In the case of a central 

 force the only possible circular trajectories 

 are, in general, those whose centers are at 

 the origin of force. If, however, the force 

 varies according to a function of the form 

 br{r^ — a)—^, then a qiiadruple infinity of 

 the trajectories are circular. In the sim- 

 plest case, arising when a vanishes, the 

 force varies inversely as the fifth power of 

 the distance, and the circles all pass through 

 the origin. In the general case they are 

 orthogonal or diametral to a fixed sphere. 



Solar Photographs: Professor F. H. Loud, 

 Colorado College, Colorado Springs, 

 Colo. Presented by title. 



The Groups of Order p™ which contain 

 Exactly p Cyclic Subgroups of Order 

 p"-: Professor G. A. Millee, Stanford 

 University, Cal. 



The main theorems proved in this paper 

 may be stated as follows: If a group of 

 order p"*, p being . any odd prime, con- 

 tains exactly p cyclic subgroups of order 

 p", a > 2, it contains exactly p cyclic sub- 

 groups of every order which exceeds p 

 and divides p'"~^. Hence it is one of the 

 two non-cyclic groups of order p™ which 

 contain operators of order p'"^^. "When 

 o ^ 2 and p > 3 the theorem is still true. 

 In fact, the only possible exception occurs 

 when a = 2, p ^ 3 and m ^ 4. In this 

 special case there are three groups which 

 contain exactly p cyclic subgroups of or- 

 der p". 



When p = 2 the preceding theorem is re- 



placed by the following: If a group of 

 order 2™ contains exactly two cyclic sub- 

 groups of order 2«, a > 2 it can not contain 

 more than two cyclic subgroups of any 

 higher order. If a group of order 2™ con- 

 tains exactly two cyclic subgroups of order 

 2^ but does not contain any cyclic subgroup 

 of order 2^+^, then m can not exceed 2^+^. 

 These theorems involve the fundamental 

 properties of all possible groups whose 

 order is a power of any prime p and which 

 involve exactly p cyclic subgroups of any 

 given order p". From a well known 

 theorem it follows that a is not unity, but 

 it can have every other possible value less 

 than in. 



Inversion and Inversors: Professor J. J. 



QuiNN, Warren, Pa. 



In this paper are presented two new 

 theorems relating to inversion, besides an 

 explanation of the construction of certain 

 linkages exhibiting the operation of in- 

 version. 



Observations of the Total Solar Eclipse of 

 190.'), August 30, at Tripoli, Barbary: 

 Professor David Todd, Amherst College, 

 Amherst, Mass. 

 Observations were undertaken under six 



different heads, as follows: 



1. Observations of the geometric con- 

 tacts. 



2. Coronal photography with a twelve- 

 inch Clacey lens photographically cor- 

 rected. Professor Todd's modified form 

 of Burckhalter revolving occulter was em- 

 ployed. The corona was photographed to 

 30'. Bailey's 'beads' were also photo- 

 gTaphed before the second contact. 



3. A duplex Clark lens of three inches 

 was used for long exposures on the circum- 

 solar stars and the outer coronal streamers. 

 Neither these nor any intra-Mercurial 

 planets were revealed, although 14 x 17 

 plates of the highest sensitiveness were 

 used. 



