MATHEMATICS. 361 



complete recasting of my work, a fresh collecting of material, as well 

 as a considerable delay in consequence of having to wait for the 

 pubUcation of the successive parts of the dictionary, which is now 

 fortunately almost completed. The delay has caused less inconven- 

 ience than might have been expected, for the reason that since 1912 

 I have been working on the Fall of Princes and have consequently 

 been able to turn from one task to the other as circumstances permitted. 



MATHEMATICS. 



Morley, Frank, Johns Hopkins University, Baltimore, Maryland. Appli- 

 cation of Cremona groups to the solution of algebraic equations. (For 

 previous reports see Year Books Nos. 9-14.) 



The second paper embodying Professor Coble's investigation of 

 point sets and Cremona groups will appear in the October number of 

 the Transactions of the American Mathematical Society. An abstract 

 of this paper was given in the Proceedings of the National Academy 

 of Sciences, volume 2, April 1916. 



The third part has been submitted for pubUcation m the Transac- 

 tions. It deals with the determination of the lines on a cubic surface. 

 The treatment differs from that of Klein and Burkhardt in three 

 important points: (1) the transition from the given surface to the 

 coUineation group arising from the theta functions is accomplished 

 through the intervention of hnear systems of irrational invariants 

 of the surface; (2) these groups are introduced by means of a certain 

 normal form of the hyperelhptic surface analogous to the normal 

 form of the elhptic curve, the hne being replaced by a Kiimmer sur- 

 face; (3) the most convenient final problem is the form problem of 

 the Burkhardt group, for which a solution (hitherto lacking) is found. 



In the entire discussion no use is made of an equation of degree 

 27 or other resolvent equation. 



A number of questions suggested by the above, which in a sense 

 are grouped by their connection with the properties of ''elhptic 

 quintics in 64," are considered under that title in a paper to be 

 submitted to the American Journal. 



The attempt to connect the general theta functions with point 

 sets and thereby to get a geometric grip on their properties has been 

 successful up to a certain point. The general tactical correspondence 

 between the two is established in the article, "An isomorphism between 

 theta characteristics and the (2p + 2) -point" which appeared in the 

 Annals of Mathematics, series 2, volume 17, March 1916. Here, also, 

 when p = 3 theta-relations are set forth which vitahze the isomorph- 

 ism. Similar relations have been obtained for p = 4. There remains 

 the problem of obtaining these relations for any value of p and of 

 identifying the theta moduh with the irrational invariants of the 

 (2p -F 2)-pomt. 



