April 27, 1917] 



SCIENCE 



411 



ber is one that would be called in general 

 algebra a multiplex, as (5, 3) or (3, 6, 2, 7). 



Several tables of homogeneous functions, 

 distribution functions, and enumerations close 

 the first volume. 



Section seven is devoted to the algebraic 

 side of the partition of numbers, giving in 

 some detail the present state of the theory, 

 but omitting the purely arithmetic side. 

 There are given here further developments 

 connected with symmetric functions. 



Section eight considers the theory of parti- 

 tions as based upon Diophantine inequalities, 

 generalizing the whole treatment. A chapter 

 is devoted to the further study by this means 

 of magic squares, the object being their enu- 

 meration rather than construction. 



Sections nine and ten study partitions in 

 two dimensions, including a complete solu- 

 tion of another long-standing problem. The 

 problem of three-dimensional partitions relat- 

 ing to a cubic lattice is also attacked. 



Section eleven relates to symmetric func- 

 tions of several systems of quantities with 

 applications to distribution functions. 



The second volume closes with tables of 

 symmetric functions of two systems, and 

 enumeration of solid graphs. 



If one were to undertake to characterize the 

 treatise of Professor MacMahon briefly he 

 would probably best state its field by saying 

 that it is a development of the algebra of 

 symmetric functions with applicatin to vari- 

 ous generating expressions whose coefficients 

 find use in enumeration problems of distribu- 

 tion. Professor MacMahon has occupied him- 

 self with the development of this theory for 

 some years and the treatise is a systematic 

 presentation of his results. 'No brief account 

 can be given of the very skilful methods em- 

 ployed. It shows amply that alongside of the 

 alternating functions so long studied in deter- 

 minant forms, the symmetric functions are 

 equally important and have their field of 

 application. It exemplifies how different 

 branches of mathematics can be correlated so 

 as to be useful in reducing problems. It also 

 draws strongly attention to the fact that there 

 still remains in the field of algebraic form 



plenty of opportunity for the interested stu- 

 dent to do research work of high order. In- 

 deed it would seem that courses on symmetric 

 functions at least should be offered alongside 

 of other courses in algebra, such as theory 

 of equations, determinants, groups, and the 

 like. The whole theory of the construction of 

 algebraic forms for certain specific purposes 

 has been enriched here with a valuable con- 

 tribution. 



James Byenie Shaw 



SPECIAL ARTICLES 

 INHERITANCE OF OIL IN COTTON 



The table of oil percentages given below 

 suggests the possibility of producing divergent 

 strains or biotypes from a " variety " of cot- 

 ton, the one having seeds relatively high in oil 

 content, the other relatively low in oil content. 



The top line of figures gives the analysis 

 (ether extract) of the seed from several mother 

 plants, followed in column by the analysis of 

 the seed of three of their progeny plants, 

 respectively. 



Progeny. 



17.33 20.64 16.58 18.97 16.79 18.92 16,87 



18.2l!22.00 



18.67J20.82 

 19.1321.06 



17.16l22.10 

 18.40121. 17 

 17.8621.36 



17.75 

 17.86 

 17..52 



19.3718.47 

 19.45; 18.92 

 19.19 18.40 



The three " high " parents have an average 

 of 19.51 per cent, oil, and their nine progeny 

 plants an average of 20.Y2 per cent. oil. 



The four " low " parents have an average of 

 16.89 per cent, oil, and their twelve progeny 

 plants an average of 18.20 per cent. oil. 



The maximum difference between parents is 

 4.06 per cent, oil, and the maximum difference 

 between plants of the progeny generation is 

 4.94 per cent. oil. 



A seasonal variation raising the oil content 

 of all plants in the progeny year is noted. 



A later report will give the correlation be- 

 tween oil content of the seed and other char- 

 acters. 



E. P. Humbert 



Agricultural Experiment Station, 

 College Station, Texas 



