MATHEMATICS.—MATHEMATICAL PHYSICS. 297 



MATHEMATICS. 



Morley, Frank, Johns Hoi)kins University, Baltimore, Maryland. Grant 

 No. 834, allotted December 13, 1912. Application of Cremona 

 groups to the solution of algebraic equations. (For previous reports 

 see Year Books Nos. 9-11.) $1,200 



The theory of restricted systems of equations when applied to a 

 particular problem failed to yield a result. Professor Coble accord- 

 ingly undertook a revision of the theory. The attempt was made to 

 handle the various phases in their greatest generality. The results 

 are given in a forthcoming paper. 



The first question set was : Given an M„ in Sn , n spreads containing 

 ilf p meet in how many points outside M^ ? This was solved by the 

 use of y+ 1 "index-numbers " attached to M„ . These index-numbers 

 were determined in special cases. A more general problem is : Given 

 an Af r_t on an M„ mSn,v spreads on Mv_k meet If „ in how many 

 points outside iW„_fc? This was solved by the introduction of the 

 v — k+1, ''relative index-numbers" of M^-u as to M^. 



A spread of v dimension composed of an ikf „ and Ms , s<v, has for 

 index-numbers the sums of the index-numbers of M„ and Ms, pro- 

 vided M^ and Ms have no common point. The necessary modification 

 when M^ and Ms have in common an M^ was discovered. Again, in 

 Sn spreads M^-vi, where ^^1+^2+ ' " ' =^^ ^11 contain M„; in how 

 many points outside ilf „ do they meet? 



The above are specimens of the questions which have been set. 

 Not all have been completely answered, but enough has been done to 

 give hope of completely general results. 



Another class of problems under way, which is central in the theory 

 of rational curves and equally so in that of rational surfaces, is the 

 following: Given a system of forms, calculate all their linear com- 

 binants, determine the relations which these linear forms satisfy, and 

 select those which define them as combinants of the original forms. 



MATHEMATICAL PHYSICS. 



Moulton, F. R., University of Chicago, Chicago, Illinois. Grant No. 852, 

 allotted December 13, 1912. Investigations in costnogony and celes- 

 tial mechanics. (For previous reports see Year Books Nos. 4, 5, 

 8-11.) $2,000 



The following work has been completed during the last year and 

 has been accepted for publication : 

 (1) On the solution of linear equations having small determinants: 



This paper treats of linear equations whose coefficients and right 

 members are given by -experiments or observations to a limited 

 number of places. The degree of precision of the solutions is deter- 

 mined. It is proved that when the determinant of the coefficients is 

 small, the solution is determined to a smaller number of places than 

 are given in the coefficients; the precise extent of the uncertainty is 



